june 28, 2004aiaa 37th thermophysics conference1 applications of kinetic fluxes to hybrid...

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

June 28, 2004 AIAA 37th Thermophysics Conference

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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

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.

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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

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)

June 28, 2004 AIAA 37th Thermophysics Conference

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Applications For NS-DSMC Hybrid

(by Craig R. Duttweiler at Stanford)

June 28, 2004 AIAA 37th Thermophysics Conference

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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