**Direct Simulation Monte Carlo:** A Particle Method for Nonequilibrium Gas Flows Iain D. BoydDepartment of Aerospace EngineeringUniversity of MichiganAnn Arbor, MI 48109Support Provided By:MSI, AFOSR, DARPA, NASA

**Physical characteristics of nonequilibrium gas flow.** Direct simulation Monte Carlo (DSMC) method. The MONACO DSMC code: data structure; scalar/parallel optimization. Illustrative DSMC applications: hypersonic aerothermodynamics; materials processing; spacecraft propulsion. Summary and future directions. Overview

**Physical characteristics of nonequilibrium gas systems:** low density and/or small length scales; high altitude hypersonics (n=1020 m-3, L=1 m); space propulsion (n=1018 m-3, L=1 cm); micro-fluidics (n=1025 m-3, L=1 m). Gas dynamics: rarefied flow (high Knudsen number); collisions still important; continuum equations physically inaccurate. Modeling Considerations

**Characterization ofNonequilibrium Gas Flows** Flow Regimes: transitional slip free-molecular continuum Kn 0.1 0.01 10 DSMC Boltzmann Equation Control equations: Collisionless Boltzmann Eqn Navier-Stokes Euler Burnett

**Direct Simulation Monte Carlo** • Particle method for nonequilibrium gas flows: • developed by Bird (1960’s); • particles move/collide in physical space; • particles possess microscopic properties, e.g. u’ (thermal velocity); • cell size Dx ~ l, time step Dt ~ t=1/n; • collisions handled statistically (not MD); • ideal for supersonic/hypersonic flows; • may be combined with other methods (CFD, PIC, MD) for complex systems. { u’, v’, w’ x, y, z m, erot, evib

**Direct Simulation Monte Carlo**

**The DSMC Algorithm** • MOVE: • translate particles Dx = uDt; • apply boundary conditions (walls, sources, sinks). • SORT: • generate list of particles in each cell. • COLLIDE: • statistically determine particles that collide in each cell; • apply collision dynamics. • SAMPLE: • update sums of various particle properties in each cell.

**Hypersonics:** vehicle aerodynamics (NASA-URETI); hybrid particle-continuum method (AFOSR); TOMEX flight experiment (Aerospace Corp). Space propulsion: NEXT ion thruster, FEEP (NASA); Hall thrusters (DOE, NASA); micro-ablation thrusters (AFOSR); two-phase plume flows (AFRL). Micro-scale flows: low-speed rarefied flow (DOE). Current DSMC-Related Projects

**The DSMC Code MONACO** • MONACO: a general purpose 2D/3D DSMC code. • Physical models: • Variable Soft Sphere (Koura & Matsumoto, 1992); • rotational relaxation (Boyd, 1990); • vibrational relaxation (Vijayakumar et al., 1999); • chemistry (dissociation, recombination, exchange). • Applications: • hypersonic vehicle aerodynamics; • spacecraft propulsion systems; • micro-scale gas flows, space physics; • materials processing (deposition, etching).

**MONACO: Data Structure** • Novel DSMC data structure: • basic unit of the algorithm is the cell; • all data associated with a cell are stored in cache; • particles sorted automatically.

**MONACO: Scalar Optimization** • Inexpensive cache memory system used on workstations: • data localization leads to performance enhancement. • Optimization for specific processor: • e.g. overlap *add*, *multiply* and *logical* instructions.

**MONACO: Parallel Implementation** • Grid geometry reflected in the code data structure: • domain decomposition employed. • When a particle crosses a cell edge: • particle pointed to new cell; • thus, particles sorted-by-cell automatically. • When a particle crosses a domain edge: • communication link employed; • linked lists of particles sent as matrix; • inter-processor communication minimized; • no explicit synchronization required.

**MONACO: Parallel Implementation**

**MONACO: The Software System** • Consists of four modular libraries: • KERN, GEOM, PHYS, UTIL.

**MONACO: Code Performance** • MONACO performance on IBM SP (Cornell, 1996): • largest DSMC computation at the time; • best performance with many particles/processor; • parallel performance ~ 90%; • serial performance 30-40%.

**MONACO: Unstructured Grids** Hypersonic flow around a planetary probe 3D Surface geometry of TOMEX flight experiment

**DSMC Applications:** 1. Hypersonic Aerothermodynamics • Hypersonic vehicles encounter a variety of flow regimes: • flights/experiments are difficult and expensive; • continuum: modeled accurately and efficiently using CFD; • rarefied: modeled accurately and efficiently using DSMC. NASA’s Hyper-X DSMC: particle approach high altitude sharp edges uses kinetic theory CFD: continuum approach low altitude long length scales solves NS equations

**Hypersonic Viscous Interaction** • Flow separation configuration: • N2 at M~10 over double cone; • data from LENS (Holden).

**Shock-Shock Interactions** • Cowl lip configuration: • N2 at M~14; • data from LENS (Holden).

**Complex 3D Flows** • TRIO flight experiment: • analysis of pressure gauges; • external/internal flows.

**Aerothermodynamics** Of Sharp Leading Edges • Computations of hypersonic flow around several power-law leading edge configurations performed using MONACO at high altitude. • Advanced physical modeling: • vibrational relaxation and air chemistry; • incomplete wall accommodation. • Effects of sharpening the leading edge: • reductions in overall drag coefficient and shock standoff distance; • increases in peak heat transfer coefficient.

**Flow Fields ** Temperature Ratio (T / T∞) Cylinder at 7.5 km/s n=0.7 at 7.5 km/s

**Aerothermodynamic Assessment** Drag Coefficient Shock Standoff Distance/Heat Transfer Coefficient

**DSMC Applications:** 2. Materials Processing 3M experimental chamber for YBCO deposition Top view Side view • Effect of atomic collisions: • – between the same species; • – between different species.

**3D MONACO Modeling** • 20x60x50 cuboid cells. • Non-uniform cell sizes. • 2,000,000 particles. • Overnight solution time

**Yttrium Evaporation** Source flux: 9.95x10-5 moles/sec Number density Z-component of velocity

**Yttrium Evaporation** • Comparison of calculated and measured film deposition thickness. • Significant effect of atomic collisions.

**Yttrium Evaporation** Calculated and measured atomic absorption spectra: – along an aperture close to the substrate symmetry line; – at frequencies of 668 nm (left) and 679 nm (right).

**Co-evaporation of Yt, Ba, and Cu** Source fluxes (10-5 moles/cm2/sec) Y : Ba :Cu = 0.84 : 1.68 : 2.52 Total Number Density

**Co-evaporation of Yt, Ba, and Cu** Flux (moles/cm2/s) across the substrate Yt Cu Ba

**Tasks for spacecraft propulsion systems:** orbit transfer (e.g. planetary exploration); orbit maintenance (e.g. station-keeping); attitude control. Motivations for development of accurate models: simulations less expensive than testing; improve our understanding of existing systems; optimize engine performance and lifetime; assessment of spacecraft integration concerns. DSMC Applications: 3. Spacecraft Propulsion

**Spacecraft Propulsion** Gridded ion thruster (UK-10) Arcjet (Stanford) Pulsed Plasma Thruster (EOS-1) Hall:stationary plasma thruster (SPT-100)

**Two Russian GEO spacecraft launched in 2000:** SPT-100 Hall thrusters used for station-keeping; in-flight characterization program managed by NASA; first in-flight plume data for Hall thrusters. Express Spacecraft • Diagnostics employed on spacecraft: • electric field sensors; • Faraday probes (ion current density); • retarding potential analyzers, RPA’s (ion current density, ion energy distribution function); • pressure sensors; • disturbance torques (from telemetry data).

**Express Spacecraft**

**Particle In Cell (PIC)** • Particle method for nonequilibrium plasma: • developed since the 1960’s; • charged particles move in physical space; • particles possess microscopic properties, e.g. u’ (thermal velocity); • cell size Dx ~ d, time step Dt ~ 1/w; • self-consistent electric fields, E; • may be combined with DSMC for collisional plasmas. E3 E4 E2 E1 { u’, v’, w’ x, y, z m, q

**Hybrid DSMC-PIC** • Particle model for ions, fluid model for electrons. • Boltzmann relation for electrons provides potential: • currentless, isothermal, un-magnetized, collisionless; • quasi-neutrality provides potential from ion density: • Collision mechanisms: • charge exchange; • momentum exchange.

**Number Densities (m-3)** Xe+ ion Xe atom

**Ion Current Density**

**Ion Energy Distributions** Beam plasma (15 deg.) CEX plasma (77 deg.)

**Direct simulation Monte Carlo:** now a mature, well-established technique; statistical simulation of particle dynamics; applied in many areas of engineering/physics; use growing due to improved computer power. Some advantages of DSMC: accurate simulation of nonequilibrium gas; framework for detailed physical modeling; can handle geometric complexity; can be combined with other methods for multi-scale and multi-process systems. Summary

**Development of MONACO:** unsteady and 3D flows; user help: “DSMC for dummies”; dynamic domain decomposition; more detailed physical models. Extensions of DSMC: hybrid DSMC-CFD (using IP interface); generalized hybrid DSMC-PIC; 2-phase DSMC (gas and solid particles); speedup: implicit DSMC, variance reduction. Future Directions