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Assessing turbulence models for large-eddy simulationusing exact solutions to the Navier-Stokes equations
Bachelor Project Mathematics
July 5 2016
Student Daniel Ward
Daily Supervisor Maurits H Silvis MSc
First Supervisor Prof dr ir RWCP Verstappen
Second Supervisor Dr AE Sterk
Abstract
We study the behaviour of turbulent fluid flows behaviour that is governed by the Navier-Stokes equations Due to the extensive detail entailed in a turbulent flow it is difficult tosolve the Navier-Stokes equations numerically A large-eddy simulation (LES) using a filteringoperation solves for the large-scale motions in a flow and smooths over small-scale motionhence requiring a turbulence model for these small-scale motions Using conditions derivedfrom and exact solutions to the Navier-Stokes equations we investigate a number of turbulencemodels for LES both with and without explicit filtering We found that the Vortex-Stretching-based eddy-viscosity model was most endorsed in the case without explicit filtering and thatthe Vreman QR Gradient and Vortex-Stretching-based models were equally endorsed by thecase with explicit filtering with the Smagorinsky model being the least endorsed To reducethe bias that may arise when exact solutions are similar we then considered classes of flowsto further determine which models are the most endorsed We considered classes based onVremanrsquos flow classes and on the principal and combined invariants of flows Once again theVortex-Stretching-based eddy-viscosity model was usually the most endorsed As more exactsolutions to the Navier-Stokes equations are found they can be easily added to the report tofurther endorse or oppose the models We also considered Vremanrsquos paper [16] and did notfind anything to contradict his results We did extend his results slightly by finding a solutionoutside of his zero-subfilter-dissipation classes with zero subfilter dissipation
2
Contents
1 Introduction 4
2 The Navier-Stokes Equations 521 Introduction 522 Exact Solutions of the Navier-Stokes Equations 623 Characterisation of Flows 20
231 Characterising Solutions using Invariants 20232 Vremanrsquos Characterisation of Flows 21
3 Conditions for Large-Eddy Simulation Models 2231 Large-Eddy Simulation 2232 Subfilter-Scale Models 2233 Conditions for Large-Eddy Simulation without Explicit Filtering 2334 Conditions for Large-Eddy Simulation with Explicit Filtering 24
4 Analysis of Large-Eddy Simulation Models 2541 Large-Eddy Simulation Without Explicit Filtering 2542 Large-Eddy Simulation With Explicit Filtering 26
421 Example of a Test for a Model 27422 Testing the Models 28
43 Vremanrsquos True Subfilter Dissipation Claim 2944 Classes of Flows 29
441 Vremanrsquos Velocity Gradient Flow Classes 30442 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes 31443 Classes using the Third Principal Invariant 33444 Two-component Flow Classes 35
5 Conclusions 37
Appendices 39
A Invariants of Flows 39
B Vremanrsquos Flow Classes 46
C Extended Tables for Flow Classes 47C1 Vremanrsquos Velocity Gradient Flow Classes 47C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes 48C3 Classes using the Third Principal Invariant 49C4 Two-Component Flow Classes 50
3
1 Introduction
In this project we will study and test models for turbulent fluid flows There is no all-encompassingdefinition of a turbulent flow there are however a number of characterising properties A turbulentflow is said to be irregular in that it appears to be random Turbulent flows are chaotic and henceusually described statistically due to the difficulty in a deterministic approach Turbulent flowsare diffusive in that there is a high rate of momentum heat and mass transfer They require apersistent source of input energy in order to sustain the turbulent nature of the flow as kineticenergy is quickly converted into internal energy A large Reynolds number is characteristic of aturbulent flow [5]
The Navier-Stokes equations are a set of fundamental governing equations for any Newtonian fluidAn exact solution to the Navier-Stokes equations is in the form of a flow velocity and a pressureterm and hence a fluid flow can be completely described by an exact solution to the Navier-Stokesequations It is still a non-trivial task to find such solutions although there are a number in literature[17]
The Navier-Stokes equations have no closed form solution resulting in difficulty when analysingcomplicated flows in complex geometries This results in difficulty when solving the Navier-Stokesequations explicitly So a mathematical model known as Large Eddy Simulation (LES) can insteadbe used to model turbulent flows LES uses what is known as a filtering operation to reduce thecomputational cost of a simulation by ignoring small-scale motion What this means however isthat information about these small-scale motions is lost This information is often non-trivial sosubfilter-scale models (or turbulence models) are used in order to approximate the effects of small-scale motions on the large-scale fluid flow
In this project we will use the behaviour of exact solutions to the Navier-Stokes equations to testsubfilter-scale models Applying a subfilter-scale model to a flow has two effects One is a bodyforce on the velocity field called a subfilter force and the other is a transfer of energy from largeto small scales of motion called a subfilter dissipation This results in respectively an extra termbeing added to the Navier-Stokes equations and to the equation of the kinetic energy of a flow
We will then consider conditions for subfilter-scale models in two different cases In LES withoutexplicit filtering we will consider how a subfilter-scale model behaves for an exact solution to theNavier-Stokes equations by checking if the subfilter force and subfilter dissipation are zero for anexact solution If so this can be interpreted as the model being inactive and hence this exactsolution can be said to lsquoendorsersquo the model In LES with explicit filtering we compare a model termwith a term in the filtered Navier-Stokes equations (called the true turbulent stress of a flow) forexact solutions If the two terms are both zero we will conclude that the exact solution lsquostronglyrsquoendorses the model and if the two terms are both non-zero we will conclude that the exact solutionlsquoweaklyrsquo endorses the model
We will also consider the classification of fluid flows in order to further inspect the models andreduce the bias that may occur when solutions have similar properties We will use properties of theinvariants of a flow to characterise flows in a coordinate-independent way Using Vremanrsquos paperldquoAn eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory and applicationsrdquo[16] we will place the fluid flows into Vremanrsquos flow classes We will also determine whether ourresults say anything beyond Vremanrsquos work
4
2 The Navier-Stokes Equations
21 Introduction
The incompressible Navier-Stokes equations are a set of four equations that model the flow of anyfluid in three-dimensional space The standard set of Navier-Stokes equations can be written in twoforms namely by using Einsteinrsquos summation convention in what we call index notation and byusing standard vector notation In index notation they are given by
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
i = 1 2 3 j = 1 2 3 (1)
These equations have four variables given by the velocity of the flow u = (u1 u2 u3) and thepressure p The vector x = (x1 x2 x3) represents position t represents time ρ corresponds to themass density and ν is the kinematic viscosity of the fluid
The Navier-Stokes equations can be written in vector notation in the following way First considerthe convection term
ujpartuipartxjequiv
3sumj=1
ujpartuipartxj
= u1partuipartx1
+ u2partuipartx2
+ u3partuipartx3
=
(u1
part
partx1+ u2
part
partx2+ u3
part
partx3
)ui
= (u middot nabla)ui
(2)
Similarly for the diffusion term
νpart2uipartxjpartxj
equiv ν3sumj=1
part2uipartxjpartxj
= νpart2uipartx1partx1
+part2uipartx2partx2
+part2uipartx3partx3
= ν
(part2
partx1partx1+
part2
partx2partx2+
part2
partx3partx3
)ui
= νnabla2ui
(3)
It therefore holds that in vector notation the Navier-Stokes equations are given by
partu
partt+ (u middot nabla)u = minus1
ρnablap+ νnabla2u (4)
It is clear that there is a problem here there are three equations and four unknowns In order toremedy this the equation for conservation of mass or the incompressibility condition is consideredThis is given by
partuipartxi
= 0 (5)
in index notation and
nabla middot u = 0 (6)
5
in vector notation
There are now four equations in four unknowns and it is possible to find solutions Althoughthere is no known general solution there are specific cases in which exact solutions to the Navier-Stokes equations can be found In order to make finding solutions slightly simpler the Navier-Stokesequations can be rescaled by introducing dimensionless variables
Let
ulowasti =uiv xlowasti =
xi` tlowast =
t
`v plowast =
p
ρv2 (7)
where ` and v are typical length and velocity scales respectively Then dropping the star notationfor ease of reading the scaled incompressible Navier-Stokes equations are
partuipartt
+ ujpartuipartxj
= minus partp
partxi+
1
Re
part2uipartxjpartxj
i = 1 2 3 j = 1 2 3
partuipartxi
= 0
(8)
where Re is the Reynolds number and is defined by the dimensionless Re =v`
ν Similarly in vector
notation the scaled incompressible Navier-Stokes equations are
partu
partt+ (u middot nabla)u = minusnablap+
1
Renabla2u
nabla middot u = 0(9)
By multiplying the incompressible Navier-Stokes equations Eq (8) by ui and grouping terms thedifferential form of the evolution of the kinetic energy of a flow can be found
part
partt
(1
2uiui
)+ uj
part
partxj
(1
2uiui
)= minusui
partp
partxi+
2
Reui
part2uipartxjpartxj
(10)
The differential form of the evolution of the kinetic energy of a flow will be used later on in thisreport to derive conditions for the subfilter-scale models
22 Exact Solutions of the Navier-Stokes Equations
By using mathematical inspection and taking results found in literature we found a number of exactsolutions to the Navier-Stokes equations The majority are steady-state solutions in that they areindependent of time We decided to consider exact solutions to the Navier-Stokes equations withoutboundary conditions in an attempt to keep the results general and applicable to a larger numberof situations All of the stated exact solutions below have been checked using a code written inMathematica In the solutions below the notation (x y z) for the Cartesian coordinate system isused analogously with (x1 x2 x3) to aid in reading comprehension and in the conversion betweenindex and vector notation
Solution 1 Linear Velocity
If the diffusion term in the Navier-Stokes equations Eq (1) is taken to be zero ie
part2uipartxjpartxj
= 0 (11)
6
then each component of the velocity can be written as a linear function of x1 x2 and x3 In indexnotation
ui = aijxj + βi (12)
If pressure is assumed to be a function of time t (so that the pressure terms in the Navier-Stokesequations are zero) then conditions can be found on the coefficients to find that the following is asolution
u1(x y z t) = 0
u2(x y z t) = ax+ b
u3(x y z t) = cx+ d
p(x y z t) = f(t)
(13)
where a b c d isin R and f Rrarr R In fact this solution is actually a specific case of Sol 2 (PlanarFlow)
Figure 1 An illustration of the flow in the xy-plane for Sol 1 (Linear Velocity) described inEq (13) with parameter values a = b = c = d = 1
Solution 2 Planar Flow
Using the text by Barbato et al [2 p62] and modifying the solution such that it now takes threedimensions into account another solution is found
u1(x y z t) = Az2 +Bz + C
u2(x y z t) = Dz2 + Ez + F
u3(x y z t) = 0
p(x y z t) =2A
Rex+
2D
Rey + f(t)
(14)
7
where ABCDE F isin R and f R rarr R It is clear to see that if A = D = 0 then this is in factthe same as Eq (13) in Sol 1 In fact we believe (although not shown here) that this is the mostgeneral solution of the form
ui(xi xj xk t) = Ax2k +Bxk + C
uj(xi xj xk t) = Dx2k + Exk + F
uk(xi xj xk t) = 0
(15)
for i 6= j 6= k 6= i
Figure 2 An illustration of the flow in the yz-plane for Sol 2 (Planar Flow) described inEq (14) with parameter values A = B = C =D = E = F = 1
Solution 3 Velocity as a function of time t
If each component of the velocity is taken to be a function of time t and independent of positionthen the incompressibility condition is automatically satisfied A condition can be imposed on thepressure term in order to satisfy the momentum equations Thus a solution is as follows
u1(x y z t) = f(t)
u2(x y z t) = g(t)
u3(x y z t) = h(t)
p(x y z t) = minusxpartf(t)
parttminus y partg(t)
parttminus z parth(t)
partt
(16)
where f g h Rrarr R In fact what is written here is the generalised Galilean transformation of thezero solution [10] We cannot produce a plot for this flow as the functions are completely unspecified
8
Solution 4 Axisymmetric Flow
Using the text by Irmay [7] a solution is defined as follows
u1(x y z t) = kxz
u2(x y z t) = kyz
u3(x y z t) = minuskz2
p(x y z t) = minus2kz
Reminus k2z4
2
(17)
where k isin R This is said to represent an axisymmetric flow with hyperbolic streamlines asymptoticto the z-axis and to the boundary plane z = 0 These asymptotes are visible in Fig 3
Figure 3 An illustration of the flow in theyz-plane for Sol 4 (Axisymmetric Flow)described in Eq (17) with parameter k = 1
Figure 4 An illustration of the flow in thexy-plane for Sol 4 (Axisymmetric Flow)described in Eq (17) with parameter k = 1
Solution 5 2D Taylor Solutions
Again using the text by Barbato et al [2 p63ndash64] a solution is defined using a stream functionψ = ψ(x y t) and by writing
u1 = minuspartψparty
u2 =partψ
partx u3 = 0 (18)
In the text the stream function is not given however it was worked out that
ψ = minuseminusπ2t
Re (cos(πx) + cos(πy))
π (19)
The solution used is of the following form note however that in the text the pressure term differsin that cos(πy) is replaced by sin(πy) (when testing this solution in Mathematica the solution below
9
was instead found)
u1(x y z t) = minus sin(πy) eminus(π2t)Re
u2(x y z t) = sin(πx) eminus(π2t)Re
u3(x y z t) = 0
p(x y z t) = minus cos(πx) cos(πy) eminus(2π2t)Re
(20)
Figure 5 An illustration of the flow in the xy-plane for Sol 5 (2D Taylor) described in Eq (20)with Re = 1 and time t = 0
Solution 6 Generalized Beltrami Flow
Again using the text by Barbato et al [2 p65] and now considering solutions with nonndashtrivialterms in all dimensions a particular form of the generalized Beltrami flow is found namely
u1(x y z t) = (A sin(πz) + C cos(πy)) eminus(π2t)Re
u2(x y z t) = (B sin(πx) +A cos(πz)) eminus(π2t)Re
u3(x y z t) = (C sin(πy) +B cos(πx)) eminus(π2t)Re
p(x y z t) = minus[(AB cos(πz) sin(πx) +BC cos(πx) sin(πy)
+AC cos(πy) sin(πz)] eminus(2π2t)Re
(21)
where ABC isin R
10
Figure 6 An illustration of the flow in the yz-plane for Sol 6 (Generalized Beltrami Flow)described in Eq (21) with parameters A = B =C = 1 and time t = 0
Solution 7 Stagnation Flow
In the book by Drazin and Riley [6 p40] an exact solution is considered which uses a stream functionψ = ψ(y z) They suggest that the following velocity field is a solution
u1(x y z t) =a
kx
u2(x y z t) = minusaky +
k
Re
partψ(y z)
partz
u3(x y z t) = minus k
Re
partψ(y z)
party
(22)
Here we choose the stream function to be ψ(y z) = yz Then a solution is given by
u1(x y z t) =a
kx
u2(x y z t) =
(minusak
+k
Re
)y
u3(x y z t) = minus kzRe
p(x y z t) = minus1
2
((axk
)2minus((
k
Reminus a
k
)y
)2
minus(kz
Re
)2)
(23)
where a is a measure of the strength of the stagnation flow and k is a constant chosen in the textto specify the distance between vortices There are no vortices in this particular solution however
11
Figure 7 An illustration of the flow in the yz-plane for Sol 7 (Stagnation Flow) describedin Eq (23) with parameter values a = Re = 1and k = 2
Figure 8 An illustration of the flow in the xy-plane for Sol 7 (Stagnation Flow) describedin Eq (23) with parameter values a = Re = 1and k = 2
Solution 8 Vortex in Stagnation Flow
As in Sol 7 (Stagnation Flow) we now consider the same solution with a different stream functionnamely ψ(z) = cos z Then given the condition that k = aRe and adding a pressure term Eq (22)becomes
u1(x y z t) =1
Rex
u2(x y z t) = minus 1
Rey minus k sin z
u3(x y z t) = 0
p(x y z t) = minus 1
2Re2(x2 + y2)
(24)
where a and k are as in Sol 7 (Stagnation Flow)
12
Figure 9 An illustration of the flow in the yz-plane for Sol 8 (Vortex in Stagnation Flow)described in Eq (24) with parameter valuesk = 3 and Re = 10
Figure 10 An illustration of the flow in thexy-plane for Sol 8 (Vortex in StagnationFlow) described in Eq (24) with parametervalues k = 3 and Re = 10
Solution 9 Burgers Vortex
The Burgers Vortex solution is given in cylindrical coordinates in the book by Lautrup [8 p 385]as follows
ur(r ϕ z t) = minus 2r
c2Re
uϕ(r ϕ z t) =Ωc2
r
(1minus eminusr
2c2)
uz(r ϕ z t) =4z
c2Re
p(r ϕ z t) = minusc2Ω2F
(r2c2
)2
minus 2
Re2r2 + 4z2
c4
(25)
where
F (ξ) =
infinintξ
(1minus eminusx)2
x2dx (26)
This solution is not particularly nice to look at however most of the unpleasantness comes fromthe pressure term which (in this project) is only required in order to validate the solution For thisproject all solutions are required in Cartesian coordinates the velocity field of the Burgers Vortexhas hence been converted into Cartesian coordinates
u1(x y z t) = minus 2x
c2Reminus Ω
h(x y)(1minus exp (minush(x y))) y
u2(x y z t) = minus 2y
c2Re+
Ω
h(x y)(1minus exp (minush(x y)))x
u3(x y z t) =4z
c2Re
(27)
13
where
h(x y) =x2 + y2
c2 (28)
and Ω and c are constants
Figure 11 An illustration of the flow for Sol9 (Burgers Vortex) described in Eq (27) withparameter values c = 15 Re = 20 and Ω = 10A slice of the xy-plane at z = 0 is shown
Solution 10 Another Vortex Solution
There is another vortex solution more simply formulated than the Burgers Vortex yet with asingularity at r = 0 as a result of the uϕ term It is again given in cylindrical coordinates
ur(r ϕ z t) = minus 2r
c2Re
uϕ(r ϕ z t) =a
r
uz(r ϕ z t) =4z
c2Re
p(r ϕ z t) = minus a2
2r2minus 2r2 + 8z2
c2Re2
(29)
which can also be converted into Cartesian coordinates
14
u1(x y z t) = minus 2x
c2Reminus ay
x2 + y2
u2(x y z t) = minus 2y
c2Re+
ax
x2 + y2
u3(x y z t) =4z
c2Re
(30)
where a and c are constants
Figure 12 An illustration of the flow for Sol9 (Burgers Vortex) described in Eq (27) withparameter values c = 15 Re = 20 and a = 10 Aslice of the xy-plane at z = 0 is shown
Solution 11 Pseudo-Motion Shots of the Second Kind
In the book by Berker [3 p 94] a solution is given in cylindrical coordinates by
ur(r ϕ z t) =2
Re
1
r
uϕ(r ϕ z t) = A1r3 +B1r +
C1
r
uz(r ϕ z t) = A2r2 log(r) +B2r
2 + C2
p(r ϕ z t) =A2
1r6
6+A1B1r
4
2+A1C1r
2 +B2
1r2
2minus C2
1
2r2minus 2
r2Re2+ 2B1C1 log(r)
+2A2z
Reminus 4B1ϕ
Re
(31)
where A1 A2 B1 B2 C1 C2 are constants Eq (31) when converted into Cartesian coordinates is
15
u1(x y z t) =1
Re
2xminus C1Rey
x2 + y2minusA1y
(x2 + y2
)minusB1y
u2(x y z t) =1
Re
2y + C1Rex
x2 + y2+A1x
(x2 + y2
)+B1x
u3(x y z t) =1
2A2
(x2 + y2
)log(x2 + y2
)+B2
(x2 + y2
)+ C2
(32)
Figure 13 An illustration of the flow in thexy-plane for Sol 11 (Pseudo-Motion Shots)described in Eqs (31) and (32) with parametervalues A1 = B1 = C1 = 1 A2 = B2 = C2 = 2and Re = 10
Figure 14 An illustration of the flow in theyz-plane for Sol 11 (Pseudo-Motion Shots)described in Eqs (31) and (32) with parametervalues A1 = B1 = C1 = 1 A2 = B2 = C2 = 2and Re = 10
Solution 12 Flows With Constant Whirl
In the book by Berker [3 p 143] another solution is given in cylindrical coordinates by
ur(r ϕ z t) = minus 1
Re
1
r
uϕ(r ϕ z t) =1
rradicte(minus
12 g(t))
uz(r ϕ z t) = 0
p(r ϕ z t) =1
4
2(minus e
minusg(t)
t minus 1Re2
)r2
minus 1
t2Re
infinintminusg(t)
eminust
tdt
(33)
where we define g(t) =r2Re
2t When converted into Cartesian coordinates we find
16
u1(x y z t) = minus 1
x2 + y2
(yradict
exp
(minus
Re(x2 + y2
)4t
)+
x
Re
)
u2(x y z t) =1
x2 + y2
(xradict
exp
(minus
Re(x2 + y2
)4t
)minus y
Re
)
u3(x y z t) = 0
(34)
Figure 15 An illustration of the flow for Sol12 (Flows with Constant Whirl) described inEqs (33) and (34) with time t = 1 and Re = 1 Aslice of the xy-plane at z = 0 is shown as velocityin the z-direction is zero this is the same for anyvalue of z
17
Solution 13 Rectilinear and Uniform Translation of a Sphere
In the book by Berker [3 p 228] another solution is given in Cartesian coordinates by
u1(x y z t) =3
4
axz
R3
(a2
R2minus 1
)
u2(x y z t) =3
4
ayz
R3
(a2
R2minus 1
)
u3(x y z t) = 1minus 3
4
a
Rminus 1
4
a3
R3minus 3
4
az2
R3
(a2
R2minus 1
)
p(x y z t) =3a(aminusR)(a+R)
32R10Re
((R2Re
(x2 + y2
)minus 2R2Rez2
) (a3 + 3aR2 minus 4R3
)+ 3aRez4
(R2 minus a2
)minus 16R5z
)
(35)
where a and R are constants
Figure 16 An illustration of the flow in the yz-plane for Sol 13 (Sphere Translation) describedin Eq (35) with parameter values a = minus2 andRe = 3
Solution 14 Radially Dependent Flow
A solution with velocity in the z-direction dependent only on the distance from the z-axis is as
18
follows in cylindrical coordinates
ur(r ϕ z t) = 0
uϕ(r ϕ z t) = 0
uz(r ϕ z t) = A(r2 minus a2
)
p(r ϕ z t) =4Az
Re
(36)
where A and a are constants In Cartesian coordinates the solution is just the same with u1(x y z t)and u2(x y z t) both zero and the r2 term replaced with x2 + y2 in u3(x y z t)
Figure 17 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus1 1] withparameter values a = 1 A = 4 and Re = 10
Figure 18 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus2 2] withparameter values a = 1 A = 4 and Re = 10
Solution 15 Rankine Vortex
A solution discovered by Rankine [1] is a vortex flow of the following form
ur(r ϕ z t) = 0
uϕ(r ϕ z t) =ar
2πR2
uz(r ϕ z t) = 0
p(r ϕ z t) =a2r2
8π2R4
(37)
19
where a and R are constants In Cartesian coordinates the flow has the form
u1(x y z t) = minus ay
2πR2
u2(x y z t) =ax
2πR2
u3(x y z t) = 0
p(x y z t) =a2(x2 + y2
)8π2R4
(38)
This flow is actually a rigid body rotational flow in that the angular rotational velocity is uniformand the velocity of the flow increases proportionally with distance from the z-axis
Figure 19 An illustration of the flow in the xy-plane for Sol 15 (Rankine Vortex) described inEq (37) and Eq (38) with parameter values a = 1and R = 1
radic2π
23 Characterisation of Flows
It is important to be able to classify flows in a way that sorts them into distinct groups This aidsin the clear identification and general study of flows It is also a very important way of recognisingflows with similar properties In this subsection we introduce two distinct ways in which flows canbe characterised
231 Characterising Solutions using Invariants
A principal invariant of a tensor is a coefficient of the characteristic polynomial of that tensorInvariants as their name suggests are invariant under rotations of the coordinate system in thattheir values do not change when the tensor is lsquorotatedrsquo into a different set of coordinates Thevelocity gradient of a flow G is defined as the Jacobian of the velocity vector given by
Gij =
[partuipartxj
] (39)
20
for i j = 1 2 3 There are three principal invariants of the velocity gradient of a fluid flow G givenby
PG = tr(G)
QG =1
2
((tr(G))2 minus tr
(G2))
RG = det(G)
(40)
Incompressibility of flows (Eq (5)) ensures that PG = 0 hence the first principal invariant is zerofor all of the flows in Section 22 We now define the rate-of-strain tensor S by
Sij =1
2
(partuipartxj
+partujpartxi
) (41)
and the rate-of-rotation tensor Ω by
Ωij =1
2
(partuipartxjminus partujpartxi
) (42)
Note that the first principal invariant of S and the first and third principal invariants of Ω are zeroAlso note that the velocity gradient of a flow can simply be written G = S + Ω
Rather than considering the principal invariants we can also consider the combined invariants
I1 = tr(S2) I2 = tr(Ω2) I3 = tr(S3) I4 = tr(SΩ2) I5 = tr(S2Ω2) (43)
We can hence write the three principal invariants for a velocity field G in terms of the combinedinvariants
PG = 0
QG = minus1
2(I1 + I2)
RG =1
3(I3 + 3I4)
(44)
In this report the flows considered are the exact solutions to the incompressible Navier-Stokesequations described in Section 22 The combined invariants in Eq (43) are used to characterisethese solutions into different classes in Section 44 Both the principal invariants and combinedinvariants for each exact solution in Section 22 are given in Appendix A
Each of the subfilter-scale eddy-viscosity models described in Section 32 are written in terms ofthese five combined invariants
232 Vremanrsquos Characterisation of Flows
Vreman [16] classifies flows using the gradient of the velocity field of a flow Each lsquoclassrsquo of flows hasthe same number of non-zero elements in the gradient matrix For example
[partuipartxj
]=
0 0 0lowast 0 0lowast 0 0
(45)
where a lowast denotes a non-zero term represents a group of flows that lie within the class denoted byQ7 (see Appendix B) where the subscript represents the number of non-zero terms in the velocity
21
gradient Flows of the form in Eq (45) have a velocity field of the form
u1 = c1 (constant)
u2 = f2(x1)
u3 = f3(x1)
(46)
where f2(x1) and f3(x1) are such that the Navier Stokes equations Eq (1) are satisfied Notethat as the elements on the leading diagonal are zero the incompressibility condition Eq (5) isautomatically satisfied Some simple flow classes are given in Appendix B and contain a number ofthe solutions discussed in Section 22 This is mentioned in greater detail in Section 441
3 Conditions for Large-Eddy Simulation Models
31 Large-Eddy Simulation
Due to the nature of numerical simulation and the complexity of turbulent flows small details aremissed and information is lost when solving the Navier-Stokes equations The computational costof solving a turbulent flow is extremely high so a filtering operation is applied to the Navier-Stokesequations represented by a bar to reduce the computational cost of the simulation This is a linearoperation and preserves constants ie ρ = ρ and is also assumed to commute with differentiation
partuipartxj
=partuipartxj
(47)
The filtered Navier-Stokes equations are found by filtering (putting a bar over) the entire equations(see Eq (1)) and after evaluating are given by
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (48)
where
τij = uiuj minus uiuj (49)
is called the lsquotrue stressrsquo of the flow This however poses a problem as it is not clear what is meantby the term uiuj for a general filtering operation The true stress τij therefore must be replaced bya model τmod
ij in practical applications
32 Subfilter-Scale Models
We will now define the subflter-scale models denoted by τmodij to be used in place of the lsquodifficultrsquo
true stress τij There are a number of models available to use in the literature [11 p7] These willbe tested using the exact solutions to the Navier-Stokes equations and by checking conditions whichare defined in Sections 33 and 34
All the models barring one are of the same form That is each of the eddy-viscosity models arewritten
τmodij minus 1
3τmodkk δij = minus2νeSij (50)
with νe representing the eddy viscosity of a specific model For simplicity we usually just writeτmodij = minus2νeSij the extra term is included here to emphasise that the models are traceless Each
22
of the models we consider here can be expressed in terms of the rate of strain tensor S defined inEq (41) the rate of rotation tensor Ω defined in Eq (42) and the combined invariants defined inEq (43) The models are then defined as follows
Smagorinsky [12] νSe = (CSδ)2radic
2I1 (51)
Vreman [16] νVe = (CV δ)2
radicQGGT
PGGT (52)
QR [13 14 15] νQRe = (CQRδ)2 max0minusI3
I1 (53)
Vortex-Stretching [11] νV Se = (CV Sδ)2radic
2I1
(I5 minus 1
2I1I2
minusI1I2
)32
(54)
with the quantities
PGGT = I1 minus I2 QGGT =1
4(I1 + I2)2 + 4
(I5 minus
1
2I1I2
) (55)
There is a fifth and final model to be considered here It is of a different form to the other four andnot an eddy-viscosity model It is known as the Gradient model and is defined by
Gradient [4 9] τmodG = CGδ
(S2 minus Ω2 minus (SΩminus ΩS)
) (56)
The values C are model constants specific to each model and δ represents the mesh size of the LESBoth of these quantities are included here to properly define the models however in this projectthey will be ignored as they are constants and have no influence on the conditions to be defined
33 Conditions for Large-Eddy Simulation without Explicit Filtering
The filtering operation introduced in Eq (48) is just a formality for LES without explicit filteringThe idea here is that filtering does not actually take place but the term lsquoaddedrsquo to the Navier-Stokesequations in fact lsquosubtractsrsquo the difficult to simulate small-scale turbulent motion of the flow Inorder to consider this properly the terms ui and p in Eq (48) are replaced by vi and q It is hopedthat τmod
ij approximates τij so that vi and q approximate ui and p respectively So LES withoutexplicit filtering is given by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (57)
It is clear to see that the first four terms of this equation are exactly the same as the Navier-Stokesequations so for the exact solutions discussed in Section 22 the combination of these terms willequal zero What is then desirable is that the model term is also zero for these exact solutions sothat Eq (57) still holds In order to test models for exact solutions the condition
part
partxjτmodij (v) = 0 (58)
is checked If this condition were to hold it would imply that the model does not influence thevelocity field of a flow hence we call it the Velocity Field Condition There are a number of otherrestrictions that can be tested One such test comes from the differential form of the evolution ofthe kinetic energy of a flow given in Eq (10) In order to find this Eq (57) is multiplied by vi andterms are grouped to find
part
partt
(1
2vivi
)+ vj
part
partxj
(1
2vivi
)= minusvi
1
ρ
partq
partxi+ νvi
part2vipartxjpartxj
minus vipart
partxjτmodij (v) (59)
23
Then similarly to Eq (58) another condition is to check whether the model affects the kineticenergy of exact solutions this is equivalent to checking whether
vipart
partxjτmodij (v) = 0 (60)
We call this condition the Kinetic Energy Condition The third and final condition considered herecan also be found using the integral form for the equation for kinetic energy
d
dt
intV
1
2vividV =
intS
[minusvj
(1
2vivi
)minus vj
q
ρ+ 2νviSij + viτ
modij (v)
]njdS
+
intV
[minus2νSijSij minus
(minusSijτmod
ij (v))]dV
(61)
where the rate-of-strain tensor Sij was defined in Eq (41) V is a volume of fluid and dS meansthe integral is over the surface of that volume with unit normal nj Then this final condition checkswhether the model affects the total kinetic energy of exact solutions to the Navier-Stokes equationsthat is
Sijτmodij (v) = 0 (62)
We will call this condition the Dissipation Condition because Sijτmodij (v) is often referred to as the
subgrid dissipation
34 Conditions for Large-Eddy Simulation with Explicit Filtering
As in Eq (48) the Navier-Stokes equations can be filtered by applying the filtering operation to thewhole equation (putting a bar over all terms) and evaluating to find again that
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (63)
In Eq (57) the filtering operation was ignored and the elements ui p and τij(u) were replacedwith vi q and τmod
ij (v) respectively Now in LES with explicit filtering the filtering operation itselfsmooths over the small-scale turbulent motion of the simulated flow In this case the elements uip and τij(u) will be replaced with vi q and τmod
ij (v) respectively LES with explicit filtering is thengiven by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (64)
Note that Eqs (63) and (64) despite their similarity to Eq (57) no longer contain the Navier-Stokesequations as Eq (57) did The bar (filtering operation) over each term results in each exact solutionfound in Section 22 not necessarily satisfying the first four terms It is hence unknown whether ornot these terms will vanish for exact solutions Terms can still be compared similarly to Eqs (58)(60) and (62) However instead of comparing to zero we can now compare to the relevant termfrom the filtered Navier-Stokes equations So the terms
part
partxjτij (65)
and
part
partxjτmodij (v) (66)
24
are to be compared Although the latter term is relatively simple to compute the former could leadto some difficulties due to the differentiation of the true stress τij = uiuj minus uiuj So although someinformation could be gleaned from the inspection of these two terms more information can be foundby considering the analogue of Eq (62) So the true subfilter dissipation
Sijτij (67)
(where Sij is shorthand for Sij(v)) and the modelled subfilter dissipation
Sij(v)τmodij (v) (68)
are considered and have to be equal for the condition to hold These can again be derived from theintegral equation for kinetic energy of a filtered velocity field similarly to Eq (61)
4 Analysis of Large-Eddy Simulation Models
In this section we use a code in Mathematica the theory described in Section 3 and the exactsolutions to the Navier-Stokes equations found in Section 22 to check whether or not each of theexact solutions lsquoendorsesrsquo each model for LES both without and with explicit filtering
41 Large-Eddy Simulation Without Explicit Filtering
We start by using a code in Mathematica to test the Velocity Field condition in Eq (58) and theKinetic Energy condition in Eq (60) For all exact solutions to the Navier-Stokes equations inSection 22 the outputs for both conditions were identical so we have combined them into one table
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 ndash ndash 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 1 For the Kinetic Energy condition a lsquo0rsquo result means that vipartpartxj
τmodij (v) = 0 for respective
models and exact solutions whereas a lsquolowastrsquo result means that vipartpartxj
τmodij (v) 6= 0 For the Velocity
Field condition a lsquo0rsquo result means that partpartxj
τmodij (v) = 0 for respective models and exact solutions
whereas a lsquolowastrsquo result means that partpartxj
τmodij (v) 6= 0 The Gradient model can take both positive and
negative values so when the sign of the model is known it is included in the table instead of a lsquorsquoBoth the conditions are included in one table because for all cases the outputs were identical
We then test the Dissipation condition in Eq (62) for all exact solutions to the Navier-Stokesequations in Section 22
25
Dissipation Condition [Does Sijτmodij (v) = 0]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 + 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 2 A lsquo0rsquo result means that Sijτmodij (v) = 0 for respective models and exact solutions whereas
a lsquolowastrsquo result means that Sijτmodij (v) 6= 0 The Gradient model can take both positive and negative
values so when the sign of the model is known it is included in the table instead of a lsquorsquo
We then sum all the results to find the total number of endorsements for each model described inSection 32
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
4 6 8 9 9
lsquoDissipationrsquoEndorsements
2 5 7 7 9
Total Endorsements 6 11 15 16 18
Table 3 The results from Tables 1 and 2 are summarised to give a general overview of how manyendorsements each model receives This makes it much clearer to see the models most supported bythe respective and combined conditions with the Vortex-Stretching model being the most supportedand the Smagorinsky model the least
Testing the models in Section 32 using LES without explicit filtering has mixed results It isclear that the Smagorinksy model is not well endorsed by the flows in Section 22 with the onlyendorsements coming from the simplest flows that is Sol 1 (Linear Velocity) Sol 3 (Velocity as afunction of time) Sol 7 (Stagnation Flow) and Sol 15 (Rankine Vortex) The Vortex-Stretching-based model is the most endorsed flow with eighteen out of a possible thirty endorsements Thisagrees with our foreknowledge as this model was constructed to be as physically consistent as possiblein the paper by Silvis amp Verstappen [11]
42 Large-Eddy Simulation With Explicit Filtering
In this section we will compare the true subfilter dissipation Sijτij and the modelled subfilterdissipation Sij(v)τmod
ij (v) given in Eqs (67) and (68) respectively for each model τmodij and each
exact solution to the Navier-Stokes equation given in Section 22 We will not consider the conditionsinvolving differentiation in Eqs (65) and (66) due to the difficulty in calculating the derivative ofthe τij term in Eq (65)
26
421 Example of a Test for a Model
The true subfilter dissipation Sijτij of a flow given in Eq (67) cannot be explicitly computed asthe filtering operation is kept general What can be done however is to partially evaluate theexpressions using the attributes of a filtering operation mentioned in Eq (48)
The calculations of Sijτij for Sol 1 (Linear Velocity) are provided to illustrate the general ideaRecall that the velocity field of Sol 1 (Linear Velocity) is written in Eq (13) in vector notation as
u(x y z t) =
0ax+ bcx+ d
(69)
Then Sij = Sij(u) is computed simply by using its definition in Eq (41) Note that as filteringpreserves constants if there is an a it is replaced with a meaning that (in this case) there are alarge number of zero terms due to the frequency of 0 Hence
[Sij(u)
]=
[1
2
(partuipartxj
+partujpartxi
)]=
0 a
2c2
a2 0 0
c2 0 0
=
0 a
2c2
a2 0 0
c2 0 0
(70)
Note that as Sijτij is being computed which is equal to the sum of the element-wise multiplicationof the two matrices (or the trace of the matrix multiplication) that we donrsquot need to consider theelements of τij(u) with corresponding zero entry in Sij As τij(u) = uiuj minus uiuj and again asfiltering preserves constants it is found that
[τij(u)] =[uiuj minus uiuj
]=
lowast 0 middot ax+ b 0 middot cx+ d
ax+ b middot 0 lowast lowastcx+ d middot 0 lowast lowast
=
lowast 0 00 lowast lowast0 lowast lowast
(71)
where a lsquolowastrsquo just represents some element (not necessarily non-zero) Therefore the true subfilterdissipation
Sijτij = tr
0 a2
c2
a2 0 0c2 0 0
middotlowast 0 0
0 lowast lowast0 lowast lowast
= 0 (72)
Now the modelled subfilter dissipation Sij(u)τmodij (u) needs to be computed (recall that Sij =
Sij(u)) In this case let us use the QR model stated in Eq (53) Once again Sij(u) calculated inEq (70) needs to be used So we have that
τmodij = minus2(CQRδ)
2 max0minusI3I1
Sij (73)
As
I1 = tr(S2) =
1
2(a2 + c2) and I3 = tr(S
3) = 0 (74)
it holds that τmodij (u) = 0 so Sij(u)τmod
ij (u) = 0 This agrees with Eq (72) and hence Sol 1(Linear Velocity) endorses the QR model
27
422 Testing the Models
Now using a Mathematica code the exact solutions in Section 22 are again used to evaluate thecondition for the models using the process just described in Section 421 The results are describedin Table 4
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 4 All entries with a ldquo0rdquo represent a value of zero and a ldquolowastrdquo represents a value that is non-zeroThe ldquoExactrdquo row represents the value of the true subfilter dissipation Sijτij Every other row witha model name represents the value of the modelled subfilter dissipation Sij(u)τmod
ij (u) respectiveto that model A ldquordquo means that it is unclear as to the value - the sign of these values is oftenunknown and the lsquoMaxrsquo function in the definition of the QR model (in Eq (53)) makes it difficultto evaluate
Reading from Table 4 the Smagorinsky model is clearly endorsed by Sol 3 (Velocity as a function oftime) and Sol 15 (Rankine Vortex) as in both cases both Sijτij = 0 = Sij(u)τmod
ij (u) In this casewe say that Sol 3 (Velocity as a function of time) and Sol 15 (Rankine Vortex) lsquostronglyrsquo endorsethe Smagorinsky model The Smagorinsky model is not endorsed by Sol 1 (Linear Velocity) Sol 2(Planar Flow) or Sol 14 (Radially Dependent) however because Sij(u)τmod
ij (u) 6= 0 For Sol 4 to
13 both Sijτij 6= 0 6= Sij(u)τmodij (u) and so we will say that these exact solutions lsquoweaklyrsquo endorse
the Smagorinsky model
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 10 10 9 6
Table 5 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Interestingly the lsquostrongrsquo and lsquoweakrsquo endorsements give different results However more stock shouldbe put in the results of the lsquostrongerrsquo condition as it is completely clear that zero is equal to zeroIt is not necessarily clear whether or not non-zero equals non-zero as the values will depend on theparameters in each solution hence the lsquoweakerrsquo endorsement
The lsquostrongrsquo condition equally endorses the Vreman QR Gradient and Vortex-Stretching-basedmodels whereas the lsquoweakrsquo condition endorses Vreman and Smagorinsky the most closely followedby the Gradient model with the Vortex-Stretching-based model being the most opposed As this
28
completely contradicts the lsquostrongrsquo condition and the results in Section 41 it makes us question thevalidity of the weaker test The number of endorsements for the QR model is unknown because of thedefinition of the model in Eq (53) and our evaluation of the true subfilter dissipation Eq (67) Dueto the exact solutions we have found it is never clear whether or not the true subfilter dissipationis strictly positive or negative This means that when the true subfilter dissipation is found to benon-zero the lsquoMaxrsquo function in the QR model cannot be evaluated
43 Vremanrsquos True Subfilter Dissipation Claim
Using the theory mentioned in Sections 232 and 34 we can place the exact solutions to the Navier-Stokes equations into different classes defined by Vreman In doing this we can also attempt to verifyone of Vremanrsquos results [16] In Vremanrsquos paper he defines the class Qn to be the set of flows withn zero entries in the velocity gradient of the flow (the velocity gradient is defined in Eq (39))Vreman then claims that the only flows with zero true subfilter dissipation are a certain selection ofthose in the classesQ7Q8 orQ9 shown in appendix B and is generally non-zero for any other flows
The true subfilter dissipation of Sol 1 (Linear Velocity) was computed in Section 421 and wasfound to be zero in Eq (72) The velocity gradient of Sol 1 is as follows
GLV =
0 0 0a 0 0c 0 0
(75)
and hence Sol 1 (Linear Velocity) is in the required subset of Q7 which agrees with Vremanrsquos claimIn Section 422 the true subfilter dissipation was computed (determined to be zero or non-zero) forall of the exact solutions to the Navier Stokes equations in Section 22 These solutions can now belsquoplacedrsquo into each class Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a functionof time) and Sol 14 (Radially Dependent) were all found to have zero true subfilter dissipation andare all in the appropriate subsets of either Q7Q8 or Q9 This agrees with Vremanrsquos results
We can extend Vremanrsquos results however We found that Sol 15 (Rankine Vortex) had zero truesubfilter dissipation and yet is not in the subset of Q7 defined in Appendix B It is important tonote that the rate of strain tensor Sij in Eq (41) for Sol 15 (Rankine Vortex) is zero This is thereason the true subfilter dissipation is zero and the same reason that all models are zero for thisflow as all models are based on Sij
We also found that all of the remaining exact solutions to the Navier-Stokes equations are not inVremanrsquos specified classes (in Appendix Section B) They also have a non-zero subfilter dissipationwhich agrees with Vremanrsquos results in his paper [16]
44 Classes of Flows
It is important to consider that some exact solutions to the Navier-Stokes equations may have similarproperties For example Sol 1 (Linear Velocity) and Sol 2 (Planar Flow) are both very similarIn fact Sol 2 (Planar Flow) is a more general case of Sol 1 (Linear Velocity) We therefore mustgroup the different flows into classes in an attempt to reduce the bias that may occur when solutionsare similar This should aid us to further determine which of the models are most endorsed We willdo this in a number of different ways using the theory discussed in Section 23 We will group theflows with respect to Vremanrsquos Flow Classes given in appendix B and with respect to the Invariantsgiven in appendix A
29
441 Vremanrsquos Velocity Gradient Flow Classes
We now put our exact solutions to the Navier-Stokes equations into one of two classes Either Q0minus6or Q7minus9 where each class is defined intuitively as follows
Q0minus6 = Q0 cupQ1 cup middot middot middot cup Q6 (76)
Q7minus9 = Q7 cupQ8 cupQ9 (77)
where Qn is defined in Section 43 By looking at the gradient of each exact solution we can see thatQ7minus9 contains the six solutions Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as afunction of time) Sol 5 (2D Taylor) Sol 14 (Radially Dependent) and Sol 15 (Rankine Vortex)and that Q0minus6 contains the remaining nine We now check the models for LES without explicitfiltering by considering endorsements within each class Tables 6 and 7 are analogues of Table 3 andillustrate the results for each flow class The analogues of Tables 1 and 2 are Tables 22 and 23 andare in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 6 6 6
lsquoDissipationrsquoEndorsements
2 5 6 6 6
Percentage Endorsed 417 833 100 100 100
Table 6 Each entry in the top two rows represents the number of solutions in the class Q7minus9 thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Q0minus6 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 2 3 3
lsquoDissipationrsquoEndorsements
0 0 1 0 3
Percentage Endorsed 56 56 167 167 333
Table 7 Each entry in the top two rows represents the number of solutions in the class Q1minus6 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking results to 1 decimal place if necessary
All of the flows in the class Q7minus9 endorse the QR Gradient and Vortex-Stretching-based subfilter-scale models (see Table 6) We can hence see that this class fully endorses these three models equallyand does not endorse the Smagorinsky model The Vortex-Stretching eddy viscosity model has themost endorsements from any of the flows in the class Q0minus6 and the Smagorinksy and Vreman modelshave the fewest endorsements (see Table 7) We can hence say that this class mostly endorses theVortex-Stretching-based eddy-viscosity model note however that the percentage endorsement is very
30
low
Now let us consider LES with explicit filtering Tables 8 and 9 are analogues of Table 5 and illustratethe results for each flow class The analogue of Table 4 is Table 24 and is in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 1 1 0 0 0
Table 8 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Q0minus6 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 9 9 9 6
Table 9 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We have grouped Vremanrsquos classes into Q7minus9 and Q0minus6 this is equivalent to saying that we havegrouped Vremanrsquos classes into those with two or fewer non-zero velocity gradient elements and thosewith three or more non-zero velocity gradient elements respectively Therefore the solutions in Q7minus9are exactly those with a lsquo0rsquo in the lsquoExactrsquo column of Table 4 and Sol 5 (2D Taylor) ExcludingSol 5 (2D Taylor) these are the same as those five that endorse the Vreman QR Gradient andVortex-Stretching-based models in Table 5 We can see in Table 8 that the results for the lsquostrongrsquoendorsements are exactly the same as in Table 5 This is because all of the flows with zero subfilterdissipation are contained in the class Q7minus9 There are hence no lsquostrongrsquo endorsements in Table 9The majority of the lsquoweakrsquo endorsements come from Q0minus6 as this class contains all but one of thesolutions with non-zero true subfilter dissipation (given in Eq (67))
442 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
We now separate our exact solutions into classes again using Vremanrsquos flow classes This timehowever we define V0 to be the class of flows Vreman claims will always have zero true subfilterdissipation namely those classes mentioned in Appendix B We define Vlowast to be all the remaining
31
classes Then
V0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow)
Sol 3 (Velocity as a function of time) Sol 14 (Radially Dependent)(78)
Vlowast = Sol 4 (Axisymmetric Flow) Sol 5 (2D Taylor) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 13 (Sphere Translation) Sol 15 (Rankine Vortex)
(79)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 10 and 11 are analogues of Table 3 and illustrate the results for each flow class Theanalogues of Tables 1 and 2 are Tables 25 and 26 and are in Appendix C2
Endorsements for models using solutions in V0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 4 4 4 4
lsquoDissipationrsquoEndorsements
1 4 4 4 4
Percentage Endorsed 375 100 100 100 100
Table 10 Each entry in the top two rows represents the number of solutions in the class V0 thatendorse each model There are 4 solutions in this class so the percentages are worked out by dividingthe total of each column by 8 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Vlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 2 4 5 5
lsquoDissipationrsquoEndorsements
1 1 3 2 5
Percentage Endorsed 136 136 318 318 455
Table 11 Each entry in the top two rows represents the number of solutions in the class Vlowast thatendorse each model There are 11 solutions in this class so the percentages are worked out bydividing the total of each column by 22 and taking results to 1 decimal place if necessary
All of the flows in the class V0 endorse the Vreman QR Gradient and Vortex-Stretching-basedsubfilter-scale models (see Table 6) This class completely endorses these four models equally anddoes not endorse the Smagorinsky model The Vortex-Stretching-based eddy-viscosity model hasthe most endorsements from the class Vlowast and the Smagorinksy and Vreman models again have thefewest endorsements (see Table 7) As in Section 441 we say that this class mostly endorses the
32
Vortex-Stretching-based eddy-viscosity model but once again the percentage endorsement is ratherlow
Now let us consider LES with explicit filtering Tables 12 and 13 are analogues of Table 5 andillustrate the results for each flow class The analogue of Table 4 is Table 27 and is in Appendix C2
Endorsements for models using solutions in V0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 4 4 4 4
lsquoWeakrsquo Endorsements 0 0 0 0 0
Table 12 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Vlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 1 1 1 1
lsquoWeakrsquo Endorsements 10 10 9 6
Table 13 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We can see in Table 12 that the results for the lsquostrongrsquo endorsements are similar to those in Table 5excluding Sol 15 (Rankine Vortex) The four solutions in V0 with zero true subfilter dissipationnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)and Sol 14 (Radially Dependent) are hence the four that endorse the Vreman QR Gradientand Vortex-Stretching-based models in Table 12 Sol 15 (Rankine Vortex) is in the class Vlowast andhence there is one lsquostrongrsquo endorsement in Table 13 The class Vlowast contains all of the flows with anon-zero true subfilter dissipation (given in Eq (67)) and hence Table 13 contains all of the lsquoweakrsquoendorsements
443 Classes using the Third Principal Invariant
Sorting the classes using the invariants is a non-trivial task in itself There are a number of waysin which to do this using both the principal invariants in Eq (40) and the combined invariants inEq (43)
We start by looking at flows with third principal invariant RG equal to zero We again define twoclasses R0 and Rlowast where R0 consists of all flows with RG = 0 and Rlowast consists of all flows withRG 6= 0 Then using the information in Appendix A we see that there are nine flows in R0 That
33
is
R0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 8 (Vortex in Stagnation Flow)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)
(80)
Rlowast = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 13 (Sphere Translation)(81)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 14 and 15 are analogues of Table 3 and they illustrate the results for LES withoutexplicit filtering for each flow class The analogues of Tables 1 and 2 are Tables 28 and 29 and arein Appendix C3
Endorsements for models using solutions in R0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 7 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 278 556 778 722 778
Table 14 Each entry in the top two rows represents the number of solutions in the class R0 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking the result to one decimal place if necessary
Endorsements for models using solutions in Rlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 2 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 83 83 83 167 333
Table 15 Each entry in the top two rows represents the number of solutions in the class Rlowast thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking the result to one decimal place if necessary
The class R0 endorses both the Vortex-Stretching-based model and the QR model the most andSmagorinsky the least in the case for LES without explicit filtering In the Rlowast class the Vortex-Stretching-based model is again the most endorsed followed by the Gradient model however thepercentages are again small
34
Now let us consider LES with explicit filtering Tables 16 and 17 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 30 and is in Appendix C3
Endorsements for models using solutions in R0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 4 4 3 2
Table 16 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Rlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 6 6 6 4
Table 17 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 we can see in Table 16 that the results for the lsquostrongrsquo endorsements are exactly thesame as in Table 5 This is because all of the flows with zero subfilter dissipation are contained in theclass R0 As in Table 9 there are no lsquostrongrsquo endorsements in Table 17 The lsquoweakrsquo endorsementshowever are split more evenly between the two classes now than in Vremanrsquos classes in Sections 441and 442 We find that the Smagorinsky and Vreman models still seem to be the most endorsed inboth classes
444 Two-component Flow Classes
Another way in which the flows can be grouped into classes is by determining whether a flow is atwo-component flow or not We define a two-component flow to have the property that the combinedprincipal invariants of a flow in Eq (43) satisfy
I3 = I4 = I5 minus1
2I1I2 = 0 (82)
We are also interested in considering flows with zero gradient model subgrid dissipation These flowshave the property that
I3 minus I4 = 0 (83)
What we find though is that for all the solutions in Section 22 if a flow is a two-component flowthen it also satisfies I3 minus I4 = 0 It may be interesting to note that although I3 minus I4 6= 0 for manysolutions there are a number with I3 equal to a multiple of I4 Although not considered in thisreport this may also be a class of interest We find that the two-component flows are
C2 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)(84)
35
and we call the remaining flows three-component flows They are
C3 = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex) Sol 11 (Pseudo-Motion Shots)
Sol 13 (Sphere Translation)
(85)
Note that Sol 8 (Vortex in Stagnation Flow) is defined to be a planar flow this does not mean that itis a two-component flow in this sense however because its two non-zero velocity components dependon three coordinates We now check the models for LES without explicit filtering by consideringendorsements within each class Tables 18 and 19 are analogues of Table 3 and they illustrate theresults for LES without explicit filtering for each flow class The analogues of Tables 1 and 2 areTables 31 and 32 and are in Appendix C4
Endorsements for models using solutions in C2 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 6 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 357 714 100 857 100
Table 18 Each entry in the top two rows represents the number of solutions in the class C2 thatendorse each model There are 7 solutions in this class so the percentages are worked out by dividingthe total of each column by 14 and taking the result to one decimal place if necessary
Endorsements for models using solutions in C3 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 3 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 63 63 63 188 250
Table 19 Each entry in the top two rows represents the number of solutions in the class C3 thatendorse each model There are 8 solutions in this class so the percentages are worked out by dividingthe total of each column by 16 and taking the result to one decimal place if necessary
The class C2 equally endorses the QR and Vortex-Stretching-based models and once again theSmagorinsky model is the least endorsed As with the previous classes the class C3 has such lowpercentages that it does not seem to endorse any models We start to see a trend appearing here inthat the flow classes that contain the more complicated flows namely Q0minus6 (Eq (76)) Vlowast (Eq (79))Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages The classes that tendto contain the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78)) R0 (Eq (80)) and C2 (Eq (84))all seem to favour the Vortex-Stretching-based model or the QR model
36
Now let us consider LES with explicit filtering Tables 20 and 21 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 33 and is in Appendix C4
Endorsements for models using solutions in C2 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 2 2 0 1 0
Table 20 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in C3 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 8 8 8 6
Table 21 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 and Table 16 we can see in Table 20 that the results for the lsquostrongrsquo endorsements areexactly the same as in Table 5 This is again because all of the flows with zero subfilter dissipationare contained in the class C2 There are two flows with non-zero true subfilter dissipation (givenin Eq (67)) in C2 and this shows in the lsquoweakrsquo endorsements in Table 20 There are no lsquostrongrsquoendorsements in Table 21 as all the zero subfilter dissipation flows are in C2
5 Conclusions
In this report we studied incompressible turbulent flows We found a number of exact solutions to theNavier-Stokes equations each representing a relatively simplistic fluid flow Large-eddy simulation(LES) is used to model turbulent flows by smoothing over the complex small-scale motion Weintroduced a number of subfilter-scale models to approximate this small-scale motion and testedthem using the found exact solutions to the Navier-Stokes equations We first considered LESwithout explicit filtering and used the Navier-Stokes equations to derive conditions for the subfilter-scale models namely the Velocity Field condition (Eq (58)) the Kinetic Energy condition (Eq (60))and the Dissipation condition (Eq (62)) Similarly we considered LES with explicit filtering andagain used the Navier-Stokes equations to derive more conditions for the models in this case wedecided to just take one condition the Dissipation condition (Eqs (67) and (68)) as the others leadto difficulty in their evaluation By lsquorunningrsquo the exact solutions to the Navier-Stokes equationsthrough each condition for the subfilter-scale models we found that the Vortex-Stretching-basededdy-viscosity model in Eq (54) was most endorsed by the tests in LES without explicit filteringWe found that four out of the five subfilter-scale models namely the Vreman QR Gradient andVortex-Stretching-based models were equally endorsed by the lsquostrongrsquo tests in LES with explicitfiltering with the Smagorinsky model being the least endorsed The results we found for the lsquoweakrsquocondition were completely opposite to those found in both the lsquostrongrsquo condition and in LES withoutexplicit filtering This led us to conclude that this test was not one in which we should put our trust
37
We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses
In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes
This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold
38
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
Abstract
We study the behaviour of turbulent fluid flows behaviour that is governed by the Navier-Stokes equations Due to the extensive detail entailed in a turbulent flow it is difficult tosolve the Navier-Stokes equations numerically A large-eddy simulation (LES) using a filteringoperation solves for the large-scale motions in a flow and smooths over small-scale motionhence requiring a turbulence model for these small-scale motions Using conditions derivedfrom and exact solutions to the Navier-Stokes equations we investigate a number of turbulencemodels for LES both with and without explicit filtering We found that the Vortex-Stretching-based eddy-viscosity model was most endorsed in the case without explicit filtering and thatthe Vreman QR Gradient and Vortex-Stretching-based models were equally endorsed by thecase with explicit filtering with the Smagorinsky model being the least endorsed To reducethe bias that may arise when exact solutions are similar we then considered classes of flowsto further determine which models are the most endorsed We considered classes based onVremanrsquos flow classes and on the principal and combined invariants of flows Once again theVortex-Stretching-based eddy-viscosity model was usually the most endorsed As more exactsolutions to the Navier-Stokes equations are found they can be easily added to the report tofurther endorse or oppose the models We also considered Vremanrsquos paper [16] and did notfind anything to contradict his results We did extend his results slightly by finding a solutionoutside of his zero-subfilter-dissipation classes with zero subfilter dissipation
2
Contents
1 Introduction 4
2 The Navier-Stokes Equations 521 Introduction 522 Exact Solutions of the Navier-Stokes Equations 623 Characterisation of Flows 20
231 Characterising Solutions using Invariants 20232 Vremanrsquos Characterisation of Flows 21
3 Conditions for Large-Eddy Simulation Models 2231 Large-Eddy Simulation 2232 Subfilter-Scale Models 2233 Conditions for Large-Eddy Simulation without Explicit Filtering 2334 Conditions for Large-Eddy Simulation with Explicit Filtering 24
4 Analysis of Large-Eddy Simulation Models 2541 Large-Eddy Simulation Without Explicit Filtering 2542 Large-Eddy Simulation With Explicit Filtering 26
421 Example of a Test for a Model 27422 Testing the Models 28
43 Vremanrsquos True Subfilter Dissipation Claim 2944 Classes of Flows 29
441 Vremanrsquos Velocity Gradient Flow Classes 30442 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes 31443 Classes using the Third Principal Invariant 33444 Two-component Flow Classes 35
5 Conclusions 37
Appendices 39
A Invariants of Flows 39
B Vremanrsquos Flow Classes 46
C Extended Tables for Flow Classes 47C1 Vremanrsquos Velocity Gradient Flow Classes 47C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes 48C3 Classes using the Third Principal Invariant 49C4 Two-Component Flow Classes 50
3
1 Introduction
In this project we will study and test models for turbulent fluid flows There is no all-encompassingdefinition of a turbulent flow there are however a number of characterising properties A turbulentflow is said to be irregular in that it appears to be random Turbulent flows are chaotic and henceusually described statistically due to the difficulty in a deterministic approach Turbulent flowsare diffusive in that there is a high rate of momentum heat and mass transfer They require apersistent source of input energy in order to sustain the turbulent nature of the flow as kineticenergy is quickly converted into internal energy A large Reynolds number is characteristic of aturbulent flow [5]
The Navier-Stokes equations are a set of fundamental governing equations for any Newtonian fluidAn exact solution to the Navier-Stokes equations is in the form of a flow velocity and a pressureterm and hence a fluid flow can be completely described by an exact solution to the Navier-Stokesequations It is still a non-trivial task to find such solutions although there are a number in literature[17]
The Navier-Stokes equations have no closed form solution resulting in difficulty when analysingcomplicated flows in complex geometries This results in difficulty when solving the Navier-Stokesequations explicitly So a mathematical model known as Large Eddy Simulation (LES) can insteadbe used to model turbulent flows LES uses what is known as a filtering operation to reduce thecomputational cost of a simulation by ignoring small-scale motion What this means however isthat information about these small-scale motions is lost This information is often non-trivial sosubfilter-scale models (or turbulence models) are used in order to approximate the effects of small-scale motions on the large-scale fluid flow
In this project we will use the behaviour of exact solutions to the Navier-Stokes equations to testsubfilter-scale models Applying a subfilter-scale model to a flow has two effects One is a bodyforce on the velocity field called a subfilter force and the other is a transfer of energy from largeto small scales of motion called a subfilter dissipation This results in respectively an extra termbeing added to the Navier-Stokes equations and to the equation of the kinetic energy of a flow
We will then consider conditions for subfilter-scale models in two different cases In LES withoutexplicit filtering we will consider how a subfilter-scale model behaves for an exact solution to theNavier-Stokes equations by checking if the subfilter force and subfilter dissipation are zero for anexact solution If so this can be interpreted as the model being inactive and hence this exactsolution can be said to lsquoendorsersquo the model In LES with explicit filtering we compare a model termwith a term in the filtered Navier-Stokes equations (called the true turbulent stress of a flow) forexact solutions If the two terms are both zero we will conclude that the exact solution lsquostronglyrsquoendorses the model and if the two terms are both non-zero we will conclude that the exact solutionlsquoweaklyrsquo endorses the model
We will also consider the classification of fluid flows in order to further inspect the models andreduce the bias that may occur when solutions have similar properties We will use properties of theinvariants of a flow to characterise flows in a coordinate-independent way Using Vremanrsquos paperldquoAn eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory and applicationsrdquo[16] we will place the fluid flows into Vremanrsquos flow classes We will also determine whether ourresults say anything beyond Vremanrsquos work
4
2 The Navier-Stokes Equations
21 Introduction
The incompressible Navier-Stokes equations are a set of four equations that model the flow of anyfluid in three-dimensional space The standard set of Navier-Stokes equations can be written in twoforms namely by using Einsteinrsquos summation convention in what we call index notation and byusing standard vector notation In index notation they are given by
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
i = 1 2 3 j = 1 2 3 (1)
These equations have four variables given by the velocity of the flow u = (u1 u2 u3) and thepressure p The vector x = (x1 x2 x3) represents position t represents time ρ corresponds to themass density and ν is the kinematic viscosity of the fluid
The Navier-Stokes equations can be written in vector notation in the following way First considerthe convection term
ujpartuipartxjequiv
3sumj=1
ujpartuipartxj
= u1partuipartx1
+ u2partuipartx2
+ u3partuipartx3
=
(u1
part
partx1+ u2
part
partx2+ u3
part
partx3
)ui
= (u middot nabla)ui
(2)
Similarly for the diffusion term
νpart2uipartxjpartxj
equiv ν3sumj=1
part2uipartxjpartxj
= νpart2uipartx1partx1
+part2uipartx2partx2
+part2uipartx3partx3
= ν
(part2
partx1partx1+
part2
partx2partx2+
part2
partx3partx3
)ui
= νnabla2ui
(3)
It therefore holds that in vector notation the Navier-Stokes equations are given by
partu
partt+ (u middot nabla)u = minus1
ρnablap+ νnabla2u (4)
It is clear that there is a problem here there are three equations and four unknowns In order toremedy this the equation for conservation of mass or the incompressibility condition is consideredThis is given by
partuipartxi
= 0 (5)
in index notation and
nabla middot u = 0 (6)
5
in vector notation
There are now four equations in four unknowns and it is possible to find solutions Althoughthere is no known general solution there are specific cases in which exact solutions to the Navier-Stokes equations can be found In order to make finding solutions slightly simpler the Navier-Stokesequations can be rescaled by introducing dimensionless variables
Let
ulowasti =uiv xlowasti =
xi` tlowast =
t
`v plowast =
p
ρv2 (7)
where ` and v are typical length and velocity scales respectively Then dropping the star notationfor ease of reading the scaled incompressible Navier-Stokes equations are
partuipartt
+ ujpartuipartxj
= minus partp
partxi+
1
Re
part2uipartxjpartxj
i = 1 2 3 j = 1 2 3
partuipartxi
= 0
(8)
where Re is the Reynolds number and is defined by the dimensionless Re =v`
ν Similarly in vector
notation the scaled incompressible Navier-Stokes equations are
partu
partt+ (u middot nabla)u = minusnablap+
1
Renabla2u
nabla middot u = 0(9)
By multiplying the incompressible Navier-Stokes equations Eq (8) by ui and grouping terms thedifferential form of the evolution of the kinetic energy of a flow can be found
part
partt
(1
2uiui
)+ uj
part
partxj
(1
2uiui
)= minusui
partp
partxi+
2
Reui
part2uipartxjpartxj
(10)
The differential form of the evolution of the kinetic energy of a flow will be used later on in thisreport to derive conditions for the subfilter-scale models
22 Exact Solutions of the Navier-Stokes Equations
By using mathematical inspection and taking results found in literature we found a number of exactsolutions to the Navier-Stokes equations The majority are steady-state solutions in that they areindependent of time We decided to consider exact solutions to the Navier-Stokes equations withoutboundary conditions in an attempt to keep the results general and applicable to a larger numberof situations All of the stated exact solutions below have been checked using a code written inMathematica In the solutions below the notation (x y z) for the Cartesian coordinate system isused analogously with (x1 x2 x3) to aid in reading comprehension and in the conversion betweenindex and vector notation
Solution 1 Linear Velocity
If the diffusion term in the Navier-Stokes equations Eq (1) is taken to be zero ie
part2uipartxjpartxj
= 0 (11)
6
then each component of the velocity can be written as a linear function of x1 x2 and x3 In indexnotation
ui = aijxj + βi (12)
If pressure is assumed to be a function of time t (so that the pressure terms in the Navier-Stokesequations are zero) then conditions can be found on the coefficients to find that the following is asolution
u1(x y z t) = 0
u2(x y z t) = ax+ b
u3(x y z t) = cx+ d
p(x y z t) = f(t)
(13)
where a b c d isin R and f Rrarr R In fact this solution is actually a specific case of Sol 2 (PlanarFlow)
Figure 1 An illustration of the flow in the xy-plane for Sol 1 (Linear Velocity) described inEq (13) with parameter values a = b = c = d = 1
Solution 2 Planar Flow
Using the text by Barbato et al [2 p62] and modifying the solution such that it now takes threedimensions into account another solution is found
u1(x y z t) = Az2 +Bz + C
u2(x y z t) = Dz2 + Ez + F
u3(x y z t) = 0
p(x y z t) =2A
Rex+
2D
Rey + f(t)
(14)
7
where ABCDE F isin R and f R rarr R It is clear to see that if A = D = 0 then this is in factthe same as Eq (13) in Sol 1 In fact we believe (although not shown here) that this is the mostgeneral solution of the form
ui(xi xj xk t) = Ax2k +Bxk + C
uj(xi xj xk t) = Dx2k + Exk + F
uk(xi xj xk t) = 0
(15)
for i 6= j 6= k 6= i
Figure 2 An illustration of the flow in the yz-plane for Sol 2 (Planar Flow) described inEq (14) with parameter values A = B = C =D = E = F = 1
Solution 3 Velocity as a function of time t
If each component of the velocity is taken to be a function of time t and independent of positionthen the incompressibility condition is automatically satisfied A condition can be imposed on thepressure term in order to satisfy the momentum equations Thus a solution is as follows
u1(x y z t) = f(t)
u2(x y z t) = g(t)
u3(x y z t) = h(t)
p(x y z t) = minusxpartf(t)
parttminus y partg(t)
parttminus z parth(t)
partt
(16)
where f g h Rrarr R In fact what is written here is the generalised Galilean transformation of thezero solution [10] We cannot produce a plot for this flow as the functions are completely unspecified
8
Solution 4 Axisymmetric Flow
Using the text by Irmay [7] a solution is defined as follows
u1(x y z t) = kxz
u2(x y z t) = kyz
u3(x y z t) = minuskz2
p(x y z t) = minus2kz
Reminus k2z4
2
(17)
where k isin R This is said to represent an axisymmetric flow with hyperbolic streamlines asymptoticto the z-axis and to the boundary plane z = 0 These asymptotes are visible in Fig 3
Figure 3 An illustration of the flow in theyz-plane for Sol 4 (Axisymmetric Flow)described in Eq (17) with parameter k = 1
Figure 4 An illustration of the flow in thexy-plane for Sol 4 (Axisymmetric Flow)described in Eq (17) with parameter k = 1
Solution 5 2D Taylor Solutions
Again using the text by Barbato et al [2 p63ndash64] a solution is defined using a stream functionψ = ψ(x y t) and by writing
u1 = minuspartψparty
u2 =partψ
partx u3 = 0 (18)
In the text the stream function is not given however it was worked out that
ψ = minuseminusπ2t
Re (cos(πx) + cos(πy))
π (19)
The solution used is of the following form note however that in the text the pressure term differsin that cos(πy) is replaced by sin(πy) (when testing this solution in Mathematica the solution below
9
was instead found)
u1(x y z t) = minus sin(πy) eminus(π2t)Re
u2(x y z t) = sin(πx) eminus(π2t)Re
u3(x y z t) = 0
p(x y z t) = minus cos(πx) cos(πy) eminus(2π2t)Re
(20)
Figure 5 An illustration of the flow in the xy-plane for Sol 5 (2D Taylor) described in Eq (20)with Re = 1 and time t = 0
Solution 6 Generalized Beltrami Flow
Again using the text by Barbato et al [2 p65] and now considering solutions with nonndashtrivialterms in all dimensions a particular form of the generalized Beltrami flow is found namely
u1(x y z t) = (A sin(πz) + C cos(πy)) eminus(π2t)Re
u2(x y z t) = (B sin(πx) +A cos(πz)) eminus(π2t)Re
u3(x y z t) = (C sin(πy) +B cos(πx)) eminus(π2t)Re
p(x y z t) = minus[(AB cos(πz) sin(πx) +BC cos(πx) sin(πy)
+AC cos(πy) sin(πz)] eminus(2π2t)Re
(21)
where ABC isin R
10
Figure 6 An illustration of the flow in the yz-plane for Sol 6 (Generalized Beltrami Flow)described in Eq (21) with parameters A = B =C = 1 and time t = 0
Solution 7 Stagnation Flow
In the book by Drazin and Riley [6 p40] an exact solution is considered which uses a stream functionψ = ψ(y z) They suggest that the following velocity field is a solution
u1(x y z t) =a
kx
u2(x y z t) = minusaky +
k
Re
partψ(y z)
partz
u3(x y z t) = minus k
Re
partψ(y z)
party
(22)
Here we choose the stream function to be ψ(y z) = yz Then a solution is given by
u1(x y z t) =a
kx
u2(x y z t) =
(minusak
+k
Re
)y
u3(x y z t) = minus kzRe
p(x y z t) = minus1
2
((axk
)2minus((
k
Reminus a
k
)y
)2
minus(kz
Re
)2)
(23)
where a is a measure of the strength of the stagnation flow and k is a constant chosen in the textto specify the distance between vortices There are no vortices in this particular solution however
11
Figure 7 An illustration of the flow in the yz-plane for Sol 7 (Stagnation Flow) describedin Eq (23) with parameter values a = Re = 1and k = 2
Figure 8 An illustration of the flow in the xy-plane for Sol 7 (Stagnation Flow) describedin Eq (23) with parameter values a = Re = 1and k = 2
Solution 8 Vortex in Stagnation Flow
As in Sol 7 (Stagnation Flow) we now consider the same solution with a different stream functionnamely ψ(z) = cos z Then given the condition that k = aRe and adding a pressure term Eq (22)becomes
u1(x y z t) =1
Rex
u2(x y z t) = minus 1
Rey minus k sin z
u3(x y z t) = 0
p(x y z t) = minus 1
2Re2(x2 + y2)
(24)
where a and k are as in Sol 7 (Stagnation Flow)
12
Figure 9 An illustration of the flow in the yz-plane for Sol 8 (Vortex in Stagnation Flow)described in Eq (24) with parameter valuesk = 3 and Re = 10
Figure 10 An illustration of the flow in thexy-plane for Sol 8 (Vortex in StagnationFlow) described in Eq (24) with parametervalues k = 3 and Re = 10
Solution 9 Burgers Vortex
The Burgers Vortex solution is given in cylindrical coordinates in the book by Lautrup [8 p 385]as follows
ur(r ϕ z t) = minus 2r
c2Re
uϕ(r ϕ z t) =Ωc2
r
(1minus eminusr
2c2)
uz(r ϕ z t) =4z
c2Re
p(r ϕ z t) = minusc2Ω2F
(r2c2
)2
minus 2
Re2r2 + 4z2
c4
(25)
where
F (ξ) =
infinintξ
(1minus eminusx)2
x2dx (26)
This solution is not particularly nice to look at however most of the unpleasantness comes fromthe pressure term which (in this project) is only required in order to validate the solution For thisproject all solutions are required in Cartesian coordinates the velocity field of the Burgers Vortexhas hence been converted into Cartesian coordinates
u1(x y z t) = minus 2x
c2Reminus Ω
h(x y)(1minus exp (minush(x y))) y
u2(x y z t) = minus 2y
c2Re+
Ω
h(x y)(1minus exp (minush(x y)))x
u3(x y z t) =4z
c2Re
(27)
13
where
h(x y) =x2 + y2
c2 (28)
and Ω and c are constants
Figure 11 An illustration of the flow for Sol9 (Burgers Vortex) described in Eq (27) withparameter values c = 15 Re = 20 and Ω = 10A slice of the xy-plane at z = 0 is shown
Solution 10 Another Vortex Solution
There is another vortex solution more simply formulated than the Burgers Vortex yet with asingularity at r = 0 as a result of the uϕ term It is again given in cylindrical coordinates
ur(r ϕ z t) = minus 2r
c2Re
uϕ(r ϕ z t) =a
r
uz(r ϕ z t) =4z
c2Re
p(r ϕ z t) = minus a2
2r2minus 2r2 + 8z2
c2Re2
(29)
which can also be converted into Cartesian coordinates
14
u1(x y z t) = minus 2x
c2Reminus ay
x2 + y2
u2(x y z t) = minus 2y
c2Re+
ax
x2 + y2
u3(x y z t) =4z
c2Re
(30)
where a and c are constants
Figure 12 An illustration of the flow for Sol9 (Burgers Vortex) described in Eq (27) withparameter values c = 15 Re = 20 and a = 10 Aslice of the xy-plane at z = 0 is shown
Solution 11 Pseudo-Motion Shots of the Second Kind
In the book by Berker [3 p 94] a solution is given in cylindrical coordinates by
ur(r ϕ z t) =2
Re
1
r
uϕ(r ϕ z t) = A1r3 +B1r +
C1
r
uz(r ϕ z t) = A2r2 log(r) +B2r
2 + C2
p(r ϕ z t) =A2
1r6
6+A1B1r
4
2+A1C1r
2 +B2
1r2
2minus C2
1
2r2minus 2
r2Re2+ 2B1C1 log(r)
+2A2z
Reminus 4B1ϕ
Re
(31)
where A1 A2 B1 B2 C1 C2 are constants Eq (31) when converted into Cartesian coordinates is
15
u1(x y z t) =1
Re
2xminus C1Rey
x2 + y2minusA1y
(x2 + y2
)minusB1y
u2(x y z t) =1
Re
2y + C1Rex
x2 + y2+A1x
(x2 + y2
)+B1x
u3(x y z t) =1
2A2
(x2 + y2
)log(x2 + y2
)+B2
(x2 + y2
)+ C2
(32)
Figure 13 An illustration of the flow in thexy-plane for Sol 11 (Pseudo-Motion Shots)described in Eqs (31) and (32) with parametervalues A1 = B1 = C1 = 1 A2 = B2 = C2 = 2and Re = 10
Figure 14 An illustration of the flow in theyz-plane for Sol 11 (Pseudo-Motion Shots)described in Eqs (31) and (32) with parametervalues A1 = B1 = C1 = 1 A2 = B2 = C2 = 2and Re = 10
Solution 12 Flows With Constant Whirl
In the book by Berker [3 p 143] another solution is given in cylindrical coordinates by
ur(r ϕ z t) = minus 1
Re
1
r
uϕ(r ϕ z t) =1
rradicte(minus
12 g(t))
uz(r ϕ z t) = 0
p(r ϕ z t) =1
4
2(minus e
minusg(t)
t minus 1Re2
)r2
minus 1
t2Re
infinintminusg(t)
eminust
tdt
(33)
where we define g(t) =r2Re
2t When converted into Cartesian coordinates we find
16
u1(x y z t) = minus 1
x2 + y2
(yradict
exp
(minus
Re(x2 + y2
)4t
)+
x
Re
)
u2(x y z t) =1
x2 + y2
(xradict
exp
(minus
Re(x2 + y2
)4t
)minus y
Re
)
u3(x y z t) = 0
(34)
Figure 15 An illustration of the flow for Sol12 (Flows with Constant Whirl) described inEqs (33) and (34) with time t = 1 and Re = 1 Aslice of the xy-plane at z = 0 is shown as velocityin the z-direction is zero this is the same for anyvalue of z
17
Solution 13 Rectilinear and Uniform Translation of a Sphere
In the book by Berker [3 p 228] another solution is given in Cartesian coordinates by
u1(x y z t) =3
4
axz
R3
(a2
R2minus 1
)
u2(x y z t) =3
4
ayz
R3
(a2
R2minus 1
)
u3(x y z t) = 1minus 3
4
a
Rminus 1
4
a3
R3minus 3
4
az2
R3
(a2
R2minus 1
)
p(x y z t) =3a(aminusR)(a+R)
32R10Re
((R2Re
(x2 + y2
)minus 2R2Rez2
) (a3 + 3aR2 minus 4R3
)+ 3aRez4
(R2 minus a2
)minus 16R5z
)
(35)
where a and R are constants
Figure 16 An illustration of the flow in the yz-plane for Sol 13 (Sphere Translation) describedin Eq (35) with parameter values a = minus2 andRe = 3
Solution 14 Radially Dependent Flow
A solution with velocity in the z-direction dependent only on the distance from the z-axis is as
18
follows in cylindrical coordinates
ur(r ϕ z t) = 0
uϕ(r ϕ z t) = 0
uz(r ϕ z t) = A(r2 minus a2
)
p(r ϕ z t) =4Az
Re
(36)
where A and a are constants In Cartesian coordinates the solution is just the same with u1(x y z t)and u2(x y z t) both zero and the r2 term replaced with x2 + y2 in u3(x y z t)
Figure 17 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus1 1] withparameter values a = 1 A = 4 and Re = 10
Figure 18 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus2 2] withparameter values a = 1 A = 4 and Re = 10
Solution 15 Rankine Vortex
A solution discovered by Rankine [1] is a vortex flow of the following form
ur(r ϕ z t) = 0
uϕ(r ϕ z t) =ar
2πR2
uz(r ϕ z t) = 0
p(r ϕ z t) =a2r2
8π2R4
(37)
19
where a and R are constants In Cartesian coordinates the flow has the form
u1(x y z t) = minus ay
2πR2
u2(x y z t) =ax
2πR2
u3(x y z t) = 0
p(x y z t) =a2(x2 + y2
)8π2R4
(38)
This flow is actually a rigid body rotational flow in that the angular rotational velocity is uniformand the velocity of the flow increases proportionally with distance from the z-axis
Figure 19 An illustration of the flow in the xy-plane for Sol 15 (Rankine Vortex) described inEq (37) and Eq (38) with parameter values a = 1and R = 1
radic2π
23 Characterisation of Flows
It is important to be able to classify flows in a way that sorts them into distinct groups This aidsin the clear identification and general study of flows It is also a very important way of recognisingflows with similar properties In this subsection we introduce two distinct ways in which flows canbe characterised
231 Characterising Solutions using Invariants
A principal invariant of a tensor is a coefficient of the characteristic polynomial of that tensorInvariants as their name suggests are invariant under rotations of the coordinate system in thattheir values do not change when the tensor is lsquorotatedrsquo into a different set of coordinates Thevelocity gradient of a flow G is defined as the Jacobian of the velocity vector given by
Gij =
[partuipartxj
] (39)
20
for i j = 1 2 3 There are three principal invariants of the velocity gradient of a fluid flow G givenby
PG = tr(G)
QG =1
2
((tr(G))2 minus tr
(G2))
RG = det(G)
(40)
Incompressibility of flows (Eq (5)) ensures that PG = 0 hence the first principal invariant is zerofor all of the flows in Section 22 We now define the rate-of-strain tensor S by
Sij =1
2
(partuipartxj
+partujpartxi
) (41)
and the rate-of-rotation tensor Ω by
Ωij =1
2
(partuipartxjminus partujpartxi
) (42)
Note that the first principal invariant of S and the first and third principal invariants of Ω are zeroAlso note that the velocity gradient of a flow can simply be written G = S + Ω
Rather than considering the principal invariants we can also consider the combined invariants
I1 = tr(S2) I2 = tr(Ω2) I3 = tr(S3) I4 = tr(SΩ2) I5 = tr(S2Ω2) (43)
We can hence write the three principal invariants for a velocity field G in terms of the combinedinvariants
PG = 0
QG = minus1
2(I1 + I2)
RG =1
3(I3 + 3I4)
(44)
In this report the flows considered are the exact solutions to the incompressible Navier-Stokesequations described in Section 22 The combined invariants in Eq (43) are used to characterisethese solutions into different classes in Section 44 Both the principal invariants and combinedinvariants for each exact solution in Section 22 are given in Appendix A
Each of the subfilter-scale eddy-viscosity models described in Section 32 are written in terms ofthese five combined invariants
232 Vremanrsquos Characterisation of Flows
Vreman [16] classifies flows using the gradient of the velocity field of a flow Each lsquoclassrsquo of flows hasthe same number of non-zero elements in the gradient matrix For example
[partuipartxj
]=
0 0 0lowast 0 0lowast 0 0
(45)
where a lowast denotes a non-zero term represents a group of flows that lie within the class denoted byQ7 (see Appendix B) where the subscript represents the number of non-zero terms in the velocity
21
gradient Flows of the form in Eq (45) have a velocity field of the form
u1 = c1 (constant)
u2 = f2(x1)
u3 = f3(x1)
(46)
where f2(x1) and f3(x1) are such that the Navier Stokes equations Eq (1) are satisfied Notethat as the elements on the leading diagonal are zero the incompressibility condition Eq (5) isautomatically satisfied Some simple flow classes are given in Appendix B and contain a number ofthe solutions discussed in Section 22 This is mentioned in greater detail in Section 441
3 Conditions for Large-Eddy Simulation Models
31 Large-Eddy Simulation
Due to the nature of numerical simulation and the complexity of turbulent flows small details aremissed and information is lost when solving the Navier-Stokes equations The computational costof solving a turbulent flow is extremely high so a filtering operation is applied to the Navier-Stokesequations represented by a bar to reduce the computational cost of the simulation This is a linearoperation and preserves constants ie ρ = ρ and is also assumed to commute with differentiation
partuipartxj
=partuipartxj
(47)
The filtered Navier-Stokes equations are found by filtering (putting a bar over) the entire equations(see Eq (1)) and after evaluating are given by
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (48)
where
τij = uiuj minus uiuj (49)
is called the lsquotrue stressrsquo of the flow This however poses a problem as it is not clear what is meantby the term uiuj for a general filtering operation The true stress τij therefore must be replaced bya model τmod
ij in practical applications
32 Subfilter-Scale Models
We will now define the subflter-scale models denoted by τmodij to be used in place of the lsquodifficultrsquo
true stress τij There are a number of models available to use in the literature [11 p7] These willbe tested using the exact solutions to the Navier-Stokes equations and by checking conditions whichare defined in Sections 33 and 34
All the models barring one are of the same form That is each of the eddy-viscosity models arewritten
τmodij minus 1
3τmodkk δij = minus2νeSij (50)
with νe representing the eddy viscosity of a specific model For simplicity we usually just writeτmodij = minus2νeSij the extra term is included here to emphasise that the models are traceless Each
22
of the models we consider here can be expressed in terms of the rate of strain tensor S defined inEq (41) the rate of rotation tensor Ω defined in Eq (42) and the combined invariants defined inEq (43) The models are then defined as follows
Smagorinsky [12] νSe = (CSδ)2radic
2I1 (51)
Vreman [16] νVe = (CV δ)2
radicQGGT
PGGT (52)
QR [13 14 15] νQRe = (CQRδ)2 max0minusI3
I1 (53)
Vortex-Stretching [11] νV Se = (CV Sδ)2radic
2I1
(I5 minus 1
2I1I2
minusI1I2
)32
(54)
with the quantities
PGGT = I1 minus I2 QGGT =1
4(I1 + I2)2 + 4
(I5 minus
1
2I1I2
) (55)
There is a fifth and final model to be considered here It is of a different form to the other four andnot an eddy-viscosity model It is known as the Gradient model and is defined by
Gradient [4 9] τmodG = CGδ
(S2 minus Ω2 minus (SΩminus ΩS)
) (56)
The values C are model constants specific to each model and δ represents the mesh size of the LESBoth of these quantities are included here to properly define the models however in this projectthey will be ignored as they are constants and have no influence on the conditions to be defined
33 Conditions for Large-Eddy Simulation without Explicit Filtering
The filtering operation introduced in Eq (48) is just a formality for LES without explicit filteringThe idea here is that filtering does not actually take place but the term lsquoaddedrsquo to the Navier-Stokesequations in fact lsquosubtractsrsquo the difficult to simulate small-scale turbulent motion of the flow Inorder to consider this properly the terms ui and p in Eq (48) are replaced by vi and q It is hopedthat τmod
ij approximates τij so that vi and q approximate ui and p respectively So LES withoutexplicit filtering is given by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (57)
It is clear to see that the first four terms of this equation are exactly the same as the Navier-Stokesequations so for the exact solutions discussed in Section 22 the combination of these terms willequal zero What is then desirable is that the model term is also zero for these exact solutions sothat Eq (57) still holds In order to test models for exact solutions the condition
part
partxjτmodij (v) = 0 (58)
is checked If this condition were to hold it would imply that the model does not influence thevelocity field of a flow hence we call it the Velocity Field Condition There are a number of otherrestrictions that can be tested One such test comes from the differential form of the evolution ofthe kinetic energy of a flow given in Eq (10) In order to find this Eq (57) is multiplied by vi andterms are grouped to find
part
partt
(1
2vivi
)+ vj
part
partxj
(1
2vivi
)= minusvi
1
ρ
partq
partxi+ νvi
part2vipartxjpartxj
minus vipart
partxjτmodij (v) (59)
23
Then similarly to Eq (58) another condition is to check whether the model affects the kineticenergy of exact solutions this is equivalent to checking whether
vipart
partxjτmodij (v) = 0 (60)
We call this condition the Kinetic Energy Condition The third and final condition considered herecan also be found using the integral form for the equation for kinetic energy
d
dt
intV
1
2vividV =
intS
[minusvj
(1
2vivi
)minus vj
q
ρ+ 2νviSij + viτ
modij (v)
]njdS
+
intV
[minus2νSijSij minus
(minusSijτmod
ij (v))]dV
(61)
where the rate-of-strain tensor Sij was defined in Eq (41) V is a volume of fluid and dS meansthe integral is over the surface of that volume with unit normal nj Then this final condition checkswhether the model affects the total kinetic energy of exact solutions to the Navier-Stokes equationsthat is
Sijτmodij (v) = 0 (62)
We will call this condition the Dissipation Condition because Sijτmodij (v) is often referred to as the
subgrid dissipation
34 Conditions for Large-Eddy Simulation with Explicit Filtering
As in Eq (48) the Navier-Stokes equations can be filtered by applying the filtering operation to thewhole equation (putting a bar over all terms) and evaluating to find again that
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (63)
In Eq (57) the filtering operation was ignored and the elements ui p and τij(u) were replacedwith vi q and τmod
ij (v) respectively Now in LES with explicit filtering the filtering operation itselfsmooths over the small-scale turbulent motion of the simulated flow In this case the elements uip and τij(u) will be replaced with vi q and τmod
ij (v) respectively LES with explicit filtering is thengiven by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (64)
Note that Eqs (63) and (64) despite their similarity to Eq (57) no longer contain the Navier-Stokesequations as Eq (57) did The bar (filtering operation) over each term results in each exact solutionfound in Section 22 not necessarily satisfying the first four terms It is hence unknown whether ornot these terms will vanish for exact solutions Terms can still be compared similarly to Eqs (58)(60) and (62) However instead of comparing to zero we can now compare to the relevant termfrom the filtered Navier-Stokes equations So the terms
part
partxjτij (65)
and
part
partxjτmodij (v) (66)
24
are to be compared Although the latter term is relatively simple to compute the former could leadto some difficulties due to the differentiation of the true stress τij = uiuj minus uiuj So although someinformation could be gleaned from the inspection of these two terms more information can be foundby considering the analogue of Eq (62) So the true subfilter dissipation
Sijτij (67)
(where Sij is shorthand for Sij(v)) and the modelled subfilter dissipation
Sij(v)τmodij (v) (68)
are considered and have to be equal for the condition to hold These can again be derived from theintegral equation for kinetic energy of a filtered velocity field similarly to Eq (61)
4 Analysis of Large-Eddy Simulation Models
In this section we use a code in Mathematica the theory described in Section 3 and the exactsolutions to the Navier-Stokes equations found in Section 22 to check whether or not each of theexact solutions lsquoendorsesrsquo each model for LES both without and with explicit filtering
41 Large-Eddy Simulation Without Explicit Filtering
We start by using a code in Mathematica to test the Velocity Field condition in Eq (58) and theKinetic Energy condition in Eq (60) For all exact solutions to the Navier-Stokes equations inSection 22 the outputs for both conditions were identical so we have combined them into one table
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 ndash ndash 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 1 For the Kinetic Energy condition a lsquo0rsquo result means that vipartpartxj
τmodij (v) = 0 for respective
models and exact solutions whereas a lsquolowastrsquo result means that vipartpartxj
τmodij (v) 6= 0 For the Velocity
Field condition a lsquo0rsquo result means that partpartxj
τmodij (v) = 0 for respective models and exact solutions
whereas a lsquolowastrsquo result means that partpartxj
τmodij (v) 6= 0 The Gradient model can take both positive and
negative values so when the sign of the model is known it is included in the table instead of a lsquorsquoBoth the conditions are included in one table because for all cases the outputs were identical
We then test the Dissipation condition in Eq (62) for all exact solutions to the Navier-Stokesequations in Section 22
25
Dissipation Condition [Does Sijτmodij (v) = 0]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 + 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 2 A lsquo0rsquo result means that Sijτmodij (v) = 0 for respective models and exact solutions whereas
a lsquolowastrsquo result means that Sijτmodij (v) 6= 0 The Gradient model can take both positive and negative
values so when the sign of the model is known it is included in the table instead of a lsquorsquo
We then sum all the results to find the total number of endorsements for each model described inSection 32
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
4 6 8 9 9
lsquoDissipationrsquoEndorsements
2 5 7 7 9
Total Endorsements 6 11 15 16 18
Table 3 The results from Tables 1 and 2 are summarised to give a general overview of how manyendorsements each model receives This makes it much clearer to see the models most supported bythe respective and combined conditions with the Vortex-Stretching model being the most supportedand the Smagorinsky model the least
Testing the models in Section 32 using LES without explicit filtering has mixed results It isclear that the Smagorinksy model is not well endorsed by the flows in Section 22 with the onlyendorsements coming from the simplest flows that is Sol 1 (Linear Velocity) Sol 3 (Velocity as afunction of time) Sol 7 (Stagnation Flow) and Sol 15 (Rankine Vortex) The Vortex-Stretching-based model is the most endorsed flow with eighteen out of a possible thirty endorsements Thisagrees with our foreknowledge as this model was constructed to be as physically consistent as possiblein the paper by Silvis amp Verstappen [11]
42 Large-Eddy Simulation With Explicit Filtering
In this section we will compare the true subfilter dissipation Sijτij and the modelled subfilterdissipation Sij(v)τmod
ij (v) given in Eqs (67) and (68) respectively for each model τmodij and each
exact solution to the Navier-Stokes equation given in Section 22 We will not consider the conditionsinvolving differentiation in Eqs (65) and (66) due to the difficulty in calculating the derivative ofthe τij term in Eq (65)
26
421 Example of a Test for a Model
The true subfilter dissipation Sijτij of a flow given in Eq (67) cannot be explicitly computed asthe filtering operation is kept general What can be done however is to partially evaluate theexpressions using the attributes of a filtering operation mentioned in Eq (48)
The calculations of Sijτij for Sol 1 (Linear Velocity) are provided to illustrate the general ideaRecall that the velocity field of Sol 1 (Linear Velocity) is written in Eq (13) in vector notation as
u(x y z t) =
0ax+ bcx+ d
(69)
Then Sij = Sij(u) is computed simply by using its definition in Eq (41) Note that as filteringpreserves constants if there is an a it is replaced with a meaning that (in this case) there are alarge number of zero terms due to the frequency of 0 Hence
[Sij(u)
]=
[1
2
(partuipartxj
+partujpartxi
)]=
0 a
2c2
a2 0 0
c2 0 0
=
0 a
2c2
a2 0 0
c2 0 0
(70)
Note that as Sijτij is being computed which is equal to the sum of the element-wise multiplicationof the two matrices (or the trace of the matrix multiplication) that we donrsquot need to consider theelements of τij(u) with corresponding zero entry in Sij As τij(u) = uiuj minus uiuj and again asfiltering preserves constants it is found that
[τij(u)] =[uiuj minus uiuj
]=
lowast 0 middot ax+ b 0 middot cx+ d
ax+ b middot 0 lowast lowastcx+ d middot 0 lowast lowast
=
lowast 0 00 lowast lowast0 lowast lowast
(71)
where a lsquolowastrsquo just represents some element (not necessarily non-zero) Therefore the true subfilterdissipation
Sijτij = tr
0 a2
c2
a2 0 0c2 0 0
middotlowast 0 0
0 lowast lowast0 lowast lowast
= 0 (72)
Now the modelled subfilter dissipation Sij(u)τmodij (u) needs to be computed (recall that Sij =
Sij(u)) In this case let us use the QR model stated in Eq (53) Once again Sij(u) calculated inEq (70) needs to be used So we have that
τmodij = minus2(CQRδ)
2 max0minusI3I1
Sij (73)
As
I1 = tr(S2) =
1
2(a2 + c2) and I3 = tr(S
3) = 0 (74)
it holds that τmodij (u) = 0 so Sij(u)τmod
ij (u) = 0 This agrees with Eq (72) and hence Sol 1(Linear Velocity) endorses the QR model
27
422 Testing the Models
Now using a Mathematica code the exact solutions in Section 22 are again used to evaluate thecondition for the models using the process just described in Section 421 The results are describedin Table 4
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 4 All entries with a ldquo0rdquo represent a value of zero and a ldquolowastrdquo represents a value that is non-zeroThe ldquoExactrdquo row represents the value of the true subfilter dissipation Sijτij Every other row witha model name represents the value of the modelled subfilter dissipation Sij(u)τmod
ij (u) respectiveto that model A ldquordquo means that it is unclear as to the value - the sign of these values is oftenunknown and the lsquoMaxrsquo function in the definition of the QR model (in Eq (53)) makes it difficultto evaluate
Reading from Table 4 the Smagorinsky model is clearly endorsed by Sol 3 (Velocity as a function oftime) and Sol 15 (Rankine Vortex) as in both cases both Sijτij = 0 = Sij(u)τmod
ij (u) In this casewe say that Sol 3 (Velocity as a function of time) and Sol 15 (Rankine Vortex) lsquostronglyrsquo endorsethe Smagorinsky model The Smagorinsky model is not endorsed by Sol 1 (Linear Velocity) Sol 2(Planar Flow) or Sol 14 (Radially Dependent) however because Sij(u)τmod
ij (u) 6= 0 For Sol 4 to
13 both Sijτij 6= 0 6= Sij(u)τmodij (u) and so we will say that these exact solutions lsquoweaklyrsquo endorse
the Smagorinsky model
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 10 10 9 6
Table 5 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Interestingly the lsquostrongrsquo and lsquoweakrsquo endorsements give different results However more stock shouldbe put in the results of the lsquostrongerrsquo condition as it is completely clear that zero is equal to zeroIt is not necessarily clear whether or not non-zero equals non-zero as the values will depend on theparameters in each solution hence the lsquoweakerrsquo endorsement
The lsquostrongrsquo condition equally endorses the Vreman QR Gradient and Vortex-Stretching-basedmodels whereas the lsquoweakrsquo condition endorses Vreman and Smagorinsky the most closely followedby the Gradient model with the Vortex-Stretching-based model being the most opposed As this
28
completely contradicts the lsquostrongrsquo condition and the results in Section 41 it makes us question thevalidity of the weaker test The number of endorsements for the QR model is unknown because of thedefinition of the model in Eq (53) and our evaluation of the true subfilter dissipation Eq (67) Dueto the exact solutions we have found it is never clear whether or not the true subfilter dissipationis strictly positive or negative This means that when the true subfilter dissipation is found to benon-zero the lsquoMaxrsquo function in the QR model cannot be evaluated
43 Vremanrsquos True Subfilter Dissipation Claim
Using the theory mentioned in Sections 232 and 34 we can place the exact solutions to the Navier-Stokes equations into different classes defined by Vreman In doing this we can also attempt to verifyone of Vremanrsquos results [16] In Vremanrsquos paper he defines the class Qn to be the set of flows withn zero entries in the velocity gradient of the flow (the velocity gradient is defined in Eq (39))Vreman then claims that the only flows with zero true subfilter dissipation are a certain selection ofthose in the classesQ7Q8 orQ9 shown in appendix B and is generally non-zero for any other flows
The true subfilter dissipation of Sol 1 (Linear Velocity) was computed in Section 421 and wasfound to be zero in Eq (72) The velocity gradient of Sol 1 is as follows
GLV =
0 0 0a 0 0c 0 0
(75)
and hence Sol 1 (Linear Velocity) is in the required subset of Q7 which agrees with Vremanrsquos claimIn Section 422 the true subfilter dissipation was computed (determined to be zero or non-zero) forall of the exact solutions to the Navier Stokes equations in Section 22 These solutions can now belsquoplacedrsquo into each class Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a functionof time) and Sol 14 (Radially Dependent) were all found to have zero true subfilter dissipation andare all in the appropriate subsets of either Q7Q8 or Q9 This agrees with Vremanrsquos results
We can extend Vremanrsquos results however We found that Sol 15 (Rankine Vortex) had zero truesubfilter dissipation and yet is not in the subset of Q7 defined in Appendix B It is important tonote that the rate of strain tensor Sij in Eq (41) for Sol 15 (Rankine Vortex) is zero This is thereason the true subfilter dissipation is zero and the same reason that all models are zero for thisflow as all models are based on Sij
We also found that all of the remaining exact solutions to the Navier-Stokes equations are not inVremanrsquos specified classes (in Appendix Section B) They also have a non-zero subfilter dissipationwhich agrees with Vremanrsquos results in his paper [16]
44 Classes of Flows
It is important to consider that some exact solutions to the Navier-Stokes equations may have similarproperties For example Sol 1 (Linear Velocity) and Sol 2 (Planar Flow) are both very similarIn fact Sol 2 (Planar Flow) is a more general case of Sol 1 (Linear Velocity) We therefore mustgroup the different flows into classes in an attempt to reduce the bias that may occur when solutionsare similar This should aid us to further determine which of the models are most endorsed We willdo this in a number of different ways using the theory discussed in Section 23 We will group theflows with respect to Vremanrsquos Flow Classes given in appendix B and with respect to the Invariantsgiven in appendix A
29
441 Vremanrsquos Velocity Gradient Flow Classes
We now put our exact solutions to the Navier-Stokes equations into one of two classes Either Q0minus6or Q7minus9 where each class is defined intuitively as follows
Q0minus6 = Q0 cupQ1 cup middot middot middot cup Q6 (76)
Q7minus9 = Q7 cupQ8 cupQ9 (77)
where Qn is defined in Section 43 By looking at the gradient of each exact solution we can see thatQ7minus9 contains the six solutions Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as afunction of time) Sol 5 (2D Taylor) Sol 14 (Radially Dependent) and Sol 15 (Rankine Vortex)and that Q0minus6 contains the remaining nine We now check the models for LES without explicitfiltering by considering endorsements within each class Tables 6 and 7 are analogues of Table 3 andillustrate the results for each flow class The analogues of Tables 1 and 2 are Tables 22 and 23 andare in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 6 6 6
lsquoDissipationrsquoEndorsements
2 5 6 6 6
Percentage Endorsed 417 833 100 100 100
Table 6 Each entry in the top two rows represents the number of solutions in the class Q7minus9 thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Q0minus6 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 2 3 3
lsquoDissipationrsquoEndorsements
0 0 1 0 3
Percentage Endorsed 56 56 167 167 333
Table 7 Each entry in the top two rows represents the number of solutions in the class Q1minus6 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking results to 1 decimal place if necessary
All of the flows in the class Q7minus9 endorse the QR Gradient and Vortex-Stretching-based subfilter-scale models (see Table 6) We can hence see that this class fully endorses these three models equallyand does not endorse the Smagorinsky model The Vortex-Stretching eddy viscosity model has themost endorsements from any of the flows in the class Q0minus6 and the Smagorinksy and Vreman modelshave the fewest endorsements (see Table 7) We can hence say that this class mostly endorses theVortex-Stretching-based eddy-viscosity model note however that the percentage endorsement is very
30
low
Now let us consider LES with explicit filtering Tables 8 and 9 are analogues of Table 5 and illustratethe results for each flow class The analogue of Table 4 is Table 24 and is in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 1 1 0 0 0
Table 8 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Q0minus6 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 9 9 9 6
Table 9 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We have grouped Vremanrsquos classes into Q7minus9 and Q0minus6 this is equivalent to saying that we havegrouped Vremanrsquos classes into those with two or fewer non-zero velocity gradient elements and thosewith three or more non-zero velocity gradient elements respectively Therefore the solutions in Q7minus9are exactly those with a lsquo0rsquo in the lsquoExactrsquo column of Table 4 and Sol 5 (2D Taylor) ExcludingSol 5 (2D Taylor) these are the same as those five that endorse the Vreman QR Gradient andVortex-Stretching-based models in Table 5 We can see in Table 8 that the results for the lsquostrongrsquoendorsements are exactly the same as in Table 5 This is because all of the flows with zero subfilterdissipation are contained in the class Q7minus9 There are hence no lsquostrongrsquo endorsements in Table 9The majority of the lsquoweakrsquo endorsements come from Q0minus6 as this class contains all but one of thesolutions with non-zero true subfilter dissipation (given in Eq (67))
442 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
We now separate our exact solutions into classes again using Vremanrsquos flow classes This timehowever we define V0 to be the class of flows Vreman claims will always have zero true subfilterdissipation namely those classes mentioned in Appendix B We define Vlowast to be all the remaining
31
classes Then
V0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow)
Sol 3 (Velocity as a function of time) Sol 14 (Radially Dependent)(78)
Vlowast = Sol 4 (Axisymmetric Flow) Sol 5 (2D Taylor) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 13 (Sphere Translation) Sol 15 (Rankine Vortex)
(79)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 10 and 11 are analogues of Table 3 and illustrate the results for each flow class Theanalogues of Tables 1 and 2 are Tables 25 and 26 and are in Appendix C2
Endorsements for models using solutions in V0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 4 4 4 4
lsquoDissipationrsquoEndorsements
1 4 4 4 4
Percentage Endorsed 375 100 100 100 100
Table 10 Each entry in the top two rows represents the number of solutions in the class V0 thatendorse each model There are 4 solutions in this class so the percentages are worked out by dividingthe total of each column by 8 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Vlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 2 4 5 5
lsquoDissipationrsquoEndorsements
1 1 3 2 5
Percentage Endorsed 136 136 318 318 455
Table 11 Each entry in the top two rows represents the number of solutions in the class Vlowast thatendorse each model There are 11 solutions in this class so the percentages are worked out bydividing the total of each column by 22 and taking results to 1 decimal place if necessary
All of the flows in the class V0 endorse the Vreman QR Gradient and Vortex-Stretching-basedsubfilter-scale models (see Table 6) This class completely endorses these four models equally anddoes not endorse the Smagorinsky model The Vortex-Stretching-based eddy-viscosity model hasthe most endorsements from the class Vlowast and the Smagorinksy and Vreman models again have thefewest endorsements (see Table 7) As in Section 441 we say that this class mostly endorses the
32
Vortex-Stretching-based eddy-viscosity model but once again the percentage endorsement is ratherlow
Now let us consider LES with explicit filtering Tables 12 and 13 are analogues of Table 5 andillustrate the results for each flow class The analogue of Table 4 is Table 27 and is in Appendix C2
Endorsements for models using solutions in V0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 4 4 4 4
lsquoWeakrsquo Endorsements 0 0 0 0 0
Table 12 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Vlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 1 1 1 1
lsquoWeakrsquo Endorsements 10 10 9 6
Table 13 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We can see in Table 12 that the results for the lsquostrongrsquo endorsements are similar to those in Table 5excluding Sol 15 (Rankine Vortex) The four solutions in V0 with zero true subfilter dissipationnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)and Sol 14 (Radially Dependent) are hence the four that endorse the Vreman QR Gradientand Vortex-Stretching-based models in Table 12 Sol 15 (Rankine Vortex) is in the class Vlowast andhence there is one lsquostrongrsquo endorsement in Table 13 The class Vlowast contains all of the flows with anon-zero true subfilter dissipation (given in Eq (67)) and hence Table 13 contains all of the lsquoweakrsquoendorsements
443 Classes using the Third Principal Invariant
Sorting the classes using the invariants is a non-trivial task in itself There are a number of waysin which to do this using both the principal invariants in Eq (40) and the combined invariants inEq (43)
We start by looking at flows with third principal invariant RG equal to zero We again define twoclasses R0 and Rlowast where R0 consists of all flows with RG = 0 and Rlowast consists of all flows withRG 6= 0 Then using the information in Appendix A we see that there are nine flows in R0 That
33
is
R0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 8 (Vortex in Stagnation Flow)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)
(80)
Rlowast = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 13 (Sphere Translation)(81)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 14 and 15 are analogues of Table 3 and they illustrate the results for LES withoutexplicit filtering for each flow class The analogues of Tables 1 and 2 are Tables 28 and 29 and arein Appendix C3
Endorsements for models using solutions in R0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 7 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 278 556 778 722 778
Table 14 Each entry in the top two rows represents the number of solutions in the class R0 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking the result to one decimal place if necessary
Endorsements for models using solutions in Rlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 2 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 83 83 83 167 333
Table 15 Each entry in the top two rows represents the number of solutions in the class Rlowast thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking the result to one decimal place if necessary
The class R0 endorses both the Vortex-Stretching-based model and the QR model the most andSmagorinsky the least in the case for LES without explicit filtering In the Rlowast class the Vortex-Stretching-based model is again the most endorsed followed by the Gradient model however thepercentages are again small
34
Now let us consider LES with explicit filtering Tables 16 and 17 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 30 and is in Appendix C3
Endorsements for models using solutions in R0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 4 4 3 2
Table 16 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Rlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 6 6 6 4
Table 17 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 we can see in Table 16 that the results for the lsquostrongrsquo endorsements are exactly thesame as in Table 5 This is because all of the flows with zero subfilter dissipation are contained in theclass R0 As in Table 9 there are no lsquostrongrsquo endorsements in Table 17 The lsquoweakrsquo endorsementshowever are split more evenly between the two classes now than in Vremanrsquos classes in Sections 441and 442 We find that the Smagorinsky and Vreman models still seem to be the most endorsed inboth classes
444 Two-component Flow Classes
Another way in which the flows can be grouped into classes is by determining whether a flow is atwo-component flow or not We define a two-component flow to have the property that the combinedprincipal invariants of a flow in Eq (43) satisfy
I3 = I4 = I5 minus1
2I1I2 = 0 (82)
We are also interested in considering flows with zero gradient model subgrid dissipation These flowshave the property that
I3 minus I4 = 0 (83)
What we find though is that for all the solutions in Section 22 if a flow is a two-component flowthen it also satisfies I3 minus I4 = 0 It may be interesting to note that although I3 minus I4 6= 0 for manysolutions there are a number with I3 equal to a multiple of I4 Although not considered in thisreport this may also be a class of interest We find that the two-component flows are
C2 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)(84)
35
and we call the remaining flows three-component flows They are
C3 = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex) Sol 11 (Pseudo-Motion Shots)
Sol 13 (Sphere Translation)
(85)
Note that Sol 8 (Vortex in Stagnation Flow) is defined to be a planar flow this does not mean that itis a two-component flow in this sense however because its two non-zero velocity components dependon three coordinates We now check the models for LES without explicit filtering by consideringendorsements within each class Tables 18 and 19 are analogues of Table 3 and they illustrate theresults for LES without explicit filtering for each flow class The analogues of Tables 1 and 2 areTables 31 and 32 and are in Appendix C4
Endorsements for models using solutions in C2 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 6 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 357 714 100 857 100
Table 18 Each entry in the top two rows represents the number of solutions in the class C2 thatendorse each model There are 7 solutions in this class so the percentages are worked out by dividingthe total of each column by 14 and taking the result to one decimal place if necessary
Endorsements for models using solutions in C3 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 3 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 63 63 63 188 250
Table 19 Each entry in the top two rows represents the number of solutions in the class C3 thatendorse each model There are 8 solutions in this class so the percentages are worked out by dividingthe total of each column by 16 and taking the result to one decimal place if necessary
The class C2 equally endorses the QR and Vortex-Stretching-based models and once again theSmagorinsky model is the least endorsed As with the previous classes the class C3 has such lowpercentages that it does not seem to endorse any models We start to see a trend appearing here inthat the flow classes that contain the more complicated flows namely Q0minus6 (Eq (76)) Vlowast (Eq (79))Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages The classes that tendto contain the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78)) R0 (Eq (80)) and C2 (Eq (84))all seem to favour the Vortex-Stretching-based model or the QR model
36
Now let us consider LES with explicit filtering Tables 20 and 21 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 33 and is in Appendix C4
Endorsements for models using solutions in C2 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 2 2 0 1 0
Table 20 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in C3 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 8 8 8 6
Table 21 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 and Table 16 we can see in Table 20 that the results for the lsquostrongrsquo endorsements areexactly the same as in Table 5 This is again because all of the flows with zero subfilter dissipationare contained in the class C2 There are two flows with non-zero true subfilter dissipation (givenin Eq (67)) in C2 and this shows in the lsquoweakrsquo endorsements in Table 20 There are no lsquostrongrsquoendorsements in Table 21 as all the zero subfilter dissipation flows are in C2
5 Conclusions
In this report we studied incompressible turbulent flows We found a number of exact solutions to theNavier-Stokes equations each representing a relatively simplistic fluid flow Large-eddy simulation(LES) is used to model turbulent flows by smoothing over the complex small-scale motion Weintroduced a number of subfilter-scale models to approximate this small-scale motion and testedthem using the found exact solutions to the Navier-Stokes equations We first considered LESwithout explicit filtering and used the Navier-Stokes equations to derive conditions for the subfilter-scale models namely the Velocity Field condition (Eq (58)) the Kinetic Energy condition (Eq (60))and the Dissipation condition (Eq (62)) Similarly we considered LES with explicit filtering andagain used the Navier-Stokes equations to derive more conditions for the models in this case wedecided to just take one condition the Dissipation condition (Eqs (67) and (68)) as the others leadto difficulty in their evaluation By lsquorunningrsquo the exact solutions to the Navier-Stokes equationsthrough each condition for the subfilter-scale models we found that the Vortex-Stretching-basededdy-viscosity model in Eq (54) was most endorsed by the tests in LES without explicit filteringWe found that four out of the five subfilter-scale models namely the Vreman QR Gradient andVortex-Stretching-based models were equally endorsed by the lsquostrongrsquo tests in LES with explicitfiltering with the Smagorinsky model being the least endorsed The results we found for the lsquoweakrsquocondition were completely opposite to those found in both the lsquostrongrsquo condition and in LES withoutexplicit filtering This led us to conclude that this test was not one in which we should put our trust
37
We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses
In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes
This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold
38
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
Contents
1 Introduction 4
2 The Navier-Stokes Equations 521 Introduction 522 Exact Solutions of the Navier-Stokes Equations 623 Characterisation of Flows 20
231 Characterising Solutions using Invariants 20232 Vremanrsquos Characterisation of Flows 21
3 Conditions for Large-Eddy Simulation Models 2231 Large-Eddy Simulation 2232 Subfilter-Scale Models 2233 Conditions for Large-Eddy Simulation without Explicit Filtering 2334 Conditions for Large-Eddy Simulation with Explicit Filtering 24
4 Analysis of Large-Eddy Simulation Models 2541 Large-Eddy Simulation Without Explicit Filtering 2542 Large-Eddy Simulation With Explicit Filtering 26
421 Example of a Test for a Model 27422 Testing the Models 28
43 Vremanrsquos True Subfilter Dissipation Claim 2944 Classes of Flows 29
441 Vremanrsquos Velocity Gradient Flow Classes 30442 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes 31443 Classes using the Third Principal Invariant 33444 Two-component Flow Classes 35
5 Conclusions 37
Appendices 39
A Invariants of Flows 39
B Vremanrsquos Flow Classes 46
C Extended Tables for Flow Classes 47C1 Vremanrsquos Velocity Gradient Flow Classes 47C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes 48C3 Classes using the Third Principal Invariant 49C4 Two-Component Flow Classes 50
3
1 Introduction
In this project we will study and test models for turbulent fluid flows There is no all-encompassingdefinition of a turbulent flow there are however a number of characterising properties A turbulentflow is said to be irregular in that it appears to be random Turbulent flows are chaotic and henceusually described statistically due to the difficulty in a deterministic approach Turbulent flowsare diffusive in that there is a high rate of momentum heat and mass transfer They require apersistent source of input energy in order to sustain the turbulent nature of the flow as kineticenergy is quickly converted into internal energy A large Reynolds number is characteristic of aturbulent flow [5]
The Navier-Stokes equations are a set of fundamental governing equations for any Newtonian fluidAn exact solution to the Navier-Stokes equations is in the form of a flow velocity and a pressureterm and hence a fluid flow can be completely described by an exact solution to the Navier-Stokesequations It is still a non-trivial task to find such solutions although there are a number in literature[17]
The Navier-Stokes equations have no closed form solution resulting in difficulty when analysingcomplicated flows in complex geometries This results in difficulty when solving the Navier-Stokesequations explicitly So a mathematical model known as Large Eddy Simulation (LES) can insteadbe used to model turbulent flows LES uses what is known as a filtering operation to reduce thecomputational cost of a simulation by ignoring small-scale motion What this means however isthat information about these small-scale motions is lost This information is often non-trivial sosubfilter-scale models (or turbulence models) are used in order to approximate the effects of small-scale motions on the large-scale fluid flow
In this project we will use the behaviour of exact solutions to the Navier-Stokes equations to testsubfilter-scale models Applying a subfilter-scale model to a flow has two effects One is a bodyforce on the velocity field called a subfilter force and the other is a transfer of energy from largeto small scales of motion called a subfilter dissipation This results in respectively an extra termbeing added to the Navier-Stokes equations and to the equation of the kinetic energy of a flow
We will then consider conditions for subfilter-scale models in two different cases In LES withoutexplicit filtering we will consider how a subfilter-scale model behaves for an exact solution to theNavier-Stokes equations by checking if the subfilter force and subfilter dissipation are zero for anexact solution If so this can be interpreted as the model being inactive and hence this exactsolution can be said to lsquoendorsersquo the model In LES with explicit filtering we compare a model termwith a term in the filtered Navier-Stokes equations (called the true turbulent stress of a flow) forexact solutions If the two terms are both zero we will conclude that the exact solution lsquostronglyrsquoendorses the model and if the two terms are both non-zero we will conclude that the exact solutionlsquoweaklyrsquo endorses the model
We will also consider the classification of fluid flows in order to further inspect the models andreduce the bias that may occur when solutions have similar properties We will use properties of theinvariants of a flow to characterise flows in a coordinate-independent way Using Vremanrsquos paperldquoAn eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory and applicationsrdquo[16] we will place the fluid flows into Vremanrsquos flow classes We will also determine whether ourresults say anything beyond Vremanrsquos work
4
2 The Navier-Stokes Equations
21 Introduction
The incompressible Navier-Stokes equations are a set of four equations that model the flow of anyfluid in three-dimensional space The standard set of Navier-Stokes equations can be written in twoforms namely by using Einsteinrsquos summation convention in what we call index notation and byusing standard vector notation In index notation they are given by
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
i = 1 2 3 j = 1 2 3 (1)
These equations have four variables given by the velocity of the flow u = (u1 u2 u3) and thepressure p The vector x = (x1 x2 x3) represents position t represents time ρ corresponds to themass density and ν is the kinematic viscosity of the fluid
The Navier-Stokes equations can be written in vector notation in the following way First considerthe convection term
ujpartuipartxjequiv
3sumj=1
ujpartuipartxj
= u1partuipartx1
+ u2partuipartx2
+ u3partuipartx3
=
(u1
part
partx1+ u2
part
partx2+ u3
part
partx3
)ui
= (u middot nabla)ui
(2)
Similarly for the diffusion term
νpart2uipartxjpartxj
equiv ν3sumj=1
part2uipartxjpartxj
= νpart2uipartx1partx1
+part2uipartx2partx2
+part2uipartx3partx3
= ν
(part2
partx1partx1+
part2
partx2partx2+
part2
partx3partx3
)ui
= νnabla2ui
(3)
It therefore holds that in vector notation the Navier-Stokes equations are given by
partu
partt+ (u middot nabla)u = minus1
ρnablap+ νnabla2u (4)
It is clear that there is a problem here there are three equations and four unknowns In order toremedy this the equation for conservation of mass or the incompressibility condition is consideredThis is given by
partuipartxi
= 0 (5)
in index notation and
nabla middot u = 0 (6)
5
in vector notation
There are now four equations in four unknowns and it is possible to find solutions Althoughthere is no known general solution there are specific cases in which exact solutions to the Navier-Stokes equations can be found In order to make finding solutions slightly simpler the Navier-Stokesequations can be rescaled by introducing dimensionless variables
Let
ulowasti =uiv xlowasti =
xi` tlowast =
t
`v plowast =
p
ρv2 (7)
where ` and v are typical length and velocity scales respectively Then dropping the star notationfor ease of reading the scaled incompressible Navier-Stokes equations are
partuipartt
+ ujpartuipartxj
= minus partp
partxi+
1
Re
part2uipartxjpartxj
i = 1 2 3 j = 1 2 3
partuipartxi
= 0
(8)
where Re is the Reynolds number and is defined by the dimensionless Re =v`
ν Similarly in vector
notation the scaled incompressible Navier-Stokes equations are
partu
partt+ (u middot nabla)u = minusnablap+
1
Renabla2u
nabla middot u = 0(9)
By multiplying the incompressible Navier-Stokes equations Eq (8) by ui and grouping terms thedifferential form of the evolution of the kinetic energy of a flow can be found
part
partt
(1
2uiui
)+ uj
part
partxj
(1
2uiui
)= minusui
partp
partxi+
2
Reui
part2uipartxjpartxj
(10)
The differential form of the evolution of the kinetic energy of a flow will be used later on in thisreport to derive conditions for the subfilter-scale models
22 Exact Solutions of the Navier-Stokes Equations
By using mathematical inspection and taking results found in literature we found a number of exactsolutions to the Navier-Stokes equations The majority are steady-state solutions in that they areindependent of time We decided to consider exact solutions to the Navier-Stokes equations withoutboundary conditions in an attempt to keep the results general and applicable to a larger numberof situations All of the stated exact solutions below have been checked using a code written inMathematica In the solutions below the notation (x y z) for the Cartesian coordinate system isused analogously with (x1 x2 x3) to aid in reading comprehension and in the conversion betweenindex and vector notation
Solution 1 Linear Velocity
If the diffusion term in the Navier-Stokes equations Eq (1) is taken to be zero ie
part2uipartxjpartxj
= 0 (11)
6
then each component of the velocity can be written as a linear function of x1 x2 and x3 In indexnotation
ui = aijxj + βi (12)
If pressure is assumed to be a function of time t (so that the pressure terms in the Navier-Stokesequations are zero) then conditions can be found on the coefficients to find that the following is asolution
u1(x y z t) = 0
u2(x y z t) = ax+ b
u3(x y z t) = cx+ d
p(x y z t) = f(t)
(13)
where a b c d isin R and f Rrarr R In fact this solution is actually a specific case of Sol 2 (PlanarFlow)
Figure 1 An illustration of the flow in the xy-plane for Sol 1 (Linear Velocity) described inEq (13) with parameter values a = b = c = d = 1
Solution 2 Planar Flow
Using the text by Barbato et al [2 p62] and modifying the solution such that it now takes threedimensions into account another solution is found
u1(x y z t) = Az2 +Bz + C
u2(x y z t) = Dz2 + Ez + F
u3(x y z t) = 0
p(x y z t) =2A
Rex+
2D
Rey + f(t)
(14)
7
where ABCDE F isin R and f R rarr R It is clear to see that if A = D = 0 then this is in factthe same as Eq (13) in Sol 1 In fact we believe (although not shown here) that this is the mostgeneral solution of the form
ui(xi xj xk t) = Ax2k +Bxk + C
uj(xi xj xk t) = Dx2k + Exk + F
uk(xi xj xk t) = 0
(15)
for i 6= j 6= k 6= i
Figure 2 An illustration of the flow in the yz-plane for Sol 2 (Planar Flow) described inEq (14) with parameter values A = B = C =D = E = F = 1
Solution 3 Velocity as a function of time t
If each component of the velocity is taken to be a function of time t and independent of positionthen the incompressibility condition is automatically satisfied A condition can be imposed on thepressure term in order to satisfy the momentum equations Thus a solution is as follows
u1(x y z t) = f(t)
u2(x y z t) = g(t)
u3(x y z t) = h(t)
p(x y z t) = minusxpartf(t)
parttminus y partg(t)
parttminus z parth(t)
partt
(16)
where f g h Rrarr R In fact what is written here is the generalised Galilean transformation of thezero solution [10] We cannot produce a plot for this flow as the functions are completely unspecified
8
Solution 4 Axisymmetric Flow
Using the text by Irmay [7] a solution is defined as follows
u1(x y z t) = kxz
u2(x y z t) = kyz
u3(x y z t) = minuskz2
p(x y z t) = minus2kz
Reminus k2z4
2
(17)
where k isin R This is said to represent an axisymmetric flow with hyperbolic streamlines asymptoticto the z-axis and to the boundary plane z = 0 These asymptotes are visible in Fig 3
Figure 3 An illustration of the flow in theyz-plane for Sol 4 (Axisymmetric Flow)described in Eq (17) with parameter k = 1
Figure 4 An illustration of the flow in thexy-plane for Sol 4 (Axisymmetric Flow)described in Eq (17) with parameter k = 1
Solution 5 2D Taylor Solutions
Again using the text by Barbato et al [2 p63ndash64] a solution is defined using a stream functionψ = ψ(x y t) and by writing
u1 = minuspartψparty
u2 =partψ
partx u3 = 0 (18)
In the text the stream function is not given however it was worked out that
ψ = minuseminusπ2t
Re (cos(πx) + cos(πy))
π (19)
The solution used is of the following form note however that in the text the pressure term differsin that cos(πy) is replaced by sin(πy) (when testing this solution in Mathematica the solution below
9
was instead found)
u1(x y z t) = minus sin(πy) eminus(π2t)Re
u2(x y z t) = sin(πx) eminus(π2t)Re
u3(x y z t) = 0
p(x y z t) = minus cos(πx) cos(πy) eminus(2π2t)Re
(20)
Figure 5 An illustration of the flow in the xy-plane for Sol 5 (2D Taylor) described in Eq (20)with Re = 1 and time t = 0
Solution 6 Generalized Beltrami Flow
Again using the text by Barbato et al [2 p65] and now considering solutions with nonndashtrivialterms in all dimensions a particular form of the generalized Beltrami flow is found namely
u1(x y z t) = (A sin(πz) + C cos(πy)) eminus(π2t)Re
u2(x y z t) = (B sin(πx) +A cos(πz)) eminus(π2t)Re
u3(x y z t) = (C sin(πy) +B cos(πx)) eminus(π2t)Re
p(x y z t) = minus[(AB cos(πz) sin(πx) +BC cos(πx) sin(πy)
+AC cos(πy) sin(πz)] eminus(2π2t)Re
(21)
where ABC isin R
10
Figure 6 An illustration of the flow in the yz-plane for Sol 6 (Generalized Beltrami Flow)described in Eq (21) with parameters A = B =C = 1 and time t = 0
Solution 7 Stagnation Flow
In the book by Drazin and Riley [6 p40] an exact solution is considered which uses a stream functionψ = ψ(y z) They suggest that the following velocity field is a solution
u1(x y z t) =a
kx
u2(x y z t) = minusaky +
k
Re
partψ(y z)
partz
u3(x y z t) = minus k
Re
partψ(y z)
party
(22)
Here we choose the stream function to be ψ(y z) = yz Then a solution is given by
u1(x y z t) =a
kx
u2(x y z t) =
(minusak
+k
Re
)y
u3(x y z t) = minus kzRe
p(x y z t) = minus1
2
((axk
)2minus((
k
Reminus a
k
)y
)2
minus(kz
Re
)2)
(23)
where a is a measure of the strength of the stagnation flow and k is a constant chosen in the textto specify the distance between vortices There are no vortices in this particular solution however
11
Figure 7 An illustration of the flow in the yz-plane for Sol 7 (Stagnation Flow) describedin Eq (23) with parameter values a = Re = 1and k = 2
Figure 8 An illustration of the flow in the xy-plane for Sol 7 (Stagnation Flow) describedin Eq (23) with parameter values a = Re = 1and k = 2
Solution 8 Vortex in Stagnation Flow
As in Sol 7 (Stagnation Flow) we now consider the same solution with a different stream functionnamely ψ(z) = cos z Then given the condition that k = aRe and adding a pressure term Eq (22)becomes
u1(x y z t) =1
Rex
u2(x y z t) = minus 1
Rey minus k sin z
u3(x y z t) = 0
p(x y z t) = minus 1
2Re2(x2 + y2)
(24)
where a and k are as in Sol 7 (Stagnation Flow)
12
Figure 9 An illustration of the flow in the yz-plane for Sol 8 (Vortex in Stagnation Flow)described in Eq (24) with parameter valuesk = 3 and Re = 10
Figure 10 An illustration of the flow in thexy-plane for Sol 8 (Vortex in StagnationFlow) described in Eq (24) with parametervalues k = 3 and Re = 10
Solution 9 Burgers Vortex
The Burgers Vortex solution is given in cylindrical coordinates in the book by Lautrup [8 p 385]as follows
ur(r ϕ z t) = minus 2r
c2Re
uϕ(r ϕ z t) =Ωc2
r
(1minus eminusr
2c2)
uz(r ϕ z t) =4z
c2Re
p(r ϕ z t) = minusc2Ω2F
(r2c2
)2
minus 2
Re2r2 + 4z2
c4
(25)
where
F (ξ) =
infinintξ
(1minus eminusx)2
x2dx (26)
This solution is not particularly nice to look at however most of the unpleasantness comes fromthe pressure term which (in this project) is only required in order to validate the solution For thisproject all solutions are required in Cartesian coordinates the velocity field of the Burgers Vortexhas hence been converted into Cartesian coordinates
u1(x y z t) = minus 2x
c2Reminus Ω
h(x y)(1minus exp (minush(x y))) y
u2(x y z t) = minus 2y
c2Re+
Ω
h(x y)(1minus exp (minush(x y)))x
u3(x y z t) =4z
c2Re
(27)
13
where
h(x y) =x2 + y2
c2 (28)
and Ω and c are constants
Figure 11 An illustration of the flow for Sol9 (Burgers Vortex) described in Eq (27) withparameter values c = 15 Re = 20 and Ω = 10A slice of the xy-plane at z = 0 is shown
Solution 10 Another Vortex Solution
There is another vortex solution more simply formulated than the Burgers Vortex yet with asingularity at r = 0 as a result of the uϕ term It is again given in cylindrical coordinates
ur(r ϕ z t) = minus 2r
c2Re
uϕ(r ϕ z t) =a
r
uz(r ϕ z t) =4z
c2Re
p(r ϕ z t) = minus a2
2r2minus 2r2 + 8z2
c2Re2
(29)
which can also be converted into Cartesian coordinates
14
u1(x y z t) = minus 2x
c2Reminus ay
x2 + y2
u2(x y z t) = minus 2y
c2Re+
ax
x2 + y2
u3(x y z t) =4z
c2Re
(30)
where a and c are constants
Figure 12 An illustration of the flow for Sol9 (Burgers Vortex) described in Eq (27) withparameter values c = 15 Re = 20 and a = 10 Aslice of the xy-plane at z = 0 is shown
Solution 11 Pseudo-Motion Shots of the Second Kind
In the book by Berker [3 p 94] a solution is given in cylindrical coordinates by
ur(r ϕ z t) =2
Re
1
r
uϕ(r ϕ z t) = A1r3 +B1r +
C1
r
uz(r ϕ z t) = A2r2 log(r) +B2r
2 + C2
p(r ϕ z t) =A2
1r6
6+A1B1r
4
2+A1C1r
2 +B2
1r2
2minus C2
1
2r2minus 2
r2Re2+ 2B1C1 log(r)
+2A2z
Reminus 4B1ϕ
Re
(31)
where A1 A2 B1 B2 C1 C2 are constants Eq (31) when converted into Cartesian coordinates is
15
u1(x y z t) =1
Re
2xminus C1Rey
x2 + y2minusA1y
(x2 + y2
)minusB1y
u2(x y z t) =1
Re
2y + C1Rex
x2 + y2+A1x
(x2 + y2
)+B1x
u3(x y z t) =1
2A2
(x2 + y2
)log(x2 + y2
)+B2
(x2 + y2
)+ C2
(32)
Figure 13 An illustration of the flow in thexy-plane for Sol 11 (Pseudo-Motion Shots)described in Eqs (31) and (32) with parametervalues A1 = B1 = C1 = 1 A2 = B2 = C2 = 2and Re = 10
Figure 14 An illustration of the flow in theyz-plane for Sol 11 (Pseudo-Motion Shots)described in Eqs (31) and (32) with parametervalues A1 = B1 = C1 = 1 A2 = B2 = C2 = 2and Re = 10
Solution 12 Flows With Constant Whirl
In the book by Berker [3 p 143] another solution is given in cylindrical coordinates by
ur(r ϕ z t) = minus 1
Re
1
r
uϕ(r ϕ z t) =1
rradicte(minus
12 g(t))
uz(r ϕ z t) = 0
p(r ϕ z t) =1
4
2(minus e
minusg(t)
t minus 1Re2
)r2
minus 1
t2Re
infinintminusg(t)
eminust
tdt
(33)
where we define g(t) =r2Re
2t When converted into Cartesian coordinates we find
16
u1(x y z t) = minus 1
x2 + y2
(yradict
exp
(minus
Re(x2 + y2
)4t
)+
x
Re
)
u2(x y z t) =1
x2 + y2
(xradict
exp
(minus
Re(x2 + y2
)4t
)minus y
Re
)
u3(x y z t) = 0
(34)
Figure 15 An illustration of the flow for Sol12 (Flows with Constant Whirl) described inEqs (33) and (34) with time t = 1 and Re = 1 Aslice of the xy-plane at z = 0 is shown as velocityin the z-direction is zero this is the same for anyvalue of z
17
Solution 13 Rectilinear and Uniform Translation of a Sphere
In the book by Berker [3 p 228] another solution is given in Cartesian coordinates by
u1(x y z t) =3
4
axz
R3
(a2
R2minus 1
)
u2(x y z t) =3
4
ayz
R3
(a2
R2minus 1
)
u3(x y z t) = 1minus 3
4
a
Rminus 1
4
a3
R3minus 3
4
az2
R3
(a2
R2minus 1
)
p(x y z t) =3a(aminusR)(a+R)
32R10Re
((R2Re
(x2 + y2
)minus 2R2Rez2
) (a3 + 3aR2 minus 4R3
)+ 3aRez4
(R2 minus a2
)minus 16R5z
)
(35)
where a and R are constants
Figure 16 An illustration of the flow in the yz-plane for Sol 13 (Sphere Translation) describedin Eq (35) with parameter values a = minus2 andRe = 3
Solution 14 Radially Dependent Flow
A solution with velocity in the z-direction dependent only on the distance from the z-axis is as
18
follows in cylindrical coordinates
ur(r ϕ z t) = 0
uϕ(r ϕ z t) = 0
uz(r ϕ z t) = A(r2 minus a2
)
p(r ϕ z t) =4Az
Re
(36)
where A and a are constants In Cartesian coordinates the solution is just the same with u1(x y z t)and u2(x y z t) both zero and the r2 term replaced with x2 + y2 in u3(x y z t)
Figure 17 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus1 1] withparameter values a = 1 A = 4 and Re = 10
Figure 18 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus2 2] withparameter values a = 1 A = 4 and Re = 10
Solution 15 Rankine Vortex
A solution discovered by Rankine [1] is a vortex flow of the following form
ur(r ϕ z t) = 0
uϕ(r ϕ z t) =ar
2πR2
uz(r ϕ z t) = 0
p(r ϕ z t) =a2r2
8π2R4
(37)
19
where a and R are constants In Cartesian coordinates the flow has the form
u1(x y z t) = minus ay
2πR2
u2(x y z t) =ax
2πR2
u3(x y z t) = 0
p(x y z t) =a2(x2 + y2
)8π2R4
(38)
This flow is actually a rigid body rotational flow in that the angular rotational velocity is uniformand the velocity of the flow increases proportionally with distance from the z-axis
Figure 19 An illustration of the flow in the xy-plane for Sol 15 (Rankine Vortex) described inEq (37) and Eq (38) with parameter values a = 1and R = 1
radic2π
23 Characterisation of Flows
It is important to be able to classify flows in a way that sorts them into distinct groups This aidsin the clear identification and general study of flows It is also a very important way of recognisingflows with similar properties In this subsection we introduce two distinct ways in which flows canbe characterised
231 Characterising Solutions using Invariants
A principal invariant of a tensor is a coefficient of the characteristic polynomial of that tensorInvariants as their name suggests are invariant under rotations of the coordinate system in thattheir values do not change when the tensor is lsquorotatedrsquo into a different set of coordinates Thevelocity gradient of a flow G is defined as the Jacobian of the velocity vector given by
Gij =
[partuipartxj
] (39)
20
for i j = 1 2 3 There are three principal invariants of the velocity gradient of a fluid flow G givenby
PG = tr(G)
QG =1
2
((tr(G))2 minus tr
(G2))
RG = det(G)
(40)
Incompressibility of flows (Eq (5)) ensures that PG = 0 hence the first principal invariant is zerofor all of the flows in Section 22 We now define the rate-of-strain tensor S by
Sij =1
2
(partuipartxj
+partujpartxi
) (41)
and the rate-of-rotation tensor Ω by
Ωij =1
2
(partuipartxjminus partujpartxi
) (42)
Note that the first principal invariant of S and the first and third principal invariants of Ω are zeroAlso note that the velocity gradient of a flow can simply be written G = S + Ω
Rather than considering the principal invariants we can also consider the combined invariants
I1 = tr(S2) I2 = tr(Ω2) I3 = tr(S3) I4 = tr(SΩ2) I5 = tr(S2Ω2) (43)
We can hence write the three principal invariants for a velocity field G in terms of the combinedinvariants
PG = 0
QG = minus1
2(I1 + I2)
RG =1
3(I3 + 3I4)
(44)
In this report the flows considered are the exact solutions to the incompressible Navier-Stokesequations described in Section 22 The combined invariants in Eq (43) are used to characterisethese solutions into different classes in Section 44 Both the principal invariants and combinedinvariants for each exact solution in Section 22 are given in Appendix A
Each of the subfilter-scale eddy-viscosity models described in Section 32 are written in terms ofthese five combined invariants
232 Vremanrsquos Characterisation of Flows
Vreman [16] classifies flows using the gradient of the velocity field of a flow Each lsquoclassrsquo of flows hasthe same number of non-zero elements in the gradient matrix For example
[partuipartxj
]=
0 0 0lowast 0 0lowast 0 0
(45)
where a lowast denotes a non-zero term represents a group of flows that lie within the class denoted byQ7 (see Appendix B) where the subscript represents the number of non-zero terms in the velocity
21
gradient Flows of the form in Eq (45) have a velocity field of the form
u1 = c1 (constant)
u2 = f2(x1)
u3 = f3(x1)
(46)
where f2(x1) and f3(x1) are such that the Navier Stokes equations Eq (1) are satisfied Notethat as the elements on the leading diagonal are zero the incompressibility condition Eq (5) isautomatically satisfied Some simple flow classes are given in Appendix B and contain a number ofthe solutions discussed in Section 22 This is mentioned in greater detail in Section 441
3 Conditions for Large-Eddy Simulation Models
31 Large-Eddy Simulation
Due to the nature of numerical simulation and the complexity of turbulent flows small details aremissed and information is lost when solving the Navier-Stokes equations The computational costof solving a turbulent flow is extremely high so a filtering operation is applied to the Navier-Stokesequations represented by a bar to reduce the computational cost of the simulation This is a linearoperation and preserves constants ie ρ = ρ and is also assumed to commute with differentiation
partuipartxj
=partuipartxj
(47)
The filtered Navier-Stokes equations are found by filtering (putting a bar over) the entire equations(see Eq (1)) and after evaluating are given by
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (48)
where
τij = uiuj minus uiuj (49)
is called the lsquotrue stressrsquo of the flow This however poses a problem as it is not clear what is meantby the term uiuj for a general filtering operation The true stress τij therefore must be replaced bya model τmod
ij in practical applications
32 Subfilter-Scale Models
We will now define the subflter-scale models denoted by τmodij to be used in place of the lsquodifficultrsquo
true stress τij There are a number of models available to use in the literature [11 p7] These willbe tested using the exact solutions to the Navier-Stokes equations and by checking conditions whichare defined in Sections 33 and 34
All the models barring one are of the same form That is each of the eddy-viscosity models arewritten
τmodij minus 1
3τmodkk δij = minus2νeSij (50)
with νe representing the eddy viscosity of a specific model For simplicity we usually just writeτmodij = minus2νeSij the extra term is included here to emphasise that the models are traceless Each
22
of the models we consider here can be expressed in terms of the rate of strain tensor S defined inEq (41) the rate of rotation tensor Ω defined in Eq (42) and the combined invariants defined inEq (43) The models are then defined as follows
Smagorinsky [12] νSe = (CSδ)2radic
2I1 (51)
Vreman [16] νVe = (CV δ)2
radicQGGT
PGGT (52)
QR [13 14 15] νQRe = (CQRδ)2 max0minusI3
I1 (53)
Vortex-Stretching [11] νV Se = (CV Sδ)2radic
2I1
(I5 minus 1
2I1I2
minusI1I2
)32
(54)
with the quantities
PGGT = I1 minus I2 QGGT =1
4(I1 + I2)2 + 4
(I5 minus
1
2I1I2
) (55)
There is a fifth and final model to be considered here It is of a different form to the other four andnot an eddy-viscosity model It is known as the Gradient model and is defined by
Gradient [4 9] τmodG = CGδ
(S2 minus Ω2 minus (SΩminus ΩS)
) (56)
The values C are model constants specific to each model and δ represents the mesh size of the LESBoth of these quantities are included here to properly define the models however in this projectthey will be ignored as they are constants and have no influence on the conditions to be defined
33 Conditions for Large-Eddy Simulation without Explicit Filtering
The filtering operation introduced in Eq (48) is just a formality for LES without explicit filteringThe idea here is that filtering does not actually take place but the term lsquoaddedrsquo to the Navier-Stokesequations in fact lsquosubtractsrsquo the difficult to simulate small-scale turbulent motion of the flow Inorder to consider this properly the terms ui and p in Eq (48) are replaced by vi and q It is hopedthat τmod
ij approximates τij so that vi and q approximate ui and p respectively So LES withoutexplicit filtering is given by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (57)
It is clear to see that the first four terms of this equation are exactly the same as the Navier-Stokesequations so for the exact solutions discussed in Section 22 the combination of these terms willequal zero What is then desirable is that the model term is also zero for these exact solutions sothat Eq (57) still holds In order to test models for exact solutions the condition
part
partxjτmodij (v) = 0 (58)
is checked If this condition were to hold it would imply that the model does not influence thevelocity field of a flow hence we call it the Velocity Field Condition There are a number of otherrestrictions that can be tested One such test comes from the differential form of the evolution ofthe kinetic energy of a flow given in Eq (10) In order to find this Eq (57) is multiplied by vi andterms are grouped to find
part
partt
(1
2vivi
)+ vj
part
partxj
(1
2vivi
)= minusvi
1
ρ
partq
partxi+ νvi
part2vipartxjpartxj
minus vipart
partxjτmodij (v) (59)
23
Then similarly to Eq (58) another condition is to check whether the model affects the kineticenergy of exact solutions this is equivalent to checking whether
vipart
partxjτmodij (v) = 0 (60)
We call this condition the Kinetic Energy Condition The third and final condition considered herecan also be found using the integral form for the equation for kinetic energy
d
dt
intV
1
2vividV =
intS
[minusvj
(1
2vivi
)minus vj
q
ρ+ 2νviSij + viτ
modij (v)
]njdS
+
intV
[minus2νSijSij minus
(minusSijτmod
ij (v))]dV
(61)
where the rate-of-strain tensor Sij was defined in Eq (41) V is a volume of fluid and dS meansthe integral is over the surface of that volume with unit normal nj Then this final condition checkswhether the model affects the total kinetic energy of exact solutions to the Navier-Stokes equationsthat is
Sijτmodij (v) = 0 (62)
We will call this condition the Dissipation Condition because Sijτmodij (v) is often referred to as the
subgrid dissipation
34 Conditions for Large-Eddy Simulation with Explicit Filtering
As in Eq (48) the Navier-Stokes equations can be filtered by applying the filtering operation to thewhole equation (putting a bar over all terms) and evaluating to find again that
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (63)
In Eq (57) the filtering operation was ignored and the elements ui p and τij(u) were replacedwith vi q and τmod
ij (v) respectively Now in LES with explicit filtering the filtering operation itselfsmooths over the small-scale turbulent motion of the simulated flow In this case the elements uip and τij(u) will be replaced with vi q and τmod
ij (v) respectively LES with explicit filtering is thengiven by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (64)
Note that Eqs (63) and (64) despite their similarity to Eq (57) no longer contain the Navier-Stokesequations as Eq (57) did The bar (filtering operation) over each term results in each exact solutionfound in Section 22 not necessarily satisfying the first four terms It is hence unknown whether ornot these terms will vanish for exact solutions Terms can still be compared similarly to Eqs (58)(60) and (62) However instead of comparing to zero we can now compare to the relevant termfrom the filtered Navier-Stokes equations So the terms
part
partxjτij (65)
and
part
partxjτmodij (v) (66)
24
are to be compared Although the latter term is relatively simple to compute the former could leadto some difficulties due to the differentiation of the true stress τij = uiuj minus uiuj So although someinformation could be gleaned from the inspection of these two terms more information can be foundby considering the analogue of Eq (62) So the true subfilter dissipation
Sijτij (67)
(where Sij is shorthand for Sij(v)) and the modelled subfilter dissipation
Sij(v)τmodij (v) (68)
are considered and have to be equal for the condition to hold These can again be derived from theintegral equation for kinetic energy of a filtered velocity field similarly to Eq (61)
4 Analysis of Large-Eddy Simulation Models
In this section we use a code in Mathematica the theory described in Section 3 and the exactsolutions to the Navier-Stokes equations found in Section 22 to check whether or not each of theexact solutions lsquoendorsesrsquo each model for LES both without and with explicit filtering
41 Large-Eddy Simulation Without Explicit Filtering
We start by using a code in Mathematica to test the Velocity Field condition in Eq (58) and theKinetic Energy condition in Eq (60) For all exact solutions to the Navier-Stokes equations inSection 22 the outputs for both conditions were identical so we have combined them into one table
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 ndash ndash 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 1 For the Kinetic Energy condition a lsquo0rsquo result means that vipartpartxj
τmodij (v) = 0 for respective
models and exact solutions whereas a lsquolowastrsquo result means that vipartpartxj
τmodij (v) 6= 0 For the Velocity
Field condition a lsquo0rsquo result means that partpartxj
τmodij (v) = 0 for respective models and exact solutions
whereas a lsquolowastrsquo result means that partpartxj
τmodij (v) 6= 0 The Gradient model can take both positive and
negative values so when the sign of the model is known it is included in the table instead of a lsquorsquoBoth the conditions are included in one table because for all cases the outputs were identical
We then test the Dissipation condition in Eq (62) for all exact solutions to the Navier-Stokesequations in Section 22
25
Dissipation Condition [Does Sijτmodij (v) = 0]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 + 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 2 A lsquo0rsquo result means that Sijτmodij (v) = 0 for respective models and exact solutions whereas
a lsquolowastrsquo result means that Sijτmodij (v) 6= 0 The Gradient model can take both positive and negative
values so when the sign of the model is known it is included in the table instead of a lsquorsquo
We then sum all the results to find the total number of endorsements for each model described inSection 32
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
4 6 8 9 9
lsquoDissipationrsquoEndorsements
2 5 7 7 9
Total Endorsements 6 11 15 16 18
Table 3 The results from Tables 1 and 2 are summarised to give a general overview of how manyendorsements each model receives This makes it much clearer to see the models most supported bythe respective and combined conditions with the Vortex-Stretching model being the most supportedand the Smagorinsky model the least
Testing the models in Section 32 using LES without explicit filtering has mixed results It isclear that the Smagorinksy model is not well endorsed by the flows in Section 22 with the onlyendorsements coming from the simplest flows that is Sol 1 (Linear Velocity) Sol 3 (Velocity as afunction of time) Sol 7 (Stagnation Flow) and Sol 15 (Rankine Vortex) The Vortex-Stretching-based model is the most endorsed flow with eighteen out of a possible thirty endorsements Thisagrees with our foreknowledge as this model was constructed to be as physically consistent as possiblein the paper by Silvis amp Verstappen [11]
42 Large-Eddy Simulation With Explicit Filtering
In this section we will compare the true subfilter dissipation Sijτij and the modelled subfilterdissipation Sij(v)τmod
ij (v) given in Eqs (67) and (68) respectively for each model τmodij and each
exact solution to the Navier-Stokes equation given in Section 22 We will not consider the conditionsinvolving differentiation in Eqs (65) and (66) due to the difficulty in calculating the derivative ofthe τij term in Eq (65)
26
421 Example of a Test for a Model
The true subfilter dissipation Sijτij of a flow given in Eq (67) cannot be explicitly computed asthe filtering operation is kept general What can be done however is to partially evaluate theexpressions using the attributes of a filtering operation mentioned in Eq (48)
The calculations of Sijτij for Sol 1 (Linear Velocity) are provided to illustrate the general ideaRecall that the velocity field of Sol 1 (Linear Velocity) is written in Eq (13) in vector notation as
u(x y z t) =
0ax+ bcx+ d
(69)
Then Sij = Sij(u) is computed simply by using its definition in Eq (41) Note that as filteringpreserves constants if there is an a it is replaced with a meaning that (in this case) there are alarge number of zero terms due to the frequency of 0 Hence
[Sij(u)
]=
[1
2
(partuipartxj
+partujpartxi
)]=
0 a
2c2
a2 0 0
c2 0 0
=
0 a
2c2
a2 0 0
c2 0 0
(70)
Note that as Sijτij is being computed which is equal to the sum of the element-wise multiplicationof the two matrices (or the trace of the matrix multiplication) that we donrsquot need to consider theelements of τij(u) with corresponding zero entry in Sij As τij(u) = uiuj minus uiuj and again asfiltering preserves constants it is found that
[τij(u)] =[uiuj minus uiuj
]=
lowast 0 middot ax+ b 0 middot cx+ d
ax+ b middot 0 lowast lowastcx+ d middot 0 lowast lowast
=
lowast 0 00 lowast lowast0 lowast lowast
(71)
where a lsquolowastrsquo just represents some element (not necessarily non-zero) Therefore the true subfilterdissipation
Sijτij = tr
0 a2
c2
a2 0 0c2 0 0
middotlowast 0 0
0 lowast lowast0 lowast lowast
= 0 (72)
Now the modelled subfilter dissipation Sij(u)τmodij (u) needs to be computed (recall that Sij =
Sij(u)) In this case let us use the QR model stated in Eq (53) Once again Sij(u) calculated inEq (70) needs to be used So we have that
τmodij = minus2(CQRδ)
2 max0minusI3I1
Sij (73)
As
I1 = tr(S2) =
1
2(a2 + c2) and I3 = tr(S
3) = 0 (74)
it holds that τmodij (u) = 0 so Sij(u)τmod
ij (u) = 0 This agrees with Eq (72) and hence Sol 1(Linear Velocity) endorses the QR model
27
422 Testing the Models
Now using a Mathematica code the exact solutions in Section 22 are again used to evaluate thecondition for the models using the process just described in Section 421 The results are describedin Table 4
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 4 All entries with a ldquo0rdquo represent a value of zero and a ldquolowastrdquo represents a value that is non-zeroThe ldquoExactrdquo row represents the value of the true subfilter dissipation Sijτij Every other row witha model name represents the value of the modelled subfilter dissipation Sij(u)τmod
ij (u) respectiveto that model A ldquordquo means that it is unclear as to the value - the sign of these values is oftenunknown and the lsquoMaxrsquo function in the definition of the QR model (in Eq (53)) makes it difficultto evaluate
Reading from Table 4 the Smagorinsky model is clearly endorsed by Sol 3 (Velocity as a function oftime) and Sol 15 (Rankine Vortex) as in both cases both Sijτij = 0 = Sij(u)τmod
ij (u) In this casewe say that Sol 3 (Velocity as a function of time) and Sol 15 (Rankine Vortex) lsquostronglyrsquo endorsethe Smagorinsky model The Smagorinsky model is not endorsed by Sol 1 (Linear Velocity) Sol 2(Planar Flow) or Sol 14 (Radially Dependent) however because Sij(u)τmod
ij (u) 6= 0 For Sol 4 to
13 both Sijτij 6= 0 6= Sij(u)τmodij (u) and so we will say that these exact solutions lsquoweaklyrsquo endorse
the Smagorinsky model
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 10 10 9 6
Table 5 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Interestingly the lsquostrongrsquo and lsquoweakrsquo endorsements give different results However more stock shouldbe put in the results of the lsquostrongerrsquo condition as it is completely clear that zero is equal to zeroIt is not necessarily clear whether or not non-zero equals non-zero as the values will depend on theparameters in each solution hence the lsquoweakerrsquo endorsement
The lsquostrongrsquo condition equally endorses the Vreman QR Gradient and Vortex-Stretching-basedmodels whereas the lsquoweakrsquo condition endorses Vreman and Smagorinsky the most closely followedby the Gradient model with the Vortex-Stretching-based model being the most opposed As this
28
completely contradicts the lsquostrongrsquo condition and the results in Section 41 it makes us question thevalidity of the weaker test The number of endorsements for the QR model is unknown because of thedefinition of the model in Eq (53) and our evaluation of the true subfilter dissipation Eq (67) Dueto the exact solutions we have found it is never clear whether or not the true subfilter dissipationis strictly positive or negative This means that when the true subfilter dissipation is found to benon-zero the lsquoMaxrsquo function in the QR model cannot be evaluated
43 Vremanrsquos True Subfilter Dissipation Claim
Using the theory mentioned in Sections 232 and 34 we can place the exact solutions to the Navier-Stokes equations into different classes defined by Vreman In doing this we can also attempt to verifyone of Vremanrsquos results [16] In Vremanrsquos paper he defines the class Qn to be the set of flows withn zero entries in the velocity gradient of the flow (the velocity gradient is defined in Eq (39))Vreman then claims that the only flows with zero true subfilter dissipation are a certain selection ofthose in the classesQ7Q8 orQ9 shown in appendix B and is generally non-zero for any other flows
The true subfilter dissipation of Sol 1 (Linear Velocity) was computed in Section 421 and wasfound to be zero in Eq (72) The velocity gradient of Sol 1 is as follows
GLV =
0 0 0a 0 0c 0 0
(75)
and hence Sol 1 (Linear Velocity) is in the required subset of Q7 which agrees with Vremanrsquos claimIn Section 422 the true subfilter dissipation was computed (determined to be zero or non-zero) forall of the exact solutions to the Navier Stokes equations in Section 22 These solutions can now belsquoplacedrsquo into each class Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a functionof time) and Sol 14 (Radially Dependent) were all found to have zero true subfilter dissipation andare all in the appropriate subsets of either Q7Q8 or Q9 This agrees with Vremanrsquos results
We can extend Vremanrsquos results however We found that Sol 15 (Rankine Vortex) had zero truesubfilter dissipation and yet is not in the subset of Q7 defined in Appendix B It is important tonote that the rate of strain tensor Sij in Eq (41) for Sol 15 (Rankine Vortex) is zero This is thereason the true subfilter dissipation is zero and the same reason that all models are zero for thisflow as all models are based on Sij
We also found that all of the remaining exact solutions to the Navier-Stokes equations are not inVremanrsquos specified classes (in Appendix Section B) They also have a non-zero subfilter dissipationwhich agrees with Vremanrsquos results in his paper [16]
44 Classes of Flows
It is important to consider that some exact solutions to the Navier-Stokes equations may have similarproperties For example Sol 1 (Linear Velocity) and Sol 2 (Planar Flow) are both very similarIn fact Sol 2 (Planar Flow) is a more general case of Sol 1 (Linear Velocity) We therefore mustgroup the different flows into classes in an attempt to reduce the bias that may occur when solutionsare similar This should aid us to further determine which of the models are most endorsed We willdo this in a number of different ways using the theory discussed in Section 23 We will group theflows with respect to Vremanrsquos Flow Classes given in appendix B and with respect to the Invariantsgiven in appendix A
29
441 Vremanrsquos Velocity Gradient Flow Classes
We now put our exact solutions to the Navier-Stokes equations into one of two classes Either Q0minus6or Q7minus9 where each class is defined intuitively as follows
Q0minus6 = Q0 cupQ1 cup middot middot middot cup Q6 (76)
Q7minus9 = Q7 cupQ8 cupQ9 (77)
where Qn is defined in Section 43 By looking at the gradient of each exact solution we can see thatQ7minus9 contains the six solutions Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as afunction of time) Sol 5 (2D Taylor) Sol 14 (Radially Dependent) and Sol 15 (Rankine Vortex)and that Q0minus6 contains the remaining nine We now check the models for LES without explicitfiltering by considering endorsements within each class Tables 6 and 7 are analogues of Table 3 andillustrate the results for each flow class The analogues of Tables 1 and 2 are Tables 22 and 23 andare in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 6 6 6
lsquoDissipationrsquoEndorsements
2 5 6 6 6
Percentage Endorsed 417 833 100 100 100
Table 6 Each entry in the top two rows represents the number of solutions in the class Q7minus9 thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Q0minus6 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 2 3 3
lsquoDissipationrsquoEndorsements
0 0 1 0 3
Percentage Endorsed 56 56 167 167 333
Table 7 Each entry in the top two rows represents the number of solutions in the class Q1minus6 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking results to 1 decimal place if necessary
All of the flows in the class Q7minus9 endorse the QR Gradient and Vortex-Stretching-based subfilter-scale models (see Table 6) We can hence see that this class fully endorses these three models equallyand does not endorse the Smagorinsky model The Vortex-Stretching eddy viscosity model has themost endorsements from any of the flows in the class Q0minus6 and the Smagorinksy and Vreman modelshave the fewest endorsements (see Table 7) We can hence say that this class mostly endorses theVortex-Stretching-based eddy-viscosity model note however that the percentage endorsement is very
30
low
Now let us consider LES with explicit filtering Tables 8 and 9 are analogues of Table 5 and illustratethe results for each flow class The analogue of Table 4 is Table 24 and is in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 1 1 0 0 0
Table 8 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Q0minus6 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 9 9 9 6
Table 9 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We have grouped Vremanrsquos classes into Q7minus9 and Q0minus6 this is equivalent to saying that we havegrouped Vremanrsquos classes into those with two or fewer non-zero velocity gradient elements and thosewith three or more non-zero velocity gradient elements respectively Therefore the solutions in Q7minus9are exactly those with a lsquo0rsquo in the lsquoExactrsquo column of Table 4 and Sol 5 (2D Taylor) ExcludingSol 5 (2D Taylor) these are the same as those five that endorse the Vreman QR Gradient andVortex-Stretching-based models in Table 5 We can see in Table 8 that the results for the lsquostrongrsquoendorsements are exactly the same as in Table 5 This is because all of the flows with zero subfilterdissipation are contained in the class Q7minus9 There are hence no lsquostrongrsquo endorsements in Table 9The majority of the lsquoweakrsquo endorsements come from Q0minus6 as this class contains all but one of thesolutions with non-zero true subfilter dissipation (given in Eq (67))
442 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
We now separate our exact solutions into classes again using Vremanrsquos flow classes This timehowever we define V0 to be the class of flows Vreman claims will always have zero true subfilterdissipation namely those classes mentioned in Appendix B We define Vlowast to be all the remaining
31
classes Then
V0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow)
Sol 3 (Velocity as a function of time) Sol 14 (Radially Dependent)(78)
Vlowast = Sol 4 (Axisymmetric Flow) Sol 5 (2D Taylor) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 13 (Sphere Translation) Sol 15 (Rankine Vortex)
(79)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 10 and 11 are analogues of Table 3 and illustrate the results for each flow class Theanalogues of Tables 1 and 2 are Tables 25 and 26 and are in Appendix C2
Endorsements for models using solutions in V0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 4 4 4 4
lsquoDissipationrsquoEndorsements
1 4 4 4 4
Percentage Endorsed 375 100 100 100 100
Table 10 Each entry in the top two rows represents the number of solutions in the class V0 thatendorse each model There are 4 solutions in this class so the percentages are worked out by dividingthe total of each column by 8 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Vlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 2 4 5 5
lsquoDissipationrsquoEndorsements
1 1 3 2 5
Percentage Endorsed 136 136 318 318 455
Table 11 Each entry in the top two rows represents the number of solutions in the class Vlowast thatendorse each model There are 11 solutions in this class so the percentages are worked out bydividing the total of each column by 22 and taking results to 1 decimal place if necessary
All of the flows in the class V0 endorse the Vreman QR Gradient and Vortex-Stretching-basedsubfilter-scale models (see Table 6) This class completely endorses these four models equally anddoes not endorse the Smagorinsky model The Vortex-Stretching-based eddy-viscosity model hasthe most endorsements from the class Vlowast and the Smagorinksy and Vreman models again have thefewest endorsements (see Table 7) As in Section 441 we say that this class mostly endorses the
32
Vortex-Stretching-based eddy-viscosity model but once again the percentage endorsement is ratherlow
Now let us consider LES with explicit filtering Tables 12 and 13 are analogues of Table 5 andillustrate the results for each flow class The analogue of Table 4 is Table 27 and is in Appendix C2
Endorsements for models using solutions in V0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 4 4 4 4
lsquoWeakrsquo Endorsements 0 0 0 0 0
Table 12 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Vlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 1 1 1 1
lsquoWeakrsquo Endorsements 10 10 9 6
Table 13 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We can see in Table 12 that the results for the lsquostrongrsquo endorsements are similar to those in Table 5excluding Sol 15 (Rankine Vortex) The four solutions in V0 with zero true subfilter dissipationnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)and Sol 14 (Radially Dependent) are hence the four that endorse the Vreman QR Gradientand Vortex-Stretching-based models in Table 12 Sol 15 (Rankine Vortex) is in the class Vlowast andhence there is one lsquostrongrsquo endorsement in Table 13 The class Vlowast contains all of the flows with anon-zero true subfilter dissipation (given in Eq (67)) and hence Table 13 contains all of the lsquoweakrsquoendorsements
443 Classes using the Third Principal Invariant
Sorting the classes using the invariants is a non-trivial task in itself There are a number of waysin which to do this using both the principal invariants in Eq (40) and the combined invariants inEq (43)
We start by looking at flows with third principal invariant RG equal to zero We again define twoclasses R0 and Rlowast where R0 consists of all flows with RG = 0 and Rlowast consists of all flows withRG 6= 0 Then using the information in Appendix A we see that there are nine flows in R0 That
33
is
R0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 8 (Vortex in Stagnation Flow)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)
(80)
Rlowast = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 13 (Sphere Translation)(81)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 14 and 15 are analogues of Table 3 and they illustrate the results for LES withoutexplicit filtering for each flow class The analogues of Tables 1 and 2 are Tables 28 and 29 and arein Appendix C3
Endorsements for models using solutions in R0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 7 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 278 556 778 722 778
Table 14 Each entry in the top two rows represents the number of solutions in the class R0 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking the result to one decimal place if necessary
Endorsements for models using solutions in Rlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 2 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 83 83 83 167 333
Table 15 Each entry in the top two rows represents the number of solutions in the class Rlowast thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking the result to one decimal place if necessary
The class R0 endorses both the Vortex-Stretching-based model and the QR model the most andSmagorinsky the least in the case for LES without explicit filtering In the Rlowast class the Vortex-Stretching-based model is again the most endorsed followed by the Gradient model however thepercentages are again small
34
Now let us consider LES with explicit filtering Tables 16 and 17 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 30 and is in Appendix C3
Endorsements for models using solutions in R0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 4 4 3 2
Table 16 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Rlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 6 6 6 4
Table 17 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 we can see in Table 16 that the results for the lsquostrongrsquo endorsements are exactly thesame as in Table 5 This is because all of the flows with zero subfilter dissipation are contained in theclass R0 As in Table 9 there are no lsquostrongrsquo endorsements in Table 17 The lsquoweakrsquo endorsementshowever are split more evenly between the two classes now than in Vremanrsquos classes in Sections 441and 442 We find that the Smagorinsky and Vreman models still seem to be the most endorsed inboth classes
444 Two-component Flow Classes
Another way in which the flows can be grouped into classes is by determining whether a flow is atwo-component flow or not We define a two-component flow to have the property that the combinedprincipal invariants of a flow in Eq (43) satisfy
I3 = I4 = I5 minus1
2I1I2 = 0 (82)
We are also interested in considering flows with zero gradient model subgrid dissipation These flowshave the property that
I3 minus I4 = 0 (83)
What we find though is that for all the solutions in Section 22 if a flow is a two-component flowthen it also satisfies I3 minus I4 = 0 It may be interesting to note that although I3 minus I4 6= 0 for manysolutions there are a number with I3 equal to a multiple of I4 Although not considered in thisreport this may also be a class of interest We find that the two-component flows are
C2 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)(84)
35
and we call the remaining flows three-component flows They are
C3 = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex) Sol 11 (Pseudo-Motion Shots)
Sol 13 (Sphere Translation)
(85)
Note that Sol 8 (Vortex in Stagnation Flow) is defined to be a planar flow this does not mean that itis a two-component flow in this sense however because its two non-zero velocity components dependon three coordinates We now check the models for LES without explicit filtering by consideringendorsements within each class Tables 18 and 19 are analogues of Table 3 and they illustrate theresults for LES without explicit filtering for each flow class The analogues of Tables 1 and 2 areTables 31 and 32 and are in Appendix C4
Endorsements for models using solutions in C2 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 6 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 357 714 100 857 100
Table 18 Each entry in the top two rows represents the number of solutions in the class C2 thatendorse each model There are 7 solutions in this class so the percentages are worked out by dividingthe total of each column by 14 and taking the result to one decimal place if necessary
Endorsements for models using solutions in C3 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 3 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 63 63 63 188 250
Table 19 Each entry in the top two rows represents the number of solutions in the class C3 thatendorse each model There are 8 solutions in this class so the percentages are worked out by dividingthe total of each column by 16 and taking the result to one decimal place if necessary
The class C2 equally endorses the QR and Vortex-Stretching-based models and once again theSmagorinsky model is the least endorsed As with the previous classes the class C3 has such lowpercentages that it does not seem to endorse any models We start to see a trend appearing here inthat the flow classes that contain the more complicated flows namely Q0minus6 (Eq (76)) Vlowast (Eq (79))Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages The classes that tendto contain the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78)) R0 (Eq (80)) and C2 (Eq (84))all seem to favour the Vortex-Stretching-based model or the QR model
36
Now let us consider LES with explicit filtering Tables 20 and 21 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 33 and is in Appendix C4
Endorsements for models using solutions in C2 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 2 2 0 1 0
Table 20 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in C3 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 8 8 8 6
Table 21 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 and Table 16 we can see in Table 20 that the results for the lsquostrongrsquo endorsements areexactly the same as in Table 5 This is again because all of the flows with zero subfilter dissipationare contained in the class C2 There are two flows with non-zero true subfilter dissipation (givenin Eq (67)) in C2 and this shows in the lsquoweakrsquo endorsements in Table 20 There are no lsquostrongrsquoendorsements in Table 21 as all the zero subfilter dissipation flows are in C2
5 Conclusions
In this report we studied incompressible turbulent flows We found a number of exact solutions to theNavier-Stokes equations each representing a relatively simplistic fluid flow Large-eddy simulation(LES) is used to model turbulent flows by smoothing over the complex small-scale motion Weintroduced a number of subfilter-scale models to approximate this small-scale motion and testedthem using the found exact solutions to the Navier-Stokes equations We first considered LESwithout explicit filtering and used the Navier-Stokes equations to derive conditions for the subfilter-scale models namely the Velocity Field condition (Eq (58)) the Kinetic Energy condition (Eq (60))and the Dissipation condition (Eq (62)) Similarly we considered LES with explicit filtering andagain used the Navier-Stokes equations to derive more conditions for the models in this case wedecided to just take one condition the Dissipation condition (Eqs (67) and (68)) as the others leadto difficulty in their evaluation By lsquorunningrsquo the exact solutions to the Navier-Stokes equationsthrough each condition for the subfilter-scale models we found that the Vortex-Stretching-basededdy-viscosity model in Eq (54) was most endorsed by the tests in LES without explicit filteringWe found that four out of the five subfilter-scale models namely the Vreman QR Gradient andVortex-Stretching-based models were equally endorsed by the lsquostrongrsquo tests in LES with explicitfiltering with the Smagorinsky model being the least endorsed The results we found for the lsquoweakrsquocondition were completely opposite to those found in both the lsquostrongrsquo condition and in LES withoutexplicit filtering This led us to conclude that this test was not one in which we should put our trust
37
We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses
In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes
This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold
38
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
1 Introduction
In this project we will study and test models for turbulent fluid flows There is no all-encompassingdefinition of a turbulent flow there are however a number of characterising properties A turbulentflow is said to be irregular in that it appears to be random Turbulent flows are chaotic and henceusually described statistically due to the difficulty in a deterministic approach Turbulent flowsare diffusive in that there is a high rate of momentum heat and mass transfer They require apersistent source of input energy in order to sustain the turbulent nature of the flow as kineticenergy is quickly converted into internal energy A large Reynolds number is characteristic of aturbulent flow [5]
The Navier-Stokes equations are a set of fundamental governing equations for any Newtonian fluidAn exact solution to the Navier-Stokes equations is in the form of a flow velocity and a pressureterm and hence a fluid flow can be completely described by an exact solution to the Navier-Stokesequations It is still a non-trivial task to find such solutions although there are a number in literature[17]
The Navier-Stokes equations have no closed form solution resulting in difficulty when analysingcomplicated flows in complex geometries This results in difficulty when solving the Navier-Stokesequations explicitly So a mathematical model known as Large Eddy Simulation (LES) can insteadbe used to model turbulent flows LES uses what is known as a filtering operation to reduce thecomputational cost of a simulation by ignoring small-scale motion What this means however isthat information about these small-scale motions is lost This information is often non-trivial sosubfilter-scale models (or turbulence models) are used in order to approximate the effects of small-scale motions on the large-scale fluid flow
In this project we will use the behaviour of exact solutions to the Navier-Stokes equations to testsubfilter-scale models Applying a subfilter-scale model to a flow has two effects One is a bodyforce on the velocity field called a subfilter force and the other is a transfer of energy from largeto small scales of motion called a subfilter dissipation This results in respectively an extra termbeing added to the Navier-Stokes equations and to the equation of the kinetic energy of a flow
We will then consider conditions for subfilter-scale models in two different cases In LES withoutexplicit filtering we will consider how a subfilter-scale model behaves for an exact solution to theNavier-Stokes equations by checking if the subfilter force and subfilter dissipation are zero for anexact solution If so this can be interpreted as the model being inactive and hence this exactsolution can be said to lsquoendorsersquo the model In LES with explicit filtering we compare a model termwith a term in the filtered Navier-Stokes equations (called the true turbulent stress of a flow) forexact solutions If the two terms are both zero we will conclude that the exact solution lsquostronglyrsquoendorses the model and if the two terms are both non-zero we will conclude that the exact solutionlsquoweaklyrsquo endorses the model
We will also consider the classification of fluid flows in order to further inspect the models andreduce the bias that may occur when solutions have similar properties We will use properties of theinvariants of a flow to characterise flows in a coordinate-independent way Using Vremanrsquos paperldquoAn eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory and applicationsrdquo[16] we will place the fluid flows into Vremanrsquos flow classes We will also determine whether ourresults say anything beyond Vremanrsquos work
4
2 The Navier-Stokes Equations
21 Introduction
The incompressible Navier-Stokes equations are a set of four equations that model the flow of anyfluid in three-dimensional space The standard set of Navier-Stokes equations can be written in twoforms namely by using Einsteinrsquos summation convention in what we call index notation and byusing standard vector notation In index notation they are given by
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
i = 1 2 3 j = 1 2 3 (1)
These equations have four variables given by the velocity of the flow u = (u1 u2 u3) and thepressure p The vector x = (x1 x2 x3) represents position t represents time ρ corresponds to themass density and ν is the kinematic viscosity of the fluid
The Navier-Stokes equations can be written in vector notation in the following way First considerthe convection term
ujpartuipartxjequiv
3sumj=1
ujpartuipartxj
= u1partuipartx1
+ u2partuipartx2
+ u3partuipartx3
=
(u1
part
partx1+ u2
part
partx2+ u3
part
partx3
)ui
= (u middot nabla)ui
(2)
Similarly for the diffusion term
νpart2uipartxjpartxj
equiv ν3sumj=1
part2uipartxjpartxj
= νpart2uipartx1partx1
+part2uipartx2partx2
+part2uipartx3partx3
= ν
(part2
partx1partx1+
part2
partx2partx2+
part2
partx3partx3
)ui
= νnabla2ui
(3)
It therefore holds that in vector notation the Navier-Stokes equations are given by
partu
partt+ (u middot nabla)u = minus1
ρnablap+ νnabla2u (4)
It is clear that there is a problem here there are three equations and four unknowns In order toremedy this the equation for conservation of mass or the incompressibility condition is consideredThis is given by
partuipartxi
= 0 (5)
in index notation and
nabla middot u = 0 (6)
5
in vector notation
There are now four equations in four unknowns and it is possible to find solutions Althoughthere is no known general solution there are specific cases in which exact solutions to the Navier-Stokes equations can be found In order to make finding solutions slightly simpler the Navier-Stokesequations can be rescaled by introducing dimensionless variables
Let
ulowasti =uiv xlowasti =
xi` tlowast =
t
`v plowast =
p
ρv2 (7)
where ` and v are typical length and velocity scales respectively Then dropping the star notationfor ease of reading the scaled incompressible Navier-Stokes equations are
partuipartt
+ ujpartuipartxj
= minus partp
partxi+
1
Re
part2uipartxjpartxj
i = 1 2 3 j = 1 2 3
partuipartxi
= 0
(8)
where Re is the Reynolds number and is defined by the dimensionless Re =v`
ν Similarly in vector
notation the scaled incompressible Navier-Stokes equations are
partu
partt+ (u middot nabla)u = minusnablap+
1
Renabla2u
nabla middot u = 0(9)
By multiplying the incompressible Navier-Stokes equations Eq (8) by ui and grouping terms thedifferential form of the evolution of the kinetic energy of a flow can be found
part
partt
(1
2uiui
)+ uj
part
partxj
(1
2uiui
)= minusui
partp
partxi+
2
Reui
part2uipartxjpartxj
(10)
The differential form of the evolution of the kinetic energy of a flow will be used later on in thisreport to derive conditions for the subfilter-scale models
22 Exact Solutions of the Navier-Stokes Equations
By using mathematical inspection and taking results found in literature we found a number of exactsolutions to the Navier-Stokes equations The majority are steady-state solutions in that they areindependent of time We decided to consider exact solutions to the Navier-Stokes equations withoutboundary conditions in an attempt to keep the results general and applicable to a larger numberof situations All of the stated exact solutions below have been checked using a code written inMathematica In the solutions below the notation (x y z) for the Cartesian coordinate system isused analogously with (x1 x2 x3) to aid in reading comprehension and in the conversion betweenindex and vector notation
Solution 1 Linear Velocity
If the diffusion term in the Navier-Stokes equations Eq (1) is taken to be zero ie
part2uipartxjpartxj
= 0 (11)
6
then each component of the velocity can be written as a linear function of x1 x2 and x3 In indexnotation
ui = aijxj + βi (12)
If pressure is assumed to be a function of time t (so that the pressure terms in the Navier-Stokesequations are zero) then conditions can be found on the coefficients to find that the following is asolution
u1(x y z t) = 0
u2(x y z t) = ax+ b
u3(x y z t) = cx+ d
p(x y z t) = f(t)
(13)
where a b c d isin R and f Rrarr R In fact this solution is actually a specific case of Sol 2 (PlanarFlow)
Figure 1 An illustration of the flow in the xy-plane for Sol 1 (Linear Velocity) described inEq (13) with parameter values a = b = c = d = 1
Solution 2 Planar Flow
Using the text by Barbato et al [2 p62] and modifying the solution such that it now takes threedimensions into account another solution is found
u1(x y z t) = Az2 +Bz + C
u2(x y z t) = Dz2 + Ez + F
u3(x y z t) = 0
p(x y z t) =2A
Rex+
2D
Rey + f(t)
(14)
7
where ABCDE F isin R and f R rarr R It is clear to see that if A = D = 0 then this is in factthe same as Eq (13) in Sol 1 In fact we believe (although not shown here) that this is the mostgeneral solution of the form
ui(xi xj xk t) = Ax2k +Bxk + C
uj(xi xj xk t) = Dx2k + Exk + F
uk(xi xj xk t) = 0
(15)
for i 6= j 6= k 6= i
Figure 2 An illustration of the flow in the yz-plane for Sol 2 (Planar Flow) described inEq (14) with parameter values A = B = C =D = E = F = 1
Solution 3 Velocity as a function of time t
If each component of the velocity is taken to be a function of time t and independent of positionthen the incompressibility condition is automatically satisfied A condition can be imposed on thepressure term in order to satisfy the momentum equations Thus a solution is as follows
u1(x y z t) = f(t)
u2(x y z t) = g(t)
u3(x y z t) = h(t)
p(x y z t) = minusxpartf(t)
parttminus y partg(t)
parttminus z parth(t)
partt
(16)
where f g h Rrarr R In fact what is written here is the generalised Galilean transformation of thezero solution [10] We cannot produce a plot for this flow as the functions are completely unspecified
8
Solution 4 Axisymmetric Flow
Using the text by Irmay [7] a solution is defined as follows
u1(x y z t) = kxz
u2(x y z t) = kyz
u3(x y z t) = minuskz2
p(x y z t) = minus2kz
Reminus k2z4
2
(17)
where k isin R This is said to represent an axisymmetric flow with hyperbolic streamlines asymptoticto the z-axis and to the boundary plane z = 0 These asymptotes are visible in Fig 3
Figure 3 An illustration of the flow in theyz-plane for Sol 4 (Axisymmetric Flow)described in Eq (17) with parameter k = 1
Figure 4 An illustration of the flow in thexy-plane for Sol 4 (Axisymmetric Flow)described in Eq (17) with parameter k = 1
Solution 5 2D Taylor Solutions
Again using the text by Barbato et al [2 p63ndash64] a solution is defined using a stream functionψ = ψ(x y t) and by writing
u1 = minuspartψparty
u2 =partψ
partx u3 = 0 (18)
In the text the stream function is not given however it was worked out that
ψ = minuseminusπ2t
Re (cos(πx) + cos(πy))
π (19)
The solution used is of the following form note however that in the text the pressure term differsin that cos(πy) is replaced by sin(πy) (when testing this solution in Mathematica the solution below
9
was instead found)
u1(x y z t) = minus sin(πy) eminus(π2t)Re
u2(x y z t) = sin(πx) eminus(π2t)Re
u3(x y z t) = 0
p(x y z t) = minus cos(πx) cos(πy) eminus(2π2t)Re
(20)
Figure 5 An illustration of the flow in the xy-plane for Sol 5 (2D Taylor) described in Eq (20)with Re = 1 and time t = 0
Solution 6 Generalized Beltrami Flow
Again using the text by Barbato et al [2 p65] and now considering solutions with nonndashtrivialterms in all dimensions a particular form of the generalized Beltrami flow is found namely
u1(x y z t) = (A sin(πz) + C cos(πy)) eminus(π2t)Re
u2(x y z t) = (B sin(πx) +A cos(πz)) eminus(π2t)Re
u3(x y z t) = (C sin(πy) +B cos(πx)) eminus(π2t)Re
p(x y z t) = minus[(AB cos(πz) sin(πx) +BC cos(πx) sin(πy)
+AC cos(πy) sin(πz)] eminus(2π2t)Re
(21)
where ABC isin R
10
Figure 6 An illustration of the flow in the yz-plane for Sol 6 (Generalized Beltrami Flow)described in Eq (21) with parameters A = B =C = 1 and time t = 0
Solution 7 Stagnation Flow
In the book by Drazin and Riley [6 p40] an exact solution is considered which uses a stream functionψ = ψ(y z) They suggest that the following velocity field is a solution
u1(x y z t) =a
kx
u2(x y z t) = minusaky +
k
Re
partψ(y z)
partz
u3(x y z t) = minus k
Re
partψ(y z)
party
(22)
Here we choose the stream function to be ψ(y z) = yz Then a solution is given by
u1(x y z t) =a
kx
u2(x y z t) =
(minusak
+k
Re
)y
u3(x y z t) = minus kzRe
p(x y z t) = minus1
2
((axk
)2minus((
k
Reminus a
k
)y
)2
minus(kz
Re
)2)
(23)
where a is a measure of the strength of the stagnation flow and k is a constant chosen in the textto specify the distance between vortices There are no vortices in this particular solution however
11
Figure 7 An illustration of the flow in the yz-plane for Sol 7 (Stagnation Flow) describedin Eq (23) with parameter values a = Re = 1and k = 2
Figure 8 An illustration of the flow in the xy-plane for Sol 7 (Stagnation Flow) describedin Eq (23) with parameter values a = Re = 1and k = 2
Solution 8 Vortex in Stagnation Flow
As in Sol 7 (Stagnation Flow) we now consider the same solution with a different stream functionnamely ψ(z) = cos z Then given the condition that k = aRe and adding a pressure term Eq (22)becomes
u1(x y z t) =1
Rex
u2(x y z t) = minus 1
Rey minus k sin z
u3(x y z t) = 0
p(x y z t) = minus 1
2Re2(x2 + y2)
(24)
where a and k are as in Sol 7 (Stagnation Flow)
12
Figure 9 An illustration of the flow in the yz-plane for Sol 8 (Vortex in Stagnation Flow)described in Eq (24) with parameter valuesk = 3 and Re = 10
Figure 10 An illustration of the flow in thexy-plane for Sol 8 (Vortex in StagnationFlow) described in Eq (24) with parametervalues k = 3 and Re = 10
Solution 9 Burgers Vortex
The Burgers Vortex solution is given in cylindrical coordinates in the book by Lautrup [8 p 385]as follows
ur(r ϕ z t) = minus 2r
c2Re
uϕ(r ϕ z t) =Ωc2
r
(1minus eminusr
2c2)
uz(r ϕ z t) =4z
c2Re
p(r ϕ z t) = minusc2Ω2F
(r2c2
)2
minus 2
Re2r2 + 4z2
c4
(25)
where
F (ξ) =
infinintξ
(1minus eminusx)2
x2dx (26)
This solution is not particularly nice to look at however most of the unpleasantness comes fromthe pressure term which (in this project) is only required in order to validate the solution For thisproject all solutions are required in Cartesian coordinates the velocity field of the Burgers Vortexhas hence been converted into Cartesian coordinates
u1(x y z t) = minus 2x
c2Reminus Ω
h(x y)(1minus exp (minush(x y))) y
u2(x y z t) = minus 2y
c2Re+
Ω
h(x y)(1minus exp (minush(x y)))x
u3(x y z t) =4z
c2Re
(27)
13
where
h(x y) =x2 + y2
c2 (28)
and Ω and c are constants
Figure 11 An illustration of the flow for Sol9 (Burgers Vortex) described in Eq (27) withparameter values c = 15 Re = 20 and Ω = 10A slice of the xy-plane at z = 0 is shown
Solution 10 Another Vortex Solution
There is another vortex solution more simply formulated than the Burgers Vortex yet with asingularity at r = 0 as a result of the uϕ term It is again given in cylindrical coordinates
ur(r ϕ z t) = minus 2r
c2Re
uϕ(r ϕ z t) =a
r
uz(r ϕ z t) =4z
c2Re
p(r ϕ z t) = minus a2
2r2minus 2r2 + 8z2
c2Re2
(29)
which can also be converted into Cartesian coordinates
14
u1(x y z t) = minus 2x
c2Reminus ay
x2 + y2
u2(x y z t) = minus 2y
c2Re+
ax
x2 + y2
u3(x y z t) =4z
c2Re
(30)
where a and c are constants
Figure 12 An illustration of the flow for Sol9 (Burgers Vortex) described in Eq (27) withparameter values c = 15 Re = 20 and a = 10 Aslice of the xy-plane at z = 0 is shown
Solution 11 Pseudo-Motion Shots of the Second Kind
In the book by Berker [3 p 94] a solution is given in cylindrical coordinates by
ur(r ϕ z t) =2
Re
1
r
uϕ(r ϕ z t) = A1r3 +B1r +
C1
r
uz(r ϕ z t) = A2r2 log(r) +B2r
2 + C2
p(r ϕ z t) =A2
1r6
6+A1B1r
4
2+A1C1r
2 +B2
1r2
2minus C2
1
2r2minus 2
r2Re2+ 2B1C1 log(r)
+2A2z
Reminus 4B1ϕ
Re
(31)
where A1 A2 B1 B2 C1 C2 are constants Eq (31) when converted into Cartesian coordinates is
15
u1(x y z t) =1
Re
2xminus C1Rey
x2 + y2minusA1y
(x2 + y2
)minusB1y
u2(x y z t) =1
Re
2y + C1Rex
x2 + y2+A1x
(x2 + y2
)+B1x
u3(x y z t) =1
2A2
(x2 + y2
)log(x2 + y2
)+B2
(x2 + y2
)+ C2
(32)
Figure 13 An illustration of the flow in thexy-plane for Sol 11 (Pseudo-Motion Shots)described in Eqs (31) and (32) with parametervalues A1 = B1 = C1 = 1 A2 = B2 = C2 = 2and Re = 10
Figure 14 An illustration of the flow in theyz-plane for Sol 11 (Pseudo-Motion Shots)described in Eqs (31) and (32) with parametervalues A1 = B1 = C1 = 1 A2 = B2 = C2 = 2and Re = 10
Solution 12 Flows With Constant Whirl
In the book by Berker [3 p 143] another solution is given in cylindrical coordinates by
ur(r ϕ z t) = minus 1
Re
1
r
uϕ(r ϕ z t) =1
rradicte(minus
12 g(t))
uz(r ϕ z t) = 0
p(r ϕ z t) =1
4
2(minus e
minusg(t)
t minus 1Re2
)r2
minus 1
t2Re
infinintminusg(t)
eminust
tdt
(33)
where we define g(t) =r2Re
2t When converted into Cartesian coordinates we find
16
u1(x y z t) = minus 1
x2 + y2
(yradict
exp
(minus
Re(x2 + y2
)4t
)+
x
Re
)
u2(x y z t) =1
x2 + y2
(xradict
exp
(minus
Re(x2 + y2
)4t
)minus y
Re
)
u3(x y z t) = 0
(34)
Figure 15 An illustration of the flow for Sol12 (Flows with Constant Whirl) described inEqs (33) and (34) with time t = 1 and Re = 1 Aslice of the xy-plane at z = 0 is shown as velocityin the z-direction is zero this is the same for anyvalue of z
17
Solution 13 Rectilinear and Uniform Translation of a Sphere
In the book by Berker [3 p 228] another solution is given in Cartesian coordinates by
u1(x y z t) =3
4
axz
R3
(a2
R2minus 1
)
u2(x y z t) =3
4
ayz
R3
(a2
R2minus 1
)
u3(x y z t) = 1minus 3
4
a
Rminus 1
4
a3
R3minus 3
4
az2
R3
(a2
R2minus 1
)
p(x y z t) =3a(aminusR)(a+R)
32R10Re
((R2Re
(x2 + y2
)minus 2R2Rez2
) (a3 + 3aR2 minus 4R3
)+ 3aRez4
(R2 minus a2
)minus 16R5z
)
(35)
where a and R are constants
Figure 16 An illustration of the flow in the yz-plane for Sol 13 (Sphere Translation) describedin Eq (35) with parameter values a = minus2 andRe = 3
Solution 14 Radially Dependent Flow
A solution with velocity in the z-direction dependent only on the distance from the z-axis is as
18
follows in cylindrical coordinates
ur(r ϕ z t) = 0
uϕ(r ϕ z t) = 0
uz(r ϕ z t) = A(r2 minus a2
)
p(r ϕ z t) =4Az
Re
(36)
where A and a are constants In Cartesian coordinates the solution is just the same with u1(x y z t)and u2(x y z t) both zero and the r2 term replaced with x2 + y2 in u3(x y z t)
Figure 17 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus1 1] withparameter values a = 1 A = 4 and Re = 10
Figure 18 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus2 2] withparameter values a = 1 A = 4 and Re = 10
Solution 15 Rankine Vortex
A solution discovered by Rankine [1] is a vortex flow of the following form
ur(r ϕ z t) = 0
uϕ(r ϕ z t) =ar
2πR2
uz(r ϕ z t) = 0
p(r ϕ z t) =a2r2
8π2R4
(37)
19
where a and R are constants In Cartesian coordinates the flow has the form
u1(x y z t) = minus ay
2πR2
u2(x y z t) =ax
2πR2
u3(x y z t) = 0
p(x y z t) =a2(x2 + y2
)8π2R4
(38)
This flow is actually a rigid body rotational flow in that the angular rotational velocity is uniformand the velocity of the flow increases proportionally with distance from the z-axis
Figure 19 An illustration of the flow in the xy-plane for Sol 15 (Rankine Vortex) described inEq (37) and Eq (38) with parameter values a = 1and R = 1
radic2π
23 Characterisation of Flows
It is important to be able to classify flows in a way that sorts them into distinct groups This aidsin the clear identification and general study of flows It is also a very important way of recognisingflows with similar properties In this subsection we introduce two distinct ways in which flows canbe characterised
231 Characterising Solutions using Invariants
A principal invariant of a tensor is a coefficient of the characteristic polynomial of that tensorInvariants as their name suggests are invariant under rotations of the coordinate system in thattheir values do not change when the tensor is lsquorotatedrsquo into a different set of coordinates Thevelocity gradient of a flow G is defined as the Jacobian of the velocity vector given by
Gij =
[partuipartxj
] (39)
20
for i j = 1 2 3 There are three principal invariants of the velocity gradient of a fluid flow G givenby
PG = tr(G)
QG =1
2
((tr(G))2 minus tr
(G2))
RG = det(G)
(40)
Incompressibility of flows (Eq (5)) ensures that PG = 0 hence the first principal invariant is zerofor all of the flows in Section 22 We now define the rate-of-strain tensor S by
Sij =1
2
(partuipartxj
+partujpartxi
) (41)
and the rate-of-rotation tensor Ω by
Ωij =1
2
(partuipartxjminus partujpartxi
) (42)
Note that the first principal invariant of S and the first and third principal invariants of Ω are zeroAlso note that the velocity gradient of a flow can simply be written G = S + Ω
Rather than considering the principal invariants we can also consider the combined invariants
I1 = tr(S2) I2 = tr(Ω2) I3 = tr(S3) I4 = tr(SΩ2) I5 = tr(S2Ω2) (43)
We can hence write the three principal invariants for a velocity field G in terms of the combinedinvariants
PG = 0
QG = minus1
2(I1 + I2)
RG =1
3(I3 + 3I4)
(44)
In this report the flows considered are the exact solutions to the incompressible Navier-Stokesequations described in Section 22 The combined invariants in Eq (43) are used to characterisethese solutions into different classes in Section 44 Both the principal invariants and combinedinvariants for each exact solution in Section 22 are given in Appendix A
Each of the subfilter-scale eddy-viscosity models described in Section 32 are written in terms ofthese five combined invariants
232 Vremanrsquos Characterisation of Flows
Vreman [16] classifies flows using the gradient of the velocity field of a flow Each lsquoclassrsquo of flows hasthe same number of non-zero elements in the gradient matrix For example
[partuipartxj
]=
0 0 0lowast 0 0lowast 0 0
(45)
where a lowast denotes a non-zero term represents a group of flows that lie within the class denoted byQ7 (see Appendix B) where the subscript represents the number of non-zero terms in the velocity
21
gradient Flows of the form in Eq (45) have a velocity field of the form
u1 = c1 (constant)
u2 = f2(x1)
u3 = f3(x1)
(46)
where f2(x1) and f3(x1) are such that the Navier Stokes equations Eq (1) are satisfied Notethat as the elements on the leading diagonal are zero the incompressibility condition Eq (5) isautomatically satisfied Some simple flow classes are given in Appendix B and contain a number ofthe solutions discussed in Section 22 This is mentioned in greater detail in Section 441
3 Conditions for Large-Eddy Simulation Models
31 Large-Eddy Simulation
Due to the nature of numerical simulation and the complexity of turbulent flows small details aremissed and information is lost when solving the Navier-Stokes equations The computational costof solving a turbulent flow is extremely high so a filtering operation is applied to the Navier-Stokesequations represented by a bar to reduce the computational cost of the simulation This is a linearoperation and preserves constants ie ρ = ρ and is also assumed to commute with differentiation
partuipartxj
=partuipartxj
(47)
The filtered Navier-Stokes equations are found by filtering (putting a bar over) the entire equations(see Eq (1)) and after evaluating are given by
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (48)
where
τij = uiuj minus uiuj (49)
is called the lsquotrue stressrsquo of the flow This however poses a problem as it is not clear what is meantby the term uiuj for a general filtering operation The true stress τij therefore must be replaced bya model τmod
ij in practical applications
32 Subfilter-Scale Models
We will now define the subflter-scale models denoted by τmodij to be used in place of the lsquodifficultrsquo
true stress τij There are a number of models available to use in the literature [11 p7] These willbe tested using the exact solutions to the Navier-Stokes equations and by checking conditions whichare defined in Sections 33 and 34
All the models barring one are of the same form That is each of the eddy-viscosity models arewritten
τmodij minus 1
3τmodkk δij = minus2νeSij (50)
with νe representing the eddy viscosity of a specific model For simplicity we usually just writeτmodij = minus2νeSij the extra term is included here to emphasise that the models are traceless Each
22
of the models we consider here can be expressed in terms of the rate of strain tensor S defined inEq (41) the rate of rotation tensor Ω defined in Eq (42) and the combined invariants defined inEq (43) The models are then defined as follows
Smagorinsky [12] νSe = (CSδ)2radic
2I1 (51)
Vreman [16] νVe = (CV δ)2
radicQGGT
PGGT (52)
QR [13 14 15] νQRe = (CQRδ)2 max0minusI3
I1 (53)
Vortex-Stretching [11] νV Se = (CV Sδ)2radic
2I1
(I5 minus 1
2I1I2
minusI1I2
)32
(54)
with the quantities
PGGT = I1 minus I2 QGGT =1
4(I1 + I2)2 + 4
(I5 minus
1
2I1I2
) (55)
There is a fifth and final model to be considered here It is of a different form to the other four andnot an eddy-viscosity model It is known as the Gradient model and is defined by
Gradient [4 9] τmodG = CGδ
(S2 minus Ω2 minus (SΩminus ΩS)
) (56)
The values C are model constants specific to each model and δ represents the mesh size of the LESBoth of these quantities are included here to properly define the models however in this projectthey will be ignored as they are constants and have no influence on the conditions to be defined
33 Conditions for Large-Eddy Simulation without Explicit Filtering
The filtering operation introduced in Eq (48) is just a formality for LES without explicit filteringThe idea here is that filtering does not actually take place but the term lsquoaddedrsquo to the Navier-Stokesequations in fact lsquosubtractsrsquo the difficult to simulate small-scale turbulent motion of the flow Inorder to consider this properly the terms ui and p in Eq (48) are replaced by vi and q It is hopedthat τmod
ij approximates τij so that vi and q approximate ui and p respectively So LES withoutexplicit filtering is given by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (57)
It is clear to see that the first four terms of this equation are exactly the same as the Navier-Stokesequations so for the exact solutions discussed in Section 22 the combination of these terms willequal zero What is then desirable is that the model term is also zero for these exact solutions sothat Eq (57) still holds In order to test models for exact solutions the condition
part
partxjτmodij (v) = 0 (58)
is checked If this condition were to hold it would imply that the model does not influence thevelocity field of a flow hence we call it the Velocity Field Condition There are a number of otherrestrictions that can be tested One such test comes from the differential form of the evolution ofthe kinetic energy of a flow given in Eq (10) In order to find this Eq (57) is multiplied by vi andterms are grouped to find
part
partt
(1
2vivi
)+ vj
part
partxj
(1
2vivi
)= minusvi
1
ρ
partq
partxi+ νvi
part2vipartxjpartxj
minus vipart
partxjτmodij (v) (59)
23
Then similarly to Eq (58) another condition is to check whether the model affects the kineticenergy of exact solutions this is equivalent to checking whether
vipart
partxjτmodij (v) = 0 (60)
We call this condition the Kinetic Energy Condition The third and final condition considered herecan also be found using the integral form for the equation for kinetic energy
d
dt
intV
1
2vividV =
intS
[minusvj
(1
2vivi
)minus vj
q
ρ+ 2νviSij + viτ
modij (v)
]njdS
+
intV
[minus2νSijSij minus
(minusSijτmod
ij (v))]dV
(61)
where the rate-of-strain tensor Sij was defined in Eq (41) V is a volume of fluid and dS meansthe integral is over the surface of that volume with unit normal nj Then this final condition checkswhether the model affects the total kinetic energy of exact solutions to the Navier-Stokes equationsthat is
Sijτmodij (v) = 0 (62)
We will call this condition the Dissipation Condition because Sijτmodij (v) is often referred to as the
subgrid dissipation
34 Conditions for Large-Eddy Simulation with Explicit Filtering
As in Eq (48) the Navier-Stokes equations can be filtered by applying the filtering operation to thewhole equation (putting a bar over all terms) and evaluating to find again that
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (63)
In Eq (57) the filtering operation was ignored and the elements ui p and τij(u) were replacedwith vi q and τmod
ij (v) respectively Now in LES with explicit filtering the filtering operation itselfsmooths over the small-scale turbulent motion of the simulated flow In this case the elements uip and τij(u) will be replaced with vi q and τmod
ij (v) respectively LES with explicit filtering is thengiven by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (64)
Note that Eqs (63) and (64) despite their similarity to Eq (57) no longer contain the Navier-Stokesequations as Eq (57) did The bar (filtering operation) over each term results in each exact solutionfound in Section 22 not necessarily satisfying the first four terms It is hence unknown whether ornot these terms will vanish for exact solutions Terms can still be compared similarly to Eqs (58)(60) and (62) However instead of comparing to zero we can now compare to the relevant termfrom the filtered Navier-Stokes equations So the terms
part
partxjτij (65)
and
part
partxjτmodij (v) (66)
24
are to be compared Although the latter term is relatively simple to compute the former could leadto some difficulties due to the differentiation of the true stress τij = uiuj minus uiuj So although someinformation could be gleaned from the inspection of these two terms more information can be foundby considering the analogue of Eq (62) So the true subfilter dissipation
Sijτij (67)
(where Sij is shorthand for Sij(v)) and the modelled subfilter dissipation
Sij(v)τmodij (v) (68)
are considered and have to be equal for the condition to hold These can again be derived from theintegral equation for kinetic energy of a filtered velocity field similarly to Eq (61)
4 Analysis of Large-Eddy Simulation Models
In this section we use a code in Mathematica the theory described in Section 3 and the exactsolutions to the Navier-Stokes equations found in Section 22 to check whether or not each of theexact solutions lsquoendorsesrsquo each model for LES both without and with explicit filtering
41 Large-Eddy Simulation Without Explicit Filtering
We start by using a code in Mathematica to test the Velocity Field condition in Eq (58) and theKinetic Energy condition in Eq (60) For all exact solutions to the Navier-Stokes equations inSection 22 the outputs for both conditions were identical so we have combined them into one table
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 ndash ndash 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 1 For the Kinetic Energy condition a lsquo0rsquo result means that vipartpartxj
τmodij (v) = 0 for respective
models and exact solutions whereas a lsquolowastrsquo result means that vipartpartxj
τmodij (v) 6= 0 For the Velocity
Field condition a lsquo0rsquo result means that partpartxj
τmodij (v) = 0 for respective models and exact solutions
whereas a lsquolowastrsquo result means that partpartxj
τmodij (v) 6= 0 The Gradient model can take both positive and
negative values so when the sign of the model is known it is included in the table instead of a lsquorsquoBoth the conditions are included in one table because for all cases the outputs were identical
We then test the Dissipation condition in Eq (62) for all exact solutions to the Navier-Stokesequations in Section 22
25
Dissipation Condition [Does Sijτmodij (v) = 0]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 + 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 2 A lsquo0rsquo result means that Sijτmodij (v) = 0 for respective models and exact solutions whereas
a lsquolowastrsquo result means that Sijτmodij (v) 6= 0 The Gradient model can take both positive and negative
values so when the sign of the model is known it is included in the table instead of a lsquorsquo
We then sum all the results to find the total number of endorsements for each model described inSection 32
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
4 6 8 9 9
lsquoDissipationrsquoEndorsements
2 5 7 7 9
Total Endorsements 6 11 15 16 18
Table 3 The results from Tables 1 and 2 are summarised to give a general overview of how manyendorsements each model receives This makes it much clearer to see the models most supported bythe respective and combined conditions with the Vortex-Stretching model being the most supportedand the Smagorinsky model the least
Testing the models in Section 32 using LES without explicit filtering has mixed results It isclear that the Smagorinksy model is not well endorsed by the flows in Section 22 with the onlyendorsements coming from the simplest flows that is Sol 1 (Linear Velocity) Sol 3 (Velocity as afunction of time) Sol 7 (Stagnation Flow) and Sol 15 (Rankine Vortex) The Vortex-Stretching-based model is the most endorsed flow with eighteen out of a possible thirty endorsements Thisagrees with our foreknowledge as this model was constructed to be as physically consistent as possiblein the paper by Silvis amp Verstappen [11]
42 Large-Eddy Simulation With Explicit Filtering
In this section we will compare the true subfilter dissipation Sijτij and the modelled subfilterdissipation Sij(v)τmod
ij (v) given in Eqs (67) and (68) respectively for each model τmodij and each
exact solution to the Navier-Stokes equation given in Section 22 We will not consider the conditionsinvolving differentiation in Eqs (65) and (66) due to the difficulty in calculating the derivative ofthe τij term in Eq (65)
26
421 Example of a Test for a Model
The true subfilter dissipation Sijτij of a flow given in Eq (67) cannot be explicitly computed asthe filtering operation is kept general What can be done however is to partially evaluate theexpressions using the attributes of a filtering operation mentioned in Eq (48)
The calculations of Sijτij for Sol 1 (Linear Velocity) are provided to illustrate the general ideaRecall that the velocity field of Sol 1 (Linear Velocity) is written in Eq (13) in vector notation as
u(x y z t) =
0ax+ bcx+ d
(69)
Then Sij = Sij(u) is computed simply by using its definition in Eq (41) Note that as filteringpreserves constants if there is an a it is replaced with a meaning that (in this case) there are alarge number of zero terms due to the frequency of 0 Hence
[Sij(u)
]=
[1
2
(partuipartxj
+partujpartxi
)]=
0 a
2c2
a2 0 0
c2 0 0
=
0 a
2c2
a2 0 0
c2 0 0
(70)
Note that as Sijτij is being computed which is equal to the sum of the element-wise multiplicationof the two matrices (or the trace of the matrix multiplication) that we donrsquot need to consider theelements of τij(u) with corresponding zero entry in Sij As τij(u) = uiuj minus uiuj and again asfiltering preserves constants it is found that
[τij(u)] =[uiuj minus uiuj
]=
lowast 0 middot ax+ b 0 middot cx+ d
ax+ b middot 0 lowast lowastcx+ d middot 0 lowast lowast
=
lowast 0 00 lowast lowast0 lowast lowast
(71)
where a lsquolowastrsquo just represents some element (not necessarily non-zero) Therefore the true subfilterdissipation
Sijτij = tr
0 a2
c2
a2 0 0c2 0 0
middotlowast 0 0
0 lowast lowast0 lowast lowast
= 0 (72)
Now the modelled subfilter dissipation Sij(u)τmodij (u) needs to be computed (recall that Sij =
Sij(u)) In this case let us use the QR model stated in Eq (53) Once again Sij(u) calculated inEq (70) needs to be used So we have that
τmodij = minus2(CQRδ)
2 max0minusI3I1
Sij (73)
As
I1 = tr(S2) =
1
2(a2 + c2) and I3 = tr(S
3) = 0 (74)
it holds that τmodij (u) = 0 so Sij(u)τmod
ij (u) = 0 This agrees with Eq (72) and hence Sol 1(Linear Velocity) endorses the QR model
27
422 Testing the Models
Now using a Mathematica code the exact solutions in Section 22 are again used to evaluate thecondition for the models using the process just described in Section 421 The results are describedin Table 4
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 4 All entries with a ldquo0rdquo represent a value of zero and a ldquolowastrdquo represents a value that is non-zeroThe ldquoExactrdquo row represents the value of the true subfilter dissipation Sijτij Every other row witha model name represents the value of the modelled subfilter dissipation Sij(u)τmod
ij (u) respectiveto that model A ldquordquo means that it is unclear as to the value - the sign of these values is oftenunknown and the lsquoMaxrsquo function in the definition of the QR model (in Eq (53)) makes it difficultto evaluate
Reading from Table 4 the Smagorinsky model is clearly endorsed by Sol 3 (Velocity as a function oftime) and Sol 15 (Rankine Vortex) as in both cases both Sijτij = 0 = Sij(u)τmod
ij (u) In this casewe say that Sol 3 (Velocity as a function of time) and Sol 15 (Rankine Vortex) lsquostronglyrsquo endorsethe Smagorinsky model The Smagorinsky model is not endorsed by Sol 1 (Linear Velocity) Sol 2(Planar Flow) or Sol 14 (Radially Dependent) however because Sij(u)τmod
ij (u) 6= 0 For Sol 4 to
13 both Sijτij 6= 0 6= Sij(u)τmodij (u) and so we will say that these exact solutions lsquoweaklyrsquo endorse
the Smagorinsky model
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 10 10 9 6
Table 5 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Interestingly the lsquostrongrsquo and lsquoweakrsquo endorsements give different results However more stock shouldbe put in the results of the lsquostrongerrsquo condition as it is completely clear that zero is equal to zeroIt is not necessarily clear whether or not non-zero equals non-zero as the values will depend on theparameters in each solution hence the lsquoweakerrsquo endorsement
The lsquostrongrsquo condition equally endorses the Vreman QR Gradient and Vortex-Stretching-basedmodels whereas the lsquoweakrsquo condition endorses Vreman and Smagorinsky the most closely followedby the Gradient model with the Vortex-Stretching-based model being the most opposed As this
28
completely contradicts the lsquostrongrsquo condition and the results in Section 41 it makes us question thevalidity of the weaker test The number of endorsements for the QR model is unknown because of thedefinition of the model in Eq (53) and our evaluation of the true subfilter dissipation Eq (67) Dueto the exact solutions we have found it is never clear whether or not the true subfilter dissipationis strictly positive or negative This means that when the true subfilter dissipation is found to benon-zero the lsquoMaxrsquo function in the QR model cannot be evaluated
43 Vremanrsquos True Subfilter Dissipation Claim
Using the theory mentioned in Sections 232 and 34 we can place the exact solutions to the Navier-Stokes equations into different classes defined by Vreman In doing this we can also attempt to verifyone of Vremanrsquos results [16] In Vremanrsquos paper he defines the class Qn to be the set of flows withn zero entries in the velocity gradient of the flow (the velocity gradient is defined in Eq (39))Vreman then claims that the only flows with zero true subfilter dissipation are a certain selection ofthose in the classesQ7Q8 orQ9 shown in appendix B and is generally non-zero for any other flows
The true subfilter dissipation of Sol 1 (Linear Velocity) was computed in Section 421 and wasfound to be zero in Eq (72) The velocity gradient of Sol 1 is as follows
GLV =
0 0 0a 0 0c 0 0
(75)
and hence Sol 1 (Linear Velocity) is in the required subset of Q7 which agrees with Vremanrsquos claimIn Section 422 the true subfilter dissipation was computed (determined to be zero or non-zero) forall of the exact solutions to the Navier Stokes equations in Section 22 These solutions can now belsquoplacedrsquo into each class Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a functionof time) and Sol 14 (Radially Dependent) were all found to have zero true subfilter dissipation andare all in the appropriate subsets of either Q7Q8 or Q9 This agrees with Vremanrsquos results
We can extend Vremanrsquos results however We found that Sol 15 (Rankine Vortex) had zero truesubfilter dissipation and yet is not in the subset of Q7 defined in Appendix B It is important tonote that the rate of strain tensor Sij in Eq (41) for Sol 15 (Rankine Vortex) is zero This is thereason the true subfilter dissipation is zero and the same reason that all models are zero for thisflow as all models are based on Sij
We also found that all of the remaining exact solutions to the Navier-Stokes equations are not inVremanrsquos specified classes (in Appendix Section B) They also have a non-zero subfilter dissipationwhich agrees with Vremanrsquos results in his paper [16]
44 Classes of Flows
It is important to consider that some exact solutions to the Navier-Stokes equations may have similarproperties For example Sol 1 (Linear Velocity) and Sol 2 (Planar Flow) are both very similarIn fact Sol 2 (Planar Flow) is a more general case of Sol 1 (Linear Velocity) We therefore mustgroup the different flows into classes in an attempt to reduce the bias that may occur when solutionsare similar This should aid us to further determine which of the models are most endorsed We willdo this in a number of different ways using the theory discussed in Section 23 We will group theflows with respect to Vremanrsquos Flow Classes given in appendix B and with respect to the Invariantsgiven in appendix A
29
441 Vremanrsquos Velocity Gradient Flow Classes
We now put our exact solutions to the Navier-Stokes equations into one of two classes Either Q0minus6or Q7minus9 where each class is defined intuitively as follows
Q0minus6 = Q0 cupQ1 cup middot middot middot cup Q6 (76)
Q7minus9 = Q7 cupQ8 cupQ9 (77)
where Qn is defined in Section 43 By looking at the gradient of each exact solution we can see thatQ7minus9 contains the six solutions Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as afunction of time) Sol 5 (2D Taylor) Sol 14 (Radially Dependent) and Sol 15 (Rankine Vortex)and that Q0minus6 contains the remaining nine We now check the models for LES without explicitfiltering by considering endorsements within each class Tables 6 and 7 are analogues of Table 3 andillustrate the results for each flow class The analogues of Tables 1 and 2 are Tables 22 and 23 andare in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 6 6 6
lsquoDissipationrsquoEndorsements
2 5 6 6 6
Percentage Endorsed 417 833 100 100 100
Table 6 Each entry in the top two rows represents the number of solutions in the class Q7minus9 thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Q0minus6 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 2 3 3
lsquoDissipationrsquoEndorsements
0 0 1 0 3
Percentage Endorsed 56 56 167 167 333
Table 7 Each entry in the top two rows represents the number of solutions in the class Q1minus6 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking results to 1 decimal place if necessary
All of the flows in the class Q7minus9 endorse the QR Gradient and Vortex-Stretching-based subfilter-scale models (see Table 6) We can hence see that this class fully endorses these three models equallyand does not endorse the Smagorinsky model The Vortex-Stretching eddy viscosity model has themost endorsements from any of the flows in the class Q0minus6 and the Smagorinksy and Vreman modelshave the fewest endorsements (see Table 7) We can hence say that this class mostly endorses theVortex-Stretching-based eddy-viscosity model note however that the percentage endorsement is very
30
low
Now let us consider LES with explicit filtering Tables 8 and 9 are analogues of Table 5 and illustratethe results for each flow class The analogue of Table 4 is Table 24 and is in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 1 1 0 0 0
Table 8 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Q0minus6 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 9 9 9 6
Table 9 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We have grouped Vremanrsquos classes into Q7minus9 and Q0minus6 this is equivalent to saying that we havegrouped Vremanrsquos classes into those with two or fewer non-zero velocity gradient elements and thosewith three or more non-zero velocity gradient elements respectively Therefore the solutions in Q7minus9are exactly those with a lsquo0rsquo in the lsquoExactrsquo column of Table 4 and Sol 5 (2D Taylor) ExcludingSol 5 (2D Taylor) these are the same as those five that endorse the Vreman QR Gradient andVortex-Stretching-based models in Table 5 We can see in Table 8 that the results for the lsquostrongrsquoendorsements are exactly the same as in Table 5 This is because all of the flows with zero subfilterdissipation are contained in the class Q7minus9 There are hence no lsquostrongrsquo endorsements in Table 9The majority of the lsquoweakrsquo endorsements come from Q0minus6 as this class contains all but one of thesolutions with non-zero true subfilter dissipation (given in Eq (67))
442 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
We now separate our exact solutions into classes again using Vremanrsquos flow classes This timehowever we define V0 to be the class of flows Vreman claims will always have zero true subfilterdissipation namely those classes mentioned in Appendix B We define Vlowast to be all the remaining
31
classes Then
V0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow)
Sol 3 (Velocity as a function of time) Sol 14 (Radially Dependent)(78)
Vlowast = Sol 4 (Axisymmetric Flow) Sol 5 (2D Taylor) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 13 (Sphere Translation) Sol 15 (Rankine Vortex)
(79)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 10 and 11 are analogues of Table 3 and illustrate the results for each flow class Theanalogues of Tables 1 and 2 are Tables 25 and 26 and are in Appendix C2
Endorsements for models using solutions in V0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 4 4 4 4
lsquoDissipationrsquoEndorsements
1 4 4 4 4
Percentage Endorsed 375 100 100 100 100
Table 10 Each entry in the top two rows represents the number of solutions in the class V0 thatendorse each model There are 4 solutions in this class so the percentages are worked out by dividingthe total of each column by 8 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Vlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 2 4 5 5
lsquoDissipationrsquoEndorsements
1 1 3 2 5
Percentage Endorsed 136 136 318 318 455
Table 11 Each entry in the top two rows represents the number of solutions in the class Vlowast thatendorse each model There are 11 solutions in this class so the percentages are worked out bydividing the total of each column by 22 and taking results to 1 decimal place if necessary
All of the flows in the class V0 endorse the Vreman QR Gradient and Vortex-Stretching-basedsubfilter-scale models (see Table 6) This class completely endorses these four models equally anddoes not endorse the Smagorinsky model The Vortex-Stretching-based eddy-viscosity model hasthe most endorsements from the class Vlowast and the Smagorinksy and Vreman models again have thefewest endorsements (see Table 7) As in Section 441 we say that this class mostly endorses the
32
Vortex-Stretching-based eddy-viscosity model but once again the percentage endorsement is ratherlow
Now let us consider LES with explicit filtering Tables 12 and 13 are analogues of Table 5 andillustrate the results for each flow class The analogue of Table 4 is Table 27 and is in Appendix C2
Endorsements for models using solutions in V0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 4 4 4 4
lsquoWeakrsquo Endorsements 0 0 0 0 0
Table 12 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Vlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 1 1 1 1
lsquoWeakrsquo Endorsements 10 10 9 6
Table 13 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We can see in Table 12 that the results for the lsquostrongrsquo endorsements are similar to those in Table 5excluding Sol 15 (Rankine Vortex) The four solutions in V0 with zero true subfilter dissipationnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)and Sol 14 (Radially Dependent) are hence the four that endorse the Vreman QR Gradientand Vortex-Stretching-based models in Table 12 Sol 15 (Rankine Vortex) is in the class Vlowast andhence there is one lsquostrongrsquo endorsement in Table 13 The class Vlowast contains all of the flows with anon-zero true subfilter dissipation (given in Eq (67)) and hence Table 13 contains all of the lsquoweakrsquoendorsements
443 Classes using the Third Principal Invariant
Sorting the classes using the invariants is a non-trivial task in itself There are a number of waysin which to do this using both the principal invariants in Eq (40) and the combined invariants inEq (43)
We start by looking at flows with third principal invariant RG equal to zero We again define twoclasses R0 and Rlowast where R0 consists of all flows with RG = 0 and Rlowast consists of all flows withRG 6= 0 Then using the information in Appendix A we see that there are nine flows in R0 That
33
is
R0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 8 (Vortex in Stagnation Flow)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)
(80)
Rlowast = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 13 (Sphere Translation)(81)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 14 and 15 are analogues of Table 3 and they illustrate the results for LES withoutexplicit filtering for each flow class The analogues of Tables 1 and 2 are Tables 28 and 29 and arein Appendix C3
Endorsements for models using solutions in R0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 7 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 278 556 778 722 778
Table 14 Each entry in the top two rows represents the number of solutions in the class R0 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking the result to one decimal place if necessary
Endorsements for models using solutions in Rlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 2 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 83 83 83 167 333
Table 15 Each entry in the top two rows represents the number of solutions in the class Rlowast thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking the result to one decimal place if necessary
The class R0 endorses both the Vortex-Stretching-based model and the QR model the most andSmagorinsky the least in the case for LES without explicit filtering In the Rlowast class the Vortex-Stretching-based model is again the most endorsed followed by the Gradient model however thepercentages are again small
34
Now let us consider LES with explicit filtering Tables 16 and 17 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 30 and is in Appendix C3
Endorsements for models using solutions in R0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 4 4 3 2
Table 16 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Rlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 6 6 6 4
Table 17 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 we can see in Table 16 that the results for the lsquostrongrsquo endorsements are exactly thesame as in Table 5 This is because all of the flows with zero subfilter dissipation are contained in theclass R0 As in Table 9 there are no lsquostrongrsquo endorsements in Table 17 The lsquoweakrsquo endorsementshowever are split more evenly between the two classes now than in Vremanrsquos classes in Sections 441and 442 We find that the Smagorinsky and Vreman models still seem to be the most endorsed inboth classes
444 Two-component Flow Classes
Another way in which the flows can be grouped into classes is by determining whether a flow is atwo-component flow or not We define a two-component flow to have the property that the combinedprincipal invariants of a flow in Eq (43) satisfy
I3 = I4 = I5 minus1
2I1I2 = 0 (82)
We are also interested in considering flows with zero gradient model subgrid dissipation These flowshave the property that
I3 minus I4 = 0 (83)
What we find though is that for all the solutions in Section 22 if a flow is a two-component flowthen it also satisfies I3 minus I4 = 0 It may be interesting to note that although I3 minus I4 6= 0 for manysolutions there are a number with I3 equal to a multiple of I4 Although not considered in thisreport this may also be a class of interest We find that the two-component flows are
C2 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)(84)
35
and we call the remaining flows three-component flows They are
C3 = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex) Sol 11 (Pseudo-Motion Shots)
Sol 13 (Sphere Translation)
(85)
Note that Sol 8 (Vortex in Stagnation Flow) is defined to be a planar flow this does not mean that itis a two-component flow in this sense however because its two non-zero velocity components dependon three coordinates We now check the models for LES without explicit filtering by consideringendorsements within each class Tables 18 and 19 are analogues of Table 3 and they illustrate theresults for LES without explicit filtering for each flow class The analogues of Tables 1 and 2 areTables 31 and 32 and are in Appendix C4
Endorsements for models using solutions in C2 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 6 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 357 714 100 857 100
Table 18 Each entry in the top two rows represents the number of solutions in the class C2 thatendorse each model There are 7 solutions in this class so the percentages are worked out by dividingthe total of each column by 14 and taking the result to one decimal place if necessary
Endorsements for models using solutions in C3 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 3 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 63 63 63 188 250
Table 19 Each entry in the top two rows represents the number of solutions in the class C3 thatendorse each model There are 8 solutions in this class so the percentages are worked out by dividingthe total of each column by 16 and taking the result to one decimal place if necessary
The class C2 equally endorses the QR and Vortex-Stretching-based models and once again theSmagorinsky model is the least endorsed As with the previous classes the class C3 has such lowpercentages that it does not seem to endorse any models We start to see a trend appearing here inthat the flow classes that contain the more complicated flows namely Q0minus6 (Eq (76)) Vlowast (Eq (79))Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages The classes that tendto contain the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78)) R0 (Eq (80)) and C2 (Eq (84))all seem to favour the Vortex-Stretching-based model or the QR model
36
Now let us consider LES with explicit filtering Tables 20 and 21 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 33 and is in Appendix C4
Endorsements for models using solutions in C2 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 2 2 0 1 0
Table 20 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in C3 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 8 8 8 6
Table 21 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 and Table 16 we can see in Table 20 that the results for the lsquostrongrsquo endorsements areexactly the same as in Table 5 This is again because all of the flows with zero subfilter dissipationare contained in the class C2 There are two flows with non-zero true subfilter dissipation (givenin Eq (67)) in C2 and this shows in the lsquoweakrsquo endorsements in Table 20 There are no lsquostrongrsquoendorsements in Table 21 as all the zero subfilter dissipation flows are in C2
5 Conclusions
In this report we studied incompressible turbulent flows We found a number of exact solutions to theNavier-Stokes equations each representing a relatively simplistic fluid flow Large-eddy simulation(LES) is used to model turbulent flows by smoothing over the complex small-scale motion Weintroduced a number of subfilter-scale models to approximate this small-scale motion and testedthem using the found exact solutions to the Navier-Stokes equations We first considered LESwithout explicit filtering and used the Navier-Stokes equations to derive conditions for the subfilter-scale models namely the Velocity Field condition (Eq (58)) the Kinetic Energy condition (Eq (60))and the Dissipation condition (Eq (62)) Similarly we considered LES with explicit filtering andagain used the Navier-Stokes equations to derive more conditions for the models in this case wedecided to just take one condition the Dissipation condition (Eqs (67) and (68)) as the others leadto difficulty in their evaluation By lsquorunningrsquo the exact solutions to the Navier-Stokes equationsthrough each condition for the subfilter-scale models we found that the Vortex-Stretching-basededdy-viscosity model in Eq (54) was most endorsed by the tests in LES without explicit filteringWe found that four out of the five subfilter-scale models namely the Vreman QR Gradient andVortex-Stretching-based models were equally endorsed by the lsquostrongrsquo tests in LES with explicitfiltering with the Smagorinsky model being the least endorsed The results we found for the lsquoweakrsquocondition were completely opposite to those found in both the lsquostrongrsquo condition and in LES withoutexplicit filtering This led us to conclude that this test was not one in which we should put our trust
37
We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses
In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes
This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold
38
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
2 The Navier-Stokes Equations
21 Introduction
The incompressible Navier-Stokes equations are a set of four equations that model the flow of anyfluid in three-dimensional space The standard set of Navier-Stokes equations can be written in twoforms namely by using Einsteinrsquos summation convention in what we call index notation and byusing standard vector notation In index notation they are given by
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
i = 1 2 3 j = 1 2 3 (1)
These equations have four variables given by the velocity of the flow u = (u1 u2 u3) and thepressure p The vector x = (x1 x2 x3) represents position t represents time ρ corresponds to themass density and ν is the kinematic viscosity of the fluid
The Navier-Stokes equations can be written in vector notation in the following way First considerthe convection term
ujpartuipartxjequiv
3sumj=1
ujpartuipartxj
= u1partuipartx1
+ u2partuipartx2
+ u3partuipartx3
=
(u1
part
partx1+ u2
part
partx2+ u3
part
partx3
)ui
= (u middot nabla)ui
(2)
Similarly for the diffusion term
νpart2uipartxjpartxj
equiv ν3sumj=1
part2uipartxjpartxj
= νpart2uipartx1partx1
+part2uipartx2partx2
+part2uipartx3partx3
= ν
(part2
partx1partx1+
part2
partx2partx2+
part2
partx3partx3
)ui
= νnabla2ui
(3)
It therefore holds that in vector notation the Navier-Stokes equations are given by
partu
partt+ (u middot nabla)u = minus1
ρnablap+ νnabla2u (4)
It is clear that there is a problem here there are three equations and four unknowns In order toremedy this the equation for conservation of mass or the incompressibility condition is consideredThis is given by
partuipartxi
= 0 (5)
in index notation and
nabla middot u = 0 (6)
5
in vector notation
There are now four equations in four unknowns and it is possible to find solutions Althoughthere is no known general solution there are specific cases in which exact solutions to the Navier-Stokes equations can be found In order to make finding solutions slightly simpler the Navier-Stokesequations can be rescaled by introducing dimensionless variables
Let
ulowasti =uiv xlowasti =
xi` tlowast =
t
`v plowast =
p
ρv2 (7)
where ` and v are typical length and velocity scales respectively Then dropping the star notationfor ease of reading the scaled incompressible Navier-Stokes equations are
partuipartt
+ ujpartuipartxj
= minus partp
partxi+
1
Re
part2uipartxjpartxj
i = 1 2 3 j = 1 2 3
partuipartxi
= 0
(8)
where Re is the Reynolds number and is defined by the dimensionless Re =v`
ν Similarly in vector
notation the scaled incompressible Navier-Stokes equations are
partu
partt+ (u middot nabla)u = minusnablap+
1
Renabla2u
nabla middot u = 0(9)
By multiplying the incompressible Navier-Stokes equations Eq (8) by ui and grouping terms thedifferential form of the evolution of the kinetic energy of a flow can be found
part
partt
(1
2uiui
)+ uj
part
partxj
(1
2uiui
)= minusui
partp
partxi+
2
Reui
part2uipartxjpartxj
(10)
The differential form of the evolution of the kinetic energy of a flow will be used later on in thisreport to derive conditions for the subfilter-scale models
22 Exact Solutions of the Navier-Stokes Equations
By using mathematical inspection and taking results found in literature we found a number of exactsolutions to the Navier-Stokes equations The majority are steady-state solutions in that they areindependent of time We decided to consider exact solutions to the Navier-Stokes equations withoutboundary conditions in an attempt to keep the results general and applicable to a larger numberof situations All of the stated exact solutions below have been checked using a code written inMathematica In the solutions below the notation (x y z) for the Cartesian coordinate system isused analogously with (x1 x2 x3) to aid in reading comprehension and in the conversion betweenindex and vector notation
Solution 1 Linear Velocity
If the diffusion term in the Navier-Stokes equations Eq (1) is taken to be zero ie
part2uipartxjpartxj
= 0 (11)
6
then each component of the velocity can be written as a linear function of x1 x2 and x3 In indexnotation
ui = aijxj + βi (12)
If pressure is assumed to be a function of time t (so that the pressure terms in the Navier-Stokesequations are zero) then conditions can be found on the coefficients to find that the following is asolution
u1(x y z t) = 0
u2(x y z t) = ax+ b
u3(x y z t) = cx+ d
p(x y z t) = f(t)
(13)
where a b c d isin R and f Rrarr R In fact this solution is actually a specific case of Sol 2 (PlanarFlow)
Figure 1 An illustration of the flow in the xy-plane for Sol 1 (Linear Velocity) described inEq (13) with parameter values a = b = c = d = 1
Solution 2 Planar Flow
Using the text by Barbato et al [2 p62] and modifying the solution such that it now takes threedimensions into account another solution is found
u1(x y z t) = Az2 +Bz + C
u2(x y z t) = Dz2 + Ez + F
u3(x y z t) = 0
p(x y z t) =2A
Rex+
2D
Rey + f(t)
(14)
7
where ABCDE F isin R and f R rarr R It is clear to see that if A = D = 0 then this is in factthe same as Eq (13) in Sol 1 In fact we believe (although not shown here) that this is the mostgeneral solution of the form
ui(xi xj xk t) = Ax2k +Bxk + C
uj(xi xj xk t) = Dx2k + Exk + F
uk(xi xj xk t) = 0
(15)
for i 6= j 6= k 6= i
Figure 2 An illustration of the flow in the yz-plane for Sol 2 (Planar Flow) described inEq (14) with parameter values A = B = C =D = E = F = 1
Solution 3 Velocity as a function of time t
If each component of the velocity is taken to be a function of time t and independent of positionthen the incompressibility condition is automatically satisfied A condition can be imposed on thepressure term in order to satisfy the momentum equations Thus a solution is as follows
u1(x y z t) = f(t)
u2(x y z t) = g(t)
u3(x y z t) = h(t)
p(x y z t) = minusxpartf(t)
parttminus y partg(t)
parttminus z parth(t)
partt
(16)
where f g h Rrarr R In fact what is written here is the generalised Galilean transformation of thezero solution [10] We cannot produce a plot for this flow as the functions are completely unspecified
8
Solution 4 Axisymmetric Flow
Using the text by Irmay [7] a solution is defined as follows
u1(x y z t) = kxz
u2(x y z t) = kyz
u3(x y z t) = minuskz2
p(x y z t) = minus2kz
Reminus k2z4
2
(17)
where k isin R This is said to represent an axisymmetric flow with hyperbolic streamlines asymptoticto the z-axis and to the boundary plane z = 0 These asymptotes are visible in Fig 3
Figure 3 An illustration of the flow in theyz-plane for Sol 4 (Axisymmetric Flow)described in Eq (17) with parameter k = 1
Figure 4 An illustration of the flow in thexy-plane for Sol 4 (Axisymmetric Flow)described in Eq (17) with parameter k = 1
Solution 5 2D Taylor Solutions
Again using the text by Barbato et al [2 p63ndash64] a solution is defined using a stream functionψ = ψ(x y t) and by writing
u1 = minuspartψparty
u2 =partψ
partx u3 = 0 (18)
In the text the stream function is not given however it was worked out that
ψ = minuseminusπ2t
Re (cos(πx) + cos(πy))
π (19)
The solution used is of the following form note however that in the text the pressure term differsin that cos(πy) is replaced by sin(πy) (when testing this solution in Mathematica the solution below
9
was instead found)
u1(x y z t) = minus sin(πy) eminus(π2t)Re
u2(x y z t) = sin(πx) eminus(π2t)Re
u3(x y z t) = 0
p(x y z t) = minus cos(πx) cos(πy) eminus(2π2t)Re
(20)
Figure 5 An illustration of the flow in the xy-plane for Sol 5 (2D Taylor) described in Eq (20)with Re = 1 and time t = 0
Solution 6 Generalized Beltrami Flow
Again using the text by Barbato et al [2 p65] and now considering solutions with nonndashtrivialterms in all dimensions a particular form of the generalized Beltrami flow is found namely
u1(x y z t) = (A sin(πz) + C cos(πy)) eminus(π2t)Re
u2(x y z t) = (B sin(πx) +A cos(πz)) eminus(π2t)Re
u3(x y z t) = (C sin(πy) +B cos(πx)) eminus(π2t)Re
p(x y z t) = minus[(AB cos(πz) sin(πx) +BC cos(πx) sin(πy)
+AC cos(πy) sin(πz)] eminus(2π2t)Re
(21)
where ABC isin R
10
Figure 6 An illustration of the flow in the yz-plane for Sol 6 (Generalized Beltrami Flow)described in Eq (21) with parameters A = B =C = 1 and time t = 0
Solution 7 Stagnation Flow
In the book by Drazin and Riley [6 p40] an exact solution is considered which uses a stream functionψ = ψ(y z) They suggest that the following velocity field is a solution
u1(x y z t) =a
kx
u2(x y z t) = minusaky +
k
Re
partψ(y z)
partz
u3(x y z t) = minus k
Re
partψ(y z)
party
(22)
Here we choose the stream function to be ψ(y z) = yz Then a solution is given by
u1(x y z t) =a
kx
u2(x y z t) =
(minusak
+k
Re
)y
u3(x y z t) = minus kzRe
p(x y z t) = minus1
2
((axk
)2minus((
k
Reminus a
k
)y
)2
minus(kz
Re
)2)
(23)
where a is a measure of the strength of the stagnation flow and k is a constant chosen in the textto specify the distance between vortices There are no vortices in this particular solution however
11
Figure 7 An illustration of the flow in the yz-plane for Sol 7 (Stagnation Flow) describedin Eq (23) with parameter values a = Re = 1and k = 2
Figure 8 An illustration of the flow in the xy-plane for Sol 7 (Stagnation Flow) describedin Eq (23) with parameter values a = Re = 1and k = 2
Solution 8 Vortex in Stagnation Flow
As in Sol 7 (Stagnation Flow) we now consider the same solution with a different stream functionnamely ψ(z) = cos z Then given the condition that k = aRe and adding a pressure term Eq (22)becomes
u1(x y z t) =1
Rex
u2(x y z t) = minus 1
Rey minus k sin z
u3(x y z t) = 0
p(x y z t) = minus 1
2Re2(x2 + y2)
(24)
where a and k are as in Sol 7 (Stagnation Flow)
12
Figure 9 An illustration of the flow in the yz-plane for Sol 8 (Vortex in Stagnation Flow)described in Eq (24) with parameter valuesk = 3 and Re = 10
Figure 10 An illustration of the flow in thexy-plane for Sol 8 (Vortex in StagnationFlow) described in Eq (24) with parametervalues k = 3 and Re = 10
Solution 9 Burgers Vortex
The Burgers Vortex solution is given in cylindrical coordinates in the book by Lautrup [8 p 385]as follows
ur(r ϕ z t) = minus 2r
c2Re
uϕ(r ϕ z t) =Ωc2
r
(1minus eminusr
2c2)
uz(r ϕ z t) =4z
c2Re
p(r ϕ z t) = minusc2Ω2F
(r2c2
)2
minus 2
Re2r2 + 4z2
c4
(25)
where
F (ξ) =
infinintξ
(1minus eminusx)2
x2dx (26)
This solution is not particularly nice to look at however most of the unpleasantness comes fromthe pressure term which (in this project) is only required in order to validate the solution For thisproject all solutions are required in Cartesian coordinates the velocity field of the Burgers Vortexhas hence been converted into Cartesian coordinates
u1(x y z t) = minus 2x
c2Reminus Ω
h(x y)(1minus exp (minush(x y))) y
u2(x y z t) = minus 2y
c2Re+
Ω
h(x y)(1minus exp (minush(x y)))x
u3(x y z t) =4z
c2Re
(27)
13
where
h(x y) =x2 + y2
c2 (28)
and Ω and c are constants
Figure 11 An illustration of the flow for Sol9 (Burgers Vortex) described in Eq (27) withparameter values c = 15 Re = 20 and Ω = 10A slice of the xy-plane at z = 0 is shown
Solution 10 Another Vortex Solution
There is another vortex solution more simply formulated than the Burgers Vortex yet with asingularity at r = 0 as a result of the uϕ term It is again given in cylindrical coordinates
ur(r ϕ z t) = minus 2r
c2Re
uϕ(r ϕ z t) =a
r
uz(r ϕ z t) =4z
c2Re
p(r ϕ z t) = minus a2
2r2minus 2r2 + 8z2
c2Re2
(29)
which can also be converted into Cartesian coordinates
14
u1(x y z t) = minus 2x
c2Reminus ay
x2 + y2
u2(x y z t) = minus 2y
c2Re+
ax
x2 + y2
u3(x y z t) =4z
c2Re
(30)
where a and c are constants
Figure 12 An illustration of the flow for Sol9 (Burgers Vortex) described in Eq (27) withparameter values c = 15 Re = 20 and a = 10 Aslice of the xy-plane at z = 0 is shown
Solution 11 Pseudo-Motion Shots of the Second Kind
In the book by Berker [3 p 94] a solution is given in cylindrical coordinates by
ur(r ϕ z t) =2
Re
1
r
uϕ(r ϕ z t) = A1r3 +B1r +
C1
r
uz(r ϕ z t) = A2r2 log(r) +B2r
2 + C2
p(r ϕ z t) =A2
1r6
6+A1B1r
4
2+A1C1r
2 +B2
1r2
2minus C2
1
2r2minus 2
r2Re2+ 2B1C1 log(r)
+2A2z
Reminus 4B1ϕ
Re
(31)
where A1 A2 B1 B2 C1 C2 are constants Eq (31) when converted into Cartesian coordinates is
15
u1(x y z t) =1
Re
2xminus C1Rey
x2 + y2minusA1y
(x2 + y2
)minusB1y
u2(x y z t) =1
Re
2y + C1Rex
x2 + y2+A1x
(x2 + y2
)+B1x
u3(x y z t) =1
2A2
(x2 + y2
)log(x2 + y2
)+B2
(x2 + y2
)+ C2
(32)
Figure 13 An illustration of the flow in thexy-plane for Sol 11 (Pseudo-Motion Shots)described in Eqs (31) and (32) with parametervalues A1 = B1 = C1 = 1 A2 = B2 = C2 = 2and Re = 10
Figure 14 An illustration of the flow in theyz-plane for Sol 11 (Pseudo-Motion Shots)described in Eqs (31) and (32) with parametervalues A1 = B1 = C1 = 1 A2 = B2 = C2 = 2and Re = 10
Solution 12 Flows With Constant Whirl
In the book by Berker [3 p 143] another solution is given in cylindrical coordinates by
ur(r ϕ z t) = minus 1
Re
1
r
uϕ(r ϕ z t) =1
rradicte(minus
12 g(t))
uz(r ϕ z t) = 0
p(r ϕ z t) =1
4
2(minus e
minusg(t)
t minus 1Re2
)r2
minus 1
t2Re
infinintminusg(t)
eminust
tdt
(33)
where we define g(t) =r2Re
2t When converted into Cartesian coordinates we find
16
u1(x y z t) = minus 1
x2 + y2
(yradict
exp
(minus
Re(x2 + y2
)4t
)+
x
Re
)
u2(x y z t) =1
x2 + y2
(xradict
exp
(minus
Re(x2 + y2
)4t
)minus y
Re
)
u3(x y z t) = 0
(34)
Figure 15 An illustration of the flow for Sol12 (Flows with Constant Whirl) described inEqs (33) and (34) with time t = 1 and Re = 1 Aslice of the xy-plane at z = 0 is shown as velocityin the z-direction is zero this is the same for anyvalue of z
17
Solution 13 Rectilinear and Uniform Translation of a Sphere
In the book by Berker [3 p 228] another solution is given in Cartesian coordinates by
u1(x y z t) =3
4
axz
R3
(a2
R2minus 1
)
u2(x y z t) =3
4
ayz
R3
(a2
R2minus 1
)
u3(x y z t) = 1minus 3
4
a
Rminus 1
4
a3
R3minus 3
4
az2
R3
(a2
R2minus 1
)
p(x y z t) =3a(aminusR)(a+R)
32R10Re
((R2Re
(x2 + y2
)minus 2R2Rez2
) (a3 + 3aR2 minus 4R3
)+ 3aRez4
(R2 minus a2
)minus 16R5z
)
(35)
where a and R are constants
Figure 16 An illustration of the flow in the yz-plane for Sol 13 (Sphere Translation) describedin Eq (35) with parameter values a = minus2 andRe = 3
Solution 14 Radially Dependent Flow
A solution with velocity in the z-direction dependent only on the distance from the z-axis is as
18
follows in cylindrical coordinates
ur(r ϕ z t) = 0
uϕ(r ϕ z t) = 0
uz(r ϕ z t) = A(r2 minus a2
)
p(r ϕ z t) =4Az
Re
(36)
where A and a are constants In Cartesian coordinates the solution is just the same with u1(x y z t)and u2(x y z t) both zero and the r2 term replaced with x2 + y2 in u3(x y z t)
Figure 17 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus1 1] withparameter values a = 1 A = 4 and Re = 10
Figure 18 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus2 2] withparameter values a = 1 A = 4 and Re = 10
Solution 15 Rankine Vortex
A solution discovered by Rankine [1] is a vortex flow of the following form
ur(r ϕ z t) = 0
uϕ(r ϕ z t) =ar
2πR2
uz(r ϕ z t) = 0
p(r ϕ z t) =a2r2
8π2R4
(37)
19
where a and R are constants In Cartesian coordinates the flow has the form
u1(x y z t) = minus ay
2πR2
u2(x y z t) =ax
2πR2
u3(x y z t) = 0
p(x y z t) =a2(x2 + y2
)8π2R4
(38)
This flow is actually a rigid body rotational flow in that the angular rotational velocity is uniformand the velocity of the flow increases proportionally with distance from the z-axis
Figure 19 An illustration of the flow in the xy-plane for Sol 15 (Rankine Vortex) described inEq (37) and Eq (38) with parameter values a = 1and R = 1
radic2π
23 Characterisation of Flows
It is important to be able to classify flows in a way that sorts them into distinct groups This aidsin the clear identification and general study of flows It is also a very important way of recognisingflows with similar properties In this subsection we introduce two distinct ways in which flows canbe characterised
231 Characterising Solutions using Invariants
A principal invariant of a tensor is a coefficient of the characteristic polynomial of that tensorInvariants as their name suggests are invariant under rotations of the coordinate system in thattheir values do not change when the tensor is lsquorotatedrsquo into a different set of coordinates Thevelocity gradient of a flow G is defined as the Jacobian of the velocity vector given by
Gij =
[partuipartxj
] (39)
20
for i j = 1 2 3 There are three principal invariants of the velocity gradient of a fluid flow G givenby
PG = tr(G)
QG =1
2
((tr(G))2 minus tr
(G2))
RG = det(G)
(40)
Incompressibility of flows (Eq (5)) ensures that PG = 0 hence the first principal invariant is zerofor all of the flows in Section 22 We now define the rate-of-strain tensor S by
Sij =1
2
(partuipartxj
+partujpartxi
) (41)
and the rate-of-rotation tensor Ω by
Ωij =1
2
(partuipartxjminus partujpartxi
) (42)
Note that the first principal invariant of S and the first and third principal invariants of Ω are zeroAlso note that the velocity gradient of a flow can simply be written G = S + Ω
Rather than considering the principal invariants we can also consider the combined invariants
I1 = tr(S2) I2 = tr(Ω2) I3 = tr(S3) I4 = tr(SΩ2) I5 = tr(S2Ω2) (43)
We can hence write the three principal invariants for a velocity field G in terms of the combinedinvariants
PG = 0
QG = minus1
2(I1 + I2)
RG =1
3(I3 + 3I4)
(44)
In this report the flows considered are the exact solutions to the incompressible Navier-Stokesequations described in Section 22 The combined invariants in Eq (43) are used to characterisethese solutions into different classes in Section 44 Both the principal invariants and combinedinvariants for each exact solution in Section 22 are given in Appendix A
Each of the subfilter-scale eddy-viscosity models described in Section 32 are written in terms ofthese five combined invariants
232 Vremanrsquos Characterisation of Flows
Vreman [16] classifies flows using the gradient of the velocity field of a flow Each lsquoclassrsquo of flows hasthe same number of non-zero elements in the gradient matrix For example
[partuipartxj
]=
0 0 0lowast 0 0lowast 0 0
(45)
where a lowast denotes a non-zero term represents a group of flows that lie within the class denoted byQ7 (see Appendix B) where the subscript represents the number of non-zero terms in the velocity
21
gradient Flows of the form in Eq (45) have a velocity field of the form
u1 = c1 (constant)
u2 = f2(x1)
u3 = f3(x1)
(46)
where f2(x1) and f3(x1) are such that the Navier Stokes equations Eq (1) are satisfied Notethat as the elements on the leading diagonal are zero the incompressibility condition Eq (5) isautomatically satisfied Some simple flow classes are given in Appendix B and contain a number ofthe solutions discussed in Section 22 This is mentioned in greater detail in Section 441
3 Conditions for Large-Eddy Simulation Models
31 Large-Eddy Simulation
Due to the nature of numerical simulation and the complexity of turbulent flows small details aremissed and information is lost when solving the Navier-Stokes equations The computational costof solving a turbulent flow is extremely high so a filtering operation is applied to the Navier-Stokesequations represented by a bar to reduce the computational cost of the simulation This is a linearoperation and preserves constants ie ρ = ρ and is also assumed to commute with differentiation
partuipartxj
=partuipartxj
(47)
The filtered Navier-Stokes equations are found by filtering (putting a bar over) the entire equations(see Eq (1)) and after evaluating are given by
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (48)
where
τij = uiuj minus uiuj (49)
is called the lsquotrue stressrsquo of the flow This however poses a problem as it is not clear what is meantby the term uiuj for a general filtering operation The true stress τij therefore must be replaced bya model τmod
ij in practical applications
32 Subfilter-Scale Models
We will now define the subflter-scale models denoted by τmodij to be used in place of the lsquodifficultrsquo
true stress τij There are a number of models available to use in the literature [11 p7] These willbe tested using the exact solutions to the Navier-Stokes equations and by checking conditions whichare defined in Sections 33 and 34
All the models barring one are of the same form That is each of the eddy-viscosity models arewritten
τmodij minus 1
3τmodkk δij = minus2νeSij (50)
with νe representing the eddy viscosity of a specific model For simplicity we usually just writeτmodij = minus2νeSij the extra term is included here to emphasise that the models are traceless Each
22
of the models we consider here can be expressed in terms of the rate of strain tensor S defined inEq (41) the rate of rotation tensor Ω defined in Eq (42) and the combined invariants defined inEq (43) The models are then defined as follows
Smagorinsky [12] νSe = (CSδ)2radic
2I1 (51)
Vreman [16] νVe = (CV δ)2
radicQGGT
PGGT (52)
QR [13 14 15] νQRe = (CQRδ)2 max0minusI3
I1 (53)
Vortex-Stretching [11] νV Se = (CV Sδ)2radic
2I1
(I5 minus 1
2I1I2
minusI1I2
)32
(54)
with the quantities
PGGT = I1 minus I2 QGGT =1
4(I1 + I2)2 + 4
(I5 minus
1
2I1I2
) (55)
There is a fifth and final model to be considered here It is of a different form to the other four andnot an eddy-viscosity model It is known as the Gradient model and is defined by
Gradient [4 9] τmodG = CGδ
(S2 minus Ω2 minus (SΩminus ΩS)
) (56)
The values C are model constants specific to each model and δ represents the mesh size of the LESBoth of these quantities are included here to properly define the models however in this projectthey will be ignored as they are constants and have no influence on the conditions to be defined
33 Conditions for Large-Eddy Simulation without Explicit Filtering
The filtering operation introduced in Eq (48) is just a formality for LES without explicit filteringThe idea here is that filtering does not actually take place but the term lsquoaddedrsquo to the Navier-Stokesequations in fact lsquosubtractsrsquo the difficult to simulate small-scale turbulent motion of the flow Inorder to consider this properly the terms ui and p in Eq (48) are replaced by vi and q It is hopedthat τmod
ij approximates τij so that vi and q approximate ui and p respectively So LES withoutexplicit filtering is given by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (57)
It is clear to see that the first four terms of this equation are exactly the same as the Navier-Stokesequations so for the exact solutions discussed in Section 22 the combination of these terms willequal zero What is then desirable is that the model term is also zero for these exact solutions sothat Eq (57) still holds In order to test models for exact solutions the condition
part
partxjτmodij (v) = 0 (58)
is checked If this condition were to hold it would imply that the model does not influence thevelocity field of a flow hence we call it the Velocity Field Condition There are a number of otherrestrictions that can be tested One such test comes from the differential form of the evolution ofthe kinetic energy of a flow given in Eq (10) In order to find this Eq (57) is multiplied by vi andterms are grouped to find
part
partt
(1
2vivi
)+ vj
part
partxj
(1
2vivi
)= minusvi
1
ρ
partq
partxi+ νvi
part2vipartxjpartxj
minus vipart
partxjτmodij (v) (59)
23
Then similarly to Eq (58) another condition is to check whether the model affects the kineticenergy of exact solutions this is equivalent to checking whether
vipart
partxjτmodij (v) = 0 (60)
We call this condition the Kinetic Energy Condition The third and final condition considered herecan also be found using the integral form for the equation for kinetic energy
d
dt
intV
1
2vividV =
intS
[minusvj
(1
2vivi
)minus vj
q
ρ+ 2νviSij + viτ
modij (v)
]njdS
+
intV
[minus2νSijSij minus
(minusSijτmod
ij (v))]dV
(61)
where the rate-of-strain tensor Sij was defined in Eq (41) V is a volume of fluid and dS meansthe integral is over the surface of that volume with unit normal nj Then this final condition checkswhether the model affects the total kinetic energy of exact solutions to the Navier-Stokes equationsthat is
Sijτmodij (v) = 0 (62)
We will call this condition the Dissipation Condition because Sijτmodij (v) is often referred to as the
subgrid dissipation
34 Conditions for Large-Eddy Simulation with Explicit Filtering
As in Eq (48) the Navier-Stokes equations can be filtered by applying the filtering operation to thewhole equation (putting a bar over all terms) and evaluating to find again that
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (63)
In Eq (57) the filtering operation was ignored and the elements ui p and τij(u) were replacedwith vi q and τmod
ij (v) respectively Now in LES with explicit filtering the filtering operation itselfsmooths over the small-scale turbulent motion of the simulated flow In this case the elements uip and τij(u) will be replaced with vi q and τmod
ij (v) respectively LES with explicit filtering is thengiven by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (64)
Note that Eqs (63) and (64) despite their similarity to Eq (57) no longer contain the Navier-Stokesequations as Eq (57) did The bar (filtering operation) over each term results in each exact solutionfound in Section 22 not necessarily satisfying the first four terms It is hence unknown whether ornot these terms will vanish for exact solutions Terms can still be compared similarly to Eqs (58)(60) and (62) However instead of comparing to zero we can now compare to the relevant termfrom the filtered Navier-Stokes equations So the terms
part
partxjτij (65)
and
part
partxjτmodij (v) (66)
24
are to be compared Although the latter term is relatively simple to compute the former could leadto some difficulties due to the differentiation of the true stress τij = uiuj minus uiuj So although someinformation could be gleaned from the inspection of these two terms more information can be foundby considering the analogue of Eq (62) So the true subfilter dissipation
Sijτij (67)
(where Sij is shorthand for Sij(v)) and the modelled subfilter dissipation
Sij(v)τmodij (v) (68)
are considered and have to be equal for the condition to hold These can again be derived from theintegral equation for kinetic energy of a filtered velocity field similarly to Eq (61)
4 Analysis of Large-Eddy Simulation Models
In this section we use a code in Mathematica the theory described in Section 3 and the exactsolutions to the Navier-Stokes equations found in Section 22 to check whether or not each of theexact solutions lsquoendorsesrsquo each model for LES both without and with explicit filtering
41 Large-Eddy Simulation Without Explicit Filtering
We start by using a code in Mathematica to test the Velocity Field condition in Eq (58) and theKinetic Energy condition in Eq (60) For all exact solutions to the Navier-Stokes equations inSection 22 the outputs for both conditions were identical so we have combined them into one table
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 ndash ndash 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 1 For the Kinetic Energy condition a lsquo0rsquo result means that vipartpartxj
τmodij (v) = 0 for respective
models and exact solutions whereas a lsquolowastrsquo result means that vipartpartxj
τmodij (v) 6= 0 For the Velocity
Field condition a lsquo0rsquo result means that partpartxj
τmodij (v) = 0 for respective models and exact solutions
whereas a lsquolowastrsquo result means that partpartxj
τmodij (v) 6= 0 The Gradient model can take both positive and
negative values so when the sign of the model is known it is included in the table instead of a lsquorsquoBoth the conditions are included in one table because for all cases the outputs were identical
We then test the Dissipation condition in Eq (62) for all exact solutions to the Navier-Stokesequations in Section 22
25
Dissipation Condition [Does Sijτmodij (v) = 0]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 + 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 2 A lsquo0rsquo result means that Sijτmodij (v) = 0 for respective models and exact solutions whereas
a lsquolowastrsquo result means that Sijτmodij (v) 6= 0 The Gradient model can take both positive and negative
values so when the sign of the model is known it is included in the table instead of a lsquorsquo
We then sum all the results to find the total number of endorsements for each model described inSection 32
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
4 6 8 9 9
lsquoDissipationrsquoEndorsements
2 5 7 7 9
Total Endorsements 6 11 15 16 18
Table 3 The results from Tables 1 and 2 are summarised to give a general overview of how manyendorsements each model receives This makes it much clearer to see the models most supported bythe respective and combined conditions with the Vortex-Stretching model being the most supportedand the Smagorinsky model the least
Testing the models in Section 32 using LES without explicit filtering has mixed results It isclear that the Smagorinksy model is not well endorsed by the flows in Section 22 with the onlyendorsements coming from the simplest flows that is Sol 1 (Linear Velocity) Sol 3 (Velocity as afunction of time) Sol 7 (Stagnation Flow) and Sol 15 (Rankine Vortex) The Vortex-Stretching-based model is the most endorsed flow with eighteen out of a possible thirty endorsements Thisagrees with our foreknowledge as this model was constructed to be as physically consistent as possiblein the paper by Silvis amp Verstappen [11]
42 Large-Eddy Simulation With Explicit Filtering
In this section we will compare the true subfilter dissipation Sijτij and the modelled subfilterdissipation Sij(v)τmod
ij (v) given in Eqs (67) and (68) respectively for each model τmodij and each
exact solution to the Navier-Stokes equation given in Section 22 We will not consider the conditionsinvolving differentiation in Eqs (65) and (66) due to the difficulty in calculating the derivative ofthe τij term in Eq (65)
26
421 Example of a Test for a Model
The true subfilter dissipation Sijτij of a flow given in Eq (67) cannot be explicitly computed asthe filtering operation is kept general What can be done however is to partially evaluate theexpressions using the attributes of a filtering operation mentioned in Eq (48)
The calculations of Sijτij for Sol 1 (Linear Velocity) are provided to illustrate the general ideaRecall that the velocity field of Sol 1 (Linear Velocity) is written in Eq (13) in vector notation as
u(x y z t) =
0ax+ bcx+ d
(69)
Then Sij = Sij(u) is computed simply by using its definition in Eq (41) Note that as filteringpreserves constants if there is an a it is replaced with a meaning that (in this case) there are alarge number of zero terms due to the frequency of 0 Hence
[Sij(u)
]=
[1
2
(partuipartxj
+partujpartxi
)]=
0 a
2c2
a2 0 0
c2 0 0
=
0 a
2c2
a2 0 0
c2 0 0
(70)
Note that as Sijτij is being computed which is equal to the sum of the element-wise multiplicationof the two matrices (or the trace of the matrix multiplication) that we donrsquot need to consider theelements of τij(u) with corresponding zero entry in Sij As τij(u) = uiuj minus uiuj and again asfiltering preserves constants it is found that
[τij(u)] =[uiuj minus uiuj
]=
lowast 0 middot ax+ b 0 middot cx+ d
ax+ b middot 0 lowast lowastcx+ d middot 0 lowast lowast
=
lowast 0 00 lowast lowast0 lowast lowast
(71)
where a lsquolowastrsquo just represents some element (not necessarily non-zero) Therefore the true subfilterdissipation
Sijτij = tr
0 a2
c2
a2 0 0c2 0 0
middotlowast 0 0
0 lowast lowast0 lowast lowast
= 0 (72)
Now the modelled subfilter dissipation Sij(u)τmodij (u) needs to be computed (recall that Sij =
Sij(u)) In this case let us use the QR model stated in Eq (53) Once again Sij(u) calculated inEq (70) needs to be used So we have that
τmodij = minus2(CQRδ)
2 max0minusI3I1
Sij (73)
As
I1 = tr(S2) =
1
2(a2 + c2) and I3 = tr(S
3) = 0 (74)
it holds that τmodij (u) = 0 so Sij(u)τmod
ij (u) = 0 This agrees with Eq (72) and hence Sol 1(Linear Velocity) endorses the QR model
27
422 Testing the Models
Now using a Mathematica code the exact solutions in Section 22 are again used to evaluate thecondition for the models using the process just described in Section 421 The results are describedin Table 4
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 4 All entries with a ldquo0rdquo represent a value of zero and a ldquolowastrdquo represents a value that is non-zeroThe ldquoExactrdquo row represents the value of the true subfilter dissipation Sijτij Every other row witha model name represents the value of the modelled subfilter dissipation Sij(u)τmod
ij (u) respectiveto that model A ldquordquo means that it is unclear as to the value - the sign of these values is oftenunknown and the lsquoMaxrsquo function in the definition of the QR model (in Eq (53)) makes it difficultto evaluate
Reading from Table 4 the Smagorinsky model is clearly endorsed by Sol 3 (Velocity as a function oftime) and Sol 15 (Rankine Vortex) as in both cases both Sijτij = 0 = Sij(u)τmod
ij (u) In this casewe say that Sol 3 (Velocity as a function of time) and Sol 15 (Rankine Vortex) lsquostronglyrsquo endorsethe Smagorinsky model The Smagorinsky model is not endorsed by Sol 1 (Linear Velocity) Sol 2(Planar Flow) or Sol 14 (Radially Dependent) however because Sij(u)τmod
ij (u) 6= 0 For Sol 4 to
13 both Sijτij 6= 0 6= Sij(u)τmodij (u) and so we will say that these exact solutions lsquoweaklyrsquo endorse
the Smagorinsky model
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 10 10 9 6
Table 5 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Interestingly the lsquostrongrsquo and lsquoweakrsquo endorsements give different results However more stock shouldbe put in the results of the lsquostrongerrsquo condition as it is completely clear that zero is equal to zeroIt is not necessarily clear whether or not non-zero equals non-zero as the values will depend on theparameters in each solution hence the lsquoweakerrsquo endorsement
The lsquostrongrsquo condition equally endorses the Vreman QR Gradient and Vortex-Stretching-basedmodels whereas the lsquoweakrsquo condition endorses Vreman and Smagorinsky the most closely followedby the Gradient model with the Vortex-Stretching-based model being the most opposed As this
28
completely contradicts the lsquostrongrsquo condition and the results in Section 41 it makes us question thevalidity of the weaker test The number of endorsements for the QR model is unknown because of thedefinition of the model in Eq (53) and our evaluation of the true subfilter dissipation Eq (67) Dueto the exact solutions we have found it is never clear whether or not the true subfilter dissipationis strictly positive or negative This means that when the true subfilter dissipation is found to benon-zero the lsquoMaxrsquo function in the QR model cannot be evaluated
43 Vremanrsquos True Subfilter Dissipation Claim
Using the theory mentioned in Sections 232 and 34 we can place the exact solutions to the Navier-Stokes equations into different classes defined by Vreman In doing this we can also attempt to verifyone of Vremanrsquos results [16] In Vremanrsquos paper he defines the class Qn to be the set of flows withn zero entries in the velocity gradient of the flow (the velocity gradient is defined in Eq (39))Vreman then claims that the only flows with zero true subfilter dissipation are a certain selection ofthose in the classesQ7Q8 orQ9 shown in appendix B and is generally non-zero for any other flows
The true subfilter dissipation of Sol 1 (Linear Velocity) was computed in Section 421 and wasfound to be zero in Eq (72) The velocity gradient of Sol 1 is as follows
GLV =
0 0 0a 0 0c 0 0
(75)
and hence Sol 1 (Linear Velocity) is in the required subset of Q7 which agrees with Vremanrsquos claimIn Section 422 the true subfilter dissipation was computed (determined to be zero or non-zero) forall of the exact solutions to the Navier Stokes equations in Section 22 These solutions can now belsquoplacedrsquo into each class Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a functionof time) and Sol 14 (Radially Dependent) were all found to have zero true subfilter dissipation andare all in the appropriate subsets of either Q7Q8 or Q9 This agrees with Vremanrsquos results
We can extend Vremanrsquos results however We found that Sol 15 (Rankine Vortex) had zero truesubfilter dissipation and yet is not in the subset of Q7 defined in Appendix B It is important tonote that the rate of strain tensor Sij in Eq (41) for Sol 15 (Rankine Vortex) is zero This is thereason the true subfilter dissipation is zero and the same reason that all models are zero for thisflow as all models are based on Sij
We also found that all of the remaining exact solutions to the Navier-Stokes equations are not inVremanrsquos specified classes (in Appendix Section B) They also have a non-zero subfilter dissipationwhich agrees with Vremanrsquos results in his paper [16]
44 Classes of Flows
It is important to consider that some exact solutions to the Navier-Stokes equations may have similarproperties For example Sol 1 (Linear Velocity) and Sol 2 (Planar Flow) are both very similarIn fact Sol 2 (Planar Flow) is a more general case of Sol 1 (Linear Velocity) We therefore mustgroup the different flows into classes in an attempt to reduce the bias that may occur when solutionsare similar This should aid us to further determine which of the models are most endorsed We willdo this in a number of different ways using the theory discussed in Section 23 We will group theflows with respect to Vremanrsquos Flow Classes given in appendix B and with respect to the Invariantsgiven in appendix A
29
441 Vremanrsquos Velocity Gradient Flow Classes
We now put our exact solutions to the Navier-Stokes equations into one of two classes Either Q0minus6or Q7minus9 where each class is defined intuitively as follows
Q0minus6 = Q0 cupQ1 cup middot middot middot cup Q6 (76)
Q7minus9 = Q7 cupQ8 cupQ9 (77)
where Qn is defined in Section 43 By looking at the gradient of each exact solution we can see thatQ7minus9 contains the six solutions Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as afunction of time) Sol 5 (2D Taylor) Sol 14 (Radially Dependent) and Sol 15 (Rankine Vortex)and that Q0minus6 contains the remaining nine We now check the models for LES without explicitfiltering by considering endorsements within each class Tables 6 and 7 are analogues of Table 3 andillustrate the results for each flow class The analogues of Tables 1 and 2 are Tables 22 and 23 andare in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 6 6 6
lsquoDissipationrsquoEndorsements
2 5 6 6 6
Percentage Endorsed 417 833 100 100 100
Table 6 Each entry in the top two rows represents the number of solutions in the class Q7minus9 thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Q0minus6 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 2 3 3
lsquoDissipationrsquoEndorsements
0 0 1 0 3
Percentage Endorsed 56 56 167 167 333
Table 7 Each entry in the top two rows represents the number of solutions in the class Q1minus6 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking results to 1 decimal place if necessary
All of the flows in the class Q7minus9 endorse the QR Gradient and Vortex-Stretching-based subfilter-scale models (see Table 6) We can hence see that this class fully endorses these three models equallyand does not endorse the Smagorinsky model The Vortex-Stretching eddy viscosity model has themost endorsements from any of the flows in the class Q0minus6 and the Smagorinksy and Vreman modelshave the fewest endorsements (see Table 7) We can hence say that this class mostly endorses theVortex-Stretching-based eddy-viscosity model note however that the percentage endorsement is very
30
low
Now let us consider LES with explicit filtering Tables 8 and 9 are analogues of Table 5 and illustratethe results for each flow class The analogue of Table 4 is Table 24 and is in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 1 1 0 0 0
Table 8 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Q0minus6 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 9 9 9 6
Table 9 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We have grouped Vremanrsquos classes into Q7minus9 and Q0minus6 this is equivalent to saying that we havegrouped Vremanrsquos classes into those with two or fewer non-zero velocity gradient elements and thosewith three or more non-zero velocity gradient elements respectively Therefore the solutions in Q7minus9are exactly those with a lsquo0rsquo in the lsquoExactrsquo column of Table 4 and Sol 5 (2D Taylor) ExcludingSol 5 (2D Taylor) these are the same as those five that endorse the Vreman QR Gradient andVortex-Stretching-based models in Table 5 We can see in Table 8 that the results for the lsquostrongrsquoendorsements are exactly the same as in Table 5 This is because all of the flows with zero subfilterdissipation are contained in the class Q7minus9 There are hence no lsquostrongrsquo endorsements in Table 9The majority of the lsquoweakrsquo endorsements come from Q0minus6 as this class contains all but one of thesolutions with non-zero true subfilter dissipation (given in Eq (67))
442 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
We now separate our exact solutions into classes again using Vremanrsquos flow classes This timehowever we define V0 to be the class of flows Vreman claims will always have zero true subfilterdissipation namely those classes mentioned in Appendix B We define Vlowast to be all the remaining
31
classes Then
V0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow)
Sol 3 (Velocity as a function of time) Sol 14 (Radially Dependent)(78)
Vlowast = Sol 4 (Axisymmetric Flow) Sol 5 (2D Taylor) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 13 (Sphere Translation) Sol 15 (Rankine Vortex)
(79)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 10 and 11 are analogues of Table 3 and illustrate the results for each flow class Theanalogues of Tables 1 and 2 are Tables 25 and 26 and are in Appendix C2
Endorsements for models using solutions in V0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 4 4 4 4
lsquoDissipationrsquoEndorsements
1 4 4 4 4
Percentage Endorsed 375 100 100 100 100
Table 10 Each entry in the top two rows represents the number of solutions in the class V0 thatendorse each model There are 4 solutions in this class so the percentages are worked out by dividingthe total of each column by 8 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Vlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 2 4 5 5
lsquoDissipationrsquoEndorsements
1 1 3 2 5
Percentage Endorsed 136 136 318 318 455
Table 11 Each entry in the top two rows represents the number of solutions in the class Vlowast thatendorse each model There are 11 solutions in this class so the percentages are worked out bydividing the total of each column by 22 and taking results to 1 decimal place if necessary
All of the flows in the class V0 endorse the Vreman QR Gradient and Vortex-Stretching-basedsubfilter-scale models (see Table 6) This class completely endorses these four models equally anddoes not endorse the Smagorinsky model The Vortex-Stretching-based eddy-viscosity model hasthe most endorsements from the class Vlowast and the Smagorinksy and Vreman models again have thefewest endorsements (see Table 7) As in Section 441 we say that this class mostly endorses the
32
Vortex-Stretching-based eddy-viscosity model but once again the percentage endorsement is ratherlow
Now let us consider LES with explicit filtering Tables 12 and 13 are analogues of Table 5 andillustrate the results for each flow class The analogue of Table 4 is Table 27 and is in Appendix C2
Endorsements for models using solutions in V0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 4 4 4 4
lsquoWeakrsquo Endorsements 0 0 0 0 0
Table 12 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Vlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 1 1 1 1
lsquoWeakrsquo Endorsements 10 10 9 6
Table 13 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We can see in Table 12 that the results for the lsquostrongrsquo endorsements are similar to those in Table 5excluding Sol 15 (Rankine Vortex) The four solutions in V0 with zero true subfilter dissipationnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)and Sol 14 (Radially Dependent) are hence the four that endorse the Vreman QR Gradientand Vortex-Stretching-based models in Table 12 Sol 15 (Rankine Vortex) is in the class Vlowast andhence there is one lsquostrongrsquo endorsement in Table 13 The class Vlowast contains all of the flows with anon-zero true subfilter dissipation (given in Eq (67)) and hence Table 13 contains all of the lsquoweakrsquoendorsements
443 Classes using the Third Principal Invariant
Sorting the classes using the invariants is a non-trivial task in itself There are a number of waysin which to do this using both the principal invariants in Eq (40) and the combined invariants inEq (43)
We start by looking at flows with third principal invariant RG equal to zero We again define twoclasses R0 and Rlowast where R0 consists of all flows with RG = 0 and Rlowast consists of all flows withRG 6= 0 Then using the information in Appendix A we see that there are nine flows in R0 That
33
is
R0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 8 (Vortex in Stagnation Flow)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)
(80)
Rlowast = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 13 (Sphere Translation)(81)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 14 and 15 are analogues of Table 3 and they illustrate the results for LES withoutexplicit filtering for each flow class The analogues of Tables 1 and 2 are Tables 28 and 29 and arein Appendix C3
Endorsements for models using solutions in R0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 7 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 278 556 778 722 778
Table 14 Each entry in the top two rows represents the number of solutions in the class R0 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking the result to one decimal place if necessary
Endorsements for models using solutions in Rlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 2 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 83 83 83 167 333
Table 15 Each entry in the top two rows represents the number of solutions in the class Rlowast thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking the result to one decimal place if necessary
The class R0 endorses both the Vortex-Stretching-based model and the QR model the most andSmagorinsky the least in the case for LES without explicit filtering In the Rlowast class the Vortex-Stretching-based model is again the most endorsed followed by the Gradient model however thepercentages are again small
34
Now let us consider LES with explicit filtering Tables 16 and 17 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 30 and is in Appendix C3
Endorsements for models using solutions in R0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 4 4 3 2
Table 16 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Rlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 6 6 6 4
Table 17 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 we can see in Table 16 that the results for the lsquostrongrsquo endorsements are exactly thesame as in Table 5 This is because all of the flows with zero subfilter dissipation are contained in theclass R0 As in Table 9 there are no lsquostrongrsquo endorsements in Table 17 The lsquoweakrsquo endorsementshowever are split more evenly between the two classes now than in Vremanrsquos classes in Sections 441and 442 We find that the Smagorinsky and Vreman models still seem to be the most endorsed inboth classes
444 Two-component Flow Classes
Another way in which the flows can be grouped into classes is by determining whether a flow is atwo-component flow or not We define a two-component flow to have the property that the combinedprincipal invariants of a flow in Eq (43) satisfy
I3 = I4 = I5 minus1
2I1I2 = 0 (82)
We are also interested in considering flows with zero gradient model subgrid dissipation These flowshave the property that
I3 minus I4 = 0 (83)
What we find though is that for all the solutions in Section 22 if a flow is a two-component flowthen it also satisfies I3 minus I4 = 0 It may be interesting to note that although I3 minus I4 6= 0 for manysolutions there are a number with I3 equal to a multiple of I4 Although not considered in thisreport this may also be a class of interest We find that the two-component flows are
C2 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)(84)
35
and we call the remaining flows three-component flows They are
C3 = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex) Sol 11 (Pseudo-Motion Shots)
Sol 13 (Sphere Translation)
(85)
Note that Sol 8 (Vortex in Stagnation Flow) is defined to be a planar flow this does not mean that itis a two-component flow in this sense however because its two non-zero velocity components dependon three coordinates We now check the models for LES without explicit filtering by consideringendorsements within each class Tables 18 and 19 are analogues of Table 3 and they illustrate theresults for LES without explicit filtering for each flow class The analogues of Tables 1 and 2 areTables 31 and 32 and are in Appendix C4
Endorsements for models using solutions in C2 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 6 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 357 714 100 857 100
Table 18 Each entry in the top two rows represents the number of solutions in the class C2 thatendorse each model There are 7 solutions in this class so the percentages are worked out by dividingthe total of each column by 14 and taking the result to one decimal place if necessary
Endorsements for models using solutions in C3 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 3 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 63 63 63 188 250
Table 19 Each entry in the top two rows represents the number of solutions in the class C3 thatendorse each model There are 8 solutions in this class so the percentages are worked out by dividingthe total of each column by 16 and taking the result to one decimal place if necessary
The class C2 equally endorses the QR and Vortex-Stretching-based models and once again theSmagorinsky model is the least endorsed As with the previous classes the class C3 has such lowpercentages that it does not seem to endorse any models We start to see a trend appearing here inthat the flow classes that contain the more complicated flows namely Q0minus6 (Eq (76)) Vlowast (Eq (79))Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages The classes that tendto contain the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78)) R0 (Eq (80)) and C2 (Eq (84))all seem to favour the Vortex-Stretching-based model or the QR model
36
Now let us consider LES with explicit filtering Tables 20 and 21 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 33 and is in Appendix C4
Endorsements for models using solutions in C2 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 2 2 0 1 0
Table 20 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in C3 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 8 8 8 6
Table 21 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 and Table 16 we can see in Table 20 that the results for the lsquostrongrsquo endorsements areexactly the same as in Table 5 This is again because all of the flows with zero subfilter dissipationare contained in the class C2 There are two flows with non-zero true subfilter dissipation (givenin Eq (67)) in C2 and this shows in the lsquoweakrsquo endorsements in Table 20 There are no lsquostrongrsquoendorsements in Table 21 as all the zero subfilter dissipation flows are in C2
5 Conclusions
In this report we studied incompressible turbulent flows We found a number of exact solutions to theNavier-Stokes equations each representing a relatively simplistic fluid flow Large-eddy simulation(LES) is used to model turbulent flows by smoothing over the complex small-scale motion Weintroduced a number of subfilter-scale models to approximate this small-scale motion and testedthem using the found exact solutions to the Navier-Stokes equations We first considered LESwithout explicit filtering and used the Navier-Stokes equations to derive conditions for the subfilter-scale models namely the Velocity Field condition (Eq (58)) the Kinetic Energy condition (Eq (60))and the Dissipation condition (Eq (62)) Similarly we considered LES with explicit filtering andagain used the Navier-Stokes equations to derive more conditions for the models in this case wedecided to just take one condition the Dissipation condition (Eqs (67) and (68)) as the others leadto difficulty in their evaluation By lsquorunningrsquo the exact solutions to the Navier-Stokes equationsthrough each condition for the subfilter-scale models we found that the Vortex-Stretching-basededdy-viscosity model in Eq (54) was most endorsed by the tests in LES without explicit filteringWe found that four out of the five subfilter-scale models namely the Vreman QR Gradient andVortex-Stretching-based models were equally endorsed by the lsquostrongrsquo tests in LES with explicitfiltering with the Smagorinsky model being the least endorsed The results we found for the lsquoweakrsquocondition were completely opposite to those found in both the lsquostrongrsquo condition and in LES withoutexplicit filtering This led us to conclude that this test was not one in which we should put our trust
37
We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses
In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes
This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold
38
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
in vector notation
There are now four equations in four unknowns and it is possible to find solutions Althoughthere is no known general solution there are specific cases in which exact solutions to the Navier-Stokes equations can be found In order to make finding solutions slightly simpler the Navier-Stokesequations can be rescaled by introducing dimensionless variables
Let
ulowasti =uiv xlowasti =
xi` tlowast =
t
`v plowast =
p
ρv2 (7)
where ` and v are typical length and velocity scales respectively Then dropping the star notationfor ease of reading the scaled incompressible Navier-Stokes equations are
partuipartt
+ ujpartuipartxj
= minus partp
partxi+
1
Re
part2uipartxjpartxj
i = 1 2 3 j = 1 2 3
partuipartxi
= 0
(8)
where Re is the Reynolds number and is defined by the dimensionless Re =v`
ν Similarly in vector
notation the scaled incompressible Navier-Stokes equations are
partu
partt+ (u middot nabla)u = minusnablap+
1
Renabla2u
nabla middot u = 0(9)
By multiplying the incompressible Navier-Stokes equations Eq (8) by ui and grouping terms thedifferential form of the evolution of the kinetic energy of a flow can be found
part
partt
(1
2uiui
)+ uj
part
partxj
(1
2uiui
)= minusui
partp
partxi+
2
Reui
part2uipartxjpartxj
(10)
The differential form of the evolution of the kinetic energy of a flow will be used later on in thisreport to derive conditions for the subfilter-scale models
22 Exact Solutions of the Navier-Stokes Equations
By using mathematical inspection and taking results found in literature we found a number of exactsolutions to the Navier-Stokes equations The majority are steady-state solutions in that they areindependent of time We decided to consider exact solutions to the Navier-Stokes equations withoutboundary conditions in an attempt to keep the results general and applicable to a larger numberof situations All of the stated exact solutions below have been checked using a code written inMathematica In the solutions below the notation (x y z) for the Cartesian coordinate system isused analogously with (x1 x2 x3) to aid in reading comprehension and in the conversion betweenindex and vector notation
Solution 1 Linear Velocity
If the diffusion term in the Navier-Stokes equations Eq (1) is taken to be zero ie
part2uipartxjpartxj
= 0 (11)
6
then each component of the velocity can be written as a linear function of x1 x2 and x3 In indexnotation
ui = aijxj + βi (12)
If pressure is assumed to be a function of time t (so that the pressure terms in the Navier-Stokesequations are zero) then conditions can be found on the coefficients to find that the following is asolution
u1(x y z t) = 0
u2(x y z t) = ax+ b
u3(x y z t) = cx+ d
p(x y z t) = f(t)
(13)
where a b c d isin R and f Rrarr R In fact this solution is actually a specific case of Sol 2 (PlanarFlow)
Figure 1 An illustration of the flow in the xy-plane for Sol 1 (Linear Velocity) described inEq (13) with parameter values a = b = c = d = 1
Solution 2 Planar Flow
Using the text by Barbato et al [2 p62] and modifying the solution such that it now takes threedimensions into account another solution is found
u1(x y z t) = Az2 +Bz + C
u2(x y z t) = Dz2 + Ez + F
u3(x y z t) = 0
p(x y z t) =2A
Rex+
2D
Rey + f(t)
(14)
7
where ABCDE F isin R and f R rarr R It is clear to see that if A = D = 0 then this is in factthe same as Eq (13) in Sol 1 In fact we believe (although not shown here) that this is the mostgeneral solution of the form
ui(xi xj xk t) = Ax2k +Bxk + C
uj(xi xj xk t) = Dx2k + Exk + F
uk(xi xj xk t) = 0
(15)
for i 6= j 6= k 6= i
Figure 2 An illustration of the flow in the yz-plane for Sol 2 (Planar Flow) described inEq (14) with parameter values A = B = C =D = E = F = 1
Solution 3 Velocity as a function of time t
If each component of the velocity is taken to be a function of time t and independent of positionthen the incompressibility condition is automatically satisfied A condition can be imposed on thepressure term in order to satisfy the momentum equations Thus a solution is as follows
u1(x y z t) = f(t)
u2(x y z t) = g(t)
u3(x y z t) = h(t)
p(x y z t) = minusxpartf(t)
parttminus y partg(t)
parttminus z parth(t)
partt
(16)
where f g h Rrarr R In fact what is written here is the generalised Galilean transformation of thezero solution [10] We cannot produce a plot for this flow as the functions are completely unspecified
8
Solution 4 Axisymmetric Flow
Using the text by Irmay [7] a solution is defined as follows
u1(x y z t) = kxz
u2(x y z t) = kyz
u3(x y z t) = minuskz2
p(x y z t) = minus2kz
Reminus k2z4
2
(17)
where k isin R This is said to represent an axisymmetric flow with hyperbolic streamlines asymptoticto the z-axis and to the boundary plane z = 0 These asymptotes are visible in Fig 3
Figure 3 An illustration of the flow in theyz-plane for Sol 4 (Axisymmetric Flow)described in Eq (17) with parameter k = 1
Figure 4 An illustration of the flow in thexy-plane for Sol 4 (Axisymmetric Flow)described in Eq (17) with parameter k = 1
Solution 5 2D Taylor Solutions
Again using the text by Barbato et al [2 p63ndash64] a solution is defined using a stream functionψ = ψ(x y t) and by writing
u1 = minuspartψparty
u2 =partψ
partx u3 = 0 (18)
In the text the stream function is not given however it was worked out that
ψ = minuseminusπ2t
Re (cos(πx) + cos(πy))
π (19)
The solution used is of the following form note however that in the text the pressure term differsin that cos(πy) is replaced by sin(πy) (when testing this solution in Mathematica the solution below
9
was instead found)
u1(x y z t) = minus sin(πy) eminus(π2t)Re
u2(x y z t) = sin(πx) eminus(π2t)Re
u3(x y z t) = 0
p(x y z t) = minus cos(πx) cos(πy) eminus(2π2t)Re
(20)
Figure 5 An illustration of the flow in the xy-plane for Sol 5 (2D Taylor) described in Eq (20)with Re = 1 and time t = 0
Solution 6 Generalized Beltrami Flow
Again using the text by Barbato et al [2 p65] and now considering solutions with nonndashtrivialterms in all dimensions a particular form of the generalized Beltrami flow is found namely
u1(x y z t) = (A sin(πz) + C cos(πy)) eminus(π2t)Re
u2(x y z t) = (B sin(πx) +A cos(πz)) eminus(π2t)Re
u3(x y z t) = (C sin(πy) +B cos(πx)) eminus(π2t)Re
p(x y z t) = minus[(AB cos(πz) sin(πx) +BC cos(πx) sin(πy)
+AC cos(πy) sin(πz)] eminus(2π2t)Re
(21)
where ABC isin R
10
Figure 6 An illustration of the flow in the yz-plane for Sol 6 (Generalized Beltrami Flow)described in Eq (21) with parameters A = B =C = 1 and time t = 0
Solution 7 Stagnation Flow
In the book by Drazin and Riley [6 p40] an exact solution is considered which uses a stream functionψ = ψ(y z) They suggest that the following velocity field is a solution
u1(x y z t) =a
kx
u2(x y z t) = minusaky +
k
Re
partψ(y z)
partz
u3(x y z t) = minus k
Re
partψ(y z)
party
(22)
Here we choose the stream function to be ψ(y z) = yz Then a solution is given by
u1(x y z t) =a
kx
u2(x y z t) =
(minusak
+k
Re
)y
u3(x y z t) = minus kzRe
p(x y z t) = minus1
2
((axk
)2minus((
k
Reminus a
k
)y
)2
minus(kz
Re
)2)
(23)
where a is a measure of the strength of the stagnation flow and k is a constant chosen in the textto specify the distance between vortices There are no vortices in this particular solution however
11
Figure 7 An illustration of the flow in the yz-plane for Sol 7 (Stagnation Flow) describedin Eq (23) with parameter values a = Re = 1and k = 2
Figure 8 An illustration of the flow in the xy-plane for Sol 7 (Stagnation Flow) describedin Eq (23) with parameter values a = Re = 1and k = 2
Solution 8 Vortex in Stagnation Flow
As in Sol 7 (Stagnation Flow) we now consider the same solution with a different stream functionnamely ψ(z) = cos z Then given the condition that k = aRe and adding a pressure term Eq (22)becomes
u1(x y z t) =1
Rex
u2(x y z t) = minus 1
Rey minus k sin z
u3(x y z t) = 0
p(x y z t) = minus 1
2Re2(x2 + y2)
(24)
where a and k are as in Sol 7 (Stagnation Flow)
12
Figure 9 An illustration of the flow in the yz-plane for Sol 8 (Vortex in Stagnation Flow)described in Eq (24) with parameter valuesk = 3 and Re = 10
Figure 10 An illustration of the flow in thexy-plane for Sol 8 (Vortex in StagnationFlow) described in Eq (24) with parametervalues k = 3 and Re = 10
Solution 9 Burgers Vortex
The Burgers Vortex solution is given in cylindrical coordinates in the book by Lautrup [8 p 385]as follows
ur(r ϕ z t) = minus 2r
c2Re
uϕ(r ϕ z t) =Ωc2
r
(1minus eminusr
2c2)
uz(r ϕ z t) =4z
c2Re
p(r ϕ z t) = minusc2Ω2F
(r2c2
)2
minus 2
Re2r2 + 4z2
c4
(25)
where
F (ξ) =
infinintξ
(1minus eminusx)2
x2dx (26)
This solution is not particularly nice to look at however most of the unpleasantness comes fromthe pressure term which (in this project) is only required in order to validate the solution For thisproject all solutions are required in Cartesian coordinates the velocity field of the Burgers Vortexhas hence been converted into Cartesian coordinates
u1(x y z t) = minus 2x
c2Reminus Ω
h(x y)(1minus exp (minush(x y))) y
u2(x y z t) = minus 2y
c2Re+
Ω
h(x y)(1minus exp (minush(x y)))x
u3(x y z t) =4z
c2Re
(27)
13
where
h(x y) =x2 + y2
c2 (28)
and Ω and c are constants
Figure 11 An illustration of the flow for Sol9 (Burgers Vortex) described in Eq (27) withparameter values c = 15 Re = 20 and Ω = 10A slice of the xy-plane at z = 0 is shown
Solution 10 Another Vortex Solution
There is another vortex solution more simply formulated than the Burgers Vortex yet with asingularity at r = 0 as a result of the uϕ term It is again given in cylindrical coordinates
ur(r ϕ z t) = minus 2r
c2Re
uϕ(r ϕ z t) =a
r
uz(r ϕ z t) =4z
c2Re
p(r ϕ z t) = minus a2
2r2minus 2r2 + 8z2
c2Re2
(29)
which can also be converted into Cartesian coordinates
14
u1(x y z t) = minus 2x
c2Reminus ay
x2 + y2
u2(x y z t) = minus 2y
c2Re+
ax
x2 + y2
u3(x y z t) =4z
c2Re
(30)
where a and c are constants
Figure 12 An illustration of the flow for Sol9 (Burgers Vortex) described in Eq (27) withparameter values c = 15 Re = 20 and a = 10 Aslice of the xy-plane at z = 0 is shown
Solution 11 Pseudo-Motion Shots of the Second Kind
In the book by Berker [3 p 94] a solution is given in cylindrical coordinates by
ur(r ϕ z t) =2
Re
1
r
uϕ(r ϕ z t) = A1r3 +B1r +
C1
r
uz(r ϕ z t) = A2r2 log(r) +B2r
2 + C2
p(r ϕ z t) =A2
1r6
6+A1B1r
4
2+A1C1r
2 +B2
1r2
2minus C2
1
2r2minus 2
r2Re2+ 2B1C1 log(r)
+2A2z
Reminus 4B1ϕ
Re
(31)
where A1 A2 B1 B2 C1 C2 are constants Eq (31) when converted into Cartesian coordinates is
15
u1(x y z t) =1
Re
2xminus C1Rey
x2 + y2minusA1y
(x2 + y2
)minusB1y
u2(x y z t) =1
Re
2y + C1Rex
x2 + y2+A1x
(x2 + y2
)+B1x
u3(x y z t) =1
2A2
(x2 + y2
)log(x2 + y2
)+B2
(x2 + y2
)+ C2
(32)
Figure 13 An illustration of the flow in thexy-plane for Sol 11 (Pseudo-Motion Shots)described in Eqs (31) and (32) with parametervalues A1 = B1 = C1 = 1 A2 = B2 = C2 = 2and Re = 10
Figure 14 An illustration of the flow in theyz-plane for Sol 11 (Pseudo-Motion Shots)described in Eqs (31) and (32) with parametervalues A1 = B1 = C1 = 1 A2 = B2 = C2 = 2and Re = 10
Solution 12 Flows With Constant Whirl
In the book by Berker [3 p 143] another solution is given in cylindrical coordinates by
ur(r ϕ z t) = minus 1
Re
1
r
uϕ(r ϕ z t) =1
rradicte(minus
12 g(t))
uz(r ϕ z t) = 0
p(r ϕ z t) =1
4
2(minus e
minusg(t)
t minus 1Re2
)r2
minus 1
t2Re
infinintminusg(t)
eminust
tdt
(33)
where we define g(t) =r2Re
2t When converted into Cartesian coordinates we find
16
u1(x y z t) = minus 1
x2 + y2
(yradict
exp
(minus
Re(x2 + y2
)4t
)+
x
Re
)
u2(x y z t) =1
x2 + y2
(xradict
exp
(minus
Re(x2 + y2
)4t
)minus y
Re
)
u3(x y z t) = 0
(34)
Figure 15 An illustration of the flow for Sol12 (Flows with Constant Whirl) described inEqs (33) and (34) with time t = 1 and Re = 1 Aslice of the xy-plane at z = 0 is shown as velocityin the z-direction is zero this is the same for anyvalue of z
17
Solution 13 Rectilinear and Uniform Translation of a Sphere
In the book by Berker [3 p 228] another solution is given in Cartesian coordinates by
u1(x y z t) =3
4
axz
R3
(a2
R2minus 1
)
u2(x y z t) =3
4
ayz
R3
(a2
R2minus 1
)
u3(x y z t) = 1minus 3
4
a
Rminus 1
4
a3
R3minus 3
4
az2
R3
(a2
R2minus 1
)
p(x y z t) =3a(aminusR)(a+R)
32R10Re
((R2Re
(x2 + y2
)minus 2R2Rez2
) (a3 + 3aR2 minus 4R3
)+ 3aRez4
(R2 minus a2
)minus 16R5z
)
(35)
where a and R are constants
Figure 16 An illustration of the flow in the yz-plane for Sol 13 (Sphere Translation) describedin Eq (35) with parameter values a = minus2 andRe = 3
Solution 14 Radially Dependent Flow
A solution with velocity in the z-direction dependent only on the distance from the z-axis is as
18
follows in cylindrical coordinates
ur(r ϕ z t) = 0
uϕ(r ϕ z t) = 0
uz(r ϕ z t) = A(r2 minus a2
)
p(r ϕ z t) =4Az
Re
(36)
where A and a are constants In Cartesian coordinates the solution is just the same with u1(x y z t)and u2(x y z t) both zero and the r2 term replaced with x2 + y2 in u3(x y z t)
Figure 17 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus1 1] withparameter values a = 1 A = 4 and Re = 10
Figure 18 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus2 2] withparameter values a = 1 A = 4 and Re = 10
Solution 15 Rankine Vortex
A solution discovered by Rankine [1] is a vortex flow of the following form
ur(r ϕ z t) = 0
uϕ(r ϕ z t) =ar
2πR2
uz(r ϕ z t) = 0
p(r ϕ z t) =a2r2
8π2R4
(37)
19
where a and R are constants In Cartesian coordinates the flow has the form
u1(x y z t) = minus ay
2πR2
u2(x y z t) =ax
2πR2
u3(x y z t) = 0
p(x y z t) =a2(x2 + y2
)8π2R4
(38)
This flow is actually a rigid body rotational flow in that the angular rotational velocity is uniformand the velocity of the flow increases proportionally with distance from the z-axis
Figure 19 An illustration of the flow in the xy-plane for Sol 15 (Rankine Vortex) described inEq (37) and Eq (38) with parameter values a = 1and R = 1
radic2π
23 Characterisation of Flows
It is important to be able to classify flows in a way that sorts them into distinct groups This aidsin the clear identification and general study of flows It is also a very important way of recognisingflows with similar properties In this subsection we introduce two distinct ways in which flows canbe characterised
231 Characterising Solutions using Invariants
A principal invariant of a tensor is a coefficient of the characteristic polynomial of that tensorInvariants as their name suggests are invariant under rotations of the coordinate system in thattheir values do not change when the tensor is lsquorotatedrsquo into a different set of coordinates Thevelocity gradient of a flow G is defined as the Jacobian of the velocity vector given by
Gij =
[partuipartxj
] (39)
20
for i j = 1 2 3 There are three principal invariants of the velocity gradient of a fluid flow G givenby
PG = tr(G)
QG =1
2
((tr(G))2 minus tr
(G2))
RG = det(G)
(40)
Incompressibility of flows (Eq (5)) ensures that PG = 0 hence the first principal invariant is zerofor all of the flows in Section 22 We now define the rate-of-strain tensor S by
Sij =1
2
(partuipartxj
+partujpartxi
) (41)
and the rate-of-rotation tensor Ω by
Ωij =1
2
(partuipartxjminus partujpartxi
) (42)
Note that the first principal invariant of S and the first and third principal invariants of Ω are zeroAlso note that the velocity gradient of a flow can simply be written G = S + Ω
Rather than considering the principal invariants we can also consider the combined invariants
I1 = tr(S2) I2 = tr(Ω2) I3 = tr(S3) I4 = tr(SΩ2) I5 = tr(S2Ω2) (43)
We can hence write the three principal invariants for a velocity field G in terms of the combinedinvariants
PG = 0
QG = minus1
2(I1 + I2)
RG =1
3(I3 + 3I4)
(44)
In this report the flows considered are the exact solutions to the incompressible Navier-Stokesequations described in Section 22 The combined invariants in Eq (43) are used to characterisethese solutions into different classes in Section 44 Both the principal invariants and combinedinvariants for each exact solution in Section 22 are given in Appendix A
Each of the subfilter-scale eddy-viscosity models described in Section 32 are written in terms ofthese five combined invariants
232 Vremanrsquos Characterisation of Flows
Vreman [16] classifies flows using the gradient of the velocity field of a flow Each lsquoclassrsquo of flows hasthe same number of non-zero elements in the gradient matrix For example
[partuipartxj
]=
0 0 0lowast 0 0lowast 0 0
(45)
where a lowast denotes a non-zero term represents a group of flows that lie within the class denoted byQ7 (see Appendix B) where the subscript represents the number of non-zero terms in the velocity
21
gradient Flows of the form in Eq (45) have a velocity field of the form
u1 = c1 (constant)
u2 = f2(x1)
u3 = f3(x1)
(46)
where f2(x1) and f3(x1) are such that the Navier Stokes equations Eq (1) are satisfied Notethat as the elements on the leading diagonal are zero the incompressibility condition Eq (5) isautomatically satisfied Some simple flow classes are given in Appendix B and contain a number ofthe solutions discussed in Section 22 This is mentioned in greater detail in Section 441
3 Conditions for Large-Eddy Simulation Models
31 Large-Eddy Simulation
Due to the nature of numerical simulation and the complexity of turbulent flows small details aremissed and information is lost when solving the Navier-Stokes equations The computational costof solving a turbulent flow is extremely high so a filtering operation is applied to the Navier-Stokesequations represented by a bar to reduce the computational cost of the simulation This is a linearoperation and preserves constants ie ρ = ρ and is also assumed to commute with differentiation
partuipartxj
=partuipartxj
(47)
The filtered Navier-Stokes equations are found by filtering (putting a bar over) the entire equations(see Eq (1)) and after evaluating are given by
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (48)
where
τij = uiuj minus uiuj (49)
is called the lsquotrue stressrsquo of the flow This however poses a problem as it is not clear what is meantby the term uiuj for a general filtering operation The true stress τij therefore must be replaced bya model τmod
ij in practical applications
32 Subfilter-Scale Models
We will now define the subflter-scale models denoted by τmodij to be used in place of the lsquodifficultrsquo
true stress τij There are a number of models available to use in the literature [11 p7] These willbe tested using the exact solutions to the Navier-Stokes equations and by checking conditions whichare defined in Sections 33 and 34
All the models barring one are of the same form That is each of the eddy-viscosity models arewritten
τmodij minus 1
3τmodkk δij = minus2νeSij (50)
with νe representing the eddy viscosity of a specific model For simplicity we usually just writeτmodij = minus2νeSij the extra term is included here to emphasise that the models are traceless Each
22
of the models we consider here can be expressed in terms of the rate of strain tensor S defined inEq (41) the rate of rotation tensor Ω defined in Eq (42) and the combined invariants defined inEq (43) The models are then defined as follows
Smagorinsky [12] νSe = (CSδ)2radic
2I1 (51)
Vreman [16] νVe = (CV δ)2
radicQGGT
PGGT (52)
QR [13 14 15] νQRe = (CQRδ)2 max0minusI3
I1 (53)
Vortex-Stretching [11] νV Se = (CV Sδ)2radic
2I1
(I5 minus 1
2I1I2
minusI1I2
)32
(54)
with the quantities
PGGT = I1 minus I2 QGGT =1
4(I1 + I2)2 + 4
(I5 minus
1
2I1I2
) (55)
There is a fifth and final model to be considered here It is of a different form to the other four andnot an eddy-viscosity model It is known as the Gradient model and is defined by
Gradient [4 9] τmodG = CGδ
(S2 minus Ω2 minus (SΩminus ΩS)
) (56)
The values C are model constants specific to each model and δ represents the mesh size of the LESBoth of these quantities are included here to properly define the models however in this projectthey will be ignored as they are constants and have no influence on the conditions to be defined
33 Conditions for Large-Eddy Simulation without Explicit Filtering
The filtering operation introduced in Eq (48) is just a formality for LES without explicit filteringThe idea here is that filtering does not actually take place but the term lsquoaddedrsquo to the Navier-Stokesequations in fact lsquosubtractsrsquo the difficult to simulate small-scale turbulent motion of the flow Inorder to consider this properly the terms ui and p in Eq (48) are replaced by vi and q It is hopedthat τmod
ij approximates τij so that vi and q approximate ui and p respectively So LES withoutexplicit filtering is given by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (57)
It is clear to see that the first four terms of this equation are exactly the same as the Navier-Stokesequations so for the exact solutions discussed in Section 22 the combination of these terms willequal zero What is then desirable is that the model term is also zero for these exact solutions sothat Eq (57) still holds In order to test models for exact solutions the condition
part
partxjτmodij (v) = 0 (58)
is checked If this condition were to hold it would imply that the model does not influence thevelocity field of a flow hence we call it the Velocity Field Condition There are a number of otherrestrictions that can be tested One such test comes from the differential form of the evolution ofthe kinetic energy of a flow given in Eq (10) In order to find this Eq (57) is multiplied by vi andterms are grouped to find
part
partt
(1
2vivi
)+ vj
part
partxj
(1
2vivi
)= minusvi
1
ρ
partq
partxi+ νvi
part2vipartxjpartxj
minus vipart
partxjτmodij (v) (59)
23
Then similarly to Eq (58) another condition is to check whether the model affects the kineticenergy of exact solutions this is equivalent to checking whether
vipart
partxjτmodij (v) = 0 (60)
We call this condition the Kinetic Energy Condition The third and final condition considered herecan also be found using the integral form for the equation for kinetic energy
d
dt
intV
1
2vividV =
intS
[minusvj
(1
2vivi
)minus vj
q
ρ+ 2νviSij + viτ
modij (v)
]njdS
+
intV
[minus2νSijSij minus
(minusSijτmod
ij (v))]dV
(61)
where the rate-of-strain tensor Sij was defined in Eq (41) V is a volume of fluid and dS meansthe integral is over the surface of that volume with unit normal nj Then this final condition checkswhether the model affects the total kinetic energy of exact solutions to the Navier-Stokes equationsthat is
Sijτmodij (v) = 0 (62)
We will call this condition the Dissipation Condition because Sijτmodij (v) is often referred to as the
subgrid dissipation
34 Conditions for Large-Eddy Simulation with Explicit Filtering
As in Eq (48) the Navier-Stokes equations can be filtered by applying the filtering operation to thewhole equation (putting a bar over all terms) and evaluating to find again that
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (63)
In Eq (57) the filtering operation was ignored and the elements ui p and τij(u) were replacedwith vi q and τmod
ij (v) respectively Now in LES with explicit filtering the filtering operation itselfsmooths over the small-scale turbulent motion of the simulated flow In this case the elements uip and τij(u) will be replaced with vi q and τmod
ij (v) respectively LES with explicit filtering is thengiven by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (64)
Note that Eqs (63) and (64) despite their similarity to Eq (57) no longer contain the Navier-Stokesequations as Eq (57) did The bar (filtering operation) over each term results in each exact solutionfound in Section 22 not necessarily satisfying the first four terms It is hence unknown whether ornot these terms will vanish for exact solutions Terms can still be compared similarly to Eqs (58)(60) and (62) However instead of comparing to zero we can now compare to the relevant termfrom the filtered Navier-Stokes equations So the terms
part
partxjτij (65)
and
part
partxjτmodij (v) (66)
24
are to be compared Although the latter term is relatively simple to compute the former could leadto some difficulties due to the differentiation of the true stress τij = uiuj minus uiuj So although someinformation could be gleaned from the inspection of these two terms more information can be foundby considering the analogue of Eq (62) So the true subfilter dissipation
Sijτij (67)
(where Sij is shorthand for Sij(v)) and the modelled subfilter dissipation
Sij(v)τmodij (v) (68)
are considered and have to be equal for the condition to hold These can again be derived from theintegral equation for kinetic energy of a filtered velocity field similarly to Eq (61)
4 Analysis of Large-Eddy Simulation Models
In this section we use a code in Mathematica the theory described in Section 3 and the exactsolutions to the Navier-Stokes equations found in Section 22 to check whether or not each of theexact solutions lsquoendorsesrsquo each model for LES both without and with explicit filtering
41 Large-Eddy Simulation Without Explicit Filtering
We start by using a code in Mathematica to test the Velocity Field condition in Eq (58) and theKinetic Energy condition in Eq (60) For all exact solutions to the Navier-Stokes equations inSection 22 the outputs for both conditions were identical so we have combined them into one table
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 ndash ndash 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 1 For the Kinetic Energy condition a lsquo0rsquo result means that vipartpartxj
τmodij (v) = 0 for respective
models and exact solutions whereas a lsquolowastrsquo result means that vipartpartxj
τmodij (v) 6= 0 For the Velocity
Field condition a lsquo0rsquo result means that partpartxj
τmodij (v) = 0 for respective models and exact solutions
whereas a lsquolowastrsquo result means that partpartxj
τmodij (v) 6= 0 The Gradient model can take both positive and
negative values so when the sign of the model is known it is included in the table instead of a lsquorsquoBoth the conditions are included in one table because for all cases the outputs were identical
We then test the Dissipation condition in Eq (62) for all exact solutions to the Navier-Stokesequations in Section 22
25
Dissipation Condition [Does Sijτmodij (v) = 0]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 + 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 2 A lsquo0rsquo result means that Sijτmodij (v) = 0 for respective models and exact solutions whereas
a lsquolowastrsquo result means that Sijτmodij (v) 6= 0 The Gradient model can take both positive and negative
values so when the sign of the model is known it is included in the table instead of a lsquorsquo
We then sum all the results to find the total number of endorsements for each model described inSection 32
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
4 6 8 9 9
lsquoDissipationrsquoEndorsements
2 5 7 7 9
Total Endorsements 6 11 15 16 18
Table 3 The results from Tables 1 and 2 are summarised to give a general overview of how manyendorsements each model receives This makes it much clearer to see the models most supported bythe respective and combined conditions with the Vortex-Stretching model being the most supportedand the Smagorinsky model the least
Testing the models in Section 32 using LES without explicit filtering has mixed results It isclear that the Smagorinksy model is not well endorsed by the flows in Section 22 with the onlyendorsements coming from the simplest flows that is Sol 1 (Linear Velocity) Sol 3 (Velocity as afunction of time) Sol 7 (Stagnation Flow) and Sol 15 (Rankine Vortex) The Vortex-Stretching-based model is the most endorsed flow with eighteen out of a possible thirty endorsements Thisagrees with our foreknowledge as this model was constructed to be as physically consistent as possiblein the paper by Silvis amp Verstappen [11]
42 Large-Eddy Simulation With Explicit Filtering
In this section we will compare the true subfilter dissipation Sijτij and the modelled subfilterdissipation Sij(v)τmod
ij (v) given in Eqs (67) and (68) respectively for each model τmodij and each
exact solution to the Navier-Stokes equation given in Section 22 We will not consider the conditionsinvolving differentiation in Eqs (65) and (66) due to the difficulty in calculating the derivative ofthe τij term in Eq (65)
26
421 Example of a Test for a Model
The true subfilter dissipation Sijτij of a flow given in Eq (67) cannot be explicitly computed asthe filtering operation is kept general What can be done however is to partially evaluate theexpressions using the attributes of a filtering operation mentioned in Eq (48)
The calculations of Sijτij for Sol 1 (Linear Velocity) are provided to illustrate the general ideaRecall that the velocity field of Sol 1 (Linear Velocity) is written in Eq (13) in vector notation as
u(x y z t) =
0ax+ bcx+ d
(69)
Then Sij = Sij(u) is computed simply by using its definition in Eq (41) Note that as filteringpreserves constants if there is an a it is replaced with a meaning that (in this case) there are alarge number of zero terms due to the frequency of 0 Hence
[Sij(u)
]=
[1
2
(partuipartxj
+partujpartxi
)]=
0 a
2c2
a2 0 0
c2 0 0
=
0 a
2c2
a2 0 0
c2 0 0
(70)
Note that as Sijτij is being computed which is equal to the sum of the element-wise multiplicationof the two matrices (or the trace of the matrix multiplication) that we donrsquot need to consider theelements of τij(u) with corresponding zero entry in Sij As τij(u) = uiuj minus uiuj and again asfiltering preserves constants it is found that
[τij(u)] =[uiuj minus uiuj
]=
lowast 0 middot ax+ b 0 middot cx+ d
ax+ b middot 0 lowast lowastcx+ d middot 0 lowast lowast
=
lowast 0 00 lowast lowast0 lowast lowast
(71)
where a lsquolowastrsquo just represents some element (not necessarily non-zero) Therefore the true subfilterdissipation
Sijτij = tr
0 a2
c2
a2 0 0c2 0 0
middotlowast 0 0
0 lowast lowast0 lowast lowast
= 0 (72)
Now the modelled subfilter dissipation Sij(u)τmodij (u) needs to be computed (recall that Sij =
Sij(u)) In this case let us use the QR model stated in Eq (53) Once again Sij(u) calculated inEq (70) needs to be used So we have that
τmodij = minus2(CQRδ)
2 max0minusI3I1
Sij (73)
As
I1 = tr(S2) =
1
2(a2 + c2) and I3 = tr(S
3) = 0 (74)
it holds that τmodij (u) = 0 so Sij(u)τmod
ij (u) = 0 This agrees with Eq (72) and hence Sol 1(Linear Velocity) endorses the QR model
27
422 Testing the Models
Now using a Mathematica code the exact solutions in Section 22 are again used to evaluate thecondition for the models using the process just described in Section 421 The results are describedin Table 4
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 4 All entries with a ldquo0rdquo represent a value of zero and a ldquolowastrdquo represents a value that is non-zeroThe ldquoExactrdquo row represents the value of the true subfilter dissipation Sijτij Every other row witha model name represents the value of the modelled subfilter dissipation Sij(u)τmod
ij (u) respectiveto that model A ldquordquo means that it is unclear as to the value - the sign of these values is oftenunknown and the lsquoMaxrsquo function in the definition of the QR model (in Eq (53)) makes it difficultto evaluate
Reading from Table 4 the Smagorinsky model is clearly endorsed by Sol 3 (Velocity as a function oftime) and Sol 15 (Rankine Vortex) as in both cases both Sijτij = 0 = Sij(u)τmod
ij (u) In this casewe say that Sol 3 (Velocity as a function of time) and Sol 15 (Rankine Vortex) lsquostronglyrsquo endorsethe Smagorinsky model The Smagorinsky model is not endorsed by Sol 1 (Linear Velocity) Sol 2(Planar Flow) or Sol 14 (Radially Dependent) however because Sij(u)τmod
ij (u) 6= 0 For Sol 4 to
13 both Sijτij 6= 0 6= Sij(u)τmodij (u) and so we will say that these exact solutions lsquoweaklyrsquo endorse
the Smagorinsky model
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 10 10 9 6
Table 5 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Interestingly the lsquostrongrsquo and lsquoweakrsquo endorsements give different results However more stock shouldbe put in the results of the lsquostrongerrsquo condition as it is completely clear that zero is equal to zeroIt is not necessarily clear whether or not non-zero equals non-zero as the values will depend on theparameters in each solution hence the lsquoweakerrsquo endorsement
The lsquostrongrsquo condition equally endorses the Vreman QR Gradient and Vortex-Stretching-basedmodels whereas the lsquoweakrsquo condition endorses Vreman and Smagorinsky the most closely followedby the Gradient model with the Vortex-Stretching-based model being the most opposed As this
28
completely contradicts the lsquostrongrsquo condition and the results in Section 41 it makes us question thevalidity of the weaker test The number of endorsements for the QR model is unknown because of thedefinition of the model in Eq (53) and our evaluation of the true subfilter dissipation Eq (67) Dueto the exact solutions we have found it is never clear whether or not the true subfilter dissipationis strictly positive or negative This means that when the true subfilter dissipation is found to benon-zero the lsquoMaxrsquo function in the QR model cannot be evaluated
43 Vremanrsquos True Subfilter Dissipation Claim
Using the theory mentioned in Sections 232 and 34 we can place the exact solutions to the Navier-Stokes equations into different classes defined by Vreman In doing this we can also attempt to verifyone of Vremanrsquos results [16] In Vremanrsquos paper he defines the class Qn to be the set of flows withn zero entries in the velocity gradient of the flow (the velocity gradient is defined in Eq (39))Vreman then claims that the only flows with zero true subfilter dissipation are a certain selection ofthose in the classesQ7Q8 orQ9 shown in appendix B and is generally non-zero for any other flows
The true subfilter dissipation of Sol 1 (Linear Velocity) was computed in Section 421 and wasfound to be zero in Eq (72) The velocity gradient of Sol 1 is as follows
GLV =
0 0 0a 0 0c 0 0
(75)
and hence Sol 1 (Linear Velocity) is in the required subset of Q7 which agrees with Vremanrsquos claimIn Section 422 the true subfilter dissipation was computed (determined to be zero or non-zero) forall of the exact solutions to the Navier Stokes equations in Section 22 These solutions can now belsquoplacedrsquo into each class Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a functionof time) and Sol 14 (Radially Dependent) were all found to have zero true subfilter dissipation andare all in the appropriate subsets of either Q7Q8 or Q9 This agrees with Vremanrsquos results
We can extend Vremanrsquos results however We found that Sol 15 (Rankine Vortex) had zero truesubfilter dissipation and yet is not in the subset of Q7 defined in Appendix B It is important tonote that the rate of strain tensor Sij in Eq (41) for Sol 15 (Rankine Vortex) is zero This is thereason the true subfilter dissipation is zero and the same reason that all models are zero for thisflow as all models are based on Sij
We also found that all of the remaining exact solutions to the Navier-Stokes equations are not inVremanrsquos specified classes (in Appendix Section B) They also have a non-zero subfilter dissipationwhich agrees with Vremanrsquos results in his paper [16]
44 Classes of Flows
It is important to consider that some exact solutions to the Navier-Stokes equations may have similarproperties For example Sol 1 (Linear Velocity) and Sol 2 (Planar Flow) are both very similarIn fact Sol 2 (Planar Flow) is a more general case of Sol 1 (Linear Velocity) We therefore mustgroup the different flows into classes in an attempt to reduce the bias that may occur when solutionsare similar This should aid us to further determine which of the models are most endorsed We willdo this in a number of different ways using the theory discussed in Section 23 We will group theflows with respect to Vremanrsquos Flow Classes given in appendix B and with respect to the Invariantsgiven in appendix A
29
441 Vremanrsquos Velocity Gradient Flow Classes
We now put our exact solutions to the Navier-Stokes equations into one of two classes Either Q0minus6or Q7minus9 where each class is defined intuitively as follows
Q0minus6 = Q0 cupQ1 cup middot middot middot cup Q6 (76)
Q7minus9 = Q7 cupQ8 cupQ9 (77)
where Qn is defined in Section 43 By looking at the gradient of each exact solution we can see thatQ7minus9 contains the six solutions Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as afunction of time) Sol 5 (2D Taylor) Sol 14 (Radially Dependent) and Sol 15 (Rankine Vortex)and that Q0minus6 contains the remaining nine We now check the models for LES without explicitfiltering by considering endorsements within each class Tables 6 and 7 are analogues of Table 3 andillustrate the results for each flow class The analogues of Tables 1 and 2 are Tables 22 and 23 andare in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 6 6 6
lsquoDissipationrsquoEndorsements
2 5 6 6 6
Percentage Endorsed 417 833 100 100 100
Table 6 Each entry in the top two rows represents the number of solutions in the class Q7minus9 thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Q0minus6 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 2 3 3
lsquoDissipationrsquoEndorsements
0 0 1 0 3
Percentage Endorsed 56 56 167 167 333
Table 7 Each entry in the top two rows represents the number of solutions in the class Q1minus6 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking results to 1 decimal place if necessary
All of the flows in the class Q7minus9 endorse the QR Gradient and Vortex-Stretching-based subfilter-scale models (see Table 6) We can hence see that this class fully endorses these three models equallyand does not endorse the Smagorinsky model The Vortex-Stretching eddy viscosity model has themost endorsements from any of the flows in the class Q0minus6 and the Smagorinksy and Vreman modelshave the fewest endorsements (see Table 7) We can hence say that this class mostly endorses theVortex-Stretching-based eddy-viscosity model note however that the percentage endorsement is very
30
low
Now let us consider LES with explicit filtering Tables 8 and 9 are analogues of Table 5 and illustratethe results for each flow class The analogue of Table 4 is Table 24 and is in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 1 1 0 0 0
Table 8 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Q0minus6 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 9 9 9 6
Table 9 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We have grouped Vremanrsquos classes into Q7minus9 and Q0minus6 this is equivalent to saying that we havegrouped Vremanrsquos classes into those with two or fewer non-zero velocity gradient elements and thosewith three or more non-zero velocity gradient elements respectively Therefore the solutions in Q7minus9are exactly those with a lsquo0rsquo in the lsquoExactrsquo column of Table 4 and Sol 5 (2D Taylor) ExcludingSol 5 (2D Taylor) these are the same as those five that endorse the Vreman QR Gradient andVortex-Stretching-based models in Table 5 We can see in Table 8 that the results for the lsquostrongrsquoendorsements are exactly the same as in Table 5 This is because all of the flows with zero subfilterdissipation are contained in the class Q7minus9 There are hence no lsquostrongrsquo endorsements in Table 9The majority of the lsquoweakrsquo endorsements come from Q0minus6 as this class contains all but one of thesolutions with non-zero true subfilter dissipation (given in Eq (67))
442 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
We now separate our exact solutions into classes again using Vremanrsquos flow classes This timehowever we define V0 to be the class of flows Vreman claims will always have zero true subfilterdissipation namely those classes mentioned in Appendix B We define Vlowast to be all the remaining
31
classes Then
V0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow)
Sol 3 (Velocity as a function of time) Sol 14 (Radially Dependent)(78)
Vlowast = Sol 4 (Axisymmetric Flow) Sol 5 (2D Taylor) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 13 (Sphere Translation) Sol 15 (Rankine Vortex)
(79)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 10 and 11 are analogues of Table 3 and illustrate the results for each flow class Theanalogues of Tables 1 and 2 are Tables 25 and 26 and are in Appendix C2
Endorsements for models using solutions in V0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 4 4 4 4
lsquoDissipationrsquoEndorsements
1 4 4 4 4
Percentage Endorsed 375 100 100 100 100
Table 10 Each entry in the top two rows represents the number of solutions in the class V0 thatendorse each model There are 4 solutions in this class so the percentages are worked out by dividingthe total of each column by 8 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Vlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 2 4 5 5
lsquoDissipationrsquoEndorsements
1 1 3 2 5
Percentage Endorsed 136 136 318 318 455
Table 11 Each entry in the top two rows represents the number of solutions in the class Vlowast thatendorse each model There are 11 solutions in this class so the percentages are worked out bydividing the total of each column by 22 and taking results to 1 decimal place if necessary
All of the flows in the class V0 endorse the Vreman QR Gradient and Vortex-Stretching-basedsubfilter-scale models (see Table 6) This class completely endorses these four models equally anddoes not endorse the Smagorinsky model The Vortex-Stretching-based eddy-viscosity model hasthe most endorsements from the class Vlowast and the Smagorinksy and Vreman models again have thefewest endorsements (see Table 7) As in Section 441 we say that this class mostly endorses the
32
Vortex-Stretching-based eddy-viscosity model but once again the percentage endorsement is ratherlow
Now let us consider LES with explicit filtering Tables 12 and 13 are analogues of Table 5 andillustrate the results for each flow class The analogue of Table 4 is Table 27 and is in Appendix C2
Endorsements for models using solutions in V0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 4 4 4 4
lsquoWeakrsquo Endorsements 0 0 0 0 0
Table 12 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Vlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 1 1 1 1
lsquoWeakrsquo Endorsements 10 10 9 6
Table 13 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We can see in Table 12 that the results for the lsquostrongrsquo endorsements are similar to those in Table 5excluding Sol 15 (Rankine Vortex) The four solutions in V0 with zero true subfilter dissipationnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)and Sol 14 (Radially Dependent) are hence the four that endorse the Vreman QR Gradientand Vortex-Stretching-based models in Table 12 Sol 15 (Rankine Vortex) is in the class Vlowast andhence there is one lsquostrongrsquo endorsement in Table 13 The class Vlowast contains all of the flows with anon-zero true subfilter dissipation (given in Eq (67)) and hence Table 13 contains all of the lsquoweakrsquoendorsements
443 Classes using the Third Principal Invariant
Sorting the classes using the invariants is a non-trivial task in itself There are a number of waysin which to do this using both the principal invariants in Eq (40) and the combined invariants inEq (43)
We start by looking at flows with third principal invariant RG equal to zero We again define twoclasses R0 and Rlowast where R0 consists of all flows with RG = 0 and Rlowast consists of all flows withRG 6= 0 Then using the information in Appendix A we see that there are nine flows in R0 That
33
is
R0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 8 (Vortex in Stagnation Flow)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)
(80)
Rlowast = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 13 (Sphere Translation)(81)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 14 and 15 are analogues of Table 3 and they illustrate the results for LES withoutexplicit filtering for each flow class The analogues of Tables 1 and 2 are Tables 28 and 29 and arein Appendix C3
Endorsements for models using solutions in R0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 7 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 278 556 778 722 778
Table 14 Each entry in the top two rows represents the number of solutions in the class R0 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking the result to one decimal place if necessary
Endorsements for models using solutions in Rlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 2 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 83 83 83 167 333
Table 15 Each entry in the top two rows represents the number of solutions in the class Rlowast thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking the result to one decimal place if necessary
The class R0 endorses both the Vortex-Stretching-based model and the QR model the most andSmagorinsky the least in the case for LES without explicit filtering In the Rlowast class the Vortex-Stretching-based model is again the most endorsed followed by the Gradient model however thepercentages are again small
34
Now let us consider LES with explicit filtering Tables 16 and 17 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 30 and is in Appendix C3
Endorsements for models using solutions in R0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 4 4 3 2
Table 16 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Rlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 6 6 6 4
Table 17 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 we can see in Table 16 that the results for the lsquostrongrsquo endorsements are exactly thesame as in Table 5 This is because all of the flows with zero subfilter dissipation are contained in theclass R0 As in Table 9 there are no lsquostrongrsquo endorsements in Table 17 The lsquoweakrsquo endorsementshowever are split more evenly between the two classes now than in Vremanrsquos classes in Sections 441and 442 We find that the Smagorinsky and Vreman models still seem to be the most endorsed inboth classes
444 Two-component Flow Classes
Another way in which the flows can be grouped into classes is by determining whether a flow is atwo-component flow or not We define a two-component flow to have the property that the combinedprincipal invariants of a flow in Eq (43) satisfy
I3 = I4 = I5 minus1
2I1I2 = 0 (82)
We are also interested in considering flows with zero gradient model subgrid dissipation These flowshave the property that
I3 minus I4 = 0 (83)
What we find though is that for all the solutions in Section 22 if a flow is a two-component flowthen it also satisfies I3 minus I4 = 0 It may be interesting to note that although I3 minus I4 6= 0 for manysolutions there are a number with I3 equal to a multiple of I4 Although not considered in thisreport this may also be a class of interest We find that the two-component flows are
C2 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)(84)
35
and we call the remaining flows three-component flows They are
C3 = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex) Sol 11 (Pseudo-Motion Shots)
Sol 13 (Sphere Translation)
(85)
Note that Sol 8 (Vortex in Stagnation Flow) is defined to be a planar flow this does not mean that itis a two-component flow in this sense however because its two non-zero velocity components dependon three coordinates We now check the models for LES without explicit filtering by consideringendorsements within each class Tables 18 and 19 are analogues of Table 3 and they illustrate theresults for LES without explicit filtering for each flow class The analogues of Tables 1 and 2 areTables 31 and 32 and are in Appendix C4
Endorsements for models using solutions in C2 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 6 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 357 714 100 857 100
Table 18 Each entry in the top two rows represents the number of solutions in the class C2 thatendorse each model There are 7 solutions in this class so the percentages are worked out by dividingthe total of each column by 14 and taking the result to one decimal place if necessary
Endorsements for models using solutions in C3 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 3 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 63 63 63 188 250
Table 19 Each entry in the top two rows represents the number of solutions in the class C3 thatendorse each model There are 8 solutions in this class so the percentages are worked out by dividingthe total of each column by 16 and taking the result to one decimal place if necessary
The class C2 equally endorses the QR and Vortex-Stretching-based models and once again theSmagorinsky model is the least endorsed As with the previous classes the class C3 has such lowpercentages that it does not seem to endorse any models We start to see a trend appearing here inthat the flow classes that contain the more complicated flows namely Q0minus6 (Eq (76)) Vlowast (Eq (79))Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages The classes that tendto contain the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78)) R0 (Eq (80)) and C2 (Eq (84))all seem to favour the Vortex-Stretching-based model or the QR model
36
Now let us consider LES with explicit filtering Tables 20 and 21 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 33 and is in Appendix C4
Endorsements for models using solutions in C2 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 2 2 0 1 0
Table 20 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in C3 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 8 8 8 6
Table 21 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 and Table 16 we can see in Table 20 that the results for the lsquostrongrsquo endorsements areexactly the same as in Table 5 This is again because all of the flows with zero subfilter dissipationare contained in the class C2 There are two flows with non-zero true subfilter dissipation (givenin Eq (67)) in C2 and this shows in the lsquoweakrsquo endorsements in Table 20 There are no lsquostrongrsquoendorsements in Table 21 as all the zero subfilter dissipation flows are in C2
5 Conclusions
In this report we studied incompressible turbulent flows We found a number of exact solutions to theNavier-Stokes equations each representing a relatively simplistic fluid flow Large-eddy simulation(LES) is used to model turbulent flows by smoothing over the complex small-scale motion Weintroduced a number of subfilter-scale models to approximate this small-scale motion and testedthem using the found exact solutions to the Navier-Stokes equations We first considered LESwithout explicit filtering and used the Navier-Stokes equations to derive conditions for the subfilter-scale models namely the Velocity Field condition (Eq (58)) the Kinetic Energy condition (Eq (60))and the Dissipation condition (Eq (62)) Similarly we considered LES with explicit filtering andagain used the Navier-Stokes equations to derive more conditions for the models in this case wedecided to just take one condition the Dissipation condition (Eqs (67) and (68)) as the others leadto difficulty in their evaluation By lsquorunningrsquo the exact solutions to the Navier-Stokes equationsthrough each condition for the subfilter-scale models we found that the Vortex-Stretching-basededdy-viscosity model in Eq (54) was most endorsed by the tests in LES without explicit filteringWe found that four out of the five subfilter-scale models namely the Vreman QR Gradient andVortex-Stretching-based models were equally endorsed by the lsquostrongrsquo tests in LES with explicitfiltering with the Smagorinsky model being the least endorsed The results we found for the lsquoweakrsquocondition were completely opposite to those found in both the lsquostrongrsquo condition and in LES withoutexplicit filtering This led us to conclude that this test was not one in which we should put our trust
37
We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses
In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes
This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold
38
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
then each component of the velocity can be written as a linear function of x1 x2 and x3 In indexnotation
ui = aijxj + βi (12)
If pressure is assumed to be a function of time t (so that the pressure terms in the Navier-Stokesequations are zero) then conditions can be found on the coefficients to find that the following is asolution
u1(x y z t) = 0
u2(x y z t) = ax+ b
u3(x y z t) = cx+ d
p(x y z t) = f(t)
(13)
where a b c d isin R and f Rrarr R In fact this solution is actually a specific case of Sol 2 (PlanarFlow)
Figure 1 An illustration of the flow in the xy-plane for Sol 1 (Linear Velocity) described inEq (13) with parameter values a = b = c = d = 1
Solution 2 Planar Flow
Using the text by Barbato et al [2 p62] and modifying the solution such that it now takes threedimensions into account another solution is found
u1(x y z t) = Az2 +Bz + C
u2(x y z t) = Dz2 + Ez + F
u3(x y z t) = 0
p(x y z t) =2A
Rex+
2D
Rey + f(t)
(14)
7
where ABCDE F isin R and f R rarr R It is clear to see that if A = D = 0 then this is in factthe same as Eq (13) in Sol 1 In fact we believe (although not shown here) that this is the mostgeneral solution of the form
ui(xi xj xk t) = Ax2k +Bxk + C
uj(xi xj xk t) = Dx2k + Exk + F
uk(xi xj xk t) = 0
(15)
for i 6= j 6= k 6= i
Figure 2 An illustration of the flow in the yz-plane for Sol 2 (Planar Flow) described inEq (14) with parameter values A = B = C =D = E = F = 1
Solution 3 Velocity as a function of time t
If each component of the velocity is taken to be a function of time t and independent of positionthen the incompressibility condition is automatically satisfied A condition can be imposed on thepressure term in order to satisfy the momentum equations Thus a solution is as follows
u1(x y z t) = f(t)
u2(x y z t) = g(t)
u3(x y z t) = h(t)
p(x y z t) = minusxpartf(t)
parttminus y partg(t)
parttminus z parth(t)
partt
(16)
where f g h Rrarr R In fact what is written here is the generalised Galilean transformation of thezero solution [10] We cannot produce a plot for this flow as the functions are completely unspecified
8
Solution 4 Axisymmetric Flow
Using the text by Irmay [7] a solution is defined as follows
u1(x y z t) = kxz
u2(x y z t) = kyz
u3(x y z t) = minuskz2
p(x y z t) = minus2kz
Reminus k2z4
2
(17)
where k isin R This is said to represent an axisymmetric flow with hyperbolic streamlines asymptoticto the z-axis and to the boundary plane z = 0 These asymptotes are visible in Fig 3
Figure 3 An illustration of the flow in theyz-plane for Sol 4 (Axisymmetric Flow)described in Eq (17) with parameter k = 1
Figure 4 An illustration of the flow in thexy-plane for Sol 4 (Axisymmetric Flow)described in Eq (17) with parameter k = 1
Solution 5 2D Taylor Solutions
Again using the text by Barbato et al [2 p63ndash64] a solution is defined using a stream functionψ = ψ(x y t) and by writing
u1 = minuspartψparty
u2 =partψ
partx u3 = 0 (18)
In the text the stream function is not given however it was worked out that
ψ = minuseminusπ2t
Re (cos(πx) + cos(πy))
π (19)
The solution used is of the following form note however that in the text the pressure term differsin that cos(πy) is replaced by sin(πy) (when testing this solution in Mathematica the solution below
9
was instead found)
u1(x y z t) = minus sin(πy) eminus(π2t)Re
u2(x y z t) = sin(πx) eminus(π2t)Re
u3(x y z t) = 0
p(x y z t) = minus cos(πx) cos(πy) eminus(2π2t)Re
(20)
Figure 5 An illustration of the flow in the xy-plane for Sol 5 (2D Taylor) described in Eq (20)with Re = 1 and time t = 0
Solution 6 Generalized Beltrami Flow
Again using the text by Barbato et al [2 p65] and now considering solutions with nonndashtrivialterms in all dimensions a particular form of the generalized Beltrami flow is found namely
u1(x y z t) = (A sin(πz) + C cos(πy)) eminus(π2t)Re
u2(x y z t) = (B sin(πx) +A cos(πz)) eminus(π2t)Re
u3(x y z t) = (C sin(πy) +B cos(πx)) eminus(π2t)Re
p(x y z t) = minus[(AB cos(πz) sin(πx) +BC cos(πx) sin(πy)
+AC cos(πy) sin(πz)] eminus(2π2t)Re
(21)
where ABC isin R
10
Figure 6 An illustration of the flow in the yz-plane for Sol 6 (Generalized Beltrami Flow)described in Eq (21) with parameters A = B =C = 1 and time t = 0
Solution 7 Stagnation Flow
In the book by Drazin and Riley [6 p40] an exact solution is considered which uses a stream functionψ = ψ(y z) They suggest that the following velocity field is a solution
u1(x y z t) =a
kx
u2(x y z t) = minusaky +
k
Re
partψ(y z)
partz
u3(x y z t) = minus k
Re
partψ(y z)
party
(22)
Here we choose the stream function to be ψ(y z) = yz Then a solution is given by
u1(x y z t) =a
kx
u2(x y z t) =
(minusak
+k
Re
)y
u3(x y z t) = minus kzRe
p(x y z t) = minus1
2
((axk
)2minus((
k
Reminus a
k
)y
)2
minus(kz
Re
)2)
(23)
where a is a measure of the strength of the stagnation flow and k is a constant chosen in the textto specify the distance between vortices There are no vortices in this particular solution however
11
Figure 7 An illustration of the flow in the yz-plane for Sol 7 (Stagnation Flow) describedin Eq (23) with parameter values a = Re = 1and k = 2
Figure 8 An illustration of the flow in the xy-plane for Sol 7 (Stagnation Flow) describedin Eq (23) with parameter values a = Re = 1and k = 2
Solution 8 Vortex in Stagnation Flow
As in Sol 7 (Stagnation Flow) we now consider the same solution with a different stream functionnamely ψ(z) = cos z Then given the condition that k = aRe and adding a pressure term Eq (22)becomes
u1(x y z t) =1
Rex
u2(x y z t) = minus 1
Rey minus k sin z
u3(x y z t) = 0
p(x y z t) = minus 1
2Re2(x2 + y2)
(24)
where a and k are as in Sol 7 (Stagnation Flow)
12
Figure 9 An illustration of the flow in the yz-plane for Sol 8 (Vortex in Stagnation Flow)described in Eq (24) with parameter valuesk = 3 and Re = 10
Figure 10 An illustration of the flow in thexy-plane for Sol 8 (Vortex in StagnationFlow) described in Eq (24) with parametervalues k = 3 and Re = 10
Solution 9 Burgers Vortex
The Burgers Vortex solution is given in cylindrical coordinates in the book by Lautrup [8 p 385]as follows
ur(r ϕ z t) = minus 2r
c2Re
uϕ(r ϕ z t) =Ωc2
r
(1minus eminusr
2c2)
uz(r ϕ z t) =4z
c2Re
p(r ϕ z t) = minusc2Ω2F
(r2c2
)2
minus 2
Re2r2 + 4z2
c4
(25)
where
F (ξ) =
infinintξ
(1minus eminusx)2
x2dx (26)
This solution is not particularly nice to look at however most of the unpleasantness comes fromthe pressure term which (in this project) is only required in order to validate the solution For thisproject all solutions are required in Cartesian coordinates the velocity field of the Burgers Vortexhas hence been converted into Cartesian coordinates
u1(x y z t) = minus 2x
c2Reminus Ω
h(x y)(1minus exp (minush(x y))) y
u2(x y z t) = minus 2y
c2Re+
Ω
h(x y)(1minus exp (minush(x y)))x
u3(x y z t) =4z
c2Re
(27)
13
where
h(x y) =x2 + y2
c2 (28)
and Ω and c are constants
Figure 11 An illustration of the flow for Sol9 (Burgers Vortex) described in Eq (27) withparameter values c = 15 Re = 20 and Ω = 10A slice of the xy-plane at z = 0 is shown
Solution 10 Another Vortex Solution
There is another vortex solution more simply formulated than the Burgers Vortex yet with asingularity at r = 0 as a result of the uϕ term It is again given in cylindrical coordinates
ur(r ϕ z t) = minus 2r
c2Re
uϕ(r ϕ z t) =a
r
uz(r ϕ z t) =4z
c2Re
p(r ϕ z t) = minus a2
2r2minus 2r2 + 8z2
c2Re2
(29)
which can also be converted into Cartesian coordinates
14
u1(x y z t) = minus 2x
c2Reminus ay
x2 + y2
u2(x y z t) = minus 2y
c2Re+
ax
x2 + y2
u3(x y z t) =4z
c2Re
(30)
where a and c are constants
Figure 12 An illustration of the flow for Sol9 (Burgers Vortex) described in Eq (27) withparameter values c = 15 Re = 20 and a = 10 Aslice of the xy-plane at z = 0 is shown
Solution 11 Pseudo-Motion Shots of the Second Kind
In the book by Berker [3 p 94] a solution is given in cylindrical coordinates by
ur(r ϕ z t) =2
Re
1
r
uϕ(r ϕ z t) = A1r3 +B1r +
C1
r
uz(r ϕ z t) = A2r2 log(r) +B2r
2 + C2
p(r ϕ z t) =A2
1r6
6+A1B1r
4
2+A1C1r
2 +B2
1r2
2minus C2
1
2r2minus 2
r2Re2+ 2B1C1 log(r)
+2A2z
Reminus 4B1ϕ
Re
(31)
where A1 A2 B1 B2 C1 C2 are constants Eq (31) when converted into Cartesian coordinates is
15
u1(x y z t) =1
Re
2xminus C1Rey
x2 + y2minusA1y
(x2 + y2
)minusB1y
u2(x y z t) =1
Re
2y + C1Rex
x2 + y2+A1x
(x2 + y2
)+B1x
u3(x y z t) =1
2A2
(x2 + y2
)log(x2 + y2
)+B2
(x2 + y2
)+ C2
(32)
Figure 13 An illustration of the flow in thexy-plane for Sol 11 (Pseudo-Motion Shots)described in Eqs (31) and (32) with parametervalues A1 = B1 = C1 = 1 A2 = B2 = C2 = 2and Re = 10
Figure 14 An illustration of the flow in theyz-plane for Sol 11 (Pseudo-Motion Shots)described in Eqs (31) and (32) with parametervalues A1 = B1 = C1 = 1 A2 = B2 = C2 = 2and Re = 10
Solution 12 Flows With Constant Whirl
In the book by Berker [3 p 143] another solution is given in cylindrical coordinates by
ur(r ϕ z t) = minus 1
Re
1
r
uϕ(r ϕ z t) =1
rradicte(minus
12 g(t))
uz(r ϕ z t) = 0
p(r ϕ z t) =1
4
2(minus e
minusg(t)
t minus 1Re2
)r2
minus 1
t2Re
infinintminusg(t)
eminust
tdt
(33)
where we define g(t) =r2Re
2t When converted into Cartesian coordinates we find
16
u1(x y z t) = minus 1
x2 + y2
(yradict
exp
(minus
Re(x2 + y2
)4t
)+
x
Re
)
u2(x y z t) =1
x2 + y2
(xradict
exp
(minus
Re(x2 + y2
)4t
)minus y
Re
)
u3(x y z t) = 0
(34)
Figure 15 An illustration of the flow for Sol12 (Flows with Constant Whirl) described inEqs (33) and (34) with time t = 1 and Re = 1 Aslice of the xy-plane at z = 0 is shown as velocityin the z-direction is zero this is the same for anyvalue of z
17
Solution 13 Rectilinear and Uniform Translation of a Sphere
In the book by Berker [3 p 228] another solution is given in Cartesian coordinates by
u1(x y z t) =3
4
axz
R3
(a2
R2minus 1
)
u2(x y z t) =3
4
ayz
R3
(a2
R2minus 1
)
u3(x y z t) = 1minus 3
4
a
Rminus 1
4
a3
R3minus 3
4
az2
R3
(a2
R2minus 1
)
p(x y z t) =3a(aminusR)(a+R)
32R10Re
((R2Re
(x2 + y2
)minus 2R2Rez2
) (a3 + 3aR2 minus 4R3
)+ 3aRez4
(R2 minus a2
)minus 16R5z
)
(35)
where a and R are constants
Figure 16 An illustration of the flow in the yz-plane for Sol 13 (Sphere Translation) describedin Eq (35) with parameter values a = minus2 andRe = 3
Solution 14 Radially Dependent Flow
A solution with velocity in the z-direction dependent only on the distance from the z-axis is as
18
follows in cylindrical coordinates
ur(r ϕ z t) = 0
uϕ(r ϕ z t) = 0
uz(r ϕ z t) = A(r2 minus a2
)
p(r ϕ z t) =4Az
Re
(36)
where A and a are constants In Cartesian coordinates the solution is just the same with u1(x y z t)and u2(x y z t) both zero and the r2 term replaced with x2 + y2 in u3(x y z t)
Figure 17 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus1 1] withparameter values a = 1 A = 4 and Re = 10
Figure 18 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus2 2] withparameter values a = 1 A = 4 and Re = 10
Solution 15 Rankine Vortex
A solution discovered by Rankine [1] is a vortex flow of the following form
ur(r ϕ z t) = 0
uϕ(r ϕ z t) =ar
2πR2
uz(r ϕ z t) = 0
p(r ϕ z t) =a2r2
8π2R4
(37)
19
where a and R are constants In Cartesian coordinates the flow has the form
u1(x y z t) = minus ay
2πR2
u2(x y z t) =ax
2πR2
u3(x y z t) = 0
p(x y z t) =a2(x2 + y2
)8π2R4
(38)
This flow is actually a rigid body rotational flow in that the angular rotational velocity is uniformand the velocity of the flow increases proportionally with distance from the z-axis
Figure 19 An illustration of the flow in the xy-plane for Sol 15 (Rankine Vortex) described inEq (37) and Eq (38) with parameter values a = 1and R = 1
radic2π
23 Characterisation of Flows
It is important to be able to classify flows in a way that sorts them into distinct groups This aidsin the clear identification and general study of flows It is also a very important way of recognisingflows with similar properties In this subsection we introduce two distinct ways in which flows canbe characterised
231 Characterising Solutions using Invariants
A principal invariant of a tensor is a coefficient of the characteristic polynomial of that tensorInvariants as their name suggests are invariant under rotations of the coordinate system in thattheir values do not change when the tensor is lsquorotatedrsquo into a different set of coordinates Thevelocity gradient of a flow G is defined as the Jacobian of the velocity vector given by
Gij =
[partuipartxj
] (39)
20
for i j = 1 2 3 There are three principal invariants of the velocity gradient of a fluid flow G givenby
PG = tr(G)
QG =1
2
((tr(G))2 minus tr
(G2))
RG = det(G)
(40)
Incompressibility of flows (Eq (5)) ensures that PG = 0 hence the first principal invariant is zerofor all of the flows in Section 22 We now define the rate-of-strain tensor S by
Sij =1
2
(partuipartxj
+partujpartxi
) (41)
and the rate-of-rotation tensor Ω by
Ωij =1
2
(partuipartxjminus partujpartxi
) (42)
Note that the first principal invariant of S and the first and third principal invariants of Ω are zeroAlso note that the velocity gradient of a flow can simply be written G = S + Ω
Rather than considering the principal invariants we can also consider the combined invariants
I1 = tr(S2) I2 = tr(Ω2) I3 = tr(S3) I4 = tr(SΩ2) I5 = tr(S2Ω2) (43)
We can hence write the three principal invariants for a velocity field G in terms of the combinedinvariants
PG = 0
QG = minus1
2(I1 + I2)
RG =1
3(I3 + 3I4)
(44)
In this report the flows considered are the exact solutions to the incompressible Navier-Stokesequations described in Section 22 The combined invariants in Eq (43) are used to characterisethese solutions into different classes in Section 44 Both the principal invariants and combinedinvariants for each exact solution in Section 22 are given in Appendix A
Each of the subfilter-scale eddy-viscosity models described in Section 32 are written in terms ofthese five combined invariants
232 Vremanrsquos Characterisation of Flows
Vreman [16] classifies flows using the gradient of the velocity field of a flow Each lsquoclassrsquo of flows hasthe same number of non-zero elements in the gradient matrix For example
[partuipartxj
]=
0 0 0lowast 0 0lowast 0 0
(45)
where a lowast denotes a non-zero term represents a group of flows that lie within the class denoted byQ7 (see Appendix B) where the subscript represents the number of non-zero terms in the velocity
21
gradient Flows of the form in Eq (45) have a velocity field of the form
u1 = c1 (constant)
u2 = f2(x1)
u3 = f3(x1)
(46)
where f2(x1) and f3(x1) are such that the Navier Stokes equations Eq (1) are satisfied Notethat as the elements on the leading diagonal are zero the incompressibility condition Eq (5) isautomatically satisfied Some simple flow classes are given in Appendix B and contain a number ofthe solutions discussed in Section 22 This is mentioned in greater detail in Section 441
3 Conditions for Large-Eddy Simulation Models
31 Large-Eddy Simulation
Due to the nature of numerical simulation and the complexity of turbulent flows small details aremissed and information is lost when solving the Navier-Stokes equations The computational costof solving a turbulent flow is extremely high so a filtering operation is applied to the Navier-Stokesequations represented by a bar to reduce the computational cost of the simulation This is a linearoperation and preserves constants ie ρ = ρ and is also assumed to commute with differentiation
partuipartxj
=partuipartxj
(47)
The filtered Navier-Stokes equations are found by filtering (putting a bar over) the entire equations(see Eq (1)) and after evaluating are given by
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (48)
where
τij = uiuj minus uiuj (49)
is called the lsquotrue stressrsquo of the flow This however poses a problem as it is not clear what is meantby the term uiuj for a general filtering operation The true stress τij therefore must be replaced bya model τmod
ij in practical applications
32 Subfilter-Scale Models
We will now define the subflter-scale models denoted by τmodij to be used in place of the lsquodifficultrsquo
true stress τij There are a number of models available to use in the literature [11 p7] These willbe tested using the exact solutions to the Navier-Stokes equations and by checking conditions whichare defined in Sections 33 and 34
All the models barring one are of the same form That is each of the eddy-viscosity models arewritten
τmodij minus 1
3τmodkk δij = minus2νeSij (50)
with νe representing the eddy viscosity of a specific model For simplicity we usually just writeτmodij = minus2νeSij the extra term is included here to emphasise that the models are traceless Each
22
of the models we consider here can be expressed in terms of the rate of strain tensor S defined inEq (41) the rate of rotation tensor Ω defined in Eq (42) and the combined invariants defined inEq (43) The models are then defined as follows
Smagorinsky [12] νSe = (CSδ)2radic
2I1 (51)
Vreman [16] νVe = (CV δ)2
radicQGGT
PGGT (52)
QR [13 14 15] νQRe = (CQRδ)2 max0minusI3
I1 (53)
Vortex-Stretching [11] νV Se = (CV Sδ)2radic
2I1
(I5 minus 1
2I1I2
minusI1I2
)32
(54)
with the quantities
PGGT = I1 minus I2 QGGT =1
4(I1 + I2)2 + 4
(I5 minus
1
2I1I2
) (55)
There is a fifth and final model to be considered here It is of a different form to the other four andnot an eddy-viscosity model It is known as the Gradient model and is defined by
Gradient [4 9] τmodG = CGδ
(S2 minus Ω2 minus (SΩminus ΩS)
) (56)
The values C are model constants specific to each model and δ represents the mesh size of the LESBoth of these quantities are included here to properly define the models however in this projectthey will be ignored as they are constants and have no influence on the conditions to be defined
33 Conditions for Large-Eddy Simulation without Explicit Filtering
The filtering operation introduced in Eq (48) is just a formality for LES without explicit filteringThe idea here is that filtering does not actually take place but the term lsquoaddedrsquo to the Navier-Stokesequations in fact lsquosubtractsrsquo the difficult to simulate small-scale turbulent motion of the flow Inorder to consider this properly the terms ui and p in Eq (48) are replaced by vi and q It is hopedthat τmod
ij approximates τij so that vi and q approximate ui and p respectively So LES withoutexplicit filtering is given by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (57)
It is clear to see that the first four terms of this equation are exactly the same as the Navier-Stokesequations so for the exact solutions discussed in Section 22 the combination of these terms willequal zero What is then desirable is that the model term is also zero for these exact solutions sothat Eq (57) still holds In order to test models for exact solutions the condition
part
partxjτmodij (v) = 0 (58)
is checked If this condition were to hold it would imply that the model does not influence thevelocity field of a flow hence we call it the Velocity Field Condition There are a number of otherrestrictions that can be tested One such test comes from the differential form of the evolution ofthe kinetic energy of a flow given in Eq (10) In order to find this Eq (57) is multiplied by vi andterms are grouped to find
part
partt
(1
2vivi
)+ vj
part
partxj
(1
2vivi
)= minusvi
1
ρ
partq
partxi+ νvi
part2vipartxjpartxj
minus vipart
partxjτmodij (v) (59)
23
Then similarly to Eq (58) another condition is to check whether the model affects the kineticenergy of exact solutions this is equivalent to checking whether
vipart
partxjτmodij (v) = 0 (60)
We call this condition the Kinetic Energy Condition The third and final condition considered herecan also be found using the integral form for the equation for kinetic energy
d
dt
intV
1
2vividV =
intS
[minusvj
(1
2vivi
)minus vj
q
ρ+ 2νviSij + viτ
modij (v)
]njdS
+
intV
[minus2νSijSij minus
(minusSijτmod
ij (v))]dV
(61)
where the rate-of-strain tensor Sij was defined in Eq (41) V is a volume of fluid and dS meansthe integral is over the surface of that volume with unit normal nj Then this final condition checkswhether the model affects the total kinetic energy of exact solutions to the Navier-Stokes equationsthat is
Sijτmodij (v) = 0 (62)
We will call this condition the Dissipation Condition because Sijτmodij (v) is often referred to as the
subgrid dissipation
34 Conditions for Large-Eddy Simulation with Explicit Filtering
As in Eq (48) the Navier-Stokes equations can be filtered by applying the filtering operation to thewhole equation (putting a bar over all terms) and evaluating to find again that
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (63)
In Eq (57) the filtering operation was ignored and the elements ui p and τij(u) were replacedwith vi q and τmod
ij (v) respectively Now in LES with explicit filtering the filtering operation itselfsmooths over the small-scale turbulent motion of the simulated flow In this case the elements uip and τij(u) will be replaced with vi q and τmod
ij (v) respectively LES with explicit filtering is thengiven by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (64)
Note that Eqs (63) and (64) despite their similarity to Eq (57) no longer contain the Navier-Stokesequations as Eq (57) did The bar (filtering operation) over each term results in each exact solutionfound in Section 22 not necessarily satisfying the first four terms It is hence unknown whether ornot these terms will vanish for exact solutions Terms can still be compared similarly to Eqs (58)(60) and (62) However instead of comparing to zero we can now compare to the relevant termfrom the filtered Navier-Stokes equations So the terms
part
partxjτij (65)
and
part
partxjτmodij (v) (66)
24
are to be compared Although the latter term is relatively simple to compute the former could leadto some difficulties due to the differentiation of the true stress τij = uiuj minus uiuj So although someinformation could be gleaned from the inspection of these two terms more information can be foundby considering the analogue of Eq (62) So the true subfilter dissipation
Sijτij (67)
(where Sij is shorthand for Sij(v)) and the modelled subfilter dissipation
Sij(v)τmodij (v) (68)
are considered and have to be equal for the condition to hold These can again be derived from theintegral equation for kinetic energy of a filtered velocity field similarly to Eq (61)
4 Analysis of Large-Eddy Simulation Models
In this section we use a code in Mathematica the theory described in Section 3 and the exactsolutions to the Navier-Stokes equations found in Section 22 to check whether or not each of theexact solutions lsquoendorsesrsquo each model for LES both without and with explicit filtering
41 Large-Eddy Simulation Without Explicit Filtering
We start by using a code in Mathematica to test the Velocity Field condition in Eq (58) and theKinetic Energy condition in Eq (60) For all exact solutions to the Navier-Stokes equations inSection 22 the outputs for both conditions were identical so we have combined them into one table
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 ndash ndash 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 1 For the Kinetic Energy condition a lsquo0rsquo result means that vipartpartxj
τmodij (v) = 0 for respective
models and exact solutions whereas a lsquolowastrsquo result means that vipartpartxj
τmodij (v) 6= 0 For the Velocity
Field condition a lsquo0rsquo result means that partpartxj
τmodij (v) = 0 for respective models and exact solutions
whereas a lsquolowastrsquo result means that partpartxj
τmodij (v) 6= 0 The Gradient model can take both positive and
negative values so when the sign of the model is known it is included in the table instead of a lsquorsquoBoth the conditions are included in one table because for all cases the outputs were identical
We then test the Dissipation condition in Eq (62) for all exact solutions to the Navier-Stokesequations in Section 22
25
Dissipation Condition [Does Sijτmodij (v) = 0]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 + 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 2 A lsquo0rsquo result means that Sijτmodij (v) = 0 for respective models and exact solutions whereas
a lsquolowastrsquo result means that Sijτmodij (v) 6= 0 The Gradient model can take both positive and negative
values so when the sign of the model is known it is included in the table instead of a lsquorsquo
We then sum all the results to find the total number of endorsements for each model described inSection 32
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
4 6 8 9 9
lsquoDissipationrsquoEndorsements
2 5 7 7 9
Total Endorsements 6 11 15 16 18
Table 3 The results from Tables 1 and 2 are summarised to give a general overview of how manyendorsements each model receives This makes it much clearer to see the models most supported bythe respective and combined conditions with the Vortex-Stretching model being the most supportedand the Smagorinsky model the least
Testing the models in Section 32 using LES without explicit filtering has mixed results It isclear that the Smagorinksy model is not well endorsed by the flows in Section 22 with the onlyendorsements coming from the simplest flows that is Sol 1 (Linear Velocity) Sol 3 (Velocity as afunction of time) Sol 7 (Stagnation Flow) and Sol 15 (Rankine Vortex) The Vortex-Stretching-based model is the most endorsed flow with eighteen out of a possible thirty endorsements Thisagrees with our foreknowledge as this model was constructed to be as physically consistent as possiblein the paper by Silvis amp Verstappen [11]
42 Large-Eddy Simulation With Explicit Filtering
In this section we will compare the true subfilter dissipation Sijτij and the modelled subfilterdissipation Sij(v)τmod
ij (v) given in Eqs (67) and (68) respectively for each model τmodij and each
exact solution to the Navier-Stokes equation given in Section 22 We will not consider the conditionsinvolving differentiation in Eqs (65) and (66) due to the difficulty in calculating the derivative ofthe τij term in Eq (65)
26
421 Example of a Test for a Model
The true subfilter dissipation Sijτij of a flow given in Eq (67) cannot be explicitly computed asthe filtering operation is kept general What can be done however is to partially evaluate theexpressions using the attributes of a filtering operation mentioned in Eq (48)
The calculations of Sijτij for Sol 1 (Linear Velocity) are provided to illustrate the general ideaRecall that the velocity field of Sol 1 (Linear Velocity) is written in Eq (13) in vector notation as
u(x y z t) =
0ax+ bcx+ d
(69)
Then Sij = Sij(u) is computed simply by using its definition in Eq (41) Note that as filteringpreserves constants if there is an a it is replaced with a meaning that (in this case) there are alarge number of zero terms due to the frequency of 0 Hence
[Sij(u)
]=
[1
2
(partuipartxj
+partujpartxi
)]=
0 a
2c2
a2 0 0
c2 0 0
=
0 a
2c2
a2 0 0
c2 0 0
(70)
Note that as Sijτij is being computed which is equal to the sum of the element-wise multiplicationof the two matrices (or the trace of the matrix multiplication) that we donrsquot need to consider theelements of τij(u) with corresponding zero entry in Sij As τij(u) = uiuj minus uiuj and again asfiltering preserves constants it is found that
[τij(u)] =[uiuj minus uiuj
]=
lowast 0 middot ax+ b 0 middot cx+ d
ax+ b middot 0 lowast lowastcx+ d middot 0 lowast lowast
=
lowast 0 00 lowast lowast0 lowast lowast
(71)
where a lsquolowastrsquo just represents some element (not necessarily non-zero) Therefore the true subfilterdissipation
Sijτij = tr
0 a2
c2
a2 0 0c2 0 0
middotlowast 0 0
0 lowast lowast0 lowast lowast
= 0 (72)
Now the modelled subfilter dissipation Sij(u)τmodij (u) needs to be computed (recall that Sij =
Sij(u)) In this case let us use the QR model stated in Eq (53) Once again Sij(u) calculated inEq (70) needs to be used So we have that
τmodij = minus2(CQRδ)
2 max0minusI3I1
Sij (73)
As
I1 = tr(S2) =
1
2(a2 + c2) and I3 = tr(S
3) = 0 (74)
it holds that τmodij (u) = 0 so Sij(u)τmod
ij (u) = 0 This agrees with Eq (72) and hence Sol 1(Linear Velocity) endorses the QR model
27
422 Testing the Models
Now using a Mathematica code the exact solutions in Section 22 are again used to evaluate thecondition for the models using the process just described in Section 421 The results are describedin Table 4
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 4 All entries with a ldquo0rdquo represent a value of zero and a ldquolowastrdquo represents a value that is non-zeroThe ldquoExactrdquo row represents the value of the true subfilter dissipation Sijτij Every other row witha model name represents the value of the modelled subfilter dissipation Sij(u)τmod
ij (u) respectiveto that model A ldquordquo means that it is unclear as to the value - the sign of these values is oftenunknown and the lsquoMaxrsquo function in the definition of the QR model (in Eq (53)) makes it difficultto evaluate
Reading from Table 4 the Smagorinsky model is clearly endorsed by Sol 3 (Velocity as a function oftime) and Sol 15 (Rankine Vortex) as in both cases both Sijτij = 0 = Sij(u)τmod
ij (u) In this casewe say that Sol 3 (Velocity as a function of time) and Sol 15 (Rankine Vortex) lsquostronglyrsquo endorsethe Smagorinsky model The Smagorinsky model is not endorsed by Sol 1 (Linear Velocity) Sol 2(Planar Flow) or Sol 14 (Radially Dependent) however because Sij(u)τmod
ij (u) 6= 0 For Sol 4 to
13 both Sijτij 6= 0 6= Sij(u)τmodij (u) and so we will say that these exact solutions lsquoweaklyrsquo endorse
the Smagorinsky model
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 10 10 9 6
Table 5 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Interestingly the lsquostrongrsquo and lsquoweakrsquo endorsements give different results However more stock shouldbe put in the results of the lsquostrongerrsquo condition as it is completely clear that zero is equal to zeroIt is not necessarily clear whether or not non-zero equals non-zero as the values will depend on theparameters in each solution hence the lsquoweakerrsquo endorsement
The lsquostrongrsquo condition equally endorses the Vreman QR Gradient and Vortex-Stretching-basedmodels whereas the lsquoweakrsquo condition endorses Vreman and Smagorinsky the most closely followedby the Gradient model with the Vortex-Stretching-based model being the most opposed As this
28
completely contradicts the lsquostrongrsquo condition and the results in Section 41 it makes us question thevalidity of the weaker test The number of endorsements for the QR model is unknown because of thedefinition of the model in Eq (53) and our evaluation of the true subfilter dissipation Eq (67) Dueto the exact solutions we have found it is never clear whether or not the true subfilter dissipationis strictly positive or negative This means that when the true subfilter dissipation is found to benon-zero the lsquoMaxrsquo function in the QR model cannot be evaluated
43 Vremanrsquos True Subfilter Dissipation Claim
Using the theory mentioned in Sections 232 and 34 we can place the exact solutions to the Navier-Stokes equations into different classes defined by Vreman In doing this we can also attempt to verifyone of Vremanrsquos results [16] In Vremanrsquos paper he defines the class Qn to be the set of flows withn zero entries in the velocity gradient of the flow (the velocity gradient is defined in Eq (39))Vreman then claims that the only flows with zero true subfilter dissipation are a certain selection ofthose in the classesQ7Q8 orQ9 shown in appendix B and is generally non-zero for any other flows
The true subfilter dissipation of Sol 1 (Linear Velocity) was computed in Section 421 and wasfound to be zero in Eq (72) The velocity gradient of Sol 1 is as follows
GLV =
0 0 0a 0 0c 0 0
(75)
and hence Sol 1 (Linear Velocity) is in the required subset of Q7 which agrees with Vremanrsquos claimIn Section 422 the true subfilter dissipation was computed (determined to be zero or non-zero) forall of the exact solutions to the Navier Stokes equations in Section 22 These solutions can now belsquoplacedrsquo into each class Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a functionof time) and Sol 14 (Radially Dependent) were all found to have zero true subfilter dissipation andare all in the appropriate subsets of either Q7Q8 or Q9 This agrees with Vremanrsquos results
We can extend Vremanrsquos results however We found that Sol 15 (Rankine Vortex) had zero truesubfilter dissipation and yet is not in the subset of Q7 defined in Appendix B It is important tonote that the rate of strain tensor Sij in Eq (41) for Sol 15 (Rankine Vortex) is zero This is thereason the true subfilter dissipation is zero and the same reason that all models are zero for thisflow as all models are based on Sij
We also found that all of the remaining exact solutions to the Navier-Stokes equations are not inVremanrsquos specified classes (in Appendix Section B) They also have a non-zero subfilter dissipationwhich agrees with Vremanrsquos results in his paper [16]
44 Classes of Flows
It is important to consider that some exact solutions to the Navier-Stokes equations may have similarproperties For example Sol 1 (Linear Velocity) and Sol 2 (Planar Flow) are both very similarIn fact Sol 2 (Planar Flow) is a more general case of Sol 1 (Linear Velocity) We therefore mustgroup the different flows into classes in an attempt to reduce the bias that may occur when solutionsare similar This should aid us to further determine which of the models are most endorsed We willdo this in a number of different ways using the theory discussed in Section 23 We will group theflows with respect to Vremanrsquos Flow Classes given in appendix B and with respect to the Invariantsgiven in appendix A
29
441 Vremanrsquos Velocity Gradient Flow Classes
We now put our exact solutions to the Navier-Stokes equations into one of two classes Either Q0minus6or Q7minus9 where each class is defined intuitively as follows
Q0minus6 = Q0 cupQ1 cup middot middot middot cup Q6 (76)
Q7minus9 = Q7 cupQ8 cupQ9 (77)
where Qn is defined in Section 43 By looking at the gradient of each exact solution we can see thatQ7minus9 contains the six solutions Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as afunction of time) Sol 5 (2D Taylor) Sol 14 (Radially Dependent) and Sol 15 (Rankine Vortex)and that Q0minus6 contains the remaining nine We now check the models for LES without explicitfiltering by considering endorsements within each class Tables 6 and 7 are analogues of Table 3 andillustrate the results for each flow class The analogues of Tables 1 and 2 are Tables 22 and 23 andare in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 6 6 6
lsquoDissipationrsquoEndorsements
2 5 6 6 6
Percentage Endorsed 417 833 100 100 100
Table 6 Each entry in the top two rows represents the number of solutions in the class Q7minus9 thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Q0minus6 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 2 3 3
lsquoDissipationrsquoEndorsements
0 0 1 0 3
Percentage Endorsed 56 56 167 167 333
Table 7 Each entry in the top two rows represents the number of solutions in the class Q1minus6 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking results to 1 decimal place if necessary
All of the flows in the class Q7minus9 endorse the QR Gradient and Vortex-Stretching-based subfilter-scale models (see Table 6) We can hence see that this class fully endorses these three models equallyand does not endorse the Smagorinsky model The Vortex-Stretching eddy viscosity model has themost endorsements from any of the flows in the class Q0minus6 and the Smagorinksy and Vreman modelshave the fewest endorsements (see Table 7) We can hence say that this class mostly endorses theVortex-Stretching-based eddy-viscosity model note however that the percentage endorsement is very
30
low
Now let us consider LES with explicit filtering Tables 8 and 9 are analogues of Table 5 and illustratethe results for each flow class The analogue of Table 4 is Table 24 and is in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 1 1 0 0 0
Table 8 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Q0minus6 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 9 9 9 6
Table 9 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We have grouped Vremanrsquos classes into Q7minus9 and Q0minus6 this is equivalent to saying that we havegrouped Vremanrsquos classes into those with two or fewer non-zero velocity gradient elements and thosewith three or more non-zero velocity gradient elements respectively Therefore the solutions in Q7minus9are exactly those with a lsquo0rsquo in the lsquoExactrsquo column of Table 4 and Sol 5 (2D Taylor) ExcludingSol 5 (2D Taylor) these are the same as those five that endorse the Vreman QR Gradient andVortex-Stretching-based models in Table 5 We can see in Table 8 that the results for the lsquostrongrsquoendorsements are exactly the same as in Table 5 This is because all of the flows with zero subfilterdissipation are contained in the class Q7minus9 There are hence no lsquostrongrsquo endorsements in Table 9The majority of the lsquoweakrsquo endorsements come from Q0minus6 as this class contains all but one of thesolutions with non-zero true subfilter dissipation (given in Eq (67))
442 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
We now separate our exact solutions into classes again using Vremanrsquos flow classes This timehowever we define V0 to be the class of flows Vreman claims will always have zero true subfilterdissipation namely those classes mentioned in Appendix B We define Vlowast to be all the remaining
31
classes Then
V0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow)
Sol 3 (Velocity as a function of time) Sol 14 (Radially Dependent)(78)
Vlowast = Sol 4 (Axisymmetric Flow) Sol 5 (2D Taylor) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 13 (Sphere Translation) Sol 15 (Rankine Vortex)
(79)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 10 and 11 are analogues of Table 3 and illustrate the results for each flow class Theanalogues of Tables 1 and 2 are Tables 25 and 26 and are in Appendix C2
Endorsements for models using solutions in V0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 4 4 4 4
lsquoDissipationrsquoEndorsements
1 4 4 4 4
Percentage Endorsed 375 100 100 100 100
Table 10 Each entry in the top two rows represents the number of solutions in the class V0 thatendorse each model There are 4 solutions in this class so the percentages are worked out by dividingthe total of each column by 8 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Vlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 2 4 5 5
lsquoDissipationrsquoEndorsements
1 1 3 2 5
Percentage Endorsed 136 136 318 318 455
Table 11 Each entry in the top two rows represents the number of solutions in the class Vlowast thatendorse each model There are 11 solutions in this class so the percentages are worked out bydividing the total of each column by 22 and taking results to 1 decimal place if necessary
All of the flows in the class V0 endorse the Vreman QR Gradient and Vortex-Stretching-basedsubfilter-scale models (see Table 6) This class completely endorses these four models equally anddoes not endorse the Smagorinsky model The Vortex-Stretching-based eddy-viscosity model hasthe most endorsements from the class Vlowast and the Smagorinksy and Vreman models again have thefewest endorsements (see Table 7) As in Section 441 we say that this class mostly endorses the
32
Vortex-Stretching-based eddy-viscosity model but once again the percentage endorsement is ratherlow
Now let us consider LES with explicit filtering Tables 12 and 13 are analogues of Table 5 andillustrate the results for each flow class The analogue of Table 4 is Table 27 and is in Appendix C2
Endorsements for models using solutions in V0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 4 4 4 4
lsquoWeakrsquo Endorsements 0 0 0 0 0
Table 12 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Vlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 1 1 1 1
lsquoWeakrsquo Endorsements 10 10 9 6
Table 13 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We can see in Table 12 that the results for the lsquostrongrsquo endorsements are similar to those in Table 5excluding Sol 15 (Rankine Vortex) The four solutions in V0 with zero true subfilter dissipationnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)and Sol 14 (Radially Dependent) are hence the four that endorse the Vreman QR Gradientand Vortex-Stretching-based models in Table 12 Sol 15 (Rankine Vortex) is in the class Vlowast andhence there is one lsquostrongrsquo endorsement in Table 13 The class Vlowast contains all of the flows with anon-zero true subfilter dissipation (given in Eq (67)) and hence Table 13 contains all of the lsquoweakrsquoendorsements
443 Classes using the Third Principal Invariant
Sorting the classes using the invariants is a non-trivial task in itself There are a number of waysin which to do this using both the principal invariants in Eq (40) and the combined invariants inEq (43)
We start by looking at flows with third principal invariant RG equal to zero We again define twoclasses R0 and Rlowast where R0 consists of all flows with RG = 0 and Rlowast consists of all flows withRG 6= 0 Then using the information in Appendix A we see that there are nine flows in R0 That
33
is
R0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 8 (Vortex in Stagnation Flow)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)
(80)
Rlowast = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 13 (Sphere Translation)(81)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 14 and 15 are analogues of Table 3 and they illustrate the results for LES withoutexplicit filtering for each flow class The analogues of Tables 1 and 2 are Tables 28 and 29 and arein Appendix C3
Endorsements for models using solutions in R0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 7 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 278 556 778 722 778
Table 14 Each entry in the top two rows represents the number of solutions in the class R0 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking the result to one decimal place if necessary
Endorsements for models using solutions in Rlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 2 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 83 83 83 167 333
Table 15 Each entry in the top two rows represents the number of solutions in the class Rlowast thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking the result to one decimal place if necessary
The class R0 endorses both the Vortex-Stretching-based model and the QR model the most andSmagorinsky the least in the case for LES without explicit filtering In the Rlowast class the Vortex-Stretching-based model is again the most endorsed followed by the Gradient model however thepercentages are again small
34
Now let us consider LES with explicit filtering Tables 16 and 17 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 30 and is in Appendix C3
Endorsements for models using solutions in R0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 4 4 3 2
Table 16 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Rlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 6 6 6 4
Table 17 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 we can see in Table 16 that the results for the lsquostrongrsquo endorsements are exactly thesame as in Table 5 This is because all of the flows with zero subfilter dissipation are contained in theclass R0 As in Table 9 there are no lsquostrongrsquo endorsements in Table 17 The lsquoweakrsquo endorsementshowever are split more evenly between the two classes now than in Vremanrsquos classes in Sections 441and 442 We find that the Smagorinsky and Vreman models still seem to be the most endorsed inboth classes
444 Two-component Flow Classes
Another way in which the flows can be grouped into classes is by determining whether a flow is atwo-component flow or not We define a two-component flow to have the property that the combinedprincipal invariants of a flow in Eq (43) satisfy
I3 = I4 = I5 minus1
2I1I2 = 0 (82)
We are also interested in considering flows with zero gradient model subgrid dissipation These flowshave the property that
I3 minus I4 = 0 (83)
What we find though is that for all the solutions in Section 22 if a flow is a two-component flowthen it also satisfies I3 minus I4 = 0 It may be interesting to note that although I3 minus I4 6= 0 for manysolutions there are a number with I3 equal to a multiple of I4 Although not considered in thisreport this may also be a class of interest We find that the two-component flows are
C2 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)(84)
35
and we call the remaining flows three-component flows They are
C3 = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex) Sol 11 (Pseudo-Motion Shots)
Sol 13 (Sphere Translation)
(85)
Note that Sol 8 (Vortex in Stagnation Flow) is defined to be a planar flow this does not mean that itis a two-component flow in this sense however because its two non-zero velocity components dependon three coordinates We now check the models for LES without explicit filtering by consideringendorsements within each class Tables 18 and 19 are analogues of Table 3 and they illustrate theresults for LES without explicit filtering for each flow class The analogues of Tables 1 and 2 areTables 31 and 32 and are in Appendix C4
Endorsements for models using solutions in C2 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 6 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 357 714 100 857 100
Table 18 Each entry in the top two rows represents the number of solutions in the class C2 thatendorse each model There are 7 solutions in this class so the percentages are worked out by dividingthe total of each column by 14 and taking the result to one decimal place if necessary
Endorsements for models using solutions in C3 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 3 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 63 63 63 188 250
Table 19 Each entry in the top two rows represents the number of solutions in the class C3 thatendorse each model There are 8 solutions in this class so the percentages are worked out by dividingthe total of each column by 16 and taking the result to one decimal place if necessary
The class C2 equally endorses the QR and Vortex-Stretching-based models and once again theSmagorinsky model is the least endorsed As with the previous classes the class C3 has such lowpercentages that it does not seem to endorse any models We start to see a trend appearing here inthat the flow classes that contain the more complicated flows namely Q0minus6 (Eq (76)) Vlowast (Eq (79))Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages The classes that tendto contain the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78)) R0 (Eq (80)) and C2 (Eq (84))all seem to favour the Vortex-Stretching-based model or the QR model
36
Now let us consider LES with explicit filtering Tables 20 and 21 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 33 and is in Appendix C4
Endorsements for models using solutions in C2 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 2 2 0 1 0
Table 20 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in C3 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 8 8 8 6
Table 21 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 and Table 16 we can see in Table 20 that the results for the lsquostrongrsquo endorsements areexactly the same as in Table 5 This is again because all of the flows with zero subfilter dissipationare contained in the class C2 There are two flows with non-zero true subfilter dissipation (givenin Eq (67)) in C2 and this shows in the lsquoweakrsquo endorsements in Table 20 There are no lsquostrongrsquoendorsements in Table 21 as all the zero subfilter dissipation flows are in C2
5 Conclusions
In this report we studied incompressible turbulent flows We found a number of exact solutions to theNavier-Stokes equations each representing a relatively simplistic fluid flow Large-eddy simulation(LES) is used to model turbulent flows by smoothing over the complex small-scale motion Weintroduced a number of subfilter-scale models to approximate this small-scale motion and testedthem using the found exact solutions to the Navier-Stokes equations We first considered LESwithout explicit filtering and used the Navier-Stokes equations to derive conditions for the subfilter-scale models namely the Velocity Field condition (Eq (58)) the Kinetic Energy condition (Eq (60))and the Dissipation condition (Eq (62)) Similarly we considered LES with explicit filtering andagain used the Navier-Stokes equations to derive more conditions for the models in this case wedecided to just take one condition the Dissipation condition (Eqs (67) and (68)) as the others leadto difficulty in their evaluation By lsquorunningrsquo the exact solutions to the Navier-Stokes equationsthrough each condition for the subfilter-scale models we found that the Vortex-Stretching-basededdy-viscosity model in Eq (54) was most endorsed by the tests in LES without explicit filteringWe found that four out of the five subfilter-scale models namely the Vreman QR Gradient andVortex-Stretching-based models were equally endorsed by the lsquostrongrsquo tests in LES with explicitfiltering with the Smagorinsky model being the least endorsed The results we found for the lsquoweakrsquocondition were completely opposite to those found in both the lsquostrongrsquo condition and in LES withoutexplicit filtering This led us to conclude that this test was not one in which we should put our trust
37
We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses
In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes
This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold
38
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
where ABCDE F isin R and f R rarr R It is clear to see that if A = D = 0 then this is in factthe same as Eq (13) in Sol 1 In fact we believe (although not shown here) that this is the mostgeneral solution of the form
ui(xi xj xk t) = Ax2k +Bxk + C
uj(xi xj xk t) = Dx2k + Exk + F
uk(xi xj xk t) = 0
(15)
for i 6= j 6= k 6= i
Figure 2 An illustration of the flow in the yz-plane for Sol 2 (Planar Flow) described inEq (14) with parameter values A = B = C =D = E = F = 1
Solution 3 Velocity as a function of time t
If each component of the velocity is taken to be a function of time t and independent of positionthen the incompressibility condition is automatically satisfied A condition can be imposed on thepressure term in order to satisfy the momentum equations Thus a solution is as follows
u1(x y z t) = f(t)
u2(x y z t) = g(t)
u3(x y z t) = h(t)
p(x y z t) = minusxpartf(t)
parttminus y partg(t)
parttminus z parth(t)
partt
(16)
where f g h Rrarr R In fact what is written here is the generalised Galilean transformation of thezero solution [10] We cannot produce a plot for this flow as the functions are completely unspecified
8
Solution 4 Axisymmetric Flow
Using the text by Irmay [7] a solution is defined as follows
u1(x y z t) = kxz
u2(x y z t) = kyz
u3(x y z t) = minuskz2
p(x y z t) = minus2kz
Reminus k2z4
2
(17)
where k isin R This is said to represent an axisymmetric flow with hyperbolic streamlines asymptoticto the z-axis and to the boundary plane z = 0 These asymptotes are visible in Fig 3
Figure 3 An illustration of the flow in theyz-plane for Sol 4 (Axisymmetric Flow)described in Eq (17) with parameter k = 1
Figure 4 An illustration of the flow in thexy-plane for Sol 4 (Axisymmetric Flow)described in Eq (17) with parameter k = 1
Solution 5 2D Taylor Solutions
Again using the text by Barbato et al [2 p63ndash64] a solution is defined using a stream functionψ = ψ(x y t) and by writing
u1 = minuspartψparty
u2 =partψ
partx u3 = 0 (18)
In the text the stream function is not given however it was worked out that
ψ = minuseminusπ2t
Re (cos(πx) + cos(πy))
π (19)
The solution used is of the following form note however that in the text the pressure term differsin that cos(πy) is replaced by sin(πy) (when testing this solution in Mathematica the solution below
9
was instead found)
u1(x y z t) = minus sin(πy) eminus(π2t)Re
u2(x y z t) = sin(πx) eminus(π2t)Re
u3(x y z t) = 0
p(x y z t) = minus cos(πx) cos(πy) eminus(2π2t)Re
(20)
Figure 5 An illustration of the flow in the xy-plane for Sol 5 (2D Taylor) described in Eq (20)with Re = 1 and time t = 0
Solution 6 Generalized Beltrami Flow
Again using the text by Barbato et al [2 p65] and now considering solutions with nonndashtrivialterms in all dimensions a particular form of the generalized Beltrami flow is found namely
u1(x y z t) = (A sin(πz) + C cos(πy)) eminus(π2t)Re
u2(x y z t) = (B sin(πx) +A cos(πz)) eminus(π2t)Re
u3(x y z t) = (C sin(πy) +B cos(πx)) eminus(π2t)Re
p(x y z t) = minus[(AB cos(πz) sin(πx) +BC cos(πx) sin(πy)
+AC cos(πy) sin(πz)] eminus(2π2t)Re
(21)
where ABC isin R
10
Figure 6 An illustration of the flow in the yz-plane for Sol 6 (Generalized Beltrami Flow)described in Eq (21) with parameters A = B =C = 1 and time t = 0
Solution 7 Stagnation Flow
In the book by Drazin and Riley [6 p40] an exact solution is considered which uses a stream functionψ = ψ(y z) They suggest that the following velocity field is a solution
u1(x y z t) =a
kx
u2(x y z t) = minusaky +
k
Re
partψ(y z)
partz
u3(x y z t) = minus k
Re
partψ(y z)
party
(22)
Here we choose the stream function to be ψ(y z) = yz Then a solution is given by
u1(x y z t) =a
kx
u2(x y z t) =
(minusak
+k
Re
)y
u3(x y z t) = minus kzRe
p(x y z t) = minus1
2
((axk
)2minus((
k
Reminus a
k
)y
)2
minus(kz
Re
)2)
(23)
where a is a measure of the strength of the stagnation flow and k is a constant chosen in the textto specify the distance between vortices There are no vortices in this particular solution however
11
Figure 7 An illustration of the flow in the yz-plane for Sol 7 (Stagnation Flow) describedin Eq (23) with parameter values a = Re = 1and k = 2
Figure 8 An illustration of the flow in the xy-plane for Sol 7 (Stagnation Flow) describedin Eq (23) with parameter values a = Re = 1and k = 2
Solution 8 Vortex in Stagnation Flow
As in Sol 7 (Stagnation Flow) we now consider the same solution with a different stream functionnamely ψ(z) = cos z Then given the condition that k = aRe and adding a pressure term Eq (22)becomes
u1(x y z t) =1
Rex
u2(x y z t) = minus 1
Rey minus k sin z
u3(x y z t) = 0
p(x y z t) = minus 1
2Re2(x2 + y2)
(24)
where a and k are as in Sol 7 (Stagnation Flow)
12
Figure 9 An illustration of the flow in the yz-plane for Sol 8 (Vortex in Stagnation Flow)described in Eq (24) with parameter valuesk = 3 and Re = 10
Figure 10 An illustration of the flow in thexy-plane for Sol 8 (Vortex in StagnationFlow) described in Eq (24) with parametervalues k = 3 and Re = 10
Solution 9 Burgers Vortex
The Burgers Vortex solution is given in cylindrical coordinates in the book by Lautrup [8 p 385]as follows
ur(r ϕ z t) = minus 2r
c2Re
uϕ(r ϕ z t) =Ωc2
r
(1minus eminusr
2c2)
uz(r ϕ z t) =4z
c2Re
p(r ϕ z t) = minusc2Ω2F
(r2c2
)2
minus 2
Re2r2 + 4z2
c4
(25)
where
F (ξ) =
infinintξ
(1minus eminusx)2
x2dx (26)
This solution is not particularly nice to look at however most of the unpleasantness comes fromthe pressure term which (in this project) is only required in order to validate the solution For thisproject all solutions are required in Cartesian coordinates the velocity field of the Burgers Vortexhas hence been converted into Cartesian coordinates
u1(x y z t) = minus 2x
c2Reminus Ω
h(x y)(1minus exp (minush(x y))) y
u2(x y z t) = minus 2y
c2Re+
Ω
h(x y)(1minus exp (minush(x y)))x
u3(x y z t) =4z
c2Re
(27)
13
where
h(x y) =x2 + y2
c2 (28)
and Ω and c are constants
Figure 11 An illustration of the flow for Sol9 (Burgers Vortex) described in Eq (27) withparameter values c = 15 Re = 20 and Ω = 10A slice of the xy-plane at z = 0 is shown
Solution 10 Another Vortex Solution
There is another vortex solution more simply formulated than the Burgers Vortex yet with asingularity at r = 0 as a result of the uϕ term It is again given in cylindrical coordinates
ur(r ϕ z t) = minus 2r
c2Re
uϕ(r ϕ z t) =a
r
uz(r ϕ z t) =4z
c2Re
p(r ϕ z t) = minus a2
2r2minus 2r2 + 8z2
c2Re2
(29)
which can also be converted into Cartesian coordinates
14
u1(x y z t) = minus 2x
c2Reminus ay
x2 + y2
u2(x y z t) = minus 2y
c2Re+
ax
x2 + y2
u3(x y z t) =4z
c2Re
(30)
where a and c are constants
Figure 12 An illustration of the flow for Sol9 (Burgers Vortex) described in Eq (27) withparameter values c = 15 Re = 20 and a = 10 Aslice of the xy-plane at z = 0 is shown
Solution 11 Pseudo-Motion Shots of the Second Kind
In the book by Berker [3 p 94] a solution is given in cylindrical coordinates by
ur(r ϕ z t) =2
Re
1
r
uϕ(r ϕ z t) = A1r3 +B1r +
C1
r
uz(r ϕ z t) = A2r2 log(r) +B2r
2 + C2
p(r ϕ z t) =A2
1r6
6+A1B1r
4
2+A1C1r
2 +B2
1r2
2minus C2
1
2r2minus 2
r2Re2+ 2B1C1 log(r)
+2A2z
Reminus 4B1ϕ
Re
(31)
where A1 A2 B1 B2 C1 C2 are constants Eq (31) when converted into Cartesian coordinates is
15
u1(x y z t) =1
Re
2xminus C1Rey
x2 + y2minusA1y
(x2 + y2
)minusB1y
u2(x y z t) =1
Re
2y + C1Rex
x2 + y2+A1x
(x2 + y2
)+B1x
u3(x y z t) =1
2A2
(x2 + y2
)log(x2 + y2
)+B2
(x2 + y2
)+ C2
(32)
Figure 13 An illustration of the flow in thexy-plane for Sol 11 (Pseudo-Motion Shots)described in Eqs (31) and (32) with parametervalues A1 = B1 = C1 = 1 A2 = B2 = C2 = 2and Re = 10
Figure 14 An illustration of the flow in theyz-plane for Sol 11 (Pseudo-Motion Shots)described in Eqs (31) and (32) with parametervalues A1 = B1 = C1 = 1 A2 = B2 = C2 = 2and Re = 10
Solution 12 Flows With Constant Whirl
In the book by Berker [3 p 143] another solution is given in cylindrical coordinates by
ur(r ϕ z t) = minus 1
Re
1
r
uϕ(r ϕ z t) =1
rradicte(minus
12 g(t))
uz(r ϕ z t) = 0
p(r ϕ z t) =1
4
2(minus e
minusg(t)
t minus 1Re2
)r2
minus 1
t2Re
infinintminusg(t)
eminust
tdt
(33)
where we define g(t) =r2Re
2t When converted into Cartesian coordinates we find
16
u1(x y z t) = minus 1
x2 + y2
(yradict
exp
(minus
Re(x2 + y2
)4t
)+
x
Re
)
u2(x y z t) =1
x2 + y2
(xradict
exp
(minus
Re(x2 + y2
)4t
)minus y
Re
)
u3(x y z t) = 0
(34)
Figure 15 An illustration of the flow for Sol12 (Flows with Constant Whirl) described inEqs (33) and (34) with time t = 1 and Re = 1 Aslice of the xy-plane at z = 0 is shown as velocityin the z-direction is zero this is the same for anyvalue of z
17
Solution 13 Rectilinear and Uniform Translation of a Sphere
In the book by Berker [3 p 228] another solution is given in Cartesian coordinates by
u1(x y z t) =3
4
axz
R3
(a2
R2minus 1
)
u2(x y z t) =3
4
ayz
R3
(a2
R2minus 1
)
u3(x y z t) = 1minus 3
4
a
Rminus 1
4
a3
R3minus 3
4
az2
R3
(a2
R2minus 1
)
p(x y z t) =3a(aminusR)(a+R)
32R10Re
((R2Re
(x2 + y2
)minus 2R2Rez2
) (a3 + 3aR2 minus 4R3
)+ 3aRez4
(R2 minus a2
)minus 16R5z
)
(35)
where a and R are constants
Figure 16 An illustration of the flow in the yz-plane for Sol 13 (Sphere Translation) describedin Eq (35) with parameter values a = minus2 andRe = 3
Solution 14 Radially Dependent Flow
A solution with velocity in the z-direction dependent only on the distance from the z-axis is as
18
follows in cylindrical coordinates
ur(r ϕ z t) = 0
uϕ(r ϕ z t) = 0
uz(r ϕ z t) = A(r2 minus a2
)
p(r ϕ z t) =4Az
Re
(36)
where A and a are constants In Cartesian coordinates the solution is just the same with u1(x y z t)and u2(x y z t) both zero and the r2 term replaced with x2 + y2 in u3(x y z t)
Figure 17 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus1 1] withparameter values a = 1 A = 4 and Re = 10
Figure 18 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus2 2] withparameter values a = 1 A = 4 and Re = 10
Solution 15 Rankine Vortex
A solution discovered by Rankine [1] is a vortex flow of the following form
ur(r ϕ z t) = 0
uϕ(r ϕ z t) =ar
2πR2
uz(r ϕ z t) = 0
p(r ϕ z t) =a2r2
8π2R4
(37)
19
where a and R are constants In Cartesian coordinates the flow has the form
u1(x y z t) = minus ay
2πR2
u2(x y z t) =ax
2πR2
u3(x y z t) = 0
p(x y z t) =a2(x2 + y2
)8π2R4
(38)
This flow is actually a rigid body rotational flow in that the angular rotational velocity is uniformand the velocity of the flow increases proportionally with distance from the z-axis
Figure 19 An illustration of the flow in the xy-plane for Sol 15 (Rankine Vortex) described inEq (37) and Eq (38) with parameter values a = 1and R = 1
radic2π
23 Characterisation of Flows
It is important to be able to classify flows in a way that sorts them into distinct groups This aidsin the clear identification and general study of flows It is also a very important way of recognisingflows with similar properties In this subsection we introduce two distinct ways in which flows canbe characterised
231 Characterising Solutions using Invariants
A principal invariant of a tensor is a coefficient of the characteristic polynomial of that tensorInvariants as their name suggests are invariant under rotations of the coordinate system in thattheir values do not change when the tensor is lsquorotatedrsquo into a different set of coordinates Thevelocity gradient of a flow G is defined as the Jacobian of the velocity vector given by
Gij =
[partuipartxj
] (39)
20
for i j = 1 2 3 There are three principal invariants of the velocity gradient of a fluid flow G givenby
PG = tr(G)
QG =1
2
((tr(G))2 minus tr
(G2))
RG = det(G)
(40)
Incompressibility of flows (Eq (5)) ensures that PG = 0 hence the first principal invariant is zerofor all of the flows in Section 22 We now define the rate-of-strain tensor S by
Sij =1
2
(partuipartxj
+partujpartxi
) (41)
and the rate-of-rotation tensor Ω by
Ωij =1
2
(partuipartxjminus partujpartxi
) (42)
Note that the first principal invariant of S and the first and third principal invariants of Ω are zeroAlso note that the velocity gradient of a flow can simply be written G = S + Ω
Rather than considering the principal invariants we can also consider the combined invariants
I1 = tr(S2) I2 = tr(Ω2) I3 = tr(S3) I4 = tr(SΩ2) I5 = tr(S2Ω2) (43)
We can hence write the three principal invariants for a velocity field G in terms of the combinedinvariants
PG = 0
QG = minus1
2(I1 + I2)
RG =1
3(I3 + 3I4)
(44)
In this report the flows considered are the exact solutions to the incompressible Navier-Stokesequations described in Section 22 The combined invariants in Eq (43) are used to characterisethese solutions into different classes in Section 44 Both the principal invariants and combinedinvariants for each exact solution in Section 22 are given in Appendix A
Each of the subfilter-scale eddy-viscosity models described in Section 32 are written in terms ofthese five combined invariants
232 Vremanrsquos Characterisation of Flows
Vreman [16] classifies flows using the gradient of the velocity field of a flow Each lsquoclassrsquo of flows hasthe same number of non-zero elements in the gradient matrix For example
[partuipartxj
]=
0 0 0lowast 0 0lowast 0 0
(45)
where a lowast denotes a non-zero term represents a group of flows that lie within the class denoted byQ7 (see Appendix B) where the subscript represents the number of non-zero terms in the velocity
21
gradient Flows of the form in Eq (45) have a velocity field of the form
u1 = c1 (constant)
u2 = f2(x1)
u3 = f3(x1)
(46)
where f2(x1) and f3(x1) are such that the Navier Stokes equations Eq (1) are satisfied Notethat as the elements on the leading diagonal are zero the incompressibility condition Eq (5) isautomatically satisfied Some simple flow classes are given in Appendix B and contain a number ofthe solutions discussed in Section 22 This is mentioned in greater detail in Section 441
3 Conditions for Large-Eddy Simulation Models
31 Large-Eddy Simulation
Due to the nature of numerical simulation and the complexity of turbulent flows small details aremissed and information is lost when solving the Navier-Stokes equations The computational costof solving a turbulent flow is extremely high so a filtering operation is applied to the Navier-Stokesequations represented by a bar to reduce the computational cost of the simulation This is a linearoperation and preserves constants ie ρ = ρ and is also assumed to commute with differentiation
partuipartxj
=partuipartxj
(47)
The filtered Navier-Stokes equations are found by filtering (putting a bar over) the entire equations(see Eq (1)) and after evaluating are given by
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (48)
where
τij = uiuj minus uiuj (49)
is called the lsquotrue stressrsquo of the flow This however poses a problem as it is not clear what is meantby the term uiuj for a general filtering operation The true stress τij therefore must be replaced bya model τmod
ij in practical applications
32 Subfilter-Scale Models
We will now define the subflter-scale models denoted by τmodij to be used in place of the lsquodifficultrsquo
true stress τij There are a number of models available to use in the literature [11 p7] These willbe tested using the exact solutions to the Navier-Stokes equations and by checking conditions whichare defined in Sections 33 and 34
All the models barring one are of the same form That is each of the eddy-viscosity models arewritten
τmodij minus 1
3τmodkk δij = minus2νeSij (50)
with νe representing the eddy viscosity of a specific model For simplicity we usually just writeτmodij = minus2νeSij the extra term is included here to emphasise that the models are traceless Each
22
of the models we consider here can be expressed in terms of the rate of strain tensor S defined inEq (41) the rate of rotation tensor Ω defined in Eq (42) and the combined invariants defined inEq (43) The models are then defined as follows
Smagorinsky [12] νSe = (CSδ)2radic
2I1 (51)
Vreman [16] νVe = (CV δ)2
radicQGGT
PGGT (52)
QR [13 14 15] νQRe = (CQRδ)2 max0minusI3
I1 (53)
Vortex-Stretching [11] νV Se = (CV Sδ)2radic
2I1
(I5 minus 1
2I1I2
minusI1I2
)32
(54)
with the quantities
PGGT = I1 minus I2 QGGT =1
4(I1 + I2)2 + 4
(I5 minus
1
2I1I2
) (55)
There is a fifth and final model to be considered here It is of a different form to the other four andnot an eddy-viscosity model It is known as the Gradient model and is defined by
Gradient [4 9] τmodG = CGδ
(S2 minus Ω2 minus (SΩminus ΩS)
) (56)
The values C are model constants specific to each model and δ represents the mesh size of the LESBoth of these quantities are included here to properly define the models however in this projectthey will be ignored as they are constants and have no influence on the conditions to be defined
33 Conditions for Large-Eddy Simulation without Explicit Filtering
The filtering operation introduced in Eq (48) is just a formality for LES without explicit filteringThe idea here is that filtering does not actually take place but the term lsquoaddedrsquo to the Navier-Stokesequations in fact lsquosubtractsrsquo the difficult to simulate small-scale turbulent motion of the flow Inorder to consider this properly the terms ui and p in Eq (48) are replaced by vi and q It is hopedthat τmod
ij approximates τij so that vi and q approximate ui and p respectively So LES withoutexplicit filtering is given by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (57)
It is clear to see that the first four terms of this equation are exactly the same as the Navier-Stokesequations so for the exact solutions discussed in Section 22 the combination of these terms willequal zero What is then desirable is that the model term is also zero for these exact solutions sothat Eq (57) still holds In order to test models for exact solutions the condition
part
partxjτmodij (v) = 0 (58)
is checked If this condition were to hold it would imply that the model does not influence thevelocity field of a flow hence we call it the Velocity Field Condition There are a number of otherrestrictions that can be tested One such test comes from the differential form of the evolution ofthe kinetic energy of a flow given in Eq (10) In order to find this Eq (57) is multiplied by vi andterms are grouped to find
part
partt
(1
2vivi
)+ vj
part
partxj
(1
2vivi
)= minusvi
1
ρ
partq
partxi+ νvi
part2vipartxjpartxj
minus vipart
partxjτmodij (v) (59)
23
Then similarly to Eq (58) another condition is to check whether the model affects the kineticenergy of exact solutions this is equivalent to checking whether
vipart
partxjτmodij (v) = 0 (60)
We call this condition the Kinetic Energy Condition The third and final condition considered herecan also be found using the integral form for the equation for kinetic energy
d
dt
intV
1
2vividV =
intS
[minusvj
(1
2vivi
)minus vj
q
ρ+ 2νviSij + viτ
modij (v)
]njdS
+
intV
[minus2νSijSij minus
(minusSijτmod
ij (v))]dV
(61)
where the rate-of-strain tensor Sij was defined in Eq (41) V is a volume of fluid and dS meansthe integral is over the surface of that volume with unit normal nj Then this final condition checkswhether the model affects the total kinetic energy of exact solutions to the Navier-Stokes equationsthat is
Sijτmodij (v) = 0 (62)
We will call this condition the Dissipation Condition because Sijτmodij (v) is often referred to as the
subgrid dissipation
34 Conditions for Large-Eddy Simulation with Explicit Filtering
As in Eq (48) the Navier-Stokes equations can be filtered by applying the filtering operation to thewhole equation (putting a bar over all terms) and evaluating to find again that
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (63)
In Eq (57) the filtering operation was ignored and the elements ui p and τij(u) were replacedwith vi q and τmod
ij (v) respectively Now in LES with explicit filtering the filtering operation itselfsmooths over the small-scale turbulent motion of the simulated flow In this case the elements uip and τij(u) will be replaced with vi q and τmod
ij (v) respectively LES with explicit filtering is thengiven by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (64)
Note that Eqs (63) and (64) despite their similarity to Eq (57) no longer contain the Navier-Stokesequations as Eq (57) did The bar (filtering operation) over each term results in each exact solutionfound in Section 22 not necessarily satisfying the first four terms It is hence unknown whether ornot these terms will vanish for exact solutions Terms can still be compared similarly to Eqs (58)(60) and (62) However instead of comparing to zero we can now compare to the relevant termfrom the filtered Navier-Stokes equations So the terms
part
partxjτij (65)
and
part
partxjτmodij (v) (66)
24
are to be compared Although the latter term is relatively simple to compute the former could leadto some difficulties due to the differentiation of the true stress τij = uiuj minus uiuj So although someinformation could be gleaned from the inspection of these two terms more information can be foundby considering the analogue of Eq (62) So the true subfilter dissipation
Sijτij (67)
(where Sij is shorthand for Sij(v)) and the modelled subfilter dissipation
Sij(v)τmodij (v) (68)
are considered and have to be equal for the condition to hold These can again be derived from theintegral equation for kinetic energy of a filtered velocity field similarly to Eq (61)
4 Analysis of Large-Eddy Simulation Models
In this section we use a code in Mathematica the theory described in Section 3 and the exactsolutions to the Navier-Stokes equations found in Section 22 to check whether or not each of theexact solutions lsquoendorsesrsquo each model for LES both without and with explicit filtering
41 Large-Eddy Simulation Without Explicit Filtering
We start by using a code in Mathematica to test the Velocity Field condition in Eq (58) and theKinetic Energy condition in Eq (60) For all exact solutions to the Navier-Stokes equations inSection 22 the outputs for both conditions were identical so we have combined them into one table
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 ndash ndash 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 1 For the Kinetic Energy condition a lsquo0rsquo result means that vipartpartxj
τmodij (v) = 0 for respective
models and exact solutions whereas a lsquolowastrsquo result means that vipartpartxj
τmodij (v) 6= 0 For the Velocity
Field condition a lsquo0rsquo result means that partpartxj
τmodij (v) = 0 for respective models and exact solutions
whereas a lsquolowastrsquo result means that partpartxj
τmodij (v) 6= 0 The Gradient model can take both positive and
negative values so when the sign of the model is known it is included in the table instead of a lsquorsquoBoth the conditions are included in one table because for all cases the outputs were identical
We then test the Dissipation condition in Eq (62) for all exact solutions to the Navier-Stokesequations in Section 22
25
Dissipation Condition [Does Sijτmodij (v) = 0]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 + 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 2 A lsquo0rsquo result means that Sijτmodij (v) = 0 for respective models and exact solutions whereas
a lsquolowastrsquo result means that Sijτmodij (v) 6= 0 The Gradient model can take both positive and negative
values so when the sign of the model is known it is included in the table instead of a lsquorsquo
We then sum all the results to find the total number of endorsements for each model described inSection 32
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
4 6 8 9 9
lsquoDissipationrsquoEndorsements
2 5 7 7 9
Total Endorsements 6 11 15 16 18
Table 3 The results from Tables 1 and 2 are summarised to give a general overview of how manyendorsements each model receives This makes it much clearer to see the models most supported bythe respective and combined conditions with the Vortex-Stretching model being the most supportedand the Smagorinsky model the least
Testing the models in Section 32 using LES without explicit filtering has mixed results It isclear that the Smagorinksy model is not well endorsed by the flows in Section 22 with the onlyendorsements coming from the simplest flows that is Sol 1 (Linear Velocity) Sol 3 (Velocity as afunction of time) Sol 7 (Stagnation Flow) and Sol 15 (Rankine Vortex) The Vortex-Stretching-based model is the most endorsed flow with eighteen out of a possible thirty endorsements Thisagrees with our foreknowledge as this model was constructed to be as physically consistent as possiblein the paper by Silvis amp Verstappen [11]
42 Large-Eddy Simulation With Explicit Filtering
In this section we will compare the true subfilter dissipation Sijτij and the modelled subfilterdissipation Sij(v)τmod
ij (v) given in Eqs (67) and (68) respectively for each model τmodij and each
exact solution to the Navier-Stokes equation given in Section 22 We will not consider the conditionsinvolving differentiation in Eqs (65) and (66) due to the difficulty in calculating the derivative ofthe τij term in Eq (65)
26
421 Example of a Test for a Model
The true subfilter dissipation Sijτij of a flow given in Eq (67) cannot be explicitly computed asthe filtering operation is kept general What can be done however is to partially evaluate theexpressions using the attributes of a filtering operation mentioned in Eq (48)
The calculations of Sijτij for Sol 1 (Linear Velocity) are provided to illustrate the general ideaRecall that the velocity field of Sol 1 (Linear Velocity) is written in Eq (13) in vector notation as
u(x y z t) =
0ax+ bcx+ d
(69)
Then Sij = Sij(u) is computed simply by using its definition in Eq (41) Note that as filteringpreserves constants if there is an a it is replaced with a meaning that (in this case) there are alarge number of zero terms due to the frequency of 0 Hence
[Sij(u)
]=
[1
2
(partuipartxj
+partujpartxi
)]=
0 a
2c2
a2 0 0
c2 0 0
=
0 a
2c2
a2 0 0
c2 0 0
(70)
Note that as Sijτij is being computed which is equal to the sum of the element-wise multiplicationof the two matrices (or the trace of the matrix multiplication) that we donrsquot need to consider theelements of τij(u) with corresponding zero entry in Sij As τij(u) = uiuj minus uiuj and again asfiltering preserves constants it is found that
[τij(u)] =[uiuj minus uiuj
]=
lowast 0 middot ax+ b 0 middot cx+ d
ax+ b middot 0 lowast lowastcx+ d middot 0 lowast lowast
=
lowast 0 00 lowast lowast0 lowast lowast
(71)
where a lsquolowastrsquo just represents some element (not necessarily non-zero) Therefore the true subfilterdissipation
Sijτij = tr
0 a2
c2
a2 0 0c2 0 0
middotlowast 0 0
0 lowast lowast0 lowast lowast
= 0 (72)
Now the modelled subfilter dissipation Sij(u)τmodij (u) needs to be computed (recall that Sij =
Sij(u)) In this case let us use the QR model stated in Eq (53) Once again Sij(u) calculated inEq (70) needs to be used So we have that
τmodij = minus2(CQRδ)
2 max0minusI3I1
Sij (73)
As
I1 = tr(S2) =
1
2(a2 + c2) and I3 = tr(S
3) = 0 (74)
it holds that τmodij (u) = 0 so Sij(u)τmod
ij (u) = 0 This agrees with Eq (72) and hence Sol 1(Linear Velocity) endorses the QR model
27
422 Testing the Models
Now using a Mathematica code the exact solutions in Section 22 are again used to evaluate thecondition for the models using the process just described in Section 421 The results are describedin Table 4
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 4 All entries with a ldquo0rdquo represent a value of zero and a ldquolowastrdquo represents a value that is non-zeroThe ldquoExactrdquo row represents the value of the true subfilter dissipation Sijτij Every other row witha model name represents the value of the modelled subfilter dissipation Sij(u)τmod
ij (u) respectiveto that model A ldquordquo means that it is unclear as to the value - the sign of these values is oftenunknown and the lsquoMaxrsquo function in the definition of the QR model (in Eq (53)) makes it difficultto evaluate
Reading from Table 4 the Smagorinsky model is clearly endorsed by Sol 3 (Velocity as a function oftime) and Sol 15 (Rankine Vortex) as in both cases both Sijτij = 0 = Sij(u)τmod
ij (u) In this casewe say that Sol 3 (Velocity as a function of time) and Sol 15 (Rankine Vortex) lsquostronglyrsquo endorsethe Smagorinsky model The Smagorinsky model is not endorsed by Sol 1 (Linear Velocity) Sol 2(Planar Flow) or Sol 14 (Radially Dependent) however because Sij(u)τmod
ij (u) 6= 0 For Sol 4 to
13 both Sijτij 6= 0 6= Sij(u)τmodij (u) and so we will say that these exact solutions lsquoweaklyrsquo endorse
the Smagorinsky model
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 10 10 9 6
Table 5 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Interestingly the lsquostrongrsquo and lsquoweakrsquo endorsements give different results However more stock shouldbe put in the results of the lsquostrongerrsquo condition as it is completely clear that zero is equal to zeroIt is not necessarily clear whether or not non-zero equals non-zero as the values will depend on theparameters in each solution hence the lsquoweakerrsquo endorsement
The lsquostrongrsquo condition equally endorses the Vreman QR Gradient and Vortex-Stretching-basedmodels whereas the lsquoweakrsquo condition endorses Vreman and Smagorinsky the most closely followedby the Gradient model with the Vortex-Stretching-based model being the most opposed As this
28
completely contradicts the lsquostrongrsquo condition and the results in Section 41 it makes us question thevalidity of the weaker test The number of endorsements for the QR model is unknown because of thedefinition of the model in Eq (53) and our evaluation of the true subfilter dissipation Eq (67) Dueto the exact solutions we have found it is never clear whether or not the true subfilter dissipationis strictly positive or negative This means that when the true subfilter dissipation is found to benon-zero the lsquoMaxrsquo function in the QR model cannot be evaluated
43 Vremanrsquos True Subfilter Dissipation Claim
Using the theory mentioned in Sections 232 and 34 we can place the exact solutions to the Navier-Stokes equations into different classes defined by Vreman In doing this we can also attempt to verifyone of Vremanrsquos results [16] In Vremanrsquos paper he defines the class Qn to be the set of flows withn zero entries in the velocity gradient of the flow (the velocity gradient is defined in Eq (39))Vreman then claims that the only flows with zero true subfilter dissipation are a certain selection ofthose in the classesQ7Q8 orQ9 shown in appendix B and is generally non-zero for any other flows
The true subfilter dissipation of Sol 1 (Linear Velocity) was computed in Section 421 and wasfound to be zero in Eq (72) The velocity gradient of Sol 1 is as follows
GLV =
0 0 0a 0 0c 0 0
(75)
and hence Sol 1 (Linear Velocity) is in the required subset of Q7 which agrees with Vremanrsquos claimIn Section 422 the true subfilter dissipation was computed (determined to be zero or non-zero) forall of the exact solutions to the Navier Stokes equations in Section 22 These solutions can now belsquoplacedrsquo into each class Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a functionof time) and Sol 14 (Radially Dependent) were all found to have zero true subfilter dissipation andare all in the appropriate subsets of either Q7Q8 or Q9 This agrees with Vremanrsquos results
We can extend Vremanrsquos results however We found that Sol 15 (Rankine Vortex) had zero truesubfilter dissipation and yet is not in the subset of Q7 defined in Appendix B It is important tonote that the rate of strain tensor Sij in Eq (41) for Sol 15 (Rankine Vortex) is zero This is thereason the true subfilter dissipation is zero and the same reason that all models are zero for thisflow as all models are based on Sij
We also found that all of the remaining exact solutions to the Navier-Stokes equations are not inVremanrsquos specified classes (in Appendix Section B) They also have a non-zero subfilter dissipationwhich agrees with Vremanrsquos results in his paper [16]
44 Classes of Flows
It is important to consider that some exact solutions to the Navier-Stokes equations may have similarproperties For example Sol 1 (Linear Velocity) and Sol 2 (Planar Flow) are both very similarIn fact Sol 2 (Planar Flow) is a more general case of Sol 1 (Linear Velocity) We therefore mustgroup the different flows into classes in an attempt to reduce the bias that may occur when solutionsare similar This should aid us to further determine which of the models are most endorsed We willdo this in a number of different ways using the theory discussed in Section 23 We will group theflows with respect to Vremanrsquos Flow Classes given in appendix B and with respect to the Invariantsgiven in appendix A
29
441 Vremanrsquos Velocity Gradient Flow Classes
We now put our exact solutions to the Navier-Stokes equations into one of two classes Either Q0minus6or Q7minus9 where each class is defined intuitively as follows
Q0minus6 = Q0 cupQ1 cup middot middot middot cup Q6 (76)
Q7minus9 = Q7 cupQ8 cupQ9 (77)
where Qn is defined in Section 43 By looking at the gradient of each exact solution we can see thatQ7minus9 contains the six solutions Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as afunction of time) Sol 5 (2D Taylor) Sol 14 (Radially Dependent) and Sol 15 (Rankine Vortex)and that Q0minus6 contains the remaining nine We now check the models for LES without explicitfiltering by considering endorsements within each class Tables 6 and 7 are analogues of Table 3 andillustrate the results for each flow class The analogues of Tables 1 and 2 are Tables 22 and 23 andare in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 6 6 6
lsquoDissipationrsquoEndorsements
2 5 6 6 6
Percentage Endorsed 417 833 100 100 100
Table 6 Each entry in the top two rows represents the number of solutions in the class Q7minus9 thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Q0minus6 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 2 3 3
lsquoDissipationrsquoEndorsements
0 0 1 0 3
Percentage Endorsed 56 56 167 167 333
Table 7 Each entry in the top two rows represents the number of solutions in the class Q1minus6 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking results to 1 decimal place if necessary
All of the flows in the class Q7minus9 endorse the QR Gradient and Vortex-Stretching-based subfilter-scale models (see Table 6) We can hence see that this class fully endorses these three models equallyand does not endorse the Smagorinsky model The Vortex-Stretching eddy viscosity model has themost endorsements from any of the flows in the class Q0minus6 and the Smagorinksy and Vreman modelshave the fewest endorsements (see Table 7) We can hence say that this class mostly endorses theVortex-Stretching-based eddy-viscosity model note however that the percentage endorsement is very
30
low
Now let us consider LES with explicit filtering Tables 8 and 9 are analogues of Table 5 and illustratethe results for each flow class The analogue of Table 4 is Table 24 and is in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 1 1 0 0 0
Table 8 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Q0minus6 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 9 9 9 6
Table 9 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We have grouped Vremanrsquos classes into Q7minus9 and Q0minus6 this is equivalent to saying that we havegrouped Vremanrsquos classes into those with two or fewer non-zero velocity gradient elements and thosewith three or more non-zero velocity gradient elements respectively Therefore the solutions in Q7minus9are exactly those with a lsquo0rsquo in the lsquoExactrsquo column of Table 4 and Sol 5 (2D Taylor) ExcludingSol 5 (2D Taylor) these are the same as those five that endorse the Vreman QR Gradient andVortex-Stretching-based models in Table 5 We can see in Table 8 that the results for the lsquostrongrsquoendorsements are exactly the same as in Table 5 This is because all of the flows with zero subfilterdissipation are contained in the class Q7minus9 There are hence no lsquostrongrsquo endorsements in Table 9The majority of the lsquoweakrsquo endorsements come from Q0minus6 as this class contains all but one of thesolutions with non-zero true subfilter dissipation (given in Eq (67))
442 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
We now separate our exact solutions into classes again using Vremanrsquos flow classes This timehowever we define V0 to be the class of flows Vreman claims will always have zero true subfilterdissipation namely those classes mentioned in Appendix B We define Vlowast to be all the remaining
31
classes Then
V0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow)
Sol 3 (Velocity as a function of time) Sol 14 (Radially Dependent)(78)
Vlowast = Sol 4 (Axisymmetric Flow) Sol 5 (2D Taylor) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 13 (Sphere Translation) Sol 15 (Rankine Vortex)
(79)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 10 and 11 are analogues of Table 3 and illustrate the results for each flow class Theanalogues of Tables 1 and 2 are Tables 25 and 26 and are in Appendix C2
Endorsements for models using solutions in V0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 4 4 4 4
lsquoDissipationrsquoEndorsements
1 4 4 4 4
Percentage Endorsed 375 100 100 100 100
Table 10 Each entry in the top two rows represents the number of solutions in the class V0 thatendorse each model There are 4 solutions in this class so the percentages are worked out by dividingthe total of each column by 8 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Vlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 2 4 5 5
lsquoDissipationrsquoEndorsements
1 1 3 2 5
Percentage Endorsed 136 136 318 318 455
Table 11 Each entry in the top two rows represents the number of solutions in the class Vlowast thatendorse each model There are 11 solutions in this class so the percentages are worked out bydividing the total of each column by 22 and taking results to 1 decimal place if necessary
All of the flows in the class V0 endorse the Vreman QR Gradient and Vortex-Stretching-basedsubfilter-scale models (see Table 6) This class completely endorses these four models equally anddoes not endorse the Smagorinsky model The Vortex-Stretching-based eddy-viscosity model hasthe most endorsements from the class Vlowast and the Smagorinksy and Vreman models again have thefewest endorsements (see Table 7) As in Section 441 we say that this class mostly endorses the
32
Vortex-Stretching-based eddy-viscosity model but once again the percentage endorsement is ratherlow
Now let us consider LES with explicit filtering Tables 12 and 13 are analogues of Table 5 andillustrate the results for each flow class The analogue of Table 4 is Table 27 and is in Appendix C2
Endorsements for models using solutions in V0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 4 4 4 4
lsquoWeakrsquo Endorsements 0 0 0 0 0
Table 12 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Vlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 1 1 1 1
lsquoWeakrsquo Endorsements 10 10 9 6
Table 13 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We can see in Table 12 that the results for the lsquostrongrsquo endorsements are similar to those in Table 5excluding Sol 15 (Rankine Vortex) The four solutions in V0 with zero true subfilter dissipationnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)and Sol 14 (Radially Dependent) are hence the four that endorse the Vreman QR Gradientand Vortex-Stretching-based models in Table 12 Sol 15 (Rankine Vortex) is in the class Vlowast andhence there is one lsquostrongrsquo endorsement in Table 13 The class Vlowast contains all of the flows with anon-zero true subfilter dissipation (given in Eq (67)) and hence Table 13 contains all of the lsquoweakrsquoendorsements
443 Classes using the Third Principal Invariant
Sorting the classes using the invariants is a non-trivial task in itself There are a number of waysin which to do this using both the principal invariants in Eq (40) and the combined invariants inEq (43)
We start by looking at flows with third principal invariant RG equal to zero We again define twoclasses R0 and Rlowast where R0 consists of all flows with RG = 0 and Rlowast consists of all flows withRG 6= 0 Then using the information in Appendix A we see that there are nine flows in R0 That
33
is
R0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 8 (Vortex in Stagnation Flow)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)
(80)
Rlowast = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 13 (Sphere Translation)(81)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 14 and 15 are analogues of Table 3 and they illustrate the results for LES withoutexplicit filtering for each flow class The analogues of Tables 1 and 2 are Tables 28 and 29 and arein Appendix C3
Endorsements for models using solutions in R0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 7 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 278 556 778 722 778
Table 14 Each entry in the top two rows represents the number of solutions in the class R0 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking the result to one decimal place if necessary
Endorsements for models using solutions in Rlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 2 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 83 83 83 167 333
Table 15 Each entry in the top two rows represents the number of solutions in the class Rlowast thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking the result to one decimal place if necessary
The class R0 endorses both the Vortex-Stretching-based model and the QR model the most andSmagorinsky the least in the case for LES without explicit filtering In the Rlowast class the Vortex-Stretching-based model is again the most endorsed followed by the Gradient model however thepercentages are again small
34
Now let us consider LES with explicit filtering Tables 16 and 17 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 30 and is in Appendix C3
Endorsements for models using solutions in R0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 4 4 3 2
Table 16 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Rlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 6 6 6 4
Table 17 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 we can see in Table 16 that the results for the lsquostrongrsquo endorsements are exactly thesame as in Table 5 This is because all of the flows with zero subfilter dissipation are contained in theclass R0 As in Table 9 there are no lsquostrongrsquo endorsements in Table 17 The lsquoweakrsquo endorsementshowever are split more evenly between the two classes now than in Vremanrsquos classes in Sections 441and 442 We find that the Smagorinsky and Vreman models still seem to be the most endorsed inboth classes
444 Two-component Flow Classes
Another way in which the flows can be grouped into classes is by determining whether a flow is atwo-component flow or not We define a two-component flow to have the property that the combinedprincipal invariants of a flow in Eq (43) satisfy
I3 = I4 = I5 minus1
2I1I2 = 0 (82)
We are also interested in considering flows with zero gradient model subgrid dissipation These flowshave the property that
I3 minus I4 = 0 (83)
What we find though is that for all the solutions in Section 22 if a flow is a two-component flowthen it also satisfies I3 minus I4 = 0 It may be interesting to note that although I3 minus I4 6= 0 for manysolutions there are a number with I3 equal to a multiple of I4 Although not considered in thisreport this may also be a class of interest We find that the two-component flows are
C2 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)(84)
35
and we call the remaining flows three-component flows They are
C3 = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex) Sol 11 (Pseudo-Motion Shots)
Sol 13 (Sphere Translation)
(85)
Note that Sol 8 (Vortex in Stagnation Flow) is defined to be a planar flow this does not mean that itis a two-component flow in this sense however because its two non-zero velocity components dependon three coordinates We now check the models for LES without explicit filtering by consideringendorsements within each class Tables 18 and 19 are analogues of Table 3 and they illustrate theresults for LES without explicit filtering for each flow class The analogues of Tables 1 and 2 areTables 31 and 32 and are in Appendix C4
Endorsements for models using solutions in C2 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 6 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 357 714 100 857 100
Table 18 Each entry in the top two rows represents the number of solutions in the class C2 thatendorse each model There are 7 solutions in this class so the percentages are worked out by dividingthe total of each column by 14 and taking the result to one decimal place if necessary
Endorsements for models using solutions in C3 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 3 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 63 63 63 188 250
Table 19 Each entry in the top two rows represents the number of solutions in the class C3 thatendorse each model There are 8 solutions in this class so the percentages are worked out by dividingthe total of each column by 16 and taking the result to one decimal place if necessary
The class C2 equally endorses the QR and Vortex-Stretching-based models and once again theSmagorinsky model is the least endorsed As with the previous classes the class C3 has such lowpercentages that it does not seem to endorse any models We start to see a trend appearing here inthat the flow classes that contain the more complicated flows namely Q0minus6 (Eq (76)) Vlowast (Eq (79))Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages The classes that tendto contain the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78)) R0 (Eq (80)) and C2 (Eq (84))all seem to favour the Vortex-Stretching-based model or the QR model
36
Now let us consider LES with explicit filtering Tables 20 and 21 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 33 and is in Appendix C4
Endorsements for models using solutions in C2 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 2 2 0 1 0
Table 20 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in C3 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 8 8 8 6
Table 21 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 and Table 16 we can see in Table 20 that the results for the lsquostrongrsquo endorsements areexactly the same as in Table 5 This is again because all of the flows with zero subfilter dissipationare contained in the class C2 There are two flows with non-zero true subfilter dissipation (givenin Eq (67)) in C2 and this shows in the lsquoweakrsquo endorsements in Table 20 There are no lsquostrongrsquoendorsements in Table 21 as all the zero subfilter dissipation flows are in C2
5 Conclusions
In this report we studied incompressible turbulent flows We found a number of exact solutions to theNavier-Stokes equations each representing a relatively simplistic fluid flow Large-eddy simulation(LES) is used to model turbulent flows by smoothing over the complex small-scale motion Weintroduced a number of subfilter-scale models to approximate this small-scale motion and testedthem using the found exact solutions to the Navier-Stokes equations We first considered LESwithout explicit filtering and used the Navier-Stokes equations to derive conditions for the subfilter-scale models namely the Velocity Field condition (Eq (58)) the Kinetic Energy condition (Eq (60))and the Dissipation condition (Eq (62)) Similarly we considered LES with explicit filtering andagain used the Navier-Stokes equations to derive more conditions for the models in this case wedecided to just take one condition the Dissipation condition (Eqs (67) and (68)) as the others leadto difficulty in their evaluation By lsquorunningrsquo the exact solutions to the Navier-Stokes equationsthrough each condition for the subfilter-scale models we found that the Vortex-Stretching-basededdy-viscosity model in Eq (54) was most endorsed by the tests in LES without explicit filteringWe found that four out of the five subfilter-scale models namely the Vreman QR Gradient andVortex-Stretching-based models were equally endorsed by the lsquostrongrsquo tests in LES with explicitfiltering with the Smagorinsky model being the least endorsed The results we found for the lsquoweakrsquocondition were completely opposite to those found in both the lsquostrongrsquo condition and in LES withoutexplicit filtering This led us to conclude that this test was not one in which we should put our trust
37
We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses
In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes
This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold
38
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
Solution 4 Axisymmetric Flow
Using the text by Irmay [7] a solution is defined as follows
u1(x y z t) = kxz
u2(x y z t) = kyz
u3(x y z t) = minuskz2
p(x y z t) = minus2kz
Reminus k2z4
2
(17)
where k isin R This is said to represent an axisymmetric flow with hyperbolic streamlines asymptoticto the z-axis and to the boundary plane z = 0 These asymptotes are visible in Fig 3
Figure 3 An illustration of the flow in theyz-plane for Sol 4 (Axisymmetric Flow)described in Eq (17) with parameter k = 1
Figure 4 An illustration of the flow in thexy-plane for Sol 4 (Axisymmetric Flow)described in Eq (17) with parameter k = 1
Solution 5 2D Taylor Solutions
Again using the text by Barbato et al [2 p63ndash64] a solution is defined using a stream functionψ = ψ(x y t) and by writing
u1 = minuspartψparty
u2 =partψ
partx u3 = 0 (18)
In the text the stream function is not given however it was worked out that
ψ = minuseminusπ2t
Re (cos(πx) + cos(πy))
π (19)
The solution used is of the following form note however that in the text the pressure term differsin that cos(πy) is replaced by sin(πy) (when testing this solution in Mathematica the solution below
9
was instead found)
u1(x y z t) = minus sin(πy) eminus(π2t)Re
u2(x y z t) = sin(πx) eminus(π2t)Re
u3(x y z t) = 0
p(x y z t) = minus cos(πx) cos(πy) eminus(2π2t)Re
(20)
Figure 5 An illustration of the flow in the xy-plane for Sol 5 (2D Taylor) described in Eq (20)with Re = 1 and time t = 0
Solution 6 Generalized Beltrami Flow
Again using the text by Barbato et al [2 p65] and now considering solutions with nonndashtrivialterms in all dimensions a particular form of the generalized Beltrami flow is found namely
u1(x y z t) = (A sin(πz) + C cos(πy)) eminus(π2t)Re
u2(x y z t) = (B sin(πx) +A cos(πz)) eminus(π2t)Re
u3(x y z t) = (C sin(πy) +B cos(πx)) eminus(π2t)Re
p(x y z t) = minus[(AB cos(πz) sin(πx) +BC cos(πx) sin(πy)
+AC cos(πy) sin(πz)] eminus(2π2t)Re
(21)
where ABC isin R
10
Figure 6 An illustration of the flow in the yz-plane for Sol 6 (Generalized Beltrami Flow)described in Eq (21) with parameters A = B =C = 1 and time t = 0
Solution 7 Stagnation Flow
In the book by Drazin and Riley [6 p40] an exact solution is considered which uses a stream functionψ = ψ(y z) They suggest that the following velocity field is a solution
u1(x y z t) =a
kx
u2(x y z t) = minusaky +
k
Re
partψ(y z)
partz
u3(x y z t) = minus k
Re
partψ(y z)
party
(22)
Here we choose the stream function to be ψ(y z) = yz Then a solution is given by
u1(x y z t) =a
kx
u2(x y z t) =
(minusak
+k
Re
)y
u3(x y z t) = minus kzRe
p(x y z t) = minus1
2
((axk
)2minus((
k
Reminus a
k
)y
)2
minus(kz
Re
)2)
(23)
where a is a measure of the strength of the stagnation flow and k is a constant chosen in the textto specify the distance between vortices There are no vortices in this particular solution however
11
Figure 7 An illustration of the flow in the yz-plane for Sol 7 (Stagnation Flow) describedin Eq (23) with parameter values a = Re = 1and k = 2
Figure 8 An illustration of the flow in the xy-plane for Sol 7 (Stagnation Flow) describedin Eq (23) with parameter values a = Re = 1and k = 2
Solution 8 Vortex in Stagnation Flow
As in Sol 7 (Stagnation Flow) we now consider the same solution with a different stream functionnamely ψ(z) = cos z Then given the condition that k = aRe and adding a pressure term Eq (22)becomes
u1(x y z t) =1
Rex
u2(x y z t) = minus 1
Rey minus k sin z
u3(x y z t) = 0
p(x y z t) = minus 1
2Re2(x2 + y2)
(24)
where a and k are as in Sol 7 (Stagnation Flow)
12
Figure 9 An illustration of the flow in the yz-plane for Sol 8 (Vortex in Stagnation Flow)described in Eq (24) with parameter valuesk = 3 and Re = 10
Figure 10 An illustration of the flow in thexy-plane for Sol 8 (Vortex in StagnationFlow) described in Eq (24) with parametervalues k = 3 and Re = 10
Solution 9 Burgers Vortex
The Burgers Vortex solution is given in cylindrical coordinates in the book by Lautrup [8 p 385]as follows
ur(r ϕ z t) = minus 2r
c2Re
uϕ(r ϕ z t) =Ωc2
r
(1minus eminusr
2c2)
uz(r ϕ z t) =4z
c2Re
p(r ϕ z t) = minusc2Ω2F
(r2c2
)2
minus 2
Re2r2 + 4z2
c4
(25)
where
F (ξ) =
infinintξ
(1minus eminusx)2
x2dx (26)
This solution is not particularly nice to look at however most of the unpleasantness comes fromthe pressure term which (in this project) is only required in order to validate the solution For thisproject all solutions are required in Cartesian coordinates the velocity field of the Burgers Vortexhas hence been converted into Cartesian coordinates
u1(x y z t) = minus 2x
c2Reminus Ω
h(x y)(1minus exp (minush(x y))) y
u2(x y z t) = minus 2y
c2Re+
Ω
h(x y)(1minus exp (minush(x y)))x
u3(x y z t) =4z
c2Re
(27)
13
where
h(x y) =x2 + y2
c2 (28)
and Ω and c are constants
Figure 11 An illustration of the flow for Sol9 (Burgers Vortex) described in Eq (27) withparameter values c = 15 Re = 20 and Ω = 10A slice of the xy-plane at z = 0 is shown
Solution 10 Another Vortex Solution
There is another vortex solution more simply formulated than the Burgers Vortex yet with asingularity at r = 0 as a result of the uϕ term It is again given in cylindrical coordinates
ur(r ϕ z t) = minus 2r
c2Re
uϕ(r ϕ z t) =a
r
uz(r ϕ z t) =4z
c2Re
p(r ϕ z t) = minus a2
2r2minus 2r2 + 8z2
c2Re2
(29)
which can also be converted into Cartesian coordinates
14
u1(x y z t) = minus 2x
c2Reminus ay
x2 + y2
u2(x y z t) = minus 2y
c2Re+
ax
x2 + y2
u3(x y z t) =4z
c2Re
(30)
where a and c are constants
Figure 12 An illustration of the flow for Sol9 (Burgers Vortex) described in Eq (27) withparameter values c = 15 Re = 20 and a = 10 Aslice of the xy-plane at z = 0 is shown
Solution 11 Pseudo-Motion Shots of the Second Kind
In the book by Berker [3 p 94] a solution is given in cylindrical coordinates by
ur(r ϕ z t) =2
Re
1
r
uϕ(r ϕ z t) = A1r3 +B1r +
C1
r
uz(r ϕ z t) = A2r2 log(r) +B2r
2 + C2
p(r ϕ z t) =A2
1r6
6+A1B1r
4
2+A1C1r
2 +B2
1r2
2minus C2
1
2r2minus 2
r2Re2+ 2B1C1 log(r)
+2A2z
Reminus 4B1ϕ
Re
(31)
where A1 A2 B1 B2 C1 C2 are constants Eq (31) when converted into Cartesian coordinates is
15
u1(x y z t) =1
Re
2xminus C1Rey
x2 + y2minusA1y
(x2 + y2
)minusB1y
u2(x y z t) =1
Re
2y + C1Rex
x2 + y2+A1x
(x2 + y2
)+B1x
u3(x y z t) =1
2A2
(x2 + y2
)log(x2 + y2
)+B2
(x2 + y2
)+ C2
(32)
Figure 13 An illustration of the flow in thexy-plane for Sol 11 (Pseudo-Motion Shots)described in Eqs (31) and (32) with parametervalues A1 = B1 = C1 = 1 A2 = B2 = C2 = 2and Re = 10
Figure 14 An illustration of the flow in theyz-plane for Sol 11 (Pseudo-Motion Shots)described in Eqs (31) and (32) with parametervalues A1 = B1 = C1 = 1 A2 = B2 = C2 = 2and Re = 10
Solution 12 Flows With Constant Whirl
In the book by Berker [3 p 143] another solution is given in cylindrical coordinates by
ur(r ϕ z t) = minus 1
Re
1
r
uϕ(r ϕ z t) =1
rradicte(minus
12 g(t))
uz(r ϕ z t) = 0
p(r ϕ z t) =1
4
2(minus e
minusg(t)
t minus 1Re2
)r2
minus 1
t2Re
infinintminusg(t)
eminust
tdt
(33)
where we define g(t) =r2Re
2t When converted into Cartesian coordinates we find
16
u1(x y z t) = minus 1
x2 + y2
(yradict
exp
(minus
Re(x2 + y2
)4t
)+
x
Re
)
u2(x y z t) =1
x2 + y2
(xradict
exp
(minus
Re(x2 + y2
)4t
)minus y
Re
)
u3(x y z t) = 0
(34)
Figure 15 An illustration of the flow for Sol12 (Flows with Constant Whirl) described inEqs (33) and (34) with time t = 1 and Re = 1 Aslice of the xy-plane at z = 0 is shown as velocityin the z-direction is zero this is the same for anyvalue of z
17
Solution 13 Rectilinear and Uniform Translation of a Sphere
In the book by Berker [3 p 228] another solution is given in Cartesian coordinates by
u1(x y z t) =3
4
axz
R3
(a2
R2minus 1
)
u2(x y z t) =3
4
ayz
R3
(a2
R2minus 1
)
u3(x y z t) = 1minus 3
4
a
Rminus 1
4
a3
R3minus 3
4
az2
R3
(a2
R2minus 1
)
p(x y z t) =3a(aminusR)(a+R)
32R10Re
((R2Re
(x2 + y2
)minus 2R2Rez2
) (a3 + 3aR2 minus 4R3
)+ 3aRez4
(R2 minus a2
)minus 16R5z
)
(35)
where a and R are constants
Figure 16 An illustration of the flow in the yz-plane for Sol 13 (Sphere Translation) describedin Eq (35) with parameter values a = minus2 andRe = 3
Solution 14 Radially Dependent Flow
A solution with velocity in the z-direction dependent only on the distance from the z-axis is as
18
follows in cylindrical coordinates
ur(r ϕ z t) = 0
uϕ(r ϕ z t) = 0
uz(r ϕ z t) = A(r2 minus a2
)
p(r ϕ z t) =4Az
Re
(36)
where A and a are constants In Cartesian coordinates the solution is just the same with u1(x y z t)and u2(x y z t) both zero and the r2 term replaced with x2 + y2 in u3(x y z t)
Figure 17 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus1 1] withparameter values a = 1 A = 4 and Re = 10
Figure 18 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus2 2] withparameter values a = 1 A = 4 and Re = 10
Solution 15 Rankine Vortex
A solution discovered by Rankine [1] is a vortex flow of the following form
ur(r ϕ z t) = 0
uϕ(r ϕ z t) =ar
2πR2
uz(r ϕ z t) = 0
p(r ϕ z t) =a2r2
8π2R4
(37)
19
where a and R are constants In Cartesian coordinates the flow has the form
u1(x y z t) = minus ay
2πR2
u2(x y z t) =ax
2πR2
u3(x y z t) = 0
p(x y z t) =a2(x2 + y2
)8π2R4
(38)
This flow is actually a rigid body rotational flow in that the angular rotational velocity is uniformand the velocity of the flow increases proportionally with distance from the z-axis
Figure 19 An illustration of the flow in the xy-plane for Sol 15 (Rankine Vortex) described inEq (37) and Eq (38) with parameter values a = 1and R = 1
radic2π
23 Characterisation of Flows
It is important to be able to classify flows in a way that sorts them into distinct groups This aidsin the clear identification and general study of flows It is also a very important way of recognisingflows with similar properties In this subsection we introduce two distinct ways in which flows canbe characterised
231 Characterising Solutions using Invariants
A principal invariant of a tensor is a coefficient of the characteristic polynomial of that tensorInvariants as their name suggests are invariant under rotations of the coordinate system in thattheir values do not change when the tensor is lsquorotatedrsquo into a different set of coordinates Thevelocity gradient of a flow G is defined as the Jacobian of the velocity vector given by
Gij =
[partuipartxj
] (39)
20
for i j = 1 2 3 There are three principal invariants of the velocity gradient of a fluid flow G givenby
PG = tr(G)
QG =1
2
((tr(G))2 minus tr
(G2))
RG = det(G)
(40)
Incompressibility of flows (Eq (5)) ensures that PG = 0 hence the first principal invariant is zerofor all of the flows in Section 22 We now define the rate-of-strain tensor S by
Sij =1
2
(partuipartxj
+partujpartxi
) (41)
and the rate-of-rotation tensor Ω by
Ωij =1
2
(partuipartxjminus partujpartxi
) (42)
Note that the first principal invariant of S and the first and third principal invariants of Ω are zeroAlso note that the velocity gradient of a flow can simply be written G = S + Ω
Rather than considering the principal invariants we can also consider the combined invariants
I1 = tr(S2) I2 = tr(Ω2) I3 = tr(S3) I4 = tr(SΩ2) I5 = tr(S2Ω2) (43)
We can hence write the three principal invariants for a velocity field G in terms of the combinedinvariants
PG = 0
QG = minus1
2(I1 + I2)
RG =1
3(I3 + 3I4)
(44)
In this report the flows considered are the exact solutions to the incompressible Navier-Stokesequations described in Section 22 The combined invariants in Eq (43) are used to characterisethese solutions into different classes in Section 44 Both the principal invariants and combinedinvariants for each exact solution in Section 22 are given in Appendix A
Each of the subfilter-scale eddy-viscosity models described in Section 32 are written in terms ofthese five combined invariants
232 Vremanrsquos Characterisation of Flows
Vreman [16] classifies flows using the gradient of the velocity field of a flow Each lsquoclassrsquo of flows hasthe same number of non-zero elements in the gradient matrix For example
[partuipartxj
]=
0 0 0lowast 0 0lowast 0 0
(45)
where a lowast denotes a non-zero term represents a group of flows that lie within the class denoted byQ7 (see Appendix B) where the subscript represents the number of non-zero terms in the velocity
21
gradient Flows of the form in Eq (45) have a velocity field of the form
u1 = c1 (constant)
u2 = f2(x1)
u3 = f3(x1)
(46)
where f2(x1) and f3(x1) are such that the Navier Stokes equations Eq (1) are satisfied Notethat as the elements on the leading diagonal are zero the incompressibility condition Eq (5) isautomatically satisfied Some simple flow classes are given in Appendix B and contain a number ofthe solutions discussed in Section 22 This is mentioned in greater detail in Section 441
3 Conditions for Large-Eddy Simulation Models
31 Large-Eddy Simulation
Due to the nature of numerical simulation and the complexity of turbulent flows small details aremissed and information is lost when solving the Navier-Stokes equations The computational costof solving a turbulent flow is extremely high so a filtering operation is applied to the Navier-Stokesequations represented by a bar to reduce the computational cost of the simulation This is a linearoperation and preserves constants ie ρ = ρ and is also assumed to commute with differentiation
partuipartxj
=partuipartxj
(47)
The filtered Navier-Stokes equations are found by filtering (putting a bar over) the entire equations(see Eq (1)) and after evaluating are given by
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (48)
where
τij = uiuj minus uiuj (49)
is called the lsquotrue stressrsquo of the flow This however poses a problem as it is not clear what is meantby the term uiuj for a general filtering operation The true stress τij therefore must be replaced bya model τmod
ij in practical applications
32 Subfilter-Scale Models
We will now define the subflter-scale models denoted by τmodij to be used in place of the lsquodifficultrsquo
true stress τij There are a number of models available to use in the literature [11 p7] These willbe tested using the exact solutions to the Navier-Stokes equations and by checking conditions whichare defined in Sections 33 and 34
All the models barring one are of the same form That is each of the eddy-viscosity models arewritten
τmodij minus 1
3τmodkk δij = minus2νeSij (50)
with νe representing the eddy viscosity of a specific model For simplicity we usually just writeτmodij = minus2νeSij the extra term is included here to emphasise that the models are traceless Each
22
of the models we consider here can be expressed in terms of the rate of strain tensor S defined inEq (41) the rate of rotation tensor Ω defined in Eq (42) and the combined invariants defined inEq (43) The models are then defined as follows
Smagorinsky [12] νSe = (CSδ)2radic
2I1 (51)
Vreman [16] νVe = (CV δ)2
radicQGGT
PGGT (52)
QR [13 14 15] νQRe = (CQRδ)2 max0minusI3
I1 (53)
Vortex-Stretching [11] νV Se = (CV Sδ)2radic
2I1
(I5 minus 1
2I1I2
minusI1I2
)32
(54)
with the quantities
PGGT = I1 minus I2 QGGT =1
4(I1 + I2)2 + 4
(I5 minus
1
2I1I2
) (55)
There is a fifth and final model to be considered here It is of a different form to the other four andnot an eddy-viscosity model It is known as the Gradient model and is defined by
Gradient [4 9] τmodG = CGδ
(S2 minus Ω2 minus (SΩminus ΩS)
) (56)
The values C are model constants specific to each model and δ represents the mesh size of the LESBoth of these quantities are included here to properly define the models however in this projectthey will be ignored as they are constants and have no influence on the conditions to be defined
33 Conditions for Large-Eddy Simulation without Explicit Filtering
The filtering operation introduced in Eq (48) is just a formality for LES without explicit filteringThe idea here is that filtering does not actually take place but the term lsquoaddedrsquo to the Navier-Stokesequations in fact lsquosubtractsrsquo the difficult to simulate small-scale turbulent motion of the flow Inorder to consider this properly the terms ui and p in Eq (48) are replaced by vi and q It is hopedthat τmod
ij approximates τij so that vi and q approximate ui and p respectively So LES withoutexplicit filtering is given by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (57)
It is clear to see that the first four terms of this equation are exactly the same as the Navier-Stokesequations so for the exact solutions discussed in Section 22 the combination of these terms willequal zero What is then desirable is that the model term is also zero for these exact solutions sothat Eq (57) still holds In order to test models for exact solutions the condition
part
partxjτmodij (v) = 0 (58)
is checked If this condition were to hold it would imply that the model does not influence thevelocity field of a flow hence we call it the Velocity Field Condition There are a number of otherrestrictions that can be tested One such test comes from the differential form of the evolution ofthe kinetic energy of a flow given in Eq (10) In order to find this Eq (57) is multiplied by vi andterms are grouped to find
part
partt
(1
2vivi
)+ vj
part
partxj
(1
2vivi
)= minusvi
1
ρ
partq
partxi+ νvi
part2vipartxjpartxj
minus vipart
partxjτmodij (v) (59)
23
Then similarly to Eq (58) another condition is to check whether the model affects the kineticenergy of exact solutions this is equivalent to checking whether
vipart
partxjτmodij (v) = 0 (60)
We call this condition the Kinetic Energy Condition The third and final condition considered herecan also be found using the integral form for the equation for kinetic energy
d
dt
intV
1
2vividV =
intS
[minusvj
(1
2vivi
)minus vj
q
ρ+ 2νviSij + viτ
modij (v)
]njdS
+
intV
[minus2νSijSij minus
(minusSijτmod
ij (v))]dV
(61)
where the rate-of-strain tensor Sij was defined in Eq (41) V is a volume of fluid and dS meansthe integral is over the surface of that volume with unit normal nj Then this final condition checkswhether the model affects the total kinetic energy of exact solutions to the Navier-Stokes equationsthat is
Sijτmodij (v) = 0 (62)
We will call this condition the Dissipation Condition because Sijτmodij (v) is often referred to as the
subgrid dissipation
34 Conditions for Large-Eddy Simulation with Explicit Filtering
As in Eq (48) the Navier-Stokes equations can be filtered by applying the filtering operation to thewhole equation (putting a bar over all terms) and evaluating to find again that
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (63)
In Eq (57) the filtering operation was ignored and the elements ui p and τij(u) were replacedwith vi q and τmod
ij (v) respectively Now in LES with explicit filtering the filtering operation itselfsmooths over the small-scale turbulent motion of the simulated flow In this case the elements uip and τij(u) will be replaced with vi q and τmod
ij (v) respectively LES with explicit filtering is thengiven by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (64)
Note that Eqs (63) and (64) despite their similarity to Eq (57) no longer contain the Navier-Stokesequations as Eq (57) did The bar (filtering operation) over each term results in each exact solutionfound in Section 22 not necessarily satisfying the first four terms It is hence unknown whether ornot these terms will vanish for exact solutions Terms can still be compared similarly to Eqs (58)(60) and (62) However instead of comparing to zero we can now compare to the relevant termfrom the filtered Navier-Stokes equations So the terms
part
partxjτij (65)
and
part
partxjτmodij (v) (66)
24
are to be compared Although the latter term is relatively simple to compute the former could leadto some difficulties due to the differentiation of the true stress τij = uiuj minus uiuj So although someinformation could be gleaned from the inspection of these two terms more information can be foundby considering the analogue of Eq (62) So the true subfilter dissipation
Sijτij (67)
(where Sij is shorthand for Sij(v)) and the modelled subfilter dissipation
Sij(v)τmodij (v) (68)
are considered and have to be equal for the condition to hold These can again be derived from theintegral equation for kinetic energy of a filtered velocity field similarly to Eq (61)
4 Analysis of Large-Eddy Simulation Models
In this section we use a code in Mathematica the theory described in Section 3 and the exactsolutions to the Navier-Stokes equations found in Section 22 to check whether or not each of theexact solutions lsquoendorsesrsquo each model for LES both without and with explicit filtering
41 Large-Eddy Simulation Without Explicit Filtering
We start by using a code in Mathematica to test the Velocity Field condition in Eq (58) and theKinetic Energy condition in Eq (60) For all exact solutions to the Navier-Stokes equations inSection 22 the outputs for both conditions were identical so we have combined them into one table
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 ndash ndash 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 1 For the Kinetic Energy condition a lsquo0rsquo result means that vipartpartxj
τmodij (v) = 0 for respective
models and exact solutions whereas a lsquolowastrsquo result means that vipartpartxj
τmodij (v) 6= 0 For the Velocity
Field condition a lsquo0rsquo result means that partpartxj
τmodij (v) = 0 for respective models and exact solutions
whereas a lsquolowastrsquo result means that partpartxj
τmodij (v) 6= 0 The Gradient model can take both positive and
negative values so when the sign of the model is known it is included in the table instead of a lsquorsquoBoth the conditions are included in one table because for all cases the outputs were identical
We then test the Dissipation condition in Eq (62) for all exact solutions to the Navier-Stokesequations in Section 22
25
Dissipation Condition [Does Sijτmodij (v) = 0]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 + 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 2 A lsquo0rsquo result means that Sijτmodij (v) = 0 for respective models and exact solutions whereas
a lsquolowastrsquo result means that Sijτmodij (v) 6= 0 The Gradient model can take both positive and negative
values so when the sign of the model is known it is included in the table instead of a lsquorsquo
We then sum all the results to find the total number of endorsements for each model described inSection 32
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
4 6 8 9 9
lsquoDissipationrsquoEndorsements
2 5 7 7 9
Total Endorsements 6 11 15 16 18
Table 3 The results from Tables 1 and 2 are summarised to give a general overview of how manyendorsements each model receives This makes it much clearer to see the models most supported bythe respective and combined conditions with the Vortex-Stretching model being the most supportedand the Smagorinsky model the least
Testing the models in Section 32 using LES without explicit filtering has mixed results It isclear that the Smagorinksy model is not well endorsed by the flows in Section 22 with the onlyendorsements coming from the simplest flows that is Sol 1 (Linear Velocity) Sol 3 (Velocity as afunction of time) Sol 7 (Stagnation Flow) and Sol 15 (Rankine Vortex) The Vortex-Stretching-based model is the most endorsed flow with eighteen out of a possible thirty endorsements Thisagrees with our foreknowledge as this model was constructed to be as physically consistent as possiblein the paper by Silvis amp Verstappen [11]
42 Large-Eddy Simulation With Explicit Filtering
In this section we will compare the true subfilter dissipation Sijτij and the modelled subfilterdissipation Sij(v)τmod
ij (v) given in Eqs (67) and (68) respectively for each model τmodij and each
exact solution to the Navier-Stokes equation given in Section 22 We will not consider the conditionsinvolving differentiation in Eqs (65) and (66) due to the difficulty in calculating the derivative ofthe τij term in Eq (65)
26
421 Example of a Test for a Model
The true subfilter dissipation Sijτij of a flow given in Eq (67) cannot be explicitly computed asthe filtering operation is kept general What can be done however is to partially evaluate theexpressions using the attributes of a filtering operation mentioned in Eq (48)
The calculations of Sijτij for Sol 1 (Linear Velocity) are provided to illustrate the general ideaRecall that the velocity field of Sol 1 (Linear Velocity) is written in Eq (13) in vector notation as
u(x y z t) =
0ax+ bcx+ d
(69)
Then Sij = Sij(u) is computed simply by using its definition in Eq (41) Note that as filteringpreserves constants if there is an a it is replaced with a meaning that (in this case) there are alarge number of zero terms due to the frequency of 0 Hence
[Sij(u)
]=
[1
2
(partuipartxj
+partujpartxi
)]=
0 a
2c2
a2 0 0
c2 0 0
=
0 a
2c2
a2 0 0
c2 0 0
(70)
Note that as Sijτij is being computed which is equal to the sum of the element-wise multiplicationof the two matrices (or the trace of the matrix multiplication) that we donrsquot need to consider theelements of τij(u) with corresponding zero entry in Sij As τij(u) = uiuj minus uiuj and again asfiltering preserves constants it is found that
[τij(u)] =[uiuj minus uiuj
]=
lowast 0 middot ax+ b 0 middot cx+ d
ax+ b middot 0 lowast lowastcx+ d middot 0 lowast lowast
=
lowast 0 00 lowast lowast0 lowast lowast
(71)
where a lsquolowastrsquo just represents some element (not necessarily non-zero) Therefore the true subfilterdissipation
Sijτij = tr
0 a2
c2
a2 0 0c2 0 0
middotlowast 0 0
0 lowast lowast0 lowast lowast
= 0 (72)
Now the modelled subfilter dissipation Sij(u)τmodij (u) needs to be computed (recall that Sij =
Sij(u)) In this case let us use the QR model stated in Eq (53) Once again Sij(u) calculated inEq (70) needs to be used So we have that
τmodij = minus2(CQRδ)
2 max0minusI3I1
Sij (73)
As
I1 = tr(S2) =
1
2(a2 + c2) and I3 = tr(S
3) = 0 (74)
it holds that τmodij (u) = 0 so Sij(u)τmod
ij (u) = 0 This agrees with Eq (72) and hence Sol 1(Linear Velocity) endorses the QR model
27
422 Testing the Models
Now using a Mathematica code the exact solutions in Section 22 are again used to evaluate thecondition for the models using the process just described in Section 421 The results are describedin Table 4
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 4 All entries with a ldquo0rdquo represent a value of zero and a ldquolowastrdquo represents a value that is non-zeroThe ldquoExactrdquo row represents the value of the true subfilter dissipation Sijτij Every other row witha model name represents the value of the modelled subfilter dissipation Sij(u)τmod
ij (u) respectiveto that model A ldquordquo means that it is unclear as to the value - the sign of these values is oftenunknown and the lsquoMaxrsquo function in the definition of the QR model (in Eq (53)) makes it difficultto evaluate
Reading from Table 4 the Smagorinsky model is clearly endorsed by Sol 3 (Velocity as a function oftime) and Sol 15 (Rankine Vortex) as in both cases both Sijτij = 0 = Sij(u)τmod
ij (u) In this casewe say that Sol 3 (Velocity as a function of time) and Sol 15 (Rankine Vortex) lsquostronglyrsquo endorsethe Smagorinsky model The Smagorinsky model is not endorsed by Sol 1 (Linear Velocity) Sol 2(Planar Flow) or Sol 14 (Radially Dependent) however because Sij(u)τmod
ij (u) 6= 0 For Sol 4 to
13 both Sijτij 6= 0 6= Sij(u)τmodij (u) and so we will say that these exact solutions lsquoweaklyrsquo endorse
the Smagorinsky model
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 10 10 9 6
Table 5 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Interestingly the lsquostrongrsquo and lsquoweakrsquo endorsements give different results However more stock shouldbe put in the results of the lsquostrongerrsquo condition as it is completely clear that zero is equal to zeroIt is not necessarily clear whether or not non-zero equals non-zero as the values will depend on theparameters in each solution hence the lsquoweakerrsquo endorsement
The lsquostrongrsquo condition equally endorses the Vreman QR Gradient and Vortex-Stretching-basedmodels whereas the lsquoweakrsquo condition endorses Vreman and Smagorinsky the most closely followedby the Gradient model with the Vortex-Stretching-based model being the most opposed As this
28
completely contradicts the lsquostrongrsquo condition and the results in Section 41 it makes us question thevalidity of the weaker test The number of endorsements for the QR model is unknown because of thedefinition of the model in Eq (53) and our evaluation of the true subfilter dissipation Eq (67) Dueto the exact solutions we have found it is never clear whether or not the true subfilter dissipationis strictly positive or negative This means that when the true subfilter dissipation is found to benon-zero the lsquoMaxrsquo function in the QR model cannot be evaluated
43 Vremanrsquos True Subfilter Dissipation Claim
Using the theory mentioned in Sections 232 and 34 we can place the exact solutions to the Navier-Stokes equations into different classes defined by Vreman In doing this we can also attempt to verifyone of Vremanrsquos results [16] In Vremanrsquos paper he defines the class Qn to be the set of flows withn zero entries in the velocity gradient of the flow (the velocity gradient is defined in Eq (39))Vreman then claims that the only flows with zero true subfilter dissipation are a certain selection ofthose in the classesQ7Q8 orQ9 shown in appendix B and is generally non-zero for any other flows
The true subfilter dissipation of Sol 1 (Linear Velocity) was computed in Section 421 and wasfound to be zero in Eq (72) The velocity gradient of Sol 1 is as follows
GLV =
0 0 0a 0 0c 0 0
(75)
and hence Sol 1 (Linear Velocity) is in the required subset of Q7 which agrees with Vremanrsquos claimIn Section 422 the true subfilter dissipation was computed (determined to be zero or non-zero) forall of the exact solutions to the Navier Stokes equations in Section 22 These solutions can now belsquoplacedrsquo into each class Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a functionof time) and Sol 14 (Radially Dependent) were all found to have zero true subfilter dissipation andare all in the appropriate subsets of either Q7Q8 or Q9 This agrees with Vremanrsquos results
We can extend Vremanrsquos results however We found that Sol 15 (Rankine Vortex) had zero truesubfilter dissipation and yet is not in the subset of Q7 defined in Appendix B It is important tonote that the rate of strain tensor Sij in Eq (41) for Sol 15 (Rankine Vortex) is zero This is thereason the true subfilter dissipation is zero and the same reason that all models are zero for thisflow as all models are based on Sij
We also found that all of the remaining exact solutions to the Navier-Stokes equations are not inVremanrsquos specified classes (in Appendix Section B) They also have a non-zero subfilter dissipationwhich agrees with Vremanrsquos results in his paper [16]
44 Classes of Flows
It is important to consider that some exact solutions to the Navier-Stokes equations may have similarproperties For example Sol 1 (Linear Velocity) and Sol 2 (Planar Flow) are both very similarIn fact Sol 2 (Planar Flow) is a more general case of Sol 1 (Linear Velocity) We therefore mustgroup the different flows into classes in an attempt to reduce the bias that may occur when solutionsare similar This should aid us to further determine which of the models are most endorsed We willdo this in a number of different ways using the theory discussed in Section 23 We will group theflows with respect to Vremanrsquos Flow Classes given in appendix B and with respect to the Invariantsgiven in appendix A
29
441 Vremanrsquos Velocity Gradient Flow Classes
We now put our exact solutions to the Navier-Stokes equations into one of two classes Either Q0minus6or Q7minus9 where each class is defined intuitively as follows
Q0minus6 = Q0 cupQ1 cup middot middot middot cup Q6 (76)
Q7minus9 = Q7 cupQ8 cupQ9 (77)
where Qn is defined in Section 43 By looking at the gradient of each exact solution we can see thatQ7minus9 contains the six solutions Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as afunction of time) Sol 5 (2D Taylor) Sol 14 (Radially Dependent) and Sol 15 (Rankine Vortex)and that Q0minus6 contains the remaining nine We now check the models for LES without explicitfiltering by considering endorsements within each class Tables 6 and 7 are analogues of Table 3 andillustrate the results for each flow class The analogues of Tables 1 and 2 are Tables 22 and 23 andare in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 6 6 6
lsquoDissipationrsquoEndorsements
2 5 6 6 6
Percentage Endorsed 417 833 100 100 100
Table 6 Each entry in the top two rows represents the number of solutions in the class Q7minus9 thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Q0minus6 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 2 3 3
lsquoDissipationrsquoEndorsements
0 0 1 0 3
Percentage Endorsed 56 56 167 167 333
Table 7 Each entry in the top two rows represents the number of solutions in the class Q1minus6 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking results to 1 decimal place if necessary
All of the flows in the class Q7minus9 endorse the QR Gradient and Vortex-Stretching-based subfilter-scale models (see Table 6) We can hence see that this class fully endorses these three models equallyand does not endorse the Smagorinsky model The Vortex-Stretching eddy viscosity model has themost endorsements from any of the flows in the class Q0minus6 and the Smagorinksy and Vreman modelshave the fewest endorsements (see Table 7) We can hence say that this class mostly endorses theVortex-Stretching-based eddy-viscosity model note however that the percentage endorsement is very
30
low
Now let us consider LES with explicit filtering Tables 8 and 9 are analogues of Table 5 and illustratethe results for each flow class The analogue of Table 4 is Table 24 and is in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 1 1 0 0 0
Table 8 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Q0minus6 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 9 9 9 6
Table 9 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We have grouped Vremanrsquos classes into Q7minus9 and Q0minus6 this is equivalent to saying that we havegrouped Vremanrsquos classes into those with two or fewer non-zero velocity gradient elements and thosewith three or more non-zero velocity gradient elements respectively Therefore the solutions in Q7minus9are exactly those with a lsquo0rsquo in the lsquoExactrsquo column of Table 4 and Sol 5 (2D Taylor) ExcludingSol 5 (2D Taylor) these are the same as those five that endorse the Vreman QR Gradient andVortex-Stretching-based models in Table 5 We can see in Table 8 that the results for the lsquostrongrsquoendorsements are exactly the same as in Table 5 This is because all of the flows with zero subfilterdissipation are contained in the class Q7minus9 There are hence no lsquostrongrsquo endorsements in Table 9The majority of the lsquoweakrsquo endorsements come from Q0minus6 as this class contains all but one of thesolutions with non-zero true subfilter dissipation (given in Eq (67))
442 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
We now separate our exact solutions into classes again using Vremanrsquos flow classes This timehowever we define V0 to be the class of flows Vreman claims will always have zero true subfilterdissipation namely those classes mentioned in Appendix B We define Vlowast to be all the remaining
31
classes Then
V0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow)
Sol 3 (Velocity as a function of time) Sol 14 (Radially Dependent)(78)
Vlowast = Sol 4 (Axisymmetric Flow) Sol 5 (2D Taylor) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 13 (Sphere Translation) Sol 15 (Rankine Vortex)
(79)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 10 and 11 are analogues of Table 3 and illustrate the results for each flow class Theanalogues of Tables 1 and 2 are Tables 25 and 26 and are in Appendix C2
Endorsements for models using solutions in V0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 4 4 4 4
lsquoDissipationrsquoEndorsements
1 4 4 4 4
Percentage Endorsed 375 100 100 100 100
Table 10 Each entry in the top two rows represents the number of solutions in the class V0 thatendorse each model There are 4 solutions in this class so the percentages are worked out by dividingthe total of each column by 8 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Vlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 2 4 5 5
lsquoDissipationrsquoEndorsements
1 1 3 2 5
Percentage Endorsed 136 136 318 318 455
Table 11 Each entry in the top two rows represents the number of solutions in the class Vlowast thatendorse each model There are 11 solutions in this class so the percentages are worked out bydividing the total of each column by 22 and taking results to 1 decimal place if necessary
All of the flows in the class V0 endorse the Vreman QR Gradient and Vortex-Stretching-basedsubfilter-scale models (see Table 6) This class completely endorses these four models equally anddoes not endorse the Smagorinsky model The Vortex-Stretching-based eddy-viscosity model hasthe most endorsements from the class Vlowast and the Smagorinksy and Vreman models again have thefewest endorsements (see Table 7) As in Section 441 we say that this class mostly endorses the
32
Vortex-Stretching-based eddy-viscosity model but once again the percentage endorsement is ratherlow
Now let us consider LES with explicit filtering Tables 12 and 13 are analogues of Table 5 andillustrate the results for each flow class The analogue of Table 4 is Table 27 and is in Appendix C2
Endorsements for models using solutions in V0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 4 4 4 4
lsquoWeakrsquo Endorsements 0 0 0 0 0
Table 12 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Vlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 1 1 1 1
lsquoWeakrsquo Endorsements 10 10 9 6
Table 13 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We can see in Table 12 that the results for the lsquostrongrsquo endorsements are similar to those in Table 5excluding Sol 15 (Rankine Vortex) The four solutions in V0 with zero true subfilter dissipationnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)and Sol 14 (Radially Dependent) are hence the four that endorse the Vreman QR Gradientand Vortex-Stretching-based models in Table 12 Sol 15 (Rankine Vortex) is in the class Vlowast andhence there is one lsquostrongrsquo endorsement in Table 13 The class Vlowast contains all of the flows with anon-zero true subfilter dissipation (given in Eq (67)) and hence Table 13 contains all of the lsquoweakrsquoendorsements
443 Classes using the Third Principal Invariant
Sorting the classes using the invariants is a non-trivial task in itself There are a number of waysin which to do this using both the principal invariants in Eq (40) and the combined invariants inEq (43)
We start by looking at flows with third principal invariant RG equal to zero We again define twoclasses R0 and Rlowast where R0 consists of all flows with RG = 0 and Rlowast consists of all flows withRG 6= 0 Then using the information in Appendix A we see that there are nine flows in R0 That
33
is
R0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 8 (Vortex in Stagnation Flow)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)
(80)
Rlowast = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 13 (Sphere Translation)(81)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 14 and 15 are analogues of Table 3 and they illustrate the results for LES withoutexplicit filtering for each flow class The analogues of Tables 1 and 2 are Tables 28 and 29 and arein Appendix C3
Endorsements for models using solutions in R0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 7 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 278 556 778 722 778
Table 14 Each entry in the top two rows represents the number of solutions in the class R0 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking the result to one decimal place if necessary
Endorsements for models using solutions in Rlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 2 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 83 83 83 167 333
Table 15 Each entry in the top two rows represents the number of solutions in the class Rlowast thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking the result to one decimal place if necessary
The class R0 endorses both the Vortex-Stretching-based model and the QR model the most andSmagorinsky the least in the case for LES without explicit filtering In the Rlowast class the Vortex-Stretching-based model is again the most endorsed followed by the Gradient model however thepercentages are again small
34
Now let us consider LES with explicit filtering Tables 16 and 17 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 30 and is in Appendix C3
Endorsements for models using solutions in R0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 4 4 3 2
Table 16 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Rlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 6 6 6 4
Table 17 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 we can see in Table 16 that the results for the lsquostrongrsquo endorsements are exactly thesame as in Table 5 This is because all of the flows with zero subfilter dissipation are contained in theclass R0 As in Table 9 there are no lsquostrongrsquo endorsements in Table 17 The lsquoweakrsquo endorsementshowever are split more evenly between the two classes now than in Vremanrsquos classes in Sections 441and 442 We find that the Smagorinsky and Vreman models still seem to be the most endorsed inboth classes
444 Two-component Flow Classes
Another way in which the flows can be grouped into classes is by determining whether a flow is atwo-component flow or not We define a two-component flow to have the property that the combinedprincipal invariants of a flow in Eq (43) satisfy
I3 = I4 = I5 minus1
2I1I2 = 0 (82)
We are also interested in considering flows with zero gradient model subgrid dissipation These flowshave the property that
I3 minus I4 = 0 (83)
What we find though is that for all the solutions in Section 22 if a flow is a two-component flowthen it also satisfies I3 minus I4 = 0 It may be interesting to note that although I3 minus I4 6= 0 for manysolutions there are a number with I3 equal to a multiple of I4 Although not considered in thisreport this may also be a class of interest We find that the two-component flows are
C2 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)(84)
35
and we call the remaining flows three-component flows They are
C3 = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex) Sol 11 (Pseudo-Motion Shots)
Sol 13 (Sphere Translation)
(85)
Note that Sol 8 (Vortex in Stagnation Flow) is defined to be a planar flow this does not mean that itis a two-component flow in this sense however because its two non-zero velocity components dependon three coordinates We now check the models for LES without explicit filtering by consideringendorsements within each class Tables 18 and 19 are analogues of Table 3 and they illustrate theresults for LES without explicit filtering for each flow class The analogues of Tables 1 and 2 areTables 31 and 32 and are in Appendix C4
Endorsements for models using solutions in C2 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 6 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 357 714 100 857 100
Table 18 Each entry in the top two rows represents the number of solutions in the class C2 thatendorse each model There are 7 solutions in this class so the percentages are worked out by dividingthe total of each column by 14 and taking the result to one decimal place if necessary
Endorsements for models using solutions in C3 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 3 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 63 63 63 188 250
Table 19 Each entry in the top two rows represents the number of solutions in the class C3 thatendorse each model There are 8 solutions in this class so the percentages are worked out by dividingthe total of each column by 16 and taking the result to one decimal place if necessary
The class C2 equally endorses the QR and Vortex-Stretching-based models and once again theSmagorinsky model is the least endorsed As with the previous classes the class C3 has such lowpercentages that it does not seem to endorse any models We start to see a trend appearing here inthat the flow classes that contain the more complicated flows namely Q0minus6 (Eq (76)) Vlowast (Eq (79))Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages The classes that tendto contain the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78)) R0 (Eq (80)) and C2 (Eq (84))all seem to favour the Vortex-Stretching-based model or the QR model
36
Now let us consider LES with explicit filtering Tables 20 and 21 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 33 and is in Appendix C4
Endorsements for models using solutions in C2 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 2 2 0 1 0
Table 20 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in C3 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 8 8 8 6
Table 21 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 and Table 16 we can see in Table 20 that the results for the lsquostrongrsquo endorsements areexactly the same as in Table 5 This is again because all of the flows with zero subfilter dissipationare contained in the class C2 There are two flows with non-zero true subfilter dissipation (givenin Eq (67)) in C2 and this shows in the lsquoweakrsquo endorsements in Table 20 There are no lsquostrongrsquoendorsements in Table 21 as all the zero subfilter dissipation flows are in C2
5 Conclusions
In this report we studied incompressible turbulent flows We found a number of exact solutions to theNavier-Stokes equations each representing a relatively simplistic fluid flow Large-eddy simulation(LES) is used to model turbulent flows by smoothing over the complex small-scale motion Weintroduced a number of subfilter-scale models to approximate this small-scale motion and testedthem using the found exact solutions to the Navier-Stokes equations We first considered LESwithout explicit filtering and used the Navier-Stokes equations to derive conditions for the subfilter-scale models namely the Velocity Field condition (Eq (58)) the Kinetic Energy condition (Eq (60))and the Dissipation condition (Eq (62)) Similarly we considered LES with explicit filtering andagain used the Navier-Stokes equations to derive more conditions for the models in this case wedecided to just take one condition the Dissipation condition (Eqs (67) and (68)) as the others leadto difficulty in their evaluation By lsquorunningrsquo the exact solutions to the Navier-Stokes equationsthrough each condition for the subfilter-scale models we found that the Vortex-Stretching-basededdy-viscosity model in Eq (54) was most endorsed by the tests in LES without explicit filteringWe found that four out of the five subfilter-scale models namely the Vreman QR Gradient andVortex-Stretching-based models were equally endorsed by the lsquostrongrsquo tests in LES with explicitfiltering with the Smagorinsky model being the least endorsed The results we found for the lsquoweakrsquocondition were completely opposite to those found in both the lsquostrongrsquo condition and in LES withoutexplicit filtering This led us to conclude that this test was not one in which we should put our trust
37
We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses
In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes
This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold
38
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
was instead found)
u1(x y z t) = minus sin(πy) eminus(π2t)Re
u2(x y z t) = sin(πx) eminus(π2t)Re
u3(x y z t) = 0
p(x y z t) = minus cos(πx) cos(πy) eminus(2π2t)Re
(20)
Figure 5 An illustration of the flow in the xy-plane for Sol 5 (2D Taylor) described in Eq (20)with Re = 1 and time t = 0
Solution 6 Generalized Beltrami Flow
Again using the text by Barbato et al [2 p65] and now considering solutions with nonndashtrivialterms in all dimensions a particular form of the generalized Beltrami flow is found namely
u1(x y z t) = (A sin(πz) + C cos(πy)) eminus(π2t)Re
u2(x y z t) = (B sin(πx) +A cos(πz)) eminus(π2t)Re
u3(x y z t) = (C sin(πy) +B cos(πx)) eminus(π2t)Re
p(x y z t) = minus[(AB cos(πz) sin(πx) +BC cos(πx) sin(πy)
+AC cos(πy) sin(πz)] eminus(2π2t)Re
(21)
where ABC isin R
10
Figure 6 An illustration of the flow in the yz-plane for Sol 6 (Generalized Beltrami Flow)described in Eq (21) with parameters A = B =C = 1 and time t = 0
Solution 7 Stagnation Flow
In the book by Drazin and Riley [6 p40] an exact solution is considered which uses a stream functionψ = ψ(y z) They suggest that the following velocity field is a solution
u1(x y z t) =a
kx
u2(x y z t) = minusaky +
k
Re
partψ(y z)
partz
u3(x y z t) = minus k
Re
partψ(y z)
party
(22)
Here we choose the stream function to be ψ(y z) = yz Then a solution is given by
u1(x y z t) =a
kx
u2(x y z t) =
(minusak
+k
Re
)y
u3(x y z t) = minus kzRe
p(x y z t) = minus1
2
((axk
)2minus((
k
Reminus a
k
)y
)2
minus(kz
Re
)2)
(23)
where a is a measure of the strength of the stagnation flow and k is a constant chosen in the textto specify the distance between vortices There are no vortices in this particular solution however
11
Figure 7 An illustration of the flow in the yz-plane for Sol 7 (Stagnation Flow) describedin Eq (23) with parameter values a = Re = 1and k = 2
Figure 8 An illustration of the flow in the xy-plane for Sol 7 (Stagnation Flow) describedin Eq (23) with parameter values a = Re = 1and k = 2
Solution 8 Vortex in Stagnation Flow
As in Sol 7 (Stagnation Flow) we now consider the same solution with a different stream functionnamely ψ(z) = cos z Then given the condition that k = aRe and adding a pressure term Eq (22)becomes
u1(x y z t) =1
Rex
u2(x y z t) = minus 1
Rey minus k sin z
u3(x y z t) = 0
p(x y z t) = minus 1
2Re2(x2 + y2)
(24)
where a and k are as in Sol 7 (Stagnation Flow)
12
Figure 9 An illustration of the flow in the yz-plane for Sol 8 (Vortex in Stagnation Flow)described in Eq (24) with parameter valuesk = 3 and Re = 10
Figure 10 An illustration of the flow in thexy-plane for Sol 8 (Vortex in StagnationFlow) described in Eq (24) with parametervalues k = 3 and Re = 10
Solution 9 Burgers Vortex
The Burgers Vortex solution is given in cylindrical coordinates in the book by Lautrup [8 p 385]as follows
ur(r ϕ z t) = minus 2r
c2Re
uϕ(r ϕ z t) =Ωc2
r
(1minus eminusr
2c2)
uz(r ϕ z t) =4z
c2Re
p(r ϕ z t) = minusc2Ω2F
(r2c2
)2
minus 2
Re2r2 + 4z2
c4
(25)
where
F (ξ) =
infinintξ
(1minus eminusx)2
x2dx (26)
This solution is not particularly nice to look at however most of the unpleasantness comes fromthe pressure term which (in this project) is only required in order to validate the solution For thisproject all solutions are required in Cartesian coordinates the velocity field of the Burgers Vortexhas hence been converted into Cartesian coordinates
u1(x y z t) = minus 2x
c2Reminus Ω
h(x y)(1minus exp (minush(x y))) y
u2(x y z t) = minus 2y
c2Re+
Ω
h(x y)(1minus exp (minush(x y)))x
u3(x y z t) =4z
c2Re
(27)
13
where
h(x y) =x2 + y2
c2 (28)
and Ω and c are constants
Figure 11 An illustration of the flow for Sol9 (Burgers Vortex) described in Eq (27) withparameter values c = 15 Re = 20 and Ω = 10A slice of the xy-plane at z = 0 is shown
Solution 10 Another Vortex Solution
There is another vortex solution more simply formulated than the Burgers Vortex yet with asingularity at r = 0 as a result of the uϕ term It is again given in cylindrical coordinates
ur(r ϕ z t) = minus 2r
c2Re
uϕ(r ϕ z t) =a
r
uz(r ϕ z t) =4z
c2Re
p(r ϕ z t) = minus a2
2r2minus 2r2 + 8z2
c2Re2
(29)
which can also be converted into Cartesian coordinates
14
u1(x y z t) = minus 2x
c2Reminus ay
x2 + y2
u2(x y z t) = minus 2y
c2Re+
ax
x2 + y2
u3(x y z t) =4z
c2Re
(30)
where a and c are constants
Figure 12 An illustration of the flow for Sol9 (Burgers Vortex) described in Eq (27) withparameter values c = 15 Re = 20 and a = 10 Aslice of the xy-plane at z = 0 is shown
Solution 11 Pseudo-Motion Shots of the Second Kind
In the book by Berker [3 p 94] a solution is given in cylindrical coordinates by
ur(r ϕ z t) =2
Re
1
r
uϕ(r ϕ z t) = A1r3 +B1r +
C1
r
uz(r ϕ z t) = A2r2 log(r) +B2r
2 + C2
p(r ϕ z t) =A2
1r6
6+A1B1r
4
2+A1C1r
2 +B2
1r2
2minus C2
1
2r2minus 2
r2Re2+ 2B1C1 log(r)
+2A2z
Reminus 4B1ϕ
Re
(31)
where A1 A2 B1 B2 C1 C2 are constants Eq (31) when converted into Cartesian coordinates is
15
u1(x y z t) =1
Re
2xminus C1Rey
x2 + y2minusA1y
(x2 + y2
)minusB1y
u2(x y z t) =1
Re
2y + C1Rex
x2 + y2+A1x
(x2 + y2
)+B1x
u3(x y z t) =1
2A2
(x2 + y2
)log(x2 + y2
)+B2
(x2 + y2
)+ C2
(32)
Figure 13 An illustration of the flow in thexy-plane for Sol 11 (Pseudo-Motion Shots)described in Eqs (31) and (32) with parametervalues A1 = B1 = C1 = 1 A2 = B2 = C2 = 2and Re = 10
Figure 14 An illustration of the flow in theyz-plane for Sol 11 (Pseudo-Motion Shots)described in Eqs (31) and (32) with parametervalues A1 = B1 = C1 = 1 A2 = B2 = C2 = 2and Re = 10
Solution 12 Flows With Constant Whirl
In the book by Berker [3 p 143] another solution is given in cylindrical coordinates by
ur(r ϕ z t) = minus 1
Re
1
r
uϕ(r ϕ z t) =1
rradicte(minus
12 g(t))
uz(r ϕ z t) = 0
p(r ϕ z t) =1
4
2(minus e
minusg(t)
t minus 1Re2
)r2
minus 1
t2Re
infinintminusg(t)
eminust
tdt
(33)
where we define g(t) =r2Re
2t When converted into Cartesian coordinates we find
16
u1(x y z t) = minus 1
x2 + y2
(yradict
exp
(minus
Re(x2 + y2
)4t
)+
x
Re
)
u2(x y z t) =1
x2 + y2
(xradict
exp
(minus
Re(x2 + y2
)4t
)minus y
Re
)
u3(x y z t) = 0
(34)
Figure 15 An illustration of the flow for Sol12 (Flows with Constant Whirl) described inEqs (33) and (34) with time t = 1 and Re = 1 Aslice of the xy-plane at z = 0 is shown as velocityin the z-direction is zero this is the same for anyvalue of z
17
Solution 13 Rectilinear and Uniform Translation of a Sphere
In the book by Berker [3 p 228] another solution is given in Cartesian coordinates by
u1(x y z t) =3
4
axz
R3
(a2
R2minus 1
)
u2(x y z t) =3
4
ayz
R3
(a2
R2minus 1
)
u3(x y z t) = 1minus 3
4
a
Rminus 1
4
a3
R3minus 3
4
az2
R3
(a2
R2minus 1
)
p(x y z t) =3a(aminusR)(a+R)
32R10Re
((R2Re
(x2 + y2
)minus 2R2Rez2
) (a3 + 3aR2 minus 4R3
)+ 3aRez4
(R2 minus a2
)minus 16R5z
)
(35)
where a and R are constants
Figure 16 An illustration of the flow in the yz-plane for Sol 13 (Sphere Translation) describedin Eq (35) with parameter values a = minus2 andRe = 3
Solution 14 Radially Dependent Flow
A solution with velocity in the z-direction dependent only on the distance from the z-axis is as
18
follows in cylindrical coordinates
ur(r ϕ z t) = 0
uϕ(r ϕ z t) = 0
uz(r ϕ z t) = A(r2 minus a2
)
p(r ϕ z t) =4Az
Re
(36)
where A and a are constants In Cartesian coordinates the solution is just the same with u1(x y z t)and u2(x y z t) both zero and the r2 term replaced with x2 + y2 in u3(x y z t)
Figure 17 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus1 1] withparameter values a = 1 A = 4 and Re = 10
Figure 18 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus2 2] withparameter values a = 1 A = 4 and Re = 10
Solution 15 Rankine Vortex
A solution discovered by Rankine [1] is a vortex flow of the following form
ur(r ϕ z t) = 0
uϕ(r ϕ z t) =ar
2πR2
uz(r ϕ z t) = 0
p(r ϕ z t) =a2r2
8π2R4
(37)
19
where a and R are constants In Cartesian coordinates the flow has the form
u1(x y z t) = minus ay
2πR2
u2(x y z t) =ax
2πR2
u3(x y z t) = 0
p(x y z t) =a2(x2 + y2
)8π2R4
(38)
This flow is actually a rigid body rotational flow in that the angular rotational velocity is uniformand the velocity of the flow increases proportionally with distance from the z-axis
Figure 19 An illustration of the flow in the xy-plane for Sol 15 (Rankine Vortex) described inEq (37) and Eq (38) with parameter values a = 1and R = 1
radic2π
23 Characterisation of Flows
It is important to be able to classify flows in a way that sorts them into distinct groups This aidsin the clear identification and general study of flows It is also a very important way of recognisingflows with similar properties In this subsection we introduce two distinct ways in which flows canbe characterised
231 Characterising Solutions using Invariants
A principal invariant of a tensor is a coefficient of the characteristic polynomial of that tensorInvariants as their name suggests are invariant under rotations of the coordinate system in thattheir values do not change when the tensor is lsquorotatedrsquo into a different set of coordinates Thevelocity gradient of a flow G is defined as the Jacobian of the velocity vector given by
Gij =
[partuipartxj
] (39)
20
for i j = 1 2 3 There are three principal invariants of the velocity gradient of a fluid flow G givenby
PG = tr(G)
QG =1
2
((tr(G))2 minus tr
(G2))
RG = det(G)
(40)
Incompressibility of flows (Eq (5)) ensures that PG = 0 hence the first principal invariant is zerofor all of the flows in Section 22 We now define the rate-of-strain tensor S by
Sij =1
2
(partuipartxj
+partujpartxi
) (41)
and the rate-of-rotation tensor Ω by
Ωij =1
2
(partuipartxjminus partujpartxi
) (42)
Note that the first principal invariant of S and the first and third principal invariants of Ω are zeroAlso note that the velocity gradient of a flow can simply be written G = S + Ω
Rather than considering the principal invariants we can also consider the combined invariants
I1 = tr(S2) I2 = tr(Ω2) I3 = tr(S3) I4 = tr(SΩ2) I5 = tr(S2Ω2) (43)
We can hence write the three principal invariants for a velocity field G in terms of the combinedinvariants
PG = 0
QG = minus1
2(I1 + I2)
RG =1
3(I3 + 3I4)
(44)
In this report the flows considered are the exact solutions to the incompressible Navier-Stokesequations described in Section 22 The combined invariants in Eq (43) are used to characterisethese solutions into different classes in Section 44 Both the principal invariants and combinedinvariants for each exact solution in Section 22 are given in Appendix A
Each of the subfilter-scale eddy-viscosity models described in Section 32 are written in terms ofthese five combined invariants
232 Vremanrsquos Characterisation of Flows
Vreman [16] classifies flows using the gradient of the velocity field of a flow Each lsquoclassrsquo of flows hasthe same number of non-zero elements in the gradient matrix For example
[partuipartxj
]=
0 0 0lowast 0 0lowast 0 0
(45)
where a lowast denotes a non-zero term represents a group of flows that lie within the class denoted byQ7 (see Appendix B) where the subscript represents the number of non-zero terms in the velocity
21
gradient Flows of the form in Eq (45) have a velocity field of the form
u1 = c1 (constant)
u2 = f2(x1)
u3 = f3(x1)
(46)
where f2(x1) and f3(x1) are such that the Navier Stokes equations Eq (1) are satisfied Notethat as the elements on the leading diagonal are zero the incompressibility condition Eq (5) isautomatically satisfied Some simple flow classes are given in Appendix B and contain a number ofthe solutions discussed in Section 22 This is mentioned in greater detail in Section 441
3 Conditions for Large-Eddy Simulation Models
31 Large-Eddy Simulation
Due to the nature of numerical simulation and the complexity of turbulent flows small details aremissed and information is lost when solving the Navier-Stokes equations The computational costof solving a turbulent flow is extremely high so a filtering operation is applied to the Navier-Stokesequations represented by a bar to reduce the computational cost of the simulation This is a linearoperation and preserves constants ie ρ = ρ and is also assumed to commute with differentiation
partuipartxj
=partuipartxj
(47)
The filtered Navier-Stokes equations are found by filtering (putting a bar over) the entire equations(see Eq (1)) and after evaluating are given by
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (48)
where
τij = uiuj minus uiuj (49)
is called the lsquotrue stressrsquo of the flow This however poses a problem as it is not clear what is meantby the term uiuj for a general filtering operation The true stress τij therefore must be replaced bya model τmod
ij in practical applications
32 Subfilter-Scale Models
We will now define the subflter-scale models denoted by τmodij to be used in place of the lsquodifficultrsquo
true stress τij There are a number of models available to use in the literature [11 p7] These willbe tested using the exact solutions to the Navier-Stokes equations and by checking conditions whichare defined in Sections 33 and 34
All the models barring one are of the same form That is each of the eddy-viscosity models arewritten
τmodij minus 1
3τmodkk δij = minus2νeSij (50)
with νe representing the eddy viscosity of a specific model For simplicity we usually just writeτmodij = minus2νeSij the extra term is included here to emphasise that the models are traceless Each
22
of the models we consider here can be expressed in terms of the rate of strain tensor S defined inEq (41) the rate of rotation tensor Ω defined in Eq (42) and the combined invariants defined inEq (43) The models are then defined as follows
Smagorinsky [12] νSe = (CSδ)2radic
2I1 (51)
Vreman [16] νVe = (CV δ)2
radicQGGT
PGGT (52)
QR [13 14 15] νQRe = (CQRδ)2 max0minusI3
I1 (53)
Vortex-Stretching [11] νV Se = (CV Sδ)2radic
2I1
(I5 minus 1
2I1I2
minusI1I2
)32
(54)
with the quantities
PGGT = I1 minus I2 QGGT =1
4(I1 + I2)2 + 4
(I5 minus
1
2I1I2
) (55)
There is a fifth and final model to be considered here It is of a different form to the other four andnot an eddy-viscosity model It is known as the Gradient model and is defined by
Gradient [4 9] τmodG = CGδ
(S2 minus Ω2 minus (SΩminus ΩS)
) (56)
The values C are model constants specific to each model and δ represents the mesh size of the LESBoth of these quantities are included here to properly define the models however in this projectthey will be ignored as they are constants and have no influence on the conditions to be defined
33 Conditions for Large-Eddy Simulation without Explicit Filtering
The filtering operation introduced in Eq (48) is just a formality for LES without explicit filteringThe idea here is that filtering does not actually take place but the term lsquoaddedrsquo to the Navier-Stokesequations in fact lsquosubtractsrsquo the difficult to simulate small-scale turbulent motion of the flow Inorder to consider this properly the terms ui and p in Eq (48) are replaced by vi and q It is hopedthat τmod
ij approximates τij so that vi and q approximate ui and p respectively So LES withoutexplicit filtering is given by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (57)
It is clear to see that the first four terms of this equation are exactly the same as the Navier-Stokesequations so for the exact solutions discussed in Section 22 the combination of these terms willequal zero What is then desirable is that the model term is also zero for these exact solutions sothat Eq (57) still holds In order to test models for exact solutions the condition
part
partxjτmodij (v) = 0 (58)
is checked If this condition were to hold it would imply that the model does not influence thevelocity field of a flow hence we call it the Velocity Field Condition There are a number of otherrestrictions that can be tested One such test comes from the differential form of the evolution ofthe kinetic energy of a flow given in Eq (10) In order to find this Eq (57) is multiplied by vi andterms are grouped to find
part
partt
(1
2vivi
)+ vj
part
partxj
(1
2vivi
)= minusvi
1
ρ
partq
partxi+ νvi
part2vipartxjpartxj
minus vipart
partxjτmodij (v) (59)
23
Then similarly to Eq (58) another condition is to check whether the model affects the kineticenergy of exact solutions this is equivalent to checking whether
vipart
partxjτmodij (v) = 0 (60)
We call this condition the Kinetic Energy Condition The third and final condition considered herecan also be found using the integral form for the equation for kinetic energy
d
dt
intV
1
2vividV =
intS
[minusvj
(1
2vivi
)minus vj
q
ρ+ 2νviSij + viτ
modij (v)
]njdS
+
intV
[minus2νSijSij minus
(minusSijτmod
ij (v))]dV
(61)
where the rate-of-strain tensor Sij was defined in Eq (41) V is a volume of fluid and dS meansthe integral is over the surface of that volume with unit normal nj Then this final condition checkswhether the model affects the total kinetic energy of exact solutions to the Navier-Stokes equationsthat is
Sijτmodij (v) = 0 (62)
We will call this condition the Dissipation Condition because Sijτmodij (v) is often referred to as the
subgrid dissipation
34 Conditions for Large-Eddy Simulation with Explicit Filtering
As in Eq (48) the Navier-Stokes equations can be filtered by applying the filtering operation to thewhole equation (putting a bar over all terms) and evaluating to find again that
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (63)
In Eq (57) the filtering operation was ignored and the elements ui p and τij(u) were replacedwith vi q and τmod
ij (v) respectively Now in LES with explicit filtering the filtering operation itselfsmooths over the small-scale turbulent motion of the simulated flow In this case the elements uip and τij(u) will be replaced with vi q and τmod
ij (v) respectively LES with explicit filtering is thengiven by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (64)
Note that Eqs (63) and (64) despite their similarity to Eq (57) no longer contain the Navier-Stokesequations as Eq (57) did The bar (filtering operation) over each term results in each exact solutionfound in Section 22 not necessarily satisfying the first four terms It is hence unknown whether ornot these terms will vanish for exact solutions Terms can still be compared similarly to Eqs (58)(60) and (62) However instead of comparing to zero we can now compare to the relevant termfrom the filtered Navier-Stokes equations So the terms
part
partxjτij (65)
and
part
partxjτmodij (v) (66)
24
are to be compared Although the latter term is relatively simple to compute the former could leadto some difficulties due to the differentiation of the true stress τij = uiuj minus uiuj So although someinformation could be gleaned from the inspection of these two terms more information can be foundby considering the analogue of Eq (62) So the true subfilter dissipation
Sijτij (67)
(where Sij is shorthand for Sij(v)) and the modelled subfilter dissipation
Sij(v)τmodij (v) (68)
are considered and have to be equal for the condition to hold These can again be derived from theintegral equation for kinetic energy of a filtered velocity field similarly to Eq (61)
4 Analysis of Large-Eddy Simulation Models
In this section we use a code in Mathematica the theory described in Section 3 and the exactsolutions to the Navier-Stokes equations found in Section 22 to check whether or not each of theexact solutions lsquoendorsesrsquo each model for LES both without and with explicit filtering
41 Large-Eddy Simulation Without Explicit Filtering
We start by using a code in Mathematica to test the Velocity Field condition in Eq (58) and theKinetic Energy condition in Eq (60) For all exact solutions to the Navier-Stokes equations inSection 22 the outputs for both conditions were identical so we have combined them into one table
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 ndash ndash 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 1 For the Kinetic Energy condition a lsquo0rsquo result means that vipartpartxj
τmodij (v) = 0 for respective
models and exact solutions whereas a lsquolowastrsquo result means that vipartpartxj
τmodij (v) 6= 0 For the Velocity
Field condition a lsquo0rsquo result means that partpartxj
τmodij (v) = 0 for respective models and exact solutions
whereas a lsquolowastrsquo result means that partpartxj
τmodij (v) 6= 0 The Gradient model can take both positive and
negative values so when the sign of the model is known it is included in the table instead of a lsquorsquoBoth the conditions are included in one table because for all cases the outputs were identical
We then test the Dissipation condition in Eq (62) for all exact solutions to the Navier-Stokesequations in Section 22
25
Dissipation Condition [Does Sijτmodij (v) = 0]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 + 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 2 A lsquo0rsquo result means that Sijτmodij (v) = 0 for respective models and exact solutions whereas
a lsquolowastrsquo result means that Sijτmodij (v) 6= 0 The Gradient model can take both positive and negative
values so when the sign of the model is known it is included in the table instead of a lsquorsquo
We then sum all the results to find the total number of endorsements for each model described inSection 32
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
4 6 8 9 9
lsquoDissipationrsquoEndorsements
2 5 7 7 9
Total Endorsements 6 11 15 16 18
Table 3 The results from Tables 1 and 2 are summarised to give a general overview of how manyendorsements each model receives This makes it much clearer to see the models most supported bythe respective and combined conditions with the Vortex-Stretching model being the most supportedand the Smagorinsky model the least
Testing the models in Section 32 using LES without explicit filtering has mixed results It isclear that the Smagorinksy model is not well endorsed by the flows in Section 22 with the onlyendorsements coming from the simplest flows that is Sol 1 (Linear Velocity) Sol 3 (Velocity as afunction of time) Sol 7 (Stagnation Flow) and Sol 15 (Rankine Vortex) The Vortex-Stretching-based model is the most endorsed flow with eighteen out of a possible thirty endorsements Thisagrees with our foreknowledge as this model was constructed to be as physically consistent as possiblein the paper by Silvis amp Verstappen [11]
42 Large-Eddy Simulation With Explicit Filtering
In this section we will compare the true subfilter dissipation Sijτij and the modelled subfilterdissipation Sij(v)τmod
ij (v) given in Eqs (67) and (68) respectively for each model τmodij and each
exact solution to the Navier-Stokes equation given in Section 22 We will not consider the conditionsinvolving differentiation in Eqs (65) and (66) due to the difficulty in calculating the derivative ofthe τij term in Eq (65)
26
421 Example of a Test for a Model
The true subfilter dissipation Sijτij of a flow given in Eq (67) cannot be explicitly computed asthe filtering operation is kept general What can be done however is to partially evaluate theexpressions using the attributes of a filtering operation mentioned in Eq (48)
The calculations of Sijτij for Sol 1 (Linear Velocity) are provided to illustrate the general ideaRecall that the velocity field of Sol 1 (Linear Velocity) is written in Eq (13) in vector notation as
u(x y z t) =
0ax+ bcx+ d
(69)
Then Sij = Sij(u) is computed simply by using its definition in Eq (41) Note that as filteringpreserves constants if there is an a it is replaced with a meaning that (in this case) there are alarge number of zero terms due to the frequency of 0 Hence
[Sij(u)
]=
[1
2
(partuipartxj
+partujpartxi
)]=
0 a
2c2
a2 0 0
c2 0 0
=
0 a
2c2
a2 0 0
c2 0 0
(70)
Note that as Sijτij is being computed which is equal to the sum of the element-wise multiplicationof the two matrices (or the trace of the matrix multiplication) that we donrsquot need to consider theelements of τij(u) with corresponding zero entry in Sij As τij(u) = uiuj minus uiuj and again asfiltering preserves constants it is found that
[τij(u)] =[uiuj minus uiuj
]=
lowast 0 middot ax+ b 0 middot cx+ d
ax+ b middot 0 lowast lowastcx+ d middot 0 lowast lowast
=
lowast 0 00 lowast lowast0 lowast lowast
(71)
where a lsquolowastrsquo just represents some element (not necessarily non-zero) Therefore the true subfilterdissipation
Sijτij = tr
0 a2
c2
a2 0 0c2 0 0
middotlowast 0 0
0 lowast lowast0 lowast lowast
= 0 (72)
Now the modelled subfilter dissipation Sij(u)τmodij (u) needs to be computed (recall that Sij =
Sij(u)) In this case let us use the QR model stated in Eq (53) Once again Sij(u) calculated inEq (70) needs to be used So we have that
τmodij = minus2(CQRδ)
2 max0minusI3I1
Sij (73)
As
I1 = tr(S2) =
1
2(a2 + c2) and I3 = tr(S
3) = 0 (74)
it holds that τmodij (u) = 0 so Sij(u)τmod
ij (u) = 0 This agrees with Eq (72) and hence Sol 1(Linear Velocity) endorses the QR model
27
422 Testing the Models
Now using a Mathematica code the exact solutions in Section 22 are again used to evaluate thecondition for the models using the process just described in Section 421 The results are describedin Table 4
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 4 All entries with a ldquo0rdquo represent a value of zero and a ldquolowastrdquo represents a value that is non-zeroThe ldquoExactrdquo row represents the value of the true subfilter dissipation Sijτij Every other row witha model name represents the value of the modelled subfilter dissipation Sij(u)τmod
ij (u) respectiveto that model A ldquordquo means that it is unclear as to the value - the sign of these values is oftenunknown and the lsquoMaxrsquo function in the definition of the QR model (in Eq (53)) makes it difficultto evaluate
Reading from Table 4 the Smagorinsky model is clearly endorsed by Sol 3 (Velocity as a function oftime) and Sol 15 (Rankine Vortex) as in both cases both Sijτij = 0 = Sij(u)τmod
ij (u) In this casewe say that Sol 3 (Velocity as a function of time) and Sol 15 (Rankine Vortex) lsquostronglyrsquo endorsethe Smagorinsky model The Smagorinsky model is not endorsed by Sol 1 (Linear Velocity) Sol 2(Planar Flow) or Sol 14 (Radially Dependent) however because Sij(u)τmod
ij (u) 6= 0 For Sol 4 to
13 both Sijτij 6= 0 6= Sij(u)τmodij (u) and so we will say that these exact solutions lsquoweaklyrsquo endorse
the Smagorinsky model
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 10 10 9 6
Table 5 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Interestingly the lsquostrongrsquo and lsquoweakrsquo endorsements give different results However more stock shouldbe put in the results of the lsquostrongerrsquo condition as it is completely clear that zero is equal to zeroIt is not necessarily clear whether or not non-zero equals non-zero as the values will depend on theparameters in each solution hence the lsquoweakerrsquo endorsement
The lsquostrongrsquo condition equally endorses the Vreman QR Gradient and Vortex-Stretching-basedmodels whereas the lsquoweakrsquo condition endorses Vreman and Smagorinsky the most closely followedby the Gradient model with the Vortex-Stretching-based model being the most opposed As this
28
completely contradicts the lsquostrongrsquo condition and the results in Section 41 it makes us question thevalidity of the weaker test The number of endorsements for the QR model is unknown because of thedefinition of the model in Eq (53) and our evaluation of the true subfilter dissipation Eq (67) Dueto the exact solutions we have found it is never clear whether or not the true subfilter dissipationis strictly positive or negative This means that when the true subfilter dissipation is found to benon-zero the lsquoMaxrsquo function in the QR model cannot be evaluated
43 Vremanrsquos True Subfilter Dissipation Claim
Using the theory mentioned in Sections 232 and 34 we can place the exact solutions to the Navier-Stokes equations into different classes defined by Vreman In doing this we can also attempt to verifyone of Vremanrsquos results [16] In Vremanrsquos paper he defines the class Qn to be the set of flows withn zero entries in the velocity gradient of the flow (the velocity gradient is defined in Eq (39))Vreman then claims that the only flows with zero true subfilter dissipation are a certain selection ofthose in the classesQ7Q8 orQ9 shown in appendix B and is generally non-zero for any other flows
The true subfilter dissipation of Sol 1 (Linear Velocity) was computed in Section 421 and wasfound to be zero in Eq (72) The velocity gradient of Sol 1 is as follows
GLV =
0 0 0a 0 0c 0 0
(75)
and hence Sol 1 (Linear Velocity) is in the required subset of Q7 which agrees with Vremanrsquos claimIn Section 422 the true subfilter dissipation was computed (determined to be zero or non-zero) forall of the exact solutions to the Navier Stokes equations in Section 22 These solutions can now belsquoplacedrsquo into each class Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a functionof time) and Sol 14 (Radially Dependent) were all found to have zero true subfilter dissipation andare all in the appropriate subsets of either Q7Q8 or Q9 This agrees with Vremanrsquos results
We can extend Vremanrsquos results however We found that Sol 15 (Rankine Vortex) had zero truesubfilter dissipation and yet is not in the subset of Q7 defined in Appendix B It is important tonote that the rate of strain tensor Sij in Eq (41) for Sol 15 (Rankine Vortex) is zero This is thereason the true subfilter dissipation is zero and the same reason that all models are zero for thisflow as all models are based on Sij
We also found that all of the remaining exact solutions to the Navier-Stokes equations are not inVremanrsquos specified classes (in Appendix Section B) They also have a non-zero subfilter dissipationwhich agrees with Vremanrsquos results in his paper [16]
44 Classes of Flows
It is important to consider that some exact solutions to the Navier-Stokes equations may have similarproperties For example Sol 1 (Linear Velocity) and Sol 2 (Planar Flow) are both very similarIn fact Sol 2 (Planar Flow) is a more general case of Sol 1 (Linear Velocity) We therefore mustgroup the different flows into classes in an attempt to reduce the bias that may occur when solutionsare similar This should aid us to further determine which of the models are most endorsed We willdo this in a number of different ways using the theory discussed in Section 23 We will group theflows with respect to Vremanrsquos Flow Classes given in appendix B and with respect to the Invariantsgiven in appendix A
29
441 Vremanrsquos Velocity Gradient Flow Classes
We now put our exact solutions to the Navier-Stokes equations into one of two classes Either Q0minus6or Q7minus9 where each class is defined intuitively as follows
Q0minus6 = Q0 cupQ1 cup middot middot middot cup Q6 (76)
Q7minus9 = Q7 cupQ8 cupQ9 (77)
where Qn is defined in Section 43 By looking at the gradient of each exact solution we can see thatQ7minus9 contains the six solutions Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as afunction of time) Sol 5 (2D Taylor) Sol 14 (Radially Dependent) and Sol 15 (Rankine Vortex)and that Q0minus6 contains the remaining nine We now check the models for LES without explicitfiltering by considering endorsements within each class Tables 6 and 7 are analogues of Table 3 andillustrate the results for each flow class The analogues of Tables 1 and 2 are Tables 22 and 23 andare in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 6 6 6
lsquoDissipationrsquoEndorsements
2 5 6 6 6
Percentage Endorsed 417 833 100 100 100
Table 6 Each entry in the top two rows represents the number of solutions in the class Q7minus9 thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Q0minus6 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 2 3 3
lsquoDissipationrsquoEndorsements
0 0 1 0 3
Percentage Endorsed 56 56 167 167 333
Table 7 Each entry in the top two rows represents the number of solutions in the class Q1minus6 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking results to 1 decimal place if necessary
All of the flows in the class Q7minus9 endorse the QR Gradient and Vortex-Stretching-based subfilter-scale models (see Table 6) We can hence see that this class fully endorses these three models equallyand does not endorse the Smagorinsky model The Vortex-Stretching eddy viscosity model has themost endorsements from any of the flows in the class Q0minus6 and the Smagorinksy and Vreman modelshave the fewest endorsements (see Table 7) We can hence say that this class mostly endorses theVortex-Stretching-based eddy-viscosity model note however that the percentage endorsement is very
30
low
Now let us consider LES with explicit filtering Tables 8 and 9 are analogues of Table 5 and illustratethe results for each flow class The analogue of Table 4 is Table 24 and is in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 1 1 0 0 0
Table 8 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Q0minus6 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 9 9 9 6
Table 9 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We have grouped Vremanrsquos classes into Q7minus9 and Q0minus6 this is equivalent to saying that we havegrouped Vremanrsquos classes into those with two or fewer non-zero velocity gradient elements and thosewith three or more non-zero velocity gradient elements respectively Therefore the solutions in Q7minus9are exactly those with a lsquo0rsquo in the lsquoExactrsquo column of Table 4 and Sol 5 (2D Taylor) ExcludingSol 5 (2D Taylor) these are the same as those five that endorse the Vreman QR Gradient andVortex-Stretching-based models in Table 5 We can see in Table 8 that the results for the lsquostrongrsquoendorsements are exactly the same as in Table 5 This is because all of the flows with zero subfilterdissipation are contained in the class Q7minus9 There are hence no lsquostrongrsquo endorsements in Table 9The majority of the lsquoweakrsquo endorsements come from Q0minus6 as this class contains all but one of thesolutions with non-zero true subfilter dissipation (given in Eq (67))
442 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
We now separate our exact solutions into classes again using Vremanrsquos flow classes This timehowever we define V0 to be the class of flows Vreman claims will always have zero true subfilterdissipation namely those classes mentioned in Appendix B We define Vlowast to be all the remaining
31
classes Then
V0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow)
Sol 3 (Velocity as a function of time) Sol 14 (Radially Dependent)(78)
Vlowast = Sol 4 (Axisymmetric Flow) Sol 5 (2D Taylor) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 13 (Sphere Translation) Sol 15 (Rankine Vortex)
(79)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 10 and 11 are analogues of Table 3 and illustrate the results for each flow class Theanalogues of Tables 1 and 2 are Tables 25 and 26 and are in Appendix C2
Endorsements for models using solutions in V0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 4 4 4 4
lsquoDissipationrsquoEndorsements
1 4 4 4 4
Percentage Endorsed 375 100 100 100 100
Table 10 Each entry in the top two rows represents the number of solutions in the class V0 thatendorse each model There are 4 solutions in this class so the percentages are worked out by dividingthe total of each column by 8 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Vlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 2 4 5 5
lsquoDissipationrsquoEndorsements
1 1 3 2 5
Percentage Endorsed 136 136 318 318 455
Table 11 Each entry in the top two rows represents the number of solutions in the class Vlowast thatendorse each model There are 11 solutions in this class so the percentages are worked out bydividing the total of each column by 22 and taking results to 1 decimal place if necessary
All of the flows in the class V0 endorse the Vreman QR Gradient and Vortex-Stretching-basedsubfilter-scale models (see Table 6) This class completely endorses these four models equally anddoes not endorse the Smagorinsky model The Vortex-Stretching-based eddy-viscosity model hasthe most endorsements from the class Vlowast and the Smagorinksy and Vreman models again have thefewest endorsements (see Table 7) As in Section 441 we say that this class mostly endorses the
32
Vortex-Stretching-based eddy-viscosity model but once again the percentage endorsement is ratherlow
Now let us consider LES with explicit filtering Tables 12 and 13 are analogues of Table 5 andillustrate the results for each flow class The analogue of Table 4 is Table 27 and is in Appendix C2
Endorsements for models using solutions in V0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 4 4 4 4
lsquoWeakrsquo Endorsements 0 0 0 0 0
Table 12 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Vlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 1 1 1 1
lsquoWeakrsquo Endorsements 10 10 9 6
Table 13 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We can see in Table 12 that the results for the lsquostrongrsquo endorsements are similar to those in Table 5excluding Sol 15 (Rankine Vortex) The four solutions in V0 with zero true subfilter dissipationnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)and Sol 14 (Radially Dependent) are hence the four that endorse the Vreman QR Gradientand Vortex-Stretching-based models in Table 12 Sol 15 (Rankine Vortex) is in the class Vlowast andhence there is one lsquostrongrsquo endorsement in Table 13 The class Vlowast contains all of the flows with anon-zero true subfilter dissipation (given in Eq (67)) and hence Table 13 contains all of the lsquoweakrsquoendorsements
443 Classes using the Third Principal Invariant
Sorting the classes using the invariants is a non-trivial task in itself There are a number of waysin which to do this using both the principal invariants in Eq (40) and the combined invariants inEq (43)
We start by looking at flows with third principal invariant RG equal to zero We again define twoclasses R0 and Rlowast where R0 consists of all flows with RG = 0 and Rlowast consists of all flows withRG 6= 0 Then using the information in Appendix A we see that there are nine flows in R0 That
33
is
R0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 8 (Vortex in Stagnation Flow)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)
(80)
Rlowast = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 13 (Sphere Translation)(81)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 14 and 15 are analogues of Table 3 and they illustrate the results for LES withoutexplicit filtering for each flow class The analogues of Tables 1 and 2 are Tables 28 and 29 and arein Appendix C3
Endorsements for models using solutions in R0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 7 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 278 556 778 722 778
Table 14 Each entry in the top two rows represents the number of solutions in the class R0 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking the result to one decimal place if necessary
Endorsements for models using solutions in Rlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 2 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 83 83 83 167 333
Table 15 Each entry in the top two rows represents the number of solutions in the class Rlowast thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking the result to one decimal place if necessary
The class R0 endorses both the Vortex-Stretching-based model and the QR model the most andSmagorinsky the least in the case for LES without explicit filtering In the Rlowast class the Vortex-Stretching-based model is again the most endorsed followed by the Gradient model however thepercentages are again small
34
Now let us consider LES with explicit filtering Tables 16 and 17 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 30 and is in Appendix C3
Endorsements for models using solutions in R0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 4 4 3 2
Table 16 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Rlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 6 6 6 4
Table 17 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 we can see in Table 16 that the results for the lsquostrongrsquo endorsements are exactly thesame as in Table 5 This is because all of the flows with zero subfilter dissipation are contained in theclass R0 As in Table 9 there are no lsquostrongrsquo endorsements in Table 17 The lsquoweakrsquo endorsementshowever are split more evenly between the two classes now than in Vremanrsquos classes in Sections 441and 442 We find that the Smagorinsky and Vreman models still seem to be the most endorsed inboth classes
444 Two-component Flow Classes
Another way in which the flows can be grouped into classes is by determining whether a flow is atwo-component flow or not We define a two-component flow to have the property that the combinedprincipal invariants of a flow in Eq (43) satisfy
I3 = I4 = I5 minus1
2I1I2 = 0 (82)
We are also interested in considering flows with zero gradient model subgrid dissipation These flowshave the property that
I3 minus I4 = 0 (83)
What we find though is that for all the solutions in Section 22 if a flow is a two-component flowthen it also satisfies I3 minus I4 = 0 It may be interesting to note that although I3 minus I4 6= 0 for manysolutions there are a number with I3 equal to a multiple of I4 Although not considered in thisreport this may also be a class of interest We find that the two-component flows are
C2 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)(84)
35
and we call the remaining flows three-component flows They are
C3 = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex) Sol 11 (Pseudo-Motion Shots)
Sol 13 (Sphere Translation)
(85)
Note that Sol 8 (Vortex in Stagnation Flow) is defined to be a planar flow this does not mean that itis a two-component flow in this sense however because its two non-zero velocity components dependon three coordinates We now check the models for LES without explicit filtering by consideringendorsements within each class Tables 18 and 19 are analogues of Table 3 and they illustrate theresults for LES without explicit filtering for each flow class The analogues of Tables 1 and 2 areTables 31 and 32 and are in Appendix C4
Endorsements for models using solutions in C2 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 6 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 357 714 100 857 100
Table 18 Each entry in the top two rows represents the number of solutions in the class C2 thatendorse each model There are 7 solutions in this class so the percentages are worked out by dividingthe total of each column by 14 and taking the result to one decimal place if necessary
Endorsements for models using solutions in C3 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 3 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 63 63 63 188 250
Table 19 Each entry in the top two rows represents the number of solutions in the class C3 thatendorse each model There are 8 solutions in this class so the percentages are worked out by dividingthe total of each column by 16 and taking the result to one decimal place if necessary
The class C2 equally endorses the QR and Vortex-Stretching-based models and once again theSmagorinsky model is the least endorsed As with the previous classes the class C3 has such lowpercentages that it does not seem to endorse any models We start to see a trend appearing here inthat the flow classes that contain the more complicated flows namely Q0minus6 (Eq (76)) Vlowast (Eq (79))Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages The classes that tendto contain the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78)) R0 (Eq (80)) and C2 (Eq (84))all seem to favour the Vortex-Stretching-based model or the QR model
36
Now let us consider LES with explicit filtering Tables 20 and 21 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 33 and is in Appendix C4
Endorsements for models using solutions in C2 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 2 2 0 1 0
Table 20 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in C3 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 8 8 8 6
Table 21 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 and Table 16 we can see in Table 20 that the results for the lsquostrongrsquo endorsements areexactly the same as in Table 5 This is again because all of the flows with zero subfilter dissipationare contained in the class C2 There are two flows with non-zero true subfilter dissipation (givenin Eq (67)) in C2 and this shows in the lsquoweakrsquo endorsements in Table 20 There are no lsquostrongrsquoendorsements in Table 21 as all the zero subfilter dissipation flows are in C2
5 Conclusions
In this report we studied incompressible turbulent flows We found a number of exact solutions to theNavier-Stokes equations each representing a relatively simplistic fluid flow Large-eddy simulation(LES) is used to model turbulent flows by smoothing over the complex small-scale motion Weintroduced a number of subfilter-scale models to approximate this small-scale motion and testedthem using the found exact solutions to the Navier-Stokes equations We first considered LESwithout explicit filtering and used the Navier-Stokes equations to derive conditions for the subfilter-scale models namely the Velocity Field condition (Eq (58)) the Kinetic Energy condition (Eq (60))and the Dissipation condition (Eq (62)) Similarly we considered LES with explicit filtering andagain used the Navier-Stokes equations to derive more conditions for the models in this case wedecided to just take one condition the Dissipation condition (Eqs (67) and (68)) as the others leadto difficulty in their evaluation By lsquorunningrsquo the exact solutions to the Navier-Stokes equationsthrough each condition for the subfilter-scale models we found that the Vortex-Stretching-basededdy-viscosity model in Eq (54) was most endorsed by the tests in LES without explicit filteringWe found that four out of the five subfilter-scale models namely the Vreman QR Gradient andVortex-Stretching-based models were equally endorsed by the lsquostrongrsquo tests in LES with explicitfiltering with the Smagorinsky model being the least endorsed The results we found for the lsquoweakrsquocondition were completely opposite to those found in both the lsquostrongrsquo condition and in LES withoutexplicit filtering This led us to conclude that this test was not one in which we should put our trust
37
We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses
In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes
This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold
38
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
Figure 6 An illustration of the flow in the yz-plane for Sol 6 (Generalized Beltrami Flow)described in Eq (21) with parameters A = B =C = 1 and time t = 0
Solution 7 Stagnation Flow
In the book by Drazin and Riley [6 p40] an exact solution is considered which uses a stream functionψ = ψ(y z) They suggest that the following velocity field is a solution
u1(x y z t) =a
kx
u2(x y z t) = minusaky +
k
Re
partψ(y z)
partz
u3(x y z t) = minus k
Re
partψ(y z)
party
(22)
Here we choose the stream function to be ψ(y z) = yz Then a solution is given by
u1(x y z t) =a
kx
u2(x y z t) =
(minusak
+k
Re
)y
u3(x y z t) = minus kzRe
p(x y z t) = minus1
2
((axk
)2minus((
k
Reminus a
k
)y
)2
minus(kz
Re
)2)
(23)
where a is a measure of the strength of the stagnation flow and k is a constant chosen in the textto specify the distance between vortices There are no vortices in this particular solution however
11
Figure 7 An illustration of the flow in the yz-plane for Sol 7 (Stagnation Flow) describedin Eq (23) with parameter values a = Re = 1and k = 2
Figure 8 An illustration of the flow in the xy-plane for Sol 7 (Stagnation Flow) describedin Eq (23) with parameter values a = Re = 1and k = 2
Solution 8 Vortex in Stagnation Flow
As in Sol 7 (Stagnation Flow) we now consider the same solution with a different stream functionnamely ψ(z) = cos z Then given the condition that k = aRe and adding a pressure term Eq (22)becomes
u1(x y z t) =1
Rex
u2(x y z t) = minus 1
Rey minus k sin z
u3(x y z t) = 0
p(x y z t) = minus 1
2Re2(x2 + y2)
(24)
where a and k are as in Sol 7 (Stagnation Flow)
12
Figure 9 An illustration of the flow in the yz-plane for Sol 8 (Vortex in Stagnation Flow)described in Eq (24) with parameter valuesk = 3 and Re = 10
Figure 10 An illustration of the flow in thexy-plane for Sol 8 (Vortex in StagnationFlow) described in Eq (24) with parametervalues k = 3 and Re = 10
Solution 9 Burgers Vortex
The Burgers Vortex solution is given in cylindrical coordinates in the book by Lautrup [8 p 385]as follows
ur(r ϕ z t) = minus 2r
c2Re
uϕ(r ϕ z t) =Ωc2
r
(1minus eminusr
2c2)
uz(r ϕ z t) =4z
c2Re
p(r ϕ z t) = minusc2Ω2F
(r2c2
)2
minus 2
Re2r2 + 4z2
c4
(25)
where
F (ξ) =
infinintξ
(1minus eminusx)2
x2dx (26)
This solution is not particularly nice to look at however most of the unpleasantness comes fromthe pressure term which (in this project) is only required in order to validate the solution For thisproject all solutions are required in Cartesian coordinates the velocity field of the Burgers Vortexhas hence been converted into Cartesian coordinates
u1(x y z t) = minus 2x
c2Reminus Ω
h(x y)(1minus exp (minush(x y))) y
u2(x y z t) = minus 2y
c2Re+
Ω
h(x y)(1minus exp (minush(x y)))x
u3(x y z t) =4z
c2Re
(27)
13
where
h(x y) =x2 + y2
c2 (28)
and Ω and c are constants
Figure 11 An illustration of the flow for Sol9 (Burgers Vortex) described in Eq (27) withparameter values c = 15 Re = 20 and Ω = 10A slice of the xy-plane at z = 0 is shown
Solution 10 Another Vortex Solution
There is another vortex solution more simply formulated than the Burgers Vortex yet with asingularity at r = 0 as a result of the uϕ term It is again given in cylindrical coordinates
ur(r ϕ z t) = minus 2r
c2Re
uϕ(r ϕ z t) =a
r
uz(r ϕ z t) =4z
c2Re
p(r ϕ z t) = minus a2
2r2minus 2r2 + 8z2
c2Re2
(29)
which can also be converted into Cartesian coordinates
14
u1(x y z t) = minus 2x
c2Reminus ay
x2 + y2
u2(x y z t) = minus 2y
c2Re+
ax
x2 + y2
u3(x y z t) =4z
c2Re
(30)
where a and c are constants
Figure 12 An illustration of the flow for Sol9 (Burgers Vortex) described in Eq (27) withparameter values c = 15 Re = 20 and a = 10 Aslice of the xy-plane at z = 0 is shown
Solution 11 Pseudo-Motion Shots of the Second Kind
In the book by Berker [3 p 94] a solution is given in cylindrical coordinates by
ur(r ϕ z t) =2
Re
1
r
uϕ(r ϕ z t) = A1r3 +B1r +
C1
r
uz(r ϕ z t) = A2r2 log(r) +B2r
2 + C2
p(r ϕ z t) =A2
1r6
6+A1B1r
4
2+A1C1r
2 +B2
1r2
2minus C2
1
2r2minus 2
r2Re2+ 2B1C1 log(r)
+2A2z
Reminus 4B1ϕ
Re
(31)
where A1 A2 B1 B2 C1 C2 are constants Eq (31) when converted into Cartesian coordinates is
15
u1(x y z t) =1
Re
2xminus C1Rey
x2 + y2minusA1y
(x2 + y2
)minusB1y
u2(x y z t) =1
Re
2y + C1Rex
x2 + y2+A1x
(x2 + y2
)+B1x
u3(x y z t) =1
2A2
(x2 + y2
)log(x2 + y2
)+B2
(x2 + y2
)+ C2
(32)
Figure 13 An illustration of the flow in thexy-plane for Sol 11 (Pseudo-Motion Shots)described in Eqs (31) and (32) with parametervalues A1 = B1 = C1 = 1 A2 = B2 = C2 = 2and Re = 10
Figure 14 An illustration of the flow in theyz-plane for Sol 11 (Pseudo-Motion Shots)described in Eqs (31) and (32) with parametervalues A1 = B1 = C1 = 1 A2 = B2 = C2 = 2and Re = 10
Solution 12 Flows With Constant Whirl
In the book by Berker [3 p 143] another solution is given in cylindrical coordinates by
ur(r ϕ z t) = minus 1
Re
1
r
uϕ(r ϕ z t) =1
rradicte(minus
12 g(t))
uz(r ϕ z t) = 0
p(r ϕ z t) =1
4
2(minus e
minusg(t)
t minus 1Re2
)r2
minus 1
t2Re
infinintminusg(t)
eminust
tdt
(33)
where we define g(t) =r2Re
2t When converted into Cartesian coordinates we find
16
u1(x y z t) = minus 1
x2 + y2
(yradict
exp
(minus
Re(x2 + y2
)4t
)+
x
Re
)
u2(x y z t) =1
x2 + y2
(xradict
exp
(minus
Re(x2 + y2
)4t
)minus y
Re
)
u3(x y z t) = 0
(34)
Figure 15 An illustration of the flow for Sol12 (Flows with Constant Whirl) described inEqs (33) and (34) with time t = 1 and Re = 1 Aslice of the xy-plane at z = 0 is shown as velocityin the z-direction is zero this is the same for anyvalue of z
17
Solution 13 Rectilinear and Uniform Translation of a Sphere
In the book by Berker [3 p 228] another solution is given in Cartesian coordinates by
u1(x y z t) =3
4
axz
R3
(a2
R2minus 1
)
u2(x y z t) =3
4
ayz
R3
(a2
R2minus 1
)
u3(x y z t) = 1minus 3
4
a
Rminus 1
4
a3
R3minus 3
4
az2
R3
(a2
R2minus 1
)
p(x y z t) =3a(aminusR)(a+R)
32R10Re
((R2Re
(x2 + y2
)minus 2R2Rez2
) (a3 + 3aR2 minus 4R3
)+ 3aRez4
(R2 minus a2
)minus 16R5z
)
(35)
where a and R are constants
Figure 16 An illustration of the flow in the yz-plane for Sol 13 (Sphere Translation) describedin Eq (35) with parameter values a = minus2 andRe = 3
Solution 14 Radially Dependent Flow
A solution with velocity in the z-direction dependent only on the distance from the z-axis is as
18
follows in cylindrical coordinates
ur(r ϕ z t) = 0
uϕ(r ϕ z t) = 0
uz(r ϕ z t) = A(r2 minus a2
)
p(r ϕ z t) =4Az
Re
(36)
where A and a are constants In Cartesian coordinates the solution is just the same with u1(x y z t)and u2(x y z t) both zero and the r2 term replaced with x2 + y2 in u3(x y z t)
Figure 17 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus1 1] withparameter values a = 1 A = 4 and Re = 10
Figure 18 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus2 2] withparameter values a = 1 A = 4 and Re = 10
Solution 15 Rankine Vortex
A solution discovered by Rankine [1] is a vortex flow of the following form
ur(r ϕ z t) = 0
uϕ(r ϕ z t) =ar
2πR2
uz(r ϕ z t) = 0
p(r ϕ z t) =a2r2
8π2R4
(37)
19
where a and R are constants In Cartesian coordinates the flow has the form
u1(x y z t) = minus ay
2πR2
u2(x y z t) =ax
2πR2
u3(x y z t) = 0
p(x y z t) =a2(x2 + y2
)8π2R4
(38)
This flow is actually a rigid body rotational flow in that the angular rotational velocity is uniformand the velocity of the flow increases proportionally with distance from the z-axis
Figure 19 An illustration of the flow in the xy-plane for Sol 15 (Rankine Vortex) described inEq (37) and Eq (38) with parameter values a = 1and R = 1
radic2π
23 Characterisation of Flows
It is important to be able to classify flows in a way that sorts them into distinct groups This aidsin the clear identification and general study of flows It is also a very important way of recognisingflows with similar properties In this subsection we introduce two distinct ways in which flows canbe characterised
231 Characterising Solutions using Invariants
A principal invariant of a tensor is a coefficient of the characteristic polynomial of that tensorInvariants as their name suggests are invariant under rotations of the coordinate system in thattheir values do not change when the tensor is lsquorotatedrsquo into a different set of coordinates Thevelocity gradient of a flow G is defined as the Jacobian of the velocity vector given by
Gij =
[partuipartxj
] (39)
20
for i j = 1 2 3 There are three principal invariants of the velocity gradient of a fluid flow G givenby
PG = tr(G)
QG =1
2
((tr(G))2 minus tr
(G2))
RG = det(G)
(40)
Incompressibility of flows (Eq (5)) ensures that PG = 0 hence the first principal invariant is zerofor all of the flows in Section 22 We now define the rate-of-strain tensor S by
Sij =1
2
(partuipartxj
+partujpartxi
) (41)
and the rate-of-rotation tensor Ω by
Ωij =1
2
(partuipartxjminus partujpartxi
) (42)
Note that the first principal invariant of S and the first and third principal invariants of Ω are zeroAlso note that the velocity gradient of a flow can simply be written G = S + Ω
Rather than considering the principal invariants we can also consider the combined invariants
I1 = tr(S2) I2 = tr(Ω2) I3 = tr(S3) I4 = tr(SΩ2) I5 = tr(S2Ω2) (43)
We can hence write the three principal invariants for a velocity field G in terms of the combinedinvariants
PG = 0
QG = minus1
2(I1 + I2)
RG =1
3(I3 + 3I4)
(44)
In this report the flows considered are the exact solutions to the incompressible Navier-Stokesequations described in Section 22 The combined invariants in Eq (43) are used to characterisethese solutions into different classes in Section 44 Both the principal invariants and combinedinvariants for each exact solution in Section 22 are given in Appendix A
Each of the subfilter-scale eddy-viscosity models described in Section 32 are written in terms ofthese five combined invariants
232 Vremanrsquos Characterisation of Flows
Vreman [16] classifies flows using the gradient of the velocity field of a flow Each lsquoclassrsquo of flows hasthe same number of non-zero elements in the gradient matrix For example
[partuipartxj
]=
0 0 0lowast 0 0lowast 0 0
(45)
where a lowast denotes a non-zero term represents a group of flows that lie within the class denoted byQ7 (see Appendix B) where the subscript represents the number of non-zero terms in the velocity
21
gradient Flows of the form in Eq (45) have a velocity field of the form
u1 = c1 (constant)
u2 = f2(x1)
u3 = f3(x1)
(46)
where f2(x1) and f3(x1) are such that the Navier Stokes equations Eq (1) are satisfied Notethat as the elements on the leading diagonal are zero the incompressibility condition Eq (5) isautomatically satisfied Some simple flow classes are given in Appendix B and contain a number ofthe solutions discussed in Section 22 This is mentioned in greater detail in Section 441
3 Conditions for Large-Eddy Simulation Models
31 Large-Eddy Simulation
Due to the nature of numerical simulation and the complexity of turbulent flows small details aremissed and information is lost when solving the Navier-Stokes equations The computational costof solving a turbulent flow is extremely high so a filtering operation is applied to the Navier-Stokesequations represented by a bar to reduce the computational cost of the simulation This is a linearoperation and preserves constants ie ρ = ρ and is also assumed to commute with differentiation
partuipartxj
=partuipartxj
(47)
The filtered Navier-Stokes equations are found by filtering (putting a bar over) the entire equations(see Eq (1)) and after evaluating are given by
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (48)
where
τij = uiuj minus uiuj (49)
is called the lsquotrue stressrsquo of the flow This however poses a problem as it is not clear what is meantby the term uiuj for a general filtering operation The true stress τij therefore must be replaced bya model τmod
ij in practical applications
32 Subfilter-Scale Models
We will now define the subflter-scale models denoted by τmodij to be used in place of the lsquodifficultrsquo
true stress τij There are a number of models available to use in the literature [11 p7] These willbe tested using the exact solutions to the Navier-Stokes equations and by checking conditions whichare defined in Sections 33 and 34
All the models barring one are of the same form That is each of the eddy-viscosity models arewritten
τmodij minus 1
3τmodkk δij = minus2νeSij (50)
with νe representing the eddy viscosity of a specific model For simplicity we usually just writeτmodij = minus2νeSij the extra term is included here to emphasise that the models are traceless Each
22
of the models we consider here can be expressed in terms of the rate of strain tensor S defined inEq (41) the rate of rotation tensor Ω defined in Eq (42) and the combined invariants defined inEq (43) The models are then defined as follows
Smagorinsky [12] νSe = (CSδ)2radic
2I1 (51)
Vreman [16] νVe = (CV δ)2
radicQGGT
PGGT (52)
QR [13 14 15] νQRe = (CQRδ)2 max0minusI3
I1 (53)
Vortex-Stretching [11] νV Se = (CV Sδ)2radic
2I1
(I5 minus 1
2I1I2
minusI1I2
)32
(54)
with the quantities
PGGT = I1 minus I2 QGGT =1
4(I1 + I2)2 + 4
(I5 minus
1
2I1I2
) (55)
There is a fifth and final model to be considered here It is of a different form to the other four andnot an eddy-viscosity model It is known as the Gradient model and is defined by
Gradient [4 9] τmodG = CGδ
(S2 minus Ω2 minus (SΩminus ΩS)
) (56)
The values C are model constants specific to each model and δ represents the mesh size of the LESBoth of these quantities are included here to properly define the models however in this projectthey will be ignored as they are constants and have no influence on the conditions to be defined
33 Conditions for Large-Eddy Simulation without Explicit Filtering
The filtering operation introduced in Eq (48) is just a formality for LES without explicit filteringThe idea here is that filtering does not actually take place but the term lsquoaddedrsquo to the Navier-Stokesequations in fact lsquosubtractsrsquo the difficult to simulate small-scale turbulent motion of the flow Inorder to consider this properly the terms ui and p in Eq (48) are replaced by vi and q It is hopedthat τmod
ij approximates τij so that vi and q approximate ui and p respectively So LES withoutexplicit filtering is given by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (57)
It is clear to see that the first four terms of this equation are exactly the same as the Navier-Stokesequations so for the exact solutions discussed in Section 22 the combination of these terms willequal zero What is then desirable is that the model term is also zero for these exact solutions sothat Eq (57) still holds In order to test models for exact solutions the condition
part
partxjτmodij (v) = 0 (58)
is checked If this condition were to hold it would imply that the model does not influence thevelocity field of a flow hence we call it the Velocity Field Condition There are a number of otherrestrictions that can be tested One such test comes from the differential form of the evolution ofthe kinetic energy of a flow given in Eq (10) In order to find this Eq (57) is multiplied by vi andterms are grouped to find
part
partt
(1
2vivi
)+ vj
part
partxj
(1
2vivi
)= minusvi
1
ρ
partq
partxi+ νvi
part2vipartxjpartxj
minus vipart
partxjτmodij (v) (59)
23
Then similarly to Eq (58) another condition is to check whether the model affects the kineticenergy of exact solutions this is equivalent to checking whether
vipart
partxjτmodij (v) = 0 (60)
We call this condition the Kinetic Energy Condition The third and final condition considered herecan also be found using the integral form for the equation for kinetic energy
d
dt
intV
1
2vividV =
intS
[minusvj
(1
2vivi
)minus vj
q
ρ+ 2νviSij + viτ
modij (v)
]njdS
+
intV
[minus2νSijSij minus
(minusSijτmod
ij (v))]dV
(61)
where the rate-of-strain tensor Sij was defined in Eq (41) V is a volume of fluid and dS meansthe integral is over the surface of that volume with unit normal nj Then this final condition checkswhether the model affects the total kinetic energy of exact solutions to the Navier-Stokes equationsthat is
Sijτmodij (v) = 0 (62)
We will call this condition the Dissipation Condition because Sijτmodij (v) is often referred to as the
subgrid dissipation
34 Conditions for Large-Eddy Simulation with Explicit Filtering
As in Eq (48) the Navier-Stokes equations can be filtered by applying the filtering operation to thewhole equation (putting a bar over all terms) and evaluating to find again that
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (63)
In Eq (57) the filtering operation was ignored and the elements ui p and τij(u) were replacedwith vi q and τmod
ij (v) respectively Now in LES with explicit filtering the filtering operation itselfsmooths over the small-scale turbulent motion of the simulated flow In this case the elements uip and τij(u) will be replaced with vi q and τmod
ij (v) respectively LES with explicit filtering is thengiven by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (64)
Note that Eqs (63) and (64) despite their similarity to Eq (57) no longer contain the Navier-Stokesequations as Eq (57) did The bar (filtering operation) over each term results in each exact solutionfound in Section 22 not necessarily satisfying the first four terms It is hence unknown whether ornot these terms will vanish for exact solutions Terms can still be compared similarly to Eqs (58)(60) and (62) However instead of comparing to zero we can now compare to the relevant termfrom the filtered Navier-Stokes equations So the terms
part
partxjτij (65)
and
part
partxjτmodij (v) (66)
24
are to be compared Although the latter term is relatively simple to compute the former could leadto some difficulties due to the differentiation of the true stress τij = uiuj minus uiuj So although someinformation could be gleaned from the inspection of these two terms more information can be foundby considering the analogue of Eq (62) So the true subfilter dissipation
Sijτij (67)
(where Sij is shorthand for Sij(v)) and the modelled subfilter dissipation
Sij(v)τmodij (v) (68)
are considered and have to be equal for the condition to hold These can again be derived from theintegral equation for kinetic energy of a filtered velocity field similarly to Eq (61)
4 Analysis of Large-Eddy Simulation Models
In this section we use a code in Mathematica the theory described in Section 3 and the exactsolutions to the Navier-Stokes equations found in Section 22 to check whether or not each of theexact solutions lsquoendorsesrsquo each model for LES both without and with explicit filtering
41 Large-Eddy Simulation Without Explicit Filtering
We start by using a code in Mathematica to test the Velocity Field condition in Eq (58) and theKinetic Energy condition in Eq (60) For all exact solutions to the Navier-Stokes equations inSection 22 the outputs for both conditions were identical so we have combined them into one table
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 ndash ndash 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 1 For the Kinetic Energy condition a lsquo0rsquo result means that vipartpartxj
τmodij (v) = 0 for respective
models and exact solutions whereas a lsquolowastrsquo result means that vipartpartxj
τmodij (v) 6= 0 For the Velocity
Field condition a lsquo0rsquo result means that partpartxj
τmodij (v) = 0 for respective models and exact solutions
whereas a lsquolowastrsquo result means that partpartxj
τmodij (v) 6= 0 The Gradient model can take both positive and
negative values so when the sign of the model is known it is included in the table instead of a lsquorsquoBoth the conditions are included in one table because for all cases the outputs were identical
We then test the Dissipation condition in Eq (62) for all exact solutions to the Navier-Stokesequations in Section 22
25
Dissipation Condition [Does Sijτmodij (v) = 0]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 + 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 2 A lsquo0rsquo result means that Sijτmodij (v) = 0 for respective models and exact solutions whereas
a lsquolowastrsquo result means that Sijτmodij (v) 6= 0 The Gradient model can take both positive and negative
values so when the sign of the model is known it is included in the table instead of a lsquorsquo
We then sum all the results to find the total number of endorsements for each model described inSection 32
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
4 6 8 9 9
lsquoDissipationrsquoEndorsements
2 5 7 7 9
Total Endorsements 6 11 15 16 18
Table 3 The results from Tables 1 and 2 are summarised to give a general overview of how manyendorsements each model receives This makes it much clearer to see the models most supported bythe respective and combined conditions with the Vortex-Stretching model being the most supportedand the Smagorinsky model the least
Testing the models in Section 32 using LES without explicit filtering has mixed results It isclear that the Smagorinksy model is not well endorsed by the flows in Section 22 with the onlyendorsements coming from the simplest flows that is Sol 1 (Linear Velocity) Sol 3 (Velocity as afunction of time) Sol 7 (Stagnation Flow) and Sol 15 (Rankine Vortex) The Vortex-Stretching-based model is the most endorsed flow with eighteen out of a possible thirty endorsements Thisagrees with our foreknowledge as this model was constructed to be as physically consistent as possiblein the paper by Silvis amp Verstappen [11]
42 Large-Eddy Simulation With Explicit Filtering
In this section we will compare the true subfilter dissipation Sijτij and the modelled subfilterdissipation Sij(v)τmod
ij (v) given in Eqs (67) and (68) respectively for each model τmodij and each
exact solution to the Navier-Stokes equation given in Section 22 We will not consider the conditionsinvolving differentiation in Eqs (65) and (66) due to the difficulty in calculating the derivative ofthe τij term in Eq (65)
26
421 Example of a Test for a Model
The true subfilter dissipation Sijτij of a flow given in Eq (67) cannot be explicitly computed asthe filtering operation is kept general What can be done however is to partially evaluate theexpressions using the attributes of a filtering operation mentioned in Eq (48)
The calculations of Sijτij for Sol 1 (Linear Velocity) are provided to illustrate the general ideaRecall that the velocity field of Sol 1 (Linear Velocity) is written in Eq (13) in vector notation as
u(x y z t) =
0ax+ bcx+ d
(69)
Then Sij = Sij(u) is computed simply by using its definition in Eq (41) Note that as filteringpreserves constants if there is an a it is replaced with a meaning that (in this case) there are alarge number of zero terms due to the frequency of 0 Hence
[Sij(u)
]=
[1
2
(partuipartxj
+partujpartxi
)]=
0 a
2c2
a2 0 0
c2 0 0
=
0 a
2c2
a2 0 0
c2 0 0
(70)
Note that as Sijτij is being computed which is equal to the sum of the element-wise multiplicationof the two matrices (or the trace of the matrix multiplication) that we donrsquot need to consider theelements of τij(u) with corresponding zero entry in Sij As τij(u) = uiuj minus uiuj and again asfiltering preserves constants it is found that
[τij(u)] =[uiuj minus uiuj
]=
lowast 0 middot ax+ b 0 middot cx+ d
ax+ b middot 0 lowast lowastcx+ d middot 0 lowast lowast
=
lowast 0 00 lowast lowast0 lowast lowast
(71)
where a lsquolowastrsquo just represents some element (not necessarily non-zero) Therefore the true subfilterdissipation
Sijτij = tr
0 a2
c2
a2 0 0c2 0 0
middotlowast 0 0
0 lowast lowast0 lowast lowast
= 0 (72)
Now the modelled subfilter dissipation Sij(u)τmodij (u) needs to be computed (recall that Sij =
Sij(u)) In this case let us use the QR model stated in Eq (53) Once again Sij(u) calculated inEq (70) needs to be used So we have that
τmodij = minus2(CQRδ)
2 max0minusI3I1
Sij (73)
As
I1 = tr(S2) =
1
2(a2 + c2) and I3 = tr(S
3) = 0 (74)
it holds that τmodij (u) = 0 so Sij(u)τmod
ij (u) = 0 This agrees with Eq (72) and hence Sol 1(Linear Velocity) endorses the QR model
27
422 Testing the Models
Now using a Mathematica code the exact solutions in Section 22 are again used to evaluate thecondition for the models using the process just described in Section 421 The results are describedin Table 4
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 4 All entries with a ldquo0rdquo represent a value of zero and a ldquolowastrdquo represents a value that is non-zeroThe ldquoExactrdquo row represents the value of the true subfilter dissipation Sijτij Every other row witha model name represents the value of the modelled subfilter dissipation Sij(u)τmod
ij (u) respectiveto that model A ldquordquo means that it is unclear as to the value - the sign of these values is oftenunknown and the lsquoMaxrsquo function in the definition of the QR model (in Eq (53)) makes it difficultto evaluate
Reading from Table 4 the Smagorinsky model is clearly endorsed by Sol 3 (Velocity as a function oftime) and Sol 15 (Rankine Vortex) as in both cases both Sijτij = 0 = Sij(u)τmod
ij (u) In this casewe say that Sol 3 (Velocity as a function of time) and Sol 15 (Rankine Vortex) lsquostronglyrsquo endorsethe Smagorinsky model The Smagorinsky model is not endorsed by Sol 1 (Linear Velocity) Sol 2(Planar Flow) or Sol 14 (Radially Dependent) however because Sij(u)τmod
ij (u) 6= 0 For Sol 4 to
13 both Sijτij 6= 0 6= Sij(u)τmodij (u) and so we will say that these exact solutions lsquoweaklyrsquo endorse
the Smagorinsky model
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 10 10 9 6
Table 5 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Interestingly the lsquostrongrsquo and lsquoweakrsquo endorsements give different results However more stock shouldbe put in the results of the lsquostrongerrsquo condition as it is completely clear that zero is equal to zeroIt is not necessarily clear whether or not non-zero equals non-zero as the values will depend on theparameters in each solution hence the lsquoweakerrsquo endorsement
The lsquostrongrsquo condition equally endorses the Vreman QR Gradient and Vortex-Stretching-basedmodels whereas the lsquoweakrsquo condition endorses Vreman and Smagorinsky the most closely followedby the Gradient model with the Vortex-Stretching-based model being the most opposed As this
28
completely contradicts the lsquostrongrsquo condition and the results in Section 41 it makes us question thevalidity of the weaker test The number of endorsements for the QR model is unknown because of thedefinition of the model in Eq (53) and our evaluation of the true subfilter dissipation Eq (67) Dueto the exact solutions we have found it is never clear whether or not the true subfilter dissipationis strictly positive or negative This means that when the true subfilter dissipation is found to benon-zero the lsquoMaxrsquo function in the QR model cannot be evaluated
43 Vremanrsquos True Subfilter Dissipation Claim
Using the theory mentioned in Sections 232 and 34 we can place the exact solutions to the Navier-Stokes equations into different classes defined by Vreman In doing this we can also attempt to verifyone of Vremanrsquos results [16] In Vremanrsquos paper he defines the class Qn to be the set of flows withn zero entries in the velocity gradient of the flow (the velocity gradient is defined in Eq (39))Vreman then claims that the only flows with zero true subfilter dissipation are a certain selection ofthose in the classesQ7Q8 orQ9 shown in appendix B and is generally non-zero for any other flows
The true subfilter dissipation of Sol 1 (Linear Velocity) was computed in Section 421 and wasfound to be zero in Eq (72) The velocity gradient of Sol 1 is as follows
GLV =
0 0 0a 0 0c 0 0
(75)
and hence Sol 1 (Linear Velocity) is in the required subset of Q7 which agrees with Vremanrsquos claimIn Section 422 the true subfilter dissipation was computed (determined to be zero or non-zero) forall of the exact solutions to the Navier Stokes equations in Section 22 These solutions can now belsquoplacedrsquo into each class Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a functionof time) and Sol 14 (Radially Dependent) were all found to have zero true subfilter dissipation andare all in the appropriate subsets of either Q7Q8 or Q9 This agrees with Vremanrsquos results
We can extend Vremanrsquos results however We found that Sol 15 (Rankine Vortex) had zero truesubfilter dissipation and yet is not in the subset of Q7 defined in Appendix B It is important tonote that the rate of strain tensor Sij in Eq (41) for Sol 15 (Rankine Vortex) is zero This is thereason the true subfilter dissipation is zero and the same reason that all models are zero for thisflow as all models are based on Sij
We also found that all of the remaining exact solutions to the Navier-Stokes equations are not inVremanrsquos specified classes (in Appendix Section B) They also have a non-zero subfilter dissipationwhich agrees with Vremanrsquos results in his paper [16]
44 Classes of Flows
It is important to consider that some exact solutions to the Navier-Stokes equations may have similarproperties For example Sol 1 (Linear Velocity) and Sol 2 (Planar Flow) are both very similarIn fact Sol 2 (Planar Flow) is a more general case of Sol 1 (Linear Velocity) We therefore mustgroup the different flows into classes in an attempt to reduce the bias that may occur when solutionsare similar This should aid us to further determine which of the models are most endorsed We willdo this in a number of different ways using the theory discussed in Section 23 We will group theflows with respect to Vremanrsquos Flow Classes given in appendix B and with respect to the Invariantsgiven in appendix A
29
441 Vremanrsquos Velocity Gradient Flow Classes
We now put our exact solutions to the Navier-Stokes equations into one of two classes Either Q0minus6or Q7minus9 where each class is defined intuitively as follows
Q0minus6 = Q0 cupQ1 cup middot middot middot cup Q6 (76)
Q7minus9 = Q7 cupQ8 cupQ9 (77)
where Qn is defined in Section 43 By looking at the gradient of each exact solution we can see thatQ7minus9 contains the six solutions Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as afunction of time) Sol 5 (2D Taylor) Sol 14 (Radially Dependent) and Sol 15 (Rankine Vortex)and that Q0minus6 contains the remaining nine We now check the models for LES without explicitfiltering by considering endorsements within each class Tables 6 and 7 are analogues of Table 3 andillustrate the results for each flow class The analogues of Tables 1 and 2 are Tables 22 and 23 andare in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 6 6 6
lsquoDissipationrsquoEndorsements
2 5 6 6 6
Percentage Endorsed 417 833 100 100 100
Table 6 Each entry in the top two rows represents the number of solutions in the class Q7minus9 thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Q0minus6 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 2 3 3
lsquoDissipationrsquoEndorsements
0 0 1 0 3
Percentage Endorsed 56 56 167 167 333
Table 7 Each entry in the top two rows represents the number of solutions in the class Q1minus6 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking results to 1 decimal place if necessary
All of the flows in the class Q7minus9 endorse the QR Gradient and Vortex-Stretching-based subfilter-scale models (see Table 6) We can hence see that this class fully endorses these three models equallyand does not endorse the Smagorinsky model The Vortex-Stretching eddy viscosity model has themost endorsements from any of the flows in the class Q0minus6 and the Smagorinksy and Vreman modelshave the fewest endorsements (see Table 7) We can hence say that this class mostly endorses theVortex-Stretching-based eddy-viscosity model note however that the percentage endorsement is very
30
low
Now let us consider LES with explicit filtering Tables 8 and 9 are analogues of Table 5 and illustratethe results for each flow class The analogue of Table 4 is Table 24 and is in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 1 1 0 0 0
Table 8 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Q0minus6 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 9 9 9 6
Table 9 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We have grouped Vremanrsquos classes into Q7minus9 and Q0minus6 this is equivalent to saying that we havegrouped Vremanrsquos classes into those with two or fewer non-zero velocity gradient elements and thosewith three or more non-zero velocity gradient elements respectively Therefore the solutions in Q7minus9are exactly those with a lsquo0rsquo in the lsquoExactrsquo column of Table 4 and Sol 5 (2D Taylor) ExcludingSol 5 (2D Taylor) these are the same as those five that endorse the Vreman QR Gradient andVortex-Stretching-based models in Table 5 We can see in Table 8 that the results for the lsquostrongrsquoendorsements are exactly the same as in Table 5 This is because all of the flows with zero subfilterdissipation are contained in the class Q7minus9 There are hence no lsquostrongrsquo endorsements in Table 9The majority of the lsquoweakrsquo endorsements come from Q0minus6 as this class contains all but one of thesolutions with non-zero true subfilter dissipation (given in Eq (67))
442 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
We now separate our exact solutions into classes again using Vremanrsquos flow classes This timehowever we define V0 to be the class of flows Vreman claims will always have zero true subfilterdissipation namely those classes mentioned in Appendix B We define Vlowast to be all the remaining
31
classes Then
V0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow)
Sol 3 (Velocity as a function of time) Sol 14 (Radially Dependent)(78)
Vlowast = Sol 4 (Axisymmetric Flow) Sol 5 (2D Taylor) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 13 (Sphere Translation) Sol 15 (Rankine Vortex)
(79)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 10 and 11 are analogues of Table 3 and illustrate the results for each flow class Theanalogues of Tables 1 and 2 are Tables 25 and 26 and are in Appendix C2
Endorsements for models using solutions in V0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 4 4 4 4
lsquoDissipationrsquoEndorsements
1 4 4 4 4
Percentage Endorsed 375 100 100 100 100
Table 10 Each entry in the top two rows represents the number of solutions in the class V0 thatendorse each model There are 4 solutions in this class so the percentages are worked out by dividingthe total of each column by 8 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Vlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 2 4 5 5
lsquoDissipationrsquoEndorsements
1 1 3 2 5
Percentage Endorsed 136 136 318 318 455
Table 11 Each entry in the top two rows represents the number of solutions in the class Vlowast thatendorse each model There are 11 solutions in this class so the percentages are worked out bydividing the total of each column by 22 and taking results to 1 decimal place if necessary
All of the flows in the class V0 endorse the Vreman QR Gradient and Vortex-Stretching-basedsubfilter-scale models (see Table 6) This class completely endorses these four models equally anddoes not endorse the Smagorinsky model The Vortex-Stretching-based eddy-viscosity model hasthe most endorsements from the class Vlowast and the Smagorinksy and Vreman models again have thefewest endorsements (see Table 7) As in Section 441 we say that this class mostly endorses the
32
Vortex-Stretching-based eddy-viscosity model but once again the percentage endorsement is ratherlow
Now let us consider LES with explicit filtering Tables 12 and 13 are analogues of Table 5 andillustrate the results for each flow class The analogue of Table 4 is Table 27 and is in Appendix C2
Endorsements for models using solutions in V0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 4 4 4 4
lsquoWeakrsquo Endorsements 0 0 0 0 0
Table 12 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Vlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 1 1 1 1
lsquoWeakrsquo Endorsements 10 10 9 6
Table 13 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We can see in Table 12 that the results for the lsquostrongrsquo endorsements are similar to those in Table 5excluding Sol 15 (Rankine Vortex) The four solutions in V0 with zero true subfilter dissipationnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)and Sol 14 (Radially Dependent) are hence the four that endorse the Vreman QR Gradientand Vortex-Stretching-based models in Table 12 Sol 15 (Rankine Vortex) is in the class Vlowast andhence there is one lsquostrongrsquo endorsement in Table 13 The class Vlowast contains all of the flows with anon-zero true subfilter dissipation (given in Eq (67)) and hence Table 13 contains all of the lsquoweakrsquoendorsements
443 Classes using the Third Principal Invariant
Sorting the classes using the invariants is a non-trivial task in itself There are a number of waysin which to do this using both the principal invariants in Eq (40) and the combined invariants inEq (43)
We start by looking at flows with third principal invariant RG equal to zero We again define twoclasses R0 and Rlowast where R0 consists of all flows with RG = 0 and Rlowast consists of all flows withRG 6= 0 Then using the information in Appendix A we see that there are nine flows in R0 That
33
is
R0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 8 (Vortex in Stagnation Flow)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)
(80)
Rlowast = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 13 (Sphere Translation)(81)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 14 and 15 are analogues of Table 3 and they illustrate the results for LES withoutexplicit filtering for each flow class The analogues of Tables 1 and 2 are Tables 28 and 29 and arein Appendix C3
Endorsements for models using solutions in R0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 7 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 278 556 778 722 778
Table 14 Each entry in the top two rows represents the number of solutions in the class R0 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking the result to one decimal place if necessary
Endorsements for models using solutions in Rlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 2 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 83 83 83 167 333
Table 15 Each entry in the top two rows represents the number of solutions in the class Rlowast thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking the result to one decimal place if necessary
The class R0 endorses both the Vortex-Stretching-based model and the QR model the most andSmagorinsky the least in the case for LES without explicit filtering In the Rlowast class the Vortex-Stretching-based model is again the most endorsed followed by the Gradient model however thepercentages are again small
34
Now let us consider LES with explicit filtering Tables 16 and 17 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 30 and is in Appendix C3
Endorsements for models using solutions in R0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 4 4 3 2
Table 16 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Rlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 6 6 6 4
Table 17 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 we can see in Table 16 that the results for the lsquostrongrsquo endorsements are exactly thesame as in Table 5 This is because all of the flows with zero subfilter dissipation are contained in theclass R0 As in Table 9 there are no lsquostrongrsquo endorsements in Table 17 The lsquoweakrsquo endorsementshowever are split more evenly between the two classes now than in Vremanrsquos classes in Sections 441and 442 We find that the Smagorinsky and Vreman models still seem to be the most endorsed inboth classes
444 Two-component Flow Classes
Another way in which the flows can be grouped into classes is by determining whether a flow is atwo-component flow or not We define a two-component flow to have the property that the combinedprincipal invariants of a flow in Eq (43) satisfy
I3 = I4 = I5 minus1
2I1I2 = 0 (82)
We are also interested in considering flows with zero gradient model subgrid dissipation These flowshave the property that
I3 minus I4 = 0 (83)
What we find though is that for all the solutions in Section 22 if a flow is a two-component flowthen it also satisfies I3 minus I4 = 0 It may be interesting to note that although I3 minus I4 6= 0 for manysolutions there are a number with I3 equal to a multiple of I4 Although not considered in thisreport this may also be a class of interest We find that the two-component flows are
C2 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)(84)
35
and we call the remaining flows three-component flows They are
C3 = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex) Sol 11 (Pseudo-Motion Shots)
Sol 13 (Sphere Translation)
(85)
Note that Sol 8 (Vortex in Stagnation Flow) is defined to be a planar flow this does not mean that itis a two-component flow in this sense however because its two non-zero velocity components dependon three coordinates We now check the models for LES without explicit filtering by consideringendorsements within each class Tables 18 and 19 are analogues of Table 3 and they illustrate theresults for LES without explicit filtering for each flow class The analogues of Tables 1 and 2 areTables 31 and 32 and are in Appendix C4
Endorsements for models using solutions in C2 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 6 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 357 714 100 857 100
Table 18 Each entry in the top two rows represents the number of solutions in the class C2 thatendorse each model There are 7 solutions in this class so the percentages are worked out by dividingthe total of each column by 14 and taking the result to one decimal place if necessary
Endorsements for models using solutions in C3 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 3 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 63 63 63 188 250
Table 19 Each entry in the top two rows represents the number of solutions in the class C3 thatendorse each model There are 8 solutions in this class so the percentages are worked out by dividingthe total of each column by 16 and taking the result to one decimal place if necessary
The class C2 equally endorses the QR and Vortex-Stretching-based models and once again theSmagorinsky model is the least endorsed As with the previous classes the class C3 has such lowpercentages that it does not seem to endorse any models We start to see a trend appearing here inthat the flow classes that contain the more complicated flows namely Q0minus6 (Eq (76)) Vlowast (Eq (79))Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages The classes that tendto contain the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78)) R0 (Eq (80)) and C2 (Eq (84))all seem to favour the Vortex-Stretching-based model or the QR model
36
Now let us consider LES with explicit filtering Tables 20 and 21 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 33 and is in Appendix C4
Endorsements for models using solutions in C2 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 2 2 0 1 0
Table 20 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in C3 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 8 8 8 6
Table 21 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 and Table 16 we can see in Table 20 that the results for the lsquostrongrsquo endorsements areexactly the same as in Table 5 This is again because all of the flows with zero subfilter dissipationare contained in the class C2 There are two flows with non-zero true subfilter dissipation (givenin Eq (67)) in C2 and this shows in the lsquoweakrsquo endorsements in Table 20 There are no lsquostrongrsquoendorsements in Table 21 as all the zero subfilter dissipation flows are in C2
5 Conclusions
In this report we studied incompressible turbulent flows We found a number of exact solutions to theNavier-Stokes equations each representing a relatively simplistic fluid flow Large-eddy simulation(LES) is used to model turbulent flows by smoothing over the complex small-scale motion Weintroduced a number of subfilter-scale models to approximate this small-scale motion and testedthem using the found exact solutions to the Navier-Stokes equations We first considered LESwithout explicit filtering and used the Navier-Stokes equations to derive conditions for the subfilter-scale models namely the Velocity Field condition (Eq (58)) the Kinetic Energy condition (Eq (60))and the Dissipation condition (Eq (62)) Similarly we considered LES with explicit filtering andagain used the Navier-Stokes equations to derive more conditions for the models in this case wedecided to just take one condition the Dissipation condition (Eqs (67) and (68)) as the others leadto difficulty in their evaluation By lsquorunningrsquo the exact solutions to the Navier-Stokes equationsthrough each condition for the subfilter-scale models we found that the Vortex-Stretching-basededdy-viscosity model in Eq (54) was most endorsed by the tests in LES without explicit filteringWe found that four out of the five subfilter-scale models namely the Vreman QR Gradient andVortex-Stretching-based models were equally endorsed by the lsquostrongrsquo tests in LES with explicitfiltering with the Smagorinsky model being the least endorsed The results we found for the lsquoweakrsquocondition were completely opposite to those found in both the lsquostrongrsquo condition and in LES withoutexplicit filtering This led us to conclude that this test was not one in which we should put our trust
37
We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses
In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes
This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold
38
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
Figure 7 An illustration of the flow in the yz-plane for Sol 7 (Stagnation Flow) describedin Eq (23) with parameter values a = Re = 1and k = 2
Figure 8 An illustration of the flow in the xy-plane for Sol 7 (Stagnation Flow) describedin Eq (23) with parameter values a = Re = 1and k = 2
Solution 8 Vortex in Stagnation Flow
As in Sol 7 (Stagnation Flow) we now consider the same solution with a different stream functionnamely ψ(z) = cos z Then given the condition that k = aRe and adding a pressure term Eq (22)becomes
u1(x y z t) =1
Rex
u2(x y z t) = minus 1
Rey minus k sin z
u3(x y z t) = 0
p(x y z t) = minus 1
2Re2(x2 + y2)
(24)
where a and k are as in Sol 7 (Stagnation Flow)
12
Figure 9 An illustration of the flow in the yz-plane for Sol 8 (Vortex in Stagnation Flow)described in Eq (24) with parameter valuesk = 3 and Re = 10
Figure 10 An illustration of the flow in thexy-plane for Sol 8 (Vortex in StagnationFlow) described in Eq (24) with parametervalues k = 3 and Re = 10
Solution 9 Burgers Vortex
The Burgers Vortex solution is given in cylindrical coordinates in the book by Lautrup [8 p 385]as follows
ur(r ϕ z t) = minus 2r
c2Re
uϕ(r ϕ z t) =Ωc2
r
(1minus eminusr
2c2)
uz(r ϕ z t) =4z
c2Re
p(r ϕ z t) = minusc2Ω2F
(r2c2
)2
minus 2
Re2r2 + 4z2
c4
(25)
where
F (ξ) =
infinintξ
(1minus eminusx)2
x2dx (26)
This solution is not particularly nice to look at however most of the unpleasantness comes fromthe pressure term which (in this project) is only required in order to validate the solution For thisproject all solutions are required in Cartesian coordinates the velocity field of the Burgers Vortexhas hence been converted into Cartesian coordinates
u1(x y z t) = minus 2x
c2Reminus Ω
h(x y)(1minus exp (minush(x y))) y
u2(x y z t) = minus 2y
c2Re+
Ω
h(x y)(1minus exp (minush(x y)))x
u3(x y z t) =4z
c2Re
(27)
13
where
h(x y) =x2 + y2
c2 (28)
and Ω and c are constants
Figure 11 An illustration of the flow for Sol9 (Burgers Vortex) described in Eq (27) withparameter values c = 15 Re = 20 and Ω = 10A slice of the xy-plane at z = 0 is shown
Solution 10 Another Vortex Solution
There is another vortex solution more simply formulated than the Burgers Vortex yet with asingularity at r = 0 as a result of the uϕ term It is again given in cylindrical coordinates
ur(r ϕ z t) = minus 2r
c2Re
uϕ(r ϕ z t) =a
r
uz(r ϕ z t) =4z
c2Re
p(r ϕ z t) = minus a2
2r2minus 2r2 + 8z2
c2Re2
(29)
which can also be converted into Cartesian coordinates
14
u1(x y z t) = minus 2x
c2Reminus ay
x2 + y2
u2(x y z t) = minus 2y
c2Re+
ax
x2 + y2
u3(x y z t) =4z
c2Re
(30)
where a and c are constants
Figure 12 An illustration of the flow for Sol9 (Burgers Vortex) described in Eq (27) withparameter values c = 15 Re = 20 and a = 10 Aslice of the xy-plane at z = 0 is shown
Solution 11 Pseudo-Motion Shots of the Second Kind
In the book by Berker [3 p 94] a solution is given in cylindrical coordinates by
ur(r ϕ z t) =2
Re
1
r
uϕ(r ϕ z t) = A1r3 +B1r +
C1
r
uz(r ϕ z t) = A2r2 log(r) +B2r
2 + C2
p(r ϕ z t) =A2
1r6
6+A1B1r
4
2+A1C1r
2 +B2
1r2
2minus C2
1
2r2minus 2
r2Re2+ 2B1C1 log(r)
+2A2z
Reminus 4B1ϕ
Re
(31)
where A1 A2 B1 B2 C1 C2 are constants Eq (31) when converted into Cartesian coordinates is
15
u1(x y z t) =1
Re
2xminus C1Rey
x2 + y2minusA1y
(x2 + y2
)minusB1y
u2(x y z t) =1
Re
2y + C1Rex
x2 + y2+A1x
(x2 + y2
)+B1x
u3(x y z t) =1
2A2
(x2 + y2
)log(x2 + y2
)+B2
(x2 + y2
)+ C2
(32)
Figure 13 An illustration of the flow in thexy-plane for Sol 11 (Pseudo-Motion Shots)described in Eqs (31) and (32) with parametervalues A1 = B1 = C1 = 1 A2 = B2 = C2 = 2and Re = 10
Figure 14 An illustration of the flow in theyz-plane for Sol 11 (Pseudo-Motion Shots)described in Eqs (31) and (32) with parametervalues A1 = B1 = C1 = 1 A2 = B2 = C2 = 2and Re = 10
Solution 12 Flows With Constant Whirl
In the book by Berker [3 p 143] another solution is given in cylindrical coordinates by
ur(r ϕ z t) = minus 1
Re
1
r
uϕ(r ϕ z t) =1
rradicte(minus
12 g(t))
uz(r ϕ z t) = 0
p(r ϕ z t) =1
4
2(minus e
minusg(t)
t minus 1Re2
)r2
minus 1
t2Re
infinintminusg(t)
eminust
tdt
(33)
where we define g(t) =r2Re
2t When converted into Cartesian coordinates we find
16
u1(x y z t) = minus 1
x2 + y2
(yradict
exp
(minus
Re(x2 + y2
)4t
)+
x
Re
)
u2(x y z t) =1
x2 + y2
(xradict
exp
(minus
Re(x2 + y2
)4t
)minus y
Re
)
u3(x y z t) = 0
(34)
Figure 15 An illustration of the flow for Sol12 (Flows with Constant Whirl) described inEqs (33) and (34) with time t = 1 and Re = 1 Aslice of the xy-plane at z = 0 is shown as velocityin the z-direction is zero this is the same for anyvalue of z
17
Solution 13 Rectilinear and Uniform Translation of a Sphere
In the book by Berker [3 p 228] another solution is given in Cartesian coordinates by
u1(x y z t) =3
4
axz
R3
(a2
R2minus 1
)
u2(x y z t) =3
4
ayz
R3
(a2
R2minus 1
)
u3(x y z t) = 1minus 3
4
a
Rminus 1
4
a3
R3minus 3
4
az2
R3
(a2
R2minus 1
)
p(x y z t) =3a(aminusR)(a+R)
32R10Re
((R2Re
(x2 + y2
)minus 2R2Rez2
) (a3 + 3aR2 minus 4R3
)+ 3aRez4
(R2 minus a2
)minus 16R5z
)
(35)
where a and R are constants
Figure 16 An illustration of the flow in the yz-plane for Sol 13 (Sphere Translation) describedin Eq (35) with parameter values a = minus2 andRe = 3
Solution 14 Radially Dependent Flow
A solution with velocity in the z-direction dependent only on the distance from the z-axis is as
18
follows in cylindrical coordinates
ur(r ϕ z t) = 0
uϕ(r ϕ z t) = 0
uz(r ϕ z t) = A(r2 minus a2
)
p(r ϕ z t) =4Az
Re
(36)
where A and a are constants In Cartesian coordinates the solution is just the same with u1(x y z t)and u2(x y z t) both zero and the r2 term replaced with x2 + y2 in u3(x y z t)
Figure 17 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus1 1] withparameter values a = 1 A = 4 and Re = 10
Figure 18 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus2 2] withparameter values a = 1 A = 4 and Re = 10
Solution 15 Rankine Vortex
A solution discovered by Rankine [1] is a vortex flow of the following form
ur(r ϕ z t) = 0
uϕ(r ϕ z t) =ar
2πR2
uz(r ϕ z t) = 0
p(r ϕ z t) =a2r2
8π2R4
(37)
19
where a and R are constants In Cartesian coordinates the flow has the form
u1(x y z t) = minus ay
2πR2
u2(x y z t) =ax
2πR2
u3(x y z t) = 0
p(x y z t) =a2(x2 + y2
)8π2R4
(38)
This flow is actually a rigid body rotational flow in that the angular rotational velocity is uniformand the velocity of the flow increases proportionally with distance from the z-axis
Figure 19 An illustration of the flow in the xy-plane for Sol 15 (Rankine Vortex) described inEq (37) and Eq (38) with parameter values a = 1and R = 1
radic2π
23 Characterisation of Flows
It is important to be able to classify flows in a way that sorts them into distinct groups This aidsin the clear identification and general study of flows It is also a very important way of recognisingflows with similar properties In this subsection we introduce two distinct ways in which flows canbe characterised
231 Characterising Solutions using Invariants
A principal invariant of a tensor is a coefficient of the characteristic polynomial of that tensorInvariants as their name suggests are invariant under rotations of the coordinate system in thattheir values do not change when the tensor is lsquorotatedrsquo into a different set of coordinates Thevelocity gradient of a flow G is defined as the Jacobian of the velocity vector given by
Gij =
[partuipartxj
] (39)
20
for i j = 1 2 3 There are three principal invariants of the velocity gradient of a fluid flow G givenby
PG = tr(G)
QG =1
2
((tr(G))2 minus tr
(G2))
RG = det(G)
(40)
Incompressibility of flows (Eq (5)) ensures that PG = 0 hence the first principal invariant is zerofor all of the flows in Section 22 We now define the rate-of-strain tensor S by
Sij =1
2
(partuipartxj
+partujpartxi
) (41)
and the rate-of-rotation tensor Ω by
Ωij =1
2
(partuipartxjminus partujpartxi
) (42)
Note that the first principal invariant of S and the first and third principal invariants of Ω are zeroAlso note that the velocity gradient of a flow can simply be written G = S + Ω
Rather than considering the principal invariants we can also consider the combined invariants
I1 = tr(S2) I2 = tr(Ω2) I3 = tr(S3) I4 = tr(SΩ2) I5 = tr(S2Ω2) (43)
We can hence write the three principal invariants for a velocity field G in terms of the combinedinvariants
PG = 0
QG = minus1
2(I1 + I2)
RG =1
3(I3 + 3I4)
(44)
In this report the flows considered are the exact solutions to the incompressible Navier-Stokesequations described in Section 22 The combined invariants in Eq (43) are used to characterisethese solutions into different classes in Section 44 Both the principal invariants and combinedinvariants for each exact solution in Section 22 are given in Appendix A
Each of the subfilter-scale eddy-viscosity models described in Section 32 are written in terms ofthese five combined invariants
232 Vremanrsquos Characterisation of Flows
Vreman [16] classifies flows using the gradient of the velocity field of a flow Each lsquoclassrsquo of flows hasthe same number of non-zero elements in the gradient matrix For example
[partuipartxj
]=
0 0 0lowast 0 0lowast 0 0
(45)
where a lowast denotes a non-zero term represents a group of flows that lie within the class denoted byQ7 (see Appendix B) where the subscript represents the number of non-zero terms in the velocity
21
gradient Flows of the form in Eq (45) have a velocity field of the form
u1 = c1 (constant)
u2 = f2(x1)
u3 = f3(x1)
(46)
where f2(x1) and f3(x1) are such that the Navier Stokes equations Eq (1) are satisfied Notethat as the elements on the leading diagonal are zero the incompressibility condition Eq (5) isautomatically satisfied Some simple flow classes are given in Appendix B and contain a number ofthe solutions discussed in Section 22 This is mentioned in greater detail in Section 441
3 Conditions for Large-Eddy Simulation Models
31 Large-Eddy Simulation
Due to the nature of numerical simulation and the complexity of turbulent flows small details aremissed and information is lost when solving the Navier-Stokes equations The computational costof solving a turbulent flow is extremely high so a filtering operation is applied to the Navier-Stokesequations represented by a bar to reduce the computational cost of the simulation This is a linearoperation and preserves constants ie ρ = ρ and is also assumed to commute with differentiation
partuipartxj
=partuipartxj
(47)
The filtered Navier-Stokes equations are found by filtering (putting a bar over) the entire equations(see Eq (1)) and after evaluating are given by
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (48)
where
τij = uiuj minus uiuj (49)
is called the lsquotrue stressrsquo of the flow This however poses a problem as it is not clear what is meantby the term uiuj for a general filtering operation The true stress τij therefore must be replaced bya model τmod
ij in practical applications
32 Subfilter-Scale Models
We will now define the subflter-scale models denoted by τmodij to be used in place of the lsquodifficultrsquo
true stress τij There are a number of models available to use in the literature [11 p7] These willbe tested using the exact solutions to the Navier-Stokes equations and by checking conditions whichare defined in Sections 33 and 34
All the models barring one are of the same form That is each of the eddy-viscosity models arewritten
τmodij minus 1
3τmodkk δij = minus2νeSij (50)
with νe representing the eddy viscosity of a specific model For simplicity we usually just writeτmodij = minus2νeSij the extra term is included here to emphasise that the models are traceless Each
22
of the models we consider here can be expressed in terms of the rate of strain tensor S defined inEq (41) the rate of rotation tensor Ω defined in Eq (42) and the combined invariants defined inEq (43) The models are then defined as follows
Smagorinsky [12] νSe = (CSδ)2radic
2I1 (51)
Vreman [16] νVe = (CV δ)2
radicQGGT
PGGT (52)
QR [13 14 15] νQRe = (CQRδ)2 max0minusI3
I1 (53)
Vortex-Stretching [11] νV Se = (CV Sδ)2radic
2I1
(I5 minus 1
2I1I2
minusI1I2
)32
(54)
with the quantities
PGGT = I1 minus I2 QGGT =1
4(I1 + I2)2 + 4
(I5 minus
1
2I1I2
) (55)
There is a fifth and final model to be considered here It is of a different form to the other four andnot an eddy-viscosity model It is known as the Gradient model and is defined by
Gradient [4 9] τmodG = CGδ
(S2 minus Ω2 minus (SΩminus ΩS)
) (56)
The values C are model constants specific to each model and δ represents the mesh size of the LESBoth of these quantities are included here to properly define the models however in this projectthey will be ignored as they are constants and have no influence on the conditions to be defined
33 Conditions for Large-Eddy Simulation without Explicit Filtering
The filtering operation introduced in Eq (48) is just a formality for LES without explicit filteringThe idea here is that filtering does not actually take place but the term lsquoaddedrsquo to the Navier-Stokesequations in fact lsquosubtractsrsquo the difficult to simulate small-scale turbulent motion of the flow Inorder to consider this properly the terms ui and p in Eq (48) are replaced by vi and q It is hopedthat τmod
ij approximates τij so that vi and q approximate ui and p respectively So LES withoutexplicit filtering is given by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (57)
It is clear to see that the first four terms of this equation are exactly the same as the Navier-Stokesequations so for the exact solutions discussed in Section 22 the combination of these terms willequal zero What is then desirable is that the model term is also zero for these exact solutions sothat Eq (57) still holds In order to test models for exact solutions the condition
part
partxjτmodij (v) = 0 (58)
is checked If this condition were to hold it would imply that the model does not influence thevelocity field of a flow hence we call it the Velocity Field Condition There are a number of otherrestrictions that can be tested One such test comes from the differential form of the evolution ofthe kinetic energy of a flow given in Eq (10) In order to find this Eq (57) is multiplied by vi andterms are grouped to find
part
partt
(1
2vivi
)+ vj
part
partxj
(1
2vivi
)= minusvi
1
ρ
partq
partxi+ νvi
part2vipartxjpartxj
minus vipart
partxjτmodij (v) (59)
23
Then similarly to Eq (58) another condition is to check whether the model affects the kineticenergy of exact solutions this is equivalent to checking whether
vipart
partxjτmodij (v) = 0 (60)
We call this condition the Kinetic Energy Condition The third and final condition considered herecan also be found using the integral form for the equation for kinetic energy
d
dt
intV
1
2vividV =
intS
[minusvj
(1
2vivi
)minus vj
q
ρ+ 2νviSij + viτ
modij (v)
]njdS
+
intV
[minus2νSijSij minus
(minusSijτmod
ij (v))]dV
(61)
where the rate-of-strain tensor Sij was defined in Eq (41) V is a volume of fluid and dS meansthe integral is over the surface of that volume with unit normal nj Then this final condition checkswhether the model affects the total kinetic energy of exact solutions to the Navier-Stokes equationsthat is
Sijτmodij (v) = 0 (62)
We will call this condition the Dissipation Condition because Sijτmodij (v) is often referred to as the
subgrid dissipation
34 Conditions for Large-Eddy Simulation with Explicit Filtering
As in Eq (48) the Navier-Stokes equations can be filtered by applying the filtering operation to thewhole equation (putting a bar over all terms) and evaluating to find again that
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (63)
In Eq (57) the filtering operation was ignored and the elements ui p and τij(u) were replacedwith vi q and τmod
ij (v) respectively Now in LES with explicit filtering the filtering operation itselfsmooths over the small-scale turbulent motion of the simulated flow In this case the elements uip and τij(u) will be replaced with vi q and τmod
ij (v) respectively LES with explicit filtering is thengiven by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (64)
Note that Eqs (63) and (64) despite their similarity to Eq (57) no longer contain the Navier-Stokesequations as Eq (57) did The bar (filtering operation) over each term results in each exact solutionfound in Section 22 not necessarily satisfying the first four terms It is hence unknown whether ornot these terms will vanish for exact solutions Terms can still be compared similarly to Eqs (58)(60) and (62) However instead of comparing to zero we can now compare to the relevant termfrom the filtered Navier-Stokes equations So the terms
part
partxjτij (65)
and
part
partxjτmodij (v) (66)
24
are to be compared Although the latter term is relatively simple to compute the former could leadto some difficulties due to the differentiation of the true stress τij = uiuj minus uiuj So although someinformation could be gleaned from the inspection of these two terms more information can be foundby considering the analogue of Eq (62) So the true subfilter dissipation
Sijτij (67)
(where Sij is shorthand for Sij(v)) and the modelled subfilter dissipation
Sij(v)τmodij (v) (68)
are considered and have to be equal for the condition to hold These can again be derived from theintegral equation for kinetic energy of a filtered velocity field similarly to Eq (61)
4 Analysis of Large-Eddy Simulation Models
In this section we use a code in Mathematica the theory described in Section 3 and the exactsolutions to the Navier-Stokes equations found in Section 22 to check whether or not each of theexact solutions lsquoendorsesrsquo each model for LES both without and with explicit filtering
41 Large-Eddy Simulation Without Explicit Filtering
We start by using a code in Mathematica to test the Velocity Field condition in Eq (58) and theKinetic Energy condition in Eq (60) For all exact solutions to the Navier-Stokes equations inSection 22 the outputs for both conditions were identical so we have combined them into one table
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 ndash ndash 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 1 For the Kinetic Energy condition a lsquo0rsquo result means that vipartpartxj
τmodij (v) = 0 for respective
models and exact solutions whereas a lsquolowastrsquo result means that vipartpartxj
τmodij (v) 6= 0 For the Velocity
Field condition a lsquo0rsquo result means that partpartxj
τmodij (v) = 0 for respective models and exact solutions
whereas a lsquolowastrsquo result means that partpartxj
τmodij (v) 6= 0 The Gradient model can take both positive and
negative values so when the sign of the model is known it is included in the table instead of a lsquorsquoBoth the conditions are included in one table because for all cases the outputs were identical
We then test the Dissipation condition in Eq (62) for all exact solutions to the Navier-Stokesequations in Section 22
25
Dissipation Condition [Does Sijτmodij (v) = 0]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 + 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 2 A lsquo0rsquo result means that Sijτmodij (v) = 0 for respective models and exact solutions whereas
a lsquolowastrsquo result means that Sijτmodij (v) 6= 0 The Gradient model can take both positive and negative
values so when the sign of the model is known it is included in the table instead of a lsquorsquo
We then sum all the results to find the total number of endorsements for each model described inSection 32
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
4 6 8 9 9
lsquoDissipationrsquoEndorsements
2 5 7 7 9
Total Endorsements 6 11 15 16 18
Table 3 The results from Tables 1 and 2 are summarised to give a general overview of how manyendorsements each model receives This makes it much clearer to see the models most supported bythe respective and combined conditions with the Vortex-Stretching model being the most supportedand the Smagorinsky model the least
Testing the models in Section 32 using LES without explicit filtering has mixed results It isclear that the Smagorinksy model is not well endorsed by the flows in Section 22 with the onlyendorsements coming from the simplest flows that is Sol 1 (Linear Velocity) Sol 3 (Velocity as afunction of time) Sol 7 (Stagnation Flow) and Sol 15 (Rankine Vortex) The Vortex-Stretching-based model is the most endorsed flow with eighteen out of a possible thirty endorsements Thisagrees with our foreknowledge as this model was constructed to be as physically consistent as possiblein the paper by Silvis amp Verstappen [11]
42 Large-Eddy Simulation With Explicit Filtering
In this section we will compare the true subfilter dissipation Sijτij and the modelled subfilterdissipation Sij(v)τmod
ij (v) given in Eqs (67) and (68) respectively for each model τmodij and each
exact solution to the Navier-Stokes equation given in Section 22 We will not consider the conditionsinvolving differentiation in Eqs (65) and (66) due to the difficulty in calculating the derivative ofthe τij term in Eq (65)
26
421 Example of a Test for a Model
The true subfilter dissipation Sijτij of a flow given in Eq (67) cannot be explicitly computed asthe filtering operation is kept general What can be done however is to partially evaluate theexpressions using the attributes of a filtering operation mentioned in Eq (48)
The calculations of Sijτij for Sol 1 (Linear Velocity) are provided to illustrate the general ideaRecall that the velocity field of Sol 1 (Linear Velocity) is written in Eq (13) in vector notation as
u(x y z t) =
0ax+ bcx+ d
(69)
Then Sij = Sij(u) is computed simply by using its definition in Eq (41) Note that as filteringpreserves constants if there is an a it is replaced with a meaning that (in this case) there are alarge number of zero terms due to the frequency of 0 Hence
[Sij(u)
]=
[1
2
(partuipartxj
+partujpartxi
)]=
0 a
2c2
a2 0 0
c2 0 0
=
0 a
2c2
a2 0 0
c2 0 0
(70)
Note that as Sijτij is being computed which is equal to the sum of the element-wise multiplicationof the two matrices (or the trace of the matrix multiplication) that we donrsquot need to consider theelements of τij(u) with corresponding zero entry in Sij As τij(u) = uiuj minus uiuj and again asfiltering preserves constants it is found that
[τij(u)] =[uiuj minus uiuj
]=
lowast 0 middot ax+ b 0 middot cx+ d
ax+ b middot 0 lowast lowastcx+ d middot 0 lowast lowast
=
lowast 0 00 lowast lowast0 lowast lowast
(71)
where a lsquolowastrsquo just represents some element (not necessarily non-zero) Therefore the true subfilterdissipation
Sijτij = tr
0 a2
c2
a2 0 0c2 0 0
middotlowast 0 0
0 lowast lowast0 lowast lowast
= 0 (72)
Now the modelled subfilter dissipation Sij(u)τmodij (u) needs to be computed (recall that Sij =
Sij(u)) In this case let us use the QR model stated in Eq (53) Once again Sij(u) calculated inEq (70) needs to be used So we have that
τmodij = minus2(CQRδ)
2 max0minusI3I1
Sij (73)
As
I1 = tr(S2) =
1
2(a2 + c2) and I3 = tr(S
3) = 0 (74)
it holds that τmodij (u) = 0 so Sij(u)τmod
ij (u) = 0 This agrees with Eq (72) and hence Sol 1(Linear Velocity) endorses the QR model
27
422 Testing the Models
Now using a Mathematica code the exact solutions in Section 22 are again used to evaluate thecondition for the models using the process just described in Section 421 The results are describedin Table 4
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 4 All entries with a ldquo0rdquo represent a value of zero and a ldquolowastrdquo represents a value that is non-zeroThe ldquoExactrdquo row represents the value of the true subfilter dissipation Sijτij Every other row witha model name represents the value of the modelled subfilter dissipation Sij(u)τmod
ij (u) respectiveto that model A ldquordquo means that it is unclear as to the value - the sign of these values is oftenunknown and the lsquoMaxrsquo function in the definition of the QR model (in Eq (53)) makes it difficultto evaluate
Reading from Table 4 the Smagorinsky model is clearly endorsed by Sol 3 (Velocity as a function oftime) and Sol 15 (Rankine Vortex) as in both cases both Sijτij = 0 = Sij(u)τmod
ij (u) In this casewe say that Sol 3 (Velocity as a function of time) and Sol 15 (Rankine Vortex) lsquostronglyrsquo endorsethe Smagorinsky model The Smagorinsky model is not endorsed by Sol 1 (Linear Velocity) Sol 2(Planar Flow) or Sol 14 (Radially Dependent) however because Sij(u)τmod
ij (u) 6= 0 For Sol 4 to
13 both Sijτij 6= 0 6= Sij(u)τmodij (u) and so we will say that these exact solutions lsquoweaklyrsquo endorse
the Smagorinsky model
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 10 10 9 6
Table 5 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Interestingly the lsquostrongrsquo and lsquoweakrsquo endorsements give different results However more stock shouldbe put in the results of the lsquostrongerrsquo condition as it is completely clear that zero is equal to zeroIt is not necessarily clear whether or not non-zero equals non-zero as the values will depend on theparameters in each solution hence the lsquoweakerrsquo endorsement
The lsquostrongrsquo condition equally endorses the Vreman QR Gradient and Vortex-Stretching-basedmodels whereas the lsquoweakrsquo condition endorses Vreman and Smagorinsky the most closely followedby the Gradient model with the Vortex-Stretching-based model being the most opposed As this
28
completely contradicts the lsquostrongrsquo condition and the results in Section 41 it makes us question thevalidity of the weaker test The number of endorsements for the QR model is unknown because of thedefinition of the model in Eq (53) and our evaluation of the true subfilter dissipation Eq (67) Dueto the exact solutions we have found it is never clear whether or not the true subfilter dissipationis strictly positive or negative This means that when the true subfilter dissipation is found to benon-zero the lsquoMaxrsquo function in the QR model cannot be evaluated
43 Vremanrsquos True Subfilter Dissipation Claim
Using the theory mentioned in Sections 232 and 34 we can place the exact solutions to the Navier-Stokes equations into different classes defined by Vreman In doing this we can also attempt to verifyone of Vremanrsquos results [16] In Vremanrsquos paper he defines the class Qn to be the set of flows withn zero entries in the velocity gradient of the flow (the velocity gradient is defined in Eq (39))Vreman then claims that the only flows with zero true subfilter dissipation are a certain selection ofthose in the classesQ7Q8 orQ9 shown in appendix B and is generally non-zero for any other flows
The true subfilter dissipation of Sol 1 (Linear Velocity) was computed in Section 421 and wasfound to be zero in Eq (72) The velocity gradient of Sol 1 is as follows
GLV =
0 0 0a 0 0c 0 0
(75)
and hence Sol 1 (Linear Velocity) is in the required subset of Q7 which agrees with Vremanrsquos claimIn Section 422 the true subfilter dissipation was computed (determined to be zero or non-zero) forall of the exact solutions to the Navier Stokes equations in Section 22 These solutions can now belsquoplacedrsquo into each class Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a functionof time) and Sol 14 (Radially Dependent) were all found to have zero true subfilter dissipation andare all in the appropriate subsets of either Q7Q8 or Q9 This agrees with Vremanrsquos results
We can extend Vremanrsquos results however We found that Sol 15 (Rankine Vortex) had zero truesubfilter dissipation and yet is not in the subset of Q7 defined in Appendix B It is important tonote that the rate of strain tensor Sij in Eq (41) for Sol 15 (Rankine Vortex) is zero This is thereason the true subfilter dissipation is zero and the same reason that all models are zero for thisflow as all models are based on Sij
We also found that all of the remaining exact solutions to the Navier-Stokes equations are not inVremanrsquos specified classes (in Appendix Section B) They also have a non-zero subfilter dissipationwhich agrees with Vremanrsquos results in his paper [16]
44 Classes of Flows
It is important to consider that some exact solutions to the Navier-Stokes equations may have similarproperties For example Sol 1 (Linear Velocity) and Sol 2 (Planar Flow) are both very similarIn fact Sol 2 (Planar Flow) is a more general case of Sol 1 (Linear Velocity) We therefore mustgroup the different flows into classes in an attempt to reduce the bias that may occur when solutionsare similar This should aid us to further determine which of the models are most endorsed We willdo this in a number of different ways using the theory discussed in Section 23 We will group theflows with respect to Vremanrsquos Flow Classes given in appendix B and with respect to the Invariantsgiven in appendix A
29
441 Vremanrsquos Velocity Gradient Flow Classes
We now put our exact solutions to the Navier-Stokes equations into one of two classes Either Q0minus6or Q7minus9 where each class is defined intuitively as follows
Q0minus6 = Q0 cupQ1 cup middot middot middot cup Q6 (76)
Q7minus9 = Q7 cupQ8 cupQ9 (77)
where Qn is defined in Section 43 By looking at the gradient of each exact solution we can see thatQ7minus9 contains the six solutions Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as afunction of time) Sol 5 (2D Taylor) Sol 14 (Radially Dependent) and Sol 15 (Rankine Vortex)and that Q0minus6 contains the remaining nine We now check the models for LES without explicitfiltering by considering endorsements within each class Tables 6 and 7 are analogues of Table 3 andillustrate the results for each flow class The analogues of Tables 1 and 2 are Tables 22 and 23 andare in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 6 6 6
lsquoDissipationrsquoEndorsements
2 5 6 6 6
Percentage Endorsed 417 833 100 100 100
Table 6 Each entry in the top two rows represents the number of solutions in the class Q7minus9 thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Q0minus6 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 2 3 3
lsquoDissipationrsquoEndorsements
0 0 1 0 3
Percentage Endorsed 56 56 167 167 333
Table 7 Each entry in the top two rows represents the number of solutions in the class Q1minus6 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking results to 1 decimal place if necessary
All of the flows in the class Q7minus9 endorse the QR Gradient and Vortex-Stretching-based subfilter-scale models (see Table 6) We can hence see that this class fully endorses these three models equallyand does not endorse the Smagorinsky model The Vortex-Stretching eddy viscosity model has themost endorsements from any of the flows in the class Q0minus6 and the Smagorinksy and Vreman modelshave the fewest endorsements (see Table 7) We can hence say that this class mostly endorses theVortex-Stretching-based eddy-viscosity model note however that the percentage endorsement is very
30
low
Now let us consider LES with explicit filtering Tables 8 and 9 are analogues of Table 5 and illustratethe results for each flow class The analogue of Table 4 is Table 24 and is in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 1 1 0 0 0
Table 8 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Q0minus6 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 9 9 9 6
Table 9 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We have grouped Vremanrsquos classes into Q7minus9 and Q0minus6 this is equivalent to saying that we havegrouped Vremanrsquos classes into those with two or fewer non-zero velocity gradient elements and thosewith three or more non-zero velocity gradient elements respectively Therefore the solutions in Q7minus9are exactly those with a lsquo0rsquo in the lsquoExactrsquo column of Table 4 and Sol 5 (2D Taylor) ExcludingSol 5 (2D Taylor) these are the same as those five that endorse the Vreman QR Gradient andVortex-Stretching-based models in Table 5 We can see in Table 8 that the results for the lsquostrongrsquoendorsements are exactly the same as in Table 5 This is because all of the flows with zero subfilterdissipation are contained in the class Q7minus9 There are hence no lsquostrongrsquo endorsements in Table 9The majority of the lsquoweakrsquo endorsements come from Q0minus6 as this class contains all but one of thesolutions with non-zero true subfilter dissipation (given in Eq (67))
442 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
We now separate our exact solutions into classes again using Vremanrsquos flow classes This timehowever we define V0 to be the class of flows Vreman claims will always have zero true subfilterdissipation namely those classes mentioned in Appendix B We define Vlowast to be all the remaining
31
classes Then
V0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow)
Sol 3 (Velocity as a function of time) Sol 14 (Radially Dependent)(78)
Vlowast = Sol 4 (Axisymmetric Flow) Sol 5 (2D Taylor) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 13 (Sphere Translation) Sol 15 (Rankine Vortex)
(79)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 10 and 11 are analogues of Table 3 and illustrate the results for each flow class Theanalogues of Tables 1 and 2 are Tables 25 and 26 and are in Appendix C2
Endorsements for models using solutions in V0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 4 4 4 4
lsquoDissipationrsquoEndorsements
1 4 4 4 4
Percentage Endorsed 375 100 100 100 100
Table 10 Each entry in the top two rows represents the number of solutions in the class V0 thatendorse each model There are 4 solutions in this class so the percentages are worked out by dividingthe total of each column by 8 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Vlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 2 4 5 5
lsquoDissipationrsquoEndorsements
1 1 3 2 5
Percentage Endorsed 136 136 318 318 455
Table 11 Each entry in the top two rows represents the number of solutions in the class Vlowast thatendorse each model There are 11 solutions in this class so the percentages are worked out bydividing the total of each column by 22 and taking results to 1 decimal place if necessary
All of the flows in the class V0 endorse the Vreman QR Gradient and Vortex-Stretching-basedsubfilter-scale models (see Table 6) This class completely endorses these four models equally anddoes not endorse the Smagorinsky model The Vortex-Stretching-based eddy-viscosity model hasthe most endorsements from the class Vlowast and the Smagorinksy and Vreman models again have thefewest endorsements (see Table 7) As in Section 441 we say that this class mostly endorses the
32
Vortex-Stretching-based eddy-viscosity model but once again the percentage endorsement is ratherlow
Now let us consider LES with explicit filtering Tables 12 and 13 are analogues of Table 5 andillustrate the results for each flow class The analogue of Table 4 is Table 27 and is in Appendix C2
Endorsements for models using solutions in V0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 4 4 4 4
lsquoWeakrsquo Endorsements 0 0 0 0 0
Table 12 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Vlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 1 1 1 1
lsquoWeakrsquo Endorsements 10 10 9 6
Table 13 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We can see in Table 12 that the results for the lsquostrongrsquo endorsements are similar to those in Table 5excluding Sol 15 (Rankine Vortex) The four solutions in V0 with zero true subfilter dissipationnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)and Sol 14 (Radially Dependent) are hence the four that endorse the Vreman QR Gradientand Vortex-Stretching-based models in Table 12 Sol 15 (Rankine Vortex) is in the class Vlowast andhence there is one lsquostrongrsquo endorsement in Table 13 The class Vlowast contains all of the flows with anon-zero true subfilter dissipation (given in Eq (67)) and hence Table 13 contains all of the lsquoweakrsquoendorsements
443 Classes using the Third Principal Invariant
Sorting the classes using the invariants is a non-trivial task in itself There are a number of waysin which to do this using both the principal invariants in Eq (40) and the combined invariants inEq (43)
We start by looking at flows with third principal invariant RG equal to zero We again define twoclasses R0 and Rlowast where R0 consists of all flows with RG = 0 and Rlowast consists of all flows withRG 6= 0 Then using the information in Appendix A we see that there are nine flows in R0 That
33
is
R0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 8 (Vortex in Stagnation Flow)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)
(80)
Rlowast = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 13 (Sphere Translation)(81)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 14 and 15 are analogues of Table 3 and they illustrate the results for LES withoutexplicit filtering for each flow class The analogues of Tables 1 and 2 are Tables 28 and 29 and arein Appendix C3
Endorsements for models using solutions in R0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 7 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 278 556 778 722 778
Table 14 Each entry in the top two rows represents the number of solutions in the class R0 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking the result to one decimal place if necessary
Endorsements for models using solutions in Rlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 2 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 83 83 83 167 333
Table 15 Each entry in the top two rows represents the number of solutions in the class Rlowast thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking the result to one decimal place if necessary
The class R0 endorses both the Vortex-Stretching-based model and the QR model the most andSmagorinsky the least in the case for LES without explicit filtering In the Rlowast class the Vortex-Stretching-based model is again the most endorsed followed by the Gradient model however thepercentages are again small
34
Now let us consider LES with explicit filtering Tables 16 and 17 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 30 and is in Appendix C3
Endorsements for models using solutions in R0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 4 4 3 2
Table 16 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Rlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 6 6 6 4
Table 17 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 we can see in Table 16 that the results for the lsquostrongrsquo endorsements are exactly thesame as in Table 5 This is because all of the flows with zero subfilter dissipation are contained in theclass R0 As in Table 9 there are no lsquostrongrsquo endorsements in Table 17 The lsquoweakrsquo endorsementshowever are split more evenly between the two classes now than in Vremanrsquos classes in Sections 441and 442 We find that the Smagorinsky and Vreman models still seem to be the most endorsed inboth classes
444 Two-component Flow Classes
Another way in which the flows can be grouped into classes is by determining whether a flow is atwo-component flow or not We define a two-component flow to have the property that the combinedprincipal invariants of a flow in Eq (43) satisfy
I3 = I4 = I5 minus1
2I1I2 = 0 (82)
We are also interested in considering flows with zero gradient model subgrid dissipation These flowshave the property that
I3 minus I4 = 0 (83)
What we find though is that for all the solutions in Section 22 if a flow is a two-component flowthen it also satisfies I3 minus I4 = 0 It may be interesting to note that although I3 minus I4 6= 0 for manysolutions there are a number with I3 equal to a multiple of I4 Although not considered in thisreport this may also be a class of interest We find that the two-component flows are
C2 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)(84)
35
and we call the remaining flows three-component flows They are
C3 = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex) Sol 11 (Pseudo-Motion Shots)
Sol 13 (Sphere Translation)
(85)
Note that Sol 8 (Vortex in Stagnation Flow) is defined to be a planar flow this does not mean that itis a two-component flow in this sense however because its two non-zero velocity components dependon three coordinates We now check the models for LES without explicit filtering by consideringendorsements within each class Tables 18 and 19 are analogues of Table 3 and they illustrate theresults for LES without explicit filtering for each flow class The analogues of Tables 1 and 2 areTables 31 and 32 and are in Appendix C4
Endorsements for models using solutions in C2 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 6 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 357 714 100 857 100
Table 18 Each entry in the top two rows represents the number of solutions in the class C2 thatendorse each model There are 7 solutions in this class so the percentages are worked out by dividingthe total of each column by 14 and taking the result to one decimal place if necessary
Endorsements for models using solutions in C3 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 3 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 63 63 63 188 250
Table 19 Each entry in the top two rows represents the number of solutions in the class C3 thatendorse each model There are 8 solutions in this class so the percentages are worked out by dividingthe total of each column by 16 and taking the result to one decimal place if necessary
The class C2 equally endorses the QR and Vortex-Stretching-based models and once again theSmagorinsky model is the least endorsed As with the previous classes the class C3 has such lowpercentages that it does not seem to endorse any models We start to see a trend appearing here inthat the flow classes that contain the more complicated flows namely Q0minus6 (Eq (76)) Vlowast (Eq (79))Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages The classes that tendto contain the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78)) R0 (Eq (80)) and C2 (Eq (84))all seem to favour the Vortex-Stretching-based model or the QR model
36
Now let us consider LES with explicit filtering Tables 20 and 21 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 33 and is in Appendix C4
Endorsements for models using solutions in C2 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 2 2 0 1 0
Table 20 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in C3 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 8 8 8 6
Table 21 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 and Table 16 we can see in Table 20 that the results for the lsquostrongrsquo endorsements areexactly the same as in Table 5 This is again because all of the flows with zero subfilter dissipationare contained in the class C2 There are two flows with non-zero true subfilter dissipation (givenin Eq (67)) in C2 and this shows in the lsquoweakrsquo endorsements in Table 20 There are no lsquostrongrsquoendorsements in Table 21 as all the zero subfilter dissipation flows are in C2
5 Conclusions
In this report we studied incompressible turbulent flows We found a number of exact solutions to theNavier-Stokes equations each representing a relatively simplistic fluid flow Large-eddy simulation(LES) is used to model turbulent flows by smoothing over the complex small-scale motion Weintroduced a number of subfilter-scale models to approximate this small-scale motion and testedthem using the found exact solutions to the Navier-Stokes equations We first considered LESwithout explicit filtering and used the Navier-Stokes equations to derive conditions for the subfilter-scale models namely the Velocity Field condition (Eq (58)) the Kinetic Energy condition (Eq (60))and the Dissipation condition (Eq (62)) Similarly we considered LES with explicit filtering andagain used the Navier-Stokes equations to derive more conditions for the models in this case wedecided to just take one condition the Dissipation condition (Eqs (67) and (68)) as the others leadto difficulty in their evaluation By lsquorunningrsquo the exact solutions to the Navier-Stokes equationsthrough each condition for the subfilter-scale models we found that the Vortex-Stretching-basededdy-viscosity model in Eq (54) was most endorsed by the tests in LES without explicit filteringWe found that four out of the five subfilter-scale models namely the Vreman QR Gradient andVortex-Stretching-based models were equally endorsed by the lsquostrongrsquo tests in LES with explicitfiltering with the Smagorinsky model being the least endorsed The results we found for the lsquoweakrsquocondition were completely opposite to those found in both the lsquostrongrsquo condition and in LES withoutexplicit filtering This led us to conclude that this test was not one in which we should put our trust
37
We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses
In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes
This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold
38
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
Figure 9 An illustration of the flow in the yz-plane for Sol 8 (Vortex in Stagnation Flow)described in Eq (24) with parameter valuesk = 3 and Re = 10
Figure 10 An illustration of the flow in thexy-plane for Sol 8 (Vortex in StagnationFlow) described in Eq (24) with parametervalues k = 3 and Re = 10
Solution 9 Burgers Vortex
The Burgers Vortex solution is given in cylindrical coordinates in the book by Lautrup [8 p 385]as follows
ur(r ϕ z t) = minus 2r
c2Re
uϕ(r ϕ z t) =Ωc2
r
(1minus eminusr
2c2)
uz(r ϕ z t) =4z
c2Re
p(r ϕ z t) = minusc2Ω2F
(r2c2
)2
minus 2
Re2r2 + 4z2
c4
(25)
where
F (ξ) =
infinintξ
(1minus eminusx)2
x2dx (26)
This solution is not particularly nice to look at however most of the unpleasantness comes fromthe pressure term which (in this project) is only required in order to validate the solution For thisproject all solutions are required in Cartesian coordinates the velocity field of the Burgers Vortexhas hence been converted into Cartesian coordinates
u1(x y z t) = minus 2x
c2Reminus Ω
h(x y)(1minus exp (minush(x y))) y
u2(x y z t) = minus 2y
c2Re+
Ω
h(x y)(1minus exp (minush(x y)))x
u3(x y z t) =4z
c2Re
(27)
13
where
h(x y) =x2 + y2
c2 (28)
and Ω and c are constants
Figure 11 An illustration of the flow for Sol9 (Burgers Vortex) described in Eq (27) withparameter values c = 15 Re = 20 and Ω = 10A slice of the xy-plane at z = 0 is shown
Solution 10 Another Vortex Solution
There is another vortex solution more simply formulated than the Burgers Vortex yet with asingularity at r = 0 as a result of the uϕ term It is again given in cylindrical coordinates
ur(r ϕ z t) = minus 2r
c2Re
uϕ(r ϕ z t) =a
r
uz(r ϕ z t) =4z
c2Re
p(r ϕ z t) = minus a2
2r2minus 2r2 + 8z2
c2Re2
(29)
which can also be converted into Cartesian coordinates
14
u1(x y z t) = minus 2x
c2Reminus ay
x2 + y2
u2(x y z t) = minus 2y
c2Re+
ax
x2 + y2
u3(x y z t) =4z
c2Re
(30)
where a and c are constants
Figure 12 An illustration of the flow for Sol9 (Burgers Vortex) described in Eq (27) withparameter values c = 15 Re = 20 and a = 10 Aslice of the xy-plane at z = 0 is shown
Solution 11 Pseudo-Motion Shots of the Second Kind
In the book by Berker [3 p 94] a solution is given in cylindrical coordinates by
ur(r ϕ z t) =2
Re
1
r
uϕ(r ϕ z t) = A1r3 +B1r +
C1
r
uz(r ϕ z t) = A2r2 log(r) +B2r
2 + C2
p(r ϕ z t) =A2
1r6
6+A1B1r
4
2+A1C1r
2 +B2
1r2
2minus C2
1
2r2minus 2
r2Re2+ 2B1C1 log(r)
+2A2z
Reminus 4B1ϕ
Re
(31)
where A1 A2 B1 B2 C1 C2 are constants Eq (31) when converted into Cartesian coordinates is
15
u1(x y z t) =1
Re
2xminus C1Rey
x2 + y2minusA1y
(x2 + y2
)minusB1y
u2(x y z t) =1
Re
2y + C1Rex
x2 + y2+A1x
(x2 + y2
)+B1x
u3(x y z t) =1
2A2
(x2 + y2
)log(x2 + y2
)+B2
(x2 + y2
)+ C2
(32)
Figure 13 An illustration of the flow in thexy-plane for Sol 11 (Pseudo-Motion Shots)described in Eqs (31) and (32) with parametervalues A1 = B1 = C1 = 1 A2 = B2 = C2 = 2and Re = 10
Figure 14 An illustration of the flow in theyz-plane for Sol 11 (Pseudo-Motion Shots)described in Eqs (31) and (32) with parametervalues A1 = B1 = C1 = 1 A2 = B2 = C2 = 2and Re = 10
Solution 12 Flows With Constant Whirl
In the book by Berker [3 p 143] another solution is given in cylindrical coordinates by
ur(r ϕ z t) = minus 1
Re
1
r
uϕ(r ϕ z t) =1
rradicte(minus
12 g(t))
uz(r ϕ z t) = 0
p(r ϕ z t) =1
4
2(minus e
minusg(t)
t minus 1Re2
)r2
minus 1
t2Re
infinintminusg(t)
eminust
tdt
(33)
where we define g(t) =r2Re
2t When converted into Cartesian coordinates we find
16
u1(x y z t) = minus 1
x2 + y2
(yradict
exp
(minus
Re(x2 + y2
)4t
)+
x
Re
)
u2(x y z t) =1
x2 + y2
(xradict
exp
(minus
Re(x2 + y2
)4t
)minus y
Re
)
u3(x y z t) = 0
(34)
Figure 15 An illustration of the flow for Sol12 (Flows with Constant Whirl) described inEqs (33) and (34) with time t = 1 and Re = 1 Aslice of the xy-plane at z = 0 is shown as velocityin the z-direction is zero this is the same for anyvalue of z
17
Solution 13 Rectilinear and Uniform Translation of a Sphere
In the book by Berker [3 p 228] another solution is given in Cartesian coordinates by
u1(x y z t) =3
4
axz
R3
(a2
R2minus 1
)
u2(x y z t) =3
4
ayz
R3
(a2
R2minus 1
)
u3(x y z t) = 1minus 3
4
a
Rminus 1
4
a3
R3minus 3
4
az2
R3
(a2
R2minus 1
)
p(x y z t) =3a(aminusR)(a+R)
32R10Re
((R2Re
(x2 + y2
)minus 2R2Rez2
) (a3 + 3aR2 minus 4R3
)+ 3aRez4
(R2 minus a2
)minus 16R5z
)
(35)
where a and R are constants
Figure 16 An illustration of the flow in the yz-plane for Sol 13 (Sphere Translation) describedin Eq (35) with parameter values a = minus2 andRe = 3
Solution 14 Radially Dependent Flow
A solution with velocity in the z-direction dependent only on the distance from the z-axis is as
18
follows in cylindrical coordinates
ur(r ϕ z t) = 0
uϕ(r ϕ z t) = 0
uz(r ϕ z t) = A(r2 minus a2
)
p(r ϕ z t) =4Az
Re
(36)
where A and a are constants In Cartesian coordinates the solution is just the same with u1(x y z t)and u2(x y z t) both zero and the r2 term replaced with x2 + y2 in u3(x y z t)
Figure 17 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus1 1] withparameter values a = 1 A = 4 and Re = 10
Figure 18 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus2 2] withparameter values a = 1 A = 4 and Re = 10
Solution 15 Rankine Vortex
A solution discovered by Rankine [1] is a vortex flow of the following form
ur(r ϕ z t) = 0
uϕ(r ϕ z t) =ar
2πR2
uz(r ϕ z t) = 0
p(r ϕ z t) =a2r2
8π2R4
(37)
19
where a and R are constants In Cartesian coordinates the flow has the form
u1(x y z t) = minus ay
2πR2
u2(x y z t) =ax
2πR2
u3(x y z t) = 0
p(x y z t) =a2(x2 + y2
)8π2R4
(38)
This flow is actually a rigid body rotational flow in that the angular rotational velocity is uniformand the velocity of the flow increases proportionally with distance from the z-axis
Figure 19 An illustration of the flow in the xy-plane for Sol 15 (Rankine Vortex) described inEq (37) and Eq (38) with parameter values a = 1and R = 1
radic2π
23 Characterisation of Flows
It is important to be able to classify flows in a way that sorts them into distinct groups This aidsin the clear identification and general study of flows It is also a very important way of recognisingflows with similar properties In this subsection we introduce two distinct ways in which flows canbe characterised
231 Characterising Solutions using Invariants
A principal invariant of a tensor is a coefficient of the characteristic polynomial of that tensorInvariants as their name suggests are invariant under rotations of the coordinate system in thattheir values do not change when the tensor is lsquorotatedrsquo into a different set of coordinates Thevelocity gradient of a flow G is defined as the Jacobian of the velocity vector given by
Gij =
[partuipartxj
] (39)
20
for i j = 1 2 3 There are three principal invariants of the velocity gradient of a fluid flow G givenby
PG = tr(G)
QG =1
2
((tr(G))2 minus tr
(G2))
RG = det(G)
(40)
Incompressibility of flows (Eq (5)) ensures that PG = 0 hence the first principal invariant is zerofor all of the flows in Section 22 We now define the rate-of-strain tensor S by
Sij =1
2
(partuipartxj
+partujpartxi
) (41)
and the rate-of-rotation tensor Ω by
Ωij =1
2
(partuipartxjminus partujpartxi
) (42)
Note that the first principal invariant of S and the first and third principal invariants of Ω are zeroAlso note that the velocity gradient of a flow can simply be written G = S + Ω
Rather than considering the principal invariants we can also consider the combined invariants
I1 = tr(S2) I2 = tr(Ω2) I3 = tr(S3) I4 = tr(SΩ2) I5 = tr(S2Ω2) (43)
We can hence write the three principal invariants for a velocity field G in terms of the combinedinvariants
PG = 0
QG = minus1
2(I1 + I2)
RG =1
3(I3 + 3I4)
(44)
In this report the flows considered are the exact solutions to the incompressible Navier-Stokesequations described in Section 22 The combined invariants in Eq (43) are used to characterisethese solutions into different classes in Section 44 Both the principal invariants and combinedinvariants for each exact solution in Section 22 are given in Appendix A
Each of the subfilter-scale eddy-viscosity models described in Section 32 are written in terms ofthese five combined invariants
232 Vremanrsquos Characterisation of Flows
Vreman [16] classifies flows using the gradient of the velocity field of a flow Each lsquoclassrsquo of flows hasthe same number of non-zero elements in the gradient matrix For example
[partuipartxj
]=
0 0 0lowast 0 0lowast 0 0
(45)
where a lowast denotes a non-zero term represents a group of flows that lie within the class denoted byQ7 (see Appendix B) where the subscript represents the number of non-zero terms in the velocity
21
gradient Flows of the form in Eq (45) have a velocity field of the form
u1 = c1 (constant)
u2 = f2(x1)
u3 = f3(x1)
(46)
where f2(x1) and f3(x1) are such that the Navier Stokes equations Eq (1) are satisfied Notethat as the elements on the leading diagonal are zero the incompressibility condition Eq (5) isautomatically satisfied Some simple flow classes are given in Appendix B and contain a number ofthe solutions discussed in Section 22 This is mentioned in greater detail in Section 441
3 Conditions for Large-Eddy Simulation Models
31 Large-Eddy Simulation
Due to the nature of numerical simulation and the complexity of turbulent flows small details aremissed and information is lost when solving the Navier-Stokes equations The computational costof solving a turbulent flow is extremely high so a filtering operation is applied to the Navier-Stokesequations represented by a bar to reduce the computational cost of the simulation This is a linearoperation and preserves constants ie ρ = ρ and is also assumed to commute with differentiation
partuipartxj
=partuipartxj
(47)
The filtered Navier-Stokes equations are found by filtering (putting a bar over) the entire equations(see Eq (1)) and after evaluating are given by
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (48)
where
τij = uiuj minus uiuj (49)
is called the lsquotrue stressrsquo of the flow This however poses a problem as it is not clear what is meantby the term uiuj for a general filtering operation The true stress τij therefore must be replaced bya model τmod
ij in practical applications
32 Subfilter-Scale Models
We will now define the subflter-scale models denoted by τmodij to be used in place of the lsquodifficultrsquo
true stress τij There are a number of models available to use in the literature [11 p7] These willbe tested using the exact solutions to the Navier-Stokes equations and by checking conditions whichare defined in Sections 33 and 34
All the models barring one are of the same form That is each of the eddy-viscosity models arewritten
τmodij minus 1
3τmodkk δij = minus2νeSij (50)
with νe representing the eddy viscosity of a specific model For simplicity we usually just writeτmodij = minus2νeSij the extra term is included here to emphasise that the models are traceless Each
22
of the models we consider here can be expressed in terms of the rate of strain tensor S defined inEq (41) the rate of rotation tensor Ω defined in Eq (42) and the combined invariants defined inEq (43) The models are then defined as follows
Smagorinsky [12] νSe = (CSδ)2radic
2I1 (51)
Vreman [16] νVe = (CV δ)2
radicQGGT
PGGT (52)
QR [13 14 15] νQRe = (CQRδ)2 max0minusI3
I1 (53)
Vortex-Stretching [11] νV Se = (CV Sδ)2radic
2I1
(I5 minus 1
2I1I2
minusI1I2
)32
(54)
with the quantities
PGGT = I1 minus I2 QGGT =1
4(I1 + I2)2 + 4
(I5 minus
1
2I1I2
) (55)
There is a fifth and final model to be considered here It is of a different form to the other four andnot an eddy-viscosity model It is known as the Gradient model and is defined by
Gradient [4 9] τmodG = CGδ
(S2 minus Ω2 minus (SΩminus ΩS)
) (56)
The values C are model constants specific to each model and δ represents the mesh size of the LESBoth of these quantities are included here to properly define the models however in this projectthey will be ignored as they are constants and have no influence on the conditions to be defined
33 Conditions for Large-Eddy Simulation without Explicit Filtering
The filtering operation introduced in Eq (48) is just a formality for LES without explicit filteringThe idea here is that filtering does not actually take place but the term lsquoaddedrsquo to the Navier-Stokesequations in fact lsquosubtractsrsquo the difficult to simulate small-scale turbulent motion of the flow Inorder to consider this properly the terms ui and p in Eq (48) are replaced by vi and q It is hopedthat τmod
ij approximates τij so that vi and q approximate ui and p respectively So LES withoutexplicit filtering is given by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (57)
It is clear to see that the first four terms of this equation are exactly the same as the Navier-Stokesequations so for the exact solutions discussed in Section 22 the combination of these terms willequal zero What is then desirable is that the model term is also zero for these exact solutions sothat Eq (57) still holds In order to test models for exact solutions the condition
part
partxjτmodij (v) = 0 (58)
is checked If this condition were to hold it would imply that the model does not influence thevelocity field of a flow hence we call it the Velocity Field Condition There are a number of otherrestrictions that can be tested One such test comes from the differential form of the evolution ofthe kinetic energy of a flow given in Eq (10) In order to find this Eq (57) is multiplied by vi andterms are grouped to find
part
partt
(1
2vivi
)+ vj
part
partxj
(1
2vivi
)= minusvi
1
ρ
partq
partxi+ νvi
part2vipartxjpartxj
minus vipart
partxjτmodij (v) (59)
23
Then similarly to Eq (58) another condition is to check whether the model affects the kineticenergy of exact solutions this is equivalent to checking whether
vipart
partxjτmodij (v) = 0 (60)
We call this condition the Kinetic Energy Condition The third and final condition considered herecan also be found using the integral form for the equation for kinetic energy
d
dt
intV
1
2vividV =
intS
[minusvj
(1
2vivi
)minus vj
q
ρ+ 2νviSij + viτ
modij (v)
]njdS
+
intV
[minus2νSijSij minus
(minusSijτmod
ij (v))]dV
(61)
where the rate-of-strain tensor Sij was defined in Eq (41) V is a volume of fluid and dS meansthe integral is over the surface of that volume with unit normal nj Then this final condition checkswhether the model affects the total kinetic energy of exact solutions to the Navier-Stokes equationsthat is
Sijτmodij (v) = 0 (62)
We will call this condition the Dissipation Condition because Sijτmodij (v) is often referred to as the
subgrid dissipation
34 Conditions for Large-Eddy Simulation with Explicit Filtering
As in Eq (48) the Navier-Stokes equations can be filtered by applying the filtering operation to thewhole equation (putting a bar over all terms) and evaluating to find again that
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (63)
In Eq (57) the filtering operation was ignored and the elements ui p and τij(u) were replacedwith vi q and τmod
ij (v) respectively Now in LES with explicit filtering the filtering operation itselfsmooths over the small-scale turbulent motion of the simulated flow In this case the elements uip and τij(u) will be replaced with vi q and τmod
ij (v) respectively LES with explicit filtering is thengiven by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (64)
Note that Eqs (63) and (64) despite their similarity to Eq (57) no longer contain the Navier-Stokesequations as Eq (57) did The bar (filtering operation) over each term results in each exact solutionfound in Section 22 not necessarily satisfying the first four terms It is hence unknown whether ornot these terms will vanish for exact solutions Terms can still be compared similarly to Eqs (58)(60) and (62) However instead of comparing to zero we can now compare to the relevant termfrom the filtered Navier-Stokes equations So the terms
part
partxjτij (65)
and
part
partxjτmodij (v) (66)
24
are to be compared Although the latter term is relatively simple to compute the former could leadto some difficulties due to the differentiation of the true stress τij = uiuj minus uiuj So although someinformation could be gleaned from the inspection of these two terms more information can be foundby considering the analogue of Eq (62) So the true subfilter dissipation
Sijτij (67)
(where Sij is shorthand for Sij(v)) and the modelled subfilter dissipation
Sij(v)τmodij (v) (68)
are considered and have to be equal for the condition to hold These can again be derived from theintegral equation for kinetic energy of a filtered velocity field similarly to Eq (61)
4 Analysis of Large-Eddy Simulation Models
In this section we use a code in Mathematica the theory described in Section 3 and the exactsolutions to the Navier-Stokes equations found in Section 22 to check whether or not each of theexact solutions lsquoendorsesrsquo each model for LES both without and with explicit filtering
41 Large-Eddy Simulation Without Explicit Filtering
We start by using a code in Mathematica to test the Velocity Field condition in Eq (58) and theKinetic Energy condition in Eq (60) For all exact solutions to the Navier-Stokes equations inSection 22 the outputs for both conditions were identical so we have combined them into one table
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 ndash ndash 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 1 For the Kinetic Energy condition a lsquo0rsquo result means that vipartpartxj
τmodij (v) = 0 for respective
models and exact solutions whereas a lsquolowastrsquo result means that vipartpartxj
τmodij (v) 6= 0 For the Velocity
Field condition a lsquo0rsquo result means that partpartxj
τmodij (v) = 0 for respective models and exact solutions
whereas a lsquolowastrsquo result means that partpartxj
τmodij (v) 6= 0 The Gradient model can take both positive and
negative values so when the sign of the model is known it is included in the table instead of a lsquorsquoBoth the conditions are included in one table because for all cases the outputs were identical
We then test the Dissipation condition in Eq (62) for all exact solutions to the Navier-Stokesequations in Section 22
25
Dissipation Condition [Does Sijτmodij (v) = 0]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 + 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 2 A lsquo0rsquo result means that Sijτmodij (v) = 0 for respective models and exact solutions whereas
a lsquolowastrsquo result means that Sijτmodij (v) 6= 0 The Gradient model can take both positive and negative
values so when the sign of the model is known it is included in the table instead of a lsquorsquo
We then sum all the results to find the total number of endorsements for each model described inSection 32
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
4 6 8 9 9
lsquoDissipationrsquoEndorsements
2 5 7 7 9
Total Endorsements 6 11 15 16 18
Table 3 The results from Tables 1 and 2 are summarised to give a general overview of how manyendorsements each model receives This makes it much clearer to see the models most supported bythe respective and combined conditions with the Vortex-Stretching model being the most supportedand the Smagorinsky model the least
Testing the models in Section 32 using LES without explicit filtering has mixed results It isclear that the Smagorinksy model is not well endorsed by the flows in Section 22 with the onlyendorsements coming from the simplest flows that is Sol 1 (Linear Velocity) Sol 3 (Velocity as afunction of time) Sol 7 (Stagnation Flow) and Sol 15 (Rankine Vortex) The Vortex-Stretching-based model is the most endorsed flow with eighteen out of a possible thirty endorsements Thisagrees with our foreknowledge as this model was constructed to be as physically consistent as possiblein the paper by Silvis amp Verstappen [11]
42 Large-Eddy Simulation With Explicit Filtering
In this section we will compare the true subfilter dissipation Sijτij and the modelled subfilterdissipation Sij(v)τmod
ij (v) given in Eqs (67) and (68) respectively for each model τmodij and each
exact solution to the Navier-Stokes equation given in Section 22 We will not consider the conditionsinvolving differentiation in Eqs (65) and (66) due to the difficulty in calculating the derivative ofthe τij term in Eq (65)
26
421 Example of a Test for a Model
The true subfilter dissipation Sijτij of a flow given in Eq (67) cannot be explicitly computed asthe filtering operation is kept general What can be done however is to partially evaluate theexpressions using the attributes of a filtering operation mentioned in Eq (48)
The calculations of Sijτij for Sol 1 (Linear Velocity) are provided to illustrate the general ideaRecall that the velocity field of Sol 1 (Linear Velocity) is written in Eq (13) in vector notation as
u(x y z t) =
0ax+ bcx+ d
(69)
Then Sij = Sij(u) is computed simply by using its definition in Eq (41) Note that as filteringpreserves constants if there is an a it is replaced with a meaning that (in this case) there are alarge number of zero terms due to the frequency of 0 Hence
[Sij(u)
]=
[1
2
(partuipartxj
+partujpartxi
)]=
0 a
2c2
a2 0 0
c2 0 0
=
0 a
2c2
a2 0 0
c2 0 0
(70)
Note that as Sijτij is being computed which is equal to the sum of the element-wise multiplicationof the two matrices (or the trace of the matrix multiplication) that we donrsquot need to consider theelements of τij(u) with corresponding zero entry in Sij As τij(u) = uiuj minus uiuj and again asfiltering preserves constants it is found that
[τij(u)] =[uiuj minus uiuj
]=
lowast 0 middot ax+ b 0 middot cx+ d
ax+ b middot 0 lowast lowastcx+ d middot 0 lowast lowast
=
lowast 0 00 lowast lowast0 lowast lowast
(71)
where a lsquolowastrsquo just represents some element (not necessarily non-zero) Therefore the true subfilterdissipation
Sijτij = tr
0 a2
c2
a2 0 0c2 0 0
middotlowast 0 0
0 lowast lowast0 lowast lowast
= 0 (72)
Now the modelled subfilter dissipation Sij(u)τmodij (u) needs to be computed (recall that Sij =
Sij(u)) In this case let us use the QR model stated in Eq (53) Once again Sij(u) calculated inEq (70) needs to be used So we have that
τmodij = minus2(CQRδ)
2 max0minusI3I1
Sij (73)
As
I1 = tr(S2) =
1
2(a2 + c2) and I3 = tr(S
3) = 0 (74)
it holds that τmodij (u) = 0 so Sij(u)τmod
ij (u) = 0 This agrees with Eq (72) and hence Sol 1(Linear Velocity) endorses the QR model
27
422 Testing the Models
Now using a Mathematica code the exact solutions in Section 22 are again used to evaluate thecondition for the models using the process just described in Section 421 The results are describedin Table 4
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 4 All entries with a ldquo0rdquo represent a value of zero and a ldquolowastrdquo represents a value that is non-zeroThe ldquoExactrdquo row represents the value of the true subfilter dissipation Sijτij Every other row witha model name represents the value of the modelled subfilter dissipation Sij(u)τmod
ij (u) respectiveto that model A ldquordquo means that it is unclear as to the value - the sign of these values is oftenunknown and the lsquoMaxrsquo function in the definition of the QR model (in Eq (53)) makes it difficultto evaluate
Reading from Table 4 the Smagorinsky model is clearly endorsed by Sol 3 (Velocity as a function oftime) and Sol 15 (Rankine Vortex) as in both cases both Sijτij = 0 = Sij(u)τmod
ij (u) In this casewe say that Sol 3 (Velocity as a function of time) and Sol 15 (Rankine Vortex) lsquostronglyrsquo endorsethe Smagorinsky model The Smagorinsky model is not endorsed by Sol 1 (Linear Velocity) Sol 2(Planar Flow) or Sol 14 (Radially Dependent) however because Sij(u)τmod
ij (u) 6= 0 For Sol 4 to
13 both Sijτij 6= 0 6= Sij(u)τmodij (u) and so we will say that these exact solutions lsquoweaklyrsquo endorse
the Smagorinsky model
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 10 10 9 6
Table 5 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Interestingly the lsquostrongrsquo and lsquoweakrsquo endorsements give different results However more stock shouldbe put in the results of the lsquostrongerrsquo condition as it is completely clear that zero is equal to zeroIt is not necessarily clear whether or not non-zero equals non-zero as the values will depend on theparameters in each solution hence the lsquoweakerrsquo endorsement
The lsquostrongrsquo condition equally endorses the Vreman QR Gradient and Vortex-Stretching-basedmodels whereas the lsquoweakrsquo condition endorses Vreman and Smagorinsky the most closely followedby the Gradient model with the Vortex-Stretching-based model being the most opposed As this
28
completely contradicts the lsquostrongrsquo condition and the results in Section 41 it makes us question thevalidity of the weaker test The number of endorsements for the QR model is unknown because of thedefinition of the model in Eq (53) and our evaluation of the true subfilter dissipation Eq (67) Dueto the exact solutions we have found it is never clear whether or not the true subfilter dissipationis strictly positive or negative This means that when the true subfilter dissipation is found to benon-zero the lsquoMaxrsquo function in the QR model cannot be evaluated
43 Vremanrsquos True Subfilter Dissipation Claim
Using the theory mentioned in Sections 232 and 34 we can place the exact solutions to the Navier-Stokes equations into different classes defined by Vreman In doing this we can also attempt to verifyone of Vremanrsquos results [16] In Vremanrsquos paper he defines the class Qn to be the set of flows withn zero entries in the velocity gradient of the flow (the velocity gradient is defined in Eq (39))Vreman then claims that the only flows with zero true subfilter dissipation are a certain selection ofthose in the classesQ7Q8 orQ9 shown in appendix B and is generally non-zero for any other flows
The true subfilter dissipation of Sol 1 (Linear Velocity) was computed in Section 421 and wasfound to be zero in Eq (72) The velocity gradient of Sol 1 is as follows
GLV =
0 0 0a 0 0c 0 0
(75)
and hence Sol 1 (Linear Velocity) is in the required subset of Q7 which agrees with Vremanrsquos claimIn Section 422 the true subfilter dissipation was computed (determined to be zero or non-zero) forall of the exact solutions to the Navier Stokes equations in Section 22 These solutions can now belsquoplacedrsquo into each class Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a functionof time) and Sol 14 (Radially Dependent) were all found to have zero true subfilter dissipation andare all in the appropriate subsets of either Q7Q8 or Q9 This agrees with Vremanrsquos results
We can extend Vremanrsquos results however We found that Sol 15 (Rankine Vortex) had zero truesubfilter dissipation and yet is not in the subset of Q7 defined in Appendix B It is important tonote that the rate of strain tensor Sij in Eq (41) for Sol 15 (Rankine Vortex) is zero This is thereason the true subfilter dissipation is zero and the same reason that all models are zero for thisflow as all models are based on Sij
We also found that all of the remaining exact solutions to the Navier-Stokes equations are not inVremanrsquos specified classes (in Appendix Section B) They also have a non-zero subfilter dissipationwhich agrees with Vremanrsquos results in his paper [16]
44 Classes of Flows
It is important to consider that some exact solutions to the Navier-Stokes equations may have similarproperties For example Sol 1 (Linear Velocity) and Sol 2 (Planar Flow) are both very similarIn fact Sol 2 (Planar Flow) is a more general case of Sol 1 (Linear Velocity) We therefore mustgroup the different flows into classes in an attempt to reduce the bias that may occur when solutionsare similar This should aid us to further determine which of the models are most endorsed We willdo this in a number of different ways using the theory discussed in Section 23 We will group theflows with respect to Vremanrsquos Flow Classes given in appendix B and with respect to the Invariantsgiven in appendix A
29
441 Vremanrsquos Velocity Gradient Flow Classes
We now put our exact solutions to the Navier-Stokes equations into one of two classes Either Q0minus6or Q7minus9 where each class is defined intuitively as follows
Q0minus6 = Q0 cupQ1 cup middot middot middot cup Q6 (76)
Q7minus9 = Q7 cupQ8 cupQ9 (77)
where Qn is defined in Section 43 By looking at the gradient of each exact solution we can see thatQ7minus9 contains the six solutions Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as afunction of time) Sol 5 (2D Taylor) Sol 14 (Radially Dependent) and Sol 15 (Rankine Vortex)and that Q0minus6 contains the remaining nine We now check the models for LES without explicitfiltering by considering endorsements within each class Tables 6 and 7 are analogues of Table 3 andillustrate the results for each flow class The analogues of Tables 1 and 2 are Tables 22 and 23 andare in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 6 6 6
lsquoDissipationrsquoEndorsements
2 5 6 6 6
Percentage Endorsed 417 833 100 100 100
Table 6 Each entry in the top two rows represents the number of solutions in the class Q7minus9 thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Q0minus6 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 2 3 3
lsquoDissipationrsquoEndorsements
0 0 1 0 3
Percentage Endorsed 56 56 167 167 333
Table 7 Each entry in the top two rows represents the number of solutions in the class Q1minus6 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking results to 1 decimal place if necessary
All of the flows in the class Q7minus9 endorse the QR Gradient and Vortex-Stretching-based subfilter-scale models (see Table 6) We can hence see that this class fully endorses these three models equallyand does not endorse the Smagorinsky model The Vortex-Stretching eddy viscosity model has themost endorsements from any of the flows in the class Q0minus6 and the Smagorinksy and Vreman modelshave the fewest endorsements (see Table 7) We can hence say that this class mostly endorses theVortex-Stretching-based eddy-viscosity model note however that the percentage endorsement is very
30
low
Now let us consider LES with explicit filtering Tables 8 and 9 are analogues of Table 5 and illustratethe results for each flow class The analogue of Table 4 is Table 24 and is in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 1 1 0 0 0
Table 8 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Q0minus6 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 9 9 9 6
Table 9 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We have grouped Vremanrsquos classes into Q7minus9 and Q0minus6 this is equivalent to saying that we havegrouped Vremanrsquos classes into those with two or fewer non-zero velocity gradient elements and thosewith three or more non-zero velocity gradient elements respectively Therefore the solutions in Q7minus9are exactly those with a lsquo0rsquo in the lsquoExactrsquo column of Table 4 and Sol 5 (2D Taylor) ExcludingSol 5 (2D Taylor) these are the same as those five that endorse the Vreman QR Gradient andVortex-Stretching-based models in Table 5 We can see in Table 8 that the results for the lsquostrongrsquoendorsements are exactly the same as in Table 5 This is because all of the flows with zero subfilterdissipation are contained in the class Q7minus9 There are hence no lsquostrongrsquo endorsements in Table 9The majority of the lsquoweakrsquo endorsements come from Q0minus6 as this class contains all but one of thesolutions with non-zero true subfilter dissipation (given in Eq (67))
442 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
We now separate our exact solutions into classes again using Vremanrsquos flow classes This timehowever we define V0 to be the class of flows Vreman claims will always have zero true subfilterdissipation namely those classes mentioned in Appendix B We define Vlowast to be all the remaining
31
classes Then
V0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow)
Sol 3 (Velocity as a function of time) Sol 14 (Radially Dependent)(78)
Vlowast = Sol 4 (Axisymmetric Flow) Sol 5 (2D Taylor) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 13 (Sphere Translation) Sol 15 (Rankine Vortex)
(79)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 10 and 11 are analogues of Table 3 and illustrate the results for each flow class Theanalogues of Tables 1 and 2 are Tables 25 and 26 and are in Appendix C2
Endorsements for models using solutions in V0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 4 4 4 4
lsquoDissipationrsquoEndorsements
1 4 4 4 4
Percentage Endorsed 375 100 100 100 100
Table 10 Each entry in the top two rows represents the number of solutions in the class V0 thatendorse each model There are 4 solutions in this class so the percentages are worked out by dividingthe total of each column by 8 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Vlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 2 4 5 5
lsquoDissipationrsquoEndorsements
1 1 3 2 5
Percentage Endorsed 136 136 318 318 455
Table 11 Each entry in the top two rows represents the number of solutions in the class Vlowast thatendorse each model There are 11 solutions in this class so the percentages are worked out bydividing the total of each column by 22 and taking results to 1 decimal place if necessary
All of the flows in the class V0 endorse the Vreman QR Gradient and Vortex-Stretching-basedsubfilter-scale models (see Table 6) This class completely endorses these four models equally anddoes not endorse the Smagorinsky model The Vortex-Stretching-based eddy-viscosity model hasthe most endorsements from the class Vlowast and the Smagorinksy and Vreman models again have thefewest endorsements (see Table 7) As in Section 441 we say that this class mostly endorses the
32
Vortex-Stretching-based eddy-viscosity model but once again the percentage endorsement is ratherlow
Now let us consider LES with explicit filtering Tables 12 and 13 are analogues of Table 5 andillustrate the results for each flow class The analogue of Table 4 is Table 27 and is in Appendix C2
Endorsements for models using solutions in V0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 4 4 4 4
lsquoWeakrsquo Endorsements 0 0 0 0 0
Table 12 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Vlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 1 1 1 1
lsquoWeakrsquo Endorsements 10 10 9 6
Table 13 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We can see in Table 12 that the results for the lsquostrongrsquo endorsements are similar to those in Table 5excluding Sol 15 (Rankine Vortex) The four solutions in V0 with zero true subfilter dissipationnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)and Sol 14 (Radially Dependent) are hence the four that endorse the Vreman QR Gradientand Vortex-Stretching-based models in Table 12 Sol 15 (Rankine Vortex) is in the class Vlowast andhence there is one lsquostrongrsquo endorsement in Table 13 The class Vlowast contains all of the flows with anon-zero true subfilter dissipation (given in Eq (67)) and hence Table 13 contains all of the lsquoweakrsquoendorsements
443 Classes using the Third Principal Invariant
Sorting the classes using the invariants is a non-trivial task in itself There are a number of waysin which to do this using both the principal invariants in Eq (40) and the combined invariants inEq (43)
We start by looking at flows with third principal invariant RG equal to zero We again define twoclasses R0 and Rlowast where R0 consists of all flows with RG = 0 and Rlowast consists of all flows withRG 6= 0 Then using the information in Appendix A we see that there are nine flows in R0 That
33
is
R0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 8 (Vortex in Stagnation Flow)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)
(80)
Rlowast = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 13 (Sphere Translation)(81)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 14 and 15 are analogues of Table 3 and they illustrate the results for LES withoutexplicit filtering for each flow class The analogues of Tables 1 and 2 are Tables 28 and 29 and arein Appendix C3
Endorsements for models using solutions in R0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 7 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 278 556 778 722 778
Table 14 Each entry in the top two rows represents the number of solutions in the class R0 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking the result to one decimal place if necessary
Endorsements for models using solutions in Rlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 2 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 83 83 83 167 333
Table 15 Each entry in the top two rows represents the number of solutions in the class Rlowast thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking the result to one decimal place if necessary
The class R0 endorses both the Vortex-Stretching-based model and the QR model the most andSmagorinsky the least in the case for LES without explicit filtering In the Rlowast class the Vortex-Stretching-based model is again the most endorsed followed by the Gradient model however thepercentages are again small
34
Now let us consider LES with explicit filtering Tables 16 and 17 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 30 and is in Appendix C3
Endorsements for models using solutions in R0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 4 4 3 2
Table 16 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Rlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 6 6 6 4
Table 17 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 we can see in Table 16 that the results for the lsquostrongrsquo endorsements are exactly thesame as in Table 5 This is because all of the flows with zero subfilter dissipation are contained in theclass R0 As in Table 9 there are no lsquostrongrsquo endorsements in Table 17 The lsquoweakrsquo endorsementshowever are split more evenly between the two classes now than in Vremanrsquos classes in Sections 441and 442 We find that the Smagorinsky and Vreman models still seem to be the most endorsed inboth classes
444 Two-component Flow Classes
Another way in which the flows can be grouped into classes is by determining whether a flow is atwo-component flow or not We define a two-component flow to have the property that the combinedprincipal invariants of a flow in Eq (43) satisfy
I3 = I4 = I5 minus1
2I1I2 = 0 (82)
We are also interested in considering flows with zero gradient model subgrid dissipation These flowshave the property that
I3 minus I4 = 0 (83)
What we find though is that for all the solutions in Section 22 if a flow is a two-component flowthen it also satisfies I3 minus I4 = 0 It may be interesting to note that although I3 minus I4 6= 0 for manysolutions there are a number with I3 equal to a multiple of I4 Although not considered in thisreport this may also be a class of interest We find that the two-component flows are
C2 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)(84)
35
and we call the remaining flows three-component flows They are
C3 = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex) Sol 11 (Pseudo-Motion Shots)
Sol 13 (Sphere Translation)
(85)
Note that Sol 8 (Vortex in Stagnation Flow) is defined to be a planar flow this does not mean that itis a two-component flow in this sense however because its two non-zero velocity components dependon three coordinates We now check the models for LES without explicit filtering by consideringendorsements within each class Tables 18 and 19 are analogues of Table 3 and they illustrate theresults for LES without explicit filtering for each flow class The analogues of Tables 1 and 2 areTables 31 and 32 and are in Appendix C4
Endorsements for models using solutions in C2 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 6 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 357 714 100 857 100
Table 18 Each entry in the top two rows represents the number of solutions in the class C2 thatendorse each model There are 7 solutions in this class so the percentages are worked out by dividingthe total of each column by 14 and taking the result to one decimal place if necessary
Endorsements for models using solutions in C3 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 3 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 63 63 63 188 250
Table 19 Each entry in the top two rows represents the number of solutions in the class C3 thatendorse each model There are 8 solutions in this class so the percentages are worked out by dividingthe total of each column by 16 and taking the result to one decimal place if necessary
The class C2 equally endorses the QR and Vortex-Stretching-based models and once again theSmagorinsky model is the least endorsed As with the previous classes the class C3 has such lowpercentages that it does not seem to endorse any models We start to see a trend appearing here inthat the flow classes that contain the more complicated flows namely Q0minus6 (Eq (76)) Vlowast (Eq (79))Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages The classes that tendto contain the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78)) R0 (Eq (80)) and C2 (Eq (84))all seem to favour the Vortex-Stretching-based model or the QR model
36
Now let us consider LES with explicit filtering Tables 20 and 21 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 33 and is in Appendix C4
Endorsements for models using solutions in C2 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 2 2 0 1 0
Table 20 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in C3 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 8 8 8 6
Table 21 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 and Table 16 we can see in Table 20 that the results for the lsquostrongrsquo endorsements areexactly the same as in Table 5 This is again because all of the flows with zero subfilter dissipationare contained in the class C2 There are two flows with non-zero true subfilter dissipation (givenin Eq (67)) in C2 and this shows in the lsquoweakrsquo endorsements in Table 20 There are no lsquostrongrsquoendorsements in Table 21 as all the zero subfilter dissipation flows are in C2
5 Conclusions
In this report we studied incompressible turbulent flows We found a number of exact solutions to theNavier-Stokes equations each representing a relatively simplistic fluid flow Large-eddy simulation(LES) is used to model turbulent flows by smoothing over the complex small-scale motion Weintroduced a number of subfilter-scale models to approximate this small-scale motion and testedthem using the found exact solutions to the Navier-Stokes equations We first considered LESwithout explicit filtering and used the Navier-Stokes equations to derive conditions for the subfilter-scale models namely the Velocity Field condition (Eq (58)) the Kinetic Energy condition (Eq (60))and the Dissipation condition (Eq (62)) Similarly we considered LES with explicit filtering andagain used the Navier-Stokes equations to derive more conditions for the models in this case wedecided to just take one condition the Dissipation condition (Eqs (67) and (68)) as the others leadto difficulty in their evaluation By lsquorunningrsquo the exact solutions to the Navier-Stokes equationsthrough each condition for the subfilter-scale models we found that the Vortex-Stretching-basededdy-viscosity model in Eq (54) was most endorsed by the tests in LES without explicit filteringWe found that four out of the five subfilter-scale models namely the Vreman QR Gradient andVortex-Stretching-based models were equally endorsed by the lsquostrongrsquo tests in LES with explicitfiltering with the Smagorinsky model being the least endorsed The results we found for the lsquoweakrsquocondition were completely opposite to those found in both the lsquostrongrsquo condition and in LES withoutexplicit filtering This led us to conclude that this test was not one in which we should put our trust
37
We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses
In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes
This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold
38
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
where
h(x y) =x2 + y2
c2 (28)
and Ω and c are constants
Figure 11 An illustration of the flow for Sol9 (Burgers Vortex) described in Eq (27) withparameter values c = 15 Re = 20 and Ω = 10A slice of the xy-plane at z = 0 is shown
Solution 10 Another Vortex Solution
There is another vortex solution more simply formulated than the Burgers Vortex yet with asingularity at r = 0 as a result of the uϕ term It is again given in cylindrical coordinates
ur(r ϕ z t) = minus 2r
c2Re
uϕ(r ϕ z t) =a
r
uz(r ϕ z t) =4z
c2Re
p(r ϕ z t) = minus a2
2r2minus 2r2 + 8z2
c2Re2
(29)
which can also be converted into Cartesian coordinates
14
u1(x y z t) = minus 2x
c2Reminus ay
x2 + y2
u2(x y z t) = minus 2y
c2Re+
ax
x2 + y2
u3(x y z t) =4z
c2Re
(30)
where a and c are constants
Figure 12 An illustration of the flow for Sol9 (Burgers Vortex) described in Eq (27) withparameter values c = 15 Re = 20 and a = 10 Aslice of the xy-plane at z = 0 is shown
Solution 11 Pseudo-Motion Shots of the Second Kind
In the book by Berker [3 p 94] a solution is given in cylindrical coordinates by
ur(r ϕ z t) =2
Re
1
r
uϕ(r ϕ z t) = A1r3 +B1r +
C1
r
uz(r ϕ z t) = A2r2 log(r) +B2r
2 + C2
p(r ϕ z t) =A2
1r6
6+A1B1r
4
2+A1C1r
2 +B2
1r2
2minus C2
1
2r2minus 2
r2Re2+ 2B1C1 log(r)
+2A2z
Reminus 4B1ϕ
Re
(31)
where A1 A2 B1 B2 C1 C2 are constants Eq (31) when converted into Cartesian coordinates is
15
u1(x y z t) =1
Re
2xminus C1Rey
x2 + y2minusA1y
(x2 + y2
)minusB1y
u2(x y z t) =1
Re
2y + C1Rex
x2 + y2+A1x
(x2 + y2
)+B1x
u3(x y z t) =1
2A2
(x2 + y2
)log(x2 + y2
)+B2
(x2 + y2
)+ C2
(32)
Figure 13 An illustration of the flow in thexy-plane for Sol 11 (Pseudo-Motion Shots)described in Eqs (31) and (32) with parametervalues A1 = B1 = C1 = 1 A2 = B2 = C2 = 2and Re = 10
Figure 14 An illustration of the flow in theyz-plane for Sol 11 (Pseudo-Motion Shots)described in Eqs (31) and (32) with parametervalues A1 = B1 = C1 = 1 A2 = B2 = C2 = 2and Re = 10
Solution 12 Flows With Constant Whirl
In the book by Berker [3 p 143] another solution is given in cylindrical coordinates by
ur(r ϕ z t) = minus 1
Re
1
r
uϕ(r ϕ z t) =1
rradicte(minus
12 g(t))
uz(r ϕ z t) = 0
p(r ϕ z t) =1
4
2(minus e
minusg(t)
t minus 1Re2
)r2
minus 1
t2Re
infinintminusg(t)
eminust
tdt
(33)
where we define g(t) =r2Re
2t When converted into Cartesian coordinates we find
16
u1(x y z t) = minus 1
x2 + y2
(yradict
exp
(minus
Re(x2 + y2
)4t
)+
x
Re
)
u2(x y z t) =1
x2 + y2
(xradict
exp
(minus
Re(x2 + y2
)4t
)minus y
Re
)
u3(x y z t) = 0
(34)
Figure 15 An illustration of the flow for Sol12 (Flows with Constant Whirl) described inEqs (33) and (34) with time t = 1 and Re = 1 Aslice of the xy-plane at z = 0 is shown as velocityin the z-direction is zero this is the same for anyvalue of z
17
Solution 13 Rectilinear and Uniform Translation of a Sphere
In the book by Berker [3 p 228] another solution is given in Cartesian coordinates by
u1(x y z t) =3
4
axz
R3
(a2
R2minus 1
)
u2(x y z t) =3
4
ayz
R3
(a2
R2minus 1
)
u3(x y z t) = 1minus 3
4
a
Rminus 1
4
a3
R3minus 3
4
az2
R3
(a2
R2minus 1
)
p(x y z t) =3a(aminusR)(a+R)
32R10Re
((R2Re
(x2 + y2
)minus 2R2Rez2
) (a3 + 3aR2 minus 4R3
)+ 3aRez4
(R2 minus a2
)minus 16R5z
)
(35)
where a and R are constants
Figure 16 An illustration of the flow in the yz-plane for Sol 13 (Sphere Translation) describedin Eq (35) with parameter values a = minus2 andRe = 3
Solution 14 Radially Dependent Flow
A solution with velocity in the z-direction dependent only on the distance from the z-axis is as
18
follows in cylindrical coordinates
ur(r ϕ z t) = 0
uϕ(r ϕ z t) = 0
uz(r ϕ z t) = A(r2 minus a2
)
p(r ϕ z t) =4Az
Re
(36)
where A and a are constants In Cartesian coordinates the solution is just the same with u1(x y z t)and u2(x y z t) both zero and the r2 term replaced with x2 + y2 in u3(x y z t)
Figure 17 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus1 1] withparameter values a = 1 A = 4 and Re = 10
Figure 18 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus2 2] withparameter values a = 1 A = 4 and Re = 10
Solution 15 Rankine Vortex
A solution discovered by Rankine [1] is a vortex flow of the following form
ur(r ϕ z t) = 0
uϕ(r ϕ z t) =ar
2πR2
uz(r ϕ z t) = 0
p(r ϕ z t) =a2r2
8π2R4
(37)
19
where a and R are constants In Cartesian coordinates the flow has the form
u1(x y z t) = minus ay
2πR2
u2(x y z t) =ax
2πR2
u3(x y z t) = 0
p(x y z t) =a2(x2 + y2
)8π2R4
(38)
This flow is actually a rigid body rotational flow in that the angular rotational velocity is uniformand the velocity of the flow increases proportionally with distance from the z-axis
Figure 19 An illustration of the flow in the xy-plane for Sol 15 (Rankine Vortex) described inEq (37) and Eq (38) with parameter values a = 1and R = 1
radic2π
23 Characterisation of Flows
It is important to be able to classify flows in a way that sorts them into distinct groups This aidsin the clear identification and general study of flows It is also a very important way of recognisingflows with similar properties In this subsection we introduce two distinct ways in which flows canbe characterised
231 Characterising Solutions using Invariants
A principal invariant of a tensor is a coefficient of the characteristic polynomial of that tensorInvariants as their name suggests are invariant under rotations of the coordinate system in thattheir values do not change when the tensor is lsquorotatedrsquo into a different set of coordinates Thevelocity gradient of a flow G is defined as the Jacobian of the velocity vector given by
Gij =
[partuipartxj
] (39)
20
for i j = 1 2 3 There are three principal invariants of the velocity gradient of a fluid flow G givenby
PG = tr(G)
QG =1
2
((tr(G))2 minus tr
(G2))
RG = det(G)
(40)
Incompressibility of flows (Eq (5)) ensures that PG = 0 hence the first principal invariant is zerofor all of the flows in Section 22 We now define the rate-of-strain tensor S by
Sij =1
2
(partuipartxj
+partujpartxi
) (41)
and the rate-of-rotation tensor Ω by
Ωij =1
2
(partuipartxjminus partujpartxi
) (42)
Note that the first principal invariant of S and the first and third principal invariants of Ω are zeroAlso note that the velocity gradient of a flow can simply be written G = S + Ω
Rather than considering the principal invariants we can also consider the combined invariants
I1 = tr(S2) I2 = tr(Ω2) I3 = tr(S3) I4 = tr(SΩ2) I5 = tr(S2Ω2) (43)
We can hence write the three principal invariants for a velocity field G in terms of the combinedinvariants
PG = 0
QG = minus1
2(I1 + I2)
RG =1
3(I3 + 3I4)
(44)
In this report the flows considered are the exact solutions to the incompressible Navier-Stokesequations described in Section 22 The combined invariants in Eq (43) are used to characterisethese solutions into different classes in Section 44 Both the principal invariants and combinedinvariants for each exact solution in Section 22 are given in Appendix A
Each of the subfilter-scale eddy-viscosity models described in Section 32 are written in terms ofthese five combined invariants
232 Vremanrsquos Characterisation of Flows
Vreman [16] classifies flows using the gradient of the velocity field of a flow Each lsquoclassrsquo of flows hasthe same number of non-zero elements in the gradient matrix For example
[partuipartxj
]=
0 0 0lowast 0 0lowast 0 0
(45)
where a lowast denotes a non-zero term represents a group of flows that lie within the class denoted byQ7 (see Appendix B) where the subscript represents the number of non-zero terms in the velocity
21
gradient Flows of the form in Eq (45) have a velocity field of the form
u1 = c1 (constant)
u2 = f2(x1)
u3 = f3(x1)
(46)
where f2(x1) and f3(x1) are such that the Navier Stokes equations Eq (1) are satisfied Notethat as the elements on the leading diagonal are zero the incompressibility condition Eq (5) isautomatically satisfied Some simple flow classes are given in Appendix B and contain a number ofthe solutions discussed in Section 22 This is mentioned in greater detail in Section 441
3 Conditions for Large-Eddy Simulation Models
31 Large-Eddy Simulation
Due to the nature of numerical simulation and the complexity of turbulent flows small details aremissed and information is lost when solving the Navier-Stokes equations The computational costof solving a turbulent flow is extremely high so a filtering operation is applied to the Navier-Stokesequations represented by a bar to reduce the computational cost of the simulation This is a linearoperation and preserves constants ie ρ = ρ and is also assumed to commute with differentiation
partuipartxj
=partuipartxj
(47)
The filtered Navier-Stokes equations are found by filtering (putting a bar over) the entire equations(see Eq (1)) and after evaluating are given by
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (48)
where
τij = uiuj minus uiuj (49)
is called the lsquotrue stressrsquo of the flow This however poses a problem as it is not clear what is meantby the term uiuj for a general filtering operation The true stress τij therefore must be replaced bya model τmod
ij in practical applications
32 Subfilter-Scale Models
We will now define the subflter-scale models denoted by τmodij to be used in place of the lsquodifficultrsquo
true stress τij There are a number of models available to use in the literature [11 p7] These willbe tested using the exact solutions to the Navier-Stokes equations and by checking conditions whichare defined in Sections 33 and 34
All the models barring one are of the same form That is each of the eddy-viscosity models arewritten
τmodij minus 1
3τmodkk δij = minus2νeSij (50)
with νe representing the eddy viscosity of a specific model For simplicity we usually just writeτmodij = minus2νeSij the extra term is included here to emphasise that the models are traceless Each
22
of the models we consider here can be expressed in terms of the rate of strain tensor S defined inEq (41) the rate of rotation tensor Ω defined in Eq (42) and the combined invariants defined inEq (43) The models are then defined as follows
Smagorinsky [12] νSe = (CSδ)2radic
2I1 (51)
Vreman [16] νVe = (CV δ)2
radicQGGT
PGGT (52)
QR [13 14 15] νQRe = (CQRδ)2 max0minusI3
I1 (53)
Vortex-Stretching [11] νV Se = (CV Sδ)2radic
2I1
(I5 minus 1
2I1I2
minusI1I2
)32
(54)
with the quantities
PGGT = I1 minus I2 QGGT =1
4(I1 + I2)2 + 4
(I5 minus
1
2I1I2
) (55)
There is a fifth and final model to be considered here It is of a different form to the other four andnot an eddy-viscosity model It is known as the Gradient model and is defined by
Gradient [4 9] τmodG = CGδ
(S2 minus Ω2 minus (SΩminus ΩS)
) (56)
The values C are model constants specific to each model and δ represents the mesh size of the LESBoth of these quantities are included here to properly define the models however in this projectthey will be ignored as they are constants and have no influence on the conditions to be defined
33 Conditions for Large-Eddy Simulation without Explicit Filtering
The filtering operation introduced in Eq (48) is just a formality for LES without explicit filteringThe idea here is that filtering does not actually take place but the term lsquoaddedrsquo to the Navier-Stokesequations in fact lsquosubtractsrsquo the difficult to simulate small-scale turbulent motion of the flow Inorder to consider this properly the terms ui and p in Eq (48) are replaced by vi and q It is hopedthat τmod
ij approximates τij so that vi and q approximate ui and p respectively So LES withoutexplicit filtering is given by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (57)
It is clear to see that the first four terms of this equation are exactly the same as the Navier-Stokesequations so for the exact solutions discussed in Section 22 the combination of these terms willequal zero What is then desirable is that the model term is also zero for these exact solutions sothat Eq (57) still holds In order to test models for exact solutions the condition
part
partxjτmodij (v) = 0 (58)
is checked If this condition were to hold it would imply that the model does not influence thevelocity field of a flow hence we call it the Velocity Field Condition There are a number of otherrestrictions that can be tested One such test comes from the differential form of the evolution ofthe kinetic energy of a flow given in Eq (10) In order to find this Eq (57) is multiplied by vi andterms are grouped to find
part
partt
(1
2vivi
)+ vj
part
partxj
(1
2vivi
)= minusvi
1
ρ
partq
partxi+ νvi
part2vipartxjpartxj
minus vipart
partxjτmodij (v) (59)
23
Then similarly to Eq (58) another condition is to check whether the model affects the kineticenergy of exact solutions this is equivalent to checking whether
vipart
partxjτmodij (v) = 0 (60)
We call this condition the Kinetic Energy Condition The third and final condition considered herecan also be found using the integral form for the equation for kinetic energy
d
dt
intV
1
2vividV =
intS
[minusvj
(1
2vivi
)minus vj
q
ρ+ 2νviSij + viτ
modij (v)
]njdS
+
intV
[minus2νSijSij minus
(minusSijτmod
ij (v))]dV
(61)
where the rate-of-strain tensor Sij was defined in Eq (41) V is a volume of fluid and dS meansthe integral is over the surface of that volume with unit normal nj Then this final condition checkswhether the model affects the total kinetic energy of exact solutions to the Navier-Stokes equationsthat is
Sijτmodij (v) = 0 (62)
We will call this condition the Dissipation Condition because Sijτmodij (v) is often referred to as the
subgrid dissipation
34 Conditions for Large-Eddy Simulation with Explicit Filtering
As in Eq (48) the Navier-Stokes equations can be filtered by applying the filtering operation to thewhole equation (putting a bar over all terms) and evaluating to find again that
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (63)
In Eq (57) the filtering operation was ignored and the elements ui p and τij(u) were replacedwith vi q and τmod
ij (v) respectively Now in LES with explicit filtering the filtering operation itselfsmooths over the small-scale turbulent motion of the simulated flow In this case the elements uip and τij(u) will be replaced with vi q and τmod
ij (v) respectively LES with explicit filtering is thengiven by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (64)
Note that Eqs (63) and (64) despite their similarity to Eq (57) no longer contain the Navier-Stokesequations as Eq (57) did The bar (filtering operation) over each term results in each exact solutionfound in Section 22 not necessarily satisfying the first four terms It is hence unknown whether ornot these terms will vanish for exact solutions Terms can still be compared similarly to Eqs (58)(60) and (62) However instead of comparing to zero we can now compare to the relevant termfrom the filtered Navier-Stokes equations So the terms
part
partxjτij (65)
and
part
partxjτmodij (v) (66)
24
are to be compared Although the latter term is relatively simple to compute the former could leadto some difficulties due to the differentiation of the true stress τij = uiuj minus uiuj So although someinformation could be gleaned from the inspection of these two terms more information can be foundby considering the analogue of Eq (62) So the true subfilter dissipation
Sijτij (67)
(where Sij is shorthand for Sij(v)) and the modelled subfilter dissipation
Sij(v)τmodij (v) (68)
are considered and have to be equal for the condition to hold These can again be derived from theintegral equation for kinetic energy of a filtered velocity field similarly to Eq (61)
4 Analysis of Large-Eddy Simulation Models
In this section we use a code in Mathematica the theory described in Section 3 and the exactsolutions to the Navier-Stokes equations found in Section 22 to check whether or not each of theexact solutions lsquoendorsesrsquo each model for LES both without and with explicit filtering
41 Large-Eddy Simulation Without Explicit Filtering
We start by using a code in Mathematica to test the Velocity Field condition in Eq (58) and theKinetic Energy condition in Eq (60) For all exact solutions to the Navier-Stokes equations inSection 22 the outputs for both conditions were identical so we have combined them into one table
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 ndash ndash 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 1 For the Kinetic Energy condition a lsquo0rsquo result means that vipartpartxj
τmodij (v) = 0 for respective
models and exact solutions whereas a lsquolowastrsquo result means that vipartpartxj
τmodij (v) 6= 0 For the Velocity
Field condition a lsquo0rsquo result means that partpartxj
τmodij (v) = 0 for respective models and exact solutions
whereas a lsquolowastrsquo result means that partpartxj
τmodij (v) 6= 0 The Gradient model can take both positive and
negative values so when the sign of the model is known it is included in the table instead of a lsquorsquoBoth the conditions are included in one table because for all cases the outputs were identical
We then test the Dissipation condition in Eq (62) for all exact solutions to the Navier-Stokesequations in Section 22
25
Dissipation Condition [Does Sijτmodij (v) = 0]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 + 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 2 A lsquo0rsquo result means that Sijτmodij (v) = 0 for respective models and exact solutions whereas
a lsquolowastrsquo result means that Sijτmodij (v) 6= 0 The Gradient model can take both positive and negative
values so when the sign of the model is known it is included in the table instead of a lsquorsquo
We then sum all the results to find the total number of endorsements for each model described inSection 32
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
4 6 8 9 9
lsquoDissipationrsquoEndorsements
2 5 7 7 9
Total Endorsements 6 11 15 16 18
Table 3 The results from Tables 1 and 2 are summarised to give a general overview of how manyendorsements each model receives This makes it much clearer to see the models most supported bythe respective and combined conditions with the Vortex-Stretching model being the most supportedand the Smagorinsky model the least
Testing the models in Section 32 using LES without explicit filtering has mixed results It isclear that the Smagorinksy model is not well endorsed by the flows in Section 22 with the onlyendorsements coming from the simplest flows that is Sol 1 (Linear Velocity) Sol 3 (Velocity as afunction of time) Sol 7 (Stagnation Flow) and Sol 15 (Rankine Vortex) The Vortex-Stretching-based model is the most endorsed flow with eighteen out of a possible thirty endorsements Thisagrees with our foreknowledge as this model was constructed to be as physically consistent as possiblein the paper by Silvis amp Verstappen [11]
42 Large-Eddy Simulation With Explicit Filtering
In this section we will compare the true subfilter dissipation Sijτij and the modelled subfilterdissipation Sij(v)τmod
ij (v) given in Eqs (67) and (68) respectively for each model τmodij and each
exact solution to the Navier-Stokes equation given in Section 22 We will not consider the conditionsinvolving differentiation in Eqs (65) and (66) due to the difficulty in calculating the derivative ofthe τij term in Eq (65)
26
421 Example of a Test for a Model
The true subfilter dissipation Sijτij of a flow given in Eq (67) cannot be explicitly computed asthe filtering operation is kept general What can be done however is to partially evaluate theexpressions using the attributes of a filtering operation mentioned in Eq (48)
The calculations of Sijτij for Sol 1 (Linear Velocity) are provided to illustrate the general ideaRecall that the velocity field of Sol 1 (Linear Velocity) is written in Eq (13) in vector notation as
u(x y z t) =
0ax+ bcx+ d
(69)
Then Sij = Sij(u) is computed simply by using its definition in Eq (41) Note that as filteringpreserves constants if there is an a it is replaced with a meaning that (in this case) there are alarge number of zero terms due to the frequency of 0 Hence
[Sij(u)
]=
[1
2
(partuipartxj
+partujpartxi
)]=
0 a
2c2
a2 0 0
c2 0 0
=
0 a
2c2
a2 0 0
c2 0 0
(70)
Note that as Sijτij is being computed which is equal to the sum of the element-wise multiplicationof the two matrices (or the trace of the matrix multiplication) that we donrsquot need to consider theelements of τij(u) with corresponding zero entry in Sij As τij(u) = uiuj minus uiuj and again asfiltering preserves constants it is found that
[τij(u)] =[uiuj minus uiuj
]=
lowast 0 middot ax+ b 0 middot cx+ d
ax+ b middot 0 lowast lowastcx+ d middot 0 lowast lowast
=
lowast 0 00 lowast lowast0 lowast lowast
(71)
where a lsquolowastrsquo just represents some element (not necessarily non-zero) Therefore the true subfilterdissipation
Sijτij = tr
0 a2
c2
a2 0 0c2 0 0
middotlowast 0 0
0 lowast lowast0 lowast lowast
= 0 (72)
Now the modelled subfilter dissipation Sij(u)τmodij (u) needs to be computed (recall that Sij =
Sij(u)) In this case let us use the QR model stated in Eq (53) Once again Sij(u) calculated inEq (70) needs to be used So we have that
τmodij = minus2(CQRδ)
2 max0minusI3I1
Sij (73)
As
I1 = tr(S2) =
1
2(a2 + c2) and I3 = tr(S
3) = 0 (74)
it holds that τmodij (u) = 0 so Sij(u)τmod
ij (u) = 0 This agrees with Eq (72) and hence Sol 1(Linear Velocity) endorses the QR model
27
422 Testing the Models
Now using a Mathematica code the exact solutions in Section 22 are again used to evaluate thecondition for the models using the process just described in Section 421 The results are describedin Table 4
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 4 All entries with a ldquo0rdquo represent a value of zero and a ldquolowastrdquo represents a value that is non-zeroThe ldquoExactrdquo row represents the value of the true subfilter dissipation Sijτij Every other row witha model name represents the value of the modelled subfilter dissipation Sij(u)τmod
ij (u) respectiveto that model A ldquordquo means that it is unclear as to the value - the sign of these values is oftenunknown and the lsquoMaxrsquo function in the definition of the QR model (in Eq (53)) makes it difficultto evaluate
Reading from Table 4 the Smagorinsky model is clearly endorsed by Sol 3 (Velocity as a function oftime) and Sol 15 (Rankine Vortex) as in both cases both Sijτij = 0 = Sij(u)τmod
ij (u) In this casewe say that Sol 3 (Velocity as a function of time) and Sol 15 (Rankine Vortex) lsquostronglyrsquo endorsethe Smagorinsky model The Smagorinsky model is not endorsed by Sol 1 (Linear Velocity) Sol 2(Planar Flow) or Sol 14 (Radially Dependent) however because Sij(u)τmod
ij (u) 6= 0 For Sol 4 to
13 both Sijτij 6= 0 6= Sij(u)τmodij (u) and so we will say that these exact solutions lsquoweaklyrsquo endorse
the Smagorinsky model
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 10 10 9 6
Table 5 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Interestingly the lsquostrongrsquo and lsquoweakrsquo endorsements give different results However more stock shouldbe put in the results of the lsquostrongerrsquo condition as it is completely clear that zero is equal to zeroIt is not necessarily clear whether or not non-zero equals non-zero as the values will depend on theparameters in each solution hence the lsquoweakerrsquo endorsement
The lsquostrongrsquo condition equally endorses the Vreman QR Gradient and Vortex-Stretching-basedmodels whereas the lsquoweakrsquo condition endorses Vreman and Smagorinsky the most closely followedby the Gradient model with the Vortex-Stretching-based model being the most opposed As this
28
completely contradicts the lsquostrongrsquo condition and the results in Section 41 it makes us question thevalidity of the weaker test The number of endorsements for the QR model is unknown because of thedefinition of the model in Eq (53) and our evaluation of the true subfilter dissipation Eq (67) Dueto the exact solutions we have found it is never clear whether or not the true subfilter dissipationis strictly positive or negative This means that when the true subfilter dissipation is found to benon-zero the lsquoMaxrsquo function in the QR model cannot be evaluated
43 Vremanrsquos True Subfilter Dissipation Claim
Using the theory mentioned in Sections 232 and 34 we can place the exact solutions to the Navier-Stokes equations into different classes defined by Vreman In doing this we can also attempt to verifyone of Vremanrsquos results [16] In Vremanrsquos paper he defines the class Qn to be the set of flows withn zero entries in the velocity gradient of the flow (the velocity gradient is defined in Eq (39))Vreman then claims that the only flows with zero true subfilter dissipation are a certain selection ofthose in the classesQ7Q8 orQ9 shown in appendix B and is generally non-zero for any other flows
The true subfilter dissipation of Sol 1 (Linear Velocity) was computed in Section 421 and wasfound to be zero in Eq (72) The velocity gradient of Sol 1 is as follows
GLV =
0 0 0a 0 0c 0 0
(75)
and hence Sol 1 (Linear Velocity) is in the required subset of Q7 which agrees with Vremanrsquos claimIn Section 422 the true subfilter dissipation was computed (determined to be zero or non-zero) forall of the exact solutions to the Navier Stokes equations in Section 22 These solutions can now belsquoplacedrsquo into each class Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a functionof time) and Sol 14 (Radially Dependent) were all found to have zero true subfilter dissipation andare all in the appropriate subsets of either Q7Q8 or Q9 This agrees with Vremanrsquos results
We can extend Vremanrsquos results however We found that Sol 15 (Rankine Vortex) had zero truesubfilter dissipation and yet is not in the subset of Q7 defined in Appendix B It is important tonote that the rate of strain tensor Sij in Eq (41) for Sol 15 (Rankine Vortex) is zero This is thereason the true subfilter dissipation is zero and the same reason that all models are zero for thisflow as all models are based on Sij
We also found that all of the remaining exact solutions to the Navier-Stokes equations are not inVremanrsquos specified classes (in Appendix Section B) They also have a non-zero subfilter dissipationwhich agrees with Vremanrsquos results in his paper [16]
44 Classes of Flows
It is important to consider that some exact solutions to the Navier-Stokes equations may have similarproperties For example Sol 1 (Linear Velocity) and Sol 2 (Planar Flow) are both very similarIn fact Sol 2 (Planar Flow) is a more general case of Sol 1 (Linear Velocity) We therefore mustgroup the different flows into classes in an attempt to reduce the bias that may occur when solutionsare similar This should aid us to further determine which of the models are most endorsed We willdo this in a number of different ways using the theory discussed in Section 23 We will group theflows with respect to Vremanrsquos Flow Classes given in appendix B and with respect to the Invariantsgiven in appendix A
29
441 Vremanrsquos Velocity Gradient Flow Classes
We now put our exact solutions to the Navier-Stokes equations into one of two classes Either Q0minus6or Q7minus9 where each class is defined intuitively as follows
Q0minus6 = Q0 cupQ1 cup middot middot middot cup Q6 (76)
Q7minus9 = Q7 cupQ8 cupQ9 (77)
where Qn is defined in Section 43 By looking at the gradient of each exact solution we can see thatQ7minus9 contains the six solutions Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as afunction of time) Sol 5 (2D Taylor) Sol 14 (Radially Dependent) and Sol 15 (Rankine Vortex)and that Q0minus6 contains the remaining nine We now check the models for LES without explicitfiltering by considering endorsements within each class Tables 6 and 7 are analogues of Table 3 andillustrate the results for each flow class The analogues of Tables 1 and 2 are Tables 22 and 23 andare in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 6 6 6
lsquoDissipationrsquoEndorsements
2 5 6 6 6
Percentage Endorsed 417 833 100 100 100
Table 6 Each entry in the top two rows represents the number of solutions in the class Q7minus9 thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Q0minus6 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 2 3 3
lsquoDissipationrsquoEndorsements
0 0 1 0 3
Percentage Endorsed 56 56 167 167 333
Table 7 Each entry in the top two rows represents the number of solutions in the class Q1minus6 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking results to 1 decimal place if necessary
All of the flows in the class Q7minus9 endorse the QR Gradient and Vortex-Stretching-based subfilter-scale models (see Table 6) We can hence see that this class fully endorses these three models equallyand does not endorse the Smagorinsky model The Vortex-Stretching eddy viscosity model has themost endorsements from any of the flows in the class Q0minus6 and the Smagorinksy and Vreman modelshave the fewest endorsements (see Table 7) We can hence say that this class mostly endorses theVortex-Stretching-based eddy-viscosity model note however that the percentage endorsement is very
30
low
Now let us consider LES with explicit filtering Tables 8 and 9 are analogues of Table 5 and illustratethe results for each flow class The analogue of Table 4 is Table 24 and is in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 1 1 0 0 0
Table 8 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Q0minus6 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 9 9 9 6
Table 9 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We have grouped Vremanrsquos classes into Q7minus9 and Q0minus6 this is equivalent to saying that we havegrouped Vremanrsquos classes into those with two or fewer non-zero velocity gradient elements and thosewith three or more non-zero velocity gradient elements respectively Therefore the solutions in Q7minus9are exactly those with a lsquo0rsquo in the lsquoExactrsquo column of Table 4 and Sol 5 (2D Taylor) ExcludingSol 5 (2D Taylor) these are the same as those five that endorse the Vreman QR Gradient andVortex-Stretching-based models in Table 5 We can see in Table 8 that the results for the lsquostrongrsquoendorsements are exactly the same as in Table 5 This is because all of the flows with zero subfilterdissipation are contained in the class Q7minus9 There are hence no lsquostrongrsquo endorsements in Table 9The majority of the lsquoweakrsquo endorsements come from Q0minus6 as this class contains all but one of thesolutions with non-zero true subfilter dissipation (given in Eq (67))
442 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
We now separate our exact solutions into classes again using Vremanrsquos flow classes This timehowever we define V0 to be the class of flows Vreman claims will always have zero true subfilterdissipation namely those classes mentioned in Appendix B We define Vlowast to be all the remaining
31
classes Then
V0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow)
Sol 3 (Velocity as a function of time) Sol 14 (Radially Dependent)(78)
Vlowast = Sol 4 (Axisymmetric Flow) Sol 5 (2D Taylor) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 13 (Sphere Translation) Sol 15 (Rankine Vortex)
(79)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 10 and 11 are analogues of Table 3 and illustrate the results for each flow class Theanalogues of Tables 1 and 2 are Tables 25 and 26 and are in Appendix C2
Endorsements for models using solutions in V0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 4 4 4 4
lsquoDissipationrsquoEndorsements
1 4 4 4 4
Percentage Endorsed 375 100 100 100 100
Table 10 Each entry in the top two rows represents the number of solutions in the class V0 thatendorse each model There are 4 solutions in this class so the percentages are worked out by dividingthe total of each column by 8 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Vlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 2 4 5 5
lsquoDissipationrsquoEndorsements
1 1 3 2 5
Percentage Endorsed 136 136 318 318 455
Table 11 Each entry in the top two rows represents the number of solutions in the class Vlowast thatendorse each model There are 11 solutions in this class so the percentages are worked out bydividing the total of each column by 22 and taking results to 1 decimal place if necessary
All of the flows in the class V0 endorse the Vreman QR Gradient and Vortex-Stretching-basedsubfilter-scale models (see Table 6) This class completely endorses these four models equally anddoes not endorse the Smagorinsky model The Vortex-Stretching-based eddy-viscosity model hasthe most endorsements from the class Vlowast and the Smagorinksy and Vreman models again have thefewest endorsements (see Table 7) As in Section 441 we say that this class mostly endorses the
32
Vortex-Stretching-based eddy-viscosity model but once again the percentage endorsement is ratherlow
Now let us consider LES with explicit filtering Tables 12 and 13 are analogues of Table 5 andillustrate the results for each flow class The analogue of Table 4 is Table 27 and is in Appendix C2
Endorsements for models using solutions in V0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 4 4 4 4
lsquoWeakrsquo Endorsements 0 0 0 0 0
Table 12 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Vlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 1 1 1 1
lsquoWeakrsquo Endorsements 10 10 9 6
Table 13 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We can see in Table 12 that the results for the lsquostrongrsquo endorsements are similar to those in Table 5excluding Sol 15 (Rankine Vortex) The four solutions in V0 with zero true subfilter dissipationnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)and Sol 14 (Radially Dependent) are hence the four that endorse the Vreman QR Gradientand Vortex-Stretching-based models in Table 12 Sol 15 (Rankine Vortex) is in the class Vlowast andhence there is one lsquostrongrsquo endorsement in Table 13 The class Vlowast contains all of the flows with anon-zero true subfilter dissipation (given in Eq (67)) and hence Table 13 contains all of the lsquoweakrsquoendorsements
443 Classes using the Third Principal Invariant
Sorting the classes using the invariants is a non-trivial task in itself There are a number of waysin which to do this using both the principal invariants in Eq (40) and the combined invariants inEq (43)
We start by looking at flows with third principal invariant RG equal to zero We again define twoclasses R0 and Rlowast where R0 consists of all flows with RG = 0 and Rlowast consists of all flows withRG 6= 0 Then using the information in Appendix A we see that there are nine flows in R0 That
33
is
R0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 8 (Vortex in Stagnation Flow)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)
(80)
Rlowast = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 13 (Sphere Translation)(81)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 14 and 15 are analogues of Table 3 and they illustrate the results for LES withoutexplicit filtering for each flow class The analogues of Tables 1 and 2 are Tables 28 and 29 and arein Appendix C3
Endorsements for models using solutions in R0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 7 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 278 556 778 722 778
Table 14 Each entry in the top two rows represents the number of solutions in the class R0 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking the result to one decimal place if necessary
Endorsements for models using solutions in Rlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 2 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 83 83 83 167 333
Table 15 Each entry in the top two rows represents the number of solutions in the class Rlowast thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking the result to one decimal place if necessary
The class R0 endorses both the Vortex-Stretching-based model and the QR model the most andSmagorinsky the least in the case for LES without explicit filtering In the Rlowast class the Vortex-Stretching-based model is again the most endorsed followed by the Gradient model however thepercentages are again small
34
Now let us consider LES with explicit filtering Tables 16 and 17 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 30 and is in Appendix C3
Endorsements for models using solutions in R0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 4 4 3 2
Table 16 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Rlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 6 6 6 4
Table 17 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 we can see in Table 16 that the results for the lsquostrongrsquo endorsements are exactly thesame as in Table 5 This is because all of the flows with zero subfilter dissipation are contained in theclass R0 As in Table 9 there are no lsquostrongrsquo endorsements in Table 17 The lsquoweakrsquo endorsementshowever are split more evenly between the two classes now than in Vremanrsquos classes in Sections 441and 442 We find that the Smagorinsky and Vreman models still seem to be the most endorsed inboth classes
444 Two-component Flow Classes
Another way in which the flows can be grouped into classes is by determining whether a flow is atwo-component flow or not We define a two-component flow to have the property that the combinedprincipal invariants of a flow in Eq (43) satisfy
I3 = I4 = I5 minus1
2I1I2 = 0 (82)
We are also interested in considering flows with zero gradient model subgrid dissipation These flowshave the property that
I3 minus I4 = 0 (83)
What we find though is that for all the solutions in Section 22 if a flow is a two-component flowthen it also satisfies I3 minus I4 = 0 It may be interesting to note that although I3 minus I4 6= 0 for manysolutions there are a number with I3 equal to a multiple of I4 Although not considered in thisreport this may also be a class of interest We find that the two-component flows are
C2 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)(84)
35
and we call the remaining flows three-component flows They are
C3 = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex) Sol 11 (Pseudo-Motion Shots)
Sol 13 (Sphere Translation)
(85)
Note that Sol 8 (Vortex in Stagnation Flow) is defined to be a planar flow this does not mean that itis a two-component flow in this sense however because its two non-zero velocity components dependon three coordinates We now check the models for LES without explicit filtering by consideringendorsements within each class Tables 18 and 19 are analogues of Table 3 and they illustrate theresults for LES without explicit filtering for each flow class The analogues of Tables 1 and 2 areTables 31 and 32 and are in Appendix C4
Endorsements for models using solutions in C2 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 6 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 357 714 100 857 100
Table 18 Each entry in the top two rows represents the number of solutions in the class C2 thatendorse each model There are 7 solutions in this class so the percentages are worked out by dividingthe total of each column by 14 and taking the result to one decimal place if necessary
Endorsements for models using solutions in C3 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 3 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 63 63 63 188 250
Table 19 Each entry in the top two rows represents the number of solutions in the class C3 thatendorse each model There are 8 solutions in this class so the percentages are worked out by dividingthe total of each column by 16 and taking the result to one decimal place if necessary
The class C2 equally endorses the QR and Vortex-Stretching-based models and once again theSmagorinsky model is the least endorsed As with the previous classes the class C3 has such lowpercentages that it does not seem to endorse any models We start to see a trend appearing here inthat the flow classes that contain the more complicated flows namely Q0minus6 (Eq (76)) Vlowast (Eq (79))Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages The classes that tendto contain the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78)) R0 (Eq (80)) and C2 (Eq (84))all seem to favour the Vortex-Stretching-based model or the QR model
36
Now let us consider LES with explicit filtering Tables 20 and 21 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 33 and is in Appendix C4
Endorsements for models using solutions in C2 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 2 2 0 1 0
Table 20 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in C3 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 8 8 8 6
Table 21 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 and Table 16 we can see in Table 20 that the results for the lsquostrongrsquo endorsements areexactly the same as in Table 5 This is again because all of the flows with zero subfilter dissipationare contained in the class C2 There are two flows with non-zero true subfilter dissipation (givenin Eq (67)) in C2 and this shows in the lsquoweakrsquo endorsements in Table 20 There are no lsquostrongrsquoendorsements in Table 21 as all the zero subfilter dissipation flows are in C2
5 Conclusions
In this report we studied incompressible turbulent flows We found a number of exact solutions to theNavier-Stokes equations each representing a relatively simplistic fluid flow Large-eddy simulation(LES) is used to model turbulent flows by smoothing over the complex small-scale motion Weintroduced a number of subfilter-scale models to approximate this small-scale motion and testedthem using the found exact solutions to the Navier-Stokes equations We first considered LESwithout explicit filtering and used the Navier-Stokes equations to derive conditions for the subfilter-scale models namely the Velocity Field condition (Eq (58)) the Kinetic Energy condition (Eq (60))and the Dissipation condition (Eq (62)) Similarly we considered LES with explicit filtering andagain used the Navier-Stokes equations to derive more conditions for the models in this case wedecided to just take one condition the Dissipation condition (Eqs (67) and (68)) as the others leadto difficulty in their evaluation By lsquorunningrsquo the exact solutions to the Navier-Stokes equationsthrough each condition for the subfilter-scale models we found that the Vortex-Stretching-basededdy-viscosity model in Eq (54) was most endorsed by the tests in LES without explicit filteringWe found that four out of the five subfilter-scale models namely the Vreman QR Gradient andVortex-Stretching-based models were equally endorsed by the lsquostrongrsquo tests in LES with explicitfiltering with the Smagorinsky model being the least endorsed The results we found for the lsquoweakrsquocondition were completely opposite to those found in both the lsquostrongrsquo condition and in LES withoutexplicit filtering This led us to conclude that this test was not one in which we should put our trust
37
We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses
In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes
This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold
38
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
u1(x y z t) = minus 2x
c2Reminus ay
x2 + y2
u2(x y z t) = minus 2y
c2Re+
ax
x2 + y2
u3(x y z t) =4z
c2Re
(30)
where a and c are constants
Figure 12 An illustration of the flow for Sol9 (Burgers Vortex) described in Eq (27) withparameter values c = 15 Re = 20 and a = 10 Aslice of the xy-plane at z = 0 is shown
Solution 11 Pseudo-Motion Shots of the Second Kind
In the book by Berker [3 p 94] a solution is given in cylindrical coordinates by
ur(r ϕ z t) =2
Re
1
r
uϕ(r ϕ z t) = A1r3 +B1r +
C1
r
uz(r ϕ z t) = A2r2 log(r) +B2r
2 + C2
p(r ϕ z t) =A2
1r6
6+A1B1r
4
2+A1C1r
2 +B2
1r2
2minus C2
1
2r2minus 2
r2Re2+ 2B1C1 log(r)
+2A2z
Reminus 4B1ϕ
Re
(31)
where A1 A2 B1 B2 C1 C2 are constants Eq (31) when converted into Cartesian coordinates is
15
u1(x y z t) =1
Re
2xminus C1Rey
x2 + y2minusA1y
(x2 + y2
)minusB1y
u2(x y z t) =1
Re
2y + C1Rex
x2 + y2+A1x
(x2 + y2
)+B1x
u3(x y z t) =1
2A2
(x2 + y2
)log(x2 + y2
)+B2
(x2 + y2
)+ C2
(32)
Figure 13 An illustration of the flow in thexy-plane for Sol 11 (Pseudo-Motion Shots)described in Eqs (31) and (32) with parametervalues A1 = B1 = C1 = 1 A2 = B2 = C2 = 2and Re = 10
Figure 14 An illustration of the flow in theyz-plane for Sol 11 (Pseudo-Motion Shots)described in Eqs (31) and (32) with parametervalues A1 = B1 = C1 = 1 A2 = B2 = C2 = 2and Re = 10
Solution 12 Flows With Constant Whirl
In the book by Berker [3 p 143] another solution is given in cylindrical coordinates by
ur(r ϕ z t) = minus 1
Re
1
r
uϕ(r ϕ z t) =1
rradicte(minus
12 g(t))
uz(r ϕ z t) = 0
p(r ϕ z t) =1
4
2(minus e
minusg(t)
t minus 1Re2
)r2
minus 1
t2Re
infinintminusg(t)
eminust
tdt
(33)
where we define g(t) =r2Re
2t When converted into Cartesian coordinates we find
16
u1(x y z t) = minus 1
x2 + y2
(yradict
exp
(minus
Re(x2 + y2
)4t
)+
x
Re
)
u2(x y z t) =1
x2 + y2
(xradict
exp
(minus
Re(x2 + y2
)4t
)minus y
Re
)
u3(x y z t) = 0
(34)
Figure 15 An illustration of the flow for Sol12 (Flows with Constant Whirl) described inEqs (33) and (34) with time t = 1 and Re = 1 Aslice of the xy-plane at z = 0 is shown as velocityin the z-direction is zero this is the same for anyvalue of z
17
Solution 13 Rectilinear and Uniform Translation of a Sphere
In the book by Berker [3 p 228] another solution is given in Cartesian coordinates by
u1(x y z t) =3
4
axz
R3
(a2
R2minus 1
)
u2(x y z t) =3
4
ayz
R3
(a2
R2minus 1
)
u3(x y z t) = 1minus 3
4
a
Rminus 1
4
a3
R3minus 3
4
az2
R3
(a2
R2minus 1
)
p(x y z t) =3a(aminusR)(a+R)
32R10Re
((R2Re
(x2 + y2
)minus 2R2Rez2
) (a3 + 3aR2 minus 4R3
)+ 3aRez4
(R2 minus a2
)minus 16R5z
)
(35)
where a and R are constants
Figure 16 An illustration of the flow in the yz-plane for Sol 13 (Sphere Translation) describedin Eq (35) with parameter values a = minus2 andRe = 3
Solution 14 Radially Dependent Flow
A solution with velocity in the z-direction dependent only on the distance from the z-axis is as
18
follows in cylindrical coordinates
ur(r ϕ z t) = 0
uϕ(r ϕ z t) = 0
uz(r ϕ z t) = A(r2 minus a2
)
p(r ϕ z t) =4Az
Re
(36)
where A and a are constants In Cartesian coordinates the solution is just the same with u1(x y z t)and u2(x y z t) both zero and the r2 term replaced with x2 + y2 in u3(x y z t)
Figure 17 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus1 1] withparameter values a = 1 A = 4 and Re = 10
Figure 18 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus2 2] withparameter values a = 1 A = 4 and Re = 10
Solution 15 Rankine Vortex
A solution discovered by Rankine [1] is a vortex flow of the following form
ur(r ϕ z t) = 0
uϕ(r ϕ z t) =ar
2πR2
uz(r ϕ z t) = 0
p(r ϕ z t) =a2r2
8π2R4
(37)
19
where a and R are constants In Cartesian coordinates the flow has the form
u1(x y z t) = minus ay
2πR2
u2(x y z t) =ax
2πR2
u3(x y z t) = 0
p(x y z t) =a2(x2 + y2
)8π2R4
(38)
This flow is actually a rigid body rotational flow in that the angular rotational velocity is uniformand the velocity of the flow increases proportionally with distance from the z-axis
Figure 19 An illustration of the flow in the xy-plane for Sol 15 (Rankine Vortex) described inEq (37) and Eq (38) with parameter values a = 1and R = 1
radic2π
23 Characterisation of Flows
It is important to be able to classify flows in a way that sorts them into distinct groups This aidsin the clear identification and general study of flows It is also a very important way of recognisingflows with similar properties In this subsection we introduce two distinct ways in which flows canbe characterised
231 Characterising Solutions using Invariants
A principal invariant of a tensor is a coefficient of the characteristic polynomial of that tensorInvariants as their name suggests are invariant under rotations of the coordinate system in thattheir values do not change when the tensor is lsquorotatedrsquo into a different set of coordinates Thevelocity gradient of a flow G is defined as the Jacobian of the velocity vector given by
Gij =
[partuipartxj
] (39)
20
for i j = 1 2 3 There are three principal invariants of the velocity gradient of a fluid flow G givenby
PG = tr(G)
QG =1
2
((tr(G))2 minus tr
(G2))
RG = det(G)
(40)
Incompressibility of flows (Eq (5)) ensures that PG = 0 hence the first principal invariant is zerofor all of the flows in Section 22 We now define the rate-of-strain tensor S by
Sij =1
2
(partuipartxj
+partujpartxi
) (41)
and the rate-of-rotation tensor Ω by
Ωij =1
2
(partuipartxjminus partujpartxi
) (42)
Note that the first principal invariant of S and the first and third principal invariants of Ω are zeroAlso note that the velocity gradient of a flow can simply be written G = S + Ω
Rather than considering the principal invariants we can also consider the combined invariants
I1 = tr(S2) I2 = tr(Ω2) I3 = tr(S3) I4 = tr(SΩ2) I5 = tr(S2Ω2) (43)
We can hence write the three principal invariants for a velocity field G in terms of the combinedinvariants
PG = 0
QG = minus1
2(I1 + I2)
RG =1
3(I3 + 3I4)
(44)
In this report the flows considered are the exact solutions to the incompressible Navier-Stokesequations described in Section 22 The combined invariants in Eq (43) are used to characterisethese solutions into different classes in Section 44 Both the principal invariants and combinedinvariants for each exact solution in Section 22 are given in Appendix A
Each of the subfilter-scale eddy-viscosity models described in Section 32 are written in terms ofthese five combined invariants
232 Vremanrsquos Characterisation of Flows
Vreman [16] classifies flows using the gradient of the velocity field of a flow Each lsquoclassrsquo of flows hasthe same number of non-zero elements in the gradient matrix For example
[partuipartxj
]=
0 0 0lowast 0 0lowast 0 0
(45)
where a lowast denotes a non-zero term represents a group of flows that lie within the class denoted byQ7 (see Appendix B) where the subscript represents the number of non-zero terms in the velocity
21
gradient Flows of the form in Eq (45) have a velocity field of the form
u1 = c1 (constant)
u2 = f2(x1)
u3 = f3(x1)
(46)
where f2(x1) and f3(x1) are such that the Navier Stokes equations Eq (1) are satisfied Notethat as the elements on the leading diagonal are zero the incompressibility condition Eq (5) isautomatically satisfied Some simple flow classes are given in Appendix B and contain a number ofthe solutions discussed in Section 22 This is mentioned in greater detail in Section 441
3 Conditions for Large-Eddy Simulation Models
31 Large-Eddy Simulation
Due to the nature of numerical simulation and the complexity of turbulent flows small details aremissed and information is lost when solving the Navier-Stokes equations The computational costof solving a turbulent flow is extremely high so a filtering operation is applied to the Navier-Stokesequations represented by a bar to reduce the computational cost of the simulation This is a linearoperation and preserves constants ie ρ = ρ and is also assumed to commute with differentiation
partuipartxj
=partuipartxj
(47)
The filtered Navier-Stokes equations are found by filtering (putting a bar over) the entire equations(see Eq (1)) and after evaluating are given by
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (48)
where
τij = uiuj minus uiuj (49)
is called the lsquotrue stressrsquo of the flow This however poses a problem as it is not clear what is meantby the term uiuj for a general filtering operation The true stress τij therefore must be replaced bya model τmod
ij in practical applications
32 Subfilter-Scale Models
We will now define the subflter-scale models denoted by τmodij to be used in place of the lsquodifficultrsquo
true stress τij There are a number of models available to use in the literature [11 p7] These willbe tested using the exact solutions to the Navier-Stokes equations and by checking conditions whichare defined in Sections 33 and 34
All the models barring one are of the same form That is each of the eddy-viscosity models arewritten
τmodij minus 1
3τmodkk δij = minus2νeSij (50)
with νe representing the eddy viscosity of a specific model For simplicity we usually just writeτmodij = minus2νeSij the extra term is included here to emphasise that the models are traceless Each
22
of the models we consider here can be expressed in terms of the rate of strain tensor S defined inEq (41) the rate of rotation tensor Ω defined in Eq (42) and the combined invariants defined inEq (43) The models are then defined as follows
Smagorinsky [12] νSe = (CSδ)2radic
2I1 (51)
Vreman [16] νVe = (CV δ)2
radicQGGT
PGGT (52)
QR [13 14 15] νQRe = (CQRδ)2 max0minusI3
I1 (53)
Vortex-Stretching [11] νV Se = (CV Sδ)2radic
2I1
(I5 minus 1
2I1I2
minusI1I2
)32
(54)
with the quantities
PGGT = I1 minus I2 QGGT =1
4(I1 + I2)2 + 4
(I5 minus
1
2I1I2
) (55)
There is a fifth and final model to be considered here It is of a different form to the other four andnot an eddy-viscosity model It is known as the Gradient model and is defined by
Gradient [4 9] τmodG = CGδ
(S2 minus Ω2 minus (SΩminus ΩS)
) (56)
The values C are model constants specific to each model and δ represents the mesh size of the LESBoth of these quantities are included here to properly define the models however in this projectthey will be ignored as they are constants and have no influence on the conditions to be defined
33 Conditions for Large-Eddy Simulation without Explicit Filtering
The filtering operation introduced in Eq (48) is just a formality for LES without explicit filteringThe idea here is that filtering does not actually take place but the term lsquoaddedrsquo to the Navier-Stokesequations in fact lsquosubtractsrsquo the difficult to simulate small-scale turbulent motion of the flow Inorder to consider this properly the terms ui and p in Eq (48) are replaced by vi and q It is hopedthat τmod
ij approximates τij so that vi and q approximate ui and p respectively So LES withoutexplicit filtering is given by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (57)
It is clear to see that the first four terms of this equation are exactly the same as the Navier-Stokesequations so for the exact solutions discussed in Section 22 the combination of these terms willequal zero What is then desirable is that the model term is also zero for these exact solutions sothat Eq (57) still holds In order to test models for exact solutions the condition
part
partxjτmodij (v) = 0 (58)
is checked If this condition were to hold it would imply that the model does not influence thevelocity field of a flow hence we call it the Velocity Field Condition There are a number of otherrestrictions that can be tested One such test comes from the differential form of the evolution ofthe kinetic energy of a flow given in Eq (10) In order to find this Eq (57) is multiplied by vi andterms are grouped to find
part
partt
(1
2vivi
)+ vj
part
partxj
(1
2vivi
)= minusvi
1
ρ
partq
partxi+ νvi
part2vipartxjpartxj
minus vipart
partxjτmodij (v) (59)
23
Then similarly to Eq (58) another condition is to check whether the model affects the kineticenergy of exact solutions this is equivalent to checking whether
vipart
partxjτmodij (v) = 0 (60)
We call this condition the Kinetic Energy Condition The third and final condition considered herecan also be found using the integral form for the equation for kinetic energy
d
dt
intV
1
2vividV =
intS
[minusvj
(1
2vivi
)minus vj
q
ρ+ 2νviSij + viτ
modij (v)
]njdS
+
intV
[minus2νSijSij minus
(minusSijτmod
ij (v))]dV
(61)
where the rate-of-strain tensor Sij was defined in Eq (41) V is a volume of fluid and dS meansthe integral is over the surface of that volume with unit normal nj Then this final condition checkswhether the model affects the total kinetic energy of exact solutions to the Navier-Stokes equationsthat is
Sijτmodij (v) = 0 (62)
We will call this condition the Dissipation Condition because Sijτmodij (v) is often referred to as the
subgrid dissipation
34 Conditions for Large-Eddy Simulation with Explicit Filtering
As in Eq (48) the Navier-Stokes equations can be filtered by applying the filtering operation to thewhole equation (putting a bar over all terms) and evaluating to find again that
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (63)
In Eq (57) the filtering operation was ignored and the elements ui p and τij(u) were replacedwith vi q and τmod
ij (v) respectively Now in LES with explicit filtering the filtering operation itselfsmooths over the small-scale turbulent motion of the simulated flow In this case the elements uip and τij(u) will be replaced with vi q and τmod
ij (v) respectively LES with explicit filtering is thengiven by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (64)
Note that Eqs (63) and (64) despite their similarity to Eq (57) no longer contain the Navier-Stokesequations as Eq (57) did The bar (filtering operation) over each term results in each exact solutionfound in Section 22 not necessarily satisfying the first four terms It is hence unknown whether ornot these terms will vanish for exact solutions Terms can still be compared similarly to Eqs (58)(60) and (62) However instead of comparing to zero we can now compare to the relevant termfrom the filtered Navier-Stokes equations So the terms
part
partxjτij (65)
and
part
partxjτmodij (v) (66)
24
are to be compared Although the latter term is relatively simple to compute the former could leadto some difficulties due to the differentiation of the true stress τij = uiuj minus uiuj So although someinformation could be gleaned from the inspection of these two terms more information can be foundby considering the analogue of Eq (62) So the true subfilter dissipation
Sijτij (67)
(where Sij is shorthand for Sij(v)) and the modelled subfilter dissipation
Sij(v)τmodij (v) (68)
are considered and have to be equal for the condition to hold These can again be derived from theintegral equation for kinetic energy of a filtered velocity field similarly to Eq (61)
4 Analysis of Large-Eddy Simulation Models
In this section we use a code in Mathematica the theory described in Section 3 and the exactsolutions to the Navier-Stokes equations found in Section 22 to check whether or not each of theexact solutions lsquoendorsesrsquo each model for LES both without and with explicit filtering
41 Large-Eddy Simulation Without Explicit Filtering
We start by using a code in Mathematica to test the Velocity Field condition in Eq (58) and theKinetic Energy condition in Eq (60) For all exact solutions to the Navier-Stokes equations inSection 22 the outputs for both conditions were identical so we have combined them into one table
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 ndash ndash 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 1 For the Kinetic Energy condition a lsquo0rsquo result means that vipartpartxj
τmodij (v) = 0 for respective
models and exact solutions whereas a lsquolowastrsquo result means that vipartpartxj
τmodij (v) 6= 0 For the Velocity
Field condition a lsquo0rsquo result means that partpartxj
τmodij (v) = 0 for respective models and exact solutions
whereas a lsquolowastrsquo result means that partpartxj
τmodij (v) 6= 0 The Gradient model can take both positive and
negative values so when the sign of the model is known it is included in the table instead of a lsquorsquoBoth the conditions are included in one table because for all cases the outputs were identical
We then test the Dissipation condition in Eq (62) for all exact solutions to the Navier-Stokesequations in Section 22
25
Dissipation Condition [Does Sijτmodij (v) = 0]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 + 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 2 A lsquo0rsquo result means that Sijτmodij (v) = 0 for respective models and exact solutions whereas
a lsquolowastrsquo result means that Sijτmodij (v) 6= 0 The Gradient model can take both positive and negative
values so when the sign of the model is known it is included in the table instead of a lsquorsquo
We then sum all the results to find the total number of endorsements for each model described inSection 32
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
4 6 8 9 9
lsquoDissipationrsquoEndorsements
2 5 7 7 9
Total Endorsements 6 11 15 16 18
Table 3 The results from Tables 1 and 2 are summarised to give a general overview of how manyendorsements each model receives This makes it much clearer to see the models most supported bythe respective and combined conditions with the Vortex-Stretching model being the most supportedand the Smagorinsky model the least
Testing the models in Section 32 using LES without explicit filtering has mixed results It isclear that the Smagorinksy model is not well endorsed by the flows in Section 22 with the onlyendorsements coming from the simplest flows that is Sol 1 (Linear Velocity) Sol 3 (Velocity as afunction of time) Sol 7 (Stagnation Flow) and Sol 15 (Rankine Vortex) The Vortex-Stretching-based model is the most endorsed flow with eighteen out of a possible thirty endorsements Thisagrees with our foreknowledge as this model was constructed to be as physically consistent as possiblein the paper by Silvis amp Verstappen [11]
42 Large-Eddy Simulation With Explicit Filtering
In this section we will compare the true subfilter dissipation Sijτij and the modelled subfilterdissipation Sij(v)τmod
ij (v) given in Eqs (67) and (68) respectively for each model τmodij and each
exact solution to the Navier-Stokes equation given in Section 22 We will not consider the conditionsinvolving differentiation in Eqs (65) and (66) due to the difficulty in calculating the derivative ofthe τij term in Eq (65)
26
421 Example of a Test for a Model
The true subfilter dissipation Sijτij of a flow given in Eq (67) cannot be explicitly computed asthe filtering operation is kept general What can be done however is to partially evaluate theexpressions using the attributes of a filtering operation mentioned in Eq (48)
The calculations of Sijτij for Sol 1 (Linear Velocity) are provided to illustrate the general ideaRecall that the velocity field of Sol 1 (Linear Velocity) is written in Eq (13) in vector notation as
u(x y z t) =
0ax+ bcx+ d
(69)
Then Sij = Sij(u) is computed simply by using its definition in Eq (41) Note that as filteringpreserves constants if there is an a it is replaced with a meaning that (in this case) there are alarge number of zero terms due to the frequency of 0 Hence
[Sij(u)
]=
[1
2
(partuipartxj
+partujpartxi
)]=
0 a
2c2
a2 0 0
c2 0 0
=
0 a
2c2
a2 0 0
c2 0 0
(70)
Note that as Sijτij is being computed which is equal to the sum of the element-wise multiplicationof the two matrices (or the trace of the matrix multiplication) that we donrsquot need to consider theelements of τij(u) with corresponding zero entry in Sij As τij(u) = uiuj minus uiuj and again asfiltering preserves constants it is found that
[τij(u)] =[uiuj minus uiuj
]=
lowast 0 middot ax+ b 0 middot cx+ d
ax+ b middot 0 lowast lowastcx+ d middot 0 lowast lowast
=
lowast 0 00 lowast lowast0 lowast lowast
(71)
where a lsquolowastrsquo just represents some element (not necessarily non-zero) Therefore the true subfilterdissipation
Sijτij = tr
0 a2
c2
a2 0 0c2 0 0
middotlowast 0 0
0 lowast lowast0 lowast lowast
= 0 (72)
Now the modelled subfilter dissipation Sij(u)τmodij (u) needs to be computed (recall that Sij =
Sij(u)) In this case let us use the QR model stated in Eq (53) Once again Sij(u) calculated inEq (70) needs to be used So we have that
τmodij = minus2(CQRδ)
2 max0minusI3I1
Sij (73)
As
I1 = tr(S2) =
1
2(a2 + c2) and I3 = tr(S
3) = 0 (74)
it holds that τmodij (u) = 0 so Sij(u)τmod
ij (u) = 0 This agrees with Eq (72) and hence Sol 1(Linear Velocity) endorses the QR model
27
422 Testing the Models
Now using a Mathematica code the exact solutions in Section 22 are again used to evaluate thecondition for the models using the process just described in Section 421 The results are describedin Table 4
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 4 All entries with a ldquo0rdquo represent a value of zero and a ldquolowastrdquo represents a value that is non-zeroThe ldquoExactrdquo row represents the value of the true subfilter dissipation Sijτij Every other row witha model name represents the value of the modelled subfilter dissipation Sij(u)τmod
ij (u) respectiveto that model A ldquordquo means that it is unclear as to the value - the sign of these values is oftenunknown and the lsquoMaxrsquo function in the definition of the QR model (in Eq (53)) makes it difficultto evaluate
Reading from Table 4 the Smagorinsky model is clearly endorsed by Sol 3 (Velocity as a function oftime) and Sol 15 (Rankine Vortex) as in both cases both Sijτij = 0 = Sij(u)τmod
ij (u) In this casewe say that Sol 3 (Velocity as a function of time) and Sol 15 (Rankine Vortex) lsquostronglyrsquo endorsethe Smagorinsky model The Smagorinsky model is not endorsed by Sol 1 (Linear Velocity) Sol 2(Planar Flow) or Sol 14 (Radially Dependent) however because Sij(u)τmod
ij (u) 6= 0 For Sol 4 to
13 both Sijτij 6= 0 6= Sij(u)τmodij (u) and so we will say that these exact solutions lsquoweaklyrsquo endorse
the Smagorinsky model
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 10 10 9 6
Table 5 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Interestingly the lsquostrongrsquo and lsquoweakrsquo endorsements give different results However more stock shouldbe put in the results of the lsquostrongerrsquo condition as it is completely clear that zero is equal to zeroIt is not necessarily clear whether or not non-zero equals non-zero as the values will depend on theparameters in each solution hence the lsquoweakerrsquo endorsement
The lsquostrongrsquo condition equally endorses the Vreman QR Gradient and Vortex-Stretching-basedmodels whereas the lsquoweakrsquo condition endorses Vreman and Smagorinsky the most closely followedby the Gradient model with the Vortex-Stretching-based model being the most opposed As this
28
completely contradicts the lsquostrongrsquo condition and the results in Section 41 it makes us question thevalidity of the weaker test The number of endorsements for the QR model is unknown because of thedefinition of the model in Eq (53) and our evaluation of the true subfilter dissipation Eq (67) Dueto the exact solutions we have found it is never clear whether or not the true subfilter dissipationis strictly positive or negative This means that when the true subfilter dissipation is found to benon-zero the lsquoMaxrsquo function in the QR model cannot be evaluated
43 Vremanrsquos True Subfilter Dissipation Claim
Using the theory mentioned in Sections 232 and 34 we can place the exact solutions to the Navier-Stokes equations into different classes defined by Vreman In doing this we can also attempt to verifyone of Vremanrsquos results [16] In Vremanrsquos paper he defines the class Qn to be the set of flows withn zero entries in the velocity gradient of the flow (the velocity gradient is defined in Eq (39))Vreman then claims that the only flows with zero true subfilter dissipation are a certain selection ofthose in the classesQ7Q8 orQ9 shown in appendix B and is generally non-zero for any other flows
The true subfilter dissipation of Sol 1 (Linear Velocity) was computed in Section 421 and wasfound to be zero in Eq (72) The velocity gradient of Sol 1 is as follows
GLV =
0 0 0a 0 0c 0 0
(75)
and hence Sol 1 (Linear Velocity) is in the required subset of Q7 which agrees with Vremanrsquos claimIn Section 422 the true subfilter dissipation was computed (determined to be zero or non-zero) forall of the exact solutions to the Navier Stokes equations in Section 22 These solutions can now belsquoplacedrsquo into each class Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a functionof time) and Sol 14 (Radially Dependent) were all found to have zero true subfilter dissipation andare all in the appropriate subsets of either Q7Q8 or Q9 This agrees with Vremanrsquos results
We can extend Vremanrsquos results however We found that Sol 15 (Rankine Vortex) had zero truesubfilter dissipation and yet is not in the subset of Q7 defined in Appendix B It is important tonote that the rate of strain tensor Sij in Eq (41) for Sol 15 (Rankine Vortex) is zero This is thereason the true subfilter dissipation is zero and the same reason that all models are zero for thisflow as all models are based on Sij
We also found that all of the remaining exact solutions to the Navier-Stokes equations are not inVremanrsquos specified classes (in Appendix Section B) They also have a non-zero subfilter dissipationwhich agrees with Vremanrsquos results in his paper [16]
44 Classes of Flows
It is important to consider that some exact solutions to the Navier-Stokes equations may have similarproperties For example Sol 1 (Linear Velocity) and Sol 2 (Planar Flow) are both very similarIn fact Sol 2 (Planar Flow) is a more general case of Sol 1 (Linear Velocity) We therefore mustgroup the different flows into classes in an attempt to reduce the bias that may occur when solutionsare similar This should aid us to further determine which of the models are most endorsed We willdo this in a number of different ways using the theory discussed in Section 23 We will group theflows with respect to Vremanrsquos Flow Classes given in appendix B and with respect to the Invariantsgiven in appendix A
29
441 Vremanrsquos Velocity Gradient Flow Classes
We now put our exact solutions to the Navier-Stokes equations into one of two classes Either Q0minus6or Q7minus9 where each class is defined intuitively as follows
Q0minus6 = Q0 cupQ1 cup middot middot middot cup Q6 (76)
Q7minus9 = Q7 cupQ8 cupQ9 (77)
where Qn is defined in Section 43 By looking at the gradient of each exact solution we can see thatQ7minus9 contains the six solutions Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as afunction of time) Sol 5 (2D Taylor) Sol 14 (Radially Dependent) and Sol 15 (Rankine Vortex)and that Q0minus6 contains the remaining nine We now check the models for LES without explicitfiltering by considering endorsements within each class Tables 6 and 7 are analogues of Table 3 andillustrate the results for each flow class The analogues of Tables 1 and 2 are Tables 22 and 23 andare in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 6 6 6
lsquoDissipationrsquoEndorsements
2 5 6 6 6
Percentage Endorsed 417 833 100 100 100
Table 6 Each entry in the top two rows represents the number of solutions in the class Q7minus9 thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Q0minus6 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 2 3 3
lsquoDissipationrsquoEndorsements
0 0 1 0 3
Percentage Endorsed 56 56 167 167 333
Table 7 Each entry in the top two rows represents the number of solutions in the class Q1minus6 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking results to 1 decimal place if necessary
All of the flows in the class Q7minus9 endorse the QR Gradient and Vortex-Stretching-based subfilter-scale models (see Table 6) We can hence see that this class fully endorses these three models equallyand does not endorse the Smagorinsky model The Vortex-Stretching eddy viscosity model has themost endorsements from any of the flows in the class Q0minus6 and the Smagorinksy and Vreman modelshave the fewest endorsements (see Table 7) We can hence say that this class mostly endorses theVortex-Stretching-based eddy-viscosity model note however that the percentage endorsement is very
30
low
Now let us consider LES with explicit filtering Tables 8 and 9 are analogues of Table 5 and illustratethe results for each flow class The analogue of Table 4 is Table 24 and is in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 1 1 0 0 0
Table 8 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Q0minus6 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 9 9 9 6
Table 9 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We have grouped Vremanrsquos classes into Q7minus9 and Q0minus6 this is equivalent to saying that we havegrouped Vremanrsquos classes into those with two or fewer non-zero velocity gradient elements and thosewith three or more non-zero velocity gradient elements respectively Therefore the solutions in Q7minus9are exactly those with a lsquo0rsquo in the lsquoExactrsquo column of Table 4 and Sol 5 (2D Taylor) ExcludingSol 5 (2D Taylor) these are the same as those five that endorse the Vreman QR Gradient andVortex-Stretching-based models in Table 5 We can see in Table 8 that the results for the lsquostrongrsquoendorsements are exactly the same as in Table 5 This is because all of the flows with zero subfilterdissipation are contained in the class Q7minus9 There are hence no lsquostrongrsquo endorsements in Table 9The majority of the lsquoweakrsquo endorsements come from Q0minus6 as this class contains all but one of thesolutions with non-zero true subfilter dissipation (given in Eq (67))
442 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
We now separate our exact solutions into classes again using Vremanrsquos flow classes This timehowever we define V0 to be the class of flows Vreman claims will always have zero true subfilterdissipation namely those classes mentioned in Appendix B We define Vlowast to be all the remaining
31
classes Then
V0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow)
Sol 3 (Velocity as a function of time) Sol 14 (Radially Dependent)(78)
Vlowast = Sol 4 (Axisymmetric Flow) Sol 5 (2D Taylor) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 13 (Sphere Translation) Sol 15 (Rankine Vortex)
(79)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 10 and 11 are analogues of Table 3 and illustrate the results for each flow class Theanalogues of Tables 1 and 2 are Tables 25 and 26 and are in Appendix C2
Endorsements for models using solutions in V0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 4 4 4 4
lsquoDissipationrsquoEndorsements
1 4 4 4 4
Percentage Endorsed 375 100 100 100 100
Table 10 Each entry in the top two rows represents the number of solutions in the class V0 thatendorse each model There are 4 solutions in this class so the percentages are worked out by dividingthe total of each column by 8 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Vlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 2 4 5 5
lsquoDissipationrsquoEndorsements
1 1 3 2 5
Percentage Endorsed 136 136 318 318 455
Table 11 Each entry in the top two rows represents the number of solutions in the class Vlowast thatendorse each model There are 11 solutions in this class so the percentages are worked out bydividing the total of each column by 22 and taking results to 1 decimal place if necessary
All of the flows in the class V0 endorse the Vreman QR Gradient and Vortex-Stretching-basedsubfilter-scale models (see Table 6) This class completely endorses these four models equally anddoes not endorse the Smagorinsky model The Vortex-Stretching-based eddy-viscosity model hasthe most endorsements from the class Vlowast and the Smagorinksy and Vreman models again have thefewest endorsements (see Table 7) As in Section 441 we say that this class mostly endorses the
32
Vortex-Stretching-based eddy-viscosity model but once again the percentage endorsement is ratherlow
Now let us consider LES with explicit filtering Tables 12 and 13 are analogues of Table 5 andillustrate the results for each flow class The analogue of Table 4 is Table 27 and is in Appendix C2
Endorsements for models using solutions in V0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 4 4 4 4
lsquoWeakrsquo Endorsements 0 0 0 0 0
Table 12 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Vlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 1 1 1 1
lsquoWeakrsquo Endorsements 10 10 9 6
Table 13 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We can see in Table 12 that the results for the lsquostrongrsquo endorsements are similar to those in Table 5excluding Sol 15 (Rankine Vortex) The four solutions in V0 with zero true subfilter dissipationnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)and Sol 14 (Radially Dependent) are hence the four that endorse the Vreman QR Gradientand Vortex-Stretching-based models in Table 12 Sol 15 (Rankine Vortex) is in the class Vlowast andhence there is one lsquostrongrsquo endorsement in Table 13 The class Vlowast contains all of the flows with anon-zero true subfilter dissipation (given in Eq (67)) and hence Table 13 contains all of the lsquoweakrsquoendorsements
443 Classes using the Third Principal Invariant
Sorting the classes using the invariants is a non-trivial task in itself There are a number of waysin which to do this using both the principal invariants in Eq (40) and the combined invariants inEq (43)
We start by looking at flows with third principal invariant RG equal to zero We again define twoclasses R0 and Rlowast where R0 consists of all flows with RG = 0 and Rlowast consists of all flows withRG 6= 0 Then using the information in Appendix A we see that there are nine flows in R0 That
33
is
R0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 8 (Vortex in Stagnation Flow)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)
(80)
Rlowast = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 13 (Sphere Translation)(81)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 14 and 15 are analogues of Table 3 and they illustrate the results for LES withoutexplicit filtering for each flow class The analogues of Tables 1 and 2 are Tables 28 and 29 and arein Appendix C3
Endorsements for models using solutions in R0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 7 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 278 556 778 722 778
Table 14 Each entry in the top two rows represents the number of solutions in the class R0 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking the result to one decimal place if necessary
Endorsements for models using solutions in Rlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 2 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 83 83 83 167 333
Table 15 Each entry in the top two rows represents the number of solutions in the class Rlowast thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking the result to one decimal place if necessary
The class R0 endorses both the Vortex-Stretching-based model and the QR model the most andSmagorinsky the least in the case for LES without explicit filtering In the Rlowast class the Vortex-Stretching-based model is again the most endorsed followed by the Gradient model however thepercentages are again small
34
Now let us consider LES with explicit filtering Tables 16 and 17 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 30 and is in Appendix C3
Endorsements for models using solutions in R0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 4 4 3 2
Table 16 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Rlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 6 6 6 4
Table 17 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 we can see in Table 16 that the results for the lsquostrongrsquo endorsements are exactly thesame as in Table 5 This is because all of the flows with zero subfilter dissipation are contained in theclass R0 As in Table 9 there are no lsquostrongrsquo endorsements in Table 17 The lsquoweakrsquo endorsementshowever are split more evenly between the two classes now than in Vremanrsquos classes in Sections 441and 442 We find that the Smagorinsky and Vreman models still seem to be the most endorsed inboth classes
444 Two-component Flow Classes
Another way in which the flows can be grouped into classes is by determining whether a flow is atwo-component flow or not We define a two-component flow to have the property that the combinedprincipal invariants of a flow in Eq (43) satisfy
I3 = I4 = I5 minus1
2I1I2 = 0 (82)
We are also interested in considering flows with zero gradient model subgrid dissipation These flowshave the property that
I3 minus I4 = 0 (83)
What we find though is that for all the solutions in Section 22 if a flow is a two-component flowthen it also satisfies I3 minus I4 = 0 It may be interesting to note that although I3 minus I4 6= 0 for manysolutions there are a number with I3 equal to a multiple of I4 Although not considered in thisreport this may also be a class of interest We find that the two-component flows are
C2 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)(84)
35
and we call the remaining flows three-component flows They are
C3 = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex) Sol 11 (Pseudo-Motion Shots)
Sol 13 (Sphere Translation)
(85)
Note that Sol 8 (Vortex in Stagnation Flow) is defined to be a planar flow this does not mean that itis a two-component flow in this sense however because its two non-zero velocity components dependon three coordinates We now check the models for LES without explicit filtering by consideringendorsements within each class Tables 18 and 19 are analogues of Table 3 and they illustrate theresults for LES without explicit filtering for each flow class The analogues of Tables 1 and 2 areTables 31 and 32 and are in Appendix C4
Endorsements for models using solutions in C2 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 6 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 357 714 100 857 100
Table 18 Each entry in the top two rows represents the number of solutions in the class C2 thatendorse each model There are 7 solutions in this class so the percentages are worked out by dividingthe total of each column by 14 and taking the result to one decimal place if necessary
Endorsements for models using solutions in C3 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 3 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 63 63 63 188 250
Table 19 Each entry in the top two rows represents the number of solutions in the class C3 thatendorse each model There are 8 solutions in this class so the percentages are worked out by dividingthe total of each column by 16 and taking the result to one decimal place if necessary
The class C2 equally endorses the QR and Vortex-Stretching-based models and once again theSmagorinsky model is the least endorsed As with the previous classes the class C3 has such lowpercentages that it does not seem to endorse any models We start to see a trend appearing here inthat the flow classes that contain the more complicated flows namely Q0minus6 (Eq (76)) Vlowast (Eq (79))Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages The classes that tendto contain the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78)) R0 (Eq (80)) and C2 (Eq (84))all seem to favour the Vortex-Stretching-based model or the QR model
36
Now let us consider LES with explicit filtering Tables 20 and 21 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 33 and is in Appendix C4
Endorsements for models using solutions in C2 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 2 2 0 1 0
Table 20 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in C3 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 8 8 8 6
Table 21 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 and Table 16 we can see in Table 20 that the results for the lsquostrongrsquo endorsements areexactly the same as in Table 5 This is again because all of the flows with zero subfilter dissipationare contained in the class C2 There are two flows with non-zero true subfilter dissipation (givenin Eq (67)) in C2 and this shows in the lsquoweakrsquo endorsements in Table 20 There are no lsquostrongrsquoendorsements in Table 21 as all the zero subfilter dissipation flows are in C2
5 Conclusions
In this report we studied incompressible turbulent flows We found a number of exact solutions to theNavier-Stokes equations each representing a relatively simplistic fluid flow Large-eddy simulation(LES) is used to model turbulent flows by smoothing over the complex small-scale motion Weintroduced a number of subfilter-scale models to approximate this small-scale motion and testedthem using the found exact solutions to the Navier-Stokes equations We first considered LESwithout explicit filtering and used the Navier-Stokes equations to derive conditions for the subfilter-scale models namely the Velocity Field condition (Eq (58)) the Kinetic Energy condition (Eq (60))and the Dissipation condition (Eq (62)) Similarly we considered LES with explicit filtering andagain used the Navier-Stokes equations to derive more conditions for the models in this case wedecided to just take one condition the Dissipation condition (Eqs (67) and (68)) as the others leadto difficulty in their evaluation By lsquorunningrsquo the exact solutions to the Navier-Stokes equationsthrough each condition for the subfilter-scale models we found that the Vortex-Stretching-basededdy-viscosity model in Eq (54) was most endorsed by the tests in LES without explicit filteringWe found that four out of the five subfilter-scale models namely the Vreman QR Gradient andVortex-Stretching-based models were equally endorsed by the lsquostrongrsquo tests in LES with explicitfiltering with the Smagorinsky model being the least endorsed The results we found for the lsquoweakrsquocondition were completely opposite to those found in both the lsquostrongrsquo condition and in LES withoutexplicit filtering This led us to conclude that this test was not one in which we should put our trust
37
We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses
In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes
This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold
38
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
u1(x y z t) =1
Re
2xminus C1Rey
x2 + y2minusA1y
(x2 + y2
)minusB1y
u2(x y z t) =1
Re
2y + C1Rex
x2 + y2+A1x
(x2 + y2
)+B1x
u3(x y z t) =1
2A2
(x2 + y2
)log(x2 + y2
)+B2
(x2 + y2
)+ C2
(32)
Figure 13 An illustration of the flow in thexy-plane for Sol 11 (Pseudo-Motion Shots)described in Eqs (31) and (32) with parametervalues A1 = B1 = C1 = 1 A2 = B2 = C2 = 2and Re = 10
Figure 14 An illustration of the flow in theyz-plane for Sol 11 (Pseudo-Motion Shots)described in Eqs (31) and (32) with parametervalues A1 = B1 = C1 = 1 A2 = B2 = C2 = 2and Re = 10
Solution 12 Flows With Constant Whirl
In the book by Berker [3 p 143] another solution is given in cylindrical coordinates by
ur(r ϕ z t) = minus 1
Re
1
r
uϕ(r ϕ z t) =1
rradicte(minus
12 g(t))
uz(r ϕ z t) = 0
p(r ϕ z t) =1
4
2(minus e
minusg(t)
t minus 1Re2
)r2
minus 1
t2Re
infinintminusg(t)
eminust
tdt
(33)
where we define g(t) =r2Re
2t When converted into Cartesian coordinates we find
16
u1(x y z t) = minus 1
x2 + y2
(yradict
exp
(minus
Re(x2 + y2
)4t
)+
x
Re
)
u2(x y z t) =1
x2 + y2
(xradict
exp
(minus
Re(x2 + y2
)4t
)minus y
Re
)
u3(x y z t) = 0
(34)
Figure 15 An illustration of the flow for Sol12 (Flows with Constant Whirl) described inEqs (33) and (34) with time t = 1 and Re = 1 Aslice of the xy-plane at z = 0 is shown as velocityin the z-direction is zero this is the same for anyvalue of z
17
Solution 13 Rectilinear and Uniform Translation of a Sphere
In the book by Berker [3 p 228] another solution is given in Cartesian coordinates by
u1(x y z t) =3
4
axz
R3
(a2
R2minus 1
)
u2(x y z t) =3
4
ayz
R3
(a2
R2minus 1
)
u3(x y z t) = 1minus 3
4
a
Rminus 1
4
a3
R3minus 3
4
az2
R3
(a2
R2minus 1
)
p(x y z t) =3a(aminusR)(a+R)
32R10Re
((R2Re
(x2 + y2
)minus 2R2Rez2
) (a3 + 3aR2 minus 4R3
)+ 3aRez4
(R2 minus a2
)minus 16R5z
)
(35)
where a and R are constants
Figure 16 An illustration of the flow in the yz-plane for Sol 13 (Sphere Translation) describedin Eq (35) with parameter values a = minus2 andRe = 3
Solution 14 Radially Dependent Flow
A solution with velocity in the z-direction dependent only on the distance from the z-axis is as
18
follows in cylindrical coordinates
ur(r ϕ z t) = 0
uϕ(r ϕ z t) = 0
uz(r ϕ z t) = A(r2 minus a2
)
p(r ϕ z t) =4Az
Re
(36)
where A and a are constants In Cartesian coordinates the solution is just the same with u1(x y z t)and u2(x y z t) both zero and the r2 term replaced with x2 + y2 in u3(x y z t)
Figure 17 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus1 1] withparameter values a = 1 A = 4 and Re = 10
Figure 18 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus2 2] withparameter values a = 1 A = 4 and Re = 10
Solution 15 Rankine Vortex
A solution discovered by Rankine [1] is a vortex flow of the following form
ur(r ϕ z t) = 0
uϕ(r ϕ z t) =ar
2πR2
uz(r ϕ z t) = 0
p(r ϕ z t) =a2r2
8π2R4
(37)
19
where a and R are constants In Cartesian coordinates the flow has the form
u1(x y z t) = minus ay
2πR2
u2(x y z t) =ax
2πR2
u3(x y z t) = 0
p(x y z t) =a2(x2 + y2
)8π2R4
(38)
This flow is actually a rigid body rotational flow in that the angular rotational velocity is uniformand the velocity of the flow increases proportionally with distance from the z-axis
Figure 19 An illustration of the flow in the xy-plane for Sol 15 (Rankine Vortex) described inEq (37) and Eq (38) with parameter values a = 1and R = 1
radic2π
23 Characterisation of Flows
It is important to be able to classify flows in a way that sorts them into distinct groups This aidsin the clear identification and general study of flows It is also a very important way of recognisingflows with similar properties In this subsection we introduce two distinct ways in which flows canbe characterised
231 Characterising Solutions using Invariants
A principal invariant of a tensor is a coefficient of the characteristic polynomial of that tensorInvariants as their name suggests are invariant under rotations of the coordinate system in thattheir values do not change when the tensor is lsquorotatedrsquo into a different set of coordinates Thevelocity gradient of a flow G is defined as the Jacobian of the velocity vector given by
Gij =
[partuipartxj
] (39)
20
for i j = 1 2 3 There are three principal invariants of the velocity gradient of a fluid flow G givenby
PG = tr(G)
QG =1
2
((tr(G))2 minus tr
(G2))
RG = det(G)
(40)
Incompressibility of flows (Eq (5)) ensures that PG = 0 hence the first principal invariant is zerofor all of the flows in Section 22 We now define the rate-of-strain tensor S by
Sij =1
2
(partuipartxj
+partujpartxi
) (41)
and the rate-of-rotation tensor Ω by
Ωij =1
2
(partuipartxjminus partujpartxi
) (42)
Note that the first principal invariant of S and the first and third principal invariants of Ω are zeroAlso note that the velocity gradient of a flow can simply be written G = S + Ω
Rather than considering the principal invariants we can also consider the combined invariants
I1 = tr(S2) I2 = tr(Ω2) I3 = tr(S3) I4 = tr(SΩ2) I5 = tr(S2Ω2) (43)
We can hence write the three principal invariants for a velocity field G in terms of the combinedinvariants
PG = 0
QG = minus1
2(I1 + I2)
RG =1
3(I3 + 3I4)
(44)
In this report the flows considered are the exact solutions to the incompressible Navier-Stokesequations described in Section 22 The combined invariants in Eq (43) are used to characterisethese solutions into different classes in Section 44 Both the principal invariants and combinedinvariants for each exact solution in Section 22 are given in Appendix A
Each of the subfilter-scale eddy-viscosity models described in Section 32 are written in terms ofthese five combined invariants
232 Vremanrsquos Characterisation of Flows
Vreman [16] classifies flows using the gradient of the velocity field of a flow Each lsquoclassrsquo of flows hasthe same number of non-zero elements in the gradient matrix For example
[partuipartxj
]=
0 0 0lowast 0 0lowast 0 0
(45)
where a lowast denotes a non-zero term represents a group of flows that lie within the class denoted byQ7 (see Appendix B) where the subscript represents the number of non-zero terms in the velocity
21
gradient Flows of the form in Eq (45) have a velocity field of the form
u1 = c1 (constant)
u2 = f2(x1)
u3 = f3(x1)
(46)
where f2(x1) and f3(x1) are such that the Navier Stokes equations Eq (1) are satisfied Notethat as the elements on the leading diagonal are zero the incompressibility condition Eq (5) isautomatically satisfied Some simple flow classes are given in Appendix B and contain a number ofthe solutions discussed in Section 22 This is mentioned in greater detail in Section 441
3 Conditions for Large-Eddy Simulation Models
31 Large-Eddy Simulation
Due to the nature of numerical simulation and the complexity of turbulent flows small details aremissed and information is lost when solving the Navier-Stokes equations The computational costof solving a turbulent flow is extremely high so a filtering operation is applied to the Navier-Stokesequations represented by a bar to reduce the computational cost of the simulation This is a linearoperation and preserves constants ie ρ = ρ and is also assumed to commute with differentiation
partuipartxj
=partuipartxj
(47)
The filtered Navier-Stokes equations are found by filtering (putting a bar over) the entire equations(see Eq (1)) and after evaluating are given by
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (48)
where
τij = uiuj minus uiuj (49)
is called the lsquotrue stressrsquo of the flow This however poses a problem as it is not clear what is meantby the term uiuj for a general filtering operation The true stress τij therefore must be replaced bya model τmod
ij in practical applications
32 Subfilter-Scale Models
We will now define the subflter-scale models denoted by τmodij to be used in place of the lsquodifficultrsquo
true stress τij There are a number of models available to use in the literature [11 p7] These willbe tested using the exact solutions to the Navier-Stokes equations and by checking conditions whichare defined in Sections 33 and 34
All the models barring one are of the same form That is each of the eddy-viscosity models arewritten
τmodij minus 1
3τmodkk δij = minus2νeSij (50)
with νe representing the eddy viscosity of a specific model For simplicity we usually just writeτmodij = minus2νeSij the extra term is included here to emphasise that the models are traceless Each
22
of the models we consider here can be expressed in terms of the rate of strain tensor S defined inEq (41) the rate of rotation tensor Ω defined in Eq (42) and the combined invariants defined inEq (43) The models are then defined as follows
Smagorinsky [12] νSe = (CSδ)2radic
2I1 (51)
Vreman [16] νVe = (CV δ)2
radicQGGT
PGGT (52)
QR [13 14 15] νQRe = (CQRδ)2 max0minusI3
I1 (53)
Vortex-Stretching [11] νV Se = (CV Sδ)2radic
2I1
(I5 minus 1
2I1I2
minusI1I2
)32
(54)
with the quantities
PGGT = I1 minus I2 QGGT =1
4(I1 + I2)2 + 4
(I5 minus
1
2I1I2
) (55)
There is a fifth and final model to be considered here It is of a different form to the other four andnot an eddy-viscosity model It is known as the Gradient model and is defined by
Gradient [4 9] τmodG = CGδ
(S2 minus Ω2 minus (SΩminus ΩS)
) (56)
The values C are model constants specific to each model and δ represents the mesh size of the LESBoth of these quantities are included here to properly define the models however in this projectthey will be ignored as they are constants and have no influence on the conditions to be defined
33 Conditions for Large-Eddy Simulation without Explicit Filtering
The filtering operation introduced in Eq (48) is just a formality for LES without explicit filteringThe idea here is that filtering does not actually take place but the term lsquoaddedrsquo to the Navier-Stokesequations in fact lsquosubtractsrsquo the difficult to simulate small-scale turbulent motion of the flow Inorder to consider this properly the terms ui and p in Eq (48) are replaced by vi and q It is hopedthat τmod
ij approximates τij so that vi and q approximate ui and p respectively So LES withoutexplicit filtering is given by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (57)
It is clear to see that the first four terms of this equation are exactly the same as the Navier-Stokesequations so for the exact solutions discussed in Section 22 the combination of these terms willequal zero What is then desirable is that the model term is also zero for these exact solutions sothat Eq (57) still holds In order to test models for exact solutions the condition
part
partxjτmodij (v) = 0 (58)
is checked If this condition were to hold it would imply that the model does not influence thevelocity field of a flow hence we call it the Velocity Field Condition There are a number of otherrestrictions that can be tested One such test comes from the differential form of the evolution ofthe kinetic energy of a flow given in Eq (10) In order to find this Eq (57) is multiplied by vi andterms are grouped to find
part
partt
(1
2vivi
)+ vj
part
partxj
(1
2vivi
)= minusvi
1
ρ
partq
partxi+ νvi
part2vipartxjpartxj
minus vipart
partxjτmodij (v) (59)
23
Then similarly to Eq (58) another condition is to check whether the model affects the kineticenergy of exact solutions this is equivalent to checking whether
vipart
partxjτmodij (v) = 0 (60)
We call this condition the Kinetic Energy Condition The third and final condition considered herecan also be found using the integral form for the equation for kinetic energy
d
dt
intV
1
2vividV =
intS
[minusvj
(1
2vivi
)minus vj
q
ρ+ 2νviSij + viτ
modij (v)
]njdS
+
intV
[minus2νSijSij minus
(minusSijτmod
ij (v))]dV
(61)
where the rate-of-strain tensor Sij was defined in Eq (41) V is a volume of fluid and dS meansthe integral is over the surface of that volume with unit normal nj Then this final condition checkswhether the model affects the total kinetic energy of exact solutions to the Navier-Stokes equationsthat is
Sijτmodij (v) = 0 (62)
We will call this condition the Dissipation Condition because Sijτmodij (v) is often referred to as the
subgrid dissipation
34 Conditions for Large-Eddy Simulation with Explicit Filtering
As in Eq (48) the Navier-Stokes equations can be filtered by applying the filtering operation to thewhole equation (putting a bar over all terms) and evaluating to find again that
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (63)
In Eq (57) the filtering operation was ignored and the elements ui p and τij(u) were replacedwith vi q and τmod
ij (v) respectively Now in LES with explicit filtering the filtering operation itselfsmooths over the small-scale turbulent motion of the simulated flow In this case the elements uip and τij(u) will be replaced with vi q and τmod
ij (v) respectively LES with explicit filtering is thengiven by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (64)
Note that Eqs (63) and (64) despite their similarity to Eq (57) no longer contain the Navier-Stokesequations as Eq (57) did The bar (filtering operation) over each term results in each exact solutionfound in Section 22 not necessarily satisfying the first four terms It is hence unknown whether ornot these terms will vanish for exact solutions Terms can still be compared similarly to Eqs (58)(60) and (62) However instead of comparing to zero we can now compare to the relevant termfrom the filtered Navier-Stokes equations So the terms
part
partxjτij (65)
and
part
partxjτmodij (v) (66)
24
are to be compared Although the latter term is relatively simple to compute the former could leadto some difficulties due to the differentiation of the true stress τij = uiuj minus uiuj So although someinformation could be gleaned from the inspection of these two terms more information can be foundby considering the analogue of Eq (62) So the true subfilter dissipation
Sijτij (67)
(where Sij is shorthand for Sij(v)) and the modelled subfilter dissipation
Sij(v)τmodij (v) (68)
are considered and have to be equal for the condition to hold These can again be derived from theintegral equation for kinetic energy of a filtered velocity field similarly to Eq (61)
4 Analysis of Large-Eddy Simulation Models
In this section we use a code in Mathematica the theory described in Section 3 and the exactsolutions to the Navier-Stokes equations found in Section 22 to check whether or not each of theexact solutions lsquoendorsesrsquo each model for LES both without and with explicit filtering
41 Large-Eddy Simulation Without Explicit Filtering
We start by using a code in Mathematica to test the Velocity Field condition in Eq (58) and theKinetic Energy condition in Eq (60) For all exact solutions to the Navier-Stokes equations inSection 22 the outputs for both conditions were identical so we have combined them into one table
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 ndash ndash 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 1 For the Kinetic Energy condition a lsquo0rsquo result means that vipartpartxj
τmodij (v) = 0 for respective
models and exact solutions whereas a lsquolowastrsquo result means that vipartpartxj
τmodij (v) 6= 0 For the Velocity
Field condition a lsquo0rsquo result means that partpartxj
τmodij (v) = 0 for respective models and exact solutions
whereas a lsquolowastrsquo result means that partpartxj
τmodij (v) 6= 0 The Gradient model can take both positive and
negative values so when the sign of the model is known it is included in the table instead of a lsquorsquoBoth the conditions are included in one table because for all cases the outputs were identical
We then test the Dissipation condition in Eq (62) for all exact solutions to the Navier-Stokesequations in Section 22
25
Dissipation Condition [Does Sijτmodij (v) = 0]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 + 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 2 A lsquo0rsquo result means that Sijτmodij (v) = 0 for respective models and exact solutions whereas
a lsquolowastrsquo result means that Sijτmodij (v) 6= 0 The Gradient model can take both positive and negative
values so when the sign of the model is known it is included in the table instead of a lsquorsquo
We then sum all the results to find the total number of endorsements for each model described inSection 32
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
4 6 8 9 9
lsquoDissipationrsquoEndorsements
2 5 7 7 9
Total Endorsements 6 11 15 16 18
Table 3 The results from Tables 1 and 2 are summarised to give a general overview of how manyendorsements each model receives This makes it much clearer to see the models most supported bythe respective and combined conditions with the Vortex-Stretching model being the most supportedand the Smagorinsky model the least
Testing the models in Section 32 using LES without explicit filtering has mixed results It isclear that the Smagorinksy model is not well endorsed by the flows in Section 22 with the onlyendorsements coming from the simplest flows that is Sol 1 (Linear Velocity) Sol 3 (Velocity as afunction of time) Sol 7 (Stagnation Flow) and Sol 15 (Rankine Vortex) The Vortex-Stretching-based model is the most endorsed flow with eighteen out of a possible thirty endorsements Thisagrees with our foreknowledge as this model was constructed to be as physically consistent as possiblein the paper by Silvis amp Verstappen [11]
42 Large-Eddy Simulation With Explicit Filtering
In this section we will compare the true subfilter dissipation Sijτij and the modelled subfilterdissipation Sij(v)τmod
ij (v) given in Eqs (67) and (68) respectively for each model τmodij and each
exact solution to the Navier-Stokes equation given in Section 22 We will not consider the conditionsinvolving differentiation in Eqs (65) and (66) due to the difficulty in calculating the derivative ofthe τij term in Eq (65)
26
421 Example of a Test for a Model
The true subfilter dissipation Sijτij of a flow given in Eq (67) cannot be explicitly computed asthe filtering operation is kept general What can be done however is to partially evaluate theexpressions using the attributes of a filtering operation mentioned in Eq (48)
The calculations of Sijτij for Sol 1 (Linear Velocity) are provided to illustrate the general ideaRecall that the velocity field of Sol 1 (Linear Velocity) is written in Eq (13) in vector notation as
u(x y z t) =
0ax+ bcx+ d
(69)
Then Sij = Sij(u) is computed simply by using its definition in Eq (41) Note that as filteringpreserves constants if there is an a it is replaced with a meaning that (in this case) there are alarge number of zero terms due to the frequency of 0 Hence
[Sij(u)
]=
[1
2
(partuipartxj
+partujpartxi
)]=
0 a
2c2
a2 0 0
c2 0 0
=
0 a
2c2
a2 0 0
c2 0 0
(70)
Note that as Sijτij is being computed which is equal to the sum of the element-wise multiplicationof the two matrices (or the trace of the matrix multiplication) that we donrsquot need to consider theelements of τij(u) with corresponding zero entry in Sij As τij(u) = uiuj minus uiuj and again asfiltering preserves constants it is found that
[τij(u)] =[uiuj minus uiuj
]=
lowast 0 middot ax+ b 0 middot cx+ d
ax+ b middot 0 lowast lowastcx+ d middot 0 lowast lowast
=
lowast 0 00 lowast lowast0 lowast lowast
(71)
where a lsquolowastrsquo just represents some element (not necessarily non-zero) Therefore the true subfilterdissipation
Sijτij = tr
0 a2
c2
a2 0 0c2 0 0
middotlowast 0 0
0 lowast lowast0 lowast lowast
= 0 (72)
Now the modelled subfilter dissipation Sij(u)τmodij (u) needs to be computed (recall that Sij =
Sij(u)) In this case let us use the QR model stated in Eq (53) Once again Sij(u) calculated inEq (70) needs to be used So we have that
τmodij = minus2(CQRδ)
2 max0minusI3I1
Sij (73)
As
I1 = tr(S2) =
1
2(a2 + c2) and I3 = tr(S
3) = 0 (74)
it holds that τmodij (u) = 0 so Sij(u)τmod
ij (u) = 0 This agrees with Eq (72) and hence Sol 1(Linear Velocity) endorses the QR model
27
422 Testing the Models
Now using a Mathematica code the exact solutions in Section 22 are again used to evaluate thecondition for the models using the process just described in Section 421 The results are describedin Table 4
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 4 All entries with a ldquo0rdquo represent a value of zero and a ldquolowastrdquo represents a value that is non-zeroThe ldquoExactrdquo row represents the value of the true subfilter dissipation Sijτij Every other row witha model name represents the value of the modelled subfilter dissipation Sij(u)τmod
ij (u) respectiveto that model A ldquordquo means that it is unclear as to the value - the sign of these values is oftenunknown and the lsquoMaxrsquo function in the definition of the QR model (in Eq (53)) makes it difficultto evaluate
Reading from Table 4 the Smagorinsky model is clearly endorsed by Sol 3 (Velocity as a function oftime) and Sol 15 (Rankine Vortex) as in both cases both Sijτij = 0 = Sij(u)τmod
ij (u) In this casewe say that Sol 3 (Velocity as a function of time) and Sol 15 (Rankine Vortex) lsquostronglyrsquo endorsethe Smagorinsky model The Smagorinsky model is not endorsed by Sol 1 (Linear Velocity) Sol 2(Planar Flow) or Sol 14 (Radially Dependent) however because Sij(u)τmod
ij (u) 6= 0 For Sol 4 to
13 both Sijτij 6= 0 6= Sij(u)τmodij (u) and so we will say that these exact solutions lsquoweaklyrsquo endorse
the Smagorinsky model
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 10 10 9 6
Table 5 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Interestingly the lsquostrongrsquo and lsquoweakrsquo endorsements give different results However more stock shouldbe put in the results of the lsquostrongerrsquo condition as it is completely clear that zero is equal to zeroIt is not necessarily clear whether or not non-zero equals non-zero as the values will depend on theparameters in each solution hence the lsquoweakerrsquo endorsement
The lsquostrongrsquo condition equally endorses the Vreman QR Gradient and Vortex-Stretching-basedmodels whereas the lsquoweakrsquo condition endorses Vreman and Smagorinsky the most closely followedby the Gradient model with the Vortex-Stretching-based model being the most opposed As this
28
completely contradicts the lsquostrongrsquo condition and the results in Section 41 it makes us question thevalidity of the weaker test The number of endorsements for the QR model is unknown because of thedefinition of the model in Eq (53) and our evaluation of the true subfilter dissipation Eq (67) Dueto the exact solutions we have found it is never clear whether or not the true subfilter dissipationis strictly positive or negative This means that when the true subfilter dissipation is found to benon-zero the lsquoMaxrsquo function in the QR model cannot be evaluated
43 Vremanrsquos True Subfilter Dissipation Claim
Using the theory mentioned in Sections 232 and 34 we can place the exact solutions to the Navier-Stokes equations into different classes defined by Vreman In doing this we can also attempt to verifyone of Vremanrsquos results [16] In Vremanrsquos paper he defines the class Qn to be the set of flows withn zero entries in the velocity gradient of the flow (the velocity gradient is defined in Eq (39))Vreman then claims that the only flows with zero true subfilter dissipation are a certain selection ofthose in the classesQ7Q8 orQ9 shown in appendix B and is generally non-zero for any other flows
The true subfilter dissipation of Sol 1 (Linear Velocity) was computed in Section 421 and wasfound to be zero in Eq (72) The velocity gradient of Sol 1 is as follows
GLV =
0 0 0a 0 0c 0 0
(75)
and hence Sol 1 (Linear Velocity) is in the required subset of Q7 which agrees with Vremanrsquos claimIn Section 422 the true subfilter dissipation was computed (determined to be zero or non-zero) forall of the exact solutions to the Navier Stokes equations in Section 22 These solutions can now belsquoplacedrsquo into each class Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a functionof time) and Sol 14 (Radially Dependent) were all found to have zero true subfilter dissipation andare all in the appropriate subsets of either Q7Q8 or Q9 This agrees with Vremanrsquos results
We can extend Vremanrsquos results however We found that Sol 15 (Rankine Vortex) had zero truesubfilter dissipation and yet is not in the subset of Q7 defined in Appendix B It is important tonote that the rate of strain tensor Sij in Eq (41) for Sol 15 (Rankine Vortex) is zero This is thereason the true subfilter dissipation is zero and the same reason that all models are zero for thisflow as all models are based on Sij
We also found that all of the remaining exact solutions to the Navier-Stokes equations are not inVremanrsquos specified classes (in Appendix Section B) They also have a non-zero subfilter dissipationwhich agrees with Vremanrsquos results in his paper [16]
44 Classes of Flows
It is important to consider that some exact solutions to the Navier-Stokes equations may have similarproperties For example Sol 1 (Linear Velocity) and Sol 2 (Planar Flow) are both very similarIn fact Sol 2 (Planar Flow) is a more general case of Sol 1 (Linear Velocity) We therefore mustgroup the different flows into classes in an attempt to reduce the bias that may occur when solutionsare similar This should aid us to further determine which of the models are most endorsed We willdo this in a number of different ways using the theory discussed in Section 23 We will group theflows with respect to Vremanrsquos Flow Classes given in appendix B and with respect to the Invariantsgiven in appendix A
29
441 Vremanrsquos Velocity Gradient Flow Classes
We now put our exact solutions to the Navier-Stokes equations into one of two classes Either Q0minus6or Q7minus9 where each class is defined intuitively as follows
Q0minus6 = Q0 cupQ1 cup middot middot middot cup Q6 (76)
Q7minus9 = Q7 cupQ8 cupQ9 (77)
where Qn is defined in Section 43 By looking at the gradient of each exact solution we can see thatQ7minus9 contains the six solutions Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as afunction of time) Sol 5 (2D Taylor) Sol 14 (Radially Dependent) and Sol 15 (Rankine Vortex)and that Q0minus6 contains the remaining nine We now check the models for LES without explicitfiltering by considering endorsements within each class Tables 6 and 7 are analogues of Table 3 andillustrate the results for each flow class The analogues of Tables 1 and 2 are Tables 22 and 23 andare in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 6 6 6
lsquoDissipationrsquoEndorsements
2 5 6 6 6
Percentage Endorsed 417 833 100 100 100
Table 6 Each entry in the top two rows represents the number of solutions in the class Q7minus9 thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Q0minus6 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 2 3 3
lsquoDissipationrsquoEndorsements
0 0 1 0 3
Percentage Endorsed 56 56 167 167 333
Table 7 Each entry in the top two rows represents the number of solutions in the class Q1minus6 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking results to 1 decimal place if necessary
All of the flows in the class Q7minus9 endorse the QR Gradient and Vortex-Stretching-based subfilter-scale models (see Table 6) We can hence see that this class fully endorses these three models equallyand does not endorse the Smagorinsky model The Vortex-Stretching eddy viscosity model has themost endorsements from any of the flows in the class Q0minus6 and the Smagorinksy and Vreman modelshave the fewest endorsements (see Table 7) We can hence say that this class mostly endorses theVortex-Stretching-based eddy-viscosity model note however that the percentage endorsement is very
30
low
Now let us consider LES with explicit filtering Tables 8 and 9 are analogues of Table 5 and illustratethe results for each flow class The analogue of Table 4 is Table 24 and is in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 1 1 0 0 0
Table 8 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Q0minus6 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 9 9 9 6
Table 9 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We have grouped Vremanrsquos classes into Q7minus9 and Q0minus6 this is equivalent to saying that we havegrouped Vremanrsquos classes into those with two or fewer non-zero velocity gradient elements and thosewith three or more non-zero velocity gradient elements respectively Therefore the solutions in Q7minus9are exactly those with a lsquo0rsquo in the lsquoExactrsquo column of Table 4 and Sol 5 (2D Taylor) ExcludingSol 5 (2D Taylor) these are the same as those five that endorse the Vreman QR Gradient andVortex-Stretching-based models in Table 5 We can see in Table 8 that the results for the lsquostrongrsquoendorsements are exactly the same as in Table 5 This is because all of the flows with zero subfilterdissipation are contained in the class Q7minus9 There are hence no lsquostrongrsquo endorsements in Table 9The majority of the lsquoweakrsquo endorsements come from Q0minus6 as this class contains all but one of thesolutions with non-zero true subfilter dissipation (given in Eq (67))
442 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
We now separate our exact solutions into classes again using Vremanrsquos flow classes This timehowever we define V0 to be the class of flows Vreman claims will always have zero true subfilterdissipation namely those classes mentioned in Appendix B We define Vlowast to be all the remaining
31
classes Then
V0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow)
Sol 3 (Velocity as a function of time) Sol 14 (Radially Dependent)(78)
Vlowast = Sol 4 (Axisymmetric Flow) Sol 5 (2D Taylor) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 13 (Sphere Translation) Sol 15 (Rankine Vortex)
(79)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 10 and 11 are analogues of Table 3 and illustrate the results for each flow class Theanalogues of Tables 1 and 2 are Tables 25 and 26 and are in Appendix C2
Endorsements for models using solutions in V0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 4 4 4 4
lsquoDissipationrsquoEndorsements
1 4 4 4 4
Percentage Endorsed 375 100 100 100 100
Table 10 Each entry in the top two rows represents the number of solutions in the class V0 thatendorse each model There are 4 solutions in this class so the percentages are worked out by dividingthe total of each column by 8 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Vlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 2 4 5 5
lsquoDissipationrsquoEndorsements
1 1 3 2 5
Percentage Endorsed 136 136 318 318 455
Table 11 Each entry in the top two rows represents the number of solutions in the class Vlowast thatendorse each model There are 11 solutions in this class so the percentages are worked out bydividing the total of each column by 22 and taking results to 1 decimal place if necessary
All of the flows in the class V0 endorse the Vreman QR Gradient and Vortex-Stretching-basedsubfilter-scale models (see Table 6) This class completely endorses these four models equally anddoes not endorse the Smagorinsky model The Vortex-Stretching-based eddy-viscosity model hasthe most endorsements from the class Vlowast and the Smagorinksy and Vreman models again have thefewest endorsements (see Table 7) As in Section 441 we say that this class mostly endorses the
32
Vortex-Stretching-based eddy-viscosity model but once again the percentage endorsement is ratherlow
Now let us consider LES with explicit filtering Tables 12 and 13 are analogues of Table 5 andillustrate the results for each flow class The analogue of Table 4 is Table 27 and is in Appendix C2
Endorsements for models using solutions in V0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 4 4 4 4
lsquoWeakrsquo Endorsements 0 0 0 0 0
Table 12 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Vlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 1 1 1 1
lsquoWeakrsquo Endorsements 10 10 9 6
Table 13 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We can see in Table 12 that the results for the lsquostrongrsquo endorsements are similar to those in Table 5excluding Sol 15 (Rankine Vortex) The four solutions in V0 with zero true subfilter dissipationnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)and Sol 14 (Radially Dependent) are hence the four that endorse the Vreman QR Gradientand Vortex-Stretching-based models in Table 12 Sol 15 (Rankine Vortex) is in the class Vlowast andhence there is one lsquostrongrsquo endorsement in Table 13 The class Vlowast contains all of the flows with anon-zero true subfilter dissipation (given in Eq (67)) and hence Table 13 contains all of the lsquoweakrsquoendorsements
443 Classes using the Third Principal Invariant
Sorting the classes using the invariants is a non-trivial task in itself There are a number of waysin which to do this using both the principal invariants in Eq (40) and the combined invariants inEq (43)
We start by looking at flows with third principal invariant RG equal to zero We again define twoclasses R0 and Rlowast where R0 consists of all flows with RG = 0 and Rlowast consists of all flows withRG 6= 0 Then using the information in Appendix A we see that there are nine flows in R0 That
33
is
R0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 8 (Vortex in Stagnation Flow)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)
(80)
Rlowast = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 13 (Sphere Translation)(81)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 14 and 15 are analogues of Table 3 and they illustrate the results for LES withoutexplicit filtering for each flow class The analogues of Tables 1 and 2 are Tables 28 and 29 and arein Appendix C3
Endorsements for models using solutions in R0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 7 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 278 556 778 722 778
Table 14 Each entry in the top two rows represents the number of solutions in the class R0 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking the result to one decimal place if necessary
Endorsements for models using solutions in Rlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 2 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 83 83 83 167 333
Table 15 Each entry in the top two rows represents the number of solutions in the class Rlowast thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking the result to one decimal place if necessary
The class R0 endorses both the Vortex-Stretching-based model and the QR model the most andSmagorinsky the least in the case for LES without explicit filtering In the Rlowast class the Vortex-Stretching-based model is again the most endorsed followed by the Gradient model however thepercentages are again small
34
Now let us consider LES with explicit filtering Tables 16 and 17 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 30 and is in Appendix C3
Endorsements for models using solutions in R0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 4 4 3 2
Table 16 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Rlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 6 6 6 4
Table 17 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 we can see in Table 16 that the results for the lsquostrongrsquo endorsements are exactly thesame as in Table 5 This is because all of the flows with zero subfilter dissipation are contained in theclass R0 As in Table 9 there are no lsquostrongrsquo endorsements in Table 17 The lsquoweakrsquo endorsementshowever are split more evenly between the two classes now than in Vremanrsquos classes in Sections 441and 442 We find that the Smagorinsky and Vreman models still seem to be the most endorsed inboth classes
444 Two-component Flow Classes
Another way in which the flows can be grouped into classes is by determining whether a flow is atwo-component flow or not We define a two-component flow to have the property that the combinedprincipal invariants of a flow in Eq (43) satisfy
I3 = I4 = I5 minus1
2I1I2 = 0 (82)
We are also interested in considering flows with zero gradient model subgrid dissipation These flowshave the property that
I3 minus I4 = 0 (83)
What we find though is that for all the solutions in Section 22 if a flow is a two-component flowthen it also satisfies I3 minus I4 = 0 It may be interesting to note that although I3 minus I4 6= 0 for manysolutions there are a number with I3 equal to a multiple of I4 Although not considered in thisreport this may also be a class of interest We find that the two-component flows are
C2 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)(84)
35
and we call the remaining flows three-component flows They are
C3 = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex) Sol 11 (Pseudo-Motion Shots)
Sol 13 (Sphere Translation)
(85)
Note that Sol 8 (Vortex in Stagnation Flow) is defined to be a planar flow this does not mean that itis a two-component flow in this sense however because its two non-zero velocity components dependon three coordinates We now check the models for LES without explicit filtering by consideringendorsements within each class Tables 18 and 19 are analogues of Table 3 and they illustrate theresults for LES without explicit filtering for each flow class The analogues of Tables 1 and 2 areTables 31 and 32 and are in Appendix C4
Endorsements for models using solutions in C2 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 6 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 357 714 100 857 100
Table 18 Each entry in the top two rows represents the number of solutions in the class C2 thatendorse each model There are 7 solutions in this class so the percentages are worked out by dividingthe total of each column by 14 and taking the result to one decimal place if necessary
Endorsements for models using solutions in C3 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 3 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 63 63 63 188 250
Table 19 Each entry in the top two rows represents the number of solutions in the class C3 thatendorse each model There are 8 solutions in this class so the percentages are worked out by dividingthe total of each column by 16 and taking the result to one decimal place if necessary
The class C2 equally endorses the QR and Vortex-Stretching-based models and once again theSmagorinsky model is the least endorsed As with the previous classes the class C3 has such lowpercentages that it does not seem to endorse any models We start to see a trend appearing here inthat the flow classes that contain the more complicated flows namely Q0minus6 (Eq (76)) Vlowast (Eq (79))Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages The classes that tendto contain the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78)) R0 (Eq (80)) and C2 (Eq (84))all seem to favour the Vortex-Stretching-based model or the QR model
36
Now let us consider LES with explicit filtering Tables 20 and 21 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 33 and is in Appendix C4
Endorsements for models using solutions in C2 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 2 2 0 1 0
Table 20 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in C3 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 8 8 8 6
Table 21 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 and Table 16 we can see in Table 20 that the results for the lsquostrongrsquo endorsements areexactly the same as in Table 5 This is again because all of the flows with zero subfilter dissipationare contained in the class C2 There are two flows with non-zero true subfilter dissipation (givenin Eq (67)) in C2 and this shows in the lsquoweakrsquo endorsements in Table 20 There are no lsquostrongrsquoendorsements in Table 21 as all the zero subfilter dissipation flows are in C2
5 Conclusions
In this report we studied incompressible turbulent flows We found a number of exact solutions to theNavier-Stokes equations each representing a relatively simplistic fluid flow Large-eddy simulation(LES) is used to model turbulent flows by smoothing over the complex small-scale motion Weintroduced a number of subfilter-scale models to approximate this small-scale motion and testedthem using the found exact solutions to the Navier-Stokes equations We first considered LESwithout explicit filtering and used the Navier-Stokes equations to derive conditions for the subfilter-scale models namely the Velocity Field condition (Eq (58)) the Kinetic Energy condition (Eq (60))and the Dissipation condition (Eq (62)) Similarly we considered LES with explicit filtering andagain used the Navier-Stokes equations to derive more conditions for the models in this case wedecided to just take one condition the Dissipation condition (Eqs (67) and (68)) as the others leadto difficulty in their evaluation By lsquorunningrsquo the exact solutions to the Navier-Stokes equationsthrough each condition for the subfilter-scale models we found that the Vortex-Stretching-basededdy-viscosity model in Eq (54) was most endorsed by the tests in LES without explicit filteringWe found that four out of the five subfilter-scale models namely the Vreman QR Gradient andVortex-Stretching-based models were equally endorsed by the lsquostrongrsquo tests in LES with explicitfiltering with the Smagorinsky model being the least endorsed The results we found for the lsquoweakrsquocondition were completely opposite to those found in both the lsquostrongrsquo condition and in LES withoutexplicit filtering This led us to conclude that this test was not one in which we should put our trust
37
We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses
In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes
This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold
38
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
u1(x y z t) = minus 1
x2 + y2
(yradict
exp
(minus
Re(x2 + y2
)4t
)+
x
Re
)
u2(x y z t) =1
x2 + y2
(xradict
exp
(minus
Re(x2 + y2
)4t
)minus y
Re
)
u3(x y z t) = 0
(34)
Figure 15 An illustration of the flow for Sol12 (Flows with Constant Whirl) described inEqs (33) and (34) with time t = 1 and Re = 1 Aslice of the xy-plane at z = 0 is shown as velocityin the z-direction is zero this is the same for anyvalue of z
17
Solution 13 Rectilinear and Uniform Translation of a Sphere
In the book by Berker [3 p 228] another solution is given in Cartesian coordinates by
u1(x y z t) =3
4
axz
R3
(a2
R2minus 1
)
u2(x y z t) =3
4
ayz
R3
(a2
R2minus 1
)
u3(x y z t) = 1minus 3
4
a
Rminus 1
4
a3
R3minus 3
4
az2
R3
(a2
R2minus 1
)
p(x y z t) =3a(aminusR)(a+R)
32R10Re
((R2Re
(x2 + y2
)minus 2R2Rez2
) (a3 + 3aR2 minus 4R3
)+ 3aRez4
(R2 minus a2
)minus 16R5z
)
(35)
where a and R are constants
Figure 16 An illustration of the flow in the yz-plane for Sol 13 (Sphere Translation) describedin Eq (35) with parameter values a = minus2 andRe = 3
Solution 14 Radially Dependent Flow
A solution with velocity in the z-direction dependent only on the distance from the z-axis is as
18
follows in cylindrical coordinates
ur(r ϕ z t) = 0
uϕ(r ϕ z t) = 0
uz(r ϕ z t) = A(r2 minus a2
)
p(r ϕ z t) =4Az
Re
(36)
where A and a are constants In Cartesian coordinates the solution is just the same with u1(x y z t)and u2(x y z t) both zero and the r2 term replaced with x2 + y2 in u3(x y z t)
Figure 17 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus1 1] withparameter values a = 1 A = 4 and Re = 10
Figure 18 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus2 2] withparameter values a = 1 A = 4 and Re = 10
Solution 15 Rankine Vortex
A solution discovered by Rankine [1] is a vortex flow of the following form
ur(r ϕ z t) = 0
uϕ(r ϕ z t) =ar
2πR2
uz(r ϕ z t) = 0
p(r ϕ z t) =a2r2
8π2R4
(37)
19
where a and R are constants In Cartesian coordinates the flow has the form
u1(x y z t) = minus ay
2πR2
u2(x y z t) =ax
2πR2
u3(x y z t) = 0
p(x y z t) =a2(x2 + y2
)8π2R4
(38)
This flow is actually a rigid body rotational flow in that the angular rotational velocity is uniformand the velocity of the flow increases proportionally with distance from the z-axis
Figure 19 An illustration of the flow in the xy-plane for Sol 15 (Rankine Vortex) described inEq (37) and Eq (38) with parameter values a = 1and R = 1
radic2π
23 Characterisation of Flows
It is important to be able to classify flows in a way that sorts them into distinct groups This aidsin the clear identification and general study of flows It is also a very important way of recognisingflows with similar properties In this subsection we introduce two distinct ways in which flows canbe characterised
231 Characterising Solutions using Invariants
A principal invariant of a tensor is a coefficient of the characteristic polynomial of that tensorInvariants as their name suggests are invariant under rotations of the coordinate system in thattheir values do not change when the tensor is lsquorotatedrsquo into a different set of coordinates Thevelocity gradient of a flow G is defined as the Jacobian of the velocity vector given by
Gij =
[partuipartxj
] (39)
20
for i j = 1 2 3 There are three principal invariants of the velocity gradient of a fluid flow G givenby
PG = tr(G)
QG =1
2
((tr(G))2 minus tr
(G2))
RG = det(G)
(40)
Incompressibility of flows (Eq (5)) ensures that PG = 0 hence the first principal invariant is zerofor all of the flows in Section 22 We now define the rate-of-strain tensor S by
Sij =1
2
(partuipartxj
+partujpartxi
) (41)
and the rate-of-rotation tensor Ω by
Ωij =1
2
(partuipartxjminus partujpartxi
) (42)
Note that the first principal invariant of S and the first and third principal invariants of Ω are zeroAlso note that the velocity gradient of a flow can simply be written G = S + Ω
Rather than considering the principal invariants we can also consider the combined invariants
I1 = tr(S2) I2 = tr(Ω2) I3 = tr(S3) I4 = tr(SΩ2) I5 = tr(S2Ω2) (43)
We can hence write the three principal invariants for a velocity field G in terms of the combinedinvariants
PG = 0
QG = minus1
2(I1 + I2)
RG =1
3(I3 + 3I4)
(44)
In this report the flows considered are the exact solutions to the incompressible Navier-Stokesequations described in Section 22 The combined invariants in Eq (43) are used to characterisethese solutions into different classes in Section 44 Both the principal invariants and combinedinvariants for each exact solution in Section 22 are given in Appendix A
Each of the subfilter-scale eddy-viscosity models described in Section 32 are written in terms ofthese five combined invariants
232 Vremanrsquos Characterisation of Flows
Vreman [16] classifies flows using the gradient of the velocity field of a flow Each lsquoclassrsquo of flows hasthe same number of non-zero elements in the gradient matrix For example
[partuipartxj
]=
0 0 0lowast 0 0lowast 0 0
(45)
where a lowast denotes a non-zero term represents a group of flows that lie within the class denoted byQ7 (see Appendix B) where the subscript represents the number of non-zero terms in the velocity
21
gradient Flows of the form in Eq (45) have a velocity field of the form
u1 = c1 (constant)
u2 = f2(x1)
u3 = f3(x1)
(46)
where f2(x1) and f3(x1) are such that the Navier Stokes equations Eq (1) are satisfied Notethat as the elements on the leading diagonal are zero the incompressibility condition Eq (5) isautomatically satisfied Some simple flow classes are given in Appendix B and contain a number ofthe solutions discussed in Section 22 This is mentioned in greater detail in Section 441
3 Conditions for Large-Eddy Simulation Models
31 Large-Eddy Simulation
Due to the nature of numerical simulation and the complexity of turbulent flows small details aremissed and information is lost when solving the Navier-Stokes equations The computational costof solving a turbulent flow is extremely high so a filtering operation is applied to the Navier-Stokesequations represented by a bar to reduce the computational cost of the simulation This is a linearoperation and preserves constants ie ρ = ρ and is also assumed to commute with differentiation
partuipartxj
=partuipartxj
(47)
The filtered Navier-Stokes equations are found by filtering (putting a bar over) the entire equations(see Eq (1)) and after evaluating are given by
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (48)
where
τij = uiuj minus uiuj (49)
is called the lsquotrue stressrsquo of the flow This however poses a problem as it is not clear what is meantby the term uiuj for a general filtering operation The true stress τij therefore must be replaced bya model τmod
ij in practical applications
32 Subfilter-Scale Models
We will now define the subflter-scale models denoted by τmodij to be used in place of the lsquodifficultrsquo
true stress τij There are a number of models available to use in the literature [11 p7] These willbe tested using the exact solutions to the Navier-Stokes equations and by checking conditions whichare defined in Sections 33 and 34
All the models barring one are of the same form That is each of the eddy-viscosity models arewritten
τmodij minus 1
3τmodkk δij = minus2νeSij (50)
with νe representing the eddy viscosity of a specific model For simplicity we usually just writeτmodij = minus2νeSij the extra term is included here to emphasise that the models are traceless Each
22
of the models we consider here can be expressed in terms of the rate of strain tensor S defined inEq (41) the rate of rotation tensor Ω defined in Eq (42) and the combined invariants defined inEq (43) The models are then defined as follows
Smagorinsky [12] νSe = (CSδ)2radic
2I1 (51)
Vreman [16] νVe = (CV δ)2
radicQGGT
PGGT (52)
QR [13 14 15] νQRe = (CQRδ)2 max0minusI3
I1 (53)
Vortex-Stretching [11] νV Se = (CV Sδ)2radic
2I1
(I5 minus 1
2I1I2
minusI1I2
)32
(54)
with the quantities
PGGT = I1 minus I2 QGGT =1
4(I1 + I2)2 + 4
(I5 minus
1
2I1I2
) (55)
There is a fifth and final model to be considered here It is of a different form to the other four andnot an eddy-viscosity model It is known as the Gradient model and is defined by
Gradient [4 9] τmodG = CGδ
(S2 minus Ω2 minus (SΩminus ΩS)
) (56)
The values C are model constants specific to each model and δ represents the mesh size of the LESBoth of these quantities are included here to properly define the models however in this projectthey will be ignored as they are constants and have no influence on the conditions to be defined
33 Conditions for Large-Eddy Simulation without Explicit Filtering
The filtering operation introduced in Eq (48) is just a formality for LES without explicit filteringThe idea here is that filtering does not actually take place but the term lsquoaddedrsquo to the Navier-Stokesequations in fact lsquosubtractsrsquo the difficult to simulate small-scale turbulent motion of the flow Inorder to consider this properly the terms ui and p in Eq (48) are replaced by vi and q It is hopedthat τmod
ij approximates τij so that vi and q approximate ui and p respectively So LES withoutexplicit filtering is given by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (57)
It is clear to see that the first four terms of this equation are exactly the same as the Navier-Stokesequations so for the exact solutions discussed in Section 22 the combination of these terms willequal zero What is then desirable is that the model term is also zero for these exact solutions sothat Eq (57) still holds In order to test models for exact solutions the condition
part
partxjτmodij (v) = 0 (58)
is checked If this condition were to hold it would imply that the model does not influence thevelocity field of a flow hence we call it the Velocity Field Condition There are a number of otherrestrictions that can be tested One such test comes from the differential form of the evolution ofthe kinetic energy of a flow given in Eq (10) In order to find this Eq (57) is multiplied by vi andterms are grouped to find
part
partt
(1
2vivi
)+ vj
part
partxj
(1
2vivi
)= minusvi
1
ρ
partq
partxi+ νvi
part2vipartxjpartxj
minus vipart
partxjτmodij (v) (59)
23
Then similarly to Eq (58) another condition is to check whether the model affects the kineticenergy of exact solutions this is equivalent to checking whether
vipart
partxjτmodij (v) = 0 (60)
We call this condition the Kinetic Energy Condition The third and final condition considered herecan also be found using the integral form for the equation for kinetic energy
d
dt
intV
1
2vividV =
intS
[minusvj
(1
2vivi
)minus vj
q
ρ+ 2νviSij + viτ
modij (v)
]njdS
+
intV
[minus2νSijSij minus
(minusSijτmod
ij (v))]dV
(61)
where the rate-of-strain tensor Sij was defined in Eq (41) V is a volume of fluid and dS meansthe integral is over the surface of that volume with unit normal nj Then this final condition checkswhether the model affects the total kinetic energy of exact solutions to the Navier-Stokes equationsthat is
Sijτmodij (v) = 0 (62)
We will call this condition the Dissipation Condition because Sijτmodij (v) is often referred to as the
subgrid dissipation
34 Conditions for Large-Eddy Simulation with Explicit Filtering
As in Eq (48) the Navier-Stokes equations can be filtered by applying the filtering operation to thewhole equation (putting a bar over all terms) and evaluating to find again that
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (63)
In Eq (57) the filtering operation was ignored and the elements ui p and τij(u) were replacedwith vi q and τmod
ij (v) respectively Now in LES with explicit filtering the filtering operation itselfsmooths over the small-scale turbulent motion of the simulated flow In this case the elements uip and τij(u) will be replaced with vi q and τmod
ij (v) respectively LES with explicit filtering is thengiven by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (64)
Note that Eqs (63) and (64) despite their similarity to Eq (57) no longer contain the Navier-Stokesequations as Eq (57) did The bar (filtering operation) over each term results in each exact solutionfound in Section 22 not necessarily satisfying the first four terms It is hence unknown whether ornot these terms will vanish for exact solutions Terms can still be compared similarly to Eqs (58)(60) and (62) However instead of comparing to zero we can now compare to the relevant termfrom the filtered Navier-Stokes equations So the terms
part
partxjτij (65)
and
part
partxjτmodij (v) (66)
24
are to be compared Although the latter term is relatively simple to compute the former could leadto some difficulties due to the differentiation of the true stress τij = uiuj minus uiuj So although someinformation could be gleaned from the inspection of these two terms more information can be foundby considering the analogue of Eq (62) So the true subfilter dissipation
Sijτij (67)
(where Sij is shorthand for Sij(v)) and the modelled subfilter dissipation
Sij(v)τmodij (v) (68)
are considered and have to be equal for the condition to hold These can again be derived from theintegral equation for kinetic energy of a filtered velocity field similarly to Eq (61)
4 Analysis of Large-Eddy Simulation Models
In this section we use a code in Mathematica the theory described in Section 3 and the exactsolutions to the Navier-Stokes equations found in Section 22 to check whether or not each of theexact solutions lsquoendorsesrsquo each model for LES both without and with explicit filtering
41 Large-Eddy Simulation Without Explicit Filtering
We start by using a code in Mathematica to test the Velocity Field condition in Eq (58) and theKinetic Energy condition in Eq (60) For all exact solutions to the Navier-Stokes equations inSection 22 the outputs for both conditions were identical so we have combined them into one table
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 ndash ndash 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 1 For the Kinetic Energy condition a lsquo0rsquo result means that vipartpartxj
τmodij (v) = 0 for respective
models and exact solutions whereas a lsquolowastrsquo result means that vipartpartxj
τmodij (v) 6= 0 For the Velocity
Field condition a lsquo0rsquo result means that partpartxj
τmodij (v) = 0 for respective models and exact solutions
whereas a lsquolowastrsquo result means that partpartxj
τmodij (v) 6= 0 The Gradient model can take both positive and
negative values so when the sign of the model is known it is included in the table instead of a lsquorsquoBoth the conditions are included in one table because for all cases the outputs were identical
We then test the Dissipation condition in Eq (62) for all exact solutions to the Navier-Stokesequations in Section 22
25
Dissipation Condition [Does Sijτmodij (v) = 0]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 + 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 2 A lsquo0rsquo result means that Sijτmodij (v) = 0 for respective models and exact solutions whereas
a lsquolowastrsquo result means that Sijτmodij (v) 6= 0 The Gradient model can take both positive and negative
values so when the sign of the model is known it is included in the table instead of a lsquorsquo
We then sum all the results to find the total number of endorsements for each model described inSection 32
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
4 6 8 9 9
lsquoDissipationrsquoEndorsements
2 5 7 7 9
Total Endorsements 6 11 15 16 18
Table 3 The results from Tables 1 and 2 are summarised to give a general overview of how manyendorsements each model receives This makes it much clearer to see the models most supported bythe respective and combined conditions with the Vortex-Stretching model being the most supportedand the Smagorinsky model the least
Testing the models in Section 32 using LES without explicit filtering has mixed results It isclear that the Smagorinksy model is not well endorsed by the flows in Section 22 with the onlyendorsements coming from the simplest flows that is Sol 1 (Linear Velocity) Sol 3 (Velocity as afunction of time) Sol 7 (Stagnation Flow) and Sol 15 (Rankine Vortex) The Vortex-Stretching-based model is the most endorsed flow with eighteen out of a possible thirty endorsements Thisagrees with our foreknowledge as this model was constructed to be as physically consistent as possiblein the paper by Silvis amp Verstappen [11]
42 Large-Eddy Simulation With Explicit Filtering
In this section we will compare the true subfilter dissipation Sijτij and the modelled subfilterdissipation Sij(v)τmod
ij (v) given in Eqs (67) and (68) respectively for each model τmodij and each
exact solution to the Navier-Stokes equation given in Section 22 We will not consider the conditionsinvolving differentiation in Eqs (65) and (66) due to the difficulty in calculating the derivative ofthe τij term in Eq (65)
26
421 Example of a Test for a Model
The true subfilter dissipation Sijτij of a flow given in Eq (67) cannot be explicitly computed asthe filtering operation is kept general What can be done however is to partially evaluate theexpressions using the attributes of a filtering operation mentioned in Eq (48)
The calculations of Sijτij for Sol 1 (Linear Velocity) are provided to illustrate the general ideaRecall that the velocity field of Sol 1 (Linear Velocity) is written in Eq (13) in vector notation as
u(x y z t) =
0ax+ bcx+ d
(69)
Then Sij = Sij(u) is computed simply by using its definition in Eq (41) Note that as filteringpreserves constants if there is an a it is replaced with a meaning that (in this case) there are alarge number of zero terms due to the frequency of 0 Hence
[Sij(u)
]=
[1
2
(partuipartxj
+partujpartxi
)]=
0 a
2c2
a2 0 0
c2 0 0
=
0 a
2c2
a2 0 0
c2 0 0
(70)
Note that as Sijτij is being computed which is equal to the sum of the element-wise multiplicationof the two matrices (or the trace of the matrix multiplication) that we donrsquot need to consider theelements of τij(u) with corresponding zero entry in Sij As τij(u) = uiuj minus uiuj and again asfiltering preserves constants it is found that
[τij(u)] =[uiuj minus uiuj
]=
lowast 0 middot ax+ b 0 middot cx+ d
ax+ b middot 0 lowast lowastcx+ d middot 0 lowast lowast
=
lowast 0 00 lowast lowast0 lowast lowast
(71)
where a lsquolowastrsquo just represents some element (not necessarily non-zero) Therefore the true subfilterdissipation
Sijτij = tr
0 a2
c2
a2 0 0c2 0 0
middotlowast 0 0
0 lowast lowast0 lowast lowast
= 0 (72)
Now the modelled subfilter dissipation Sij(u)τmodij (u) needs to be computed (recall that Sij =
Sij(u)) In this case let us use the QR model stated in Eq (53) Once again Sij(u) calculated inEq (70) needs to be used So we have that
τmodij = minus2(CQRδ)
2 max0minusI3I1
Sij (73)
As
I1 = tr(S2) =
1
2(a2 + c2) and I3 = tr(S
3) = 0 (74)
it holds that τmodij (u) = 0 so Sij(u)τmod
ij (u) = 0 This agrees with Eq (72) and hence Sol 1(Linear Velocity) endorses the QR model
27
422 Testing the Models
Now using a Mathematica code the exact solutions in Section 22 are again used to evaluate thecondition for the models using the process just described in Section 421 The results are describedin Table 4
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 4 All entries with a ldquo0rdquo represent a value of zero and a ldquolowastrdquo represents a value that is non-zeroThe ldquoExactrdquo row represents the value of the true subfilter dissipation Sijτij Every other row witha model name represents the value of the modelled subfilter dissipation Sij(u)τmod
ij (u) respectiveto that model A ldquordquo means that it is unclear as to the value - the sign of these values is oftenunknown and the lsquoMaxrsquo function in the definition of the QR model (in Eq (53)) makes it difficultto evaluate
Reading from Table 4 the Smagorinsky model is clearly endorsed by Sol 3 (Velocity as a function oftime) and Sol 15 (Rankine Vortex) as in both cases both Sijτij = 0 = Sij(u)τmod
ij (u) In this casewe say that Sol 3 (Velocity as a function of time) and Sol 15 (Rankine Vortex) lsquostronglyrsquo endorsethe Smagorinsky model The Smagorinsky model is not endorsed by Sol 1 (Linear Velocity) Sol 2(Planar Flow) or Sol 14 (Radially Dependent) however because Sij(u)τmod
ij (u) 6= 0 For Sol 4 to
13 both Sijτij 6= 0 6= Sij(u)τmodij (u) and so we will say that these exact solutions lsquoweaklyrsquo endorse
the Smagorinsky model
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 10 10 9 6
Table 5 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Interestingly the lsquostrongrsquo and lsquoweakrsquo endorsements give different results However more stock shouldbe put in the results of the lsquostrongerrsquo condition as it is completely clear that zero is equal to zeroIt is not necessarily clear whether or not non-zero equals non-zero as the values will depend on theparameters in each solution hence the lsquoweakerrsquo endorsement
The lsquostrongrsquo condition equally endorses the Vreman QR Gradient and Vortex-Stretching-basedmodels whereas the lsquoweakrsquo condition endorses Vreman and Smagorinsky the most closely followedby the Gradient model with the Vortex-Stretching-based model being the most opposed As this
28
completely contradicts the lsquostrongrsquo condition and the results in Section 41 it makes us question thevalidity of the weaker test The number of endorsements for the QR model is unknown because of thedefinition of the model in Eq (53) and our evaluation of the true subfilter dissipation Eq (67) Dueto the exact solutions we have found it is never clear whether or not the true subfilter dissipationis strictly positive or negative This means that when the true subfilter dissipation is found to benon-zero the lsquoMaxrsquo function in the QR model cannot be evaluated
43 Vremanrsquos True Subfilter Dissipation Claim
Using the theory mentioned in Sections 232 and 34 we can place the exact solutions to the Navier-Stokes equations into different classes defined by Vreman In doing this we can also attempt to verifyone of Vremanrsquos results [16] In Vremanrsquos paper he defines the class Qn to be the set of flows withn zero entries in the velocity gradient of the flow (the velocity gradient is defined in Eq (39))Vreman then claims that the only flows with zero true subfilter dissipation are a certain selection ofthose in the classesQ7Q8 orQ9 shown in appendix B and is generally non-zero for any other flows
The true subfilter dissipation of Sol 1 (Linear Velocity) was computed in Section 421 and wasfound to be zero in Eq (72) The velocity gradient of Sol 1 is as follows
GLV =
0 0 0a 0 0c 0 0
(75)
and hence Sol 1 (Linear Velocity) is in the required subset of Q7 which agrees with Vremanrsquos claimIn Section 422 the true subfilter dissipation was computed (determined to be zero or non-zero) forall of the exact solutions to the Navier Stokes equations in Section 22 These solutions can now belsquoplacedrsquo into each class Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a functionof time) and Sol 14 (Radially Dependent) were all found to have zero true subfilter dissipation andare all in the appropriate subsets of either Q7Q8 or Q9 This agrees with Vremanrsquos results
We can extend Vremanrsquos results however We found that Sol 15 (Rankine Vortex) had zero truesubfilter dissipation and yet is not in the subset of Q7 defined in Appendix B It is important tonote that the rate of strain tensor Sij in Eq (41) for Sol 15 (Rankine Vortex) is zero This is thereason the true subfilter dissipation is zero and the same reason that all models are zero for thisflow as all models are based on Sij
We also found that all of the remaining exact solutions to the Navier-Stokes equations are not inVremanrsquos specified classes (in Appendix Section B) They also have a non-zero subfilter dissipationwhich agrees with Vremanrsquos results in his paper [16]
44 Classes of Flows
It is important to consider that some exact solutions to the Navier-Stokes equations may have similarproperties For example Sol 1 (Linear Velocity) and Sol 2 (Planar Flow) are both very similarIn fact Sol 2 (Planar Flow) is a more general case of Sol 1 (Linear Velocity) We therefore mustgroup the different flows into classes in an attempt to reduce the bias that may occur when solutionsare similar This should aid us to further determine which of the models are most endorsed We willdo this in a number of different ways using the theory discussed in Section 23 We will group theflows with respect to Vremanrsquos Flow Classes given in appendix B and with respect to the Invariantsgiven in appendix A
29
441 Vremanrsquos Velocity Gradient Flow Classes
We now put our exact solutions to the Navier-Stokes equations into one of two classes Either Q0minus6or Q7minus9 where each class is defined intuitively as follows
Q0minus6 = Q0 cupQ1 cup middot middot middot cup Q6 (76)
Q7minus9 = Q7 cupQ8 cupQ9 (77)
where Qn is defined in Section 43 By looking at the gradient of each exact solution we can see thatQ7minus9 contains the six solutions Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as afunction of time) Sol 5 (2D Taylor) Sol 14 (Radially Dependent) and Sol 15 (Rankine Vortex)and that Q0minus6 contains the remaining nine We now check the models for LES without explicitfiltering by considering endorsements within each class Tables 6 and 7 are analogues of Table 3 andillustrate the results for each flow class The analogues of Tables 1 and 2 are Tables 22 and 23 andare in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 6 6 6
lsquoDissipationrsquoEndorsements
2 5 6 6 6
Percentage Endorsed 417 833 100 100 100
Table 6 Each entry in the top two rows represents the number of solutions in the class Q7minus9 thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Q0minus6 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 2 3 3
lsquoDissipationrsquoEndorsements
0 0 1 0 3
Percentage Endorsed 56 56 167 167 333
Table 7 Each entry in the top two rows represents the number of solutions in the class Q1minus6 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking results to 1 decimal place if necessary
All of the flows in the class Q7minus9 endorse the QR Gradient and Vortex-Stretching-based subfilter-scale models (see Table 6) We can hence see that this class fully endorses these three models equallyand does not endorse the Smagorinsky model The Vortex-Stretching eddy viscosity model has themost endorsements from any of the flows in the class Q0minus6 and the Smagorinksy and Vreman modelshave the fewest endorsements (see Table 7) We can hence say that this class mostly endorses theVortex-Stretching-based eddy-viscosity model note however that the percentage endorsement is very
30
low
Now let us consider LES with explicit filtering Tables 8 and 9 are analogues of Table 5 and illustratethe results for each flow class The analogue of Table 4 is Table 24 and is in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 1 1 0 0 0
Table 8 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Q0minus6 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 9 9 9 6
Table 9 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We have grouped Vremanrsquos classes into Q7minus9 and Q0minus6 this is equivalent to saying that we havegrouped Vremanrsquos classes into those with two or fewer non-zero velocity gradient elements and thosewith three or more non-zero velocity gradient elements respectively Therefore the solutions in Q7minus9are exactly those with a lsquo0rsquo in the lsquoExactrsquo column of Table 4 and Sol 5 (2D Taylor) ExcludingSol 5 (2D Taylor) these are the same as those five that endorse the Vreman QR Gradient andVortex-Stretching-based models in Table 5 We can see in Table 8 that the results for the lsquostrongrsquoendorsements are exactly the same as in Table 5 This is because all of the flows with zero subfilterdissipation are contained in the class Q7minus9 There are hence no lsquostrongrsquo endorsements in Table 9The majority of the lsquoweakrsquo endorsements come from Q0minus6 as this class contains all but one of thesolutions with non-zero true subfilter dissipation (given in Eq (67))
442 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
We now separate our exact solutions into classes again using Vremanrsquos flow classes This timehowever we define V0 to be the class of flows Vreman claims will always have zero true subfilterdissipation namely those classes mentioned in Appendix B We define Vlowast to be all the remaining
31
classes Then
V0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow)
Sol 3 (Velocity as a function of time) Sol 14 (Radially Dependent)(78)
Vlowast = Sol 4 (Axisymmetric Flow) Sol 5 (2D Taylor) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 13 (Sphere Translation) Sol 15 (Rankine Vortex)
(79)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 10 and 11 are analogues of Table 3 and illustrate the results for each flow class Theanalogues of Tables 1 and 2 are Tables 25 and 26 and are in Appendix C2
Endorsements for models using solutions in V0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 4 4 4 4
lsquoDissipationrsquoEndorsements
1 4 4 4 4
Percentage Endorsed 375 100 100 100 100
Table 10 Each entry in the top two rows represents the number of solutions in the class V0 thatendorse each model There are 4 solutions in this class so the percentages are worked out by dividingthe total of each column by 8 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Vlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 2 4 5 5
lsquoDissipationrsquoEndorsements
1 1 3 2 5
Percentage Endorsed 136 136 318 318 455
Table 11 Each entry in the top two rows represents the number of solutions in the class Vlowast thatendorse each model There are 11 solutions in this class so the percentages are worked out bydividing the total of each column by 22 and taking results to 1 decimal place if necessary
All of the flows in the class V0 endorse the Vreman QR Gradient and Vortex-Stretching-basedsubfilter-scale models (see Table 6) This class completely endorses these four models equally anddoes not endorse the Smagorinsky model The Vortex-Stretching-based eddy-viscosity model hasthe most endorsements from the class Vlowast and the Smagorinksy and Vreman models again have thefewest endorsements (see Table 7) As in Section 441 we say that this class mostly endorses the
32
Vortex-Stretching-based eddy-viscosity model but once again the percentage endorsement is ratherlow
Now let us consider LES with explicit filtering Tables 12 and 13 are analogues of Table 5 andillustrate the results for each flow class The analogue of Table 4 is Table 27 and is in Appendix C2
Endorsements for models using solutions in V0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 4 4 4 4
lsquoWeakrsquo Endorsements 0 0 0 0 0
Table 12 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Vlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 1 1 1 1
lsquoWeakrsquo Endorsements 10 10 9 6
Table 13 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We can see in Table 12 that the results for the lsquostrongrsquo endorsements are similar to those in Table 5excluding Sol 15 (Rankine Vortex) The four solutions in V0 with zero true subfilter dissipationnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)and Sol 14 (Radially Dependent) are hence the four that endorse the Vreman QR Gradientand Vortex-Stretching-based models in Table 12 Sol 15 (Rankine Vortex) is in the class Vlowast andhence there is one lsquostrongrsquo endorsement in Table 13 The class Vlowast contains all of the flows with anon-zero true subfilter dissipation (given in Eq (67)) and hence Table 13 contains all of the lsquoweakrsquoendorsements
443 Classes using the Third Principal Invariant
Sorting the classes using the invariants is a non-trivial task in itself There are a number of waysin which to do this using both the principal invariants in Eq (40) and the combined invariants inEq (43)
We start by looking at flows with third principal invariant RG equal to zero We again define twoclasses R0 and Rlowast where R0 consists of all flows with RG = 0 and Rlowast consists of all flows withRG 6= 0 Then using the information in Appendix A we see that there are nine flows in R0 That
33
is
R0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 8 (Vortex in Stagnation Flow)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)
(80)
Rlowast = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 13 (Sphere Translation)(81)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 14 and 15 are analogues of Table 3 and they illustrate the results for LES withoutexplicit filtering for each flow class The analogues of Tables 1 and 2 are Tables 28 and 29 and arein Appendix C3
Endorsements for models using solutions in R0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 7 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 278 556 778 722 778
Table 14 Each entry in the top two rows represents the number of solutions in the class R0 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking the result to one decimal place if necessary
Endorsements for models using solutions in Rlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 2 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 83 83 83 167 333
Table 15 Each entry in the top two rows represents the number of solutions in the class Rlowast thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking the result to one decimal place if necessary
The class R0 endorses both the Vortex-Stretching-based model and the QR model the most andSmagorinsky the least in the case for LES without explicit filtering In the Rlowast class the Vortex-Stretching-based model is again the most endorsed followed by the Gradient model however thepercentages are again small
34
Now let us consider LES with explicit filtering Tables 16 and 17 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 30 and is in Appendix C3
Endorsements for models using solutions in R0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 4 4 3 2
Table 16 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Rlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 6 6 6 4
Table 17 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 we can see in Table 16 that the results for the lsquostrongrsquo endorsements are exactly thesame as in Table 5 This is because all of the flows with zero subfilter dissipation are contained in theclass R0 As in Table 9 there are no lsquostrongrsquo endorsements in Table 17 The lsquoweakrsquo endorsementshowever are split more evenly between the two classes now than in Vremanrsquos classes in Sections 441and 442 We find that the Smagorinsky and Vreman models still seem to be the most endorsed inboth classes
444 Two-component Flow Classes
Another way in which the flows can be grouped into classes is by determining whether a flow is atwo-component flow or not We define a two-component flow to have the property that the combinedprincipal invariants of a flow in Eq (43) satisfy
I3 = I4 = I5 minus1
2I1I2 = 0 (82)
We are also interested in considering flows with zero gradient model subgrid dissipation These flowshave the property that
I3 minus I4 = 0 (83)
What we find though is that for all the solutions in Section 22 if a flow is a two-component flowthen it also satisfies I3 minus I4 = 0 It may be interesting to note that although I3 minus I4 6= 0 for manysolutions there are a number with I3 equal to a multiple of I4 Although not considered in thisreport this may also be a class of interest We find that the two-component flows are
C2 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)(84)
35
and we call the remaining flows three-component flows They are
C3 = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex) Sol 11 (Pseudo-Motion Shots)
Sol 13 (Sphere Translation)
(85)
Note that Sol 8 (Vortex in Stagnation Flow) is defined to be a planar flow this does not mean that itis a two-component flow in this sense however because its two non-zero velocity components dependon three coordinates We now check the models for LES without explicit filtering by consideringendorsements within each class Tables 18 and 19 are analogues of Table 3 and they illustrate theresults for LES without explicit filtering for each flow class The analogues of Tables 1 and 2 areTables 31 and 32 and are in Appendix C4
Endorsements for models using solutions in C2 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 6 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 357 714 100 857 100
Table 18 Each entry in the top two rows represents the number of solutions in the class C2 thatendorse each model There are 7 solutions in this class so the percentages are worked out by dividingthe total of each column by 14 and taking the result to one decimal place if necessary
Endorsements for models using solutions in C3 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 3 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 63 63 63 188 250
Table 19 Each entry in the top two rows represents the number of solutions in the class C3 thatendorse each model There are 8 solutions in this class so the percentages are worked out by dividingthe total of each column by 16 and taking the result to one decimal place if necessary
The class C2 equally endorses the QR and Vortex-Stretching-based models and once again theSmagorinsky model is the least endorsed As with the previous classes the class C3 has such lowpercentages that it does not seem to endorse any models We start to see a trend appearing here inthat the flow classes that contain the more complicated flows namely Q0minus6 (Eq (76)) Vlowast (Eq (79))Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages The classes that tendto contain the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78)) R0 (Eq (80)) and C2 (Eq (84))all seem to favour the Vortex-Stretching-based model or the QR model
36
Now let us consider LES with explicit filtering Tables 20 and 21 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 33 and is in Appendix C4
Endorsements for models using solutions in C2 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 2 2 0 1 0
Table 20 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in C3 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 8 8 8 6
Table 21 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 and Table 16 we can see in Table 20 that the results for the lsquostrongrsquo endorsements areexactly the same as in Table 5 This is again because all of the flows with zero subfilter dissipationare contained in the class C2 There are two flows with non-zero true subfilter dissipation (givenin Eq (67)) in C2 and this shows in the lsquoweakrsquo endorsements in Table 20 There are no lsquostrongrsquoendorsements in Table 21 as all the zero subfilter dissipation flows are in C2
5 Conclusions
In this report we studied incompressible turbulent flows We found a number of exact solutions to theNavier-Stokes equations each representing a relatively simplistic fluid flow Large-eddy simulation(LES) is used to model turbulent flows by smoothing over the complex small-scale motion Weintroduced a number of subfilter-scale models to approximate this small-scale motion and testedthem using the found exact solutions to the Navier-Stokes equations We first considered LESwithout explicit filtering and used the Navier-Stokes equations to derive conditions for the subfilter-scale models namely the Velocity Field condition (Eq (58)) the Kinetic Energy condition (Eq (60))and the Dissipation condition (Eq (62)) Similarly we considered LES with explicit filtering andagain used the Navier-Stokes equations to derive more conditions for the models in this case wedecided to just take one condition the Dissipation condition (Eqs (67) and (68)) as the others leadto difficulty in their evaluation By lsquorunningrsquo the exact solutions to the Navier-Stokes equationsthrough each condition for the subfilter-scale models we found that the Vortex-Stretching-basededdy-viscosity model in Eq (54) was most endorsed by the tests in LES without explicit filteringWe found that four out of the five subfilter-scale models namely the Vreman QR Gradient andVortex-Stretching-based models were equally endorsed by the lsquostrongrsquo tests in LES with explicitfiltering with the Smagorinsky model being the least endorsed The results we found for the lsquoweakrsquocondition were completely opposite to those found in both the lsquostrongrsquo condition and in LES withoutexplicit filtering This led us to conclude that this test was not one in which we should put our trust
37
We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses
In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes
This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold
38
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
Solution 13 Rectilinear and Uniform Translation of a Sphere
In the book by Berker [3 p 228] another solution is given in Cartesian coordinates by
u1(x y z t) =3
4
axz
R3
(a2
R2minus 1
)
u2(x y z t) =3
4
ayz
R3
(a2
R2minus 1
)
u3(x y z t) = 1minus 3
4
a
Rminus 1
4
a3
R3minus 3
4
az2
R3
(a2
R2minus 1
)
p(x y z t) =3a(aminusR)(a+R)
32R10Re
((R2Re
(x2 + y2
)minus 2R2Rez2
) (a3 + 3aR2 minus 4R3
)+ 3aRez4
(R2 minus a2
)minus 16R5z
)
(35)
where a and R are constants
Figure 16 An illustration of the flow in the yz-plane for Sol 13 (Sphere Translation) describedin Eq (35) with parameter values a = minus2 andRe = 3
Solution 14 Radially Dependent Flow
A solution with velocity in the z-direction dependent only on the distance from the z-axis is as
18
follows in cylindrical coordinates
ur(r ϕ z t) = 0
uϕ(r ϕ z t) = 0
uz(r ϕ z t) = A(r2 minus a2
)
p(r ϕ z t) =4Az
Re
(36)
where A and a are constants In Cartesian coordinates the solution is just the same with u1(x y z t)and u2(x y z t) both zero and the r2 term replaced with x2 + y2 in u3(x y z t)
Figure 17 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus1 1] withparameter values a = 1 A = 4 and Re = 10
Figure 18 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus2 2] withparameter values a = 1 A = 4 and Re = 10
Solution 15 Rankine Vortex
A solution discovered by Rankine [1] is a vortex flow of the following form
ur(r ϕ z t) = 0
uϕ(r ϕ z t) =ar
2πR2
uz(r ϕ z t) = 0
p(r ϕ z t) =a2r2
8π2R4
(37)
19
where a and R are constants In Cartesian coordinates the flow has the form
u1(x y z t) = minus ay
2πR2
u2(x y z t) =ax
2πR2
u3(x y z t) = 0
p(x y z t) =a2(x2 + y2
)8π2R4
(38)
This flow is actually a rigid body rotational flow in that the angular rotational velocity is uniformand the velocity of the flow increases proportionally with distance from the z-axis
Figure 19 An illustration of the flow in the xy-plane for Sol 15 (Rankine Vortex) described inEq (37) and Eq (38) with parameter values a = 1and R = 1
radic2π
23 Characterisation of Flows
It is important to be able to classify flows in a way that sorts them into distinct groups This aidsin the clear identification and general study of flows It is also a very important way of recognisingflows with similar properties In this subsection we introduce two distinct ways in which flows canbe characterised
231 Characterising Solutions using Invariants
A principal invariant of a tensor is a coefficient of the characteristic polynomial of that tensorInvariants as their name suggests are invariant under rotations of the coordinate system in thattheir values do not change when the tensor is lsquorotatedrsquo into a different set of coordinates Thevelocity gradient of a flow G is defined as the Jacobian of the velocity vector given by
Gij =
[partuipartxj
] (39)
20
for i j = 1 2 3 There are three principal invariants of the velocity gradient of a fluid flow G givenby
PG = tr(G)
QG =1
2
((tr(G))2 minus tr
(G2))
RG = det(G)
(40)
Incompressibility of flows (Eq (5)) ensures that PG = 0 hence the first principal invariant is zerofor all of the flows in Section 22 We now define the rate-of-strain tensor S by
Sij =1
2
(partuipartxj
+partujpartxi
) (41)
and the rate-of-rotation tensor Ω by
Ωij =1
2
(partuipartxjminus partujpartxi
) (42)
Note that the first principal invariant of S and the first and third principal invariants of Ω are zeroAlso note that the velocity gradient of a flow can simply be written G = S + Ω
Rather than considering the principal invariants we can also consider the combined invariants
I1 = tr(S2) I2 = tr(Ω2) I3 = tr(S3) I4 = tr(SΩ2) I5 = tr(S2Ω2) (43)
We can hence write the three principal invariants for a velocity field G in terms of the combinedinvariants
PG = 0
QG = minus1
2(I1 + I2)
RG =1
3(I3 + 3I4)
(44)
In this report the flows considered are the exact solutions to the incompressible Navier-Stokesequations described in Section 22 The combined invariants in Eq (43) are used to characterisethese solutions into different classes in Section 44 Both the principal invariants and combinedinvariants for each exact solution in Section 22 are given in Appendix A
Each of the subfilter-scale eddy-viscosity models described in Section 32 are written in terms ofthese five combined invariants
232 Vremanrsquos Characterisation of Flows
Vreman [16] classifies flows using the gradient of the velocity field of a flow Each lsquoclassrsquo of flows hasthe same number of non-zero elements in the gradient matrix For example
[partuipartxj
]=
0 0 0lowast 0 0lowast 0 0
(45)
where a lowast denotes a non-zero term represents a group of flows that lie within the class denoted byQ7 (see Appendix B) where the subscript represents the number of non-zero terms in the velocity
21
gradient Flows of the form in Eq (45) have a velocity field of the form
u1 = c1 (constant)
u2 = f2(x1)
u3 = f3(x1)
(46)
where f2(x1) and f3(x1) are such that the Navier Stokes equations Eq (1) are satisfied Notethat as the elements on the leading diagonal are zero the incompressibility condition Eq (5) isautomatically satisfied Some simple flow classes are given in Appendix B and contain a number ofthe solutions discussed in Section 22 This is mentioned in greater detail in Section 441
3 Conditions for Large-Eddy Simulation Models
31 Large-Eddy Simulation
Due to the nature of numerical simulation and the complexity of turbulent flows small details aremissed and information is lost when solving the Navier-Stokes equations The computational costof solving a turbulent flow is extremely high so a filtering operation is applied to the Navier-Stokesequations represented by a bar to reduce the computational cost of the simulation This is a linearoperation and preserves constants ie ρ = ρ and is also assumed to commute with differentiation
partuipartxj
=partuipartxj
(47)
The filtered Navier-Stokes equations are found by filtering (putting a bar over) the entire equations(see Eq (1)) and after evaluating are given by
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (48)
where
τij = uiuj minus uiuj (49)
is called the lsquotrue stressrsquo of the flow This however poses a problem as it is not clear what is meantby the term uiuj for a general filtering operation The true stress τij therefore must be replaced bya model τmod
ij in practical applications
32 Subfilter-Scale Models
We will now define the subflter-scale models denoted by τmodij to be used in place of the lsquodifficultrsquo
true stress τij There are a number of models available to use in the literature [11 p7] These willbe tested using the exact solutions to the Navier-Stokes equations and by checking conditions whichare defined in Sections 33 and 34
All the models barring one are of the same form That is each of the eddy-viscosity models arewritten
τmodij minus 1
3τmodkk δij = minus2νeSij (50)
with νe representing the eddy viscosity of a specific model For simplicity we usually just writeτmodij = minus2νeSij the extra term is included here to emphasise that the models are traceless Each
22
of the models we consider here can be expressed in terms of the rate of strain tensor S defined inEq (41) the rate of rotation tensor Ω defined in Eq (42) and the combined invariants defined inEq (43) The models are then defined as follows
Smagorinsky [12] νSe = (CSδ)2radic
2I1 (51)
Vreman [16] νVe = (CV δ)2
radicQGGT
PGGT (52)
QR [13 14 15] νQRe = (CQRδ)2 max0minusI3
I1 (53)
Vortex-Stretching [11] νV Se = (CV Sδ)2radic
2I1
(I5 minus 1
2I1I2
minusI1I2
)32
(54)
with the quantities
PGGT = I1 minus I2 QGGT =1
4(I1 + I2)2 + 4
(I5 minus
1
2I1I2
) (55)
There is a fifth and final model to be considered here It is of a different form to the other four andnot an eddy-viscosity model It is known as the Gradient model and is defined by
Gradient [4 9] τmodG = CGδ
(S2 minus Ω2 minus (SΩminus ΩS)
) (56)
The values C are model constants specific to each model and δ represents the mesh size of the LESBoth of these quantities are included here to properly define the models however in this projectthey will be ignored as they are constants and have no influence on the conditions to be defined
33 Conditions for Large-Eddy Simulation without Explicit Filtering
The filtering operation introduced in Eq (48) is just a formality for LES without explicit filteringThe idea here is that filtering does not actually take place but the term lsquoaddedrsquo to the Navier-Stokesequations in fact lsquosubtractsrsquo the difficult to simulate small-scale turbulent motion of the flow Inorder to consider this properly the terms ui and p in Eq (48) are replaced by vi and q It is hopedthat τmod
ij approximates τij so that vi and q approximate ui and p respectively So LES withoutexplicit filtering is given by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (57)
It is clear to see that the first four terms of this equation are exactly the same as the Navier-Stokesequations so for the exact solutions discussed in Section 22 the combination of these terms willequal zero What is then desirable is that the model term is also zero for these exact solutions sothat Eq (57) still holds In order to test models for exact solutions the condition
part
partxjτmodij (v) = 0 (58)
is checked If this condition were to hold it would imply that the model does not influence thevelocity field of a flow hence we call it the Velocity Field Condition There are a number of otherrestrictions that can be tested One such test comes from the differential form of the evolution ofthe kinetic energy of a flow given in Eq (10) In order to find this Eq (57) is multiplied by vi andterms are grouped to find
part
partt
(1
2vivi
)+ vj
part
partxj
(1
2vivi
)= minusvi
1
ρ
partq
partxi+ νvi
part2vipartxjpartxj
minus vipart
partxjτmodij (v) (59)
23
Then similarly to Eq (58) another condition is to check whether the model affects the kineticenergy of exact solutions this is equivalent to checking whether
vipart
partxjτmodij (v) = 0 (60)
We call this condition the Kinetic Energy Condition The third and final condition considered herecan also be found using the integral form for the equation for kinetic energy
d
dt
intV
1
2vividV =
intS
[minusvj
(1
2vivi
)minus vj
q
ρ+ 2νviSij + viτ
modij (v)
]njdS
+
intV
[minus2νSijSij minus
(minusSijτmod
ij (v))]dV
(61)
where the rate-of-strain tensor Sij was defined in Eq (41) V is a volume of fluid and dS meansthe integral is over the surface of that volume with unit normal nj Then this final condition checkswhether the model affects the total kinetic energy of exact solutions to the Navier-Stokes equationsthat is
Sijτmodij (v) = 0 (62)
We will call this condition the Dissipation Condition because Sijτmodij (v) is often referred to as the
subgrid dissipation
34 Conditions for Large-Eddy Simulation with Explicit Filtering
As in Eq (48) the Navier-Stokes equations can be filtered by applying the filtering operation to thewhole equation (putting a bar over all terms) and evaluating to find again that
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (63)
In Eq (57) the filtering operation was ignored and the elements ui p and τij(u) were replacedwith vi q and τmod
ij (v) respectively Now in LES with explicit filtering the filtering operation itselfsmooths over the small-scale turbulent motion of the simulated flow In this case the elements uip and τij(u) will be replaced with vi q and τmod
ij (v) respectively LES with explicit filtering is thengiven by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (64)
Note that Eqs (63) and (64) despite their similarity to Eq (57) no longer contain the Navier-Stokesequations as Eq (57) did The bar (filtering operation) over each term results in each exact solutionfound in Section 22 not necessarily satisfying the first four terms It is hence unknown whether ornot these terms will vanish for exact solutions Terms can still be compared similarly to Eqs (58)(60) and (62) However instead of comparing to zero we can now compare to the relevant termfrom the filtered Navier-Stokes equations So the terms
part
partxjτij (65)
and
part
partxjτmodij (v) (66)
24
are to be compared Although the latter term is relatively simple to compute the former could leadto some difficulties due to the differentiation of the true stress τij = uiuj minus uiuj So although someinformation could be gleaned from the inspection of these two terms more information can be foundby considering the analogue of Eq (62) So the true subfilter dissipation
Sijτij (67)
(where Sij is shorthand for Sij(v)) and the modelled subfilter dissipation
Sij(v)τmodij (v) (68)
are considered and have to be equal for the condition to hold These can again be derived from theintegral equation for kinetic energy of a filtered velocity field similarly to Eq (61)
4 Analysis of Large-Eddy Simulation Models
In this section we use a code in Mathematica the theory described in Section 3 and the exactsolutions to the Navier-Stokes equations found in Section 22 to check whether or not each of theexact solutions lsquoendorsesrsquo each model for LES both without and with explicit filtering
41 Large-Eddy Simulation Without Explicit Filtering
We start by using a code in Mathematica to test the Velocity Field condition in Eq (58) and theKinetic Energy condition in Eq (60) For all exact solutions to the Navier-Stokes equations inSection 22 the outputs for both conditions were identical so we have combined them into one table
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 ndash ndash 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 1 For the Kinetic Energy condition a lsquo0rsquo result means that vipartpartxj
τmodij (v) = 0 for respective
models and exact solutions whereas a lsquolowastrsquo result means that vipartpartxj
τmodij (v) 6= 0 For the Velocity
Field condition a lsquo0rsquo result means that partpartxj
τmodij (v) = 0 for respective models and exact solutions
whereas a lsquolowastrsquo result means that partpartxj
τmodij (v) 6= 0 The Gradient model can take both positive and
negative values so when the sign of the model is known it is included in the table instead of a lsquorsquoBoth the conditions are included in one table because for all cases the outputs were identical
We then test the Dissipation condition in Eq (62) for all exact solutions to the Navier-Stokesequations in Section 22
25
Dissipation Condition [Does Sijτmodij (v) = 0]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 + 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 2 A lsquo0rsquo result means that Sijτmodij (v) = 0 for respective models and exact solutions whereas
a lsquolowastrsquo result means that Sijτmodij (v) 6= 0 The Gradient model can take both positive and negative
values so when the sign of the model is known it is included in the table instead of a lsquorsquo
We then sum all the results to find the total number of endorsements for each model described inSection 32
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
4 6 8 9 9
lsquoDissipationrsquoEndorsements
2 5 7 7 9
Total Endorsements 6 11 15 16 18
Table 3 The results from Tables 1 and 2 are summarised to give a general overview of how manyendorsements each model receives This makes it much clearer to see the models most supported bythe respective and combined conditions with the Vortex-Stretching model being the most supportedand the Smagorinsky model the least
Testing the models in Section 32 using LES without explicit filtering has mixed results It isclear that the Smagorinksy model is not well endorsed by the flows in Section 22 with the onlyendorsements coming from the simplest flows that is Sol 1 (Linear Velocity) Sol 3 (Velocity as afunction of time) Sol 7 (Stagnation Flow) and Sol 15 (Rankine Vortex) The Vortex-Stretching-based model is the most endorsed flow with eighteen out of a possible thirty endorsements Thisagrees with our foreknowledge as this model was constructed to be as physically consistent as possiblein the paper by Silvis amp Verstappen [11]
42 Large-Eddy Simulation With Explicit Filtering
In this section we will compare the true subfilter dissipation Sijτij and the modelled subfilterdissipation Sij(v)τmod
ij (v) given in Eqs (67) and (68) respectively for each model τmodij and each
exact solution to the Navier-Stokes equation given in Section 22 We will not consider the conditionsinvolving differentiation in Eqs (65) and (66) due to the difficulty in calculating the derivative ofthe τij term in Eq (65)
26
421 Example of a Test for a Model
The true subfilter dissipation Sijτij of a flow given in Eq (67) cannot be explicitly computed asthe filtering operation is kept general What can be done however is to partially evaluate theexpressions using the attributes of a filtering operation mentioned in Eq (48)
The calculations of Sijτij for Sol 1 (Linear Velocity) are provided to illustrate the general ideaRecall that the velocity field of Sol 1 (Linear Velocity) is written in Eq (13) in vector notation as
u(x y z t) =
0ax+ bcx+ d
(69)
Then Sij = Sij(u) is computed simply by using its definition in Eq (41) Note that as filteringpreserves constants if there is an a it is replaced with a meaning that (in this case) there are alarge number of zero terms due to the frequency of 0 Hence
[Sij(u)
]=
[1
2
(partuipartxj
+partujpartxi
)]=
0 a
2c2
a2 0 0
c2 0 0
=
0 a
2c2
a2 0 0
c2 0 0
(70)
Note that as Sijτij is being computed which is equal to the sum of the element-wise multiplicationof the two matrices (or the trace of the matrix multiplication) that we donrsquot need to consider theelements of τij(u) with corresponding zero entry in Sij As τij(u) = uiuj minus uiuj and again asfiltering preserves constants it is found that
[τij(u)] =[uiuj minus uiuj
]=
lowast 0 middot ax+ b 0 middot cx+ d
ax+ b middot 0 lowast lowastcx+ d middot 0 lowast lowast
=
lowast 0 00 lowast lowast0 lowast lowast
(71)
where a lsquolowastrsquo just represents some element (not necessarily non-zero) Therefore the true subfilterdissipation
Sijτij = tr
0 a2
c2
a2 0 0c2 0 0
middotlowast 0 0
0 lowast lowast0 lowast lowast
= 0 (72)
Now the modelled subfilter dissipation Sij(u)τmodij (u) needs to be computed (recall that Sij =
Sij(u)) In this case let us use the QR model stated in Eq (53) Once again Sij(u) calculated inEq (70) needs to be used So we have that
τmodij = minus2(CQRδ)
2 max0minusI3I1
Sij (73)
As
I1 = tr(S2) =
1
2(a2 + c2) and I3 = tr(S
3) = 0 (74)
it holds that τmodij (u) = 0 so Sij(u)τmod
ij (u) = 0 This agrees with Eq (72) and hence Sol 1(Linear Velocity) endorses the QR model
27
422 Testing the Models
Now using a Mathematica code the exact solutions in Section 22 are again used to evaluate thecondition for the models using the process just described in Section 421 The results are describedin Table 4
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 4 All entries with a ldquo0rdquo represent a value of zero and a ldquolowastrdquo represents a value that is non-zeroThe ldquoExactrdquo row represents the value of the true subfilter dissipation Sijτij Every other row witha model name represents the value of the modelled subfilter dissipation Sij(u)τmod
ij (u) respectiveto that model A ldquordquo means that it is unclear as to the value - the sign of these values is oftenunknown and the lsquoMaxrsquo function in the definition of the QR model (in Eq (53)) makes it difficultto evaluate
Reading from Table 4 the Smagorinsky model is clearly endorsed by Sol 3 (Velocity as a function oftime) and Sol 15 (Rankine Vortex) as in both cases both Sijτij = 0 = Sij(u)τmod
ij (u) In this casewe say that Sol 3 (Velocity as a function of time) and Sol 15 (Rankine Vortex) lsquostronglyrsquo endorsethe Smagorinsky model The Smagorinsky model is not endorsed by Sol 1 (Linear Velocity) Sol 2(Planar Flow) or Sol 14 (Radially Dependent) however because Sij(u)τmod
ij (u) 6= 0 For Sol 4 to
13 both Sijτij 6= 0 6= Sij(u)τmodij (u) and so we will say that these exact solutions lsquoweaklyrsquo endorse
the Smagorinsky model
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 10 10 9 6
Table 5 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Interestingly the lsquostrongrsquo and lsquoweakrsquo endorsements give different results However more stock shouldbe put in the results of the lsquostrongerrsquo condition as it is completely clear that zero is equal to zeroIt is not necessarily clear whether or not non-zero equals non-zero as the values will depend on theparameters in each solution hence the lsquoweakerrsquo endorsement
The lsquostrongrsquo condition equally endorses the Vreman QR Gradient and Vortex-Stretching-basedmodels whereas the lsquoweakrsquo condition endorses Vreman and Smagorinsky the most closely followedby the Gradient model with the Vortex-Stretching-based model being the most opposed As this
28
completely contradicts the lsquostrongrsquo condition and the results in Section 41 it makes us question thevalidity of the weaker test The number of endorsements for the QR model is unknown because of thedefinition of the model in Eq (53) and our evaluation of the true subfilter dissipation Eq (67) Dueto the exact solutions we have found it is never clear whether or not the true subfilter dissipationis strictly positive or negative This means that when the true subfilter dissipation is found to benon-zero the lsquoMaxrsquo function in the QR model cannot be evaluated
43 Vremanrsquos True Subfilter Dissipation Claim
Using the theory mentioned in Sections 232 and 34 we can place the exact solutions to the Navier-Stokes equations into different classes defined by Vreman In doing this we can also attempt to verifyone of Vremanrsquos results [16] In Vremanrsquos paper he defines the class Qn to be the set of flows withn zero entries in the velocity gradient of the flow (the velocity gradient is defined in Eq (39))Vreman then claims that the only flows with zero true subfilter dissipation are a certain selection ofthose in the classesQ7Q8 orQ9 shown in appendix B and is generally non-zero for any other flows
The true subfilter dissipation of Sol 1 (Linear Velocity) was computed in Section 421 and wasfound to be zero in Eq (72) The velocity gradient of Sol 1 is as follows
GLV =
0 0 0a 0 0c 0 0
(75)
and hence Sol 1 (Linear Velocity) is in the required subset of Q7 which agrees with Vremanrsquos claimIn Section 422 the true subfilter dissipation was computed (determined to be zero or non-zero) forall of the exact solutions to the Navier Stokes equations in Section 22 These solutions can now belsquoplacedrsquo into each class Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a functionof time) and Sol 14 (Radially Dependent) were all found to have zero true subfilter dissipation andare all in the appropriate subsets of either Q7Q8 or Q9 This agrees with Vremanrsquos results
We can extend Vremanrsquos results however We found that Sol 15 (Rankine Vortex) had zero truesubfilter dissipation and yet is not in the subset of Q7 defined in Appendix B It is important tonote that the rate of strain tensor Sij in Eq (41) for Sol 15 (Rankine Vortex) is zero This is thereason the true subfilter dissipation is zero and the same reason that all models are zero for thisflow as all models are based on Sij
We also found that all of the remaining exact solutions to the Navier-Stokes equations are not inVremanrsquos specified classes (in Appendix Section B) They also have a non-zero subfilter dissipationwhich agrees with Vremanrsquos results in his paper [16]
44 Classes of Flows
It is important to consider that some exact solutions to the Navier-Stokes equations may have similarproperties For example Sol 1 (Linear Velocity) and Sol 2 (Planar Flow) are both very similarIn fact Sol 2 (Planar Flow) is a more general case of Sol 1 (Linear Velocity) We therefore mustgroup the different flows into classes in an attempt to reduce the bias that may occur when solutionsare similar This should aid us to further determine which of the models are most endorsed We willdo this in a number of different ways using the theory discussed in Section 23 We will group theflows with respect to Vremanrsquos Flow Classes given in appendix B and with respect to the Invariantsgiven in appendix A
29
441 Vremanrsquos Velocity Gradient Flow Classes
We now put our exact solutions to the Navier-Stokes equations into one of two classes Either Q0minus6or Q7minus9 where each class is defined intuitively as follows
Q0minus6 = Q0 cupQ1 cup middot middot middot cup Q6 (76)
Q7minus9 = Q7 cupQ8 cupQ9 (77)
where Qn is defined in Section 43 By looking at the gradient of each exact solution we can see thatQ7minus9 contains the six solutions Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as afunction of time) Sol 5 (2D Taylor) Sol 14 (Radially Dependent) and Sol 15 (Rankine Vortex)and that Q0minus6 contains the remaining nine We now check the models for LES without explicitfiltering by considering endorsements within each class Tables 6 and 7 are analogues of Table 3 andillustrate the results for each flow class The analogues of Tables 1 and 2 are Tables 22 and 23 andare in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 6 6 6
lsquoDissipationrsquoEndorsements
2 5 6 6 6
Percentage Endorsed 417 833 100 100 100
Table 6 Each entry in the top two rows represents the number of solutions in the class Q7minus9 thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Q0minus6 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 2 3 3
lsquoDissipationrsquoEndorsements
0 0 1 0 3
Percentage Endorsed 56 56 167 167 333
Table 7 Each entry in the top two rows represents the number of solutions in the class Q1minus6 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking results to 1 decimal place if necessary
All of the flows in the class Q7minus9 endorse the QR Gradient and Vortex-Stretching-based subfilter-scale models (see Table 6) We can hence see that this class fully endorses these three models equallyand does not endorse the Smagorinsky model The Vortex-Stretching eddy viscosity model has themost endorsements from any of the flows in the class Q0minus6 and the Smagorinksy and Vreman modelshave the fewest endorsements (see Table 7) We can hence say that this class mostly endorses theVortex-Stretching-based eddy-viscosity model note however that the percentage endorsement is very
30
low
Now let us consider LES with explicit filtering Tables 8 and 9 are analogues of Table 5 and illustratethe results for each flow class The analogue of Table 4 is Table 24 and is in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 1 1 0 0 0
Table 8 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Q0minus6 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 9 9 9 6
Table 9 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We have grouped Vremanrsquos classes into Q7minus9 and Q0minus6 this is equivalent to saying that we havegrouped Vremanrsquos classes into those with two or fewer non-zero velocity gradient elements and thosewith three or more non-zero velocity gradient elements respectively Therefore the solutions in Q7minus9are exactly those with a lsquo0rsquo in the lsquoExactrsquo column of Table 4 and Sol 5 (2D Taylor) ExcludingSol 5 (2D Taylor) these are the same as those five that endorse the Vreman QR Gradient andVortex-Stretching-based models in Table 5 We can see in Table 8 that the results for the lsquostrongrsquoendorsements are exactly the same as in Table 5 This is because all of the flows with zero subfilterdissipation are contained in the class Q7minus9 There are hence no lsquostrongrsquo endorsements in Table 9The majority of the lsquoweakrsquo endorsements come from Q0minus6 as this class contains all but one of thesolutions with non-zero true subfilter dissipation (given in Eq (67))
442 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
We now separate our exact solutions into classes again using Vremanrsquos flow classes This timehowever we define V0 to be the class of flows Vreman claims will always have zero true subfilterdissipation namely those classes mentioned in Appendix B We define Vlowast to be all the remaining
31
classes Then
V0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow)
Sol 3 (Velocity as a function of time) Sol 14 (Radially Dependent)(78)
Vlowast = Sol 4 (Axisymmetric Flow) Sol 5 (2D Taylor) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 13 (Sphere Translation) Sol 15 (Rankine Vortex)
(79)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 10 and 11 are analogues of Table 3 and illustrate the results for each flow class Theanalogues of Tables 1 and 2 are Tables 25 and 26 and are in Appendix C2
Endorsements for models using solutions in V0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 4 4 4 4
lsquoDissipationrsquoEndorsements
1 4 4 4 4
Percentage Endorsed 375 100 100 100 100
Table 10 Each entry in the top two rows represents the number of solutions in the class V0 thatendorse each model There are 4 solutions in this class so the percentages are worked out by dividingthe total of each column by 8 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Vlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 2 4 5 5
lsquoDissipationrsquoEndorsements
1 1 3 2 5
Percentage Endorsed 136 136 318 318 455
Table 11 Each entry in the top two rows represents the number of solutions in the class Vlowast thatendorse each model There are 11 solutions in this class so the percentages are worked out bydividing the total of each column by 22 and taking results to 1 decimal place if necessary
All of the flows in the class V0 endorse the Vreman QR Gradient and Vortex-Stretching-basedsubfilter-scale models (see Table 6) This class completely endorses these four models equally anddoes not endorse the Smagorinsky model The Vortex-Stretching-based eddy-viscosity model hasthe most endorsements from the class Vlowast and the Smagorinksy and Vreman models again have thefewest endorsements (see Table 7) As in Section 441 we say that this class mostly endorses the
32
Vortex-Stretching-based eddy-viscosity model but once again the percentage endorsement is ratherlow
Now let us consider LES with explicit filtering Tables 12 and 13 are analogues of Table 5 andillustrate the results for each flow class The analogue of Table 4 is Table 27 and is in Appendix C2
Endorsements for models using solutions in V0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 4 4 4 4
lsquoWeakrsquo Endorsements 0 0 0 0 0
Table 12 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Vlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 1 1 1 1
lsquoWeakrsquo Endorsements 10 10 9 6
Table 13 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We can see in Table 12 that the results for the lsquostrongrsquo endorsements are similar to those in Table 5excluding Sol 15 (Rankine Vortex) The four solutions in V0 with zero true subfilter dissipationnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)and Sol 14 (Radially Dependent) are hence the four that endorse the Vreman QR Gradientand Vortex-Stretching-based models in Table 12 Sol 15 (Rankine Vortex) is in the class Vlowast andhence there is one lsquostrongrsquo endorsement in Table 13 The class Vlowast contains all of the flows with anon-zero true subfilter dissipation (given in Eq (67)) and hence Table 13 contains all of the lsquoweakrsquoendorsements
443 Classes using the Third Principal Invariant
Sorting the classes using the invariants is a non-trivial task in itself There are a number of waysin which to do this using both the principal invariants in Eq (40) and the combined invariants inEq (43)
We start by looking at flows with third principal invariant RG equal to zero We again define twoclasses R0 and Rlowast where R0 consists of all flows with RG = 0 and Rlowast consists of all flows withRG 6= 0 Then using the information in Appendix A we see that there are nine flows in R0 That
33
is
R0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 8 (Vortex in Stagnation Flow)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)
(80)
Rlowast = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 13 (Sphere Translation)(81)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 14 and 15 are analogues of Table 3 and they illustrate the results for LES withoutexplicit filtering for each flow class The analogues of Tables 1 and 2 are Tables 28 and 29 and arein Appendix C3
Endorsements for models using solutions in R0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 7 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 278 556 778 722 778
Table 14 Each entry in the top two rows represents the number of solutions in the class R0 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking the result to one decimal place if necessary
Endorsements for models using solutions in Rlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 2 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 83 83 83 167 333
Table 15 Each entry in the top two rows represents the number of solutions in the class Rlowast thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking the result to one decimal place if necessary
The class R0 endorses both the Vortex-Stretching-based model and the QR model the most andSmagorinsky the least in the case for LES without explicit filtering In the Rlowast class the Vortex-Stretching-based model is again the most endorsed followed by the Gradient model however thepercentages are again small
34
Now let us consider LES with explicit filtering Tables 16 and 17 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 30 and is in Appendix C3
Endorsements for models using solutions in R0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 4 4 3 2
Table 16 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Rlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 6 6 6 4
Table 17 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 we can see in Table 16 that the results for the lsquostrongrsquo endorsements are exactly thesame as in Table 5 This is because all of the flows with zero subfilter dissipation are contained in theclass R0 As in Table 9 there are no lsquostrongrsquo endorsements in Table 17 The lsquoweakrsquo endorsementshowever are split more evenly between the two classes now than in Vremanrsquos classes in Sections 441and 442 We find that the Smagorinsky and Vreman models still seem to be the most endorsed inboth classes
444 Two-component Flow Classes
Another way in which the flows can be grouped into classes is by determining whether a flow is atwo-component flow or not We define a two-component flow to have the property that the combinedprincipal invariants of a flow in Eq (43) satisfy
I3 = I4 = I5 minus1
2I1I2 = 0 (82)
We are also interested in considering flows with zero gradient model subgrid dissipation These flowshave the property that
I3 minus I4 = 0 (83)
What we find though is that for all the solutions in Section 22 if a flow is a two-component flowthen it also satisfies I3 minus I4 = 0 It may be interesting to note that although I3 minus I4 6= 0 for manysolutions there are a number with I3 equal to a multiple of I4 Although not considered in thisreport this may also be a class of interest We find that the two-component flows are
C2 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)(84)
35
and we call the remaining flows three-component flows They are
C3 = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex) Sol 11 (Pseudo-Motion Shots)
Sol 13 (Sphere Translation)
(85)
Note that Sol 8 (Vortex in Stagnation Flow) is defined to be a planar flow this does not mean that itis a two-component flow in this sense however because its two non-zero velocity components dependon three coordinates We now check the models for LES without explicit filtering by consideringendorsements within each class Tables 18 and 19 are analogues of Table 3 and they illustrate theresults for LES without explicit filtering for each flow class The analogues of Tables 1 and 2 areTables 31 and 32 and are in Appendix C4
Endorsements for models using solutions in C2 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 6 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 357 714 100 857 100
Table 18 Each entry in the top two rows represents the number of solutions in the class C2 thatendorse each model There are 7 solutions in this class so the percentages are worked out by dividingthe total of each column by 14 and taking the result to one decimal place if necessary
Endorsements for models using solutions in C3 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 3 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 63 63 63 188 250
Table 19 Each entry in the top two rows represents the number of solutions in the class C3 thatendorse each model There are 8 solutions in this class so the percentages are worked out by dividingthe total of each column by 16 and taking the result to one decimal place if necessary
The class C2 equally endorses the QR and Vortex-Stretching-based models and once again theSmagorinsky model is the least endorsed As with the previous classes the class C3 has such lowpercentages that it does not seem to endorse any models We start to see a trend appearing here inthat the flow classes that contain the more complicated flows namely Q0minus6 (Eq (76)) Vlowast (Eq (79))Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages The classes that tendto contain the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78)) R0 (Eq (80)) and C2 (Eq (84))all seem to favour the Vortex-Stretching-based model or the QR model
36
Now let us consider LES with explicit filtering Tables 20 and 21 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 33 and is in Appendix C4
Endorsements for models using solutions in C2 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 2 2 0 1 0
Table 20 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in C3 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 8 8 8 6
Table 21 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 and Table 16 we can see in Table 20 that the results for the lsquostrongrsquo endorsements areexactly the same as in Table 5 This is again because all of the flows with zero subfilter dissipationare contained in the class C2 There are two flows with non-zero true subfilter dissipation (givenin Eq (67)) in C2 and this shows in the lsquoweakrsquo endorsements in Table 20 There are no lsquostrongrsquoendorsements in Table 21 as all the zero subfilter dissipation flows are in C2
5 Conclusions
In this report we studied incompressible turbulent flows We found a number of exact solutions to theNavier-Stokes equations each representing a relatively simplistic fluid flow Large-eddy simulation(LES) is used to model turbulent flows by smoothing over the complex small-scale motion Weintroduced a number of subfilter-scale models to approximate this small-scale motion and testedthem using the found exact solutions to the Navier-Stokes equations We first considered LESwithout explicit filtering and used the Navier-Stokes equations to derive conditions for the subfilter-scale models namely the Velocity Field condition (Eq (58)) the Kinetic Energy condition (Eq (60))and the Dissipation condition (Eq (62)) Similarly we considered LES with explicit filtering andagain used the Navier-Stokes equations to derive more conditions for the models in this case wedecided to just take one condition the Dissipation condition (Eqs (67) and (68)) as the others leadto difficulty in their evaluation By lsquorunningrsquo the exact solutions to the Navier-Stokes equationsthrough each condition for the subfilter-scale models we found that the Vortex-Stretching-basededdy-viscosity model in Eq (54) was most endorsed by the tests in LES without explicit filteringWe found that four out of the five subfilter-scale models namely the Vreman QR Gradient andVortex-Stretching-based models were equally endorsed by the lsquostrongrsquo tests in LES with explicitfiltering with the Smagorinsky model being the least endorsed The results we found for the lsquoweakrsquocondition were completely opposite to those found in both the lsquostrongrsquo condition and in LES withoutexplicit filtering This led us to conclude that this test was not one in which we should put our trust
37
We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses
In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes
This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold
38
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
follows in cylindrical coordinates
ur(r ϕ z t) = 0
uϕ(r ϕ z t) = 0
uz(r ϕ z t) = A(r2 minus a2
)
p(r ϕ z t) =4Az
Re
(36)
where A and a are constants In Cartesian coordinates the solution is just the same with u1(x y z t)and u2(x y z t) both zero and the r2 term replaced with x2 + y2 in u3(x y z t)
Figure 17 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus1 1] withparameter values a = 1 A = 4 and Re = 10
Figure 18 An illustration of the flow in 3Dfor Sol 14 (Radially Dependent) describedin Eq (36) for x y z isin [minus2 2] withparameter values a = 1 A = 4 and Re = 10
Solution 15 Rankine Vortex
A solution discovered by Rankine [1] is a vortex flow of the following form
ur(r ϕ z t) = 0
uϕ(r ϕ z t) =ar
2πR2
uz(r ϕ z t) = 0
p(r ϕ z t) =a2r2
8π2R4
(37)
19
where a and R are constants In Cartesian coordinates the flow has the form
u1(x y z t) = minus ay
2πR2
u2(x y z t) =ax
2πR2
u3(x y z t) = 0
p(x y z t) =a2(x2 + y2
)8π2R4
(38)
This flow is actually a rigid body rotational flow in that the angular rotational velocity is uniformand the velocity of the flow increases proportionally with distance from the z-axis
Figure 19 An illustration of the flow in the xy-plane for Sol 15 (Rankine Vortex) described inEq (37) and Eq (38) with parameter values a = 1and R = 1
radic2π
23 Characterisation of Flows
It is important to be able to classify flows in a way that sorts them into distinct groups This aidsin the clear identification and general study of flows It is also a very important way of recognisingflows with similar properties In this subsection we introduce two distinct ways in which flows canbe characterised
231 Characterising Solutions using Invariants
A principal invariant of a tensor is a coefficient of the characteristic polynomial of that tensorInvariants as their name suggests are invariant under rotations of the coordinate system in thattheir values do not change when the tensor is lsquorotatedrsquo into a different set of coordinates Thevelocity gradient of a flow G is defined as the Jacobian of the velocity vector given by
Gij =
[partuipartxj
] (39)
20
for i j = 1 2 3 There are three principal invariants of the velocity gradient of a fluid flow G givenby
PG = tr(G)
QG =1
2
((tr(G))2 minus tr
(G2))
RG = det(G)
(40)
Incompressibility of flows (Eq (5)) ensures that PG = 0 hence the first principal invariant is zerofor all of the flows in Section 22 We now define the rate-of-strain tensor S by
Sij =1
2
(partuipartxj
+partujpartxi
) (41)
and the rate-of-rotation tensor Ω by
Ωij =1
2
(partuipartxjminus partujpartxi
) (42)
Note that the first principal invariant of S and the first and third principal invariants of Ω are zeroAlso note that the velocity gradient of a flow can simply be written G = S + Ω
Rather than considering the principal invariants we can also consider the combined invariants
I1 = tr(S2) I2 = tr(Ω2) I3 = tr(S3) I4 = tr(SΩ2) I5 = tr(S2Ω2) (43)
We can hence write the three principal invariants for a velocity field G in terms of the combinedinvariants
PG = 0
QG = minus1
2(I1 + I2)
RG =1
3(I3 + 3I4)
(44)
In this report the flows considered are the exact solutions to the incompressible Navier-Stokesequations described in Section 22 The combined invariants in Eq (43) are used to characterisethese solutions into different classes in Section 44 Both the principal invariants and combinedinvariants for each exact solution in Section 22 are given in Appendix A
Each of the subfilter-scale eddy-viscosity models described in Section 32 are written in terms ofthese five combined invariants
232 Vremanrsquos Characterisation of Flows
Vreman [16] classifies flows using the gradient of the velocity field of a flow Each lsquoclassrsquo of flows hasthe same number of non-zero elements in the gradient matrix For example
[partuipartxj
]=
0 0 0lowast 0 0lowast 0 0
(45)
where a lowast denotes a non-zero term represents a group of flows that lie within the class denoted byQ7 (see Appendix B) where the subscript represents the number of non-zero terms in the velocity
21
gradient Flows of the form in Eq (45) have a velocity field of the form
u1 = c1 (constant)
u2 = f2(x1)
u3 = f3(x1)
(46)
where f2(x1) and f3(x1) are such that the Navier Stokes equations Eq (1) are satisfied Notethat as the elements on the leading diagonal are zero the incompressibility condition Eq (5) isautomatically satisfied Some simple flow classes are given in Appendix B and contain a number ofthe solutions discussed in Section 22 This is mentioned in greater detail in Section 441
3 Conditions for Large-Eddy Simulation Models
31 Large-Eddy Simulation
Due to the nature of numerical simulation and the complexity of turbulent flows small details aremissed and information is lost when solving the Navier-Stokes equations The computational costof solving a turbulent flow is extremely high so a filtering operation is applied to the Navier-Stokesequations represented by a bar to reduce the computational cost of the simulation This is a linearoperation and preserves constants ie ρ = ρ and is also assumed to commute with differentiation
partuipartxj
=partuipartxj
(47)
The filtered Navier-Stokes equations are found by filtering (putting a bar over) the entire equations(see Eq (1)) and after evaluating are given by
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (48)
where
τij = uiuj minus uiuj (49)
is called the lsquotrue stressrsquo of the flow This however poses a problem as it is not clear what is meantby the term uiuj for a general filtering operation The true stress τij therefore must be replaced bya model τmod
ij in practical applications
32 Subfilter-Scale Models
We will now define the subflter-scale models denoted by τmodij to be used in place of the lsquodifficultrsquo
true stress τij There are a number of models available to use in the literature [11 p7] These willbe tested using the exact solutions to the Navier-Stokes equations and by checking conditions whichare defined in Sections 33 and 34
All the models barring one are of the same form That is each of the eddy-viscosity models arewritten
τmodij minus 1
3τmodkk δij = minus2νeSij (50)
with νe representing the eddy viscosity of a specific model For simplicity we usually just writeτmodij = minus2νeSij the extra term is included here to emphasise that the models are traceless Each
22
of the models we consider here can be expressed in terms of the rate of strain tensor S defined inEq (41) the rate of rotation tensor Ω defined in Eq (42) and the combined invariants defined inEq (43) The models are then defined as follows
Smagorinsky [12] νSe = (CSδ)2radic
2I1 (51)
Vreman [16] νVe = (CV δ)2
radicQGGT
PGGT (52)
QR [13 14 15] νQRe = (CQRδ)2 max0minusI3
I1 (53)
Vortex-Stretching [11] νV Se = (CV Sδ)2radic
2I1
(I5 minus 1
2I1I2
minusI1I2
)32
(54)
with the quantities
PGGT = I1 minus I2 QGGT =1
4(I1 + I2)2 + 4
(I5 minus
1
2I1I2
) (55)
There is a fifth and final model to be considered here It is of a different form to the other four andnot an eddy-viscosity model It is known as the Gradient model and is defined by
Gradient [4 9] τmodG = CGδ
(S2 minus Ω2 minus (SΩminus ΩS)
) (56)
The values C are model constants specific to each model and δ represents the mesh size of the LESBoth of these quantities are included here to properly define the models however in this projectthey will be ignored as they are constants and have no influence on the conditions to be defined
33 Conditions for Large-Eddy Simulation without Explicit Filtering
The filtering operation introduced in Eq (48) is just a formality for LES without explicit filteringThe idea here is that filtering does not actually take place but the term lsquoaddedrsquo to the Navier-Stokesequations in fact lsquosubtractsrsquo the difficult to simulate small-scale turbulent motion of the flow Inorder to consider this properly the terms ui and p in Eq (48) are replaced by vi and q It is hopedthat τmod
ij approximates τij so that vi and q approximate ui and p respectively So LES withoutexplicit filtering is given by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (57)
It is clear to see that the first four terms of this equation are exactly the same as the Navier-Stokesequations so for the exact solutions discussed in Section 22 the combination of these terms willequal zero What is then desirable is that the model term is also zero for these exact solutions sothat Eq (57) still holds In order to test models for exact solutions the condition
part
partxjτmodij (v) = 0 (58)
is checked If this condition were to hold it would imply that the model does not influence thevelocity field of a flow hence we call it the Velocity Field Condition There are a number of otherrestrictions that can be tested One such test comes from the differential form of the evolution ofthe kinetic energy of a flow given in Eq (10) In order to find this Eq (57) is multiplied by vi andterms are grouped to find
part
partt
(1
2vivi
)+ vj
part
partxj
(1
2vivi
)= minusvi
1
ρ
partq
partxi+ νvi
part2vipartxjpartxj
minus vipart
partxjτmodij (v) (59)
23
Then similarly to Eq (58) another condition is to check whether the model affects the kineticenergy of exact solutions this is equivalent to checking whether
vipart
partxjτmodij (v) = 0 (60)
We call this condition the Kinetic Energy Condition The third and final condition considered herecan also be found using the integral form for the equation for kinetic energy
d
dt
intV
1
2vividV =
intS
[minusvj
(1
2vivi
)minus vj
q
ρ+ 2νviSij + viτ
modij (v)
]njdS
+
intV
[minus2νSijSij minus
(minusSijτmod
ij (v))]dV
(61)
where the rate-of-strain tensor Sij was defined in Eq (41) V is a volume of fluid and dS meansthe integral is over the surface of that volume with unit normal nj Then this final condition checkswhether the model affects the total kinetic energy of exact solutions to the Navier-Stokes equationsthat is
Sijτmodij (v) = 0 (62)
We will call this condition the Dissipation Condition because Sijτmodij (v) is often referred to as the
subgrid dissipation
34 Conditions for Large-Eddy Simulation with Explicit Filtering
As in Eq (48) the Navier-Stokes equations can be filtered by applying the filtering operation to thewhole equation (putting a bar over all terms) and evaluating to find again that
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (63)
In Eq (57) the filtering operation was ignored and the elements ui p and τij(u) were replacedwith vi q and τmod
ij (v) respectively Now in LES with explicit filtering the filtering operation itselfsmooths over the small-scale turbulent motion of the simulated flow In this case the elements uip and τij(u) will be replaced with vi q and τmod
ij (v) respectively LES with explicit filtering is thengiven by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (64)
Note that Eqs (63) and (64) despite their similarity to Eq (57) no longer contain the Navier-Stokesequations as Eq (57) did The bar (filtering operation) over each term results in each exact solutionfound in Section 22 not necessarily satisfying the first four terms It is hence unknown whether ornot these terms will vanish for exact solutions Terms can still be compared similarly to Eqs (58)(60) and (62) However instead of comparing to zero we can now compare to the relevant termfrom the filtered Navier-Stokes equations So the terms
part
partxjτij (65)
and
part
partxjτmodij (v) (66)
24
are to be compared Although the latter term is relatively simple to compute the former could leadto some difficulties due to the differentiation of the true stress τij = uiuj minus uiuj So although someinformation could be gleaned from the inspection of these two terms more information can be foundby considering the analogue of Eq (62) So the true subfilter dissipation
Sijτij (67)
(where Sij is shorthand for Sij(v)) and the modelled subfilter dissipation
Sij(v)τmodij (v) (68)
are considered and have to be equal for the condition to hold These can again be derived from theintegral equation for kinetic energy of a filtered velocity field similarly to Eq (61)
4 Analysis of Large-Eddy Simulation Models
In this section we use a code in Mathematica the theory described in Section 3 and the exactsolutions to the Navier-Stokes equations found in Section 22 to check whether or not each of theexact solutions lsquoendorsesrsquo each model for LES both without and with explicit filtering
41 Large-Eddy Simulation Without Explicit Filtering
We start by using a code in Mathematica to test the Velocity Field condition in Eq (58) and theKinetic Energy condition in Eq (60) For all exact solutions to the Navier-Stokes equations inSection 22 the outputs for both conditions were identical so we have combined them into one table
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 ndash ndash 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 1 For the Kinetic Energy condition a lsquo0rsquo result means that vipartpartxj
τmodij (v) = 0 for respective
models and exact solutions whereas a lsquolowastrsquo result means that vipartpartxj
τmodij (v) 6= 0 For the Velocity
Field condition a lsquo0rsquo result means that partpartxj
τmodij (v) = 0 for respective models and exact solutions
whereas a lsquolowastrsquo result means that partpartxj
τmodij (v) 6= 0 The Gradient model can take both positive and
negative values so when the sign of the model is known it is included in the table instead of a lsquorsquoBoth the conditions are included in one table because for all cases the outputs were identical
We then test the Dissipation condition in Eq (62) for all exact solutions to the Navier-Stokesequations in Section 22
25
Dissipation Condition [Does Sijτmodij (v) = 0]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 + 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 2 A lsquo0rsquo result means that Sijτmodij (v) = 0 for respective models and exact solutions whereas
a lsquolowastrsquo result means that Sijτmodij (v) 6= 0 The Gradient model can take both positive and negative
values so when the sign of the model is known it is included in the table instead of a lsquorsquo
We then sum all the results to find the total number of endorsements for each model described inSection 32
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
4 6 8 9 9
lsquoDissipationrsquoEndorsements
2 5 7 7 9
Total Endorsements 6 11 15 16 18
Table 3 The results from Tables 1 and 2 are summarised to give a general overview of how manyendorsements each model receives This makes it much clearer to see the models most supported bythe respective and combined conditions with the Vortex-Stretching model being the most supportedand the Smagorinsky model the least
Testing the models in Section 32 using LES without explicit filtering has mixed results It isclear that the Smagorinksy model is not well endorsed by the flows in Section 22 with the onlyendorsements coming from the simplest flows that is Sol 1 (Linear Velocity) Sol 3 (Velocity as afunction of time) Sol 7 (Stagnation Flow) and Sol 15 (Rankine Vortex) The Vortex-Stretching-based model is the most endorsed flow with eighteen out of a possible thirty endorsements Thisagrees with our foreknowledge as this model was constructed to be as physically consistent as possiblein the paper by Silvis amp Verstappen [11]
42 Large-Eddy Simulation With Explicit Filtering
In this section we will compare the true subfilter dissipation Sijτij and the modelled subfilterdissipation Sij(v)τmod
ij (v) given in Eqs (67) and (68) respectively for each model τmodij and each
exact solution to the Navier-Stokes equation given in Section 22 We will not consider the conditionsinvolving differentiation in Eqs (65) and (66) due to the difficulty in calculating the derivative ofthe τij term in Eq (65)
26
421 Example of a Test for a Model
The true subfilter dissipation Sijτij of a flow given in Eq (67) cannot be explicitly computed asthe filtering operation is kept general What can be done however is to partially evaluate theexpressions using the attributes of a filtering operation mentioned in Eq (48)
The calculations of Sijτij for Sol 1 (Linear Velocity) are provided to illustrate the general ideaRecall that the velocity field of Sol 1 (Linear Velocity) is written in Eq (13) in vector notation as
u(x y z t) =
0ax+ bcx+ d
(69)
Then Sij = Sij(u) is computed simply by using its definition in Eq (41) Note that as filteringpreserves constants if there is an a it is replaced with a meaning that (in this case) there are alarge number of zero terms due to the frequency of 0 Hence
[Sij(u)
]=
[1
2
(partuipartxj
+partujpartxi
)]=
0 a
2c2
a2 0 0
c2 0 0
=
0 a
2c2
a2 0 0
c2 0 0
(70)
Note that as Sijτij is being computed which is equal to the sum of the element-wise multiplicationof the two matrices (or the trace of the matrix multiplication) that we donrsquot need to consider theelements of τij(u) with corresponding zero entry in Sij As τij(u) = uiuj minus uiuj and again asfiltering preserves constants it is found that
[τij(u)] =[uiuj minus uiuj
]=
lowast 0 middot ax+ b 0 middot cx+ d
ax+ b middot 0 lowast lowastcx+ d middot 0 lowast lowast
=
lowast 0 00 lowast lowast0 lowast lowast
(71)
where a lsquolowastrsquo just represents some element (not necessarily non-zero) Therefore the true subfilterdissipation
Sijτij = tr
0 a2
c2
a2 0 0c2 0 0
middotlowast 0 0
0 lowast lowast0 lowast lowast
= 0 (72)
Now the modelled subfilter dissipation Sij(u)τmodij (u) needs to be computed (recall that Sij =
Sij(u)) In this case let us use the QR model stated in Eq (53) Once again Sij(u) calculated inEq (70) needs to be used So we have that
τmodij = minus2(CQRδ)
2 max0minusI3I1
Sij (73)
As
I1 = tr(S2) =
1
2(a2 + c2) and I3 = tr(S
3) = 0 (74)
it holds that τmodij (u) = 0 so Sij(u)τmod
ij (u) = 0 This agrees with Eq (72) and hence Sol 1(Linear Velocity) endorses the QR model
27
422 Testing the Models
Now using a Mathematica code the exact solutions in Section 22 are again used to evaluate thecondition for the models using the process just described in Section 421 The results are describedin Table 4
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 4 All entries with a ldquo0rdquo represent a value of zero and a ldquolowastrdquo represents a value that is non-zeroThe ldquoExactrdquo row represents the value of the true subfilter dissipation Sijτij Every other row witha model name represents the value of the modelled subfilter dissipation Sij(u)τmod
ij (u) respectiveto that model A ldquordquo means that it is unclear as to the value - the sign of these values is oftenunknown and the lsquoMaxrsquo function in the definition of the QR model (in Eq (53)) makes it difficultto evaluate
Reading from Table 4 the Smagorinsky model is clearly endorsed by Sol 3 (Velocity as a function oftime) and Sol 15 (Rankine Vortex) as in both cases both Sijτij = 0 = Sij(u)τmod
ij (u) In this casewe say that Sol 3 (Velocity as a function of time) and Sol 15 (Rankine Vortex) lsquostronglyrsquo endorsethe Smagorinsky model The Smagorinsky model is not endorsed by Sol 1 (Linear Velocity) Sol 2(Planar Flow) or Sol 14 (Radially Dependent) however because Sij(u)τmod
ij (u) 6= 0 For Sol 4 to
13 both Sijτij 6= 0 6= Sij(u)τmodij (u) and so we will say that these exact solutions lsquoweaklyrsquo endorse
the Smagorinsky model
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 10 10 9 6
Table 5 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Interestingly the lsquostrongrsquo and lsquoweakrsquo endorsements give different results However more stock shouldbe put in the results of the lsquostrongerrsquo condition as it is completely clear that zero is equal to zeroIt is not necessarily clear whether or not non-zero equals non-zero as the values will depend on theparameters in each solution hence the lsquoweakerrsquo endorsement
The lsquostrongrsquo condition equally endorses the Vreman QR Gradient and Vortex-Stretching-basedmodels whereas the lsquoweakrsquo condition endorses Vreman and Smagorinsky the most closely followedby the Gradient model with the Vortex-Stretching-based model being the most opposed As this
28
completely contradicts the lsquostrongrsquo condition and the results in Section 41 it makes us question thevalidity of the weaker test The number of endorsements for the QR model is unknown because of thedefinition of the model in Eq (53) and our evaluation of the true subfilter dissipation Eq (67) Dueto the exact solutions we have found it is never clear whether or not the true subfilter dissipationis strictly positive or negative This means that when the true subfilter dissipation is found to benon-zero the lsquoMaxrsquo function in the QR model cannot be evaluated
43 Vremanrsquos True Subfilter Dissipation Claim
Using the theory mentioned in Sections 232 and 34 we can place the exact solutions to the Navier-Stokes equations into different classes defined by Vreman In doing this we can also attempt to verifyone of Vremanrsquos results [16] In Vremanrsquos paper he defines the class Qn to be the set of flows withn zero entries in the velocity gradient of the flow (the velocity gradient is defined in Eq (39))Vreman then claims that the only flows with zero true subfilter dissipation are a certain selection ofthose in the classesQ7Q8 orQ9 shown in appendix B and is generally non-zero for any other flows
The true subfilter dissipation of Sol 1 (Linear Velocity) was computed in Section 421 and wasfound to be zero in Eq (72) The velocity gradient of Sol 1 is as follows
GLV =
0 0 0a 0 0c 0 0
(75)
and hence Sol 1 (Linear Velocity) is in the required subset of Q7 which agrees with Vremanrsquos claimIn Section 422 the true subfilter dissipation was computed (determined to be zero or non-zero) forall of the exact solutions to the Navier Stokes equations in Section 22 These solutions can now belsquoplacedrsquo into each class Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a functionof time) and Sol 14 (Radially Dependent) were all found to have zero true subfilter dissipation andare all in the appropriate subsets of either Q7Q8 or Q9 This agrees with Vremanrsquos results
We can extend Vremanrsquos results however We found that Sol 15 (Rankine Vortex) had zero truesubfilter dissipation and yet is not in the subset of Q7 defined in Appendix B It is important tonote that the rate of strain tensor Sij in Eq (41) for Sol 15 (Rankine Vortex) is zero This is thereason the true subfilter dissipation is zero and the same reason that all models are zero for thisflow as all models are based on Sij
We also found that all of the remaining exact solutions to the Navier-Stokes equations are not inVremanrsquos specified classes (in Appendix Section B) They also have a non-zero subfilter dissipationwhich agrees with Vremanrsquos results in his paper [16]
44 Classes of Flows
It is important to consider that some exact solutions to the Navier-Stokes equations may have similarproperties For example Sol 1 (Linear Velocity) and Sol 2 (Planar Flow) are both very similarIn fact Sol 2 (Planar Flow) is a more general case of Sol 1 (Linear Velocity) We therefore mustgroup the different flows into classes in an attempt to reduce the bias that may occur when solutionsare similar This should aid us to further determine which of the models are most endorsed We willdo this in a number of different ways using the theory discussed in Section 23 We will group theflows with respect to Vremanrsquos Flow Classes given in appendix B and with respect to the Invariantsgiven in appendix A
29
441 Vremanrsquos Velocity Gradient Flow Classes
We now put our exact solutions to the Navier-Stokes equations into one of two classes Either Q0minus6or Q7minus9 where each class is defined intuitively as follows
Q0minus6 = Q0 cupQ1 cup middot middot middot cup Q6 (76)
Q7minus9 = Q7 cupQ8 cupQ9 (77)
where Qn is defined in Section 43 By looking at the gradient of each exact solution we can see thatQ7minus9 contains the six solutions Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as afunction of time) Sol 5 (2D Taylor) Sol 14 (Radially Dependent) and Sol 15 (Rankine Vortex)and that Q0minus6 contains the remaining nine We now check the models for LES without explicitfiltering by considering endorsements within each class Tables 6 and 7 are analogues of Table 3 andillustrate the results for each flow class The analogues of Tables 1 and 2 are Tables 22 and 23 andare in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 6 6 6
lsquoDissipationrsquoEndorsements
2 5 6 6 6
Percentage Endorsed 417 833 100 100 100
Table 6 Each entry in the top two rows represents the number of solutions in the class Q7minus9 thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Q0minus6 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 2 3 3
lsquoDissipationrsquoEndorsements
0 0 1 0 3
Percentage Endorsed 56 56 167 167 333
Table 7 Each entry in the top two rows represents the number of solutions in the class Q1minus6 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking results to 1 decimal place if necessary
All of the flows in the class Q7minus9 endorse the QR Gradient and Vortex-Stretching-based subfilter-scale models (see Table 6) We can hence see that this class fully endorses these three models equallyand does not endorse the Smagorinsky model The Vortex-Stretching eddy viscosity model has themost endorsements from any of the flows in the class Q0minus6 and the Smagorinksy and Vreman modelshave the fewest endorsements (see Table 7) We can hence say that this class mostly endorses theVortex-Stretching-based eddy-viscosity model note however that the percentage endorsement is very
30
low
Now let us consider LES with explicit filtering Tables 8 and 9 are analogues of Table 5 and illustratethe results for each flow class The analogue of Table 4 is Table 24 and is in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 1 1 0 0 0
Table 8 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Q0minus6 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 9 9 9 6
Table 9 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We have grouped Vremanrsquos classes into Q7minus9 and Q0minus6 this is equivalent to saying that we havegrouped Vremanrsquos classes into those with two or fewer non-zero velocity gradient elements and thosewith three or more non-zero velocity gradient elements respectively Therefore the solutions in Q7minus9are exactly those with a lsquo0rsquo in the lsquoExactrsquo column of Table 4 and Sol 5 (2D Taylor) ExcludingSol 5 (2D Taylor) these are the same as those five that endorse the Vreman QR Gradient andVortex-Stretching-based models in Table 5 We can see in Table 8 that the results for the lsquostrongrsquoendorsements are exactly the same as in Table 5 This is because all of the flows with zero subfilterdissipation are contained in the class Q7minus9 There are hence no lsquostrongrsquo endorsements in Table 9The majority of the lsquoweakrsquo endorsements come from Q0minus6 as this class contains all but one of thesolutions with non-zero true subfilter dissipation (given in Eq (67))
442 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
We now separate our exact solutions into classes again using Vremanrsquos flow classes This timehowever we define V0 to be the class of flows Vreman claims will always have zero true subfilterdissipation namely those classes mentioned in Appendix B We define Vlowast to be all the remaining
31
classes Then
V0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow)
Sol 3 (Velocity as a function of time) Sol 14 (Radially Dependent)(78)
Vlowast = Sol 4 (Axisymmetric Flow) Sol 5 (2D Taylor) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 13 (Sphere Translation) Sol 15 (Rankine Vortex)
(79)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 10 and 11 are analogues of Table 3 and illustrate the results for each flow class Theanalogues of Tables 1 and 2 are Tables 25 and 26 and are in Appendix C2
Endorsements for models using solutions in V0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 4 4 4 4
lsquoDissipationrsquoEndorsements
1 4 4 4 4
Percentage Endorsed 375 100 100 100 100
Table 10 Each entry in the top two rows represents the number of solutions in the class V0 thatendorse each model There are 4 solutions in this class so the percentages are worked out by dividingthe total of each column by 8 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Vlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 2 4 5 5
lsquoDissipationrsquoEndorsements
1 1 3 2 5
Percentage Endorsed 136 136 318 318 455
Table 11 Each entry in the top two rows represents the number of solutions in the class Vlowast thatendorse each model There are 11 solutions in this class so the percentages are worked out bydividing the total of each column by 22 and taking results to 1 decimal place if necessary
All of the flows in the class V0 endorse the Vreman QR Gradient and Vortex-Stretching-basedsubfilter-scale models (see Table 6) This class completely endorses these four models equally anddoes not endorse the Smagorinsky model The Vortex-Stretching-based eddy-viscosity model hasthe most endorsements from the class Vlowast and the Smagorinksy and Vreman models again have thefewest endorsements (see Table 7) As in Section 441 we say that this class mostly endorses the
32
Vortex-Stretching-based eddy-viscosity model but once again the percentage endorsement is ratherlow
Now let us consider LES with explicit filtering Tables 12 and 13 are analogues of Table 5 andillustrate the results for each flow class The analogue of Table 4 is Table 27 and is in Appendix C2
Endorsements for models using solutions in V0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 4 4 4 4
lsquoWeakrsquo Endorsements 0 0 0 0 0
Table 12 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Vlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 1 1 1 1
lsquoWeakrsquo Endorsements 10 10 9 6
Table 13 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We can see in Table 12 that the results for the lsquostrongrsquo endorsements are similar to those in Table 5excluding Sol 15 (Rankine Vortex) The four solutions in V0 with zero true subfilter dissipationnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)and Sol 14 (Radially Dependent) are hence the four that endorse the Vreman QR Gradientand Vortex-Stretching-based models in Table 12 Sol 15 (Rankine Vortex) is in the class Vlowast andhence there is one lsquostrongrsquo endorsement in Table 13 The class Vlowast contains all of the flows with anon-zero true subfilter dissipation (given in Eq (67)) and hence Table 13 contains all of the lsquoweakrsquoendorsements
443 Classes using the Third Principal Invariant
Sorting the classes using the invariants is a non-trivial task in itself There are a number of waysin which to do this using both the principal invariants in Eq (40) and the combined invariants inEq (43)
We start by looking at flows with third principal invariant RG equal to zero We again define twoclasses R0 and Rlowast where R0 consists of all flows with RG = 0 and Rlowast consists of all flows withRG 6= 0 Then using the information in Appendix A we see that there are nine flows in R0 That
33
is
R0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 8 (Vortex in Stagnation Flow)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)
(80)
Rlowast = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 13 (Sphere Translation)(81)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 14 and 15 are analogues of Table 3 and they illustrate the results for LES withoutexplicit filtering for each flow class The analogues of Tables 1 and 2 are Tables 28 and 29 and arein Appendix C3
Endorsements for models using solutions in R0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 7 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 278 556 778 722 778
Table 14 Each entry in the top two rows represents the number of solutions in the class R0 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking the result to one decimal place if necessary
Endorsements for models using solutions in Rlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 2 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 83 83 83 167 333
Table 15 Each entry in the top two rows represents the number of solutions in the class Rlowast thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking the result to one decimal place if necessary
The class R0 endorses both the Vortex-Stretching-based model and the QR model the most andSmagorinsky the least in the case for LES without explicit filtering In the Rlowast class the Vortex-Stretching-based model is again the most endorsed followed by the Gradient model however thepercentages are again small
34
Now let us consider LES with explicit filtering Tables 16 and 17 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 30 and is in Appendix C3
Endorsements for models using solutions in R0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 4 4 3 2
Table 16 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Rlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 6 6 6 4
Table 17 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 we can see in Table 16 that the results for the lsquostrongrsquo endorsements are exactly thesame as in Table 5 This is because all of the flows with zero subfilter dissipation are contained in theclass R0 As in Table 9 there are no lsquostrongrsquo endorsements in Table 17 The lsquoweakrsquo endorsementshowever are split more evenly between the two classes now than in Vremanrsquos classes in Sections 441and 442 We find that the Smagorinsky and Vreman models still seem to be the most endorsed inboth classes
444 Two-component Flow Classes
Another way in which the flows can be grouped into classes is by determining whether a flow is atwo-component flow or not We define a two-component flow to have the property that the combinedprincipal invariants of a flow in Eq (43) satisfy
I3 = I4 = I5 minus1
2I1I2 = 0 (82)
We are also interested in considering flows with zero gradient model subgrid dissipation These flowshave the property that
I3 minus I4 = 0 (83)
What we find though is that for all the solutions in Section 22 if a flow is a two-component flowthen it also satisfies I3 minus I4 = 0 It may be interesting to note that although I3 minus I4 6= 0 for manysolutions there are a number with I3 equal to a multiple of I4 Although not considered in thisreport this may also be a class of interest We find that the two-component flows are
C2 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)(84)
35
and we call the remaining flows three-component flows They are
C3 = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex) Sol 11 (Pseudo-Motion Shots)
Sol 13 (Sphere Translation)
(85)
Note that Sol 8 (Vortex in Stagnation Flow) is defined to be a planar flow this does not mean that itis a two-component flow in this sense however because its two non-zero velocity components dependon three coordinates We now check the models for LES without explicit filtering by consideringendorsements within each class Tables 18 and 19 are analogues of Table 3 and they illustrate theresults for LES without explicit filtering for each flow class The analogues of Tables 1 and 2 areTables 31 and 32 and are in Appendix C4
Endorsements for models using solutions in C2 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 6 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 357 714 100 857 100
Table 18 Each entry in the top two rows represents the number of solutions in the class C2 thatendorse each model There are 7 solutions in this class so the percentages are worked out by dividingthe total of each column by 14 and taking the result to one decimal place if necessary
Endorsements for models using solutions in C3 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 3 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 63 63 63 188 250
Table 19 Each entry in the top two rows represents the number of solutions in the class C3 thatendorse each model There are 8 solutions in this class so the percentages are worked out by dividingthe total of each column by 16 and taking the result to one decimal place if necessary
The class C2 equally endorses the QR and Vortex-Stretching-based models and once again theSmagorinsky model is the least endorsed As with the previous classes the class C3 has such lowpercentages that it does not seem to endorse any models We start to see a trend appearing here inthat the flow classes that contain the more complicated flows namely Q0minus6 (Eq (76)) Vlowast (Eq (79))Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages The classes that tendto contain the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78)) R0 (Eq (80)) and C2 (Eq (84))all seem to favour the Vortex-Stretching-based model or the QR model
36
Now let us consider LES with explicit filtering Tables 20 and 21 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 33 and is in Appendix C4
Endorsements for models using solutions in C2 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 2 2 0 1 0
Table 20 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in C3 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 8 8 8 6
Table 21 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 and Table 16 we can see in Table 20 that the results for the lsquostrongrsquo endorsements areexactly the same as in Table 5 This is again because all of the flows with zero subfilter dissipationare contained in the class C2 There are two flows with non-zero true subfilter dissipation (givenin Eq (67)) in C2 and this shows in the lsquoweakrsquo endorsements in Table 20 There are no lsquostrongrsquoendorsements in Table 21 as all the zero subfilter dissipation flows are in C2
5 Conclusions
In this report we studied incompressible turbulent flows We found a number of exact solutions to theNavier-Stokes equations each representing a relatively simplistic fluid flow Large-eddy simulation(LES) is used to model turbulent flows by smoothing over the complex small-scale motion Weintroduced a number of subfilter-scale models to approximate this small-scale motion and testedthem using the found exact solutions to the Navier-Stokes equations We first considered LESwithout explicit filtering and used the Navier-Stokes equations to derive conditions for the subfilter-scale models namely the Velocity Field condition (Eq (58)) the Kinetic Energy condition (Eq (60))and the Dissipation condition (Eq (62)) Similarly we considered LES with explicit filtering andagain used the Navier-Stokes equations to derive more conditions for the models in this case wedecided to just take one condition the Dissipation condition (Eqs (67) and (68)) as the others leadto difficulty in their evaluation By lsquorunningrsquo the exact solutions to the Navier-Stokes equationsthrough each condition for the subfilter-scale models we found that the Vortex-Stretching-basededdy-viscosity model in Eq (54) was most endorsed by the tests in LES without explicit filteringWe found that four out of the five subfilter-scale models namely the Vreman QR Gradient andVortex-Stretching-based models were equally endorsed by the lsquostrongrsquo tests in LES with explicitfiltering with the Smagorinsky model being the least endorsed The results we found for the lsquoweakrsquocondition were completely opposite to those found in both the lsquostrongrsquo condition and in LES withoutexplicit filtering This led us to conclude that this test was not one in which we should put our trust
37
We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses
In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes
This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold
38
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
where a and R are constants In Cartesian coordinates the flow has the form
u1(x y z t) = minus ay
2πR2
u2(x y z t) =ax
2πR2
u3(x y z t) = 0
p(x y z t) =a2(x2 + y2
)8π2R4
(38)
This flow is actually a rigid body rotational flow in that the angular rotational velocity is uniformand the velocity of the flow increases proportionally with distance from the z-axis
Figure 19 An illustration of the flow in the xy-plane for Sol 15 (Rankine Vortex) described inEq (37) and Eq (38) with parameter values a = 1and R = 1
radic2π
23 Characterisation of Flows
It is important to be able to classify flows in a way that sorts them into distinct groups This aidsin the clear identification and general study of flows It is also a very important way of recognisingflows with similar properties In this subsection we introduce two distinct ways in which flows canbe characterised
231 Characterising Solutions using Invariants
A principal invariant of a tensor is a coefficient of the characteristic polynomial of that tensorInvariants as their name suggests are invariant under rotations of the coordinate system in thattheir values do not change when the tensor is lsquorotatedrsquo into a different set of coordinates Thevelocity gradient of a flow G is defined as the Jacobian of the velocity vector given by
Gij =
[partuipartxj
] (39)
20
for i j = 1 2 3 There are three principal invariants of the velocity gradient of a fluid flow G givenby
PG = tr(G)
QG =1
2
((tr(G))2 minus tr
(G2))
RG = det(G)
(40)
Incompressibility of flows (Eq (5)) ensures that PG = 0 hence the first principal invariant is zerofor all of the flows in Section 22 We now define the rate-of-strain tensor S by
Sij =1
2
(partuipartxj
+partujpartxi
) (41)
and the rate-of-rotation tensor Ω by
Ωij =1
2
(partuipartxjminus partujpartxi
) (42)
Note that the first principal invariant of S and the first and third principal invariants of Ω are zeroAlso note that the velocity gradient of a flow can simply be written G = S + Ω
Rather than considering the principal invariants we can also consider the combined invariants
I1 = tr(S2) I2 = tr(Ω2) I3 = tr(S3) I4 = tr(SΩ2) I5 = tr(S2Ω2) (43)
We can hence write the three principal invariants for a velocity field G in terms of the combinedinvariants
PG = 0
QG = minus1
2(I1 + I2)
RG =1
3(I3 + 3I4)
(44)
In this report the flows considered are the exact solutions to the incompressible Navier-Stokesequations described in Section 22 The combined invariants in Eq (43) are used to characterisethese solutions into different classes in Section 44 Both the principal invariants and combinedinvariants for each exact solution in Section 22 are given in Appendix A
Each of the subfilter-scale eddy-viscosity models described in Section 32 are written in terms ofthese five combined invariants
232 Vremanrsquos Characterisation of Flows
Vreman [16] classifies flows using the gradient of the velocity field of a flow Each lsquoclassrsquo of flows hasthe same number of non-zero elements in the gradient matrix For example
[partuipartxj
]=
0 0 0lowast 0 0lowast 0 0
(45)
where a lowast denotes a non-zero term represents a group of flows that lie within the class denoted byQ7 (see Appendix B) where the subscript represents the number of non-zero terms in the velocity
21
gradient Flows of the form in Eq (45) have a velocity field of the form
u1 = c1 (constant)
u2 = f2(x1)
u3 = f3(x1)
(46)
where f2(x1) and f3(x1) are such that the Navier Stokes equations Eq (1) are satisfied Notethat as the elements on the leading diagonal are zero the incompressibility condition Eq (5) isautomatically satisfied Some simple flow classes are given in Appendix B and contain a number ofthe solutions discussed in Section 22 This is mentioned in greater detail in Section 441
3 Conditions for Large-Eddy Simulation Models
31 Large-Eddy Simulation
Due to the nature of numerical simulation and the complexity of turbulent flows small details aremissed and information is lost when solving the Navier-Stokes equations The computational costof solving a turbulent flow is extremely high so a filtering operation is applied to the Navier-Stokesequations represented by a bar to reduce the computational cost of the simulation This is a linearoperation and preserves constants ie ρ = ρ and is also assumed to commute with differentiation
partuipartxj
=partuipartxj
(47)
The filtered Navier-Stokes equations are found by filtering (putting a bar over) the entire equations(see Eq (1)) and after evaluating are given by
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (48)
where
τij = uiuj minus uiuj (49)
is called the lsquotrue stressrsquo of the flow This however poses a problem as it is not clear what is meantby the term uiuj for a general filtering operation The true stress τij therefore must be replaced bya model τmod
ij in practical applications
32 Subfilter-Scale Models
We will now define the subflter-scale models denoted by τmodij to be used in place of the lsquodifficultrsquo
true stress τij There are a number of models available to use in the literature [11 p7] These willbe tested using the exact solutions to the Navier-Stokes equations and by checking conditions whichare defined in Sections 33 and 34
All the models barring one are of the same form That is each of the eddy-viscosity models arewritten
τmodij minus 1
3τmodkk δij = minus2νeSij (50)
with νe representing the eddy viscosity of a specific model For simplicity we usually just writeτmodij = minus2νeSij the extra term is included here to emphasise that the models are traceless Each
22
of the models we consider here can be expressed in terms of the rate of strain tensor S defined inEq (41) the rate of rotation tensor Ω defined in Eq (42) and the combined invariants defined inEq (43) The models are then defined as follows
Smagorinsky [12] νSe = (CSδ)2radic
2I1 (51)
Vreman [16] νVe = (CV δ)2
radicQGGT
PGGT (52)
QR [13 14 15] νQRe = (CQRδ)2 max0minusI3
I1 (53)
Vortex-Stretching [11] νV Se = (CV Sδ)2radic
2I1
(I5 minus 1
2I1I2
minusI1I2
)32
(54)
with the quantities
PGGT = I1 minus I2 QGGT =1
4(I1 + I2)2 + 4
(I5 minus
1
2I1I2
) (55)
There is a fifth and final model to be considered here It is of a different form to the other four andnot an eddy-viscosity model It is known as the Gradient model and is defined by
Gradient [4 9] τmodG = CGδ
(S2 minus Ω2 minus (SΩminus ΩS)
) (56)
The values C are model constants specific to each model and δ represents the mesh size of the LESBoth of these quantities are included here to properly define the models however in this projectthey will be ignored as they are constants and have no influence on the conditions to be defined
33 Conditions for Large-Eddy Simulation without Explicit Filtering
The filtering operation introduced in Eq (48) is just a formality for LES without explicit filteringThe idea here is that filtering does not actually take place but the term lsquoaddedrsquo to the Navier-Stokesequations in fact lsquosubtractsrsquo the difficult to simulate small-scale turbulent motion of the flow Inorder to consider this properly the terms ui and p in Eq (48) are replaced by vi and q It is hopedthat τmod
ij approximates τij so that vi and q approximate ui and p respectively So LES withoutexplicit filtering is given by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (57)
It is clear to see that the first four terms of this equation are exactly the same as the Navier-Stokesequations so for the exact solutions discussed in Section 22 the combination of these terms willequal zero What is then desirable is that the model term is also zero for these exact solutions sothat Eq (57) still holds In order to test models for exact solutions the condition
part
partxjτmodij (v) = 0 (58)
is checked If this condition were to hold it would imply that the model does not influence thevelocity field of a flow hence we call it the Velocity Field Condition There are a number of otherrestrictions that can be tested One such test comes from the differential form of the evolution ofthe kinetic energy of a flow given in Eq (10) In order to find this Eq (57) is multiplied by vi andterms are grouped to find
part
partt
(1
2vivi
)+ vj
part
partxj
(1
2vivi
)= minusvi
1
ρ
partq
partxi+ νvi
part2vipartxjpartxj
minus vipart
partxjτmodij (v) (59)
23
Then similarly to Eq (58) another condition is to check whether the model affects the kineticenergy of exact solutions this is equivalent to checking whether
vipart
partxjτmodij (v) = 0 (60)
We call this condition the Kinetic Energy Condition The third and final condition considered herecan also be found using the integral form for the equation for kinetic energy
d
dt
intV
1
2vividV =
intS
[minusvj
(1
2vivi
)minus vj
q
ρ+ 2νviSij + viτ
modij (v)
]njdS
+
intV
[minus2νSijSij minus
(minusSijτmod
ij (v))]dV
(61)
where the rate-of-strain tensor Sij was defined in Eq (41) V is a volume of fluid and dS meansthe integral is over the surface of that volume with unit normal nj Then this final condition checkswhether the model affects the total kinetic energy of exact solutions to the Navier-Stokes equationsthat is
Sijτmodij (v) = 0 (62)
We will call this condition the Dissipation Condition because Sijτmodij (v) is often referred to as the
subgrid dissipation
34 Conditions for Large-Eddy Simulation with Explicit Filtering
As in Eq (48) the Navier-Stokes equations can be filtered by applying the filtering operation to thewhole equation (putting a bar over all terms) and evaluating to find again that
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (63)
In Eq (57) the filtering operation was ignored and the elements ui p and τij(u) were replacedwith vi q and τmod
ij (v) respectively Now in LES with explicit filtering the filtering operation itselfsmooths over the small-scale turbulent motion of the simulated flow In this case the elements uip and τij(u) will be replaced with vi q and τmod
ij (v) respectively LES with explicit filtering is thengiven by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (64)
Note that Eqs (63) and (64) despite their similarity to Eq (57) no longer contain the Navier-Stokesequations as Eq (57) did The bar (filtering operation) over each term results in each exact solutionfound in Section 22 not necessarily satisfying the first four terms It is hence unknown whether ornot these terms will vanish for exact solutions Terms can still be compared similarly to Eqs (58)(60) and (62) However instead of comparing to zero we can now compare to the relevant termfrom the filtered Navier-Stokes equations So the terms
part
partxjτij (65)
and
part
partxjτmodij (v) (66)
24
are to be compared Although the latter term is relatively simple to compute the former could leadto some difficulties due to the differentiation of the true stress τij = uiuj minus uiuj So although someinformation could be gleaned from the inspection of these two terms more information can be foundby considering the analogue of Eq (62) So the true subfilter dissipation
Sijτij (67)
(where Sij is shorthand for Sij(v)) and the modelled subfilter dissipation
Sij(v)τmodij (v) (68)
are considered and have to be equal for the condition to hold These can again be derived from theintegral equation for kinetic energy of a filtered velocity field similarly to Eq (61)
4 Analysis of Large-Eddy Simulation Models
In this section we use a code in Mathematica the theory described in Section 3 and the exactsolutions to the Navier-Stokes equations found in Section 22 to check whether or not each of theexact solutions lsquoendorsesrsquo each model for LES both without and with explicit filtering
41 Large-Eddy Simulation Without Explicit Filtering
We start by using a code in Mathematica to test the Velocity Field condition in Eq (58) and theKinetic Energy condition in Eq (60) For all exact solutions to the Navier-Stokes equations inSection 22 the outputs for both conditions were identical so we have combined them into one table
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 ndash ndash 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 1 For the Kinetic Energy condition a lsquo0rsquo result means that vipartpartxj
τmodij (v) = 0 for respective
models and exact solutions whereas a lsquolowastrsquo result means that vipartpartxj
τmodij (v) 6= 0 For the Velocity
Field condition a lsquo0rsquo result means that partpartxj
τmodij (v) = 0 for respective models and exact solutions
whereas a lsquolowastrsquo result means that partpartxj
τmodij (v) 6= 0 The Gradient model can take both positive and
negative values so when the sign of the model is known it is included in the table instead of a lsquorsquoBoth the conditions are included in one table because for all cases the outputs were identical
We then test the Dissipation condition in Eq (62) for all exact solutions to the Navier-Stokesequations in Section 22
25
Dissipation Condition [Does Sijτmodij (v) = 0]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 + 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 2 A lsquo0rsquo result means that Sijτmodij (v) = 0 for respective models and exact solutions whereas
a lsquolowastrsquo result means that Sijτmodij (v) 6= 0 The Gradient model can take both positive and negative
values so when the sign of the model is known it is included in the table instead of a lsquorsquo
We then sum all the results to find the total number of endorsements for each model described inSection 32
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
4 6 8 9 9
lsquoDissipationrsquoEndorsements
2 5 7 7 9
Total Endorsements 6 11 15 16 18
Table 3 The results from Tables 1 and 2 are summarised to give a general overview of how manyendorsements each model receives This makes it much clearer to see the models most supported bythe respective and combined conditions with the Vortex-Stretching model being the most supportedand the Smagorinsky model the least
Testing the models in Section 32 using LES without explicit filtering has mixed results It isclear that the Smagorinksy model is not well endorsed by the flows in Section 22 with the onlyendorsements coming from the simplest flows that is Sol 1 (Linear Velocity) Sol 3 (Velocity as afunction of time) Sol 7 (Stagnation Flow) and Sol 15 (Rankine Vortex) The Vortex-Stretching-based model is the most endorsed flow with eighteen out of a possible thirty endorsements Thisagrees with our foreknowledge as this model was constructed to be as physically consistent as possiblein the paper by Silvis amp Verstappen [11]
42 Large-Eddy Simulation With Explicit Filtering
In this section we will compare the true subfilter dissipation Sijτij and the modelled subfilterdissipation Sij(v)τmod
ij (v) given in Eqs (67) and (68) respectively for each model τmodij and each
exact solution to the Navier-Stokes equation given in Section 22 We will not consider the conditionsinvolving differentiation in Eqs (65) and (66) due to the difficulty in calculating the derivative ofthe τij term in Eq (65)
26
421 Example of a Test for a Model
The true subfilter dissipation Sijτij of a flow given in Eq (67) cannot be explicitly computed asthe filtering operation is kept general What can be done however is to partially evaluate theexpressions using the attributes of a filtering operation mentioned in Eq (48)
The calculations of Sijτij for Sol 1 (Linear Velocity) are provided to illustrate the general ideaRecall that the velocity field of Sol 1 (Linear Velocity) is written in Eq (13) in vector notation as
u(x y z t) =
0ax+ bcx+ d
(69)
Then Sij = Sij(u) is computed simply by using its definition in Eq (41) Note that as filteringpreserves constants if there is an a it is replaced with a meaning that (in this case) there are alarge number of zero terms due to the frequency of 0 Hence
[Sij(u)
]=
[1
2
(partuipartxj
+partujpartxi
)]=
0 a
2c2
a2 0 0
c2 0 0
=
0 a
2c2
a2 0 0
c2 0 0
(70)
Note that as Sijτij is being computed which is equal to the sum of the element-wise multiplicationof the two matrices (or the trace of the matrix multiplication) that we donrsquot need to consider theelements of τij(u) with corresponding zero entry in Sij As τij(u) = uiuj minus uiuj and again asfiltering preserves constants it is found that
[τij(u)] =[uiuj minus uiuj
]=
lowast 0 middot ax+ b 0 middot cx+ d
ax+ b middot 0 lowast lowastcx+ d middot 0 lowast lowast
=
lowast 0 00 lowast lowast0 lowast lowast
(71)
where a lsquolowastrsquo just represents some element (not necessarily non-zero) Therefore the true subfilterdissipation
Sijτij = tr
0 a2
c2
a2 0 0c2 0 0
middotlowast 0 0
0 lowast lowast0 lowast lowast
= 0 (72)
Now the modelled subfilter dissipation Sij(u)τmodij (u) needs to be computed (recall that Sij =
Sij(u)) In this case let us use the QR model stated in Eq (53) Once again Sij(u) calculated inEq (70) needs to be used So we have that
τmodij = minus2(CQRδ)
2 max0minusI3I1
Sij (73)
As
I1 = tr(S2) =
1
2(a2 + c2) and I3 = tr(S
3) = 0 (74)
it holds that τmodij (u) = 0 so Sij(u)τmod
ij (u) = 0 This agrees with Eq (72) and hence Sol 1(Linear Velocity) endorses the QR model
27
422 Testing the Models
Now using a Mathematica code the exact solutions in Section 22 are again used to evaluate thecondition for the models using the process just described in Section 421 The results are describedin Table 4
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 4 All entries with a ldquo0rdquo represent a value of zero and a ldquolowastrdquo represents a value that is non-zeroThe ldquoExactrdquo row represents the value of the true subfilter dissipation Sijτij Every other row witha model name represents the value of the modelled subfilter dissipation Sij(u)τmod
ij (u) respectiveto that model A ldquordquo means that it is unclear as to the value - the sign of these values is oftenunknown and the lsquoMaxrsquo function in the definition of the QR model (in Eq (53)) makes it difficultto evaluate
Reading from Table 4 the Smagorinsky model is clearly endorsed by Sol 3 (Velocity as a function oftime) and Sol 15 (Rankine Vortex) as in both cases both Sijτij = 0 = Sij(u)τmod
ij (u) In this casewe say that Sol 3 (Velocity as a function of time) and Sol 15 (Rankine Vortex) lsquostronglyrsquo endorsethe Smagorinsky model The Smagorinsky model is not endorsed by Sol 1 (Linear Velocity) Sol 2(Planar Flow) or Sol 14 (Radially Dependent) however because Sij(u)τmod
ij (u) 6= 0 For Sol 4 to
13 both Sijτij 6= 0 6= Sij(u)τmodij (u) and so we will say that these exact solutions lsquoweaklyrsquo endorse
the Smagorinsky model
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 10 10 9 6
Table 5 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Interestingly the lsquostrongrsquo and lsquoweakrsquo endorsements give different results However more stock shouldbe put in the results of the lsquostrongerrsquo condition as it is completely clear that zero is equal to zeroIt is not necessarily clear whether or not non-zero equals non-zero as the values will depend on theparameters in each solution hence the lsquoweakerrsquo endorsement
The lsquostrongrsquo condition equally endorses the Vreman QR Gradient and Vortex-Stretching-basedmodels whereas the lsquoweakrsquo condition endorses Vreman and Smagorinsky the most closely followedby the Gradient model with the Vortex-Stretching-based model being the most opposed As this
28
completely contradicts the lsquostrongrsquo condition and the results in Section 41 it makes us question thevalidity of the weaker test The number of endorsements for the QR model is unknown because of thedefinition of the model in Eq (53) and our evaluation of the true subfilter dissipation Eq (67) Dueto the exact solutions we have found it is never clear whether or not the true subfilter dissipationis strictly positive or negative This means that when the true subfilter dissipation is found to benon-zero the lsquoMaxrsquo function in the QR model cannot be evaluated
43 Vremanrsquos True Subfilter Dissipation Claim
Using the theory mentioned in Sections 232 and 34 we can place the exact solutions to the Navier-Stokes equations into different classes defined by Vreman In doing this we can also attempt to verifyone of Vremanrsquos results [16] In Vremanrsquos paper he defines the class Qn to be the set of flows withn zero entries in the velocity gradient of the flow (the velocity gradient is defined in Eq (39))Vreman then claims that the only flows with zero true subfilter dissipation are a certain selection ofthose in the classesQ7Q8 orQ9 shown in appendix B and is generally non-zero for any other flows
The true subfilter dissipation of Sol 1 (Linear Velocity) was computed in Section 421 and wasfound to be zero in Eq (72) The velocity gradient of Sol 1 is as follows
GLV =
0 0 0a 0 0c 0 0
(75)
and hence Sol 1 (Linear Velocity) is in the required subset of Q7 which agrees with Vremanrsquos claimIn Section 422 the true subfilter dissipation was computed (determined to be zero or non-zero) forall of the exact solutions to the Navier Stokes equations in Section 22 These solutions can now belsquoplacedrsquo into each class Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a functionof time) and Sol 14 (Radially Dependent) were all found to have zero true subfilter dissipation andare all in the appropriate subsets of either Q7Q8 or Q9 This agrees with Vremanrsquos results
We can extend Vremanrsquos results however We found that Sol 15 (Rankine Vortex) had zero truesubfilter dissipation and yet is not in the subset of Q7 defined in Appendix B It is important tonote that the rate of strain tensor Sij in Eq (41) for Sol 15 (Rankine Vortex) is zero This is thereason the true subfilter dissipation is zero and the same reason that all models are zero for thisflow as all models are based on Sij
We also found that all of the remaining exact solutions to the Navier-Stokes equations are not inVremanrsquos specified classes (in Appendix Section B) They also have a non-zero subfilter dissipationwhich agrees with Vremanrsquos results in his paper [16]
44 Classes of Flows
It is important to consider that some exact solutions to the Navier-Stokes equations may have similarproperties For example Sol 1 (Linear Velocity) and Sol 2 (Planar Flow) are both very similarIn fact Sol 2 (Planar Flow) is a more general case of Sol 1 (Linear Velocity) We therefore mustgroup the different flows into classes in an attempt to reduce the bias that may occur when solutionsare similar This should aid us to further determine which of the models are most endorsed We willdo this in a number of different ways using the theory discussed in Section 23 We will group theflows with respect to Vremanrsquos Flow Classes given in appendix B and with respect to the Invariantsgiven in appendix A
29
441 Vremanrsquos Velocity Gradient Flow Classes
We now put our exact solutions to the Navier-Stokes equations into one of two classes Either Q0minus6or Q7minus9 where each class is defined intuitively as follows
Q0minus6 = Q0 cupQ1 cup middot middot middot cup Q6 (76)
Q7minus9 = Q7 cupQ8 cupQ9 (77)
where Qn is defined in Section 43 By looking at the gradient of each exact solution we can see thatQ7minus9 contains the six solutions Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as afunction of time) Sol 5 (2D Taylor) Sol 14 (Radially Dependent) and Sol 15 (Rankine Vortex)and that Q0minus6 contains the remaining nine We now check the models for LES without explicitfiltering by considering endorsements within each class Tables 6 and 7 are analogues of Table 3 andillustrate the results for each flow class The analogues of Tables 1 and 2 are Tables 22 and 23 andare in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 6 6 6
lsquoDissipationrsquoEndorsements
2 5 6 6 6
Percentage Endorsed 417 833 100 100 100
Table 6 Each entry in the top two rows represents the number of solutions in the class Q7minus9 thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Q0minus6 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 2 3 3
lsquoDissipationrsquoEndorsements
0 0 1 0 3
Percentage Endorsed 56 56 167 167 333
Table 7 Each entry in the top two rows represents the number of solutions in the class Q1minus6 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking results to 1 decimal place if necessary
All of the flows in the class Q7minus9 endorse the QR Gradient and Vortex-Stretching-based subfilter-scale models (see Table 6) We can hence see that this class fully endorses these three models equallyand does not endorse the Smagorinsky model The Vortex-Stretching eddy viscosity model has themost endorsements from any of the flows in the class Q0minus6 and the Smagorinksy and Vreman modelshave the fewest endorsements (see Table 7) We can hence say that this class mostly endorses theVortex-Stretching-based eddy-viscosity model note however that the percentage endorsement is very
30
low
Now let us consider LES with explicit filtering Tables 8 and 9 are analogues of Table 5 and illustratethe results for each flow class The analogue of Table 4 is Table 24 and is in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 1 1 0 0 0
Table 8 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Q0minus6 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 9 9 9 6
Table 9 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We have grouped Vremanrsquos classes into Q7minus9 and Q0minus6 this is equivalent to saying that we havegrouped Vremanrsquos classes into those with two or fewer non-zero velocity gradient elements and thosewith three or more non-zero velocity gradient elements respectively Therefore the solutions in Q7minus9are exactly those with a lsquo0rsquo in the lsquoExactrsquo column of Table 4 and Sol 5 (2D Taylor) ExcludingSol 5 (2D Taylor) these are the same as those five that endorse the Vreman QR Gradient andVortex-Stretching-based models in Table 5 We can see in Table 8 that the results for the lsquostrongrsquoendorsements are exactly the same as in Table 5 This is because all of the flows with zero subfilterdissipation are contained in the class Q7minus9 There are hence no lsquostrongrsquo endorsements in Table 9The majority of the lsquoweakrsquo endorsements come from Q0minus6 as this class contains all but one of thesolutions with non-zero true subfilter dissipation (given in Eq (67))
442 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
We now separate our exact solutions into classes again using Vremanrsquos flow classes This timehowever we define V0 to be the class of flows Vreman claims will always have zero true subfilterdissipation namely those classes mentioned in Appendix B We define Vlowast to be all the remaining
31
classes Then
V0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow)
Sol 3 (Velocity as a function of time) Sol 14 (Radially Dependent)(78)
Vlowast = Sol 4 (Axisymmetric Flow) Sol 5 (2D Taylor) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 13 (Sphere Translation) Sol 15 (Rankine Vortex)
(79)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 10 and 11 are analogues of Table 3 and illustrate the results for each flow class Theanalogues of Tables 1 and 2 are Tables 25 and 26 and are in Appendix C2
Endorsements for models using solutions in V0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 4 4 4 4
lsquoDissipationrsquoEndorsements
1 4 4 4 4
Percentage Endorsed 375 100 100 100 100
Table 10 Each entry in the top two rows represents the number of solutions in the class V0 thatendorse each model There are 4 solutions in this class so the percentages are worked out by dividingthe total of each column by 8 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Vlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 2 4 5 5
lsquoDissipationrsquoEndorsements
1 1 3 2 5
Percentage Endorsed 136 136 318 318 455
Table 11 Each entry in the top two rows represents the number of solutions in the class Vlowast thatendorse each model There are 11 solutions in this class so the percentages are worked out bydividing the total of each column by 22 and taking results to 1 decimal place if necessary
All of the flows in the class V0 endorse the Vreman QR Gradient and Vortex-Stretching-basedsubfilter-scale models (see Table 6) This class completely endorses these four models equally anddoes not endorse the Smagorinsky model The Vortex-Stretching-based eddy-viscosity model hasthe most endorsements from the class Vlowast and the Smagorinksy and Vreman models again have thefewest endorsements (see Table 7) As in Section 441 we say that this class mostly endorses the
32
Vortex-Stretching-based eddy-viscosity model but once again the percentage endorsement is ratherlow
Now let us consider LES with explicit filtering Tables 12 and 13 are analogues of Table 5 andillustrate the results for each flow class The analogue of Table 4 is Table 27 and is in Appendix C2
Endorsements for models using solutions in V0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 4 4 4 4
lsquoWeakrsquo Endorsements 0 0 0 0 0
Table 12 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Vlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 1 1 1 1
lsquoWeakrsquo Endorsements 10 10 9 6
Table 13 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We can see in Table 12 that the results for the lsquostrongrsquo endorsements are similar to those in Table 5excluding Sol 15 (Rankine Vortex) The four solutions in V0 with zero true subfilter dissipationnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)and Sol 14 (Radially Dependent) are hence the four that endorse the Vreman QR Gradientand Vortex-Stretching-based models in Table 12 Sol 15 (Rankine Vortex) is in the class Vlowast andhence there is one lsquostrongrsquo endorsement in Table 13 The class Vlowast contains all of the flows with anon-zero true subfilter dissipation (given in Eq (67)) and hence Table 13 contains all of the lsquoweakrsquoendorsements
443 Classes using the Third Principal Invariant
Sorting the classes using the invariants is a non-trivial task in itself There are a number of waysin which to do this using both the principal invariants in Eq (40) and the combined invariants inEq (43)
We start by looking at flows with third principal invariant RG equal to zero We again define twoclasses R0 and Rlowast where R0 consists of all flows with RG = 0 and Rlowast consists of all flows withRG 6= 0 Then using the information in Appendix A we see that there are nine flows in R0 That
33
is
R0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 8 (Vortex in Stagnation Flow)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)
(80)
Rlowast = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 13 (Sphere Translation)(81)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 14 and 15 are analogues of Table 3 and they illustrate the results for LES withoutexplicit filtering for each flow class The analogues of Tables 1 and 2 are Tables 28 and 29 and arein Appendix C3
Endorsements for models using solutions in R0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 7 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 278 556 778 722 778
Table 14 Each entry in the top two rows represents the number of solutions in the class R0 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking the result to one decimal place if necessary
Endorsements for models using solutions in Rlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 2 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 83 83 83 167 333
Table 15 Each entry in the top two rows represents the number of solutions in the class Rlowast thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking the result to one decimal place if necessary
The class R0 endorses both the Vortex-Stretching-based model and the QR model the most andSmagorinsky the least in the case for LES without explicit filtering In the Rlowast class the Vortex-Stretching-based model is again the most endorsed followed by the Gradient model however thepercentages are again small
34
Now let us consider LES with explicit filtering Tables 16 and 17 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 30 and is in Appendix C3
Endorsements for models using solutions in R0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 4 4 3 2
Table 16 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Rlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 6 6 6 4
Table 17 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 we can see in Table 16 that the results for the lsquostrongrsquo endorsements are exactly thesame as in Table 5 This is because all of the flows with zero subfilter dissipation are contained in theclass R0 As in Table 9 there are no lsquostrongrsquo endorsements in Table 17 The lsquoweakrsquo endorsementshowever are split more evenly between the two classes now than in Vremanrsquos classes in Sections 441and 442 We find that the Smagorinsky and Vreman models still seem to be the most endorsed inboth classes
444 Two-component Flow Classes
Another way in which the flows can be grouped into classes is by determining whether a flow is atwo-component flow or not We define a two-component flow to have the property that the combinedprincipal invariants of a flow in Eq (43) satisfy
I3 = I4 = I5 minus1
2I1I2 = 0 (82)
We are also interested in considering flows with zero gradient model subgrid dissipation These flowshave the property that
I3 minus I4 = 0 (83)
What we find though is that for all the solutions in Section 22 if a flow is a two-component flowthen it also satisfies I3 minus I4 = 0 It may be interesting to note that although I3 minus I4 6= 0 for manysolutions there are a number with I3 equal to a multiple of I4 Although not considered in thisreport this may also be a class of interest We find that the two-component flows are
C2 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)(84)
35
and we call the remaining flows three-component flows They are
C3 = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex) Sol 11 (Pseudo-Motion Shots)
Sol 13 (Sphere Translation)
(85)
Note that Sol 8 (Vortex in Stagnation Flow) is defined to be a planar flow this does not mean that itis a two-component flow in this sense however because its two non-zero velocity components dependon three coordinates We now check the models for LES without explicit filtering by consideringendorsements within each class Tables 18 and 19 are analogues of Table 3 and they illustrate theresults for LES without explicit filtering for each flow class The analogues of Tables 1 and 2 areTables 31 and 32 and are in Appendix C4
Endorsements for models using solutions in C2 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 6 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 357 714 100 857 100
Table 18 Each entry in the top two rows represents the number of solutions in the class C2 thatendorse each model There are 7 solutions in this class so the percentages are worked out by dividingthe total of each column by 14 and taking the result to one decimal place if necessary
Endorsements for models using solutions in C3 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 3 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 63 63 63 188 250
Table 19 Each entry in the top two rows represents the number of solutions in the class C3 thatendorse each model There are 8 solutions in this class so the percentages are worked out by dividingthe total of each column by 16 and taking the result to one decimal place if necessary
The class C2 equally endorses the QR and Vortex-Stretching-based models and once again theSmagorinsky model is the least endorsed As with the previous classes the class C3 has such lowpercentages that it does not seem to endorse any models We start to see a trend appearing here inthat the flow classes that contain the more complicated flows namely Q0minus6 (Eq (76)) Vlowast (Eq (79))Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages The classes that tendto contain the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78)) R0 (Eq (80)) and C2 (Eq (84))all seem to favour the Vortex-Stretching-based model or the QR model
36
Now let us consider LES with explicit filtering Tables 20 and 21 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 33 and is in Appendix C4
Endorsements for models using solutions in C2 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 2 2 0 1 0
Table 20 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in C3 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 8 8 8 6
Table 21 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 and Table 16 we can see in Table 20 that the results for the lsquostrongrsquo endorsements areexactly the same as in Table 5 This is again because all of the flows with zero subfilter dissipationare contained in the class C2 There are two flows with non-zero true subfilter dissipation (givenin Eq (67)) in C2 and this shows in the lsquoweakrsquo endorsements in Table 20 There are no lsquostrongrsquoendorsements in Table 21 as all the zero subfilter dissipation flows are in C2
5 Conclusions
In this report we studied incompressible turbulent flows We found a number of exact solutions to theNavier-Stokes equations each representing a relatively simplistic fluid flow Large-eddy simulation(LES) is used to model turbulent flows by smoothing over the complex small-scale motion Weintroduced a number of subfilter-scale models to approximate this small-scale motion and testedthem using the found exact solutions to the Navier-Stokes equations We first considered LESwithout explicit filtering and used the Navier-Stokes equations to derive conditions for the subfilter-scale models namely the Velocity Field condition (Eq (58)) the Kinetic Energy condition (Eq (60))and the Dissipation condition (Eq (62)) Similarly we considered LES with explicit filtering andagain used the Navier-Stokes equations to derive more conditions for the models in this case wedecided to just take one condition the Dissipation condition (Eqs (67) and (68)) as the others leadto difficulty in their evaluation By lsquorunningrsquo the exact solutions to the Navier-Stokes equationsthrough each condition for the subfilter-scale models we found that the Vortex-Stretching-basededdy-viscosity model in Eq (54) was most endorsed by the tests in LES without explicit filteringWe found that four out of the five subfilter-scale models namely the Vreman QR Gradient andVortex-Stretching-based models were equally endorsed by the lsquostrongrsquo tests in LES with explicitfiltering with the Smagorinsky model being the least endorsed The results we found for the lsquoweakrsquocondition were completely opposite to those found in both the lsquostrongrsquo condition and in LES withoutexplicit filtering This led us to conclude that this test was not one in which we should put our trust
37
We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses
In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes
This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold
38
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
for i j = 1 2 3 There are three principal invariants of the velocity gradient of a fluid flow G givenby
PG = tr(G)
QG =1
2
((tr(G))2 minus tr
(G2))
RG = det(G)
(40)
Incompressibility of flows (Eq (5)) ensures that PG = 0 hence the first principal invariant is zerofor all of the flows in Section 22 We now define the rate-of-strain tensor S by
Sij =1
2
(partuipartxj
+partujpartxi
) (41)
and the rate-of-rotation tensor Ω by
Ωij =1
2
(partuipartxjminus partujpartxi
) (42)
Note that the first principal invariant of S and the first and third principal invariants of Ω are zeroAlso note that the velocity gradient of a flow can simply be written G = S + Ω
Rather than considering the principal invariants we can also consider the combined invariants
I1 = tr(S2) I2 = tr(Ω2) I3 = tr(S3) I4 = tr(SΩ2) I5 = tr(S2Ω2) (43)
We can hence write the three principal invariants for a velocity field G in terms of the combinedinvariants
PG = 0
QG = minus1
2(I1 + I2)
RG =1
3(I3 + 3I4)
(44)
In this report the flows considered are the exact solutions to the incompressible Navier-Stokesequations described in Section 22 The combined invariants in Eq (43) are used to characterisethese solutions into different classes in Section 44 Both the principal invariants and combinedinvariants for each exact solution in Section 22 are given in Appendix A
Each of the subfilter-scale eddy-viscosity models described in Section 32 are written in terms ofthese five combined invariants
232 Vremanrsquos Characterisation of Flows
Vreman [16] classifies flows using the gradient of the velocity field of a flow Each lsquoclassrsquo of flows hasthe same number of non-zero elements in the gradient matrix For example
[partuipartxj
]=
0 0 0lowast 0 0lowast 0 0
(45)
where a lowast denotes a non-zero term represents a group of flows that lie within the class denoted byQ7 (see Appendix B) where the subscript represents the number of non-zero terms in the velocity
21
gradient Flows of the form in Eq (45) have a velocity field of the form
u1 = c1 (constant)
u2 = f2(x1)
u3 = f3(x1)
(46)
where f2(x1) and f3(x1) are such that the Navier Stokes equations Eq (1) are satisfied Notethat as the elements on the leading diagonal are zero the incompressibility condition Eq (5) isautomatically satisfied Some simple flow classes are given in Appendix B and contain a number ofthe solutions discussed in Section 22 This is mentioned in greater detail in Section 441
3 Conditions for Large-Eddy Simulation Models
31 Large-Eddy Simulation
Due to the nature of numerical simulation and the complexity of turbulent flows small details aremissed and information is lost when solving the Navier-Stokes equations The computational costof solving a turbulent flow is extremely high so a filtering operation is applied to the Navier-Stokesequations represented by a bar to reduce the computational cost of the simulation This is a linearoperation and preserves constants ie ρ = ρ and is also assumed to commute with differentiation
partuipartxj
=partuipartxj
(47)
The filtered Navier-Stokes equations are found by filtering (putting a bar over) the entire equations(see Eq (1)) and after evaluating are given by
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (48)
where
τij = uiuj minus uiuj (49)
is called the lsquotrue stressrsquo of the flow This however poses a problem as it is not clear what is meantby the term uiuj for a general filtering operation The true stress τij therefore must be replaced bya model τmod
ij in practical applications
32 Subfilter-Scale Models
We will now define the subflter-scale models denoted by τmodij to be used in place of the lsquodifficultrsquo
true stress τij There are a number of models available to use in the literature [11 p7] These willbe tested using the exact solutions to the Navier-Stokes equations and by checking conditions whichare defined in Sections 33 and 34
All the models barring one are of the same form That is each of the eddy-viscosity models arewritten
τmodij minus 1
3τmodkk δij = minus2νeSij (50)
with νe representing the eddy viscosity of a specific model For simplicity we usually just writeτmodij = minus2νeSij the extra term is included here to emphasise that the models are traceless Each
22
of the models we consider here can be expressed in terms of the rate of strain tensor S defined inEq (41) the rate of rotation tensor Ω defined in Eq (42) and the combined invariants defined inEq (43) The models are then defined as follows
Smagorinsky [12] νSe = (CSδ)2radic
2I1 (51)
Vreman [16] νVe = (CV δ)2
radicQGGT
PGGT (52)
QR [13 14 15] νQRe = (CQRδ)2 max0minusI3
I1 (53)
Vortex-Stretching [11] νV Se = (CV Sδ)2radic
2I1
(I5 minus 1
2I1I2
minusI1I2
)32
(54)
with the quantities
PGGT = I1 minus I2 QGGT =1
4(I1 + I2)2 + 4
(I5 minus
1
2I1I2
) (55)
There is a fifth and final model to be considered here It is of a different form to the other four andnot an eddy-viscosity model It is known as the Gradient model and is defined by
Gradient [4 9] τmodG = CGδ
(S2 minus Ω2 minus (SΩminus ΩS)
) (56)
The values C are model constants specific to each model and δ represents the mesh size of the LESBoth of these quantities are included here to properly define the models however in this projectthey will be ignored as they are constants and have no influence on the conditions to be defined
33 Conditions for Large-Eddy Simulation without Explicit Filtering
The filtering operation introduced in Eq (48) is just a formality for LES without explicit filteringThe idea here is that filtering does not actually take place but the term lsquoaddedrsquo to the Navier-Stokesequations in fact lsquosubtractsrsquo the difficult to simulate small-scale turbulent motion of the flow Inorder to consider this properly the terms ui and p in Eq (48) are replaced by vi and q It is hopedthat τmod
ij approximates τij so that vi and q approximate ui and p respectively So LES withoutexplicit filtering is given by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (57)
It is clear to see that the first four terms of this equation are exactly the same as the Navier-Stokesequations so for the exact solutions discussed in Section 22 the combination of these terms willequal zero What is then desirable is that the model term is also zero for these exact solutions sothat Eq (57) still holds In order to test models for exact solutions the condition
part
partxjτmodij (v) = 0 (58)
is checked If this condition were to hold it would imply that the model does not influence thevelocity field of a flow hence we call it the Velocity Field Condition There are a number of otherrestrictions that can be tested One such test comes from the differential form of the evolution ofthe kinetic energy of a flow given in Eq (10) In order to find this Eq (57) is multiplied by vi andterms are grouped to find
part
partt
(1
2vivi
)+ vj
part
partxj
(1
2vivi
)= minusvi
1
ρ
partq
partxi+ νvi
part2vipartxjpartxj
minus vipart
partxjτmodij (v) (59)
23
Then similarly to Eq (58) another condition is to check whether the model affects the kineticenergy of exact solutions this is equivalent to checking whether
vipart
partxjτmodij (v) = 0 (60)
We call this condition the Kinetic Energy Condition The third and final condition considered herecan also be found using the integral form for the equation for kinetic energy
d
dt
intV
1
2vividV =
intS
[minusvj
(1
2vivi
)minus vj
q
ρ+ 2νviSij + viτ
modij (v)
]njdS
+
intV
[minus2νSijSij minus
(minusSijτmod
ij (v))]dV
(61)
where the rate-of-strain tensor Sij was defined in Eq (41) V is a volume of fluid and dS meansthe integral is over the surface of that volume with unit normal nj Then this final condition checkswhether the model affects the total kinetic energy of exact solutions to the Navier-Stokes equationsthat is
Sijτmodij (v) = 0 (62)
We will call this condition the Dissipation Condition because Sijτmodij (v) is often referred to as the
subgrid dissipation
34 Conditions for Large-Eddy Simulation with Explicit Filtering
As in Eq (48) the Navier-Stokes equations can be filtered by applying the filtering operation to thewhole equation (putting a bar over all terms) and evaluating to find again that
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (63)
In Eq (57) the filtering operation was ignored and the elements ui p and τij(u) were replacedwith vi q and τmod
ij (v) respectively Now in LES with explicit filtering the filtering operation itselfsmooths over the small-scale turbulent motion of the simulated flow In this case the elements uip and τij(u) will be replaced with vi q and τmod
ij (v) respectively LES with explicit filtering is thengiven by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (64)
Note that Eqs (63) and (64) despite their similarity to Eq (57) no longer contain the Navier-Stokesequations as Eq (57) did The bar (filtering operation) over each term results in each exact solutionfound in Section 22 not necessarily satisfying the first four terms It is hence unknown whether ornot these terms will vanish for exact solutions Terms can still be compared similarly to Eqs (58)(60) and (62) However instead of comparing to zero we can now compare to the relevant termfrom the filtered Navier-Stokes equations So the terms
part
partxjτij (65)
and
part
partxjτmodij (v) (66)
24
are to be compared Although the latter term is relatively simple to compute the former could leadto some difficulties due to the differentiation of the true stress τij = uiuj minus uiuj So although someinformation could be gleaned from the inspection of these two terms more information can be foundby considering the analogue of Eq (62) So the true subfilter dissipation
Sijτij (67)
(where Sij is shorthand for Sij(v)) and the modelled subfilter dissipation
Sij(v)τmodij (v) (68)
are considered and have to be equal for the condition to hold These can again be derived from theintegral equation for kinetic energy of a filtered velocity field similarly to Eq (61)
4 Analysis of Large-Eddy Simulation Models
In this section we use a code in Mathematica the theory described in Section 3 and the exactsolutions to the Navier-Stokes equations found in Section 22 to check whether or not each of theexact solutions lsquoendorsesrsquo each model for LES both without and with explicit filtering
41 Large-Eddy Simulation Without Explicit Filtering
We start by using a code in Mathematica to test the Velocity Field condition in Eq (58) and theKinetic Energy condition in Eq (60) For all exact solutions to the Navier-Stokes equations inSection 22 the outputs for both conditions were identical so we have combined them into one table
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 ndash ndash 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 1 For the Kinetic Energy condition a lsquo0rsquo result means that vipartpartxj
τmodij (v) = 0 for respective
models and exact solutions whereas a lsquolowastrsquo result means that vipartpartxj
τmodij (v) 6= 0 For the Velocity
Field condition a lsquo0rsquo result means that partpartxj
τmodij (v) = 0 for respective models and exact solutions
whereas a lsquolowastrsquo result means that partpartxj
τmodij (v) 6= 0 The Gradient model can take both positive and
negative values so when the sign of the model is known it is included in the table instead of a lsquorsquoBoth the conditions are included in one table because for all cases the outputs were identical
We then test the Dissipation condition in Eq (62) for all exact solutions to the Navier-Stokesequations in Section 22
25
Dissipation Condition [Does Sijτmodij (v) = 0]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 + 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 2 A lsquo0rsquo result means that Sijτmodij (v) = 0 for respective models and exact solutions whereas
a lsquolowastrsquo result means that Sijτmodij (v) 6= 0 The Gradient model can take both positive and negative
values so when the sign of the model is known it is included in the table instead of a lsquorsquo
We then sum all the results to find the total number of endorsements for each model described inSection 32
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
4 6 8 9 9
lsquoDissipationrsquoEndorsements
2 5 7 7 9
Total Endorsements 6 11 15 16 18
Table 3 The results from Tables 1 and 2 are summarised to give a general overview of how manyendorsements each model receives This makes it much clearer to see the models most supported bythe respective and combined conditions with the Vortex-Stretching model being the most supportedand the Smagorinsky model the least
Testing the models in Section 32 using LES without explicit filtering has mixed results It isclear that the Smagorinksy model is not well endorsed by the flows in Section 22 with the onlyendorsements coming from the simplest flows that is Sol 1 (Linear Velocity) Sol 3 (Velocity as afunction of time) Sol 7 (Stagnation Flow) and Sol 15 (Rankine Vortex) The Vortex-Stretching-based model is the most endorsed flow with eighteen out of a possible thirty endorsements Thisagrees with our foreknowledge as this model was constructed to be as physically consistent as possiblein the paper by Silvis amp Verstappen [11]
42 Large-Eddy Simulation With Explicit Filtering
In this section we will compare the true subfilter dissipation Sijτij and the modelled subfilterdissipation Sij(v)τmod
ij (v) given in Eqs (67) and (68) respectively for each model τmodij and each
exact solution to the Navier-Stokes equation given in Section 22 We will not consider the conditionsinvolving differentiation in Eqs (65) and (66) due to the difficulty in calculating the derivative ofthe τij term in Eq (65)
26
421 Example of a Test for a Model
The true subfilter dissipation Sijτij of a flow given in Eq (67) cannot be explicitly computed asthe filtering operation is kept general What can be done however is to partially evaluate theexpressions using the attributes of a filtering operation mentioned in Eq (48)
The calculations of Sijτij for Sol 1 (Linear Velocity) are provided to illustrate the general ideaRecall that the velocity field of Sol 1 (Linear Velocity) is written in Eq (13) in vector notation as
u(x y z t) =
0ax+ bcx+ d
(69)
Then Sij = Sij(u) is computed simply by using its definition in Eq (41) Note that as filteringpreserves constants if there is an a it is replaced with a meaning that (in this case) there are alarge number of zero terms due to the frequency of 0 Hence
[Sij(u)
]=
[1
2
(partuipartxj
+partujpartxi
)]=
0 a
2c2
a2 0 0
c2 0 0
=
0 a
2c2
a2 0 0
c2 0 0
(70)
Note that as Sijτij is being computed which is equal to the sum of the element-wise multiplicationof the two matrices (or the trace of the matrix multiplication) that we donrsquot need to consider theelements of τij(u) with corresponding zero entry in Sij As τij(u) = uiuj minus uiuj and again asfiltering preserves constants it is found that
[τij(u)] =[uiuj minus uiuj
]=
lowast 0 middot ax+ b 0 middot cx+ d
ax+ b middot 0 lowast lowastcx+ d middot 0 lowast lowast
=
lowast 0 00 lowast lowast0 lowast lowast
(71)
where a lsquolowastrsquo just represents some element (not necessarily non-zero) Therefore the true subfilterdissipation
Sijτij = tr
0 a2
c2
a2 0 0c2 0 0
middotlowast 0 0
0 lowast lowast0 lowast lowast
= 0 (72)
Now the modelled subfilter dissipation Sij(u)τmodij (u) needs to be computed (recall that Sij =
Sij(u)) In this case let us use the QR model stated in Eq (53) Once again Sij(u) calculated inEq (70) needs to be used So we have that
τmodij = minus2(CQRδ)
2 max0minusI3I1
Sij (73)
As
I1 = tr(S2) =
1
2(a2 + c2) and I3 = tr(S
3) = 0 (74)
it holds that τmodij (u) = 0 so Sij(u)τmod
ij (u) = 0 This agrees with Eq (72) and hence Sol 1(Linear Velocity) endorses the QR model
27
422 Testing the Models
Now using a Mathematica code the exact solutions in Section 22 are again used to evaluate thecondition for the models using the process just described in Section 421 The results are describedin Table 4
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 4 All entries with a ldquo0rdquo represent a value of zero and a ldquolowastrdquo represents a value that is non-zeroThe ldquoExactrdquo row represents the value of the true subfilter dissipation Sijτij Every other row witha model name represents the value of the modelled subfilter dissipation Sij(u)τmod
ij (u) respectiveto that model A ldquordquo means that it is unclear as to the value - the sign of these values is oftenunknown and the lsquoMaxrsquo function in the definition of the QR model (in Eq (53)) makes it difficultto evaluate
Reading from Table 4 the Smagorinsky model is clearly endorsed by Sol 3 (Velocity as a function oftime) and Sol 15 (Rankine Vortex) as in both cases both Sijτij = 0 = Sij(u)τmod
ij (u) In this casewe say that Sol 3 (Velocity as a function of time) and Sol 15 (Rankine Vortex) lsquostronglyrsquo endorsethe Smagorinsky model The Smagorinsky model is not endorsed by Sol 1 (Linear Velocity) Sol 2(Planar Flow) or Sol 14 (Radially Dependent) however because Sij(u)τmod
ij (u) 6= 0 For Sol 4 to
13 both Sijτij 6= 0 6= Sij(u)τmodij (u) and so we will say that these exact solutions lsquoweaklyrsquo endorse
the Smagorinsky model
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 10 10 9 6
Table 5 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Interestingly the lsquostrongrsquo and lsquoweakrsquo endorsements give different results However more stock shouldbe put in the results of the lsquostrongerrsquo condition as it is completely clear that zero is equal to zeroIt is not necessarily clear whether or not non-zero equals non-zero as the values will depend on theparameters in each solution hence the lsquoweakerrsquo endorsement
The lsquostrongrsquo condition equally endorses the Vreman QR Gradient and Vortex-Stretching-basedmodels whereas the lsquoweakrsquo condition endorses Vreman and Smagorinsky the most closely followedby the Gradient model with the Vortex-Stretching-based model being the most opposed As this
28
completely contradicts the lsquostrongrsquo condition and the results in Section 41 it makes us question thevalidity of the weaker test The number of endorsements for the QR model is unknown because of thedefinition of the model in Eq (53) and our evaluation of the true subfilter dissipation Eq (67) Dueto the exact solutions we have found it is never clear whether or not the true subfilter dissipationis strictly positive or negative This means that when the true subfilter dissipation is found to benon-zero the lsquoMaxrsquo function in the QR model cannot be evaluated
43 Vremanrsquos True Subfilter Dissipation Claim
Using the theory mentioned in Sections 232 and 34 we can place the exact solutions to the Navier-Stokes equations into different classes defined by Vreman In doing this we can also attempt to verifyone of Vremanrsquos results [16] In Vremanrsquos paper he defines the class Qn to be the set of flows withn zero entries in the velocity gradient of the flow (the velocity gradient is defined in Eq (39))Vreman then claims that the only flows with zero true subfilter dissipation are a certain selection ofthose in the classesQ7Q8 orQ9 shown in appendix B and is generally non-zero for any other flows
The true subfilter dissipation of Sol 1 (Linear Velocity) was computed in Section 421 and wasfound to be zero in Eq (72) The velocity gradient of Sol 1 is as follows
GLV =
0 0 0a 0 0c 0 0
(75)
and hence Sol 1 (Linear Velocity) is in the required subset of Q7 which agrees with Vremanrsquos claimIn Section 422 the true subfilter dissipation was computed (determined to be zero or non-zero) forall of the exact solutions to the Navier Stokes equations in Section 22 These solutions can now belsquoplacedrsquo into each class Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a functionof time) and Sol 14 (Radially Dependent) were all found to have zero true subfilter dissipation andare all in the appropriate subsets of either Q7Q8 or Q9 This agrees with Vremanrsquos results
We can extend Vremanrsquos results however We found that Sol 15 (Rankine Vortex) had zero truesubfilter dissipation and yet is not in the subset of Q7 defined in Appendix B It is important tonote that the rate of strain tensor Sij in Eq (41) for Sol 15 (Rankine Vortex) is zero This is thereason the true subfilter dissipation is zero and the same reason that all models are zero for thisflow as all models are based on Sij
We also found that all of the remaining exact solutions to the Navier-Stokes equations are not inVremanrsquos specified classes (in Appendix Section B) They also have a non-zero subfilter dissipationwhich agrees with Vremanrsquos results in his paper [16]
44 Classes of Flows
It is important to consider that some exact solutions to the Navier-Stokes equations may have similarproperties For example Sol 1 (Linear Velocity) and Sol 2 (Planar Flow) are both very similarIn fact Sol 2 (Planar Flow) is a more general case of Sol 1 (Linear Velocity) We therefore mustgroup the different flows into classes in an attempt to reduce the bias that may occur when solutionsare similar This should aid us to further determine which of the models are most endorsed We willdo this in a number of different ways using the theory discussed in Section 23 We will group theflows with respect to Vremanrsquos Flow Classes given in appendix B and with respect to the Invariantsgiven in appendix A
29
441 Vremanrsquos Velocity Gradient Flow Classes
We now put our exact solutions to the Navier-Stokes equations into one of two classes Either Q0minus6or Q7minus9 where each class is defined intuitively as follows
Q0minus6 = Q0 cupQ1 cup middot middot middot cup Q6 (76)
Q7minus9 = Q7 cupQ8 cupQ9 (77)
where Qn is defined in Section 43 By looking at the gradient of each exact solution we can see thatQ7minus9 contains the six solutions Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as afunction of time) Sol 5 (2D Taylor) Sol 14 (Radially Dependent) and Sol 15 (Rankine Vortex)and that Q0minus6 contains the remaining nine We now check the models for LES without explicitfiltering by considering endorsements within each class Tables 6 and 7 are analogues of Table 3 andillustrate the results for each flow class The analogues of Tables 1 and 2 are Tables 22 and 23 andare in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 6 6 6
lsquoDissipationrsquoEndorsements
2 5 6 6 6
Percentage Endorsed 417 833 100 100 100
Table 6 Each entry in the top two rows represents the number of solutions in the class Q7minus9 thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Q0minus6 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 2 3 3
lsquoDissipationrsquoEndorsements
0 0 1 0 3
Percentage Endorsed 56 56 167 167 333
Table 7 Each entry in the top two rows represents the number of solutions in the class Q1minus6 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking results to 1 decimal place if necessary
All of the flows in the class Q7minus9 endorse the QR Gradient and Vortex-Stretching-based subfilter-scale models (see Table 6) We can hence see that this class fully endorses these three models equallyand does not endorse the Smagorinsky model The Vortex-Stretching eddy viscosity model has themost endorsements from any of the flows in the class Q0minus6 and the Smagorinksy and Vreman modelshave the fewest endorsements (see Table 7) We can hence say that this class mostly endorses theVortex-Stretching-based eddy-viscosity model note however that the percentage endorsement is very
30
low
Now let us consider LES with explicit filtering Tables 8 and 9 are analogues of Table 5 and illustratethe results for each flow class The analogue of Table 4 is Table 24 and is in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 1 1 0 0 0
Table 8 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Q0minus6 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 9 9 9 6
Table 9 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We have grouped Vremanrsquos classes into Q7minus9 and Q0minus6 this is equivalent to saying that we havegrouped Vremanrsquos classes into those with two or fewer non-zero velocity gradient elements and thosewith three or more non-zero velocity gradient elements respectively Therefore the solutions in Q7minus9are exactly those with a lsquo0rsquo in the lsquoExactrsquo column of Table 4 and Sol 5 (2D Taylor) ExcludingSol 5 (2D Taylor) these are the same as those five that endorse the Vreman QR Gradient andVortex-Stretching-based models in Table 5 We can see in Table 8 that the results for the lsquostrongrsquoendorsements are exactly the same as in Table 5 This is because all of the flows with zero subfilterdissipation are contained in the class Q7minus9 There are hence no lsquostrongrsquo endorsements in Table 9The majority of the lsquoweakrsquo endorsements come from Q0minus6 as this class contains all but one of thesolutions with non-zero true subfilter dissipation (given in Eq (67))
442 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
We now separate our exact solutions into classes again using Vremanrsquos flow classes This timehowever we define V0 to be the class of flows Vreman claims will always have zero true subfilterdissipation namely those classes mentioned in Appendix B We define Vlowast to be all the remaining
31
classes Then
V0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow)
Sol 3 (Velocity as a function of time) Sol 14 (Radially Dependent)(78)
Vlowast = Sol 4 (Axisymmetric Flow) Sol 5 (2D Taylor) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 13 (Sphere Translation) Sol 15 (Rankine Vortex)
(79)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 10 and 11 are analogues of Table 3 and illustrate the results for each flow class Theanalogues of Tables 1 and 2 are Tables 25 and 26 and are in Appendix C2
Endorsements for models using solutions in V0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 4 4 4 4
lsquoDissipationrsquoEndorsements
1 4 4 4 4
Percentage Endorsed 375 100 100 100 100
Table 10 Each entry in the top two rows represents the number of solutions in the class V0 thatendorse each model There are 4 solutions in this class so the percentages are worked out by dividingthe total of each column by 8 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Vlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 2 4 5 5
lsquoDissipationrsquoEndorsements
1 1 3 2 5
Percentage Endorsed 136 136 318 318 455
Table 11 Each entry in the top two rows represents the number of solutions in the class Vlowast thatendorse each model There are 11 solutions in this class so the percentages are worked out bydividing the total of each column by 22 and taking results to 1 decimal place if necessary
All of the flows in the class V0 endorse the Vreman QR Gradient and Vortex-Stretching-basedsubfilter-scale models (see Table 6) This class completely endorses these four models equally anddoes not endorse the Smagorinsky model The Vortex-Stretching-based eddy-viscosity model hasthe most endorsements from the class Vlowast and the Smagorinksy and Vreman models again have thefewest endorsements (see Table 7) As in Section 441 we say that this class mostly endorses the
32
Vortex-Stretching-based eddy-viscosity model but once again the percentage endorsement is ratherlow
Now let us consider LES with explicit filtering Tables 12 and 13 are analogues of Table 5 andillustrate the results for each flow class The analogue of Table 4 is Table 27 and is in Appendix C2
Endorsements for models using solutions in V0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 4 4 4 4
lsquoWeakrsquo Endorsements 0 0 0 0 0
Table 12 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Vlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 1 1 1 1
lsquoWeakrsquo Endorsements 10 10 9 6
Table 13 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We can see in Table 12 that the results for the lsquostrongrsquo endorsements are similar to those in Table 5excluding Sol 15 (Rankine Vortex) The four solutions in V0 with zero true subfilter dissipationnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)and Sol 14 (Radially Dependent) are hence the four that endorse the Vreman QR Gradientand Vortex-Stretching-based models in Table 12 Sol 15 (Rankine Vortex) is in the class Vlowast andhence there is one lsquostrongrsquo endorsement in Table 13 The class Vlowast contains all of the flows with anon-zero true subfilter dissipation (given in Eq (67)) and hence Table 13 contains all of the lsquoweakrsquoendorsements
443 Classes using the Third Principal Invariant
Sorting the classes using the invariants is a non-trivial task in itself There are a number of waysin which to do this using both the principal invariants in Eq (40) and the combined invariants inEq (43)
We start by looking at flows with third principal invariant RG equal to zero We again define twoclasses R0 and Rlowast where R0 consists of all flows with RG = 0 and Rlowast consists of all flows withRG 6= 0 Then using the information in Appendix A we see that there are nine flows in R0 That
33
is
R0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 8 (Vortex in Stagnation Flow)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)
(80)
Rlowast = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 13 (Sphere Translation)(81)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 14 and 15 are analogues of Table 3 and they illustrate the results for LES withoutexplicit filtering for each flow class The analogues of Tables 1 and 2 are Tables 28 and 29 and arein Appendix C3
Endorsements for models using solutions in R0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 7 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 278 556 778 722 778
Table 14 Each entry in the top two rows represents the number of solutions in the class R0 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking the result to one decimal place if necessary
Endorsements for models using solutions in Rlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 2 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 83 83 83 167 333
Table 15 Each entry in the top two rows represents the number of solutions in the class Rlowast thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking the result to one decimal place if necessary
The class R0 endorses both the Vortex-Stretching-based model and the QR model the most andSmagorinsky the least in the case for LES without explicit filtering In the Rlowast class the Vortex-Stretching-based model is again the most endorsed followed by the Gradient model however thepercentages are again small
34
Now let us consider LES with explicit filtering Tables 16 and 17 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 30 and is in Appendix C3
Endorsements for models using solutions in R0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 4 4 3 2
Table 16 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Rlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 6 6 6 4
Table 17 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 we can see in Table 16 that the results for the lsquostrongrsquo endorsements are exactly thesame as in Table 5 This is because all of the flows with zero subfilter dissipation are contained in theclass R0 As in Table 9 there are no lsquostrongrsquo endorsements in Table 17 The lsquoweakrsquo endorsementshowever are split more evenly between the two classes now than in Vremanrsquos classes in Sections 441and 442 We find that the Smagorinsky and Vreman models still seem to be the most endorsed inboth classes
444 Two-component Flow Classes
Another way in which the flows can be grouped into classes is by determining whether a flow is atwo-component flow or not We define a two-component flow to have the property that the combinedprincipal invariants of a flow in Eq (43) satisfy
I3 = I4 = I5 minus1
2I1I2 = 0 (82)
We are also interested in considering flows with zero gradient model subgrid dissipation These flowshave the property that
I3 minus I4 = 0 (83)
What we find though is that for all the solutions in Section 22 if a flow is a two-component flowthen it also satisfies I3 minus I4 = 0 It may be interesting to note that although I3 minus I4 6= 0 for manysolutions there are a number with I3 equal to a multiple of I4 Although not considered in thisreport this may also be a class of interest We find that the two-component flows are
C2 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)(84)
35
and we call the remaining flows three-component flows They are
C3 = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex) Sol 11 (Pseudo-Motion Shots)
Sol 13 (Sphere Translation)
(85)
Note that Sol 8 (Vortex in Stagnation Flow) is defined to be a planar flow this does not mean that itis a two-component flow in this sense however because its two non-zero velocity components dependon three coordinates We now check the models for LES without explicit filtering by consideringendorsements within each class Tables 18 and 19 are analogues of Table 3 and they illustrate theresults for LES without explicit filtering for each flow class The analogues of Tables 1 and 2 areTables 31 and 32 and are in Appendix C4
Endorsements for models using solutions in C2 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 6 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 357 714 100 857 100
Table 18 Each entry in the top two rows represents the number of solutions in the class C2 thatendorse each model There are 7 solutions in this class so the percentages are worked out by dividingthe total of each column by 14 and taking the result to one decimal place if necessary
Endorsements for models using solutions in C3 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 3 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 63 63 63 188 250
Table 19 Each entry in the top two rows represents the number of solutions in the class C3 thatendorse each model There are 8 solutions in this class so the percentages are worked out by dividingthe total of each column by 16 and taking the result to one decimal place if necessary
The class C2 equally endorses the QR and Vortex-Stretching-based models and once again theSmagorinsky model is the least endorsed As with the previous classes the class C3 has such lowpercentages that it does not seem to endorse any models We start to see a trend appearing here inthat the flow classes that contain the more complicated flows namely Q0minus6 (Eq (76)) Vlowast (Eq (79))Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages The classes that tendto contain the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78)) R0 (Eq (80)) and C2 (Eq (84))all seem to favour the Vortex-Stretching-based model or the QR model
36
Now let us consider LES with explicit filtering Tables 20 and 21 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 33 and is in Appendix C4
Endorsements for models using solutions in C2 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 2 2 0 1 0
Table 20 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in C3 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 8 8 8 6
Table 21 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 and Table 16 we can see in Table 20 that the results for the lsquostrongrsquo endorsements areexactly the same as in Table 5 This is again because all of the flows with zero subfilter dissipationare contained in the class C2 There are two flows with non-zero true subfilter dissipation (givenin Eq (67)) in C2 and this shows in the lsquoweakrsquo endorsements in Table 20 There are no lsquostrongrsquoendorsements in Table 21 as all the zero subfilter dissipation flows are in C2
5 Conclusions
In this report we studied incompressible turbulent flows We found a number of exact solutions to theNavier-Stokes equations each representing a relatively simplistic fluid flow Large-eddy simulation(LES) is used to model turbulent flows by smoothing over the complex small-scale motion Weintroduced a number of subfilter-scale models to approximate this small-scale motion and testedthem using the found exact solutions to the Navier-Stokes equations We first considered LESwithout explicit filtering and used the Navier-Stokes equations to derive conditions for the subfilter-scale models namely the Velocity Field condition (Eq (58)) the Kinetic Energy condition (Eq (60))and the Dissipation condition (Eq (62)) Similarly we considered LES with explicit filtering andagain used the Navier-Stokes equations to derive more conditions for the models in this case wedecided to just take one condition the Dissipation condition (Eqs (67) and (68)) as the others leadto difficulty in their evaluation By lsquorunningrsquo the exact solutions to the Navier-Stokes equationsthrough each condition for the subfilter-scale models we found that the Vortex-Stretching-basededdy-viscosity model in Eq (54) was most endorsed by the tests in LES without explicit filteringWe found that four out of the five subfilter-scale models namely the Vreman QR Gradient andVortex-Stretching-based models were equally endorsed by the lsquostrongrsquo tests in LES with explicitfiltering with the Smagorinsky model being the least endorsed The results we found for the lsquoweakrsquocondition were completely opposite to those found in both the lsquostrongrsquo condition and in LES withoutexplicit filtering This led us to conclude that this test was not one in which we should put our trust
37
We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses
In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes
This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold
38
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
gradient Flows of the form in Eq (45) have a velocity field of the form
u1 = c1 (constant)
u2 = f2(x1)
u3 = f3(x1)
(46)
where f2(x1) and f3(x1) are such that the Navier Stokes equations Eq (1) are satisfied Notethat as the elements on the leading diagonal are zero the incompressibility condition Eq (5) isautomatically satisfied Some simple flow classes are given in Appendix B and contain a number ofthe solutions discussed in Section 22 This is mentioned in greater detail in Section 441
3 Conditions for Large-Eddy Simulation Models
31 Large-Eddy Simulation
Due to the nature of numerical simulation and the complexity of turbulent flows small details aremissed and information is lost when solving the Navier-Stokes equations The computational costof solving a turbulent flow is extremely high so a filtering operation is applied to the Navier-Stokesequations represented by a bar to reduce the computational cost of the simulation This is a linearoperation and preserves constants ie ρ = ρ and is also assumed to commute with differentiation
partuipartxj
=partuipartxj
(47)
The filtered Navier-Stokes equations are found by filtering (putting a bar over) the entire equations(see Eq (1)) and after evaluating are given by
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (48)
where
τij = uiuj minus uiuj (49)
is called the lsquotrue stressrsquo of the flow This however poses a problem as it is not clear what is meantby the term uiuj for a general filtering operation The true stress τij therefore must be replaced bya model τmod
ij in practical applications
32 Subfilter-Scale Models
We will now define the subflter-scale models denoted by τmodij to be used in place of the lsquodifficultrsquo
true stress τij There are a number of models available to use in the literature [11 p7] These willbe tested using the exact solutions to the Navier-Stokes equations and by checking conditions whichare defined in Sections 33 and 34
All the models barring one are of the same form That is each of the eddy-viscosity models arewritten
τmodij minus 1
3τmodkk δij = minus2νeSij (50)
with νe representing the eddy viscosity of a specific model For simplicity we usually just writeτmodij = minus2νeSij the extra term is included here to emphasise that the models are traceless Each
22
of the models we consider here can be expressed in terms of the rate of strain tensor S defined inEq (41) the rate of rotation tensor Ω defined in Eq (42) and the combined invariants defined inEq (43) The models are then defined as follows
Smagorinsky [12] νSe = (CSδ)2radic
2I1 (51)
Vreman [16] νVe = (CV δ)2
radicQGGT
PGGT (52)
QR [13 14 15] νQRe = (CQRδ)2 max0minusI3
I1 (53)
Vortex-Stretching [11] νV Se = (CV Sδ)2radic
2I1
(I5 minus 1
2I1I2
minusI1I2
)32
(54)
with the quantities
PGGT = I1 minus I2 QGGT =1
4(I1 + I2)2 + 4
(I5 minus
1
2I1I2
) (55)
There is a fifth and final model to be considered here It is of a different form to the other four andnot an eddy-viscosity model It is known as the Gradient model and is defined by
Gradient [4 9] τmodG = CGδ
(S2 minus Ω2 minus (SΩminus ΩS)
) (56)
The values C are model constants specific to each model and δ represents the mesh size of the LESBoth of these quantities are included here to properly define the models however in this projectthey will be ignored as they are constants and have no influence on the conditions to be defined
33 Conditions for Large-Eddy Simulation without Explicit Filtering
The filtering operation introduced in Eq (48) is just a formality for LES without explicit filteringThe idea here is that filtering does not actually take place but the term lsquoaddedrsquo to the Navier-Stokesequations in fact lsquosubtractsrsquo the difficult to simulate small-scale turbulent motion of the flow Inorder to consider this properly the terms ui and p in Eq (48) are replaced by vi and q It is hopedthat τmod
ij approximates τij so that vi and q approximate ui and p respectively So LES withoutexplicit filtering is given by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (57)
It is clear to see that the first four terms of this equation are exactly the same as the Navier-Stokesequations so for the exact solutions discussed in Section 22 the combination of these terms willequal zero What is then desirable is that the model term is also zero for these exact solutions sothat Eq (57) still holds In order to test models for exact solutions the condition
part
partxjτmodij (v) = 0 (58)
is checked If this condition were to hold it would imply that the model does not influence thevelocity field of a flow hence we call it the Velocity Field Condition There are a number of otherrestrictions that can be tested One such test comes from the differential form of the evolution ofthe kinetic energy of a flow given in Eq (10) In order to find this Eq (57) is multiplied by vi andterms are grouped to find
part
partt
(1
2vivi
)+ vj
part
partxj
(1
2vivi
)= minusvi
1
ρ
partq
partxi+ νvi
part2vipartxjpartxj
minus vipart
partxjτmodij (v) (59)
23
Then similarly to Eq (58) another condition is to check whether the model affects the kineticenergy of exact solutions this is equivalent to checking whether
vipart
partxjτmodij (v) = 0 (60)
We call this condition the Kinetic Energy Condition The third and final condition considered herecan also be found using the integral form for the equation for kinetic energy
d
dt
intV
1
2vividV =
intS
[minusvj
(1
2vivi
)minus vj
q
ρ+ 2νviSij + viτ
modij (v)
]njdS
+
intV
[minus2νSijSij minus
(minusSijτmod
ij (v))]dV
(61)
where the rate-of-strain tensor Sij was defined in Eq (41) V is a volume of fluid and dS meansthe integral is over the surface of that volume with unit normal nj Then this final condition checkswhether the model affects the total kinetic energy of exact solutions to the Navier-Stokes equationsthat is
Sijτmodij (v) = 0 (62)
We will call this condition the Dissipation Condition because Sijτmodij (v) is often referred to as the
subgrid dissipation
34 Conditions for Large-Eddy Simulation with Explicit Filtering
As in Eq (48) the Navier-Stokes equations can be filtered by applying the filtering operation to thewhole equation (putting a bar over all terms) and evaluating to find again that
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (63)
In Eq (57) the filtering operation was ignored and the elements ui p and τij(u) were replacedwith vi q and τmod
ij (v) respectively Now in LES with explicit filtering the filtering operation itselfsmooths over the small-scale turbulent motion of the simulated flow In this case the elements uip and τij(u) will be replaced with vi q and τmod
ij (v) respectively LES with explicit filtering is thengiven by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (64)
Note that Eqs (63) and (64) despite their similarity to Eq (57) no longer contain the Navier-Stokesequations as Eq (57) did The bar (filtering operation) over each term results in each exact solutionfound in Section 22 not necessarily satisfying the first four terms It is hence unknown whether ornot these terms will vanish for exact solutions Terms can still be compared similarly to Eqs (58)(60) and (62) However instead of comparing to zero we can now compare to the relevant termfrom the filtered Navier-Stokes equations So the terms
part
partxjτij (65)
and
part
partxjτmodij (v) (66)
24
are to be compared Although the latter term is relatively simple to compute the former could leadto some difficulties due to the differentiation of the true stress τij = uiuj minus uiuj So although someinformation could be gleaned from the inspection of these two terms more information can be foundby considering the analogue of Eq (62) So the true subfilter dissipation
Sijτij (67)
(where Sij is shorthand for Sij(v)) and the modelled subfilter dissipation
Sij(v)τmodij (v) (68)
are considered and have to be equal for the condition to hold These can again be derived from theintegral equation for kinetic energy of a filtered velocity field similarly to Eq (61)
4 Analysis of Large-Eddy Simulation Models
In this section we use a code in Mathematica the theory described in Section 3 and the exactsolutions to the Navier-Stokes equations found in Section 22 to check whether or not each of theexact solutions lsquoendorsesrsquo each model for LES both without and with explicit filtering
41 Large-Eddy Simulation Without Explicit Filtering
We start by using a code in Mathematica to test the Velocity Field condition in Eq (58) and theKinetic Energy condition in Eq (60) For all exact solutions to the Navier-Stokes equations inSection 22 the outputs for both conditions were identical so we have combined them into one table
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 ndash ndash 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 1 For the Kinetic Energy condition a lsquo0rsquo result means that vipartpartxj
τmodij (v) = 0 for respective
models and exact solutions whereas a lsquolowastrsquo result means that vipartpartxj
τmodij (v) 6= 0 For the Velocity
Field condition a lsquo0rsquo result means that partpartxj
τmodij (v) = 0 for respective models and exact solutions
whereas a lsquolowastrsquo result means that partpartxj
τmodij (v) 6= 0 The Gradient model can take both positive and
negative values so when the sign of the model is known it is included in the table instead of a lsquorsquoBoth the conditions are included in one table because for all cases the outputs were identical
We then test the Dissipation condition in Eq (62) for all exact solutions to the Navier-Stokesequations in Section 22
25
Dissipation Condition [Does Sijτmodij (v) = 0]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 + 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 2 A lsquo0rsquo result means that Sijτmodij (v) = 0 for respective models and exact solutions whereas
a lsquolowastrsquo result means that Sijτmodij (v) 6= 0 The Gradient model can take both positive and negative
values so when the sign of the model is known it is included in the table instead of a lsquorsquo
We then sum all the results to find the total number of endorsements for each model described inSection 32
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
4 6 8 9 9
lsquoDissipationrsquoEndorsements
2 5 7 7 9
Total Endorsements 6 11 15 16 18
Table 3 The results from Tables 1 and 2 are summarised to give a general overview of how manyendorsements each model receives This makes it much clearer to see the models most supported bythe respective and combined conditions with the Vortex-Stretching model being the most supportedand the Smagorinsky model the least
Testing the models in Section 32 using LES without explicit filtering has mixed results It isclear that the Smagorinksy model is not well endorsed by the flows in Section 22 with the onlyendorsements coming from the simplest flows that is Sol 1 (Linear Velocity) Sol 3 (Velocity as afunction of time) Sol 7 (Stagnation Flow) and Sol 15 (Rankine Vortex) The Vortex-Stretching-based model is the most endorsed flow with eighteen out of a possible thirty endorsements Thisagrees with our foreknowledge as this model was constructed to be as physically consistent as possiblein the paper by Silvis amp Verstappen [11]
42 Large-Eddy Simulation With Explicit Filtering
In this section we will compare the true subfilter dissipation Sijτij and the modelled subfilterdissipation Sij(v)τmod
ij (v) given in Eqs (67) and (68) respectively for each model τmodij and each
exact solution to the Navier-Stokes equation given in Section 22 We will not consider the conditionsinvolving differentiation in Eqs (65) and (66) due to the difficulty in calculating the derivative ofthe τij term in Eq (65)
26
421 Example of a Test for a Model
The true subfilter dissipation Sijτij of a flow given in Eq (67) cannot be explicitly computed asthe filtering operation is kept general What can be done however is to partially evaluate theexpressions using the attributes of a filtering operation mentioned in Eq (48)
The calculations of Sijτij for Sol 1 (Linear Velocity) are provided to illustrate the general ideaRecall that the velocity field of Sol 1 (Linear Velocity) is written in Eq (13) in vector notation as
u(x y z t) =
0ax+ bcx+ d
(69)
Then Sij = Sij(u) is computed simply by using its definition in Eq (41) Note that as filteringpreserves constants if there is an a it is replaced with a meaning that (in this case) there are alarge number of zero terms due to the frequency of 0 Hence
[Sij(u)
]=
[1
2
(partuipartxj
+partujpartxi
)]=
0 a
2c2
a2 0 0
c2 0 0
=
0 a
2c2
a2 0 0
c2 0 0
(70)
Note that as Sijτij is being computed which is equal to the sum of the element-wise multiplicationof the two matrices (or the trace of the matrix multiplication) that we donrsquot need to consider theelements of τij(u) with corresponding zero entry in Sij As τij(u) = uiuj minus uiuj and again asfiltering preserves constants it is found that
[τij(u)] =[uiuj minus uiuj
]=
lowast 0 middot ax+ b 0 middot cx+ d
ax+ b middot 0 lowast lowastcx+ d middot 0 lowast lowast
=
lowast 0 00 lowast lowast0 lowast lowast
(71)
where a lsquolowastrsquo just represents some element (not necessarily non-zero) Therefore the true subfilterdissipation
Sijτij = tr
0 a2
c2
a2 0 0c2 0 0
middotlowast 0 0
0 lowast lowast0 lowast lowast
= 0 (72)
Now the modelled subfilter dissipation Sij(u)τmodij (u) needs to be computed (recall that Sij =
Sij(u)) In this case let us use the QR model stated in Eq (53) Once again Sij(u) calculated inEq (70) needs to be used So we have that
τmodij = minus2(CQRδ)
2 max0minusI3I1
Sij (73)
As
I1 = tr(S2) =
1
2(a2 + c2) and I3 = tr(S
3) = 0 (74)
it holds that τmodij (u) = 0 so Sij(u)τmod
ij (u) = 0 This agrees with Eq (72) and hence Sol 1(Linear Velocity) endorses the QR model
27
422 Testing the Models
Now using a Mathematica code the exact solutions in Section 22 are again used to evaluate thecondition for the models using the process just described in Section 421 The results are describedin Table 4
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 4 All entries with a ldquo0rdquo represent a value of zero and a ldquolowastrdquo represents a value that is non-zeroThe ldquoExactrdquo row represents the value of the true subfilter dissipation Sijτij Every other row witha model name represents the value of the modelled subfilter dissipation Sij(u)τmod
ij (u) respectiveto that model A ldquordquo means that it is unclear as to the value - the sign of these values is oftenunknown and the lsquoMaxrsquo function in the definition of the QR model (in Eq (53)) makes it difficultto evaluate
Reading from Table 4 the Smagorinsky model is clearly endorsed by Sol 3 (Velocity as a function oftime) and Sol 15 (Rankine Vortex) as in both cases both Sijτij = 0 = Sij(u)τmod
ij (u) In this casewe say that Sol 3 (Velocity as a function of time) and Sol 15 (Rankine Vortex) lsquostronglyrsquo endorsethe Smagorinsky model The Smagorinsky model is not endorsed by Sol 1 (Linear Velocity) Sol 2(Planar Flow) or Sol 14 (Radially Dependent) however because Sij(u)τmod
ij (u) 6= 0 For Sol 4 to
13 both Sijτij 6= 0 6= Sij(u)τmodij (u) and so we will say that these exact solutions lsquoweaklyrsquo endorse
the Smagorinsky model
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 10 10 9 6
Table 5 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Interestingly the lsquostrongrsquo and lsquoweakrsquo endorsements give different results However more stock shouldbe put in the results of the lsquostrongerrsquo condition as it is completely clear that zero is equal to zeroIt is not necessarily clear whether or not non-zero equals non-zero as the values will depend on theparameters in each solution hence the lsquoweakerrsquo endorsement
The lsquostrongrsquo condition equally endorses the Vreman QR Gradient and Vortex-Stretching-basedmodels whereas the lsquoweakrsquo condition endorses Vreman and Smagorinsky the most closely followedby the Gradient model with the Vortex-Stretching-based model being the most opposed As this
28
completely contradicts the lsquostrongrsquo condition and the results in Section 41 it makes us question thevalidity of the weaker test The number of endorsements for the QR model is unknown because of thedefinition of the model in Eq (53) and our evaluation of the true subfilter dissipation Eq (67) Dueto the exact solutions we have found it is never clear whether or not the true subfilter dissipationis strictly positive or negative This means that when the true subfilter dissipation is found to benon-zero the lsquoMaxrsquo function in the QR model cannot be evaluated
43 Vremanrsquos True Subfilter Dissipation Claim
Using the theory mentioned in Sections 232 and 34 we can place the exact solutions to the Navier-Stokes equations into different classes defined by Vreman In doing this we can also attempt to verifyone of Vremanrsquos results [16] In Vremanrsquos paper he defines the class Qn to be the set of flows withn zero entries in the velocity gradient of the flow (the velocity gradient is defined in Eq (39))Vreman then claims that the only flows with zero true subfilter dissipation are a certain selection ofthose in the classesQ7Q8 orQ9 shown in appendix B and is generally non-zero for any other flows
The true subfilter dissipation of Sol 1 (Linear Velocity) was computed in Section 421 and wasfound to be zero in Eq (72) The velocity gradient of Sol 1 is as follows
GLV =
0 0 0a 0 0c 0 0
(75)
and hence Sol 1 (Linear Velocity) is in the required subset of Q7 which agrees with Vremanrsquos claimIn Section 422 the true subfilter dissipation was computed (determined to be zero or non-zero) forall of the exact solutions to the Navier Stokes equations in Section 22 These solutions can now belsquoplacedrsquo into each class Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a functionof time) and Sol 14 (Radially Dependent) were all found to have zero true subfilter dissipation andare all in the appropriate subsets of either Q7Q8 or Q9 This agrees with Vremanrsquos results
We can extend Vremanrsquos results however We found that Sol 15 (Rankine Vortex) had zero truesubfilter dissipation and yet is not in the subset of Q7 defined in Appendix B It is important tonote that the rate of strain tensor Sij in Eq (41) for Sol 15 (Rankine Vortex) is zero This is thereason the true subfilter dissipation is zero and the same reason that all models are zero for thisflow as all models are based on Sij
We also found that all of the remaining exact solutions to the Navier-Stokes equations are not inVremanrsquos specified classes (in Appendix Section B) They also have a non-zero subfilter dissipationwhich agrees with Vremanrsquos results in his paper [16]
44 Classes of Flows
It is important to consider that some exact solutions to the Navier-Stokes equations may have similarproperties For example Sol 1 (Linear Velocity) and Sol 2 (Planar Flow) are both very similarIn fact Sol 2 (Planar Flow) is a more general case of Sol 1 (Linear Velocity) We therefore mustgroup the different flows into classes in an attempt to reduce the bias that may occur when solutionsare similar This should aid us to further determine which of the models are most endorsed We willdo this in a number of different ways using the theory discussed in Section 23 We will group theflows with respect to Vremanrsquos Flow Classes given in appendix B and with respect to the Invariantsgiven in appendix A
29
441 Vremanrsquos Velocity Gradient Flow Classes
We now put our exact solutions to the Navier-Stokes equations into one of two classes Either Q0minus6or Q7minus9 where each class is defined intuitively as follows
Q0minus6 = Q0 cupQ1 cup middot middot middot cup Q6 (76)
Q7minus9 = Q7 cupQ8 cupQ9 (77)
where Qn is defined in Section 43 By looking at the gradient of each exact solution we can see thatQ7minus9 contains the six solutions Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as afunction of time) Sol 5 (2D Taylor) Sol 14 (Radially Dependent) and Sol 15 (Rankine Vortex)and that Q0minus6 contains the remaining nine We now check the models for LES without explicitfiltering by considering endorsements within each class Tables 6 and 7 are analogues of Table 3 andillustrate the results for each flow class The analogues of Tables 1 and 2 are Tables 22 and 23 andare in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 6 6 6
lsquoDissipationrsquoEndorsements
2 5 6 6 6
Percentage Endorsed 417 833 100 100 100
Table 6 Each entry in the top two rows represents the number of solutions in the class Q7minus9 thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Q0minus6 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 2 3 3
lsquoDissipationrsquoEndorsements
0 0 1 0 3
Percentage Endorsed 56 56 167 167 333
Table 7 Each entry in the top two rows represents the number of solutions in the class Q1minus6 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking results to 1 decimal place if necessary
All of the flows in the class Q7minus9 endorse the QR Gradient and Vortex-Stretching-based subfilter-scale models (see Table 6) We can hence see that this class fully endorses these three models equallyand does not endorse the Smagorinsky model The Vortex-Stretching eddy viscosity model has themost endorsements from any of the flows in the class Q0minus6 and the Smagorinksy and Vreman modelshave the fewest endorsements (see Table 7) We can hence say that this class mostly endorses theVortex-Stretching-based eddy-viscosity model note however that the percentage endorsement is very
30
low
Now let us consider LES with explicit filtering Tables 8 and 9 are analogues of Table 5 and illustratethe results for each flow class The analogue of Table 4 is Table 24 and is in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 1 1 0 0 0
Table 8 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Q0minus6 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 9 9 9 6
Table 9 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We have grouped Vremanrsquos classes into Q7minus9 and Q0minus6 this is equivalent to saying that we havegrouped Vremanrsquos classes into those with two or fewer non-zero velocity gradient elements and thosewith three or more non-zero velocity gradient elements respectively Therefore the solutions in Q7minus9are exactly those with a lsquo0rsquo in the lsquoExactrsquo column of Table 4 and Sol 5 (2D Taylor) ExcludingSol 5 (2D Taylor) these are the same as those five that endorse the Vreman QR Gradient andVortex-Stretching-based models in Table 5 We can see in Table 8 that the results for the lsquostrongrsquoendorsements are exactly the same as in Table 5 This is because all of the flows with zero subfilterdissipation are contained in the class Q7minus9 There are hence no lsquostrongrsquo endorsements in Table 9The majority of the lsquoweakrsquo endorsements come from Q0minus6 as this class contains all but one of thesolutions with non-zero true subfilter dissipation (given in Eq (67))
442 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
We now separate our exact solutions into classes again using Vremanrsquos flow classes This timehowever we define V0 to be the class of flows Vreman claims will always have zero true subfilterdissipation namely those classes mentioned in Appendix B We define Vlowast to be all the remaining
31
classes Then
V0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow)
Sol 3 (Velocity as a function of time) Sol 14 (Radially Dependent)(78)
Vlowast = Sol 4 (Axisymmetric Flow) Sol 5 (2D Taylor) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 13 (Sphere Translation) Sol 15 (Rankine Vortex)
(79)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 10 and 11 are analogues of Table 3 and illustrate the results for each flow class Theanalogues of Tables 1 and 2 are Tables 25 and 26 and are in Appendix C2
Endorsements for models using solutions in V0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 4 4 4 4
lsquoDissipationrsquoEndorsements
1 4 4 4 4
Percentage Endorsed 375 100 100 100 100
Table 10 Each entry in the top two rows represents the number of solutions in the class V0 thatendorse each model There are 4 solutions in this class so the percentages are worked out by dividingthe total of each column by 8 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Vlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 2 4 5 5
lsquoDissipationrsquoEndorsements
1 1 3 2 5
Percentage Endorsed 136 136 318 318 455
Table 11 Each entry in the top two rows represents the number of solutions in the class Vlowast thatendorse each model There are 11 solutions in this class so the percentages are worked out bydividing the total of each column by 22 and taking results to 1 decimal place if necessary
All of the flows in the class V0 endorse the Vreman QR Gradient and Vortex-Stretching-basedsubfilter-scale models (see Table 6) This class completely endorses these four models equally anddoes not endorse the Smagorinsky model The Vortex-Stretching-based eddy-viscosity model hasthe most endorsements from the class Vlowast and the Smagorinksy and Vreman models again have thefewest endorsements (see Table 7) As in Section 441 we say that this class mostly endorses the
32
Vortex-Stretching-based eddy-viscosity model but once again the percentage endorsement is ratherlow
Now let us consider LES with explicit filtering Tables 12 and 13 are analogues of Table 5 andillustrate the results for each flow class The analogue of Table 4 is Table 27 and is in Appendix C2
Endorsements for models using solutions in V0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 4 4 4 4
lsquoWeakrsquo Endorsements 0 0 0 0 0
Table 12 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Vlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 1 1 1 1
lsquoWeakrsquo Endorsements 10 10 9 6
Table 13 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We can see in Table 12 that the results for the lsquostrongrsquo endorsements are similar to those in Table 5excluding Sol 15 (Rankine Vortex) The four solutions in V0 with zero true subfilter dissipationnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)and Sol 14 (Radially Dependent) are hence the four that endorse the Vreman QR Gradientand Vortex-Stretching-based models in Table 12 Sol 15 (Rankine Vortex) is in the class Vlowast andhence there is one lsquostrongrsquo endorsement in Table 13 The class Vlowast contains all of the flows with anon-zero true subfilter dissipation (given in Eq (67)) and hence Table 13 contains all of the lsquoweakrsquoendorsements
443 Classes using the Third Principal Invariant
Sorting the classes using the invariants is a non-trivial task in itself There are a number of waysin which to do this using both the principal invariants in Eq (40) and the combined invariants inEq (43)
We start by looking at flows with third principal invariant RG equal to zero We again define twoclasses R0 and Rlowast where R0 consists of all flows with RG = 0 and Rlowast consists of all flows withRG 6= 0 Then using the information in Appendix A we see that there are nine flows in R0 That
33
is
R0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 8 (Vortex in Stagnation Flow)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)
(80)
Rlowast = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 13 (Sphere Translation)(81)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 14 and 15 are analogues of Table 3 and they illustrate the results for LES withoutexplicit filtering for each flow class The analogues of Tables 1 and 2 are Tables 28 and 29 and arein Appendix C3
Endorsements for models using solutions in R0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 7 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 278 556 778 722 778
Table 14 Each entry in the top two rows represents the number of solutions in the class R0 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking the result to one decimal place if necessary
Endorsements for models using solutions in Rlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 2 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 83 83 83 167 333
Table 15 Each entry in the top two rows represents the number of solutions in the class Rlowast thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking the result to one decimal place if necessary
The class R0 endorses both the Vortex-Stretching-based model and the QR model the most andSmagorinsky the least in the case for LES without explicit filtering In the Rlowast class the Vortex-Stretching-based model is again the most endorsed followed by the Gradient model however thepercentages are again small
34
Now let us consider LES with explicit filtering Tables 16 and 17 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 30 and is in Appendix C3
Endorsements for models using solutions in R0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 4 4 3 2
Table 16 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Rlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 6 6 6 4
Table 17 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 we can see in Table 16 that the results for the lsquostrongrsquo endorsements are exactly thesame as in Table 5 This is because all of the flows with zero subfilter dissipation are contained in theclass R0 As in Table 9 there are no lsquostrongrsquo endorsements in Table 17 The lsquoweakrsquo endorsementshowever are split more evenly between the two classes now than in Vremanrsquos classes in Sections 441and 442 We find that the Smagorinsky and Vreman models still seem to be the most endorsed inboth classes
444 Two-component Flow Classes
Another way in which the flows can be grouped into classes is by determining whether a flow is atwo-component flow or not We define a two-component flow to have the property that the combinedprincipal invariants of a flow in Eq (43) satisfy
I3 = I4 = I5 minus1
2I1I2 = 0 (82)
We are also interested in considering flows with zero gradient model subgrid dissipation These flowshave the property that
I3 minus I4 = 0 (83)
What we find though is that for all the solutions in Section 22 if a flow is a two-component flowthen it also satisfies I3 minus I4 = 0 It may be interesting to note that although I3 minus I4 6= 0 for manysolutions there are a number with I3 equal to a multiple of I4 Although not considered in thisreport this may also be a class of interest We find that the two-component flows are
C2 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)(84)
35
and we call the remaining flows three-component flows They are
C3 = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex) Sol 11 (Pseudo-Motion Shots)
Sol 13 (Sphere Translation)
(85)
Note that Sol 8 (Vortex in Stagnation Flow) is defined to be a planar flow this does not mean that itis a two-component flow in this sense however because its two non-zero velocity components dependon three coordinates We now check the models for LES without explicit filtering by consideringendorsements within each class Tables 18 and 19 are analogues of Table 3 and they illustrate theresults for LES without explicit filtering for each flow class The analogues of Tables 1 and 2 areTables 31 and 32 and are in Appendix C4
Endorsements for models using solutions in C2 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 6 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 357 714 100 857 100
Table 18 Each entry in the top two rows represents the number of solutions in the class C2 thatendorse each model There are 7 solutions in this class so the percentages are worked out by dividingthe total of each column by 14 and taking the result to one decimal place if necessary
Endorsements for models using solutions in C3 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 3 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 63 63 63 188 250
Table 19 Each entry in the top two rows represents the number of solutions in the class C3 thatendorse each model There are 8 solutions in this class so the percentages are worked out by dividingthe total of each column by 16 and taking the result to one decimal place if necessary
The class C2 equally endorses the QR and Vortex-Stretching-based models and once again theSmagorinsky model is the least endorsed As with the previous classes the class C3 has such lowpercentages that it does not seem to endorse any models We start to see a trend appearing here inthat the flow classes that contain the more complicated flows namely Q0minus6 (Eq (76)) Vlowast (Eq (79))Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages The classes that tendto contain the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78)) R0 (Eq (80)) and C2 (Eq (84))all seem to favour the Vortex-Stretching-based model or the QR model
36
Now let us consider LES with explicit filtering Tables 20 and 21 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 33 and is in Appendix C4
Endorsements for models using solutions in C2 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 2 2 0 1 0
Table 20 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in C3 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 8 8 8 6
Table 21 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 and Table 16 we can see in Table 20 that the results for the lsquostrongrsquo endorsements areexactly the same as in Table 5 This is again because all of the flows with zero subfilter dissipationare contained in the class C2 There are two flows with non-zero true subfilter dissipation (givenin Eq (67)) in C2 and this shows in the lsquoweakrsquo endorsements in Table 20 There are no lsquostrongrsquoendorsements in Table 21 as all the zero subfilter dissipation flows are in C2
5 Conclusions
In this report we studied incompressible turbulent flows We found a number of exact solutions to theNavier-Stokes equations each representing a relatively simplistic fluid flow Large-eddy simulation(LES) is used to model turbulent flows by smoothing over the complex small-scale motion Weintroduced a number of subfilter-scale models to approximate this small-scale motion and testedthem using the found exact solutions to the Navier-Stokes equations We first considered LESwithout explicit filtering and used the Navier-Stokes equations to derive conditions for the subfilter-scale models namely the Velocity Field condition (Eq (58)) the Kinetic Energy condition (Eq (60))and the Dissipation condition (Eq (62)) Similarly we considered LES with explicit filtering andagain used the Navier-Stokes equations to derive more conditions for the models in this case wedecided to just take one condition the Dissipation condition (Eqs (67) and (68)) as the others leadto difficulty in their evaluation By lsquorunningrsquo the exact solutions to the Navier-Stokes equationsthrough each condition for the subfilter-scale models we found that the Vortex-Stretching-basededdy-viscosity model in Eq (54) was most endorsed by the tests in LES without explicit filteringWe found that four out of the five subfilter-scale models namely the Vreman QR Gradient andVortex-Stretching-based models were equally endorsed by the lsquostrongrsquo tests in LES with explicitfiltering with the Smagorinsky model being the least endorsed The results we found for the lsquoweakrsquocondition were completely opposite to those found in both the lsquostrongrsquo condition and in LES withoutexplicit filtering This led us to conclude that this test was not one in which we should put our trust
37
We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses
In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes
This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold
38
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
of the models we consider here can be expressed in terms of the rate of strain tensor S defined inEq (41) the rate of rotation tensor Ω defined in Eq (42) and the combined invariants defined inEq (43) The models are then defined as follows
Smagorinsky [12] νSe = (CSδ)2radic
2I1 (51)
Vreman [16] νVe = (CV δ)2
radicQGGT
PGGT (52)
QR [13 14 15] νQRe = (CQRδ)2 max0minusI3
I1 (53)
Vortex-Stretching [11] νV Se = (CV Sδ)2radic
2I1
(I5 minus 1
2I1I2
minusI1I2
)32
(54)
with the quantities
PGGT = I1 minus I2 QGGT =1
4(I1 + I2)2 + 4
(I5 minus
1
2I1I2
) (55)
There is a fifth and final model to be considered here It is of a different form to the other four andnot an eddy-viscosity model It is known as the Gradient model and is defined by
Gradient [4 9] τmodG = CGδ
(S2 minus Ω2 minus (SΩminus ΩS)
) (56)
The values C are model constants specific to each model and δ represents the mesh size of the LESBoth of these quantities are included here to properly define the models however in this projectthey will be ignored as they are constants and have no influence on the conditions to be defined
33 Conditions for Large-Eddy Simulation without Explicit Filtering
The filtering operation introduced in Eq (48) is just a formality for LES without explicit filteringThe idea here is that filtering does not actually take place but the term lsquoaddedrsquo to the Navier-Stokesequations in fact lsquosubtractsrsquo the difficult to simulate small-scale turbulent motion of the flow Inorder to consider this properly the terms ui and p in Eq (48) are replaced by vi and q It is hopedthat τmod
ij approximates τij so that vi and q approximate ui and p respectively So LES withoutexplicit filtering is given by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (57)
It is clear to see that the first four terms of this equation are exactly the same as the Navier-Stokesequations so for the exact solutions discussed in Section 22 the combination of these terms willequal zero What is then desirable is that the model term is also zero for these exact solutions sothat Eq (57) still holds In order to test models for exact solutions the condition
part
partxjτmodij (v) = 0 (58)
is checked If this condition were to hold it would imply that the model does not influence thevelocity field of a flow hence we call it the Velocity Field Condition There are a number of otherrestrictions that can be tested One such test comes from the differential form of the evolution ofthe kinetic energy of a flow given in Eq (10) In order to find this Eq (57) is multiplied by vi andterms are grouped to find
part
partt
(1
2vivi
)+ vj
part
partxj
(1
2vivi
)= minusvi
1
ρ
partq
partxi+ νvi
part2vipartxjpartxj
minus vipart
partxjτmodij (v) (59)
23
Then similarly to Eq (58) another condition is to check whether the model affects the kineticenergy of exact solutions this is equivalent to checking whether
vipart
partxjτmodij (v) = 0 (60)
We call this condition the Kinetic Energy Condition The third and final condition considered herecan also be found using the integral form for the equation for kinetic energy
d
dt
intV
1
2vividV =
intS
[minusvj
(1
2vivi
)minus vj
q
ρ+ 2νviSij + viτ
modij (v)
]njdS
+
intV
[minus2νSijSij minus
(minusSijτmod
ij (v))]dV
(61)
where the rate-of-strain tensor Sij was defined in Eq (41) V is a volume of fluid and dS meansthe integral is over the surface of that volume with unit normal nj Then this final condition checkswhether the model affects the total kinetic energy of exact solutions to the Navier-Stokes equationsthat is
Sijτmodij (v) = 0 (62)
We will call this condition the Dissipation Condition because Sijτmodij (v) is often referred to as the
subgrid dissipation
34 Conditions for Large-Eddy Simulation with Explicit Filtering
As in Eq (48) the Navier-Stokes equations can be filtered by applying the filtering operation to thewhole equation (putting a bar over all terms) and evaluating to find again that
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (63)
In Eq (57) the filtering operation was ignored and the elements ui p and τij(u) were replacedwith vi q and τmod
ij (v) respectively Now in LES with explicit filtering the filtering operation itselfsmooths over the small-scale turbulent motion of the simulated flow In this case the elements uip and τij(u) will be replaced with vi q and τmod
ij (v) respectively LES with explicit filtering is thengiven by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (64)
Note that Eqs (63) and (64) despite their similarity to Eq (57) no longer contain the Navier-Stokesequations as Eq (57) did The bar (filtering operation) over each term results in each exact solutionfound in Section 22 not necessarily satisfying the first four terms It is hence unknown whether ornot these terms will vanish for exact solutions Terms can still be compared similarly to Eqs (58)(60) and (62) However instead of comparing to zero we can now compare to the relevant termfrom the filtered Navier-Stokes equations So the terms
part
partxjτij (65)
and
part
partxjτmodij (v) (66)
24
are to be compared Although the latter term is relatively simple to compute the former could leadto some difficulties due to the differentiation of the true stress τij = uiuj minus uiuj So although someinformation could be gleaned from the inspection of these two terms more information can be foundby considering the analogue of Eq (62) So the true subfilter dissipation
Sijτij (67)
(where Sij is shorthand for Sij(v)) and the modelled subfilter dissipation
Sij(v)τmodij (v) (68)
are considered and have to be equal for the condition to hold These can again be derived from theintegral equation for kinetic energy of a filtered velocity field similarly to Eq (61)
4 Analysis of Large-Eddy Simulation Models
In this section we use a code in Mathematica the theory described in Section 3 and the exactsolutions to the Navier-Stokes equations found in Section 22 to check whether or not each of theexact solutions lsquoendorsesrsquo each model for LES both without and with explicit filtering
41 Large-Eddy Simulation Without Explicit Filtering
We start by using a code in Mathematica to test the Velocity Field condition in Eq (58) and theKinetic Energy condition in Eq (60) For all exact solutions to the Navier-Stokes equations inSection 22 the outputs for both conditions were identical so we have combined them into one table
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 ndash ndash 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 1 For the Kinetic Energy condition a lsquo0rsquo result means that vipartpartxj
τmodij (v) = 0 for respective
models and exact solutions whereas a lsquolowastrsquo result means that vipartpartxj
τmodij (v) 6= 0 For the Velocity
Field condition a lsquo0rsquo result means that partpartxj
τmodij (v) = 0 for respective models and exact solutions
whereas a lsquolowastrsquo result means that partpartxj
τmodij (v) 6= 0 The Gradient model can take both positive and
negative values so when the sign of the model is known it is included in the table instead of a lsquorsquoBoth the conditions are included in one table because for all cases the outputs were identical
We then test the Dissipation condition in Eq (62) for all exact solutions to the Navier-Stokesequations in Section 22
25
Dissipation Condition [Does Sijτmodij (v) = 0]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 + 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 2 A lsquo0rsquo result means that Sijτmodij (v) = 0 for respective models and exact solutions whereas
a lsquolowastrsquo result means that Sijτmodij (v) 6= 0 The Gradient model can take both positive and negative
values so when the sign of the model is known it is included in the table instead of a lsquorsquo
We then sum all the results to find the total number of endorsements for each model described inSection 32
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
4 6 8 9 9
lsquoDissipationrsquoEndorsements
2 5 7 7 9
Total Endorsements 6 11 15 16 18
Table 3 The results from Tables 1 and 2 are summarised to give a general overview of how manyendorsements each model receives This makes it much clearer to see the models most supported bythe respective and combined conditions with the Vortex-Stretching model being the most supportedand the Smagorinsky model the least
Testing the models in Section 32 using LES without explicit filtering has mixed results It isclear that the Smagorinksy model is not well endorsed by the flows in Section 22 with the onlyendorsements coming from the simplest flows that is Sol 1 (Linear Velocity) Sol 3 (Velocity as afunction of time) Sol 7 (Stagnation Flow) and Sol 15 (Rankine Vortex) The Vortex-Stretching-based model is the most endorsed flow with eighteen out of a possible thirty endorsements Thisagrees with our foreknowledge as this model was constructed to be as physically consistent as possiblein the paper by Silvis amp Verstappen [11]
42 Large-Eddy Simulation With Explicit Filtering
In this section we will compare the true subfilter dissipation Sijτij and the modelled subfilterdissipation Sij(v)τmod
ij (v) given in Eqs (67) and (68) respectively for each model τmodij and each
exact solution to the Navier-Stokes equation given in Section 22 We will not consider the conditionsinvolving differentiation in Eqs (65) and (66) due to the difficulty in calculating the derivative ofthe τij term in Eq (65)
26
421 Example of a Test for a Model
The true subfilter dissipation Sijτij of a flow given in Eq (67) cannot be explicitly computed asthe filtering operation is kept general What can be done however is to partially evaluate theexpressions using the attributes of a filtering operation mentioned in Eq (48)
The calculations of Sijτij for Sol 1 (Linear Velocity) are provided to illustrate the general ideaRecall that the velocity field of Sol 1 (Linear Velocity) is written in Eq (13) in vector notation as
u(x y z t) =
0ax+ bcx+ d
(69)
Then Sij = Sij(u) is computed simply by using its definition in Eq (41) Note that as filteringpreserves constants if there is an a it is replaced with a meaning that (in this case) there are alarge number of zero terms due to the frequency of 0 Hence
[Sij(u)
]=
[1
2
(partuipartxj
+partujpartxi
)]=
0 a
2c2
a2 0 0
c2 0 0
=
0 a
2c2
a2 0 0
c2 0 0
(70)
Note that as Sijτij is being computed which is equal to the sum of the element-wise multiplicationof the two matrices (or the trace of the matrix multiplication) that we donrsquot need to consider theelements of τij(u) with corresponding zero entry in Sij As τij(u) = uiuj minus uiuj and again asfiltering preserves constants it is found that
[τij(u)] =[uiuj minus uiuj
]=
lowast 0 middot ax+ b 0 middot cx+ d
ax+ b middot 0 lowast lowastcx+ d middot 0 lowast lowast
=
lowast 0 00 lowast lowast0 lowast lowast
(71)
where a lsquolowastrsquo just represents some element (not necessarily non-zero) Therefore the true subfilterdissipation
Sijτij = tr
0 a2
c2
a2 0 0c2 0 0
middotlowast 0 0
0 lowast lowast0 lowast lowast
= 0 (72)
Now the modelled subfilter dissipation Sij(u)τmodij (u) needs to be computed (recall that Sij =
Sij(u)) In this case let us use the QR model stated in Eq (53) Once again Sij(u) calculated inEq (70) needs to be used So we have that
τmodij = minus2(CQRδ)
2 max0minusI3I1
Sij (73)
As
I1 = tr(S2) =
1
2(a2 + c2) and I3 = tr(S
3) = 0 (74)
it holds that τmodij (u) = 0 so Sij(u)τmod
ij (u) = 0 This agrees with Eq (72) and hence Sol 1(Linear Velocity) endorses the QR model
27
422 Testing the Models
Now using a Mathematica code the exact solutions in Section 22 are again used to evaluate thecondition for the models using the process just described in Section 421 The results are describedin Table 4
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 4 All entries with a ldquo0rdquo represent a value of zero and a ldquolowastrdquo represents a value that is non-zeroThe ldquoExactrdquo row represents the value of the true subfilter dissipation Sijτij Every other row witha model name represents the value of the modelled subfilter dissipation Sij(u)τmod
ij (u) respectiveto that model A ldquordquo means that it is unclear as to the value - the sign of these values is oftenunknown and the lsquoMaxrsquo function in the definition of the QR model (in Eq (53)) makes it difficultto evaluate
Reading from Table 4 the Smagorinsky model is clearly endorsed by Sol 3 (Velocity as a function oftime) and Sol 15 (Rankine Vortex) as in both cases both Sijτij = 0 = Sij(u)τmod
ij (u) In this casewe say that Sol 3 (Velocity as a function of time) and Sol 15 (Rankine Vortex) lsquostronglyrsquo endorsethe Smagorinsky model The Smagorinsky model is not endorsed by Sol 1 (Linear Velocity) Sol 2(Planar Flow) or Sol 14 (Radially Dependent) however because Sij(u)τmod
ij (u) 6= 0 For Sol 4 to
13 both Sijτij 6= 0 6= Sij(u)τmodij (u) and so we will say that these exact solutions lsquoweaklyrsquo endorse
the Smagorinsky model
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 10 10 9 6
Table 5 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Interestingly the lsquostrongrsquo and lsquoweakrsquo endorsements give different results However more stock shouldbe put in the results of the lsquostrongerrsquo condition as it is completely clear that zero is equal to zeroIt is not necessarily clear whether or not non-zero equals non-zero as the values will depend on theparameters in each solution hence the lsquoweakerrsquo endorsement
The lsquostrongrsquo condition equally endorses the Vreman QR Gradient and Vortex-Stretching-basedmodels whereas the lsquoweakrsquo condition endorses Vreman and Smagorinsky the most closely followedby the Gradient model with the Vortex-Stretching-based model being the most opposed As this
28
completely contradicts the lsquostrongrsquo condition and the results in Section 41 it makes us question thevalidity of the weaker test The number of endorsements for the QR model is unknown because of thedefinition of the model in Eq (53) and our evaluation of the true subfilter dissipation Eq (67) Dueto the exact solutions we have found it is never clear whether or not the true subfilter dissipationis strictly positive or negative This means that when the true subfilter dissipation is found to benon-zero the lsquoMaxrsquo function in the QR model cannot be evaluated
43 Vremanrsquos True Subfilter Dissipation Claim
Using the theory mentioned in Sections 232 and 34 we can place the exact solutions to the Navier-Stokes equations into different classes defined by Vreman In doing this we can also attempt to verifyone of Vremanrsquos results [16] In Vremanrsquos paper he defines the class Qn to be the set of flows withn zero entries in the velocity gradient of the flow (the velocity gradient is defined in Eq (39))Vreman then claims that the only flows with zero true subfilter dissipation are a certain selection ofthose in the classesQ7Q8 orQ9 shown in appendix B and is generally non-zero for any other flows
The true subfilter dissipation of Sol 1 (Linear Velocity) was computed in Section 421 and wasfound to be zero in Eq (72) The velocity gradient of Sol 1 is as follows
GLV =
0 0 0a 0 0c 0 0
(75)
and hence Sol 1 (Linear Velocity) is in the required subset of Q7 which agrees with Vremanrsquos claimIn Section 422 the true subfilter dissipation was computed (determined to be zero or non-zero) forall of the exact solutions to the Navier Stokes equations in Section 22 These solutions can now belsquoplacedrsquo into each class Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a functionof time) and Sol 14 (Radially Dependent) were all found to have zero true subfilter dissipation andare all in the appropriate subsets of either Q7Q8 or Q9 This agrees with Vremanrsquos results
We can extend Vremanrsquos results however We found that Sol 15 (Rankine Vortex) had zero truesubfilter dissipation and yet is not in the subset of Q7 defined in Appendix B It is important tonote that the rate of strain tensor Sij in Eq (41) for Sol 15 (Rankine Vortex) is zero This is thereason the true subfilter dissipation is zero and the same reason that all models are zero for thisflow as all models are based on Sij
We also found that all of the remaining exact solutions to the Navier-Stokes equations are not inVremanrsquos specified classes (in Appendix Section B) They also have a non-zero subfilter dissipationwhich agrees with Vremanrsquos results in his paper [16]
44 Classes of Flows
It is important to consider that some exact solutions to the Navier-Stokes equations may have similarproperties For example Sol 1 (Linear Velocity) and Sol 2 (Planar Flow) are both very similarIn fact Sol 2 (Planar Flow) is a more general case of Sol 1 (Linear Velocity) We therefore mustgroup the different flows into classes in an attempt to reduce the bias that may occur when solutionsare similar This should aid us to further determine which of the models are most endorsed We willdo this in a number of different ways using the theory discussed in Section 23 We will group theflows with respect to Vremanrsquos Flow Classes given in appendix B and with respect to the Invariantsgiven in appendix A
29
441 Vremanrsquos Velocity Gradient Flow Classes
We now put our exact solutions to the Navier-Stokes equations into one of two classes Either Q0minus6or Q7minus9 where each class is defined intuitively as follows
Q0minus6 = Q0 cupQ1 cup middot middot middot cup Q6 (76)
Q7minus9 = Q7 cupQ8 cupQ9 (77)
where Qn is defined in Section 43 By looking at the gradient of each exact solution we can see thatQ7minus9 contains the six solutions Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as afunction of time) Sol 5 (2D Taylor) Sol 14 (Radially Dependent) and Sol 15 (Rankine Vortex)and that Q0minus6 contains the remaining nine We now check the models for LES without explicitfiltering by considering endorsements within each class Tables 6 and 7 are analogues of Table 3 andillustrate the results for each flow class The analogues of Tables 1 and 2 are Tables 22 and 23 andare in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 6 6 6
lsquoDissipationrsquoEndorsements
2 5 6 6 6
Percentage Endorsed 417 833 100 100 100
Table 6 Each entry in the top two rows represents the number of solutions in the class Q7minus9 thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Q0minus6 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 2 3 3
lsquoDissipationrsquoEndorsements
0 0 1 0 3
Percentage Endorsed 56 56 167 167 333
Table 7 Each entry in the top two rows represents the number of solutions in the class Q1minus6 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking results to 1 decimal place if necessary
All of the flows in the class Q7minus9 endorse the QR Gradient and Vortex-Stretching-based subfilter-scale models (see Table 6) We can hence see that this class fully endorses these three models equallyand does not endorse the Smagorinsky model The Vortex-Stretching eddy viscosity model has themost endorsements from any of the flows in the class Q0minus6 and the Smagorinksy and Vreman modelshave the fewest endorsements (see Table 7) We can hence say that this class mostly endorses theVortex-Stretching-based eddy-viscosity model note however that the percentage endorsement is very
30
low
Now let us consider LES with explicit filtering Tables 8 and 9 are analogues of Table 5 and illustratethe results for each flow class The analogue of Table 4 is Table 24 and is in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 1 1 0 0 0
Table 8 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Q0minus6 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 9 9 9 6
Table 9 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We have grouped Vremanrsquos classes into Q7minus9 and Q0minus6 this is equivalent to saying that we havegrouped Vremanrsquos classes into those with two or fewer non-zero velocity gradient elements and thosewith three or more non-zero velocity gradient elements respectively Therefore the solutions in Q7minus9are exactly those with a lsquo0rsquo in the lsquoExactrsquo column of Table 4 and Sol 5 (2D Taylor) ExcludingSol 5 (2D Taylor) these are the same as those five that endorse the Vreman QR Gradient andVortex-Stretching-based models in Table 5 We can see in Table 8 that the results for the lsquostrongrsquoendorsements are exactly the same as in Table 5 This is because all of the flows with zero subfilterdissipation are contained in the class Q7minus9 There are hence no lsquostrongrsquo endorsements in Table 9The majority of the lsquoweakrsquo endorsements come from Q0minus6 as this class contains all but one of thesolutions with non-zero true subfilter dissipation (given in Eq (67))
442 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
We now separate our exact solutions into classes again using Vremanrsquos flow classes This timehowever we define V0 to be the class of flows Vreman claims will always have zero true subfilterdissipation namely those classes mentioned in Appendix B We define Vlowast to be all the remaining
31
classes Then
V0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow)
Sol 3 (Velocity as a function of time) Sol 14 (Radially Dependent)(78)
Vlowast = Sol 4 (Axisymmetric Flow) Sol 5 (2D Taylor) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 13 (Sphere Translation) Sol 15 (Rankine Vortex)
(79)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 10 and 11 are analogues of Table 3 and illustrate the results for each flow class Theanalogues of Tables 1 and 2 are Tables 25 and 26 and are in Appendix C2
Endorsements for models using solutions in V0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 4 4 4 4
lsquoDissipationrsquoEndorsements
1 4 4 4 4
Percentage Endorsed 375 100 100 100 100
Table 10 Each entry in the top two rows represents the number of solutions in the class V0 thatendorse each model There are 4 solutions in this class so the percentages are worked out by dividingthe total of each column by 8 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Vlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 2 4 5 5
lsquoDissipationrsquoEndorsements
1 1 3 2 5
Percentage Endorsed 136 136 318 318 455
Table 11 Each entry in the top two rows represents the number of solutions in the class Vlowast thatendorse each model There are 11 solutions in this class so the percentages are worked out bydividing the total of each column by 22 and taking results to 1 decimal place if necessary
All of the flows in the class V0 endorse the Vreman QR Gradient and Vortex-Stretching-basedsubfilter-scale models (see Table 6) This class completely endorses these four models equally anddoes not endorse the Smagorinsky model The Vortex-Stretching-based eddy-viscosity model hasthe most endorsements from the class Vlowast and the Smagorinksy and Vreman models again have thefewest endorsements (see Table 7) As in Section 441 we say that this class mostly endorses the
32
Vortex-Stretching-based eddy-viscosity model but once again the percentage endorsement is ratherlow
Now let us consider LES with explicit filtering Tables 12 and 13 are analogues of Table 5 andillustrate the results for each flow class The analogue of Table 4 is Table 27 and is in Appendix C2
Endorsements for models using solutions in V0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 4 4 4 4
lsquoWeakrsquo Endorsements 0 0 0 0 0
Table 12 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Vlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 1 1 1 1
lsquoWeakrsquo Endorsements 10 10 9 6
Table 13 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We can see in Table 12 that the results for the lsquostrongrsquo endorsements are similar to those in Table 5excluding Sol 15 (Rankine Vortex) The four solutions in V0 with zero true subfilter dissipationnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)and Sol 14 (Radially Dependent) are hence the four that endorse the Vreman QR Gradientand Vortex-Stretching-based models in Table 12 Sol 15 (Rankine Vortex) is in the class Vlowast andhence there is one lsquostrongrsquo endorsement in Table 13 The class Vlowast contains all of the flows with anon-zero true subfilter dissipation (given in Eq (67)) and hence Table 13 contains all of the lsquoweakrsquoendorsements
443 Classes using the Third Principal Invariant
Sorting the classes using the invariants is a non-trivial task in itself There are a number of waysin which to do this using both the principal invariants in Eq (40) and the combined invariants inEq (43)
We start by looking at flows with third principal invariant RG equal to zero We again define twoclasses R0 and Rlowast where R0 consists of all flows with RG = 0 and Rlowast consists of all flows withRG 6= 0 Then using the information in Appendix A we see that there are nine flows in R0 That
33
is
R0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 8 (Vortex in Stagnation Flow)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)
(80)
Rlowast = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 13 (Sphere Translation)(81)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 14 and 15 are analogues of Table 3 and they illustrate the results for LES withoutexplicit filtering for each flow class The analogues of Tables 1 and 2 are Tables 28 and 29 and arein Appendix C3
Endorsements for models using solutions in R0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 7 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 278 556 778 722 778
Table 14 Each entry in the top two rows represents the number of solutions in the class R0 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking the result to one decimal place if necessary
Endorsements for models using solutions in Rlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 2 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 83 83 83 167 333
Table 15 Each entry in the top two rows represents the number of solutions in the class Rlowast thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking the result to one decimal place if necessary
The class R0 endorses both the Vortex-Stretching-based model and the QR model the most andSmagorinsky the least in the case for LES without explicit filtering In the Rlowast class the Vortex-Stretching-based model is again the most endorsed followed by the Gradient model however thepercentages are again small
34
Now let us consider LES with explicit filtering Tables 16 and 17 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 30 and is in Appendix C3
Endorsements for models using solutions in R0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 4 4 3 2
Table 16 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Rlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 6 6 6 4
Table 17 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 we can see in Table 16 that the results for the lsquostrongrsquo endorsements are exactly thesame as in Table 5 This is because all of the flows with zero subfilter dissipation are contained in theclass R0 As in Table 9 there are no lsquostrongrsquo endorsements in Table 17 The lsquoweakrsquo endorsementshowever are split more evenly between the two classes now than in Vremanrsquos classes in Sections 441and 442 We find that the Smagorinsky and Vreman models still seem to be the most endorsed inboth classes
444 Two-component Flow Classes
Another way in which the flows can be grouped into classes is by determining whether a flow is atwo-component flow or not We define a two-component flow to have the property that the combinedprincipal invariants of a flow in Eq (43) satisfy
I3 = I4 = I5 minus1
2I1I2 = 0 (82)
We are also interested in considering flows with zero gradient model subgrid dissipation These flowshave the property that
I3 minus I4 = 0 (83)
What we find though is that for all the solutions in Section 22 if a flow is a two-component flowthen it also satisfies I3 minus I4 = 0 It may be interesting to note that although I3 minus I4 6= 0 for manysolutions there are a number with I3 equal to a multiple of I4 Although not considered in thisreport this may also be a class of interest We find that the two-component flows are
C2 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)(84)
35
and we call the remaining flows three-component flows They are
C3 = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex) Sol 11 (Pseudo-Motion Shots)
Sol 13 (Sphere Translation)
(85)
Note that Sol 8 (Vortex in Stagnation Flow) is defined to be a planar flow this does not mean that itis a two-component flow in this sense however because its two non-zero velocity components dependon three coordinates We now check the models for LES without explicit filtering by consideringendorsements within each class Tables 18 and 19 are analogues of Table 3 and they illustrate theresults for LES without explicit filtering for each flow class The analogues of Tables 1 and 2 areTables 31 and 32 and are in Appendix C4
Endorsements for models using solutions in C2 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 6 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 357 714 100 857 100
Table 18 Each entry in the top two rows represents the number of solutions in the class C2 thatendorse each model There are 7 solutions in this class so the percentages are worked out by dividingthe total of each column by 14 and taking the result to one decimal place if necessary
Endorsements for models using solutions in C3 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 3 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 63 63 63 188 250
Table 19 Each entry in the top two rows represents the number of solutions in the class C3 thatendorse each model There are 8 solutions in this class so the percentages are worked out by dividingthe total of each column by 16 and taking the result to one decimal place if necessary
The class C2 equally endorses the QR and Vortex-Stretching-based models and once again theSmagorinsky model is the least endorsed As with the previous classes the class C3 has such lowpercentages that it does not seem to endorse any models We start to see a trend appearing here inthat the flow classes that contain the more complicated flows namely Q0minus6 (Eq (76)) Vlowast (Eq (79))Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages The classes that tendto contain the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78)) R0 (Eq (80)) and C2 (Eq (84))all seem to favour the Vortex-Stretching-based model or the QR model
36
Now let us consider LES with explicit filtering Tables 20 and 21 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 33 and is in Appendix C4
Endorsements for models using solutions in C2 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 2 2 0 1 0
Table 20 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in C3 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 8 8 8 6
Table 21 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 and Table 16 we can see in Table 20 that the results for the lsquostrongrsquo endorsements areexactly the same as in Table 5 This is again because all of the flows with zero subfilter dissipationare contained in the class C2 There are two flows with non-zero true subfilter dissipation (givenin Eq (67)) in C2 and this shows in the lsquoweakrsquo endorsements in Table 20 There are no lsquostrongrsquoendorsements in Table 21 as all the zero subfilter dissipation flows are in C2
5 Conclusions
In this report we studied incompressible turbulent flows We found a number of exact solutions to theNavier-Stokes equations each representing a relatively simplistic fluid flow Large-eddy simulation(LES) is used to model turbulent flows by smoothing over the complex small-scale motion Weintroduced a number of subfilter-scale models to approximate this small-scale motion and testedthem using the found exact solutions to the Navier-Stokes equations We first considered LESwithout explicit filtering and used the Navier-Stokes equations to derive conditions for the subfilter-scale models namely the Velocity Field condition (Eq (58)) the Kinetic Energy condition (Eq (60))and the Dissipation condition (Eq (62)) Similarly we considered LES with explicit filtering andagain used the Navier-Stokes equations to derive more conditions for the models in this case wedecided to just take one condition the Dissipation condition (Eqs (67) and (68)) as the others leadto difficulty in their evaluation By lsquorunningrsquo the exact solutions to the Navier-Stokes equationsthrough each condition for the subfilter-scale models we found that the Vortex-Stretching-basededdy-viscosity model in Eq (54) was most endorsed by the tests in LES without explicit filteringWe found that four out of the five subfilter-scale models namely the Vreman QR Gradient andVortex-Stretching-based models were equally endorsed by the lsquostrongrsquo tests in LES with explicitfiltering with the Smagorinsky model being the least endorsed The results we found for the lsquoweakrsquocondition were completely opposite to those found in both the lsquostrongrsquo condition and in LES withoutexplicit filtering This led us to conclude that this test was not one in which we should put our trust
37
We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses
In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes
This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold
38
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
Then similarly to Eq (58) another condition is to check whether the model affects the kineticenergy of exact solutions this is equivalent to checking whether
vipart
partxjτmodij (v) = 0 (60)
We call this condition the Kinetic Energy Condition The third and final condition considered herecan also be found using the integral form for the equation for kinetic energy
d
dt
intV
1
2vividV =
intS
[minusvj
(1
2vivi
)minus vj
q
ρ+ 2νviSij + viτ
modij (v)
]njdS
+
intV
[minus2νSijSij minus
(minusSijτmod
ij (v))]dV
(61)
where the rate-of-strain tensor Sij was defined in Eq (41) V is a volume of fluid and dS meansthe integral is over the surface of that volume with unit normal nj Then this final condition checkswhether the model affects the total kinetic energy of exact solutions to the Navier-Stokes equationsthat is
Sijτmodij (v) = 0 (62)
We will call this condition the Dissipation Condition because Sijτmodij (v) is often referred to as the
subgrid dissipation
34 Conditions for Large-Eddy Simulation with Explicit Filtering
As in Eq (48) the Navier-Stokes equations can be filtered by applying the filtering operation to thewhole equation (putting a bar over all terms) and evaluating to find again that
partuipartt
+ ujpartuipartxj
= minus1
ρ
partp
partxi+ ν
part2uipartxjpartxj
minus part
partxjτij (63)
In Eq (57) the filtering operation was ignored and the elements ui p and τij(u) were replacedwith vi q and τmod
ij (v) respectively Now in LES with explicit filtering the filtering operation itselfsmooths over the small-scale turbulent motion of the simulated flow In this case the elements uip and τij(u) will be replaced with vi q and τmod
ij (v) respectively LES with explicit filtering is thengiven by
partvipartt
+ vjpartvipartxj
= minus1
ρ
partq
partxi+ ν
part2vipartxjpartxj
minus part
partxjτmodij (v) (64)
Note that Eqs (63) and (64) despite their similarity to Eq (57) no longer contain the Navier-Stokesequations as Eq (57) did The bar (filtering operation) over each term results in each exact solutionfound in Section 22 not necessarily satisfying the first four terms It is hence unknown whether ornot these terms will vanish for exact solutions Terms can still be compared similarly to Eqs (58)(60) and (62) However instead of comparing to zero we can now compare to the relevant termfrom the filtered Navier-Stokes equations So the terms
part
partxjτij (65)
and
part
partxjτmodij (v) (66)
24
are to be compared Although the latter term is relatively simple to compute the former could leadto some difficulties due to the differentiation of the true stress τij = uiuj minus uiuj So although someinformation could be gleaned from the inspection of these two terms more information can be foundby considering the analogue of Eq (62) So the true subfilter dissipation
Sijτij (67)
(where Sij is shorthand for Sij(v)) and the modelled subfilter dissipation
Sij(v)τmodij (v) (68)
are considered and have to be equal for the condition to hold These can again be derived from theintegral equation for kinetic energy of a filtered velocity field similarly to Eq (61)
4 Analysis of Large-Eddy Simulation Models
In this section we use a code in Mathematica the theory described in Section 3 and the exactsolutions to the Navier-Stokes equations found in Section 22 to check whether or not each of theexact solutions lsquoendorsesrsquo each model for LES both without and with explicit filtering
41 Large-Eddy Simulation Without Explicit Filtering
We start by using a code in Mathematica to test the Velocity Field condition in Eq (58) and theKinetic Energy condition in Eq (60) For all exact solutions to the Navier-Stokes equations inSection 22 the outputs for both conditions were identical so we have combined them into one table
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 ndash ndash 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 1 For the Kinetic Energy condition a lsquo0rsquo result means that vipartpartxj
τmodij (v) = 0 for respective
models and exact solutions whereas a lsquolowastrsquo result means that vipartpartxj
τmodij (v) 6= 0 For the Velocity
Field condition a lsquo0rsquo result means that partpartxj
τmodij (v) = 0 for respective models and exact solutions
whereas a lsquolowastrsquo result means that partpartxj
τmodij (v) 6= 0 The Gradient model can take both positive and
negative values so when the sign of the model is known it is included in the table instead of a lsquorsquoBoth the conditions are included in one table because for all cases the outputs were identical
We then test the Dissipation condition in Eq (62) for all exact solutions to the Navier-Stokesequations in Section 22
25
Dissipation Condition [Does Sijτmodij (v) = 0]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 + 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 2 A lsquo0rsquo result means that Sijτmodij (v) = 0 for respective models and exact solutions whereas
a lsquolowastrsquo result means that Sijτmodij (v) 6= 0 The Gradient model can take both positive and negative
values so when the sign of the model is known it is included in the table instead of a lsquorsquo
We then sum all the results to find the total number of endorsements for each model described inSection 32
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
4 6 8 9 9
lsquoDissipationrsquoEndorsements
2 5 7 7 9
Total Endorsements 6 11 15 16 18
Table 3 The results from Tables 1 and 2 are summarised to give a general overview of how manyendorsements each model receives This makes it much clearer to see the models most supported bythe respective and combined conditions with the Vortex-Stretching model being the most supportedand the Smagorinsky model the least
Testing the models in Section 32 using LES without explicit filtering has mixed results It isclear that the Smagorinksy model is not well endorsed by the flows in Section 22 with the onlyendorsements coming from the simplest flows that is Sol 1 (Linear Velocity) Sol 3 (Velocity as afunction of time) Sol 7 (Stagnation Flow) and Sol 15 (Rankine Vortex) The Vortex-Stretching-based model is the most endorsed flow with eighteen out of a possible thirty endorsements Thisagrees with our foreknowledge as this model was constructed to be as physically consistent as possiblein the paper by Silvis amp Verstappen [11]
42 Large-Eddy Simulation With Explicit Filtering
In this section we will compare the true subfilter dissipation Sijτij and the modelled subfilterdissipation Sij(v)τmod
ij (v) given in Eqs (67) and (68) respectively for each model τmodij and each
exact solution to the Navier-Stokes equation given in Section 22 We will not consider the conditionsinvolving differentiation in Eqs (65) and (66) due to the difficulty in calculating the derivative ofthe τij term in Eq (65)
26
421 Example of a Test for a Model
The true subfilter dissipation Sijτij of a flow given in Eq (67) cannot be explicitly computed asthe filtering operation is kept general What can be done however is to partially evaluate theexpressions using the attributes of a filtering operation mentioned in Eq (48)
The calculations of Sijτij for Sol 1 (Linear Velocity) are provided to illustrate the general ideaRecall that the velocity field of Sol 1 (Linear Velocity) is written in Eq (13) in vector notation as
u(x y z t) =
0ax+ bcx+ d
(69)
Then Sij = Sij(u) is computed simply by using its definition in Eq (41) Note that as filteringpreserves constants if there is an a it is replaced with a meaning that (in this case) there are alarge number of zero terms due to the frequency of 0 Hence
[Sij(u)
]=
[1
2
(partuipartxj
+partujpartxi
)]=
0 a
2c2
a2 0 0
c2 0 0
=
0 a
2c2
a2 0 0
c2 0 0
(70)
Note that as Sijτij is being computed which is equal to the sum of the element-wise multiplicationof the two matrices (or the trace of the matrix multiplication) that we donrsquot need to consider theelements of τij(u) with corresponding zero entry in Sij As τij(u) = uiuj minus uiuj and again asfiltering preserves constants it is found that
[τij(u)] =[uiuj minus uiuj
]=
lowast 0 middot ax+ b 0 middot cx+ d
ax+ b middot 0 lowast lowastcx+ d middot 0 lowast lowast
=
lowast 0 00 lowast lowast0 lowast lowast
(71)
where a lsquolowastrsquo just represents some element (not necessarily non-zero) Therefore the true subfilterdissipation
Sijτij = tr
0 a2
c2
a2 0 0c2 0 0
middotlowast 0 0
0 lowast lowast0 lowast lowast
= 0 (72)
Now the modelled subfilter dissipation Sij(u)τmodij (u) needs to be computed (recall that Sij =
Sij(u)) In this case let us use the QR model stated in Eq (53) Once again Sij(u) calculated inEq (70) needs to be used So we have that
τmodij = minus2(CQRδ)
2 max0minusI3I1
Sij (73)
As
I1 = tr(S2) =
1
2(a2 + c2) and I3 = tr(S
3) = 0 (74)
it holds that τmodij (u) = 0 so Sij(u)τmod
ij (u) = 0 This agrees with Eq (72) and hence Sol 1(Linear Velocity) endorses the QR model
27
422 Testing the Models
Now using a Mathematica code the exact solutions in Section 22 are again used to evaluate thecondition for the models using the process just described in Section 421 The results are describedin Table 4
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 4 All entries with a ldquo0rdquo represent a value of zero and a ldquolowastrdquo represents a value that is non-zeroThe ldquoExactrdquo row represents the value of the true subfilter dissipation Sijτij Every other row witha model name represents the value of the modelled subfilter dissipation Sij(u)τmod
ij (u) respectiveto that model A ldquordquo means that it is unclear as to the value - the sign of these values is oftenunknown and the lsquoMaxrsquo function in the definition of the QR model (in Eq (53)) makes it difficultto evaluate
Reading from Table 4 the Smagorinsky model is clearly endorsed by Sol 3 (Velocity as a function oftime) and Sol 15 (Rankine Vortex) as in both cases both Sijτij = 0 = Sij(u)τmod
ij (u) In this casewe say that Sol 3 (Velocity as a function of time) and Sol 15 (Rankine Vortex) lsquostronglyrsquo endorsethe Smagorinsky model The Smagorinsky model is not endorsed by Sol 1 (Linear Velocity) Sol 2(Planar Flow) or Sol 14 (Radially Dependent) however because Sij(u)τmod
ij (u) 6= 0 For Sol 4 to
13 both Sijτij 6= 0 6= Sij(u)τmodij (u) and so we will say that these exact solutions lsquoweaklyrsquo endorse
the Smagorinsky model
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 10 10 9 6
Table 5 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Interestingly the lsquostrongrsquo and lsquoweakrsquo endorsements give different results However more stock shouldbe put in the results of the lsquostrongerrsquo condition as it is completely clear that zero is equal to zeroIt is not necessarily clear whether or not non-zero equals non-zero as the values will depend on theparameters in each solution hence the lsquoweakerrsquo endorsement
The lsquostrongrsquo condition equally endorses the Vreman QR Gradient and Vortex-Stretching-basedmodels whereas the lsquoweakrsquo condition endorses Vreman and Smagorinsky the most closely followedby the Gradient model with the Vortex-Stretching-based model being the most opposed As this
28
completely contradicts the lsquostrongrsquo condition and the results in Section 41 it makes us question thevalidity of the weaker test The number of endorsements for the QR model is unknown because of thedefinition of the model in Eq (53) and our evaluation of the true subfilter dissipation Eq (67) Dueto the exact solutions we have found it is never clear whether or not the true subfilter dissipationis strictly positive or negative This means that when the true subfilter dissipation is found to benon-zero the lsquoMaxrsquo function in the QR model cannot be evaluated
43 Vremanrsquos True Subfilter Dissipation Claim
Using the theory mentioned in Sections 232 and 34 we can place the exact solutions to the Navier-Stokes equations into different classes defined by Vreman In doing this we can also attempt to verifyone of Vremanrsquos results [16] In Vremanrsquos paper he defines the class Qn to be the set of flows withn zero entries in the velocity gradient of the flow (the velocity gradient is defined in Eq (39))Vreman then claims that the only flows with zero true subfilter dissipation are a certain selection ofthose in the classesQ7Q8 orQ9 shown in appendix B and is generally non-zero for any other flows
The true subfilter dissipation of Sol 1 (Linear Velocity) was computed in Section 421 and wasfound to be zero in Eq (72) The velocity gradient of Sol 1 is as follows
GLV =
0 0 0a 0 0c 0 0
(75)
and hence Sol 1 (Linear Velocity) is in the required subset of Q7 which agrees with Vremanrsquos claimIn Section 422 the true subfilter dissipation was computed (determined to be zero or non-zero) forall of the exact solutions to the Navier Stokes equations in Section 22 These solutions can now belsquoplacedrsquo into each class Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a functionof time) and Sol 14 (Radially Dependent) were all found to have zero true subfilter dissipation andare all in the appropriate subsets of either Q7Q8 or Q9 This agrees with Vremanrsquos results
We can extend Vremanrsquos results however We found that Sol 15 (Rankine Vortex) had zero truesubfilter dissipation and yet is not in the subset of Q7 defined in Appendix B It is important tonote that the rate of strain tensor Sij in Eq (41) for Sol 15 (Rankine Vortex) is zero This is thereason the true subfilter dissipation is zero and the same reason that all models are zero for thisflow as all models are based on Sij
We also found that all of the remaining exact solutions to the Navier-Stokes equations are not inVremanrsquos specified classes (in Appendix Section B) They also have a non-zero subfilter dissipationwhich agrees with Vremanrsquos results in his paper [16]
44 Classes of Flows
It is important to consider that some exact solutions to the Navier-Stokes equations may have similarproperties For example Sol 1 (Linear Velocity) and Sol 2 (Planar Flow) are both very similarIn fact Sol 2 (Planar Flow) is a more general case of Sol 1 (Linear Velocity) We therefore mustgroup the different flows into classes in an attempt to reduce the bias that may occur when solutionsare similar This should aid us to further determine which of the models are most endorsed We willdo this in a number of different ways using the theory discussed in Section 23 We will group theflows with respect to Vremanrsquos Flow Classes given in appendix B and with respect to the Invariantsgiven in appendix A
29
441 Vremanrsquos Velocity Gradient Flow Classes
We now put our exact solutions to the Navier-Stokes equations into one of two classes Either Q0minus6or Q7minus9 where each class is defined intuitively as follows
Q0minus6 = Q0 cupQ1 cup middot middot middot cup Q6 (76)
Q7minus9 = Q7 cupQ8 cupQ9 (77)
where Qn is defined in Section 43 By looking at the gradient of each exact solution we can see thatQ7minus9 contains the six solutions Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as afunction of time) Sol 5 (2D Taylor) Sol 14 (Radially Dependent) and Sol 15 (Rankine Vortex)and that Q0minus6 contains the remaining nine We now check the models for LES without explicitfiltering by considering endorsements within each class Tables 6 and 7 are analogues of Table 3 andillustrate the results for each flow class The analogues of Tables 1 and 2 are Tables 22 and 23 andare in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 6 6 6
lsquoDissipationrsquoEndorsements
2 5 6 6 6
Percentage Endorsed 417 833 100 100 100
Table 6 Each entry in the top two rows represents the number of solutions in the class Q7minus9 thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Q0minus6 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 2 3 3
lsquoDissipationrsquoEndorsements
0 0 1 0 3
Percentage Endorsed 56 56 167 167 333
Table 7 Each entry in the top two rows represents the number of solutions in the class Q1minus6 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking results to 1 decimal place if necessary
All of the flows in the class Q7minus9 endorse the QR Gradient and Vortex-Stretching-based subfilter-scale models (see Table 6) We can hence see that this class fully endorses these three models equallyand does not endorse the Smagorinsky model The Vortex-Stretching eddy viscosity model has themost endorsements from any of the flows in the class Q0minus6 and the Smagorinksy and Vreman modelshave the fewest endorsements (see Table 7) We can hence say that this class mostly endorses theVortex-Stretching-based eddy-viscosity model note however that the percentage endorsement is very
30
low
Now let us consider LES with explicit filtering Tables 8 and 9 are analogues of Table 5 and illustratethe results for each flow class The analogue of Table 4 is Table 24 and is in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 1 1 0 0 0
Table 8 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Q0minus6 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 9 9 9 6
Table 9 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We have grouped Vremanrsquos classes into Q7minus9 and Q0minus6 this is equivalent to saying that we havegrouped Vremanrsquos classes into those with two or fewer non-zero velocity gradient elements and thosewith three or more non-zero velocity gradient elements respectively Therefore the solutions in Q7minus9are exactly those with a lsquo0rsquo in the lsquoExactrsquo column of Table 4 and Sol 5 (2D Taylor) ExcludingSol 5 (2D Taylor) these are the same as those five that endorse the Vreman QR Gradient andVortex-Stretching-based models in Table 5 We can see in Table 8 that the results for the lsquostrongrsquoendorsements are exactly the same as in Table 5 This is because all of the flows with zero subfilterdissipation are contained in the class Q7minus9 There are hence no lsquostrongrsquo endorsements in Table 9The majority of the lsquoweakrsquo endorsements come from Q0minus6 as this class contains all but one of thesolutions with non-zero true subfilter dissipation (given in Eq (67))
442 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
We now separate our exact solutions into classes again using Vremanrsquos flow classes This timehowever we define V0 to be the class of flows Vreman claims will always have zero true subfilterdissipation namely those classes mentioned in Appendix B We define Vlowast to be all the remaining
31
classes Then
V0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow)
Sol 3 (Velocity as a function of time) Sol 14 (Radially Dependent)(78)
Vlowast = Sol 4 (Axisymmetric Flow) Sol 5 (2D Taylor) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 13 (Sphere Translation) Sol 15 (Rankine Vortex)
(79)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 10 and 11 are analogues of Table 3 and illustrate the results for each flow class Theanalogues of Tables 1 and 2 are Tables 25 and 26 and are in Appendix C2
Endorsements for models using solutions in V0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 4 4 4 4
lsquoDissipationrsquoEndorsements
1 4 4 4 4
Percentage Endorsed 375 100 100 100 100
Table 10 Each entry in the top two rows represents the number of solutions in the class V0 thatendorse each model There are 4 solutions in this class so the percentages are worked out by dividingthe total of each column by 8 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Vlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 2 4 5 5
lsquoDissipationrsquoEndorsements
1 1 3 2 5
Percentage Endorsed 136 136 318 318 455
Table 11 Each entry in the top two rows represents the number of solutions in the class Vlowast thatendorse each model There are 11 solutions in this class so the percentages are worked out bydividing the total of each column by 22 and taking results to 1 decimal place if necessary
All of the flows in the class V0 endorse the Vreman QR Gradient and Vortex-Stretching-basedsubfilter-scale models (see Table 6) This class completely endorses these four models equally anddoes not endorse the Smagorinsky model The Vortex-Stretching-based eddy-viscosity model hasthe most endorsements from the class Vlowast and the Smagorinksy and Vreman models again have thefewest endorsements (see Table 7) As in Section 441 we say that this class mostly endorses the
32
Vortex-Stretching-based eddy-viscosity model but once again the percentage endorsement is ratherlow
Now let us consider LES with explicit filtering Tables 12 and 13 are analogues of Table 5 andillustrate the results for each flow class The analogue of Table 4 is Table 27 and is in Appendix C2
Endorsements for models using solutions in V0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 4 4 4 4
lsquoWeakrsquo Endorsements 0 0 0 0 0
Table 12 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Vlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 1 1 1 1
lsquoWeakrsquo Endorsements 10 10 9 6
Table 13 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We can see in Table 12 that the results for the lsquostrongrsquo endorsements are similar to those in Table 5excluding Sol 15 (Rankine Vortex) The four solutions in V0 with zero true subfilter dissipationnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)and Sol 14 (Radially Dependent) are hence the four that endorse the Vreman QR Gradientand Vortex-Stretching-based models in Table 12 Sol 15 (Rankine Vortex) is in the class Vlowast andhence there is one lsquostrongrsquo endorsement in Table 13 The class Vlowast contains all of the flows with anon-zero true subfilter dissipation (given in Eq (67)) and hence Table 13 contains all of the lsquoweakrsquoendorsements
443 Classes using the Third Principal Invariant
Sorting the classes using the invariants is a non-trivial task in itself There are a number of waysin which to do this using both the principal invariants in Eq (40) and the combined invariants inEq (43)
We start by looking at flows with third principal invariant RG equal to zero We again define twoclasses R0 and Rlowast where R0 consists of all flows with RG = 0 and Rlowast consists of all flows withRG 6= 0 Then using the information in Appendix A we see that there are nine flows in R0 That
33
is
R0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 8 (Vortex in Stagnation Flow)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)
(80)
Rlowast = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 13 (Sphere Translation)(81)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 14 and 15 are analogues of Table 3 and they illustrate the results for LES withoutexplicit filtering for each flow class The analogues of Tables 1 and 2 are Tables 28 and 29 and arein Appendix C3
Endorsements for models using solutions in R0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 7 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 278 556 778 722 778
Table 14 Each entry in the top two rows represents the number of solutions in the class R0 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking the result to one decimal place if necessary
Endorsements for models using solutions in Rlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 2 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 83 83 83 167 333
Table 15 Each entry in the top two rows represents the number of solutions in the class Rlowast thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking the result to one decimal place if necessary
The class R0 endorses both the Vortex-Stretching-based model and the QR model the most andSmagorinsky the least in the case for LES without explicit filtering In the Rlowast class the Vortex-Stretching-based model is again the most endorsed followed by the Gradient model however thepercentages are again small
34
Now let us consider LES with explicit filtering Tables 16 and 17 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 30 and is in Appendix C3
Endorsements for models using solutions in R0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 4 4 3 2
Table 16 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Rlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 6 6 6 4
Table 17 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 we can see in Table 16 that the results for the lsquostrongrsquo endorsements are exactly thesame as in Table 5 This is because all of the flows with zero subfilter dissipation are contained in theclass R0 As in Table 9 there are no lsquostrongrsquo endorsements in Table 17 The lsquoweakrsquo endorsementshowever are split more evenly between the two classes now than in Vremanrsquos classes in Sections 441and 442 We find that the Smagorinsky and Vreman models still seem to be the most endorsed inboth classes
444 Two-component Flow Classes
Another way in which the flows can be grouped into classes is by determining whether a flow is atwo-component flow or not We define a two-component flow to have the property that the combinedprincipal invariants of a flow in Eq (43) satisfy
I3 = I4 = I5 minus1
2I1I2 = 0 (82)
We are also interested in considering flows with zero gradient model subgrid dissipation These flowshave the property that
I3 minus I4 = 0 (83)
What we find though is that for all the solutions in Section 22 if a flow is a two-component flowthen it also satisfies I3 minus I4 = 0 It may be interesting to note that although I3 minus I4 6= 0 for manysolutions there are a number with I3 equal to a multiple of I4 Although not considered in thisreport this may also be a class of interest We find that the two-component flows are
C2 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)(84)
35
and we call the remaining flows three-component flows They are
C3 = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex) Sol 11 (Pseudo-Motion Shots)
Sol 13 (Sphere Translation)
(85)
Note that Sol 8 (Vortex in Stagnation Flow) is defined to be a planar flow this does not mean that itis a two-component flow in this sense however because its two non-zero velocity components dependon three coordinates We now check the models for LES without explicit filtering by consideringendorsements within each class Tables 18 and 19 are analogues of Table 3 and they illustrate theresults for LES without explicit filtering for each flow class The analogues of Tables 1 and 2 areTables 31 and 32 and are in Appendix C4
Endorsements for models using solutions in C2 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 6 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 357 714 100 857 100
Table 18 Each entry in the top two rows represents the number of solutions in the class C2 thatendorse each model There are 7 solutions in this class so the percentages are worked out by dividingthe total of each column by 14 and taking the result to one decimal place if necessary
Endorsements for models using solutions in C3 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 3 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 63 63 63 188 250
Table 19 Each entry in the top two rows represents the number of solutions in the class C3 thatendorse each model There are 8 solutions in this class so the percentages are worked out by dividingthe total of each column by 16 and taking the result to one decimal place if necessary
The class C2 equally endorses the QR and Vortex-Stretching-based models and once again theSmagorinsky model is the least endorsed As with the previous classes the class C3 has such lowpercentages that it does not seem to endorse any models We start to see a trend appearing here inthat the flow classes that contain the more complicated flows namely Q0minus6 (Eq (76)) Vlowast (Eq (79))Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages The classes that tendto contain the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78)) R0 (Eq (80)) and C2 (Eq (84))all seem to favour the Vortex-Stretching-based model or the QR model
36
Now let us consider LES with explicit filtering Tables 20 and 21 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 33 and is in Appendix C4
Endorsements for models using solutions in C2 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 2 2 0 1 0
Table 20 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in C3 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 8 8 8 6
Table 21 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 and Table 16 we can see in Table 20 that the results for the lsquostrongrsquo endorsements areexactly the same as in Table 5 This is again because all of the flows with zero subfilter dissipationare contained in the class C2 There are two flows with non-zero true subfilter dissipation (givenin Eq (67)) in C2 and this shows in the lsquoweakrsquo endorsements in Table 20 There are no lsquostrongrsquoendorsements in Table 21 as all the zero subfilter dissipation flows are in C2
5 Conclusions
In this report we studied incompressible turbulent flows We found a number of exact solutions to theNavier-Stokes equations each representing a relatively simplistic fluid flow Large-eddy simulation(LES) is used to model turbulent flows by smoothing over the complex small-scale motion Weintroduced a number of subfilter-scale models to approximate this small-scale motion and testedthem using the found exact solutions to the Navier-Stokes equations We first considered LESwithout explicit filtering and used the Navier-Stokes equations to derive conditions for the subfilter-scale models namely the Velocity Field condition (Eq (58)) the Kinetic Energy condition (Eq (60))and the Dissipation condition (Eq (62)) Similarly we considered LES with explicit filtering andagain used the Navier-Stokes equations to derive more conditions for the models in this case wedecided to just take one condition the Dissipation condition (Eqs (67) and (68)) as the others leadto difficulty in their evaluation By lsquorunningrsquo the exact solutions to the Navier-Stokes equationsthrough each condition for the subfilter-scale models we found that the Vortex-Stretching-basededdy-viscosity model in Eq (54) was most endorsed by the tests in LES without explicit filteringWe found that four out of the five subfilter-scale models namely the Vreman QR Gradient andVortex-Stretching-based models were equally endorsed by the lsquostrongrsquo tests in LES with explicitfiltering with the Smagorinsky model being the least endorsed The results we found for the lsquoweakrsquocondition were completely opposite to those found in both the lsquostrongrsquo condition and in LES withoutexplicit filtering This led us to conclude that this test was not one in which we should put our trust
37
We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses
In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes
This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold
38
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
are to be compared Although the latter term is relatively simple to compute the former could leadto some difficulties due to the differentiation of the true stress τij = uiuj minus uiuj So although someinformation could be gleaned from the inspection of these two terms more information can be foundby considering the analogue of Eq (62) So the true subfilter dissipation
Sijτij (67)
(where Sij is shorthand for Sij(v)) and the modelled subfilter dissipation
Sij(v)τmodij (v) (68)
are considered and have to be equal for the condition to hold These can again be derived from theintegral equation for kinetic energy of a filtered velocity field similarly to Eq (61)
4 Analysis of Large-Eddy Simulation Models
In this section we use a code in Mathematica the theory described in Section 3 and the exactsolutions to the Navier-Stokes equations found in Section 22 to check whether or not each of theexact solutions lsquoendorsesrsquo each model for LES both without and with explicit filtering
41 Large-Eddy Simulation Without Explicit Filtering
We start by using a code in Mathematica to test the Velocity Field condition in Eq (58) and theKinetic Energy condition in Eq (60) For all exact solutions to the Navier-Stokes equations inSection 22 the outputs for both conditions were identical so we have combined them into one table
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 ndash ndash 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 1 For the Kinetic Energy condition a lsquo0rsquo result means that vipartpartxj
τmodij (v) = 0 for respective
models and exact solutions whereas a lsquolowastrsquo result means that vipartpartxj
τmodij (v) 6= 0 For the Velocity
Field condition a lsquo0rsquo result means that partpartxj
τmodij (v) = 0 for respective models and exact solutions
whereas a lsquolowastrsquo result means that partpartxj
τmodij (v) 6= 0 The Gradient model can take both positive and
negative values so when the sign of the model is known it is included in the table instead of a lsquorsquoBoth the conditions are included in one table because for all cases the outputs were identical
We then test the Dissipation condition in Eq (62) for all exact solutions to the Navier-Stokesequations in Section 22
25
Dissipation Condition [Does Sijτmodij (v) = 0]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 + 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 2 A lsquo0rsquo result means that Sijτmodij (v) = 0 for respective models and exact solutions whereas
a lsquolowastrsquo result means that Sijτmodij (v) 6= 0 The Gradient model can take both positive and negative
values so when the sign of the model is known it is included in the table instead of a lsquorsquo
We then sum all the results to find the total number of endorsements for each model described inSection 32
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
4 6 8 9 9
lsquoDissipationrsquoEndorsements
2 5 7 7 9
Total Endorsements 6 11 15 16 18
Table 3 The results from Tables 1 and 2 are summarised to give a general overview of how manyendorsements each model receives This makes it much clearer to see the models most supported bythe respective and combined conditions with the Vortex-Stretching model being the most supportedand the Smagorinsky model the least
Testing the models in Section 32 using LES without explicit filtering has mixed results It isclear that the Smagorinksy model is not well endorsed by the flows in Section 22 with the onlyendorsements coming from the simplest flows that is Sol 1 (Linear Velocity) Sol 3 (Velocity as afunction of time) Sol 7 (Stagnation Flow) and Sol 15 (Rankine Vortex) The Vortex-Stretching-based model is the most endorsed flow with eighteen out of a possible thirty endorsements Thisagrees with our foreknowledge as this model was constructed to be as physically consistent as possiblein the paper by Silvis amp Verstappen [11]
42 Large-Eddy Simulation With Explicit Filtering
In this section we will compare the true subfilter dissipation Sijτij and the modelled subfilterdissipation Sij(v)τmod
ij (v) given in Eqs (67) and (68) respectively for each model τmodij and each
exact solution to the Navier-Stokes equation given in Section 22 We will not consider the conditionsinvolving differentiation in Eqs (65) and (66) due to the difficulty in calculating the derivative ofthe τij term in Eq (65)
26
421 Example of a Test for a Model
The true subfilter dissipation Sijτij of a flow given in Eq (67) cannot be explicitly computed asthe filtering operation is kept general What can be done however is to partially evaluate theexpressions using the attributes of a filtering operation mentioned in Eq (48)
The calculations of Sijτij for Sol 1 (Linear Velocity) are provided to illustrate the general ideaRecall that the velocity field of Sol 1 (Linear Velocity) is written in Eq (13) in vector notation as
u(x y z t) =
0ax+ bcx+ d
(69)
Then Sij = Sij(u) is computed simply by using its definition in Eq (41) Note that as filteringpreserves constants if there is an a it is replaced with a meaning that (in this case) there are alarge number of zero terms due to the frequency of 0 Hence
[Sij(u)
]=
[1
2
(partuipartxj
+partujpartxi
)]=
0 a
2c2
a2 0 0
c2 0 0
=
0 a
2c2
a2 0 0
c2 0 0
(70)
Note that as Sijτij is being computed which is equal to the sum of the element-wise multiplicationof the two matrices (or the trace of the matrix multiplication) that we donrsquot need to consider theelements of τij(u) with corresponding zero entry in Sij As τij(u) = uiuj minus uiuj and again asfiltering preserves constants it is found that
[τij(u)] =[uiuj minus uiuj
]=
lowast 0 middot ax+ b 0 middot cx+ d
ax+ b middot 0 lowast lowastcx+ d middot 0 lowast lowast
=
lowast 0 00 lowast lowast0 lowast lowast
(71)
where a lsquolowastrsquo just represents some element (not necessarily non-zero) Therefore the true subfilterdissipation
Sijτij = tr
0 a2
c2
a2 0 0c2 0 0
middotlowast 0 0
0 lowast lowast0 lowast lowast
= 0 (72)
Now the modelled subfilter dissipation Sij(u)τmodij (u) needs to be computed (recall that Sij =
Sij(u)) In this case let us use the QR model stated in Eq (53) Once again Sij(u) calculated inEq (70) needs to be used So we have that
τmodij = minus2(CQRδ)
2 max0minusI3I1
Sij (73)
As
I1 = tr(S2) =
1
2(a2 + c2) and I3 = tr(S
3) = 0 (74)
it holds that τmodij (u) = 0 so Sij(u)τmod
ij (u) = 0 This agrees with Eq (72) and hence Sol 1(Linear Velocity) endorses the QR model
27
422 Testing the Models
Now using a Mathematica code the exact solutions in Section 22 are again used to evaluate thecondition for the models using the process just described in Section 421 The results are describedin Table 4
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 4 All entries with a ldquo0rdquo represent a value of zero and a ldquolowastrdquo represents a value that is non-zeroThe ldquoExactrdquo row represents the value of the true subfilter dissipation Sijτij Every other row witha model name represents the value of the modelled subfilter dissipation Sij(u)τmod
ij (u) respectiveto that model A ldquordquo means that it is unclear as to the value - the sign of these values is oftenunknown and the lsquoMaxrsquo function in the definition of the QR model (in Eq (53)) makes it difficultto evaluate
Reading from Table 4 the Smagorinsky model is clearly endorsed by Sol 3 (Velocity as a function oftime) and Sol 15 (Rankine Vortex) as in both cases both Sijτij = 0 = Sij(u)τmod
ij (u) In this casewe say that Sol 3 (Velocity as a function of time) and Sol 15 (Rankine Vortex) lsquostronglyrsquo endorsethe Smagorinsky model The Smagorinsky model is not endorsed by Sol 1 (Linear Velocity) Sol 2(Planar Flow) or Sol 14 (Radially Dependent) however because Sij(u)τmod
ij (u) 6= 0 For Sol 4 to
13 both Sijτij 6= 0 6= Sij(u)τmodij (u) and so we will say that these exact solutions lsquoweaklyrsquo endorse
the Smagorinsky model
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 10 10 9 6
Table 5 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Interestingly the lsquostrongrsquo and lsquoweakrsquo endorsements give different results However more stock shouldbe put in the results of the lsquostrongerrsquo condition as it is completely clear that zero is equal to zeroIt is not necessarily clear whether or not non-zero equals non-zero as the values will depend on theparameters in each solution hence the lsquoweakerrsquo endorsement
The lsquostrongrsquo condition equally endorses the Vreman QR Gradient and Vortex-Stretching-basedmodels whereas the lsquoweakrsquo condition endorses Vreman and Smagorinsky the most closely followedby the Gradient model with the Vortex-Stretching-based model being the most opposed As this
28
completely contradicts the lsquostrongrsquo condition and the results in Section 41 it makes us question thevalidity of the weaker test The number of endorsements for the QR model is unknown because of thedefinition of the model in Eq (53) and our evaluation of the true subfilter dissipation Eq (67) Dueto the exact solutions we have found it is never clear whether or not the true subfilter dissipationis strictly positive or negative This means that when the true subfilter dissipation is found to benon-zero the lsquoMaxrsquo function in the QR model cannot be evaluated
43 Vremanrsquos True Subfilter Dissipation Claim
Using the theory mentioned in Sections 232 and 34 we can place the exact solutions to the Navier-Stokes equations into different classes defined by Vreman In doing this we can also attempt to verifyone of Vremanrsquos results [16] In Vremanrsquos paper he defines the class Qn to be the set of flows withn zero entries in the velocity gradient of the flow (the velocity gradient is defined in Eq (39))Vreman then claims that the only flows with zero true subfilter dissipation are a certain selection ofthose in the classesQ7Q8 orQ9 shown in appendix B and is generally non-zero for any other flows
The true subfilter dissipation of Sol 1 (Linear Velocity) was computed in Section 421 and wasfound to be zero in Eq (72) The velocity gradient of Sol 1 is as follows
GLV =
0 0 0a 0 0c 0 0
(75)
and hence Sol 1 (Linear Velocity) is in the required subset of Q7 which agrees with Vremanrsquos claimIn Section 422 the true subfilter dissipation was computed (determined to be zero or non-zero) forall of the exact solutions to the Navier Stokes equations in Section 22 These solutions can now belsquoplacedrsquo into each class Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a functionof time) and Sol 14 (Radially Dependent) were all found to have zero true subfilter dissipation andare all in the appropriate subsets of either Q7Q8 or Q9 This agrees with Vremanrsquos results
We can extend Vremanrsquos results however We found that Sol 15 (Rankine Vortex) had zero truesubfilter dissipation and yet is not in the subset of Q7 defined in Appendix B It is important tonote that the rate of strain tensor Sij in Eq (41) for Sol 15 (Rankine Vortex) is zero This is thereason the true subfilter dissipation is zero and the same reason that all models are zero for thisflow as all models are based on Sij
We also found that all of the remaining exact solutions to the Navier-Stokes equations are not inVremanrsquos specified classes (in Appendix Section B) They also have a non-zero subfilter dissipationwhich agrees with Vremanrsquos results in his paper [16]
44 Classes of Flows
It is important to consider that some exact solutions to the Navier-Stokes equations may have similarproperties For example Sol 1 (Linear Velocity) and Sol 2 (Planar Flow) are both very similarIn fact Sol 2 (Planar Flow) is a more general case of Sol 1 (Linear Velocity) We therefore mustgroup the different flows into classes in an attempt to reduce the bias that may occur when solutionsare similar This should aid us to further determine which of the models are most endorsed We willdo this in a number of different ways using the theory discussed in Section 23 We will group theflows with respect to Vremanrsquos Flow Classes given in appendix B and with respect to the Invariantsgiven in appendix A
29
441 Vremanrsquos Velocity Gradient Flow Classes
We now put our exact solutions to the Navier-Stokes equations into one of two classes Either Q0minus6or Q7minus9 where each class is defined intuitively as follows
Q0minus6 = Q0 cupQ1 cup middot middot middot cup Q6 (76)
Q7minus9 = Q7 cupQ8 cupQ9 (77)
where Qn is defined in Section 43 By looking at the gradient of each exact solution we can see thatQ7minus9 contains the six solutions Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as afunction of time) Sol 5 (2D Taylor) Sol 14 (Radially Dependent) and Sol 15 (Rankine Vortex)and that Q0minus6 contains the remaining nine We now check the models for LES without explicitfiltering by considering endorsements within each class Tables 6 and 7 are analogues of Table 3 andillustrate the results for each flow class The analogues of Tables 1 and 2 are Tables 22 and 23 andare in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 6 6 6
lsquoDissipationrsquoEndorsements
2 5 6 6 6
Percentage Endorsed 417 833 100 100 100
Table 6 Each entry in the top two rows represents the number of solutions in the class Q7minus9 thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Q0minus6 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 2 3 3
lsquoDissipationrsquoEndorsements
0 0 1 0 3
Percentage Endorsed 56 56 167 167 333
Table 7 Each entry in the top two rows represents the number of solutions in the class Q1minus6 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking results to 1 decimal place if necessary
All of the flows in the class Q7minus9 endorse the QR Gradient and Vortex-Stretching-based subfilter-scale models (see Table 6) We can hence see that this class fully endorses these three models equallyand does not endorse the Smagorinsky model The Vortex-Stretching eddy viscosity model has themost endorsements from any of the flows in the class Q0minus6 and the Smagorinksy and Vreman modelshave the fewest endorsements (see Table 7) We can hence say that this class mostly endorses theVortex-Stretching-based eddy-viscosity model note however that the percentage endorsement is very
30
low
Now let us consider LES with explicit filtering Tables 8 and 9 are analogues of Table 5 and illustratethe results for each flow class The analogue of Table 4 is Table 24 and is in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 1 1 0 0 0
Table 8 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Q0minus6 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 9 9 9 6
Table 9 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We have grouped Vremanrsquos classes into Q7minus9 and Q0minus6 this is equivalent to saying that we havegrouped Vremanrsquos classes into those with two or fewer non-zero velocity gradient elements and thosewith three or more non-zero velocity gradient elements respectively Therefore the solutions in Q7minus9are exactly those with a lsquo0rsquo in the lsquoExactrsquo column of Table 4 and Sol 5 (2D Taylor) ExcludingSol 5 (2D Taylor) these are the same as those five that endorse the Vreman QR Gradient andVortex-Stretching-based models in Table 5 We can see in Table 8 that the results for the lsquostrongrsquoendorsements are exactly the same as in Table 5 This is because all of the flows with zero subfilterdissipation are contained in the class Q7minus9 There are hence no lsquostrongrsquo endorsements in Table 9The majority of the lsquoweakrsquo endorsements come from Q0minus6 as this class contains all but one of thesolutions with non-zero true subfilter dissipation (given in Eq (67))
442 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
We now separate our exact solutions into classes again using Vremanrsquos flow classes This timehowever we define V0 to be the class of flows Vreman claims will always have zero true subfilterdissipation namely those classes mentioned in Appendix B We define Vlowast to be all the remaining
31
classes Then
V0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow)
Sol 3 (Velocity as a function of time) Sol 14 (Radially Dependent)(78)
Vlowast = Sol 4 (Axisymmetric Flow) Sol 5 (2D Taylor) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 13 (Sphere Translation) Sol 15 (Rankine Vortex)
(79)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 10 and 11 are analogues of Table 3 and illustrate the results for each flow class Theanalogues of Tables 1 and 2 are Tables 25 and 26 and are in Appendix C2
Endorsements for models using solutions in V0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 4 4 4 4
lsquoDissipationrsquoEndorsements
1 4 4 4 4
Percentage Endorsed 375 100 100 100 100
Table 10 Each entry in the top two rows represents the number of solutions in the class V0 thatendorse each model There are 4 solutions in this class so the percentages are worked out by dividingthe total of each column by 8 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Vlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 2 4 5 5
lsquoDissipationrsquoEndorsements
1 1 3 2 5
Percentage Endorsed 136 136 318 318 455
Table 11 Each entry in the top two rows represents the number of solutions in the class Vlowast thatendorse each model There are 11 solutions in this class so the percentages are worked out bydividing the total of each column by 22 and taking results to 1 decimal place if necessary
All of the flows in the class V0 endorse the Vreman QR Gradient and Vortex-Stretching-basedsubfilter-scale models (see Table 6) This class completely endorses these four models equally anddoes not endorse the Smagorinsky model The Vortex-Stretching-based eddy-viscosity model hasthe most endorsements from the class Vlowast and the Smagorinksy and Vreman models again have thefewest endorsements (see Table 7) As in Section 441 we say that this class mostly endorses the
32
Vortex-Stretching-based eddy-viscosity model but once again the percentage endorsement is ratherlow
Now let us consider LES with explicit filtering Tables 12 and 13 are analogues of Table 5 andillustrate the results for each flow class The analogue of Table 4 is Table 27 and is in Appendix C2
Endorsements for models using solutions in V0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 4 4 4 4
lsquoWeakrsquo Endorsements 0 0 0 0 0
Table 12 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Vlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 1 1 1 1
lsquoWeakrsquo Endorsements 10 10 9 6
Table 13 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We can see in Table 12 that the results for the lsquostrongrsquo endorsements are similar to those in Table 5excluding Sol 15 (Rankine Vortex) The four solutions in V0 with zero true subfilter dissipationnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)and Sol 14 (Radially Dependent) are hence the four that endorse the Vreman QR Gradientand Vortex-Stretching-based models in Table 12 Sol 15 (Rankine Vortex) is in the class Vlowast andhence there is one lsquostrongrsquo endorsement in Table 13 The class Vlowast contains all of the flows with anon-zero true subfilter dissipation (given in Eq (67)) and hence Table 13 contains all of the lsquoweakrsquoendorsements
443 Classes using the Third Principal Invariant
Sorting the classes using the invariants is a non-trivial task in itself There are a number of waysin which to do this using both the principal invariants in Eq (40) and the combined invariants inEq (43)
We start by looking at flows with third principal invariant RG equal to zero We again define twoclasses R0 and Rlowast where R0 consists of all flows with RG = 0 and Rlowast consists of all flows withRG 6= 0 Then using the information in Appendix A we see that there are nine flows in R0 That
33
is
R0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 8 (Vortex in Stagnation Flow)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)
(80)
Rlowast = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 13 (Sphere Translation)(81)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 14 and 15 are analogues of Table 3 and they illustrate the results for LES withoutexplicit filtering for each flow class The analogues of Tables 1 and 2 are Tables 28 and 29 and arein Appendix C3
Endorsements for models using solutions in R0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 7 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 278 556 778 722 778
Table 14 Each entry in the top two rows represents the number of solutions in the class R0 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking the result to one decimal place if necessary
Endorsements for models using solutions in Rlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 2 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 83 83 83 167 333
Table 15 Each entry in the top two rows represents the number of solutions in the class Rlowast thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking the result to one decimal place if necessary
The class R0 endorses both the Vortex-Stretching-based model and the QR model the most andSmagorinsky the least in the case for LES without explicit filtering In the Rlowast class the Vortex-Stretching-based model is again the most endorsed followed by the Gradient model however thepercentages are again small
34
Now let us consider LES with explicit filtering Tables 16 and 17 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 30 and is in Appendix C3
Endorsements for models using solutions in R0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 4 4 3 2
Table 16 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Rlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 6 6 6 4
Table 17 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 we can see in Table 16 that the results for the lsquostrongrsquo endorsements are exactly thesame as in Table 5 This is because all of the flows with zero subfilter dissipation are contained in theclass R0 As in Table 9 there are no lsquostrongrsquo endorsements in Table 17 The lsquoweakrsquo endorsementshowever are split more evenly between the two classes now than in Vremanrsquos classes in Sections 441and 442 We find that the Smagorinsky and Vreman models still seem to be the most endorsed inboth classes
444 Two-component Flow Classes
Another way in which the flows can be grouped into classes is by determining whether a flow is atwo-component flow or not We define a two-component flow to have the property that the combinedprincipal invariants of a flow in Eq (43) satisfy
I3 = I4 = I5 minus1
2I1I2 = 0 (82)
We are also interested in considering flows with zero gradient model subgrid dissipation These flowshave the property that
I3 minus I4 = 0 (83)
What we find though is that for all the solutions in Section 22 if a flow is a two-component flowthen it also satisfies I3 minus I4 = 0 It may be interesting to note that although I3 minus I4 6= 0 for manysolutions there are a number with I3 equal to a multiple of I4 Although not considered in thisreport this may also be a class of interest We find that the two-component flows are
C2 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)(84)
35
and we call the remaining flows three-component flows They are
C3 = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex) Sol 11 (Pseudo-Motion Shots)
Sol 13 (Sphere Translation)
(85)
Note that Sol 8 (Vortex in Stagnation Flow) is defined to be a planar flow this does not mean that itis a two-component flow in this sense however because its two non-zero velocity components dependon three coordinates We now check the models for LES without explicit filtering by consideringendorsements within each class Tables 18 and 19 are analogues of Table 3 and they illustrate theresults for LES without explicit filtering for each flow class The analogues of Tables 1 and 2 areTables 31 and 32 and are in Appendix C4
Endorsements for models using solutions in C2 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 6 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 357 714 100 857 100
Table 18 Each entry in the top two rows represents the number of solutions in the class C2 thatendorse each model There are 7 solutions in this class so the percentages are worked out by dividingthe total of each column by 14 and taking the result to one decimal place if necessary
Endorsements for models using solutions in C3 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 3 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 63 63 63 188 250
Table 19 Each entry in the top two rows represents the number of solutions in the class C3 thatendorse each model There are 8 solutions in this class so the percentages are worked out by dividingthe total of each column by 16 and taking the result to one decimal place if necessary
The class C2 equally endorses the QR and Vortex-Stretching-based models and once again theSmagorinsky model is the least endorsed As with the previous classes the class C3 has such lowpercentages that it does not seem to endorse any models We start to see a trend appearing here inthat the flow classes that contain the more complicated flows namely Q0minus6 (Eq (76)) Vlowast (Eq (79))Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages The classes that tendto contain the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78)) R0 (Eq (80)) and C2 (Eq (84))all seem to favour the Vortex-Stretching-based model or the QR model
36
Now let us consider LES with explicit filtering Tables 20 and 21 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 33 and is in Appendix C4
Endorsements for models using solutions in C2 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 2 2 0 1 0
Table 20 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in C3 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 8 8 8 6
Table 21 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 and Table 16 we can see in Table 20 that the results for the lsquostrongrsquo endorsements areexactly the same as in Table 5 This is again because all of the flows with zero subfilter dissipationare contained in the class C2 There are two flows with non-zero true subfilter dissipation (givenin Eq (67)) in C2 and this shows in the lsquoweakrsquo endorsements in Table 20 There are no lsquostrongrsquoendorsements in Table 21 as all the zero subfilter dissipation flows are in C2
5 Conclusions
In this report we studied incompressible turbulent flows We found a number of exact solutions to theNavier-Stokes equations each representing a relatively simplistic fluid flow Large-eddy simulation(LES) is used to model turbulent flows by smoothing over the complex small-scale motion Weintroduced a number of subfilter-scale models to approximate this small-scale motion and testedthem using the found exact solutions to the Navier-Stokes equations We first considered LESwithout explicit filtering and used the Navier-Stokes equations to derive conditions for the subfilter-scale models namely the Velocity Field condition (Eq (58)) the Kinetic Energy condition (Eq (60))and the Dissipation condition (Eq (62)) Similarly we considered LES with explicit filtering andagain used the Navier-Stokes equations to derive more conditions for the models in this case wedecided to just take one condition the Dissipation condition (Eqs (67) and (68)) as the others leadto difficulty in their evaluation By lsquorunningrsquo the exact solutions to the Navier-Stokes equationsthrough each condition for the subfilter-scale models we found that the Vortex-Stretching-basededdy-viscosity model in Eq (54) was most endorsed by the tests in LES without explicit filteringWe found that four out of the five subfilter-scale models namely the Vreman QR Gradient andVortex-Stretching-based models were equally endorsed by the lsquostrongrsquo tests in LES with explicitfiltering with the Smagorinsky model being the least endorsed The results we found for the lsquoweakrsquocondition were completely opposite to those found in both the lsquostrongrsquo condition and in LES withoutexplicit filtering This led us to conclude that this test was not one in which we should put our trust
37
We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses
In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes
This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold
38
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
Dissipation Condition [Does Sijτmodij (v) = 0]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 + 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 2 A lsquo0rsquo result means that Sijτmodij (v) = 0 for respective models and exact solutions whereas
a lsquolowastrsquo result means that Sijτmodij (v) 6= 0 The Gradient model can take both positive and negative
values so when the sign of the model is known it is included in the table instead of a lsquorsquo
We then sum all the results to find the total number of endorsements for each model described inSection 32
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
4 6 8 9 9
lsquoDissipationrsquoEndorsements
2 5 7 7 9
Total Endorsements 6 11 15 16 18
Table 3 The results from Tables 1 and 2 are summarised to give a general overview of how manyendorsements each model receives This makes it much clearer to see the models most supported bythe respective and combined conditions with the Vortex-Stretching model being the most supportedand the Smagorinsky model the least
Testing the models in Section 32 using LES without explicit filtering has mixed results It isclear that the Smagorinksy model is not well endorsed by the flows in Section 22 with the onlyendorsements coming from the simplest flows that is Sol 1 (Linear Velocity) Sol 3 (Velocity as afunction of time) Sol 7 (Stagnation Flow) and Sol 15 (Rankine Vortex) The Vortex-Stretching-based model is the most endorsed flow with eighteen out of a possible thirty endorsements Thisagrees with our foreknowledge as this model was constructed to be as physically consistent as possiblein the paper by Silvis amp Verstappen [11]
42 Large-Eddy Simulation With Explicit Filtering
In this section we will compare the true subfilter dissipation Sijτij and the modelled subfilterdissipation Sij(v)τmod
ij (v) given in Eqs (67) and (68) respectively for each model τmodij and each
exact solution to the Navier-Stokes equation given in Section 22 We will not consider the conditionsinvolving differentiation in Eqs (65) and (66) due to the difficulty in calculating the derivative ofthe τij term in Eq (65)
26
421 Example of a Test for a Model
The true subfilter dissipation Sijτij of a flow given in Eq (67) cannot be explicitly computed asthe filtering operation is kept general What can be done however is to partially evaluate theexpressions using the attributes of a filtering operation mentioned in Eq (48)
The calculations of Sijτij for Sol 1 (Linear Velocity) are provided to illustrate the general ideaRecall that the velocity field of Sol 1 (Linear Velocity) is written in Eq (13) in vector notation as
u(x y z t) =
0ax+ bcx+ d
(69)
Then Sij = Sij(u) is computed simply by using its definition in Eq (41) Note that as filteringpreserves constants if there is an a it is replaced with a meaning that (in this case) there are alarge number of zero terms due to the frequency of 0 Hence
[Sij(u)
]=
[1
2
(partuipartxj
+partujpartxi
)]=
0 a
2c2
a2 0 0
c2 0 0
=
0 a
2c2
a2 0 0
c2 0 0
(70)
Note that as Sijτij is being computed which is equal to the sum of the element-wise multiplicationof the two matrices (or the trace of the matrix multiplication) that we donrsquot need to consider theelements of τij(u) with corresponding zero entry in Sij As τij(u) = uiuj minus uiuj and again asfiltering preserves constants it is found that
[τij(u)] =[uiuj minus uiuj
]=
lowast 0 middot ax+ b 0 middot cx+ d
ax+ b middot 0 lowast lowastcx+ d middot 0 lowast lowast
=
lowast 0 00 lowast lowast0 lowast lowast
(71)
where a lsquolowastrsquo just represents some element (not necessarily non-zero) Therefore the true subfilterdissipation
Sijτij = tr
0 a2
c2
a2 0 0c2 0 0
middotlowast 0 0
0 lowast lowast0 lowast lowast
= 0 (72)
Now the modelled subfilter dissipation Sij(u)τmodij (u) needs to be computed (recall that Sij =
Sij(u)) In this case let us use the QR model stated in Eq (53) Once again Sij(u) calculated inEq (70) needs to be used So we have that
τmodij = minus2(CQRδ)
2 max0minusI3I1
Sij (73)
As
I1 = tr(S2) =
1
2(a2 + c2) and I3 = tr(S
3) = 0 (74)
it holds that τmodij (u) = 0 so Sij(u)τmod
ij (u) = 0 This agrees with Eq (72) and hence Sol 1(Linear Velocity) endorses the QR model
27
422 Testing the Models
Now using a Mathematica code the exact solutions in Section 22 are again used to evaluate thecondition for the models using the process just described in Section 421 The results are describedin Table 4
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 4 All entries with a ldquo0rdquo represent a value of zero and a ldquolowastrdquo represents a value that is non-zeroThe ldquoExactrdquo row represents the value of the true subfilter dissipation Sijτij Every other row witha model name represents the value of the modelled subfilter dissipation Sij(u)τmod
ij (u) respectiveto that model A ldquordquo means that it is unclear as to the value - the sign of these values is oftenunknown and the lsquoMaxrsquo function in the definition of the QR model (in Eq (53)) makes it difficultto evaluate
Reading from Table 4 the Smagorinsky model is clearly endorsed by Sol 3 (Velocity as a function oftime) and Sol 15 (Rankine Vortex) as in both cases both Sijτij = 0 = Sij(u)τmod
ij (u) In this casewe say that Sol 3 (Velocity as a function of time) and Sol 15 (Rankine Vortex) lsquostronglyrsquo endorsethe Smagorinsky model The Smagorinsky model is not endorsed by Sol 1 (Linear Velocity) Sol 2(Planar Flow) or Sol 14 (Radially Dependent) however because Sij(u)τmod
ij (u) 6= 0 For Sol 4 to
13 both Sijτij 6= 0 6= Sij(u)τmodij (u) and so we will say that these exact solutions lsquoweaklyrsquo endorse
the Smagorinsky model
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 10 10 9 6
Table 5 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Interestingly the lsquostrongrsquo and lsquoweakrsquo endorsements give different results However more stock shouldbe put in the results of the lsquostrongerrsquo condition as it is completely clear that zero is equal to zeroIt is not necessarily clear whether or not non-zero equals non-zero as the values will depend on theparameters in each solution hence the lsquoweakerrsquo endorsement
The lsquostrongrsquo condition equally endorses the Vreman QR Gradient and Vortex-Stretching-basedmodels whereas the lsquoweakrsquo condition endorses Vreman and Smagorinsky the most closely followedby the Gradient model with the Vortex-Stretching-based model being the most opposed As this
28
completely contradicts the lsquostrongrsquo condition and the results in Section 41 it makes us question thevalidity of the weaker test The number of endorsements for the QR model is unknown because of thedefinition of the model in Eq (53) and our evaluation of the true subfilter dissipation Eq (67) Dueto the exact solutions we have found it is never clear whether or not the true subfilter dissipationis strictly positive or negative This means that when the true subfilter dissipation is found to benon-zero the lsquoMaxrsquo function in the QR model cannot be evaluated
43 Vremanrsquos True Subfilter Dissipation Claim
Using the theory mentioned in Sections 232 and 34 we can place the exact solutions to the Navier-Stokes equations into different classes defined by Vreman In doing this we can also attempt to verifyone of Vremanrsquos results [16] In Vremanrsquos paper he defines the class Qn to be the set of flows withn zero entries in the velocity gradient of the flow (the velocity gradient is defined in Eq (39))Vreman then claims that the only flows with zero true subfilter dissipation are a certain selection ofthose in the classesQ7Q8 orQ9 shown in appendix B and is generally non-zero for any other flows
The true subfilter dissipation of Sol 1 (Linear Velocity) was computed in Section 421 and wasfound to be zero in Eq (72) The velocity gradient of Sol 1 is as follows
GLV =
0 0 0a 0 0c 0 0
(75)
and hence Sol 1 (Linear Velocity) is in the required subset of Q7 which agrees with Vremanrsquos claimIn Section 422 the true subfilter dissipation was computed (determined to be zero or non-zero) forall of the exact solutions to the Navier Stokes equations in Section 22 These solutions can now belsquoplacedrsquo into each class Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a functionof time) and Sol 14 (Radially Dependent) were all found to have zero true subfilter dissipation andare all in the appropriate subsets of either Q7Q8 or Q9 This agrees with Vremanrsquos results
We can extend Vremanrsquos results however We found that Sol 15 (Rankine Vortex) had zero truesubfilter dissipation and yet is not in the subset of Q7 defined in Appendix B It is important tonote that the rate of strain tensor Sij in Eq (41) for Sol 15 (Rankine Vortex) is zero This is thereason the true subfilter dissipation is zero and the same reason that all models are zero for thisflow as all models are based on Sij
We also found that all of the remaining exact solutions to the Navier-Stokes equations are not inVremanrsquos specified classes (in Appendix Section B) They also have a non-zero subfilter dissipationwhich agrees with Vremanrsquos results in his paper [16]
44 Classes of Flows
It is important to consider that some exact solutions to the Navier-Stokes equations may have similarproperties For example Sol 1 (Linear Velocity) and Sol 2 (Planar Flow) are both very similarIn fact Sol 2 (Planar Flow) is a more general case of Sol 1 (Linear Velocity) We therefore mustgroup the different flows into classes in an attempt to reduce the bias that may occur when solutionsare similar This should aid us to further determine which of the models are most endorsed We willdo this in a number of different ways using the theory discussed in Section 23 We will group theflows with respect to Vremanrsquos Flow Classes given in appendix B and with respect to the Invariantsgiven in appendix A
29
441 Vremanrsquos Velocity Gradient Flow Classes
We now put our exact solutions to the Navier-Stokes equations into one of two classes Either Q0minus6or Q7minus9 where each class is defined intuitively as follows
Q0minus6 = Q0 cupQ1 cup middot middot middot cup Q6 (76)
Q7minus9 = Q7 cupQ8 cupQ9 (77)
where Qn is defined in Section 43 By looking at the gradient of each exact solution we can see thatQ7minus9 contains the six solutions Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as afunction of time) Sol 5 (2D Taylor) Sol 14 (Radially Dependent) and Sol 15 (Rankine Vortex)and that Q0minus6 contains the remaining nine We now check the models for LES without explicitfiltering by considering endorsements within each class Tables 6 and 7 are analogues of Table 3 andillustrate the results for each flow class The analogues of Tables 1 and 2 are Tables 22 and 23 andare in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 6 6 6
lsquoDissipationrsquoEndorsements
2 5 6 6 6
Percentage Endorsed 417 833 100 100 100
Table 6 Each entry in the top two rows represents the number of solutions in the class Q7minus9 thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Q0minus6 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 2 3 3
lsquoDissipationrsquoEndorsements
0 0 1 0 3
Percentage Endorsed 56 56 167 167 333
Table 7 Each entry in the top two rows represents the number of solutions in the class Q1minus6 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking results to 1 decimal place if necessary
All of the flows in the class Q7minus9 endorse the QR Gradient and Vortex-Stretching-based subfilter-scale models (see Table 6) We can hence see that this class fully endorses these three models equallyand does not endorse the Smagorinsky model The Vortex-Stretching eddy viscosity model has themost endorsements from any of the flows in the class Q0minus6 and the Smagorinksy and Vreman modelshave the fewest endorsements (see Table 7) We can hence say that this class mostly endorses theVortex-Stretching-based eddy-viscosity model note however that the percentage endorsement is very
30
low
Now let us consider LES with explicit filtering Tables 8 and 9 are analogues of Table 5 and illustratethe results for each flow class The analogue of Table 4 is Table 24 and is in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 1 1 0 0 0
Table 8 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Q0minus6 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 9 9 9 6
Table 9 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We have grouped Vremanrsquos classes into Q7minus9 and Q0minus6 this is equivalent to saying that we havegrouped Vremanrsquos classes into those with two or fewer non-zero velocity gradient elements and thosewith three or more non-zero velocity gradient elements respectively Therefore the solutions in Q7minus9are exactly those with a lsquo0rsquo in the lsquoExactrsquo column of Table 4 and Sol 5 (2D Taylor) ExcludingSol 5 (2D Taylor) these are the same as those five that endorse the Vreman QR Gradient andVortex-Stretching-based models in Table 5 We can see in Table 8 that the results for the lsquostrongrsquoendorsements are exactly the same as in Table 5 This is because all of the flows with zero subfilterdissipation are contained in the class Q7minus9 There are hence no lsquostrongrsquo endorsements in Table 9The majority of the lsquoweakrsquo endorsements come from Q0minus6 as this class contains all but one of thesolutions with non-zero true subfilter dissipation (given in Eq (67))
442 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
We now separate our exact solutions into classes again using Vremanrsquos flow classes This timehowever we define V0 to be the class of flows Vreman claims will always have zero true subfilterdissipation namely those classes mentioned in Appendix B We define Vlowast to be all the remaining
31
classes Then
V0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow)
Sol 3 (Velocity as a function of time) Sol 14 (Radially Dependent)(78)
Vlowast = Sol 4 (Axisymmetric Flow) Sol 5 (2D Taylor) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 13 (Sphere Translation) Sol 15 (Rankine Vortex)
(79)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 10 and 11 are analogues of Table 3 and illustrate the results for each flow class Theanalogues of Tables 1 and 2 are Tables 25 and 26 and are in Appendix C2
Endorsements for models using solutions in V0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 4 4 4 4
lsquoDissipationrsquoEndorsements
1 4 4 4 4
Percentage Endorsed 375 100 100 100 100
Table 10 Each entry in the top two rows represents the number of solutions in the class V0 thatendorse each model There are 4 solutions in this class so the percentages are worked out by dividingthe total of each column by 8 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Vlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 2 4 5 5
lsquoDissipationrsquoEndorsements
1 1 3 2 5
Percentage Endorsed 136 136 318 318 455
Table 11 Each entry in the top two rows represents the number of solutions in the class Vlowast thatendorse each model There are 11 solutions in this class so the percentages are worked out bydividing the total of each column by 22 and taking results to 1 decimal place if necessary
All of the flows in the class V0 endorse the Vreman QR Gradient and Vortex-Stretching-basedsubfilter-scale models (see Table 6) This class completely endorses these four models equally anddoes not endorse the Smagorinsky model The Vortex-Stretching-based eddy-viscosity model hasthe most endorsements from the class Vlowast and the Smagorinksy and Vreman models again have thefewest endorsements (see Table 7) As in Section 441 we say that this class mostly endorses the
32
Vortex-Stretching-based eddy-viscosity model but once again the percentage endorsement is ratherlow
Now let us consider LES with explicit filtering Tables 12 and 13 are analogues of Table 5 andillustrate the results for each flow class The analogue of Table 4 is Table 27 and is in Appendix C2
Endorsements for models using solutions in V0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 4 4 4 4
lsquoWeakrsquo Endorsements 0 0 0 0 0
Table 12 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Vlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 1 1 1 1
lsquoWeakrsquo Endorsements 10 10 9 6
Table 13 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We can see in Table 12 that the results for the lsquostrongrsquo endorsements are similar to those in Table 5excluding Sol 15 (Rankine Vortex) The four solutions in V0 with zero true subfilter dissipationnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)and Sol 14 (Radially Dependent) are hence the four that endorse the Vreman QR Gradientand Vortex-Stretching-based models in Table 12 Sol 15 (Rankine Vortex) is in the class Vlowast andhence there is one lsquostrongrsquo endorsement in Table 13 The class Vlowast contains all of the flows with anon-zero true subfilter dissipation (given in Eq (67)) and hence Table 13 contains all of the lsquoweakrsquoendorsements
443 Classes using the Third Principal Invariant
Sorting the classes using the invariants is a non-trivial task in itself There are a number of waysin which to do this using both the principal invariants in Eq (40) and the combined invariants inEq (43)
We start by looking at flows with third principal invariant RG equal to zero We again define twoclasses R0 and Rlowast where R0 consists of all flows with RG = 0 and Rlowast consists of all flows withRG 6= 0 Then using the information in Appendix A we see that there are nine flows in R0 That
33
is
R0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 8 (Vortex in Stagnation Flow)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)
(80)
Rlowast = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 13 (Sphere Translation)(81)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 14 and 15 are analogues of Table 3 and they illustrate the results for LES withoutexplicit filtering for each flow class The analogues of Tables 1 and 2 are Tables 28 and 29 and arein Appendix C3
Endorsements for models using solutions in R0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 7 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 278 556 778 722 778
Table 14 Each entry in the top two rows represents the number of solutions in the class R0 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking the result to one decimal place if necessary
Endorsements for models using solutions in Rlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 2 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 83 83 83 167 333
Table 15 Each entry in the top two rows represents the number of solutions in the class Rlowast thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking the result to one decimal place if necessary
The class R0 endorses both the Vortex-Stretching-based model and the QR model the most andSmagorinsky the least in the case for LES without explicit filtering In the Rlowast class the Vortex-Stretching-based model is again the most endorsed followed by the Gradient model however thepercentages are again small
34
Now let us consider LES with explicit filtering Tables 16 and 17 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 30 and is in Appendix C3
Endorsements for models using solutions in R0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 4 4 3 2
Table 16 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Rlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 6 6 6 4
Table 17 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 we can see in Table 16 that the results for the lsquostrongrsquo endorsements are exactly thesame as in Table 5 This is because all of the flows with zero subfilter dissipation are contained in theclass R0 As in Table 9 there are no lsquostrongrsquo endorsements in Table 17 The lsquoweakrsquo endorsementshowever are split more evenly between the two classes now than in Vremanrsquos classes in Sections 441and 442 We find that the Smagorinsky and Vreman models still seem to be the most endorsed inboth classes
444 Two-component Flow Classes
Another way in which the flows can be grouped into classes is by determining whether a flow is atwo-component flow or not We define a two-component flow to have the property that the combinedprincipal invariants of a flow in Eq (43) satisfy
I3 = I4 = I5 minus1
2I1I2 = 0 (82)
We are also interested in considering flows with zero gradient model subgrid dissipation These flowshave the property that
I3 minus I4 = 0 (83)
What we find though is that for all the solutions in Section 22 if a flow is a two-component flowthen it also satisfies I3 minus I4 = 0 It may be interesting to note that although I3 minus I4 6= 0 for manysolutions there are a number with I3 equal to a multiple of I4 Although not considered in thisreport this may also be a class of interest We find that the two-component flows are
C2 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)(84)
35
and we call the remaining flows three-component flows They are
C3 = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex) Sol 11 (Pseudo-Motion Shots)
Sol 13 (Sphere Translation)
(85)
Note that Sol 8 (Vortex in Stagnation Flow) is defined to be a planar flow this does not mean that itis a two-component flow in this sense however because its two non-zero velocity components dependon three coordinates We now check the models for LES without explicit filtering by consideringendorsements within each class Tables 18 and 19 are analogues of Table 3 and they illustrate theresults for LES without explicit filtering for each flow class The analogues of Tables 1 and 2 areTables 31 and 32 and are in Appendix C4
Endorsements for models using solutions in C2 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 6 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 357 714 100 857 100
Table 18 Each entry in the top two rows represents the number of solutions in the class C2 thatendorse each model There are 7 solutions in this class so the percentages are worked out by dividingthe total of each column by 14 and taking the result to one decimal place if necessary
Endorsements for models using solutions in C3 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 3 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 63 63 63 188 250
Table 19 Each entry in the top two rows represents the number of solutions in the class C3 thatendorse each model There are 8 solutions in this class so the percentages are worked out by dividingthe total of each column by 16 and taking the result to one decimal place if necessary
The class C2 equally endorses the QR and Vortex-Stretching-based models and once again theSmagorinsky model is the least endorsed As with the previous classes the class C3 has such lowpercentages that it does not seem to endorse any models We start to see a trend appearing here inthat the flow classes that contain the more complicated flows namely Q0minus6 (Eq (76)) Vlowast (Eq (79))Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages The classes that tendto contain the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78)) R0 (Eq (80)) and C2 (Eq (84))all seem to favour the Vortex-Stretching-based model or the QR model
36
Now let us consider LES with explicit filtering Tables 20 and 21 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 33 and is in Appendix C4
Endorsements for models using solutions in C2 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 2 2 0 1 0
Table 20 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in C3 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 8 8 8 6
Table 21 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 and Table 16 we can see in Table 20 that the results for the lsquostrongrsquo endorsements areexactly the same as in Table 5 This is again because all of the flows with zero subfilter dissipationare contained in the class C2 There are two flows with non-zero true subfilter dissipation (givenin Eq (67)) in C2 and this shows in the lsquoweakrsquo endorsements in Table 20 There are no lsquostrongrsquoendorsements in Table 21 as all the zero subfilter dissipation flows are in C2
5 Conclusions
In this report we studied incompressible turbulent flows We found a number of exact solutions to theNavier-Stokes equations each representing a relatively simplistic fluid flow Large-eddy simulation(LES) is used to model turbulent flows by smoothing over the complex small-scale motion Weintroduced a number of subfilter-scale models to approximate this small-scale motion and testedthem using the found exact solutions to the Navier-Stokes equations We first considered LESwithout explicit filtering and used the Navier-Stokes equations to derive conditions for the subfilter-scale models namely the Velocity Field condition (Eq (58)) the Kinetic Energy condition (Eq (60))and the Dissipation condition (Eq (62)) Similarly we considered LES with explicit filtering andagain used the Navier-Stokes equations to derive more conditions for the models in this case wedecided to just take one condition the Dissipation condition (Eqs (67) and (68)) as the others leadto difficulty in their evaluation By lsquorunningrsquo the exact solutions to the Navier-Stokes equationsthrough each condition for the subfilter-scale models we found that the Vortex-Stretching-basededdy-viscosity model in Eq (54) was most endorsed by the tests in LES without explicit filteringWe found that four out of the five subfilter-scale models namely the Vreman QR Gradient andVortex-Stretching-based models were equally endorsed by the lsquostrongrsquo tests in LES with explicitfiltering with the Smagorinsky model being the least endorsed The results we found for the lsquoweakrsquocondition were completely opposite to those found in both the lsquostrongrsquo condition and in LES withoutexplicit filtering This led us to conclude that this test was not one in which we should put our trust
37
We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses
In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes
This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold
38
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
421 Example of a Test for a Model
The true subfilter dissipation Sijτij of a flow given in Eq (67) cannot be explicitly computed asthe filtering operation is kept general What can be done however is to partially evaluate theexpressions using the attributes of a filtering operation mentioned in Eq (48)
The calculations of Sijτij for Sol 1 (Linear Velocity) are provided to illustrate the general ideaRecall that the velocity field of Sol 1 (Linear Velocity) is written in Eq (13) in vector notation as
u(x y z t) =
0ax+ bcx+ d
(69)
Then Sij = Sij(u) is computed simply by using its definition in Eq (41) Note that as filteringpreserves constants if there is an a it is replaced with a meaning that (in this case) there are alarge number of zero terms due to the frequency of 0 Hence
[Sij(u)
]=
[1
2
(partuipartxj
+partujpartxi
)]=
0 a
2c2
a2 0 0
c2 0 0
=
0 a
2c2
a2 0 0
c2 0 0
(70)
Note that as Sijτij is being computed which is equal to the sum of the element-wise multiplicationof the two matrices (or the trace of the matrix multiplication) that we donrsquot need to consider theelements of τij(u) with corresponding zero entry in Sij As τij(u) = uiuj minus uiuj and again asfiltering preserves constants it is found that
[τij(u)] =[uiuj minus uiuj
]=
lowast 0 middot ax+ b 0 middot cx+ d
ax+ b middot 0 lowast lowastcx+ d middot 0 lowast lowast
=
lowast 0 00 lowast lowast0 lowast lowast
(71)
where a lsquolowastrsquo just represents some element (not necessarily non-zero) Therefore the true subfilterdissipation
Sijτij = tr
0 a2
c2
a2 0 0c2 0 0
middotlowast 0 0
0 lowast lowast0 lowast lowast
= 0 (72)
Now the modelled subfilter dissipation Sij(u)τmodij (u) needs to be computed (recall that Sij =
Sij(u)) In this case let us use the QR model stated in Eq (53) Once again Sij(u) calculated inEq (70) needs to be used So we have that
τmodij = minus2(CQRδ)
2 max0minusI3I1
Sij (73)
As
I1 = tr(S2) =
1
2(a2 + c2) and I3 = tr(S
3) = 0 (74)
it holds that τmodij (u) = 0 so Sij(u)τmod
ij (u) = 0 This agrees with Eq (72) and hence Sol 1(Linear Velocity) endorses the QR model
27
422 Testing the Models
Now using a Mathematica code the exact solutions in Section 22 are again used to evaluate thecondition for the models using the process just described in Section 421 The results are describedin Table 4
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 4 All entries with a ldquo0rdquo represent a value of zero and a ldquolowastrdquo represents a value that is non-zeroThe ldquoExactrdquo row represents the value of the true subfilter dissipation Sijτij Every other row witha model name represents the value of the modelled subfilter dissipation Sij(u)τmod
ij (u) respectiveto that model A ldquordquo means that it is unclear as to the value - the sign of these values is oftenunknown and the lsquoMaxrsquo function in the definition of the QR model (in Eq (53)) makes it difficultto evaluate
Reading from Table 4 the Smagorinsky model is clearly endorsed by Sol 3 (Velocity as a function oftime) and Sol 15 (Rankine Vortex) as in both cases both Sijτij = 0 = Sij(u)τmod
ij (u) In this casewe say that Sol 3 (Velocity as a function of time) and Sol 15 (Rankine Vortex) lsquostronglyrsquo endorsethe Smagorinsky model The Smagorinsky model is not endorsed by Sol 1 (Linear Velocity) Sol 2(Planar Flow) or Sol 14 (Radially Dependent) however because Sij(u)τmod
ij (u) 6= 0 For Sol 4 to
13 both Sijτij 6= 0 6= Sij(u)τmodij (u) and so we will say that these exact solutions lsquoweaklyrsquo endorse
the Smagorinsky model
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 10 10 9 6
Table 5 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Interestingly the lsquostrongrsquo and lsquoweakrsquo endorsements give different results However more stock shouldbe put in the results of the lsquostrongerrsquo condition as it is completely clear that zero is equal to zeroIt is not necessarily clear whether or not non-zero equals non-zero as the values will depend on theparameters in each solution hence the lsquoweakerrsquo endorsement
The lsquostrongrsquo condition equally endorses the Vreman QR Gradient and Vortex-Stretching-basedmodels whereas the lsquoweakrsquo condition endorses Vreman and Smagorinsky the most closely followedby the Gradient model with the Vortex-Stretching-based model being the most opposed As this
28
completely contradicts the lsquostrongrsquo condition and the results in Section 41 it makes us question thevalidity of the weaker test The number of endorsements for the QR model is unknown because of thedefinition of the model in Eq (53) and our evaluation of the true subfilter dissipation Eq (67) Dueto the exact solutions we have found it is never clear whether or not the true subfilter dissipationis strictly positive or negative This means that when the true subfilter dissipation is found to benon-zero the lsquoMaxrsquo function in the QR model cannot be evaluated
43 Vremanrsquos True Subfilter Dissipation Claim
Using the theory mentioned in Sections 232 and 34 we can place the exact solutions to the Navier-Stokes equations into different classes defined by Vreman In doing this we can also attempt to verifyone of Vremanrsquos results [16] In Vremanrsquos paper he defines the class Qn to be the set of flows withn zero entries in the velocity gradient of the flow (the velocity gradient is defined in Eq (39))Vreman then claims that the only flows with zero true subfilter dissipation are a certain selection ofthose in the classesQ7Q8 orQ9 shown in appendix B and is generally non-zero for any other flows
The true subfilter dissipation of Sol 1 (Linear Velocity) was computed in Section 421 and wasfound to be zero in Eq (72) The velocity gradient of Sol 1 is as follows
GLV =
0 0 0a 0 0c 0 0
(75)
and hence Sol 1 (Linear Velocity) is in the required subset of Q7 which agrees with Vremanrsquos claimIn Section 422 the true subfilter dissipation was computed (determined to be zero or non-zero) forall of the exact solutions to the Navier Stokes equations in Section 22 These solutions can now belsquoplacedrsquo into each class Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a functionof time) and Sol 14 (Radially Dependent) were all found to have zero true subfilter dissipation andare all in the appropriate subsets of either Q7Q8 or Q9 This agrees with Vremanrsquos results
We can extend Vremanrsquos results however We found that Sol 15 (Rankine Vortex) had zero truesubfilter dissipation and yet is not in the subset of Q7 defined in Appendix B It is important tonote that the rate of strain tensor Sij in Eq (41) for Sol 15 (Rankine Vortex) is zero This is thereason the true subfilter dissipation is zero and the same reason that all models are zero for thisflow as all models are based on Sij
We also found that all of the remaining exact solutions to the Navier-Stokes equations are not inVremanrsquos specified classes (in Appendix Section B) They also have a non-zero subfilter dissipationwhich agrees with Vremanrsquos results in his paper [16]
44 Classes of Flows
It is important to consider that some exact solutions to the Navier-Stokes equations may have similarproperties For example Sol 1 (Linear Velocity) and Sol 2 (Planar Flow) are both very similarIn fact Sol 2 (Planar Flow) is a more general case of Sol 1 (Linear Velocity) We therefore mustgroup the different flows into classes in an attempt to reduce the bias that may occur when solutionsare similar This should aid us to further determine which of the models are most endorsed We willdo this in a number of different ways using the theory discussed in Section 23 We will group theflows with respect to Vremanrsquos Flow Classes given in appendix B and with respect to the Invariantsgiven in appendix A
29
441 Vremanrsquos Velocity Gradient Flow Classes
We now put our exact solutions to the Navier-Stokes equations into one of two classes Either Q0minus6or Q7minus9 where each class is defined intuitively as follows
Q0minus6 = Q0 cupQ1 cup middot middot middot cup Q6 (76)
Q7minus9 = Q7 cupQ8 cupQ9 (77)
where Qn is defined in Section 43 By looking at the gradient of each exact solution we can see thatQ7minus9 contains the six solutions Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as afunction of time) Sol 5 (2D Taylor) Sol 14 (Radially Dependent) and Sol 15 (Rankine Vortex)and that Q0minus6 contains the remaining nine We now check the models for LES without explicitfiltering by considering endorsements within each class Tables 6 and 7 are analogues of Table 3 andillustrate the results for each flow class The analogues of Tables 1 and 2 are Tables 22 and 23 andare in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 6 6 6
lsquoDissipationrsquoEndorsements
2 5 6 6 6
Percentage Endorsed 417 833 100 100 100
Table 6 Each entry in the top two rows represents the number of solutions in the class Q7minus9 thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Q0minus6 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 2 3 3
lsquoDissipationrsquoEndorsements
0 0 1 0 3
Percentage Endorsed 56 56 167 167 333
Table 7 Each entry in the top two rows represents the number of solutions in the class Q1minus6 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking results to 1 decimal place if necessary
All of the flows in the class Q7minus9 endorse the QR Gradient and Vortex-Stretching-based subfilter-scale models (see Table 6) We can hence see that this class fully endorses these three models equallyand does not endorse the Smagorinsky model The Vortex-Stretching eddy viscosity model has themost endorsements from any of the flows in the class Q0minus6 and the Smagorinksy and Vreman modelshave the fewest endorsements (see Table 7) We can hence say that this class mostly endorses theVortex-Stretching-based eddy-viscosity model note however that the percentage endorsement is very
30
low
Now let us consider LES with explicit filtering Tables 8 and 9 are analogues of Table 5 and illustratethe results for each flow class The analogue of Table 4 is Table 24 and is in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 1 1 0 0 0
Table 8 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Q0minus6 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 9 9 9 6
Table 9 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We have grouped Vremanrsquos classes into Q7minus9 and Q0minus6 this is equivalent to saying that we havegrouped Vremanrsquos classes into those with two or fewer non-zero velocity gradient elements and thosewith three or more non-zero velocity gradient elements respectively Therefore the solutions in Q7minus9are exactly those with a lsquo0rsquo in the lsquoExactrsquo column of Table 4 and Sol 5 (2D Taylor) ExcludingSol 5 (2D Taylor) these are the same as those five that endorse the Vreman QR Gradient andVortex-Stretching-based models in Table 5 We can see in Table 8 that the results for the lsquostrongrsquoendorsements are exactly the same as in Table 5 This is because all of the flows with zero subfilterdissipation are contained in the class Q7minus9 There are hence no lsquostrongrsquo endorsements in Table 9The majority of the lsquoweakrsquo endorsements come from Q0minus6 as this class contains all but one of thesolutions with non-zero true subfilter dissipation (given in Eq (67))
442 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
We now separate our exact solutions into classes again using Vremanrsquos flow classes This timehowever we define V0 to be the class of flows Vreman claims will always have zero true subfilterdissipation namely those classes mentioned in Appendix B We define Vlowast to be all the remaining
31
classes Then
V0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow)
Sol 3 (Velocity as a function of time) Sol 14 (Radially Dependent)(78)
Vlowast = Sol 4 (Axisymmetric Flow) Sol 5 (2D Taylor) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 13 (Sphere Translation) Sol 15 (Rankine Vortex)
(79)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 10 and 11 are analogues of Table 3 and illustrate the results for each flow class Theanalogues of Tables 1 and 2 are Tables 25 and 26 and are in Appendix C2
Endorsements for models using solutions in V0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 4 4 4 4
lsquoDissipationrsquoEndorsements
1 4 4 4 4
Percentage Endorsed 375 100 100 100 100
Table 10 Each entry in the top two rows represents the number of solutions in the class V0 thatendorse each model There are 4 solutions in this class so the percentages are worked out by dividingthe total of each column by 8 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Vlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 2 4 5 5
lsquoDissipationrsquoEndorsements
1 1 3 2 5
Percentage Endorsed 136 136 318 318 455
Table 11 Each entry in the top two rows represents the number of solutions in the class Vlowast thatendorse each model There are 11 solutions in this class so the percentages are worked out bydividing the total of each column by 22 and taking results to 1 decimal place if necessary
All of the flows in the class V0 endorse the Vreman QR Gradient and Vortex-Stretching-basedsubfilter-scale models (see Table 6) This class completely endorses these four models equally anddoes not endorse the Smagorinsky model The Vortex-Stretching-based eddy-viscosity model hasthe most endorsements from the class Vlowast and the Smagorinksy and Vreman models again have thefewest endorsements (see Table 7) As in Section 441 we say that this class mostly endorses the
32
Vortex-Stretching-based eddy-viscosity model but once again the percentage endorsement is ratherlow
Now let us consider LES with explicit filtering Tables 12 and 13 are analogues of Table 5 andillustrate the results for each flow class The analogue of Table 4 is Table 27 and is in Appendix C2
Endorsements for models using solutions in V0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 4 4 4 4
lsquoWeakrsquo Endorsements 0 0 0 0 0
Table 12 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Vlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 1 1 1 1
lsquoWeakrsquo Endorsements 10 10 9 6
Table 13 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We can see in Table 12 that the results for the lsquostrongrsquo endorsements are similar to those in Table 5excluding Sol 15 (Rankine Vortex) The four solutions in V0 with zero true subfilter dissipationnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)and Sol 14 (Radially Dependent) are hence the four that endorse the Vreman QR Gradientand Vortex-Stretching-based models in Table 12 Sol 15 (Rankine Vortex) is in the class Vlowast andhence there is one lsquostrongrsquo endorsement in Table 13 The class Vlowast contains all of the flows with anon-zero true subfilter dissipation (given in Eq (67)) and hence Table 13 contains all of the lsquoweakrsquoendorsements
443 Classes using the Third Principal Invariant
Sorting the classes using the invariants is a non-trivial task in itself There are a number of waysin which to do this using both the principal invariants in Eq (40) and the combined invariants inEq (43)
We start by looking at flows with third principal invariant RG equal to zero We again define twoclasses R0 and Rlowast where R0 consists of all flows with RG = 0 and Rlowast consists of all flows withRG 6= 0 Then using the information in Appendix A we see that there are nine flows in R0 That
33
is
R0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 8 (Vortex in Stagnation Flow)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)
(80)
Rlowast = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 13 (Sphere Translation)(81)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 14 and 15 are analogues of Table 3 and they illustrate the results for LES withoutexplicit filtering for each flow class The analogues of Tables 1 and 2 are Tables 28 and 29 and arein Appendix C3
Endorsements for models using solutions in R0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 7 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 278 556 778 722 778
Table 14 Each entry in the top two rows represents the number of solutions in the class R0 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking the result to one decimal place if necessary
Endorsements for models using solutions in Rlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 2 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 83 83 83 167 333
Table 15 Each entry in the top two rows represents the number of solutions in the class Rlowast thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking the result to one decimal place if necessary
The class R0 endorses both the Vortex-Stretching-based model and the QR model the most andSmagorinsky the least in the case for LES without explicit filtering In the Rlowast class the Vortex-Stretching-based model is again the most endorsed followed by the Gradient model however thepercentages are again small
34
Now let us consider LES with explicit filtering Tables 16 and 17 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 30 and is in Appendix C3
Endorsements for models using solutions in R0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 4 4 3 2
Table 16 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Rlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 6 6 6 4
Table 17 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 we can see in Table 16 that the results for the lsquostrongrsquo endorsements are exactly thesame as in Table 5 This is because all of the flows with zero subfilter dissipation are contained in theclass R0 As in Table 9 there are no lsquostrongrsquo endorsements in Table 17 The lsquoweakrsquo endorsementshowever are split more evenly between the two classes now than in Vremanrsquos classes in Sections 441and 442 We find that the Smagorinsky and Vreman models still seem to be the most endorsed inboth classes
444 Two-component Flow Classes
Another way in which the flows can be grouped into classes is by determining whether a flow is atwo-component flow or not We define a two-component flow to have the property that the combinedprincipal invariants of a flow in Eq (43) satisfy
I3 = I4 = I5 minus1
2I1I2 = 0 (82)
We are also interested in considering flows with zero gradient model subgrid dissipation These flowshave the property that
I3 minus I4 = 0 (83)
What we find though is that for all the solutions in Section 22 if a flow is a two-component flowthen it also satisfies I3 minus I4 = 0 It may be interesting to note that although I3 minus I4 6= 0 for manysolutions there are a number with I3 equal to a multiple of I4 Although not considered in thisreport this may also be a class of interest We find that the two-component flows are
C2 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)(84)
35
and we call the remaining flows three-component flows They are
C3 = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex) Sol 11 (Pseudo-Motion Shots)
Sol 13 (Sphere Translation)
(85)
Note that Sol 8 (Vortex in Stagnation Flow) is defined to be a planar flow this does not mean that itis a two-component flow in this sense however because its two non-zero velocity components dependon three coordinates We now check the models for LES without explicit filtering by consideringendorsements within each class Tables 18 and 19 are analogues of Table 3 and they illustrate theresults for LES without explicit filtering for each flow class The analogues of Tables 1 and 2 areTables 31 and 32 and are in Appendix C4
Endorsements for models using solutions in C2 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 6 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 357 714 100 857 100
Table 18 Each entry in the top two rows represents the number of solutions in the class C2 thatendorse each model There are 7 solutions in this class so the percentages are worked out by dividingthe total of each column by 14 and taking the result to one decimal place if necessary
Endorsements for models using solutions in C3 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 3 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 63 63 63 188 250
Table 19 Each entry in the top two rows represents the number of solutions in the class C3 thatendorse each model There are 8 solutions in this class so the percentages are worked out by dividingthe total of each column by 16 and taking the result to one decimal place if necessary
The class C2 equally endorses the QR and Vortex-Stretching-based models and once again theSmagorinsky model is the least endorsed As with the previous classes the class C3 has such lowpercentages that it does not seem to endorse any models We start to see a trend appearing here inthat the flow classes that contain the more complicated flows namely Q0minus6 (Eq (76)) Vlowast (Eq (79))Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages The classes that tendto contain the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78)) R0 (Eq (80)) and C2 (Eq (84))all seem to favour the Vortex-Stretching-based model or the QR model
36
Now let us consider LES with explicit filtering Tables 20 and 21 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 33 and is in Appendix C4
Endorsements for models using solutions in C2 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 2 2 0 1 0
Table 20 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in C3 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 8 8 8 6
Table 21 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 and Table 16 we can see in Table 20 that the results for the lsquostrongrsquo endorsements areexactly the same as in Table 5 This is again because all of the flows with zero subfilter dissipationare contained in the class C2 There are two flows with non-zero true subfilter dissipation (givenin Eq (67)) in C2 and this shows in the lsquoweakrsquo endorsements in Table 20 There are no lsquostrongrsquoendorsements in Table 21 as all the zero subfilter dissipation flows are in C2
5 Conclusions
In this report we studied incompressible turbulent flows We found a number of exact solutions to theNavier-Stokes equations each representing a relatively simplistic fluid flow Large-eddy simulation(LES) is used to model turbulent flows by smoothing over the complex small-scale motion Weintroduced a number of subfilter-scale models to approximate this small-scale motion and testedthem using the found exact solutions to the Navier-Stokes equations We first considered LESwithout explicit filtering and used the Navier-Stokes equations to derive conditions for the subfilter-scale models namely the Velocity Field condition (Eq (58)) the Kinetic Energy condition (Eq (60))and the Dissipation condition (Eq (62)) Similarly we considered LES with explicit filtering andagain used the Navier-Stokes equations to derive more conditions for the models in this case wedecided to just take one condition the Dissipation condition (Eqs (67) and (68)) as the others leadto difficulty in their evaluation By lsquorunningrsquo the exact solutions to the Navier-Stokes equationsthrough each condition for the subfilter-scale models we found that the Vortex-Stretching-basededdy-viscosity model in Eq (54) was most endorsed by the tests in LES without explicit filteringWe found that four out of the five subfilter-scale models namely the Vreman QR Gradient andVortex-Stretching-based models were equally endorsed by the lsquostrongrsquo tests in LES with explicitfiltering with the Smagorinsky model being the least endorsed The results we found for the lsquoweakrsquocondition were completely opposite to those found in both the lsquostrongrsquo condition and in LES withoutexplicit filtering This led us to conclude that this test was not one in which we should put our trust
37
We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses
In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes
This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold
38
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
422 Testing the Models
Now using a Mathematica code the exact solutions in Section 22 are again used to evaluate thecondition for the models using the process just described in Section 421 The results are describedin Table 4
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Solution 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 4 All entries with a ldquo0rdquo represent a value of zero and a ldquolowastrdquo represents a value that is non-zeroThe ldquoExactrdquo row represents the value of the true subfilter dissipation Sijτij Every other row witha model name represents the value of the modelled subfilter dissipation Sij(u)τmod
ij (u) respectiveto that model A ldquordquo means that it is unclear as to the value - the sign of these values is oftenunknown and the lsquoMaxrsquo function in the definition of the QR model (in Eq (53)) makes it difficultto evaluate
Reading from Table 4 the Smagorinsky model is clearly endorsed by Sol 3 (Velocity as a function oftime) and Sol 15 (Rankine Vortex) as in both cases both Sijτij = 0 = Sij(u)τmod
ij (u) In this casewe say that Sol 3 (Velocity as a function of time) and Sol 15 (Rankine Vortex) lsquostronglyrsquo endorsethe Smagorinsky model The Smagorinsky model is not endorsed by Sol 1 (Linear Velocity) Sol 2(Planar Flow) or Sol 14 (Radially Dependent) however because Sij(u)τmod
ij (u) 6= 0 For Sol 4 to
13 both Sijτij 6= 0 6= Sij(u)τmodij (u) and so we will say that these exact solutions lsquoweaklyrsquo endorse
the Smagorinsky model
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 10 10 9 6
Table 5 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Interestingly the lsquostrongrsquo and lsquoweakrsquo endorsements give different results However more stock shouldbe put in the results of the lsquostrongerrsquo condition as it is completely clear that zero is equal to zeroIt is not necessarily clear whether or not non-zero equals non-zero as the values will depend on theparameters in each solution hence the lsquoweakerrsquo endorsement
The lsquostrongrsquo condition equally endorses the Vreman QR Gradient and Vortex-Stretching-basedmodels whereas the lsquoweakrsquo condition endorses Vreman and Smagorinsky the most closely followedby the Gradient model with the Vortex-Stretching-based model being the most opposed As this
28
completely contradicts the lsquostrongrsquo condition and the results in Section 41 it makes us question thevalidity of the weaker test The number of endorsements for the QR model is unknown because of thedefinition of the model in Eq (53) and our evaluation of the true subfilter dissipation Eq (67) Dueto the exact solutions we have found it is never clear whether or not the true subfilter dissipationis strictly positive or negative This means that when the true subfilter dissipation is found to benon-zero the lsquoMaxrsquo function in the QR model cannot be evaluated
43 Vremanrsquos True Subfilter Dissipation Claim
Using the theory mentioned in Sections 232 and 34 we can place the exact solutions to the Navier-Stokes equations into different classes defined by Vreman In doing this we can also attempt to verifyone of Vremanrsquos results [16] In Vremanrsquos paper he defines the class Qn to be the set of flows withn zero entries in the velocity gradient of the flow (the velocity gradient is defined in Eq (39))Vreman then claims that the only flows with zero true subfilter dissipation are a certain selection ofthose in the classesQ7Q8 orQ9 shown in appendix B and is generally non-zero for any other flows
The true subfilter dissipation of Sol 1 (Linear Velocity) was computed in Section 421 and wasfound to be zero in Eq (72) The velocity gradient of Sol 1 is as follows
GLV =
0 0 0a 0 0c 0 0
(75)
and hence Sol 1 (Linear Velocity) is in the required subset of Q7 which agrees with Vremanrsquos claimIn Section 422 the true subfilter dissipation was computed (determined to be zero or non-zero) forall of the exact solutions to the Navier Stokes equations in Section 22 These solutions can now belsquoplacedrsquo into each class Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a functionof time) and Sol 14 (Radially Dependent) were all found to have zero true subfilter dissipation andare all in the appropriate subsets of either Q7Q8 or Q9 This agrees with Vremanrsquos results
We can extend Vremanrsquos results however We found that Sol 15 (Rankine Vortex) had zero truesubfilter dissipation and yet is not in the subset of Q7 defined in Appendix B It is important tonote that the rate of strain tensor Sij in Eq (41) for Sol 15 (Rankine Vortex) is zero This is thereason the true subfilter dissipation is zero and the same reason that all models are zero for thisflow as all models are based on Sij
We also found that all of the remaining exact solutions to the Navier-Stokes equations are not inVremanrsquos specified classes (in Appendix Section B) They also have a non-zero subfilter dissipationwhich agrees with Vremanrsquos results in his paper [16]
44 Classes of Flows
It is important to consider that some exact solutions to the Navier-Stokes equations may have similarproperties For example Sol 1 (Linear Velocity) and Sol 2 (Planar Flow) are both very similarIn fact Sol 2 (Planar Flow) is a more general case of Sol 1 (Linear Velocity) We therefore mustgroup the different flows into classes in an attempt to reduce the bias that may occur when solutionsare similar This should aid us to further determine which of the models are most endorsed We willdo this in a number of different ways using the theory discussed in Section 23 We will group theflows with respect to Vremanrsquos Flow Classes given in appendix B and with respect to the Invariantsgiven in appendix A
29
441 Vremanrsquos Velocity Gradient Flow Classes
We now put our exact solutions to the Navier-Stokes equations into one of two classes Either Q0minus6or Q7minus9 where each class is defined intuitively as follows
Q0minus6 = Q0 cupQ1 cup middot middot middot cup Q6 (76)
Q7minus9 = Q7 cupQ8 cupQ9 (77)
where Qn is defined in Section 43 By looking at the gradient of each exact solution we can see thatQ7minus9 contains the six solutions Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as afunction of time) Sol 5 (2D Taylor) Sol 14 (Radially Dependent) and Sol 15 (Rankine Vortex)and that Q0minus6 contains the remaining nine We now check the models for LES without explicitfiltering by considering endorsements within each class Tables 6 and 7 are analogues of Table 3 andillustrate the results for each flow class The analogues of Tables 1 and 2 are Tables 22 and 23 andare in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 6 6 6
lsquoDissipationrsquoEndorsements
2 5 6 6 6
Percentage Endorsed 417 833 100 100 100
Table 6 Each entry in the top two rows represents the number of solutions in the class Q7minus9 thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Q0minus6 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 2 3 3
lsquoDissipationrsquoEndorsements
0 0 1 0 3
Percentage Endorsed 56 56 167 167 333
Table 7 Each entry in the top two rows represents the number of solutions in the class Q1minus6 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking results to 1 decimal place if necessary
All of the flows in the class Q7minus9 endorse the QR Gradient and Vortex-Stretching-based subfilter-scale models (see Table 6) We can hence see that this class fully endorses these three models equallyand does not endorse the Smagorinsky model The Vortex-Stretching eddy viscosity model has themost endorsements from any of the flows in the class Q0minus6 and the Smagorinksy and Vreman modelshave the fewest endorsements (see Table 7) We can hence say that this class mostly endorses theVortex-Stretching-based eddy-viscosity model note however that the percentage endorsement is very
30
low
Now let us consider LES with explicit filtering Tables 8 and 9 are analogues of Table 5 and illustratethe results for each flow class The analogue of Table 4 is Table 24 and is in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 1 1 0 0 0
Table 8 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Q0minus6 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 9 9 9 6
Table 9 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We have grouped Vremanrsquos classes into Q7minus9 and Q0minus6 this is equivalent to saying that we havegrouped Vremanrsquos classes into those with two or fewer non-zero velocity gradient elements and thosewith three or more non-zero velocity gradient elements respectively Therefore the solutions in Q7minus9are exactly those with a lsquo0rsquo in the lsquoExactrsquo column of Table 4 and Sol 5 (2D Taylor) ExcludingSol 5 (2D Taylor) these are the same as those five that endorse the Vreman QR Gradient andVortex-Stretching-based models in Table 5 We can see in Table 8 that the results for the lsquostrongrsquoendorsements are exactly the same as in Table 5 This is because all of the flows with zero subfilterdissipation are contained in the class Q7minus9 There are hence no lsquostrongrsquo endorsements in Table 9The majority of the lsquoweakrsquo endorsements come from Q0minus6 as this class contains all but one of thesolutions with non-zero true subfilter dissipation (given in Eq (67))
442 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
We now separate our exact solutions into classes again using Vremanrsquos flow classes This timehowever we define V0 to be the class of flows Vreman claims will always have zero true subfilterdissipation namely those classes mentioned in Appendix B We define Vlowast to be all the remaining
31
classes Then
V0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow)
Sol 3 (Velocity as a function of time) Sol 14 (Radially Dependent)(78)
Vlowast = Sol 4 (Axisymmetric Flow) Sol 5 (2D Taylor) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 13 (Sphere Translation) Sol 15 (Rankine Vortex)
(79)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 10 and 11 are analogues of Table 3 and illustrate the results for each flow class Theanalogues of Tables 1 and 2 are Tables 25 and 26 and are in Appendix C2
Endorsements for models using solutions in V0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 4 4 4 4
lsquoDissipationrsquoEndorsements
1 4 4 4 4
Percentage Endorsed 375 100 100 100 100
Table 10 Each entry in the top two rows represents the number of solutions in the class V0 thatendorse each model There are 4 solutions in this class so the percentages are worked out by dividingthe total of each column by 8 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Vlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 2 4 5 5
lsquoDissipationrsquoEndorsements
1 1 3 2 5
Percentage Endorsed 136 136 318 318 455
Table 11 Each entry in the top two rows represents the number of solutions in the class Vlowast thatendorse each model There are 11 solutions in this class so the percentages are worked out bydividing the total of each column by 22 and taking results to 1 decimal place if necessary
All of the flows in the class V0 endorse the Vreman QR Gradient and Vortex-Stretching-basedsubfilter-scale models (see Table 6) This class completely endorses these four models equally anddoes not endorse the Smagorinsky model The Vortex-Stretching-based eddy-viscosity model hasthe most endorsements from the class Vlowast and the Smagorinksy and Vreman models again have thefewest endorsements (see Table 7) As in Section 441 we say that this class mostly endorses the
32
Vortex-Stretching-based eddy-viscosity model but once again the percentage endorsement is ratherlow
Now let us consider LES with explicit filtering Tables 12 and 13 are analogues of Table 5 andillustrate the results for each flow class The analogue of Table 4 is Table 27 and is in Appendix C2
Endorsements for models using solutions in V0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 4 4 4 4
lsquoWeakrsquo Endorsements 0 0 0 0 0
Table 12 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Vlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 1 1 1 1
lsquoWeakrsquo Endorsements 10 10 9 6
Table 13 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We can see in Table 12 that the results for the lsquostrongrsquo endorsements are similar to those in Table 5excluding Sol 15 (Rankine Vortex) The four solutions in V0 with zero true subfilter dissipationnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)and Sol 14 (Radially Dependent) are hence the four that endorse the Vreman QR Gradientand Vortex-Stretching-based models in Table 12 Sol 15 (Rankine Vortex) is in the class Vlowast andhence there is one lsquostrongrsquo endorsement in Table 13 The class Vlowast contains all of the flows with anon-zero true subfilter dissipation (given in Eq (67)) and hence Table 13 contains all of the lsquoweakrsquoendorsements
443 Classes using the Third Principal Invariant
Sorting the classes using the invariants is a non-trivial task in itself There are a number of waysin which to do this using both the principal invariants in Eq (40) and the combined invariants inEq (43)
We start by looking at flows with third principal invariant RG equal to zero We again define twoclasses R0 and Rlowast where R0 consists of all flows with RG = 0 and Rlowast consists of all flows withRG 6= 0 Then using the information in Appendix A we see that there are nine flows in R0 That
33
is
R0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 8 (Vortex in Stagnation Flow)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)
(80)
Rlowast = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 13 (Sphere Translation)(81)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 14 and 15 are analogues of Table 3 and they illustrate the results for LES withoutexplicit filtering for each flow class The analogues of Tables 1 and 2 are Tables 28 and 29 and arein Appendix C3
Endorsements for models using solutions in R0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 7 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 278 556 778 722 778
Table 14 Each entry in the top two rows represents the number of solutions in the class R0 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking the result to one decimal place if necessary
Endorsements for models using solutions in Rlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 2 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 83 83 83 167 333
Table 15 Each entry in the top two rows represents the number of solutions in the class Rlowast thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking the result to one decimal place if necessary
The class R0 endorses both the Vortex-Stretching-based model and the QR model the most andSmagorinsky the least in the case for LES without explicit filtering In the Rlowast class the Vortex-Stretching-based model is again the most endorsed followed by the Gradient model however thepercentages are again small
34
Now let us consider LES with explicit filtering Tables 16 and 17 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 30 and is in Appendix C3
Endorsements for models using solutions in R0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 4 4 3 2
Table 16 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Rlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 6 6 6 4
Table 17 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 we can see in Table 16 that the results for the lsquostrongrsquo endorsements are exactly thesame as in Table 5 This is because all of the flows with zero subfilter dissipation are contained in theclass R0 As in Table 9 there are no lsquostrongrsquo endorsements in Table 17 The lsquoweakrsquo endorsementshowever are split more evenly between the two classes now than in Vremanrsquos classes in Sections 441and 442 We find that the Smagorinsky and Vreman models still seem to be the most endorsed inboth classes
444 Two-component Flow Classes
Another way in which the flows can be grouped into classes is by determining whether a flow is atwo-component flow or not We define a two-component flow to have the property that the combinedprincipal invariants of a flow in Eq (43) satisfy
I3 = I4 = I5 minus1
2I1I2 = 0 (82)
We are also interested in considering flows with zero gradient model subgrid dissipation These flowshave the property that
I3 minus I4 = 0 (83)
What we find though is that for all the solutions in Section 22 if a flow is a two-component flowthen it also satisfies I3 minus I4 = 0 It may be interesting to note that although I3 minus I4 6= 0 for manysolutions there are a number with I3 equal to a multiple of I4 Although not considered in thisreport this may also be a class of interest We find that the two-component flows are
C2 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)(84)
35
and we call the remaining flows three-component flows They are
C3 = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex) Sol 11 (Pseudo-Motion Shots)
Sol 13 (Sphere Translation)
(85)
Note that Sol 8 (Vortex in Stagnation Flow) is defined to be a planar flow this does not mean that itis a two-component flow in this sense however because its two non-zero velocity components dependon three coordinates We now check the models for LES without explicit filtering by consideringendorsements within each class Tables 18 and 19 are analogues of Table 3 and they illustrate theresults for LES without explicit filtering for each flow class The analogues of Tables 1 and 2 areTables 31 and 32 and are in Appendix C4
Endorsements for models using solutions in C2 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 6 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 357 714 100 857 100
Table 18 Each entry in the top two rows represents the number of solutions in the class C2 thatendorse each model There are 7 solutions in this class so the percentages are worked out by dividingthe total of each column by 14 and taking the result to one decimal place if necessary
Endorsements for models using solutions in C3 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 3 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 63 63 63 188 250
Table 19 Each entry in the top two rows represents the number of solutions in the class C3 thatendorse each model There are 8 solutions in this class so the percentages are worked out by dividingthe total of each column by 16 and taking the result to one decimal place if necessary
The class C2 equally endorses the QR and Vortex-Stretching-based models and once again theSmagorinsky model is the least endorsed As with the previous classes the class C3 has such lowpercentages that it does not seem to endorse any models We start to see a trend appearing here inthat the flow classes that contain the more complicated flows namely Q0minus6 (Eq (76)) Vlowast (Eq (79))Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages The classes that tendto contain the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78)) R0 (Eq (80)) and C2 (Eq (84))all seem to favour the Vortex-Stretching-based model or the QR model
36
Now let us consider LES with explicit filtering Tables 20 and 21 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 33 and is in Appendix C4
Endorsements for models using solutions in C2 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 2 2 0 1 0
Table 20 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in C3 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 8 8 8 6
Table 21 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 and Table 16 we can see in Table 20 that the results for the lsquostrongrsquo endorsements areexactly the same as in Table 5 This is again because all of the flows with zero subfilter dissipationare contained in the class C2 There are two flows with non-zero true subfilter dissipation (givenin Eq (67)) in C2 and this shows in the lsquoweakrsquo endorsements in Table 20 There are no lsquostrongrsquoendorsements in Table 21 as all the zero subfilter dissipation flows are in C2
5 Conclusions
In this report we studied incompressible turbulent flows We found a number of exact solutions to theNavier-Stokes equations each representing a relatively simplistic fluid flow Large-eddy simulation(LES) is used to model turbulent flows by smoothing over the complex small-scale motion Weintroduced a number of subfilter-scale models to approximate this small-scale motion and testedthem using the found exact solutions to the Navier-Stokes equations We first considered LESwithout explicit filtering and used the Navier-Stokes equations to derive conditions for the subfilter-scale models namely the Velocity Field condition (Eq (58)) the Kinetic Energy condition (Eq (60))and the Dissipation condition (Eq (62)) Similarly we considered LES with explicit filtering andagain used the Navier-Stokes equations to derive more conditions for the models in this case wedecided to just take one condition the Dissipation condition (Eqs (67) and (68)) as the others leadto difficulty in their evaluation By lsquorunningrsquo the exact solutions to the Navier-Stokes equationsthrough each condition for the subfilter-scale models we found that the Vortex-Stretching-basededdy-viscosity model in Eq (54) was most endorsed by the tests in LES without explicit filteringWe found that four out of the five subfilter-scale models namely the Vreman QR Gradient andVortex-Stretching-based models were equally endorsed by the lsquostrongrsquo tests in LES with explicitfiltering with the Smagorinsky model being the least endorsed The results we found for the lsquoweakrsquocondition were completely opposite to those found in both the lsquostrongrsquo condition and in LES withoutexplicit filtering This led us to conclude that this test was not one in which we should put our trust
37
We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses
In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes
This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold
38
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
completely contradicts the lsquostrongrsquo condition and the results in Section 41 it makes us question thevalidity of the weaker test The number of endorsements for the QR model is unknown because of thedefinition of the model in Eq (53) and our evaluation of the true subfilter dissipation Eq (67) Dueto the exact solutions we have found it is never clear whether or not the true subfilter dissipationis strictly positive or negative This means that when the true subfilter dissipation is found to benon-zero the lsquoMaxrsquo function in the QR model cannot be evaluated
43 Vremanrsquos True Subfilter Dissipation Claim
Using the theory mentioned in Sections 232 and 34 we can place the exact solutions to the Navier-Stokes equations into different classes defined by Vreman In doing this we can also attempt to verifyone of Vremanrsquos results [16] In Vremanrsquos paper he defines the class Qn to be the set of flows withn zero entries in the velocity gradient of the flow (the velocity gradient is defined in Eq (39))Vreman then claims that the only flows with zero true subfilter dissipation are a certain selection ofthose in the classesQ7Q8 orQ9 shown in appendix B and is generally non-zero for any other flows
The true subfilter dissipation of Sol 1 (Linear Velocity) was computed in Section 421 and wasfound to be zero in Eq (72) The velocity gradient of Sol 1 is as follows
GLV =
0 0 0a 0 0c 0 0
(75)
and hence Sol 1 (Linear Velocity) is in the required subset of Q7 which agrees with Vremanrsquos claimIn Section 422 the true subfilter dissipation was computed (determined to be zero or non-zero) forall of the exact solutions to the Navier Stokes equations in Section 22 These solutions can now belsquoplacedrsquo into each class Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a functionof time) and Sol 14 (Radially Dependent) were all found to have zero true subfilter dissipation andare all in the appropriate subsets of either Q7Q8 or Q9 This agrees with Vremanrsquos results
We can extend Vremanrsquos results however We found that Sol 15 (Rankine Vortex) had zero truesubfilter dissipation and yet is not in the subset of Q7 defined in Appendix B It is important tonote that the rate of strain tensor Sij in Eq (41) for Sol 15 (Rankine Vortex) is zero This is thereason the true subfilter dissipation is zero and the same reason that all models are zero for thisflow as all models are based on Sij
We also found that all of the remaining exact solutions to the Navier-Stokes equations are not inVremanrsquos specified classes (in Appendix Section B) They also have a non-zero subfilter dissipationwhich agrees with Vremanrsquos results in his paper [16]
44 Classes of Flows
It is important to consider that some exact solutions to the Navier-Stokes equations may have similarproperties For example Sol 1 (Linear Velocity) and Sol 2 (Planar Flow) are both very similarIn fact Sol 2 (Planar Flow) is a more general case of Sol 1 (Linear Velocity) We therefore mustgroup the different flows into classes in an attempt to reduce the bias that may occur when solutionsare similar This should aid us to further determine which of the models are most endorsed We willdo this in a number of different ways using the theory discussed in Section 23 We will group theflows with respect to Vremanrsquos Flow Classes given in appendix B and with respect to the Invariantsgiven in appendix A
29
441 Vremanrsquos Velocity Gradient Flow Classes
We now put our exact solutions to the Navier-Stokes equations into one of two classes Either Q0minus6or Q7minus9 where each class is defined intuitively as follows
Q0minus6 = Q0 cupQ1 cup middot middot middot cup Q6 (76)
Q7minus9 = Q7 cupQ8 cupQ9 (77)
where Qn is defined in Section 43 By looking at the gradient of each exact solution we can see thatQ7minus9 contains the six solutions Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as afunction of time) Sol 5 (2D Taylor) Sol 14 (Radially Dependent) and Sol 15 (Rankine Vortex)and that Q0minus6 contains the remaining nine We now check the models for LES without explicitfiltering by considering endorsements within each class Tables 6 and 7 are analogues of Table 3 andillustrate the results for each flow class The analogues of Tables 1 and 2 are Tables 22 and 23 andare in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 6 6 6
lsquoDissipationrsquoEndorsements
2 5 6 6 6
Percentage Endorsed 417 833 100 100 100
Table 6 Each entry in the top two rows represents the number of solutions in the class Q7minus9 thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Q0minus6 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 2 3 3
lsquoDissipationrsquoEndorsements
0 0 1 0 3
Percentage Endorsed 56 56 167 167 333
Table 7 Each entry in the top two rows represents the number of solutions in the class Q1minus6 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking results to 1 decimal place if necessary
All of the flows in the class Q7minus9 endorse the QR Gradient and Vortex-Stretching-based subfilter-scale models (see Table 6) We can hence see that this class fully endorses these three models equallyand does not endorse the Smagorinsky model The Vortex-Stretching eddy viscosity model has themost endorsements from any of the flows in the class Q0minus6 and the Smagorinksy and Vreman modelshave the fewest endorsements (see Table 7) We can hence say that this class mostly endorses theVortex-Stretching-based eddy-viscosity model note however that the percentage endorsement is very
30
low
Now let us consider LES with explicit filtering Tables 8 and 9 are analogues of Table 5 and illustratethe results for each flow class The analogue of Table 4 is Table 24 and is in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 1 1 0 0 0
Table 8 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Q0minus6 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 9 9 9 6
Table 9 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We have grouped Vremanrsquos classes into Q7minus9 and Q0minus6 this is equivalent to saying that we havegrouped Vremanrsquos classes into those with two or fewer non-zero velocity gradient elements and thosewith three or more non-zero velocity gradient elements respectively Therefore the solutions in Q7minus9are exactly those with a lsquo0rsquo in the lsquoExactrsquo column of Table 4 and Sol 5 (2D Taylor) ExcludingSol 5 (2D Taylor) these are the same as those five that endorse the Vreman QR Gradient andVortex-Stretching-based models in Table 5 We can see in Table 8 that the results for the lsquostrongrsquoendorsements are exactly the same as in Table 5 This is because all of the flows with zero subfilterdissipation are contained in the class Q7minus9 There are hence no lsquostrongrsquo endorsements in Table 9The majority of the lsquoweakrsquo endorsements come from Q0minus6 as this class contains all but one of thesolutions with non-zero true subfilter dissipation (given in Eq (67))
442 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
We now separate our exact solutions into classes again using Vremanrsquos flow classes This timehowever we define V0 to be the class of flows Vreman claims will always have zero true subfilterdissipation namely those classes mentioned in Appendix B We define Vlowast to be all the remaining
31
classes Then
V0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow)
Sol 3 (Velocity as a function of time) Sol 14 (Radially Dependent)(78)
Vlowast = Sol 4 (Axisymmetric Flow) Sol 5 (2D Taylor) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 13 (Sphere Translation) Sol 15 (Rankine Vortex)
(79)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 10 and 11 are analogues of Table 3 and illustrate the results for each flow class Theanalogues of Tables 1 and 2 are Tables 25 and 26 and are in Appendix C2
Endorsements for models using solutions in V0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 4 4 4 4
lsquoDissipationrsquoEndorsements
1 4 4 4 4
Percentage Endorsed 375 100 100 100 100
Table 10 Each entry in the top two rows represents the number of solutions in the class V0 thatendorse each model There are 4 solutions in this class so the percentages are worked out by dividingthe total of each column by 8 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Vlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 2 4 5 5
lsquoDissipationrsquoEndorsements
1 1 3 2 5
Percentage Endorsed 136 136 318 318 455
Table 11 Each entry in the top two rows represents the number of solutions in the class Vlowast thatendorse each model There are 11 solutions in this class so the percentages are worked out bydividing the total of each column by 22 and taking results to 1 decimal place if necessary
All of the flows in the class V0 endorse the Vreman QR Gradient and Vortex-Stretching-basedsubfilter-scale models (see Table 6) This class completely endorses these four models equally anddoes not endorse the Smagorinsky model The Vortex-Stretching-based eddy-viscosity model hasthe most endorsements from the class Vlowast and the Smagorinksy and Vreman models again have thefewest endorsements (see Table 7) As in Section 441 we say that this class mostly endorses the
32
Vortex-Stretching-based eddy-viscosity model but once again the percentage endorsement is ratherlow
Now let us consider LES with explicit filtering Tables 12 and 13 are analogues of Table 5 andillustrate the results for each flow class The analogue of Table 4 is Table 27 and is in Appendix C2
Endorsements for models using solutions in V0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 4 4 4 4
lsquoWeakrsquo Endorsements 0 0 0 0 0
Table 12 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Vlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 1 1 1 1
lsquoWeakrsquo Endorsements 10 10 9 6
Table 13 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We can see in Table 12 that the results for the lsquostrongrsquo endorsements are similar to those in Table 5excluding Sol 15 (Rankine Vortex) The four solutions in V0 with zero true subfilter dissipationnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)and Sol 14 (Radially Dependent) are hence the four that endorse the Vreman QR Gradientand Vortex-Stretching-based models in Table 12 Sol 15 (Rankine Vortex) is in the class Vlowast andhence there is one lsquostrongrsquo endorsement in Table 13 The class Vlowast contains all of the flows with anon-zero true subfilter dissipation (given in Eq (67)) and hence Table 13 contains all of the lsquoweakrsquoendorsements
443 Classes using the Third Principal Invariant
Sorting the classes using the invariants is a non-trivial task in itself There are a number of waysin which to do this using both the principal invariants in Eq (40) and the combined invariants inEq (43)
We start by looking at flows with third principal invariant RG equal to zero We again define twoclasses R0 and Rlowast where R0 consists of all flows with RG = 0 and Rlowast consists of all flows withRG 6= 0 Then using the information in Appendix A we see that there are nine flows in R0 That
33
is
R0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 8 (Vortex in Stagnation Flow)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)
(80)
Rlowast = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 13 (Sphere Translation)(81)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 14 and 15 are analogues of Table 3 and they illustrate the results for LES withoutexplicit filtering for each flow class The analogues of Tables 1 and 2 are Tables 28 and 29 and arein Appendix C3
Endorsements for models using solutions in R0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 7 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 278 556 778 722 778
Table 14 Each entry in the top two rows represents the number of solutions in the class R0 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking the result to one decimal place if necessary
Endorsements for models using solutions in Rlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 2 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 83 83 83 167 333
Table 15 Each entry in the top two rows represents the number of solutions in the class Rlowast thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking the result to one decimal place if necessary
The class R0 endorses both the Vortex-Stretching-based model and the QR model the most andSmagorinsky the least in the case for LES without explicit filtering In the Rlowast class the Vortex-Stretching-based model is again the most endorsed followed by the Gradient model however thepercentages are again small
34
Now let us consider LES with explicit filtering Tables 16 and 17 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 30 and is in Appendix C3
Endorsements for models using solutions in R0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 4 4 3 2
Table 16 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Rlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 6 6 6 4
Table 17 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 we can see in Table 16 that the results for the lsquostrongrsquo endorsements are exactly thesame as in Table 5 This is because all of the flows with zero subfilter dissipation are contained in theclass R0 As in Table 9 there are no lsquostrongrsquo endorsements in Table 17 The lsquoweakrsquo endorsementshowever are split more evenly between the two classes now than in Vremanrsquos classes in Sections 441and 442 We find that the Smagorinsky and Vreman models still seem to be the most endorsed inboth classes
444 Two-component Flow Classes
Another way in which the flows can be grouped into classes is by determining whether a flow is atwo-component flow or not We define a two-component flow to have the property that the combinedprincipal invariants of a flow in Eq (43) satisfy
I3 = I4 = I5 minus1
2I1I2 = 0 (82)
We are also interested in considering flows with zero gradient model subgrid dissipation These flowshave the property that
I3 minus I4 = 0 (83)
What we find though is that for all the solutions in Section 22 if a flow is a two-component flowthen it also satisfies I3 minus I4 = 0 It may be interesting to note that although I3 minus I4 6= 0 for manysolutions there are a number with I3 equal to a multiple of I4 Although not considered in thisreport this may also be a class of interest We find that the two-component flows are
C2 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)(84)
35
and we call the remaining flows three-component flows They are
C3 = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex) Sol 11 (Pseudo-Motion Shots)
Sol 13 (Sphere Translation)
(85)
Note that Sol 8 (Vortex in Stagnation Flow) is defined to be a planar flow this does not mean that itis a two-component flow in this sense however because its two non-zero velocity components dependon three coordinates We now check the models for LES without explicit filtering by consideringendorsements within each class Tables 18 and 19 are analogues of Table 3 and they illustrate theresults for LES without explicit filtering for each flow class The analogues of Tables 1 and 2 areTables 31 and 32 and are in Appendix C4
Endorsements for models using solutions in C2 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 6 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 357 714 100 857 100
Table 18 Each entry in the top two rows represents the number of solutions in the class C2 thatendorse each model There are 7 solutions in this class so the percentages are worked out by dividingthe total of each column by 14 and taking the result to one decimal place if necessary
Endorsements for models using solutions in C3 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 3 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 63 63 63 188 250
Table 19 Each entry in the top two rows represents the number of solutions in the class C3 thatendorse each model There are 8 solutions in this class so the percentages are worked out by dividingthe total of each column by 16 and taking the result to one decimal place if necessary
The class C2 equally endorses the QR and Vortex-Stretching-based models and once again theSmagorinsky model is the least endorsed As with the previous classes the class C3 has such lowpercentages that it does not seem to endorse any models We start to see a trend appearing here inthat the flow classes that contain the more complicated flows namely Q0minus6 (Eq (76)) Vlowast (Eq (79))Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages The classes that tendto contain the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78)) R0 (Eq (80)) and C2 (Eq (84))all seem to favour the Vortex-Stretching-based model or the QR model
36
Now let us consider LES with explicit filtering Tables 20 and 21 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 33 and is in Appendix C4
Endorsements for models using solutions in C2 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 2 2 0 1 0
Table 20 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in C3 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 8 8 8 6
Table 21 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 and Table 16 we can see in Table 20 that the results for the lsquostrongrsquo endorsements areexactly the same as in Table 5 This is again because all of the flows with zero subfilter dissipationare contained in the class C2 There are two flows with non-zero true subfilter dissipation (givenin Eq (67)) in C2 and this shows in the lsquoweakrsquo endorsements in Table 20 There are no lsquostrongrsquoendorsements in Table 21 as all the zero subfilter dissipation flows are in C2
5 Conclusions
In this report we studied incompressible turbulent flows We found a number of exact solutions to theNavier-Stokes equations each representing a relatively simplistic fluid flow Large-eddy simulation(LES) is used to model turbulent flows by smoothing over the complex small-scale motion Weintroduced a number of subfilter-scale models to approximate this small-scale motion and testedthem using the found exact solutions to the Navier-Stokes equations We first considered LESwithout explicit filtering and used the Navier-Stokes equations to derive conditions for the subfilter-scale models namely the Velocity Field condition (Eq (58)) the Kinetic Energy condition (Eq (60))and the Dissipation condition (Eq (62)) Similarly we considered LES with explicit filtering andagain used the Navier-Stokes equations to derive more conditions for the models in this case wedecided to just take one condition the Dissipation condition (Eqs (67) and (68)) as the others leadto difficulty in their evaluation By lsquorunningrsquo the exact solutions to the Navier-Stokes equationsthrough each condition for the subfilter-scale models we found that the Vortex-Stretching-basededdy-viscosity model in Eq (54) was most endorsed by the tests in LES without explicit filteringWe found that four out of the five subfilter-scale models namely the Vreman QR Gradient andVortex-Stretching-based models were equally endorsed by the lsquostrongrsquo tests in LES with explicitfiltering with the Smagorinsky model being the least endorsed The results we found for the lsquoweakrsquocondition were completely opposite to those found in both the lsquostrongrsquo condition and in LES withoutexplicit filtering This led us to conclude that this test was not one in which we should put our trust
37
We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses
In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes
This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold
38
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
441 Vremanrsquos Velocity Gradient Flow Classes
We now put our exact solutions to the Navier-Stokes equations into one of two classes Either Q0minus6or Q7minus9 where each class is defined intuitively as follows
Q0minus6 = Q0 cupQ1 cup middot middot middot cup Q6 (76)
Q7minus9 = Q7 cupQ8 cupQ9 (77)
where Qn is defined in Section 43 By looking at the gradient of each exact solution we can see thatQ7minus9 contains the six solutions Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as afunction of time) Sol 5 (2D Taylor) Sol 14 (Radially Dependent) and Sol 15 (Rankine Vortex)and that Q0minus6 contains the remaining nine We now check the models for LES without explicitfiltering by considering endorsements within each class Tables 6 and 7 are analogues of Table 3 andillustrate the results for each flow class The analogues of Tables 1 and 2 are Tables 22 and 23 andare in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 6 6 6
lsquoDissipationrsquoEndorsements
2 5 6 6 6
Percentage Endorsed 417 833 100 100 100
Table 6 Each entry in the top two rows represents the number of solutions in the class Q7minus9 thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Q0minus6 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 2 3 3
lsquoDissipationrsquoEndorsements
0 0 1 0 3
Percentage Endorsed 56 56 167 167 333
Table 7 Each entry in the top two rows represents the number of solutions in the class Q1minus6 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking results to 1 decimal place if necessary
All of the flows in the class Q7minus9 endorse the QR Gradient and Vortex-Stretching-based subfilter-scale models (see Table 6) We can hence see that this class fully endorses these three models equallyand does not endorse the Smagorinsky model The Vortex-Stretching eddy viscosity model has themost endorsements from any of the flows in the class Q0minus6 and the Smagorinksy and Vreman modelshave the fewest endorsements (see Table 7) We can hence say that this class mostly endorses theVortex-Stretching-based eddy-viscosity model note however that the percentage endorsement is very
30
low
Now let us consider LES with explicit filtering Tables 8 and 9 are analogues of Table 5 and illustratethe results for each flow class The analogue of Table 4 is Table 24 and is in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 1 1 0 0 0
Table 8 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Q0minus6 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 9 9 9 6
Table 9 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We have grouped Vremanrsquos classes into Q7minus9 and Q0minus6 this is equivalent to saying that we havegrouped Vremanrsquos classes into those with two or fewer non-zero velocity gradient elements and thosewith three or more non-zero velocity gradient elements respectively Therefore the solutions in Q7minus9are exactly those with a lsquo0rsquo in the lsquoExactrsquo column of Table 4 and Sol 5 (2D Taylor) ExcludingSol 5 (2D Taylor) these are the same as those five that endorse the Vreman QR Gradient andVortex-Stretching-based models in Table 5 We can see in Table 8 that the results for the lsquostrongrsquoendorsements are exactly the same as in Table 5 This is because all of the flows with zero subfilterdissipation are contained in the class Q7minus9 There are hence no lsquostrongrsquo endorsements in Table 9The majority of the lsquoweakrsquo endorsements come from Q0minus6 as this class contains all but one of thesolutions with non-zero true subfilter dissipation (given in Eq (67))
442 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
We now separate our exact solutions into classes again using Vremanrsquos flow classes This timehowever we define V0 to be the class of flows Vreman claims will always have zero true subfilterdissipation namely those classes mentioned in Appendix B We define Vlowast to be all the remaining
31
classes Then
V0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow)
Sol 3 (Velocity as a function of time) Sol 14 (Radially Dependent)(78)
Vlowast = Sol 4 (Axisymmetric Flow) Sol 5 (2D Taylor) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 13 (Sphere Translation) Sol 15 (Rankine Vortex)
(79)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 10 and 11 are analogues of Table 3 and illustrate the results for each flow class Theanalogues of Tables 1 and 2 are Tables 25 and 26 and are in Appendix C2
Endorsements for models using solutions in V0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 4 4 4 4
lsquoDissipationrsquoEndorsements
1 4 4 4 4
Percentage Endorsed 375 100 100 100 100
Table 10 Each entry in the top two rows represents the number of solutions in the class V0 thatendorse each model There are 4 solutions in this class so the percentages are worked out by dividingthe total of each column by 8 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Vlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 2 4 5 5
lsquoDissipationrsquoEndorsements
1 1 3 2 5
Percentage Endorsed 136 136 318 318 455
Table 11 Each entry in the top two rows represents the number of solutions in the class Vlowast thatendorse each model There are 11 solutions in this class so the percentages are worked out bydividing the total of each column by 22 and taking results to 1 decimal place if necessary
All of the flows in the class V0 endorse the Vreman QR Gradient and Vortex-Stretching-basedsubfilter-scale models (see Table 6) This class completely endorses these four models equally anddoes not endorse the Smagorinsky model The Vortex-Stretching-based eddy-viscosity model hasthe most endorsements from the class Vlowast and the Smagorinksy and Vreman models again have thefewest endorsements (see Table 7) As in Section 441 we say that this class mostly endorses the
32
Vortex-Stretching-based eddy-viscosity model but once again the percentage endorsement is ratherlow
Now let us consider LES with explicit filtering Tables 12 and 13 are analogues of Table 5 andillustrate the results for each flow class The analogue of Table 4 is Table 27 and is in Appendix C2
Endorsements for models using solutions in V0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 4 4 4 4
lsquoWeakrsquo Endorsements 0 0 0 0 0
Table 12 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Vlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 1 1 1 1
lsquoWeakrsquo Endorsements 10 10 9 6
Table 13 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We can see in Table 12 that the results for the lsquostrongrsquo endorsements are similar to those in Table 5excluding Sol 15 (Rankine Vortex) The four solutions in V0 with zero true subfilter dissipationnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)and Sol 14 (Radially Dependent) are hence the four that endorse the Vreman QR Gradientand Vortex-Stretching-based models in Table 12 Sol 15 (Rankine Vortex) is in the class Vlowast andhence there is one lsquostrongrsquo endorsement in Table 13 The class Vlowast contains all of the flows with anon-zero true subfilter dissipation (given in Eq (67)) and hence Table 13 contains all of the lsquoweakrsquoendorsements
443 Classes using the Third Principal Invariant
Sorting the classes using the invariants is a non-trivial task in itself There are a number of waysin which to do this using both the principal invariants in Eq (40) and the combined invariants inEq (43)
We start by looking at flows with third principal invariant RG equal to zero We again define twoclasses R0 and Rlowast where R0 consists of all flows with RG = 0 and Rlowast consists of all flows withRG 6= 0 Then using the information in Appendix A we see that there are nine flows in R0 That
33
is
R0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 8 (Vortex in Stagnation Flow)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)
(80)
Rlowast = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 13 (Sphere Translation)(81)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 14 and 15 are analogues of Table 3 and they illustrate the results for LES withoutexplicit filtering for each flow class The analogues of Tables 1 and 2 are Tables 28 and 29 and arein Appendix C3
Endorsements for models using solutions in R0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 7 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 278 556 778 722 778
Table 14 Each entry in the top two rows represents the number of solutions in the class R0 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking the result to one decimal place if necessary
Endorsements for models using solutions in Rlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 2 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 83 83 83 167 333
Table 15 Each entry in the top two rows represents the number of solutions in the class Rlowast thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking the result to one decimal place if necessary
The class R0 endorses both the Vortex-Stretching-based model and the QR model the most andSmagorinsky the least in the case for LES without explicit filtering In the Rlowast class the Vortex-Stretching-based model is again the most endorsed followed by the Gradient model however thepercentages are again small
34
Now let us consider LES with explicit filtering Tables 16 and 17 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 30 and is in Appendix C3
Endorsements for models using solutions in R0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 4 4 3 2
Table 16 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Rlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 6 6 6 4
Table 17 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 we can see in Table 16 that the results for the lsquostrongrsquo endorsements are exactly thesame as in Table 5 This is because all of the flows with zero subfilter dissipation are contained in theclass R0 As in Table 9 there are no lsquostrongrsquo endorsements in Table 17 The lsquoweakrsquo endorsementshowever are split more evenly between the two classes now than in Vremanrsquos classes in Sections 441and 442 We find that the Smagorinsky and Vreman models still seem to be the most endorsed inboth classes
444 Two-component Flow Classes
Another way in which the flows can be grouped into classes is by determining whether a flow is atwo-component flow or not We define a two-component flow to have the property that the combinedprincipal invariants of a flow in Eq (43) satisfy
I3 = I4 = I5 minus1
2I1I2 = 0 (82)
We are also interested in considering flows with zero gradient model subgrid dissipation These flowshave the property that
I3 minus I4 = 0 (83)
What we find though is that for all the solutions in Section 22 if a flow is a two-component flowthen it also satisfies I3 minus I4 = 0 It may be interesting to note that although I3 minus I4 6= 0 for manysolutions there are a number with I3 equal to a multiple of I4 Although not considered in thisreport this may also be a class of interest We find that the two-component flows are
C2 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)(84)
35
and we call the remaining flows three-component flows They are
C3 = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex) Sol 11 (Pseudo-Motion Shots)
Sol 13 (Sphere Translation)
(85)
Note that Sol 8 (Vortex in Stagnation Flow) is defined to be a planar flow this does not mean that itis a two-component flow in this sense however because its two non-zero velocity components dependon three coordinates We now check the models for LES without explicit filtering by consideringendorsements within each class Tables 18 and 19 are analogues of Table 3 and they illustrate theresults for LES without explicit filtering for each flow class The analogues of Tables 1 and 2 areTables 31 and 32 and are in Appendix C4
Endorsements for models using solutions in C2 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 6 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 357 714 100 857 100
Table 18 Each entry in the top two rows represents the number of solutions in the class C2 thatendorse each model There are 7 solutions in this class so the percentages are worked out by dividingthe total of each column by 14 and taking the result to one decimal place if necessary
Endorsements for models using solutions in C3 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 3 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 63 63 63 188 250
Table 19 Each entry in the top two rows represents the number of solutions in the class C3 thatendorse each model There are 8 solutions in this class so the percentages are worked out by dividingthe total of each column by 16 and taking the result to one decimal place if necessary
The class C2 equally endorses the QR and Vortex-Stretching-based models and once again theSmagorinsky model is the least endorsed As with the previous classes the class C3 has such lowpercentages that it does not seem to endorse any models We start to see a trend appearing here inthat the flow classes that contain the more complicated flows namely Q0minus6 (Eq (76)) Vlowast (Eq (79))Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages The classes that tendto contain the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78)) R0 (Eq (80)) and C2 (Eq (84))all seem to favour the Vortex-Stretching-based model or the QR model
36
Now let us consider LES with explicit filtering Tables 20 and 21 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 33 and is in Appendix C4
Endorsements for models using solutions in C2 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 2 2 0 1 0
Table 20 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in C3 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 8 8 8 6
Table 21 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 and Table 16 we can see in Table 20 that the results for the lsquostrongrsquo endorsements areexactly the same as in Table 5 This is again because all of the flows with zero subfilter dissipationare contained in the class C2 There are two flows with non-zero true subfilter dissipation (givenin Eq (67)) in C2 and this shows in the lsquoweakrsquo endorsements in Table 20 There are no lsquostrongrsquoendorsements in Table 21 as all the zero subfilter dissipation flows are in C2
5 Conclusions
In this report we studied incompressible turbulent flows We found a number of exact solutions to theNavier-Stokes equations each representing a relatively simplistic fluid flow Large-eddy simulation(LES) is used to model turbulent flows by smoothing over the complex small-scale motion Weintroduced a number of subfilter-scale models to approximate this small-scale motion and testedthem using the found exact solutions to the Navier-Stokes equations We first considered LESwithout explicit filtering and used the Navier-Stokes equations to derive conditions for the subfilter-scale models namely the Velocity Field condition (Eq (58)) the Kinetic Energy condition (Eq (60))and the Dissipation condition (Eq (62)) Similarly we considered LES with explicit filtering andagain used the Navier-Stokes equations to derive more conditions for the models in this case wedecided to just take one condition the Dissipation condition (Eqs (67) and (68)) as the others leadto difficulty in their evaluation By lsquorunningrsquo the exact solutions to the Navier-Stokes equationsthrough each condition for the subfilter-scale models we found that the Vortex-Stretching-basededdy-viscosity model in Eq (54) was most endorsed by the tests in LES without explicit filteringWe found that four out of the five subfilter-scale models namely the Vreman QR Gradient andVortex-Stretching-based models were equally endorsed by the lsquostrongrsquo tests in LES with explicitfiltering with the Smagorinsky model being the least endorsed The results we found for the lsquoweakrsquocondition were completely opposite to those found in both the lsquostrongrsquo condition and in LES withoutexplicit filtering This led us to conclude that this test was not one in which we should put our trust
37
We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses
In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes
This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold
38
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
low
Now let us consider LES with explicit filtering Tables 8 and 9 are analogues of Table 5 and illustratethe results for each flow class The analogue of Table 4 is Table 24 and is in Appendix C1
Endorsements for models using solutions in Q7minus9 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 1 1 0 0 0
Table 8 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Q0minus6 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 9 9 9 6
Table 9 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We have grouped Vremanrsquos classes into Q7minus9 and Q0minus6 this is equivalent to saying that we havegrouped Vremanrsquos classes into those with two or fewer non-zero velocity gradient elements and thosewith three or more non-zero velocity gradient elements respectively Therefore the solutions in Q7minus9are exactly those with a lsquo0rsquo in the lsquoExactrsquo column of Table 4 and Sol 5 (2D Taylor) ExcludingSol 5 (2D Taylor) these are the same as those five that endorse the Vreman QR Gradient andVortex-Stretching-based models in Table 5 We can see in Table 8 that the results for the lsquostrongrsquoendorsements are exactly the same as in Table 5 This is because all of the flows with zero subfilterdissipation are contained in the class Q7minus9 There are hence no lsquostrongrsquo endorsements in Table 9The majority of the lsquoweakrsquo endorsements come from Q0minus6 as this class contains all but one of thesolutions with non-zero true subfilter dissipation (given in Eq (67))
442 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
We now separate our exact solutions into classes again using Vremanrsquos flow classes This timehowever we define V0 to be the class of flows Vreman claims will always have zero true subfilterdissipation namely those classes mentioned in Appendix B We define Vlowast to be all the remaining
31
classes Then
V0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow)
Sol 3 (Velocity as a function of time) Sol 14 (Radially Dependent)(78)
Vlowast = Sol 4 (Axisymmetric Flow) Sol 5 (2D Taylor) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 13 (Sphere Translation) Sol 15 (Rankine Vortex)
(79)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 10 and 11 are analogues of Table 3 and illustrate the results for each flow class Theanalogues of Tables 1 and 2 are Tables 25 and 26 and are in Appendix C2
Endorsements for models using solutions in V0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 4 4 4 4
lsquoDissipationrsquoEndorsements
1 4 4 4 4
Percentage Endorsed 375 100 100 100 100
Table 10 Each entry in the top two rows represents the number of solutions in the class V0 thatendorse each model There are 4 solutions in this class so the percentages are worked out by dividingthe total of each column by 8 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Vlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 2 4 5 5
lsquoDissipationrsquoEndorsements
1 1 3 2 5
Percentage Endorsed 136 136 318 318 455
Table 11 Each entry in the top two rows represents the number of solutions in the class Vlowast thatendorse each model There are 11 solutions in this class so the percentages are worked out bydividing the total of each column by 22 and taking results to 1 decimal place if necessary
All of the flows in the class V0 endorse the Vreman QR Gradient and Vortex-Stretching-basedsubfilter-scale models (see Table 6) This class completely endorses these four models equally anddoes not endorse the Smagorinsky model The Vortex-Stretching-based eddy-viscosity model hasthe most endorsements from the class Vlowast and the Smagorinksy and Vreman models again have thefewest endorsements (see Table 7) As in Section 441 we say that this class mostly endorses the
32
Vortex-Stretching-based eddy-viscosity model but once again the percentage endorsement is ratherlow
Now let us consider LES with explicit filtering Tables 12 and 13 are analogues of Table 5 andillustrate the results for each flow class The analogue of Table 4 is Table 27 and is in Appendix C2
Endorsements for models using solutions in V0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 4 4 4 4
lsquoWeakrsquo Endorsements 0 0 0 0 0
Table 12 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Vlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 1 1 1 1
lsquoWeakrsquo Endorsements 10 10 9 6
Table 13 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We can see in Table 12 that the results for the lsquostrongrsquo endorsements are similar to those in Table 5excluding Sol 15 (Rankine Vortex) The four solutions in V0 with zero true subfilter dissipationnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)and Sol 14 (Radially Dependent) are hence the four that endorse the Vreman QR Gradientand Vortex-Stretching-based models in Table 12 Sol 15 (Rankine Vortex) is in the class Vlowast andhence there is one lsquostrongrsquo endorsement in Table 13 The class Vlowast contains all of the flows with anon-zero true subfilter dissipation (given in Eq (67)) and hence Table 13 contains all of the lsquoweakrsquoendorsements
443 Classes using the Third Principal Invariant
Sorting the classes using the invariants is a non-trivial task in itself There are a number of waysin which to do this using both the principal invariants in Eq (40) and the combined invariants inEq (43)
We start by looking at flows with third principal invariant RG equal to zero We again define twoclasses R0 and Rlowast where R0 consists of all flows with RG = 0 and Rlowast consists of all flows withRG 6= 0 Then using the information in Appendix A we see that there are nine flows in R0 That
33
is
R0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 8 (Vortex in Stagnation Flow)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)
(80)
Rlowast = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 13 (Sphere Translation)(81)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 14 and 15 are analogues of Table 3 and they illustrate the results for LES withoutexplicit filtering for each flow class The analogues of Tables 1 and 2 are Tables 28 and 29 and arein Appendix C3
Endorsements for models using solutions in R0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 7 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 278 556 778 722 778
Table 14 Each entry in the top two rows represents the number of solutions in the class R0 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking the result to one decimal place if necessary
Endorsements for models using solutions in Rlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 2 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 83 83 83 167 333
Table 15 Each entry in the top two rows represents the number of solutions in the class Rlowast thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking the result to one decimal place if necessary
The class R0 endorses both the Vortex-Stretching-based model and the QR model the most andSmagorinsky the least in the case for LES without explicit filtering In the Rlowast class the Vortex-Stretching-based model is again the most endorsed followed by the Gradient model however thepercentages are again small
34
Now let us consider LES with explicit filtering Tables 16 and 17 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 30 and is in Appendix C3
Endorsements for models using solutions in R0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 4 4 3 2
Table 16 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Rlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 6 6 6 4
Table 17 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 we can see in Table 16 that the results for the lsquostrongrsquo endorsements are exactly thesame as in Table 5 This is because all of the flows with zero subfilter dissipation are contained in theclass R0 As in Table 9 there are no lsquostrongrsquo endorsements in Table 17 The lsquoweakrsquo endorsementshowever are split more evenly between the two classes now than in Vremanrsquos classes in Sections 441and 442 We find that the Smagorinsky and Vreman models still seem to be the most endorsed inboth classes
444 Two-component Flow Classes
Another way in which the flows can be grouped into classes is by determining whether a flow is atwo-component flow or not We define a two-component flow to have the property that the combinedprincipal invariants of a flow in Eq (43) satisfy
I3 = I4 = I5 minus1
2I1I2 = 0 (82)
We are also interested in considering flows with zero gradient model subgrid dissipation These flowshave the property that
I3 minus I4 = 0 (83)
What we find though is that for all the solutions in Section 22 if a flow is a two-component flowthen it also satisfies I3 minus I4 = 0 It may be interesting to note that although I3 minus I4 6= 0 for manysolutions there are a number with I3 equal to a multiple of I4 Although not considered in thisreport this may also be a class of interest We find that the two-component flows are
C2 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)(84)
35
and we call the remaining flows three-component flows They are
C3 = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex) Sol 11 (Pseudo-Motion Shots)
Sol 13 (Sphere Translation)
(85)
Note that Sol 8 (Vortex in Stagnation Flow) is defined to be a planar flow this does not mean that itis a two-component flow in this sense however because its two non-zero velocity components dependon three coordinates We now check the models for LES without explicit filtering by consideringendorsements within each class Tables 18 and 19 are analogues of Table 3 and they illustrate theresults for LES without explicit filtering for each flow class The analogues of Tables 1 and 2 areTables 31 and 32 and are in Appendix C4
Endorsements for models using solutions in C2 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 6 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 357 714 100 857 100
Table 18 Each entry in the top two rows represents the number of solutions in the class C2 thatendorse each model There are 7 solutions in this class so the percentages are worked out by dividingthe total of each column by 14 and taking the result to one decimal place if necessary
Endorsements for models using solutions in C3 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 3 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 63 63 63 188 250
Table 19 Each entry in the top two rows represents the number of solutions in the class C3 thatendorse each model There are 8 solutions in this class so the percentages are worked out by dividingthe total of each column by 16 and taking the result to one decimal place if necessary
The class C2 equally endorses the QR and Vortex-Stretching-based models and once again theSmagorinsky model is the least endorsed As with the previous classes the class C3 has such lowpercentages that it does not seem to endorse any models We start to see a trend appearing here inthat the flow classes that contain the more complicated flows namely Q0minus6 (Eq (76)) Vlowast (Eq (79))Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages The classes that tendto contain the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78)) R0 (Eq (80)) and C2 (Eq (84))all seem to favour the Vortex-Stretching-based model or the QR model
36
Now let us consider LES with explicit filtering Tables 20 and 21 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 33 and is in Appendix C4
Endorsements for models using solutions in C2 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 2 2 0 1 0
Table 20 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in C3 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 8 8 8 6
Table 21 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 and Table 16 we can see in Table 20 that the results for the lsquostrongrsquo endorsements areexactly the same as in Table 5 This is again because all of the flows with zero subfilter dissipationare contained in the class C2 There are two flows with non-zero true subfilter dissipation (givenin Eq (67)) in C2 and this shows in the lsquoweakrsquo endorsements in Table 20 There are no lsquostrongrsquoendorsements in Table 21 as all the zero subfilter dissipation flows are in C2
5 Conclusions
In this report we studied incompressible turbulent flows We found a number of exact solutions to theNavier-Stokes equations each representing a relatively simplistic fluid flow Large-eddy simulation(LES) is used to model turbulent flows by smoothing over the complex small-scale motion Weintroduced a number of subfilter-scale models to approximate this small-scale motion and testedthem using the found exact solutions to the Navier-Stokes equations We first considered LESwithout explicit filtering and used the Navier-Stokes equations to derive conditions for the subfilter-scale models namely the Velocity Field condition (Eq (58)) the Kinetic Energy condition (Eq (60))and the Dissipation condition (Eq (62)) Similarly we considered LES with explicit filtering andagain used the Navier-Stokes equations to derive more conditions for the models in this case wedecided to just take one condition the Dissipation condition (Eqs (67) and (68)) as the others leadto difficulty in their evaluation By lsquorunningrsquo the exact solutions to the Navier-Stokes equationsthrough each condition for the subfilter-scale models we found that the Vortex-Stretching-basededdy-viscosity model in Eq (54) was most endorsed by the tests in LES without explicit filteringWe found that four out of the five subfilter-scale models namely the Vreman QR Gradient andVortex-Stretching-based models were equally endorsed by the lsquostrongrsquo tests in LES with explicitfiltering with the Smagorinsky model being the least endorsed The results we found for the lsquoweakrsquocondition were completely opposite to those found in both the lsquostrongrsquo condition and in LES withoutexplicit filtering This led us to conclude that this test was not one in which we should put our trust
37
We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses
In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes
This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold
38
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
classes Then
V0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow)
Sol 3 (Velocity as a function of time) Sol 14 (Radially Dependent)(78)
Vlowast = Sol 4 (Axisymmetric Flow) Sol 5 (2D Taylor) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 13 (Sphere Translation) Sol 15 (Rankine Vortex)
(79)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 10 and 11 are analogues of Table 3 and illustrate the results for each flow class Theanalogues of Tables 1 and 2 are Tables 25 and 26 and are in Appendix C2
Endorsements for models using solutions in V0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 4 4 4 4
lsquoDissipationrsquoEndorsements
1 4 4 4 4
Percentage Endorsed 375 100 100 100 100
Table 10 Each entry in the top two rows represents the number of solutions in the class V0 thatendorse each model There are 4 solutions in this class so the percentages are worked out by dividingthe total of each column by 8 and taking results to 1 decimal place if necessary
Endorsements for models using solutions in Vlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
2 2 4 5 5
lsquoDissipationrsquoEndorsements
1 1 3 2 5
Percentage Endorsed 136 136 318 318 455
Table 11 Each entry in the top two rows represents the number of solutions in the class Vlowast thatendorse each model There are 11 solutions in this class so the percentages are worked out bydividing the total of each column by 22 and taking results to 1 decimal place if necessary
All of the flows in the class V0 endorse the Vreman QR Gradient and Vortex-Stretching-basedsubfilter-scale models (see Table 6) This class completely endorses these four models equally anddoes not endorse the Smagorinsky model The Vortex-Stretching-based eddy-viscosity model hasthe most endorsements from the class Vlowast and the Smagorinksy and Vreman models again have thefewest endorsements (see Table 7) As in Section 441 we say that this class mostly endorses the
32
Vortex-Stretching-based eddy-viscosity model but once again the percentage endorsement is ratherlow
Now let us consider LES with explicit filtering Tables 12 and 13 are analogues of Table 5 andillustrate the results for each flow class The analogue of Table 4 is Table 27 and is in Appendix C2
Endorsements for models using solutions in V0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 4 4 4 4
lsquoWeakrsquo Endorsements 0 0 0 0 0
Table 12 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Vlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 1 1 1 1
lsquoWeakrsquo Endorsements 10 10 9 6
Table 13 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We can see in Table 12 that the results for the lsquostrongrsquo endorsements are similar to those in Table 5excluding Sol 15 (Rankine Vortex) The four solutions in V0 with zero true subfilter dissipationnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)and Sol 14 (Radially Dependent) are hence the four that endorse the Vreman QR Gradientand Vortex-Stretching-based models in Table 12 Sol 15 (Rankine Vortex) is in the class Vlowast andhence there is one lsquostrongrsquo endorsement in Table 13 The class Vlowast contains all of the flows with anon-zero true subfilter dissipation (given in Eq (67)) and hence Table 13 contains all of the lsquoweakrsquoendorsements
443 Classes using the Third Principal Invariant
Sorting the classes using the invariants is a non-trivial task in itself There are a number of waysin which to do this using both the principal invariants in Eq (40) and the combined invariants inEq (43)
We start by looking at flows with third principal invariant RG equal to zero We again define twoclasses R0 and Rlowast where R0 consists of all flows with RG = 0 and Rlowast consists of all flows withRG 6= 0 Then using the information in Appendix A we see that there are nine flows in R0 That
33
is
R0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 8 (Vortex in Stagnation Flow)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)
(80)
Rlowast = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 13 (Sphere Translation)(81)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 14 and 15 are analogues of Table 3 and they illustrate the results for LES withoutexplicit filtering for each flow class The analogues of Tables 1 and 2 are Tables 28 and 29 and arein Appendix C3
Endorsements for models using solutions in R0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 7 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 278 556 778 722 778
Table 14 Each entry in the top two rows represents the number of solutions in the class R0 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking the result to one decimal place if necessary
Endorsements for models using solutions in Rlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 2 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 83 83 83 167 333
Table 15 Each entry in the top two rows represents the number of solutions in the class Rlowast thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking the result to one decimal place if necessary
The class R0 endorses both the Vortex-Stretching-based model and the QR model the most andSmagorinsky the least in the case for LES without explicit filtering In the Rlowast class the Vortex-Stretching-based model is again the most endorsed followed by the Gradient model however thepercentages are again small
34
Now let us consider LES with explicit filtering Tables 16 and 17 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 30 and is in Appendix C3
Endorsements for models using solutions in R0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 4 4 3 2
Table 16 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Rlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 6 6 6 4
Table 17 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 we can see in Table 16 that the results for the lsquostrongrsquo endorsements are exactly thesame as in Table 5 This is because all of the flows with zero subfilter dissipation are contained in theclass R0 As in Table 9 there are no lsquostrongrsquo endorsements in Table 17 The lsquoweakrsquo endorsementshowever are split more evenly between the two classes now than in Vremanrsquos classes in Sections 441and 442 We find that the Smagorinsky and Vreman models still seem to be the most endorsed inboth classes
444 Two-component Flow Classes
Another way in which the flows can be grouped into classes is by determining whether a flow is atwo-component flow or not We define a two-component flow to have the property that the combinedprincipal invariants of a flow in Eq (43) satisfy
I3 = I4 = I5 minus1
2I1I2 = 0 (82)
We are also interested in considering flows with zero gradient model subgrid dissipation These flowshave the property that
I3 minus I4 = 0 (83)
What we find though is that for all the solutions in Section 22 if a flow is a two-component flowthen it also satisfies I3 minus I4 = 0 It may be interesting to note that although I3 minus I4 6= 0 for manysolutions there are a number with I3 equal to a multiple of I4 Although not considered in thisreport this may also be a class of interest We find that the two-component flows are
C2 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)(84)
35
and we call the remaining flows three-component flows They are
C3 = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex) Sol 11 (Pseudo-Motion Shots)
Sol 13 (Sphere Translation)
(85)
Note that Sol 8 (Vortex in Stagnation Flow) is defined to be a planar flow this does not mean that itis a two-component flow in this sense however because its two non-zero velocity components dependon three coordinates We now check the models for LES without explicit filtering by consideringendorsements within each class Tables 18 and 19 are analogues of Table 3 and they illustrate theresults for LES without explicit filtering for each flow class The analogues of Tables 1 and 2 areTables 31 and 32 and are in Appendix C4
Endorsements for models using solutions in C2 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 6 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 357 714 100 857 100
Table 18 Each entry in the top two rows represents the number of solutions in the class C2 thatendorse each model There are 7 solutions in this class so the percentages are worked out by dividingthe total of each column by 14 and taking the result to one decimal place if necessary
Endorsements for models using solutions in C3 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 3 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 63 63 63 188 250
Table 19 Each entry in the top two rows represents the number of solutions in the class C3 thatendorse each model There are 8 solutions in this class so the percentages are worked out by dividingthe total of each column by 16 and taking the result to one decimal place if necessary
The class C2 equally endorses the QR and Vortex-Stretching-based models and once again theSmagorinsky model is the least endorsed As with the previous classes the class C3 has such lowpercentages that it does not seem to endorse any models We start to see a trend appearing here inthat the flow classes that contain the more complicated flows namely Q0minus6 (Eq (76)) Vlowast (Eq (79))Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages The classes that tendto contain the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78)) R0 (Eq (80)) and C2 (Eq (84))all seem to favour the Vortex-Stretching-based model or the QR model
36
Now let us consider LES with explicit filtering Tables 20 and 21 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 33 and is in Appendix C4
Endorsements for models using solutions in C2 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 2 2 0 1 0
Table 20 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in C3 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 8 8 8 6
Table 21 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 and Table 16 we can see in Table 20 that the results for the lsquostrongrsquo endorsements areexactly the same as in Table 5 This is again because all of the flows with zero subfilter dissipationare contained in the class C2 There are two flows with non-zero true subfilter dissipation (givenin Eq (67)) in C2 and this shows in the lsquoweakrsquo endorsements in Table 20 There are no lsquostrongrsquoendorsements in Table 21 as all the zero subfilter dissipation flows are in C2
5 Conclusions
In this report we studied incompressible turbulent flows We found a number of exact solutions to theNavier-Stokes equations each representing a relatively simplistic fluid flow Large-eddy simulation(LES) is used to model turbulent flows by smoothing over the complex small-scale motion Weintroduced a number of subfilter-scale models to approximate this small-scale motion and testedthem using the found exact solutions to the Navier-Stokes equations We first considered LESwithout explicit filtering and used the Navier-Stokes equations to derive conditions for the subfilter-scale models namely the Velocity Field condition (Eq (58)) the Kinetic Energy condition (Eq (60))and the Dissipation condition (Eq (62)) Similarly we considered LES with explicit filtering andagain used the Navier-Stokes equations to derive more conditions for the models in this case wedecided to just take one condition the Dissipation condition (Eqs (67) and (68)) as the others leadto difficulty in their evaluation By lsquorunningrsquo the exact solutions to the Navier-Stokes equationsthrough each condition for the subfilter-scale models we found that the Vortex-Stretching-basededdy-viscosity model in Eq (54) was most endorsed by the tests in LES without explicit filteringWe found that four out of the five subfilter-scale models namely the Vreman QR Gradient andVortex-Stretching-based models were equally endorsed by the lsquostrongrsquo tests in LES with explicitfiltering with the Smagorinsky model being the least endorsed The results we found for the lsquoweakrsquocondition were completely opposite to those found in both the lsquostrongrsquo condition and in LES withoutexplicit filtering This led us to conclude that this test was not one in which we should put our trust
37
We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses
In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes
This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold
38
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
Vortex-Stretching-based eddy-viscosity model but once again the percentage endorsement is ratherlow
Now let us consider LES with explicit filtering Tables 12 and 13 are analogues of Table 5 andillustrate the results for each flow class The analogue of Table 4 is Table 27 and is in Appendix C2
Endorsements for models using solutions in V0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 4 4 4 4
lsquoWeakrsquo Endorsements 0 0 0 0 0
Table 12 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Vlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 1 1 1 1 1
lsquoWeakrsquo Endorsements 10 10 9 6
Table 13 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
We can see in Table 12 that the results for the lsquostrongrsquo endorsements are similar to those in Table 5excluding Sol 15 (Rankine Vortex) The four solutions in V0 with zero true subfilter dissipationnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)and Sol 14 (Radially Dependent) are hence the four that endorse the Vreman QR Gradientand Vortex-Stretching-based models in Table 12 Sol 15 (Rankine Vortex) is in the class Vlowast andhence there is one lsquostrongrsquo endorsement in Table 13 The class Vlowast contains all of the flows with anon-zero true subfilter dissipation (given in Eq (67)) and hence Table 13 contains all of the lsquoweakrsquoendorsements
443 Classes using the Third Principal Invariant
Sorting the classes using the invariants is a non-trivial task in itself There are a number of waysin which to do this using both the principal invariants in Eq (40) and the combined invariants inEq (43)
We start by looking at flows with third principal invariant RG equal to zero We again define twoclasses R0 and Rlowast where R0 consists of all flows with RG = 0 and Rlowast consists of all flows withRG 6= 0 Then using the information in Appendix A we see that there are nine flows in R0 That
33
is
R0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 8 (Vortex in Stagnation Flow)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)
(80)
Rlowast = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 13 (Sphere Translation)(81)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 14 and 15 are analogues of Table 3 and they illustrate the results for LES withoutexplicit filtering for each flow class The analogues of Tables 1 and 2 are Tables 28 and 29 and arein Appendix C3
Endorsements for models using solutions in R0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 7 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 278 556 778 722 778
Table 14 Each entry in the top two rows represents the number of solutions in the class R0 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking the result to one decimal place if necessary
Endorsements for models using solutions in Rlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 2 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 83 83 83 167 333
Table 15 Each entry in the top two rows represents the number of solutions in the class Rlowast thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking the result to one decimal place if necessary
The class R0 endorses both the Vortex-Stretching-based model and the QR model the most andSmagorinsky the least in the case for LES without explicit filtering In the Rlowast class the Vortex-Stretching-based model is again the most endorsed followed by the Gradient model however thepercentages are again small
34
Now let us consider LES with explicit filtering Tables 16 and 17 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 30 and is in Appendix C3
Endorsements for models using solutions in R0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 4 4 3 2
Table 16 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Rlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 6 6 6 4
Table 17 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 we can see in Table 16 that the results for the lsquostrongrsquo endorsements are exactly thesame as in Table 5 This is because all of the flows with zero subfilter dissipation are contained in theclass R0 As in Table 9 there are no lsquostrongrsquo endorsements in Table 17 The lsquoweakrsquo endorsementshowever are split more evenly between the two classes now than in Vremanrsquos classes in Sections 441and 442 We find that the Smagorinsky and Vreman models still seem to be the most endorsed inboth classes
444 Two-component Flow Classes
Another way in which the flows can be grouped into classes is by determining whether a flow is atwo-component flow or not We define a two-component flow to have the property that the combinedprincipal invariants of a flow in Eq (43) satisfy
I3 = I4 = I5 minus1
2I1I2 = 0 (82)
We are also interested in considering flows with zero gradient model subgrid dissipation These flowshave the property that
I3 minus I4 = 0 (83)
What we find though is that for all the solutions in Section 22 if a flow is a two-component flowthen it also satisfies I3 minus I4 = 0 It may be interesting to note that although I3 minus I4 6= 0 for manysolutions there are a number with I3 equal to a multiple of I4 Although not considered in thisreport this may also be a class of interest We find that the two-component flows are
C2 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)(84)
35
and we call the remaining flows three-component flows They are
C3 = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex) Sol 11 (Pseudo-Motion Shots)
Sol 13 (Sphere Translation)
(85)
Note that Sol 8 (Vortex in Stagnation Flow) is defined to be a planar flow this does not mean that itis a two-component flow in this sense however because its two non-zero velocity components dependon three coordinates We now check the models for LES without explicit filtering by consideringendorsements within each class Tables 18 and 19 are analogues of Table 3 and they illustrate theresults for LES without explicit filtering for each flow class The analogues of Tables 1 and 2 areTables 31 and 32 and are in Appendix C4
Endorsements for models using solutions in C2 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 6 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 357 714 100 857 100
Table 18 Each entry in the top two rows represents the number of solutions in the class C2 thatendorse each model There are 7 solutions in this class so the percentages are worked out by dividingthe total of each column by 14 and taking the result to one decimal place if necessary
Endorsements for models using solutions in C3 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 3 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 63 63 63 188 250
Table 19 Each entry in the top two rows represents the number of solutions in the class C3 thatendorse each model There are 8 solutions in this class so the percentages are worked out by dividingthe total of each column by 16 and taking the result to one decimal place if necessary
The class C2 equally endorses the QR and Vortex-Stretching-based models and once again theSmagorinsky model is the least endorsed As with the previous classes the class C3 has such lowpercentages that it does not seem to endorse any models We start to see a trend appearing here inthat the flow classes that contain the more complicated flows namely Q0minus6 (Eq (76)) Vlowast (Eq (79))Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages The classes that tendto contain the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78)) R0 (Eq (80)) and C2 (Eq (84))all seem to favour the Vortex-Stretching-based model or the QR model
36
Now let us consider LES with explicit filtering Tables 20 and 21 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 33 and is in Appendix C4
Endorsements for models using solutions in C2 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 2 2 0 1 0
Table 20 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in C3 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 8 8 8 6
Table 21 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 and Table 16 we can see in Table 20 that the results for the lsquostrongrsquo endorsements areexactly the same as in Table 5 This is again because all of the flows with zero subfilter dissipationare contained in the class C2 There are two flows with non-zero true subfilter dissipation (givenin Eq (67)) in C2 and this shows in the lsquoweakrsquo endorsements in Table 20 There are no lsquostrongrsquoendorsements in Table 21 as all the zero subfilter dissipation flows are in C2
5 Conclusions
In this report we studied incompressible turbulent flows We found a number of exact solutions to theNavier-Stokes equations each representing a relatively simplistic fluid flow Large-eddy simulation(LES) is used to model turbulent flows by smoothing over the complex small-scale motion Weintroduced a number of subfilter-scale models to approximate this small-scale motion and testedthem using the found exact solutions to the Navier-Stokes equations We first considered LESwithout explicit filtering and used the Navier-Stokes equations to derive conditions for the subfilter-scale models namely the Velocity Field condition (Eq (58)) the Kinetic Energy condition (Eq (60))and the Dissipation condition (Eq (62)) Similarly we considered LES with explicit filtering andagain used the Navier-Stokes equations to derive more conditions for the models in this case wedecided to just take one condition the Dissipation condition (Eqs (67) and (68)) as the others leadto difficulty in their evaluation By lsquorunningrsquo the exact solutions to the Navier-Stokes equationsthrough each condition for the subfilter-scale models we found that the Vortex-Stretching-basededdy-viscosity model in Eq (54) was most endorsed by the tests in LES without explicit filteringWe found that four out of the five subfilter-scale models namely the Vreman QR Gradient andVortex-Stretching-based models were equally endorsed by the lsquostrongrsquo tests in LES with explicitfiltering with the Smagorinsky model being the least endorsed The results we found for the lsquoweakrsquocondition were completely opposite to those found in both the lsquostrongrsquo condition and in LES withoutexplicit filtering This led us to conclude that this test was not one in which we should put our trust
37
We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses
In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes
This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold
38
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
is
R0 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 8 (Vortex in Stagnation Flow)
Sol 11 (Pseudo-Motion Shots) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)
(80)
Rlowast = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 9 (Burgers Vortex) Sol 10 (Another Vortex)
Sol 13 (Sphere Translation)(81)
We now check the models for LES without explicit filtering by considering endorsements within eachclass Tables 14 and 15 are analogues of Table 3 and they illustrate the results for LES withoutexplicit filtering for each flow class The analogues of Tables 1 and 2 are Tables 28 and 29 and arein Appendix C3
Endorsements for models using solutions in R0 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 7 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 278 556 778 722 778
Table 14 Each entry in the top two rows represents the number of solutions in the class R0 thatendorse each model There are 9 solutions in this class so the percentages are worked out by dividingthe total of each column by 18 and taking the result to one decimal place if necessary
Endorsements for models using solutions in Rlowast for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 2 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 83 83 83 167 333
Table 15 Each entry in the top two rows represents the number of solutions in the class Rlowast thatendorse each model There are 6 solutions in this class so the percentages are worked out by dividingthe total of each column by 12 and taking the result to one decimal place if necessary
The class R0 endorses both the Vortex-Stretching-based model and the QR model the most andSmagorinsky the least in the case for LES without explicit filtering In the Rlowast class the Vortex-Stretching-based model is again the most endorsed followed by the Gradient model however thepercentages are again small
34
Now let us consider LES with explicit filtering Tables 16 and 17 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 30 and is in Appendix C3
Endorsements for models using solutions in R0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 4 4 3 2
Table 16 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Rlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 6 6 6 4
Table 17 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 we can see in Table 16 that the results for the lsquostrongrsquo endorsements are exactly thesame as in Table 5 This is because all of the flows with zero subfilter dissipation are contained in theclass R0 As in Table 9 there are no lsquostrongrsquo endorsements in Table 17 The lsquoweakrsquo endorsementshowever are split more evenly between the two classes now than in Vremanrsquos classes in Sections 441and 442 We find that the Smagorinsky and Vreman models still seem to be the most endorsed inboth classes
444 Two-component Flow Classes
Another way in which the flows can be grouped into classes is by determining whether a flow is atwo-component flow or not We define a two-component flow to have the property that the combinedprincipal invariants of a flow in Eq (43) satisfy
I3 = I4 = I5 minus1
2I1I2 = 0 (82)
We are also interested in considering flows with zero gradient model subgrid dissipation These flowshave the property that
I3 minus I4 = 0 (83)
What we find though is that for all the solutions in Section 22 if a flow is a two-component flowthen it also satisfies I3 minus I4 = 0 It may be interesting to note that although I3 minus I4 6= 0 for manysolutions there are a number with I3 equal to a multiple of I4 Although not considered in thisreport this may also be a class of interest We find that the two-component flows are
C2 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)(84)
35
and we call the remaining flows three-component flows They are
C3 = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex) Sol 11 (Pseudo-Motion Shots)
Sol 13 (Sphere Translation)
(85)
Note that Sol 8 (Vortex in Stagnation Flow) is defined to be a planar flow this does not mean that itis a two-component flow in this sense however because its two non-zero velocity components dependon three coordinates We now check the models for LES without explicit filtering by consideringendorsements within each class Tables 18 and 19 are analogues of Table 3 and they illustrate theresults for LES without explicit filtering for each flow class The analogues of Tables 1 and 2 areTables 31 and 32 and are in Appendix C4
Endorsements for models using solutions in C2 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 6 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 357 714 100 857 100
Table 18 Each entry in the top two rows represents the number of solutions in the class C2 thatendorse each model There are 7 solutions in this class so the percentages are worked out by dividingthe total of each column by 14 and taking the result to one decimal place if necessary
Endorsements for models using solutions in C3 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 3 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 63 63 63 188 250
Table 19 Each entry in the top two rows represents the number of solutions in the class C3 thatendorse each model There are 8 solutions in this class so the percentages are worked out by dividingthe total of each column by 16 and taking the result to one decimal place if necessary
The class C2 equally endorses the QR and Vortex-Stretching-based models and once again theSmagorinsky model is the least endorsed As with the previous classes the class C3 has such lowpercentages that it does not seem to endorse any models We start to see a trend appearing here inthat the flow classes that contain the more complicated flows namely Q0minus6 (Eq (76)) Vlowast (Eq (79))Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages The classes that tendto contain the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78)) R0 (Eq (80)) and C2 (Eq (84))all seem to favour the Vortex-Stretching-based model or the QR model
36
Now let us consider LES with explicit filtering Tables 20 and 21 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 33 and is in Appendix C4
Endorsements for models using solutions in C2 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 2 2 0 1 0
Table 20 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in C3 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 8 8 8 6
Table 21 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 and Table 16 we can see in Table 20 that the results for the lsquostrongrsquo endorsements areexactly the same as in Table 5 This is again because all of the flows with zero subfilter dissipationare contained in the class C2 There are two flows with non-zero true subfilter dissipation (givenin Eq (67)) in C2 and this shows in the lsquoweakrsquo endorsements in Table 20 There are no lsquostrongrsquoendorsements in Table 21 as all the zero subfilter dissipation flows are in C2
5 Conclusions
In this report we studied incompressible turbulent flows We found a number of exact solutions to theNavier-Stokes equations each representing a relatively simplistic fluid flow Large-eddy simulation(LES) is used to model turbulent flows by smoothing over the complex small-scale motion Weintroduced a number of subfilter-scale models to approximate this small-scale motion and testedthem using the found exact solutions to the Navier-Stokes equations We first considered LESwithout explicit filtering and used the Navier-Stokes equations to derive conditions for the subfilter-scale models namely the Velocity Field condition (Eq (58)) the Kinetic Energy condition (Eq (60))and the Dissipation condition (Eq (62)) Similarly we considered LES with explicit filtering andagain used the Navier-Stokes equations to derive more conditions for the models in this case wedecided to just take one condition the Dissipation condition (Eqs (67) and (68)) as the others leadto difficulty in their evaluation By lsquorunningrsquo the exact solutions to the Navier-Stokes equationsthrough each condition for the subfilter-scale models we found that the Vortex-Stretching-basededdy-viscosity model in Eq (54) was most endorsed by the tests in LES without explicit filteringWe found that four out of the five subfilter-scale models namely the Vreman QR Gradient andVortex-Stretching-based models were equally endorsed by the lsquostrongrsquo tests in LES with explicitfiltering with the Smagorinsky model being the least endorsed The results we found for the lsquoweakrsquocondition were completely opposite to those found in both the lsquostrongrsquo condition and in LES withoutexplicit filtering This led us to conclude that this test was not one in which we should put our trust
37
We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses
In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes
This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold
38
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
Now let us consider LES with explicit filtering Tables 16 and 17 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 30 and is in Appendix C3
Endorsements for models using solutions in R0 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 4 4 3 2
Table 16 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in Rlowast for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 6 6 6 4
Table 17 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 we can see in Table 16 that the results for the lsquostrongrsquo endorsements are exactly thesame as in Table 5 This is because all of the flows with zero subfilter dissipation are contained in theclass R0 As in Table 9 there are no lsquostrongrsquo endorsements in Table 17 The lsquoweakrsquo endorsementshowever are split more evenly between the two classes now than in Vremanrsquos classes in Sections 441and 442 We find that the Smagorinsky and Vreman models still seem to be the most endorsed inboth classes
444 Two-component Flow Classes
Another way in which the flows can be grouped into classes is by determining whether a flow is atwo-component flow or not We define a two-component flow to have the property that the combinedprincipal invariants of a flow in Eq (43) satisfy
I3 = I4 = I5 minus1
2I1I2 = 0 (82)
We are also interested in considering flows with zero gradient model subgrid dissipation These flowshave the property that
I3 minus I4 = 0 (83)
What we find though is that for all the solutions in Section 22 if a flow is a two-component flowthen it also satisfies I3 minus I4 = 0 It may be interesting to note that although I3 minus I4 6= 0 for manysolutions there are a number with I3 equal to a multiple of I4 Although not considered in thisreport this may also be a class of interest We find that the two-component flows are
C2 = Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time)
Sol 5 (2D Taylor) Sol 12 (Flows with Constant Whirl)
Sol 14 (Radially Dependent) Sol 15 (Rankine Vortex)(84)
35
and we call the remaining flows three-component flows They are
C3 = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex) Sol 11 (Pseudo-Motion Shots)
Sol 13 (Sphere Translation)
(85)
Note that Sol 8 (Vortex in Stagnation Flow) is defined to be a planar flow this does not mean that itis a two-component flow in this sense however because its two non-zero velocity components dependon three coordinates We now check the models for LES without explicit filtering by consideringendorsements within each class Tables 18 and 19 are analogues of Table 3 and they illustrate theresults for LES without explicit filtering for each flow class The analogues of Tables 1 and 2 areTables 31 and 32 and are in Appendix C4
Endorsements for models using solutions in C2 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 6 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 357 714 100 857 100
Table 18 Each entry in the top two rows represents the number of solutions in the class C2 thatendorse each model There are 7 solutions in this class so the percentages are worked out by dividingthe total of each column by 14 and taking the result to one decimal place if necessary
Endorsements for models using solutions in C3 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 3 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 63 63 63 188 250
Table 19 Each entry in the top two rows represents the number of solutions in the class C3 thatendorse each model There are 8 solutions in this class so the percentages are worked out by dividingthe total of each column by 16 and taking the result to one decimal place if necessary
The class C2 equally endorses the QR and Vortex-Stretching-based models and once again theSmagorinsky model is the least endorsed As with the previous classes the class C3 has such lowpercentages that it does not seem to endorse any models We start to see a trend appearing here inthat the flow classes that contain the more complicated flows namely Q0minus6 (Eq (76)) Vlowast (Eq (79))Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages The classes that tendto contain the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78)) R0 (Eq (80)) and C2 (Eq (84))all seem to favour the Vortex-Stretching-based model or the QR model
36
Now let us consider LES with explicit filtering Tables 20 and 21 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 33 and is in Appendix C4
Endorsements for models using solutions in C2 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 2 2 0 1 0
Table 20 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in C3 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 8 8 8 6
Table 21 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 and Table 16 we can see in Table 20 that the results for the lsquostrongrsquo endorsements areexactly the same as in Table 5 This is again because all of the flows with zero subfilter dissipationare contained in the class C2 There are two flows with non-zero true subfilter dissipation (givenin Eq (67)) in C2 and this shows in the lsquoweakrsquo endorsements in Table 20 There are no lsquostrongrsquoendorsements in Table 21 as all the zero subfilter dissipation flows are in C2
5 Conclusions
In this report we studied incompressible turbulent flows We found a number of exact solutions to theNavier-Stokes equations each representing a relatively simplistic fluid flow Large-eddy simulation(LES) is used to model turbulent flows by smoothing over the complex small-scale motion Weintroduced a number of subfilter-scale models to approximate this small-scale motion and testedthem using the found exact solutions to the Navier-Stokes equations We first considered LESwithout explicit filtering and used the Navier-Stokes equations to derive conditions for the subfilter-scale models namely the Velocity Field condition (Eq (58)) the Kinetic Energy condition (Eq (60))and the Dissipation condition (Eq (62)) Similarly we considered LES with explicit filtering andagain used the Navier-Stokes equations to derive more conditions for the models in this case wedecided to just take one condition the Dissipation condition (Eqs (67) and (68)) as the others leadto difficulty in their evaluation By lsquorunningrsquo the exact solutions to the Navier-Stokes equationsthrough each condition for the subfilter-scale models we found that the Vortex-Stretching-basededdy-viscosity model in Eq (54) was most endorsed by the tests in LES without explicit filteringWe found that four out of the five subfilter-scale models namely the Vreman QR Gradient andVortex-Stretching-based models were equally endorsed by the lsquostrongrsquo tests in LES with explicitfiltering with the Smagorinsky model being the least endorsed The results we found for the lsquoweakrsquocondition were completely opposite to those found in both the lsquostrongrsquo condition and in LES withoutexplicit filtering This led us to conclude that this test was not one in which we should put our trust
37
We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses
In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes
This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold
38
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
and we call the remaining flows three-component flows They are
C3 = Sol 4 (Axisymmetric Flow) Sol 6 (Generalized Beltrami Flow)
Sol 7 (Stagnation Flow) Sol 8 (Vortex in Stagnation Flow)
Sol 9 (Burgers Vortex) Sol 10 (Another Vortex) Sol 11 (Pseudo-Motion Shots)
Sol 13 (Sphere Translation)
(85)
Note that Sol 8 (Vortex in Stagnation Flow) is defined to be a planar flow this does not mean that itis a two-component flow in this sense however because its two non-zero velocity components dependon three coordinates We now check the models for LES without explicit filtering by consideringendorsements within each class Tables 18 and 19 are analogues of Table 3 and they illustrate theresults for LES without explicit filtering for each flow class The analogues of Tables 1 and 2 areTables 31 and 32 and are in Appendix C4
Endorsements for models using solutions in C2 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
3 5 7 6 7
lsquoDissipationrsquoEndorsements
2 5 7 6 7
Percentage Endorsed 357 714 100 857 100
Table 18 Each entry in the top two rows represents the number of solutions in the class C2 thatendorse each model There are 7 solutions in this class so the percentages are worked out by dividingthe total of each column by 14 and taking the result to one decimal place if necessary
Endorsements for models using solutions in C3 for LES without explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoVelocityKineticEnergyrsquo Endorsements
1 1 1 3 2
lsquoDissipationrsquoEndorsements
0 0 0 0 2
Percentage Endorsed 63 63 63 188 250
Table 19 Each entry in the top two rows represents the number of solutions in the class C3 thatendorse each model There are 8 solutions in this class so the percentages are worked out by dividingthe total of each column by 16 and taking the result to one decimal place if necessary
The class C2 equally endorses the QR and Vortex-Stretching-based models and once again theSmagorinsky model is the least endorsed As with the previous classes the class C3 has such lowpercentages that it does not seem to endorse any models We start to see a trend appearing here inthat the flow classes that contain the more complicated flows namely Q0minus6 (Eq (76)) Vlowast (Eq (79))Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages The classes that tendto contain the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78)) R0 (Eq (80)) and C2 (Eq (84))all seem to favour the Vortex-Stretching-based model or the QR model
36
Now let us consider LES with explicit filtering Tables 20 and 21 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 33 and is in Appendix C4
Endorsements for models using solutions in C2 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 2 2 0 1 0
Table 20 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in C3 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 8 8 8 6
Table 21 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 and Table 16 we can see in Table 20 that the results for the lsquostrongrsquo endorsements areexactly the same as in Table 5 This is again because all of the flows with zero subfilter dissipationare contained in the class C2 There are two flows with non-zero true subfilter dissipation (givenin Eq (67)) in C2 and this shows in the lsquoweakrsquo endorsements in Table 20 There are no lsquostrongrsquoendorsements in Table 21 as all the zero subfilter dissipation flows are in C2
5 Conclusions
In this report we studied incompressible turbulent flows We found a number of exact solutions to theNavier-Stokes equations each representing a relatively simplistic fluid flow Large-eddy simulation(LES) is used to model turbulent flows by smoothing over the complex small-scale motion Weintroduced a number of subfilter-scale models to approximate this small-scale motion and testedthem using the found exact solutions to the Navier-Stokes equations We first considered LESwithout explicit filtering and used the Navier-Stokes equations to derive conditions for the subfilter-scale models namely the Velocity Field condition (Eq (58)) the Kinetic Energy condition (Eq (60))and the Dissipation condition (Eq (62)) Similarly we considered LES with explicit filtering andagain used the Navier-Stokes equations to derive more conditions for the models in this case wedecided to just take one condition the Dissipation condition (Eqs (67) and (68)) as the others leadto difficulty in their evaluation By lsquorunningrsquo the exact solutions to the Navier-Stokes equationsthrough each condition for the subfilter-scale models we found that the Vortex-Stretching-basededdy-viscosity model in Eq (54) was most endorsed by the tests in LES without explicit filteringWe found that four out of the five subfilter-scale models namely the Vreman QR Gradient andVortex-Stretching-based models were equally endorsed by the lsquostrongrsquo tests in LES with explicitfiltering with the Smagorinsky model being the least endorsed The results we found for the lsquoweakrsquocondition were completely opposite to those found in both the lsquostrongrsquo condition and in LES withoutexplicit filtering This led us to conclude that this test was not one in which we should put our trust
37
We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses
In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes
This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold
38
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
Now let us consider LES with explicit filtering Tables 20 and 21 are analogues of Table 5 and theyillustrate the results for LES with explicit filtering for each flow class The analogue of Table 4 isTable 33 and is in Appendix C4
Endorsements for models using solutions in C2 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 2 5 5 5 5
lsquoWeakrsquo Endorsements 2 2 0 1 0
Table 20 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
Endorsements for models using solutions in C3 for LES with explicit filtering
Smagorinsky Vreman QR Gradient Vortex-Stretching
lsquoStrongrsquo Endorsements 0 0 0 0 0
lsquoWeakrsquo Endorsements 8 8 8 6
Table 21 A lsquostrongrsquo endorsement means that both Sijτij and Sij(u)τmodij (u) were found to be zero
A lsquoweakrsquo endorsement means that both were found to be non-zero
As in Table 8 and Table 16 we can see in Table 20 that the results for the lsquostrongrsquo endorsements areexactly the same as in Table 5 This is again because all of the flows with zero subfilter dissipationare contained in the class C2 There are two flows with non-zero true subfilter dissipation (givenin Eq (67)) in C2 and this shows in the lsquoweakrsquo endorsements in Table 20 There are no lsquostrongrsquoendorsements in Table 21 as all the zero subfilter dissipation flows are in C2
5 Conclusions
In this report we studied incompressible turbulent flows We found a number of exact solutions to theNavier-Stokes equations each representing a relatively simplistic fluid flow Large-eddy simulation(LES) is used to model turbulent flows by smoothing over the complex small-scale motion Weintroduced a number of subfilter-scale models to approximate this small-scale motion and testedthem using the found exact solutions to the Navier-Stokes equations We first considered LESwithout explicit filtering and used the Navier-Stokes equations to derive conditions for the subfilter-scale models namely the Velocity Field condition (Eq (58)) the Kinetic Energy condition (Eq (60))and the Dissipation condition (Eq (62)) Similarly we considered LES with explicit filtering andagain used the Navier-Stokes equations to derive more conditions for the models in this case wedecided to just take one condition the Dissipation condition (Eqs (67) and (68)) as the others leadto difficulty in their evaluation By lsquorunningrsquo the exact solutions to the Navier-Stokes equationsthrough each condition for the subfilter-scale models we found that the Vortex-Stretching-basededdy-viscosity model in Eq (54) was most endorsed by the tests in LES without explicit filteringWe found that four out of the five subfilter-scale models namely the Vreman QR Gradient andVortex-Stretching-based models were equally endorsed by the lsquostrongrsquo tests in LES with explicitfiltering with the Smagorinsky model being the least endorsed The results we found for the lsquoweakrsquocondition were completely opposite to those found in both the lsquostrongrsquo condition and in LES withoutexplicit filtering This led us to conclude that this test was not one in which we should put our trust
37
We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses
In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes
This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold
38
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
We then grouped the found exact solutions into different classes in order to present more accurateresults and attempt to remove bias due to similarity of exact solutions We constructed four sets ofclasses two sets with respect to Vremanrsquos work one with respect the the third principal invariantand one with respect to two-component flows We found that in the case for LES without explicitfiltering that the classes that contained the simpler flows namely Q7minus9 (Eq (77)) V0 (Eq (78))R0 (Eq (80)) and C2 (Eq (84)) all seem to endorse the Vortex-Stretching-based model or the QRmodel The flow classes that tended to contain the more complicated flows namely Q0minus6 (Eq (76))Vlowast (Eq (79)) Rlowast (Eq (81)) and C3 (Eq (85)) all have very low endorsement percentages and henceit seems as if these classes do not endorse the flows Despite this however the majority still appearsto lie with the Vortex-Stretching-based eddy-viscosity model In the case for LES with explicitfiltering we found for all types of classes that the flows that contribute to the lsquostrongrsquo conditionnamely Sol 1 (Linear Velocity) Sol 2 (Planar Flow) Sol 3 (Velocity as a function of time) Sol14 (Radially Dependent) and Sol 15 (Rankine Vortex) were almost always all in the simpler flowclasses
In Section 43 we considered Vremanrsquos flow classes [16] and attempted to verify or extend his resultsWe found that fourteen out of our fifteen solutions all follow Vremanrsquos claim that the subfilterdissipation is zero for flows in the zero-subfilter-dissipation flow classes (Appendix B) Sol 15(Rankine Vortex) however extends Vremanrsquos results in that we found a zero-subfilter-dissipationflow that is not in one of Vremanrsquos zero-subfilter-dissipation classes
This project could be extended by adding more exact solutions to the Navier-Stokes equations Thiswould result in more data for testing the models and would also extend the flow classes found inSection 44 Finally it may also be possible to design and build new eddy-viscosity models for LESby determining functions of the combined invariants in Eq (43) for which we find that the conditionsfor large eddy simulation without explicit filtering in Section 33 and the condition for large eddysimulation with explicit filtering in Section 34 hold
38
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
Appendices
A Invariants of Flows
Sol 1 (Linear Velocity)The principal invariants for Sol 1 (Linear Velocity) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 =1
2
(a2 + c2
) I2 = minus1
2
(a2 + c2
) I3 = 0 I4 = 0 I5 = minus1
8
(a2 + c2
)2
Sol 2 (Planar Flow)The principal invariants for the Sol 2 (Planar Flow) are the same as for Sol 1
PG = 0 QG = 0 RG = 0
and the combined invariants are similar in that
I1 =1
2
((2dz + e)
2+ (2gz + h)
2)
I2 = minus1
2
((2dz + e)
2+ (2gz + h)
2)
I3 = 0
I4 = 0
I5 = minus1
8
((2dz + e)
2+ (2gz + h)
2)2
Sol 3 (Velocity as a function of time)The principal invariants for Sol 3 (Velocity as a function of time) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = 0 I3 = 0 I4 = 0 I5 = 0
Sol 4 (Axisymmetric Flow)The principal invariants for Sol 4 (Axisymmetric Flow) are as follows
PG = 0 QG = minus3a2z2 RG = minus2a2z2
39
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
and the combined invariants are
I1 =a2
2
(x2 + y2 + 12z2
)
I2 = minusa2
2(x2 + y2)
I3 = minus3a3z
4
(x2 + y2 + 8z2
)
I4 =a3z
4
(x2 + y2
)
I5 = minusa4
8
(x2 + y2
) (x2 + y2 + 10z2
)
Sol 5 (2D Taylor)The principal invariants for Sol 5 (2D Taylor) are as follows
PG = 0 QG = π2eminus2π2tR cos(πx) cos(πy) RG = 0
and the combined invariants are
I1 =π2
2eminus
2π2tR (cos(πx)minus cos(πy))2
I2 = minusπ2
2eminus
2π2tR (cos(πx) + cos(πy))2
I3 = 0
I4 = 0
I5 = minusπ4
32eminus
4π2tR (cos(2πx)minus cos(2πy))2
Sol 6 (Generalized Beltrami Flow)The principal invariants for Sol 6 (Generalized Beltrami Flow) are as follows
PG = 0
QG = π2eminus2π2tR (AB sin(πx) cos(πz) +AC cos(πy) sin(πz) +BC cos(πx) sin(πy))
RG = π3ABCeminus3π2tR (cos(πx) cos(πy) cos(πz)minus sin(πx) sin(πy) sin(πz))
40
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
and the terms for the combined invariants are
I1 =π2
2eminus
2π2tR
(A2 +B2 + C2 minus 2AB sin(πx) cos(πz)minus 2AC cos(πy) sin(πz)minus 2BC cos(πx) sin(πy)
)
I2 = minusπ2
2eminus
2π2tR
(A2 +B2 + C2 + 2AB sin(πx) cos(πz) + 2AC cos(πy) sin(πz) + 2BC cos(πx) sin(πy)
)
I3 =3π3
4eminus
3π2tR (A cos(πz)minusB sin(πx))(C cos(πy)minusA sin(πz))(B cos(πx)minus C sin(πy))
I4 =π3
4eminus
3π2tR
(B cos(πx)(C cos(πy)(3A cos(πz) +B sin(πx)) +A sin(πz)(A cos(πz)minusB sin(πx)))+
C sin(πy)(C cos(πy)(A cos(πz)minusB sin(πx))minusA sin(πz)(A cos(πz) + 3B sin(πx))))
I5 = minusπ4
8eminus
4π2tR
(A4 + C2
(B2 minusA2
)cos(2πy) +B2(Aminus C)(A+ C) cos(2πx)
minusA(2BC(A sin(πx) cos(πy) sin(2πz) +B sin(2πx) sin(πy) cos(πz) + C cos(πx) sin(2πy) sin(πz))+
A(B minus C)(B + C) cos(2πz)) +B4 + C4)
Sol 7 (Stagnation Flow)The principal invariants for Sol 7 (Stagnation Flow) are as follows
PG = 0 QG = minusa2
k2+ anminus k2n2 RG =
an(aminus k2n
)k
and the combined invariants are
I1 =2a2
k2minus 2an+ 2k2n2 I2 = 0 I3 =
3an(aminus k2n
)k
I4 = 0 I5 = 0
Sol 8 (Vortex in Stagnation Flow)The principal invariants for Sol 8 (Vortex in Stagnation Flow) are as follows
PG = 0 QG = minus 1
Re2 RG = 0
and the combined invariants are
I1 =k2
2cos2(z) +
2
Re2
I2 = minusk2
2cos2(z)
I3 = minus3k2 cos2(z)
4Re
I4 =k2 cos2(z)
4Re
I5 = minusk2 cos2(z)
(k2Re2 cos2(z) + 2
)8Re2
41
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
Sol 9 (Burgers Vortex)The principal invariants for Sol 9 (Burgers Vortex) are as follows
PG = 0
QG =1
2
(minus 24
c4Re2minus
8eminus2h(xy)(eh(xy) minus 1
)2Ω2x2y2
h(x y)2
+16eminus2h(xy)
(eh(xy) minus 1
)Ω2x2y2
h(x y) (x2 + y2)minus 8eminus2h(xy)Ω2x2y2
(x2 + y2)2
)
RG =4eminus2h(xy)
c6Re3 (x2 + y2)2
(e2h(xy)
(4(x2 + y2
)2 minus c8Re2Ω2)
+ 2c6Re2Ω2eh(xy)(c2 + x2 + y2
)minus c6Re2Ω2
(c2 + 2
(x2 + y2
)))
where h(x y) is as defined in Eq (28) The combined invariants are
I1 =2eminus2h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 12
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
I2 = minus2Ω2eminus2h(xy)
I3 =1
c6Re3 (x2 + y2)2
I4 =4Ω2eminus2h(xy)
c2Re
I5 = minus 2Ω2eminus4h(xy)
c4Re2 (x2 + y2)2
(e2h(xy)
(c8Re2Ω2 + 4
(x2 + y2
)2)minus 2c6Re2Ω2eh(xy)
(c2 + x2 + y2
)+ c4Re2Ω2
(c2 + x2 + y2
)2)
Sol 10 (Another Vortex)The principal invariants for Sol 10 (Another Vortex) are as follows
PG = 0 QG =12
c4Re2minus d2
(x2 + y2)2 RG =
16
c6Re3minus 4d2
c2Re (x2 + y2)2
and the combined invariants are
I1 =24
c4Re2+
2d2
(x2 + y2)2 I2 = 0 I3 =
48
c6Re3minus 12d2
c2Re (x2 + y2)2 I4 = 0 I5 = 0
42
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
Sol 11 (Pseudo-Motion Shots)The principal invariants for Sol 11 (Pseudo-Motion Shots) are as follows
PG = 0
QG =(A1
(x2 + y2
)+B1
) (3A1
(x2 + y2
)+B1
)+ 2A1C1 minus
C21Re2 + 4
Re2 (x2 + y2)2
RG = 0
and the combined invariants are
I1 =1
2
(Re2
((x2 + y2
)3 (4A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 8A1C1
(x2 + y2
)2+ 4C2
1
)+ 16
Re2 (x2 + y2)2
+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
))
I2 = minus2(2A1
(x2 + y2
)+B1
)2 minus 1
2x2(A2 log
(x2 + y2
)+A2 + 2B2
)2 minus 1
2y2(A2 log
(x2 + y2
)+A2 + 2B2
)2
I3 = minus3(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I4 =
(A2 log
(x2 + y2
)+A2 + 2B2
)22Re
I5 = minus 1
8Re2 (x2 + y2)
(2A2
2
(Re2
(3(x2 + y2
)3 (3A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8A2B2
(Re2
((x2 + y2
)3 (9A2
1
(x2 + y2
)+ 4B2
2
)minus 6A1C1
(x2 + y2
)2+ C2
1
)+ 4)
+ 8
(8A2
1
(x2 + y2
)(Re2
(C1 minusA1
(x2 + y2
)2)2+ 4
)+B2
2
(Re2
(C1 minus 3A1
(x2 + y2
)2)2+ 4
)+ 2B4
2Re2(x2 + y2
)3)+A4
2Re2(x2 + y2
)3+ 8A3
2B2Re2(x2 + y2
)3)minus 1
8Re2 (x2 + y2)2
(2B2
1
(Re2
((x2 + y2
)3 (8A2
1
(x2 + y2
)+ (A2 + 2B2)2
)minus 16A1C1
(x2 + y2
)2+ 8C2
1
)+ 32
)+A2
(x2 + y2
)log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) (2(Re2
( (x2 + y2
)2 ( (x2 + y2
) (9A2
1
(x2 + y2
)+ (A2 + 2B2)2
)+ 6A1B1
(x2 + y2
)+B2
1
)minus 2C1
(x2 + y2
) (3A1
(x2 + y2
)+B1
)+ C2
1
)+ 4)
+A2Re2(x2 + y2
)3log(x2 + y2
) (A2 log
(x2 + y2
)+ 2A2 + 4B2
) )+ 4B1
(x2 + y2
) (16A3
1Re2(x2 + y2
)4 minus 32A21C1Re2
(x2 + y2
)2+A1
(Re2
(3(A2 + 2B2)2
(x2 + y2
)3+ 16C2
1
)+ 64
)minus C1Re2(A2 + 2B2)2
(x2 + y2
) ))
43
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
Sol 12 (Flows with Constant Whirl)The principal invariants for Sol 12 (Flows with Constant Whirl) are as follows
PG = 0 QG = minus2
Re2+ 1
t2 eminus
Re(x2+y2)2t
(Re(x2 + y2
)+ 2t
)2 (x2 + y2)
2 RG = 0
and the combined invariants are
I1 =
eminusRe(x2+y2)
2t
(Re4
(x2 + y2
)2+ 8Re3t
(x2 + y2
)+ 16Re2t2 + 16t3e
Re(x2+y2)2t
)8Re2t3 (x2 + y2)
2
I2 = minusRe2eminusRe(x2+y2)
2t
8t3
I3 = 0
I4 = 0
I5 =
eminusRe(x2+y2)
t
(Re4
(minus(x2 + y2
)2)minus 8Re3t(x2 + y2
)minus 16Re2t2 minus 16t3e
Re(x2+y2)2t
)128t6 (x2 + y2)
2
Sol 13 (Sphere Translation)The principal invariants for Sol 13 (Sphere Translation) are as follows
PG = 0
QG = minus27a2z2
(a2 minusR2
)216R10
RG = minus27a3z3
(a2 minusR2
)332R15
and the combined invariants are
I1 =9(a3 minus aR2
)2 (x2 + y2 + 12z2
)32R10
I2 = minus9(a3 minus aR2
)2 (x2 + y2
)32R10
I3 =81z
(aR2 minus a3
)3 (x2 + y2 + 8z2
)256R15
I4 =(27(a3 minus aR2)3(x2 + y2)z)
(256R15)
I5 = minus81(a3 minus aR2
)4 (x2 + y2
) (x2 + y2 + 10z2
)2048R20
Sol 14 (Radially Dependent)The principal invariants for Sol 14 (Radially Dependent) are as follows
PG = 0 QG = 0 RG = 0
44
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
and the combined invariants are
I1 = 2(x2 + y2
) I2 = minus2
(x2 + y2
) I3 = 0 I4 = 0 I5 = minus2
(x2 + y2
)2
Sol 15 (Rankine Vortex)The principal invariants for Sol 15 (Rankine Vortex) are as follows
PG = 0 QG = 0 RG = 0
and the combined invariants are
I1 = 0 I2 = minus a2
2π2R4 I3 = 0 I4 = 0 I5 = 0
45
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
B Vremanrsquos Flow Classes
In his paper [16] Vreman claims that there are 13 classes of flows for which the true subfilterdissipation equals zero The classes are all subsets of Q7Q8 and Q9 where Qn with n isin [0 9]is defined to be the class of flows with zero entries in the velocity gradient of the flow (defined inEq (39)) Below are the flow classes claimed to have zero true subfilter dissipation0 0 0
0 0 00 0 0
sube Q9
0 0 0lowast 0 00 0 0
0 lowast 0
0 0 00 0 0
0 0 0
0 0 0lowast 0 0
0 0 lowast
0 0 00 0 0
0 0 0
0 0 00 lowast 0
0 0 0
0 0 lowast0 0 0
sube Q8
0 0 0lowast 0 0lowast 0 0
0 lowast 0
0 0 00 lowast 0
0 0 lowast
0 0 lowast0 0 0
0 lowast lowast
0 0 00 0 0
0 0 0lowast 0 lowast0 0 0
0 0 0
0 0 0lowast lowast 0
sube Q7
46
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
C Extended Tables for Flow Classes
C1 Vremanrsquos Velocity Gradient Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 22 Table 1 rearranged into the flow classes described in Section 441
Dissipation Condition [Does Sijτmodij (v) = 0]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 23 Table 2 rearranged into the flow classes described in Section 441
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
Q7minus9 Q0minus6
Solution 1 2 3 5 14 15 4 6 7 8 9 10 11 12 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 24 Table 4 rearranged into the flow classes described in Section 441
47
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
C2 Vremanrsquos Zero-True-Subfilter-Dissipation Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 25 Table 1 rearranged into the flow classes described in Section 442
Dissipation Condition [Does Sijτmodij (v) = 0]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 26 Table 2 rearranged into the flow classes described in Section 442
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
V0 Vlowast
Solution 1 2 3 14 4 5 6 7 8 9 10 11 12 13 15
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 27 Table 4 rearranged into the flow classes described in Section 442
48
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
C3 Classes using the Third Principal Invariant
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 28 Table 1 rearranged into the flow classes described in Section 443
Dissipation Condition [Does Sijτmodij (v) = 0]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 29 Table 2 rearranged into the flow classes described in Section 443
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
R0 Rlowast
Solution 1 2 3 5 8 11 12 14 15 4 6 7 9 10 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 30 Table 4 rearranged into the flow classes described in Section 443
49
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
C4 Two-Component Flow Classes
Velocity Field amp Kinetic Energy Conditions [Dopart
partxjτmodij (v) and vi
part
partxjτmodij (v) equal zero]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0 0 0
Vreman 0 0 0 0 0 0
QR 0 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 31 Table 1 rearranged into the flow classes described in Section 444
Dissipation Condition [Does Sijτmodij (v) = 0]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 32 Table 2 rearranged into the flow classes described in Section 444
Dissipation Condition 2 [Does Sijτij = Sij(v)τmodij (v)]
C2 C3
Solution 1 2 3 5 12 14 15 4 6 7 8 9 10 11 13
Exact 0 0 0 0 0
Smagorinsky 0 0
Vreman 0 0 0 0 0
QR 0 0 0 0 0 0 0
Gradient 0 0 0 0 0 0
Vortex-Stretching 0 0 0 0 0 0 0 0 0
Table 33 Table 4 rearranged into the flow classes described in Section 444
50
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
References
[1] D J Acheson Elementary Fluid Dynamics Oxford University Press 1990 ISBN 0-19-859679-0
[2] D Barbato LC Berselli and CR Grisanti Analytical and numerical results for the rationallarge eddy simulation model Journal of Mathematical Fluid Mechanics 944ndash74 2007
[3] R Berker Integration des equations du mouvement drsquoun fluide visqueux incompressibleEncyclopedia of Physics volume VIII2 Fluid Dynamics II Springer-Verlag 1963
[4] RA Clark JH Ferziger and WC Reynolds Evaluation of subgrid-scale models using anaccurately simulated turbulent flow Journal of Fluid Mechanics 911ndash16 1979 doi101017
S002211207900001X
[5] L Davidson Fluid mechanics turbulent flow and turbulence modeling 2016 httpwwwtfd
chalmersse~ladaMoFlecture_noteshtml Accessed 17th May 2016
[6] PG Drazin and N Riley The Navier-Stokes Equations A Classification of Flows and ExactSolutions Cambridge University Press 2006
[7] S Irmay New exact solutions of the Navier Stokes equations near a plane International Journalof Engineering Science 31397ndash401 1993 doi1010160020-7225(93)90014-L
[8] B Lautrup Physics of Continuous Matter Exotic and Everyday Phenomena in the MacroscopicWorld Institute of Physics Publishing 2005
[9] A Leonard Energy cascade in large-eddy simulations of turbulent fluid flows Turbulent Diffusionin Environmental Pollution Proceedings of a Symposium held at Charlottesville VirginiaAdvances in Geophysics 18 A237ndash248 1973
[10] M Oberlack Invariant modeling in large eddy simulation of turbulence Centre for turbulenceresearch 1997 arXiv151007881[physicsflu-dyn]
[11] MH Silvis and RWCP Verstappen Physically-consistent subgrid-scale models for large-eddysimulation of incompressible turbulent flows 2015 arXiv151007881[physicsflu-dyn]
[12] J Smagorinsky General circulation experiments with the primitive equations Monthly WeatherReview 9199ndash164 1963 doi1011751520-0493(1963)091lt0099GCEWTPgt23CO2
[13] RWCP Verstappen When does eddy viscosity damp subfilter scales sufficiently Journal ofScientific Computing 4994ndash110 2011 doi101007s10915-011-9504-4
[14] RWCP Verstappen ST Bose J Lee H Choi and P Moin A dynamic eddy-viscosity modelbased on the invariants of the rate-of-strain Proc Summer Program Center for TurbulenceResearch Stanford UniversityNASA Ames pages 183ndash192 2010
[15] RWCP Verstappen W Rozema and HJ Bae Numerical scale separation in large-eddysimulation Proc Summer Program Center for Turbulence Research Stanford UniversityNASAAmes pages 417ndash426 2014
[16] AW Vreman An eddy-viscosity subgrid-scale model for turbulent shear flow Algebraic theory andapplications Physics of Fluids 163670ndash3681 2004 httpdxdoiorg10106311785131
[17] CY Wang Exact solutions of the steady-state Navier-Stokes equations Annual Review of FluidMechanics 23(1)159ndash177 1991 httpdxdoiorg101146annurevfl23010191001111
51
Recommended