Discrete unified gas-kinetic scheme for compressible flows

Zhaoli Guo

(Huazhong University of Science and Technology, Wuhan, China)

Joint work with Kun Xu and Ruijie Wang (Hong Kong University of Science and Technology)

Sino-German Symposium on Advanced Numerical Methods for Compressible Fluid Mechanics and Related Problems, May 21-27, 2014, Beijing, China

Outline

• Motivation

• Formulation and properties

• Numerical results

• Summary

Motivation Non-equilibrium flows covering different flow regimes

Inhalable particles Chips Re-Entry Vehicle

Slip

Continuum

Transition

Free-m

olecular

1010-3 10-1 10010-2

Challenges in numerical simulations

• Based on Navier-Stokes equations• Efficient for continuum flows• does not work for other regimes

• Noise• Small time and cell size• Difficult for continuum flows / low-speed non-equilibrium flows

Modern CFD:

Particle Methods: (MD, DSMC… )

• Theoretical foundations• Numerical difficulties (Stability, boundary conditions, ……)• Limited to weak-nonequilibrium flows

Method based on extended hydrodynamic models :

Lockerby’s test (2005, Phys. Fluid)

= constthe most common high-order continuum equation sets (Grad’s 13

moment, Burnett, and super-Burnett equations ) cannot capture the Knudsen Layer, Variants of these equation families have, however, been proposed and some of them can qualitatively describe the Knudsen layer structure … the

quantitative agreement with kinetic theory and DSMC

data is only slight

A popular technique: hybrid method

MD NS

Limitations

Artifacts

Time coupling

Numerical rather than physical

Dynamic scale changes

Hadjiconstantinou Int J Multiscale Comput Eng 3 189-202, 2004

Hybrid method is inappropriate for problems with dynamic scale changes

Efforts based on kinetic description of flows

# Discrete Ordinate Method (DOM) [1,2]:

• Time-splitting scheme for kinetic equations (similar with DSMC)• dt (time step) < (collision time)• dx (cell size) < (mean-free-path)• numerical dissipation dt

# Asymptotic preserving (AP) scheme [3,4]:

Works well for highly non-equilibrium flows, but encounters difficult for continuum flows

Aims to solve continuum flows, but may encounter difficulties for free molecular flows

• Consistent with the Chapman-Enskog representation in the continuum limit (Kn 0)

• dt (time step) is not restricted by (collision time)• at least 2nd-order accuracy to reduce numerical dissipation [5]

[1] J. Y. Yang and J. C. Huang, J. Comput. Phys. 120, 323 (1995)[2] A. N. Kudryavtsev and A. A. Shershnev, J. Sci. Comput. 57, 42 (2013).[3] S. Pieraccini and G. Puppo, J. Sci. Comput. 32, 1 (2007).[4] M. Bennoune, M. Lemo, and L. Mieussens, J. Comput. Phys. 227, 3781 (2008).[5] K. Xu and J.-C. Huang, J. Comput. Phys. 229, 7747 (2010)

# Unified Gas-Kinetic Scheme (UGKS) [1]:

Efforts based on kinetic description of flows

A dynamic multi-scale scheme, efficient for multi-regime flows

• Coupling of collision and transport in the evolution• Dynamicly changes from collision-less to continuum according to the local

flow• The nice AP property

[1] K. Xu and J.-C. Huang, J. Comput. Phys. 229, 7747 (2010)

In this report, we will present an alternative kinetic scheme (Discrete Unified Gas-Kinetic Scheme), sharing many advantages of the UGKS method, but having some special features .

Outline

• Motivation

• Formulation and properties

• Numerical results

• Summary

# Kinetic model (BGK-type)

1 eqtff ff

té ù¶ + ×Ñ =Wº - -ë ûx

( , , )ff t= x xDistribution function

Particel velocity

[ , ( , ), ( , ),...]eq eqff t t= W x J xxEquilibrium:

Conserved variables Flux

Example: 2

(3 )/ 2exp

(2 ) 2eq M

K

cff

RT RT

r

p +

é ùê ú= = -ê úë û

Maxwell (standard BGK)

2

1 (1 Pr) 55

eq S M cff f

pRT RT

é æ öù× ÷çê ú= = + - - ÷ç ÷çè øê úë û

c q Shakhov model

ES model 1exp

2det(2 )eq ESff

r

p

é ùê ú= = - ×L ×ê úë ûL

c c

11

Prij ij ijRT d s

æ ö÷çL = + - ÷ç ÷çè ø

( )fd

é ùê úê ú= = Xê úê úê úë û

òW u

E

r

r j x

r

Conserved variables 1 22

1

( )

é ùê úê ú= ê úê úê úë û

j x x

x

[ ]10eqfd ff fdW X = - X =ò òj j

t

Conservation of the collision operator

A property: for any linear combination of f and f eq , i.e.,

( )f d¢= XòW j x

(1 ) eqff fb b¢= - +

The conservation variables can be calculated by

# Formulation: A finite-volume scheme1 eq

tff ffté ù¶ + ×Ñ =Wº - -ë ûx

1/ 2 1/ 2 11/ 2 1

1/ 2 2

n n n nj jj j

n nj jff

t tff

x+ + + +

+ -

D Dé ù é ù- + - = W +Wê ú ë ûë ûDx

j j+1

j+1/2

1. integrating in cell j:

Trapezoidal Mid-point

2. transformation:

2t

ff= -D

W% 2 22 2 2

eqt t tff

t tff+

D - D DW= +

+D D= +

+%% t

t t

1/ 2 1/ 21/ 2

,11/ 2

n nj

nnj jjf

tf

xf f++ + +

- +

D é ù= + -ê úë ûD% % x

3. update rule:

Key: distribution function at cell interface

12

Point 1: Updating rule for cell-center distribution function

1 eqtff ff

té ù¶ + ×Ñ =Wº - -ë ûxAgain

j j+1

j+1/2

1. integrating along the characteristic line

1/ 2 1/ 21/ 21/ 2 1/ 2 ( )

2n n

jn n

jj jf xfh

h+++ +

+ é ù- = W +W -ê úë ûx

explicit Implicit

2. transformation:

2f

hf= - W

2 22 2 2

eqh hff

hff

h h+ -

W= ++

++

=tt t

, 11/ 22

1/1/ 2 ,

1/ 2

, 11 1/

2

2 1 12

( ) ( ) , 0

( )

( ) ( ) , 0

nj j j j

jn

n

n

j j j

j

j

f x x x t

x h

f

f

x x

f

x t

x s x

x

x s x

+ ++

+

+ ++ +

+ ++

+ +

ìï + - - D ³ïïï= - = íïï + - - D <ïïî

So

1/ 21/ 2

njf++

2t

hD

=

1/ 2 1/ 2 1/ 21/ 2 1/ 2 1/ 2

2( )

2 2n n neqj j j

hff f

h h+ + ++ + += +

+ +W

tt t

3. original:

1/ 2 1/ 21/ 2 1/ 2( ) ( )n n

j jf dy x x x+ ++ += òW

1 eqtff ff

té ù¶ + ×Ñ =Wº - -ë ûxAgain

Point 2: Evolution of the cell-interface distribution function

How to determine

js+

j jfs+ += Ñ

Slope

# Boundary condition

·( , , ) ( , , ) 2 ,

0

i ww i

iw

i

iw i

Wf t h f t h

w RTr+ = - + +

× >n

ux x

xx x

x

Bounce-back

Diffuse Scatting

1

0 0 0

21 ( ) 2 ( ) .

i i i

i i iw i w i iW wf w fRT

r

-

×> ×= ×<

é ù é ùê ú ê ú= - × +ê ú ê úê ú ê úë û ë û

å å ån n n

ux x x

x x x

n

( , , ) ( ; , , 0)eqw i i w w if t h f r+ >= ×x u nx x x

0 0

( ) ( , , ) ( ) ( , )i i

eqw w wi i i iff t hr

×> ×<

=× × +å ån n

n u n xx x

x x x ,x

wr

n

# Properties of DUGKS1. Multi-dimensional

2. Asymptotic Preserving (AP)

1/ 2 1/ 2 1/ 21/ 2 1/ 2 1/ 2

2( )

2 2n n neqj j j

hff f

h h+ + ++ + += +

+ +W

tt t

(a) time step (t) is not limited by the particle collision time ():

(b) in the continuum limit (t >> ):

1/ 2 1/ 2 1/ 2eq eq eq

tj j jf Df h f+ + +- + ¶t

Chapman-Ensokg expansion

in the free-molecule limit: (t << ): 1/ 21/ 21/ 2 ( )n n

jjff x h+++ = - x

(c) second-order in time; space accuracy can be ensured by choosing linear or high-order reconstruction methods

• It is not easy to device a wave-based multi-dimensional scheme based on hydrodynamic equations

• In the DUGKS, the particle is tracked instead of wave in a natural way (followed by its trajectory)

Unified GKS (Xu & Huang, JCP 2010)

Starting Point:

1 eqtff ff

té ù¶ + ×Ñ =Wº - -ë ûx

0t¶ +Ñ × =W F f d= òF xy x Macroscopic flux

Updating rule:

11

1/ 21

( ) 0| |

n

n

tn nj j j

tj

t dtV

++

+- + =òW W F

j j+1

j+1/2

# Comparison with UGKS

11 1

1/ 2 11

/ 21

( ) ( ) ( , ) ( , )2

n

n

tn n n n

j jn nj j jj j j

t

tf t f t dt ff

xff x

++ +

+ -+ Dé ù é ù- + - = W +Wë û ë ûD ò W W

If the cell-interface distribution f(t) is known, the update both f and W can be accomplished

Unified GKS (cont’d)

Key Point:

j j+1

j+1/2

( )/ ( )// /( ) ( , ) ( ( ), )1 2 1 2

1n

n

teq t t t t

j j n nt

f t f x t e dt e f x t t tt t

t¢- - - -

+ +¢ ¢ ¢= + - -ò x

1 eqtff ff

té ù¶ + ×Ñ =Wº - -ë ûx

( )/ // // ( ) ( )( ) ˆˆ1 2 11 2 21 t t

j neqj jnf t e ef ft tt t

+-

+-

+ = - +

After some algebraic, the above solution can be approximated as

Integral solution:

Free transport Equilibrium

t tD ?/

ˆ ( )1 2eq

njf t+ Chapman-Enskog expansion

t tD =/

ˆ ( )1 2j nf t+ Free-transport

DUGKS vs UGKS

(a)Common:

(b) Differences: in DUGKS

• W are slave variables and are not required to update simultaneously with f

• Using a discrete (characteristic) solution instead of integral solution in the construction of cell-interface distribution function

• Finite-volume formulation; • AP property; • collision-transport coupling

Lattice Boltzmann method (LBM)

Standard LBM: time-splitting scheme

[ ]( , ) ( , ) ( , ) ( , )eqi i i i i

tf t t t f t f t f t

D+ D +D - =- -x c x x x

t

ci

( )( )

2

21

22i ieq

iif wRT RTRT

é ù× × ×= + + - +ê ú

ê úë û

c u c u u uLr

[ ]1 eqt i i i i iff ff¶ + ×Ñ =- -c

t

# Comparison with Finite-Volume LBM

Viscosity: ( )2p

trn t= -

DNumerical viscosity is absorbed into the physical one

[ ]1 eq

t i i iff ft

¶ =- -Collision [ ]( , ) ( , ) ( , ) ( , )eqi i i i

tf t f t f t f t

tD¢ = - -x x x x

( , ) ( , )i i if t t t f t¢+ D +D =x c xFree transport 0t i i iff¶ + ×Ñ =c

Evolution equation:

Limitations:

2. Low Mach incompressible flows 1. Regular lattice

[ ]1 eqt i i i i iff ff¶ + ×Ñ =- -c

t

Finite-volume LBM (Peng et al, PRE 1999; Succi et al, PCFD 2005; )

j j+1

j+1/2

[ ], ,, / , /ˆ ˆ( ) ( ( ) ( )) ( ) ( )1 2 11 2

1n n

eqj i n j i n i ni ij nj i

tf t f t f t f t

xf t f t

t+ + -

D é ù- + - =- -ê úë ûD

Limitations (Succi, PCFD, 2005):

2tD < t 2. Large numerical dissipation 1. time step is limited by collision time

# Comparison with Finite-Volume LBM

Micro-flux is reconstructed without considering collision effects

prn t=Viscosity: Numerical dissipation cannot be absorbed

DUGKS is AP, but FV-LBM not Difference between DUGKS and FV-LBM:

Outline

• Motivation

• Formulation and properties

• Numerical results

• Summary

Test cases

• 1D shock wave structure

• 1D shock tube

• 2D cavity flow

2

1 (1 Pr) 55

eq S M cff f

pRT RT

é æ öù× ÷çê ú= = + - - ÷ç ÷çè øê úë û

c qShakhov model

Collision model:

1D shock wave structureParameters: Pr=2/3, = 5/3, Tw

Left: Density and velocity profiles; Right: heat flux and stress (Ma=1.2)

DUGKS agree with UGKS excellently

Again, DUGKS agree with UGKS excellently

DUGKS as a shock capturing scheme

Density (Left) and Temperature (Right) profile with different grid resolutions (Ma=1.2, CFL=0.95)

1D shock tube problemParameters: Pr=0.72, = 1.4, T0.5

( , , ) ( . , . , . )10 00 10Lu pr =

( , , ) ( . , . , . )0125 00 01Ru pr =

Domain: 0 x 1;Mesh: 100 cell, uniformDiscrete velocity : 200 uniform gird in [-10 10]

Reference mean free path

By changing the reference viscosity at left boundary, the flow can changes from continuum to free-molecular flows

, , . , ,3 50 10 1 01 10 10m - -=

. , . , . , . , .3 50 1277 1277 01277 1277 10 1277 10l - -= ´ ´

0 0.5 10

0.2

0.4

0.6

0.8

1

UGKSpresent

0 0.5 11.4

1.6

1.8

2

2.2

UGKSpresent

0 0.5 10

0.2

0.4

0.6

0.8

1

UGKSpresent=10: Free-molecular flow

=1: transition flow

0 0.5 10

0.2

0.4

0.6

0.8

1

UGKSpresent

0 0.5 11.4

1.6

1.8

2

2.2

UGKSpresent

0 0.5 10

0.2

0.4

0.6

0.8

1

UGKSpresent

=0.1: low transition flow

0 0.5 10

0.2

0.4

0.6

0.8

1

UGKSpresent

0 0.5 11.4

1.6

1.8

2

2.2

UGKSpresent

0 0.5 10

0.2

0.4

0.6

0.8

1

UGKSpresent

=0.001: slip flow

0 0.5 10

0.2

0.4

0.6

0.8

1

UGKSpresent

0 0.5 11.4

1.6

1.8

2

2.2

UGKSpresent

0 0.5 10

0.2

0.4

0.6

0.8

1

UGKSpresent

=1.0e-5: continuum flow

0 0.5 10

0.2

0.4

0.6

0.8

1

UGKSpresent

0 0.5 11.4

1.6

1.8

2

2.2

UGKSpresent

0 0.5 10

0.2

0.4

0.6

0.8

1

UGKSpresent

2D Cavity FlowParameters: Pr=2/3, = 5/3, T0.81 Domain: 0 x, y 1;

Mesh: 60x60 cell, uniformDiscrete velocity : 28x28 Gauss-Hermite

Kn=0.075Temperature. White and background: DSMCBlack Dashed: DUGKS

Kn=0.075Heat Flux

Kn=0.075Velocity

Temperature and Heat FluxKn=1.44e-3; Re=100

UGKS: Huang, Xu, and Yu, CiCP 12 (2012) Present DUGKS

Comparison with LBM Stability:

Re=1000

LBM becomes unstable on 64 x 64 uniform mesh

UGKS is still stable on 20 x 20 uniform mesh

80 x 80 uniform mesh

LBM becomes unstable as Re=1195

UGKS is still stable as Re=4000 (CFL=0.95)

DUGKS LBM

Velocity

DUGKS

LBM

Pressure fields

Summary

• The DUGKS provides a potential tool for compressible flows in different regimes

• The DUGKS method has the nice AP property

Thank you for your attention!