annurev shock turb

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Annu. Rev. Fluid Mech. 2000. 32:309345Copyright 2000 by Annual Reviews. All rights reserved

00664189/00/01150309$12.00 309

SHOCK WAVETURBULENCE INTERACTIONS

Yiannis Andreopoulos, Juan H. Agui, andGeorge Briassulis

Experimental Aerodynamics and Fluid Mechanics Laboratory, Department of

Mechanical and Aerospace Engineering, The City College of the City University of New

York, 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, and

shape 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 physics

involved in this interaction is necessary for the development of future supersonic-

and hypersonic-transport aircraft, and advances in combustion processes as well

as high-speed rotor flows. In such flows the interaction between the shock wave

and 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 alter

the dynamics of the shock-wave motion.

Amplification of velocity fluctuations and substantial changes in length scales

are the most important outcomes of interactions of shock waves with turbulence.

This indicates that such interactions may greatly affect mixing. The use of shock

waves, for instance, has been proposed by Budzinski et al (1992) as a means of

enhancing the mixing of fuel with oxidant in ramjets. Turbulence amplification

through shock wave interactions is a direct effect of the Rankine-Hugoniot rela-

tions. However, this type of amplification should be decoupled from other effects

that also contribute to turbulence amplification such as destabilizing streamline


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310 ANDREOPOULOS AGUI BRIASSULIS

curvature, flow separation, dilatational effects, or longitudinal pressure gradients

that 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 the

incoming flow, and location and shape, (b) the state of turbulence of the incoming

flow as it is characterized by the fluctuation levels of velocity, density, pressure,

and entropy and length scales, (c) the level of compressibility of the incoming

flow, and (d) the flow geometry and boundary conditions.

Basic understanding of the physics of such complex interactions has been

obtained 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 confined

to the following cases: (a) homogeneous and isotropic turbulence interactions

with shock waves (see Figure 1a), (b) constant shear homogeneous turbulence

interactions with shock waves (see Figure 1b), (c) circular jet flows interacting

with shock waves (see Figure 1c), (d) plane shear layers interacting with oblique

shock waves (see Figure 1d), (e) wake flows interacting with oblique or normal

shock waves (see Figure 1e), and (f) boundary layer interactions with oblique

shock waves (see Figure 1f). This classification of the interactions also reflects

the 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 because

turbulence in all its complexity is characterized by instantaneous flow variables

that exhibit a variation in time and space. Shock waveboundary layer interactions

have been reviewed in the past by Green (1970), Adamson & Messiter (1980),

and Settles & Dodson (1991). Compressibility effects of turbulence, including

some shock wave interactions, have been recently reviewed by Lele (1994). Com-

pressibility effects in turbulent boundary layers and shear layers in the absence

of shock interactions have been reviewed by Spina et al (1994) and Gutmark et

al (1995), respectively. Birch & Eggers (1972) and Bradshaw (1977) reviewed

compressibility effects in shear layers, whereas Bradshaw (1996) provided a more

recent review of physical and modeling aspects of turbulence in compressible

flows. The reader is also referred to Morkovins (1962, 1992) reviews of com-pressibility effects on boundary layers and shear layers. Interactions of shock

waves with boundary layers in three-dimensional geometries have been reviewed

by Settles & Dolling (1992) and Dolling (1993). The textbook by Smits & Dus-

sauge (1996) also provides an excellent account of compressibility effects in

turbulent shear layers, with chapter 10 devoted to boundary layershock wave

interactions.

Kovazsnay (1953) suggested the decomposition of turbulent fluctuations into

the vorticity, acoustic, and entropy modes. He performed a small perturbation

analysis, valid for weak fluctuations of density, pressure, and entropy, to show

how acoustic waves may accompany vorticity fluctuations and how entropy is

changed in regions of different shear. In Kovazsnays 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) turbulent

jet, (d) free shear layer, (e) turbulent wake, and (f) two-dimensional turbulent boundary

layer.

modes of fluctuations are dynamically decoupled from each other to a first-order

approximation. However, interaction of the modes can take on a second-order

approximation of fluctuations, in which mode couplings arise as one mode can

be generated from the interaction of the other two modes (Chu & Kovazsnay

1958). An interaction of the modes can take place when a shock wave is present

in 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 the

vorticity mode dominates the incoming flow.

The fundamental concept behind interactions of shock waves with turbulence

is the transfer of vorticity through shock waves. Longitudinal vorticity, for

instance, is expected to be transferred unchanged through a normal planar shock,

whereas the other components of vorticity parallel to the shock surface should

change substantially.

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

1. How much of the amplification of turbulence in interactions with shock waves

is 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 interactions

with shock waves is also considered: Linear interaction analysis (LIA), rapid

distortion 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 steep

pressure gradient. Information from experiments and simulation of low-speed

flows with such pressure gradients indicate that rapid-distortion concepts hold

and, in the limit of extremely sharp gradients, the Reynolds stresses and turbulent

intensities 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 field

through a shock wave is attributed to Ribner (1953), who decomposed theincident

disturbance shear wave into acoustic, entropy, and vorticity waves. In his LIA,

Ribner formulated the interaction of a plane vorticity wave with a shock wave as

a boundary-value problem. Figure 2 shows a schematic of the incident plane

sinusoidal shear wave crossing the shock. The wave number vector is refracted

across the shock owing to the changes in thermodynamic properties and therefore

emerges at a different angle from the incident. The condition that the phases

should remain unchanged across the shock yields that the frequency, as well as

the component of the wave number vector parallel to the shock front, are the

same upstream and downstream of the shock. Ribners 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. His

predictions 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 Lee

et al (1991, 1993, 1994) show very similar turbulence enhancement.

Goldstein (1978) and Goldstein & Durbin (1980) formulated a decomposition

of the disturbances into a vortical and an acoustic or potential part. This

formulation was used by Durbin & Zeman (1992) to study numerically the

response of homogeneous turbulence to bulk compression. Helmholtzs 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 successful

DNS of a homogeneous and isotropic turbulence interacting with a plane shock

wave.

The time-dependent, three-dimensional equations governing the interactions

that describe the conservation of mass, momentum, and total energy are

Dq Uiq 0, (1)

Dt xi

DU p si ijq , (2)

Dt x xi j

(qE ) (qE U) (pU) (s U) qt t i i ij i i. (3)

t x x x xi i j i

Et is the total energy per unit mass, defined as Et e12 UiUi, qi is the rate of

heat 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 is

related to the bulk viscosity lb through k lb 2/3 l.

2.1 Dissipation Rate of Kinetic Energy

The transport equation for the instantaneous kinetic energy 12 Ui Ui in compress-

ible flows is

1D U Ui i 2 p sij

q U U . (4)i iDt x xi j

Equation 4 can be manipulated to yield

1D U Ui i 2 ( pU s U)j ij i

q pS s S , (5)kk ij ijDt xj

where the last term on the right-hand side contains the dissipation rate of kinetic

energy E converted into thermal/internal energy. Sij is the strain tensor, and the

term pSkk represents the work done by pressure forces during compression or

expansion of the flow. Both terms, the dissipation rate E sij Sij and the pressure

work term, also appear with opposite sign in the transport equation for internal

energy. Since the dissipation rate is always positive or zero at any given point in

space and time, the pressure-dilatation term can, in principle, be positive or

negative.

The dissipation rate is given by

U U Ui k iE s s S 2lS S k d . (6)ij ij ij ij ij ij

x x xj k j

After invoking Stokess hypothesis, which suggests that the bulk viscosity is

negligible, lb 0, Equation 6 becomes

U U2 k mE 2lS S l . (7)i j i j 3

x xk m

The second term in the right-hand side of Equation 7 represents the additional

contribution of compressibility to the dissipation rate of kinetic energy. This term

vanishes for incompressible flows.

Because Uk/ xk Um/ xm ( Uk/ xk)2, this term is always positive, and the

negative sign in front of this term in Equation 7 may erroneously suggest that

compressibility reduces dissipation. This is incorrect because the term SijSij also

contains contributions from dilatation effects, which can be revealed if one con-

siders that


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1 U Ui jS S X X ,ij ij k k

2 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 rate

then becomes

U U4 i i 2 2E l X X lS 2l S . (8)k k kk kk 3 x xj i

The second term on the right-hand side of Equation 8 describes the direct effects

of compressibility, that is, dilatation on the dissipation rate. It is obviously zero

for incompressible flows.

The first two terms on the right-hand side of Equation 8 are quadratic with

positive coefficients and positive signs, and they are, therefore, always positive

or zero. The last term on the right-hand side indicates the contributions to the

dissipation rate by the purely nonhomogeneous part of the flow. Its time-averaged

contribution disappears in homogeneous flows. This term, in principle, can result

in negative values, and thus it can reduce the dissipation rate. This does not violate

the 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 term

appears as a source term in the transport equation for entropy s, which, for ideal

gases, is

Ds qiqT E. (9)

Dt xi

It has been customary in the past [see, for instance, Zeman (1990)] to decompose

Einto a solenoidal part Es, which is the traditional incompressible dissipation and

the dilatational part Ed. In this case

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

3. VORTICITY TRANSFER THROUGH SHOCK WAVES

A better understanding of the nature of the interaction of turbulent structures and

vortex motions of turbulent flows with shock waves requires information on

important quantities such as vorticity, rate-of-strain tensor and its matrix invari-

ants, and dissipation of turbulent kinetic energy. These flow quantities, although

computationally as well as experimentally very demanding to obtain, if resolved

to proper scales are well suited for describing physical phenomena in vortical

flows. 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 Snells law. Hayes (1957) derived an equa-

tion for the vorticity jump across a surface of discontinuity in unsteady flows by

considering the thermodynamic relations upstream and downstream of it. The

vorticity changes are given as follows:

rdX 0, (10)n

r r1 (D v v D n )d( q)r t t s t rdX n (qv )d . (11)t t r q ( qv )nHere the subscripts n and t represent components in the directions normal and

tangential to the surface of discontinuity, the vector being the unit vector normalrn

to the surface of discontinuity. The vector represents the relative normal veloc-rv

ity component with respect to the surface of discontinuity that is r rv vr n. The operator Dt is the tangential part of the total derivative operator or the

rv smaterial derivative for an observer moving with and along the shock surface with

velocity , where andr r r r r r rn v v D v [d v /dt] v v d x /dts t t t t t t t t t vs and , where K is the curvature ten-

r r r rv / t v / n D n v v Kt t t t s t

sor .r

K 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 shock

surface remain unchanged through it. Equation 11 shows that the vorticity change

across the shock is determined by the density jump and by the gradients, tangential

to the surface of discontinuity, of the incoming-flow-velocity.

Although Hayess theory is very significant, in many cases it is not useful

because it provides information in reference to a local system of coordinates

attached to the shock front, which may change position and shape with time

during the interaction. The position and shape of the shock wave must emerge as

part of the solution of the mathematical system of equations and is not known a

priori. Nevertheless, this theory has shown that vorticity parallel to the shock

front 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 gjS X X S e e . (12)ik k i kk iqg iqg 2Dt q x x x q xq g q j

It describes four dynamically significant processes for the vorticity component

Xi, namely that of stretching or compression and tilting by the strain Sik, vorticity

generation through dilatation, baroclinic generation through the interaction of

pressure and density gradients, and viscous effects expressed by the viscous stress

term. 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 its

magnitude, very often, is small, then the change of vorticity of a fluid element in

a Lagrangian frame of reference can be entirely attributed to vortex stretching

and/or tilting, to dilatational effects, and to baroclinic torque.

If Equation 12 is multiplied by Xi, then the transport equation for the enstrophy12 Xi Xi can be obtained:

D(1/2X X ) 1 q pi i S X X 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 e . (13)iqg i iqg x q x q x xq j q jThe physical mechanisms associated with the time-dependent changes of enstro-

phy of a fluid element in a Lagrangian frame are similar to those responsible for

the generation or destruction of vorticity. Enstrophy is a very significant quantity

in fluid dynamics because it is related not only to the solenoidal dissipation as

was mentioned in Section 2, but also to the invariants of the strain-of-rate matrix

Sij. In addition, enstrophy is a source term in the transport equation of dilatation

Skk:

2D(S ) 1 1 q p 1 p 1 skk kqS S X X . (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, which

can be caused by the straining action of the dissipative motions (SikSik), as well

as by the rotational energy of the spinning motions as it is expressed by the

enstrophy 12XkXk. Pressure and density gradients, as well as viscous diffusion,

can also affect dilatation. It should be noted that transport Equation 14 reduces

to the well-known Poisson equation

21 p 1S S X X , (15)ik ki k k

q x x 2k k

for incompressible flows of constant density (Skk 0).

3.1 A Demonstration

The transport equations of vorticity and enstrophy can be used to gain further

insight 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, several

experiments were carried out at City College of New York by Agui (1998), in

which a subminiature multiwire probe was used to measure vorticity and other

velocity gradients in an interaction of a grid-generated turbulence with a traveling

normal 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 of1020 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 Skk 1/qDq/Dt, dilatational dissipation Ed SkkSkk,

and solenoidal dissipation/enstrophy Es XiXi. Each signal has been normalized

by its rms value upstream of the interaction so that signals with relatively large

fluctuations are reduced and signals with relatively small fluctuations are

enhanced. 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, which

is defined in terms of a representative variable Q as GQ Qrms,d/Qrms,u, where

the 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 by

multiples of 10 rms units to provide better visual aid. The shock wave location

is 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 the

actual computation of their rms values, indicates that some signals are amplified

and some are not. The longitudinal vorticity X1 and lateral velocity U3 signals

are only slightly affected by the interaction. The computed data also show a 25% reduction in the rms values, which practically indicates that within the exper-

imental uncertainty, no changes occur in the transmission of X1 and U3 through

the shock, as is expected from the Rankine-Hugoniot relations. Longitudinal

velocity fluctuations and lateral vorticity fluctuations X2 are substantially ampli-

fied through the interaction, with gains of1.4. Fluctuations of dilatational dis-

sipation SkkSkk and solenoidal dissipation also show gains of the order of1.2.

These two dissipation signals exhibit strong intermittent behaviors that are char-

acterized by bursts of high-amplitude events. These, sometimes, reach values of

up 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, only

10% of the fluctuations of vorticity. These values are typical in flows with low

fluctuations of Mach number. Consequently, baroclinic vorticity generation is

negligible because pressure fluctuations are also small. Thus, the only source ofvorticity change, in addition to viscous diffusion, is through the stretching or

compression and tilting term SikXk. A closer look at the components of this vector

indicates that the level of fluctuations of the terms S11X1, S22X2, and S33X3, which

are the terms indicating vortex stretching or compression, decreases by 30% on

average. Clearly, compression by the shock wave reduces the level of vorticity

fluctuations associated with the so-called mechanism of vortex line compression

on all vorticity components. However, tilting of vorticity components by the

action of the shear Sik increases through the interaction by various amounts on

each of the three vorticity components so as to compensate for the deficit caused

by the vortex line compression and to further increase in the case of the two

components 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 1.2.

319


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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 HOMOGENEOUS

AND ISOTROPIC TURBULENCE

The interaction of an isotropic and homogeneous turbulent flow with a strong

axisymmetric disturbance, such as a normal shock, is the best paradigm case of

a flow where the geometry is reasonably simplified and basic physics of such

interactions can be investigated. Traditionally, this flow has been used as a test

case 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 models strengths and

weaknesses.

Experimental realization of a homogeneous and isotropic flow interacting with

a 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 flow

is the second. These two problems may not be independent from each other. As

a result of these difficulties, two different categories of experiments have been

performed. The experimental arrangement in shock tubes offers the possibility of

unsteady shock interactions with isotropic turbulence of various length scales and

intensity. This category includes the experiments of Hesselink & Sturtevant

(1988), Keller & Merzkirch (1990), and Honkan & Andreopoulos (1992). In the

last experiments the induced flow behind the incident shock wave passes through

a turbulence-generating grid and then interacts with the shock reflected off the

end wall. A unique facility has been developed at City College of New York in

which the flow Mach number can, to a certain extent, be controlled independently

from the shock wave strength. This has been achieved by replacing the end wall

of 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 with

incompressible flows can be generated.

Configuring a homogeneous and isotropic turbulence interacting with a normal

shock in a supersonic wind tunnel appears to be more difficult than in a shock

tube. A turbulence-generating grid or other device is usually placed in the flow

upstream of the converging-diverging nozzle. The flow then interacts with a sta-

tionary shock produced by a suitable shock generator located farther downstream

in 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 the

shock waves are restricted to low Reynolds numbers, they have been very useful

in providing physical insights of the mechanisms involved. These computations

can provide results that sometimes are very difficult to obtain experimentally, or

they can be used as diagnostic tools. Lee et al (1992), for instance, computed

results containing pressure fluctuations inside the flow or entropy fluctuations

through shocks, which currently seem impossible to obtain experimentally

because of lack of appropriate experimental techniques for measuring these

quantities.

There are three different experiments of homogeneous isotropic turbulence

interactions with shocks all carried out in supersonic wind tunnel arrangements

in France. Debieve & Lacharme (1985) carried out an oblique shock interaction

with turbulence generated inside the plenum chamber and developed in a flow

with a Mach number of 2.3. A threefold amplification of turbulence was measured

close to the shock, which was attributed to shock oscillations and distortions

caused 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 to

generate turbulence was also acting as a sonic throat to accelerate the flow to

Mach 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 data

provided by O Leuchter (private communication, 1997) indicated a considerable

deceleration of the incoming flow before the shock. The longitudinal mean veloc-

ity from 469 m/s at x 0.03 m becomes 380 m/s at x 0.32 m. This deceleration

is 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, the

turbulence level of the incoming flow may be higher than it ought to be because

it is known that supersonic flow deceleration through gradual compression is

associated 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 exchange

between 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 flows

overestimates turbulence intensities. The probe volume of the laser Doppler ane-

mometry was 10 mm 5 mm, and it is considered rather large to resolve accu-

rately the turbulence of the flow field. In terms of Kolmogorov length scales, the

probe volume appears to be 500 250. Thus, the existence of Mach waves

inside the flow has contaminated the flow, and instrumentation problems may

have 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 of

Alem (1995) published by Barre et al (1996). A normal shock was formed by the

interaction 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 the

boundaries of the useful flow region. The velocity is 200 m/s at x 10 mm after

the shock and increases to 250 m/s at x 20 mm, which results in an acceleration

of u/ x 5400 s 1.

This level of acceleration is very strong and is expected to reduce turbulence

intensities after the interaction. Thus, the amplification levels found in this work

are probably contaminated by the additional effects of acceleration in the subsonic

flow downstream of the interaction. Another weakness of this data set is that the

level 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 shock

front is not substantially distorted and the Rankine-Hugoniot conditions can be

linearized. In interactions in which strong distortions of the shock front areevident

because of higher turbulence intensity, the use of LIA is fully justified. DNS data

in 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 computational

result. The experiments of Jacquin et al (1993) report no significant enhancement

of turbulent kinetic energy. However, as was mentioned earlier, the deceleration

in the flow upstream of the interaction because of the existence of Mach waves

may have contributed toward overestimating the level of turbulence upstream of

the interaction.

In the recent experimental work of Zwart et al (1997), a clear increase in

turbulent intensities is reported. These authors, in their study of the decay behav-

ior of compressible gridgenerated turbulence, encountered shock waves in the

range of flow Mach numbers of 0.7 M 1.0. They attributed the increased

turbulent intensities in this regime to shock wave unsteadiness.

Typically, the amplification of turbulent fluctuations depends on the shock

wave 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 velocity

and vorticity fluctuations, respectively, as a function of the shock strength Ms.

The experimental data have been obtained by Agui (1998). Two different grids

were used in these experiments at two different Mach numbers each. The data

shown in Figure 4 clearly suggest an amplification of longitudinal velocity fluc-

tuations, whereas the fluctuations of the other two components are slightly

decreased or remain unchanged through the interactions. The amplification data

of vorticity, shown in Figure 5, indicate a substantial enhancement of vorticity

fluctuations in the direction parallel to the shock front with no amplification or

small attenuation of longitudinal vorticity fluctuations. The amplification data of

the lateral component of vorticity fluctuations shown in Figure 5 represent the


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Figure 4 Amplification of velocity fluctuations in grid turbulenceshock interactions.

Experimental data of Agui (1998), LIA data of Lee et al (1993), RDT data of Jacquin et

al (1993), and DNS data of Lee et al (1993).

average of the two components parallel to the shock front. The results of the LIA

of Lee et al (1993) as well as their DNS data are also plotted in Figure 5. LIA

shows that lateral vorticity fluctuations increase with the shock strength after the

interaction with the shock, and the experimental data tend to confirm this.

The DNS data of Hannappel & Friedrich (1995) show that the amplification

of the vorticity fluctuations in the lateral direction increases with compressibility

in the upstream flow, whereas the amplification of longitudinal velocity fluctua-tions is reduced by the same effect. They also found that longitudinal vorticity

fluctuations are slightly reduced through the shock, as it has been found in the

experiments 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 C , G ,2 2u x 3

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


16/39


Figure 5 Amplification of vorticity fluctuations in grid turbulenceshock interactions.

Experimental data of Agui (1998), LIA data of Lee et al (1993), RDT data of Jacquin et

al (1993), and DNS data of Lee et al (1993).

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

between 1.25 and 1.7.

The early work of Honkan & Andreopoulos (1992) showed that amplification

ofurms, Gu, also depends on the turbulence intensity and length scale of the incom-

ing isotropic turbulence. The work of Briassulis and Andreopoulos (1996a,b) and

Agui (1998) also confirm this finding. The amplification of turbulent kinetic

energy across shocks seems to decrease with increasing Mt. Both experimentaldata from the above-mentioned studies and the DNS data of Lee et al (1993) and

Hannappel & Friedrich (1995) agree with this finding. Thus, the outcome of the

shock turbulence interaction depends also on the compressibility level of the

incoming flow.

The effect of the shock strength on the velocity fluctuations is shown in Figure

6, where the data of Briassulis (1996) are plotted. For the 1 1 (cells/in2) grid,

it appears that the amplification of turbulence fluctuations defined as Gu 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 shock

interaction are felt differently. For the 2 2 grid, for instance, the data show


17/39


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 weak

shock, amplification of turbulence occurs at x/M 35. The amplification is

greater when Mflow increases to 0.436. However, when compressibility effects in

the upstream flow start to become important, no amplification takes place (Gu is1).

Some more dramatic effects of compressibility are illustrated in Figure 7, in

which the amplification Gu is plotted for a finer grid with mesh size 5 5. The

interaction of a weak shock with a practically incompressible turbulent flow pro-

duces the highest amplification of velocity fluctuations, with Gu reaching a value

close to 2. As the Mflow increases, Gu decreases, and at Mflow 0.576 a slightattenuation occurs at downstream distances. It is, therefore, plausible to conclude

that for high shock strength (high Mach number), compressibility effects control

the velocity fluctuations, which are generated by fine grids, and no amplification

of 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 incompressible

isotropic turbulent field, predicts amplification of turbulence fluctuations.

Recently, Mahesh et al (1997) have shown that LIA as well as DNS may show a

complete suppression of amplification of kinetic energy if the upstream correla-

tion between velocity and temperature fluctuations is positive. It is therefore pos-


18/39


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 dissipation

rate of turbulent kinetic energy is high, entropy or pressure fluctuations may be

responsible 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. These

results indicate that the outcome of the interaction depends strongly on the

upstream turbulence of the flow.

4.2 Turbulence Length Scales Through the Interaction

As mentioned by Lele (1994), there is some disagreement between experimental

results and DNS data as far as how the various length scales of turbulence are

affected 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-length

scales and microscales of density fluctuations, decrease through shock interac-

tions. The experimental data of Keller & Merzkirch (1990) show that the density

microscale increases across the shock. Hannappel & Friedrich (1995) also show

that 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 results

of Lee et al (1994) have indicated a small increase of dissipative-length scales

through weak shock interactions. Thus, there is no agreement among various

researchers on how shock interactions affect the length scales. Intuitively, one

should 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 six

parallel wires and three temperature wires separated by 1 mm. To estimate the

length scales in the longitudinal n1 direction and normal n2, the cross correlation

coefficients rij(nk) / 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 direction

after invoking Taylors hypothesis.

The data show that the integral length scale L11(n1) increases with downstream

nondimensional distance x/M for all investigated cases in the flow before the

interaction. It is also shown that L11 for Mflow 0.475 is higher than for

Mflow 0.36. However, if the flow Mach number increases to Mflow 0.6 and,

therefore, stronger compressibility effects are present, then the integral-length

scale drops.After the interaction with the shock wave, the distribution of L11(n1) is more

complicated. All the scales are reduced considerably. However, the reduction of

the 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 L (n )/L (n )L 11,d 1 11,u 111the 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. It

is 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 defined

length 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 interaction

are reduced with increasing flow Mach number. This behavior is very similar to

that of L11(n1). After the interaction, however, the length scale L11(n2) increases

in the first two cases and decreases in the strongest interaction.

To investigate the effect of initial conditions on this correlation at the highest

flow and shock Mach number, where the lateral scales are shown to reduce (Figure

9), various grids were used. The data shown in Figure 10 indicate that the cor-

relation increases substantially in the finest grid, 8 8 with the lowest Rek, after

the interaction for the range of length scales investigated. However, the coarser

2 2 grid with the highest Rek 737 shows the greatest attenuation in the

lateral 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).


21/39


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 after

the interaction in all investigated flow cases. The corresponding length scales in

the normal direction increased at low Mach numbers and decreased during

stronger interactions. It appears that, in the weakest of the present interactions,

the eddies are compressed in the longitudinal direction drastically, whereas their

extent in the normal direction remains relatively the same. As the shock strength

increases, the lateral length scale increases, whereas the longitudinal-length scale

decreases. At the strongest interaction of the present cases, the eddies are com-pressed in both directions. However, even at the highest Mach number case, the

issue is more complicated because amplification of the lateral scales has been

observed in fine grids. Thus, the outcome of the interaction strongly depends on

the initial conditions.

The dissipative length scale was found to increase in several cases, and the

disagreement with the DNS remains an issue, although it may be attributed to the

difference 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 experiments

of Briassulis (1996) and may be the cause of this disagreement between experi-

ments and DNS. The difference between experiments and DNS is also noticeable

if the Reynolds number


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2(qu u )i iReT

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 timescaleimposed

by 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 of

the changes, either attenuation or amplification of quantities involved, occur forlarge x/Mdistances in which the length scales of the incoming flows are high and

turbulence intensities are low. Thus, large eddies with low-velocity fluctuations

are 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 energy

depends on anisotropy of the Reynolds stress tensor of the incoming flow and the

ratio of shearing rate to that of compression. The dependence of turbulence ampli-

fication on the initially anisotropy is reduced with the obliqueness of the shock

wave. A change in the sign of the initial shear stress has been observed for

sufficiently strong interactions. This type of idealized interaction has yet to be

configured experimentally.

6. TURBULENT JET FLOWS AND SHOCK WAVEINTERACTIONS

Interactions of jets with shock waves are considerably more complex than the

previously considered homogeneous flow interactions because inhomogeneity

and anisotropy are additional parameters to consider, which may change the flow

physics completely. The variable shear, for instance, in the shear layer emanatingfrom the jet is an additional parameter that is responsible for the production of

large amounts of turbulent kinetic energy before the interaction. One obvious

question is how this shear is affected by the interaction with the shock wave.

Jacquin & Geffroy (1997) have shown that this shear is decreased through the

interaction.

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 in

addition to the shock strength, compressibility level, and velocity and length scale

of the incoming turbulence. A schematic of the flow interaction and the param-

eters involved is shown in Figure 11. Baroclinic generation of vorticity may take

place due to the nonalignment of the density and pressure gradients. Density


23/39


Figure 11 Schematic of the turbulent jetshock 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 pressure

gradient across the shock interacts with the density gradient across the interface

between the jet and the ambient fluid to generate additional vorticity.

Two types of configurations have been used in the past to study experimentally

shock-jet interactions: steady interaction in supersonic wind tunnels and unsteady

interaction 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 and

the sonic coflowing stream were further expanded to M 1.6. Density ratios

from 0.33 to 1 were achieved through heating of the jet flows. The velocity ratio

ranged from 0.58 to 1 and the convective Mach number was 0.42. Turbulence

amplification 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 baroclinic

vorticity 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 or

reduction of turbulent kinetic energy, were identified in experiments. Their flow

visualization indicated that the shock front is not distorted by the shock. Thus,

baroclinic vorticity generation or destruction can take place in the absence of

shock front deformations. Substantial shock front deformations, Mach reflections,

and additional shocks can be developed when an overexpanded jet interacts with

a normal shock (see Jacquin et al 1991). Large amplification of turbulence has

been found in this case.


24/39


Figure 12 Helium jetshock 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 with

traveling shock waves. Density ratios of 0.0595 (helium jet) through 1.31 (CO2jet) were achieved in these shock tube experiments. Figure 12 shows an image

of the flow containing a vortical structure produced behind the traveling shock

wave for a helium jet interaction obtained at x/d 50 (from JC Hermanson &

BM Cetegen, private communication, 1999). Planar Mie scattering of planar laser

light was used to visualize the flow, which was seeded with mineral oil. The

structure contains vorticity with a sign corresponding to that of jet-like vortices

formed by jets with velocities considerably higher than the coflowing fluid veloc-

ity behind the traveling shock wave. The formation of this structure is attributed

by Hermanson & Cetegen to baroclinic generation of vorticity. There is some

apparent inconsistency between the experiments of Jacquin & Geffroy (1997),

who found that baroclinic effects reduce vorticity after the interaction with the


25/39


shock, and the experiments of Hermanson & Cetegen (1998), who found that the

same effects enhance vorticity after the interaction. This disagreement may be

resolved if the baroclinic term of the vorticity transport Equation 12

1 q p 1B e pxX iqg i 2q x x qq g

is considered. reaches maxima or minima when the two gradients p andBXi(1/q) are at 90 or 270 to each other. If only vorticity X3 is considered, then

1 q p q p 1 q pB .X 3 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 little

in 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 in

the lower half of the flow. This shows that BX is negative in the upper half, where

the initial vorticity X3 is positive, and positive in the lower half of the flow, where

X3 is negative. Therefore, for stationary shocks such as in the experiments of

Jacquin & Geffroy, baroclinic effects tend to suppress vorticity; that is, the bar-oclinic torque tends to bring the vorticity toward zero values. In the experiments

by Hermanson & Cetegen, the shock wave travels in the same direction as the jet

flow, and it introduces a negative pressure gradient p/ x1 into the flow (for a

stationary observer traveling with the shock), and, because the density gradient

is the same as in the previous case, all contributions from BX tend to enhance

vorticity 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 vorticity

of 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 jet

fluid 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 the

same direction as the air jet flow obtained by Hermening. The images were

obtained from different experiments with the same bulk flow parameters and with

the shock wave located at different positions from the jet exit. The jet flow had

a Mach number Mj 0.9, and the shock wave was traveling with a Mach number

of Ms 1.33. Talcum powder with particles 1 lm in diameter was used to

seed 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-


26/39

Figure 13 (ac) Images of the interaction of a MS 1.34 shock wave with a MJ 0.9 coflowing jet

Hermening (1999). (d) Schematic of the interaction.

334


27/39


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 passage

of the shock, which is located outside the viewing area. There are three regions

of interest in the flow during the interaction as shown in Figure 13b. The first

region is characterized by a stenosis of the jet cross section observed at a distance

between 2 and 3 diameters downstream of the exit. This region is followed by a

region closer to the exit, with a jet stream of apparently larger diameter than the

previous one, and it is proceeded by a third region immediately behind the shock

wave, where spreading of the flow can be observed. This structure of the jet can

be explained if one considers the time-dependent motion of the shock from the

time it is located at the jet exit until the time it has left the near-field region. The

shock wave is characterized by an induced flow with higher pressure and higher

density behind it than ahead of it. This changes the velocity, pressure, and density

of the ambient fluid, which is now put into motion. Thus, the imposition of higher

ambient pressure compresses the jet fluid to a higher density, which reduces its

cross 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 pressure

ratio 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 is

maintained 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 spreading

of 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 in

Figure 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 the

shock, this region is characterized by a decaying mean velocity, high shear, and

increased spreading of the jet, which results in the production of substantial levels

of turbulence. It appears that, after the interaction, considerable mixing has taken

place, and the turbulence most probably has increased. Figure 13ddepicts a sche-

matic illustration of the interaction of the jet flow with the shock wave, which

includes 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 from

right to left, whereas the shock travels in the opposite direction. The jet and shock

Mach numbers are the same as in the previous configuration. The most striking

feature of this interaction is the substantial level of spreading of the jet flow

observed 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 indicated

some similar behavior. The initial spreading rate of the helium jet before the

interaction was higher than that of the air jet. As a result, substantially more

mixing 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


28/39


Figure 14 Airjet 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 0.9, MS1.33. Images are from Hermening (1999).

air jet. The shock interaction in the coflowing configuration increased the mixing

compared with the initial jet. However, substantially more mixing was observed

in the counterflow configuration, which supports the notion explained above that

baroclinic effects, in this case of a heavier-than-ambient jet, enhance the vorticity

of the jet. In this case the interaction of the density and pressure gradients brings

about additional vorticity into the flow, which enhances the mixing.

The mixing enhancement shown in the experiments of a helium jet interacting

with a traveling shock wave has been also verified by the calculations of Obata

& Hermanson (1998), who used a Reynolds-averaged Navier-Stokesbased

model to calculate the turbulent jet before the interaction and a TVD scheme to

simulate the passage of the shock.

7. PLANE SHEAR LAYER INTERACTIONS WITH SHOCKWAVES

Interactions of plane shear layers with shock waves have been configured in

supersonic 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 distance

downstream, is used to generate an oblique shock. This configuration has been

used in the experiments by Settles et al (1982), Hayakawa et al (1984), and

Samimy et al (1986). The separated shear layer develops over the cavity as a

compressible shear flow under the influence of a recirculating zone beneath it and

subsequently undergoes compression by the shock system and then reattaches to

the 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


29/39


wave is also time dependent, which further complicates our understanding. All

measurements 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 oscillation

and the unsteady phenomena associated with the reattachment region.

Standard Reynolds-averaged Navier-Stokes methods also have been used to

calculate the flow (see Horstman et al 1982, Visbal & Knight 1984) with moderate

success in predicting the mean flow behavior. Turbulence is considerably over-

estimated in these calculations.

8. TURBULENT WAKESHOCK INTERACTIONS

This type of interaction is very often found inside compressors or turbines of

turbomachinery, where a shock may be formed and very often may travel down-

stream. The shock can subsequently interact with wakes of individual blades of

stators or rotors and alter their structure. As a result of this interaction, blade

wakes carrying more turbulence impinge on the surface of blades of downstream

stages and can cause additional fatigue of the blades. No previous experimental

or analytical work could be identified on these types of interactions.

9. BOUNDARY LAYERSHOCK WAVE INTERACTIONS

The majority of the previous work in turbulenceshock interactions belongs in

this category, and it mainly includes interactions of boundary layers with oblique

shock waves. Usually, an oblique shock wave is generated either at the opposite

wall 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 compression

turning, which can be a sharp turning as for a compression corner or a gradual

turning by a smooth change in curvature. In the first configuration the incident

shock emanating from the opposite side of the wall is also reflected over the

boundary-layer wall, and therefore the flow interacts with two oblique shocks. Inthe second configuration the flow after the interaction is further compressed by

the new boundary imposed by the ramp slope, whereas concave curvature of the

streamlines has a destabilizing effect on turbulence. Thus, turbulence can not only

be affected by the shock interaction but also by the spatiotemporal oscillation of

the shock system, the destabilizing effects of the concave streamline curvature,

and the continuing downstream compression. All these effects contribute to an

increasing level of turbulence. In addition, unsteady flow separation and flow

reversal take place at the corner region, which at large turning angles can affect

the shock system upstream through a feedback mechanism.

One of the major characteristics of the turbulent boundary-layer interaction

with an oblique shock over a compression ramp is the unsteady motion of the


30/39


Figure 15 A hypersonic compression ramp interaction is visualized using planar laser

scattering imaging of a condensed alcohol fog (NT Clemens & DS Dolling, private com-

munication, 1999). The ramp angle is 28 and the field of view is 75 mm in the streamwise

direction 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. These

two images are of a double-pulse sequence with a time delay between pulses of 30 ls. In

these images the free stream is rendered gray, boundary layer fluid is darker gray, very

low velocity fluid is black(owing to evaporation of the particles), and the separation shock

is the thin white interface, as labeled. Note that these images are somewhat difficult to

interpret owing to the potential for evaporation and condensation of the alcohol seed. For

reference, arrows point to the same structure in both images.

shock system, which has been observed to occur over a significant distance in

the 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 are

deformed by the interaction and therefore no longer remain planar. Figure 15

shows a sequence of two images obtained by a double-pulse planar laserscatteringtechnique at the University of Texas at Austin by NT Clemens & DS Dolling


31/39


(private communication, 1999). The images are from an M 5 interaction over

a 28C compression ramp. They very clearly show a substantial deformation of

the shock front, which changes significantly from the first to the second image

as a result of the mutual interaction with the incoming turbulence. A turbulent

structure going through the system is also visible. The outer part of the eddy

seems to go through the shock without substantial changes. The shock itself,

however, seems to have moved upward, although its extent toward the wall

remains strong.

The first evidence of the grossly unsteady nature of the shock wave system

came from measurements of the time-dependent pressure at the wall beneath the

region of interaction. Kistler (1959) provided one of the first measurements of

pressure fluctuations in a supersonic boundary layer approaching a forward-facing

step. Coe (1969), Dolling & Murphy (1982), Dolling & Or (1983), and Muck et

al (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 the

time-averaged system. Two possible mechanisms have been proposed to explain

the shock system unsteadiness: the first one is the turbulence of the incoming

boundary 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 subsonic

region of the separated zone.

The experimental work of Andreopoulos & Muck (1987) suggested that the

frequency of the shock wave system unsteadiness may scale on the bursting fre-

quency of the incoming boundary layer. This finding supports the notion that the

incoming turbulence plays a dominant role in triggering the shock unsteadiness.

Since this work, several other experiments have provided support for one or the

other of the two proposed mechanisms. Erengil & Dolling (1991) found that some

of the low-frequency oscillations of the shock are caused by the contraction and

expansion cycle of the separated zone. Unalmis & Dolling (1994) also showed

that the large-scale structures of the incoming boundary layer also contribute to

low-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 individual

turbulent fluctuations, whereas the large-scale motion is caused by the large-scale

structures of the boundary layer. Thus, this evidence supports the notion that the

small and large scales of the incoming boundary layer play a dominant role in

the 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 interaction

points to the supposition that Taylor-Gortler vortices formed by the concave

streamline curvature may exist and contribute to increased complexity of the

phenomena involved (see Ardonceau 1984).


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10. FINAL REMARKS

The information that is available now on the interaction of turbulence with shock

waves appears to be limited. The phenomenology associated with the interactions

is not very extensive, and the understanding of their physical attributes is not

very thorough. Experiments or simulations for instance, of shock interactions with

wake flows, do not exist, and experimental data in shock interactions with homog-

enous sheared turbulence are not available for comparison with existing DNS

data. However, the understanding of the physics has been advanced in the last

years owing to the synergy among theory, numerical simulations, and experi-

ments. Despite this progress, a lot of questions remain unanswered, and more

work is needed to illuminate the physics of the interactions. Ribners (1953)

theory, for instance, which predicts turbulent amplification through shock waves,

has, in principle, been verified by DNS and several experiments. There is new

evidence, however, that suggests that under some conditions of the upstream

compressible turbulence the interaction may not lead to increased fluctuations of

velocity. Thus, the final word is not yet out. In addition, several of the questions

posed in the introduction remain unanswered.

The major impediment to understanding the physical aspects of the interactions

is the difficulty in simulating the interactions numerically or experimentally. Forinstance, it is very challenging to generate normal shock waves or homogenous

flows inside wind tunnels. Similar difficulties are attributed to DNS applied to

flows with homogeneous or periodic boundary conditions and covering a rela-

tively low range of Reynolds numbers. In addition, the lack of availability of

proper 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 still

needed. This review of available information revealed that the interaction between

a shock wave and turbulence is mutual and very complex: Turbulence is affected

by 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 seem

to 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 Force

Office of Scientific Research in carrying out some of the experiments described

in this review and in writing up this article is greatly appreciated. The authors

also thank J Hermanson and B Cetegen for providing the images of Figure 12


33/39


and N Clemens and D Dolling for providing the images of Figure 15. Useful

comments provided by D Knight are also appreciated.

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

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