Rarefied Gas Dynamics: Kinetic Modeling and Multi-Scale Simulation

Description

This book highlights a comprehensive description of the numerical methods in rarefied gas dynamics, which has strong applications ranging from space vehicle re-entry, micro-electromechanical systems, to shale gas extraction.  The book consists of five major parts:  The fast spectral method to solve the Boltzmann collision operator for dilute monatomic gas and the Enskog collision operator for dense granular gas; The general synthetic iterative scheme to solve the kinetic equations with the properties of fast convergence and asymptotic preserving; The kinetic modeling of monatomic and molecular gases, and the extraction of critical gas parameters from the experiment of Rayleigh-Brillouin scattering; The assessment of the fluid-dynamics equations derived from the Boltzmann equation and typical kinetic gas-surface boundary conditions;The applications of the fast spectral method and general synthetic iterative scheme to reveal the dynamics in some canonical rarefied gas flows.  The book is suitable for postgraduates and researchers interested in rarefied gas dynamics and provides many numerical codes for them to begin with. 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