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June 28, 2004 AIAA 37th Thermophysics Conference
1
Applications of Kinetic Fluxes to Hybrid Continuum-Rarefied Methods
Harrison S. Y. Chou
Research Scientist
Nielsen Engineering & Research, Inc
Mountain View, California
June 28, 2004 AIAA 37th Thermophysics Conference
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Outlines
• History• Difficulties• Approaches• Applications• Concluding Remarks
June 28, 2004 AIAA 37th Thermophysics Conference
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Research at Stanford (1991~1995)
D. Baganoff
Particle MethodJ. McDonald
Continuum MethodS. Y. Chou
L. DagumB. HassA. GoswamiT. DeneryD. DahlbyT. LouC. D. DuttweilerA. Garcia (Professor at SJSU)
Kinetic Theory Study
T. LouD. DahlbyC. D. Duttweiler
June 28, 2004 AIAA 37th Thermophysics Conference
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References
1. Chou, S. Y. and Baganoff, D., "Kinetic Flux Vector Splitting for the Navier-Stokes Equations," Journal of Computational Physics, 130, Jan.
1997.2. Garcia, A. and B. Alder, "Generation of the Chapman-Enskog Distribution," Journal of Computational Physics, 140, May 1998.3. Lou, T.; Dahlby, D. C.; Baganoff, D, “A Numerical Study Comparing Kinetic Flux–Vector Splitting for the Navier–Stokes Equations with a Particle Method,” Journal of Computational Physics, 145, Sep. 1998.4. Duttweiler, C. R., “Development and Parallelization of a Hybrid particle/Continuum Method for Simulation Rarefied Flow,” Ph.D. Thesis, Stanford University, 1998.5. Chou, S. Y., "On the Mathematical Properties of Kinetic Split Fluxes," AIAA 2000-0921, AIAA 38th Aerospace Sciences Meeting & Exhibit, Jan. 2000.
June 28, 2004 AIAA 37th Thermophysics Conference
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Typical DSMC/NS Hybrid Applications
June 28, 2004 AIAA 37th Thermophysics Conference
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Difficulties
Fj
1
2Fj
1
2
DSMCFVS/FDS
(Kinetic Flux)
(Viscous Flux ???)
F CmassKinetic
1
4 ~ F cmass
S W
1
2
June 28, 2004 AIAA 37th Thermophysics Conference
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Interfaces
• From DSMC to continuum methods (a) Sum up particles across boundaries from DSMC domain. (b) Overset grid techniques.
• From continuum methods to DSMC (a) Convert fluxes into particles back to DSMC domain. (b) Sampling from Chapmann-Enskog PDF. (c) Sampling by acceptance/rejection methods.
June 28, 2004 AIAA 37th Thermophysics Conference
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Compatibilities
• Transport Properties (a) Viscosity,…
• Governing Equations (a) NS/DSMC, High-order Moment Equations.
• Numerical Methods (a) Steady-state solutions algorithms. (b) Solutions transfer between grids. (c) Boundary conditions at solid walls. (d) Computational stabilities and efficiencies.
June 28, 2004 AIAA 37th Thermophysics Conference
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Kinetic Approaches
Deshpande (1986)
Chou & Baganoff (1995)
KFVS Scheme For Euler Equations
KFVS SchemeForNavier-StokesEquations
MaxwellianPDF
Chapman-EnskogPDF
June 28, 2004 AIAA 37th Thermophysics Conference
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PDFs (1)
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PDFs (2)
-4 -3 -2 -1 0 1 2 3 4-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Molecular Thermal Speed Ratio
Pro
babi
lityMaxwellian
Chapman-Enskog
For 1-D Case
nn
n
p
q
pc
0 5
0 5
.
.
Mach 2.5
June 28, 2004 AIAA 37th Thermophysics Conference
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Moment Equations
Navier-Stokes Equations
nf
t
nfC
x
nf
tj
j coll
nQ f
t
nQ C f
x
INV INVj
j
, , 0 Q m
C
C
C
e C
INV
n
t
t
1
1
2
1
2
2int
f f CE
where
U
t
F
xj
j
0 whereU nQ fINV CE ,
F nQ C fjINV
jCE ,
Moment Equations
Boltzmann Equation
June 28, 2004 AIAA 37th Thermophysics Conference
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Mathematical Integrations
t1
n+n Fluxt2
F nQ C f dC dC dCQK INV
nCE
n t t
CCC ntt
1 2
012
June 28, 2004 AIAA 37th Thermophysics Conference
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Split Kinetic Fluxes
F c Mq
pcM
pmassK
nnCE
nnnCE
1
21 1
5 21 2
F c Mp
Mq
pcn momentumK
nnnCE
nnCE
1
21
11
2
52
12
2
Split Mass Flux
Split Normal Momentum Flux
Split Tangential Momentum Fluxes (2)
Split Energy Fluxes
-4 -3 -2 -1 0 1 2 3 4-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Speed Ratio
AL
PH
A1
-4 -3 -2 -1 0 1 2 3 4-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Speed Ratio
AL
PH
A2*
SQ
RT
(gam
ma/
2)
June 28, 2004 AIAA 37th Thermophysics Conference
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Split Kinetic Mass Fluxes
-2 -1 0 1 2-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
compressio
n
compres
si
on
Mach Number
F
cmass
June 28, 2004 AIAA 37th Thermophysics Conference
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Split Kinetic Momentum Fluxes
-2 -1 0 1 2-1
0
1
2
3
4
5
6
Mach Number
F
cn momentum
2
compr
essio
ncompression
June 28, 2004 AIAA 37th Thermophysics Conference
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Split Kinetic Energy Fluxes
-2 -1 0 1 2-10
-5
0
5
10
compr
essio
n
Mach Number
F
ctotal energy
3
compression
June 28, 2004 AIAA 37th Thermophysics Conference
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Properties of Flux Jacobian
Split Define Check
A A AS W S W S W
F A US W S W
A
F
US W
S W
Steger-Warming Flux Vector Splitting Algorithm
Split Define Check
F F FVL VL VL
A
F
UVL
VL
F A U
VL VL
Van Leer Flux Vector Splitting Algorithm
Split Define Check
F F Fkinetic kinetic kinetic
AF
Ukinetic
kinetic
F A U
kinetic kinetic
Kinetic Flux Vector Splitting Algorithm
June 28, 2004 AIAA 37th Thermophysics Conference
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Split Conservative Variables (Mass)
-3 -2 -1 0 1 2 3-0.2
0
0.2
0.4
0.6
0.8
1
1.2
Mach Number
Umass
June 28, 2004 AIAA 37th Thermophysics Conference
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Split Conservative Variables (Momentum)
-3 -2 -1 0 1 2 3-4
-3
-2
-1
0
1
2
3
4
Mach Number
U
cn momentum
June 28, 2004 AIAA 37th Thermophysics Conference
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Split Conservative Variables (Energy)
-3 -2 -1 0 1 2 3-1
0
1
2
3
4
5
6
7
8
9
Mach Number
U
cenergy
2
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Sampling Techniques
Cumulative Sampling Method
Acceptance-Rejection Sampling Method
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Kinetic-BasedNS Steady-State Solutions
Mach Number Contour Plot2-D Cylinder w/ Isothermal BC
(Mach=4.0, AOA=0 degree)
Pressure Contour Plot3-D OSC Taurus Launch Vehicle(Mach=3.98, AOA=10 degree)
June 28, 2004 AIAA 37th Thermophysics Conference
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Boundary Conditions at Wall(Slip/No Slip)
Fmass wall0
Isothermal Wall (Given
Temperature)
t1
F
Fmomentum
energy wall
Constrain Equations
n
flow
wallf
ISO-WallfChapman-Enskog
Fflow Freflect
Accomdation coefficient
June 28, 2004 AIAA 37th Thermophysics Conference
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DSMC-NS Solutions – Sliding Plate
1 1.2 1.4 1.6 1.8 2
5
10
15
20
25
30
35
40HIGH
LOW
Pressure (KFVS)
020
4060
80
020
4060
80
0.5
1
1.5
2
2.5
3
xy 1 1.2 1.4 1.6 1.8 2
5
10
15
20
25
30
35
40HIGH
LOW
020
4060
80
020
4060
80
0.5
1
1.5
2
2.5
3
xy
Pressure (DSMC)
(By T. Lou and D. Dahlby)
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Applications For NS-DSMC Hybrid
(by Craig R. Duttweiler at Stanford)
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Concluding Remarks
Compatibilities at different levels.
Efficient Kinetic-based algorithms.
Kinetic-based boundary conditions for all domains.
Dynamic NS/DSMC interfaces.
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Future Researches
High-order moment equations algorithms.
Steady-state solution techniques for DSMC.
Dynamic NS-DSMC interfaces