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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