regular and mach refn of shocks

8/3/2019 Regular and Mach Refn of Shocks

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Ann. Rev. Fluid Mech. 1986. 18: 33-58Copyright 1986 by Annual Reviews inc. All rights reserved

REGULAR AND MACHREFLECTION OFSHOCK WAVES1Hans Hornun9Institute for Experimental Fluid Mechanics, DFVLR,BunsenstraBe 10,D-3400 Grttingen, Federal Republic of Germany

INTRODUCTIONThe properties that distinguish a shock wavefrom other wavesare that itsthickness is negligible comparedwith other characteristic lengths, and thatthe state of the mediums changed irreversibly by the passage of the wave.Shocks are therefore a highly nonlinear phenomenon, and parameterchanges maybe expected to lead to numerousbifurcations.The multitude of possibilities is compoundedwhenmore than one shockoccurs, such as in the interaction of a shock with a solid surface or symmetryplane (i.e. in shock reflection). The subject of shock reflection iscomplicatedhat it is necessary o introduce it at some ength. In the interestof conciseness, this is done according to a logical rather than a historicalsequence, with explicit reference to the important authors omitted in thetext unless their work s relatively recent.Salient points in the early development of the subject should bementioned here, however. These maybe found in the experiments of Mach(1878), the theoretical work of von Neumann1943), the experimentaltheoretical work of the group around Bleakney at Princeton (e.g. Bleakney& Taub 1949, Smith 1945), the experiments of Kawamura& Saito (1956),Smith (1959), and Bryson & Gross (1961), and the theoretical workLighthill (1949) and Jones et al. (195l). A review by Pack (1964) givesdetailed references. Of the work since 1960, that of the group around Glassat Toronto (e.g. Law & Glass 1971, Ben-Dor & Glass 1979, 1980) andaround Henderson at Sydney (e.g. Henderson & Lozzi 1975, 1979,

t Dedicatedo the memoryf ErnstBecket.330066-4189/86/0115-0033502.00

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34 HORNUNGHenderson & Gray 1981) stand out. A review by Griffith (1981) gives moredetailed references.In this review the discussion is restricted to plane flow. This is becauserelatively little work has been done on three-dimensional situations, andbecause a compromise had to be made between depth and breadth of thefield covered.Material PropertiesShocks may occur in manydifferent kinds of materials. However, in themainpart of this review the discussion is restricted to situations in which hematerial may be described to a good approximation by the model of acompressible, heat-conducting Newtonian continuum. The additionalassumption that a material element is subjected only to such slow changesthat (to a good approximation) it is able to maintain (local) thermodynamicequilibrium everywhere leads to a simple expression for the thickness ofshocks in terms of the bulk and shear viscosities, thermal conductivity,density, and speed of sound of the material. If these properties are replacedby corresponding values for a real gas of discrete molecules, then thethickness of sufficiently strong shocks amounts only to a few mean freepaths. Thus, to be consistent with our assumption hat the material behavesas a continuum, we have to neglect the shock thickness and treat the shockas a surface of discontinuity.In a continuum framework, local thermodynamic equilibrium is there-fore not maintainedwithin this thin shock. In the mainpart of this review tis assumed that thermodynamic quilibrium is maintained everywhere else,and that (to a good approximation) the temperature and equilibrium statesare defined by the equation of state of a thermally and calorically perfectgas. For our purposes the material is therefore specified by the density p,shear viscosity p, thermal conductivity k, specific heat cp, and specific gasconstant R. In this framework, a typical situation whose characteristiclength, flow velocity, and wall and reservoir temperatures are d, q, Tw, andTo, respectively, leads to a dependenceof any dimensionless quantity Q onfive parameters:

Q = Q[pqd/l~,pq2/(~p), ~, pcp/k, Tw/To],or

Q = Q(Re,2, y, P r, T~/To), (1)where

~ = c~/(cp- R).The further restriction to a Situation in which shear rates and temperaturegradients are so small that shear stresses and heat flux maybe neglected


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SHOCK REFLECTION 35defines such an important limiting case that a large part of this review isdevoted o it. It is referred to somewhatoosely as "inviscid" flow. Equation(1) degenerates for inviscid flow

Q = Q(M, ~). (2)Other material properties, such as vibrational or chemicalrelaxation, areof considerable importance in the reflection of strong shock waves. They,like viscosity and heat conduction, introduce new length scales into theproblem, albeit in a different way. Relaxation effects are discussed onlybriefly at the end of this review.

The Shock-Jump RelationsIn steady flow through an oblique shock wave (see Figure 1), theconservation equations may be used to relate the state downstreamof theshock (subscript 2) to the state upstream subscript 1)

P2/Pl = (~ + 1)M~ in 2 a/J2 +(y--1)M~2 sin 2 a], (3)Pz/P~= [2yM~ in2 a--(~--1)]/(~, + (4)M2~ in 2 (a--O) = [y+ 1 +(y-- 1)(M~sin2 a-- 1)]

+ [~ + 1 + 27(M~ sin 2 a-- 1)], (5)tan 0 = tan a[M~2 cos2 a--cot 2 a]/[l+M~(7+cos 2a)]. (6)

For fixed V and Mr, Figure 2 shows the shock angle a as a function of thedeflection angle 0. As a is increased from aresin (M~- ), M2decreases fromM~ o a value of 1 (sonic point)just before 0 reaches a maximum alue(maximum-deflection point). Further increase of ct causes M2 and 0 todecrease until 0 = 0 at ~ = ~/2 (normal shock).Again for fixed ~ and M~, a similar curve maybe drawn for p2(O)/p~ byeliminating a betweenEquations (4) and (6) (see Figure 3). Both curvessymmetrical about 0 = 0, of course. Their nonlinearity is evident.

go~

q~ 1_ ~"2Oe,

Fioure (left)Fioure 2 (center)Fioure 3 (right)

~MOmax n~2=IDefining sketch.

Shock ocus in (0, a)-plane. Mr, ? constant.Shock ocus in (0, p)-plane. M , 7 constant.


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36 HORNUNGReflection of a Plane Shock From a Plane SurfaceSaEADYrCVtSID FLOW onsider a plane shock (for example, one gene-rated by a wedge n steady flow) being reflected off a wall or symmetry laneparallel to ql (see Figure 4). The incident shock I deflects the flow toangle 02, and a second shock is needed to deflect it back to 03 = 0. Thissituation maybe represented in the (0, p)-plane (see Figure 4): Conditionlies on the incident-shock locus I, below the point M2= 1. From his newcondition we may draw a second shock locus with M = Mz from (Oz, P2).Thus, the deflection is taken from 0 to 02 by the incident shock I, and from02 to 0 by the reflected shock R. This configuration is called regularreflection. The (0, p)-diagram n Figure 4 is a graphical representation of themathematical problem of solving a double application of the shock-jumprelations in a particular case.If the angle ~ is increased, a value ~t = ctd(M~,~,) is reached, beyondwhichthe second shock locus does not reach the p-axis any more (see Figure 5).The deflection in region 3 is thus still positive (toward the wall). Thisnecessitates a nonuniform buffer zone, across which the transition from0 > 0 to 0 = 0 is accommodated.A near-normal shock S allows the triple-shock point P to moveaway from the wall, with the result that a vortexsheet V separate s the gas processed by S from that processed by I and R.The density and velocity, as well as the entropy, are discontinuous across V,but the pressure and streamline deflection must be continuous. This is thereason for the choice of the (0, p)-plane for the graphical representationthe solution of shock-reflection problems: Since 03 = 04 and P3 = P,~, thepoints representing states 3 and 4 are congruent in (0, p). Thus, the upperintersection of I and R in Figure 5 gives the triple-point condition, and the"Mach tem" S is represented by the part 4-5 of the curve I. Clearly S mustbe curved as a consequence. This reflection configuration is called Machreflection, after its discoverer. (As is seen later, the transition to Machreflection mayoccur at angles smaller than ~d.)

Figure 4 Regular eflection in steady flow. Physical (left) and (0, p)-plane (right).


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SHOCK REFLECTION

Figure 5 Mach eflection in steady flow. Physical (left) and (0, p)-plane flight).

37

PSEUDOSTEADYNVISCID FLOW onsider a plane shock I moving fromright to left into a gas at rest in an inertial frame, with the speedof the shockbeing qs- Let it strike a symmetricalwedge,at rest in this inertial frame,whose eading edge T is parallel and whosesymmetryplane is normal to theplane of the shock(see Figure 6). If ~ is sufficiently small, regular reflectionoccurs. The intersection point P of I and the wall moves up the wedgesurface at a constant speedq l = q~/sin~t, (7)

so that the distance d increases linearly with time. In a frame of referencemovingwith P (which is therefore also an inertial frame), the flow in theimmediate icinity of P is steady, unless Ma < 1. In the latter case, region 3is unsteady. The (0, p)-plane mayagain be used to describe the reflectionsystem (see Figure 7).The transition from regular to Mach eflection again becomesnecessary

oFi.qure 6 (left) Regular reflection in pseudosteadyflow, viewedfrom an inertial frame fixedin T.Figure 7 (center and right) Regular reflection in pseudosteady flow, viewed from an inertialframeixed n P. Physicalcenter)nd 0, p)-planeright).


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38 HORNUNGbecauseof the failure of R to reach the p-axis (see Figure 8). The triple pointP is nowoff the wall, so that the angle ~ between he oncoming low (in theframe of reference of P) and the incident shock I is no longer equal to theangle ~ + Z betweenI and the wall. Again, the gas processed by I and R isseparated by a vortex sheet V from that processed by S. In the frame ofreference of P~, the wall has a finite componentf velocity normal o its ownplane. The gas contained in the triangular region PBA s that whichoriginally occupied the region PTA.

Flowsof the kind depicted in Figures 7 and 8 are self-similar, inasmuch sthey grow inearly with time. They are called pseudosteady. This is becausein the absence of shear stresses and heat flux, the only characteristic lengthof the boundary onditions is d, which grows inearly with time (see Jones etal. 1951). Indeed, if the space coordinates are divided by the time t elapsedfrom the instant that I hits T, the equations of motion possess similarsolutions. Thus, if the inviscid flow equations

Dq-- = -- grad p/p, (8)DtDp-- = --p div q (9)Dt

are rewritten in the variables(X, t) = (x/t, t) (10)U q-X

the explicit dependenceon t disappears for all t > 0:U" Grad U + U = - GradU" Grad p + p(Div U + 2) = (11)

Figure Macheflection in pseudosteadylow, viewedroman inertial flame ixed in P.Physicalleft) and 0, p)-planeright).


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SHOCK REFLECTION 39

Figure 9 Regular reflection in pseudo-steady similarity space, S.P. = sink pointsfor the uniform-flow egions 1, 2, and 3.

where Div and Grad denote vector operators in plane X-space. Providedthat the boundary and initial conditions are consistent with this scheme,self-similar solutions are possible. Figure 9 shows the "streamlines" inpseudosteady pace, i.e. the integral curves of U in X-space, for the case ofregular reflection. Note that a region of uniform flow in physical space hasstraight integral curves that converge onto a single "sink" point inpseudosteady space. The whole of the flow pattern is independent of time inthis space.INVISCID STEADY FLOW: CASE DISTINCTIONSIn this section wediscuss the differences in the reflection configurations thatarise as consequences f the shock-jump elations, i.e. the influences of theparameters M1,y, and ~. The result of this discussion is a definition of thedomainsn the (M1,~)-plane (for fixed ~) to whichcertain formsof reflectionare restricted.The Special Values of ~(M1, y)The condition in which the reflected-shock locus just reaches the p-axis iscalled the maximum-deflection ondition of the reflection, and the shockangle at which it occurs is

~d = ~d(M1,V). (12)The pressure at the point of tangencyof the reflected-shock locus and the p-axis may lie below or above the normal-shock value, depending on MI and~,. Accordingly,one refers to "weak" r "strong" reflection, respectively (seeFigure 10).Just below this value of ~, a value

~s = ~s(M~,~) (13)exists, at which he sonic point of the reflected-shock locus lies on the p-axis.


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40 ItORNUNG

Figure 10 The conditions c~ = % full line) and ~ = ~s (dashed line) for weak left) and strongreflection (right).

This condition is indicated by the dashed lines in Figure 10. It is referred toas the sonic condition for the reflection.A third special value of ~ is defined by the condition that the reflected-" shock locus intersects the p-axis at the normal-shockpoint (see Figure 11).This is named after yon Neumann (1943), who first recognized itsimportance:~N = ~N(M1, 7)" (14)

For values of 7 that are realistic for gases, there exists a single configurationfor which

~r~(Mt,7) = ~d(M1, 7). (15)This occurs at a particular value of M~(7) see Figure 11). There also existtwo configurations for which

~N(Mt, ) = c~(M1, (16)The less important one of these occurs in the weak-reflection range. Forthis, the sonic point on the reflected shock corresponds o a pressure that is

Figure 11 The condition ~ = ~N"Weak left) and strong reflection (right), N = % (center).


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SHOCKREFLECTION

Figure 2

M~-I

The woconditions t which

lower than that at the normal-shockpoint. The other occurs in the strong-reflection range, and for this the pressure at the sonic point of the reflectedshock coincides with the normal-shockpressure (see Figure 12). For a given),, the values of 1 for t hese two configurations lie on either s ide of t hat for~a = aN and very close to it.The Mach-Number anees for ~, = 1.4For a fixed value of y, coincidences of the special values of ct, such asEquations(15) and (16), as well as other conditions, define special valuesthe Machnumber. In this section the Mach-numberanges defined by thesespecial values are discussed for the case y = 1.4. Noqualitative differencesoccur wheny is changed o values relevant for other gases.THE ANGE< MI< 1.25 In this range there exists no value of~t for which hereflected-shock locus R can intersect the incident-shock locus I other thanat the point 2 (see Figure 13). Mach eflection is therefore not possiblethis range. If~ > ~d (dashedcurve for R), no reflected shockoccurs at all, theincident shock is curved, and a subsonic region occurs behind it. The states

~./2>I1\~--~--M = IM=1 \M


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Figure 14 Reflected shock inclined for-ward,1.25 < Ms< 1.48 (~ = 1.4).I~JR

q~ ~M~ ~d, a direct Math eflection occurs, this being he only solution.This time we have 03 < 02, and the reflected shock s inclined downstream

regulor~Fiffure15

invertedDirect and inverted Macheflection in the range 1.48 < M1 2.2 (~ -- 1.4).

I , M~


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SHOCK REFLECTION 43when is near ~d, but as ~ increases further, the reflection reverts back o theconfiguration observed in the range 1.25 < M1< 1.48, with forward-inclined R.For aN < ~t < Ctd, both regular and direct Mathreflection are possible,and since M3 < 1, downstream onditions could enforce Mach eflection inthis region.THE ANGE.2 < M~ < 2.4 At M1 = 2.2, region 3 is sonic at ~ ---- aN. This is theupper one of the two points defined by Equation (16). Above1 =2. 4 aMathreflection is characterized by the fact that V separates the supersonicregion 3 from the subsonic region 4. This is the case illustrated in Figure 5.Again, both regular and direct Mach eflection are possible in the range~N< 0~ < ~a (see also Figures 10, 11).The Reflection Regions in the (M1, o~)-PlaneThe constraints on the various reflection modesmay be delineated in adiagram of a versus M1 see Figure 16). This summarizes he discussionthe previous section. (Note that the domains of complex and double Machreflection are omittedhere. They re discussed n a later section.) It is well toremember hat these boundaries apply to inviscid flow with y = 1.4. Inaddition, it became lear at a number f points that the (0, p)-plane, whichdeals with the immediatevicinity of the point P only, cannot give any cluesabout which modeactually occurs when more than one reflection mode spossible. The downstream boundary condition needs to be invoked. Wemay herefore expect differences between steady and pseudosteady flow insuch cases. This is because though the region around P is steady in both

\ ~ Henderson& Lozzi 1975 .... ~l;~f~

)~ Homu~ Robinson 1982p~sible

a =orcsinlM~11 ~~ ~ regulor. ossible20 ~ ~2 3 4 M~ 5

Figure 6The spe~a] alues f ~ as f~cfionsf M~ fo~ ~ = ].4, d~e~ecfio~on~r~fio~.he poi~{s how bserved~il~e f ~e~l~ e~eefio~


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44 HORNUNGcases if observed from Ps reference frame, the downstream oundaryconditions maynot be influenced in pseudosteady low (without breakingthe pseudostationarity).THE INFORMATION CONDITIONAn mportantdeterminingactor in regionsof overlapwhere,on the basis ofthe shock-jumponditions, both regular and Macheflection are possible isthe fact that--unlike regular reflection--Mach eflection does display acharacteristic ength n the vicinity of P. For this to be possible, informationabout he characteristic length of the problemw for steady, d for unsteadyflow) mustreach the point P.Pseudosteady FlowIf weconsider he case of a regularreflection in pseudosteadylow, the pointP moves way rom he wedgeip T at a constant speed ql. The nformationfrom he wedgeip travels at the local speedof sound elative to the fluidandchasesP (see Figure7). Clearly, his signal is only able to keepup withif M3 < 1, i.e. if a _> as. Thus, information boutT does not reach P, andMach eflection is not possible in pseudosteady low unless a > as. Thissimple argument ll but eliminates the regions of overlap in the case ofpseudosteadyflow, because the difference ad-as is so small as to beexperimentallyninteresting. Weherefore write the transition criterion forpseudosteadylow a~,s)

a~s = as(M1, ) .~ ad(Mt,?). (17)Acommonisconceptions that viscouseffects at the wall alwaysprovideapath of information etween and P. It is made lain in a later section thatthis is not so.Theweak-reflectionregion of overlap (MI < 2) poses a problem n thisconnection.Here,a Math eflection is associatedwith a subsonic egion 3.Thus, the informationcondition does not provide an argument gainst thepersistence of a Mach eflection once it has been set up near the tip. Anexperimenthat might est this possibility is one in which he tip of thewedge s made little sharper (wedge oncavenear the tip). HendersonLozzi(1979)performeduch experiments, ut these wereonly in the strong-reflection region, where his effect is not to be expected.Steady FlowTHE AREA OF OVERLAPWITH SUPERSONICREGION3 (Mt > 2.4) Figure 5 showsaMacheflection in this range. Theexpansionwave rom he trailing edge ofthe shock-generating edges refracted by the reflected shockR and strikes


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SHOCK EFLECTION 45the vortex sheet V. If it were not for this expansion, the vortex sheet wouldasymptotically approach a line parallel to the wall. The expansion causes to curve morestrongly awayfrom the wall, thus forming a sonic throat withthe wall. Hence, an information path exists from the trailing edge along theleading characteristic to the subsonic region upstream of the sonic pointand thus to P. Information about the size of the wedge can be com-municated to P. The information condition therefore does not provide animpediment to Mach reflection in steady flow, and we thus expect thetransition criterion for strong reflection in steady flow (~) to

~ = 0~r~(M1,),). (18)This has been amply confirmed by experiment (Henderson & Lozzi 1975,1979, Hornung & Kychakoff 1977, Hornung et al. 1979, Hornung &Robinson 1982). The experimental points from some of these studies areshown in Figure 16 for comparison with Equation (18).

The distinction between ~, and ~ is ignored by most textbooks ongasdynamics, though it was already suspected by von Neumann1943) anddiscussed at some ength by Courant & Friedrichs (1948).THEAREAOFOVERLAPWITH.48 < MI < 2.4 Between M1 = 2 and M1 = 2.4, thedifference between 0~ and ~a is so small as to be experimentallyunresolvable. In the range 1.48 < M1 < 2, three reflection types arepossible. Twoof these are regular, one with weak and one with strongreflected shock, and the third is a Mach eflection [see the (0, p)-planeFigure 15]. The Math reflection has a subsonic region 3, and the regularreflection with weakshock has a supersonic region 3. In order to force theMathreflection, it is therefore necessary to constrain the flow downstream,which thus raises the back pressure (see Henderson & Lozzi 1979).unimpeded low the regular reflection persists up toDOUBLE MACH REFLECTIONDouble Mach reflection was first observed by Smith (1945) and White(1951). It was further studied in somedetail by Gvozdeva t al. (1969),related it to real-gas effects. While uch effects can be responsible for doubleMach eflection, its occurrence does not depend on them.The Reason for the Occurrence of Double MachReflectionConsider a Mach reflection in the strong shock range M1 > 2.4 inpseudosteady low. In the frame of reference of P, region 3 is supersonic andthe wall has a finite componentof velocity normal to its ownplane (seeFigure 17). Let us assume for the moment hat the flow in region 4 is


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46 HORNUNG

Figure17Pseudosteady acheflection.For 0, p)-plane,ee Figure

uniform, so that PB is a straight vortex sheet that makes an angle 0a withthe triple-point path. This angle is determined n terms of M1,~, and ~ by theshock-locus intersection :03 ---- 03(M1,~, ~t). (19)

Thus, the position of the point B and therefore its velocity qB are known nPs reference frame:qB qB(MI,, ~, X). (20)

The assumption that the flow in region 4 is uniform and the wallimpermeability condition mean that the velocity component of the wallnormal o its ownplane, q~,, is equal to the component ftl~ = q4 normal tothe wall:q,, = qa sin (03 + X). (21)

However,he velocity along the vortex sheet in region 3 (q3) is different fromthat in region 4 (qs). Therefore, the component f th normal to the wallnot the same as the corresponding componentof the walls velocity. Thisinevitably leads to the need for the flow along PB in region 3 to beaccelerated away from or toward the wall, depending on whether q3 -q4 ispositive or negative. In the frame of reference of B, this boils down o theneed for a deflection of the flow in the vicinity of B to a direction parallel tothe wall.In the strong-reflection range, we have q3-q4 ;> 0. The flow in region 3therefore has to be decelerated and deflected (mostly) toward the wedge ip.If the relative speed q3-q, is smaller than the speed of sound a_~, thedeflection takes place without a shock being generated. The resultingconfiguration is either single Mach eflection or what is sometimescalledcomplexMathreflection. If, on the other hand q3- q, > a3, the deflectionoccurs via a shock, and we speak of double Mach eflection. The shock may


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SHOCK REFLECTION 47

Figure l 8 Complex left) and double Mach cflections. The second Mach tem maybe curved(center)or straight (ri#ht).

either be straight or curved, depending on the shock strength required forthe deflection (see Figure 18).ComplexMach eflection is an interesting case, because while region 3 issupersonic in Ps reference frame, it is subsonic in Bs reference frame.Hence, the reflected shock is straight up to the point D at which he signalfrom B has caught up to it (Figure 18).The requirement that the flow pattern be pseudosteady, together with theassumption that the whole of region 4 is uniform, amounts to theassumption hat the Mach tem is straight and normal to the wall. That thisis so maybe seen by considering the sink points for the various regions ofthis reflection in pseudosteadyspace (see Figure 19). The assumption hatis normal to the wall at the triple point is made by Law& Glass (1971)in order to calculate the angle Jr. Glass and his collaborators have built

V S.P. 4" ~,~

Figure 19 Double Math reflection in pseudosteady similarity space with the sink points ofthe uniform-flow rogions indicated.


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48 HORNUNG90

ax,deg.[70

50

30 1 2 4 6 8 10Ms--M~ in ~x

Figure20The structure of the directMach-reflectionange cf. Figure16) forpseudosteadylowat ~ = 1.4. (AfterBen-Dor1978.)R = regular, D= doubleMach,C = complexMath, M= single Mach,and N = no reflection. M~s the Mathnumberelative o the secondriple point.

up a considerable body of knowledge on these more complicated Machreflections. In particular, the domains in which double and complex Machreflection are to be expected on the basis of the Law/Glassassumption havebeen mapped ut in detail by Ben-Dor 1978). This is shown n Figure 20the form of a plot of the angle between the incident shock and the wedge(~+;0 against the incident-shock Mach number (Ms = M1 sin ~). Theboundary between double and complex Mach eflection in this diagram iscalculated by Ben-Dot rom the condition that the flow in region 3 is sonicin the frame of reference of the second triple point (see also Ben-DorGlass 1979, 1980, Ando1981). This condition is slightly different from theview taken here, that region 3 should be sonic in the reference frame of B attransition.It should be pointed out that all the computations of the domains of thevarious reflection configurations and their boundaries (except for theboundary ndicating failure of regular reflection) rely directly or indirectlyon assumptions of how the wall impermeability condition influences theangles of the triple-point paths. The above description of the reasons for theoccurrence of double Mach eflection makes it clear that such assumptionslead only to relatively crude approximations. This is further illustrated inrecent work by Shirouzu & Glass (1984), whoattempted a refinement of theassumptions by taking into account the curvature of the Machstem.The Curling Upqf the Vortex SheetThe need for the flow from region 3 to be deflected at B means that thepressure is higher at B than to the left of B (see Figure 21). In the framereferenceof B, the gas to the left of B (region 4, considered niform) s at rest.The pressure distribution corresponding to this flow is as sketched inFigure 21. Such a pressure distribution causes the flow to be accelerated


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SHOCK REFLECTION 49

B " P Rp-P~

BFigure 21 Pseudosteady Mach reflection viewed from Bs reference frame. The pressuregradient to the left of B causes the vortex sheet to roll up.

away rom B in all directions (stagnation point), with the result that thevortex sheet is deflected to the left and the dividing streamline, which s thenfree of vorticity, strikes the wall at right angles. Occasionally, he pressuregradient to the left of B can be so severe as to drive a wall jet to the left sostrongly that it influences the Mach tem (see, for example,Sandemant al.1980, Collela & Glaz 1984). This phenomenonstem bulge) maybe expectedto be impededby viscous effects.Computation of the Flow FieldA tempting approach to the solution ofpseudosteady flows is the numericalsolution of Equations (11). Computationsof this kind have been presented,for example, by Schneyer (1975), Kutler & Shankar (1977), whoalso presenta review of the subject, Shankaret al. (1977), and (more recently) CollelaGlaz (1984). The results of the first three of these were comparedwithexperiments by Ben-Dor & Glass (1978). While this comparison showsgeneral agreement, it also reveals discrepancies that maybe due to viscous,real-gas, or numerical viscosity effects. The more recent work of Collela &Glaz (1984) shows that the numerical approach predicts an amazing degreeof complexity and detail, including multiple Mach eflection with stembulge and vorticity concentrations. Publications on this subject are in pressfrom this group and should be of interest to all students of Mach eflection.An important field where computation of the flow field is necessary isthat of weak eflection. This is because the downstreamlow often containssubsonic regions, so that the downstream boundary conditions matter. Ananalytical technique using the method of strained coordinates wasdeveloped for very weak shocks by Obermeier (1984). The results ofparticular computation were compared with the experiments of Henderson& Siegenthaler (1980) and showed promising agreement.


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50 HORNUNG

EFFECTS DUE TO THE WALL BOUNDARYCONDITIONSThe Effect of Viscosity and Heat Transfer in"Pseudosteady" FlowThe introduction of any length scale independent of d into a pseudosteadyflow must break the pseudostationarity, i.e. the self-similarity. Withviscosity, such an independent ength, I, = kt/(pq), is introduced. Hence hequotes in the title of this section.The experimental results on pseudosteady flow indicate that thetransition from regular to Maeh eflection occurs at values of ~ that liesignificantly above 0~a when he reflection is arranged to occur off a solidsurface (see, for example, Bleakney & Taub 1949, Henderson& Lozzi 1975,Takayama& Sekiguchi 1977). However,when the reflection occurs (as, forexample, in the case of symmetrically interacting shocks) off a symmetryplane (see Figure 22), such as in the experiments of Smith (1959)Henderson& Lozzi (1975), the no-slip condition at the reflection surfaceand associated viscous effects are avoided. In these cases the transition wasfound to occur--to within experimental accuracy--at the theoretical valueof ~a. The persistence of regular reflection to beyond~a is therefore a walleffect.Consider a regular pseudosteady reflection in the frame of reference of P(see Figure 23). There exists no shear along section EP of the wall, but sinceq3 < ql, a velocity discontinuity exists in an inviscid flow across the surfacealong PF. The no-slip condition and shear viscosity therefore cause aboundary layer to develop from P, in which the flow depends on all five ofthe parameters in Equation (1). This boundary ayer is of the type in whichthe wall velocity is greater than the free-stream velocity (in the reference

Figure 22 Arrangement or setting up pseudosteady reflection off a synunetry plane to avoidviscouseffects. (left) Early.(rioht) Later.


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SHOCK REFLECTION 51frame in which the flow is steady), so that its displacement thickness isnegative. Since the wall temperature is also usually lower than the recoverytemperature [the free stream (region 3) having been shock heated from thetemperature of region 1, i.e. the wall temperature], the magnitudeof thenegative displacement thickness is increased by heat transfer to the wall.The influence that this effect will have on the transition angle maybeestimated by the following argument. The effective shape of the wall to theright of P as seen by the far field (Figure 23) is such as to require the flowdeflection through the reflected shock to be smaller than that through theincident shock by a small angle ~ (see Hornunget al. 1979). Hence, thepossibility of a regular reflection ceases to exist not at ~d, but at a conditionwhere the maximum-deflection angle across the reflected shock is 02- e(Figure 23). This causes the transition angle to be raised by viscosity tovalue

~* > ~p~ = ~,. (22)The subscripts are intended to distinguish viscous from pseudosteady (andtherefore inviscid) values.It is sometimes rgued that the presence of a boundary layer on the wallprovides a path for the information from the wedge tip to reach thereflection point. In fact, this is not so, because he flow speed (and the Machnumber)n Ps reference frame, against which he signal has to travel, is evenhigher in the boundary layer than in the free stream.Provided that the boundary layer is laminar, its displacement thickness

Figure 23 Regular reflection viewed rom Ps reference frame. (left) lnviscid. (center) Viscousdisplacement effect as seen by flow outside the boundary layer. (right) Viscosity causestransition to Mach eflection to be delayed.


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52 HORNUNG3" may be shown to behave (see Becket 1961)

~*/x = --f(Pr, ~, M,T~,/To)(lv/x)/2, (23)where x is the distance along the wall from the point at which the boundarylayer starts (in this case P), and the parameterson the right-hand side arebe evaluated outside the boundary ayer. If the wall temperature is equal tothe static temperature n region 1, and ), and t are held f ixed, t hen M3, Toa,lva and Pra will also be constant for approximately constant ~. Undersuchconditions, it may be shown hat

~*v = O~*ps - C(lv3/l)/2, (24)whereC is a constant. The difficulty with this form is that there seems o beno local characteristic length to choose for I. Hornung& Taylor (1982)chose I to be the smallest resolvable length of the observer, on the basis thatthis is the Mach-stem ength at observed transition. Their experimentsconfirmed Equation (24) and showed that, if extrapolated to infiniteReynolds number l/lva), it gave excellent agreement with ~p*~ = ~, thoughthe maximumifference between ~* and ~s was more than 6. This result isindependentof their choice of I.In fact, the problem is not quite so simple. To understand it better,consider the question of how he length of the Mach tem is modified by thepresenceof viscous effects at the wall. Let the inviscid (pseudosteady)valueof the Mach-stem ength be Lps and its viscous value L~. For a laminarboundary layer it may be shown(see Hornung 1985) that

Lv/Lps = 1 -- 2f,~ (tan fl/tan g)I/2(Iv/d)/2, (25)where ,~ is the function in Equation 23) evaluated n region 4, fl is the anglebetweenV and the Mach tem, and ;t is the angle of the triple-point path forinviscid flow.The resulting path of the triple point of a single Mach eflection is shownin Figure 24. Whether a reflection is interpreted as a regular or a Machreflection appears to dependon whether t is observedat a distance from thewedge ip that is smaller or greater than

do = lv4(4f42 an fl/tan ~), (26)i.e. the value ofd at which Lv = 0. However, he theory leading to Equation(25) is an asymptotic theory for high Reynolds number nd assumes that thesecond term on the right of(25) is small comparedwith unity. The step from(25) to (26) is therefore incorrect because t uses the theory outside its rangeof validity. It is clear from (25), however, that the length to be usedEquation (24) is the distance d~ from the tip at which observations aremade.


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Figure 4Sketch llustrating the effectof viscosity on the triple-point path., inviscid low; --, viscous low;..... , according o observer assum-ing inviscid flow; , by the pro-cedureof HendersonGray 1981).

SHOCK EFLECTION 53

Lps / \\

Very careful experiments made by Henderson & Gray (1981) confirmedthe behavior of the triple-point path sketched in Figure 24, inasmuch asthe paths apparent origin was observed to lie below the wedge tip ifextrapolated back along a straight path chosen in such a manner as tosatisfy the shock-jump conditions at the observed geometry. Their inter-pretation of this effect was hat it mightbe due to leakage near the tip of thewedge, whose symmetrywas approximated in their experiments by contactof the sharp leading edge with a flat plate along the symmetryplane to theright.Fromour discussion of the effect of shear viscosity, it maybe recognizedthat the mechanism y which heat conduction affects shock reflection is viaits effect on the displacement hickness of the boundary ayer. Thus, a colderwall wouldact like a highe~ viscosity.The Effects of Surface Roughness and PorosityIf the wall is rough, a mechanismby which momentum ay be imparted tothe gas near the wall is possible even without shear viscosity. Thus,roughness may cause a negative displacement thickness, and regularreflection mayagain be expected to persist to higher values of e thanThis was shown to be indeed the case by Takayama et al. (1981), whoperformed experiments with different kinds of roughness. They alsoconstructed an empirical model nvolving the total pressure loss due to theroughness. Excesses of the transition angle of up to 20 beyondc~a could beobtained.Aneffect that is closely related to that of viscosity maybe obtained byletting the shock reflect off a porous wall. Since the shock system aises thepressure, the passage of the shock switches on a flow through the wall. Theresult is that gas leaks out of region 4, whichhas the sameeffect as a growingnegative displacement thickness.


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54 HORNUNGA theory describing this effect was constructed by Clarke (1984a)which he porosity of the wall is modeledby Darcys law and a second, solidwall backs up the porous material. The strength of the reflected shock is

considerably attenuated compared with solid-wall behavior, and theadjustment to final equilibrium occurs in an essentially shock-free manner.The theory gives solutions for the flow field and has been shown o agreevery well with experiments designed to test it (see Clarke 1984b). Clarke didnot examine he transition to Mach eflection, though a computation of theeffect on the transition angle appears to be within the scope of the theory.INVISCID UNSTEADY EFFECTsThe introduction of any independent length scale breaks the pseudo-stationarity. Therefore, the wall effects of the previous section all cause theflow to become ruly unsteady, so that the point P is accelerated. A frame ofreference attached to P would herefore not be an inertial frame, and bodyforces wouldneed to be introduced to correct for this. While his is only ofminor importance in the weakunsteadinesses considered so far, it becomescompletely impossible to use the pseudosteady flow ideas for truly unsteadyflows such as are encountered, for example, when a plane shock strikes acircular cylinder transversely, or when a spherical shock strikes a planesurface. It must therefore not be expected (as is sometimes done) thatpseudosteady- or steady-flow criteria have any meaningat all in unsteadysituations.Short of computing the whole flow field in a space of at least threeindependent variables, the only theoretical modelavailable for predictingthe behavior of such flows is that of Whitham 1957) for the propagationand interaction of curved shocks. Heilig (1969) performed very detailedexperiments on the interaction of a plane shock with a circular cylinder. Atand after first contact, regular reflection is observed, and at a particularangle ~1 between the shock and the cylinder surface a Machstem begins togrow. Heilig found that this value ofcq exceeds d by an amount hat varieswith Ms and reaches a maximumf 13 at Ms = 1.2. He also computed thetriple-point path using Whithams theory and found good agreement forthe case of strong shocks.A number of further experimental investigations of unsteady shockreflection have been madeon both convex and concave surfaces. Only a fewof these are mentioned for further reference: Henderson & Lozzi (1979),Itoh et al. (1981), Ben-Dor t al. (1980), Takayamat al. (1981). Onconcavesurfaces it is possible for the reverse transition from Mach o regularreflection to occur. Also, the angle at which this occurs is substantiallysmaller than ~d" Here, as in other experiments, information enabling the


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SHOCK REFLECTION 55determination of the viscous and heat-transfer parameters necessary forcorrect interpretation is often omitted.REAL-GAS EFFECTSRarefied GasesAt this point we depart from our restriction to continua in order to describean interesting effect due to gas rarefaction observed by Walenta (1983).Estimates of the effect of shock thickness on the flow in the very nearvicinity of the triple point had been made by Sternberg (1959), whoconcluded hat it was negligible in the flow regime he considered. Walentasexperiments (see also Schmidt1985) investigate the interaction of a planeshock with a wedge n a range where the typical values of d are around 100mean ree paths. He shows hat the shock angles at the triple point differmarkedly from those expected from inviscid continuum theory and that,furthermore, the constancy of 0 and p across the vortex "sheet" does notapply any more. This is not surprising, because the vortex sheet is nownot athin shear layer, but one that grows rapidly due to viscous action in thissituation. [This latter effect has been disregarded so far because ts effect ontransition in continuum flows is minor (see also Sternberg 1959, Sakurai1964).]

The most interesting aspect of Walentas results is that the triple-pointpath approximately follows a straight line that does not pass through thewedge ip but intersects the wedge urface at a distance from the tip of some50 to 100 mean ree paths. He explains this as follows (private communi-cation): The reflected shock starts at the tip as a cylindrical shock. Thespeed of a curved shock whose adius of curvature R is not very much argerthan its thickness 6 is proportional to 1 -6/R. It therefore lags behind theincident shock until R becomes arge, even in a situation that should giveMach eflection according to continuum theory. In these experiments too,the boundaryconditions at the wall as regards slip and heat transfer couldplay a significant part (see Hornung 985).High-Temperature Continuum EffectsIf the shockspeed is so high that the gas is vibrationally excited, dissociated,or ionized, we distinguish between two cases. In the first of these,thermodynamic equilibrium is assumed to be maintained outside of verythin shocks. Since the equilibrium condition is changed from that of aperfect gas, the transition angles will be modifiedalso. Nonew length scaleis introduced into the flow, however, so that the new ransition angles maybe computed for steady and pseudosteady flows just as before. This hasbeen done, for example, by Gvozdeva t al. (1969), Hornung t al. (1979),


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56 HORNUNGand Ben-Dot& Glass (1980). Where atisfactory agreementwith experi-mentwasnot observed, the direction of the discrepancymaybe explainedby viscous effects. Both ~d and gN are increased by endothermic real-gaseffects. (The opposite is true for exothermic effects, such as in detonationwaves.)

The second effect is that duc to relaxation. If the relaxation time isresolvable by the observer, then a new, independent length (the relaxationlength) is introduced into the problem. This was shown by Sandeman ct al.(1980) to have a considerable effect in dissociating nitrogen and in ionizingargon when the relaxation length is comparable with the distance d.CONCLUSIONSThe last decade has seen extensive developments in the study of shockreflection. These have led to a satisfactory understanding of the failure ofregular reflection in the strong-reflection range for inviscid steady andpse.udosteady flow. However, there remain a number of unansweredquestionseven in steady and pseudosteadylow. This is especially true forweakshock reflection, where the effect of the downstreamboundaryconditionsneeds o be examinedn moredepth. A further uncertainty ies inthe assumptions usually made to predict the triple-point paths and theresulting transitions to complex and double Mach reflection. Some of thesedifficulties may be associated with the faulty interpretation of experimentalresults that often occurs through ignoring the effects of shear viscosity.

In this review several effects have been singled out that cause thepseudostationarity of shock reflection to be broken by introducing anindependent length scale. The most important of these are the effects due toshear viscosity and heat conduction at the wall because they strongly affectthe interpretation of nominally pseudosteady results. A solution for theeffect of viscosity on the triple-point path, given in this review, illustrateshow such a misinterpretation may occur. Future experiments must take thisinto account.ACKNOWLEDGMENTSI wish to thank Wladek Fiszdon and Reinhard Niehuis for the care withwhich they read and corrected the manuscripL

Literature CitedAndo, S. 1981. Domains nd boundaries ofpseudosteady bliqueshockreflections incarbon dioxide. UTIASTech-Note No.231, Inst. Aerosp.Stud., Univ.TorontoBecker,E. 1961.Instation/ire Grenzschichten

hinter Verdichtungsst6Bennd Expansions-wellen.Prog.Aeronaut.$ci. 1 : 10~73Ben-Dor,G. 1978. Regions nd transitions ofnonstationaryobliqueshock-waveiffrac-tions in imperfect and perfect gases.


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SHOCK REFLECTION 57UTIASRep. No. 232, Inst. Aerosp. Stud.,Univ. TorontoBen-Dot, G., Glass, I. I. 1978. Non-stationary oblique shock-wave eflections:Actual isopycnics and numerical experi-ments. AIAA J. 16:1146-53Ben-Dot, G., Glass, I. I. 1979. Domainsandboundaries of nonstationary obliqueshock-wave eflexions : 1. Diatomlcgas. J.Fluid Mech. 92 : 459-96Ben-Dot, G., Glass, I. I. 1980. Domainsandboundaries of non-stationary obliqueshock-wave eflexions : 2. Monatomic as.J. Fluid Mech. 96: 735-56Ben-Dot, G., Takayama, K., Kawauchi, T.1980. The transition from regular to Mathreflexion and from Maeh to regular re-flexion in truly non-stationary flow. J.Fluid Mech. 100:147~i0Bleakney, W., Taub, A. H. 1949. Interactionof shock waves. Rev. Mod. Phys. 21 : 584-6O5Bryson, A. E., Gross, R. F. W.1961. Diffrac-tion of strong shocks by cones, cylindersand spheres. J. Fluid Mech. 10:1-16Clarke, J. F. 1984a. Regular reflection of aweak shock wave from a rigid porous wall.Q. J. Mech. Appl. Math. 37:87-111Clarke, J. F. 1984b. The reflection of weakshocks from absorbent surfaces. Prec. R.Soc. LondonSet. A 396: 365-82

Collela, P., Glaz, H. M. 1984. Numericalcalculation of complex shock reflectionsin ~lases. Presented at MaehReflectionSymp., 4th, Sendai, Jpn.Courant, R., Friedrichs, K. O. 1948. Super-sonic Flow and Shock Waves. NewYork:Interscience. 464 pp.Griffith, W. C. 1981. Shock waves. J. FluidMech. 106: 81-101Gvozdeva, L. G., Bazhenova, T. V.,Predvoditeleva, O. A., Fokeev, V. P. 1969.Mach eflection of shock waves in realgases. Astron. Acta 14: 503-8Heilig, W. 1969. Theoretische und experimen-telle Untersuchungen zur Beuaung yonStosswellen an Ku#eln und Zylindern.Dissertation. Univ. Karlsruhe. 114 pp. Seealso 1977. A result concerning the transi-tion from regular reflection to Mach eflec-tion of strong shock waves. In Shock Tubeand Shock Wave Research, ed. B. Ahlborn,A. Herzberg, D. Russell, pp. 288-96.Seattle: Univ. Wash.PressHenderson, L. F., Gray, P. M. 1981.Experiments on the diffraction of strongblast waves. Prec. R. Soc. London Ser. A377 : 363-78Henderson, L. F., Lozzi, A. 1975. Experi-ments on transition to Math reflexion.d. Fluid Mech.68 : 139-55Henderson, L. F., Lozzi, A. 1979. Furtherexperiments on transition to Mach re-flexion. J. Fluid Mech.94:541-59

Henderson, L. F., Siegenthaler, A. 1980.Experiments on the diffraction of weakblast waves. The von Neumannparadox.Prec. R. Soc. LondonSet. A 369:537-55Hornung, H. G. 1985. The effect of viscosityon the Mach stem length in unsteadyshock reflection. In Unsteady FluidMotion, Lecture Notes in Physics. In pressHornung, H. G., Oertel, H., Sandeman,R. J.1979. Transition to Math reflexion ofshock waves in steady and pseudosteadyflow with and without relaxation. J. FluidMech. 90 : 541-60Hornung, H. G., Robinson, M. L. 1982.Transition from regular to Math reflectionof shock waves. Part 2, The steady-flowcriterion. J. Fluid Mech.123:155-64Hornung, H. G., Taylor, J. R. 1982.Transition from regular to Mach eflectionof shock waves. Part 1. The effect ofviscosity in the pseudosteady ase. J. FluidMech. 123 : 143-53Hornung, H. G., Kychakoff, G. 1977.Transition from regular to Mach eflexionof shock waves n relaxing gases. In ShockTube and Shock Wave Research, ed. B.Ahlborn, A. Herzberg, D. Russell, pp. 297-302. Seattle: Univ. Wash.PressItoh, S., Okazaki, N., Itaya, M. 1981. On hetransition between regular and Math re-flection in truly non-stationary flows. J.Fluid Mech. 108 : 383-400Jones, D. M., Martin, P. M. E., Thornhill, C.K. 1951. A note on the pseudo-stationaryflow behind a strong shock diffracted orreflected at a corner. Prec. R. Soc. LondonSer. A 209 : 238-48Kawamura,R., Saito, H. 1956. Reflection ofshock waves-- 1. Pseudostationary case. J.Phys. Soc. Jpn. 11 : 584-92Kutler, P., Shankar, V. 1977. Diffraction of ashock wave by a compression corner. PartI : Regular reflection. AIAA . 5 : 197-202Law, C. K., Glass, I. I. 1971. Diffraction ofstrong shock waves by a sharp compres-sive corner. CAS! Trans. 4:2-12Lighthill, M. J. 1949. On the diffraction ofblast I. Prec. R. Soc. LondonSet. A 198:454-70Mach, E. 1878. t0ber den Verlauf yonFunkenwellen in der Ebene und imRaume. Sitzun#sber. Akad. Wiss. Wien78 : 819-38Obermeier, F. 1984. Development and propa-#ation of weak Mach-shock waves. Pre-sented at Math Reflection Symp., 4th,Sendai, Jpn.

Pack, D. C. 1964. The reflexion and diffrac-tion of shock waves. J. Fluid Mech. 18:549-76Sakurai, A. 1964. On the problem of weakMach eflection. J. Phys. Soc. Jpn. 19:1440-50Sandeman, R. J., Leitch, A., Hornung, H.


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58 HORNUNG1980. The influence of relaxation on tran-sition to Mach eflection in pseudosteadyflow. In Shock Tubes and Waves, ed. A.Lifshitz, J. Rom,pp. 298-307. Jerusalem:Magnes

Schmidt, B. 1985. Stosswellen/ibergang vonregul/irer Reflektion zur MachReflektionauf der Flanke eines Keils. Z. Angew.Math. Mech. 65 : 234-36Schneyer, G. P. 1975. Numerical simulationof regular and Mach reflection. Phys.Fluids 18 : 1119-24Shankar, V. S., Kutler, P., Anderson, D. A.1977. Diffraction of shock waves by acompression corner, Part II--Single Mathreflection. AIAA Pap. No. 77-78Shirouzu, M., Glass, I. I. 1984. An assessmentof recent results on pseudo-stationaryoblique-shock-wave reflections. Presentedat Mach Reflection Symp., 4th, Sendai,Jpn.Smith, L. G. 1945. Photographic investiga-tion of the reflection of plane shocks n air.OSRD ep. No. 6271, Off. Sci. Res. Dev.,Washington, DCSmith, W. R. 1959. Mutual reflection of twoshock waves of arbitrary strengths. Phys.Fluids 2 : 533-41Sternberg, J. 1959. Triple-shock-wave inter-

sections. Phys. Fluids 2:179-206Takayama, K., Sekiguehi, H. 1977. An ex-periment on shock diffraction by cones.Rep. Inst. High Speed Mech., TohokuUniv.336 : 53-74Takayama, K., Gotoh, J., Ben-Dor, G. 1981.Influence of surface roughness on theshock transition in quasistationary andtruly non-stationary flows. In Shock Tubesand Waves, ed. C. F. Treanor, J. G. Hall,pp. 326-34. Albany: State Univ. N.Y.Pressyon Neumann, . 1943. Oblique reflection ofshocks. Explos. Res. Rep. No. 12, Dep.Navy, Washington, DC. See also CollectedWorks, 6: 238-99. Oxford : Pergamon(1963)Walenta, Z. A. 1983. Formation of the Mach-

type reflection of shock waves. Arch. Mech.35:187-96White, D. R. 1951. An experimental survey ofthe Math reflection of shock waves.Princeton Univ. Dept. Phys. Tech. Rep. II-10, Princeton, N.J.Whitham, G. B. 1957. A new approach toproblems of shock dynamics. Part 1. Two-dimensional problems, d. Fluid Mech. 2:145-71


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