Applications of Kinetic Fluxes to Hybrid Continuum-Rarefied Methods
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June 28, 2004 AIAA 37th Thermophysics Conference
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Applications of Kinetic Fluxes to Hybrid Continuum-Rarefied Methods
Harrison S. Y. Chou
Research Scientist
Nielsen Engineering & Research, Inc
Mountain View, California
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Outlines
• History• Difficulties• Approaches• Applications• Concluding Remarks
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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
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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.
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Typical DSMC/NS Hybrid Applications
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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
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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.
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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.
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Kinetic Approaches
Deshpande (1986)
Chou & Baganoff (1995)
KFVS Scheme For Euler Equations
KFVS SchemeForNavier-StokesEquations
MaxwellianPDF
Chapman-EnskogPDF
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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
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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
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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
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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)
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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
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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
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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
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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
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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
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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
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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)
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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
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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