Gas-kinetic schemes for flow computations

Kun Xu Mathematics Department

Hong Kong University of Science and Technology

AcknowledgementsAcknowledgements: : RGC6108/02E, 6116/03E,RGC6108/02E, 6116/03E, 6102/04E,6210/05E6102/04E,6210/05E

CollaboratorsCollaborators: : Changqiu Jin, Meiliang Mao, Changqiu Jin, Meiliang Mao, Huazhong Tang, Chun-lin TianHuazhong Tang, Chun-lin Tian

Contents• Gas-kinetic BGK-NS flow solver

• Navier-Stokes equations under gravitational field

• Two component flow

• MHD

• Beyond Navier-Stokes equations

FLUID MODELING

Molecular Models Continuum Models

Euler Navier-Stokes BurnettDeterministic Statistical

MD Liouville

DSMC Boltzmann

Chapman-Enskog

0.001 0.1 10Kn

Continuum Slip flow Transition Free moleculae

Gas-kinetic BGK scheme for the Navier-Stokes equations

fluxesfluxesji ,

• Based on the gas-kinetic BGK model, a time dependent gas distribution function is obtained under the following IC,

• Update of conservative flow variables,

LeftEU ),,(

RightEU ),,(

dttFtFx

WW j

t

jnj

nj ))()((

12/1

0

2/11

Gas-kinetic Finite Volume Scheme

)})()((exp{)( 2222/)2( VvUug K

BGK model:

Equilibrium state:

Collision time: p/

/)( fgvfuff yxt

)( yxt vgugggf To the Navier-Stokes order:

in the smooth flow region !!!

A single temperature is assumed: kTm 2/

• Relation between and macroscopic variables

• Conservation constraint

f w

)1/()24( and ,...

,... , where

)(2

1,,,1(

vector theof component theis where

4,3,2,1 ,

222

21

2

21

222

K

dddddudvdd

vuvu

fd

E

V

U

K

K

T

w

.4,3,2,1 ,0)( dfg

• BGK flow solver Integral solution of the BGK model

y. trajectorparticle theis )'(' where

)('),,,','(),,,,(

2/1

0

2/10//)'(

2/1

ttuxx

utxfedtevutxgvutxf

j

t

jttt

j

0f

gnt

1nt

)'('2/1 ttuxx j

2/1jx

• Initial gas distribution function

.0)( ,0)(

,0 )),(1(

,0 )),(1(

/)(

/)(

/)(

/

0

,,

dAuagdAuag

xAuaxag

xAuaxagf

x

xE

xV

xU

x

E

V

U

E

V

U

rrrlll

rrrr

llll

rlrl

02/1 jx

LeftEVU ),,,(

RightEVU ),,,(

on both sides of a cell interface. The corresponding is

where the non-equilibrium states have no contributions to conservative macroscopic variables,

0f

• Equilibrium state

part. variation temporal theis where

),)(H))(H1(1(0

A

tAxaxxaxggrl

0f

0grg

lg

2/1jx 1jxjx

g

• Equilibrium state is determined by

0 0

00

00

00

0

2/1

.

giveswhich

),0,(at 0)(

0 u u

rl

j

dgdg

E

V

Udg

txdgf

),))(H1()1()(H)1((

)))(H1)(()(H)((

))(H1()(H)()1((

)1/()1(

),,,,(

/

/

0//

0/

0/

2/1

rrllt

rrrlllt

rltt

tt

j

guutaguutae

guAuaguAuae

uguauatee

gAetge

vutxf

Where is determined byA

.0)(0

dtdfgt

• Numerical fluxes:

• Update of flow variables:

.),,,,(

)(

1

2/1

22221

2/1

dvutxf

vu

v

uu

F

F

F

F

j

jE

V

U

.))()(( 2/12/10

11 dttFtFww jj

t

xnj

nj

Double Cones510Re,5.9 M

Attachedshock

Detached shock

Double-cone M=9.50 (RUN 28 in experiment)

Mesh: 500x100

)y,x,t( ),,(

)M,L,B,A(

Unified moving mesh method

MdBddVdy

LdAddUdx

ddt

g

g

gggg VM

,UL

,VB

,UA

Unified coordinate system ( W.H.Hui, 1999)

physical domain computational domain

geometric conservation law

The 2D BGK model under the

transformation

fg

vfuff yxt

fg f AVvBUu

fLVvMUu

f

gg

gg

)()(

)()()(

),( vu ),( VU ),(g

VUg

Particle velocity macroscopic velocity Grid velocity

The computed paths

fluttering

tumbling

-

-

-

-

computed

experiment

fluid force as functions of phasexF

fluid force as functions of phaseyF

3D cavity flow3D cavity flow

BGK model under gravitational field:

fg

ffct

fc

)(

Integral solution:

t

ttt cxfedtectxgtcxf0

00//)'( )0,,('),','(

1),,(

2/)( ,)( 200 tctxxtcc

where the trajectory is

Integral solution:

12/1

2/1

1

)(1

))()((1

2/12/11

n

n

j

j

n

n

t

t

x

x

t

t

jjnj

nj dxdtWS

xdttFtF

xWW

tt

Gravitational potentialGravitational potential

where

Rx

Lx

xfor x<0

for x>0

LeftEU ),,(

RightEU ),,(

X=0X=0

0 )],)(()(1[

0 )],)(()(1[

2/10

2/100

jrrrrrr

jllllll

xAbcatbctag

xAbcatbctagf

]')()'()'()()'()'(1[

]')()'()'()()'()'(1[

0

0

tAbttcattbttcattg

tAbttcattbttcattgg

ttttn

rnn

rn

ttttn

lnn

ln

Initial non-equilibrium state:

Equilibrium state

})(()(1{])[1(

})(()(1{][

})]1([])(])[1)()((

][))()[()(()1{(

/0

/0

/

//0

rrrrrtn

r

llllltn

l

tttttnn

rnn

rn

nn

lnn

ln

tt

AbcatbctaecHg

AbcatbctaecHg

AetbcacHbca

cHbcateegf

The gas distribution function at a cell interface:

Flux with gravitational effect:

Flux without gravitational effect (multi-dimensional):

}(1{])[1(

}(1{][

})]1([]])[1(

][)[)(()1{(

/0

/0

/

//0

rrrtn

r

llltn

l

tttnn

rn

nn

ln

tt

AcactaecHg

AcactaecHg

AetcacHca

cHcateegf

N=500000 steps

Steady state under gravitationalpotential

L

xL

2sin

20

Diamond: with gravitational force term in fluxSolid line: without G in flux

.0)

)(2

1

1

0

)(2

1

0

1

(

/)(

and

/)(

22

2

22

1

,22)2()2()2()2(

11)1()1()1()1(

dud

u

uQ

u

uQ

where

Qfguff

Qfguff

xt

xt

)1(f and )2(f have different .

Gas-kinetic scheme for multi-component flow

Gas distribution function at a cell interface:

Shock tube test:

),,,( 1111 pU

),,,( 2222 pU

= +

Sod test

A Ms=1.22 shock wave in air hits a helium cylindrical bubble

air preshock P V U W 4.1,1,0,0,1

air postshock P V U W 4.1,5698.1,0,394.0,3764.1

Helium , P , V , U ,W 67.11001358.0

Shock helium bubble interaction

(Y.S. Lian and K. Xu, JCP 2000)

Ideal Magnetohydrodynamics Equations in 1D

energy. total theis )(2

1)(

2

1

and pressure total theis )(2

1 where

,0)]()[()(

,0)()(

,0)()(

,0)()(

,0)()(

,0)()(

,0)(

222222

222*

*

2*

2

zyx

zyx

xzyxxt

xxztz

xxyty

xzxt

xyxt

xxt

xt

BBBeWVU

BBBpp

WBVBUBBUp

WBUBB

VBUBB

BBUWW

BBUVV

BpUU

U

Moments of a gas distribution function:

kTmUug 2/ ],)(exp[)( 22/1

Equilibrium state:

The macroscopic flow variables are the moments of g.

For example,

.2

1Energy , Momentum ,Density 2 gduuugduUgdu

Then, according to particle velocities, we can split flow variables as:

0

0

-0

000 ugduugdu ,

)()( Momentum ,Density

gduugduu

UUU

With the definition of moments:

022/1

0

22/1

])(exp[))((

and ])(exp[))((

duUu

duUu

We have

)exp(

2

1;

)exp(

2

1

and

)(erfc2

1);(erfc

2

1

20

20

00

UuUu

UuUu

UuUu

Recursive relation:

nnn u

nuUu

2

112

Therefore,

)(2

1)(

2

1

)()(

2

1

2

1)

2

1()

2

1(

2

1

have weSimilarly,

. :splittingenergy thermal

2

1

2

1

2

1 :splittingenergy kinetic

mean which ,2

1

2

1)(

and , )(,)(

, ,

1010

11

1212333

00

112

0

012

11

00

upuUpupuUp

pUpUpU

ueueUe

uUuUUUU

ueuee

uUuUU

ueuUgduu

uUuU

uu

Kinetic Flux vector splitting scheme(Croisille, Khanfir, and Ghanteur, 1995)

12/1 jjf

j FFF

j+1/2

free transport

Flux splitting for MHD equations:

leftzyx

x

x

zx

yx

left

z

y

f

uWBVBBupuUp

uWB

uVB

uBB

uBB

up

B

B

W

V

U

uF

010

00

0

0

0

0

00

1

)()(2

1

0

rightzyx

x

x

zx

yx

right

z

y

f

uWBVBBupuUp

uWB

uVB

uBB

uBB

up

B

B

W

V

U

uF

010

00

0

0

0

0

00

1

)()(2

1

0

Construction of equilibrium state:

1

2/1

2/1 jj

j

z

y

j qq

B

B

W

V

U

q

j

z

y

j

uUuU

uB

uB

uW

uV

u

u

q

102

0

0

0

0

1

0

2

1)

2

1(

1

102

0

0

0

0

1

0

1

2

1)

2

1(

j

z

y

j

uUuU

uB

uB

uW

uV

u

u

q

where,

j+1/2

free transport

collision

j

j+1

Equilibrium flux function:

The BGK flux is a combination of non-equilibrium andequilibrium ones:

2/1

*

2*

2

2/12/1

)()(

)(

j

zyxx

xz

xy

zx

yx

x

jej

WBVBUBBUp

WBUB

VBUBBBUW

BBUVBpU

U

qFF

,)1( 2/12/12/1ej

fjj FFF

(K. Xu, JCP159)5.0

1D Brio-Wu test case:

Left state:

Right state:

density x-component velocity

solid lines: current BGK schemedash-line: Roe-MHD solver

0.1,75.0,0.1,0.0,0.1 ,, lylxlll BBpU

0.1,75.0,1.0,0.0,125.0 ,, ryrxrrr BBpU

y-component velocity By distribution

+: BGK, o: Roe-MHD, *: KFVS

shock Contactdiscontinuity

Orszag-Tang MHD Turbulence:

t=0.5(a): density(b): gas pressure(c): magnetic pressure(d): kinetic energy

5th WENO

.3/5 where

),2sin(),sin(,)0,,(

),sin(),sin(,)0,,( 2

xByByxp

xVyUyx

yx

t=2.0(a): density(b): gas pressure(c): magnetic pressure(d): kinetic energy

5th WENO

t=3.0(a): density(b): gas pressure(c): magnetic pressure(d): kinetic energy

5th WENO

t=8.0(a): density(b): gas pressure(c): magnetic pressure(d): kinetic energy

3D examples:3D examples:

BGK (100^3)

FLUID MODELING

Molecular Models Continuum Models

Euler Navier-Stokes BurnettDeterministic Statistical

MD Liouville

DSMC Boltzmann

Chapman-Enskog

0.001 0.1 10Kn

Continuum Slip flow Transition Free moleculae

new continuum models

Generalization of Constitutive Relationship

Gas-kinetic BGK model:

.

fguff xt

Compatibility condition:

.0)( 21 ddudfg

ST and p

Constitutive relationship:

),(* xt ugggf

* is obtained by substituting the above solution into BGK eqn.

re whe,/)( sxt Tfguff

The solution becomes

,/1 2* DggD

With the assumption of closed solution of the BGK model:With the assumption of closed solution of the BGK model:

A time-dependent gas distribution function at a cell interface

)/1/( 2* DggD

),)(1( * tAAaugf where

Extended Navier-Stokes-type Equations

Viscosity and heat conduction coefficient *

Argon shock structure

Observation: '90s). (Chapman, 8.0 '70), Cowling,-(Chapman 75.0 , ssT s

Experiment: Alsmeyer (‘76), Schmidt (‘69), ...

Shock thickness:

max12 )//()( dxdLs

Mean free path (upstream):

11

11

25

16

RT

Density distribution in Mach=9 Argon shock front

Circles : experimental data (Alsmeyer, ‘76); dash-dot line: BGK-NS; solid line: BGK-Xu

Diatomic gas: N2(two temperature model: bulk viscosity is

replaced by temperature relaxation)

]))((exp[)()( 2222/3

rtrt Uug

dfdud

E

E

UW

r

energy rotational :

energy total:

rE

E

,

)2

1),(

2

1,,1( 2222 uu

.4,3,2,1,

/)(

0

0

0

)(

)(

rreq

er ZEE

ddudfg

2/4

5 and ,/

2UEE eqeqeq

r

/)( fguff xt BGK

Compatibility condition

.101shock nitrogen for 53order on the is MZZ RR

M=12.9 nitrogen shock structure

M=11 nitrogen shock structure

Efficiency: DSMC: hours Extended BGK: minutes