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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!