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; myong@gnu.ac.kr

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.