Applications of Kinetic Fluxes to Hybrid Continuum-Rarefied Methods

Applications of Kinetic Fluxes to Hybrid Continuum-Rarefied Methods Harrison S. Y. Chou Research Scientist Nielsen Engineering & Research, Inc Mountain View, California AIAA 37th Thermophysics Conference

Outlines • History • Difficulties • Approaches • Applications • Concluding Remarks AIAA 37th Thermophysics Conference

Research at Stanford (1991~1995) 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) AIAA 37th Thermophysics Conference

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 • Flux–Vector Splitting for the Navier–Stokes Equationswith 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. AIAA 37th Thermophysics Conference

Typical DSMC/NS Hybrid Applications AIAA 37th Thermophysics Conference

Difficulties FVS/FDS DSMC (Kinetic Flux) (Viscous Flux ???) AIAA 37th Thermophysics Conference

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. AIAA 37th Thermophysics Conference

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. AIAA 37th Thermophysics Conference

Kinetic Approaches Maxwellian PDF KFVS Scheme For Euler Equations Deshpande (1986) Chapman- Enskog PDF KFVS Scheme For Navier-Stokes Equations Chou & Baganoff (1995) AIAA 37th Thermophysics Conference

PDFs (1) AIAA 37th Thermophysics Conference

PDFs (2) For 1-D Case 0.7 0.6 Maxwellian 0.5 Mach 2.5 Chapman-Enskog 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 AIAA 37th Thermophysics Conference

Moment Equations Boltzmann Equation Moment Equations where Navier-Stokes Equations where AIAA 37th Thermophysics Conference

t1 n +n Flux t2 Mathematical Integrations AIAA 37th Thermophysics Conference

Split Mass Flux Split Normal Momentum Flux Split Tangential Momentum Fluxes (2) Split Energy Fluxes Split Kinetic Fluxes AIAA 37th Thermophysics Conference

compression compression Mach Number Split Kinetic Mass Fluxes AIAA 37th Thermophysics Conference

compression compression Mach Number Split Kinetic Momentum Fluxes AIAA 37th Thermophysics Conference

compression compression Mach Number Split Kinetic Energy Fluxes AIAA 37th Thermophysics Conference

± ± ± ¶ F + - F º A U A = A + A ± S - W A ¹ S - W S - W S - W S - W S - W S - W ¶ U ± ¶ F + - ± ± F ¹ A U F = F + F ± VL A º VL VL VL VL VL VL ¶ U ± ¶ F ± A º kinetic + - ± ± F = F + F F = A U kinetic ¶ U kinetic kinetic kinetic kinetic kinetic 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 AIAA 37th Thermophysics Conference

Mach Number Split Conservative Variables (Mass) AIAA 37th Thermophysics Conference

Mach Number Split Conservative Variables (Momentum) AIAA 37th Thermophysics Conference

Mach Number Split Conservative Variables (Energy) AIAA 37th Thermophysics Conference

Cumulative Sampling Method Acceptance-Rejection Sampling Method Sampling Techniques AIAA 37th Thermophysics Conference

Kinetic-BasedNS Steady-State Solutions Mach Number Contour Plot 2-D Cylinder w/ Isothermal BC (Mach=4.0, AOA=0 degree) Pressure Contour Plot 3-D OSC Taurus Launch Vehicle (Mach=3.98, AOA=10 degree) AIAA 37th Thermophysics Conference

flow wall ISO-Wall Chapman-Enskog f f n Accomdation coefficient Boundary Conditions at Wall(Slip/No Slip) t1 Isothermal Wall (Given Temperature) Constrain Equations AIAA 37th Thermophysics Conference

HIGH HIGH Pressure (KFVS) Pressure (DSMC) y x y x LOW LOW DSMC-NS Solutions – Sliding Plate (By T. Lou and D. Dahlby) AIAA 37th Thermophysics Conference

Applications For NS-DSMC Hybrid (by Craig R. Duttweiler at Stanford) AIAA 37th Thermophysics Conference

Concluding Remarks • Compatibilities at different levels. • Efficient Kinetic-based algorithms. • Kinetic-based boundary conditions for all domains. • Dynamic NS/DSMC interfaces. AIAA 37th Thermophysics Conference

Future Researches • High-order moment equations algorithms. • Steady-state solution techniques for DSMC. • Dynamic NS-DSMC interfaces AIAA 37th Thermophysics Conference

Applications of Kinetic Fluxes to Hybrid Continuum-Rarefied Methods