The Direct Simulation of Acoustic The Direct Simulation of Acoustic Wave Propagation Using a Monte Wave Propagation Using a Monte

Carlo MethodCarlo Method

Amanda Danforth HanfordAmanda Danforth HanfordNASA Graduate Student Research Fellow NASA Graduate Student Research Fellow

in Acousticsin Acousticsald227@psu.eduald227@psu.edu

April 16, 2007

ContentsContents

• Introduction to particle methods– Boltzmann equation

• Overview of Direct Simulation Monte Carlo (DSMC)

• Acoustics applications– Monatomic gases– Gases with internal energy– Gas mixtures

• Modeling the Martian atmosphere

Simulation ModelsSimulation Models

• Microscopic models– Model the fluid as individual particles– Uses Boltzmann equation as mathematical model– Uses properties such as particle velocity and particle

position as dependent variables

• Macroscopic models– Model the fluid as a continuous medium– Use Euler and Navier-Stokes as mathematical model– Use properties such as velocity, density, pressure, and

temperature as dependent variables

But continuum model breaks down!

Knudsen NumberKnudsen Number

LMFPKn =

MFP = mean free path or the average distanceMFP = mean free path or the average distancea particle travels between successive collisionsa particle travels between successive collisions

L = characteristic length scale of systemL = characteristic length scale of system

•• 50 nm in standard conditions on Earth50 nm in standard conditions on Earth•• 6000 nm on Mars6000 nm on Mars

• wavelengthwavelength

DSMCDSMC

Boltzmann EquationBoltzmann Equation

• Describes how the velocity distribution function changes as a function of time due

to intermolecular collisions• Problem is a 7n dimensional problem

– Enormous computer memory requirements for using traditional deterministic numerical methods

– Monte Carlo is one of the most efficient ways to solve these kinds of problems

( ) ( ) ( ) ( )∫∫∫ ∫ Ω−=∂∂

⋅+∂∂

⋅+∂∂

c1c

cF

rc

π

σ4

01

*1

* ddcffffffft r

( )twvuzyxf ,,,,,,

Boltzmann EquationBoltzmann Equation• Nondimensionalize the Boltzmann equation and Kn number

comes out of the collision integral• Navier-Stokes can be derived from Boltzmann for Kn < 0.1• Navier-Stokes reduces to Euler as Kn -> 0• In thermal equilibrium, the velocity distribution function

becomes a Maxwellian distribution– Nonequilibrium effects can be seen in

• Absorption of sound• High Kn applications

• Any macroscopic property can be written in terms of velocity distribution function f

∫∫∫=c

cfdmρ ∫∫∫=c

ccfdmuρ ∫∫∫=

c

cfdCRmT 2

3 ρ

DSMC IntroductionDSMC Introduction

• DSMC is a particle model developed by Bird in 1960s for use in rarefied gas dynamics and hypersonic flow1

• Direct physical modeling of particle interactions• Monte Carlo:

– Initial conditions randomly assigned– particle collisions determined stochastically– Final flow field is average over statistically independent ensemble runs

• Follow representative particles as they move and collide with other particles

• DSMC has been proven to accurately represent the Boltzmann equation given small time steps and small cell sizes2

1 G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Clarendon Press, Oxford, 1994. 2W. Wagner, “A convergence proof for Bird’s direct simulation Monte Carlo method for the Boltzmann equation,” J. Stat. Phys. 66, 1011 (1992).

DSMC IntroductionDSMC Introduction• DSMC is not limited by Knudsen number

– High Kn applications: micro and nano devices, rarefied gas dynamics, hypersonic flows, shock waves, shuttle reentry simulations, high frequency sound1

– Low Kn applications: acoustics, large scale simulations, dense gases

• DSMC can be used to model monatomic, diatomic, polyatomic molecules, mixtures, chemical reactions, and explosions2

• BUT, requires significant CPU time1A. Danforth, L. Long, “"Nonlinear Acoustic Simulations Using the Direct Simulation Monte Carlo," J. Acoust. Soc. Am., 116, 1948 (2004).2O’Connor, P. Long L. and Anderson J., “Direct simulation of ultrafast detonations in mixtures” Proceedings from the 24th International Symposium on Rarefied Gas Dynamics, Bari, Italy (2004).

DSMC AlgorithmDSMC AlgorithmInitialize and define variables

– Each particle is assigned a random position and Maxwellian velocity

– Each cell is given particles dependent on acoustic source• Main Time Stepping Loop

– Move Particles based on their velocities• Have any particles hit a boundary?

– Sort Particles according to cells– Collide Particles that are in the same cell

• Determined stochastically– Sample cells to get macroscopic quantities like density,

temperature, and pressure

DSMC Code SpecificsDSMC Code Specifics• Object Oriented Programming using C++

– 1 dimensional in space• We currently have other 3 dimensional versions

– Typically use 1000-6000 cells with 100-2000 particles per cell

• Parallel Programming using Message Passing Interface (MPI) – Ensemble averaging using a master/slave algorithm– Parallel efficiency of approximately 95%– Run on large parallel processing machines

• Mufasa (PSU): 162 2.8 GHz Processors with peak speed of 0.4 teraflops and 91 GB RAM

• Columbia (NASA):10100 1.5-GHz processors, peak speed 50 teraflops, currently ranked 8th top supercomputer in the world (www.top500.org)

• Computation on 32 processors takes 3 days for one simulation

DSMC in Argon DSMC in Argon –– Low Low KnKn

• The parallel DSMC method was compared with:– Linear Euler equations– Nonlinear Euler equations– Navier-Stokes equations– Theory

• Nonlinear acoustic effects:– Wave steepening– Harmonic generation– Shockwaves

Nonlinear Euler Equations Code Nonlinear Euler Equations Code SpecificsSpecifics

• One-dimensional in space• Finite difference method in

Matlab• MacCormack Time

Accurate Algorithm• Added artificial viscosity• Should agree with DSMC

for very small Kn even with large amplitudes TRp

xupE

tE

xp

xu

tu

xu

t

ρ

ρ

ρρ

ρρ

=

=∂+∂

+∂∂

∂∂

−=∂

∂+

∂∂

=∂∂

+∂∂

0)(

0

2

Nonlinear Euler equationsNonlinear Euler equations

Comparison in Argon Comparison in Argon •Standard atmospheric conditions: P0 = 111285 Pa, T=300 K•Cell size: ½ mfp = 25 nm•Domain size: 150 μm•Time step size: 7.5 ps•Elapsed time: 0.2 μs•Frequency: 30 MHz•Kn = 0.005

DSMCNonlinear Euler

Results: Harmonic GenerationResults: Harmonic Generation

Absorption of soundAbsorption of sound

• Mechanisms include:– classical losses associated with the transfer of

acoustic energy into heat– relaxation losses associated with the redistribution

of internal energy of molecules.

The Classical Absorption Coefficient The Classical Absorption Coefficient • The classical absorption coefficient can accurately

model the sound absorption in Argon at low Kn

• This is a low frequency approximation to the dispersion relation given by the Navier-Stokes equations

• Where μ is shear viscosity, κ is thermal conductivity, and is the complex propagation constant

( )⎟⎟⎠

⎞⎜⎜⎝

⎛ −+=

pcl cc

κγμρωα 1

34

2 300

2

034

341 4

2000

2

0020

2

0

=⎟⎟⎠

⎞⎜⎜⎝

⎛−+⎥

⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+++⎟⎟

⎠

⎞⎜⎜⎝

⎛K

ci

cK

cci

c vv ρωμ

γωρκ

ρκ

ρμωω

ciK clωα +=

Assumptions in the Classical Assumptions in the Classical Absorption CoefficientAbsorption Coefficient

• Linearized Navier-Stokes equations• Small perturbation from thermal equilibrium• Small amplitude• Small Kn -- continuum assumption

The Classical Absorption CoefficientThe Classical Absorption Coefficient

Kn = 0.05 at standard atmospheric conditions gives a frequency of 300 MHz

Absorption as a Function of Absorption as a Function of KnKn• But we know that even the full Navier-Stokes solution doesn’t

hold for large Kn because of the breakdown of the continuum approximation

Absorption in Polyatomic GasesAbsorption in Polyatomic Gases• The classical absorption coefficient doesn’t hold for

gases with internal energy– Molecular relaxation

• For low Kn (< 0.05)– αcl is the absorption due to thermal and viscous losses

– αrot is the absorption due to rotational relaxation– αvib is the absorption due to vibrational relaxation

• For high Kn ( > 0.05)– Absorption mechanisms are not additive– Direct Simulation Monte Carlo (DSMC)

vibrotcl αααα ++=

( )⎟⎟⎠

⎞⎜⎜⎝

⎛ −+=

pcl Cc

κγμρωα 1

34

2 30

2

Absorption in Polyatomic GasesAbsorption in Polyatomic Gases• Greenspan1 wrote that for a gas with rotational energy

(neglecting vibration)

• Here is the low frequency wave number

• is the wave number associated with thermal viscous losses

• is the wave number associated with rotational relaxation losses and is a function of relaxation collision number

00 kk

kk clrotrotcl ααα +=

00 c

k ω=

clk

rotk

1Greenspan, M. “Combined Translational and Relaxational Dispersion of Sound in Gases,” J. Acoust. Soc. Am, 26 (1954).

Why is This Important?Why is This Important?

• Greenspan’s theory for absorption including rotational relaxation is current theory for modeling absorption in the Earth's atmosphere at high altitudes– Sutherland, L., Bass, H., “Atmospheric absorption

in the atmosphere up to 160 km,” J. Acoust. Soc. Am., 115, (2004).

Absorption as a Function of AltitudeAbsorption as a Function of Altitude

– Sutherland, L., Bass, H., “Atmospheric absorption in the atmosphere up to 160 km,” J. Acoust. Soc. Am., 115, (2004).

US Standard AtmosphereUS Standard Atmosphere

US Standard Atmosphere, 1976 (US GPO, Washington, DC, 1976).

US Standard AtmosphereUS Standard Atmosphere

US Standard Atmosphere, 1976 (US GPO, Washington, DC, 1976).

BorgnakkeBorgnakke –– LarsonLarson11 Model for Model for Internal EnergyInternal Energy

• A phenomenological approach to simulate polyatomic molecules – A classical, harmonic oscillator model

• Handles the internal energy exchange between the molecules during collisions

• A fraction of the collisions are regarded as inelastic – Post-collision energy is redistributed in a stochastic

manner to produce physically realistic behavior at the macroscopic level

• The probability of a change in internal energy is equal to the reciprocal of the relaxation collision number – Which is specified for each collision pair

1C. Borgnakke, P. Larson, “Statistical Collision Model for Monte Carlo Simulation of Polyatomic Gas Mixture,”J. Comp. Phys. 18, (1975)

Quantum Vibration model for Internal Quantum Vibration model for Internal EnergyEnergy

• Takes into account the discrete quantum energy levels

• Gives a more realistic representation of the internal structure of diatomic and polyatomic molecules

• Because of the temperatures used, only using ground state and 1 excited state

Absorption in NitrogenAbsorption in Nitrogen

273 K

Absorption in NitrogenAbsorption in Nitrogen

2000 K

Absorption in NitrogenAbsorption in Nitrogen

4000 K

DSMC in mixtures: Model of Martian DSMC in mixtures: Model of Martian Atmosphere Atmosphere

• Pressure: 700 Pa – 0.07% of that on Earth

• Temperature: 200 – 300 K• Thin and transparent

atmosphere• Martian Day: 24.6 hours• Martian Year: 686 Earth

days

The Atmosphere on Mars: Molecular The Atmosphere on Mars: Molecular CompositionComposition

• Mars– 95.3% Carbon Dioxide– 2.7% Nitrogen– 1.6% Argon– 0.27% Water vapor– 0.13% Oxygen

• Earth – 78% Nitrogen– 20% Oxygen– Other: Water Vapor, Argon, Carbon Dioxide, ...

DSMC in mixtures: Model of Martian DSMC in mixtures: Model of Martian AtmosphereAtmosphere

• CO2, N2, Ar , H2O , and O2

12O2

42H2O

00Ar

12N2

42CO2

Vibrational DOFRotational DOFMolecular species

Results Results –– MovieMovie

• Frequency:1 MHz

• Pressure: 700 Pa

• Temperature:200 K

• Kn: 0.05

Results Results –– Absorption coefficientAbsorption coefficient

Frequency: 82000 HzBest fit: α =65.593 np/mTheoretical: α = 56.2 np/mPressure: 700 PaTemperature: 200 KKn: 0.0025

1H.E. Bass, J. Chambers, “Absorption of sound in the Martian atmosphere,” JASA, 109(6) 3069-3071 (2001).

Examples of Sound on MarsExamples of Sound on Mars11

• Characteristic impedance on Mars is 1% of that on Earth

• Absorption is 100 times greater on Mars than on Earth

• Sound radiation is down 20 dB on Mars• A Helmholtz resonator on Mars will resonate

at 66% of ω0 on Earth• Shocks form on Mars at half the distance as on

Earth1Thanks to Dr. Victor Sparrow

Results Results –– Nonequilibrium effects Nonequilibrium effects

The gas starts in equilibrium (Trot = Tvib = Ttr)

ConclusionsConclusions• DSMC is a good computational tool for simulating high

amplitude sound for all Kn– restrictions

• The absorption of sound– High Kn

• Mechanisms are not additive• Does not agree with theory based on continuum assumptions

– Low Kn• Absorption due to molecular relaxation• Good agreement with theory

• Mars– Given the difference in molecular composition and lower atmospheric

pressure:• Absorption is significantly larger on Mars than on Earth• Strong nonequilibrium effects

Questions?Questions?

• Acknowledgments– I would like to thank the NASA GSRP program; as

well as Dr Lyle Long, Dr Feri Farassat, Dr. James Anderson, Dr. Victor Sparrow, Dr Tom Gabrielson, and Patrick O’Connor.

• Email:– Amanda D. Hanford: ald227@psu.edu