The Pennsylvania State University The Graduate School NUMERICAL SIMULATIONS OF ACOUSTICS PROBLEMS USING THE DIRECT SIMULATION MONTE CARLO METHOD

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1 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

2 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. Sparrow Professor of Acoustics Anthony A. Atchley Professor of Acoustics Chair of Graduate Program in Acoustics Signatures are on file in the Graduate School.

3 Abstract In the current study, real gas effects in the propagation of sound waves are simulated 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 range of Knudsen numbers, defined as the ratio of molecular mean free path to wavelength. Continuum models such as the Euler and Navier-Stokes equations break down for flows greater than a Knudsen number of approximately Continuum models also suffer from the inability to simultaneously model nonequilibrium conditions, diatomic or polyatomic molecules, nonlinearity and relaxation effects and are limited in their range of validity. Therefore, direct simulation Monte Carlo is capable of directly simulating acoustic waves with a level of detail not possible with continuum approaches. The basis of direct simulation Monte Carlo lies within kinetic theory where representative particles are followed as they move and collide with other particles. A parallel, object-oriented DSMC solver was developed for this problem. Despite excellent parallel efficiency, computation time is considerable. Monatomic gases, gases with internal energy, planetary environments, and amplitude effects spanning a large range of Knudsen number have all been modeled with the same method and compared to existing theory. With the direct simulation method, significant deviations from continuum predictions are observed for high Knudsen number flows. iii

4 Table of Contents List of Figures List of Tables List of Symbols Acknowledgments viii xiii xiv xviii Chapter 1 Introduction Motivation Numerical models Continuum methods Particle methods Sound for all Knudsen numbers Direct Simulation Monte Carlo Introduction Chapter 2 Kinetic Theory of Gases Historical background and Ludwig Boltzmann Velocity distribution function Macroscopic properties in a simple gas Deriving the Boltzmann equation The calculation of the collision integral Deriving conservation equations from the Boltzmann equation iv

5 2.5 Solutions to the Boltzmann equation Equilibrium properties Linearized Boltzmann Equation st order solution Transport coefficients Bhatnagar, Gross, and Krook (BGK) equation Numerical solutions to the Boltzmann equation Chapter 3 Implementation of DSMC for Acoustics Simulations Algorithm Initialization Move particles and Boundary Conditions Sort particles Collision Binary collisions DSMC collision routine Sampling Diatomic and polyatomic gases DSMC tests for gases with internal energy Mixtures implementation Assumptions and Error within DSMC Chapter 4 Computing Issues Object-Oriented Programming Approach Parallel Implementation Modifications to the DSMC algorithm Small deviation from equilibrium Low Kn flows modifications Chapter 5 Absorption and Dispersion in a Monatomic Gas The speed of sound The absorption of sound Theory on the absorption and dispersion of sound in a monatomic gas DSMC Results for the absorption and dispersion of sound in a monatomic gas Simulation approach v

6 5.4.2 Results Nonequilibrium Chapter 6 Absorption and Dispersion in a Gas With Internal Energy Introduction Absorption and dispersion from a simple relaxation process Current theories including rotational relaxation Other theories for rotational relaxation Current theories including vibrational relaxation Implementation in DSMC for a gas with rotational energy for multiple collision numbers Simulation approach Results Implementation in DSMC for a gas with rotational and vibrational energy Simulation approach Results Nonequilibrium Absorption as a function of temperature Dispersion as a function of temperature Chapter 7 The Effect of Amplitude Simulation approach Breakdown of the propagation constant Nonequilibrium effects as a function of amplitude Absorption and dispersion as a function of amplitude Harmonic generation Shock coalescence Coalescence at high Kn Chapter 8 DSMC Applications: Planetary Acoustics Simulation approach vi

7 8.2 Results and Discussion Earth Mars Titan Vertical profiles Chapter 9 Conclusions Future work Appendix A Buffon s Needle Experiment 142 Appendix B Sampling a Maxwellian Distribution 144 Appendix C Deriving the Navier-Stokes Dispersion Relation 147 Bibliography 150 vii

8 List of Figures 1.1 Limits of applicability of various mathematical models to simulate fluid flow [1] Ludwig Boltzmann ( ) Maxwellian velocity distribution function for argon at A flowchart of the DSMC algorithm. There are NST total samples with NIS time steps in between samples. J and I increment NST and NIS respectively A flowchart of the collision routine in the DSMC algorithm. sigvmax 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) A flowchart of the internal energy exchange routine in the DSMC algorithm DSMC simulation of nitrogen molecules undergoing classical relaxation at 4000 K. Bird s exponential model[1] given by solid lines and DSMC results in dashed lines [2] DSMC simulation of nitrogen molecules undergoing relaxation with a coupled discrete vibration / classical rotation model at 4000 K. Bird s exponential model[1] given by solid lines and DSMC results in dashed lines [2] Equilibrium initialization of argon at 0 C after 1 ensemble Equilibrium initialization of argon at 0 C after 10 ensembles Equilibrium initialization of argon at 0 C after 100 ensembles Equilibrium initialization of argon at 0 C after 1000 ensembles A flowchart of the parallel algorithm viii

9 4.2 CPU time on Columbia and Mufasa with respect to number of processors compared to ideal CPU time Speedup on Columbia and Mufasa with respect to the number of processors compared to ideal speedup Parallel efficiency on Columbia and Mufasa with respect to the number of processors Parallel efficiency on Mufasa with respect to the number of processors for a large and small system Scaled absorption α cl /k 0 and dispersion β cl /k 0 predictions given by the linearized Navier-Stokes equations from Eq. (5.20) are plotted with the low frequency classical absorption coefficient given by Eq. (5.21) [3] Scaled absorption α cl /k 0 in argon for 273 K. DSMC results (red circles) compared to experimental results by Greenspan [4] (green triangles) and Schotter[5] (blue squares) and continuum theory predictions results from Eq. (5.20) [3] Scaled dispersion k/k 0 in argon for 273 K. DSMC results (red circles) compared to experimental results by Greenspan [4] (green triangles) and continuum theory predictions results from Eq. (5.20) [3] Translational nonequilibrium effects for Kn = 2.0 at 273 K in argon Translational nonequilibrium effects for Kn = 0.02 at 273 K in argon Scaled absorption for collision numbers Z rot =1, 10, 100, and 1000 in a nitrogen-like gas at 0 C from Eqs. (6.7) and (6.3) [3, 6] Scaled dispersion for collision numbers Z rot =1, 10, 100, and 1000 in a nitrogen-like gas at 0 C from Eqs. (6.8) and (6.2) [3, 6] Vibrational relaxation frequency for oxygen and nitrogen as a function of temperature [7, 8] Scaled absorption for relaxation collision number of 1. DSMC simulations (points) are plotted with continuum theory for the rotational relaxation given by Eq. (6.3) (dashed line) [3] and total absorption given by Eq. (6.7) (solid line) [6] Scaled absorption for relaxation collision number of 5. DSMC simulations (points) are plotted with continuum theory for the rotational relaxation given by Eq. (6.3) (dashed line) [3] and total absorption given by Eq. (6.7) (solid line) [6] ix

10 6.6 Scaled absorption for relaxation collision number of 40. DSMC simulations (points) are plotted with continuum theory for the rotational relaxation given by Eq. (6.3) (dashed line) [3] and total absorption given by Eq. (6.7) (solid line) [6] Scaled absorption for relaxation collision number of 200. DSMC simulations (points) are plotted with continuum theory for the rotational relaxation given by Eq. (6.3) (dashed line) [3] and total absorption given by Eq. (6.7) (solid line) [6] DSMC results for the scaled absorption with relaxation collision numbers of 1, 5, 40, and 200. DSMC simulations (points) are plotted with continuum theory for rotational relaxation (dashed line) [3] Nonequilibrium effects for Kn = 0.02 at 273 K with classical vibration model Nonequilibrium effects for Kn = 1.0 at 273 K with classical vibration model 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) (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] 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] 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] 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] 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] 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] Maximum pressure amplitude (Pa) in argon for Kn = 2, 40 m/s amplitude. DSMC results (points) compared to restricted exponential best fit curve for x < 10u /ω (solid line) x

11 7.2 Maximum pressure amplitude (Pa) in argon for Kn = 2, 40 m/s amplitude. DSMC results (points) compared to exponential best fit curve for x > 10u /ω (solid line) Maximum pressure amplitude (Pa) in argon for Kn = 2, 5 m/s amplitude. DSMC results (points) compared to exponential best fit curve for x > 10u /ω (solid line) Maximum pressure amplitude (Pa) in argon for Kn = 0.02, 40 m/s amplitude. DSMC results (points) compared to exponential best fit curve for x > 10u /ω (solid line) T trx /T tr for Kn = 2, 40 m/s amplitude T trx /T tr for Kn = 2, 5 m/s amplitude T trx /T tr for Kn = 0.02, 40 m/s amplitude T trx /T tr for Kn = 0.02, 5 m/s amplitude Amplitude dependence on the scaled absorption in argon at 0 as a function of Kn. DSMC results (points) compared to continuum theory given by Eq. (5.20) (line) Wave steepening at Kn = 0.02 and 40 m/s amplitude Amplitude dependence on the scaled dispersion in argon at 0 as a function of Kn based on the maximum pressure amplitude. DSMC results (points) compared to continuum theory given by Eq. (5.20) (line) Amplitude dependence on the scaled dispersion in argon at 0 as a function of Kn based on the zero crossings of the acoustic pressure. DSMC results (points) compared to continuum theory given by Eq. (5.20) (line) 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 a time of 1 nanosecond Fourier component amplitude for Kn = 0.005, 40 m/s amplitude compared to Fubini solution given by Eq. (7.4) and Burgers equation given by Eq. (7.5) [9, 10] Fourier component amplitude for Kn = 2, 40 m/s amplitude Fourier component amplitude for Kn = 2, 5 m/s amplitude Shock coalescence for Kn = 0.02 at 6 nanoseconds Shock coalescence for Kn = 0.02 at 15 nanoseconds Shock coalescence for Kn = 0.02 at 21 nanoseconds Shock coalescence for Kn = 0.02 at 46 nanoseconds Coalescence for Kn = 2 at 42 nanoseconds Coalescence for Kn = 2 at 63 nanoseconds Coalescence for Kn = 2 at 74 nanoseconds xi

12 7.24 Coalescence for Kn = 2 at 84 nanoseconds Temperature and pressure profiles as a function of altitude above Earth s surface [11] Scaled absorption for dry air on Earth. DSMC simulations (points) are plotted with continuum theory for the vibrational relaxation given by Eq. (6.3) (green) and total absorption given by Eq. (6.7) (red) Scaled absorption for humid air on Earth. DSMC simulations (points) are plotted with continuum theory for the vibrational relaxation given by Eq. (6.3) (green) and total absorption given by Eq. (6.7) (red) DSMC results (solid line) for the acoustic pressure on Earth for Kn = 0.02 compared to predicted amplitude dependence determined from the Navier-Stokes derived absorption coefficient from Eq. (5.20)(dashed line) Temperature and pressure profiles as a function of altitude above the Mars surface [12] DSMC results for the acoustic pressure amplitude as a function of distance on Mars for Kn = 0.02 compared to theoretical predictions [13] Nonequilibrium effects on Mars showing rotational (dashed line) and translational (solid line) temperatures for Kn = Temperature and pressure profiles as a function of altitude above Titan s surface [14] Kn = waveform on Earth with ɛ = Absorption dominates nonlinearity with little or no nonlinear effects visible Kn = 0.01 waveform on Titan with ɛ = Significant wave steepening can be observed Knudsen number as a function of altitude on Earth for frequencies of 100 Hz, 1000 Hz, Hz, and Hz Knudsen number as a function of altitude on Mars for frequencies of 100 Hz, 1000 Hz, Hz, and Hz Knudsen number as a function of altitude on Titan for frequencies of 100 Hz, 1000 Hz, Hz, and Hz 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) A.1 An illustration of the Buffons needle experiment xii

13 List of Tables 6.1 The Prandlt number for common gases Atmospheric conditions at the surface on Earth, Mars and Titan xiii

14 List of Symbols c Speed of sound, phase speed, [m/s], p. 63 c p Specific heat at constant pressure, [J/(kg K)], p. 16 C p Heat capacity at constant pressure, [J/(kmol K)], p. 15 c v Specific heat at constant volume, [J/(kg K)], p. 16 C v Heat capacity at constant volume, [J/(kmol K)], p. 15 c 0 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 e int 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 F num Ratio of real to simulated particles, p. 37 f 0 Maxwellian velocity distribution function, p. 23 f r Relaxation frequency, [Hz], p. 76 J Bessel function, p. 110 k Complex propagation constant, [1/m], p. 64 xiv

15 k b Boltzmann s constant, [J/K], p. 13 k cl Complex propagation constant for classical losses, [1/m], p. 64 k 0 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/m 3 ], p. 11 N Avogadro s number, [molecules/kmol], p. 13 N c 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/m 2 ], p. 15 Q Macroscopic quantity, p. 12 r Particle position vector, [m], p. 11 R Universal gas constant, [J/(kmol K)], p. 13 s Relaxation strength, p. 76 S Sonine polynomials, p. 25 t Time, [s], p. 11 T Temperature, [K], p. 13 T int Internal temperature, [K], p. 14 T tr Translational temperature, [K], p. 14 T ov Overall temperature, [K], p. 14 u Molecular velocity, [m/s], p. 11 U Mean velocity, [m/s], p. 12 xv

16 U 0 Macroscopic velocity amplitude, [m/s], p. 107 u Thermal velocity, [m/s], p. 12 u Average thermal speed, [m/s], p. 23 u m Most probable molecular thermal speed, [m/s], p. 23 u r Relative speed, [m/s], p. 18 x Shock formation distance, [m], p. 109 Z Relaxation collision number, p. 40 Z rot Rotational relaxation collision number, p. 77 Z vib Vibrational relaxation collision number, p. 87 α Absorption coefficient, [np/m], p. 64 α cr Combined absorption due to classical thermal-viscous losses and rotational relaxation, [np/m], p. 77 α r Absorption due to a single relaxation process, [np/m], p. 76 α rot Absorption due to rotational relaxation, [np/m], p. 77 α vib Absorption due to vibrational relaxation, [np/m], p. 82 β Dispersion coefficient, [1/m], p. 64 β cr Combined dispersion due to classical thermal-viscous losses and rotational relaxation, [1/m], p. 77 β NL Coefficient of nonlinearity, p. 107 β m Reciprocal of the most probable molecular thermal speed, [s/m], p. 23 β r Dispersion due to a single relaxation process, [1/m], p. 76 β rot Dispersion due to rotational relaxation, [1/m], p. 77 γ Ratio of specific heats, p. 15 Γ Gol dberg number, p. 111 δ Kronecker delta function, p. 13 xvi

17 ɛ Acoustic Mach number, p. 109 ζ Number of internal degrees of freedom, p. 14 κ Thermal conductivity, [J/K m s)], p. 26 λ m Mean free path, [m], p. 1 µ Coefficient of viscosity, [kg/(m s)], p. 26 µ b Bulk viscosity, [kg/(m s)], p. 27 ν Collision frequency, [1/s], p. 76 ξ Number of total degrees of freedom, p. 15 ρ Mass density, [kg/m 3 ], p. 11 σ Collision cross section, [kg m 2 ], p. 17 σ NL Nondimensional shock formation distance, p. 110 τ Viscous stress tensor, [Pa], p. 13 τ r Relaxation time, [1/s], p. 76 φ Macroscopic quantity, p. 64 ψ Perturbation from distribution function, p. 24 Ψ Dissipation function, p. 21 ω Frequency, [rad/s], p. 64 Ω Scattering angle, p. 17 xvii

18 Acknowledgments I would like to give my sincere thanks to Prof. Lyle N. Long for his advice and support throughout my pursuit of this degree. This project would not have been possible 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 their comments and suggestions. I would also like to acknowledge the National Science Foundation for funding the Consortium for Education in Many-Body Applications (CEMBA), Grant No. NSF-DGE , and the NASA Graduate Student Fellowship Program for funding and providing computer resources for this research project. A huge thanks to Dr. Patrick D. O Connor. I ve said it before, but I ll say it again, if I m getting a PhD, he should be getting two for all the hard work and countless hours of conversation we ve had over the years. I honestly wouldn t be where I am today without his help and friendship. Special thanks to Rebecca Sanford DeRousie and Bernadette Rakszawski for making the littleman laugh while I typed away and to Catherine Hofstetter whose friendship and motherhood I will always admire. And a huge thanks to Jen Marcovich, Mahreen George, Jen Dombroskie, Suzi Lang, Shelley Farahani, Julie Willits, Allison Bohn and Bethany Heim for being such a loving and enriching part 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 County Babywearers 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 the day. To my brother, David Danforth for always keeping me on my toes with love and big hugs. And much love and thanks to Pat, Keith and Kim Hanford for all of the loving, warm encouragement and support in all things. Many thanks to xviii

19 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 activities prove that I am not. I am indebted to my husband, Scott, for providing me motivation and encouragement at every step during this pursuit, for being with me through thick and thin, and for the joy and happiness his company brings me. And many thanks to Noah, the best littleman a mother could have, for reminding me that unconditional love triumphs any degree. xix

20 Dedication To mothers and children everywhere. Children can teach you so much more about life than books. xx

21 Chapter 1 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 inability to simultaneously model nonequilibrium conditions, diatomic or polyatomic molecules, nonlinearity and relaxation effects and are limited in their range of validity. Theory for the behavior of sound at high Knudsen numbers is inconsistent with 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 complexity 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

22 2 the state of the fluid macroscopically using quantities such as density, pressure, velocity, and temperature. The continuum approximation is valid when the characteristic length of the problem is much larger than the molecular mean free path (λ 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 the fact 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 φ 0 and a small perturbation from equilibrium φ. The governing continuum equations that describe the perturbation from equilibrium φ can be derived from linearizing the Navier-Stokes or Euler equations. During the linearization process, higher order terms are ignored by assuming the perturbation from equilibrium φ 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 φ does not necessarily have to be small to describe nonlinear phenomena such as shock wave formation and harmonic generation [10]. An overview of computational 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

23 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 contains 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 increased 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 and are 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 viscosity, 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 established, one of the most practical aeroacoustic analyses relies upon a so called wave equation using Lighthill s 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 traditional 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 governing 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 < 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].

24 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 fluid flow [1] 1.4 Particle methods Particle methods that are based on the Boltzmann equation are valid for all Knudsen numbers. Therefore, particle methods are necessary for, but not limited to, problems where the Knudsen number is greater than about 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 approach the particles move according to Newton s laws and particle interactions are calculated by force potentials. This model can even account for quantum mechanical 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 methods involving statistical techniques, such as the use of random numbers to find the

25 5 solutions to mathematical or physical problems [31]. The first documented application 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 π 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 π and is discussed further in Appendix A. Lord Kelvin used a Monte Carlo method in 1901 to perform time integrals which appeared in his kinetic 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 discrete 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]. Limitations 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 simulate 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 simulating 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 conservation. DSMC offers computational flexibility to study sound for all Kn in a wide range of systems.

26 6 1.5 Sound for all Knudsen numbers The Knudsen number is large for sound propagation in very dilute gases, in microchannels, 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 propagation properties such as nonlinear phenomena and absorption are inherent in the

27 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 Introduction The direct simulation Monte Carlo (DSMC) method is a direct particle simulation 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 gas dynamics 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]. DSMC s 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 Jupiter s moon Io [80, 81]. DSMC is a particle method that describes the state of the gas at the microscopic 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 efficient 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 situations. Without modification, DSMC is capable of simulating all physical properties of interest at the molecular level for sound propagation: absorption, dispersion,

28 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 Saturn s 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.

29 Chapter 2 Kinetic Theory of Gases 2.1 Historical background and Ludwig Boltzmann Ludwig Boltzmann ( ) was an Austrian physicist and philosopher and is pictured in Fig. (2.1). In the late 1800 s, 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, expansive or irritable moods, and the often violent disputes about his scientific theory were enough to depress Boltzmann. He began to feel that his life s 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]. Perrin s studies of colloidal suspensions ( ) [91] confirmed the values of Avogadro s number and Boltzmann s constant, and Einstein s theory on Brownian motion (1905) [92] helped convince the world that the

30 10 Figure 2.1. Ludwig Boltzmann ( ) tiny particles really exist. Ostwald later told Sommerfeld that he had been converted to a belief in atoms by Einstein s complete explanation of his theory on Brownian motion [93]. Boltzmann s equation for entropy s = k b log W as it was published in his day, is engraved on his tombstone, where k b is called the Boltzmann constant. One of Boltzmann s main contributions is the invention of the field of statistical mechanics. In addition, Boltzmann s 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.

31 Velocity distribution function Consider a simple gas in which all molecules are alike with mass m. Let dr denote a 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, ρ, are related by: ρ = nm. (2.1) The distribution of velocities among a large number of molecules in the parcel dr 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 distribution function is a function of time. Therefore, the probable number of molecules that have velocities in the neighborhood of u = (u x, u y, u z ) and positions in the neighborhood of r = (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 = fdu. (2.3) u 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 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

32 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 Qfdu. (2.4) The lower-order moments of the distribution function all have names and simple physical interpretations, where the k th moment is defined as: u mu k fdu, (2.5) Because u is a vector, higher order moments can become tensors and require tensor multiplication within the integrand. vector a = (a 1, a 2, a 3 ) and b = (b 1, b 2, b 3 ) is given as: (a 1, a 2, a 3 ) (b 1, b 2, b 3 ) = The tensor product, denoted, of a 1 b 1, a 1 b 2, a 1 b 3, a 2 b 1, a 2 b 2, a 2 b 3, a 3 b 1, a 3 b 2, a 3 b 3, (2.6) Using Eq. (2.3), the zeroth moment of the distribution function becomes: giving the mass density of the gas. nm = ρ = u mfdu, (2.7) 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: ρu = u muf du, (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 = u U. (2.9) The second moment case is of particular interest, defining the momentum flux

33 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 = ρu u = In component form, the pressure tensor is given by: p = u ρu 2 x, ρu xu y, ρu xu z, ρu yu x, ρu 2 y, ρu yu z, ρu zu x, ρu zu y, ρu 2 z, Using tensor notation, the pressure tensor is written: mu u fdu. (2.10). (2.11) p ij = ρu i u 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 ρ(u 2 x + u 2 y + u 2 z ) = 1 3 ρu 2. (2.13) The viscous stress tensor τ 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: τ = τ ij = (ρu i u j δ ijp), (2.14) where δ ij is the Kronecker delta such that δ ij = 1 if i = j and δ ij = 0 if i 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, 1 2 mu 2. The proportionality is given by the relation: 1 2 mu 2 = 3 2 k bt, (2.15) where k b = [J/K] is the Boltzmann constant which is related

34 14 to the universal gas constant R = [J/(kmol K)] by k b = R/N where N = [mol/kmol] is Avogadro s number. An immediate consequence of the definition of temperature is that the hydrostatic pressure p of a gas in equilibrium can be written in terms of the temperature T resulting in the perfect gas law: p = ρrt/m = nk b T, (2.16) where M = m/n is the molecular weight of the molecule. T is an equilibrium gas property, but Eq (2.16) will hold, even in nonequilibrium situations, for the translational kinetic temperature T tr which is defined as: 3 2 k bt tr = 1 2 mu 2. (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: k b T trx = mu 2 x, (2.18) which can therefore provide a measure of translational nonequilibrium when compared 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 T int can be defined for the internal modes associated with the rotational and vibrational energy of the molecule. Similarly to the translational temperature, the internal temperature T int can be defined as: 1 2 ζ R M T int = e int, (2.19) where ζ is the number of internal degrees of freedom and e int is the energy associated with the internal energy mode of interest. For a nonequilibrium gas, an overall kinetic temperature T ov can be defined as a weighted average of the translational and internal temperatures given by Eqs (2.17) and (2.19) given by: T ov = (3T tr + ζt int )/(3 + ζ). (2.20)