Rarefied Gas Dynamics: Kinetic Modeling and Multi-Scale Simulation

✍ Scribed by Lei Wu

• Publisher: Springer
• Year: 2022
• Tongue: English
• Leaves: 293
• Edition: 1st ed. 2022
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

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