Apsley-Turbulence modelling in CFD

CFD 7-1 David Apsley

7. TURBULENCE MODELLING IN CFD SPRING 2004 7.1 Turbulence models for general-purpose CFD 7.2 Reynolds-averaged Navier-Stokes (RANS) models 7.2.1 Linear eddy-viscosity models 7.2.2 Non-linear eddy-viscosity models 7.2.3 Differential stress models 7.3 Implementation of turbulence models in CFD 7.3.1 Transport equations 7.3.2 Wall boundary conditions 7.3.3 Effective viscosity for differential stress models 7.1 Turbulence Models For General-Purpose CFD Turbulence models for general-purpose CFD must be frame-invariant – i.e. independent of any particular coordinate system – and hence must be expressed in tensor form. This rules out simpler models of boundary-layer type (e.g. mixing-length models). Turbulent flows are computed either by solving the Reynolds-averaged Navier-Stokes equations with suitable models for turbulent fluxes or by computing the fluctuating quantities directly. The main approaches are summarised below. Reynolds-Averaged Navier-Stokes (RANS) Models • Eddy-viscosity models (EVM) – assume the (deviatoric) turbulent stress proportional to the mean rate of strain; – eddy viscosity derived from turbulent transport equations (usually k + one other). • Non-linear eddy-viscosity models (NLEVM) – turbulent stress modelled as a non-linear function of mean velocity gradients; - turbulent scales determined by solving transport equations (usually k + one other) – mimic response of turbulence to certain important types of strain. • Differential stress models (DSM) – aka Reynolds-stress transport models (RSTM) or second-order closure (SOC); – solve transport equations for all turbulent stresses. Computation of fluctuating quantities • Large-eddy simulation (LES) – compute time-varying flow, but model sub-grid-scale motions. • Direct numerical simulation (DNS) – no modelling; resolve the smallest scales of the flow.


7.2 Reynolds-Averaged Navier-Stokes (RANS) Models 7.2.1 Linear Eddy-Viscosity Models

Stress-strain constitutive relation:

iji

j

j

itji k

x

U

x

Uuu

3

2−��

�

�

��

�

�

∂∂

+∂∂

=− (1)

The eddy viscosity t (= t) is derived from turbulent variables such as turbulent kinetic energy k and its dissipation rate . These turbulent scalars are found by solving scalar-transport equations. Typical shear and normal stresses are given by

kx

Uu

x

V

y

Uuv

t

t

3

222 −

∂∂=−

��

��

�

∂∂+

∂∂=−

From these the other stress components are easily deduced by inspection. General Comments • is a physical property of the fluid and can be measured; t is a property of the flow

and must be modelled; • t varies with position; • At high Reynolds numbers, t � throughout much of the flow. Eddy-viscosity models are widely used and popular because: • they are easy to implement in viscous solvers; • extra viscosity aids stability; • they have some theoretical foundation in simple shear flows. However: • there is little turbulence physics; in particular, anisotropy and history effects are

neglected; • in such models, turbulent transport of momentum is determined by a single scalar t

and hence at most one Reynolds stress ( uv− ) can be represented accurately; thus, such models are questionable in complex flow.

2-Equation Models Most eddy-viscosity models in general-purpose CFD codes are of the 2-equation type; (i.e. scalar-transport equations are solved for 2 turbulent scalars). The commonest combinations of turbulent scalar are k- and k- , although others are used. Specifications are given below.


(i) k- models Eddy viscosity:

~

2

��k

fCt = (2)

Scalar-transport equations:

)�(

22�)(

11�)�(

)()(

~)~()~~()~(

)()()(

Sk

fCPfCt

Pkkkt

k

kk

+−=∇−•∇+∂∂

−=∇−•∇+∂∂

U

U (3)

Diffusivities (k) and (� ) are given by

)�()�(

)()( , t

ktk +=+=

The source term of each transport equation is primarily a balance between the rate of production of turbulent kinetic energy (per unit mass)

j

iji

k

x

UuuP

∂∂

−≡)( (4)

and the dissipation rate . In many models ~ is simply the same as . Others (notably Launder and Sharma, 1974) make a distinction between the homogeneous dissipation rate (which vanishes at solid boundaries) and the inhomogeneous dissipation rate (which doesn’ t). In the latter case, 22/1 )(2,~ kDD ∇=+= , (5)

which is consistent with the theoretical near-wall behaviour of (i.e. 2/2~ yk ) The factors f � , f1 and f2 are used in low-Re models to incorporate effects of molecular viscosity. An additional source term S( � ) may be used to incorporate viscous or non-equilibrium behaviour. In the standard k- model (Launder and Spalding, 1974) the coefficients take the values C � = 0.09, C � 1 = 1.92, C � 2 = 1.44, (k) = 1.0, ( � ) = 1.3 (6) Other important variants include RNG k- (Yakhot et al., 1992) and low-Re models such as Launder and Sharma (1974), Lam and Bremhorst (1981), and Lien and Leschziner (1993). (ii) k- models Eddy viscosity:

* kt = (7)

Scalar-transport equations:


)�(2)()�(

*)()(

)()()(

)()()(

SPt

kPkkkt

k

t

kk

+−=∇−•∇+∂∂

−=∇−•∇+∂∂

U

U (8)

Again, the diffusivities of k and are related to the eddy-viscosity:

)�(

)�()(

)( , tktk +=+=

There is a (rough) high-Re correspondance:

k

�

�C~ (9)

The definitive k- model is that of Wilcox (1988a) where the coefficients take the values

* = 1, 100

9* = , 9

5= , 40

3= , (k) = 2.0, ( � ) = 2.0, S( � ) = 0 (10)

Wilcox (1994) provided a revised version where the coefficients are functions of the turbulent Reynolds number R � k/� , in order to provide the correct near-wall behaviour. Menter (1994) devised a shear-stress-transport (SST) model. The model, which is expressed in k- form, blends the k- model (which is – allegedly – superior in the near-wall region), with the k- model (which is less sensitive to the level of turbulence in the free stream). All models of this type suffer from a problematic wall boundary condition ( ) Eddy-Viscosity Models in Simple Shear In simple shear flow the shear stress is

y

Uuv t ∂

∂=−

The three normal stresses are predicted to be equal:

kwvu3

2222 ===

whereas, in practice, there is considerable anisotropy; e.g. in the log-law region:

6.0:4.0:0.1:: 222 =wvu Actually, in simple shear flows, this is not a problem, since the shear stress gradient is the only non-zero turbulence term in the mean momentum equation. However, it is indicative of more serious problems in complex flows (those for which more than one stress component is dynamically significant).

uv

U(y)

y


7.5.2 Non-Linear Eddy-Viscosity Models NLEVMs posit a non-linear constitutive relation between turbulent stress and mean strain. This must satisfy certain tensorial properties (e.g. the anisotropy tensor is symmetric and traceless) and can be written down in many equivalent ways. The de facto UMIST standard (Craft et al., 1996) is a cubic model which can be written in tensor form as:

klklijtklklijt

ijmklmkllikljkljkliktkllikjljkit

ijklkljkiktikjkjkiktijklkljkikt

ijtijji

ScSSSc

SSScSSSc

cSScSSSSc

Skuu

)()(

)()()(

23

2

27

26

322

52

4

31

3231

1

++

−++++

−+++−+

−=

(11) where the eddy viscosity and turbulent timescale are defined by

~

2

��k

fCt = , ~k= (12)

and the mean strain and mean vorticity tensors are the symmetric and antisymmetric parts of the mean-velocity gradient:

)(2

1,)(

2

1

i

j

j

iij

i

j

j

iij x

U

x

U

x

U

x

US

∂∂

−∂∂

=∂

∂+

∂∂

= (13)

In the Craft et al. model, coefficients are functions of the mean-strain invariants and turbulent Reynolds number:

~,])400

()90

(exp[1

35.01

)]36.0exp(1[3.0

222/1

�

2/3

�75.0

�

kR

RRf

eC

ttt =−−−=

+−−=

(14)

where

),max(~,2,2 Sk

SSS ijijijij === (15)

The coefficients of the non-linear terms are: 2

�72

�652

�4321 40,40,0,80,04.1,4.0,4.0 CcCccCcccc =−==−===−= (16)

Non-linearity is built into both tensor products and strain-dependent coefficients – notably C � . The model is completed by transport equations for k and ~ . To avoid typographical errors associated with lots of suffices, I prefer a dimensionless matrix form of the non-linear stress-strain constitutive relationship:

)()}{}{(}{}{

)}{()()}{(

2

2243

22223

22

21

2312

322

312

1

��

ssIsssssss

IssIss

sa

−−−−+−−−−+−+−+

−= fC

(17) where the anisotropy and dimensionless mean strain and vorticity tensors are, respectively,

ijijijijijji

ij

kS

ks

k

uua ,,

3

2 ==−≡ (18)


and, in matrix notation, ijijkkijij TtraceT )(,)(}{,)( =≡≡≡ ITTT

The advantage of the particular choice of tensor bases in (17) is that the 3- and 4-related cubic terms vanish in 2-d incompressible flow (see below). Exercise. Show how to convert between coefficients in the alternative forms of constitutive relation, (11) and (17). Popular non-linear models include the quadratic NLEVMs of Gatski and Speziale (1993) and Shih et al. (1995) and the cubic NLEVM of Craft et al. (1996). You might even investigate Apsley and Leschziner (1998). In devising such NLEVMs, model developers have sought to incorporate such physically-significant properties as realisability:

)inequalitySchwartzCauchy(1

stresses) normal positive(0

2�

2�

2��

2�

−≤

≥

uu

uu

u

(19)

Theory and experiment also indicate that pure rotation generates no turbulence. This implies that 3 ought to be 0, at least in the limit 0→S . General Properties of Non-Linear Eddy-Viscosity Models (i) 2-d Incompressible Flow The non-linear combinations of s and have particularly simple forms in 2-d incompressible flow. In 2-dimensional flow:

��

�

�

��

�

�

=��

�

�

��

�

�

=000

00

00

,

000

0

0

21

12

2221

1211

ss

ss

s

Now, x

Uks

∂∂=11 and

y

Vks

∂∂=22 , so incompressibility ( 0=

∂∂+

∂∂

y

V

x

U) implies

1122 ss −= whilst the symmetry and antisymmetry properties of sij and ij imply

12211221 , −== ss Hence, in 2-d incompressible flow,

��

�

�

��

�

�

−=��

�

�

��

�

�

−=000

00

00

,

000

0

0

12

12

1112

1211

ss

ss

s

From these we find


��

�

�

��

�

�

−��

�

�

��

�

�

−=−

��

�

�

��

�

�

−=

��

�

�

��

�

�

+=

000

001

010

2

000

010

001

2

000

010

001

000

010

001

)(

11121212

212

2

212

211

2

ss

ss

(20)

Thus,

2

2212

22

212

}{

}{

I

Iss

=

= (21)

where

212

2

212

211

2

2}{

)(2}{

−=

+= ss (22)

Here, { } denotes the trace of a matrix and I2 is the ‘2-d identity matrix’ – not quite the identity matrix but one which has the same effect when multiplying matrices with all rows or columns equal. PROPERTY 1 In 2-d incompressible flow the quadratic terms do not contribute to the production of turbulent kinetic energy. Proof.

))(( 32)(

ijijijijj

iji

k Sakx

UuuP ++−=

∂∂

−=

But 0)( 32 =+ ijijija by the antisymmetry of ij, whilst incompressibility implies

0== iiijij SS . Hence,

ijijk SkaP −=)(

or

}{)(

as−=−= ijij

k

saP

This is true for any incompressible flow, but, in the 2-d case, multiplying (17) by s, taking the trace and using the results (21) it is found that the contribution of the quadratic terms to { as} is 0. PROPERTY 2 In 2-d incompressible flow the 3- and 4-related terms of the non-linear expansion (17) vanish. Proof. Substitute the results (21) for s2 and 2 into (17).


(ii) Particular Types of Strain The non-linear constitutive relationship (17) allows the model to mimic the response of turbulence to particular important types of strain. PROPERTY 3 The quadratic terms yield turbulence anisotropy in simple shear:

6)(

3

2

12)6(

3

2

12)6(

3

2

2

31

2

2

321

2

2

321

2

−−=

−−+=

−++=

k

w

k

v

k

u

where y

Uk

∂∂=

This may be deduced by substituting the results (20) into (17), noting that s11 = 0, whilst

2

1

2

11212 =

∂∂==

y

Uks

PROPERTY 4 The 1 and 2-related cubic terms yield the correct sensitivity to curvature.

In curved shear flow, c

ss

R

U

x

V

R

U

y

U −=∂∂

∂∂

=∂∂

, , where Rc is the radius of curvature. From

(22), )(2}{}{ 2

12212

22 −≡+ ss where

��

��

�+

∂∂

=��

��

�−

∂∂

=c

ss

c

ss

R

U

R

U

R

U

R

Us

2

1,

2

11212

Hence,

c

ss

R

U

R

Uk

∂∂

−≡+ 222 )(2}{}{s

Inspection of the production terms in the stress-transport equations (Section 7.2.3) shows that curvature is stabilising (reducing turbulence) if Us increases in the direction away from the centre of curvature (∂Us/∂R > 0) and destabilising (increasing turbulence) if Us decreases in the direction away from the centre of curvature (∂Us/∂R < 0). In the constitutive relation (17) the response is correct if 1 and 2 are both positive. PROPERTY 5 In 3-d flows, the 4-related term evokes the correct sensitivity to swirl.

'stable' curvature(reducing turbulence)

'unstable' curvature(increasing turbulence)

UW


7.2.3 Differential Stress Modelling Differential stress models (aka Reynolds-stress transport models or second-order closure)

solve a separate scalar-transport equation for each stress component jiuu :

)()()( ijijijijijkjikk

ji FPduuUx

uut

−++=−∂∂+

∂∂ (23)

For a derivation see the course notes for the “Boundary Layers” module. Such models, in principle, contain much more turbulence physics because the rate-of-change, advection and production terms are exact. The UMIST “basic model” is a high-Re closure based on that of Launder et al. (1975) and Gibson and Launder (1978). Term Name and role Model

)( jiuut∂

∂ RATE OF CHANGE

EXACT

jik uuU ADVECTION Transport with the flow

EXACT

ijP PRODUCTION (mean strain) Generation of turbulence energy from the mean flow

EXACT

k

ikj

k

jkiij x

Uuu

x

UuuP

∂∂

−∂∂

−≡

Fij

PRODUCTION (body forces) Generation of turbulence energy by body forces.

EXACT (in principle)

ijjiij ufufF +≡

ijkd DIFFUSION Spatial redistribution

)()( jil

lksklijk uu

x

uukCd

∂∂+=

ijΦ PRESSURE-STRAIN Redistribution of turbulence energy between components

)()2()1( wijijijij ++=

)( 32

1)1(

ijjiij kuuk

C −−=

)( 31

2)2(

ijkkijij PPC −−=

nlij

wji

wij

kijkkjikijlkklw

ij

yC

kfCuu

kC

fnnnnnn

/,

~

)~~~

(2/3

)2()(2

)(1

23

23)(

=+=

−−=

ijε DISSIPATION Removal of turbulence energy by viscosity

ijij 32=

Typical values of the constants are: 5.2,3.0,5.0,6.0,8.1 )(

2)(

121 ===== lww CCCCC (24)


Energy in Turbulent Fluctuations

PRODUCTION ADVECTION by mean flow

2u 2v 2w REDISTRIBUTION

DISSIPATION by viscosity

by pressure fluctuations


The stress-transport equations must be supplemented by a means of specifying – typically by its own transport equation, or one for a related quantity such as . As is suggested by the table, the most significant term requiring modelling is the pressure strain correlation (which is formed, in practice, by the average product of pressure fluctuations and fluctuating velocity gradients). This term is traceless and its accepted role is to restore isotropy – hence the form of model for )1(

ij and )2(ij . Near walls this isotropising

tendency must be over-ridden, necessitating a “wall-correction” term )(wij .

Where body forces are present (e.g. in buoyant or rotating flows) additional production terms must be included. General Assessment of DSMs For: • Include more turbulence physics than eddy-viscosity models. • Advection and production terms (“energy-in” terms) are exact and do not need

modelling. Against: • Models are very complex and many important terms (particularly the redistribution

and dissipation terms) require modelling. • Models are very expensive computationally (6 stress-transport equations in 3

dimensions) and tend to be numerically unstable (only the small molecular viscosity contributes to any sort of gradient diffusion term).

Other DSMs of Interest • Speziale et al. (1991) – non-linear ij formulation, eliminating wall-correction terms; • Craft (1998) – low-Re DSM, attempting to eliminate wall-dependent parameters; • Jakirli and Hanjali (1995) – low-Re DSM admitting anisotropic dissipation; • Wilcox (1988b) – low-Re DSM, with rather than as additional turbulent scalar. Excellent references for developments in Reynolds-stress transport modelling can be found in Launder (1989) and Hanjali (1994).


7.3 Implementation of Turbulence Models in CFD 7.3.1 Transport Equations The implementation of a turbulence model in CFD requires:

• a means of specifying the turbulent stresses jiuu , either by a constitutive relation

(eddy-viscosity models) or individual transport equations (differential stress models); • the solution of additional scalar-transport equations. Important considerations apply for the mean flow equations:

• jiuu represents a turbulent flux of Ui-momentum in the xj

direction. For eddy-viscosity models only a part of this can be treated implicitly as a diffusion-like term; e.g. for the U equation through a face normal to the y direction:

��

sourcetodtransferrepart

diffusive

t termslinearnonx

V

y

Uuv )()( −+

∂∂+

∂∂=−

The remainder of the flux is treated as part of the source term for that control volume. Nevertheless, it is still treated in a conservative fashion; i.e. the mean momentum lost by one cell is that gained by the adjacent cell.

• The lack of a turbulent viscosity in differential stress models leads to numerical

instability. This can be addressed by the use of “effective viscosities” – see below. Important considerations for the turbulence transport equations: • They are usually source-dominated; i.e. the most significant terms are production,

redistribution and dissipation. • Variables such as k and must be non-negative. This demands: – care in discretising the source term (see below); – use of an unconditionally-bounded advection scheme. Source-Term Linearisation For Non-Negative Quantities The general discretised scalar-transport equation for a control volume centred on node P is PPP

FFFPP sbaa φ+=φ−φ �

For stability one requires 0≤Ps

To ensure non-negative φ one requires, in addition, 0≥Pb You should, by inspection of the k and transport equations (3) be able to identify how the source term is linearised in this way.

U

uv

vu


If bP < 0 for a quantity such as k or which is always non-negative (e.g. due to transfer of non-linear parts of the advection term or non-diffusive fluxes to the source term) then, to ensure that the variable doesn’ t become negative, the source term should be rearranged as

0

)(*

→

φφ

+→

P

PP

PPP

b

bss

(25)

where * denotes the current value of a variable. 7.3.2 Wall Boundary Conditions At walls the no-slip boundary condition applies, so that both mean and fluctuating velocities vanish. At high Reynolds numbers this presents three problems: • there are very large flow gradients; • wall-normal fluctuations are suppressed (i.e. selectively damped); • viscous and turbulent stresses are of comparable magnitude. There are two main ways of handling this in turbulent flow – low-Reynolds-number turbulence models and wall functions. Low-Reynolds-Number Turbulence Models • Aim to resolve the flow right up to the boundary. • Have to include effects of molecular viscosity in the coefficients of the eddy-viscosity

formula and (or ) transport equations. • Try to ensure the theoretical near-wall behaviour:

)0(,constant~2

~, 32

2 →∝∝ yyy

kyk t (26)

• Full resolution of the flow requires the near-wall node to satisfy y+ ≤ 1, where

�yu

y ≡+ , /� wu = (27)

This can be very computationally demanding, particularly for high-speed flows. High-Reynolds-Number Turbulence Models • Bridge the near-wall region with wall functions; i.e.

assume profiles (based on equilibrium boundary-layer theory) between near-wall node and boundary.

• OK if the equilibrium assumption is reasonable (e.g. slowly-developing boundary layers); dodgy in highly non-equilibrium regions (particularly near impingement/separation/reattachment points).

• The near-wall node should optimally be placed in the region 30 < y+ < 50 (range 15 -150 just about acceptable). This means that numerical meshes cannot be arbitrarily refined close to solid boundaries.

Up

w(wall shear stress)

assumed velocityprofile

control volume

near-wallnode

τ


In the finite-volume method, various quantities are required of the wall-function approach. (i) Mean momentum These require the wall shear stress, w ( 2

�u= ). This is determined as follows.

If the near-wall node lies in the logarithmic region then

,)ln(1 �

�

uyyEy

u

U PPP

P == ++ (28)

where subscript P denotes the near-wall node. Given the value of UP and yP this can be solved (iteratively) for u� and hence the wall stress w. However, a better approach when the turbulence is clearly far from equilibrium (e.g. near separation or reattachment points) is to

estimate an “equivalent” friction velocity proportional to k and estimate the wall shear stress from the tangential velocity UP and turbulent kinetic energy kP at the near-wall node, assuming an eddy viscosity t = u0y. This gives

2/14/1�0

0

0 where,)ln(

/ PP

Pw kCu

uyE

Uu== (29)

(If the turbulence were genuinely in equilibrium, then u0 would equal u� and (28) and (29) would be equivalent). A similar approach can be applied for rough-wall boundary layers. (ii) Turbulent kinetic energy The source term of the k transport equation requires average values of production P(k) and dissipation rate over a cell. These are derived by integrating assumed profiles for these quantities:

)(,,

)(,2

,0

�

30)(

�2

)(

yyy

u

y

UP

yyy

kP

wk

k

>=∂∂=

<==

(where 2/1�� /4.20 ky ≈ is the height of the viscous sublayer) over the depth of the near-

wall cell to give:

)ln(2

)ln()/(

�

30

�

�0

2)(

y

u

y

k

yuP

av

wkav

+=

= (30)

k� , the turbulent kinetic energy at the top of the viscous sublayer, is usually assumed to be equal to kP.


(iii) Dissipation rate is fixed from its equilibrium-layer profile at the near-wall node:

P

P y

u30= (31)

(iv) Reynolds stresses For the Reynolds-stress transport equations, the values of individual stresses at the near-wall

node are fixed by the value of k and the structure functions kuu ji / , with the latter derived

from the differential stress-transport equations on the assumption of local equilibrium. For the standard model this gives (see the example sheet):

k

v

CC

CCC

k

uv

k

v

C

C

C

CCCC

k

w

k

v

C

C

C

CCCC

k

u

CC

CCCC

k

v

w

w

ww

ww

w

w

2

)(12

31

2)(

223

2

2

1

)(1

1

2)(

2212

2

1

)(1

1

2)(

2212

)(11

2)(

2212

1

1

3

2

22

3

2

2

21

3

2

��

��

�

++−

=−

+��

��

� +++−=

+��

��

� +−+=

��

��

�

+−++−

=

(32)

With the values for C1, C2, etc. from the standard model this gives

255.0,654.0,248.0,098.1222

=−===k

uv

k

w

k

v

k

u (33)

7.3.3 Effective Viscosity for Differential Stress Models DSMs contain no turbulent viscosity and have a reputation for numerical instability. An artificial means of promoting stability is to add and subtract a gradient-diffusion term to the turbulent flux:

�

� ��

� �� )(x

U

x

Uuuuu aa

∂∂

−∂∂

+= (34)

with the last part treated implicitly. The simplest choice for the effective viscosity �� is just

2

�� k

Ct == (35)

A better choice is to make use of a natural linkage between individual stresses and the corresponding mean-velocity gradient which arise from the actual stress-transport equations.


Assuming that the stress-transport equations (with no body forces) are source-dominated then 0≈−+ ijijijP

or, with the basic DSM (without wall-reflection terms),

0)()( 32

31

232

1 ≈−−−−− ijijkkijijji

ij PPCk

uuCP

Expand this, identifying the terms which contain only �� uu or �

�

x

U

∂∂

as follows.

For the normal stresses 2�u :

0)( ��32

22�1�� =+−−− �� PCu

kCP

Hence,

)2()1(

)1(

�

�2�

1

232

��

1

232

2�

�

�

+∂∂

−−

=

+−

=

x

Uu

k

C

C

Pk

C

Cu

Similarly for the shear stresses �� uu :

0��2��1�� =+−− �PCuuk

CP

whence

)(

)1(

)1(

�

�2�

1

2

��

1

2��

�

�

+∂∂

−−=

+−=

x

Uu

k

C

C

Pk

C

Cuu

Hence, from the stress-transport equations,

�

�

+∂∂

−=

+∂∂

−=

�

��

�

��

2�

x

Uuu

x

Uu

(36)

where the effective viscosities (both for the Uα component of momentum) are:

1

,1

22�

1

2� �

2�

1

232

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C

Cuk

C

C��

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Note that the effective viscosities are anisotropic, being linked to particular normal stresses. A more detailed analysis can accommodate wall-reflection terms in the pressure-strain model, but the extra complexity is not justified.


Some References for Individual Turbulence Models Apsley, D.D. and Leschziner, M.A., 1998, A new low-Reynolds-number nonlinear two-equation turbulence model for complex flows, Int. J. Heat Fluid Flow, 19, 209-222 Craft, T.J., 1998, Developments in a low-Reynolds-number second-moment closure and its application to separating and reattaching flows, Int. J. Heat Fluid Flow, 19, 541-548. Craft, T.J., Launder, B.E. and Suga, K., 1996, Development and application of a cubic eddy-viscosity model of turbulence, Int. J. Heat Fluid Flow, 17, 108-115. Gatski, T.B. and Speziale, C.G., 1993, On explicit algebraic stress models for complex turbulent flows, J. Fluid Mech., 254, 59-78. Gibson, M.M. and Launder, B.E., 1978, Ground effects on pressure fluctuations in the atmospheric boundary layer, J. Fluid Mech., 86, 491-511. Hanjali � , K., 1994, Advanced turbulence closure models: a view of current status and future prospects, Int. J. Heat Fluid Flow, 15, 178-203. Jakirli � , S. and Hanjali � , K., 1995, A second-moment closure for non-equilibrium and separating high- and low-Re-number flows, Proc. 10th Symp. Turbulent Shear Flows, Pennsylvania State University. Lam, C.K.G. and Bremhorst, K.A., 1981, Modified form of the k-e model for predicting wall turbulence, Journal of Fluids Engineering, 103, 456-460. Launder, B.E., 1989, Second-Moment Closure and its use in modelling turbulent industrial flows, Int. J. Numer. Meth. Fluids, 9, 963-985. Launder, B.E., Reece, G.J. and Rodi, W., 1975, Progress in the development of a Reynolds-stress turbulence closure, J. Fluid Mech., 68, 537-566. Launder, B.E. and Sharma, B.I., 1974, Application of the energy-dissipation model of turbulence to the calculation of flow near a spinning disc, Letters in Heat and Mass Transfer, 1, 131-138. Launder, B.E. and Spalding, D.B., 1974, The numerical computation of turbulent flows, Computer Meth. Appl. Mech. Eng., 3, 269-289. Lien, F-S. and Leschziner, M.A., 1993, Second-moment modelling of recirculating flow with a non-orthogonal collocated finite-volume algorithm, in Turbulent Shear Flows 8 (Munich, 1991), Springer-Verlag. Menter, F.R., 1994, Two-equation eddy-viscosity turbulence models for engineering applications, AIAA J., 32, 1598-1605. Shih, T-H., Liou, W.W., Shabbir, A., Yang, Z. and Zhu, J., 1995, A new k-� eddy viscosity model for high Reynolds number turbulent flows, Computers Fluids, 24, 227-238. Speziale, C.G., Sarkar, S. and Gatski, T.B., 1991, Modelling the pressure-strain correlation of turbulence: an invariant dynamical systems approach, J. Fluid Mech., 227, 245-272. Wilcox, D.C., 1988, Reassessment of the scale-determining equation for advanced turbulence models, AIAA J., 26, 1299-1310. Wilcox, D.C., 1988, Multi-scale model for turbulent flows, AIAA Journal, 26, 1311-1320. Wilcox, D.C., 1994, Simulation of transition with a two-equation turbulence model, AIAA J., 32, 247-255. Yakhot, V., Orszag, S.A., Thangam, S., Gatski, T.B. and Speziale, C.G., 1992, Development of turbulence models for shear flows by a double expansion technique, Phys. Fluids A, 7, 1510.

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Apsley-Turbulence modelling in CFD