Applications of Kinetic Fluxes to Hybrid Continuum-Rarefied Methods

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

Documents

Applications of Kinetic Fluxes to Hybrid Continuum-Rarefied Methods