This book is aimed to make careful analysis to various mathematical problems derived from shock reflection by using the theory of partial differential equations. The occurrence, propagation and reflection of shock waves are important phenomena in fluid dynamics. Comparing the plenty of studies of physical experiments and numerical simulations on this subject, this book makes main efforts to develop the related theory of mathematical analysis, which is rather incomplete so far. The book first introduces some basic knowledge on the system of compressible flow and shock waves, then presents the concept of shock polar and its properties, particularly the properties of the shock polar for potential flow equation, which are first systematically presented and proved in this book. Mathematical analysis of regular reflection and Mach reflection in steady and unsteady flow are the most essential parts of this book. To give challenges in future research, some long-standing open problems are listed in the end. This book is attractive to researchers in the fields of partial differential equations, system of conservation laws, fluid dynamics, and shock theory.
Author(s): Shuxing Chen
Series: Series in Contemporary Mathematics 4
Edition: 1
Publisher: Springer
Year: 2020
Language: English
Pages: 251
Tags: PDE, Shock Reflection, Shock Polar, Mach
Preface
Contents
1 Introduction
1.1 Physical Background of Shock Reflection
1.2 Equations and Boundary Conditions
1.2.1 Euler System and Its Simplified Models
1.2.2 Shock, Rankine-Hugoniot Conditions
1.2.3 Entropy Condition
1.2.4 Boundary Conditions
1.3 Reflection of Planar Shock
1.3.1 Normal Reflection of Planar Shock
1.3.2 Oblique Reflection of Planar Shock
References
2 Shock Polar Analysis
2.1 Shock Polar for Euler Equation
2.1.1 Shock Polar on (u,v) Plane
2.1.2 Shock Polar on (θ,p) Plane
2.2 Shock Polar for Potential Flow Equation
2.2.1 Shock Polar on (u,v) Plane
2.2.2 Shock Polar on (q,θ) Plane
2.3 Reflection of Planar Shock and Mach Configuration
2.3.1 Regular Reflection of Planar Shock
2.3.2 Mach Configuration
References
3 Perturbation of Regular Shock Reflection
3.1 Regular Reflection Containing Supersonic Shock in Two-Dimensional Space
3.1.1 Boundary Value Problems in Angular Domain
3.1.2 Results on Free Boundary Problems with Characteristic Boundary
3.1.3 Local Existence of Solution to the Problem for Shock Reflection of Isentropic Irrotational Flow
3.1.4 Local Existence of Solution to the Problem for Shock Reflection of Non-isentropic Flow
3.2 Regular Reflection Containing Supersonic Shock in Three-Dimensional Space
3.2.1 Preparation
3.2.2 Linearized Problem and Related a Priori Estimates
3.2.3 Construction of the First Approximate Solution to Nonlinear Problem
3.2.4 Newton's Iteration and Existence of Genuine Solution to Nonlinear Problem
3.3 Regular Reflection Containing Transonic Shock
References
4 Stability of Mach Configuration
4.1 Reduction and Classification of Mach Configuration
4.1.1 E-E Type and E-H Type Mach Configuration
4.1.2 System and Boundary Conditions
4.2 Lagrange Transformation and Canonical Form of Nonlinear System
4.2.1 Lagrange Transformation for Stationary Flow
4.2.2 Treatment of Shock Boundary
4.2.3 Decomposition of System
4.3 Estimates of Linearized Problem Derived from E-E Type Mach Configuration
4.3.1 Linearized Problem
4.3.2 Elliptic Sub-problem
4.3.3 Sobolev Estimate
4.3.4 Hölder Estimate
4.4 Convergence of Iterative Process and Stability of E-E Type Mach Configuration
4.4.1 Iterative Process of Solving Nonlinear Problem (NL)
4.4.2 Convergence of Iterative Scheme
4.4.3 Existence of Free Boundary Value Problem
4.5 Stability of E-H Type Mach Configuration
4.5.1 Problem and Conclusion
4.5.2 Nonlinear Lavrentiev-Bitsadze Mixed Type Equation
4.5.3 Linearization
4.5.4 Solution to Generalized Tricomi Problem of Linear Lavrentiev-Bitsadze Equation
4.5.5 Conclusion on Nonlinear Problem
References
5 Shock Reflection in Unsteady Flow
5.1 Shock Reflection by a Smooth Surface
5.1.1 Formulation
5.1.2 Reduce to a Goursat Problem with Fixed Boundary
5.1.3 Solution to Nonlinear Boundary Value Problem
5.2 Regular Reflection of Planar Shock by a Ramp
5.2.1 Formulation
5.2.2 Determine Flow Field in Pseudo-Supersonic Region
5.2.3 Nonlinear Degenerate Elliptic Boundary Value Problem
5.2.4 Elliptic Transaction
5.2.5 Nonlinear Iterative Scheme
5.2.6 Elliptic Regularization
5.2.7 Existence of Solution to Nonlinear Degenerate Elliptic Boundary Value Problem
5.3 Mach Reflection of Plane Shock by a Ramp
5.3.1 Formulation
5.3.2 Perturbation of Flat Mach Configuration
5.3.3 Main Steps of Proof
5.3.4 Proof of Theorem 5.4
References
6 Further Considerations and Open Problems
6.1 Shock Reflection by a Ramp for Non-isentropic Flow
6.2 Shock Reflection in Three-Dimensional Space
6.2.1 Reflection of a Planar Shock by a Curved Ramp
6.2.2 Reflection of Planar Shock by a Cone
6.2.3 Stability of Mach Configuration in Three-Dimensional Space
6.3 Big Perturbation and Global Solutions
6.3.1 Problems on Big Perturbation
6.3.2 Global Solutions
6.4 Further Discussions on Irregular Shock Reflection
6.4.1 Transition of Different Configurations
6.4.2 Other Modes of Irregular Reflection
6.5 Shock Reflection and Refraction on Interface of Two Media
References
Appendix A Estimates for Elliptic Equations in Curvilinear Polygon
A.1 Main Estimates for Elliptic Equations in Curvilinear Polygon
A.1.1 Estimates near Corners
A.1.2 Boundary Value Problem in an Infinite Strip
A.1.3 Expansion of Solutions According to Regularity
A.2 Estimates and Expansion of Solutions with Lower Regularity
References
Index
