shock unsteadiness in a reattaching shear layerjpoggie/papers/aiaa-2000-0140.pdf · layer,...

AIAA 2000–0140Shock Unsteadinessin a Reattaching Shear LayerJ. PoggieUS Air Force Research Laboratory, Wright-PattersonAFB, OH 45433-7521A. J. SmitsPrinceton University, Princeton, NJ 08544-5263

38th AIAA Aerospace SciencesMeeting and Exhibit

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Shock Unsteadinessin a Reattaching Shear Layer

J. Poggie∗

US Air Force Research Laboratory, Wright-Patterson AFB, OH 45433-7521

A. J. Smits†

Princeton University, Princeton, NJ 08544-5263

The origin of shock unsteadiness in a Mach 2.9 turbulent, reattaching shear layerwas investigated experimentally using temporally-resolved flow visualization and mea-surements of wall pressure fluctuations. In order to isolate the influence of disturbancesoriginating in the incoming shear layer, experiments were conducted in which artificialdisturbances were introduced into the flow through air injection in the vicinity of sepa-ration. The effect on the reattachment shock system was dramatic: the intensity of thepressure fluctuations and shock motion increased substantially and power spectra of thepressure fluctuations showed a distinct shift to lower frequency. The spectra collapsedonto a common curve in nondimensional coordinates based on a length scale derivedfrom two-point cross-correlations of the flow visualization data and a convection velocityderived from cross-correlations of the pressure measurements. This curve showed fairlygood agreement with a theory developed by Plotkin (AIAA J., Vol. 13, No. 8, 1975,pp. 1036-1040), which is based on perturbation of a shock by random fluctuations in theincoming turbulent flow. These results indicate that, unlike separated compression rampflows where shock motion is associated primarily with relatively low-frequency expansionand contraction of the separation bubble, the shock motion in the reattaching shear layeris primarily caused by organized structures in the incoming turbulent flow.

Nomenclaturef = frequencyG = power spectrumI = scattering intensityp = pressureR = correlationt,τ ,ξ = timeu = fluctuating velocity componentU = mean velocity componentx,X = coordinate along rampy,Y = coordinate perpendicular to rampδ = boundary layer or shear layer thicknessρ = densityσp = (p′2)1/2

Subscripts

c correlatione edgeh holeI intensityp pressureR shock recoveryr reattachment

∗Research Aerospace Engineer, Air Vehicles Directorate,AFRL/VAAC, 2210 Eighth Street. Senior Member AIAA.†Professor, Department of Mechanical and Aerospace Engi-

neering, P.O. Box CN5263. Associate Fellow AIAA.This paper is a work of the U.S. Government and is not subject

to copyright protection in the United States.

ref references separationu velocityx position

Superscripts

∗ = nondimensional variable· = time derivative

Introduction

THE mechanisms of shock unsteadiness in sepa-rated boundary layer flow have been a topic ofextensive research, motivated primarily by the need tomitigate aircraft fatigue loading caused by the intensefluctuations in wall pressure and heat flux that accom-pany shock oscillation. The first study of this problemwas carried out by Kistler,1 who measured wall pres-sure fluctuations in a supersonic forward-facing stepflow. Kistler observed an intermittent wall pressuresignal in which fluctuations due to turbulence in theincoming boundary layer and separation bubble weremodulated by rapid pressure jumps associated withthe motion of the separation shock back and forthacross the transducer. This pattern has since beenobserved experimentally in a wide variety of separatedsupersonic flows over two- and three-dimensional testconfigurations.2

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American Institute of Aeronautics and Astronautics Paper 2000–0140

The shock motion occurs on two scales: one becauseof perturbation of the shock by organized structuresin the flow turbulence and the other because of rel-atively large-scale, low-frequency expansion and con-traction of the separation bubble.3 The large-scalephenomenon appears to be analogous to the cyclicvortex shedding caused by instability of the separa-tion vortex system in low-speed separated flows,4–7 butmay also be connected to turbulence in the separationbubble and in the incoming flow.8–10

The small-scale shock motion has been observeddirectly11–14 using a planar visualization techniquebased on Rayleigh scattering. Images obtained withthis technique show that the shock is distorted, andoften split, by organized turbulence structures as theyconvect into the separation zone. Difficulty in seed-ing the separation zone with scattering particles hasprecluded visualization of the large-scale shock mo-tion with laser scattering techniques. Kussoy et al.,15

however, have observed the expansion and contrac-tion of the separation bubble using cinematic shadowphotography in a configuration where the line of sightaveraging inherent in this technique was minimized.

Profiles of the intensity of pressure fluctuations16

and heat transfer fluctuations17,18 along a streamwiseline at the wall of a separated boundary layer in com-pressible flow typically show two distinct maxima: onejust upstream of the mean separation line and one nearthe mean reattachment line. These peaks may be anorder of magnitude above the fluctuation levels in theincoming turbulent boundary layer flow. Conditionalaveraging has shown that, in many flows, the peaks arecaused by motion of the two legs of a lambda shocksystem (the separation and reattachment shocks) asthe separation bubble expands and contracts.10,19

Most research has focused on the motion of theseparation shock, with relatively little emphasis onreattachment shock unsteadiness. To address this lackof data, Shen et al.20 investigated the wall pressurefluctuations in a flow where the separation point wasessentially fixed at a backward-facing step, but thereattachment point was free to move along a ramp.The character of the shock unsteadiness in the reat-taching flow was found to be substantially differentfrom that of a typical separating flow.

The present study was undertaken to extend thework of Shen et al., and to search for mechanismswhich might be responsible for shock unsteadiness inthe reattaching shear layer. The fluctuations in wallpressure caused by the motion of the reattachmentshock system were measured, and the flow was visu-alized using a laser scattering technique. In order toisolate the influence on the reattachment shock sys-tem of disturbances originating in the incoming shearlayer, experiments were conducted in which artificialdisturbances were introduced into the flow through airinjection in the vicinity of separation. This controlled

perturbation of the flow makes clear the connection be-tween shock unsteadiness and incoming disturbancesin the shear layer, and raises the possibility of futuremethods of controlling shock unsteadiness.

Theoretical Model

Since the separation location is fixed at thebackward-facing step in the reattaching shear layerconfiguration, the primary mechanism for shock oscil-lation is believed to be perturbation by disturbancesin the incoming turbulent flow. A theoretical modelfor this process has been developed by Plotkin.21

Plotkin assumed that a stable location exists for theshock, the position where the shock would sit if nodisturbances were present in the incoming flow. As alarge-scale eddy convects into the vicinity of the shock,it changes the jump conditions across the shock, andcauses it to move away from the equilibrium position.After the eddy passes, the shock obtains a velocity inits new environment that tends to return it toward theequilibrium position.

For simplicity, Plotkin considered only a one-dimensional model for oscillation in the direction par-allel to the wall (x). The shock velocity (ẋ) was takento be the superposition of a random forcing function(u) and a restoring velocity that is proportional to thedisplacement (x) of the shock from its equilibrium po-sition:

ẋ = u(t)− x/τR (1)

Here τR is a time constant specifying how rapidly theshock recovers from a perturbation. The variables xand u are defined to have zero mean.

Plotkin considered the random function u to repre-sent convection of the shock by velocity fluctuationsin the boundary layer. This function is probably bet-ter viewed as the shock velocity induced by changes inthe jump conditions due to turbulent fluctuations. Forlinearized jump conditions, u would be proportional tothe velocity fluctuations in the boundary layer. Therestoring term in Eq. (1) represents the shock veloc-ity induced by changes in the jump conditions due tochanges in the local mean flow associated with dis-placement of the shock from its equilibrium location.

For a given history of velocity perturbations, Eq. (1)can be solved to give the resulting time-history ofshock position. Assuming that x(0) = 0, the solutioncan be put in the form:

x = e−t/τR∫ t

0

u(ξ)eξ/τRdξ (2)

Plotkin used Eq. (2) to relate the statistical propertiesof the shock motion to those of the fluctuations inthe turbulent boundary layer, noting an analogy tolinearly-damped Brownian motion.

An important parameter characterizing both the ve-locity fluctuations and the consequent shock motion is

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the integral time scale of the autocorrelation function.The integral time scale is defined as:

τi =∫ ∞

0

Ri(τ)dτ (3)

where i is replaced by u, x, or p to indicate the autocor-relation of the fluctuations in the turbulent boundarylayer, the shock position, or the wall pressure fluctu-ations. The autocorrelation function is defined in theusual manner for time-series data.

Assuming that the shock response is much slowerthan the turbulent fluctuations (τ � τu and τR � τu),Plotkin found an approximate expression for the meansquare shock excursion:

x2 = u2 τuτR (4)

From the same assumptions, he also derived the fol-lowing simplified form for the autocorrelation of theshock position:

Rx(t) = e−t/τR (5)

At this level of approximation, the autocorrelation ofthe shock position is independent of the detailed sta-tistical properties of the boundary layer turbulence,and the integral time scale of the shock position is thesame as the time constant of the restoring velocity:τx = τR.

Plotkin went on to assume that the pressure distri-bution induced by the oscillating shock can be approx-imated by the mean pressure distribution translatedto the instantaneous shock position. Expanding themean pressure distribution in a Taylor series aboutthe mean shock location, and retaining terms throughfirst order, he showed that the mean square fluctuat-ing pressure is proportional to the mean square shockexcursion.

p′2 =(∂p

∂x

)2x2 =

(∂p

∂x

)2u2 τuτR (6)

(Recently, a related approximation has been used suc-cessfully to predict fluctuating loads due to separationshock oscillation.22) Further, the autocorrelation ofthe pressure fluctuations is the same as that of theshock position: Rp(t) = Rx(t) and τp = τR. He thencalculated the spectrum of the pressure fluctuationsfrom the Fourier transform of the auto-correlation:

G(f) =4p′2τp

1 + (2πfτp)2(7)

where 2πfτp < 1. Plotkin also assumed that the inte-gral time scales were proportional to a characteristicboundary layer time scale δ/Ue. These results sug-gest plotting the power spectra of pressure fluctuation

Fig. 1 Diagram of the experimental model.Adapted from Baca.24

measurements in the form G∗ = UrefG/(p′2δref) ver-sus f∗ = fδref/Uref and comparing the data to theequation:

G∗(f∗) =4τ∗p

1 + (2πf∗τ∗p )2(8)

where τ∗p = Urefτp/δref .Alternative treatments of the interaction of orga-

nized structures with a shock have used linearizedversions of the Euler equations and shock jump condi-tions to predict the distortion of the shock shape andthe amplitude of downstream disturbances caused bya small oncoming perturbation. (See, for example, Er-lebacher and Hussaini23 and the references therein.)Plotkin’s work is better viewed as a conceptual modelof the shock unsteadiness than as a linearized shockevolution equation. Where a linearized theory wouldpredict the spectrum of shock unsteadiness to be thesame as that of the incoming turbulence, Plotkin’smodel is able to relate the spectrum of shock motionto the mean flow conditions through the parameter τp.In this way, the model incorporates some of the nonlin-ear aspects of the interaction of organized structuresand a shock.

Experimental MethodsThe experiments were carried out in the Princeton

University Gas Dynamics Laboratory Mach 3 blow-down wind tunnel. High-pressure air is supplied to the

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Fig. 2 Sketch of a cross-section of the mean flow-field. Mean reattachment location marked as ‘R’.Adapted from Baca.24

facility by as many as five Worthington 75 kW four-stage compressors. The compressed air passes througha chemical drying system, and is stored at pressuresof up to 20 MPa in four tanks with a total capacityof 57 m3. A hydraulically-controlled valve regulatesthe tunnel stagnation pressure as the air flows fromthe storage tanks into the settling chamber. The ex-periments were carried out in the first of the windtunnel’s three 902 mm long test sections, which hasa square cross-section of 203 mm by 203 mm. Thepressure in the settling chamber was maintained at0.69±0.03 MPa for all experiments, and the freestreamMach number was 2.92 ± 0.01. In a two-minute typi-cal run, the stagnation temperature was initially 290 Kand dropped by about 8%.

The experimental configuration for the reattachingshear layer is illustrated in Figs. 1 and 2. The exper-imental model was originally designed by Baca.24 Inthis flow, a Mach 2.9 turbulent boundary layer formson a flat plate and detaches at a backward-facing step.As a result, a free shear layer forms over a region ofrecirculating flow. The shear layer reattaches on a 20◦

ramp, passing through an oblique shock system, and aturbulent boundary layer develops on the ramp down-stream. Surveys of the basic properties of the flowfieldwere performed in earlier experimental programs atthe Princeton Mach 3 wind tunnel20,24–27 and similarconfigurations have been investigated in other facili-

ties.28–30

Measurements have indicated that a zero-pressure-gradient, equilibrium turbulent boundary layer ispresent in the vicinity of the backward-facing step witha thickness of about δs = 3 mm (based on the loca-tion of 98% of the freestream Pitot pressure) and amomentum thickness Reynolds number of about 104.Due to the position of the reattachment ramp, theboundary layer separates at the 25 mm step with aminimal change in flow direction. The mean velocityprofiles of the resulting free shear layer become self-similar about 18δs (58 mm) downstream of the step,where the nominal convective Mach number31 is about1.1 and the growth rate is in good agreement withdata from other compressible, turbulent mixing layerexperiments.25 According to surface flow visualiza-tion data, the mean reattachment line lies 67± 1 mmup the 20◦ ramp. The boundary layer on the ramp isstrongly perturbed near reattachment, but approachesthe equilibrium condition farther downstream.

In order to provide optical access to the flow, the ex-perimental model was modified by Shen et al.20 fromthe original design of Baca. The cavity of the modelwas fitted with removable inserts so that the heightof the sidewall could be set at either 15.9 mm or25.4 mm (see Figure 1). The experiments reportedhere were carried out with the sidewall inserts and theaerodynamic fences on the sides of the ramp removed.Experimental checks32 showed that the alterations didnot significantly change the flowfield, although a slightdeflection angle was introduced where the flow de-tached from the backward-facing step, changing thestatic pressure ratio across the step from 1.01 to 1.04.

Flow Visualization

A series of experiments was carried out in whichthe reattachment shock system was visualized usingRayleigh scattering from nanometer-scale contaminantparticles in the flow. Illumination was provided by anultraviolet laser beam focused into a thin sheet in thewind tunnel. The experimental arrangement used op-tics with a UV-coating, and quartz windows providedoptical access to the tunnel test section. Two laserswere used in the course of the experimental program: aLambda-Physik argon fluoride laser with a wavelengthof 193 nm and a frequency-quadrupled ContinuumNd:YAG laser with a wavelength of 266 nm. Bothlasers provided a pulse on the order of several nanosec-onds in duration at a repetition rate of 10 Hz, anddelivered 20-50 mJ of energy per pulse. A double-intensified ITT CID (Charge Integrated Device) cam-era recorded the light scattered from the laser sheet.The camera had a resolution of 388 by 244 pixels, andthe light intensifier had a resolution of 180 lines. Thevideo data were recorded on VHS video tape, and laterdigitized for analysis.

Additional experiments were carried out in which

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the Continuum Nd:YAG laser was used in double-pulse mode, producing pairs of pulses separated byan interval of 12 µs to 60 µs.13,33 For this case, twocameras, gated to record the two pulses in sequence,were mounted on opposite sides of the wind tunnel andaligned to record the same field of view. In order toensure that the cameras were properly aligned, an im-age of a precision grid in the test section was recordedbefore and after each wind tunnel run. The accuracyof positioning was found to be within 1% of the widthof the field of view in all the tests.

The scattering signal is believed to be dominatedby Rayleigh scattering from a uniform fog of clus-ters of H2O, CO2, or O2 molecules that form inthe freestream flow as the air cools in the expansionthrough the wind tunnel nozzle.34,35 Observationsfrom past work have indicated that the scattering in-tensity (and thus the particle number density) tendsto follow the air density, except where the temperaturereaches a level high enough to vaporize the contami-nant particles.12,36 Since the molecular constituents ofair have a much smaller scattering cross-section thanthe relatively larger particles, images of high temper-ature regions of a flow may appear dark.

For freestream temperatures on the order of 100 K,the fluid near the edge of a Mach 2.9 turbulent bound-ary layer or free shear layer has a temperature highenough to vaporize the molecular clusters, and a sharpvaporization interface reveals organized structures inthe two turbulent shear flows. For relatively weakshocks, the Rayleigh scattering images show an in-crease in scattering intensity due to the shock den-sity jump, but for stronger shocks, the images showa marked decrease in scattering intensity across theshock, due to vaporization of the scattering particles.

A body of evidence has been accumulated that indi-cates that the scattering images have accurately por-trayed many of the physical features of a turbulentboundary layer, a turbulent mixing layer, and severalshock-wave / boundary-layer interactions studied inthe Princeton supersonic wind tunnel.37 In partic-ular, quantitative measures of the scale, orientation,and speed of large-scale organized structures derivedfrom a statistical analysis of scattering images of theturbulent boundary layer and turbulent mixing layeragree well with results obtained in the same flows usinghotwire anemometry. Further, Nau36 has examinedsimultaneous hotwire and Rayleigh scattering mea-surements of the turbulent boundary layer, and founda strong correlation between traces of the hotwire sig-nal and the corresponding scattering profiles.

Measurements of Wall Pressure Fluctuations

Static pressure measurements were made withminiature differential pressure transducers manufac-tured by Kulite Semiconductor Products (model XCQ-72-062-25D). The transducers were calibrated stati-

cally at the operating temperature. Previous studies,and checks made before and after wind tunnel runs,have shown that the calibration is consistently linearand repeatable.

The transducers were mounted in a block that couldbe positioned at different streamwise positions alongthe reattachment ramp. The spacing between thetransducers was 5.1 mm, and they were mounted2.5 mm off the centerline of the wind tunnel. Thedata were taken with the block positioned in three lo-cations. Three of the four available transducers werefound to calibrate accurately, and were used to covera range of positions between 52 mm and 88 mm fromthe start of the ramp.

The signals from the transducers were amplified,then band-pass filtered with a four-pole Butterworthfilter. The analog data were then sampled digitallywith 10 bit resolution using a CAMAC (ComputerAutomated Measurement and Control) system fromLeCroy, Inc. Three sampling rates were used: 10 kHz,250 kHz, and 1 MHz. In all three cases the high-passfilter was set to 10 Hz, while the low-pass filter was setto 5 kHz for the 10 kHz sampling rate, and to 80 kHzotherwise. Data were obtained simultaneously for thethree channels in files of four records, each containing24576 contiguous points per record.

Artificial Disturbances

Artificial disturbances were introduced into the flowthrough air injection in order to isolate the effect of up-stream disturbances on the unsteadiness of the shocksystem. Two spanwise rows of holes for air injec-tion were added to the experimental model. One rowwas located 12.7 mm upstream of the backward-facingstep, and the other was located 12.7 mm downstreamof the step. Each row was 101.6 mm long, and con-sisted of 33 holes, 1.6 mm in diameter. Although testswere made using a number of air injection patterns,32

for the experiments described here, air was injectedfrom the holes downstream of the backward-facing stepin two configurations: one and three holes on the cen-terline of the model.

Air was supplied to the holes from the wind tunnelstorage tanks, by way of a stagnation tank. A series ofthree pressure regulators in the supply lines brought atypical storage pressure of about 20 MPa down to thedesired tank pressure of about 0.7 MPa. The blowingstagnation tank consisted of a 1.02 m length of 8 in(203 mm) nominal diameter pipe with the ends sealedwith flanges and blinds.

Air was allowed to leave the tank through sections of1.6 mm diameter stainless steel tubing. Plastic tubingof 1.6 mm inside diameter connected the stainless steeltubing to similar tubing soldered into the holes in theexperimental model.

The supply pressure was maintained at 0.69 ±0.03 MPa for the tests reported in this paper. Re-

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Fig. 3 Schlieren photographs of the flow in thevicinity of the step. (a) Sketch of field of view, notto scale. (b) Undisturbed flow. (c) Air injectionthrough three holes.

sults of calibration32 indicate that the ratio of themass flux through a hole to the mass flux in thefreestream was on the order of ρhUh/(ρeUe) = 0.07,and the momentum flux ratio was on the order ofρhU

2h/(ρeU

2e ) = 0.04.

ResultsSchlieren Photography

Schlieren photographs were taken in order to pro-vide an overview of the flowfield and to demonstratethe general effects of air injection. Images were ob-tained for the region of the flow in the vicinity of thebackward-facing step and for the region near the reat-tachment ramp. Example images are shown in Figs. 3and 4. In each case a sketch of the field of view is shownat the top of the figure, the upper image was obtainedin the undisturbed flow, and the bottom image wasobtained for the case with air injection through threeholes. The field of view for both the images of theregion near the step and the region near the ramp isapproximately 100 mm wide by 80 mm high (10.6δr by7.7δr, where the boundary layer thickness in the vicin-ity of the mean reattachment line in the undisturbedflow is δr = 10.4 mm).

Figure 3b shows a view of the undisturbed flow inthe vicinity of the backward-facing step. The unevenouter edge of the incoming turbulent boundary layer

Fig. 4 Schlieren photographs of the flow in thevicinity of the ramp. (a) Sketch of field of view, notto scale. (b) Undisturbed flow. (c) Air injectionthrough three holes.

can be seen on the flat plate at the left in the fieldof view. A Mach wave originates at the row of holesupstream of the step. The boundary layer separatesat the step with a slight change in flow direction, andan expansion fan appears centered on the edge of thebackward-facing step.

The upper and lower edges of the developing freeshear layer are clearly visible in the image. Since aschlieren image represents a spanwise average of thedensity gradients in the flow, there is relatively littleindication of the large-scale structure of the turbu-lent flow. There are, however, waves visible emanatingfrom the upper edge of the shear layer that are prob-ably associated with organized structures.

Figure 3c shows the effect of air injection on the flowin the vicinity of the step. The jets of injected air arenot visible in the image because of the orientation ofthe schlieren knife edge. A three-dimensional shocksystem forms due to the jets, and is discernible as adark curve roughly parallel to the expansion fan up-stream. One surprising observation is that the shearlayer shifts down (note the increased size of the ex-pansion fan), even though mass is being added to therecirculating region, suggesting that there is enhancedentrainment of freestream fluid into the shear layer.The shear layer also appears to thicken.

The reattachment zone for the undisturbed flow case

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is shown in Fig. 4b. The expansion fan centered onthe backward-facing step is apparent in the freestreamflow, as are two Mach waves originating from joints inthe ceiling of the test section. The free shear layer canbe seen entering from the left side of the image; mostof the shear layer is visible above the cavity sidewall.The highly turbulent nature of the shear layer in thevicinity of reattachment is plain, despite the spanwiseaveraging process inherent in the schlieren technique.

The reattachment shock system is evident in the im-age as the light, fan-like region above the redevelopingboundary layer. The shock system appears to be dis-tributed in this manner for two reasons: the primaryoblique shock is wrinkled along the spanwise direction,and the system is composed of a combination of a num-ber of shocks and compression waves (see the Rayleighscattering results below).

The effect of air injection on the reattachment regionis quite dramatic (Fig. 4c). The increased thickness ofthe shear layer and the developing boundary layer isobvious, as is the great distortion of the reattachmentshock system. In the example image shown, the pri-mary oblique shock appears to have split into severalsections. Severe curvature and strong displacementof the shock system indicate a much higher degree ofunsteadiness. The increase in unsteadiness is quitespectacular in the original video recording.

Single-Pulse Laser Scattering

The laser scattering technique used in the presentwork offers a visualization of an instantaneous planarsection of the flow, revealing cross-sections of coherentstructures and shock waves. A series of single-pulselaser scattering experiments was carried out to exam-ine the scale and orientation of organized structuresin several cross-sectional planes in the reattachmentregion of the flow, and to look for an association ofthese structures with shock waves. Two optical con-figurations were used: one with a vertical laser sheetlocated along the wind tunnel centerline and the fieldof view aligned with the reattachment ramp, and an-other with a horizontal laser sheet located in the outerpart of the redeveloping boundary layer.

The side-view images were obtained with a rela-tively small field of view that allowed the resolutionof smaller-scale features of the interaction of the shocksystem with the incoming turbulent flow. The camerawas tilted 20◦ from horizontal, in alignment with thereattachment ramp. The field of view was approxi-mately 26 mm wide by 20 mm high, and began 54 mmup the reattachment ramp. Sample images are shownin Fig. 5, with the inclined orientation of the field ofview sketched at the top of the figure. The mean reat-tachment location for the undisturbed flow is marked‘R’ in the figure, and the ramp surface is indicatedwith a line.

Figure 5b shows the undisturbed flow. The free

Fig. 5 Side-view laser scattering images of flowalong reattachment ramp. (a) Experimental con-figuration, not to scale. (b) Undisturbed flow. (c)Air injection through three holes.

shear layer can be seen entering from the upper leftof each image, and the redeveloping boundary layerexits horizontally from the right. Shocks are seen toform at the interface between slow-moving fluid in theδ-scale bulges of the redeveloping boundary layer andthe high-speed fluid of the freestream. These shocksare relatively weak, and are made visible by an in-crease in the intensity of scattered light due to theincreased density of scattering particles downstreamof the shocks. Shocks are associated with organizedstructures having a wide range of length scales, and thestrongest shocks are associated with structures withscales on the order of the thickness of the redevelopingboundary layer (δr = 10.4 mm).

Figure 5c shows an example image for the case withair injection through one hole in the cavity of the ex-perimental model. There is a dramatic increase in thethickness of the redeveloping boundary layer, the scaleof the turbulence structures, and the strength of theshock waves. Again, there is an association of shockswith large-scale structures in the shear layer.

For plan views of the reattaching flow, a horizon-tal laser sheet generated with the Spectra Physics ArFlaser was positioned 38 mm off the floor of the cavity inthe experimental model. The field of view was 31 mmfrom top to bottom, and began 121 mm downstreamof the backward-facing step (see Fig. 6a). The flow

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Fig. 6 Horizontal section through reattachmentshock system. Field of view aligned with modelcenterline. (a) Experimental configuration, not toscale. (b) Undisturbed flow. (c) Air injectionthrough one hole.

direction is from top to bottom in the images, with acomponent that passes up through the laser sheet frombelow. For these images the laser sheet is relatively farfrom the wall, where the compression due to the 20◦

turning angle occurs over a relatively short streamwisedistance. Consequently, the shocks are strong enoughhere to vaporize the scattering particles, and in the im-ages the flow appears dark downstream of the shocks.

The flow over this portion of the experimental modelis highly complex and unsteady, even in the absenceof air injection (Fig. 6b). The shock system exhibitsspanwise wrinkling with a length scale on the order ofthe reattachment boundary layer thickness. An imageof the flow with air injection through the center holedownstream of the backward-facing step is shown inFig. 6c. In this image, a bow shock is clearly seenassociated with the upstream side of a structure in theredeveloping boundary layer.

In order to quantify the increase in structure lengthscale seen in the images, two-point cross-correlationswere carried out for the set of side-view images of thereattachment region (see Fig. 5). The cross-correlationbetween the scattering intensity at a reference pointI(X,Y ) and the scattering intensity at a given point

x (mm)

y(m

m)

55 60 65 70 75 80

0

5

10

15

R

(a)

x (mm)

y(m

m)

55 60 65 70 75 80

0

5

10

15

R

(b)

Fig. 7 Two-point auto-correlation of side-view im-ages. (a) Undisturbed flow. (b) Air injection.

in the field of view I(x, y) was defined as:

RI(x, y,X, Y ) =Cov[I(x, y), I(X,Y )]√Var[I(x, y)]Var[I(X,Y )]

(9)

where the covariance and variance are defined in theusual way, e.g.: Var(I) =

∑Ni=1(Ii− I)2/N . This type

of correlation is similar to the zero time-delay cross-correlation between a pair of point probes for differentprobe separations.

The horizontal position of the reference point wastaken to be X = 60.5 mm. In order to allow aconsistent comparison between the two cases, the ver-tical coordinate was taken to be the location of max-imum root-mean-square scattering intensity (aboutY = 6 mm for the undisturbed flow and Y = 10 mmfor the case with air injection).

The results are shown in Fig. 7. Contours are shownfrom 0.4 to 0.9, with an interval of 0.1. The correla-tion contours generally have an elliptical shape, as seenin other studies of free shear layers and boundary lay-ers.11,30 One feature of note, however, is the horizontalorientation of the contours. Typically, the structureorientation is in the neighborhood of 45◦ from themean flow direction. The present results reflect theadjustment of the mean flow from a direction parallelto the freestream to a direction aligned with the ramp

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Fig. 8 A pair of images of the undisturbed flowtaken with the dual camera configuration and thelaser in single-pulse mode. (a) Field of view, notto scale. (b) Camera One. (c) Camera Two.

(see Fig. 5a): the structures are actually inclined tothe local mean flow direction.

The primary difference between the baseline caseand the perturbed flow is a dramatic increase in thestructure length scale. For example, the horizontallength scale of the RI = 0.5 contour increases fromabout δc = 6 mm for the undisturbed flow to aboutδc = 9 mm for the case with air injection.

Double-Pulse Laser Scattering

Double-pulse laser scattering experiments were car-ried out using a vertical laser sheet to obtain side viewsof the flow over the ramp. For these experiments, thefield of view was aligned with the floor of the cavityin the experimental model rather than with the reat-tachment ramp. The field of view was 58 mm wideby 43 mm high, and began 92 mm downstream of thebackward-facing step. The bottom edge of the field ofview lay 14 mm above the cavity floor.

To illustrate that the cameras were aligned and themagnifications matched, Fig. 8 shows a pair of exampleimages of the undisturbed flow for the case with zerotime delay (single-pulse mode). Both large-scale tur-bulence structures and the reattachment shock systemare visible in the images, and the fields of view are seento be closely aligned. The free shear layer enters thefield of view from the lower left side of each image, and

Fig. 9 Pair of double-pulse images obtained in theundisturbed flow using a delay of 30.2 µs. (a) Fieldof view, not to scale. (b) Camera One. (c) CameraTwo.

reattaches on the 20◦ ramp, forming a new boundarylayer which exits near the upper right of the images.The great variability of the thickness of the turbulentboundary layer visible in the reattachment region isconsistent with the high degree of intermittency foundby Hayakawa et al.27 in hotwire data obtained in thisregion of the undisturbed flow. As in the images shownabove, shocks are seen to form at the upstream sidesof the bulges in the redeveloping boundary layer.

Figure 9 shows a pair of images of the undisturbedflow taken with a time delay of 30.2 µs. The top imagewas captured with one video camera, and the bottomimage was captured after the time delay with the othercamera. This pair of images was selected because itshows a particularly large structure entering the reat-tachment zone, where a relatively strong shock formson its upstream side. Note that a shock associatedwith the upstream edge of the structure appears toconvect along with it as the structure moves down-stream.

A similar set of images was obtained in the flowperturbed by air injection through the center hole inthe cavity of the experimental model. A pair of imagesshowing a structure entering the reattachment zone isshown in Fig. 10. Although the length scale of thestructures is much larger in the disturbed flow, thequalitative behavior of the structures is similar to that

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Fig. 10 Pair of double-pulse images obtained inthe flow with air injection through one hole usinga delay of 30.2 µs. (a) Field of view, not to scale.(b) Camera One. (c) Camera Two.

seen in the undisturbed flow: a shock is present on theupstream edge of a given structure, and moves with thestructure as it convects downstream along the ramp.

Pressure Measurements

Measurements were made along the reattachmentramp of the mean and fluctuating surface pressurefor the undisturbed flow and for the case with airinjection through three holes in the cavity of the ex-perimental model. Figure 11a shows the results forthe mean pressure, which are normalized by the staticpressure in the freestream flow. The pressure is seento rise monotonically from the freestream level to thelevel downstream of the primary oblique shock. Withair injection, the pressure rise begins sooner, but thepressure increases more slowly along the streamwisedirection, and reaches approximately the same leveldownstream.

The pressure distributions were found to collapseonto one curve when plotted against the nondimen-sional distance (x− xr)/δc, where the mean reattach-ment location was taken to be xr = 67 mm for theundisturbed flow and xr = 75 mm for the case withair injection (Fig. 11b). The following equation wasfound to provide a good fit to the reduced data:

p

pref=

114

+74

erf(x− xr

4δc+ 0.1

)(10)

(x - x r)/δc

p/p

ref

-12 -8 -4 0 4 8 12 160

1

2

3

4

5

undisturbed flowair injectioncurve fit

(b)

x (mm)

p/p

ref

0 20 40 60 80 100 120 140 1600

1

2

3

4

5

undisturbed flowair injectionR

(a)

Fig. 11 Mean wall pressure distribution alongramp. (a) Dimensional distance. Mean reattach-ment location marked as ‘R’. (b) Nondimensionaldistance.

These results indicate that broader pressure distribu-tion observed with air injection reflects primarily theincrease in the thickness of the reattaching shear layer,rather than - for example - a large-scale shear layerflapping motion.

Figure 12 shows the standard deviation of the wallpressure fluctuations in the undisturbed flow plottedversus position along the ramp. The measurements ofShen et al.20 are shown for comparison. The datahave been normalized by the static pressure in thefreestream flow. An error bar of ±0.05pref indicatesthe level of scatter in the data. The intensity of thepressure fluctuations in the present flow is somewhathigher than that found by Shen et al., primarily be-cause of the removal of the aerodynamic fences andsidewall inserts to allow optical access to the flow.32

Nevertheless, both data sets show a rise in the pres-sure fluctuation levels to a peak downstream of themean reattachment line, followed by a gradual de-

10 of 14


0 20 40 60 80 100 120 140 160

x (mm)

0

0.1

0.2

0.3

0.4

0.5σ p

/pre

fShen et al. (1993)undisturbed flow

R

Fig. 12 Intensity of wall pressure fluctuationsalong ramp. Mean reattachment location markedas ‘R’.

crease downstream.Figure 13a compares the pressure fluctuation levels

in the undisturbed flow to the case with air injectioninto the flow through three holes on the cavity center-line. The distribution of the intensity of the pressurefluctuations has the same shape in the air injectioncase as in the undisturbed flow case, but is shifted up-ward significantly. There is an increase of about 50%in the standard deviation of the pressure signal at thestation farthest upstream. Given that the pressurefluctuations are caused primarily by shock motion,this change is consistent with the apparent increasein shock unsteadiness observed in the flow visualiza-tion data, and indicates a strong connection betweenthe incoming disturbances and the shock motion in theflow.

Plotkin’s model gives an estimate of the intensity ofthe pressure fluctuations, Eq. (6), provided that thestreamwise component of the mean pressure gradientand the amplitude of the shock motion are known.Here we take δc as a measure of the amplitude of theshock motion, and use the derivative of Eq. (10) to de-termine the streamwise pressure gradient: σp/pref ≈(∂p/∂x)δc/pref . The results are shown in Fig. 13b,plotted against the nondimensional streamwise posi-tion. Since the mean pressure distribution collapsed innondimensional coordinates, its derivative ∂p/∂x alsodoes, and the theory predicts the single curve shown inthe figure. The theory gives the correct order of mag-nitude for the location and value of the peak pressurefluctuation intensity, but does not predict the increasein intensity observed with air injection.

Cross-correlation calculations were made betweenadjacent pairs of pressure transducers. Figure 14shows an example cross-correlation plot for a pair oftransducers located 82.6 mm and 87.6 mm up the

x (mm)

σ p/p

ref

0 20 40 60 80 100 120 140 1600

0.1

0.2

0.3

0.4

0.5

undisturbed flowair injection

R

(a)

(x - x r)/δc

σ p/p

ref

-12 -8 -4 0 4 8 12 160

0.1

0.2

0.3

0.4

0.5

undisturbed flowair injection(dp/dx) δc/pref

(b)

Fig. 13 Effect of air injection on wall pressurefluctuations. (a) Dimensional distance. Mean reat-tachment location marked as ‘R’. (b) Nondimen-sional distance.

ramp. This location was chosen because it lies nearthe point of maximum intensity in the wall pressurefluctuations. The single peak in the cross-correlationindicates the downstream convection of the large-scalestructures in the redeveloping boundary layer. Theeffect of air injection was to increase the peak correla-tion, increase the speed of the structures (reduce theoptimum time delay), and to increase the width of thecorrelation function. With air injection, the convec-tion velocity (adjusted to a nominal stagnation tem-perature of 270 K) increased from 478 m/s to 577 m/s,and the width of the curve at the Rp = 0.5 correlationlevel increased from 14 µs to 20 µs.

An important parameter identified by Plotkin21 insetting the characteristic frequency of the power spec-trum is the integral time scale of the auto-correlation(see above). Auto-correlations for the transducer lo-

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t (µs)

R(t

)

-100 -50 0 50 100-0.2

0

0.2

0.4

0.6

0.8

1


Fig. 14 Cross-correlations of pressure signals atx = 82.6 mm with ∆x = 5.1 mm.

t (s)

Rp(

t)

0 2E-05 4E-05 6E-05 8E-05 0.00010

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


Fig. 15 Auto-correlations of pressure signals atx = 82.6 mm.

cated at x = 82.6 mm are shown in Fig. 15. Asignificant increase in the characteristic time scale ofthe auto-correlation is seen with air injection: the in-tegral time scale increases from about 11 µs to 15 µs.The nondimensional time scale, however, remains ap-proximately constant at τ∗p ≈ 0.9.

Figure 16 shows the corresponding auto-spectra ofthe fluctuating pressure signal. The data are seento have a very broad-band energy content, with noprominent peaks. The roll-off at higher frequency isa property of the flow, not the signal processing sys-tem: the fourth-order Butterworth filter passes about70% of the input amplitude at the cutoff frequency of80 kHz. Air injection is seen to shift the spectrumup, reflecting the increase in the intensity of pressurefluctuations, and to the left, reflecting a decrease in

102 103 104 105

f (Hz)

10-6

10-5

G(f

)/p r

ef2(s

)


Fig. 16 Auto-spectra for the wall pressure fluctu-ation data at x = 82.6 mm.

f*

G* (

f* )

10-3 10-2 10-1 100

100

101

undisturbed flowair injectionTheory

Fig. 17 Normalized auto-spectra for the wall pres-sure fluctuation data at x = 82.6 mm.

characteristic frequency.Figure 17 shows the spectral data plotted in the

normalized form of Eq. (8). The frequency was nondi-mensionalized by the correlation length scale δc andthe convection velocity Uc. The spectrum was dividedby the mean square fluctuating pressure p′2 and thetime scale δc/Uc. The data collapse well. Nondi-mensionalizing the frequency causes the ‘knees’ in thespectrum (present near 2 kHz and 20 kHz in the undis-turbed flow) to line up. Plotkin’s model is seen tooffer a fairly good approximation of the spectrum for0.02 < f∗ < 0.6 using the measured time scale ofτ∗p = 0.9.

ConclusionsThe origin of shock unsteadiness in a Mach 2.9 tur-

bulent, reattaching shear layer was investigated exper-

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imentally using temporally-resolved flow visualizationand measurements of wall pressure fluctuations. Theresults indicate that the shock motion in the reat-taching shear layer is primarily caused by organizedstructures in the incoming turbulent flow, in contrastto the related class of separated compression rampflows, where shock motion is associated primarily withrelatively low-frequency expansion and contraction ofthe separation bubble.

In order to isolate the influence of disturbances orig-inating in the incoming shear layer, experiments wereconducted in which artificial disturbances were intro-duced into the flow through air injection in the vicinityof separation. The effect on the reattachment shocksystem was dramatic: the intensity of the pressurefluctuations and shock motion increased substantially,and power spectra of the pressure fluctuations showeda distinct shift to lower frequency. Mean pressure pro-files on the reattachment ramp collapsed onto a com-mon curve in nondimensional coordinates based on alength scale derived from two-point cross-correlationsof the flow visualization data. Similarly, the powerspectra of the pressure fluctuations collapsed in coordi-nates based on this same length scale and a convectionvelocity derived from cross-correlations of the pressuremeasurements.

The data were compared to a theory developed byPlotkin,21 which is based on perturbation of a shockby random fluctuations in the incoming turbulent flow.Spectra of the wall pressure fluctuations predicted bythe theory showed fairly good agreement with theexperimental data over a relatively broad frequencyband. Discrepancies between the model and the dataat lower frequency (f < 1 kHz) may reflect the windtunnel background noise field or a shear layer flap-ping motion. The theory predicted the correct orderof magnitude for the location and value of the peakpressure fluctuation intensity, but did not predict theincrease in intensity observed with air injection. Thereason for this shortcoming may be the use (in thepresent work) of the length scale δc to estimate theamplitude of the shock motion (x2)1/2.

An important limitation of the theory is the re-striction of the shock motion to one dimension. Theprimary source of pressure fluctuations in the presentflow appears to be the highly three-dimensional shocksassociated with the upstream sides of δ-scale structuresconvecting through the reattachment zone.

AcknowledgmentsThis research was funded in part by grants from the

AFOSR and NASA Langley. J. Poggie received sup-port from an NSF Graduate Research Fellowship andthe USAF Palace Knight program during the courseof this research program.

W. Stokes and R. Bogart provided technical sup-port for this project. The Princeton University laser

physics group, and in particular J. Forkey, providedassistance in the laser visualization experiments. Thefirst author would like to thank R. Kimmel and A.Creese for advice in the preparation of this paper.

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