前言In the motion of continuous media,such as compressible fluid, the occurrence and propagation of shock waves are common physical phenomena. For instance, the detonation of explosives in a continuous medium will cause an shock wave propagating starting from the source of the explosion; a fast flying projectile with supersonic speed always produces a shock wave ahead of the projectile, moving with it together.Physically,the shock wave is a very thin layer in the medium,and its characteristic feature is that the state of the medium in this thin layer changes rapidly. Then the parameters describing the medium, such as velocity,density, pressure,and temperature, etc., generally may have significant change from the one side of the layer to the other side of it. Mathematically,the shock wave is often described by a surface with zero width, and the parameters of the fluid are dis- continuous on this surface. The occurrence of shock waves brings great influence to the physical state of the medium around it. Particularly,in the case when a shock hits an obstacle and then is reflected, the reflection is often powerful and produces severe damage. Therefore, it is crucial to deeply understand and give great concer on the occurrence,propagation,and reflection of shock waves. Obviously,since the obstacles could be in various way, the structure of shock waves and the flow field caused by the reflection of shocks would be quite complicated. Consequently, precisely understanding the process of shock reflection and the resulting effect is very important and rather difficult.In the motion of continuous media,such as compressible fluid, the occurrence and propagation of shock waves are common physical phenomena. For instance, the detonation of explosives in a continuous medium will cause an shock wave propagating starting from the source of the explosion; a fast flying projectile with supersonic speed always produces a shock wave ahead of the projectile, moving with it together.Physically,the shock wave is a very thin layer in the medium,and its characteristic feature is that the state of the medium in this thin layer changes rapidly. Then the parameters describing the medium, such as velocity,density, pressure,and temperature, etc., generally may have significant change from the one side of the layer to the other side of it. Mathematically,the shock wave is often described by a surface with zero width, and the parameters of the fluid are dis- continuous on this surface. The occurrence of shock waves brings great influence to the physical state of the medium around it. Particularly,in the case when a shock hits an obstacle and then is reflected, the reflection is often powerful and produces severe damage. Therefore, it is crucial to deeply understand and give great concer on the occurrence,propagation,and reflection of shock waves. Obviously,since the obstacles could be in various way, the structure of shock waves and the flow field caused by the reflection of shocks would be quite complicated. Consequently, precisely understanding the process of shock reflection and the resulting effect is very important and rather difficult.
Generally, their are three ways to study various problems in fuid dynamics： experiment investigation, numerical computation,and theoretical analysis. The theoretical analysis, especially the mathematical analysis, often predicts physical phenomena or offers qualitative characters to observed phenomena, the numerical computation offers required quantitative results in engineering technology,and the experiment investigation gives verification of obtained results or established con- clusions, and occasionally finds new phenomena to raise new research topics. In any case,the theoretical analysis are indispensable for either numerical computation or experiment investigation. For instance,rigorous theoretical analysis points out that the flow parameters on both sides of any shock wave should satisfy Rankine-Hugoniot conditions and entropy condition, then these conditions have become basic rule for numerical computation in compressible fow involving shock waves. Since the recent development of engineering technology requires more precise and accurate numerical results, then more eficient mathematical tools. particularly the theory of partial differential equations,are expected to play their role.However,we should say that though the theory of partialdifferential equations developed rapidly in recent decades, the application of the theory to the problems involving shock waves is far from enough and anticipated.The situation reminds us the words written by R. Courant and K. O. Friedrichs in their book"Supersonic flow and shock waves"[1]： The confidence of he engineer and physicist in the result of mathematical analysis should ultimately rest on the proof that the solution obtained is singled out by the data ofthe problem.A great efor will be necessary to develop the theories presentedin this book to astage where they satisfyboth the needs ofapplications and the basicrequirements of natral philosophy. This is also the purpose of our writing this book, in which I try to do some contribution to develop mathematical theory in this area.
The book is aimed to make careful analysis to various mathematical problems derived from shock reflection by using the theory of partial differential equations as main tool. It is known that the refection of shock in compressble flow is a moving process, then the related problem is generally unsteady one depending on time. Meanwhile,in some special cases the parameters of the flow can be stable with respect to time,or are independent of time in the coordinate system moving with particles.Then one can treat these problems as steady ones.Hence, in this book we will discuss shock reflection for both steady flow and unsteady flow.
In the study of phenomena of shock reflection the structure of the shock waves and the flow field near the reflection point may be quite varied, depending on the incident angle of the shock with the surface of the obstacle.Locally there are two different basic wave structures,one is similar to the structure often appearing in the reflection of linear waves called regular reflection,the other is a structure including triple intersection,which is first reported by Ernst Mach over 140 years ago in 1878,and is called Mach configuration later,so that coresponding shock reflection is called Mach reflection. The possible appearance of Mach configuration or more complicated configurationgreatly increases the complexity of the related problems. In this book we will prove the stability for both configuration for regularreflection and Mach reflection, which is necessary and fundamental to establish a complete theory on shock reflection.
The solution to a given problem on shock reflection often depends not only on the conditions near the reflection point, but also on the surrounding environment. Hence,it is often expected to find global solution for specific problems.Obviously, such a requirement causes more difficulties,in many cases even the conditions on the surrounding environment can hardly described.So far only some results in very special cases are obtained.
G.Ben-Dor in his book"Shock Waves Refection Phenomena"summarized and carefully analyzed various phenomena and results obtained in experiment investi- gation.The book also showed that the mathematical analysis basedon the theory of partial diferential equations is only at its beginning. Many prblems are just formulated and are completely open. We hope that the publicationof our book will increase people''s interest in this subject. It is desirable that the book can give a preparation in some extent, as well as offer some first resuls and promote the research in this field.
Chapter 1 of the book is an introduction, where some basic knowledge on the system of compressible flow and shock waves are presented.Welist the knowledge here for reader''s convenience, though readers can find them from other classical books e.g.[1-3].In Chap.2 we introduce the concept of shock polar and present its properties, which are useful in our future discussion but scattered in related literatures. Some properties are first presented and proved in this book, particularly the properties of the shock polar for potential flow equation. Chapter 3 is devoted to the mathematical analysis of regular reflection of steady shock waves. The math- ematical treatment on the regular shock reflection is essentially similar to that for supersonic flow past a wedge,so we cite the results and the techniques developed in[4], for shockreflection in two-dimensional space,and in [5],for three-dimensional space. Chapter 4 is devoted to the mathematical analysis of Mach reflection in steady flow.The reflection is divided to E-E Mach reflection and E-H Mach reflection according to their different physical characteristic feature. The material in this chapter is taken from [6,7]. In Chap. 5 we discuss the shock reflection in unsteady flow,including regular reflection and Mach reflection. The results are taken from [7-9]. Finally, in Chap.6,welisted a few long-standing open problems, which give big challenge in future research.
In the writing of this book besides the results obtained in my previous papers I also referred many results and techniques scattered in related literatures, which are cited in the book. Besides,I also had much discussions with my colleagues and friends, from whom Iwas much benefited.Iam very grateful to all these people for their valuable comments and suggestions. However, due to my limited knowledge and ability the book may sill contain many shortcomings and mistakes. I sincerely hope to get more help and corrections from my colleagues and readers.
Shanghai, China
Chen Shuxing
May 2020
內容試閱：
In the motion of continuous media,such as compressible fluid, the occurrence and propagation of shock waves are common physical phenomena. For instance, the detonation of explosives in a continuous medium will cause an shock wave propagating starting from the source of the explosion; a fast flying projectile with supersonic speed always produces a shock wave ahead of the projectile, moving with it together.Physically,the shock wave is a very thin layer in the medium,and its characteristic feature is that the state of the medium in this thin layer changes rapidly. Then the parameters describing the medium, such as velocity,density, pressure,and temperature, etc., generally may have significant change from the one side of the layer to the other side of it. Mathematically,the shock wave is often described by a surface with zero width, and the parameters of the fluid are dis- continuous on this surface. The occurrence of shock waves brings great influence to the physical state of the medium around it. Particularly,in the case when a shock hits an obstacle and then is reflected, the reflection is often powerful and produces severe damage. Therefore, it is crucial to deeply understand and give great concer on the occurrence,propagation,and reflection of shock waves. Obviously,since the obstacles could be in various way, the structure of shock waves and the flow field caused by the reflection of shocks would be quite complicated. Consequently, precisely understanding the process of shock reflection and the resulting effect is very important and rather difficult.
Generally, their are three ways to study various problems in fuid dynamics： experiment investigation, numerical computation,and theoretical analysis. The theoretical analysis, especially the mathematical analysis, often predicts physical phenomena or offers qualitative characters to observed phenomena, the numerical computation offers required quantitative results in engineering technology,and the experiment investigation gives verification of obtained results or established con- clusions, and occasionally finds new phenomena to raise new research topics. In any case,the theoretical analysis are indispensable for either numerical computation or experiment investigation. For instance,rigorous theoretical analysis points out that the flow parameters on both sides of any shock wave should satisfy Rankine-Hugoniot conditions and entropy condition, then these conditions have become basic rule for numerical computation in compressible fow involving shock waves. Since the recent development of engineering technology requires more precise and accurate numerical results, then more eficient mathematical tools. particularly the theory of partial differential equations,are expected to play their role.However,we should say that though the theory of partialdifferential equations developed rapidly in recent decades, the application of the theory to the problems involving shock waves is far from enough and anticipated.The situation reminds us the words written by R. Courant and K. O. Friedrichs in their book"Supersonic flow and shock waves"[1]： The confidence of he engineer and physicist in the result of mathematical analysis should ultimately rest on the proof that the solution obtained is singled out by the data ofthe problem.A great efor will be necessary to develop the theories presentedin this book to astage where they satisfyboth the needs ofapplications and the basicrequirements of natral philosophy. This is also the purpose of our writing this book, in which I try to do some contribution to develop mathematical theory in this area.
The book is aimed to make careful analysis to various mathematical problems derived from shock reflection by using the theory of partial differential equations as main tool. It is known that the refection of shock in compressble flow is a moving process, then the related problem is generally unsteady one depending on time. Meanwhile,in some special cases the parameters of the flow can be stable with respect to time,or are independent of time in the coordinate system moving with particles.Then one can treat these problems as steady ones.Hence, in this book we will discuss shock reflection for both steady flow and unsteady flow.
In the study of phenomena of shock reflection the structure of the shock waves and the flow field near the reflection point may be quite varied, depending on the incident angle of the shock with the surface of the obstacle.Locally there are two different basic wave structures,one is similar to the structure often appearing in the reflection of linear waves called regular reflection,the other is a structure including triple intersection,which is first reported by Ernst Mach over 140 years ago in 1878,and is called Mach configuration later,so that coresponding shock reflection is called Mach reflection. The possible appearance of Mach configuration or more complicated configurationgreatly increases the complexity of the related problems. In this book we will prove the stability for both configuration for regularreflection and Mach reflection, which is necessary and fundamental to establish a complete theory on shock reflection.
The solution to a given problem on shock reflection often depends not only on the conditions near the reflection point, but also on the surrounding environment. Hence,it is often expected to find global solution for specific problems.Obviously, such a requirement causes more difficulties,in many cases even the conditions on the surrounding environment can hardly described.So far only some results in very special cases are obtained.
G.Ben-Dor in his book"Shock Waves Refection Phenomena"summarized and carefully analyzed various phenomena and results obtained in experiment investi- gation.The book also showed that the mathematical analysis basedon the theory of partial diferential equations is only at its beginning. Many prblems are just formulated and are completely open. We hope that the publicationof our book will increase people''s interest in this subject. It is desirable that the book can give a preparation in some extent, as well as offer some first resuls and promote the research in this field.
Chapter 1 of the book is an introduction, where some basic knowledge on the system of compressible flow and shock waves are presented.Welist the knowledge here for reader''s convenience, though readers can find them from other classical books e.g.[1-3].In Chap.2 we introduce the concept of shock polar and present its properties, which are useful in our future discussion but scattered in related literatures. Some properties are first presented and proved in this book, particularly the properties of the shock polar for potential flow equation. Chapter 3 is devoted to the mathematical analysis of regular reflection of steady shock waves. The math- ematical treatment on the regular shock reflection is essentially similar to that for supersonic flow past a wedge,so we cite the results and the techniques developed in[4], for shockreflection in two-dimensional space,and in [5],for three-dimensional space. Chapter 4 is devoted to the mathematical analysis of Mach reflection in steady flow.The reflection is divided to E-E Mach reflection and E-H Mach reflection according to their different physical characteristic feature. The material in this chapter is taken from [6,7]. In Chap. 5 we discuss the shock reflection in unsteady flow,including regular reflection and Mach reflection. The results are taken from [7-9]. Finally, in Chap.6,welisted a few long-standing open problems, which give big challenge in future research.
In the writing of this book besides the results obtained in my previous papers I also referred many results and techniques scattered in related literatures, which are cited in the book. Besides,I also had much discussions with my colleagues and friends, from whom Iwas much benefited.Iam very grateful to all these people for their valuable comments and suggestions. However, due to my limited knowledge and ability the book may sill contain many shortcomings and mistakes. I sincerely hope to get more help and corrections from my colleagues and readers.
Shanghai, China
Chen Shuxing
May 2020
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