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. 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: Established Particle 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; DT from 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 y 1m argon gas V=0m/s, T=273K

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

Flow over flat plate of 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 Flow Regimes: continuum slip transitional free-molecular Kn 0.01 0.1 10 DSMC Model Accuracy: Boltzmann Equation Navier-Stokes Collisionless Boltzmann Eqn Euler Burnett high Numerical Performance: DSMC CFD Kn 0.01 0.1 10 low Accuracy + } hybrid approach Performance

Sometimes CFD Works Best

Sometimes DSMC Works Best

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

Domain Coupling

Continuum Breakdown Parameters Interface Location: Continuum Breakdown Parameters Local Knudsen number, hypersonic flow (Boyd et al., 1995) where Q= 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?

Cut-off Value = 0.05

Hybrid CFD/DSMC-IP Process 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.

Normal Shock Waves of Argon 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.

Mach 5 Profiles

Reciprocal Shock Thickness

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

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

Particle Domain (Knmax > 0.03)

Comparisons of Density Contours

Comparisons of Temperature Contours

Comparisons of Surface Properties

Along the Stagnation Streamline 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).