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Hybrid Particle-Continuum Computation of Nonequilibrium Multi-Scale Gas Flows. Iain D. Boyd Department of Aerospace Engineering University of Michigan Ann Arbor, MI 48109 Support Provided By: MSI, AFOSR, NASA. Overview. Problems with DSMC for low-speed MEMS flows.
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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
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.
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.
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
Subsonic Flow Over a 5% Flat Plate Total sample size:760,000 particles/cell Flat Plate Density field Pressure field
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
Outline of IP Algorithm select collision pairs perform binary collisions inject new particles Dt move particles, compute boundary interactions update information sample cell information
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
Information Preservation Method:Computational Performance • Numerically efficient implementation: • only 20% overhead above DSMC; • parallelized using domain decomposition.
Thermal Couette Flow T=373K 1m argon gas y T=173K
High Speed Couette Flow V=300m/s, T=273K 1m y argon gas V=0m/s, T=273K
Rayleigh Flow y V=0 T=273K argon gas t=0: Vx=0, Tw=273K x t>0: Vx=10m/s, Tw=373K Vx
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)
Flow over a NACA-0012 Airfoil Air: Ma = 0.8, Re = 73, Kn = 0.014, Lchord= 0.04m DSMC IP
Subsonic Flow Over a 5% Flat Plate Air: Ma = 0.1, Re = 100, Kn = 0.01, L= 100 m Density field Pressure field
Flow Over a 5% Flat Plate Pressure contours (angle of attack 10o) IP NS
Flow Over a 5% Flat Plate Surface properties IP NS pressure skin friction
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
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
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
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?
Hybrid CFD/DSMC-IP Process CFD Solution Interface & Particle Domain Determined Hybrid Calculation Final Solution No Yes End
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.
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.
Numerical Performance • Hybrid simulation employed 57% particle cells • DSMC time-step could be 20 times larger
Ma l (m) r (kg/m3) U (m/s) T (K) Tvib (K) Tw (K) 12.6 1.310-4 5.61810-4 2630.4 104.4 2680.2 297.2 Numerical Example (2) Blunted Cone
Comparisons of Temperature Contours
Comparisons of Surface Properties
Detailed Comparisons Along the Stagnation Streamline
Detailed Comparisons at x = 2 cm
Detailed Comparisons at x = 4 cm
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!
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).