Table Of ContentSeries in Contemporary Mathematics 4
Shuxing Chen
Mathematical
Analysis
of Shock Wave
Reflection
Series in Contemporary Mathematics
Volume 4
Editor-in-Chief
Tatsien Li, School of Mathematical Sciences, Fudan University, Shanghai,
Shanghai, China
Series Editors
Philippe G. Ciarlet, City University of Hong Kong, Hong Kong, China
Jean-Michel Coron, Laboratoire Jacques-Louis Lions, Université Pierre et Marie
Curie, Paris, France
Weinan E, Department of Mathematics, Princeton University, PRINCETON, NJ,
USA
JianshuLi,DepartmentofMathematics,TheHongKongUniversityofScienceand
Technology, Hong Kong, China
Jun Li, Department of Mathematics, Stanford University, STANFORD, CA, USA
Tatsien Li, School of Mathematical Sciences, Fudan University, Shanghai,
Shanghai, China
Fanghua Lin, Courant Inst. Mathematical Scienc, New York University, NEW
YORK, NY, USA
Zhi-ming Ma, Academy of Mathematics and Systems Science, Beijing, Beijing,
China
Andrew J. Majda, Department of Mathematics, New York University, New York,
NY, USA
Cédric Villani, Institut Henri Poincaré, Paris, Paris, France
Ya-xiang Yuan, Institute of Computational Mathematics and Science/Engineering
Computing, Academy of Mathematics and Systems Science, Beijing, Beijing,
China
Weiping Zhang, Chern Institute of Mathematics, Nankai University, Tianjin,
Tianjin, China
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Shuxing Chen
Mathematical Analysis
fl
of Shock Wave Re ection
123
ShuxingChen
FudanUniversity
Shanghai, China
ISSN 2364-009X ISSN 2364-0103 (electronic)
Series in ContemporaryMathematics
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Preface
In the motion of continuous media, such ascompressiblefluid,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
withittogether.Physically,theshockwaveisaverythinlayerinthemedium,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,andtemperature,etc.,generallymayhavesignificantchangefromtheone
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-
continuousonthissurface.Theoccurrenceofshockwavesbringsgreatinfluenceto
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
severedamage.Therefore,itiscrucialtodeeplyunderstandandgivegreatconcern
ontheoccurrence,propagation,andreflectionofshockwaves.Obviously,sincethe
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 fluid 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
anycase,thetheoreticalanalysisareindispensableforeithernumericalcomputation
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
v
vi Preface
becomebasicrulefornumericalcomputationincompressibleflowinvolvingshock
waves. Since the recent development of engineering technology requires more
precise and accurate numerical results, then more efficient mathematical tools,
particularly the theory of partial differential equations, are expected to play their
role.However,weshouldsaythatthoughthetheoryofpartialdifferentialequations
developed rapidly in recent decades, the application of the theory to the problems
involvingshockwavesisfarfromenoughandanticipated.Thesituationremindsus
the words written by R. Courant and K. O. Friedrichs in their book “Supersonic
flow and shock waves” [1]: The confidence of the engineer and physicist in the
resultofmathematicalanalysisshouldultimatelyrestontheproofthatthesolution
obtainedissingledoutbythedata oftheproblem.Agreat effortwill benecessary
todevelopthetheoriespresentedinthisbooktoastagewheretheysatisfyboththe
needsofapplicationsandthebasicrequirementsofnaturalphilosophy.Thisisalso
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
derivedfromshockreflectionbyusingthetheoryofpartialdifferentialequationsas
maintool.Itisknownthatthereflectionofshockincompressibleflowisamoving
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.Thenonecantreattheseproblemsassteadyones.Hence,inthisbookwe
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
differentbasicwavestructures,oneissimilartothestructureoftenappearinginthe
reflectionoflinearwavescalledregularreflection,theotherisastructureincluding
triple intersection, which is first reported by Ernst Mach over 140 years ago in
1878,andiscalledMachconfigurationlater,sothatcorrespondingshockreflection
iscalledMachreflection.ThepossibleappearanceofMachconfiguration(ormore
complicatedconfiguration)greatlyincreasesthecomplexityoftherelatedproblems.
Inthisbookwewillprovethestabilityforbothconfigurationforregularreflection
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,itisoftenexpectedtofindglobalsolutionforspecificproblems.Obviously,
such a requirement causes more difficulties, in many cases even the conditions on
thesurroundingenvironmentcanhardlydescribed.Sofaronlysomeresultsinvery
special cases are obtained.
G.Ben-Dorinhisbook“ShockWavesReflectionPhenomena”summarizedand
carefully analyzed various phenomena and results obtained in experiment investi-
gation.Thebookalsoshowedthatthemathematicalanalysisbasedonthetheoryof
partial differential equations is only at its beginning. Many problems are just
Preface vii
formulatedandarecompletelyopen.Wehopethatthepublicationofourbookwill
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 results and promote the
research in this field.
Chapter 1 of the book is an introduction, where some basic knowledge on the
systemofcompressibleflowandshockwavesarepresented.Welisttheknowledge
here for reader’s convenience, though readers can find them from other classical
books(e.g.,[1–3]).InChap.2weintroducetheconceptofshockpolarandpresent
its properties, which are useful in our future discussion but scattered in related
literatures.Somepropertiesarefirstpresentedandprovedinthisbook,particularly
thepropertiesoftheshockpolarforpotentialflowequation.Chapter3isdevotedto
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
supersonicflowpastawedge,sowecitetheresultsandthetechniquesdevelopedin
[4],forshockreflectionintwo-dimensionalspace,andin[5],forthree-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
reflectionaccordingtotheirdifferentphysicalcharacteristicfeature.Thematerialin
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
takenfrom[7–9].Finally,inChap.6,welistedafewlong-standingopenproblems,
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,fromwhomIwasmuchbenefited.Iamverygratefultoallthesepeoplefor
their valuable comments and suggestions. However, due to my limited knowledge
andabilitythebookmaystillcontainmanyshortcomingsandmistakes.Isincerely
hope to get more help and corrections from my colleagues and readers.
Shanghai, China Shuxing Chen
May 2020
References
1. R.Courant,K.O.Friedrichs,SupersonicFlowandShockWaves(IntersciencePublishersInc.,
NewYork,1948)
2. C.M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics (Springer-Verlag:
Berlin,Heidelberg,NewYork,2000)
3. J. Smoller, Shock Waves and Reaction-Diffusion Equations (Springer-Verlag, New York,
1994)
4. T-T.Li,W-C.Yu,Boundryvalueproblemsforquasi-linearhyperbolicsystems.DukeUniv.
Math.Ser.5(1985)
viii Preface
5. S.X.Chen,Existenceoflocalsolutiontosupersonicflowaroundathreedimensionalwing.
Adv.Appl.Math.13,273–304(1992)
6. S.X.Chen,Stabilityofamachconfiguration.Comm.PureAppl.Math.59,1–33(2006)
7. S.X. Chen, Mach configuration in pseudo-stationary compressible flow. Jour. Amer. Math.
Soc.21,63–100(2008)
8. G-Q. Chen, M. Feldman, Global solution to shock reflection by large-angle wedges for
potentialflow.Ann.Math.171,1067–1182(2010)
9. S.X.Chen,Onreflectionofmultidimensionalshockfront.Jour.Diff.Eqs.80,199–236(1989)
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Physical Background of Shock Reflection . . . . . . . . . . . . . . . . . . 1
1.2 Equations and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Euler System and Its Simplified Models . . . . . . . . . . . . . . 4
1.2.2 Shock, Rankine-Hugoniot Conditions . . . . . . . . . . . . . . . . 12
1.2.3 Entropy Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.2.4 Boundary Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.3 Reflection of Planar Shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.3.1 Normal Reflection of Planar Shock. . . . . . . . . . . . . . . . . . 27
1.3.2 Oblique Reflection of Planar Shock . . . . . . . . . . . . . . . . . 30
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2 Shock Polar Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.1 Shock Polar for Euler Equation. . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.1.1 Shock Polar on ðu;vÞ Plane . . . . . . . . . . . . . . . . . . . . . . . 33
2.1.2 Shock Polar on ðh;pÞ Plane . . . . . . . . . . . . . . . . . . . . . . . 41
2.2 Shock Polar for Potential Flow Equation . . . . . . . . . . . . . . . . . . . 45
2.2.1 Shock Polar on ðu;vÞ Plane . . . . . . . . . . . . . . . . . . . . . . . 45
2.2.2 Shock Polar on ðq;hÞ Plane . . . . . . . . . . . . . . . . . . . . . . . 53
2.3 Reflection of Planar Shock and Mach Configuration. . . . . . . . . . . 56
2.3.1 Regular Reflection of Planar Shock . . . . . . . . . . . . . . . . . 56
2.3.2 Mach Configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3 Perturbation of Regular Shock Reflection. . . . . . . . . . . . . . . . . . . . . 69
3.1 Regular Reflection Containing Supersonic Shock
in Two-Dimensional Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.1.1 Boundary Value Problems in Angular Domain . . . . . . . . . 69
3.1.2 Results on Free Boundary Problems
with Characteristic Boundary . . . . . . . . . . . . . . . . . . . . . . 73
ix
