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Direct Simulation Monte Carlo: A Particle Method for Nonequilibrium Gas Flows. Iain D. Boyd Department of Aerospace Engineering University of Michigan Ann Arbor, MI 48109 Support Provided By: MSI, AFOSR, DARPA, NASA. Physical characteristics of nonequilibrium gas flow.

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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

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Physical characteristics of nonequilibrium gas flow.

Direct simulation Monte Carlo (DSMC) method.


data structure;

scalar/parallel optimization.

Illustrative DSMC applications:

hypersonic aerothermodynamics;

materials processing;

spacecraft propulsion.

Summary and future directions.


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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

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Characterization ofNonequilibrium Gas Flows

Flow Regimes:










Boltzmann Equation

Control equations:


Boltzmann Eqn




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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

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The DSMC Algorithm

  • MOVE:

    • translate particles Dx = uDt;

    • apply boundary conditions (walls, sources, sinks).

  • SORT:

    • generate list of particles in each cell.


    • statistically determine particles that collide in each cell;

    • apply collision dynamics.


    • update sums of various particle properties in each cell.

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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

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  • 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).

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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.

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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.

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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.

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MONACO: The Software System

  • Consists of four modular libraries:


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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%.

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MONACO: Unstructured Grids

Hypersonic flow around

a planetary probe

3D Surface geometry of

TOMEX flight experiment

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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

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Hypersonic Viscous Interaction

  • Flow separation configuration:

    • N2 at M~10 over double cone;

    • data from LENS (Holden).

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Shock-Shock Interactions

  • Cowl lip configuration:

    • N2 at M~14;

    • data from LENS (Holden).

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Complex 3D Flows

  • TRIO flight experiment:

    • analysis of pressure gauges;

    • external/internal flows.

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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.

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    Flow Fields

    Temperature Ratio (T / T∞)

    Cylinder at 7.5 km/s

    n=0.7 at 7.5 km/s

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    Aerothermodynamic Assessment

    Drag Coefficient

    Shock Standoff Distance/Heat Transfer Coefficient

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    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.

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    3D MONACO Modeling

    • 20x60x50 cuboid cells.

    • Non-uniform cell sizes.

    • 2,000,000 particles.

    • Overnight solution time

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    Yttrium Evaporation

    Source flux: 9.95x10-5 moles/sec

    Number density

    Z-component of velocity

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    Yttrium Evaporation

    • Comparison of calculated and measured film deposition thickness.

    • Significant effect of atomic collisions.

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    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).

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    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

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    Co-evaporation of Yt, Ba, and Cu

    Flux (moles/cm2/s) across the substrate




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    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

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    Spacecraft Propulsion


    ion thruster









    plasma thruster


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    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).

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    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.






    u’, v’, w’

    x, y, z

    m, q

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    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.

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    Number Densities (m-3)

    Xe+ ion

    Xe atom

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    Ion Energy Distributions

    Beam plasma (15 deg.)

    CEX plasma (77 deg.)

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    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.


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    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