Shock wave reflection phenomena

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Shock Wave andHigh Pressure Phenomena

Shock WaveReectionPhenomena

G. Ben-Dor

Second Edition

Shock Wave andHigh Pressure Phenomena

Shock WaveReectionPhenomena

G. Ben-Dor

Second Edition


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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


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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 GeneratorsT. Antoun, D.R. Curran, G.I. Kanel, S.V. Razorenov, and A.V. Utkin: Spall Fracture J. Asay and M. Shahinpoor (Eds.): High-Pressure Shock Compression of SolidsS.S. Batsanov: Effects of Explosion on Materials: Modication and Synthesis UnderHigh-Pressure Shock Compression R. Cherét: Detonation of Condensed Explosives L. Davison, D. Grady, and M. Shahinpoor (Eds.): High-Pressure Shock Compression 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 of Solids V A.N. Dremin: Toward Detonation TheoryY. 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 ScienceV.F. Nesterenko: Dynamics of Heterogeneous Materials M. Su´ ceska: Test Methods of Explosives J.A. Zukas and W.P. Walters (Eds.): Explosive Effects and ApplicationsG.I. Kanel, S.V. Razorenov, and V.E. Fortov: Shock-Wave Phenomena and theProperties of Condensed MatterV.E. Fortov, L.V. Altshuler, R.F. Trunin, and A.I. Funtikov: High-Pressure Shock Compression of Solids VII L.C. Chhabildas, L. Davison, and Y. Horie (Eds.): High-Pressure Shock Compression of Solids VIII

D. Grady: Fragmentation of Rings and Shells M. V. Zhernokletov and B. L. Glushak (Eds.): Material Properties under Intensive

Dynamic Loading

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

G. Ben-Dor: Shock Wave Reflection Phenomena


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ABC

G. Ben-Dor

Reflection PhenomenaShock Wave

With 194 Figures

Second Edition


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Series Editors-in-Chief:Lee Davison39 Cañoncito Vista RoadTijeras, NM 87059 , USAE-mail: [email protected]

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Ben˜Gurion˜UniversityõfÑegevInstitute˜forÃpplied˜ResearchBeer-Sheva,˜IsraelE-mail: [email protected]

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Springer-Verlag Berlin Heidelberg , ˜20071991


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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 wavereection phenomena,

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

and nally,

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.


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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 ndings of my researches in the areaof shock wave reection, which are the reason for putting together this secondedition of my monograph.


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Preface

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

The distinguished philosopher Ernst Mach rst reported the phenomenonof shock wave reection 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 reection of shock waves in pseudosteady ows werecarried out. It was during this period that the four basic shock wave reec-tion congurations, regular, single-Mach, transitional-Mach and double-Machreections, were discovered. Then, for a period of about 10 years from themid-1950s until the mid-1960s, the investigation of the reection phenom-enon of shock waves was kept on a low ame 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 scientic leadership, numerous ndings 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 rstreported his discoveries on the reection phenomenon; I published my Ph.D.thesis in which, for the rst time, analytical transition criteria between the

various shock wave reection congurations were established.For reasons which for me are yet unknown, the publication of my Ph.D.

ndings triggered intensive experimental and analytical studies of the shockwave reection phenomenon over a variety of geometries and properties of thereecting 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 ow visualization techniques reached such


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VIII Preface

a stage that the phrase “ cannot be resolved experimentally ” almost ceased toexist in the scientic 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 rst journal paper related to the shock wave reection phenomenon. This paper,entitled “Nonstationary Oblique Shock Wave Reections: 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 not only 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 reection phenomenon

in the following decade (i.e., in the 1980s) was made by American compu-tational uid dynamicists, who demonstrated that almost nothing is beyondtheir simulation capability. At one time, it was feared that the computationaluid dynamicists would put the experimentalists out of business. Fortunately,this did not occur. Instead, experimentalists, computational uid dynami-cists, and theoreticians worked together in harmony under the orchestrationof Professor John Dewey, who realized, in 1981, that scientists interested inthe reection phenomenon of shock waves will benet the most if they meetonce every one/two years and exchange views and ideas. In 1981, he initiatedthe International Mach Reection Symposium, which became the frameworkfor excellent cooperation between scientists from all over the world who areinterested in better understanding the shock wave reection phenomenon.

Ten years later, in 1991, I completed writing my monograph entitled Shock Wave Reection 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 rst (in the early 1990s), was the discovery of the hysteresis phenom-enon in the reection of shock waves in steady ows.

– 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-wavereection process and the shock-induced ow deection process. Thisapproach led to the development of new analytical models for describingthe transitional- and the double-Mach reections; 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


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Preface IX

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

As a nal remark I would like to point out that this book comes as closeas possible to summarizing almost all that I know about shock wave reectionphenomena from a phenomenological point of view. Thirty-one years ago,when I rst met Professor Irvine Israel Glass, I almost knew nothing aboutthe reection 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 with

whom I have been collaborating throughout the past thirty years, and withwhom I hope to continue collaborating in the future.


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Contents

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

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

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

and Mach Reections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Reection Point of a Regular Reection . . . . . . 211.4.2 Shock-Polar Presentation of the Flow Field

Near the Triple Point of a Mach Reection . . . . . . . . . . . 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 Reections in Steady Flows . . . . . . . . . . . . . . . . . . . 392.1 Categories of Steady Reection Phenomena . . . . . . . . . . . . . . . . . 422.1.1 Curved Incident Shock Wave Reections over Straight

Reecting Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.1.2 Straight Incident Shock Wave Reections over Curved

Reecting Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432.1.3 Curved Incident Shock Wave Reections over Curved

Reecting Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44


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XII Contents

2.1.4 Straight Incident Shock Wave Reectionsover Straight Reecting Surfaces . . . . . . . . . . . . . . . . . . . . 44

2.2 Modications 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 Reection Shapeand the Mach Stem Height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.3.1 Assumptions and Concepts of the Models. . . . . . . . . . . . . 542.3.2 Governing Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.3.3 Derivation of a General Expression for a CurvedLine as a Function of Some Boundary Conditionsat Its Ends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

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

2.3.5 Geometric Relations of the Wave CongurationShown 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 Reection

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

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

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

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

3.1.1 Reection Congurations . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 Reection (SMR) . . . . . . . . . . . . . . . . . . . . . . 1613.2.5 Formation of Transitional-Mach Reection (TMR)

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


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1

General Introduction

List of symbolsLatin Letters C P Specic heat capacity at constant pressureC V Specic heat capacity at constant volumeh i Enthalpy in state ( i)

w Length scale required for the formation of an MRM i Flow Mach number in state ( i)M S Incident shock wave Mach number pi Static pressure in state ( i)T i Static temperature in state ( i)u i Flow velocity in state ( i) with respect to the reection

point, R , in RR or the triple point, T , in MR

V i Flow velocity in state ( i) in a laboratory frame of reference.V S Incident shock wave velocity in a laboratory frame of

reference.

Greek Letters χ First triple point trajectory angleχ Second triple point trajectory angleδ max (M i) Maximum ow deection angle for a ow having a Mach

number M i through an oblique shock waveφi Angle of incidence between the ow and the oblique

shock wave across which the ow enters into state ( i)γ Specic heat capacities ratio (= C P /C V )

µ Mach angleθi Angle of deection of the ow while passing across anoblique shock wave and entering into state ( i)

θW Reecting wedge angleθCW Complementary wedge angle (= 90

◦ − φ1)ρi Flow density in state ( i)


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2 1 General Introduction

ωi Angle between the incident shock wave and the reect-ing surfaceωr Angle between the reected shock wave and the reect-

ing surfaceAbbreviations (not in alphabetic order )R Reection pointT Triple pointi Incident shock waver Reected shock wavem Mach stems SlipstreamT Triple point

RR Regular reectionIR Irregular reectionMR Mach reectionWMR Weak Mach reectionvMR von Neumann reectionVR Vasilev reectionGR Guderley reectionDiMR Direct-Mach reectionStMR Stationary-Mach reectionInMR Inverse-Mach reectionTRR Transitioned regular reectionSMR Single-Mach reectionPTMR Pseudo-transitional-Mach reectionTMR Transitional-Mach reectionDMR Double-Mach reectionDMR + Positive double-Mach reectionDMR − Negative double-Mach reectionTerDMR Terminal double-Mach reection

Subscripts 0 Flow state ahead of the incident shock wave, i, or the

Mach stem, m1 Flow state behind the incident shock wave, i2 Flow state behind the reected shock wave, r3 Flow state behind the Mach stem, mm Maximum deection point (also known as the detach-

ment point) on the shock polars Sonic point on the shock polar

Superscripts R With respect to the reection point RT with respect to the triple point Ts Strong solutionw Weak solution


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1.1 Introduction and Historical Background 3

When a shock wave propagating in a medium with given acousticimpedance obliquely encounters another medium, having a different acousticimpedance, it experiences a reection that is known in the literature asoblique shock wave reection.

1.1 Introduction and Historical Background

Ernst Mach, who reported his discovery as early as 1878, was probably the rstscientist to notice and record the reection phenomenon of shock waves. Inhis ingenious experimental study, which was surveyed by Reichenbach (1983)and re-conducted by Krehl & van der Geest (1991), he recorded two different

shock wave reection congurations. The rst, a two shock wave congurationis known nowadays as regular reection, RR, and the second, a three shockwave conguration, was named after him, and is known nowadays as Machreection, MR.

Intensive research of the reection phenomena of shock waves wasre-initiated in the early 1940s by von Neumann. Since then it has beenrealized that the Mach reection wave conguration can be further dividedinto more specic wave structures. In addition, three new types of reectionwere recognized:

– The rst, a von Neumann reection, vNR, was forwarded in the early1990s.

– The second, a reection that has been named Guderley reection, GR,

after Guderley (1947) who was the rst one to hypothesize it.– The third, an intermediate wave conguration that appears for conditionsbetween those appropriate for the establishment and existence of vNR andGR. Since it was rst mentioned by Vasilev [see e.g., Vasilev & Kraiko(1999)] it will be referred to in this monograph as Vasilev reection, VR.

In general, the reection of shock waves can be divided into:

– Regular reection, RR, or– Irregular reections, IR.

The RR wave conguration consists of two shock waves, the incident shockwave, i, and the reected shock wave, r, that meet at the reection point, R,which is located on the reecting surface. A schematic illustration of the wave

conguration of an RR is shown in Fig. 1.1. All the other wave congurationsare termed irregular reections, IR.

The IR-domain is divided, in general, into four subdomains:

– A subdomain inside which the three-shock theory of von Neumann (seeSect.1.3.2) has a “standard” solution that corresponds to an MR

– A subdomain inside which the three-shock theory has a “nonstandard”solution that corresponds to a vNR


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1.1 Introduction and Historical Background 7

m

T

θW

R

r

i

Fig. 1.4. Schematic illustration of a transitioned regular reection wave congura-tion – TRR

Since the InMR is an MR in which the triple point moves towards thereecting surface, it terminates as soon as its triple point collides with thereecting surface. The termination of the InMR leads to the formation of anew wave conguration that was mentioned rst by Ben-Dor & Takayama(1986/7). The wave conguration of this reection consists of an RR followedby an MR. A schematic illustration of this wave conguration is shown inFig.1.4. Since this wave conguration is formed following a transition froman InMR, and since its main structure is an RR, it is called transitionedregular reection, TRR.

As will be shown subsequently, in pseudosteady ows the shock wave reec-tion process over the reecting surface interacts with a ow deection processaround the leading edge of the reecting wedge. This interaction results inthree different MR wave congurations, which were all discovered during theManhattan project. Until the early 1940s the only two wave congurationsthat were known to exist in pseudosteady ows were the regular reection,RR, and the Mach reection, MR, that as mentioned earlier were rst observedby Mach (1878). Smith (1945) investigated the shock wave reection phenom-

enon and noted that in some cases the reected shock wave of the MR hada kink or a reversal of curvature. However, only after White (1951) discov-ered a completely different type of reection, which he called double-Machreection, DMR, was the wave conguration observed by Smith (1945), i.e.,an MR with a kink or a reversal of curvature in the reected shock wave,recognized as a unique type of reection. Following White’s (1951) ndingthe reection that was rst observed by Mach (1878) was named simple-Mach reection, SMR, the reection that was discovered by Smith (1945) was


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named complex-Mach reection, CMR, and the reection that was discoveredby White (1951) was termed double-Mach reection, DMR, because its struc-ture (see Fig. 3.9) consisted of two triple points. In the 1970s when it wasrealized that the so-called simple-Mach reection is not simple at all, it wasrenamed and is known nowadays as single-Mach reection, SMR. Similarly,since the so-called complex-Mach reection is less complex than some of theother reection congurations, DMR for example, and since, as will be shownsubsequently, it can be viewed as an intermediate wave conguration betweenthe SMR and the DMR, it was re-named and is called nowadays transitional-Mach reection, 3 TMR. Li & Ben-Dor (1995) showed that there is anadditional wave conguration, a pseudo-transitional-Mach reection, PTMR.A PTMR is, in fact, a TMR in which the reected shock wave does not have

a reversal of curvature, and as a result, its appearance is identical to a SMR.An SMR, a TMR and a DMR are shown in Figs.3.7–3.9, respectively.In summary, the MR wave conguration consists, in pseudo steady ows,

of four types:

– A single-Mach reection, SMR– A pseudo-transitional-Mach reection, PTMR– A transitional-Mach reection, TMR– A double-Mach reection, DMR

Ben-Dor (1981) showed that, depending on the initial conditions, the trajec-tory angle of the second triple point, χ , could be either larger ( χ > χ ) orsmaller ( χ < χ ) than the trajectory angle of the rst triple point, χ . Lee &Glass (1984) termed the DMR for which χ > χ as DMR+ and the DMR forwhich χ < χ as DMR− . Photographs of a DMR + and a DMR − are shown inFig.3.11a, b, respectively. An intermediate DMR for which χ = χ is shownin Fig. 3.11c. Lee & Glass (1984) argued that there could be conditions forwhich the second triple point, T , would be located on the reecting surface,i.e., χ = 0. They termed this wave conguration as a terminal double-Machreection, TerDMR. A TerDMR is shown in Fig.3.12.

In summary, there are 13 different possible wave congurations, whichare associated with the reection of a shock wave over an oblique surface,namely: RR, WMR (i.e., vNR, VR, and GR), StMR, InMR, TRR, SMR,PTMR, TMR, DMR + , DMR− , and TerDMR. In steady ows only RR andSMR (usually referred to only as MR) are possible. Pseudosteady ows, where,as will be shown subsequently, there is an interaction between two processes,

the shock wave reection over the reecting wedge and the shock-induced owdeection around the leading edge of the reecting wedge, give rise, in additionto RR and SMR, to WMR (i.e., vNR, VR, and GR), PTMR, TMR, DMR + ,DMR − , and TerDMR. In unsteady ows three additional wave congurationsare possible: StMR, InMR, and TRR. The just mentioned 13 different wavecongurations are shown in an evolution tree type presentation in Fig. 1.5.

3 This name was originally suggested by Professor I.I. Glass.


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1.2 Reasons for the Reection 9

DMR

Types of Shock Wave Reflections

RR IR

vNR/VR/GRMR

DiMR StMR InMR

TRRSMR

PTMR

DMR −DMR +

TerDMR

TMR

Fig. 1.5. The 13 possible shock wave reection congurations

Because of the fact that different types of ow give rise to different typesof reections, the presentation of the shock wave reection phenomenon willbe divided, in this book, into three parts:

– Reection in steady ows in Chap. 2– Reection in pseudosteady ows in Chap.3– Reection in unsteady ows in Chap. 4.

1.2 Reasons for the ReectionNow that the shock wave reection phenomenon has been introduced briey,it is appropriate to explain the physical reasons for its occurrence.

The major reason for the occurrence of the reection phenomenon arisesfrom a very basic gas dynamic phenomenon. Consider Fig. 1.6 where threedifferent cases in which a ow with Mach number M 0 moves towards a wedge


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1.3 Analytical Approaches for Describing Regular and Mach Reections 13

an attached oblique shock wave emanating from the reection point R willdeect the ow away from the reecting wedge surface, while forming an RRwave conguration, and if θ1 > δ max (M R1 ), the ow deection will be achievedby a detached shock wave, which will evolve into an MR wave conguration.

If, however, M R1 < 1, the analogy to Fig.1.6a suggests that the subsonicow should negotiate the wedge surface continuously and smoothly, as shownschematically in Fig. 1.8b, without any need for a shock wave. In reality, how-ever, this is not the case. For all the combinations of M 0 and θW for whichM R1 < 1 an MR or a WMR (i.e., vNR, VR, or GR) wave conguration isobtained. The exact reason for this lies probably in the following explanation.

Consider Fig. 1.8b, where the subsonic ow obtained behind the incidentshock wave is seen to negotiate the wedge by a continuous turn. Although it

was noted earlier that this situation is analogous to the one shown in Fig.1.6a,there is one important difference. While in Fig.1.6a the ow “knows” aboutthe obstacle awaiting it when it is far away from the wedge, and hence itstarts adjusting its streamline to negotiate the obstacle long before actuallyencountering it, in the situation shown in Fig. 1.8b the ow streamline adjacentto the reecting wedge surface does not “know” about the obstacle until itpasses through the foot of the incident shock, i. Hence, upon passing throughthe foot of the incident shock wave it “nds” itself in a situation in whichit must negotiate a new boundary condition that is suddenly imposed on it.This sudden change in the boundary condition is, most probably, the reasonfor generating an additional shock wave, which in turn results in a reectionfor a situation where the ow Mach number behind the incident shock waveis subsonic with respect to the reection point R.

1.3 Analytical Approaches for Describing Regularand Mach Reections

The analytical approaches for describing the RR and the MR wave cong-urations were initiated both by von Neumann (1943a and 1943b). The onedescribing the RR is known as the two-shock theory – 2ST while the onedescribing the MR is known as the three-shock theory – 3ST . Both theoriesmake use of the inviscid conservation equations across an oblique shock wave,together with appropriate boundary conditions.

Consider Fig. 1.9 where an oblique shock wave and the associated ow

elds are illustrated. The ow states ahead and behind the oblique shock waveare ( i) and ( j ), respectively. The angle of incidence between the oncoming owand the oblique shock wave is φ j. While passing through the oblique shockwave, from state ( i) to state ( j ), the ow is deected by an angle θ j. Theconservation equations across an oblique shock wave, relating states ( i) and( j ) for a steady inviscid ow are:

– The conservation of mass:ρiu i sin φ j = ρ ju j sin (φ j − θ j) (1.1)


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( j )

u i

u j

θ j

φ j

( i )

Fig. 1.9. Denition of parameters across an oblique shock wave

– The conservation of normal momentum:

pi + ρi u2i sin2 φj = pj + ρj u2 j sin

2 (φj − θj ) . (1.2)

– The conservation of tangential momentum

ρi tan φj = ρj tan ( φj − θj ) . (1.3)

– The conservation of energy

h i + 12

u2i sin2 φj = h j +

12

u2j sin2 (φj − θ j) (1.4)

Here u is the ow velocity in a frame of reference attached to the obliqueshock wave, and ρ, p, and h are the ow density, ow static pressure and owenthalpy, respectively.

If thermodynamic equilibrium is assumed to exist on both sides of theoblique shock wave, then two thermodynamic properties are sufficient to fullydene a thermodynamic state, e.g., ρ = ρ ( p, T ) and h = h ( p, T ), where T isthe ow temperature. Consequently, under this assumption the above set of four conservation equations contains eight parameters, namely, pi , pj , T i , T j ,u i , uj , φj and θj . Thus, if four of these parameters are known, the above setof the conservation equations is solvable in principle.

1.3.1 Two-Shock Theory (2ST) for an Inviscid Flow

The two-shock theory (2ST) is the analytical model for describing the oweld near the reection point, R, of an RR. The wave conguration of an RRand some associated parameters are shown schematically in Fig. 1.10. The RRconsists of two discontinuities: the incident shock wave, i, and the reectedshock wave, r. These two shock waves intersect at the reection point, R,which is located on the reecting surface. Since the reection of shock wavesis not a linear phenomenon, the RR wave conguration is not linear either,i.e., ωi = ωr .


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If thermodynamic equilibrium is assumed in states (0), (1), and (2) then boththe density, ρ, and the enthalpy, h, could be expressed in terms of the pressure, p, and the temperature, T , [i.e., ρ = ρ( p, T ) and h = h( p, T )] and the aboveset of nine governing equations consists of only 13 parameters, namely: p0 , p1 , p2 , T 0 , T 1 , T 2 , u0 , u1 , u2 , φ1 , φ2 , θ1 and θ2 . Consequently, four of these 13parameters must be known in order to have a closed set, which, in principle,could be solved.

Henderson (1982) showed, that if the gas is assumed to obey the equationof state of a perfect gas, p = ρRT, and to be thermally perfect, h = C P T ,then (1.5)–(1.13) could be combined to a single polynomial of the order six.Although a polynomial of order six yields six roots, Henderson (1982) showedthat using simple physical considerations four of the six roots could be dis-

carded. This nding implies that equations (1.5) to (1.13) do not result ina unique solution for a given set of initial conditions. This will be furtherillustrated and discussed in Sect.1.4.1.

1.3.2 Three-Shock Theory (3ST) for an Inviscid Flow

The three-shock theory is the analytical model for describing the ow eldnear the triple point of an MR. The wave conguration and some associatedparameters of an MR are shown schematically in Fig. 1.11. The MR consistsof four discontinuities: three shock waves (the incident shock wave, i, thereected shock wave, r, and the Mach stem, m) and one slipstream, s. Thesefour discontinuities meet at a single point, known as the triple point, T, whichis located above the reecting surface. The Mach stem is usually curved alongits entire length although its curvature could be very small. Depending uponthe initial conditions it can be either concave or convex. At its foot, i.e., atthe reection point, R, it is perpendicular to the reecting surface.

(1)

(2)

(3)

(0)

R

m

i

T S

rφ1

φ3

φ2

θ1

θ2

θ3

Fig. 1.11. Schematic illustration of the wave conguration of a Mach reection –MR


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1.3 Analytical Approaches for Describing Regular and Mach Reections 17

Applying the oblique shock wave equations, given in Sect. 1.3, on the threeoblique shock waves, i, r and m, that are associated with the wave congura-tion of an MR, results in the following set of governing equations for an MRin an inviscid ow.

Across the incident shock wave, i:

ρ0u0 sin φ1 = ρ1u1 sin (φ1 − θ1) (1.14) p0 + ρ0u20 sin

2 φ1 = p1 + ρ1u21 sin2 (φ1 − θ1) (1.15)

ρ0 tan φ1 = ρ1 tan ( φ1 − θ1) (1.16)

h0 + 12

u20 sin2 φ1 = h 1 +

12

u21 sin2 (φ1 − θ1) (1.17)

Across the reected shock wave, r:

ρ1u1 sin φ2 = ρ2u2 sin (φ2 − θ2) (1.18) p1 + ρ1u21 sin

2 φ2 = p2 + ρ2u22 sin2 (φ2 − θ2) (1.19)

ρ1 tan φ2 = ρ2 tan ( φ2 − θ2) (1.20)

h1 + 12

u21 sin2 φ2 = h 2 +

12

u22 sin2 (φ2 − θ2) (1.21)

Across the Mach stem, m:ρ0u0 sin φ3 = ρ3u3 sin (φ3 − θ3) (1.22) p0 + ρ0u20 sin

2 φ3 = p3 + ρ3u23 sin2 (φ3 − θ3) (1.23)

ρ0 tan φ3 = ρ3 tan ( φ3 − θ3) (1.24)

h0 + 12u20 sin

2 φ3 = h3 + 12u23 sin

2 (φ3 − θ3) . (1.25)

In addition to these 12 conservation equations, there are also two boundaryconditions, which arise from the fact that the ow states (2) and (3) areseparated by a contact surface across which the pressure remains constant,i.e.,

p2 = p3 (1.26)Furthermore, under the assumptions of an inviscid ow and an innitely thincontact surface the streamlines on both sides of the contact surface are par-allel. This implies that:

θ1 ∓ θ2 = θ3 (1.27)Equation (1.27) gives rise to two possible three-shock theories:

θ1 − θ2 = θ3 (1.28a)

A three-shock theory fullling the requirement given by (1.28a) will be referredto in the followings as the “standard” three-shock theory , as opposed to a“nonstandard” three-shock theory , which fullls the condition:

θ1 + θ2 = θ3 . (1.28b)


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As will be shown subsequently, the solution of the standard three-shocktheory yields an MR, while the solution of the nonstandard three-shock theoryyields a vNR.

Thus, the three-shock theory (either the standard or the nonstandard),which describes the ow eld near the triple point, T, consists of 14 governingequations. Again, if thermodynamic equilibrium is assumed in states (0), (1),(2) and (3) then the set of 14 governing equations contains 18 parameters,namely: p0 , p1 , p2 , p3 , T 0 , T 1 , T 2 , T 3 , u0 , u1 , u2 , u3 , φ1 , φ2 , φ3 , θ1 , θ2 and θ3 .Consequently, four of these 18 parameters must be known in order to have aclosed set of equations, which, in principle, could be solved.

Henderson (1982) showed that if the gas is assumed to obey the equationof state of a perfect gas, p = ρRT, and to be thermally perfect, h = C P T , then

(1.14) to (1.27) could be reduced to a single polynomial of order ten, with thepressure ratio p3 /p 0 as the polynomial variable. The polynomial coefficientswere taken to be a function of the specic heat capacities ratio, γ = C P /C V ,the ow Mach number in state (0), M 0 , and the incident shock wave strengthin terms of the pressure ratio across it, p1 /p 0 . Although a polynomial of degreeten yields ten roots, Henderson (1982) showed that seven out of the ten rootscould be discarded by using simple physical considerations and the possibilityof double roots. This implies that (1.14)–(1.27) do not yield a unique solutionfor a given set of initial conditions. This is further illustrated and discussedin Sect. 1.4.2.

1.4 Shock Polars

Kawamura & Saito (1956) were the rst to suggest that owing to the factthat the boundary conditions of an RR (1.13) and an MR (1.26 and 1.27) areexpressed in terms of the ow deection angles, θ and the ow static pressures, p, the use of ( p, θ)-polars could be of great advantage in better understandingthe shock wave reection phenomenon.

The graphical presentation of the relationship between the pressure, pj ,obtained behind an oblique shock wave (see Fig. 1.9) and the angle, θj , bywhich the ow is deected while passing through an oblique shock wave, fora xed ow Mach number, M i , and different angles of incidence, φj , is knownas the a pressure-deection shock polar. A typical pressure-deection shockpolar is shown in Fig. 1.12. Four special points are indicated on the shown

shock polar:– Point “a” illustrates a situation in which the ow state behind the oblique

shock wave is identical to the ow state ahead of it. This situation isobtained when the angle of incidence between the oblique shock wave andthe oncoming ow, φ j, is equal to the Mach angle µi = sin − 1 (1/M i ).In this case the pressure does not change across the oblique shock wave, pj /p i = 1, and the ow deection is zero, θj = 0.


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Fig. 1.15. pi /p 0 , θRi -polar solution of a regular reection

(2), which is obtained from state (1) by passing through the reected shockwave, is on the R-polar. The boundary condition for an RR (1.13) impliesthat θR2 = 0, therefore, state (2) is obtained at the point where the R-polarintersects the p-axis, i.e., the line along which θR = 0.

Figure 1.15 implies that two different points, (2 w ) and (2 s), fulll the just-mentioned requirement. Each one of these two points indicates a possiblesolution of the governing equations of an RR [(1.5)–(1.13)]. Point (2 w ) isknown as the “weak-shock solution” and point (2 s) is known as the “strong-shock solution.” Note that none of these two solutions could be discarded ontheoretical grounds. However, it is an experimental fact that, unless specialmeasures are taken, the weak-shock solution is the one that usually occurs.Consequently, the ow state behind the reected shock wave is representedby point (2 w ) of Fig. 1.15. In the following this state will be labeled as (2)only. Note that the just-described situation in which the graphical solution of the governing equations of an RR, using pi /p 0 , θRi -polar, implies that thereare two possible solutions of an RR for a given set of initial conditions wasalready mentioned in Sect. 1.3.1.

1.4.2 Shock-Polar Presentation of the Flow Field Near the TriplePoint of a Mach Reection

Figure 1.16 presents the pi /p 0 , θTi -polar solution of the ow eld in thevicinity of the triple point, T, of an MR. The ow deection angles, θTi , aremeasured with respect to the direction of the oncoming ow when the frameof reference is attached to the triple point, T. State (0) at which pi = p0 ,i.e., pi /p 0 = 1, and θTi = θT0 = 0, is at the origin. The locus of all the owstates that could be obtained from state (0) by passing through any oblique


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Fig. 1.17. Three different possible MR-solutions resulting in a DiMR at point “a”,an StMR at point “b” and an InMR at point “c”

congurations as: Direct-Mach reection (DiMR) at point “a”, Stationary-Mach reection (StMR) at point “b” and Inverse-Mach reection (InMR) atpoint “c.”

Note that the (I–R II )-polars combination (in Fig. 1.17) indicates, in addi-tion to the StMR at point “b”, also a possible RR-solution at this point sincethe R II -polar intersects the p-axis at this point. Similarly, the I–R III polarscombination indicates, in addition to the InMR at point “c”, a possible RR-solution at point “d” where the R

III-polar intersects the p-axis. Thus, it is

again evident that based on the graphical solution of the governing equationsof an MR, using pi /p 0 , θTi -polar, different reection congurations can betheoretically obtained for the same initial conditions.


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1.5 Suggested RR IR Transition Criteria 27

P i P 0 P i P 0

P i P 0

3 3

2

1

m

2

0 5 10 15 0 5 10 151

3

m

2

1

R R

R

(1)(1)

(1)

II

I

(0) (0)

(0)

(2),(3)

(2),(3)

(2),(3)

(a) (b)

0 5 10 15(c)

θiT θiT

θiT

Fig. 1.19. (I–R)-polars presentation of possible solution of the three-shock theoryfor perfect nitrogen ( γ = 1 .4): (a ) θT2 = θT3 < θ T1 (M 0 = 1 .6, φ1 = 47 .88

◦ ); (b ) θT2 =θT

3 > θ T

1 (M 0 = 1 .5, φ1 = 49 .67◦ ); and ( c ) θT

2 = θT

3 = θT

1 (M 0 = 1 .55, φ1 = 41 .50◦ )

in which θT2 = θT3 < θ T1 . This situation, which implies that θ1 − θ2 = θ3(1.28a), i.e., a “standard” solution of the three-shock theory (see Sect.1.3.2),results in an MR. The (I–R)-polars combination shown in Fig. 1.19b illus-trates a different solution. It is seen that the ow that is deected towards thewedge surface while passing through the incident shock wave is not deected


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away from the wedge when in passes through the reected shock wave butit is further deected towards the wedge to result in a situation in whichθT2 = θT3 > θ T1 . This situation, which implies that θ1 + θ2 = θ3 (1.28b), i.e., a“nonstandard” solution of the three-shock theory (see Sect. 1.3.2), results ina vNR. The limiting (I–R)-polars combination between these two solutions isshown in Fig.1.19c that indicates that the ow passing through the reectedshock wave is not deected at all, i.e., θ2 = 0 and hence θT2 = θT3 = θT1 . Theboundary condition of the three-shock theory for this case is simply θ1 = θ3 .

Schematic drawings of the three possible shock wave congurationsthat correspond to the three (I–R)-polars combinations that are shown inFig.1.19a–c is given in Fig.1.20a–c, respectively. It should be noted here that

(0)(0)

(1)

(2)

(2)

(3)

(1)

(2)

(3)

(0)

(1)

(3)

i

T

m

s

r

i

i

r

T

T

m

m

s

s

r

(a) (b)

(c)

Fig. 1.20. The wave congurations of the three possible solutions of the three-shocktheory whose graphical solutions are shown in Fig.1.19a–c, respectively


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Fig. 1.21. (I–R)-polars combination illustrating the mechanical-equilibriumcriterion

these additional waves has ever been observed experimentally, Henderson &Lozzi (1975) concluded that the detachment criterion is not physical. Alter-natively, they suggested a transition, which corresponds to the polars combi-nation shown in Fig.1.21.

In this (I–R)-polars combination the R-polar intersects the p-axis exactlyat the normal shock point of the I-polar. Consequently, both an RR and anMR are theoretically possible at the intersection point. Hence, if this pointis indeed the RR IR transition point, then from the pressures point of view, the transition would be continuous and mechanical equilibrium wouldbe maintained during the transition. The mechanical equilibrium transitionline can be obtained by solving equations (1.14)–(1.28a) and requiring thatθ1 − θ2 = θ3 = 0.

1.5.3 Sonic Criterion

This transition criterion, which was also rst introduced, as a possible tran-sition criterion, by von Neumann (1943), is based on the argument that theRR IR transition depends on whether the corner-generated signals can

catch-up with the reection point, R, of the RR. Hence, as long as the owMach number behind the reected shock wave is supersonic, the reectionpoint is isolated from the corner-generated signals, and they cannot reach it.

Consider Fig. 1.22a, b where two different (I–R)-polars combinations areshown. While in Fig.1.22a, the R-polar intersects the p-axis along its “weak”portion in Fig. 1.22b the R-polar intersects the p-axis along its “strong” por-tion. Thus, while the ow behind the reected shock wave is supersonic for theformer case, it is subsonic for the latter. The limit between these two cases is


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P i P 0 P i P 0

P i P 0

6

4

2

1

6

2

1

6

3

1

0 10 20 0 10 20

0 10 20

ms s

s

(2)

R I

(1)

R I

R

I

(1)

(1)

m

m

(2)

(2)

(a) (b)

(c)

(0)

θi θi

θi

Fig. 1.22. pi /p 0 , θTi -polar solutions of three different RRs for perfect nitrogen:(a ) supersonic ow behind the reected shock wave ( M 0 = 2, φ1 = 40 .41◦ , θW =49.59◦ and M S = 1 .3); (b ) subsonic ow behind the reected shock wave ( M 0 = 2,φ1 = 42 .54◦ , θW = 47 .46◦ and M s = 1 .35); (c ) sonic ow behind the reected shockwave

shown in Fig.1.22c where the R-polar intersects the p-axis exactly at its sonicpoint, s. This (I–R)-polars combination is appropriate to the sonic criterion,or the catch-up condition, since this is the limit for which the corner-generatedsignals can catch-up with the reection point, R, of the regular reection. Thetransition line arising from the sonic criterion can be calculated by solving thegoverning equations of the two-shock theory, i.e., (1.5)–(1.13), and replacingθ2 by θ2s .

It is worthwhile noting that since the sonic and the detachment points arevery close to each other, the sonic criterion results in transition conditions thatare very close to those of the detachment criterion. In many cases the differencebetween them in terms of the value of the reecting wedge angle is only a


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fraction of a degree. For this reason, it is almost impossible to distinguishexperimentally between the sonic and detachment criteria.

Lock & Dewey (1989) developed an ingenious experimental set-up by whichthey were able to experimentally distinguish between the “sonic” and the“detachment” criteria. Their experimental investigation led to the conclusionthat, in pseudosteady ows, the RR IR transition occurs when the corner-generated signals manage to catch-up with the reection point, R, i.e., at thesonic condition rather than the detachment one.

1.5.4 Length-Scale Criterion

The length-scale criterion was introduced by Hornung et al. (1979). The phys-

ical reasoning of this criterion is based on their argument that, unlike the waveconguration of an RR that is not associated with any length scale, since boththe incident and reected shock waves extend to innity (see Fig. 1.1), thewave conguration of an MR inherently includes a length scale, namely thenite length of the Mach stem that extends from the reection point, R, onthe reecting surface to the triple point, T (see Fig. 1.2). Thus they arguedthat in order for an MR to be formed, i.e., in order for a shock wave with anite length to exist, a physical length scale must be available at the reectionpoint, namely, pressure signals must be communicated to the reection pointof the RR. This argument eventually led them to conclude that there are twodifferent conditions for the termination of the RR depending on whether theow under consideration is steady or pseudosteady.

Consider the pseudosteady RR in Fig. 1.23a and note that the length of the reecting surface, w , can be communicated to the reection point, R,only if a subsonic ow is established between points Q and R (in a frame of reference attached to R). This requirement corresponds to the polars com-bination shown in Fig.1.22c, which, as discussed earlier, corresponds also tothe sonic criterion. In a steady ow (Fig. 1.23b) the length, w , of the wedgethat is used to generate the incident shock wave can be communicated to thereection point, R, only if a propagation path exists between points Q and Rvia the expansion wave at point Q . This is possible only if the ow betweenpoints R and Q is subsonic. According to Hornung et al. (1979) this couldhappen if an MR existed, since the ow behind the Mach stem of an MR isalways subsonic. Consequently, they argued that the RR → MR transitiontakes place the very rst time the MR becomes theoretically possible. This

requirement corresponds to the (I–R)-polars combination shown in Fig.1.21,which, as discussed earlier, corresponds also to the mechanical-equilibriumcriterion.

Thus, the length-scale concept of Hornung et al. (1979) led to two differ-ent transition lines. In steady ows it predicts transition at the point pre-dicted by the mechanical-equilibrium criterion, θ1 − θ2 = θ3 = 0, and inpseudosteady ows it predicts transition at the point predicted by the sonic


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i

i

(0)(1)

(2)

(2)

(1) (0)

R Q

Q

Q’ R

r

rl w

lw

(a) (b)

Fig. 1.23. Denition of the physical length, w , which should be communicated tothe reection point, R, in order to enable the RR → MR transition: ( a ) pseudosteadyows; and ( b ) steady ows

criterion, θ1 − θ2s = 0, which is practically identical to the detachment crite-rion, θ1 − θ2m = 0.

1.5.5 Summary, Critique, and Discussion

The four foregoing suggested transition criteria yield three different RR IRtransition lines, which can be calculated in the following manner:

– The transition line arising from the detachment criterion is calculatedusing the two-shock theory while requiring that

θ2 = θ2m . (1.30)

– The transition line arising from the sonic criterion is calculated using thetwo-shock theory while requiring that

θ2 = θ2s (1.31)

– The transition line arising from the mechanical-equilibrium criterion iscalculated using the three-shock theory while requiring that

θ1 − θ2 = θ3 = 0 . (1.32)

Recall that the transition lines arising from the length-scale concept are givenby (1.31) for pseudosteady ows and by (1.32) for steady ows. It should alsobe mentioned that the transition lines as calculated by (1.30) and (1.31) arepractically identical.

Figure 1.24 illustrates three different (I–R)-polars combinations. The(I–R i)-polars combination corresponds to the mechanical-equilibrium con-dition; the (I–R iii )-polars combination corresponds to the detachment/soniccondition; and the (I–R ii )-polars combination corresponds to an intermediate


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θiT

P i P 0

(2)

(2')

(2),(3)(2),(3)

(2),(3)

9

6

3

10 10 20 30

I

(1)

(0)

(1)(1)R i

RiiR iii

Fig. 1.24. Various (I–R)-polars combinations: the (I–R i )-polars combination cor-responds to the mechanical-equilibrium criterion; the (I–R iii )-polars combinationcorresponds to the detachment/sonic criterion; the (I–R ii )-polars combination cor-responds to an intermediate situation

situation. For the latter polars combination the mechanical-equilibrium cri-terion predicts an MR at the point where the R ii -polar intersects the I-polar[points (2) and (3)] while the detachment criterion predicts an RR at thepoint where the R ii -polar intersects the p-axis [point (2 )]. For all the R-polarsbetween the R i- and the R iii -polars, two solutions, RR or MR, are theoreticallypossible.

Figure 1.25 illustrates the size of the dual-solution region in the M S , θCw -plane, where θCw is the complementary angle of φ1 , i.e., θCw = 90 ◦ − φ1 . It is seenclearly that the area of disagreement between the mechanical-equilibrium andthe detachment criteria is very large. Note that if the transition line arisingfrom the sonic criterion had been added to Fig. 1.25 it would have laid slightlyabove the detachment transition line.

Although Henderson & Lozzi (1975) reported that excellent agreement was

obtained between the mechanical-equilibrium criterion and their experimentsin steady ows, i.e., wind tunnel experiments, there are unfortunately somedifficulties associated with the physical concept upon which this criterion isbased.

First, as can be seen in Fig.1.25, the mechanical-equilibrium criterion doesnot apply over the entire range of incident shock wave Mach numbers, M S .It exists only for values of M S larger than the value where the mechanicalequilibrium transition line emanates from the detachment transition line. This


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θ CW RR

PR or MRθ1−θ2 = θ3 =0

θ2 = θ2m

80

60

40

20

01 4 7 10

MS

M e c h a n i c a l e q u i l i b r i u m c r i t e r i o nf o r R R t e r m i n a t i o n ,

D e t a c h m e n t c r i t e r i o n f o r R Rt e r m i n a t i o n

Fig. 1.25. Domains of RR and MR in the M S , θCW -plane as dened by the

mechanical-equilibrium and the detachment criteria. θC

W = 90◦

− φ1

in turn implies that since M S = M 0 sin φ1 there are combinations of M 0 andφ1 for which the condition given by (1.32) cannot be met.

Second, in their experiments in pseudosteady ows, e.g., shock tube exper-iments, they observed that RR wave congurations persisted not only insidethe dual-solution region shown in Fig.1.25 but also slightly below the detach-ment transition line, where RR is theoretically impossible. In the weak-shockwave domain the persistence was up to 5 ◦ while in the strong-shock wavedomain RR prevailed to about 2 ◦ below its theoretical limit. Other inves-tigators who also studied experimental, the RR IR transition obtainedsimilar results. Henderson & Lozzi (1975) attempted to resolve this anom-

aly by advancing a hypothesis that the RR wave congurations that wereobserved beyond the limit predicted by the mechanical-equilibrium criterionwere, in fact, undeveloped MR wave congurations in which the Mach stem,the slipstream and the triple point were too close together and too small tobe resolved as is the case in a well developed MR wave conguration. How-ever, in pseudosteady ows, the shock wave conguration grows with time.Thus the triple point should eventually show up if a long enough reecting


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wedge is used. This, unfortunately, did not occur even in experiments wherethe reecting surface was very long.

Finally, Henderson and Lozzi’s requirement of mechanical equilibrium isnot justied when the ow under consideration is either steady or pseudo-steady, since for these cases, depending upon the initial conditions, eitheran RR or an IR wave conguration is established, and the requirement of acontinuous pressure change during transition is unnecessary since transitiondoes not take place at all. If, however, the ow under consideration is unsteady,and the reection actually goes through a transition from RR to IR or from IRto RR, then their argument could apply. However, as will be shown in Chap. 4,in the case of unsteady ows, the additional waves required by Henderson &Lozzi (1975) to be associated with a transition at detachment that arises from

the sudden pressure drop do indeed appear in the ow eld (see e.g., Fig.1.4in which a normal shock wave follows the RR that was obtained when theInMR was terminated and transitioned to a TRR).

In summary, experimental results in both steady and unsteady (includingpseudosteady) ows have suggested that in steady ows the RR IR transi-tion generally agrees with the condition given by (1.32), while in pseudosteadyand unsteady ows the RR IR transition seems to agree with the conditionsgiven by either (1.30) or (1.31). Thus it can be concluded that the length-scaleconcept of Hornung et al. (1979) most likely leads to the adequate criterionfor the RR IR transition because it results in the correct transition linesin steady, pseudosteady, and unsteady ows.

As will be shown subsequently, the agreement between this transition cri-terion and careful experimental investigation was never satisfactory enough inthe close vicinity of the transition lines. This fact has been motivating inves-tigators to continue searching for the “correct” RR IR transition criterion.However, one must recall that the transition criteria are based on the two-and three-shock theories which were developed under the assumption that allthe discontinuities are straight in the vicinity of their intersection points andhence that the various ow states bounded by them are uniform. In addi-tion to this assumption, which introduces inherent errors into the transitionlines that are calculated based on the two- and three-shock theories, it will beshown subsequently that the inclusion of viscous effects and real gas effectsdoes improve the agreement between the experimental results and predictionsbased on these two fundamental theories.

References

Ben-Dor, G., “Relations between rst and second triple point trajectory anglesin double Mach reection”, AIAA J., 19, 531–533, 1981.

Ben-Dor, G. & Takayama, K., “The dynamics of the transition from Mach toregular reection over concave cylinders”, Israel J. Tech., 23, 71–74, 1986/7.


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References 37

Colella, P. & Henderson, L.F., “The von Neumann paradox for the diffractionof weak shock waves”, J. Fluid Mech., 213, 71–94, 1990.

Courant, R. & Freidrichs, K.O., Hypersonic Flow and Shock Waves , WileyInterscience, New York, N.Y., USA, 1948.

Guderley, K.G., “Considerations on the structure of mixed subsonic-supersonic ow patterns”, Tech. Rep. F-TR-2168-ND, Wright Field, USA,1947.

Henderson, L.F., “Exact Expressions for Shock Reection Transition Criteriain a Perfect Gas”, ZAMM, 62, 258–261, 1982.

Henderson, L.F. & Lozzi, A., “Experiments on transition of Mach reection”,J. Fluid Mech., 68, 139–155, 1975.

Hornung, H.G., Oertel, H. Jr. & Sandeman, R.J., “Transition to Mach reec-

tion of shock waves in steady and pseudo-steady ows with and withoutrelaxation”, J. Fluid Mech., 90, 541–560, 1979.Kawamura, R. & Saito, H., “Reection of shock waves-1. Pseudo-stationary

case”, J. Phys. Soc. Japan, 11, 584–592, 1956.Krehl, P. & van der Geest, “The discovery of the Mach reection effect and

its demonstration in an auditorium”, Shock Waves, 1, 3–15, 1991.Lee, J.-H. & Glass, I.I., “Pseudo-stationary oblique-shock wave reections in

frozen and equilibrium air”, Prog. Aerospace Sci., 21, 33–80, 1984.Li, H. & Ben-Dor, G., “Reconsideration of pseudo-steady shock wave reec-

tions and the transition criteria between them”, Shock Waves, 5(1/2),59–73, 1995.

Liepmann, H.W. & Roshko, A., Elements of Gasdynamics , John Wiley &Sons, New York, N.Y., USA., 1957.

Lock, G. & Dewey, J.M., “An experimental investigation of the sonic criterionfor transition from regular to Mach reection of weak shock waves”, Exp.Fluids, 7, 289–292, 1989.

Mach, E., “Uber den verlauf von funkenwellen in der ebene und im raume”,Sitzungsbr. Akad. Wiss. Wien, 78, 819–838, 1878.

Neumann, J. von, “Oblique reection of shocks”, Explos. Res. Rep. 12, NavyDept., Bureau of Ordinance, Washington, DC, USA., 1943a.

Neumann, J. von, “Refraction, intersection and reection of shock waves”,NAVORD Rep. 203-45, Navy Dept., Bureau of Ordinance, Washington,DC, U.S.A., 1943b.

Olim, M. & Dewey, J.M., “A revised three-shock solution for the Mach reec-tion of weak shock waves”, Shock Waves, 2, 167–176, 1992.

Reichenbach, H., “Contribution of Ernst Mach to uid dynamics”, Ann. Rev.Fluid Mech., 15, 1–28, 1983.Skews, B. & Ashworth J.T., “The physical nature of weak shock wave reec-

tion”, J. Fluid Mech., 542, 105–114, 2005.Smith, L.G., “Photographic investigation of the reection of plane shocks in

air”, OSRD Rep. 6271, Off. Sci. Res. Dev., Washington, DC., USA., orNDRC Rep. A-350, 1945.


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40 2 Shock Wave Reections in Steady Flows

L Length of the reecting wedgeL Width (see Fig.2.41)M Average ow Mach number behind a curved Mach stemM 0C Critical ow Mach number below which the mechanical

equilibrium does not existM i Flow Mach number in state ( i)M f Flight Mach numberM tr Flight Mach number at which a transition takes place pi Static pressure in state ( i) pw Wake pressure behind the tail of the reecting wedgeR Specic gas constantR Reection coefficient (see (2.42))R

I Intensity reection coefficient (see (2.43))

S Distance (see Fig.2.41)t TimeT i Temperature in state ( i)u i Flow velocity in state ( i)u Average ow velocity behind a curved Mach stemw Length of the reecting wedgex CoordinateX Nondimensional horizontal distance (= S/L )y CoordinateZ i Acoustic impedance at state ( i)

Greek Letters

α Flow direction relative to a horizontal directionβ i Incident shock wave angleβ r Reected shock wave angleβ Di Incident shock wave angle at the detachment conditionβ Ni Incident shock wave angle at the von Neumann conditionβ Si Limiting incident shock wave angle for a stable RRδ max (M i) Maximum deection angle for a ow having Mach number

M i through an oblique shock waveφ Angle of incidenceφi Angle of incidence between the ow and the oblique shock

wave across which the ow enters state ( i)φ∗ Limiting angle of incidence (see (2.1))γ Specic heat capacities ratioµ Dynamic viscosityµi Mach angle of the ow having a Mach number M iν (M ) Prandtl–Meyer functionθi Angle of deection of the ow while passing across an

oblique shock wave and entering into state ( i)θw Reecting wedge angleθCw Complementary wedge angle


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2 Shock Wave Reections in Steady Flows 41

θDw Reecting wedge angle at the detachment conditionθEw Reecting wedge angle at the condition analogous

to detachment condition in the reection of asymmetric shock waves

θNw Reecting wedge angle at the von Neumanncondition

θTw Reecting wedge angle at the condition analogousto von Neumann condition in the reection of asymmetric shock waves

ρi Flow density in state ( i)ρ̄ Average ow density behind a curved Mach stemτ Nondimensional time

Subscripts G Foot of the Mach stemT Triple point0 Flow state ahead of the incident shock wave, i1 Flow state behind the incident shock wave, i2 Flow state behind the reected shock wave, r3 Flow state behind the Mach stem, mSuperscripts D At detachment criterionN At von Neumann criterionAbbreviations Waves and Pointsi Incident shock wavem Mach stemr Reected shock waves SlipstreamT Triple pointR Reection pointWave CongurationIR Irregular reectionRR Regular reectionsRR Strong regular reection (an RR with a strong

reected shock wave)wRR Weak regular reection (an RR with a weak

reected shock wave)MR Mach reectionNR No reectionOverall Wave CongurationoMR Overall Mach reectionoMR[DiMR + DiMR] An oMR that consists of two DiMRsoMR[DiMR + StMR] An oMR that consists of one DiMR and one StMRoMR[DiMR + InMR] An oMR that consists of one DiMR and one InMR


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oRR Overall regular reectionoRR[wRR + sRR] An oRR that consists of one wRR and one sRRoRR[wRR + wRR] An oRR that consists of two wRRs

Types of Mach ReectionDiMR Direct-Mach reectionInMR Inverse-Mach reectionStMR Stationary-Mach reection

As mentioned in Chap. 1 only regular reection (RR) and Mach reections(MR) are possible in steady ows. Hence, the reection phenomenon in steadyows is less complicated than in pseudosteady or unsteady ows, and itsanalytical investigation is much simpler.

Unfortunately, in spite of this obvious advantage, not too many exper-imental studies on the reection of shock waves in steady ows have beenreported thus far. Furthermore, most of the available basic experimentaldata (excluding the new data regarding the recently discovered hysteresisphenomena) were obtained more than three decades ago, with experimen-tal equipment and diagnostic technique less accurate than those existingnowadays.

2.1 Categories of Steady Reection Phenomena

The shock wave reection phenomenon in steady ows could be divided, in

general, into four different categories:– Reection of a curved incident shock wave from a straight surface– Reection of a straight incident shock wave from a curved surface– Reection of a curved incident shock wave from a curved surface– Reection of a straight incident shock wave from a straight surface

2.1.1 Curved Incident Shock Wave Reections over StraightReecting Surfaces

If a supersonic ow, M 0 > 1, encounters a concave or a convex reectingwedge, then the shock wave which results in, to enable the ow to negotiatethe wedge, is also concave or convex. The regular reections of the incident

shock wave for these two possibilities are shown schematically in Fig.2.1a, b,respectively. The intermediate case of a straight reecting wedge, of course,results in a straight attached oblique shock wave provided that the reectingwedge angle is smaller than the maximum ow deection appropriate to M 0 ,as shown in Fig.1.6b [i.e., θw < δ max (M 0)]. If, however, the reecting wedgeangle is greater than the maximum ow deection, as shown in Fig. 1.6c, thenthe straight reecting wedge results in a detached bow shock wave whichresults in a situation similar to that shown in Fig.2.1b.


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2.1 Categories of Steady Reection Phenomena 43

M 0>1 M 0>1(1)

(1)

i

irr

(2) (0) (0)(2)

(a) (b)

φ

R R

Fig. 2.1. Schematic illustrations of the RR wave congurations of a curved incidentshock wave reection over a straight surface: ( a ) concave incident shock wave; ( b )convex incident shock wave

Pant (1971), who analytically studied the reection of steady curved shockwaves, showed that for weak incident shock waves there is a wave angle, φ,(see Fig. 2.1b) for which the reected shock wave is straight. This specicwave angle, φ∗, which was found to be independent of the incident ow Machnumber, M 0 , could be obtained from:

φ∗ = cos − 112 γ + 1 (2.1)

Thus, in the regular reection of weak shock waves of all strengths for φ < φ∗the incident and the reected shock waves have curvatures of opposite sign.As the wave angle in the vicinity of the reection point approaches φ∗ thereected shock wave straightens out until it becomes straight at φ = φ∗. Forφ > φ∗ the curvatures of the incident and the reected shock waves have thesame sign.

Molder (1971) numerically investigated this type of steady ow reection.In the case of an RR a zero downstream curvature on the streamline behindthe reected shock wave near the reection point, R, was imposed, and inthe case of an MR the pressure gradients and curvatures of the streamlinesalong the slipstream, in the vicinity of the triple point, T, were matched.Molder’s (1971) results showed many possible combinations of reected-shockcurvatures, streamline-curvatures and pressure gradients.

In addition Molder (1971) presented both theoretical arguments andexperimental evidence that the RR MR transition occurs when the Machstem is normal to the incident ow, i.e., at the point predicted by the lengthscale criterion θ1 − θ2 = θ3 = 0̃.

Although only RR wave congurations are shown in Fig. 2.1a, b, MR wavecongurations are also possible in this steady ow reection category.

2.1.2 Straight Incident Shock Wave Reections over CurvedReecting Surfaces

Two general cases, which belong to this category of RRs in steady ows, areshown schematically in Fig.2.2a, b. The incident shock waves are straight


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M 0M 0

(1) (1)

i i

r r

(2) (0) (0)(2)

(a) (b)

R R

Fig. 2.2. Schematic illustrations of the RR wave congurations of a straight incidentshock wave reecting over a curved surface: ( a ) convex surface; ( b ) concave surface

and the reecting surfaces are straight up to the reection point, R, afterwhich they are either concave or convex. Depending upon the curvature of thereecting surface downstream of the reection point, R, a concave or a convexreected shock wave is obtained. The curvature of the reected shock has thesame sign as the curvature of the reecting surface as shown schematically inFig. 2.2a, b.

Although only RR wave congurations are shown in Fig. 2.2a, b, MR wavecongurations are also possible in this steady ow reection category.

2.1.3 Curved Incident Shock Wave Reections over CurvedReecting Surfaces

Four general cases, which belong to this category of shock wave reections insteady ows, are shown schematically in Fig. 2.3a–d. The incident shock wavein each of these cases is curved and the reecting surface is straight up to thereection point, R, beyond which it is either concave or convex. The reectedshock waves assume a curvature with the same sign as the curvature of thereecting surface, as is shown schematically in Fig.2.3a–d.

It is obvious that there should be conditions in this steady-ow reectioncategory, for which MR wave congurations are obtained rather than the RRwave congurations that are shown in Fig.2.3a–d.

2.1.4 Straight Incident Shock Wave Reections over StraightReecting Surfaces

This category of shock wave reections in steady ows is undoubtedly theeasiest one to treat analytically as the incident shock wave, the reected shockwave and the reecting surface are all straight. Most of the reported analyticaland experimental studies on the reection of shock waves in steady ows fallinto this steady ow reection category.


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surface, i.e., θ1 = θw . Since the just-mentioned oblique shock wave arises fromthe weak solution (see Sect. 1.2), the ow behind the incident shock wave, i, issupersonic. The deected ow obliquely approaches the bottom surface withan angle θw (see Fig.2.4a). If θw is smaller than the maximum deection angleappropriate to the ow Mach number, M 1 , in state (1), i.e., if θw < δ max (M 1),then an RR as shown in Fig. 2.4a could be obtained. If, however, θw is greaterthan the maximum deection angle of the ow Mach number, M 1 , in state (1),i.e., if θw > δ max (M 1), then an RR is impossible, and the resulted reectionis an MR as shown in Fig. 2.4b.

RR MR Transition Criterion

Hornung & Robinson (1982) showed that the RR MR transition is the onearising from the length scale criterion (see Sect. 1.5.4). In the case of steadyows, the length scale criterion results in two different transition formulaedepending upon whether the incident ow Mach number, M 0 , is smaller orgreater than a certain critical value, M 0C . This critical value is the smallestvalue of M 0 for which the condition imposed by the mechanical equilibrium(see (1.32)) could be satised. In fact this is the value of M 0 at which thetransition line appropriate to the mechanical equilibrium criterion emanatesfrom the transition line appropriate to the detachment (see Fig. 1.25). TheI–R shock polar combination at this critical value of M 0C is shown inFig.1.18c.

For values of M 0 > M 0C the length scale criterion yields a transition at

θ1 − θ2 = θ3 = 0 . (2.2)

This, incidentally, is identical to that predicted by the mechanical equilibriumcriterion (see Sect.1.5.2). For values of M 0 < M 0C , for which (2.2) cannot besatised, the length scale criterion predicts transition at the point where theow behind the reected shock wave is sonic, i.e.,

M 2 = 1 (2.3)

Equation (2.3) which, is also known as the sonic criterion (see Sect. 1.5.3),could be rewritten as:

θ1 = θ2s (2.4)

The transition lines which result from (2.2) and (2.3) or (2.4) join at the pointM 0 = M 0C . Note that the transition line arising from (2.2) is calculated bymeans of the three-shock theory while the transition line arising from (2.3) or(2.4) is calculated by means of the two-shock theory. The exact values of M 0Cfor diatomic ( γ = 7 / 5) and monatomic ( γ = 5 / 3) perfect gases were calculatedby Molder (1979); they are M 0C = 2 .202 and M 0C = 2.470, respectively.


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Fig. 2.6. Experimental measurements of the height of the Mach stem as a functionof the angle of incidence for different ow Mach numbers and the determination of the transition angle

The results of the experimental investigation of Hornung & Robinson(1982) are shown in Fig.2.6, where the nondimensional height of the Machstem, H m /L w , as a function of the angle of incidence, φ1 , for different owMach numbers, M 0 , is shown (Lw , a characteristic dimension of the reectingwedge, is dened in Fig. 1.23b). By extrapolating their experimental resultsto H m /L w = 0, they showed that the actual RR MR transition occurs ata value of φ1 appropriate to that obtained from (2.2). The analytical valuesfor the given incident ow Mach numbers are shown in Fig. 2.6 by arrow-heads.

In spite of the excellent agreement between the experiments and the theoryregarding the RR MR transition, one must recall that the two-shocktheory as applied to obtain the transition lines shown in Fig.2.5a, b assumesthat the uid is an ideal one, i.e., inviscid ( µ = 0) and thermally nonconduc-tive ( k = 0), here µ is the dynamic viscosity and k is the thermal conductivity.These, of course, are only simplifying assumptions as a real uid always hasa nite viscosity and thermal conductivity. Discussions on these effects aregiven in the following sections.

2.2 Modications of the Perfect InviscidTwo- and Three-Shock Theories

The two- and three-shock theories that were presented in Sects.1.3.1 and1.3.2 were developed using simplifying assumptions that are not fully justied,since the reection phenomenon in steady ows might be affected by nonidealeffects. The major assumptions, upon which the two- and three-shock theorieswere based, are:


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2.2 Modications of the Perfect Inviscid Two- and Three-Shock Theories 49

(1) The ow is steady.(2) The discontinuities at the reection point of the RR and the triple point

of the MR are straight. This in turns implies that the ow elds boundedby these discontinuities are uniform.

(3) The ow obeys the equation of state of a perfect gas (p = ρRT).(4) The ow is inviscid ( µ = 0).(5) The ow is thermally nonconductive (k = 0).(6) The contact discontinuity at the triple point of the MR is innitely thin,

i.e., it is a slipstream.

Beside the rst assumption, which for the case of steady shock wave reec-tions, is fullled by denition, the other assumptions could have a meaningfuleffect. Hence, in the following, the validity of these assumptions is discussedseparately.

2.2.1 Nonstraight Discontinuities

Based on experimental observations it is clear that not all the discontinuitiesof an MR are straight. In fact both the Mach stem and the slipstream arecurved. Whether their curvature is meaningful as they approach the triplepoint is an open question. If yes, then one could assume that the use of thethree-shock theory to calculate the ow eld near the triple point introducesan inherent error into the predicted results. Note that while in the case of a pseudosteady SMR (see Fig. 3.7), where only the incident shock wave isstraight, in the case of a steady MR both the incident and the reected shock

waves are straight. This could imply that predictions based on the three-shock theory should better agree with steady MR-congurations than withpseudosteady SMR-congurations.

2.2.2 Viscous Effects

The ow in state (0) develops a boundary layer along the reecting surface,and hence the incident shock wave, i, which emanates from the leading edgeof the reecting wedge (see Fig. 2.4a), interacts with this boundary layer toresult in a relatively complex structure near the reection point on the reect-ing surface. The interaction with the boundary layer depends on whether theboundary layer is laminar or turbulent as shown in Fig.2.7, where the inter-

action of the incident shock wave with the boundary layer near the reectionpoint, R, of an RR is shown schematically. Figure 2.7 reveals that if one is tosolve accurately the ow eld near the reection point, R, of an RR then avery complex ow eld must be dealt with.

Henderson (1967) analytically investigated the reection of a shock wavefrom a rigid wall in the presence of a boundary layer by treating the prob-lem not as a reection but as a refraction process. He found that a Machstem was always present and that the bottom of this wave was bifurcated


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Expansion Wave

TurbulentBoundary Layer



TurbulentBoundary Layer Turbulent

Boundary Layer

LaminarBoundary Layer

M0 M0

M0

r r

r

i

i

r i

(a) (b)

(c)

Separated Zone

Separated Zone

Fig. 2.7. Schematic illustration of the way by which the boundary layer over thereecting surface could affect the wave structure of an RR near the reection point

(a lambda foot). The reection was said to be regular (RR) if the Mach stemand the lambda foot were conned to the boundary layer and irregular (IR) if either the Mach stem or the lambda foot extended into the main stream. Two

types of regular reection were found, one that had a reected compressionwave and the other that had both reected compression waves and expan-sion waves. Henderson (1967) presented initial conditions that enable oneto decide which type of reection would appear. Henderson (1967) reportedfurthermore, that there were two types of IR, one that had a Mach stempresent in the main stream and the other that was characterized by a four-wave conguration. There were also two processes by which the RR becameIR. One was due to the formation of a downstream shock wave that subse-quently swept upstream to establish the irregular system and the other wasdue to boundary layer separation, which forced the lambda foot into the mainstream.

There is, however, a possibility by which the above-mentioned viscouseffects could be eliminated in steady ow reections. By using a relativelysimple experimental set-up the above-illustrated interaction with the bound-ary layer developing over the reecting surface could be avoided. This is shownin Fig. 2.8 where two identical reecting wedges are placed in such a way thatthey produce two symmetrical regular (Fig. 2.8a) and Mach (Fig. 2.8b) reec-tions. In this case, the line of symmetry replaces the reecting wedge, thuscompletely eliminating the development of a boundary layer along it. Hence,it is possible to generate inviscid RR wave congurations in steady ows.


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2.2 Modications of the Perfect Inviscid Two- and Three-Shock Theories 51

r

r

r

rii

ii

(1)

(1)

T

T

(0)

(a) (b)

(0)

(2)

(2)

(3) m

Fig. 2.8. Schematic illustration of an experimental set-up for eliminating the bound-ary layer effect shown in Fig. 2.7: ( a ) RR; ( b ) MR

Things are different in the case of an MR, in which viscous effects alongthe slipstream of an MR always exist. These effects would have, in the case of a steady MR, similar inuence as in the pseudosteady MR (see Sect.3.4.4).Thus, the modication of the three-shock theory that is presented briey inSect.3.4.4 is probably applicable also to a steady MR if one is to accuratelypredict the angles between the various discontinuities near the triple point.Using an experimental set-up similar to that mentioned above, the boundarylayer with which the foot of the Mach stem interacts could also be eliminated.This is shown in Fig.2.8b.

Some excellent photographs showing the interactions of both the incidentshock wave of an RR and the Mach stem of an MR with the boundary layercould be found in Sect. 28.3 of the book The Dynamics and Thermodynamics of Compressible Fluid Flow by Shapiro (1953).

2.2.3 Thermal Conduction Effects

The fact that a real gas has a nite thermal conductivity, introduces anadditional mechanism, heat transfer, which might affect the ow elds nearthe reection point of an RR and the triple point of an SMR. (For moredetails of see Sect.3.4.5 where the effect of thermal conduction for the case

of pseudosteady reections is presented). Unfortunately, neither experimentalnor analytical studies of this effect are available.

The foregoing remark on the elimination of viscous effects along thereecting surface by usin

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Shock wave reflection phenomena