Nano Defects

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The Pennsylvania State University

The Graduate School

NUMERICAL SIMULATIONS OF ACOUSTICS PROBLEMS

USING THE DIRECT SIMULATION MONTE CARLO METHOD

A Dissertation in

Acoustics

by

Amanda Danforth Hanford

c 2008 Amanda Danforth Hanford

Submitted in Partial Fulfillment

of the Requirements

for the Degree of

Doctor of Philosophy

August 2008


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The Dissertation of Amanda Danforth Hanford was reviewed and approved by

the following:

Lyle N. Long

Distinguished Professor of Aerospace Engineering and Acoustics

Dissertation Advisor, Chair of Committee

James B. Anderson

Evan Pugh Professor of Chemistry and Physics

Feri Farassat

Senior Theoretical Aeroacoustician, NASA Langley Research Center

Special Member

Thomas B. Gabrielson

Professor of Acoustics

Victor W. SparrowProfessor of Acoustics

Anthony A. Atchley

Professor of Acoustics

Chair of Graduate Program in Acoustics

Signatures are on file in the Graduate School.


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Abstract

In the current study, real gas effects in the propagation of sound waves are simu-lated using the direct simulation Monte Carlo method for a wide range of systems.This particle method allows for treatment of acoustic phenomena for a wide rangeof Knudsen numbers, defined as the ratio of molecular mean free path to wave-length. Continuum models such as the Euler and Navier-Stokes equations breakdown for flows greater than a Knudsen number of approximately 0.05. Continuummodels also suffer from the inability to simultaneously model nonequilibrium con-ditions, diatomic or polyatomic molecules, nonlinearity and relaxation effects andare limited in their range of validity. Therefore, direct simulation Monte Carlois capable of directly simulating acoustic waves with a level of detail not possiblewith continuum approaches.

The basis of direct simulation Monte Carlo lies within kinetic theory whererepresentative particles are followed as they move and collide with other parti-cles. A parallel, object-oriented DSMC solver was developed for this problem.Despite excellent parallel efficiency, computation time is considerable. Monatomicgases, gases with internal energy, planetary environments, and amplitude effectsspanning a large range of Knudsen number have all been modeled with the samemethod and compared to existing theory. With the direct simulation method,significant deviations from continuum predictions are observed for high Knudsennumber flows.

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Table of Contents

List of Figures viii

List of Tables xiii

List of Symbols xiv

Acknowledgments xviii

Chapter 1Introduction 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Numerical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Continuum methods . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Particle methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Sound for all Knudsen numbers . . . . . . . . . . . . . . . . . . . . 61.6 Direct Simulation Monte Carlo . . . . . . . . . . . . . . . . . . . . 7

1.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Chapter 2Kinetic Theory of Gases 92.1 Historical background and

Ludwig Boltzmann . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Velocity distribution function . . . . . . . . . . . . . . . . . . . . . 112.2.1 Macroscopic properties in a simple gas . . . . . . . . . . . . 112.3 Deriving the Boltzmann equation . . . . . . . . . . . . . . . . . . . 16

2.3.1 The calculation of the collision integral . . . . . . . . . . . . 172.4 Deriving conservation equations

from the Boltzmann equation . . . . . . . . . . . . . . . . . . . . . 19

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2.5 Solutions to the Boltzmann equation . . . . . . . . . . . . . . . . . 222.5.1 Equilibrium properties . . . . . . . . . . . . . . . . . . . . . 22

2.5.2 Linearized Boltzmann Equation . . . . . . . . . . . . . . . . 242.5.2.1 1st order solution . . . . . . . . . . . . . . . . . . . 252.5.2.2 Transport coefficients . . . . . . . . . . . . . . . . 26

2.5.3 Bhatnagar, Gross, and Krook (BGK) equation . . . . . . . . 282.5.4 Numerical solutions to the Boltzmann equation . . . . . . . 29

Chapter 3Implementation of DSMC for Acoustics Simulations 313.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1.1 Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.1.2 Move particles and Boundary Conditions . . . . . . . . . . . 33

3.1.3 Sort particles . . . . . . . . . . . . . . . . . . . . . . . . . . 353.1.4 Collision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1.4.1 Binary collisions . . . . . . . . . . . . . . . . . . . 353.1.4.2 DSMC collision routine . . . . . . . . . . . . . . . 37

3.1.5 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2 Diatomic and polyatomic gases . . . . . . . . . . . . . . . . . . . . 40

3.2.1 DSMC tests for gases with internal energy . . . . . . . . . . 433.3 Mixtures implementation . . . . . . . . . . . . . . . . . . . . . . . . 443.4 Assumptions and Error within DSMC . . . . . . . . . . . . . . . . . 45

Chapter 4

Computing Issues 504.1 Object-Oriented Programming Approach . . . . . . . . . . . . . . . 504.2 Parallel Implementation . . . . . . . . . . . . . . . . . . . . . . . . 534.3 Modifications to the DSMC algorithm . . . . . . . . . . . . . . . . . 58

4.3.1 Small deviation from equilibrium . . . . . . . . . . . . . . . 584.3.2 Low Kn flows modifications . . . . . . . . . . . . . . . . . . 60

Chapter 5Absorption and Dispersion in a Monatomic Gas 625.1 The speed of sound . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.2 The absorption of sound . . . . . . . . . . . . . . . . . . . . . . . . 63

5.3 Theory on the absorption anddispersion of sound in a monatomic gas . . . . . . . . . . . . . . . . 64

5.4 DSMC Results for the absorptionand dispersion of sound in a monatomic gas . . . . . . . . . . . . . 695.4.1 Simulation approach . . . . . . . . . . . . . . . . . . . . . . 69

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5.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 705.4.2.1 Nonequilibrium . . . . . . . . . . . . . . . . . . . . 72

Chapter 6Absorption and Dispersion in a Gas With Internal Energy 756.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756.2 Absorption and dispersion from a simple

relaxation process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 756.3 Current theories including rotational

relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776.3.1 Other theories for rotational relaxation . . . . . . . . . . . . 80

6.4 Current theories including vibrationalrelaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

6.5 Implementation in DSMC for a gas withrotational energy for multiple collisionnumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836.5.1 Simulation approach . . . . . . . . . . . . . . . . . . . . . . 836.5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

6.6 Implementation in DSMC for a gas withrotational and vibrational energy . . . . . . . . . . . . . . . . . . . 876.6.1 Simulation approach . . . . . . . . . . . . . . . . . . . . . . 876.6.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.6.2.1 Nonequilibrium . . . . . . . . . . . . . . . . . . . . 906.6.2.2 Absorption as a function of temperature . . . . . . 92

6.6.2.3 Dispersion as a function of temperature . . . . . . 95

Chapter 7The Effect of Amplitude 987.1 Simulation approach . . . . . . . . . . . . . . . . . . . . . . . . . . 987.2 Breakdown of the propagation constant . . . . . . . . . . . . . . . . 997.3 Nonequilibrium effects as a function

of amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1037.4 Absorption and dispersion as a function of amplitude . . . . . . . . 1057.5 Harmonic generation . . . . . . . . . . . . . . . . . . . . . . . . . . 1097.6 Shock coalescence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

7.6.1 Coalescence at high Kn . . . . . . . . . . . . . . . . . . . . 116

Chapter 8DSMC Applications: Planetary Acoustics 1198.1 Simulation approach . . . . . . . . . . . . . . . . . . . . . . . . . . 121

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8.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 1218.2.1 Earth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

8.2.2 Mars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1248.2.3 Titan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1288.2.4 Vertical profiles . . . . . . . . . . . . . . . . . . . . . . . . . 131

Chapter 9Conclusions 1379.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Appendix ABuffons Needle Experiment 142

Appendix BSampling a Maxwellian Distribution 144

Appendix CDeriving the Navier-Stokes Dispersion Relation 147

Bibliography 150

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List of Figures

1.1 Limits of applicability of various mathematical models to simulatefluid flow [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 Ludwig Boltzmann (1844-1906) . . . . . . . . . . . . . . . . . . . . 10

2.2 Maxwellian velocity distribution function for argon at 0

. . . . . . 23

3.1 A flowchart of the DSMC algorithm. There are NST total sampleswith NIS time steps in between samples. J and I increment NSTand NIS respectively. . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2 A flowchart of the collision routine in the DSMC algorithm. sigV-max is calculated by looping through all the particles in the cell,the number of collisions to be performed, numColl, is given by Eq.(3.15) and the probability of collision, prob, is given by Eq. (3.14).Details of the internal energy exchange routine will be given in Sec.(3.2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3 A flowchart of the internal energy exchange routine in the DSMCalgorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.4 DSMC simulation of nitrogen molecules undergoing classical relax-ation at 4000 K. Birds exponential model[1] given by solid linesand DSMC results in dashed lines [2]. . . . . . . . . . . . . . . . . . 44

3.5 DSMC simulation of nitrogen molecules undergoing relaxation witha coupled discrete vibration / classical rotation model at 4000 K.Birds exponential model[1] given by solid lines and DSMC resultsin dashed lines [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.6 Equilibrium initialization of argon at 0 C after 1 ensemble . . . . . 473.7 Equilibrium initialization of argon at 0 C after 10 ensembles . . . . 473.8 Equilibrium initialization of argon at 0 C after 100 ensembles . . . 483.9 Equilibrium initialization of argon at 0 C after 1000 ensembles . . 48

4.1 A flowchart of the parallel algorithm . . . . . . . . . . . . . . . . . 54

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4.2 CPU time on Columbia and Mufasa with respect to number ofprocessors compared to ideal CPU time . . . . . . . . . . . . . . . . 56

4.3 Speedup on Columbia and Mufasa with respect to the number ofprocessors compared to ideal speedup . . . . . . . . . . . . . . . . . 574.4 Parallel efficiency on Columbia and Mufasa with respect to the

number of processors . . . . . . . . . . . . . . . . . . . . . . . . . . 584.5 Parallel efficiency on Mufasa with respect to the number of proces-

sors for a large and small system . . . . . . . . . . . . . . . . . . . 59

5.1 Scaled absorption cl/k0 and dispersion cl/k0 predictions given bythe linearized Navier-Stokes equations from Eq. (5.20) are plottedwith the low frequency classical absorption coefficient given by Eq.(5.21) [3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5.2 Scaled absorption cl/k0 in argon for 273 K. DSMC results (redcircles) compared to experimental results by Greenspan [4] (greentriangles) and Schotter[5] (blue squares) and continuum theory pre-dictions results from Eq. (5.20) [3] . . . . . . . . . . . . . . . . . . 71

5.3 Scaled dispersion k/k0 in argon for 273 K. DSMC results (red cir-cles) compared to experimental results by Greenspan [4] (green tri-angles) and continuum theory predictions results from Eq. (5.20)[3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.4 Translational nonequilibrium effects for Kn = 2.0 at 273 K in argon. 735.5 Translational nonequilibrium effects for Kn = 0.02 at 273 K in argon. 74

6.1 Scaled absorption for collision numbers Zrot =1, 10, 100, and 1000in a nitrogen-like gas at 0 C from Eqs. (6.7) and (6.3) [3, 6] . . . . 79

6.2 Scaled dispersion for collision numbers Zrot =1, 10, 100, and 1000in a nitrogen-like gas at 0 C from Eqs. (6.8) and (6.2) [3, 6] . . . . 79

6.3 Vibrational relaxation frequency for oxygen and nitrogen as a func-tion of temperature [7, 8] . . . . . . . . . . . . . . . . . . . . . . . 82

6.4 Scaled absorption for relaxation collision number of 1. DSMC simu-lations (points) are plotted with continuum theory for the rotationalrelaxation given by Eq. (6.3) (dashed line) [3] and total absorptiongiven by Eq. (6.7) (solid line) [6] . . . . . . . . . . . . . . . . . . . 85

6.5 Scaled absorption for relaxation collision number of 5. DSMC simu-

lations (points) are plotted with continuum theory for the rotationalrelaxation given by Eq. (6.3) (dashed line) [3] and total absorptiongiven by Eq. (6.7) (solid line) [6] . . . . . . . . . . . . . . . . . . . 85

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6.6 Scaled absorption for relaxation collision number of 40. DSMCsimulations (points) are plotted with continuum theory for the ro-

tational relaxation given by Eq. (6.3) (dashed line) [3] and totalabsorption given by Eq. (6.7) (solid line) [6] . . . . . . . . . . . . . 866.7 Scaled absorption for relaxation collision number of 200. DSMC

simulations (points) are plotted with continuum theory for the ro-tational relaxation given by Eq. (6.3) (dashed line) [3] and totalabsorption given by Eq. (6.7) (solid line) [6] . . . . . . . . . . . . . 86

6.8 DSMC results for the scaled absorption with relaxation collisionnumbers of 1, 5, 40, and 200. DSMC simulations (points) are plot-ted with continuum theory for rotational relaxation (dashed line)[3] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6.9 Nonequilibrium effects for Kn = 0.02 at 273 K with classical vibra-

tion model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916.10 Nonequilibrium effects for Kn = 1.0 at 273 K with classical vibra-

tion model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 916.11 Fraction of molecules in the excited state for Kn = 0.02 at 273 K

(red square), 2000 K (green triangle) and 4000 K (blue circle). . . . 926.12 (Scaled absorption in nitrogen for 273 K. DSMC results (symbols)

compared to continuum theory predictions given by Eqs. (5.20),(5.21), and (6.7) (lines) [3, 6]. . . . . . . . . . . . . . . . . . . . . . 93

6.13 Scaled absorption in nitrogen for 2000 K. DSMC results (symbols)compared to continuum theory predictions given by Eqs. (5.20),(5.21), and (6.7) (lines) [3, 6]. . . . . . . . . . . . . . . . . . . . . . 94

6.14 Scaled absorption in nitrogen for 4000 K. DSMC results (symbols)compared to continuum theory predictions given by Eqs. (5.20),(5.21), and (6.7) (lines) [3, 6]. . . . . . . . . . . . . . . . . . . . . . 95

6.15 Scaled dispersion in nitrogen for 273 K. DSMC results (symbols)compared to continuum theory predictions given by Eqs. (5.20),(6.2), and (6.8) (lines) [3, 6]. . . . . . . . . . . . . . . . . . . . . . . 96

6.16 Scaled dispersion in nitrogen for 2000 K. DSMC results (symbols)compared to continuum theory predictions given by Eqs. (5.20),(6.2), and (6.8) (lines) [3, 6]. . . . . . . . . . . . . . . . . . . . . . . 97

6.17 Scaled dispersion in nitrogen for 4000 K. DSMC results (symbols)

compared to continuum theory predictions given by Eqs. (5.20),(6.2), and (6.8) (lines) [3, 6]. . . . . . . . . . . . . . . . . . . . . . . 97

7.1 Maximum pressure amplitude (Pa) in argon for Kn = 2, 40 m/samplitude. DSMC results (points) compared to restricted exponen-tial best fit curve for x < 10u/ (solid line) . . . . . . . . . . . . . 100

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7.2 Maximum pressure amplitude (Pa) in argon for Kn = 2, 40 m/samplitude. DSMC results (points) compared to exponential best

fit curve for x > 10u

/ (solid line) . . . . . . . . . . . . . . . . . . 1017.3 Maximum pressure amplitude (Pa) in argon for Kn = 2, 5 m/samplitude. DSMC results (points) compared to exponential bestfit curve for x > 10u/ (solid line) . . . . . . . . . . . . . . . . . . 101

7.4 Maximum pressure amplitude (Pa) in argon for Kn = 0.02, 40 m/samplitude. DSMC results (points) compared to exponential bestfit curve for x > 10u/ (solid line) . . . . . . . . . . . . . . . . . . 102

7.5 Ttrx/Ttr for Kn = 2, 40 m/s amplitude. . . . . . . . . . . . . . . . . 1037.6 Ttrx/Ttr for Kn = 2, 5 m/s amplitude. . . . . . . . . . . . . . . . . 1047.7 Ttrx/Ttr for Kn = 0.02, 40 m/s amplitude. . . . . . . . . . . . . . . 1047.8 Ttrx/Ttr for Kn = 0.02, 5 m/s amplitude. . . . . . . . . . . . . . . . 105

7.9 Amplitude dependence on the scaled absorption in argon at 0

asa function of Kn. DSMC results (points) compared to continuumtheory given by Eq. (5.20) (line) . . . . . . . . . . . . . . . . . . . . 106

7.10 Wave steepening at Kn = 0.02 and 40 m/s amplitude . . . . . . . . 1077.11 Amplitude dependence on the scaled dispersion in argon at 0 as a

function ofKn based on the maximum pressure amplitude. DSMCresults (points) compared to continuum theory given by Eq. (5.20)(line) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

7.12 Amplitude dependence on the scaled dispersion in argon at 0 as afunction ofKn based on the zero crossings of the acoustic pressure.DSMC results (points) compared to continuum theory given by Eq.(5.20) (line) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7.13 Number of collisions performed in each cell for a 40 m/s amplitude(dashed line) and 5 m/s amplitude (solid line) at Kn = 0.2 after atime of 1 nanosecond . . . . . . . . . . . . . . . . . . . . . . . . . . 110

7.14 Fourier component amplitude for Kn = 0.005, 40 m/s amplitudecompared to Fubini solution given by Eq. (7.4) and Burgers equa-tion given by Eq. (7.5) [9, 10]. . . . . . . . . . . . . . . . . . . . . . 112

7.15 Fourier component amplitude for Kn = 2, 40 m/s amplitude. . . . . 1137.16 Fourier component amplitude for Kn = 2, 5 m/s amplitude. . . . . 1137.17 Shock coalescence for Kn = 0.02 at 6 nanoseconds . . . . . . . . . . 114

7.18 Shock coalescence for Kn = 0.02 at 15 nanoseconds . . . . . . . . . 1157.19 Shock coalescence for Kn = 0.02 at 21 nanoseconds . . . . . . . . . 1157.20 Shock coalescence for Kn = 0.02 at 46 nanoseconds . . . . . . . . . 1167.21 Coalescence for Kn = 2 at 42 nanoseconds . . . . . . . . . . . . . . 1177.22 Coalescence for Kn = 2 at 63 nanoseconds . . . . . . . . . . . . . . 1177.23 Coalescence for Kn = 2 at 74 nanoseconds . . . . . . . . . . . . . . 118

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7.24 Coalescence for Kn = 2 at 84 nanoseconds . . . . . . . . . . . . . . 118

8.1 Temperature and pressure profiles as a function of altitude above

Earths surface [11] . . . . . . . . . . . . . . . . . . . . . . . . . . . 1228.2 Scaled absorption for dry air on Earth. DSMC simulations (points)

are plotted with continuum theory for the vibrational relaxationgiven by Eq. (6.3) (green) and total absorption given by Eq. (6.7)(red) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

8.3 Scaled absorption for humid air on Earth. DSMC simulations(points) are plotted with continuum theory for the vibrational re-laxation given by Eq. (6.3) (green) and total absorption given byEq. (6.7) (red) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

8.4 DSMC results (solid line) for the acoustic pressure on Earth for

Kn = 0.02 compared to predicted amplitude dependence deter-mined from the Navier-Stokes derived absorption coefficient fromEq. (5.20)(dashed line) . . . . . . . . . . . . . . . . . . . . . . . . . 125

8.5 Temperature and pressure profiles as a function of altitude abovethe Mars surface [12] . . . . . . . . . . . . . . . . . . . . . . . . . . 126

8.6 DSMC results for the acoustic pressure amplitude as a function ofdistance on Mars for Kn = 0.02 compared to theoretical predictions[13] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

8.7 Nonequilibrium effects on Mars showing rotational (dashed line)and translational (solid line) temperatures for Kn = 2 . . . . . . . . 128

8.8 Temperature and pressure profiles as a function of altitude above

Titans surface [14] . . . . . . . . . . . . . . . . . . . . . . . . . . . 1298.9 Kn = 0.042 waveform on Earth with = 0.14. Absorption domi-

nates nonlinearity with little or no nonlinear effects visible. . . . . . 1 318.10 Kn = 0.01 waveform on Titan with = 0.14. Significant wave

steepening can be observed. . . . . . . . . . . . . . . . . . . . . . . 1328.11 Knudsen number as a function of altitude on Earth for frequencies

of 100 Hz, 1000 Hz, 10000 Hz, and 1000000 Hz . . . . . . . . . . . . 1338.12 Knudsen number as a function of altitude on Mars for frequencies

of 100 Hz, 1000 Hz, 10000 Hz, and 1000000 Hz . . . . . . . . . . . . 1348.13 Knudsen number as a function of altitude on Titan for frequencies

of 100 Hz, 1000 Hz, 10000 Hz, and 1000000 Hz . . . . . . . . . . . . 1358.14 The scaled absorption for a 70 MHz signal on Earth (blue), Mars(red), and Titan (green) at an altitude of 25 km compared to Navier-Stokes predicted thermal-viscous losses (black line) . . . . . . . . . 136

A.1 An illustration of the Buffons needle experiment. . . . . . . . . . . 142

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List of Tables

6.1 The Prandlt number for common gases . . . . . . . . . . . . . . . . 78

8.1 Atmospheric conditions at the surface on Earth, Mars and Titan . . 120

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List of Symbols

c Speed of sound, phase speed, [m/s], p. 63

cp Specific heat at constant pressure, [J/(kg K)], p. 16

Cp Heat capacity at constant pressure, [J/(kmol K)], p. 15

cv Specific heat at constant volume, [J/(kg K)], p. 16

Cv Heat capacity at constant volume, [J/(kmol K)], p. 15

c0 Low amplitude, low frequency speed of sound, [m/s], p. 63

C Heat capacity of relaxing mode, [J/(kmol K)], p. 76

d Distance, [m], p. 142

eint Internal energy per unit mass, [J/kg], p. 14

f Velocity distribution function, p. 11

f Frequency, [Hz], p. 76

F Force field per unit mass, p. 16

Fnum Ratio of real to simulated particles, p. 37

f0 Maxwellian velocity distribution function, p. 23

fr Relaxation frequency, [Hz], p. 76

J Bessel function, p. 110

k Complex propagation constant, [1/m], p. 64

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kb Boltzmanns constant, [J/K], p. 13

kcl Complex propagation constant for classical losses, [1/m], p. 64

k0 Wave number, [1/m], p. 67

Kn Knudsen number, p. 1

m Molecular mass, [kg/molecule], 11

M Molecular weight, [kg/kmol], 14

n Number density, [molecules/m3], p. 11

N Avogadros number, [molecules/kmol], p. 13Nc Number of molecules in a cell, p. 37

p Pressure tensor, [Pa], p. 13

p Hydrostatic pressure, [Pa], p. 13

q Heat flux vector, [W/m2], p. 15

Q Macroscopic quantity, p. 12

r Particle position vector, [m], p. 11

R Universal gas constant, [J/(kmol K)], p. 13s Relaxation strength, p. 76

S Sonine polynomials, p. 25

t Time, [s], p. 11

T Temperature, [K], p. 13

Tint Internal temperature, [K], p. 14

Ttr Translational temperature, [K], p. 14Tov Overall temperature, [K], p. 14

u Molecular velocity, [m/s], p. 11

U Mean velocity, [m/s], p. 12

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U0 Macroscopic velocity amplitude, [m/s], p. 107

u Thermal velocity, [m/s], p. 12

u Average thermal speed, [m/s], p. 23

um Most probable molecular thermal speed, [m/s], p. 23

ur Relative speed, [m/s], p. 18

x Shock formation distance, [m], p. 109

Z Relaxation collision number, p. 40

Zrot Rotational relaxation collision number, p. 77

Zvib Vibrational relaxation collision number, p. 87

Absorption coefficient, [np/m], p. 64

cr Combined absorption due to classical thermal-viscous losses and rotationalrelaxation, [np/m], p. 77

r Absorption due to a single relaxation process, [np/m], p. 76

rot Absorption due to rotational relaxation, [np/m], p. 77

vib Absorption due to vibrational relaxation, [np/m], p. 82

Dispersion coefficient, [1/m], p. 64

cr Combined dispersion due to classical thermal-viscous losses and rotationalrelaxation, [1/m], p. 77

NL Coefficient of nonlinearity, p. 107

m Reciprocal of the most probable molecular thermal speed, [s/m], p. 23

r Dispersion due to a single relaxation process, [1/m], p. 76

rot Dispersion due to rotational relaxation, [1/m], p. 77

Ratio of specific heats, p. 15

Goldberg number, p. 111

Kronecker delta function, p. 13

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Acoustic Mach number, p. 109

Number of internal degrees of freedom, p. 14

Thermal conductivity, [J/K m s)], p. 26

m Mean free path, [m], p. 1

Coefficient of viscosity, [kg/(m s)], p. 26

b Bulk viscosity, [kg/(m s)], p. 27

Collision frequency, [1/s], p. 76

Number of total degrees of freedom, p. 15

Mass density, [kg/m3], p. 11

Collision cross section, [kg m2], p. 17

NL Nondimensional shock formation distance, p. 110

Viscous stress tensor, [Pa], p. 13

r Relaxation time, [1/s], p. 76

Macroscopic quantity, p. 64

Perturbation from distribution function, p. 24

Dissipation function, p. 21

Frequency, [rad/s], p. 64

Scattering angle, p. 17

xvii


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Acknowledgments

I would like to give my sincere thanks to Prof. Lyle N. Long for his advice andsupport throughout my pursuit of this degree. This project would not have beenpossible without his guidance.

I express my gratitude to my other committee members, Dr. James Anderson,Dr. Thomas Gabrielson, Dr. Victor Sparrow, and Dr. Feri Farassat for theircomments and suggestions.

I would also like to acknowledge the National Science Foundation for fund-ing the Consortium for Education in Many-Body Applications (CEMBA), GrantNo. NSF-DGE-9987589, and the NASA Graduate Student Fellowship Program forfunding and providing computer resources for this research project.

A huge thanks to Dr. Patrick D. OConnor. Ive said it before, but Ill say itagain, if Im getting a PhD, he should be getting two for all the hard work andcountless hours of conversation weve had over the years. I honestly wouldnt be

where I am today without his help and friendship.Special thanks to Rebecca Sanford DeRousie and Bernadette Rakszawski formaking the littleman laugh while I typed away and to Catherine Hofstetter whosefriendship and motherhood I will always admire. And a huge thanks to JenMarcovich, Mahreen George, Jen Dombroskie, Suzi Lang, Shelley Farahani, JulieWillits, Allison Bohn and Bethany Heim for being such a loving and enrichingpart of our lives and whose support and friendship have meant so much to me.And thanks also to the La Leche League of State College and Centre CountyBabywearers for enriching our lives and making me the mother I am today.

Of course, I also owe a large debt of gratitude to my family. My parents,Rhoda and Larry Danforth have given me countless words of encouragement and

support over the years. To my sister and her family Becky, Hans, Chloe, Paige,and Peanut Watz for plenty of phone calls and stories to help me through theday. To my brother, David Danforth for always keeping me on my toes with loveand big hugs. And much love and thanks to Pat, Keith and Kim Hanford for allof the loving, warm encouragement and support in all things. Many thanks to

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Karl, Melissa, Alice and Liz Sweitzer for being such an amazing part of my life,giving me rides, talking Matlab, making cookies and hair dying. Thank you all for

making me feel like the smartest human alive, even when my day to day activitiesprove that I am not.I am indebted to my husband, Scott, for providing me motivation and encour-

agement at every step during this pursuit, for being with me through thick andthin, and for the joy and happiness his company brings me. And many thanks toNoah, the best littleman a mother could have, for reminding me that unconditionallove triumphs any degree.

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Dedication

To mothers and children everywhere. Children can teach you so much more aboutlife than books.

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Chapter1

Introduction

1.1 Motivation

Although popular in acoustics, continuum models such as the Euler and Navier-

Stokes equations break down for flows approaching the well-known continuum limit

defined by the Knudsen number. Continuum models also suffer from the inabil-

ity to simultaneously model nonequilibrium conditions, diatomic or polyatomic

molecules, nonlinearity and relaxation effects and are limited in their range of va-

lidity. Theory for the behavior of sound at high Knudsen numbers is inconsistentwith experimental results and needs to be resolved. Therefore, a single method

that spans the entire Knudsen number range is desirable. The best approach,

one of the few available, is to use a particle-based model based on the Boltzmann

equation to bypass continuum limitations while also minimizing the numerical com-

plexity of the model itself. Therefore, a parallel, object-oriented direct simulation

Monte Carlo (DSMC) method was chosen for this study in the application to the

simulation of acoustic wave propagation.

1.2 Numerical models

Fluid dynamics models for a gas can be categorized into two groups: continuum

methods and particle methods. Continuum methods, which are widely used for

acoustic problems, model the fluid as a continuous medium. This model describes


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2

the state of the fluid macroscopically using quantities such as density, pressure,

velocity, and temperature. The continuum approximation is valid when the char-

acteristic length of the problem is much larger than the molecular mean free path

(m). The Knudsen number (Kn) is a nondimensional parameter defined as the

mean free path divided by a characteristic length and is a measure of the thermal

nonequilibrium in a gas [15]. The continuum condition (Kn < 0.05) is satisfied

for many engineering problems, which can be described using continuum equations

such as the Navier-Stokes or Euler equations.

Particle methods are based on molecular models that describe the state of the

gas at the microscopic level. The mathematical model at this level is the Boltzmann

equation which will be derived and discussed in detail in Chapter (2). Despite thefact that the Boltzmann equation has been derived using a microscopic approach

it will also been shown in Chapter (2) that the Boltzmann equation will reduce to

the continuum conservation equations (e.g., Navier-Stokes) for low Kn [1, 15].

1.3 Continuum methods

The classical linear theory of sound propagation assumes that all acoustic fields

can be written as the sum of an equilibrium value 0 and a small perturbationfrom equilibrium . The governing continuum equations that describe the per-

turbation from equilibrium can be derived from linearizing the Navier-Stokes

or Euler equations. During the linearization process, higher order terms are ig-

nored by assuming the perturbation from equilibrium is small. Many valuable

acoustic phenomenon can be developed under this assumption (see reference [9]).

However, using the linearized versions of the continuum equations is only valid for

small acoustic amplitudes. Computational methods in linear acoustics are fast and

simple but the restriction to low acoustic amplitudes is undesirable.

Keeping higher order terms allows for exploration in nonlinear acoustics where

does not necessarily have to be small to describe nonlinear phenomena such

as shock wave formation and harmonic generation [10]. An overview of compu-

tational approaches in nonlinear acoustics reveals that nonlinear effects are often

treated separately using the nonlinear Euler equations or Burgers equation, and

then superimposed to produce a final result, often going back and forth between


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3

the time and frequency domains [16, 17]. These methods are often tailored for quite

specific cases, and usually model only one-dimensional propagation, hindering the

general utility of such models. A method by Sparrow and Raspet [18] that con-

tains up to second-order nonlinear terms provides a multidimensional simulation

tool for nonlinear acoustics, but the method did not explicitly model relaxation

mechanisms.

The full Navier-Stokes equations are considered a viable alternative for many

systems as they include more physics than the Euler equations. However, the in-

creased complexity of the Navier-Stokes equations makes it much more difficult to

simulate than the Euler equations. Computational fluid dynamics algorithms most

often used in aeroacoustics, can be implemented in two or three dimensions andare nonlinear, but they do not include the effects of molecular relaxation, and are

computationally expensive [19, 20, 21, 22, 23]. Wochner et al. [24] use nonlinear

fluid dynamics equations written in terms the total fluid dynamic variables rather

than the acoustic variables to include the effects of shear viscosity, bulk viscos-

ity, thermal conductivity, and molecular relaxation to simulate nonlinear acoustic

phenomena.

Computational aeroacoustics studies noise generation via either turbulent fluid

motion or aerodynamic forces interacting with surfaces. Although no complete

scientific theory of the generation of noise by aerodynamic flows has been estab-

lished, one of the most practical aeroacoustic analyses relies upon a so called wave

equation using Lighthills aeroacoustic analogy. Lighthill rearranged the Navier-

Stokes equations into an inhomogeneous wave equation, thereby making an analogy

between fluid mechanics and acoustics [25, 26].

However, these approaches are also limited because of their reliance on tradi-

tional finite difference algorithms, requiring considerable computational resources,

subject to numerical instability, and are all based on the continuum assumption.

The Knudsen number is used to distinguish the regimes where different gov-

erning equations of fluid dynamics are applicable. Fig. (1.1) schematically draws

the limits of applicability of different continuum and particle methods. Continuum

methods are applicable in the small range where Kn < 0.05. For larger Kn, the

momentum and heat fluxes in the Navier-Stokes equations cannot be written in

terms of macroscopic quantities and hence the set of equations is incomplete [ 27].


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4

For even larger Kn, the fluid can no longer be considered a continuum, thus the

Navier-Stokes equations do not hold. The Boltzmann equation is valid over the

entire Kn range.

Figure 1.1. Limits of applicability of various mathematical models to simulate fluidflow [1]

1.4 Particle methods

Particle methods that are based on the Boltzmann equation are valid for all Knud-sen numbers. Therefore, particle methods are necessary for, but not limited to,

problems where the Knudsen number is greater than about 0.05.

Some of the different particle methods include the Molecular Dynamics (MD)

model [28, 29, 30], Monte Carlo methods [31], cellular automata [32, 33], discrete

velocity [34, 35], and the Lattice Boltzmann Method (LBM) [36, 37, 38, 39, 40].

The most fundamental approach is the Molecular Dynamics model. In this ap-

proach the particles move according to Newtons laws and particle interactions

are calculated by force potentials. This model can even account for quantum me-chanical effects. Molecular dynamics is usually required to simulate dense gases or

liquids, therefore the expensive intermolecular force calculations make it unsuitable

for dilute gases.

The Monte Carlo approaches and the Boltzmann equation are derived from the

Liouville equation. The term Monte Carlo was adopted in the 1940s for meth-

ods involving statistical techniques, such as the use of random numbers to find the


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5

solutions to mathematical or physical problems [31]. The first documented appli-

cation of using a Monte Carlo method is given by Georges-Louis Leclerc, Comte

de Buffon in 1777 when he describes a Monte Carlo procedure that can be used

to evaluate from the throws of a needle onto a floor ruled with parallel straight

lines [41]. The probability that the needle would intersect a line is calculated by

randomly throwing the needle many times and calculating the ratio of the number

of throws intersecting a line to the total number of throws. This probability gives

an estimation of and is discussed further in Appendix A. Lord Kelvin used a

Monte Carlo method in 1901 to perform time integrals which appeared in his ki-

netic theory of gases [42]. However, it was only in 1940s that Monte Carlo methods

were developed enough to be used for solving engineering problems.The cellular automata method considers the evolution of idealized particles

which move at unit speed from one node to another on a discretized temporal

grid [32, 33]. The discrete velocity method allows molecules to move in continuous

space, with velocities that are discrete, and non-uniformly distributed [34, 35]. The

Lattice Boltzmann method (LBM) was developed from cellular automata and dis-

crete velocity methods, and applies Boltzmann equation approximations to cellular

automata models. Particle motion using the LBM is restricted to a lattice, with

simple collision rules to conserve mass and momentum [36, 37, 38]. The LBM has

been successful in the study of one-dimensional sound propagation [39, 40]. Lim-

itations of some of these methods can be significant due to the simplification and

discretization of physical or velocity space, and can include the inability to simu-

late heat transfer effects or the effect of temperature on the transport properties.

These limitations are discussed further by Long et al. [33].

The direct simulation Monte Carlo (DSMC) is a stochastic, particle-based

method first introduced by G. A. Bird in 1963 [43] which is capable of simulat-

ing real gas effects for all values of Kn that traditional continuum models cannot

offer [1]. Particles in a DSMC simulation are not restricted to a grid and move

according to their velocities. While the collisions between particles are determined

statistically, they are required to satisfy mass, momentum, and energy conserva-

tion. DSMC offers computational flexibility to study sound for all Kn in a wide

range of systems.


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6

1.5 Sound for all Knudsen numbers

The Knudsen number is large for sound propagation in very dilute gases, in micro-channels, or at high frequencies, and thus requires a particle method, or kinetic

theory solution. There have been numerous attempts to study sound propagation

for high Kn based on kinetic theory in simple monatomic gases [44, 45, 46, 47, 48,

49, 50, 51]. Most of these attempts use approximations that replace the Boltzmann

collision integral with concomitant losses in accuracy. All attempts report that the

speed and absorption of sound depend heavily on Kn which deviates significantly

from traditional continuum theory at high Kn. Experimental results show close

agreement to kinetic theory results for Kn < 1, but for larger Kn, the results are

poor and inconsistent [4, 5, 52, 53, 54]. Detailed descriptions of continuum and

kinetic theory predictions for the absorption and dispersion of sound will be given

in Chapter (5).

Several authors have conducted theoretical studies of sound waves in rarefied

polyatomic gases to include internal energy effects on the propagation of sound.

These studies are based either on classical or kinetic theory. Classical contributions

include Greenspan [6], Herzfeld and Rice [55], Herzfeld and Litovitz [3], and Kneser

[56]. The kinetic theory studies of interest are by Wang Chang and Uhlenbeck [ 57],

Monchick et al. [58], Mason and Monchick [59], Hanson and Morse [60], Hanson et

al. [61], McCormack and Creech [62], McCormack [63], Banankhah [64]. As in the

monatomic gas case, most of these attempts use approximations that replace the

Boltzmann collision integral. Results vary in degrees of success with comparison

to experimental studies in gases with internal energy [52, 53] and will be discussed

further in Chapter (6).

Because of the inconsistencies between theory and experiment at high Kn and

the limitations of continuum methods at low Kn, a computational method that

spans the whole Kn range is desirable. The best approach, one of the few available,is to use the particle-based method DSMC to bypass continuum limitations while

also minimizing the numerical complexity of the model itself. DSMC has become

the de facto standard computational approach for high Kn due to its success at

accurately simulating a wide range of phenomena. In particular, sound propaga-

tion properties such as nonlinear phenomena and absorption are inherent in the


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7

algorithm because of the particle nature of the algorithm. Therefore, the DSMC

method was chosen for study in its application to the simulation of the acoustic

wave propagation.

1.6 Direct Simulation Monte Carlo

1.6.1 Introduction

The direct simulation Monte Carlo (DSMC) method is a direct particle simula-

tion tool that describes the dynamics of a gas through direct physical modeling

of particle motions and collisions. DSMC is based on the kinetic theory of gasdynamics modeled on the Boltzmann equation, where representative particles are

followed as they move and collide with other particles. Introductory [65, 66] and

detailed [1] descriptions of DSMC, as well as formal derivations [67] can be found

in the literature and will be discussed further in Chapter (3). Due to the particle

nature of the method, DSMC offers considerable flexibility with regard to the type

of system available for modeling: rarefied gas dynamics [1, 68, 69, 70], hypersonic

flows [1, 71, 72], and acoustics [65, 2, 73, 74]. DSMCs origins are based upon

the Boltzmann equation, but the applications of and the extensions to DSMC

method have now gone beyond the range of validity of this equation by simulating

chemical reactions [75, 76, 77] detonations [78, 79], and volcanic plumes and upper

atmospheric winds on Jupiters moon Io [80, 81].

DSMC is a particle method that describes the state of the gas at the micro-

scopic level, and is valid beyond the continuum assumption. In the case of one

dimensional acoustic wave propagation, the characteristic length is the acoustic

wavelength so the Knudsen number is directly proportional to the frequency of

oscillation. Despite the fact that DSMC is valid for all Kn, DSMC is most ef-

ficient for high Kn flows. DSMC has traditionally been used in regimes where

continuum methods fail, but in fact compares well with Navier-Stokes calculations

and experimental results within the continuum regime [72]. Moreover, DSMC has

many advantages over traditional continuum approaches even for low Kn situa-

tions. Without modification, DSMC is capable of simulating all physical properties

of interest at the molecular level for sound propagation: absorption, dispersion,


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8

nonlinearity, and molecular relaxation.

A parallel, object oriented DSMC solver was developed for this problem. The

code was written in C++ and is designed using the benefits of an object oriented

approach [82, 83]. The parallelization is achieved by performing parallel ensemble

averaging, using the Message Passing Interface (MPI) library for inter-processor

communications [84, 85]. The reason for implementing a parallel model is to reduce

the high computation time, and make the problem more efficient. An investigation

of this is discussed in Chapter 4.

DSMC simulations in a simple gas at low amplitudes are described in Chapter

(5) and as a function of amplitude in Chapter (7) to investigate absorption and

dispersion as a function of Kn.The DSMC program contains several types of energy models to treat molecules

in gas mixtures with internal energy. Internal energy is represented by rotational

and vibrational modes (electronic energy is ignored) and has been programmed

to simulate either classical or quantum behavior. Details on the internal energy

models used in DSMC will be given in Chapter (3). DSMC simulation results

implementing these internal energy models along with their effect on acoustics is

shown for a wide range of temperatures in Chapter (6) [2].

The flexibility of the DSMC algorithm allows for modeling of sound in specific

gas mixtures including models for Earth, Mars and Saturns moon Titan [73, 86,

87, 88]. Little modification to the method is needed to change the molecular and

ambient atmospheric properties in order to simulate sound on the different planets.

This feature of DSMC makes it beneficial for use in planetary acoustics where

atmospheric conditions are dependent on planet, time of year, altitude, etc. In

addition, the Kn is high in upper atmospheric conditions, thus requiring a particle

method solution. DSMC simulation results describing acoustic phenomenon on

Earth, Mars and Titan for all Kn is given in Chapter (8).

Chapter (9) is a summary of the work presented along with general conclusions

regarding this research. Potential applications and modifications of the method

are also described.


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Chapter2

Kinetic Theory of Gases

2.1 Historical background and

Ludwig Boltzmann

Ludwig Boltzmann (1844-1906) was an Austrian physicist and philosopher and is

pictured in Fig. (2.1). In the late 1800s, Boltzmann claimed that matter was

made up of tiny particles, that is atoms and molecules. Much of the physics

establishment at the time did not share his belief, which was subject to much

debate. Boltzmann had many scientific opponents, and was even on bad personal

terms with some of his colleagues. One of the leaders of the anti-atom school,

Wilhelm Ostwald, often battled Boltzmann fiercely on some of his theories. One

of the battles between the two was described by a colleague Sommerfeld [89].

The battle between Boltzmann and Ostwald resembled the battle of the bull

with the supple fighter. However, this time the bull was victorious...

Boltzmann suffered from an alternation of depressed moods with elevated, ex-

pansive or irritable moods, and the often violent disputes about his scientific theory

were enough to depress Boltzmann. He began to feel that his lifes work was about

to collapse despite his attempts to defend his theories. He attempted suicide once

in 1900, only to succeed in taking his life in 1906 just before experimental work

verified his theories [90]. Perrins studies of colloidal suspensions (1908-1909) [91]

confirmed the values of Avogadros number and Boltzmanns constant, and Ein-

steins theory on Brownian motion (1905) [92] helped convince the world that the


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10

Figure 2.1. Ludwig Boltzmann (1844-1906)

tiny particles really exist. Ostwald later told Sommerfeld that he had been con-

verted to a belief in atoms by Einsteins complete explanation of his theory on

Brownian motion [93].

Boltzmanns equation for entropy s = kb log W as it was published in his day,

is engraved on his tombstone, where kb is called the Boltzmann constant.

One of Boltzmanns main contributions is the invention of the field of statistical

mechanics. In addition, Boltzmanns work in the foundations of kinetic theory

still remains widely popular as his theories including the distribution for molecular

speeds in a gas are applicable to many phenomena in gas dynamics, and will now

be discussed.


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11

2.2 Velocity distribution function

Consider a simple gas in which all molecules are alike with mass m. Let dr denotea small parcel of this gas surrounding the point in space r. This parcel is large

enough to contain many molecules but small enough so that ambient macroscopic

quantities such as pressure or temperature do not vary throughout the parcel. The

number of molecules within the volume dr averaged over a time dt is then given

by ndr where n is the number density. The number density, n, and mass density,

, are related by:

= nm . (2.1)

The distribution of velocities among a large number of molecules in the parceldr can be represented by the velocity distribution function, f, that describes the

state of the gas. As molecules pass in and out of the volume dr, the distribu-

tion function is a function of time. Therefore, the probable number of molecules

that have velocities in the neighborhood of u = (ux, uy, uz) and positions in the

neighborhood ofr = (x,y,z), at the time t, is equal to:

f(u, r, t)dudr . (2.2)

Moreover, the whole number of molecules in the parcel dr is given by integrating

Eq (2.2) throughout the whole velocity space. Therefore,

n =

u

f du . (2.3)

In this case of a simple gas with no internal energy, the independent variables

are the three components of particle velocity and position, and time, which define

phase space. The particle velocity and position ranges from

to

in each

direction.

2.2.1 Macroscopic properties in a simple gas

Macroscopic flow quantities such as temperature and pressure can be calculated

in terms of averages over particle velocities. Therefore, we can use the velocity


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12

distribution function to calculate all macroscopic quantities. For any quantity

Q(u) that is a function of molecular velocity u, the mean value of Q, denoted Q,

can be written as:

Q =1

n

u

Qf du . (2.4)

The lower-order moments of the distribution function all have names and simple

physical interpretations, where the kth moment is defined as:

u

mukfdu , (2.5)

Because u is a vector, higher order moments can become tensors and requiretensor multiplication within the integrand. The tensor product, denoted , ofvector a = (a1, a2, a3) and b = (b1, b2, b3) is given as:

(a1, a2, a3) (b1, b2, b3) =

a1b1, a1b2, a1b3,

a2b1, a2b2, a2b3,

a3b1, a3b2, a3b3,

(2.6)

Using Eq. (2.3), the zeroth moment of the distribution function becomes:

nm = =

u

mfdu , (2.7)

giving the mass density of the gas .

The first order moment of the distribution function defines the particle flux

density associated with the transport of mass. Using Eqs. (2.4) and (2.5), this is

written as:

u = u

mufdu , (2.8)

where u = U is defined as the mean or stream velocity. The velocity of a molecule

relative to the stream velocity is call the thermal velocity, denoted by u, is given

by:

u = uU . (2.9)

The second moment case is of particular interest, defining the momentum flux


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13

by the thermal motion of the gas. Since momentum is a vector quantity, the

resulting expression is a tensor with nine Cartesian components, called the pressure

tensor p and given by:

p = uu =

u

muuf du . (2.10)

In component form, the pressure tensor is given by:

p =

u2x , u

xu

y, u

xu

z,

uyu

x, u2y , u

yu

z,

u

zu

x, u

zu

y, u2z ,

. (2.11)

Using tensor notation, the pressure tensor is written:

pij = uiu

j . (2.12)

The mean or hydrostatic pressure p is usually then defined as the average of

the three normal components of the pressure tensor, that is:

p =

1

3(u2

x + u2

y + u2

z ) =

1

3u2

. (2.13)

The viscous stress tensor is defined as the negative of the pressure tensor

p with the scalar pressure p subtracted from the normal components. It can be

written in tensor notation as:

= ij = (uiuj ijp) , (2.14)

where ij is the Kronecker delta such that ij = 1 ifi = j and ij = 0 ifi = j.The temperature T of a gas in uniform steady state is directly proportional to

the average kinetic energy associated with the translational motion of a molecule,12mu

2. The proportionality is given by the relation:

1

2mu2 =

3

2kbT , (2.15)

where kb = 1.3806504 1023 [J/K] is the Boltzmann constant which is related


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14

to the universal gas constant R = 8314.34 [J/(kmol K)] by kb = R/N whereN= 6.02214179 1026 [mol/kmol] is Avogadros number.

An immediate consequence of the definition of temperature is that the hydro-

static pressure p of a gas in equilibrium can be written in terms of the temperature

T resulting in the perfect gas law:

p = RT /M = nkbT , (2.16)

where M = m/N is the molecular weight of the molecule. T is an equilibriumgas property, but Eq (2.16) will hold, even in nonequilibrium situations, for the

translational kinetic temperature Ttr which is defined as:

3

2kbTtr =

1

2mu2 . (2.17)

In addition, the translational temperature may also be defined for each velocity

component separately. That is, the translational temperature in the x direction

can be defined as:

kbTtrx = mu2x , (2.18)

which can therefore provide a measure of translational nonequilibrium when com-

pared to Eq. (2.17). Measuring the amount of nonequilibrium in a system is a

useful tool that is unavailable in traditional continuum methods.

For diatomic or polyatomic gases that possess internal energy, a temperature

Tint can be defined for the internal modes associated with the rotational and vi-

brational energy of the molecule. Similarly to the translational temperature, the

internal temperature Tint can be defined as:

1

2RM

Tint = eint , (2.19)

where is the number of internal degrees of freedom and eint is the energy associ-

ated with the internal energy mode of interest. For a nonequilibrium gas, an overall

kinetic temperature Tov can be defined as a weighted average of the translational

and internal temperatures given by Eqs (2.17) and (2.19) given by:

Tov = (3Ttr + Tint)/(3 + ) . (2.20)


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15

Finally, the third moment of the velocity distribution function, the heat flux

vector q, is obtained. For a simple gas, the molecular energy is given by 12

mu2

and the heat flux vector is then given by:

q =1

2u2u =

u

1

2mu2uf du . (2.21)

For a gas in equilibrium at constant volume, the amount of heat that is added

to the system when the temperature changes is called heat capacity at constant

volume, denoted Cv, and is related to the universal gas constant by the equation:

Cv = R

2 , (2.22)

where is the number of degrees of freedom.

For a gas in equilibrium at constant pressure, the amount of heat that is added

to the system when the temperature changes is called heat capacity at constant

pressure, denoted Cp, and is related to the heat capacity at constant volume by

the equation:

Cp = R+ Cv . (2.23)

The specific heat at constant volume and pressure, cv and cp respectively, arerelated to the molecular weight of the gas and are defined as:

cv =CvM

(2.24)

and

cp =CpM

. (2.25)

The ratio of specific heats is denoted by . Thus:

=cpcv

=CpCv

= 1 +2

. (2.26)

where is the total number of degrees of freedom.


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16

Eqs. (2.22) and (2.23) can now be written as:

Cv = R 1 , (2.27)

and

Cp = R

1 +1

1

. (2.28)

In addition, the specific heats can also be written:

cv =kb

2m. (2.29)

cp =kbm

1 +

2

. (2.30)

2.3 Deriving the Boltzmann equation

Consider once more a small parcel of fluid. At any particular instant in time, the

number of molecules in the phase space element dudr is given by Eq. (2.2). If the

location and shape of the fluid element does not vary with time, the rate of change

of the number of molecules in the element is given by the expression:

f

tdudr . (2.31)

There are three processes that can contribute to the change in the number of

molecules within the phase space element dudr. They are:

1. The movement of molecules by the molecular velocity u.

2. The convection of molecules as a result of an external force per unit mass,

F, which may be a function ofr and t but not u.

3. The scattering of molecules as a result of intermolecular collisions.

When considering the first two processes during the times between t and t + dt,

any molecule with velocity u will change to u+Fdt and its position r will change

to r+udt. At time t, there are f(u,r, t)dudr molecules in this set. Without taking


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intermolecular collisions into account, after the time interval dt, these molecules

now occupy the fluid parcel r + udt with velocities u + Fdt. Applying Eq. (2.2),

the number of such molecules in this set is given by:

f(u + Fdt,r + udt,t + dt)dudr . (2.32)

The difference between these sets of molecules is the portion of Eq. (2.31) due to

scattering of molecules as the result of intermolecular collisions, which is denoted

(colf/t)dudr. Therefore,

f(u + Fdt, r + udt,t + dt)dudr

f(u, r, t)dudr =

colf

t

dudrdt . (2.33)

Dividing both sides by dudrdt gives:

f(u + Fdt, r + udt,t + dt) f(u, r, t)dt

=colf

t. (2.34)

Letting dt go to zero results in the equation to be written in terms of partial

derivatives:

f

t+ ux

f

x+ uy

f

y+ uz

f

z+ Fx

f

ux+ Fy

f

uy+ Fz

f

uz=

colf

t(2.35)

Written in vector notation, this results in the Boltzmann equation:

f

t+ u f

r+ F f

u=

colf

t. (2.36)

2.3.1 The calculation of the collision integral

In the evaluation of the collision integral, it is assumed that only binary collisions

between molecules are considered and collisions occupy a very small part of thelifetime of a molecule. Consider at the moment, the collision of a molecule with

velocity u and a molecule with velocity u1. Their post collision velocities are

given by u and u1 respectively. Let a molecule with velocity u be chosen as a

test particle moving with speed ur = u u1 amongst stationary molecules withvelocity u1. The number of molecules in the fluid element dr that have velocities

ofu1 is given by f1du1. The volume swept out in physical space per unit time


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that accounts for the cross section for this collision is given by urd where is

the collision cross sectional area and d is the scattering angle. Therefore, the

number of collisions between molecules with velocity u1 with the test particle per

unit time is given by:

f1urddu1 , (2.37)

where f denotes the value of the velocity distribution function f at u. Similarly,

f1, f and f1 denote the values of f at u1, u

and u1 respectively. Since the

number of molecules with velocity u in the phase space element dudr is f dudr,

the number of collisions between molecules with velocity u and u1 resulting in

post collision velocities ofu and u1 is:

f f1urddu1dudr . (2.38)

This expression represents the loss of molecules with velocity u due to collisions

with molecules with velocity u1. There are, however, inverse collisions that result

in molecules with post collision velocities ofu. Similarly, the number of these such

encounters is:

ff1 urddu1dudr . (2.39)

The rate of increase of molecules that have velocity u in the phase space element

dudr as a result of direct and inverse collisions with molecules with velocity u1 is

obtained by combining equations (2.38) and (2.39). This gives:

(ff1 f f1)urddu1dudr . (2.40)

The net gain of molecules per unit time that have velocity u is obtained by in-

tegrating this expression over the cross section for its collision with molecules of

velocity u1 and then integrating over all velocity space. This results in:

dudr

u1

40

(ff1 f f1)urddu1 , (2.41)


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which can be denoted by (colf/t)dudr. Hence, dividing by dudr results in the

collision integral:

colf

t=

u1

40

(ff1 f f1)urddu1 . (2.42)

Combining Eq. (2.36) and (2.42), the full expression for the Boltzmann equation

becomes:

f

t+ u f

r+ F f

u=

u1

4

0

(ff1 f f1)urddu1 . (2.43)

2.4 Deriving conservation equations

from the Boltzmann equation

Despite the fact that the Boltzmann equation was derived using a microscopic

approach, it can also be shown that the continuum conservation equations are

moments of the Boltzmann equation. To see this, begin by multiplying both sides of

the Boltzmann equation by a quantity Q(u) that is a function of molecular velocity

and then integrating over velocity space. The multiplication of the Boltzmann

equation (2.36) gives:

Qf

t+ Qu f

r+ QF f

u= Q

colf

t. (2.44)

Q is a function of molecular velocity and not of distance, thus when integrating

both sides over velocity space we can write:

u

t (Qf)du +u

(Quf)du + u

QF f

udu =u

Qcolf

tdu . (2.45)


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The third integral term can be rewritten using the chain rule. In addition, by

assumption, F is independent ofu, thus, Eq. (2.45) can be simplified as:

u

t(Qf)du+

u

(Quf)duu

FQ

uf du =

u

Qcolf

tdu . (2.46)

By using Eq. (2.4) this can be rewritten as:

t(nQ) + (nQu) nF Q

u=

u

nQcolf

tdu . (2.47)

If the quantity Q is either the mass m, momentum mu

or energy

1

2mu

2

ofa molecule, then the collision integral colft on the right hand side of Eq. (2.47)

is zero because the details of binary collisions say that mass, momentum, and

energy are conserved during collisions. The quantities m,mu and 12

mu2 are called

summational invariants. It can be shown that these, or linear combinations of

these, are the only summational invariants [94].

The equation for the conservation of mass is obtained for substituting Q = m

into Eq. (2.47). This gives:

t

+ (U) = 0 . (2.48)

Next, by setting Q = mu the conservation of momentum equation becomes:

U

t+ (U2) +p F = 0 . (2.49)

Eq. (2.49) can be rewritten in terms of the material derivative D/Dt = t

+U and the definitions of pressure and viscous stress tensor from Eqs. (2.13) and

(2.14). This gives:

DU

Dt+p F = 0 . (2.50)

Finally, inserting 12mu2 in for Q results in the conservation of energy equation:

t(

1

2U2) + ( 1

2uu2) U F = 0 . (2.51)


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The averages over molecular velocities in Eq. (2.51) may be converted to aver-

ages over the thermal velcoities. Introducing the heat flux vector defined by Eq.

(2.21), the hydrostatic pressure p given by Eq. (2.13) , and which is called the

dissipation function and is written in Cartesian coordinates as:

= xxUxx

+ yyUyy

+ zzUzz

+ xy

Uxy

+Uyx

+yz

Uyz

+Uzy

+ zx

Uzx

+Uxz

, (2.52)

the conservation of energy equation can be rewritten as:

DeDt

= p U q + . (2.53)

Here, e is the intrinsic energy per unit mass.

Equations (2.48), (2.50), and (2.53) make up the Navier-Stokes equations. Note

that the original Navier-Stokes equation derived by Navier in 1822 and by Stokes

in 1845 is the momentum equation shown in Eq. (2.50) for a simple gas [93].

Due mainly to the popularity of computational fluid dynamics, citations of the

Navier-Stokes equations have grown to include the mass and energy equations to

form the complete system of equations described above. Because the momentumconservation equation (2.50) is a vector equation, the the Navier-Stokes equations

constitute five equations. Together with the equation of state, in this case, the

perfect gas law given by Eq. (2.16), the set of equations is closed and can describe

the continuum behavior of a fluid. If both and q are zero, then the momentum

equation reduces to Eulers equation given by:

DU

Dt+p = 0 . (2.54)

Together with Eqs. (2.16) and (2.48), Eulers equation can describe continuum

behavior in the limit of Kn approaching zero.


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2.5 Solutions to the Boltzmann equation

Because the Boltzmann equation is an integro-differential equation, and becauseparticle velocities and positions must be known for all particles at all times, exact

solutions to the Boltzmann equation are complex and only exist in physically

simple situations. For problems in nontrivial gases or multidimensional flows, an

analytical solution to the Boltzmann equation is improbable, and even proves to

be difficult to obtain a computational solution using standard numerical methods

[95, 96]. Approximations and limited solutions to the Boltzmann equation are

discussed below.

2.5.1 Equilibrium properties

For a simple, monatomic gas that is subject to no external force field and is sta-

tionary so that the velocity distribution is independent ofr, Boltzmanns equation

(2.36) reduces to:

f

t=

u1

40

(ff1 f f1)urddu1 . (2.55)

Boltzmanns H-theorem [15] shows that in order for this to be true, then:

u1

40

(ff1 f f1)urddu1 = 0 , (2.56)

or:

ff1 = f f1 . (2.57)

This implies that the probable number of molecules in any element of velocity

space remains constant in time, that is, collisions between molecules do not affect

the velocity distribution function. Therefore, the velocity distribution function

here describes the equilibrium state of the gas.

It can be shown that solution to Eq. (2.57) requires f in its equilibrium state,


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denoted f0, to take the form:

f0 = n

m2kbT

3/2emu

2/2kbT , (2.58)

which is called the Maxwellian velocity distribution function. E q . (2.58) can be

written in terms of the most probable molecular thermal speed um =

2kbT /m as:

f0 = n3m

(3/2)e

2mu

2

, (2.59)

where m = 1/u

m. The average thermal speed u is related to the most probable

molecular thermal speed by u

= (2/)u

m.Fig. (2.2) shows the Maxwellian distribution function for the monatomic gas

argon (molecular weight M = 39.9 kg/kmol) in equilibrium at 0 C with pressure

of 101,000 Pa. The Maxwellian distribution plotted in Fig. (2.2) gives the number

of molecules with speed u in a volume element around r.

Figure 2.2. Maxwellian velocity distribution function for argon at 0


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As Kn goes to 0, the Maxwellian velocity distribution function provides a

solution to the Boltzmann equation. The range of validity of this solution is when

the gas is in equilibrium and can also be governed by the continuum equations of

Euler.

2.5.2 Linearized Boltzmann Equation

The theory based on work by Chapman [97] and Enksog [98] provides a perturba-

tion solution to the Boltzmann equation for a restricted set of problems [15]. In

this theory, a solution to the Boltzmann equation is represented by an asymptotic

power series expansion of succesive approximation. The expansion is carried out

assuming small deviations from the Maxwellian velocity distribution function f0

defined by Eq. (2.58). The solution to the Boltzmann equation can be written in

the form of the power series:

f = f0 + f1 + 2f2 + . . . , (2.60)

where in this case is taken to be either the mean collision time or the Knudsen

number. Equation (2.60) is commonly rewritten as:

f = f0(1 + 1 + 2 + 3 + . . .) . (2.61)

Upon assuming a solution of zeroth order, that is, f = f0, the Euler equations

are obtained. The Navier-Stokes equations are obtained by assuming a first order

solution, f = f0(1 + 1), and a second order approximation results in the Burnett

equations [99, 100].

Because the method of successive approximation implies that higher order

terms are dependent on lower order terms, the complexity of successive terms in-

creases rapidly with each iteration. In addition, the range of validity is restricted

at any order. For example, numerical instabilities and violations of the second law

of thermodynamics can occur with higher order solutions at high Kn [99]. How-

ever, to describe the gas dynamics in the Earths atmosphere near the ground, a

first order solution is a sufficiently good approximation for most purposes [15].


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2.5.2.1 1st order solution

Substitution of f = f0(1 + ) into the Boltzmann equation with no force field

(2.36) yields:

t+ u

r=

u1

40

f0f01 ( + 1 1)urddu1 , (2.62)

which can be shown [15] to have a solution of the form:

= 1mn

A ln T 2nB : U (2.63)

where B : U denotes the scalar product ofB and U. This can also be writtenin tensor notation Bi,j

Uixj

. A and B are functions of T and U and are usually

written as an infinite series of Sonine polynomials Si [15]. That is:

A =i=1

aiSiU , (2.64)

and

B =

i=1

biSi1UiUj U

2

3 ij

, (2.65)

where U=

m2kbT

1/2U

Sonine polynomials arise in the study of Bessel functions and are solutions to

the differential equation [101, 102]:

xy + (m + 1 x)y + (nm)y = 0 , (2.66)

where m = 0 and n is a real number. Sonine polynomials were first used inthe kinetic theory of gases by Burnett [103]. The ease of certain orthogonality

properties and fast convergence makes Sonine polynomials ideal for use in this

context [103]. The results by Burnett are essentially identical to the earlier work

by Chapman [97] and Enskog [98] without the elaborate expansions that were

necessary in Chapman and Enskogs theory.


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2.5.2.2 Transport coefficients

The infinite sums A and B from Eqs. (2.64) and (2.65) can be rewritten since all

terms vanish except for i = 1 using properties of Sonine polynomials:

A = a1

5

2 U2

U , (2.67)

and

B = b1

UiUj U

2

3ij

. (2.68)

The coefficient of thermal conductivity and the coefficient of viscosity can be

written in terms ofA and B given above. It can be shown that first approximationto the coefficient of thermal conductivity for a simple monatomic gas is:

(1) = 5cvkbT3

a1 . (2.69)

Similarly, the first approximation to the coefficient of viscosity is written:

(1) = b1kbT . (2.70)

Evaluating a1 and b1 from the Sonine polynomial expansion for a hard spheremolecular model, Eq. (2.69) can be written in terms of the coefficient of viscosity

by:

(1) =5

2cv

(1) . (2.71)

For a gas with molecular diameter d, the first approximation to the coefficient of

viscosity is then:

(1) =5

16d2

kbmT

1/2. (2.72)

Later approximations can be determined by evaluating the Sonine polynomialsfurther. To the fourth degree of approximation:

(4) = 1.02513(1) , (2.73)

and

(4) = 1.01600(1) . (2.74)


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Under the hard sphere molecular model approximation, the viscous stress tensor

can now be written:

i,j =

Uixj

+Ujxi

+ 2i,j

Ukxk

, (2.75)

where 2 is the second coefficient of viscosity, which is related by the bulk viscosity

defined as b =23 + 2.

The heat flux vector also now becomes:

q = T . (2.76)

In the case of a dilute monatomic gas with no internal energy, the bulk viscosity

is zero, so the viscous stress tensor can be rewritten as:

i,j =

Uixj

+Ujxi

2

3i,j

Ukxk

. (2.77)

Inserting Eqs. (2.67) and (2.68) into Eq. (2.63), the Chapman Enskog solution

for can be written:

=a1

mn5

2 U2

U ln T2b1nUiUj U

2

3 ijUi

xj (2.78)

Written in terms of Eqs. (2.69) and (2.70), Eq. (2.78) can be rewritten as:

= 1mn

3(1)

5cvkbT

U2 5

2

U ln T 2

n

(1)

kbT

UiUj U

2

3ij

Uixj

. (2.79)

Using the perfect gas law and Eq. (2.71) along with f = f0(1 + ), Eq. (2.79) can

be simplified to:

f = f0

1

1

m

3(1)

2p

U2 5

2

U ln T+ 2

(1)

p

UiUjU

2

3ij

Uixj

. (2.80)

Eq. (2.80) is referred to as the first order Chapman-Enskog solution to the Boltz-

mann equation for a perfect monatomic gas.

Many assumptions and approximations have gone into the theory for the first

order solution which is equivalent to the Navier-Stokes equations. To summarize,


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these approximations are listed below:

1. Small perturbation from equilibrium

2. Monatomic gas

3. Perfect gas

4. Hard sphere gas

5. Continuum approximation

6. Bulk viscosity is zero

The following excerpt from Dorrance [104] appropriately describes the limi-

tations of using the continuum approach: The Navier-Stokes equations and the

remaining equations can be used to describe many situations with remarkable suc-

cess over a wide range of gas pressures and temperatures for the flow of mixtures

of polyatomic gases. We must not expect too much of the theory, however. There

is no a priori reason to expect that the theory can be applied to ionized gas flows

or the flow of liquids with any degree of success, nor is uncritical application of

the low-density or nearly free molecule flow of gases warranted.

2.5.3 Bhatnagar, Gross, and Krook (BGK) equation

Some other techniques have evolved which provide approximations to the Boltz-

mann equation as a method of finding a solution. One such technique is simplifying

the collision integral, as was done by Bhatnagar, Gross, and Krook [105]. In this

theory, it is assumed that collisions between molecules cause the velocity distribu-

tion function f to approach a Maxwellian value. This collision theory assumes, as

its basic approximation, that during time dt, a fraction dt/c of molecules undergocollisions, which alter their velocity distribution function from f to f0 where c

is the time between collisions. In this case, the collision term of the Boltzmann

equation given by Eq. (2.42) can be approximated by the sum of two parts: the

loss term and the gain term. In the absence of an external force field, this is written

as:f

t+ u f

r= G(f) fL(f) . (2.81)


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The simplest model for the gain and loss terms is:

G(f) = An , (2.82)

and

L(f) = Anf0 , (2.83)

where A = m/um for a hard sphere monatomic gas in equilibrium. um is themean thermal speed of the gas. In this case, the Boltzmann equation reduces to:

f

t+ u f

r= An(f0 f) = f

0 fc

. (2.84)

Eq. (2.84), often referred to as the BGK equation, follows theory that says the

collision term should vanish when in equilibrium. This approximation to the Boltz-

mann equation is much simpler to model and solutions are more readily available

[96, 95]. However, its range of validity is much smaller than that of the full Boltz-

mann and Navier-Stokes equations and should only be used in near equilibrium

conditions [105].

2.5.4 Numerical solutions to the Boltzmann equationWhen analytical solutions to the Boltzmann equation are not available or too

complex, numerical solutions are considered. Standard numerical method solutions

only exist for simplified geometries or approximations to the Boltzmann collision

integral by use of the BGK equation [95, 96].

As an illustration of the difficulty involved in the computational problem for

solving the Boltzmann or BGK equation, consider a simple gas in which the Boltz-

mann equation is written in terms of phase space. This means that the computa-

tional problem is seven dimensional (space, velocity space and time) and discretiz-ing phase space quickly becomes intractable. If one needs a 100 by 100 by 100 grid

in space and if even a small 20 by 20 by 20 grid could be used in velocity space,

this would require 8 billion grid points. Even if each grid point requires only 500

bytes of memory, this would mean 4 terabytes of memory for the whole system. In

addition, the computer operations would be on the order ofN7 and that computa-


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tional cost is prohibitive. In addition, in order to model multiple molecular species

and chemical reactions, one would need separate Boltzmann equations for each

species. This curse of dimensionality is the reason why Monte Carlo methods

are the best approach for problems involving many dimensions [106].

The Direct Simulation Monte Carlo (DSMC) method has been shown to be

equivalent to a numerical solution of the Boltzmann equation [67]. DSMC was

derived through physical reasoning just as the Boltzmann equation was derived.

Because of its particle nature, DSMC can be applied to complex situations that

would not be accessible with standard numerical models. Chapter (3) presents the

DSMC method in detail.


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Chapter3

Implementation of DSMC for

Acoustics Simulations

3.1 Algorithm

The DSMC approach used for the simulation of acoustic wave propagation consists

of essentially four subroutines which are uncoupled during each time step t:

1. Move particles according to the time step and their velocity.

2. Apply boundary conditions, sort and index particles into cells.

3. Collide particles locally and allow for redistribution of internal energy if

simulating diatomic or polyatomic gases.

4. Sample macroscopic properties of interest in each cell.

Fig. (3.1) is a flowchart of a typical DSMC simulation. Each step is called

systematically during every time step and each run is completed independently

with a new set of random numbers generated for each additional run. The number

of samples and the number of time steps between samples is determined the the

user defined values ofNST and NIS respectively. So there are NST NIS total numberof time steps. The individual processes within the algorithm are described below.


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Figure 3.1. A flowchart of the DSMC algorithm. There are NST total samples with NIStime steps in between samples. J and I increment NST and NIS respectively.


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3.1.1 Initialization

Similar to a conventional numerical fluid dynamics scheme, a computational do-

main in physical space is defined for DSMC simulations. An overlaying grid of cells

is used for selecting collision pairs and also for sampling macroscopic properties.

The first step in the DSMC routine is to specify the initial state of the gas by

the initialization of cells and particles. Each cell in the computational domain is

initialized to contain a fraction of the total number of particles in the simulation.

In the cases considered here, every cell initially contains the same number of par-

ticles. The particles are initialized by specifying their molecular weight, diameter,

number of internal degrees of freedom, internal energy, location in space and veloc-

ity. The hard sphere molecular diameter is usually calculated by using Eq. (2.72)

and experimental values of viscosity. In the cases considered here all particles

within the domain are initialized to have random velocities sampled from the equi-

librium Maxwellian velocity distribution function defined by Eq.(2.58) for a given

temperature. The procedure for randomly sampling a Maxwellian distribution is

shown in Appendix (B). Each particle is also assigned internal energy depending

on how many internal degrees of freedom are active for the particular gas species

simulated. In most cases considered in this work, the gas was initialized to be in

thermal equilibrium so that the internal energy was dependent on the equilibriumgas temperature.

3.1.2 Move particles and Boundary Conditions

Each particle is moved a distance ut every time step, where u is the velocity of

the particle and t is the global time step. Appropriate conditions

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