Direct Simulation Monte Carlo:A Particle Method for Nonequilibrium Gas Flows

Iain D. BoydDepartment of Aerospace Engineering

University of MichiganAnn Arbor, MI 48109

Support Provided By:MSI, AFOSR, DARPA, NASA

Overview

• 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.

Modeling Considerations

• 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 ).

• Gas dynamics:– rarefied flow (high Knudsen number);– collisions still important;– continuum equations physically inaccurate.

Characterization ofNonequilibrium Gas Flows

Kn 0.01 0.1 10

continuum slip transitional free-molecular

Euler

Navier-StokesBoltzmann EquationControl

equations:

Flow Regimes:

Collisionless Boltzmann EqnBurnett

DSMC

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 x ~ , time step t ~ =1/;– 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, zm, erot, evib

Direct Simulation Monte Carlo

The DSMC Algorithm

• MOVE:– translate particles x = u t;– 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.

Current DSMC-Related Projects

• 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).

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 arounda planetary probe

3D Surface geometry ofTOMEX 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.

DSMC: particle approachhigh altitudesharp edgesuses kinetic theory

CFD: continuum approachlow altitudelong length scalessolves NS equations

NASA’s Hyper-X

• Flow separation configuration:

– N2 at M~10 over double cone;

– data from LENS (Holden).

Hypersonic Viscous Interaction

• Cowl lip configuration:– N2 at M~14;– data from LENS (Holden).

Shock-Shock Interactions

• TRIO flight experiment:– analysis of pressure gauges;– external/internal flows.

Complex 3D Flows

• 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.

AerothermodynamicsOf Sharp Leading Edges

Flow Fields

Temperature Ratio (T / T∞)

Cylinder at 7.5 km/s n=0.7 at 7.5 km/s

Drag Coefficient Shock Standoff Distance/

Heat Transfer Coefficient

Aerothermodynamic Assessment

DSMC Applications:2. Materials Processing

• Effect of atomic collisions: – between the same species; – between different species.

Top view

Side view

3M experimental chamber for YBCO deposition

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

• 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).

Yttrium Evaporation

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

Ba CuYt

Flux (moles/cm2/s) across the substrate

Co-evaporation of Yt, Ba, and Cu

DSMC Applications:3. Spacecraft Propulsion

• 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.

Spacecraft Propulsion

Griddedion thruster(UK-10)

Arcjet (Stanford)

Hall:stationaryplasma thruster(SPT-100)

PulsedPlasmaThruster(EOS-1)

Express Spacecraft

• 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.

• 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)

{u’, v’, w’x, y, zm, q

E3

E2

E1

E4

• 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 x ~ , time step t ~ 1/;– self-consistent electric fields, E;– may be combined with DSMC for

collisional plasmas.

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:

φ−φ* =kTe

lnnn*

⎛ ⎝ ⎜

⎞ ⎠ ⎟

• 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.)

Summary

• 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.

Future Directions

• 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.