On the onset of postshock flow instabilities over concave surfacesM. S. Shadloo, A. Hadjadj, and A. Chaudhuri

Citation: Physics of Fluids (1994-present) 26, 076101 (2014); doi: 10.1063/1.4890482 View online: http://dx.doi.org/10.1063/1.4890482 View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/26/7?ver=pdfcov Published by the AIP Publishing

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.66.116.143 On: Mon, 28 Jul 2014 12:18:10

PHYSICS OF FLUIDS 26, 076101 (2014)

On the onset of postshock flow instabilities overconcave surfaces

M. S. Shadloo, A. Hadjadj,a) and A. ChaudhuriCORIA-UMR 6614, Normandie University, CNRS-University & INSA of Rouen,76800 Saint Etienne du Rouvray, France

(Received 9 October 2013; accepted 28 April 2014; published online 28 July 2014)

This work reports a numerical investigation of shock focusing phenomena over con-cave surfaces. The study focuses on the effects of Reynolds and Mach numberson the detailed behavior of flow features related to shear-layer instabilities and jetformation in the post-shock region. Computations are done for four incident-shockMach numbers covering subsonic and transonic flow regimes and a wide range ofReynolds numbers. The simulations reveal a number of interesting wave featuresstarting from early stage of shock interaction and transition from inverse-Mach re-flection to transitioned regular reflection followed by very complex flow patternsat focusing and post focusing stages. Different subsequent flow characteristics de-velop as a result of multiple shock/shear layer interactions. During the later stageof the flow interaction, a formation of two opposing jets is predicted by the simula-tion in accordance with the experiments. It is shown that the formation of primaryopposing jets as well as the development of Kelvin-Helmholtz instabilities can behindered for low Mach and Reynolds numbers. However, for high flow regimes asecond pair of opposing jets appears and develops far from the wall, exhibiting sim-ilar features as the primary pair of opposing jets at moderate Mach numbers. Twonew bifurcations in flow patterns are observed at this stage which promote furtherdevelopment of vortex structures and shear-layer rollup. C© 2014 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4890482]

I. INTRODUCTION

Understanding of complex flow features associated with the interaction of shock waves withreflector surfaces is important for various applications, such as aviation safety, detonation ignition,mining, and biomedical research. When a shock wave encounters a reflector surface, a region ofhigh pressure and temperature develops in the vicinity of the symmetry axis of the reflector, wherea shock focusing phenomenon occurs. Many researchers contributed towards the knowledge ofunsteady shock waves interacting with solid obstacles and associated complex wave structures. Acomprehensive discussion is given by Ben-Dor1 for regular and irregular shock-wave reflections insteady and unsteady flows. In particular, Izumi et al.2 showed that depending upon the shape of thereflector (curvature) and the speed of the shock, transition from one reflection pattern to anothertakes place.

Studies presented in Refs. 2–6, dealing with shock focusing phenomena, reveal a highly unsteadycharacter of the shocked flow past various reflector configurations. Numerical simulations predictingsuch phenomena2, 7–11 essentially relied on inviscid assumption, while successfully captured the basicshock structures (early stage of inverse-Mach reflection (invMR) → transitioned regular reflection(TRR) transition). Other aspects of energy concentration by spherical converging shocks producedin a conventional shock tube with a circular cross-section are studied by Kjellander et al.12

a)Author to whom correspondence should be addressed. Electronic mail: abdellah.hadjadj@coria.fr

1070-6631/2014/26(7)/076101/11/$30.00 C©2014 AIP Publishing LLC26, 076101-1

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.66.116.143 On: Mon, 28 Jul 2014 12:18:10

076101-2 Shadloo, Hadjadj, and Chaudhuri Phys. Fluids 26, 076101 (2014)

Modeling of such complex flows requires accurate descriptions of both non-stationary shocksand unsteady shear layers. In the present study, viscous effects are included by means of directnumerical simulations of two-dimensional compressible flows with particular emphasis on the long-time flow evolution. The study follows the recent experiments by Skews and Kleine6 on shockfocusing over a cylindrical reflector, which highlighted a formation of new flow features during thelater time using time-resolved imaging. In particular, the development of two opposite jets are cap-tured in the sheared flows for weak Mach numbers. The authors also acknowledged the necessity ofperforming numerical simulations for further understanding of these new unsteady features. Similartrends are also observed by Gefland et al.13 who studied experimentally the detonation/deflagrationprocesses initiated by semi-cylinder and parabolic focusing elements. In terms of mixing, linearstability analysis of both temporally- and spatially evolving compressible shear layers highlightedthe two-dimensional nature of the most amplified disturbances up to convective Mach numbers of0.6. Although, for higher Mach numbers the Kelvin-Helmholtz instabilities are inhibited and theflow becomes strongly three-dimensional, the two-dimensional solutions for such problems stilloffer new insights into the shock focusing phenomena.

The objective of the present work is to investigate the effects of both Reynolds and Machnumbers on the post-shock flow instabilities. In this regard, the effect of pressure on the appearanceof the secondary (backward) jet and the speed of the primary (forward) jet as well as vortex formationon the slipstreams are studied, while keeping the incident shock Mach number, Ms, constant. This,in turn, will depict the dependency of the flow evolution on the Reynolds number. Furthermore, theeffect of Ms is investigated for a given Reynolds number. The values of Ms are chosen such thatthe shocked flow can be either subsonic or transonic. In the later case, the formation of a stationaryshock wave at the cavity entrance creates additional complexities that act on the post-focusing flow.

The paper is organized as follows. A brief description of the problem setup together with therelevant boundary conditions is presented in Sec. II. Time evolution of the incident shock reflectedoff a concave surface is discussed in Sec. III. Furthermore, the effects of Reynolds and Mach numberson the post-focusing features are investigated and illustrated in Sec. III. Finally, the main conclusionsare drawn in Sec. IV.

II. PROBLEM SETUP

A planar moving shock wave at Ms is allowed to impact a cylindrical concave surface of 64 mmradius. The computations are carried out on a domain of 390 mm × 64 mm, which fits the dimensionsof Skews and Kleine’s6 experiments. Direct numerical simulations are used to compute fully solutionof compressible Navier-Stokes equations. The set of equations is solved using a fifth-order WeightedEssentially Non-Oscillatory (WENO) scheme in conjunction with a fourth-order compact centraldifferencing formula for the diffusive terms. Time is advanced using a third-order Runge-Kutta TotalVariation Diminishing scheme. To take advantage of the high accuracy of the numerical scheme,a Cartesian grid based immersed boundary method is used. Detailed descriptions of the numericalprocedure can be found in the authors previous works.14–16

Free-slip and adiabatic boundary conditions are applied at the body surface, while the bottomboundary is fixed with a slip condition to avoid wave reflections. Since only half of the reflectoris computed, symmetry conditions are imposed at the top boundary. The initial flow conditionsfor the shocked gas (left state, subscripted as “2”) and the stagnant gas (right state, subscriptedas “1”) are specified using Rankine-Hugoniot relations for a given Mach number. The Reynoldsnumber is defined based on the initial shocked-gas state and the radius, r, of the concave reflector asRe = u2 r/ν2, where u and ν denote velocity and kinematic viscosity, respectively. The simulationsstart when the shock reaches the leading edge of the reflector.

The results are reported for a mesh having 11.5 × 106 grid points (i.e., 4800 × 2400 in x andy directions, respectively). The mesh is uniformly distributed in the cavity domain (including thereflector and an upstream extent of 2 r) with �y ≈ 27 μm and �x ≈ 40 μm and stretched in theupstream direction over a distance of 3 r. This provides a higher spatial resolution in the desiredportion of the computational domain. A grid sensitivity study using coarser (1200 × 600 and 2400× 1200) and finer (9600 × 4800) meshes showed that the most relevant flow features are well

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.66.116.143 On: Mon, 28 Jul 2014 12:18:10

076101-3 Shadloo, Hadjadj, and Chaudhuri Phys. Fluids 26, 076101 (2014)

captured with the medium grid. For flow visualizations, numerical schlieren pictures, based ondensity gradient portrayed on a non-linear scale, are generated following the procedure describedby Hadjadj and Kudryavtsev.17

III. RESULTS AND DISCUSSIONS

A. Time evolution

The various stages of shock reflection and shear layer formation for Ms = 1.35 and Re = 9.33× 105 are shown in Fig. 1. Since the entrance wall of the reflector is parallel to the streamwisedirection, there is no initial Mach reflection in this region (Fig. 1(a)). As the shock progresses,continues compression waves exerted by compressive acoustic signals (C) cause the formation ofa kink on the main incident shock (I). The acoustic waves are generated due to the steepening ofthe wall angle with respect to the incident shock. Different configurations of shock reflection areobserved. The simple structure of Mach reflection is shown in Fig. 1(b) which turns into three-shockinteractions and a triple point (TP1). Due to the entropy unbalance, a slipstream (L1) is generatedfrom TP1.

Although the triple point initially moves away from the wall (see Skews and Kleine6), it goestowards the concave surface as the incident shock moves forward. An invMR is created and remainsuntil the triple point reaches the surface body (i.e., the length of the Mach stem becomes zero).Subsequently TRR takes place (see Ben-Dor and Elperin18). At this stage, a second triple point(TP2) appears, along with a new slipstream (S1) that links TP2 to the initial slipstream L1 (seeFig. 1(c)).

The new reflected shock (F) is shown to emanate from the bottom of the cavity which interactswith the existing shocks and forms a rhombus-like pattern at the end-right of the domain (Fig. 1(d)).

FIG. 1. Numerical schlieren pictures for Ms = 1.35 and Re = 9.33 × 105 at (a) t = 50 μs, (b) t = 100 μs, (c) t = 132 μs,(d) t = 150 μs, (e) t = 250 μs, (f) t = 300 μs, (g) t = 350 μs, and (h) t = 1050 μs. C: compressive acoustic waves, I:incident shock, R1: primary reflected shock, L1: primary slipstream, TP1 and TP2: first and second triple points, F: newreflected shock, S1: secondary slipstream, W: shock at the wall, M: main reflected shock, J1 and J2: primary and secondaryjets.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.66.116.143 On: Mon, 28 Jul 2014 12:18:10

076101-4 Shadloo, Hadjadj, and Chaudhuri Phys. Fluids 26, 076101 (2014)

Subsequently, the merged shock, having a concave shape, moves towards the cavity entrance, whilethe secondary triple point propagates towards the symmetry axis. When TP2 hits the symmetryplane, a quadruple point known as the focal point, is produced.

As M progresses, it becomes more curved and the associated triple points gradually vanishwhile propagating towards the wall (Figs. 1(e)–1(g)). Consequently, S1 is detached from it. Being ashear layer, a short distance behind the shock, it forms a mushroom-like shape with the opposite-sidelayer, hereafter called primary jet (J1), with a pair of vortices and two side tails (Figs. 1(f)–1(h)).

On the other side and in the distance closer to the cavity bottom, a bifurcated flow patternbecomes apparent, and a small reversal jet (J2) appears at the junction of the main shear layer(Fig. 1(h)). It is found that the primary jet moves towards the cavity entrance with a nearly constantvelocity (will be shown in detail in Sec. III B). Both J1 and J2 spread in y direction in addition totheir elongation in the streamwise direction. Unlike the primary jet, the streamwise elongation of J2is mainly due to the movement of its stem, while its tip is almost stationary. It should be mentionedthat after the focusing and as long as the main reflected wave gets bended and interacts with thewall, an ephemeral triple point associated with an apparent invMR is observed at the wall (Figs. 1(f)and 1(g)).

For further analysis, the numerical results are directly compared to the experiments of Skewsand Kleine.6 Figure 2 illustrates the experimental shadowgraph pictures and their correspondingnumerical schlieren images. The exposure time interval is �t = 48 μs. As it can be seen, excellentagreement is found between the current simulations and the experimental data in terms of size,shape, and location of different waves and flow structures. It is worth mentioning that for comparisonpurpose, a small inlet ramp is added to the entrance of the cavity, to better fit with the geometry “A”of Skews and Kleine’s6 experiments. The ramp has a 3 mm length and a 1 mm height, and causes asmall lip shock visible in Figs. 2(a) and 2(b).

B. Effect of Reynolds number

The Reynolds number is varied from 9.33 × 103 to 9.33 × 107 by changing both P1 and P2

while keeping their ratio constant for a given Ms, taken here as 1.35 (see Table I). The temperaturesin both sides are kept constant. The simulation showed that the main flow features at early stage ofthe interaction remain unchanged with respect to Re. Therefore, the results are mainly analyzed forthe late stage of the focusing process. As shown in Fig. 3, the post shock instabilities are weakenedwhen decreasing Re. In fact, at very low Reynolds numbers, the secondary jet vanishes and theprimary jet is barely visible due to the low rate of density change across the slipstreams. It can alsobe noted that the primary jet moves at a relatively low speed and the associated Kelvin-Helmholtzvortices are inhibited as the Reynolds number decreases. Evidently, the relative decrease of Re hasa dissipation effect (viscous damping) on the long-term evolution of both jets and their associatedshear layers (Figs. 3(a) and 3(b)).

On the other hand and with increasing the Reynolds number, the primary jet becomes strongerand reaches asymptotically a steady-state velocity (with the rolling up of the tails towards thecenterline, see Figs. 3(c) and 3(d)). Space-time evolution of the jet-leading edge positions is plottedfor different Re as a distance from the cavity bottom against time. The slope of each line representsthe velocity of the jet in the laboratory frame for a given Reynolds number. As it can be seen fromFig. 4, the jet is moving at a very low speed compared to the sound speed in the shocked flow andthis velocity reaches a plateau for high Reynolds numbers.

C. Effect of Mach number

The relationship between the Mach number in the shocked flow, M2, and the incident Mach num-

ber, Ms, is given by1 M2 = (M2s − 1)/

√(1 + γ−1

2 M2s )(γ M2

s − γ−12 ). Depending on the operating gas

(here is air with γ = 1.4) and Ms, the shocked flow can be subsonic, transonic, or supersonic. Uponsetting M2 = 1, one can determine the critical Mach number, Ms,cr, above which the shocked flow issupersonic (i.e., M2 > 1). It is noted that the (M2 − Ms) relation has two roots, distinguishing thereby

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.66.116.143 On: Mon, 28 Jul 2014 12:18:10

076101-5 Shadloo, Hadjadj, and Chaudhuri Phys. Fluids 26, 076101 (2014)

FIG. 2. Experimental shadowgraph images of the post-shock process for Ms = 1.35 and their corresponding numericalschlieren pictures taken at (a) t = 158 μs, (b) t = 206 μs, (c) t = 254 μs, (d) t = 302 μs, (e) t = 350 μs, and (f) t = 398 μs.The time gap between the frames is �t = 48 μs. Experimental data reprinted with permission from B. W. Skews and H.Kleine, J. Fluid Mech. 580, 481–493 (2007). Copyright 2007 Cambridge University Press.

weak and strong solutions, which are in our case Ms,cr = 0.624 and Ms,cr = 2.068, respectively. Thelater value is of interest since it provides the condition for which the shock can exist (i.e., Ms > 1).Note that the maximum reachable Mach number for the shocked flow is M2 = 1.75 for the experi-mental achievable pressure ratios.

In the present section, four different incident Mach numbers, Ms =1.05, 1.35, 1.7, and 2 areconsidered to cover both subsonic and transonic regimes (see Table II). It is worth noting that

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.66.116.143 On: Mon, 28 Jul 2014 12:18:10

076101-6 Shadloo, Hadjadj, and Chaudhuri Phys. Fluids 26, 076101 (2014)

TABLE I. Flow conditions for Ms = 1.35 with different Reynolds numbers.

Case P1 (kPa) T1 (K) P2 (kPa) T2 (K) Re × 105

1 0.95 300 1.8616 366.78 0.009332 1.9 300 3.7232 366.78 0.09333 9.5 300 18.616 366.78 0.9334 95 300 186.16 366.78 9.335 950 300 1861.6 366.78 93.36 9500 300 18616 366.78 933

FIG. 3. Numerical schlieren pictures at t = 1500 μs for Ms = 1.35 and (a) Re = 9.33 × 103, (b) Re = 9.33 × 104, (c) Re =9.33 × 105, and (d) Re = 9.33 × 107.

(a) (b)

FIG. 4. (a) Space-time evolution of the primary jet for different Reynolds numbers, (b) velocity of the primary jet scaled bythe speed of sound in the shocked flow, c2, as a function of Reynolds number. xj is the distance of the jet leading front fromthe cavity bottom and uj = dxj/dt.

TABLE II. Flow conditions for different incident Mach numbers.

Case Ms P2 (kPa) T2 (K) c2 (m/s) M2 Re × 105

7 1.05 106.36 309.85 347.19 0.08 1.1468 1.35 186.16 366.77 383.89 0.45 9.3289 1.7 304.47 437.50 419.27 0.77 20.5710 2.0 427.50 506.25 451.01 0.96 30.45

the study of supersonic regimes (Ms > 2) involves dominating compressibility effects with three-dimensional flow organization. To avoid such complex flows, the current investigation is limited toMs < 2. In the following, two different flow regimes will be considered; subsonic (M2 < 1) andtransonic (M2 ≈ 1).

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.66.116.143 On: Mon, 28 Jul 2014 12:18:10

076101-7 Shadloo, Hadjadj, and Chaudhuri Phys. Fluids 26, 076101 (2014)

FIG. 5. Numerical schlieren pictures for Ms = 1.05 and Re = 1.15 × 105 at (a) t = 50 μs, (b) t = 100 μs, (c) t = 200 μs,(d) t = 250 μs, (e) t = 300 μs, (f) t = 350 μs, and (g) t = 400 μs. For notations, see Fig. 1.

1. Subsonic shocked-flow

For weak and moderate incident shock waves, the shocked gas flow remains weakly compressible(cases 7 and 8 in Table II). Since the flow at Ms = 1.35 was discussed in detail in Secs. III A and III B,only the case of Ms = 1.05 will be considered here. First, the interaction starts with a regular reflectionand the incident shock remains straight until a TRR occurs (Fig. 5(b)). This transition happens atthe position closer to the cavity entrance and later in time compared to case 8.

While traveling towards the cavity entrance, the reflected shock (M) hits the wall first in a regularway, before the transition to a double regular reflection occurs (Figs. 5(f) and 5(g), also see Skewsand Kleine’s6 Fig. 9(a)). It is interesting to note that the slipstreams of the three-shock interactionfor Ms = 1.05 are weaker compared to Ms = 1.35 and they did not exhibit strong vortices. Thedeveloped jet is relatively small in size, slow in speed and weak on its rotational strength comparedto that of case 8. Consequently, the main flow stream is not capable of producing a secondary jet.

2. Transonic shocked-flow

For stronger incident shock waves, an auxiliary moving shock (T) appears at the cavity entrance(Fig. 6(a)). This shock becomes stronger with slowly upstream-running ahead of the obstacle as Ms

increases and finally transforms into a stationary shock for Ms > Ms,cr.As the incident shock wave (I) progresses, the triple point appears sooner in time and at a

position closer to the cavity entrance compared to the subsonic cases. However, the creation of TRRoccurs at a position closer to the cavity bottom. Soon after TRR, S1 interacts with L1 to form apair of vortices at their junctions. These vortices grow mainly on the slipstream S1, initially in adirection perpendicular to its elongation and create a mushroom shape structure, here after calledS1-mushroom, which moves towards the cavity bottom (see Figs. 6(b)–6(f)).

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.66.116.143 On: Mon, 28 Jul 2014 12:18:10

076101-8 Shadloo, Hadjadj, and Chaudhuri Phys. Fluids 26, 076101 (2014)

FIG. 6. Numerical schlieren pictures for Ms = 1.7 and Re = 2.06 × 106 at (a) t = 50 μs, (b) t = 125 μs, (c) t = 150 μs,(d) t = 175 μs, (e) t = 200 μs, (f) t = 300 μs, (g) t = 375 μs, and (h) t = 500 μs. T: shock at the entrance, J2′: jet-likestructure. For the remaining notations, see Fig. 1.

It is known from earlier studies10 that the focal point forms at the distance closer to the cavitybottom for higher Ms. As a consequence, both J1 and J2 take place close to the cavity bottom(Figs. 6(d) and 6(e)). The primary jet has a higher velocity and exhibits strong vorticity magnitudecompared to the previous cases. On each side of the main-jet stem, two new vortices appear. Thesevortical structures, which are not visible in weak incident Mach numbers, pair and create a cat-eyelike shape (Figs. 6(f)–6(h)). The secondary jet, on the other hand, moves towards the cavity entranceunlike the subsonic cases. Meanwhile, the S1-mushroom growing eddy interacts with other flowinstabilities and promotes the emergence of large-scale structures at the cavity bottom. This resultsin a creation of a jet-like structure (here after called jet J2′), which moves in the same direction asJ1 (Fig. 6(h)). As mentioned before, for high-Mach number flow regime, the shock (T) stands atthe cavity entrance and interacts with the main reflected shock (M) (Fig. 7). This interaction causesthe formation of a new triple point with an associate slipstream (S2). As the main reflected shockprogresses downstream of the cavity bottom, the size of T decreases (Fig. 7(d)). While they arealigned and have comparable height, the reflected shock stem (H) and the shock at the entrance Tmerge to create a new combined shock (TH) with a vertical slipstream (V) behind. Additionally,

FIG. 7. Numerical schlieren pictures for Ms = 2.0 and Re = 3.04 × 106 at (a) t = 130 μs, (b) t = 170 μs, (c) t = 230 μs,and (d) t = 280 μs. H: new Mach stem. For the remaining notation see Figs. 6 and 1.

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.66.116.143 On: Mon, 28 Jul 2014 12:18:10

076101-9 Shadloo, Hadjadj, and Chaudhuri Phys. Fluids 26, 076101 (2014)

FIG. 8. Numerical schlieren pictures for Ms = 2.0 and Re = 3.04 × 106 at (a) t = 320 μs, (b) t = 400 μs, (c) t = 600 μs,and (d) t = 800 μs. V: vertical slipstream, TH: combined shock. For the remaining notation see Figs. 6 and 7.

three other slipstreams are depicted at the cavity entrance: (i) L2 which was initially connected toH, (ii) S2 which was created at the intersection of M and T, and (ii) the new generated S3, which isconnected to the new triple point that comes from the intersection of M, TH, and the new reflectedshock, R2, created after shocks merging.

The slipstream (L2) elongates and stays connected to the slipstream (S2). The slipstream (S2),which is connected from one side to the slipstream (L2), elongates and joins the stem of primaryjet (J1) from the other side. Later in time the Kelvin-Helmholtz kind of instabilities form on thisslipstream (Fig. 8(b) and 8(c)). The slipstreams V and S3 remain connected and create vortices ontheir joint branches. The slipstream (S3) elongates from its side connected to the triple point, TP4(Fig. 8(b)). TP4s from either side collide at the symmetric axis and create the second focal point(see also Figs. 4– 6(a) in Ref. 13). At this point, once more two slipstreams pass each other and asa consequence the second cross-like shape forms (Fig. 8(c)). From this point, the second cross-likeshape acts similar to those in subsonic flow. The slipstream (S3) is divided into two parts. Two newjets appear and move in opposite direction. However, the sizes of both jets are almost the same andtheir absolute velocities are more close to each other. The Kelvin-Helmholtz instabilities becamevisible on either slipstreams which grow faster on the part of S3 that are connected to J4 (Fig. 8(d)).The existence of at least three stagnation points on the centerline are apparent at this time, whichare between: (i) J3 and J4, (ii) J4 and J1, and (iii) J2′ and the cavity bottom. Finally, J1 exhibits avery complex behavior and J2′ joins J1 from its beneath to act as a stem for it (Figs. 8(c) and 8(d)).

Space-time evolution of the jet-leading edge for different incident Mach numbers is plotted inFig. 9. Unlike the Reynolds variation, the velocity of the primary jet does not reach a plateau butincreases linearly with an increment in Ms (Fig. 9(b)). It is noted that the evolution of the primaryjet-leading edge is not linear in time for higher values of Ms. Hence, for Ms = 1.7, the slope of theline continuously decreases, however, for Ms = 2, it initially decreases and then increases. The dropin the slope is due to a deceleration caused by an increment in the jet cross-sectional area, which isa consequence of sequential growing and pairing of vortices followed by their interaction with the

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.66.116.143 On: Mon, 28 Jul 2014 12:18:10

076101-10 Shadloo, Hadjadj, and Chaudhuri Phys. Fluids 26, 076101 (2014)

FIG. 9. (a) Space-time evolution of the primary jet for different Mach numbers, (b) velocity of the primary jet scaled by thespeed of sound in the shocked flow, c2, as a function of Mach number. xj is the distance of the jet leading front from thecavity bottom and uj = dxj/dt.

primary jet. On the other hand, the slope increment is a result of an acceleration caused by joininghigh-speed J2′ to J1 from its beneath (see Fig. 9(a) and also Figs. 8(c) and 8(d)). Note that if weconsider an averaged jet velocity for higher values of Ms, a quasi-linear relationship exists betweenthe jet speed and the incident Mach number (see Fig. 9(b)).

IV. CONCLUSIONS

Direct numerical simulations are carried out to study the main flow features occurring duringa shock focusing process over a cylindrical cavity, with particular emphasis on the post-shock flowinstabilities. The simulations accurately reproduced the early stage of shock-reflection patterns andsubsequent complex shear-layer formations. In all considered cases, the post-shock instabilitiesare seen to be driven by a complex mechanism of vortex generation through a main jet streamthat flows towards the cavity entrance at a very low speed. The velocity of this jet is shown tobe linearly dependent on the incident shock-Mach number and asymptotically constant for highReynolds numbers. Also, the onset of the jet instabilities is found to be governed mainly by theinviscid Kelvin-Helmholtz mechanism. Simulations reveal the progressive rollup and pairing of thejet shear layer with increasing the Reynolds number.

The effect of the incident Mach number is even more evident on both pre- and post-focusingphenomena. For weak shocks (Ms � 1.1), the initial reflection is regular and the focal point appearsfar from the cavity bottom. Similar to the low-Reynolds number cases, the formation of a secondaryjet and shear-layer instabilities is hindered. However, with increasing the incident Mach number,the primary jet grows rapidly and the secondary jet moves progressively towards the cavity entranceto finally become a part of the main jet. The shear-layer instabilities also emerge faster when Ms

increases. In the transonic regime, an oblique shock is formed at the cavity lip, which interactsindirectly with the jet creating thereby a complex mixing zone with sequence of vortex growing andpairing.

Furthermore, the results highlighted the appearance of a second focal point as a consequenceof the interaction between the main reflected wave and the stationary shock at the entrance. Thisfocal point becomes closer to the cavity bottom with increasing Ms. A second pair of jets is alsoseen to develop at the second focal point, which exhibits similar behavior as the first pair of jet insubsonic flow regimes. At this stage, two new bifurcations in flow patterns are observed which fosteradditional vortices and shear-layer rollup.

As an outlook, three-dimensional simulations including boundary-layer effects will be consid-ered in the future to study the influence of the near-wall effects on the post-shock flow instabilities.

ACKNOWLEDGMENTS

The authors acknowledge the financial support of the french National Research Agency(ANR), through the program “Investissements d’Avenir” (ANR-10-LABX-09-01), LabEx EMC3

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.66.116.143 On: Mon, 28 Jul 2014 12:18:10

076101-11 Shadloo, Hadjadj, and Chaudhuri Phys. Fluids 26, 076101 (2014)

“Energy Materials and Clean Combustion Center.” This work was performed using HPC resourcesfrom GENCI [CCRT/CINES/IDRIS] (Grant No. 2013-0211640) and from CRIHAN (Centre deRessources Informatiques de Haute-Normandie, Rouen).

1 G. Ben-Dor, Shock Wave Reflection Phenomena (Springer Verlag, 1992).2 K. Izumi, S. Aso, and M. Nishida, “Experimental and computational studies focusing processes of shock waves reflected

from parabolic reflectors,” Shock Waves 3, 213–222 (1994).3 H. Gronig, “Shock wave focusing phenomena,” Shock Waves and Shock Tubes (Stanford University Press, 1986), p. 43.4 B. Sturtevant and V. Kulkarny, “The focusing of weak shock waves,” J. Fluid Mech. 73, 651–671 (1976).5 F. Shugaev, A. Serov, L. Shtemenko, H. Kishige, and M. Nishida, “Formation of a jet and vortices behind a shock wave

reflected from a concave body,” Shock Waves 9, 31–35 (1999).6 B. W. Skews and H. Kleine, “Flow features resulting from shock wave impact on a cylindrical cavity,” J. Fluid Mech. 580,

481–493 (2007).7 H.-S. Choi and J.-H. Baek, “Computations of nonlinear wave interaction in shock-wave focusing process using finite

volume TVD schemes,” Comput. Fluids 25, 509–525 (1996).8 S. Liang, C. Wu, F. Yu, and L. Wu, “Numerical simulation of shock wave focusing over parabolic reflectors,” Shock Waves

5, 139–148 (1995).9 D. Taieb, G. Ribert, and A. Hadjadj, “Numerical simulations of shock focusing over concave surfaces,” AIAA J. 48,

1739–1747 (2010).10 Y. Jung and K. Chang, “Shock focusing flow field simulated by a high-resolution numerical algorithm,” Shock Waves 22,

641–645 (2012).11 C. Bond, D. Hill, D. Meiron, and P. Dimotakis, “Shock focusing in a planar convergent geometry: experiment and

simulation,” J. Fluid Mech. 641, 297–333 (2009).12 M. K. Kjellander, N. Tillmark, and N. Apazidis, “Energy concentration by spherical converging shocks generated in a

shock tube,” Phys. Fluids 24, 126103 (2012).13 B. Gelfand, S. Khomik, A. Bartenev, S. Medvedev, H. Gronig, and H. Olivier, “Detonation and deflagration initiation at

the focusing of shock waves in combustible gaseous mixture,” Shock Waves 10, 197–204 (2000).14 A. Chaudhuri, A. Hadjadj, and A. Chinnayya, “On the use of immersed boundary methods for shock/obstacle interactions,”

J. Comput. Phys. 230, 1731–1748 (2011).15 A. Chaudhuri, A. Hadjadj, O. Sadot, and E. Glazer, “Computational study of shock-wave interaction with solid obstacles

using immersed boundary methods,” Int. J. Numer. Methods Eng. 89, 975–990 (2012).16 D. Ngomo, A. Chaudhuri, A. Chinnayya, and A. Hadjadj, “Numerical study of shock propagation and attenuation in narrow

tubes including friction and heat losses,” Comput. Fluids 39, 1711–1721 (2010).17 A. Hadjadj and A. Kudryavtsev, “Computation and flow visualization in high-speed aero- dynamics,” J. Turbul. 6, 1–25

(2005).18 G. Ben-Dor and T. Elperin, “Analysis of the wave configuration resulting from the termination of an inverse Mach

reflection,” Shock Waves 1, 237–241 (1991).

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.66.116.143 On: Mon, 28 Jul 2014 12:18:10