S W –TURBULENCE INTERACTIONS - hypersonic...

Annu. Rev. Fluid Mech. 2000. 32:309–345Copyright q 2000 by Annual Reviews. All rights reserved

0066–4189/00/0115–0309$12.00 309

SHOCK WAVE–TURBULENCE INTERACTIONS

Yiannis Andreopoulos, Juan H. Agui, andGeorge BriassulisExperimental Aerodynamics and Fluid Mechanics Laboratory, Department ofMechanical and Aerospace Engineering, The City College of the City University of NewYork, New York, New York 10031; e-mail: [email protected]

Key Words compressible flows, flow structure

Abstract The idealized interactions of shock waves with homogeneous and iso-tropic turbulence, homogeneous sheared turbulence, turbulent jets, shear layers, tur-bulent wake flows, and two-dimensional boundary layers have been reviewed. Theinteraction between a shock wave and turbulence is mutual. A shock wave exhibitssubstantial unsteadiness and deformation as a result of the interaction, whereas thecharacteristic velocity, timescales and length scales of turbulence change considerably.The outcomes of the interaction depend on the strength, orientation, location, andshape of the shock wave, as well as the flow geometry and boundary conditions. Thestate of turbulence and the compressibility of the incoming flow are two additionalparameters that also affect the interaction.

1. INTRODUCTION

The interaction of shock waves with turbulent flows is of great practical impor-tance in engineering applications. A fundamental understanding of the physicsinvolved in this interaction is necessary for the development of future supersonic-and hypersonic-transport aircraft, and advances in combustion processes as wellas high-speed rotor flows. In such flows the interaction between the shock waveand turbulent flow is mutual, and the coupling between them is very strong.Complex linear and nonlinear mechanisms are involved, that can cause consid-erable changes in the structure of turbulence and its statistical properties and alterthe dynamics of the shock-wave motion.

Amplification of velocity fluctuations and substantial changes in length scalesare the most important outcomes of interactions of shock waves with turbulence.This indicates that such interactions may greatly affect mixing. The use of shockwaves, for instance, has been proposed by Budzinski et al (1992) as a means ofenhancing the mixing of fuel with oxidant in ramjets. Turbulence amplificationthrough shock wave interactions is a direct effect of the Rankine-Hugoniot rela-tions. However, this type of amplification should be decoupled from other effectsthat also contribute to turbulence amplification such as destabilizing streamline

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310 ANDREOPOULOS n AGUI n BRIASSULIS

curvature, flow separation, dilatational effects, or longitudinal pressure gradientsthat may be present in flow before or after the interaction with the shock.

The outcomes of the shock-turbulence interaction depend on (a) the charac-teristics of the interacting shock-wave-like strength, relative orientation to theincoming flow, and location and shape, (b) the state of turbulence of the incomingflow as it is characterized by the fluctuation levels of velocity, density, pressure,and entropy and length scales, (c) the level of compressibility of the incomingflow, and (d) the flow geometry and boundary conditions.

Basic understanding of the physics of such complex interactions has beenobtained through investigations of conveniently selected and reasonably simpli-fied flow configurations. The flows to be considered here include shear free flows,shear layers, and wall-bounded flows. Most of the work in this review is confinedto the following cases: (a) homogeneous and isotropic turbulence interactionswith shock waves (see Figure 1a), (b) constant shear homogeneous turbulenceinteractions with shock waves (see Figure 1b), (c) circular jet flows interactingwith shock waves (see Figure 1c), (d) plane shear layers interacting with obliqueshock waves (see Figure 1d), (e) wake flows interacting with oblique or normalshock waves (see Figure 1e), and ( f ) boundary layer interactions with obliqueshock waves (see Figure 1f). This classification of the interactions also reflectsthe level of increasing complexity from the first to the last flow configuration.

We restrict our review to nominally two-dimensional interactions, albeit keep-ing in mind that all of these interactions are three-dimensional in nature becauseturbulence in all its complexity is characterized by instantaneous flow variablesthat exhibit a variation in time and space. Shock wave–boundary layer interactionshave been reviewed in the past by Green (1970), Adamson & Messiter (1980),and Settles & Dodson (1991). Compressibility effects of turbulence, includingsome shock wave interactions, have been recently reviewed by Lele (1994). Com-pressibility effects in turbulent boundary layers and shear layers in the absenceof shock interactions have been reviewed by Spina et al (1994) and Gutmark etal (1995), respectively. Birch & Eggers (1972) and Bradshaw (1977) reviewedcompressibility effects in shear layers, whereas Bradshaw (1996) provided a morerecent review of physical and modeling aspects of turbulence in compressibleflows. The reader is also referred to Morkovin’s (1962, 1992) reviews of com-pressibility effects on boundary layers and shear layers. Interactions of shockwaves with boundary layers in three-dimensional geometries have been reviewedby Settles & Dolling (1992) and Dolling (1993). The textbook by Smits & Dus-sauge (1996) also provides an excellent account of compressibility effects inturbulent shear layers, with chapter 10 devoted to boundary layer–shock waveinteractions.

Kovazsnay (1953) suggested the decomposition of turbulent fluctuations intothe vorticity, acoustic, and entropy modes. He performed a small perturbationanalysis, valid for weak fluctuations of density, pressure, and entropy, to showhow acoustic waves may accompany vorticity fluctuations and how entropy ischanged in regions of different shear. In Kovazsnay’s decomposition the three

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SHOCK-TURBULENCE INTERACTIONS 311

Figure 1 Schematic of flow configuration interactions with shock waves. (a) Homoge-neous and isotropic turbulence, (b) constant shear homogeneous turbulence, (c) turbulentjet, (d) free shear layer, (e) turbulent wake, and (f) two-dimensional turbulent boundarylayer.

modes of fluctuations are dynamically decoupled from each other to a first-orderapproximation. However, interaction of the modes can take on a second-orderapproximation of fluctuations, in which mode couplings arise as one mode canbe generated from the interaction of the other two modes (Chu & Kovazsnay1958). An interaction of the modes can take place when a shock wave is presentin the flow if conversion of one mode into either of the other two is possible. In

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the present work, particular emphasis is given to interactions during which thevorticity mode dominates the incoming flow.

The fundamental concept behind interactions of shock waves with turbulenceis the transfer of vorticity through shock waves. Longitudinal vorticity, forinstance, is expected to be transferred unchanged through a normal planar shock,whereas the other components of vorticity parallel to the shock surface shouldchange substantially.

Discussion of the following unanswered questions is attempted in this review:

1. How much of the amplification of turbulence in interactions with shock wavesis caused entirely by the Rankine-Hugoniot conditions?

2. Why are small-size eddies amplified more than large eddies?3. Are the length scales of the incoming turbulence reduced or amplified through

such interactions?4. Is the dissipation rate of turbulent kinetic energy also reduced?5. Why are vorticity fluctuations amplified more than velocity fluctuations?

The ability of existing theoretical models to predict the flow after interactionswith shock waves is also considered: Linear interaction analysis (LIA), rapiddistortion theory (RDT), large eddy simulations, direct numerical simulations(DNS), and Reynolds-averaged Navier-Stokes give different results even in thesimplest of flow.

2. BACKGROUND INFORMATION

The shock wave, from a simplistic point of view, can be considered as a steeppressure gradient. Information from experiments and simulation of low-speedflows with such pressure gradients indicate that ‘‘rapid-distortion’’ concepts holdand, in the limit of extremely sharp gradients, the Reynolds stresses and turbulentintensities are ‘‘frozen,’’ because there is insufficient residence time in the gra-dient for the turbulence to alter these quantities at all (Hunt 1973, 1978; Hunt &Carruthers 1990).

The first attempt to consider theoretically the passing of a turbulent fieldthrough a shock wave is attributed to Ribner (1953), who decomposed the incidentdisturbance shear wave into acoustic, entropy, and vorticity waves. In his LIA,Ribner formulated the interaction of a plane vorticity wave with a shock wave asa boundary-value problem. Figure 2 shows a schematic of the incident planesinusoidal shear wave crossing the shock. The wave number vector is refractedacross the shock owing to the changes in thermodynamic properties and thereforeemerges at a different angle from the incident. The condition that the phasesshould remain unchanged across the shock yields that the frequency, as well asthe component of the wave number vector parallel to the shock front, are thesame upstream and downstream of the shock. Ribner’s analysis obtained the first

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Figure 2 Schematic of the incident shear wave crossing the shock.

evidence of turbulence enhancement through interactions with shock waves. Hispredictions were verified by Sekundov (1974) and Dosanjh & Weeks (1964).Several analytical and numerical studies of this phenomenon by Morkovin (1960,1962), Zang et al (1982), Anyiwo & Bushnell (1982), Rotman (1991), and Leeet al (1991, 1993, 1994) show very similar turbulence enhancement.

Goldstein (1978) and Goldstein & Durbin (1980) formulated a decompositionof the disturbances into a ‘‘vortical’’ and an ‘‘acoustic’’ or ‘‘potential’’ part. Thisformulation was used by Durbin & Zeman (1992) to study numerically theresponse of homogeneous turbulence to bulk compression. Helmholtz’s decom-position into a ‘‘vortical’’ part, which is rotational, and a ‘‘dilatational’’ part,which is irrotational, has been also used extensively. Finally Lee et al (1992,1993, 1994) and Hannappel & Friedrich (1995) demonstrated a very successfulDNS of a homogeneous and isotropic turbulence interacting with a plane shockwave.

The time-dependent, three-dimensional equations governing the interactionsthat describe the conservation of mass, momentum, and total energy are

Dq ]Ui` q 4 0, (1)

Dt ]xi

DU ]p ]si ijq 4 1 ` , (2)Dt ]x ]xi j

](qE ) ](qE U ) ](pU ) ](s U ) ]qt t i i ij i i` 4 1 ` 1 . (3)

]t ]x ]x ]x ]xi i j i

Et is the total energy per unit mass, defined as Et 4 e ` 1⁄2 UiUi, qi is the rate ofheat added by conduction, and e is the internal energy, sij is the stress tensor,

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equal to 2lSij ` kdijSkk, where k is the second coefficient of viscosity that isrelated to the bulk viscosity lb through k 4 lb 1 2/3 l.

2.1 Dissipation Rate of Kinetic Energy

The transport equation for the instantaneous kinetic energy 1⁄2 Ui Ui in compress-ible flows is

1D U Ui i1 22 ]p ]sijq 4 1U ` U . (4)i iDt ]x ]xi j

Equation 4 can be manipulated to yield

1D U Ui i1 22 ](1pU ` s U )j ij iq 4 ` pS 1 s S , (5)kk ij ijDt ]xj

where the last term on the right-hand side contains the dissipation rate of kineticenergy E converted into thermal/internal energy. Sij is the strain tensor, and theterm pSkk represents the work done by pressure forces during compression orexpansion of the flow. Both terms, the dissipation rate E 4 sij Sij and the pressurework term, also appear with opposite sign in the transport equation for internalenergy. Since the dissipation rate is always positive or zero at any given point inspace and time, the pressure-dilatation term can, in principle, be positive ornegative.

The dissipation rate is given by

]U ]U ]Ui k iE 4 s 4 s S 4 2lS S ` k d . (6)ij ij ij ij ij ij]x ]x ]xj k j

After invoking Stokes’s hypothesis, which suggests that the bulk viscosity isnegligible, lb ' 0, Equation 6 becomes

]U ]U2 k moE 4 2lS S 1 l . (7)ij ij 3 ]x ]xk m

The second term in the right-hand side of Equation 7 represents the additionalcontribution of compressibility to the dissipation rate of kinetic energy. This termvanishes for incompressible flows.

Because ]Uk/]xk ]Um/]xm 4 (]Uk/]xk)2, this term is always positive, and thenegative sign in front of this term in Equation 7 may erroneously suggest thatcompressibility reduces dissipation. This is incorrect because the term SijSij alsocontains contributions from dilatation effects, which can be revealed if one con-siders that

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1 ]U ]Ui jS S 4 X X ` ,ij ij k k2 ]x ]xj i

where Xk Xk is the enstrophy rate. The second term in the right-hand side rep-resents the inhomogeneous contribution for incompressible flows. For compress-ible flows, terms related to dilatation can be extracted, and the dissipation ratethen becomes

]U ]U4 i io 2 2E 4 l X X ` lS ` 2l 1 S . (8)k k kk kk3 3 4]x ]xj i

The second term on the right-hand side of Equation 8 describes the direct effectsof compressibility, that is, dilatation on the dissipation rate. It is obviously zerofor incompressible flows.

The first two terms on the right-hand side of Equation 8 are quadratic withpositive coefficients and positive signs, and they are, therefore, always positiveor zero. The last term on the right-hand side indicates the contributions to thedissipation rate by the purely nonhomogeneous part of the flow. Its time-averagedcontribution disappears in homogeneous flows. This term, in principle, can resultin negative values, and thus it can reduce the dissipation rate. This does not violatethe second law of thermodynamics as long as the total dissipation remains positiveor zero at any point in space and time. It should be noted that the dissipation termappears as a source term in the transport equation for entropy s, which, for idealgases, is

Ds ]qiqT 4 ` E. (9)Dt ]xi

It has been customary in the past [see, for instance, Zeman (1990)] to decomposeE into a solenoidal part Es, which is the traditional incompressible dissipation andthe dilatational part Ed. In this case

]U ]Ui j 2 2E 4 E ` E , E 4 l X X ` 2l 1 S , E 4 4/3 lS .s d s k k kk d kk3 4]x ]xj j

3. VORTICITY TRANSFER THROUGH SHOCK WAVES

A better understanding of the nature of the interaction of turbulent structures andvortex motions of turbulent flows with shock waves requires information onimportant quantities such as vorticity, rate-of-strain tensor and its matrix invari-ants, and dissipation of turbulent kinetic energy. These flow quantities, althoughcomputationally as well as experimentally very demanding to obtain, if resolvedto proper scales are well suited for describing physical phenomena in vorticalflows. One of the fundamental questions is how vorticity is transferred through

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shock waves. Ribner (1953) considered the case of a vorticity wave that is trans-mitted through the shock governed by Snell’s law. Hayes (1957) derived an equa-tion for the vorticity jump across a surface of discontinuity in unsteady flows byconsidering the thermodynamic relations upstream and downstream of it. Thevorticity changes are given as follows:

rdX 4 0, (10)n

r r1 (D v ` v D n)d(q)r t t s trdX 4 n 2 ¹ (qv )d 1 . (11)t t r3 1 2 4q (qv )n

Here the subscripts n and t represent components in the directions normal andtangential to the surface of discontinuity, the vector being the unit vector normalrnto the surface of discontinuity. The vector represents the relative normal veloc-rvity component with respect to the surface of discontinuity that is r rv 4 v 1r n

. The operator Dt is the tangential part of the total derivative operator or therv s

material derivative for an observer moving with and along the shock surface withvelocity , where andr r r r r r rnv ` v D v 4 [dv /dt] ` v • ¹ v dx /dt 4s t t t t t t t t t

` vs and , where K is the curvature ten-r r r r]v /]t ]v /]n D n 4 1¹ v 1 v • Kt t t t s t

sor .rK 4 1¹ nThe first relation (Equation 10) indicates that the surface normal-vorticity com-

ponent is not affected by the shock. This is a direct outcome of the Rankine-Hugoniot relations, which show that velocity components parallel to the shocksurface remain unchanged through it. Equation 11 shows that the vorticity changeacross the shock is determined by the density jump and by the gradients, tangentialto the surface of discontinuity, of the incoming-flow-velocity.

Although Hayes’s theory is very significant, in many cases it is not usefulbecause it provides information in reference to a local system of coordinatesattached to the shock front, which may change position and shape with timeduring the interaction. The position and shape of the shock wave must emerge aspart of the solution of the mathematical system of equations and is not known apriori. Nevertheless, this theory has shown that vorticity parallel to the shockfront will be affected by the interaction with the shock.

The transport equation of vorticity is very useful in understanding the mech-anisms associated with compressible fluid motions (in tensor notation):

DX 1 ]q ]p ] 1 ]si gj4 S X 1 X S ` e ` e . (12)ik k i kk iqg iqg 1 22Dt q ]x ]x ]x q ]xq g q j

It describes four dynamically significant processes for the vorticity componentXi, namely that of stretching or compression and tilting by the strain Sik, vorticitygeneration through dilatation, baroclinic generation through the interaction ofpressure and density gradients, and viscous effects expressed by the viscous stressterm. The viscous term may also describe reconnection of vortex lines at very

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small scales owing to viscosity. If the viscous term can be ignored because itsmagnitude, very often, is small, then the change of vorticity of a fluid element ina Lagrangian frame of reference can be entirely attributed to vortex stretchingand/or tilting, to dilatational effects, and to baroclinic torque.

If Equation 12 is multiplied by Xi, then the transport equation for the enstrophy1⁄2 Xi Xi can be obtained:

D(1/2X X ) 1 ]q ]pi i4 S X X 1 X X S ` e Xik k i i i kk iqg i2Dt q ]x ]xq g

] 1 ]s 1 ]X ]sgj j gj` e X 1 e . (13)iqg i iqg1 2]x q ]x q ]x ]xq j q j

The physical mechanisms associated with the time-dependent changes of enstro-phy of a fluid element in a Lagrangian frame are similar to those responsible forthe generation or destruction of vorticity. Enstrophy is a very significant quantityin fluid dynamics because it is related not only to the solenoidal dissipation aswas mentioned in Section 2, but also to the invariants of the strain-of-rate matrixSij. In addition, enstrophy is a source term in the transport equation of dilatationSkk:

2D(S ) 1 1 ]q ]p 1 ] p ] 1 ]skk kq4 1S S ` X X ` 1 ` . (14)ik ki k k 2Dt 2 q ]x ]x q ]x ]x ]x q ]xk k k k k q

This transport equation relates the change of dilatation along a particle path, whichcan be caused by the straining action of the dissipative motions (SikSik), as wellas by the rotational energy of the spinning motions as it is expressed by theenstrophy 1⁄2XkXk. Pressure and density gradients, as well as viscous diffusion,can also affect dilatation. It should be noted that transport Equation 14 reducesto the well-known Poisson equation

21 ] p 14 1S S ` X X , (15)ik ki k kq ]x ]x 2k k

for incompressible flows of constant density (Skk 4 0).

3.1 A Demonstration

The transport equations of vorticity and enstrophy can be used to gain furtherinsight into the processes involved in the interactions of shock waves with tur-bulence, by looking at the instantaneous signals of the various quantities involved.To demonstrate how vorticity is being transmitted through shock waves, severalexperiments were carried out at City College of New York by Agui (1998), inwhich a subminiature multiwire probe was used to measure vorticity and othervelocity gradients in an interaction of a grid-generated turbulence with a travelingnormal shock. This probe, a modification of the probe used by Honkan & Andreo-poulos (1997), is capable of measuring time-dependent velocity gradients with a

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resolution of ;10–20 Kolmogorov viscous scales in these high-speed flows. Fig-ure 3 shows signals of longitudinal velocity component U1, lateral velocity com-ponent U3, and longitudinal and lateral vorticity components X1 and X2,respectively, dilatation 1Skk 4 1/qDq/Dt, dilatational dissipation Ed 4 SkkSkk,and solenoidal dissipation/enstrophy Es 4 XiXi. Each signal has been normalizedby its rms value upstream of the interaction so that signals with relatively largefluctuations are reduced and signals with relatively small fluctuations areenhanced. Thus all signals have been brought to about the same level. This nor-malization also helps to observe whether the signal is amplified after the inter-action because the rms value of this portion of the signals is the gain G, whichis defined in terms of a representative variable Q as GQ 4 Qrms,d /Qrms,u, wherethe superscripts u and d denote upstream and downstream of the interactions,respectively.

Each of the signals, with the exception of that of U1, has been displaced bymultiples of 10 rms units to provide better visual aid. The shock wave locationis evident on the longitudinal velocity signal, where its value drops substantially.An inspection of the level of fluctuations, after the passage of the shock and theactual computation of their rms values, indicates that some signals are amplifiedand some are not. The longitudinal vorticity X1 and lateral velocity U3 signalsare only slightly affected by the interaction. The computed data also show a 2–5% reduction in the rms values, which practically indicates that within the exper-imental uncertainty, no changes occur in the transmission of X1 and U3 throughthe shock, as is expected from the Rankine-Hugoniot relations. Longitudinalvelocity fluctuations and lateral vorticity fluctuations X2 are substantially ampli-fied through the interaction, with gains of ;1.4. Fluctuations of dilatational dis-sipation SkkSkk and solenoidal dissipation also show gains of the order of ;1.2.These two dissipation signals exhibit strong intermittent behaviors that are char-acterized by bursts of high-amplitude events. These, sometimes, reach values ofup to 8 rms units after which less violent time periods occur. It should be men-tioned that fluctuations of dilatation Skk are very small in this experiment, only10% of the fluctuations of vorticity. These values are typical in flows with lowfluctuations of Mach number. Consequently, baroclinic vorticity generation isnegligible because pressure fluctuations are also small. Thus, the only source ofvorticity change, in addition to viscous diffusion, is through the stretching orcompression and tilting term SikXk. A closer look at the components of this vectorindicates that the level of fluctuations of the terms S11X1, S22X2, and S33X3, whichare the terms indicating vortex stretching or compression, decreases by ;30% onaverage. Clearly, compression by the shock wave reduces the level of vorticityfluctuations associated with the so-called mechanism of vortex line compressionon all vorticity components. However, tilting of vorticity components by theaction of the shear Sik increases through the interaction by various amounts oneach of the three vorticity components so as to compensate for the deficit causedby the vortex line compression and to further increase in the case of the twocomponents parallel to the shock front. Thus, the complete stretching term

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Figure 3 Time-dependent signals in grid turbulence-shock interactions at Ms 4 1.2.

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increases after the shock. It should be noted that the DNS of Lee et al (1992)agree with this finding.

4. SHOCK INTERACTIONS WITH HOMOGENEOUSAND ISOTROPIC TURBULENCE

The interaction of an isotropic and homogeneous turbulent flow with a strongaxisymmetric disturbance, such as a normal shock, is the best paradigm case ofa flow where the geometry is reasonably simplified and basic physics of suchinteractions can be investigated. Traditionally, this flow has been used as a testcase where a turbulence model of the large eddy simulations or Reynolds-averaged Navier-Stokes class can be evaluated. The absence of turbulence pro-duction and the simplified flow geometry can expose the model’s strengths andweaknesses.

Experimental realization of a homogeneous and isotropic flow interacting witha normal shock in the laboratory is a formidable task. There are two major dif-ficulties associated with this: setting up a compressible and isotropic turbulentflow is the first one, and the generation of a normal shock interacting with flowis the second. These two problems may not be independent from each other. Asa result of these difficulties, two different categories of experiments have beenperformed. The experimental arrangement in shock tubes offers the possibility ofunsteady shock interactions with isotropic turbulence of various length scales andintensity. This category includes the experiments of Hesselink & Sturtevant(1988), Keller & Merzkirch (1990), and Honkan & Andreopoulos (1992). In thelast experiments the induced flow behind the incident shock wave passes througha turbulence-generating grid and then interacts with the shock reflected off theend wall. A unique facility has been developed at City College of New York inwhich the flow Mach number can, to a certain extent, be controlled independentlyfrom the shock wave strength. This has been achieved by replacing the end wallof the shock tube with a porous wall of variable porosity. The large size of thisfacility allows measurements of turbulence with high spatial and temporal reso-lution (see Briassulis & Andreopoulos 1996; G Briassulis, JH Agui and J Andreo-poulos, submitted for publication; Agui 1998). Thus, even shock interactions withincompressible flows can be generated.

Configuring a homogeneous and isotropic turbulence interacting with a normalshock in a supersonic wind tunnel appears to be more difficult than in a shocktube. A turbulence-generating grid or other device is usually placed in the flowupstream of the converging-diverging nozzle. The flow then interacts with a sta-tionary shock produced by a suitable shock generator located farther downstreamin the working section. The problem with such configurations is that the flow an-isotropy is substantially increased through the nozzle.

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Although DNS of homogeneous and isotropic turbulence interacting with theshock waves are restricted to low Reynolds numbers, they have been very usefulin providing physical insights of the mechanisms involved. These computationscan provide results that sometimes are very difficult to obtain experimentally, orthey can be used as diagnostic tools. Lee et al (1992), for instance, computedresults containing pressure fluctuations inside the flow or entropy fluctuationsthrough shocks, which currently seem impossible to obtain experimentallybecause of lack of appropriate experimental techniques for measuring thesequantities.

There are three different experiments of homogeneous isotropic turbulenceinteractions with shocks all carried out in supersonic wind tunnel arrangementsin France. Debieve & Lacharme (1985) carried out an oblique shock interactionwith turbulence generated inside the plenum chamber and developed in a flowwith a Mach number of 2.3. A threefold amplification of turbulence was measuredclose to the shock, which was attributed to shock oscillations and distortionscaused by the incoming turbulence. Farther downstream, a residual gain of tur-bulence fluctuations of 1.25 was measured.

In the work of Blin (1993), described by Jacquin et al (1993), the grid used togenerate turbulence was also acting as a sonic throat to accelerate the flow toMach number 1.4. The position of the shock was controlled by a second throatdownstream and by the suction of the wall boundary layer. The tabulated dataprovided by O Leuchter (private communication, 1997) indicated a considerabledeceleration of the incoming flow before the shock. The longitudinal mean veloc-ity from 469 m/s at x 4 0.03 m becomes 380 m/s at x 4 0.32 m. This decelerationis most probably caused by Mach waves present in the flow or by pressure waves,as they are called by Jacquin et al, which emanate from the grid. As a result, theturbulence level of the incoming flow may be higher than it ought to be becauseit is known that supersonic flow deceleration through gradual compression isassociated with turbulence augmentation. This may also explain the lack of tur-bulence amplification through the shock found in this work. Jacquin et al, how-ever, attribute this to the role of pressure fluctuations in the energy exchangebetween kinetic energy and potential energy in pressure fluctuations. In addition,the use of laser Doppler anemometry to obtain the data may have contributed tothis because it is known that laser Doppler anemometry in compressible flowsoverestimates turbulence intensities. The probe volume of the laser Doppler ane-mometry was 10 mm 2 5 mm, and it is considered rather large to resolve accu-rately the turbulence of the flow field. In terms of Kolmogorov length scales, theprobe volume appears to be 500 2 250. Thus, the existence of Mach wavesinside the flow has contaminated the flow, and instrumentation problems mayhave biased the results, which did not show the expected amplification of tur-bulence after the interaction with the shock.

A multinozzle turbulence generator was used in the Mach 3 experiments ofAlem (1995) published by Barre et al (1996). A normal shock was formed by theinteraction of two oblique shock waves of opposite directions. The flow after the

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interaction is highly accelerated because of the two shear layers/slip lines in theboundaries of the useful flow region. The velocity is 200 m/s at x 4 10 mm afterthe shock and increases to 250 m/s at x 4 20 mm, which results in an accelerationof ]u/]x 4 5400 s11.

This level of acceleration is very strong and is expected to reduce turbulenceintensities after the interaction. Thus, the amplification levels found in this workare probably contaminated by the additional effects of acceleration in the subsonicflow downstream of the interaction. Another weakness of this data set is that thelevel of turbulence intensity at the location of the shock is extremely low, '0.4%.

4.1 Turbulence Amplification Through the Interaction

Amplification of turbulence is one of the major features of shock-turbulence inter-actions. LIA predicts amplification of turbulence as long as fluctuations of pres-sure, velocity, and temperature upstream of the shock are small so that the shockfront is not substantially distorted and the Rankine-Hugoniot conditions can belinearized. In interactions in which strong distortions of the shock front are evidentbecause of higher turbulence intensity, the use of LIA is fully justified. DNS datain interactions with weak shocks (Lee et al 1993, Hannappel & Friedrich 1995)and Euler simulations (Rotman 1991) also predict amplification of turbulence.Almost all experiments confirm qualitatively this analytical and computationalresult. The experiments of Jacquin et al (1993) report no significant enhancementof turbulent kinetic energy. However, as was mentioned earlier, the decelerationin the flow upstream of the interaction because of the existence of Mach wavesmay have contributed toward overestimating the level of turbulence upstream ofthe interaction.

In the recent experimental work of Zwart et al (1997), a clear increase inturbulent intensities is reported. These authors, in their study of the decay behav-ior of compressible grid–generated turbulence, encountered shock waves in therange of flow Mach numbers of 0.7 , M , 1.0. They attributed the increasedturbulent intensities in this regime to shock wave unsteadiness.

Typically, the amplification of turbulent fluctuations depends on the shockwave strength, the state of turbulence of the incoming flow before the interaction,and its level of compressibility. Figures 4 and 5 show the amplification of velocityand vorticity fluctuations, respectively, as a function of the shock strength Ms.The experimental data have been obtained by Agui (1998). Two different gridswere used in these experiments at two different Mach numbers each. The datashown in Figure 4 clearly suggest an amplification of longitudinal velocity fluc-tuations, whereas the fluctuations of the other two components are slightlydecreased or remain unchanged through the interactions. The amplification dataof vorticity, shown in Figure 5, indicate a substantial enhancement of vorticityfluctuations in the direction parallel to the shock front with no amplification orsmall attenuation of longitudinal vorticity fluctuations. The amplification data ofthe lateral component of vorticity fluctuations shown in Figure 5 represent the

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Figure 4 Amplification of velocity fluctuations in grid turbulence–shock interactions.Experimental data of Agui (1998), LIA data of Lee et al (1993), RDT data of Jacquin etal (1993), and DNS data of Lee et al (1993).

average of the two components parallel to the shock front. The results of the LIAof Lee et al (1993) as well as their DNS data are also plotted in Figure 5. LIAshows that lateral vorticity fluctuations increase with the shock strength after theinteraction with the shock, and the experimental data tend to confirm this.

The DNS data of Hannappel & Friedrich (1995) show that the amplificationof the vorticity fluctuations in the lateral direction increases with compressibilityin the upstream flow, whereas the amplification of longitudinal velocity fluctua-tions is reduced by the same effect. They also found that longitudinal vorticityfluctuations are slightly reduced through the shock, as it has been found in theexperiments of Agui (1998).

If the damping effects of pressure are ignored, the RDT of Jacquin et al (1993)leads to the following simple relations for the amplifications of longitudinal veloc-ity and lateral vorticity fluctuations:

21 ` 2C2G 4 C , G 4 ,2 2u x 3 43

where C is the density ratio C 4 qd /qu. The results of the RDT are also plotted

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Figure 5 Amplification of vorticity fluctuations in grid turbulence–shock interactions.Experimental data of Agui (1998), LIA data of Lee et al (1993), RDT data of Jacquin etal (1993), and DNS data of Lee et al (1993).

in Figures 4 and 5. The density ratio in the experiments by Agui (1998) wasbetween 1.25 and 1.7.

The early work of Honkan & Andreopoulos (1992) showed that amplificationof urms, Gu, also depends on the turbulence intensity and length scale of the incom-ing isotropic turbulence. The work of Briassulis and Andreopoulos (1996a,b) andAgui (1998) also confirm this finding. The amplification of turbulent kineticenergy across shocks seems to decrease with increasing Mt. Both experimentaldata from the above-mentioned studies and the DNS data of Lee et al (1993) andHannappel & Friedrich (1995) agree with this finding. Thus, the outcome of theshock turbulence interaction depends also on the compressibility level of theincoming flow.

The effect of the shock strength on the velocity fluctuations is shown in Figure6, where the data of Briassulis (1996) are plotted. For the 1 2 1 (cells/in2) grid,it appears that the amplification of turbulence fluctuations defined as Gu 4 urms,d/urms,u increases with downstream distance for a given flow case and interaction.Thus, turbulence amplification depends on the evolution of the flow downstream.As the Mflow increases, Gu also increases. For finer grids the effects of shockinteraction are felt differently. For the 2 2 2 grid, for instance, the data show

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Figure 6 Amplification of velocity fluctuations in interactions of grid-generated turbu-lence with a normal shock. Experimental data of Briassulis (1996).

that, for a practically incompressible upstream flow interacting with a rather weakshock, amplification of turbulence occurs at x/M . 35. The amplification isgreater when Mflow increases to 0.436. However, when compressibility effects inthe upstream flow start to become important, no amplification takes place (Gu is;1).

Some more dramatic effects of compressibility are illustrated in Figure 7, inwhich the amplification Gu is plotted for a finer grid with mesh size 5 2 5. Theinteraction of a weak shock with a practically incompressible turbulent flow pro-duces the highest amplification of velocity fluctuations, with Gu reaching a valueclose to 2. As the Mflow increases, Gu decreases, and at Mflow 4 0.576 a slightattenuation occurs at downstream distances. It is, therefore, plausible to concludethat for high shock strength (high Mach number), compressibility effects controlthe velocity fluctuations, which are generated by fine grids, and no amplificationof turbulent kinetic energy is observed.

Hannappel & Friedrich (1995) have also shown in their DNS work that com-pressibility effects in the upstream reduce turbulence amplification significantly.The LIA of Ribner (1953), which was initially developed for an incompressibleisotropic turbulent field, predicts amplification of turbulence fluctuations.Recently, Mahesh et al (1997) have shown that LIA as well as DNS may show acomplete suppression of amplification of kinetic energy if the upstream correla-tion between velocity and temperature fluctuations is positive. It is therefore pos-

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Figure 7 Amplification or attenuation of velocity fluctuations in grid-generated turbu-lence with a normal shock. Experimental data of Briassulis (1996).

sible that, for very fine grids and high Mach number flows in which the dissipationrate of turbulent kinetic energy is high, entropy or pressure fluctuations may beresponsible for completely suppressing turbulence amplification.

Briassulis (1996) found amplification of velocity fluctuations after the inter-action in all cases involving turbulence produced by coarse grids. This amplifi-cation increases with shock strength and flow Mach number. For fine grids,amplification was found in all interactions with low Mflow, whereas at higher Mflow,amplification was reduced, or no amplification of turbulence was evident. Theseresults indicate that the outcome of the interaction depends strongly on theupstream turbulence of the flow.

4.2 Turbulence Length Scales Through the Interaction

As mentioned by Lele (1994), there is some disagreement between experimentalresults and DNS data as far as how the various length scales of turbulence areaffected by the interaction with the shock wave. The DNS data of Lee et al (1993)and Hannappel & Friedrich (1995) indicate that all characteristic length scales,namely longitudinal- and lateral-velocity, integral-length scales, longitudinal-velocity microscales, and dissipation-length scales, as well as integral-lengthscales and microscales of density fluctuations, decrease through shock interac-tions. The experimental data of Keller & Merzkirch (1990) show that the densitymicroscale increases across the shock. Hannappel & Friedrich (1995) also showthat the reduction in the Taylor microscale parallel to the shock front is weaker

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by a factor of 2 when the compressibility level of the incoming turbulence is high.On the other hand, the dissipation length scale in the experiment of Honkan &Andreopoulos (1992) was also found to increase after the interaction. DNS resultsof Lee et al (1994) have indicated a small increase of dissipative-length scalesthrough weak shock interactions. Thus, there is no agreement among variousresearchers on how shock interactions affect the length scales. Intuitively, oneshould expect that compression should reduce length scales.

To resolve the disagreement between experiments and DNS, Briassulis (1996)carried out detailed space-time correlation measurements by using a rake of sixparallel wires and three temperature wires separated by 1 mm. To estimate thelength scales in the longitudinal n1 direction and normal n2, the cross correlation

coefficients rij(nk) 4 / were evaluated by two-2 2u (x)u (x ` n ) u (x) u (x ` n )! !i j k i j k

point measurement in the n2 direction and from autocorrelations in the n1 directionafter invoking Taylor’s hypothesis.

The data show that the integral length scale L11(n1) increases with downstreamnondimensional distance x/M for all investigated cases in the flow before theinteraction. It is also shown that L11 for Mflow 4 0.475 is higher than forMflow 4 0.36. However, if the flow Mach number increases to Mflow 4 0.6 and,therefore, stronger compressibility effects are present, then the integral-lengthscale drops.

After the interaction with the shock wave, the distribution of L11(n1) is morecomplicated. All the scales are reduced considerably. However, the reduction ofthe larger scales is greater. This is shown in Figure 8, where the attenuation ratio

is plotted. At large x/M, where the initial scales wereG 4 L (n )/L (n )L 11,d 1 11,u 111

the largest, the reduction is dramatic. Thus, once again, it is found that amplifi-cation or attenuation is not the same for all initial length and velocity scales. Itis interesting to observe that the stronger the shock strength, the greater the atten-uation of the longitudinal-length scales.

The two-point correlation r11(n2) in the lateral direction n2 of the longitudinal-velocity fluctuations is shown in Figure 9. Not all the curves cross the zero line,and, therefore, it is very difficult to integrate them to obtain the classically definedlength scale in the lateral direction. However, the slopes of these curves are indic-ative of their trend. It is rather obvious that the length scales before interactionare reduced with increasing flow Mach number. This behavior is very similar tothat of L11(n1). After the interaction, however, the length scale L11(n2) increasesin the first two cases and decreases in the strongest interaction.

To investigate the effect of initial conditions on this correlation at the highestflow and shock Mach number, where the lateral scales are shown to reduce (Figure9), various grids were used. The data shown in Figure 10 indicate that the cor-relation increases substantially in the finest grid, 8 2 8 with the lowest Rek, afterthe interaction for the range of length scales investigated. However, the coarser2 2 2 grid with the highest Rek 4 737 shows the greatest attenuation in thelateral integral scale of turbulence after the interaction.

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Figure 8 Ratio of the longitudinal integral-length scales for various experiments in grid-generated turbulence interactions with a normal shock. Experimental data of Briassulis(1996).

Figure 9 Spatial correlation in the lateral direction for three different flow cases in grid-generated turbulence interactions with a normal shock. Experimental data of Briassulis(1996).

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Figure 10 Spatial correlation in the lateral direction for three different downstream loca-tions in grid-generated turbulence interactions with a normal shock. Experimental data ofBriassulis (1996).

Thus, integral-length scales in the longitudinal direction were reduced afterthe interaction in all investigated flow cases. The corresponding length scales inthe normal direction increased at low Mach numbers and decreased duringstronger interactions. It appears that, in the weakest of the present interactions,the eddies are compressed in the longitudinal direction drastically, whereas theirextent in the normal direction remains relatively the same. As the shock strengthincreases, the lateral length scale increases, whereas the longitudinal-length scaledecreases. At the strongest interaction of the present cases, the eddies are com-pressed in both directions. However, even at the highest Mach number case, theissue is more complicated because amplification of the lateral scales has beenobserved in fine grids. Thus, the outcome of the interaction strongly depends onthe initial conditions.

The dissipative length scale was found to increase in several cases, and thedisagreement with the DNS remains an issue, although it may be attributed to thedifference in Rek. The Rek of the DNS was between 12 and 22, which is consid-erably lower than the values between 200 and 750 achieved in the experimentsof Briassulis (1996) and may be the cause of this disagreement between experi-ments and DNS. The difference between experiments and DNS is also noticeableif the Reynolds number

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2(qu u )i iRe 4T le

is considered. ReT in the experiments is ;4000, whereas in DNS it is only 750.As a result, the timescale of turbulence uiui/e and its ratio to the timescale imposedby the shock 1/S11, uiuiS11/e, are quite different between DNS and experiments.

The results of the experiments of Briassulis (1996) clearly show that most ofthe changes, either attenuation or amplification of quantities involved, occur forlarge x/M distances in which the length scales of the incoming flows are high andturbulence intensities are low. Thus, large eddies with low-velocity fluctuationsare affected the most by the interaction with the shock.

5. HOMOGENEOUS SHEARED TURBULENCETHROUGH SHOCKS

The only work available in this type of interactions is that of Mahesh et al (1997),who used RDT to demonstrate that the amplification of turbulent kinetic energydepends on anisotropy of the Reynolds stress tensor of the incoming flow and theratio of shearing rate to that of compression. The dependence of turbulence ampli-fication on the initially anisotropy is reduced with the obliqueness of the shockwave. A change in the sign of the initial shear stress has been observed forsufficiently strong interactions. This type of idealized interaction has yet to beconfigured experimentally.

6. TURBULENT JET FLOWS AND SHOCK WAVEINTERACTIONS

Interactions of jets with shock waves are considerably more complex than thepreviously considered homogeneous flow interactions because inhomogeneityand anisotropy are additional parameters to consider, which may change the flowphysics completely. The variable shear, for instance, in the shear layer emanatingfrom the jet is an additional parameter that is responsible for the production oflarge amounts of turbulent kinetic energy before the interaction. One obviousquestion is how this shear is affected by the interaction with the shock wave.Jacquin & Geffroy (1997) have shown that this shear is decreased through theinteraction.

Parameters such as the convective Mach numbers Mc, the velocity ratio UJ/Ua, and density ratio qJ/qa have been considered in this type of interaction inaddition to the shock strength, compressibility level, and velocity and length scaleof the incoming turbulence. A schematic of the flow interaction and the param-eters involved is shown in Figure 11. Baroclinic generation of vorticity may takeplace due to the nonalignment of the density and pressure gradients. Density

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Figure 11 Schematic of the turbulent jet–shock interaction.

gradients can be generated by heating the jet flow, as in the experiments of Jacquin& Geffroy (1997), or by using different gases, as in the case of Cetegen & Her-manson (1995), Hermanson & Cetegen (1998) or Hermening (1999). The pressuregradient across the shock interacts with the density gradient across the interfacebetween the jet and the ambient fluid to generate additional vorticity.

Two types of configurations have been used in the past to study experimentallyshock-jet interactions: steady interaction in supersonic wind tunnels and unsteadyinteraction with traveling shock waves in shock tube. Jacquin & Geffroy (1997)designed a facility with a jet flow exiting at the throat of a nozzle. The jet andthe sonic coflowing stream were further expanded to M 4 1.6. Density ratiosfrom 0.33 to 1 were achieved through heating of the jet flows. The velocity ratioranged from 0.58 to 1 and the convective Mach number was #0.42. Turbulenceamplification was measured in the quasihomogeneous region of the jet, whereasturbulence kinetic energy was decreased in the mixing layer region of the flows.According to Jacquin and Geffroy, this is, most probably, caused by baroclinicvorticity effects resulting from a reduction in the near shear by a factor of 2.0.No substantial effects of jet temperature, that is, density ratio on amplification orreduction of turbulent kinetic energy, were identified in experiments. Their flowvisualization indicated that the shock front is not distorted by the shock. Thus,baroclinic vorticity generation or destruction can take place in the absence ofshock front deformations. Substantial shock front deformations, Mach reflections,and additional shocks can be developed when an overexpanded jet interacts witha normal shock (see Jacquin et al 1991). Large amplification of turbulence hasbeen found in this case.

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Figure 12 Helium jet–shock interaction in a shock tube (JC Hermanson & BM Cetegen,private communication, 1999).

Considerable generation of baroclinic vorticity is reported by Hermanson &Cetegen (1998) in their experiments on nonuniform-density jets interacting withtraveling shock waves. Density ratios of 0.0595 (helium jet) through 1.31 (CO2

jet) were achieved in these shock tube experiments. Figure 12 shows an imageof the flow containing a vortical structure produced behind the traveling shockwave for a helium jet interaction obtained at x/d 4 50 (from JC Hermanson &BM Cetegen, private communication, 1999). Planar Mie scattering of planar laserlight was used to visualize the flow, which was seeded with mineral oil. Thestructure contains vorticity with a sign corresponding to that of ‘‘jet-like’’ vorticesformed by jets with velocities considerably higher than the coflowing fluid veloc-ity behind the traveling shock wave. The formation of this structure is attributedby Hermanson & Cetegen to baroclinic generation of vorticity. There is someapparent inconsistency between the experiments of Jacquin & Geffroy (1997),who found that baroclinic effects reduce vorticity after the interaction with the

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shock, and the experiments of Hermanson & Cetegen (1998), who found that thesame effects enhance vorticity after the interaction. This disagreement may beresolved if the baroclinic term of the vorticity transport Equation 12

1 ]q ]p 1B 4 e 4 ¹px¹X iqg 1 2i 2q ]x ]x qq g

is considered. reaches maxima or minima when the two gradients ¹p andBXi

¹(1/q) are at 908 or 2708 to each other. If only vorticity X3 is considered, then

1 ]q ]p ]q ]p 1 ]q ]pB 4 1 ù 1 .X 3 43 2 2q ]x ]x ]x ]x q ]x ]x1 2 2 1 2 1

The first term can be neglected because pressure is expected to change very littlein the normal direction owing to the thin shear layer approximation. In the exper-iments of Jacquin & Geffroy (1997), ]p/]x1 is positive across the shock and ]q/]x2 is positive in the upper half of the flow (see also Figure 11) and negative inthe lower half of the flow. This shows that BX is negative in the upper half, wherethe initial vorticity X3 is positive, and positive in the lower half of the flow, whereX3 is negative. Therefore, for stationary shocks such as in the experiments ofJacquin & Geffroy, baroclinic effects tend to suppress vorticity; that is, the bar-oclinic torque tends to bring the vorticity toward zero values. In the experimentsby Hermanson & Cetegen, the shock wave travels in the same direction as the jetflow, and it introduces a negative pressure gradient ]p/]x1 into the flow (for astationary observer traveling with the shock), and, because the density gradientis the same as in the previous case, all contributions from BX tend to enhancevorticity X3. In that respect, there is no disagreement between the two experi-ments. Hermanson & Cetegen found no such vortical structures in their experi-ments with air or CO2 jets.

No distinct vortex structures were observed in the experiments by Hermening(1999), who used helium, air, and krypton gases to generate baroclinic vorticityof various signs because helium is lighter than air and krypton is heavier than air.Two jet configurations were used in these experiments. In the first one, the jetfluid flowed in the same direction as the traveling shock wave. In the secondconfiguration the shock traveled in the opposite direction to that of the jet flow.Figure 13 shows several images of the interaction of a shock wave moving in thesame direction as the air jet flow obtained by Hermening. The images wereobtained from different experiments with the same bulk flow parameters and withthe shock wave located at different positions from the jet exit. The jet flow hada Mach number Mj 4 0.9, and the shock wave was traveling with a Mach numberof Ms 4 1.33. Talcum powder with particles ,1 lm in diameter was used toseed the flow, and a Nd:YAG laser was used to visualize the flow. The first image(Figure 13a) shows the jet flow before the interaction. The jet flows downstream,while it spreads laterally and its velocity decreases. The complexity of the inter-action is shown in Figure 13b and c. Figure 13b shows the flow during the inter-

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Figure 13 (a–c) Images of the interaction of a MS 4 1.34 shock wave with a MJ 4 0.9 coflowing jet. Flows from left to right. Images are fromHermening (1999). (d) Schematic of the interaction.

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action with the shock wave located at 5 diameters downstream of the jet exit;traveling from left to right. Figure 13c shows the flow considerably after passageof the shock, which is located outside the viewing area. There are three regionsof interest in the flow during the interaction as shown in Figure 13b. The firstregion is characterized by a stenosis of the jet cross section observed at a distancebetween 2 and 3 diameters downstream of the exit. This region is followed by aregion closer to the exit, with a jet stream of apparently larger diameter than theprevious one, and it is proceeded by a third region immediately behind the shockwave, where spreading of the flow can be observed. This structure of the jet canbe explained if one considers the time-dependent motion of the shock from thetime it is located at the jet exit until the time it has left the near-field region. Theshock wave is characterized by an induced flow with higher pressure and higherdensity behind it than ahead of it. This changes the velocity, pressure, and densityof the ambient fluid, which is now put into motion. Thus, the imposition of higherambient pressure compresses the jet fluid to a higher density, which reduces itscross section and causes the apparent stenosis. At the same time, the higher ambi-ent pressure requires a reduction of the exit Mach number Mj because the pressureratio of the ambient pressure to the jet stagnation pressure, pa/poJ, has increased.Thus, the jet velocity UJ is reduced because the stagnation pressure of the jet ismaintained as constant by a plenum pressure controller. The new velocity of thejet is sightly lower than the new ambient velocity Ua, which results in no spreadingof the jet into the ambient coflowing fluid and the formation of a ‘‘wakelike’’vortex between the readjusted jet and the region with the stenosis, as shown inFigure 13c.

The downstream region, however, exhibits a different behavior, which indi-cates no stenosis of the jet flow after the shock. Before the interaction with theshock, this region is characterized by a decaying mean velocity, high shear, andincreased spreading of the jet, which results in the production of substantial levelsof turbulence. It appears that, after the interaction, considerable mixing has takenplace, and the turbulence most probably has increased. Figure 13d depicts a sche-matic illustration of the interaction of the jet flow with the shock wave, whichincludes all three regions of the flow discussed above.

Images of the counterflow configuration of a jet-shock interaction, obtainedby Hermening (1999), are shown in Figures 14a and b. The air jet flows fromright to left, whereas the shock travels in the opposite direction. The jet and shockMach numbers are the same as in the previous configuration. The most strikingfeature of this interaction is the substantial level of spreading of the jet flowobserved after the shock, which is accompanied by high levels of turbulence.Flow reversal is also expected at some distance from the jet exit.

The experiments of Hermening (1999) with helium and krypton indicatedsome similar behavior. The initial spreading rate of the helium jet before theinteraction was higher than that of the air jet. As a result, substantially moremixing was observed after the interaction with the shock than in the case of air.The spreading rate of the krypton jet, which was initially smaller than that of the

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Figure 14 Air–jet interaction with a traveling shock wave in a counterflow configuration.Jet flows from right to left, while shock wave travels from left to right; MJ 4 0.9, MS 41.33. Images are from Hermening (1999).

air jet. The shock interaction in the coflowing configuration increased the mixingcompared with the initial jet. However, substantially more mixing was observedin the counterflow configuration, which supports the notion explained above thatbaroclinic effects, in this case of a heavier-than-ambient jet, enhance the vorticityof the jet. In this case the interaction of the density and pressure gradients bringsabout additional vorticity into the flow, which enhances the mixing.

The mixing enhancement shown in the experiments of a helium jet interactingwith a traveling shock wave has been also verified by the calculations of Obata& Hermanson (1998), who used a Reynolds-averaged Navier-Stokes–basedmodel to calculate the turbulent jet before the interaction and a TVD scheme tosimulate the passage of the shock.

7. PLANE SHEAR LAYER INTERACTIONS WITH SHOCKWAVES

Interactions of plane shear layers with shock waves have been configured insupersonic wind tunnels where a boundary layer flow separates over a backward-facing step formed by a cavity in the tunnel wall. A ramp, located at some distancedownstream, is used to generate an oblique shock. This configuration has beenused in the experiments by Settles et al (1982), Hayakawa et al (1984), andSamimy et al (1986). The separated shear layer develops over the cavity as acompressible shear flow under the influence of a recirculating zone beneath it andsubsequently undergoes compression by the shock system and then reattaches tothe ramp. The phenomena appear complicated within the reattachment region,showing a great degree of unsteadiness, which is characterized by large fluctua-tions of wall pressure and mass flux in the flow above. The location of the shock

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wave is also time dependent, which further complicates our understanding. Allmeasurements of turbulence reported in the papers mentioned above indicate sub-stantial amplification of turbulence intensities that depend on the Mach number.It is not clear, however, how these results are affected by the shock oscillationand the unsteady phenomena associated with the reattachment region.

Standard Reynolds-averaged Navier-Stokes methods also have been used tocalculate the flow (see Horstman et al 1982, Visbal & Knight 1984) with moderatesuccess in predicting the mean flow behavior. Turbulence is considerably over-estimated in these calculations.

8. TURBULENT WAKE–SHOCK INTERACTIONS

This type of interaction is very often found inside compressors or turbines ofturbomachinery, where a shock may be formed and very often may travel down-stream. The shock can subsequently interact with wakes of individual blades ofstators or rotors and alter their structure. As a result of this interaction, bladewakes carrying more turbulence impinge on the surface of blades of downstreamstages and can cause additional fatigue of the blades. No previous experimentalor analytical work could be identified on these types of interactions.

9. BOUNDARY LAYER–SHOCK WAVE INTERACTIONS

The majority of the previous work in turbulence–shock interactions belongs inthis category, and it mainly includes interactions of boundary layers with obliqueshock waves. Usually, an oblique shock wave is generated either at the oppositewall to that of the boundary layer by a sharp edge or protrusion (see Rose &Johnson 1975) or on the same wall of the boundary-layer flow by a compressionturning, which can be a sharp turning as for a compression corner or a gradualturning by a smooth change in curvature. In the first configuration the incidentshock emanating from the opposite side of the wall is also reflected over theboundary-layer wall, and therefore the flow interacts with two oblique shocks. Inthe second configuration the flow after the interaction is further compressed bythe new boundary imposed by the ramp slope, whereas concave curvature of thestreamlines has a destabilizing effect on turbulence. Thus, turbulence can not onlybe affected by the shock interaction but also by the spatiotemporal oscillation ofthe shock system, the destabilizing effects of the concave streamline curvature,and the continuing downstream compression. All these effects contribute to anincreasing level of turbulence. In addition, unsteady flow separation and flowreversal take place at the corner region, which at large turning angles can affectthe shock system upstream through a feedback mechanism.

One of the major characteristics of the turbulent boundary-layer interactionwith an oblique shock over a compression ramp is the unsteady motion of the

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Figure 15 A hypersonic compression ramp interaction is visualized using planar laserscattering imaging of a condensed alcohol fog (NT Clemens & DS Dolling, private com-munication, 1999). The ramp angle is 288 and the field of view is 75 mm in the streamwisedirection by 36 mm in the transverse direction. The upstream boundary layer is turbulent,and the free stream Mach number and velocity are 4.95 and 765 m/s, respectively. Thesetwo images are of a double-pulse sequence with a time delay between pulses of 30 ls. Inthese images the free stream is rendered gray, boundary layer fluid is darker gray, verylow velocity fluid is black (owing to evaporation of the particles), and the separation shockis the thin white interface, as labeled. Note that these images are somewhat difficult tointerpret owing to the potential for evaporation and condensation of the alcohol seed. Forreference, arrows point to the same structure in both images.

shock system, which has been observed to occur over a significant distance inthe longitudinal and lateral directions. The shock system itself may at times con-sist of multiple shocks, a K shock, or a single shock. In addition, shocks aredeformed by the interaction and therefore no longer remain planar. Figure 15shows a sequence of two images obtained by a double-pulse planar laser scatteringtechnique at the University of Texas at Austin by NT Clemens & DS Dolling

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(private communication, 1999). The images are from an M 4 5 interaction overa 288C compression ramp. They very clearly show a substantial deformation ofthe shock front, which changes significantly from the first to the second imageas a result of the mutual interaction with the incoming turbulence. A turbulentstructure going through the system is also visible. The outer part of the eddyseems to go through the shock without substantial changes. The shock itself,however, seems to have moved upward, although its extent toward the wallremains strong.

The first evidence of the grossly unsteady nature of the shock wave systemcame from measurements of the time-dependent pressure at the wall beneath theregion of interaction. Kistler (1959) provided one of the first measurements ofpressure fluctuations in a supersonic boundary layer approaching a forward-facingstep. Coe (1969), Dolling & Murphy (1982), Dolling & Or (1983), and Muck etal (1985) measured the fluctuating wall pressure in compression corners of dif-ferent geometries. These studies, as well as the studies of Smits & Muck (1987),Muck et al (1988), and Selig et al (1989), clearly demonstrated that the instan-taneous structure of the shock wave system is quite different from that of thetime-averaged system. Two possible mechanisms have been proposed to explainthe shock system unsteadiness: the first one is the turbulence of the incomingboundary layer (Plotkin 1975, Andreopoulos & Muck 1987), and the second isthat the separated shear layer amplifies the low frequencies of the contracting/expanding bubble motion, which are felt upstream through the large subsonicregion of the separated zone.

The experimental work of Andreopoulos & Muck (1987) suggested that thefrequency of the shock wave system unsteadiness may scale on the bursting fre-quency of the incoming boundary layer. This finding supports the notion that theincoming turbulence plays a dominant role in triggering the shock unsteadiness.Since this work, several other experiments have provided support for one or theother of the two proposed mechanisms. Erengil & Dolling (1991) found that someof the low-frequency oscillations of the shock are caused by the contraction andexpansion cycle of the separated zone. Unalmis & Dolling (1994) also showedthat the large-scale structures of the incoming boundary layer also contribute tolow-frequency oscillations of the shock. The latest work of Beresh et al (1999)concludes that the small-scale motion of the shock results from the individualturbulent fluctuations, whereas the large-scale motion is caused by the large-scalestructures of the boundary layer. Thus, this evidence supports the notion that thesmall and large scales of the incoming boundary layer play a dominant role inthe shock system unsteadiness.

Spanwise variation of several time-averaged quantities, such as the intermit-tency factor or correlation coefficient, for instance, in the region of the interactionpoints to the supposition that Taylor-Gortler vortices formed by the concavestreamline curvature may exist and contribute to increased complexity of thephenomena involved (see Ardonceau 1984).

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10. FINAL REMARKS

The information that is available now on the interaction of turbulence with shockwaves appears to be limited. The phenomenology associated with the interactionsis not very extensive, and the understanding of their physical attributes is notvery thorough. Experiments or simulations for instance, of shock interactions withwake flows, do not exist, and experimental data in shock interactions with homog-enous sheared turbulence are not available for comparison with existing DNSdata. However, the understanding of the physics has been advanced in the lastyears owing to the synergy among theory, numerical simulations, and experi-ments. Despite this progress, a lot of questions remain unanswered, and morework is needed to illuminate the physics of the interactions. Ribner’s (1953)theory, for instance, which predicts turbulent amplification through shock waves,has, in principle, been verified by DNS and several experiments. There is newevidence, however, that suggests that under some conditions of the upstreamcompressible turbulence the interaction may not lead to increased fluctuations ofvelocity. Thus, the final word is not yet out. In addition, several of the questionsposed in the introduction remain unanswered.

The major impediment to understanding the physical aspects of the interactionsis the difficulty in simulating the interactions numerically or experimentally. Forinstance, it is very challenging to generate normal shock waves or homogenousflows inside wind tunnels. Similar difficulties are attributed to DNS applied toflows with homogeneous or periodic boundary conditions and covering a rela-tively low range of Reynolds numbers. In addition, the lack of availability ofproper experimental or numerical tools with adequate temporal and spatial reso-lution remains a severe limitation. The new generation of nonintrusive flow visu-alization and quantitative information techniques holds a lot of promise (see Miles& Lempert 1997) with the prospect of further increasing our physical understand-ing, although extensive development and verification of these techniques are stillneeded. This review of available information revealed that the interaction betweena shock wave and turbulence is mutual and very complex: Turbulence is affectedby the shock wave, and the shock wave is affected by the turbulence. Compress-ibility of the incoming flow and the characteristics of turbulence in the incomingflow, such as velocity and length scales, are some additional parameters that seemto affect the outcome of the interaction.

ACKNOWLEDGMENTS

The support from NASA Langley Research Center, NASA Lewis Research Cen-ter, National Science Foundation, U.S. Army TACOM/ARDEC, and Air ForceOffice of Scientific Research in carrying out some of the experiments describedin this review and in writing up this article is greatly appreciated. The authorsalso thank J Hermanson and B Cetegen for providing the images of Figure 12

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and N Clemens and D Dolling for providing the images of Figure 15. Usefulcomments provided by D Knight are also appreciated.

Visit the Annual Reviews home page at www.AnnualReviews.org.

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Annual Review of Fluid Mechanics Volume 32, 2000

CONTENTS

Scale-Invariance and Turbulence Models for Large-Eddy Simulation, Charles Meneveau, Joseph Katz 1

Hydrodynamics of Fishlike Swimming, M. S. Triantafyllou, G. S. Triantafyllou, D. K. P. Yue 33

Mixing and Segregation of Granular Materials, J. M. Ottino, D. V. Khakhar 55

Fluid Mechanics in the Driven Cavity, P. N. Shankar, M. D. Deshpande 93

Active Control of Sound, N. Peake, D. G. Crighton 137

Laboratory Studies of Orographic Effects in Rotating and Stratified Flows, Don L. Boyer, Peter A. Davies 165

Passive Scalars in Turbulent Flows, Z. Warhaft 203

Capillary Effects on Surface Waves, Marc Perlin, William W. Schultz 241Liquid Jet Instability and Atomization in a Coaxial Gas Stream, J. C. Lasheras, E. J. Hopfinger 275

Shock Wave and Turbulence Interactions, Yiannis Andreopoulos, Juan H. Agui, George Briassulis 309

Flows in Stenotic Vessels, S. A. Berger, L-D. Jou 347

Homogeneous Dynamos in Planetary Cores and in the Laboratory, F. H. Busse 383

Magnetohydrodynamics in Rapidly Rotating spherical Systems, Keke Zhang, Gerald Schubert 409

Sonoluminescence: How Bubbles Turn Sound into Light, S. J. Putterman, K. R. Weninger 445

The Dynamics of Lava Flows, R. W. Griffiths 477

Turbulence in Plant Canopies, John Finnigan 519

Vapor Explosions, Georges Berthoud 573

Fluid Motions in the Presence of Strong Stable Stratification, James J. Riley, Marie-Pascale Lelong 613

The Motion of High-Reynolds-Number Bubbles in Inhomogeneous Flows,J. Magnaudet, I. Eames 659

Recent Developments in Rayleigh-Benard Convection, Eberhard Bodenschatz, Werner Pesch, Guenter Ahlers 709

Flows Induced by Temperature Fields in a Rarefied Gas and their Ghost Effect on the Behavior of a Gas in the Continuum Limit, Yoshio Sone 779

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S W –TURBULENCE INTERACTIONS - hypersonic...

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