Rarefied Gas Dynamics: Kinetic Modeling and Multi-Scale Simulation

Preface
Contents
1 Introduction
	1.1 Navier-Stokes-Fourier Equations
	1.2 Continuum Breakdown
		1.2.1 Reentry of Space Vehicle
		1.2.2 Microelectromechanical Systems
		1.2.3 Shale Gas Extraction
		1.2.4 Global Wind Profiling
	1.3 Simple Gas Kinetic Theory
	1.4 Knudsen Number
		1.4.1 Spatial Knudsen Number
		1.4.2 Temporal Knudsen Number
	1.5 Molecular Dynamics Simulations
	References
2 Gas Kinetic Theory
	2.1 Velocity Distribution Function
	2.2 Binary Collision
		2.2.1 Deflection Angle
		2.2.2 Differential Cross Section
		2.2.3 Grazing Collision
	2.3 Boltzmann Equation
		2.3.1 H-Theorem
		2.3.2 Equilibrium Collision Frequency
		2.3.3 Linearized Boltzmann Equation
	2.4 Wang-Chang and Uhlenbeck Equation
	2.5 Enskog Equation
		2.5.1 Liquid-Vapor Flow
		2.5.2 Granular Gas
	2.6 Gas-Surface Boundary Condition
		2.6.1 Maxwell Boundary Condition
		2.6.2 Epstein Boundary Condition
		2.6.3 Cercignani-Lampis Boundary Condition
	2.7 Numerical Methods
		2.7.1 Direct Simulation Monte Carlo
		2.7.2 Discrete Velocity Methods
		2.7.3 Multi-scale Simulation
	References
3 Fluid-Dynamic Equation
	3.1 Hilbert Expansion
	3.2 Chapman-Enskog Expansion
		3.2.1 Expansion in Sonine Polynomials
		3.2.2 Expansion to the First Order
		3.2.3 Expansion to Higher Orders
	3.3 Moment Methods
	3.4 Accuracy of Macroscopic Equations
		3.4.1 Equations from Chapman-Enskog Expansion
		3.4.2 Moment Equations
	3.5 Convergence of Moment Equations
		3.5.1 Rayleigh-Brillouin Scattering
		3.5.2 Sound Propagation
	References
4 Fast Spectral Method for Monatomic Gas Flow
	4.1 Inverse Design of Collision Kernel
		4.1.1 Power-Law Potential
		4.1.2 Lennard-Jones Potential
	4.2 Normalization
	4.3 Fast Spectral Method
		4.3.1 Carleman Representation
		4.3.2 Fourier-Galerkin Spectral Method
		4.3.3 Detailed Implementation
		4.3.4 Non-uniform Discretization of Velocity Space
	4.4 Homogeneous Relaxation
		4.4.1 Bobylev-Krook-Wu Solution
		4.4.2 Discontinuous Velocity Distribution
	4.5 Accuracy in Inhomogeneous Problems
		4.5.1 Normal Shock Waves
		4.5.2 Force-Driven Poiseuille Flows
		4.5.3 Thermal Transpiration in a Cavity
	4.6 Concluding Remarks
	References
5 Fast Spectral Method for Linear Gas Flow
	5.1 Linearization
	5.2 Poiseuille Flow
		5.2.1 Poiseuille Flow Between Parallel Plates
		5.2.2 Poiseuille Flow Through a Long Duct
	5.3 Thermal Transpiration
	5.4 Onsager-Casimir Relation
	5.5 Influence of Intermolecular Potential
		5.5.1 Lennard-Jones Potential
		5.5.2 Accurate Transport Coefficients
		5.5.3 Poiseuille Flow
		5.5.4 Planar Fourier Flow
		5.5.5 Planar Couette Flow
	5.6 Cercignani-Lampis Boundary Condition
		5.6.1 Poiseuille Flow Through Parallel Plates
		5.6.2 Poiseuille Flow Through Long Tube
	References
6 Kinetic Modeling of Monatomic Gas Flow
	6.1 Basic Rules
	6.2 Velocity-Independent Collision Frequency
		6.2.1 BGK Model
		6.2.2 Ellipsoidal-Statistical BGK Model
		6.2.3 Shakhov Model
		6.2.4 Gross-Jackson Model
		6.2.5 Nonlinearization
	6.3 Velocity-Dependent Collision Frequency
	6.4 Fokker-Planck Model
	6.5 Accuracy of Kinetic Models
		6.5.1 Normal Shock Wave
		6.5.2 Thermal Transpiration
	References
7 Kinetic Modeling of Molecular Gas Flow
	7.1 Bulk Viscosity
	7.2 Thermal Conductivity
	7.3 Thermal Relaxation Rates in DSMC
	7.4 Kinetic Models
		7.4.1 Hanson-Morse Model
		7.4.2 Rykov Model
		7.4.3 ESBGK Model
		7.4.4 Wu Model
	7.5 Accuracy of Kinetic Models
		7.5.1 Normal Shock Wave
		7.5.2 Couette Flow
		7.5.3 Maxwell's Demon
	7.6 Uncertainty Quantification
		7.6.1 Normal Shock Wave
		7.6.2 Flow Driven by Maxwell's Demon
		7.6.3 Thermal Transpiration in Cavity
	7.7 Conclusions and Discussions
	References
8 General Synthetic Iterative Scheme
	8.1 Problems of CIS
		8.1.1 Slow Convergence
		8.1.2 False Convergence
	8.2 General Synthetic Iterative Scheme
		8.2.1 Scheme-I GSIS
		8.2.2 Scheme-II GSIS
	8.3 Properties of GSIS
		8.3.1 Super Convergence
		8.3.2 Asymptotic Preserving
	8.4 Numerical Tests
		8.4.1 Coherent Rayleigh-Brillouin Scattering
		8.4.2 Planar Fourier Flow
		8.4.3 Couette Flow Between Eccentric Cylinders
	8.5 Concluding Remarks and Outlooks
	References
9 Acoustics in Rarefied Gas
	9.1 Formulation of the Problem
	9.2 Oscillatory Couette Flow
	9.3 Oscillating Lid-Driven Cavity Flow
		9.3.1 Scaling Law for Anti-resonant Frequency
	9.4 Planar Sound Propagation
	9.5 Sound Propagation in Cavity
		9.5.1 Two Types of Resonances
		9.5.2 Sound Speed
	References
10 Slip and Jump Coefficients
	10.1 State of the Problem
	10.2 Viscous Slip
		10.2.1 Viscous Slip Coefficient
		10.2.2 Knudsen Layer Function
	10.3 Thermal Slip
		10.3.1 Thermal Slip Coefficient
		10.3.2 Knudsen Layer Function
		10.3.3 Molecular Gases
	10.4 Temperature Jump
	References
11 Accuracy of Kinetic Boundary Condition
	11.1 Reynolds Lubrication Equation
	11.2 Experiments and Upscaling
	11.3 Approximate Velocity Slip Coefficients
	11.4 Comparison with Experiment
		11.4.1 Poiseuille Flow Through a Rectangular Duct
		11.4.2 Thermal Transpiration in a Rectangular Duct
		11.4.3 Thermal Transpiration Through a Long Tube
	11.5 Implication in Hypersonic Flows
	References
12 Porous Media Flow
	12.1 Apparent Gas Permeability
	12.2 Kinetic Formulation
	12.3 Accuracy of Navier-Stokes Equations
	12.4 Interpretation of Experiment
	12.5 Asymptotic Behavior at Large KnKn
	References
13 Gas Mixture
	13.1 Boltzmann Equation for Gas Mixture
	13.2 Fast Spectral Method
		13.2.1 Accuracy Analysis
		13.2.2 Efficient Algorithm for Large Mass Ratio
	13.3 Accuracy in Inhomogeneous Problems
	13.4 Linearization and GSIS
	13.5 McCormack Model
	References
14 Dense Gas Flow
	14.1 Fast Spectral Method
	14.2 Heated Granular Gas
	14.3 Force-Driven Poiseuille Flow
		14.3.1 Mass Flow Rate of Dense Gas
		14.3.2 Influence of Restitution Coefficient
	14.4 Heat Transfer
	14.5 Kinetic Model for Dense Gas
	References
15 Fluctuation and Light Scattering
	15.1 Rayleigh-Brillouin Scattering
		15.1.1 Spontaneous RBS
		15.1.2 Coherent RBS
	15.2 Numerical Methods
		15.2.1 Monatomic Gas
		15.2.2 Molecular Gas
	15.3 Accuracy of the Tenti Model
		15.3.1 Temperature Retrieval Error
	15.4 Extraction of Gas Property
	References
Appendix A Special Functions
Appendix B Relaxation Rates of Maxwellian Molecules
Appendix C Numerical Quadratures
C.1  Gauss-Legendre Quadrature
C.2  Gauss-Hermite Quadrature
Appendix D Implementation of Fast Spectral Method
D.1  Algorithm 1: Zero-Padding
D.2  Algorithm 2: No Zero-Padding
D.3  Algorithm 3: Collision Frequency
Appendix E MATLAB Code for Normal Shock Wave
Appendix F MATLAB Code for Poiseuille Flow and Thermal Transpiration
Index