Some physical aspects of shock wave/boundary layer interactions

Shock Waves (2009) 19:453468DOI 10.1007/s00193-009-0220-z

ORIGINAL ARTICLE

Some physical aspects of shock wave/boundary layer interactions

Jean Dlery Jean-Paul Dussauge

Received: 9 February 2009 / Accepted: 29 June 2009 / Published online: 26 July 2009 Springer-Verlag 2009

Abstract When the flow past a vehicle flying at highvelocity becomes supersonic, shock waves form, causedeither by a change in the slope of a surface, a downstreamobstacle or a back pressure constraining the flow to becomesubsonic. In modern aerodynamics, one can cite a large num-ber of circumstances where shock waves are present. Theencounter of a shock wave with a boundary layer resultsin complex phenomena because of the rapid retardation ofthe boundary layer flow and the propagation of the shock ina multilayered structure. The consequence of shock wave/boundary layer interaction (SWBLI) are multiple and oftencritical for the vehicle or machine performance. The shocksubmits the boundary layer to an adverse pressure gradientwhich may strongly distort its velocity profile. At the sametime, in turbulent flows, turbulence production is enhancedwhich amplifies the viscous dissipation leading to aggravatedperformance losses. In addition, shock-induced separationmost often results in large unsteadiness which can damage thevehicle structure or, at least, severely limit its performance.The article first presents basic and well-established results onthe physics of SWBLI corresponding to a description in termsof an average two-dimensional steady flow. Such a descrip-tion allows apprehending the essential properties of SWBLIsand drawing the main features of the overall flow structureassociated with SWBLI. Then, some emphasis is placed onunsteadiness in SWBLI which constitutes a salient feature of

Communicated by A. Hadjadj.

J. DleryONERA/DAFE, Centre de Meudon, Meudon, Francee-mail: [email protected]

J.-P. Dussauge (B)IUSTI, UMR 6595 CNRS-Universit dAix Marseille,Marseille, Francee-mail: [email protected]

this phenomenon. In spite of their importance, fluctuations inSWBLI have been considered since a relatively recent datealthough they represent a domain which deserves a specialattention because of its importance for a clear physical under-standing of interactions and of its practical consequences asin aeroelasticity.

Keywords Shock wave/boundary layer interaction Shock polar Triple deck structure Rotational flow Separated flow Shock-shock interference Unsteadiness Turbulence Strouhal number

PACS 47.40Nm 47.32Ff

List of symbols(C) Designates a shockE( f ) Power spectral densityf Frequencyh Height of the separated bubbleL Interaction lengthM Mach numberMc Convective Mach numberMe Mach number at the boundary layer outer edgep Pressurepst Stagnation pressurer Density ratioR Designates the reattachment points Velocity ratioS Designates the separation pointSL Strouhal numberT Designates a triple pointUD Flow velocity on the separated flow dividing

streamlineUe Flow velocity at the boundary layer outer edgeUs Shock displacement velocity

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454 J. Dlery, J.-P. Dussauge

X0 Interaction origin(Mc) Normalized spreading rate of the mixing layer Shock induced deflection Boundary layer thickness() Designates a shock polar

1 General introduction

When the flow past a vehicle flying at high velocity becomessupersonic, shock waves inevitably form, caused either by achange in the slope of a surface, a downstream obstacle or aback pressure constraining the flow to become subsonic. Inmodern aerodynamics, one can cite a large number of circum-stances where shock waves are present. On transport aircraft,a nearly normal shock terminates the supersonic region exist-ing on the wing in certain flight conditions (see Fig. 1a). Thistransonic situation is also encountered in turbomachine cas-cades and on helicopter blades. Supersonic aircraft are muchaffected by shock waves which are of prime importance in airintakes whose purpose is to decelerate a supersonic incom-ing flow down to a low subsonic flow in the engine entrancesection (see Fig. 1b). Intense shock phenomena also occurin over-expanded propulsive nozzles where a shock formsat the nozzle lip if the exit pressure is lower than the exter-nal pressure. For hypersonic vehicles, the high-temperaturerise provoked by intense shocks influences the thermody-namic behaviour of air, causing the so-called real gas effectsand their multiple repercussions on the vehicle aerodynam-ics (see Fig. 1c). In addition, strong interactions with theboundary layers are the origin of severe aero-heating prob-lems if the shock is strong enough to provoke separation.Shocks are also met on missiles and aircraft afterbodies, asshown in Fig. 1d, as well as in space launcher nozzles andon projectiles of all kinds.

The encounter of a shock wave with a boundary layerresults in complex phenomena because of the rapid retar-dation of the boundary layer flow and the propagation ofthe shock in a multilayered structure. The consequences ofshock wave/boundary layer interaction (SWBLI) are multi-ple and often critical for the vehicle or machine performance.The shock submits the boundary layer to an adverse pressuregradient which may strongly distort its velocity profile. Atthe same time, in turbulent flows, turbulence production isenhanced which amplifies the viscous dissipation leading toaggravated efficiency loss in internal flow machines or sub-stantial drag rise for profiles and wings. This interaction, feltthrough a coupling between the boundary layer flow and thecontiguous inviscid stream, can greatly affect the flow past atransonic airfoil or inside an air-intake. The foregoing con-sequences are exacerbated when the shock is strong enoughto separate the boundary layer. The consequence can be adramatic change of the entire flow field structure with theformation of intense vortices and complex shock patterns

replacing the simple purely inviscid flow structure. Inaddition, shock-induced separation most often results inunsteadiness, damaging the vehicle structure and limiting itsperformance.

In some respect, shock-induced separation can be viewedas the compressible facet of the ubiquitous separation phe-nomenon, the shock being an epiphenomenon. Indeed, thebehaviour of the separating boundary layer is basically thesame as in incompressible separation and the overall flowtopology is identical. Perhaps, the most distinctive and salientfeature of shock-separated flows is the accompanying shockpatterns forming in the contiguous inviscid flow, whose exis-tence may have major consequences on the entire flow field.It is difficult to completely separate SWBLI and phenomenainduced by the crossing of shock waves, designated by thegeneric term shock-shock interferences.

Shock wave/boundary layer interaction is the result of aclose coupling between the boundary layer, which is sub-mitted at the shock foot or shock impact point to a suddenretardation and the outer, mostly inviscid supersonic flow.The clear understanding of this process necessitates a closeanalysis of both the inviscid flow and the boundary layerbehaviours. A large number of studies have been devotedto SWBLI since the first investigation of transonic flows inthe early 40s (for a review see Dlery and Marvin [1]). Theforthcoming sections are devoted to a reminder of some basicand well-established results on the physics of SWBLI corre-sponding to a description in terms of an average two-dimen-sional steady flow. Such a description allows apprehendingthe essential properties of SWBLIs and drawing the main fea-tures of the overall flow structure associated with SWBLI. InSect. 6 emphasis is placed on unsteadiness in SWBLI whichconstitutes a salient feature of this phenomenon. In spite oftheir importance, fluctuations in SWBLI have been consid-ered since a relatively recent date although they representa domain which deserves a special attention because of itsimportance for a clear physical understanding of interactionsand of its practical consequences as in aeroelasticity.

2 The basic shock wave/boundary layer interaction

What can be considered as the four basic interactions betweena shock wave and a boundary layer, in two-dimensional flows,are the impingingreflecting shock, the ramp flow, the normalshock, and the pressure jump.

In the oblique shock reflection, the incoming supersonicflow of Mach number M1 undergoes a deflection 1 throughthe incident shock (C1) and the necessity for the downstreamflow to be again parallel to the wall (Euler type, or slip bound-ary condition for a non-viscous fluid) entails the formationof a reflected shock (C2), the deflection 2 across (C2) beingsuch that 2 = 1 (see Fig. 2a). Such a shock occurs inside

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Some physical aspects of shock wave/boundary layer interactions 455

Fig. 1 Examples of shock waveformation in high speed flows(Onera documents). a transonicprofile, b supersonic air-intake,c hypersonic vehicle,d afterbody withunder-expanded nozzle

Fig. 2 Basic shockwave/boundary layerinteractions (Onera documents).a oblique shock reflection,b ramp induced shock wave,c normal shock wave,d adaptation shock at anozzle exit

a supersonic air-intake of the mixed compression type or atthe impact of the shock generated by any obstacle on a nearbysurface.

1. In the ramp flow, a discontinuous change in the wallinclination is the origin of a shock through which theincoming flow undergoes a deflection 1 equal to thewedge angle (see Fig. 2b). Such a shock occurs ata supersonic air-intake compression ramp, at a controlsurface or at any change in the direction of a surface.

2. A normal shock wave is produced in a supersonic flowby a back pressure forcing the flow to become subsonic.In channel flow, a normal shock is also formed when adownstream choking necessitates a stagnation pressureloss in order to satisfy mass conservation (see Fig. 2c).The distinctive feature of a normal shock is to deceleratethe flow without imparting a deflection to the velocityvector, the Mach number behind the shock being sub-sonic. However, in most practical cases, the shock isnot perfectly normal, the situation corresponding to the

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strong oblique shock solution to the Rankine-Hugoniotequations, even if the shock intensity is very weak as intransonic flows! In such situations, the velocity deflec-tion through the shock is small so that the shock issaid to be normal. Normal, or nearly normal, shocksare met in channel flows (turbomachine cascades, airintakes, supersonic diffusers), in shock tubes and overtransonic profiles where a nearly normal shock termi-nates the supersonic pocket. Such interactions wherethe downstream flow is totally (or partly) subsonic areof special interest and lead to specific problems becauseof the possibility for downstream disturbances to influ-ence the shock and initiate an interactive process at theorigin of large-scale unsteadiness involving the wholeflow, as in transonic buffeting or air-intake buzz.

3. An oblique shock is produced if a supersonic flowencounters a change in pressure as at the exit of anover-expanded nozzle (see Fig. 2d). In the present situa-tion, the pressure discontinuity induces a flow deflection,whereas in cases 1 and 2, the pressure discontinuity isinduced by a deflection. This is the mirror problem ofthe duality [deflection, pressure jump].

As far as the response of the boundary layer to the shock isconcerned, there are no basic differences between the abovesituations, except perhaps case 4 where the interacting flowcommunicates with an atmosphere. So we will not distin-guish different cases when discussing the viscous flow behav-iour in the forthcoming sections. The major distinctions arebetween interactions without and with separation.

3 The boundary layers response to a rapidpressure variation

The flow along a solid surface can be viewed as a structurecomposed of three layers (see the sketch in Fig. 3) as follows:

1. An outer inviscid layer which usually is irrotational (i.e.,isentropic) and hence obeys the Euler equations or alter-natives such as the potential equation. However, thereare exceptions where this part of the flow is rotationalas, for example, downstream of the curved shock formedahead of a blunted leading edge, where what is referredto as an entropy layer is formed. A similar rotationallayer can occur behind the near-normal but curved shockthat forms on a transonic aerofoil.

2. Closer to the surface, and deeper within the boundarylayer we come first to an outer portion where, over astreamwise distance of several boundary layer thick-nesses, the flow can be considered as inviscid but rota-tional. In this part of the flow, viscosity contributes tocreate entropy and consequently vorticity, in agreement

viscous sublayer or inner deck

non viscous rotational flow or middle deck

outer potential flow or upper deck

Fig. 3 The interacting flow multi-layer structure or triple deck (Light-hill, StewartsonWilliams)

with Croccos equation connecting the gradient ofentropy grad s with the rotational vector rot V in a steadynon-viscous flow:

Tgrad s = V rot V

More simply said, this layer is a region of variablestagnation pressure, the stagnation temperature beingnearly constant. Although varying across a boundarylayer, because of viscous effects, the stagnation enthalpyis almost constant for adiabatic walls, especially for tur-bulent boundary layers. This is even more true in theintermediate inviscid rotational layer. As the flow is con-sidered inviscid, the stagnation conditions are constantalong streamlines, since entropy is a transported quan-tity. The static pressure is constant across the bound-ary layer and hence the layer behaves like an inviscidflow through which the velocity, and hence Mach num-ber, decreases steadily from the outer value Me at theboundary layer edge (y = ) towards zero at the wall.

3. The third layer is in contact with the wall and is to insurethe transition between the previous region and the sur-face; there, viscosity has again to play a role. This vis-cous layer must be introduced to avoid inconsistenciessince it is not possible for a non-viscous flow to decreaseits velocity without a rise in the static pressure and at thewall the stagnation pressure is equal to the static pressure(the velocity being equal to zero because of the no-slipcondition).

The structure that is described above was first suggestedby Lighthill [2]. A more formal justification was proposed in1969 by Stewartson and Williams [3] for the case of a lami-nar boundary layer using an asymptotic expansion approach.They introduced the triple-deck terminology to designatesuch a structure. The outer deck is the outer irrotational flow,the middle deck the inviscid rotational layer, and the inner

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deck the viscous layer in contact with the wall. It shouldbe made clear that such a representation is valid only if theviscous forces have not contributed to modify the entropylevel of the boundary layer streamlines (except in the innerdeck). This implies that the time scale of any phenomenaconsidered with this approach is short compared to the timescale over which the viscous terms take affect. This is the casefor SWBLIs where the shock imparts a sudden retardationto the flow. Such a model is also valid for a rapid accelera-tion as in the centred expansion wave that can occur at thebase of a vehicle. In a turbulent boundary layer, the middledeck represents the greatest part of the boundary layer, evenat moderate Mach number, so that the behaviour of an inter-action can be for the most part described by considering aperfect fluid model. But, for the reasons cited above, such aninviscid model becomes inadequate close to the wall whereviscosity has to be taken into account.

During the first part of an SWBLI, most of the flow,including a greater part of the boundary layer, behaves asan inviscid flow for which the pressure and inertia terms ofthe Navier-Stokes equations are predominant compared tothe viscous terms. Thus, many aspects of the boundary layerresponse can be interpreted with perfect fluid arguments andby considering the boundary layer mean properties definedabove. Such a description of the boundary layer behaviourcalls upon the concept of rapid interaction. This is justifiedby the fact that in an SWBLI important changes occur over ashort streamwise distance, the extent of the interaction beingof the order of 10 times the boundary layer thickness forthe laminar flow and much less in the turbulent case. Thisfact has several major consequences:

Streamwise derivatives are comparable to derivatives inthe direction normal to the wall, whereas in a classicalboundary layer they are considered to be of lower order.This fact also influences the mechanism for turbulenceproduction since the normal components of the Reynoldsstress tensor may now play a role comparable to that ofthe turbulent shear stress, which, in general, is the onlyquantity considered.

The turbulent Reynolds stresses do not react instantly tochanges in the mean flow that are imparted via the pres-sure gradient. In the first phases of the interaction, there isa lag in the response of the turbulence and hence a discon-nection occurs between the mean velocity and turbulentfields. Reciprocally, the velocity field is weakly affectedby the shear stress, the action of viscosity being confinedto a thin layer in contact with the wall. Thereafter, the tur-bulence level increases and can reach very high levels ifseparation occurs. This explains the difficulty in devisingadequate turbulence models for SWBLIs, especially forthe inception part of the process.

An inviscid fluid analysis provides a way to explain somebasic features of an interaction, but it is not entirely correct,in the sense that viscous forces cannot be neglected overthe entire extent of the interaction. Viscous terms must beretained in the region in contact with the wall; otherwise,one is confronted with inconsistency as said above. Never-theless, the neglect of viscosity is justifiable in describing thepenetration of the shock into the boundary layer. But thereare a number of situations where the shock does not penetrateinto the boundary layer, as in transonic interactions (exceptif the shock is very weak) or in shock-induced interaction(except at very high Mach number). In these cases, viscositymay have sufficient time to influence the flow behaviour evenoutside the near wall region. This is also true for interactionswith large separated regions where the flow depends on itsviscous properties to determine the longitudinal extent of theinteraction.

In what follows, for conciseness we will concentrate ourattention on the interaction induced by the impact of anoblique shock on a surface. It is clear that most of the con-clusions could be applied mutatis mutandis to other kinds ofshock interactions (see Fig. 2).

4 Interactions without separationweakly interactingflows: the incident reflecting shock case

4.1 Overall flow organisation

The interaction resulting from the reflection of an obliqueshock wave from a turbulent boundary layer is illustrated bythe schlieren visualisation in Fig. 4. A similar structure wouldbe seen for a laminar boundary layer, but the streamwiseextent of the interaction domain would be greater. (The appar-ent thickening of the incident shock is due to its interactionwith the boundary layers on the test section side windows;

Fig. 4 Schlieren photograph of a shock reflection at Mach 1.95 (Oneradocument)

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( )waves ( )waves( )waves

eM

1M =

y

2

( )2C

Sonic line

Viscous sublayer

Subsonic layer

Incident shockReflected shock

1

( )1C

Fig. 5 A sketch of a turbulent shock reflection without boundary layerseparation

p

p1

Viscous flow

p2Interaction origin

X

X0

Upstream influence

Inviscid solution

Fig. 6 The corresponding pressure distribution

its true location is indicated by the sharp deflection in thesuperimposed streamline).

The flow field organisation is sketched in Fig. 5. Theincident shock (C1) can be seen penetrating into therotational inviscid part of the boundary layer where it pro-gressively bends because of the local Mach number decrease.Correspondingly, its intensity weakens and it vanishes alto-gether when it reaches the boundary layer sonic line. At thesame time, the pressure rise through (C1) is felt upstream ofwhere the incident shock would have impacted with the wallin the absence of a boundary layer. This upstream influencephenomenon is predominantly an inviscid mechanism, thepressure rise caused by the shock being transmitted upstreamthrough the subsonic part of the boundary layer. This leadsto a spreading of the wall pressure distribution over a dis-tance of the order of the boundary layer thickness, comparedwith the purely inviscid flow solution. As shown in Fig. 6,the pressure starts to rise upstream of the inviscid pressurejump, after which it steadily increases and tends towards thedownstream inviscid level. In this case, the viscous, or real,

solution does not depart far from the purely inviscid solution.Accounting for the viscous effect would be a mere correctionto a solution that is already close to reality. Such behaviouris said to be a weak interaction process in the sense that theflow is weakly affected by viscous effects. The dilatation ofthe boundary layer subsonic region is felt by the outer super-sonic flow, which constitutes the major part of the boundarylayer if the flow is turbulent. It acts like a ramp inducingcompression waves () that coalesce to form the reflectedshock (C2). The thickness of the subsonic layer depends onthe velocity distribution and hence a fuller profile, whichhas a thinner subsonic channel, also has a shorter upstreaminfluence length. In addition, a boundary layer profile witha small velocity deficit has a higher momentum, hence agreater resistance to the retardation imparted by an adversepressure gradient.

4.2 Shock penetration in a rotational layer

The propagation of a shock wave in a turbulent boundarylayer is here illustrated by perfect fluid calculations using therotational method of characteristic. This provides both highaccuracy (the shock being fitted) and a picture of the wavespropagation in the supersonic flows. Calculations were madefor a turbulent velocity distribution represented by the Coles[4] analytical expression, the outer Mach number being equalto 4. The part of the boundary layer whose Mach number isless than 1.8 has been removed (this cut-off distance fromthe wall was chosen to avoid singular shock reflection). Thebehaviour of the viscous sub layer is neglected, which is jus-tified for moderate shock strengths at high Mach number.The calculation corresponds to the reflection on a rectilinearwall of a shock producing a downward deflection of 6 inthe outer irrotational stream. The characteristic mesh repre-sented in Fig. 7 shows the bending of the shock through therotational layer and the waves coming from the wall down-stream of the reflection.

The wall pressure distribution plotted in Fig. 8 shows thatthe pressure first jumps at the impact point to an intermediatevalue and then progressively reaches the constant level cor-responding to shock reflection in a Mach 4 uniform flow.This behaviour, which is observed in high Mach numberflows, can thus be interpreted by inviscid arguments. At lowerMach number, below 2.5, an overshoot is observed in the wallpressure distributions, which cannot be explained simply byrotational effects. In these circumstances, the influence of thesubsonic layer close to the wall and also the viscous innerlayer can no longer be neglected and a purely inviscid anal-ysis captures only a part of the solution. The contours ofFig. 9 confirm that behind the shock, there is a static pres-sure decrease from the outer flow down to the wall. The invis-cid analysis proposed by Henderson [5] and the method ofcharacteristics calculations are instructive since they give a

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Fig. 7 Method of characteristiccalculation of a shock reflectionin a rotational layer. Wavesystem and shocks. Turbulentboundary layer profile(upstream Mach number 4,primary deflection 6)

Incident shock Reflected shock

Boundary layer edge

Fig. 8 Method of characteristiccalculation of a shock reflectionin a rotational layer. Entropygradient effect on the wallpressure distribution (upstreamMach number 4- primarydeflection 6)

0stpp

X

Rotational characteristics

Perfectly inviscid shock

description of the complex wave pattern which is generatedwhen a shock traverses a boundary layer considered as arotational inviscid stream. However, the above scenario doesnot take into account the upstream transmission through thesubsonic part of the boundary layer with the subsequent gen-eration of compression waves, which coalesce to produce thereflected shock.

5 Interaction with separationstrongly interactingflows: the incident reflecting shock case

5.1 Overall flow organisation

A boundary layer is a flow within which the stagnationpressure decreases when approaching the wall and where,at least for short distances, it can be considered constantalong each streamline. Neglecting compressibility (which is

Fig. 9 Method of characteristic calculation of a shock reflection in arotational layer. Static pressure contour. Turbulent boundary layer pro-file (Upstream Mach number 4, primary deflection 6)

of course an over simplification) we can write the Bernoulliequation for each streamline:

pst = p + 2 V2

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Fig. 10 Schlieren visualisation of an incident-reflecting shock at Mach2 (Onera document)

Thus any rise in p will provoke a greater retardation in regionswhere the stagnation pressure pst is lowest, that is, in theboundary layer inner part. By imposing an adverse pressuregradient a situation can be reached where the flow adjacentto the wall is stagnated or reversed so that a separated regionforms. An incident shock wave can readily induce separa-tion this way as, for example, in the Mach 2 flow for which aschlieren picture is presented in Fig. 10 (The apparent thick-ness of the shock waves is due to the interactions taking placeon the test section side windows). The structure of this flowis sketched in Fig. 11. Downstream of the separation point Sthere exists a recirculating bubble flow bounded by a divid-ing streamline (S), which separates the recirculating flowfrom the flow streaming from upstream to downstream infin-ity. The streamline (S) originates at the separation point Sand ends at the reattachment point R. Due to the action of thestrong mixing taking place in the detached shear layer ema-nating from S, a mechanical energy transfer takes place fromthe outer high-speed flow towards the separated region. As a

p

p1

Inviscid solution

p2

Interaction origin

X

S

R

Plateau pressure

S R

Viscous flow

First pressure rise at separation

Second pressure rise at reattachment

Fig. 12 Wall pressure distribution, in a shock separated flow

consequence, the velocity UD on the dividing streamline (S)steadily increases, until the deceleration associated with thereattachment process starts.

The transmitted shock (C4) penetrates into the separatedviscous flow where it is reflected as an expansion wavebecause there is near constant pressure level within the bub-ble. This causes a deflection of the shear layer towards thewall where it eventually reattaches at R. At this point, theseparation bubble vanishes and the flow on (S) is deceler-ated until it stagnates at R. This process is accompanied bya sequence of compression waves that coalesce into a reat-tachment shock in the outer stream. As shown in Fig. 12,the wall pressure distribution exhibits initially a steep rise,associated with separation, followed by a plateau typical ofseparated flows. A second more progressive pressure risetakes place during reattachment. In this situation, the flowfield structure is markedly different from what it would be for

Fig. 11 Sketch of the flowinduced by a shock reflectionwith separation ( )3C

( )

( )1C

( )2C( )4CSeparation shock

Compression waves

1

3

2

5Slip line

Expansion waves

Subsonic layer Separated bubble Dividing streamline

( )4H

S R

( )S

1

2

Sonic line

Reattachment shock

Compression waves

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the purely inviscid case and the shock reflection is said to be astrong viscous-inviscid interaction. This means that the vis-cous effects have to be taken fully into account when pre-dicting the flow. They are no longer a simple adjustment toan already nearly-correct inviscid solution, but they play acentral role in the establishment of the solution. It is evidentthat there has been a hierarchy reversal.

The free interaction theory [6] establishes that thepressure rise at separation and the extent of the first partof the interaction depend only on the flow properties at theinteraction onset and not on the downstream conditions, inparticular the shock intensity. During the first part of the inter-action, the flow is a consequence of the reciprocal and mutualinfluence, or coupling, between the local boundary layer andthe inviscid contiguous stream, and not the further develop-ment of the interaction. This important result, well verifiedby experiment, explains many features of interactions withshock induced separation.

5.2 The outer inviscid flow structure

The pressure rise at separation generates compression waveswhich coalesce to constitute the separation shock (C2). Thisshock intersects the incident shock (C1) at point H where(C1) undergoes a deflection (refraction) to become the shock(C4); the separation shock (C2) becomes in a similar waythe shock (C3). The shock (C4) meets the separated regionboundary at point I. There, to insure continuity of pressure,the pressure rise produced by (C4) must be compensatedfor by a centred expansion emanating from I. This expan-sion provokes a deflection of the separated region boundarywhich is turned towards the wall such that it impacts with it atthe reattachment point R. There a new deflection occurs withformation of the reattachment shock (C5). In addition, a slipline emanates from the intersection point H. For this case, thetwo-shock system of the perfect-fluid oblique shock reflec-tion, that comprises simply an incident plus reflected shock,is replaced by a pattern involving five shock waves.

The pattern, made by the shocks (C1), (C2), (C3) and (C4)is a Type I shock/shock interference according to Edneysclassification [7], which can be best understood by consider-ing the shock polar representation in Fig. 13. The figure corre-sponds to an incoming uniform flow of Mach number 2.5. Theseparation shock deflection is given by an adequate turbulentseparation criterion, which fixes the Mach number behind theseparation shock [8]. This angle, which is here around 14,does not depend on the intensity of the shock having causedthe separation. The polar (1) is associated with the upstreamuniform state 1 and represents any shock forming in 1, in par-ticular, the incident shock (C1). The image of the downstreamflow 3 is the point 3 on (1), the deflection imparted by (C1)being negative (the velocity is deflected towards the wall).

-30 -20 -10 0 10 20 301

2

3

4

5

6

7

8

9

10

( )1C( )2C

( )3C ( )4C

( )1

( )2( )3

1

23

1pp

( )

5 4

Fig. 13 Shock pattern interpretation in the shock polar diagram.Upstream Mach number 2. Separation shock deflection 14; incidentshock deflection 10

The separation shock (C2) is also represented by (1) sincethe upstream state is 1. The image of the downstream flow 2 isat point 2 on (1), the deflection 2 being upward. The situa-tion downstream of H is at the intersection of the polars (3)and (2) attached to the states 3 and 2, respectively. Theirintersection is the image of two states 4 and 5 having thesame pressure (p4 = p5) and the same direction (4 = 5),hence compatible with the Rankine-Hugoniot equations. Theset of successive shocks (C1)+ (C3) is different from the set(C2) + (C4) and hence the flows that have traversed eachset have undergone different entropy rises. Thus a slip line() is formed separating flows 4 and 5 which have differ-ent velocities, densities, temperatures, and Mach numbers(but identical pressures). In a real flow a shear layer developsalong () insuring a continuous variation of the flow prop-erties between states 4 and 5. The fluid that flows along astreamline passing under the point H, and belonging to theinviscid part of the field, crosses three shock waves: (C2)and (C4), plus the reattachment shock (C5). Thus, its finalentropy level is lower than for the entirely inviscid case wherethe fluid would have only traversed the incident plus reflectedshocks. This is also the case for an interaction without sepa-ration, which is close to the inviscid model at some distancefrom the wall. The conclusion is that, in an interaction withshock induced separation, entropy production is smaller thanin a non-separated interaction, or in the limiting case of theinviscid model. This result is exploited by control techniquesaiming at reducing wing drag or efficiency losses in internalflows [9].

If, for a fixed upstream Mach number, the strength of theincident shock is increased, a situation is reached where thetwo polars (2) and (3) do not intersect. Then a EdneyType II interference occurs at the crossing of shocks (C1)

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( )3C( )2

( )2C( )1

Slip lines

T2

2

3

57

( )4C4

61 ( )5C

T1( )1C

Fig. 14 Shock reflection with singular shock intersection or Machphenomenon: Schematic view of the situation in the physical plane

-20 -10 0 10 201

2

3

4

5

6

( )1

( )2( )3

1

23

1pp

( )

5

6

( )1C ( )2C

4

7

( )3C ( )4C

( )5C

Fig. 15 Shock reflection with singular shock intersection or Machphenomenon: Situation in the shock polar plane (M1 = 2, 1 = 14,2 =16)

and (C2) and a nearly normal shock, or Mach stem is formedbetween the two triple points T1 and T2 as shown in Fig. 14.The singular shock interaction of Fig. 15 is for an upstreamMach number of 2.5, the separation shock deflection of 14resulting from a separation criterion [8]. The Mach reflec-tion is obtained by increasing the incident shock deflection.The downstream states 4 and 6 located at the intersectionof the polars (1) and (2) are separated in the physicalplane by the slip line (1), whereas the downstream states5 and 7 at the intersection of (1) and (3) are separatedby the slip line (2). The subsonic channel downstreamof the Mach stem (C5) is accelerated under the influenceof the contiguous supersonic flows, so that a sonic throat

appears after which the flow is supersonic (see Fig. 14).In this case, the interaction produces a completely differentouter flow structure with the formation of a complex shockpattern replacing the simple purely inviscid flow solution.Occurrence of a Mach phenomenon can be very detrimen-tal in hypersonic air-intakes as the stagnation pressure lossbehind the normal shock is much greater than behind theoblique shocks.

Similar shock patterns are encountered in over-expandednozzle when the adaptation shock forming at the nozzle exit isstrong enough to separate the nozzle boundary layer [10]. Asshown in Fig. 16a which is relative to a two-dimensional noz-zle, separation takes place inside the nozzle, the separationshocks forming on each wall crossing to form a Type I shockpattern. The situation in the polar plane is depicted in Fig. 16bfor the following conditions: Mach number 2, shock induceddeflections 10 et 10. If separation progresses inside thenozzle, the Mach number at separation origin decreases anda situation will be reached where the intersection of the sep-aration shocks leads to a Type II interference with formationof a Mach phenomenon (a Mach disc for an axisymmetricflow), as shown in Fig. 16c. The corresponding situation inthe polar plane is shown in Fig. 16d (situation correspond-ing to Mach number 1.8, shock induced deflections 12 et12).

In some circumstances, the Type II solution of Fig. 17ais not possible, and the shock pattern sketch in Fig. 17b isobserved, to which corresponds the shock polar diagram ofFig. 17c. Then, a Mach phenomenon occurs but in such a waythan the flow angles after the triple points have the oppositesigns as compared to the situation in Fig. 17a. In this case, thearea of the subsonic channel downstream of the Mach stem(disc) increases and a situation is generally met where thesubsonic flow breaks down, a recirculation bubble form-ing as shown in Fig. 17b. Such a situation is sometimes calledan inverse Mach reflection.

6 Shock wave unsteadiness

6.1 Introduction

The previous sections have given the elements of organizationof steady interactions, showing how the shocks and mostof the perfect fluid flow properties can be connected. Thiswas essentially a steady picture. However, when the shock isstrong enough to induce the separation of a turbulent bound-ary layer, it is known that unsteadiness appears. In gen-eral, this produces strong flow oscillations which are felt fardownstream of the interaction, and can damage airframes orengines. They are generally called unsteadiness or breath-ing, because they involve very low frequencies, typically atleast at two orders of magnitude below energetic eddies of

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Fig. 16 Separation in anover-expanded nozzle (Oneradocuments). a separation in anozzle with Type I interference,b polar plane representation forType I interference, c separationin a nozzle with Type IIinterference, d polar planerepresentation for Type IIinterference

a b

c d

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the incoming boundary layer. The origin of these oscillationsraises several types of questions. What is their cause, and isthere a general way to understand them? Such interactionscan be produced in several ways depending on geometry, pro-ducing different sorts of geometry. An attempt of interpreta-tion was proposed in [11,12], under the form of a diagramreproduced in Fig. 18.

The organization of the diagram needs a comment. Theshock wave separates two parts of the flow: the upstream andthe downstream layers. Therefore, the shock wave may beconsidered as an interface between upstream and downstreamconditions, and its position and its motion will vary accord-ingly. This implies that the shock motion may be two-folded.A first class of motions can include the cases where upstreamand/or downstream conditions vary, making the shock move.The other class of motions comes from the propagation ofa perturbation, for example, at the foot of the shock, whichpropagates along the shock sheet. Having these elements inmind, the shock motion should be analysed from the pointof view of the upstream and downstream conditions, and therest of the present analysis will consist in commenting thephenomena or the flow organisation related to the differentbranches of the diagram.

6.2 The upper branch, local, and long distance influence

In the upper branch, we consider the evolution of the turbulentfield. Turbulence is subjected to a shock wave. It is distortedand generally amplified by its passage through the shock; itsanisotropy is modified in the distortion. If this passage is fastenough, the non-linear effects can be neglected, and the evo-lution of turbulence can be described by a (linear) theory ofrapid distortion or in the linear theory of Ribner [13]. Furtherevolution of such distorted turbulence depends of course onnon-linear effects. Further downstream, this non-equilibriumturbulence contributes to form a new boundary layer. This isrepresented in the diagram by the upper branch, which isrelevant in all cases, separated or non-separated. This upperbranch may contribute to the shock motion in two ways: first,as it represents cases where eddies fly through the shock, andtherefore distort them, there is a motion due to local flowvariations. Note that such a motion is taken in account inRibners linear theory. Second, distorted turbulence contrib-utes to form a non-equilibrium boundary layer downstreamof the interaction. If long-distance coupling can exist, as willbe discussed in the next paragraph, this distorted turbulencecan have an indirect influence on shock motion.

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a

b

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Fig. 17 Direct and inverse Mach phenomenon. a direct Mach phe-nomenon, b inverse Mach phenomenon, c inverse Mach phenomenon:Situation in the shock polar plane (M1 = 2, 1 = 10, 2 = 10)

6.3 The lower branch, separated flows

Now, situations in which the shock wave is strong enoughto make the boundary layer separate are considered. Theseparation can be incipient or well developed. We will term aseparation as well developed if there is an entire zone of spaceexperiencing reverse velocity during amounts of time longenough to produce an average separated bubble. The caseswhere some isolated and intermittent spots contain fluid with

Distorted turbulence

Downstreamboundary layer

Incomingturbulence

Shoc

k w

ave

Separated zone

Fig. 18 A diagrammatic representation of shock boundary layer inter-actions

negative velocity and producing no average separation will besupposed to belong to incipient separation. When an averageseparation zone exists, it can impose its specific character-istic scales of time and space as downstream conditions forthe shock wave. The two cases, incipient and well-developedproduce probably different shock dynamics. Finally, the flowexiting from the separated region is shed into the downstreamboundary layer and will contribute to form a new boundarylayer together with the turbulence coming from upstream anddistorted by the shock system.

A first remark is that the shock wave may have a particularfrequency response. This depends on the shape of the shockwave and on the flow around the shock wave [14,15]. In someparticular cases, as in the transonic experiments of Sajben andKroutil [16], the shock can be frequency selective. In gen-eral, this transfer function is not known and depends on theflow, but if they are not frequency selective, the overall trendseems that the shock waves behave rather like low-pass fil-ters, and therefore they may be expected to be more sensitiveto low frequencies.

Coming back to the diagram in Fig. 18 it may be noticedthat the lower branch is made of two-sided arrows. This mim-ics the fact that the downstream flow may control the motionof the shock, and therefore that couplings between the down-stream layer and the shock wave may exist. Such a situationcan happen if the flow downstream of the shock is mostlysubsonic, like in plane shock interactions in channels, or ininteractions on profiles in the transonic regime. This occurs,for example, when the shock motion depends on the turbu-lent flow far downstream of the interaction. This is probablythe case of transonic buffeting for which, according to clas-sical interpretations, there is an acoustic feedback betweenthe flow at the trailing edge of the profile and the shockwave [17]. Note that more recent interpretations [18] suggestthat this could also be linked to global instability properties.A consequence of such a far field influence is that it is possibleto generate shock motion by imposing far downstream condi-tions. This is classically achieved in wind tunnel experimentsby using a rotating cam to produce shock motion like in[19,20]. Another consequence for wind tunnel experiments

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Fig. 19 Dominant frequency in supersonic shock/boundary layerinteractions (from [26])

in the transonic regime is that the fluctuations in the nozzlediffuser, if strong enough, may contribute significantly to theshock motion. This represents a very particular class of flows.

If we consider now situations in which there is no feedbackwith the far downstream flow, different possibilities can befound. In a first one, the flow downstream of the shock wavedoes not impose particular conditions. Therefore, the shockmotion will be specified by the incoming turbulence. Thiscorresponds to the upper branch of the diagram in Fig. 18.This is the case of non-separated interactions, in which tur-bulence is just amplified through the shock, but no turbulentstructures with a particular dynamics are formed just down-stream of the shock wave. This is also found in shock turbu-lence interactions [2123] and in compressions by turning asstudied by Poggie and Smits [24]. This latter flow is made bythe reattachment of a free shear layer on an inclined flat plate.The fluid can flow freely downstream, and the measurementsshow very clearly that the resulting mean pressure gradientand the spectra of pressure fluctuations scale with the size ofthe incoming turbulent structures.

The question is quite different when an obstacle or a largeseparation zone is present. In this case, the downstream con-ditions often control predominantly the shock dynamics. Thiswas shown unambiguously by the experiments [25] on theinteraction produced by a cylinder normal to a plate. Theyfound that the dominant frequency of the shock unsteadi-ness varies like the inverse of the cylinder diameter. Similarresults were found in interactions produced by oblique shockreflections. More precisely, a compilation of results in dif-ferent sorts of interactions has been proposed in [26], and isrecalled in Fig. 19.

In this figure, the dominant frequency of the shockunsteadiness has been represented. It is defined as the maxi-mum of the premultiplied power spectrum f E( f ) ( f is the

frequency and E( f ) the power spectral density) of pressurefluctuations at the foot of the shock. It is normalized by thelength of interaction and the external velocity downstreamof the leading shock, leading to the Strouhal number:

SL = f LUeThe collapse of the data versus Mach number for the

different flows (shock reflection, compression ramps, bluntbodies, channel flows) was not excellent. The scatter wasabout 20%. However, no particular trend was found versusMach number, excepted for the compression ramp flow ofThomas et al. [27], which was found a little higher than theother data. This collapse, even if partial, is surprising, sincethe dominant frequency may be expected dependent on theflow geometry. This result, however, suggests that the dif-ferent flow cases share some common features for the originof the unsteadiness. The scatter was not totally satisfactory,indicating that some details were not well represented. How-ever, it was clear that all the experiments had generally aStrouhal number around 0.030.04. As for well-developedinteractions, L is much larger than the boundary layer thick-ness ; it appears without any ambiguity that the unsteadi-ness is several orders of magnitude below the frequencies Ue

produced by the energetic eddies of the incoming layer, oftypical size . Another property of the shock motion can bederived in this case. As noticed in [28] and recalled by [12],a velocity scale

Us = L fcan be deduced for the shock motion. Since SL


Fig. 20 Sketch of the separatedzone in impinging oblique shockinteractions

1. Interactions like the oblique shock reflection studied in[12] have ratios L

about 5 or 7 for Strouhal numbers about

0.030.04. Therefore, such low frequencies cannot explainthe unsteadiness observed in the shock reflection, and do notprovide a general answer to the question of the origin of theshock wave unsteadiness in separated flows.

Piponniau et al. [32] have proposed an analysis for theorigin of this unsteadiness. Their conditional measurementsof the size of the separated bubble showed that this zone isstrongly intermittent, with a few events during which back-wards flow of strong intensity engulfs into the separationpocket. These events are connected with shock motions oflarge amplitude. This has led to the assessment that the largeshock pulsations are closely related to the flapping of themixing layer formed at the edge of the separated bubble asshown in Fig. 20. Therefore, they proposed an explanationbased on considerations on air entrainment by this mixinglayer.

Their objective was to find the parametric dependence ofthe shock motion frequency rather than a complete theoreti-cal description. They considered that air entrainment drainedair from the separated zone. They evaluated the amount ofmass contained in the bubble and the rate of mass entrained.The ratio of these two quantities provides a time scale whichrepresents the time necessary to drain a significant amountof mass from the separated zone. The inverse of this timegives a frequency scale. They assumed that the dependenceof the spreading rate of the mixing layer on density and veloc-ity ratios and on convective Mach number is the same as incanonical mixing layers. The analysis provides a Strouhalnumber of the form:

SL = (Mc)g(r, s) Lhin which (Mc) in normalized spreading rate of the mixinglayer, g(r, s) is a weak function of the velocity and den-sity ratios r and s; h is the height of the separated bubble.

Essentially, this correlation suggests that for the same aspectratio Lh , the Strouhal number varies with the convective Machnumber like the spreading rate of the mixing layer. This issupported by the existing data. It should be remarked that inmost of the interactions under investigation, the convectiveMach number of the large structures in the mixing layer isclose to 1. This corresponds to values of (Mc) in the range0.20.3 and suggests that the aspect ratio of the consideredseparations is about 5 or 6. This is consistent with what isknown from the details of the geometry of these interactions.Therefore, it seems that this simple model provides a moregeneral representation of the unsteadiness. Of course, thisscheme is limited to two-dimensional situations in which areattachment point exists, as in Restricted Shock Separationfound in nozzle flows; for other types of interactions, thereader may be referred to [33]. The simple model proposedhere gives, however, indications on the leading elements foranalysing other situations, and on the way to control them.

7 Concluding remarks

The structure of shock wave/boundary layer interactions ispredominantly a consequence of the response of the bound-ary layer to the sudden local compression imparted by theshock and it reacts as a non-uniform flow in which viscousand inertial terms combine in an intricate manner. The mostsignificant result of this is the spreading of the pressure dis-continuity caused by the shock so that its influence is feltwell upstream of where this would have been located in aninviscid fluid model. When the shock is strong enough to sep-arate the boundary layer, the interaction has dramatic con-sequences for the development of the boundary layer andfor the contiguous inviscid flow field. Complex shock pat-terns are then formed which involve shock/shock interfer-ences whose nature depends on the Mach number and onthe way the primary shock is produced (whether by shock

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reflection, ramp or normal shock). In these circumstances,the most salient feature of shock-induced separation is thepattern of shocks produced even though this is a secondaryphenomenon associated with the process. The boundary layerbehaves more or less as it would for any other ordinary sepa-ration and essentially the same as in subsonic flows. It obeysthe specific laws mainly dictated by the intensity of the over-all pressure rise imparted by the shock, regardless of the wayin which this is generated. A very striking feature of theseinteractions is the overwhelming repercussion that the shockhas on the contiguous inviscid supersonic stream, which canbe spectacular for internal flows. Although their basic flowtopology is the same, laminar and turbulent interactions havedistinctly different properties that stem from the far greaterresistance of a turbulent boundary layer to flow retardationand hence separation.

One of the most detrimental consequences of SWBLI isthe occurrence of flow unsteadiness. This can be of highintensity when the shock is strong enough to induce sep-aration. Such unsteadiness can occur at high frequencieswhen associated with turbulent fluctuations and to a lesserextent with separated bubble instabilities. In other cases,unsteadiness occurs at very low frequency when the fluctuat-ing motions involve the whole aerodynamic field. This cor-responds to large length scales, as, for example, in transonicbuffeting or in air-intake buzz. Such large-scale unsteadinessseems to be a special feature of transonic interactions wherethe downstream subsonic flow allows a forward transmissionof perturbations that excite the shock wave. In fully super-sonic interactions, the higher Mach number of the outer flowfield tends to isolate the interaction domain from downstreamperturbations; the perturbations however remain at frequen-cies much lower than the energetic eddies of the incomingboundary layer. This can be related to the differences ofmass entrainment in the mixing layer of the separated bub-ble. A consequence is that the frequency range involved bythe fluctuations of the separated bubble and by the shockwave decreases with Mach number like the spreading rateof the supersonic mixing layer; this implies a reduction offrequency with respect to the subsonic situation. These pic-tures are consistent with a two-dimensional situation, andthe influence of three-dimensional geometries and/or of massbleeding, together with the free separation in which separa-tion is not followed by reattachment remain issues to lead tothe determination of unsteadiness in the general case.

In spite of a large numbers of studies over more than half acentury, the complex phenomena resulting from the interac-tion of a shock and a boundary layer remain a major concernfor high speed aerodynamicists. Important and challengingaspects of the phenomena have not been considered in thisbrief review article such as the 3D character of SWBLIs. Inreality, most of the configurations are three-dimensional lead-ing to complex flow topology and shock structure. Further-

more, even in nearly 2D situations (channel flow or nozzleflow for example), the flow inevitably adopts a 3D organisa-tion, whose influence on the overall flow is ill known. Thisaspect, which has been frequently ignored, has received onlyrecently some attention. Also, as already said, the unsteadi-ness in SWBLIs are considered since a relatively short timeand it can be anticipated that this aspect (also true for anyseparated flow) is of crucial importance in domains such asaeroacoustics, aeroelasticity, combustion and others.

Thus, there remains a large field of basic research onSWBLI to establish a still more realistic and precise physicaldescription of the flow structure. On the other hand, reliablemodelling of shock induced separation, including the predic-tion of unsteadiness, is still largely an open questionbut thisis another story.

References

1. Dlery, J., Marvin, J.G.: Shock wave/Boundary Layer Interactions.AGARDograph N 280, (1986)

2. Lighthill, M.J.: On boundary layer upstream influence. Part II:supersonic flows without separation. Proc. R. Soc. A217, 478507 (1953)

3. Stewartson, K., Williams, P.G.: Self-induced separation. Proc. R.Soc. A312, 181206 (1969)

4. Coles, D.E.: The law of the wake in the turbulent boundary layer.J. Fluid Mech. 2, 191226 (1956)

5. Henderson, L.F.: The reflection of a shock wave at a rigid will in thepresence of a boundary layer. J. Fluid Mech. 30(4), 699722 (1967)

6. Chapman, D.R., Kuhen, D.M., Larson, H.K.: Investigation of sep-arated flows in supersonic and subsonic streams with emphasis onthe effect of transition. NACA TN-3869 (1957)

7. Edney, B.: Anomalous heat transfer and pressure distributions onblunt bodies at hypersonic speeds in the presence of an imping-ing shock. Aeronautical Research Institute of Sweden, FFA Report115, Stockholm (1968)

8. Zhukoski, E.E.: Turbulent boundary layer separation in front of aforward-facing step. AIAA J. 5, 17461753 (1967) [N 10, Oct.1967]

9. Stanewsky, E., Dlery, J., Fulker, J., Geissler, W.: EUROSHOCK:Drag Reduction by Passive Shock Control. Notes on NumericalFluid Mechanics, Vieweg (1997)

10. Reijasse, P., Bouvier, F., Servel, P. : Experimental and Numer-ical Investigation of the Cap-shock Structure in Over-expandedThrust-Optimized Nozzles. In: Zeitoun, D., Prriaux, J., Dsidri,J.-A., Marini, M. (eds.) West East High Speed Flow FieldsAerospace Applications from High Subsonic to HypersonicRegime., pp. 338345. CIMNE, Barcelona (2003)

11. Dussauge, J.P.: Compressible turbulence in interactions of super-sonic flows. In: Proceedings of the Conference TI 2006. Springer,Heidelberg (2009, in press)

12. Dussauge, J.P., Piponniau, S.: Shock/ boundary layer interactions: possible sources of unsteadiness. J. Fluids Struct. 24, 11661175(2008) [N 8]

13. Ribner, H.S.: Convection of a pattern of vorticity through a shockwave. NACA TN 2864 (1953)

14. Culick, F.E.C., Rogers, T.: The response of normal shocks indiffusers. AIAA J. 21, 13821390 (1983) [N10]

15. Robinet, J.C., Casalis, G.: Shock oscillations in a diffuser modelledby a selective noise amplification. AIAA J. 37, 18 (1999) [N4]

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16. Sajben, M., Kroutil, J.C.: Effect of initial boundary layer thicknesson transonic diffuser flow. AIAA J. 19, (1981) [N11]

17. Lee, B.H.K.: Self-sustained shock oscillations on airfoils at tran-sonic speeds. Progress Aerosp. Sci. 37, 147196 (2001)

18. Crouch, J.D., Garbaruk, A., Magidov, D.: Predicting the onset offlow unsteadiness based on global instability. J. Comput. Phys.224(2), 924940 (2007)

19. Galli, A., Corbel, B., Bur, R.: Control of forced shock wave oscilla-tions and separated boundary later interaction. Aerosp. Sci. Tech-nol. 9, 653660 (2005) [N 8]

20. Bruce, P.J.K., Babinsky, H.: Unsteady shock wave dynamics.J. Fluid Mech. 603, 463473 (2008)

21. Debive, J.F., Lacharme, J.P.: A shock wave/free turbulence inter-action. In: Dlery, J. (ed.) Turbulent Shear Layer/Shock Wave Inter-actions. Springer, Heidelberg, (1985)

22. Hannapel, R., Friedrich, R.: Direct numerical simulation of aMach 2 shock interacting with isotropic turbulence. Appl. Sci.Res. 54, 205221 (1995)

23. Garnier, E., Sagaut, P., Deville, M.: Large Eddy Simulationof shock/homogeneous turbulence interaction. Comput. Fluids31(2), 245268 (2002)

24. Poggie, J., Smits, A.J.: Shock unsteadiness in a reattaching shearlayer. J. Fluid Mech. 429, 155185 (2001)

25. Dolling, D.S., Smith, D.R.: Unsteady shock_induced separationin Mach 5 cylinder interactions. AIAA J. 27, 15981706 (1989)[N12]

26. Dussauge, J.P., Dupont, P., Debive, J.F.: Unsteadiness in shockwave boundary layer interactions with sparation. Aerosp. Sci.Technol. 10, 8591 (2006)

27. Thomas, F., Putman, C., Chu, H.: On the mechanism of unsteadyshock oscillation in shock wave/ turbulent boundary layer interac-tion. Exp. Fluids 18, 6981 (1994)

28. Dupont, P., Haddad, C., Debive, J.F.: Space and time organizationin a shock induced separated boundary layer. J. Fluid Mech.559, 255277 (2006)

29. Adrian, R.J., Meinhart, C.D., Tomkins, C.D.: Vortex organizationin the outer region of the turbulent boundary layer. J. Fluid Mech.422, 153 (2000)

30. Ganapathisubramani, B., Longmire, E.K., Marusic, I.: Charac-teristics of vortex packets in turbulent boundary layers. J. FluidMech. 478, 3546 (2003)

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Some physical aspects of shock wave/boundary layer interactionsAbstract1 General introduction2 The basic shock wave/boundary layer interaction3 The boundary layer's response to a rapid pressure variation4 Interactions without separation---weakly interacting flows: the incident reflecting shock case4.1 Overall flow organisation4.2 Shock penetration in a rotational layer

5 Interaction with separation---strongly interacting flows: the incident reflecting shock case5.1 Overall flow organisation5.2 The outer inviscid flow structure

6 Shock wave unsteadiness6.1 Introduction6.2 The upper branch, local, and long distance influence6.3 The lower branch, separated flows

7 Concluding remarks

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Some physical aspects of shock wave/boundary layer interactions

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