A Review of Boltzmann-Based CFD Schemes
March 18, 2015 (10:50-11:30)
R. S. Myong, Ph.D.
Dept. of Aerospace and System Engineering
Gyeongsang National University (GNU)
South Korea
http://acml.gnu.ac.kr; [email protected]
Presented at the 18th International Conference on
Finite Elements in Flow Problems
(FEF2015; Kinetic Methods in Fluid Flow)
16-18 March 2015, Taipei, Taiwan
Special thanks to
organizers of this conference
(Profs. Chao-An Lin, Jong-Shinn Wu)
for making this opportunity possible.
Background: why kinetic methods in fluids?
(High Reynolds) fluid dynamics is difficult because:
(Low Reynolds mesoscopic) fluid dynamics is difficult because:
At extremely small scales, even turbulent flow is very
simple. It is smooth and well behaved. At larger
scales, however, a fluid is subject to very few
constraints. It can develop arbitrary levels of
complexity like the effect of turbulence on separation.
(P. L. Roe, “Future developments in CFD,” ICAAT-GNU, May 2014)
It involves microscopic collisional interactions among fluid
particles and their interplay with the kinematic motion of
particles in the macroscopic framework. This challenge is
vividly illustrated by the high Mach number shock singularity
problem (HMNP). (R. S. Myong, “On the high Mach number shock
structure singularity caused by overreach of Maxwellian molecules,”
Physics of Fluids, May 2014) Rarefied, micro- & nano- gases,
viscoelastic fluids, elastic solids
Part I.
Introduction to Boltzmann-
based computational
schemes
Continuum vs molecular
Navier & Fourier
conservation
laws and
constitutive
laws (1822)
Maxwell (1867), Boltzmann (1872)
kinetic
2( , , ) ,f t C f ft
v r v
(2)
(2)
2 [ ]
Equivalently,
and viscous stress 2 [ ]
dp
dt
dp
dt
uu
uI Π 0
Π u
How BTE is connected with continuum
2( , , ) ,f t C f ft
v r v
0d
pdt
u
I Π
( , , )f t r vBoltzmann transport equation (BTE): 1023
( , , )
Enormous reduction of information
x y z
mf t
dv dv dv
r v
),(,,,,, rQΠu tT
the statistical definition
( , , )
and
with BTE
Differentiating
m f t
with time
then combining
u v r v
Conservation laws & constitutive equations: 13
Nonlinear collision
integral
Still exact to BTE
Boltzmann-based CFD schemes (partial list)
Basically a game of reducing the degree of freedom of BTE from
1023 to 103 -1010, the level modern computers can handle
DSMC LBM Gas-kinetic Chapman-
Enskog
Moment
method A representative particle to cover real particles in order of 1013
Then describing the motion of the particles via deterministic movement and stochastic collision
No PDE
Solving BTE on discrete lattice
Introducing finite numbers of discrete velocity
Finally replace by 1st-order accurate BGK
Solving a discretized version of conservation laws
and discretized BGK-BTE in an iterative way
Only discretized PDE
Assuming
and inserting into BGK-BTE and deriving 1st, 2nd, 3rd approximations
PDE of high order
Need of extra BC due to
( , , )
( , ), ( , ), ,
f t
f t t
r v
W r W r v
Differentiating the statistical definition
with time
and combining with BTE
PDE of high order
( , , ) m f tu v r v
( , )tW r
Continuum Molecular
Boltzmann CFD schemes-continued
DSMC
( , , )
Sampling
x y zmf t dv dv dv r v
( , , ) 0
Kinematic: the collision-less
movement of molecules
f tt
v r v
* *
2 2 2 2( )
Molecular collisions
ff f ff d
t
v v v
Constraints:
t < mean collision time
< mean free path
Particles cross less than
1 cell/timestep
x
Gas-kinetic scheme
K. Xu (2014)
Some points for Boltzmann-based schemes
2
1 10
Re
dp
dt M
uI Π
I. Kn vs (Kn, M)
II. The equation below is an exact consequence of BTE, meaning
its validity regardless of the degree of non-equilibrium.
should be replaced by
Arkilic et al. (1997)
Some points for Boltzmann-based schemes-
continued
2
1 10 or 0 at steady-state
Re Sp dS
M
uu I Π F n
III. Conservation laws and boundary treatments
- A verification is possible based on the following property
- A challenge is the statement of C. Villani (2010 Field Medalist):
“The conservation laws should hold true when there are no
boundaries. In presence of boundaries, however, conservation
laws may be violated: momentum is not preserved by specular
reflection.”
In conservative PDE-based schemes, the nearest cell to the wall
always satisfies the conservation laws via
but it may not be true in the case of non-PDE schemes.
41
, ,
1,
n n
i j i j k k
ki j
tL
A
U U F
Part II.
CFD schemes based on the
continuum version of BTE:
the moment method (PDE)
Continuum version of BTE (Maxwell 1867)
2( , , ) ,f t C f ft
v r v
Boltzmann transport equation (BTE): 1023
),(,,,,, rQΠu tT
( ) the statistical definition of moments ( , , )
and with BTE
nDifferentiating h f t with time
then combining
r v
Conservation laws & constitutive equations: finite
Boltzmann collision
integral
Still exact to BTE
* *2 2 2 2( )f f ff d v v v
( )
( ) ( ) ( ) ( )2[ , ]
nn n n nd dh
h f h f f f h h C f fdt dt
c c
Two places (not one)
to closure Non-hyperbolic
Not necessarily explicit
Moment methods: Grad (1949)
(2)(0) 2
Polynomial expansion with 3 terms
1 1ˆˆ ˆ ˆ ˆ ˆ1 : 12 5
was inserted into the Maxwell's continuum version of BTE.
f f c
Π cc Q c
(2)(2) (2)
(2) (2) (2)
3
( ) ( )
3 1 1( )
1
Then we have (when viscous stress , )
( / ) 22 2 ,
5
1where 1 ( )(quadratic terms of : )
6 /
and high order
rd Approx
NS
Q
rd Approx
h f h m
d pp F
dt
F B BB m m
Π cc
ΠQ Π u u Π
Π ΠΠ
(2) (2)(2)
3
2term .
5rd Approx
h f m f
c c cc Q
( )
( ) ( ) ( ) ( )2[ , ]
nn n n nd dh
h f h f f f h h C f fdt dt
c c
Genesis of the high Mach number problem
(HMNP): Grad 1952
By arguing that ( ) ( ) ( )
1 1 1( ) / 1/ 7,QB B B
Grad ignored the quadratic terms in the dissipative collision term
(2) (2) (2)
1 1
( / ) 22 2 , where 1
5st Approx st Approx
NS
d pp F F
dt
ΠQ Π u u Π
Grad solved this equation (quadratic for left-hand side due to presence
of , while linear for right-hand side) in 1952, but found a shock
structure singularity at M=1.65 and could not figure out why.
(2)
2 Π u
The ultimate origin of Grad’s failure was found to be the unbalanced
closure; ignoring the quadratic term in the dissipation term by Myong in
2014.
While can be small near the equilibrium, it will increase
as the problem becomes away from equilibrium! So we cannot ignore it.
( ) ( ) ( )
1 1 1( ) /QB B B
Moment methods: Truesdell (1956,1980)
Re-enforcement of Grad’s unbalanced approach
( ) ( )2[ , ]n n
Exacth C f f h f
It was rigorously proven in the Maxwellian
molecule that the linear relation in the collisional
term is an exact consequence of the original
Boltzmann collision integral by Truesdell (“On the
pressures and the flux of energy in a gas according to Maxwell’s
kinetic theory,” II, J. Rational Mech. Anal. 5 (1956), 55–128.)
This misuse, the linear relation
of the Maxwellian molecule,
was never questioned by
practitioners in the field
until recently.
Summary of the origin of high Mach number
shock singularity: Procrustean bed
Procrustean bed
Enforcing conformity without regard to natural variation or individuality, just as Procrustes violently adjusted his guests to fit their bed.
In order to force the constitutive equations—not necessarily hyperbolic—into a hyperbolic system (with distinct eigenvalues), over-simplify the Boltzmann collision integral by assuming the Maxwellian molecule (which is exceptional, rather than general)
刖趾適屨 yuè zhǐ shì jù: Cut the tiptoe in order to fit the foot into
a shoe (rather than modifying the shoe)
( ) ( )2[ , ]n nh C f f h f
And this is the precious reason why the shock singularity (originally accidental) remained unsolved for several decades! (R. S. Myong, Phys. Fluids, 2014)
The constitutive theory: new balanced
closure
Then the solution to remove the HMNP is to abandon the 1st-
order accurate linear relation in dissipative collision term.
( )
( ) ( ) ( ) ( )2[ , ]
nn n n nd dh
h f h f f f h h C f fdt dt
c c , where
the RH terms represent the change due to the molecular collision.
(No approximations so far!: Approximations was deferred to the
last stage in order to minimize accumulated errors.)
(2) (2)( / )
2 2 exact
NS
d pp F
dt
ΠΨ Π u u Π
Then, when 2nd-order balanced closure is applied; we have
(2) (2)
2
( / ) 0 2 2 nd order
NS
d pp F
dt
ΠΠ u u Π
Only remaining task is to determine F2nd-order; nothing else!
Any 2nd-order expression can basically be used for the
nonlinear factor, but the hyperbolic sinh form, originally
derived by B. C. Eu in 80-90s, was found adequate.
Key ideas are; exponential canonical form, consideration of
entropy production σ, and non-polynomial expansion
called as cumulant expansion.
The constitutive theory: Exponential form and
2nd law of thermodynamics
2 ( ) ( )
1
( ) ( )
2 2
1
the distribution function in the exponential form
a thermodynamically-consistent constitutive equati
By writing
1 1exp , ,
2
1ln [ , ] [ , ] ,
n n
n B
n n
B
n
f mc X h Nk T
k f C f f X h C
f
f fT
(2) (2)( ) (2 ) ( ) ( ) ( )
12 2 1 2
1
(
on, still exact to BTE, can be derived
/ ) 12 2 ( , , ).
;
l l
l
dp R X q
dt g
Π
Π u u
The 2nd-order constitutive theory:
sinh nonlinear form
The simplest closure of LH term in next level to the linear Navier-
Fourier theory is
while the 2nd-order closure of RH term is Then
( ) 0,
(21) (1) ( )
12 2 1( ).R X q
(2) (2)
1
1/21/4 1/41
1 11
( / )2 2 ( ),
sinh ( ) :( ) , .
2
NS
B
NS NS
d pp q
dt
mk T Tq
p kd
ΠΠ u u Π
Π Π Q Q /
This new 2nd-order constitutive equation beyond the two-
century old Navier-Stokes equation, of course, recovers NS
(2)
2 ,NS
pp
u Π and reduces to an algebraic equation in
steady-state (also assuming 1-D)
2 41 1ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆΠΠ Π Π ( Π ) Π Π( Π) Π( Π) .3! 5!
NS NS q c c c
Implicit!
Grad non-classical
model (1952)
Myong non-classical
model (1999, 2014)
Navier classical model (1822)
Gaseous compression
기체팽창
Velocity gradient /pressure
Singularity
Gaseous expansion
Viscous stress /pressure
Π /NS p
/ p
NCCR: Nonlinear Coupled Constitutive Relation
The singularity due to unbalancing does not
occur in expansion flow, since the sign of
quadratic term is opposite.
1F sinh p
Fp
Grad non-classical
model (1952)
Myong non-classical
model (1999, 2014)
Navier classical
model (1822)
Velocity gradient /pressure
Viscous stress /pressure
Π /NS p
/ p
NCCR: shear-dominated flow
The singularity due to unbalancing does not occur
in velocity shear flows, since the high order
nonlinear effects are cancelled under the
constraint of the asymptotic behavior.
The DG-NCCR CFD scheme
Conservation laws (exact
consequence of BTE)
1/ 0
t
dp
dtE p
u
u I Π 0
u Π u Q
in conjunction with the 2nd-order constitutive equations
(2) (2)
1
1
( / )2 2 ( ),
( / )( ).
NS
p
p p
NS
d pp q
dt
pCd dC T C p T q
dt dt k
ΠΠ u u Π
Q uΠ Q u Π Q
Non-hyperbolic
Implicit
Hyperbolic
(inviscid)
Conservation laws
t inv vis( ) ( , ) 0 U F U F U U
Discretization in mixed (hybrid) form ( include Ts )
s
t inv vis
T 0
( ) ( , ) 0
S U
U F U F U S
NSF model (П, Q) = flinear(S(U))
NCCR model (П, Q)NCCR = fnon-linear(S(U), p, T)
The DG-NCCR CFD scheme
0 0
( , ) , ( , )k k
i i i ih j h j
i i
t U t t S t
U x x S x x
inv inv vis vis
0,
0.
s s
I I I
I I I I I
φdV T φ dV T φ d
φdV φ dV φ d φ dV φ dt
S U U n
U F F n F F n
Modal basis function proposed by Dubiner for triangular element
Boundary (∂I) integrals replaced by fluxes
The Lax-Friedrichs (LxF) flux for inviscid terms
inv1 inv1 inv1 inv1inv1
where
1, , ,
2
max ,s s
C
a aC
M M
F nx h U U nx F U F U U U
u u
The DG-NCCR CFD scheme
Central flux for viscous terms (Bassi and Rebay 1997)
Limiter proposed by K. Kontzialis et al. in 2013
Semi-discrete form resolved by explicit Runge-Kutta time integration
( )t
UL R U
Results: hypersonic case; M=5.48, Kn=0.5
Pressure
DSMC DSMC
NCCR NSF
0 1000 2000 3000 4000 5000 6000-7
-6
-5
-4
-3
-2
-1
Re
sid
ua
l (L
og
)
Iteration
Ma=5.48 Kn=0.5
NSF NCCR
Computing time
Le, N. T. P., Xiao, H., Myong, R. S., A
triangular discontinuous Galerkin method
for non-Newtonian implicit constitutive
models of rarefied and microscale gases,
Journal of Computational Physics, Vol.
273, pp. 160-184, 2014.
Results: hypersonic case; M=5.48, Kn=0.5
Temperature Mach number
DSMC DSMC DSMC DSMC
NCCR NCCR NSF NSF
Results: low Mach no. case; M=0.1, Kn=0.1
Density
DSMC DSMC
NCCR NSF
Results: airfoil case; M=2.0, Re=106
NCCR
DSMC
Experiment
NSF
Velocity
Part III. Ongoing topic
Secret of Boltzmann’s success and
pushing its boundary
(gas to elastic solid):
A unified theory for continuum
media
R. S. Myong, In Review, 2015
“On the high Weissenberg number singularity in the Maxwell-Oldroyd
model of viscoelastic fluids”
Cumulant expansion method-I
l-th moment of the distribution function and the moment-generated
function (x being the non-equilibrium variables) are
( ) , ( )l l x xx x f x dx e e f x dx
Then we have
0 1
=exp where ! !
l lx l
l
l l
e xl l
221 2
0
ln ; , , (mean, variance)l
xl l
de x x x
d
22
3
polynomial
1
2!
cumulant
12
3
2 / 2 sinh
x x
x xx xx x
e e x x x
e e e e e x
221 2 3
2 3
polynomical
1 1 1
2! 3! 2!
cumulant
1 11 ,
2! 3!
=
x
x x xx
e x x x
e e e
Cumulant expansion method-II
Therefore, the ratio of the cumulant expansion to the polynomial
expansion becomes
2 4cumulant
polynomial
/ 2sinh 1 1
13! 5!/ 2
x x
x x
e ex
x xxe e
The factor, , is responsible for removing the shock
singularity of the moment method and can handle transitional (and
even free-molecular in cavity flow as high as Kn=6.71) flow.
sinh x x
Cumulant expansion: E. Meeron, J. Chem. Phys. 27, p. 1238, 1957.
It explicitly considers terms of all orders in the perturbation expansions.
Gain – Loss = = sinh → Secret of Boltzmann’s success x xe e
Gaseous compression
Rarefied shock structure
High p
HMNP HWNP
Low p
Gas particle
Viscoelastic complex fluids
Polymer extension
Bead-Spring model
High Mach number problem High Weissenberg number problem
Fluids
category
Viscoelastic fluids (HWNP)
Viscoelastic stress
Gases in non-equilibrium
(HMNP)
Conserva-
tion laws
Constitutive
equations
Parameters
( ) ( )
1 for the Maxwell-Oldroyd model
T TD
Dt
τu τ τ u u u
τ
(2)( / )
2
1 for Maxwellian molecule
/
TD pp p
Dt
p
ΠΠ u u u
Π
τ ( ) Π τ
Wissenberg
2
/
/ Re
p
M
0D
pDt
u
I τ 0D
pDt
u
I Π
The head and tail of a coin: exact analogy
Resolving the HMNP automatically solves the HWNP.
Why analogy? Because all media follow two simple actions at a
molecular level: movement and interaction among particles!
Thank you for your attention.