Applications of Kinetic Fluxes to Hybrid ContinuumRarefied Methods

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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 (19911995)
Kinetic Theory Study
D. Baganoff
Particle Method
Continuum Method
J. McDonald
S. Y. Chou
T. Lou D. Dahlby C. D. Duttweiler
L. Dagum B. Hass A. Goswami T. Denery D.
Dahlby T. Lou C. D. Duttweiler A. Garcia
(Professor at SJSU)
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References

• 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
• FluxVector Splitting for the
  NavierStokes Equations with a Particle Method,
• Journal of Computational Physics, 145,
  Sep. 1998.
• 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
DSMC
FVS/FDS
(Kinetic Flux)
(Viscous Flux ???)
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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
KFVS Scheme For Euler Equations
Maxwellian PDF
Deshpande (1986)
KFVS Scheme For Navier-Stokes Equations
Chapman- Enskog PDF
Chou Baganoff (1995)
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PDFs (1)
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PDFs (2)
For 1-D Case
0.7
0.6
Maxwellian
0.5
Chapman-Enskog
Mach 2.5
0.4
Probability
0.3
0.2
0.1
0
-0.1
-4
-3
-2
-1
0
1
2
3
4
Molecular Thermal Speed Ratio
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Moment Equations
Boltzmann Equation

Moment Equations

where
Navier-Stokes Equations

where


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Mathematical Integrations
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Split Kinetic Fluxes
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Split Kinetic Mass Fluxes
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Split Kinetic Momentum Fluxes
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Split Kinetic Energy Fluxes
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Properties of Flux Jacobian
Define
Check
Split
Steger-Warming Flux Vector Splitting Algorithm
Define
Check
Split
Van Leer Flux Vector Splitting Algorithm
Define
Check
Split
Kinetic Flux Vector Splitting Algorithm
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Split Conservative Variables (Mass)
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Split Conservative Variables (Momentum)
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Split Conservative Variables (Energy)
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Sampling Techniques
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Kinetic-BasedNS Steady-State Solutions
Mach Number Contour Plot 2-D Cylinder w/
Isothermal BC (Mach4.0, AOA0 degree)
Pressure Contour Plot 3-D OSC Taurus Launch
Vehicle (Mach3.98, AOA10 degree)
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Boundary Conditions at Wall(Slip/No Slip)
t1
Isothermal Wall (Given Temperature)
Constrain Equations
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DSMC-NS Solutions Sliding Plate
(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