Shock Wave and High Pressure Phenomena

Series Editor-in-Chief

L. Davison, USAY. Horie, USA

Founding Editor

R. A. Graham, USA

Advisory Board

V. E. Fortov, RussiaY. M. Gupta, USAR. R. Asay, USAG. Ben-Dor, IsraelK. Takayama, JapanF. Lu, USA

Shock Wave and High Pressure Phenomena

L.L. Altgilbers, M.D.J. Brown, I. Grishnaev, B.M. Novac, I.R. Smith, I. Tkach,and Y. Tkach: Magnetocumulative Generators

T. Antoun, D.R. Curran, G.I. Kanel, S.V. Razorenov, and A.V. Utkin: Spall FractureJ. Asay and M. Shahinpoor (Eds.): High-Pressure Shock Compression of Solids

S.S. Batsanov: Effects of Explosion on Materials: Modification and Synthesis UnderHigh-Pressure Shock Compression

R. Cherét: Detonation of Condensed Explosives

L. Davison, D. Grady, and M. Shahinpoor (Eds.): High-Pressure ShockCompression of Solids II

L. Davison and M. Shahinpoor (Eds.): High-Pressure Shock Compressionof Solids III

L. Davison, Y. Horie, and M. Shahinpoor (Eds.): High-Pressure Shock Compressionof Solids IV

L. Davison, Y. Horie, and T. Sekine (Eds.): High-Pressure Shock Compression ofSolids V

A.N. Dremin: Toward Detonation Theory

Y. Horie, L. Davison, and N.N. Thadhani (Eds.): High-Pressure Shock Compressionof Solids VI

R. Graham: Solids Under High-Pressure Shock Compression

J.N. Johnson and R. Cherét (Eds.): Classic Papers in Shock Compression Science

V.F. Nesterenko: Dynamics of Heterogeneous Materials

M. Suceska: Test Methods of Explosives

J.A. Zukas and W.P. Walters (Eds.): Explosive Effects and Applications

G.I. Kanel, S.V. Razorenov, and V.E. Fortov: Shock-Wave Phenomena and theProperties of Condensed Matter

V.E. Fortov, L.V. Altshuler, R.F. Trunin, and A.I. Funtikov: High-Pressure ShockCompression of Solids VII

L.C. Chhabildas, L. Davison, and Y. Horie (Eds.): High-Pressure ShockCompression of Solids VIII

D. Grady: Fragmentation of Rings and ShellsM. V. Zhernokletov and B. L. Glushak (Eds.): Material Properties under IntensiveDynamic Loading

R.P. Drake: High-Energy-Density Physics

G. Ben-Dor: Shock Wave Reflection Phenomena

ABC

G. Ben-Dor

Reflection PhenomenaShock Wave

With 194 Figures

Second Edition

Series Editors-in-Chief:Lee Davison39 Canoncito Vista RoadTijeras, NM 87059, USAE-mail: leedavison@aol.com

Yasuyuki HorieAFRL/MNME Munitions Directorate2306 Perimeter RoadEglin AFB, FL 32542, USAE-mail: yasuyuki.horie@eglin.af.mil

ISSN 8063-7200ISBN

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Gabi Ben-Dor

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1st ed.

Springer-Verlag Berlin Heidelberg , 20071991

To Professor Ozer Igra who introduced me to the world of shock tubesand waves,

to Professor Irvine Israel Glass who led me into the world of shock wavereflection phenomena,

to my colleagues all over the world with whom I have been investigatingthe fascinating phenomena of shock wave reflection for over 30 years,

and finally,

to Ms. Edna Magen, and our three children, Shai, Lavi and Tsachit,who provided me with an excellent atmosphere and support to accomplish

all my goals.

Acknowledgment

I would like to thank Dr. Li Huaidong, currently at the Jet PropulsionLaboratory, California Institute of Technology, in Pasadena, who was myPh.D. student and Post Doctoral Fellow during the years 1992–1997, for hisinvaluable contribution to many of the findings of my researches in the areaof shock wave reflection, which are the reason for putting together this secondedition of my monograph.

Preface

Nothing is more exciting to a scientist than realizing that his/her areas ofexpertise are developing and that the state-of-the-knowledge yesterday is out-dated today.

The distinguished philosopher Ernst Mach first reported the phenomenonof shock wave reflection over 125 years ago in 1878. The study of this fasci-nating phenomenon was then abandoned for a period of about 60 years untilProfessors John von Neumann and Bleakney initiated its investigation in theearly 1940s. Under their supervision, 15 years of intensive research related tovarious aspects of the reflection of shock waves in pseudosteady flows werecarried out. It was during this period that the four basic shock wave reflec-tion configurations, regular, single-Mach, transitional-Mach and double-Machreflections, were discovered. Then, for a period of about 10 years from themid-1950s until the mid-1960s, the investigation of the reflection phenom-enon of shock waves was kept on a low flame all over the world (e.g. Australia,Japan, Canada, USA, USSR, etc.) until Professor Tatyana Bazhenova from theUSSR, Professor Irvine Israel Glass from Canada, and Professor Roy Hender-son from Australia re-initiated the study of this and related phenomena. Undertheir scientific leadership, numerous findings related to this phenomenon werereported. Probably the most productive research group in the mid-1970s wasthat led by Professor Irvine Israel Glass in the Institute of Aerospace Studiesof the University of Toronto. In 1978, exactly 100 years after Ernst Mach firstreported his discoveries on the reflection phenomenon; I published my Ph.D.thesis in which, for the first time, analytical transition criteria between thevarious shock wave reflection configurations were established.

For reasons which for me are yet unknown, the publication of my Ph.D.findings triggered intensive experimental and analytical studies of the shockwave reflection phenomenon over a variety of geometries and properties of thereflecting surface and in a variety of gases. The center of the experimentalinvestigation was shifted from Canada to Japan, in general, and to the ShockWave Research Center that was led by Professor Kazuyoshi Takayama, inparticular. Under his supervision flow visualization techniques reached such

VIII Preface

a stage that the phrase “cannot be resolved experimentally” almost ceased toexist in the scientific dictionary, especially after Dr. Harald Kleine joined hisresearch group for a couple of years.

In the same year that I published my Ph.D. thesis, I published my firstjournal paper related to the shock wave reflection phenomenon. This paper,entitled “Nonstationary Oblique Shock Wave Reflections: Actual Isopycnicsand Numerical Experiments” was co-authored with my Ph.D. supervisor, Pro-fessor Irvine Israel Glass. In the conclusion to the paper we wrote Undoubt-edly, numerical codes will evolve in the future which will reliably predict notonly RR and SMR but also CMR and DMR in real gases. I wish my lot-tery predictions were as successful as this prediction, since probably the mostremarkable progress in the study of the shock wave reflection phenomenonin the following decade (i.e., in the 1980s) was made by American compu-tational fluid dynamicists, who demonstrated that almost nothing is beyondtheir simulation capability. At one time, it was feared that the computationalfluid dynamicists would put the experimentalists out of business. Fortunately,this did not occur. Instead, experimentalists, computational fluid dynami-cists, and theoreticians worked together in harmony under the orchestrationof Professor John Dewey, who realized, in 1981, that scientists interested inthe reflection phenomenon of shock waves will benefit the most if they meetonce every one/two years and exchange views and ideas. In 1981, he initiatedthe International Mach Reflection Symposium, which became the frameworkfor excellent cooperation between scientists from all over the world who areinterested in better understanding the shock wave reflection phenomenon.

Ten years later, in 1991, I completed writing my monograph entitled ShockWave Reflection Phenomena, which summarized the state-of-the-knowledge atthat time.

Three major developments, which shattered this state-of-the-knowledge,took place in the 15 years that has passed since then.

– The first (in the early 1990s), was the discovery of the hysteresis phenom-enon in the reflection of shock waves in steady flows.

– The second (in the mid-1990s), was a re-initiation of a abandonedapproach considering an overall shock wave diffraction process thatresults from the interaction of two sub-processes, namely, the shock-wavereflection process and the shock-induced flow deflection process. Thisapproach led to the development of new analytical models for describingthe transitional- and the double-Mach reflections; and

– The third (in the late 1990s and the mid-2000s), was the resolution of thewell-known von Neumann paradox.

As a result, only one out of the four main chapters of the monograph couldbe still considered as relevant and providing updated information. Unlike thischapter, the other four are simply outdated. Consequently, the monograph hasbeen re-written, to again describe the state-of-the-knowledge of the fascinating

Preface IX

phenomena of shock wave reflection, which I have been investigating for overthree decades.

As a final remark I would like to point out that this book comes as closeas possible to summarizing almost all that I know about shock wave reflectionphenomena from a phenomenological point of view. Thirty-one years ago,when I first met Professor Irvine Israel Glass, I almost knew nothing aboutthe reflection of shock waves. When he assigned me the investigation of thisphenomenon, I thought that it would take a lifetime to understand and explainit. Now I can state wholeheartedly that I was lucky to have been assigned toinvestigate this fascinating phenomenon and to have met and worked underthe supervision of Professor Irvine Israel Glass. I have been even luckier tobecome a part of a wonderful group of scientists from all over the world withwhom I have been collaborating throughout the past thirty years, and withwhom I hope to continue collaborating in the future.

Contents

1 General Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction and Historical Background . . . . . . . . . . . . . . . . . . . . 31.2 Reasons for the Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.2.1 Reason for the Reflection in Steady Flows . . . . . . . . . . . . 111.2.2 Reasons for the Reflection in Pseudosteady

and Unsteady Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.3 Analytical Approaches for Describing Regular

and Mach Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.3.1 Two-Shock Theory (2ST) for an Inviscid Flow . . . . . . . . 141.3.2 Three-Shock Theory (3ST) for an Inviscid Flow . . . . . . . 16

1.4 Shock Polars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.4.1 Shock-Polar Presentation of the Flow Field

Near the Reflection Point of a Regular Reflection . . . . . . 211.4.2 Shock-Polar Presentation of the Flow Field

Near the Triple Point of a Mach Reflection . . . . . . . . . . . 221.5 Suggested RR ��� IR Transition Criteria . . . . . . . . . . . . . . . . . . . . 25

1.5.1 Detachment Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.5.2 Mechanical-Equilibrium Criterion . . . . . . . . . . . . . . . . . . . 291.5.3 Sonic Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301.5.4 Length-Scale Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321.5.5 Summary, Critique, and Discussion . . . . . . . . . . . . . . . . . . 33

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2 Shock Wave Reflections in Steady Flows . . . . . . . . . . . . . . . . . . . 392.1 Categories of Steady Reflection Phenomena . . . . . . . . . . . . . . . . . 42

2.1.1 Curved Incident Shock Wave Reflections over StraightReflecting Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.1.2 Straight Incident Shock Wave Reflections over CurvedReflecting Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.1.3 Curved Incident Shock Wave Reflections over CurvedReflecting Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

XII Contents

2.1.4 Straight Incident Shock Wave Reflectionsover Straight Reflecting Surfaces . . . . . . . . . . . . . . . . . . . . 44

2.2 Modifications of the Perfect InviscidTwo- and Three-Shock Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . 482.2.1 Nonstraight Discontinuities . . . . . . . . . . . . . . . . . . . . . . . . . 492.2.2 Viscous Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.2.3 Thermal Conduction Effects . . . . . . . . . . . . . . . . . . . . . . . . 512.2.4 Real Gas Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.3 Prediction of the Mach Reflection Shapeand the Mach Stem Height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.3.1 Assumptions and Concepts of the Models . . . . . . . . . . . . . 542.3.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582.3.3 Derivation of a General Expression for a Curved

Line as a Function of Some Boundary Conditionsat Its Ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

2.3.4 Estimation of the Strength of the Expansion Wavesthat are Reflected at the Slipstream . . . . . . . . . . . . . . . . . 66

2.3.5 Geometric Relations of the Wave ConfigurationShown in Figs. 2.12 and 2.15 . . . . . . . . . . . . . . . . . . . . . . . . 67

2.3.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 702.4 Hysteresis Processes in the RR � MR Transition . . . . . . . . . . . . 76

2.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 762.4.2 Hysteresis Processes in the Reflection

of Symmetric Shock Waves . . . . . . . . . . . . . . . . . . . . . . . . . 792.4.3 Hysteresis Process in the Reflection of Asymmetric

Shock Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 902.4.4 Hysteresis Process in the Reflection of Axisymmetric

(Conical) Shock Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

3 Shock Wave Reflections in Pseudosteady Flows . . . . . . . . . . . . 1353.1 “Old” State-of-the-Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

3.1.1 Reflection Configurations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1403.1.2 The Transition Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1433.1.3 Second Triple Point Trajectory and Some Critical

Remarks Regarding the Old State-of-the-Knowledge . . . 1513.2 “New” (Present) State-of-the-Knowledge . . . . . . . . . . . . . . . . . . . 156

3.2.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1563.2.2 Shock-Diffraction Process . . . . . . . . . . . . . . . . . . . . . . . . . . 1573.2.3 Transition Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1593.2.4 Single-Mach Reflection (SMR) . . . . . . . . . . . . . . . . . . . . . . 1613.2.5 Formation of Transitional-Mach Reflection (TMR)

or Double-Mach Reflection (DMR) . . . . . . . . . . . . . . . . . . 1613.2.6 Transitional-Mach Reflection (TMR) . . . . . . . . . . . . . . . . . 1623.2.7 Double-Mach Reflection – DMR. . . . . . . . . . . . . . . . . . . . . 167

Contents XIII

3.2.8 SMR � PTMR/TMR/DMR and the TMR � DMRTransition Criteria and Domains of Different Typesof Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

3.2.9 Triple-Mach Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1763.2.10 Summary of the New State-of-the-Knowledge . . . . . . . . . 1773.2.11 Domains and Transition Boundaries . . . . . . . . . . . . . . . . . 1793.2.12 Weak Shock Wave Reflection Domain . . . . . . . . . . . . . . . . 180

3.3 Summary, Critique, and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 1903.4 Modifications of the Perfect Inviscid

Two- and Three-Shock Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . 1943.4.1 Nonsteady Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1943.4.2 Nonstraight Discontinuities . . . . . . . . . . . . . . . . . . . . . . . . . 1953.4.3 Real Gas Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1963.4.4 Viscous Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2013.4.5 Thermal Conduction Effects . . . . . . . . . . . . . . . . . . . . . . . . 2223.4.6 Noninfinitely Thin Contact Discontinuity . . . . . . . . . . . . . 2243.4.7 Non-Self-Similar Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

3.5 Additional Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2303.5.1 Flow Deflection Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . 2303.5.2 Shock Wave Diffraction Domains . . . . . . . . . . . . . . . . . . . . 2323.5.3 Comparison Between Steady and Pseudosteady

Reflection Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

4 Shock Wave Reflections in Unsteady Flows . . . . . . . . . . . . . . . . 2474.1 Constant Velocity Shock Wave Reflections

Over Nonstraight Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2504.1.1 Shock Wave Reflections Over Cylindrical

Concave Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2504.1.2 Shock Wave Reflections Over Cylindrical

Convex Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2824.1.3 Shock Wave Reflections Over Double Wedges . . . . . . . . . 291

4.2 Nonconstant Velocity Shock Wave Reflections OverStraight Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

4.3 Spherical Shock Wave Reflections Over Straightand Nonstraight Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

5 Source List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3075.1 Scientific Journals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3085.2 Reports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339