Hybrid Particle-Continuum Computation of Nonequilibrium Multi-Scale Gas Flows

1. Hybrid Particle-Continuum Computation of Nonequilibrium Multi-Scale Gas Flows Iain D. BoydDepartment of Aerospace EngineeringUniversity of MichiganAnn Arbor, MI 48109Support Provided By:MSI, AFOSR, NASA

2. Overview • Problems with DSMC for low-speed MEMS flows. • The Information Preservation (IP) method: • basic ideas and algorithms; • validation studies. • Use of IP in a hybrid CFD-DSMC method: • data communication and interface location; • hypersonic flow applications. • Future directions.

3. Gas Flows at the Micro–Scale • Examples: • accelerometers (air-bags); • micro-turbines (e.g. Karlsruhe); • Knudsen pumps; • Micro-Air-Vehicles (MAV’s). • Physical characteristics: • atmospheric pressure; • characteristic length ~ 10-6 m; • Knudsen number ~ 0.01; • non-continuum effects.

4. DSMC: EstablishedParticle Method for Gas Flows • Direct simulation Monte Carlo (DSMC): • developed by Bird (1960’s); • particles move in physical space; • particles possess microscopic properties, e.g. u’ (thermal velocity); • cell size ~ l, time step ~ 1/n; • collisions handled statistically; • ideal for supersonic/hypersonic flows; • statistical scatter for subsonic flow. { u’, v’, w’ x, y, z erot, evib

5. Subsonic Flow Over a 5% Flat Plate Total sample size:760,000 particles/cell Flat Plate Density field Pressure field

6. Information Preservation (IP) • Relatively new method: • first proposed by Fan and Shen (1998); • further developed by Sun, Boyd, et al.; • evolves alongside DSMC; • particles and cells possess preserved information, e.g. n, <u>, T; • Dn from mass conservation; • D<u> from momentum conservation; • DTfrom energy conservation; • aims to reduce statistical fluctuations; • may provide connection to CFD code. } Dn D<u> DT { u’, v’, w’ n, <u>, T x, y, z erot, evib

7. Outline of IP Algorithm select collision pairs perform binary collisions inject new particles Dt move particles, compute boundary interactions update information sample cell information

8. Information Preservation Method • Statistical scatter (from a flow over a flat plate): Time step: 1 Sampling size: 20 Time step: 1000 Sampling size: 20,000 Time step: 100,000 Sampling size: 2,000,000

9. Information Preservation Method:Computational Performance • Numerically efficient implementation: • only 20% overhead above DSMC; • parallelized using domain decomposition.

10. Thermal Couette Flow T=373K 1m argon gas y T=173K

11. High Speed Couette Flow V=300m/s, T=273K 1m y argon gas V=0m/s, T=273K

12. Rayleigh Flow y V=0 T=273K argon gas t=0: Vx=0, Tw=273K x t>0: Vx=10m/s, Tw=373K Vx

13. Flow over flat plateof zero thickness M=0.2 Free-molecular theory: Cd=1.35/M Compressible experiment: Schaaf and Sherman (1954) Slip Oseen-Stokes solution: Tamada and Miura (1978) Incompressible N-S: Dennis and Dunwoody (1966) Incompressible experiment: Janour (1951)

14. Flow over a NACA-0012 Airfoil Air: Ma = 0.8, Re = 73, Kn = 0.014, Lchord= 0.04m DSMC IP

15. Subsonic Flow Over a 5% Flat Plate Air: Ma = 0.1, Re = 100, Kn = 0.01, L= 100 m Density field Pressure field

16. Flow Over a 5% Flat Plate Pressure contours (angle of attack 10o) IP NS

17. Flow Over a 5% Flat Plate Surface properties IP NS pressure skin friction

18. Hybrid Method for Hypersonic Flows • Hypersonic vehicles encounter a variety of flow regimes: • continuum: modeled accurately and efficiently with CFD; • rarefied: modeled accurately and efficiently with DSMC. • A hybrid DSMC-CFD method is attractive for mixed flows: • CFD: Navier-Stokes finite-volume algorithm; • DSMC: MONACO+ Information Preservation (DSMC/IP). Rarefied DSMC approach: based on kinetic theory high altitude sharp edges Continuum CFD approach: solve NS equations low altitude long length scales

19. Motivation for Hybrid Method transitional Flow Regimes: slip free-molecular continuum Kn 0.1 0.01 10 DSMC Boltzmann Equation Model Accuracy: Collisionless Boltzmann Eqn Navier-Stokes Euler Burnett high Numerical Performance: DSMC CFD Kn 0.1 0.01 10 low Accuracy + } hybrid approach Performance

20. Sometimes CFD Works Best

21. Sometimes DSMC Works Best

22. Hybrid Approach Sun and Boyd J. Comp. Phys., 179, 2002 MacCormack and Candler Comp. Fluids, 17, 1989 Navier-Stokes solver DSMC/IP method Macroscopic properties are preserved as well as microscopic particle information 2nd order accurate modified Steger-Warming flux-vector splitting approach (FVM) interface

23. Domain Coupling

24. Interface Location: Continuum Breakdown Parameters • Local Knudsen number, hypersonic flow (Boyd et al., 1995) • whereQ= r, T, V. • Determined through detailed DSMC versus CFD comparisons: • Parameters also under investigation for use inside DSMC: • - failure of breakdown parameters at shock front; • - use DSMC to evaluate continuum onset parameter?

25. Cut-off Value = 0.05

26. Hybrid CFD/DSMC-IP Process CFD Solution Interface & Particle Domain Determined Hybrid Calculation Final Solution No Yes End

27. Summary of Hybrid Code • Numerical methods: • 2d/axially symmetric, steady state; • CFD: explicit, finite volume solution of NS Eqs.; • DSMC: based on MONACO; • interface: Information Preservation scheme; • implementation: a single, parallel code. • Physical modeling: • simple, perfect gas (rotation, but no vibration); • walls: slip / incomplete accommodation; • breakdown parameter: local Knudsen number.

28. Numerical Example (1) Normal Shock Waves of Argon • Argon normal shocks investigated: • relatively simple hypersonic flow; • Alsmeyer experimental measurements; • well-known case for testing new algorithms. • Simulations: • modeled in 2D (400 x 5 cells); • initialized by jump conditions; • pure DSMC; • pure CFD (Navier-Stokes equations); • hybrid code initialized by CFD solution.

29. Mach 5 Profiles

30. Reciprocal Shock Thickness

31. Numerical Performance • Hybrid simulation employed 57% particle cells • DSMC time-step could be 20 times larger

32. Ma l  (m) r  (kg/m3) U (m/s) T (K) Tvib (K) Tw (K) 12.6 1.310-4 5.61810-4 2630.4 104.4 2680.2 297.2 Numerical Example (2) Blunted Cone

33. Particle Domain (Knmax > 0.03)

34. Comparisons of Density Contours

35. Comparisons of Temperature Contours

36. Comparisons of Surface Properties

37. Detailed Comparisons Along the Stagnation Streamline

38. Detailed Comparisons at x = 2 cm

39. Detailed Comparisons at x = 4 cm

40. Summary • Simulation of subsonic nonequilibrium gas flows: • difficulties caused by low speed and rarefaction; • DSMC not effective due to statistical fluctuations; • IP method developed and applied to several flows; • IP is effective at reducing statistical fluctuations; • IP provides a basis for creating a hybrid method. • Simulation of hypersonic nonequilibrium gas flows: • vehicle analysis involves continuum/rarefied flow; • hybrid CFD-DSMC/IP method looks promising; • physical modeling must be improved; • numerical performance must be improved; • much work still to be done!

41. Future Directions • Algorithm development for hybrid method: • CFD: parallel, implicit solver on unstructured mesh; • DSMC: implicit and/or other acceleration schemes. • Physics development for hybrid method: • vibrational relaxation; • chemically reacting gas mixture. • Development of hybrid methodology: • refinement of breakdown parameters; • evaluation against data (measured, computed).