THREE-DIMENSIONAL STEADY SHOCK WAVE · PDF fileIt is not easy to visualize complex 3D shapes of shock waves in experiments ... Mikhail S. Ivanov et al. the Mach stem ... non-uniqueness

European Congress on Computational Methods in Applied Sciences and EngineeringECCOMAS 2004

P. Neittaanmaki, T. Rossi, S. Korotov, E. Onate, J. Periaux, and D. Knorzer (eds.)Jyvaskyla, 24–28 July 2004

THREE-DIMENSIONAL STEADY SHOCK WAVEINTERACTIONS. NUMERICAL SIMULATIONS AND

EXPERIMENTAL VALIDATION

Mikhail S. Ivanov?, Anatoly M. Kharitonov?, Dmitry V. Khotyanovsky?,

Alexey N. Kudryavtsev?†, Stanislav B. Nikiforov?, and Alexandr A. Pavlov?

?Institute of Theoretical and Applied Mechanics Russian Academy of Sciences,Siberian Division, 4/1 Institutskaya St., 630090 Novosibirsk, Russia

† Corresponding Author: e-mail: [email protected],web page: http://www.itam.nsc.ru/users/alex

Key words: Supersonic Flows, Shock Waves, 3D Simulations, Experimental Validation.

Abstract. Numerical simulations and wind tunnel experiments have been performed toinvestigate 3D shock configurations in the supersonic flow around two symmetrical finite-width wedges. Regular and irregular (Mach) interactions of shock waves have been studiedand a number of unexpected details have been revealed. The existence of a new type of3D shock interaction, combined reflection, has been numerically predicted. Experimentsperformed with the laser sheet visualization technique have confirmed all these numericalfindings. Close qualitative and quantitative agreement between the results of computationsand experiments has been established.

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Mikhail S. Ivanov et al.

1 INTRODUCTION

The 3D interaction of steady shock waves generated by two or more solid bodies is ageneral feature of many supersonic flows. In particular, the shock wave configurationsforming around two symmetrical finite-width wedges were studied in a number of recentinvestigations dealing with the transition between regular reflection (RR) and Mach re-flection (MR) (see, e.g. [1, 2, 3]). It is not easy to visualize complex 3D shapes of shockwaves in experiments. Consequently, numerical modeling is especially important for abetter understanding of such flows.

The goal of our paper is to present results of numerical simulations of 3D shock waveinteraction and to compare them with data obtained in wind-tunnel experiments. The 3DEuler equations are solved by a high-order TVD scheme. The flow around two symmetricalfinite-width wedges is simulated at the free-stream Mach numbers 4÷5. Though inviscidnature of shock interaction phenomena facilitates their numerical modeling, the problemcontains a number of intricate and delicate features. First of all, in a certain rangeof the angles of shock incidence both RR and MR are theoretically possible, and thehysteresis phenomena are expected in the transition between them. Further, the flowhas a complicated 3D structure and includes strong shock waves, subsonic pockets, andlow-density regions.

Numerical simulations reproduce accurately the hysteresis phenomena and predict someunexpected details. For the RR configuration, a peripheral Mach stem surface with a su-personic flow behind it is observed at large spanwise distances. For the MR configuration,a non-monotonous variation of the Mach stem height in the spanwise direction is revealed,provided the wedge width/length ratio is large enough. Moreover, in the numerical sim-ulations, a new shock wave configuration is found out. In this configuration, MR existsnear the vertical plane of symmetry, is changed to RR closer to the periphery, and retainsagain at even larger spanwise distances.

Wind-tunnel experiments have been performed to validate the results of numericalsimulations. Optical diagnostics with the laser sheet method employed in the streamwisedirection enables us to visualize the flow in arbitrary spanwise cross-section and recon-struct the complete 3D flow structure. All numerical findings have been confirmed andclose quantitative agreement between computations and experiments has been established.

2 PROBLEM FORMULATION

The flow around two symmetrical wedges immersed in a supersonic stream and used asshock waves generators is investigated – see Fig. 1, where necessary notions are also given.There w is the wedge length (along the shock-generating side), b is the wedge width, θ isthe angle of attack, 2g is the distance between the trailing edges of two wedges, and 2h isthe distance between their leading edges. In addition to the test model geometry, a Machshock wave configuration in the central vertical plane is also shown in Fig. 1. Here α isthe inclination angle of the incident shocks IS; RS, MS, and SS are the reflected shocks,

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the Mach stem (a strong, close-to-normal, shock wave), and the slipstreams, originatingfrom the triple points T , respectively. The Mach stem height is denoted as 2s.

b

w

θ

SS

SS

αIS

IS

2g

M

2h

2s

θ

T

RS

RS

MS

T

Figure 1: A schematic of double wedge configuration

This double wedge configuration is exactly the same as that experimentally studied inmany recent works aimed at investigation of the transition between steady regular andMach shock wave reflections. In the Euler computations below, the wedges are replacedby two inclined flat plates to make easier generation of a body-fitted structured grid. Weobserved no essential difference between results obtained with the wedges or the plates.

Owing to the symmetry of the problem, the computations are performed only in aquarter of the domain. The downstream boundary was chosen in such a way that the flowthere was supersonic. The upper and lateral boundaries were chosen sufficiently far fromthe body so that the uniform free stream flow could be prescibed on them.

The angle of wedge-generated oblique shock wave is changed by rotating the wedgearound its trailing (in the most part of simulations and experiments) or leading edges, sothat the distance between the horizontal symmetry plane and either the trailing edge gor the leading edge h is constant during the rotation.

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3 NUMERICAL AND EXPERIMENTAL TECHNIQUES

3.1 3D Euler simulations

Shock wave interaction is an essentially inviscid phenomenon. Therefore it can be de-scribed by the Euler equations rather the the Navier-Stokes equations, which significantlyfacilitates its numerical simulation. Nevertheless, such simulations at high supersonicMach numbers pose several problems for CFD methods. These are mainly related tostrong shock waves and very low density regions observed in different areas of the flow;unsteady phenomena connected with a physical instability of the slip surface; and thenon-uniqueness of steady solution. As it is well-known now, both regular and Mach re-flections are possible configurations in some range of the angle of shock incidence, andthe transition between them is accompanied by the hysteresis phenomenon [4, 5].

The three-dimensional unsteady Euler equations can be written as

∂Q

∂t+ ∇ · F = 0 (1)

Q =

ρρuE

, F =

ρuρuu + pI(E + p)u

, p = (γ − 1)(E −

1

2ρu2). (2)

Here ρ is the density, u is the velocity vector, E is the total energy per unit mass, p isthe pressure, γ is the specific heats ratio (γ = 1.4 for air) and I is the identity matrix.

The equations are solved with the high-order total variation diminishing (TVD) schemeon a multiblock structured grid. The HLLE (Harten-Lax-van Leer-Einfeldt) approximateRiemann solver [6] is used to calculate the numerical fluxes on the grid cell boundariesbecause of its robustness for flows with strong shock waves and expansions. If the su-perscripts L and R correspond to the states on the left-hand and right-hand sides of theboundary between the grid cells (i, j, k) and (i + 1, j, k), respectively, then, omitting thesubscripts j, k, the numerical flux through the boundary Fi+1/2 given by the HLLE solveris written as

Fi+1/2 =a+FL − a−FR + a+a−(QR − QL)

a+ − a−(3)

a− = min{0, uLn − cL, un − c},

a+ = max{0, uRn + cR, un + c}

Here un is the normal-to-the-boundary component of velocity, c =√

γp/ρ is the speedof sound and the tilde marks quantities in the state which is obtained by Roe averagingfrom the states QL and QR.

The variables in the states QL and QR is reconstructed from cell averaged ones usingthe 4-th order formula [7].

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qL = qi + (DqLi + 2DqR

i )/6, (4)

qR = qi+1 − (DqRi+1 + 2DqL

i+1)/6, (5)

DqRi = m(Dqi+1/2, bDqi−1/2),

DqLi = m(Dqi−1/2, bDqi+1/2),

Dqi+1/2 = ∆qi+1/2 − ∆3qi+1/2/6,

∆3qi+1/2 = ∆αqi−1/2 − 2∆βqi+1/2 + ∆γqi+3/2,

∆αqi+1/2 = m(∆qi−1/2, b1∆qi+1/2, b1∆qi+3/2),

∆βqi+1/2 = m(∆qi+1/2, b1∆qi−1/2, b1∆qi+3/2),

∆γqi+1/2 = m(∆qi+3/2, b1∆qi−1/2, b1∆qi+1/2).

Here ∆qi+1/2 = qi+1 − qi and minmod function m is determined as

m(x1, . . . , xn) =

s min(|x1|, . . . , |xn|),if sgn(x1) = . . . = sgn(xn) = s0, otherwise

The parameters b, b1 are taken in our computation as b = 4, b1 = 2.The reconstruction is applied to the primitive variables q = (ρ,u, p)T . The use of high

order reconstruction formula allows us to decrease a large numerical diffusion inherent tothe HLLE solver and provide a high resolution of the smooth part of the solution withoutloss of robustness near strong shock waves.

A multiblock body-fitted grid is used in simulations with the number of nodes rangingfrom hundreds of thousand up to several millions. The third order explicit TVD Runge-Kutta scheme [8] is chosen as a time stepping method. It should be noted that anaccurate modeling of unsteady phenomena can prove to be important in this problemwhere hysteresis phenomena are observed.

3.2 Wind-tunnel experiments

The experiments have been performed in the T-313 facility of the Institute of The-oretical and Applied Mechanics. T-313 is a supersonic blow-down wind tunnel with aclosed rectangular closed section of 600 mm×600 mm size. The experiments have beenperformed at flow Mach number M = 4 and 5 with stagnation temperature 290K, andstagnation pressure 106Pa. Two symmetrical wedges with the length w = 80 mm mountedin the wind tunnel test section were used as shock wave generators.

The angle of attack θ and, consequently, the angle of generated shock waves can bechanged by simultaneous rotation of the wedges with a pivoting mechanism. The gapbetween the wedge trailing edges g was fixed during rotation. The wedge models ofdifferent relative spans b/w = 0.66, 1, 2, 3.75 were used to elucidate the influence of theflow three-dimensionality on the transition.

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The investigation of 3D flow structure is a challenging task for the experimental re-search. Common schlieren visualization, which is widely used in experiments, producesflow images integrated along the spanwise direction — see below Figs. 3 and 7.

It is not easy to extract the information on 3D shock wave configuration from theseintegrated images. An oblique shadowgraphy technique with the inclined optical axiswas utilized in [2] for this problem and some valuable features of the flow could then bededuced from these visualizations. More detailed information can be provided by use ofthe laser sheet imaging technique (vapor screen visualization). This technique utilizesthe laser light scattering on micron-sized droplets of water. As a result, only the flow inthe plane of the laser sheet is visualized, in contrast to common schlieren technique. Thespanwise laser sheet visualization was employed in [1] to investigate the structure of 3Dshock wave reflection, and gave some promising results. In our study we developed thelaser sheet visualization in streamwise direction. A schematic of experimental set-up andoptical diagnostic system is shown in Fig.2.

3.8o

��

��

600

mm

input

output

PCTV

CCDcamera

Wedge

4500 mm

Argon laser

Optical systemControl

Shock wave

Video tape recorder

max

min

Laser sheetx

z

Figure 2: A schematic of experimental set-up (top view) and optical diagnostic system.

As a light source, an argon-ion laser was used. The power of the laser was adjusted at3 W to provide clear pictures.

The laser beam was expanded to a light sheet 1 mm thick by means of a combinationof convex and spherical lenses. The laser sheet was introduced into the flow using a smallprism mounted on a side wall in the subsonic part of the wind tunnel, approximately1 m upstream of the nozzle throat and 4.5 m upstream of the test section. It causesonly a small distortion of the surrounding low-speed stream and has no influence on the

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supersonic flow in the test section.In order to investigate the 3D structure of shock wave reflection configurations, the

laser sheet was traversed in the spanwise direction scanning the flowfield from the centralplane of the test section up to the wall. The maximum angle between the laser sheet andthe wind tunnel axis did not exceed 3.8◦. An appropriate amount of water (less than 1.1gH2O/kgair) was seeded in the wind tunnel before the settling chamber. Slices of the flowpattern were grabbed by the CCD camera mounted outside the test section. To avoid theimage scale disturbances the movement of the CCD camera was synchronized with thelaser sheet movement. The maximum number of flow images corresponding to differentstreamwise planes, which were obtained at different spanwise locations during one passacross the test section, was equal to 20. The CCD camera was connected to both thecomputer frame grabber and the video tape recorder (VTR).

4 RESULTS

4.1 3D regular reflection

Typical experimental and numerical schlieren images of RR are shown in Fig. 3 a andb, respectively.

a

b

Figure 3: Experimental schlieren a and ”numerical schlieren” b of regular reflection at the same flowconditions, M = 4, α = 39.73◦, b/w = 0.66, g/w = 0.3.

The numerical ”schlierens” are derived from the computed 3D density flowfield. First,the spanwise averaged density gradient is calculated as

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∇ρ(x, y) =∫

|∇ρ(x, y, z)| dz, (6)

and then displayed as a half-tone grey scale image with a special nonlinear scale (see [9]).Though some details, which are absent in a pure 2D flow, are distinguishable in these flowimages, in general, no clear idea about 3D shock configuration can be derived from them.

At the same time, numerical modeling provide a lot of information about the flowstructure. In particilar, it allows us to reconstruct directly 3D shapes of shock waves.Figure 4 show them for two different wedge width/length aspect ratios b/w = 0.66 andb/w = 2. They are visualized as density isosurfaces corresponding to some density valuebetween the free-stream and post-shock values.

Figure 4: Density isosurfaces ρ/ρ∞ = 2.2 for M = 5, α = 39.73◦, b/w = 0.66, g/w = 0.3 (left) and M = 4,α = 38◦, b/w = 2, g/w = 0.3 (right).

The isosurfaces demonstrate that the interaction of wedge-generated shock is regularonly not very far from the vertical plane of symmetry. An important feature of RRconfigurations is the existence, on the flow periphery, of a peripheral Mach reflection witha supersonic flow behind the Mach stem surface. There is a point where the RR andperipheral MR co-exist in equilibrium. The region with regular interaction shrinks as b/wdecreases.

The influence of side effects, which is manifested in curving the shock wave, is clearlyseen in Fig. 4. Naturally, it is more pronounced at a smaller value of b/w. A pure 2Dflow keeps near the vertical symmetry plane (in the coreflow). A spanwise bending of theincident shock wave leads to a decrease of the Mach number based on the normal-to-its-surface component of velocity. As a result, the flow behind the reflected wave can not beparallel to the horizontal symmetry plane, and MR should be formed on the periphery.However, the flow remains supersonic behind the peripheral Mach stem surface. Anemergence of the peripheral MR seems to be a common feature in similar 3D flows withshock waves. At least, it was observed in the numerical simulation [10] at the interactionof two conic shock waves.

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Emergence of the peripheral MR is also clearly visible in Fig. 5, which shows the flowpatterns in different spanwise cross-sections.

z/w = 0 z/w = 1.26 z/w = 1.38

a b c

Figure 5: 2D slices of numerical density flowfield for RR. M = 4, α = 38◦, b/w = 2, g/w = 0.3.

z = 4 mm z = 45 mm z = 59 mm

Figure 6: Laser-sheet images of RR at three spanwise positions. M = 4, α = 35◦, b/w = 0.66, g/w = 0.15.

Experimental results support this numerical finding. The laser-sheet images in Fig. 6taken at different distances z from the central plane show the birth and growth of theperipheral Mach stem, which is quite similar to that numerically observed.

4.2 3D Mach reflection

Three-dimensional effects are particularly essential for MR when a subsonic regionbehind the Mach stem surface exists and the rarefactions from the wedge lateral edges canaffect the core flow through this region. In other words, MR is always three-dimensional.

The experimental and numerical schlieren images for MR are given in Fig. 7. A numberof details originated from 3D effects can be seen even in these integrated flow images, e.g.additional black lines indicating a variation of the triple point position with the spanwisecoordinate. Much more explicitly this behavior can be observed in Fig. 8, which showsthe 3D shape of shock waves visualized as a density isosurface, as well as in Figs. 9 and 10where numerical and experimental flow patterns in different spanwise cross-sections aregiven.

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a

b

Figure 7: Experimental schlieren a and ”numerical schlieren” b of Mach reflection at the same flowconditions, M = 4, α = 40◦, b/w = 2, g/w = 0.3.

It should be noted that though MR exists at all spanwise locations, the flow onlybehind the central part of the Mach stem surface (in the coreflow) is subsonic. On theperiphery, the same peripheral MR as for RR with the Mach stem surface convex to theoncoming flow is formed. Far from the test model, both the incident shock and Machstem degenerate into a weak conical shock wave.

An interesting feature of 3D Mach reflection is a non-monotonous behaviour of theMach stem height in the spanwise direction. It can be seen that the Mach stem heightis almost constant in the coreflow, where the influence of side effects is not significant.At larger distances from the vertical symmetry plane, the Mach stem height decreasesnoticeably, and then considerably increases on the periphery. Such behavior can be hardly

Figure 8: Density isosurfaces ρ/ρ∞ = 2.2 for M = 5, α = 40◦, b/w = 1.1, g/w = 0.24 (left) and M = 4,α = 37◦, b/w = 3.75, g/w = 0.3 (right).

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z/w = 0 z/w = 1.38

a b c

z/w = 0.78

Figure 9: 2D slices of numerical density flowfield for MR. M = 4, α = 38◦, b/w = 2, g/w = 0.3.

z = 138 mm z = 178 mmz = 0 mm

Figure 10: Laser-sheet images of MR at three spanwise positions. M = 4, α = 37◦, b/w = 3.75,g/w = 0.3.

foreseen a priori.The experimental laser sheet visualizations allow us to measure the Mach stem heights

variation along the spanwise coordinate, which is shown in Fig. 11. For comparison, theMach stem heights obtained in 3D Euler computations at the same conditions are alsogiven. It is quite evident, that experimental and numerical data are in close agreement.Thus, numerical simulations are quite successful in predicting these complicated 3D shock-dominated flows.

4.3 Combined reflection

Computations demonstrate that if the Mach stem height in the central vertical planeis small enough, then a particular shock wave configuration with a combined type ofreflection is possible: MR with a small stem in the central plane, RR at some spanwisedistance to the periphery, and MR again — even farther to periphery. This shock reflectionconfiguration, first observed numerically, is shown in Fig. 12.

In a combined shock reflection configuration there exist two points where differenttypes of reflection meet. In the first one the transition from MR to RR occurs, the triplepoints merge and disappear as well as slipstreams emanating from them. In the secondone, on the contrary, peripheral MR appears owing to shock wave bending and decrease

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0 0.5 1 1.5 2 2.5z/w

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

s/w

Experiments

2D computations3D computations

Figure 11: Spanwise variation of Mach stem height. M = 4, α = 37◦, b/w = 3.75.

in the Mach number normal to the shock wave surface.The existence of combined reflection has been also confirmed experimentally. Series

of images in Fig. 13 shows the Mach reflection at z = 0mm, then regular reflection atz = 158mm, and peripheral Mach reflection at larger distance from the central planez = 198mm.

4.4 Transition and hysteresis phenomenon

When increasing the angle of shock incidence, the transition from RR to MR in 2Dnumerical simulations occurs approximately at αD, an angle which is deduced from so-called detachment criterion [4]. For Mach numbers 4 and 5, αD = 39.2◦ and αD = 39.3◦,respectively. The reverse transition from MR to RR is observed near αN , an angle given byvon Neumann criterion, αN = 33.4◦ at M = 4 and αN = 30.8 at M = 5. The applicationof both detachment and von Neumann criteria to 3D flows is questionable [11, 12]. Inparticular, when α → αD, the Mach number behind the reflected shock wave of RR isabout unity, and the influence of side effects may extend inwards up to the vertical planeof symmetry. As it was already mentioned, 3D effects should be even more pronouncedin the case of MR when a subsonic region exists, and information from the periphery canpropagate to the coreflow.

The results of 3D numerical simulations of the transition between RR and MR alongwith those of experimental measurements are shown in Fig. 14 where the dependence of

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Figure 12: Shock wave reflection of combined type. M = 4, α = 35.5◦, b/w = 3.75, g/w = 0.3.

z = 158 mmz = 0 mm z = 198 mm

Figure 13: Laser sheet images of combined at three spanwise positions. M = 4, α = 34◦, b/w = 3.75,g/w = 0.3.

the Mach stem height in the central plane on the shock wave angle is shown.For M = 4, the transition occurs when α is increasing from 39◦ to 40◦. When the shock

angle is decreasing, MR is preserved down to 36◦, and RR is formed at α = 39◦. Thus, theresults obtained give clear evidence that the final reflection type may be really dependingon initial conditions, so that the hysteresis phenomenon is observed. The transition anglesare in reasonable agreement with the theoretical values. An earlier transition from MRto RR than in experiments can be explained by the very small Mach stem height at theangles close to αN . This height becomes comparable with the size of numerical cells andcannot be resolved without further grid refinement. It is also clear that the numerical andexperimental Mach stem heights are in beautiful agreement except the values at small αwhere the earlier transition to RR explained above is observed in numerical simulations .

Figure 15 illustrates the influence of the wedge span on the Mach stem heights. The ex-erimental data are taken from the test with b/w = 3.75. 2D numerical results (b/w = ∞)overestimate experimental values, and the difference becomes more pronounced for largeα. This is because 3D effects increase when increasing α. The results for 3D computa-tions with b/w = 3.75 coincide with the experiment very closely. Further decreasing ofthe aspect ratio leads to reduction of the Mach stem height.

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s/w0.20

0.15

0.10

0.05

0.0034 36 38 40 42αα

3D computationsExperiments

N

Figure 14: Comparison of experimental and numerical Mach stem heights at M = 4, b/w = 2, g/w = 0.3.

5 CONCLUSIONS

A study of 3D structure of regular and Mach configurations and the transition betweenthem has been performed using numerical simualtions and wind tunnel experiments. Re-construction of 3D shapes of shock waves has been performed from numerical flowfieldsand experimental laser-sheet images taken at different spanwise locations. A number ofunexpected details have been observed in numerical simualtions: formation of peripheralMach reflection far from the vertical central plane, a non-monotonous variation of Machstem height with the distance from the central plane, existence of a combined type ofshock interaction with intermittent regular/Mach/regular reflections. All these numericalfindings have been confirmed by experimental investigations. Quantitative comparisonsof numerical and experimental results show their excellent agreement. The general con-clusion can be given that modern numerical schemes applied to the 3D Euler equationsenable us to predict complicated shock wave configurations with great robustness andaccuracy.

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0.40

0.30

0.20

0.10

0.00 32 36 40 44 48

S/W

α

3D computations

2D computations

Experiments

αN

2D

, degrees

b/w = 3.75

b/w = 2.0

Figure 15: Influence of aspect ratio b/w on Mach stem heights. M = 4, g/w = 0.56.

REFERENCES

[1] N. Sudani, M. Sato, M. Watanabe, J. Noda, A. Tate, and T. Karasawa. Three-dimensional effects on shock wave reflections in steady flows. AIAA Paper 99-0148,1999.

[2] B.W. Skews. Three-dimensional effects in wind tunnel studies of shock wave reflec-tion. J. Fluid Mech., 407, 85–104, 2000.

[3] M.S. Ivanov, D. Vandromme, A.N. Kudryavtsev, V.M. Fomin, A. Hadjadj, andD.V. Khotyanovsky. Transition between regular and Mach reflection of shock waves.Shock Waves, 11, 199–207, 2001.

[4] M.S. Ivanov, S.F. Gimelshein, A.E. Beylich. Hysteresis effect in stationary reflectionof shock waves, Phys. Fluids, 7, 685–687, 1995.

[5] A. Chpoun, D. Passerel, H. Li, G. Ben-Dor. Reconsideration of oblique shock wavesreflections in steady flows. Part 1. Experimental investigation. J. Fluid Mech., 301,19–35, 1995.

[6] B. Einfeldt, C.D. Munz, P.L. Roe, B. Sjogren. On Godunov-type methods near lowdensities. J. Comput. Phys., 92, 273–295, 1991.

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[7] S. Yamamoto, H. Daiguji. Higher-order-accurate upwind schemes for solving the com-pressible Euler and Navier-Stokes equations. Computers and Fluids, 22, 259–270,1993.

[8] C.-W. Shu, S. Osher. Efficient implementation of essentially non-oscillatory shockcapturing schemes, J. Comput. Phys., 77, 439–471, 1998.

[9] J.J. Quirk. A contribution to the great Riemann solver debate. Int. J. Numer. Meth-ods in Fluids, 18, 555–574, 1994.

[10] F. Marconi. Shock reflection transition in three-dimensional steady flow about inter-fering bodies. AIAA Journ., 21, 707–713, 1983.

[11] V.M. Fomin, M.S. Ivanov, A.M. Kharitonov, et al. The study of transition betweenregular and Mach reflections of shock waves in different wind tunnels. Proc. of 12thInt. Mach Reflection Symp. (Pilanesberg, South Africa, 1996), 137–151, 1996.

[12] B.W. Skews. Aspect ratio effects in wind tunnel studies of shock wave reflectiontransition. Shock Waves, 7, 373–383, 1997.

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THREE-DIMENSIONAL STEADY SHOCK WAVE · PDF fileIt is not easy to visualize complex 3D shapes of shock waves in experiments ... Mikhail S. Ivanov et al. the Mach stem ... non-uniqueness