Waves in Flows

Preface
Contents
1 A Priori Estimates from First Principles in Gas Dynamics
1.1 Introduction
1.2 Compensated Integrability
1.2.1 Evolution Problems
A Homogeneous Estimate
Integrating in Time First
1.3 Applications to Gas Dynamics (I): Euler Equations
1.3.1 Euler Equations for a Compressible Inviscid Fluid
1.3.2 Why Do We Care?
1.3.3 Estimating the Velocity Field
1.3.4 Flows in a Bounded Domain
1.3.5 Relativistic Gas
1.3.6 Flows with External Force
1.4 Applications (II): Kinetic Models
1.4.1 Boltzmann-Like Models
The Cauchy Problem
An Extra Estimate for Boltzmann-Like Equations
1.4.2 What Should a Dissipative Model Be?
1.4.3 Discrete Velocity Models
1.5 Applications (III): Mean-Field Models
1.5.1 Vlasov-Type Models
1.5.2 The DPT of a Single Vlasov-Type Equation
1.5.3 Genuine Plasmas
1.6 Applications (IV): Molecular Dynamics
1.6.1 Mass–Momentum Tensor of a Single Particle
1.6.2 Long-Range Forces
1.6.3 Hard Spheres
References
2 Equatorial Wave–Current Interactions
2.1 Introduction
2.2 Preliminaries
2.3 The Equatorial f-Plane Approximation
2.4 The Equatorial β-Plane Approximation
2.5 An Exact β-Plane Solution (Equatorially Trapped Wave)
2.5.1 Analysis of the Equatorially Trapped Wave Motion
2.5.2 Quantitative Aspects
2.6 The Ocean Flow in the Equatorial Pacific
2.6.1 The El Niño Phenomenon
2.6.2 A Model for the Equatorial Currents
2.6.3 Equatorial Wave–Current Interactions
2.6.4 Linear Wave Theory
2.6.5 Weakly Nonlinear Models
References
3 Linear and Nonlinear Equatorial Waves in a Simple Modelof the Atmosphere
3.1 Introduction
3.2 Linear Equatorial Waves
3.3 Weakly Nonlinear Long Equatorial Waves
3.3.1 Long Linear Rossby and Kelvin Waves
3.3.2 Nonlinear Slow Dynamics of Long Waves
3.4 Equatorial Modons
3.5 Equatorial Adjustment: Initial-Value Problem on the Equatorial Beta-Plane
3.6 Brief Summary and Discussion
References
4 The Water Wave Problem and Hamiltonian TransformationTheory
4.1 Introduction
4.2 Water Waves and Hamiltonian PDEs
4.2.1 Physical Derivation of the Governing Equations
4.2.2 General Notions on Hamiltonian Systems
4.2.3 Examples of Hamiltonian PDEs
Quasilinear Wave Equation
Boussinesq System
Korteweg–de Vries Equation
Nonlinear Schrödinger Equation
4.2.4 Zakharov's Hamiltonian for Water Waves
4.3 Dirichlet–Neumann Operator and Its Analysis
4.3.1 Legendre Transform
4.3.2 Shape Derivative of H
4.3.3 Invariants of Motion
4.3.4 Taylor Expansion of G
4.4 Birkhoff Normal Forms
4.4.1 Significance of the Normal Form
4.4.2 Complex Symplectic Coordinates and Poisson Brackets
4.4.3 Resonances
4.4.4 FormalTransformationTheoryandBirkhoffNormalForm
4.4.5 Solving the Third-Order Cohomological Equation
4.4.6 Normal Forms for Gravity Waves on Infinite Depth
Third-Order Normal Form and Burgers' Equation
Fourth-Order Normal Form
Integrable Birkhoff Normal Form
4.5 Model Equations for Water Waves
4.5.1 Linearized Problem
4.5.2 Non-dimensionalization
4.5.3 Canonical Transformation Theory
4.5.4 Calculus of Transformations
Amplitude Scaling
Spatial Scaling
Surface Elevation-Velocity Coordinates
Moving Reference Frame
Characteristic Coordinates
4.5.5 Boussinesq and KdV Scaling Limits
4.5.6 Modulational Scaling Limit and the NLS Equation
Normal Form Transformation
Modulational Ansatz
Expansion and Homogenization of Multiscale Functions
NLS Equation
Reconstruction of the Free Surface
4.6 Initial Value Problems
4.6.1 Local Well-Posedness
4.6.2 Recent Results on Global Well-Posedness for Small Data
4.6.3 Water Waves in a Periodic Geometry
4.7 Numerical Simulation of Surface Gravity Waves
4.7.1 Tanaka's Method for Solitary Waves
4.7.2 High-Order Spectral Method
Space Discretization
Time Integration
4.7.3 Collision of Solitary Waves
References
5 Gravity Wave Propagation in Inhomogeneous Media
5.1 Introduction
5.2 Water Waves
5.2.1 Propagation on Uneven Bottoms: First Order StokesWaves
5.2.2 Second Order Stokes Waves
5.2.3 Propagation in the Presence of Current or Through Porous Media
Propagation in the Presence of Current
Propagation Through Porous Media
5.3 Wave Scattering: 2D Case
5.3.1 Standing Wave in a Tank: Resonance and Sloshing
5.3.2 Case of Smooth Bathymetries: Sinusoidal Beds
Perturbation Method with Multiple-scale Expansion for Sinusoidal Beds of Finite Extend
Mild-Slope and Modified Mild-Slope Equations
5.3.3 Case of Abrupt Bathymetries
General Expression of the Velocity Potentials
Integral Matching Conditions Method
5.3.4 Examples
Sloping Beds
Sinusoidal Beds
Reflection Due to Structures
5.4 Water Focusing: 3D Case
5.4.1 Refraction—Snell—Descartes' Law
5.4.2 Refraction-Diffraction
5.4.3 Diffraction
Analytic Solution: Semi-Infinite Dike
Channels of Finite Width
5.4.4 Examples
Wave Scattering in the Presence of Underwater Mound
Wave Scattering by Surface Piercing Structures
Wave Scattering by Emerging Porous Media
5.5 Application to Wave Energy Device
5.5.1 Oscillating Water Column
5.5.2 Pressure Oscillation
References
6 Physical Models for Flow: Acoustic Interaction
6.1 Introduction
6.2 Fluid Dynamics
6.2.1 Conservation Equations
Conservation of Mass
Conservation of Momentum
Conservation of Energy
6.2.2 Constitutive Equations
6.2.3 Characterization of Flows by Dimensionless Numbers
6.2.4 Vorticity
6.2.5 Towards Acoustics
Formulation for Scalar Potential
Formulation for Vector Potential
6.3 Acoustics
6.3.1 Wave Equation
6.3.2 Simple Solutions: d'Alembert
6.3.3 Impulsive Sound Sources
6.3.4 Free-Space Green's Functions
6.3.5 Monopoles, Dipoles, and Quadrupoles
6.3.6 Calculation of Acoustic Far Field
6.3.7 Compactness
6.3.8 Solution of Wave Equation Using Green's Function
6.4 Aeroacoustics
6.4.1 Lighthill's Acoustic Analogy
6.4.2 Curle's Theory
6.4.3 Vortex Sound
6.4.4 Perturbation Equations
6.4.5 Comparison of Different Formulations
6.4.6 Acoustic Feedback Mechanisms
6.5 Applications
Coupling strategy of flow and acoustics
Fluid dynamics
Acoustics
Conclusions of workflow
6.5.1 Human Phonation
6.5.2 Axial Fan
6.5.3 Cavity at Low Mach Number schoder2020numerical
6.5.4 Cavity at High Mach Number
Appendix
References