984 AIAA JOURNAL VOL. 15, NO. 7

Experimental Investigation of Oscillationsin Flows Over Shallow Cavities

Virendra Sarohia*California Institute of Technology, Pasadena, Calif.

Laminar axisymmetric flows over shallow cavities at low subsonic speeds were experimentally investigated.The results indicate that the cavity depth has little effect on oscillations in shallow cavities, except when thedepth is of the order of the thickness of the cavity shear flow. For such cavity configurations, measurementsindicate a strong stabilizing effect of depth on laminar cavity shear layer. Results of motion picture and hot-wiresurveys of the cavity shear layer show that, close to the downstream cavity corner, large lateral motion of theshear layer occurs, which results in a periodic shedding of vortices at a frequency of cavity oscillation. Meanvelocity measurements show growth rates as high as d0/djt — 0.022, where 0 is the shear layer momentumthickness and A is the streamwise coordinate. These are attributed to strong imposed velocity fluctuations on theflow, by the oscillating cavity system. Phase measurements indicate that the disturbances propagate at a constantphase speed through the cavity shear layer. The wavelength of the propagating disturbance bears an approximateintegral relation to cavity width, in each mode of cavity oscillation given by & —A(/2 + 1/2), where b is the cavitywidth, X the wavelength of the propagating disturbance, and n is an integer, which takes values 0, 1, 2, ... etc.,depending upon the mode of oscillation.

Nomenclature

b = cavity width (cavity length)d = cavity depth/ = frequency in Hzf = nose length of the modeln = an integeru' = velocity in direction xU - mean velocity in direction xUe = mean velocity at the upstream cavity cornerUc =convective speed of the disturbances in the

cavity shear layerUao = freestream velocity in front of the modelx = streamwise coordinatey - transverse coordinated(x) = shear-layer thickness at x6 (x) = shear-layer momentum thickness at xA = disturbance wavelengthv = kinematic viscosity\l/ = phase angle by which the phase at a given

location lags behind x = 0fb_ fOUe' U<

Re*o

Reeo\ /min

——, —— = nondimensional frequencies

, Reynold number based on 60

uee0 , Reynold number based on 60

corresponds to the conditions for onset ofcavity oscillations

I. Introduction

T HE problem of fluid flow (gases or liquids) over cavitieson solid surfaces has gained renewed significance. For

example, uncovered cavities on flight vehicles are necessary to

Presented as Paper 76-182 at the AIAA 14th Aerospace SciencesMeeting, Washington, D.C., Jan. 26-28, 1976; submitted Oct. 6,1976; revision received April 15, 1977.

Index categories: Jets, Wakes, and Viscid-lnviscid Flow In-teractions; Nonsteady Aerodynamics; Aeroacoustics.

*Senior Scientist, Energy and Materials Research Section, JetPropulsion Laboratory. Member AIAA.

house optical instruments. Other applications occur in thetransonic wind tunnel where slotted walls are used, in con-tinuous laser cavities, and even in cavities of ship hulls. Flowover cavities is of interest, because the presence of cavitieschanges the drag and heat transfer and may cause intenseperiodic oscillations, which in turn may lead to severe buf-feting of the aerodynamic structure and production of sound.

Periodic oscillations in cavities have been observed over alarge range of Mach numbers and Reynolds numbers, withboth laminar and turbulent boundary layers and over a widerange of length-to-depth ratios. In general, cavities aredivided into open and closed cavities as defined by Charwat eta l . ] Open cavities refer to flow over cavities where theboundary layer separates at the upstream corner and reat-taches near the downstream corner. Cavities are closed whenthe separated layer reattaches at the cavity bottom and againseparates ahead of the downstream wall. At supersonic speedsand for a turbulent layer, the dividing line between open andclosed cavities was found to be b/d—\l by Charwat et al.Present experiments at low subsonic speeds with laminarseparation show a demarcation line between an open and aclosed cavity to be approximated by b/d—l-S.

Open cavities may further be divided into shallow and deepcavities. On the basis of previous experiments,2'3 theoscillations in deep cavities are in fundamental acoustic depthmodes. Deep cavities act as resonators and the shear layerabove the cavity provides a forcing mechanism. Resonantoscillations are established under certain flow conditions,corresponding to natural acoustic depth modes of the cavities.

In contrast to deep cavities, present experiments of flowover shallow cavities at low subsonic speed show that thephenomenon of cavity oscillations is not one of the standinglongitudinal acoustic waves but one of propagating distur-bances which get amplified through the shear layer. Theimportant length scale, therefore, in shallow cavity flows, isthe width b of the cavity. On the basis of past experiments,2"4

a very rough division between shallow and deep cavities isbid- 1. For b/d> 1 the cavities may be considered shallow,and for bid< 1 the cavities may be considered deep.

Previous work on cavity oscillations has been mainly ex-perimental. Because of the practical importance ofoscillations in bomb bays, experiments were performed atBoeing,5 Douglas,6 and at the Royal Aircraft Establish-ment.7 The first experiments covering a wide range of

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JULY 1977 EXPERIMENTAL FLOWS OVER SHALLOW CAVITIES 985

geometrical and flow parameters were performed byKaramchet.8 Further extensive work was performed byEast,2 Heller et al.,9 McGregor and White,10 Plumblee etal.,3 Rossiter,7'11 and others.m5

Karamcheti studied the acoustic field of two-dimensionalshallow cavities in the range of Mach numbers from 0.25 to1.5 by schlieren and interferometric observations. Karamchetinoticed that, for a fixed freestream Mach number and depth,there exists a minimum width below which no sound emissionis noticed. For a fixed cavity, experimental results furthershowed a minimum Mach number below which no soundemission was noticed. Under a nonoscillating cavity con-figuration, schlieren and spark shadowgraph pictures showedthat the shear layer bridges the cavities without strong in-teraction with the downstream corner. No detailed study ofthe aerodynamic and geometric conditions for the onset ofcavity oscillations was undertaken.

The effects of Mach number on nondimensional frequencyfb/Ue have been studied by many investigators for bothlaminar and turbulent boundary layers. On the basis of high-speed shadowgraphs of cavity oscillations, Rossiter11

speculated that periodic vortices are shed at the upstreamcorner in sympathy with the pressure oscillation produced byinteraction of the vortices with the downstream corner. Basedon this idea Rossiter derived a formula for the oscillationfrequency. Heller et al.9 studied shallow cavities over a widerange of Mach numbers and correlated a great many ex-perimental results with Rossiter's formula. In Rossiter'sformulations of cavity oscillation frequency, the vortices shedfrom the upstream cavity corner are assumed to convect at aconstant phase velocity through the shear layer, resulting in alinear phase distribution. It is further assumed that the phasevelocity of these vortices is independent of the cavitygeometry and flow configuration. Rossiter's formula does notshed any light on the possible mode or modes in which thecavity is most likely to oscillate. Unfortunately, no systematicmeasurements of cavity shear layer have been made in the pastto verify Rossiter's assumptions.

For a given flow, the prerequisite of a minimum width forthe onset of cavity oscillations strongly suggests that themechanism of cavity oscillations depends upon the stabilitycharacteristics of the shear layer. Woolley16 and Karamchetiextended their stability analysis of nonparallel jets for edge-tone generation to explain the main features of cavityoscillations qualitatively. Their measurements17"19 of edge-tone flowfield indicated that the phase and propagation speedof the disturbances in the jet do not agree with the linearizedstability theory of the incompressible two-dimensionalparallel jet. According to the stability theory of nonparallelshear flows, the stability characteristic of the flow is a func-tion of local quantities, viz., thickness of the shear layer,mean velocity profile, etc., which result in different am-plification rates as a disturbance of fixed frequencypropagates downstream through the flow. This explains theirmeasurements of edge-tone flowfield in which, according tothem, a nonlinear distribution of the phase and convectivespeed of the disturbances occurs. They speculated that asimilar behavior would occur in cavity flows and extendedtheir results accordingly. On the basis of stability of almostparallel shear flow, they concluded that the frequency ofcavity oscillations is the one which received the maximumtotal integrated amplification through the shear layer. Theirtheory does not take into consideration the presence of thedownstream corner that this study indicates to be the keyfactor in inducing self-sustained oscillations in the cavityshear layer. Present experiments further show that thepresence of a back face, in fact, results in an integral relationbetween the wavelength of the propagating disturbance andthe cavity width, in each mode of cavity operation.

It was therefore felt that a detailed measurement of thecavity shear layer was necessary for better understanding ofthe mechanism. Flow visualization and measurements of the

cavity shear layer, from which phase distribution, phasevelocity of the propagating disturbance, etc., can be deter-mined, would be of great help in understanding thephenomenon of cavity oscillations. \

II. Experimental Arrangement

Model and Wind TunnelIn the present study, three axisymmetric models were

employed with outer diameter D of 1, 2, and 6 in. Thesemodels had an arrangement for variation of depth d in stepsand continuously varying with b. The l-in.-diam model with ahemispherical nose was used mostly for preliminary ex-perimental work. The main quantitative work was done onthe 2-in. and 6-in. models. The 2-in. diam model had ojive-shape noses, whereas an elliptic nose was used for the 6-in.model. For all these models, the boundary layer at separationwas laminar. The details of these models are given in Fig. 1.

The 2-in. model was made of aluminum with step depths dof 0.875, 0.5, 0.25, 0.125, and 0.05 in. The width b could be

SHEAR LAYER^THICKNESS 8,

D - 1.0", 2.0" AND 6.0"

b AND d ARE VARIABLEU : 0-80 ft/secRen: 2 x 104 - 105 (LAMINAR BOUNDARY LAYER

AT SEPARATION)

FREE STREAM TURBULENCE

AT 50 ft/sec- 0.3% Vu '2 /Uoo

D = 1.0" HEMISPHERICAL NOSE

I OGIVE WITH J2/D = 0.6NOSE SHAPE D = 2,0" II OGIVE WITH J0/D - 1.12

III OGIVE WITH J0/D - 2 . V 2D = 6.0" ELLIPSOID WITH RATIO OF

MAJOR TO MINOR AXIS = 3.

Fig. 1 Model of cavity oscillations with pertinent nomenclature.

0.5x 10',3-

0.3

0.2

x MODEL #1

> MODEL #2

• MODEL #3

»}MODEL WITH CYLINDRICAL•JEXTENSION

NO CAVITY OSCILLATIONS_J_____l_____I_____J.

10 15 20 25

<*/*

Fig. 2 Region of cavity oscillation.

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986 V. SAROHIA AIAA JOURNAL

varied continuously from 0.0. to 2.5 in. The accuracy ofmeasuring width b was ±0.002 in. This model had a family ofthree ogive nose shapes with HD of 0.6, 1.12, and 2.12 in. Toanalyze cavity flow, a hot-wire probe was inserted fromoutside intp the shear layer. The probe could be moved with±0.001-in. accuracy across and along the shear layer. Thismodel had a provision for injecting smoke from inside thecavity. The flow around it was investigated in a 6-in free jettunnel.

The second model tested was of 6-in. diam D. This modelwas tested in 2 x 2 ft. rectangular section, open circuit windtunnel. It had a semielliposoidal head with a major to minoraxis ratio of 3. Cavity depths of could be set at 0.5, 1, 1.5, and2 in. The model had provision for insertion of a hot-wire frominside the cavity. This probe could be moved along and acrossthe axis of the model. Another hot wire probe was broughtfrom the top of the wind tunnel and could be moved veryprecisely up and down and along the axis of the model. Thetwo hot-wire probes could be moved circumferentially relativeto each other to check the axisymmetry of the cavity flow.

InstrumentationConstant temperature hot-wire anemometry was ex-

tensively used in measuring both mean and fluctuatingstream wise velocity components. The output of the hot-wirewas fed to a wave analyzer to analyze the frequency contentsof the cavity flow. Electronic sweeping was used and theoutput of the wave-analyzer was fed to a rms voltmeter. Themean square output from the rms meter (available as d.c.) wasfed to an x-y plotter to measure the relative amplitude of thefrequency content in the hot-wire signal. For phase angle andphase velocity measurements, the output of the two hot wireswas analyzed on an SAI Correlation and ProbabilityAnalyzer. The output of the correlation measurement wasdisplayed on an oscilloscope or plotted on an x-y plotter.

Flow VisualizationFlow around the cavity was visualized by injecting smoke

from inside the cavity. This was done at various locations,viz., close to the upstream and downstream corner, just afterthe downstream corner, etc., to study the details of flowaround the cavity. The model was lighted from above. Bothstill and motion pictures of the cavity flow were taken.

III. Experimental Results

Minimum Width for Oscillating to OccurIt has been observed experimentally for flow over cavities

that, for given flow conditions, there exists a minimum width^min below which no oscillation occur. Also, no cavityoscillations occur below a minimum velocity Uemin for a givencavity geometry and boundary-layer thickness at 50 atseparation.

A detailed investigation of the effect of flow and cavitygeometry on the onset of the cavity oscillations was un-dertaken on the 2-in.-diam model. A family of three ogivenose shapes (with £/£> = 0.6, 1.12, and 2.12) were employed.For a fixed nose shape and depth d, edge velocity Ue wasdetermined for various widths b at which the cavity began tooscillate. For the same nose shape, the above experiment wasrepeated with five different depths. This gave one set ofexperimental data. Two other similar sets of data were ob-tained for the other two nose shapes. A total of 90 data pointsof known Ue, 5, b, d, and / at which oscillations began wereobtained. When the data were nondimensionalized andplotted as a nondimensional width (bm-m/d0) (Ued0/v)'/2against the nondimensional depth d/50, all the experimentalresults fell on a single curve. Figure 2 shows the results of theexperiments. No cavity oscillations occur below these points.It is clear from these results that the depth has little effect onthe nondimensional width (&min/60) ( U e b 0 / v ) 1 / J when depth

d/d0 > 2. There is a sharp increase in (bmin/50 ) (Red0 ) 1/2 when

To investigate the effect of any pressure gradient beforeseparation, the length of the ogive nose was varied by meansof two cylindrical extensions. Relatively thick boundarylayers, which were laminar at separation, were obtained. Theminimum width for a particular depth was determined againby changing edge velocity Ue. These results are also shown inFig. 2.

Nondimensional Frequency of Cavity OscillationsFor an axisymmetric model, the nondimensional frequency

fbl C/oo for low subsonic flows can be expressed as

fb =F'

/Ueo0 b a \x( — £ — , — '— , mean velocity profile at separation, etc. )

\ *> <50 00 /

Experimental results (shown later) indicate that non-dimensional frequency is independent of t/D and the mostimportant parameters defining cavity oscillations are Ue and60 instead of £/«,, D, and t Therefore, one can write non-dimensional frequency /&/ Ue as a function

-F~

x( — ̂ - , - , — , - , mean velocity profile at separation, etc. )\ v D - D D /To study the dependence of nondimensional frequency on

Reynolds number Red0 = Ued0/v, depth d/50 and width b/5 0 ,each of them was varied separately, keeping others constant.

Effect of WidthFigure 3 shows the effect of width on cavity oscillations at

Reynolds number Red0 =2.86x 103 and depth d/d0 = 10where nondimensional frequency fb/Ue is plotted againstnondimensional width b/d0. No cavity oscillations occurredbelow &/60 = 5.25, which represents the minimum width (b/60 ) min . First mode fluctuations occurred at a nondimensionalfrequency of about 0.6. There was a slow .ncrease in non-dimensional frequency as b/d0 increased. A? the critical valueof &/60 = 8.15 was reached, oscillations jumped to a highermode. At this width, two modes occurred alternately. The twomodes never occurred simultaneously. Under certain flowconditions, switching between the two modes was audible.The second mode of cavity oscillations occurred at a non-dimensional frequency of approximately 0.95. As the widthb/d0 was increased slightly beyond the critical value of 8.15,the first mode disappeared and the flow began to oscillate inthe second mode of cavity oscillation. As the width was

2nd MODE

1st MODE

1.0

0.5

Ol,

Fig. 3 Effect of width on nondimensional frequency at Red0= 2.86xl03 and cf/6. = 10.

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JULY 1977 EXPERIMENTAL FLOWS OVER SHALLOW CAVITIES 987

10 12 14 16 18 20

momentum thickness 0, defined as

U

D® i.o\2 0.8

0.6

0.4

Fig. 4 Effect of depth on nondimensional frequency at Reb0= 0.96 X 103 and b/b0 = 11.00.

decreased, the cavity oscillations jumped back to the firstmode where the width b/d0 was 8.15, and no significanthysteresis region was noticed. Beyond b/50 > 10, the flow overthe cavity became irregular and the periodic velocity fluc-tuations ceased to exist or were weaker than the turbulencefluctuations in the shear layer.

Effect of DepthFigure 4 represents the effect of depth d/d0 on non-

dimensional frequency fb/Ue at a fixed Reynolds numberRed0 = 0.96 x 103 and width b/60 = 11.0. The minimum widthbmm/&o f°r tne onset °f cavity oscillations was 9.40. Thecavity began to oscillate in the second mode of oscillation witha nondimensional frequency fb/Ue of about 0.80. The resultsin Fig. 4 show that the nondimensional frequency was in-dependent of depth d/d0 > 6. In cavity flows with a depth d/50less than 5, nondimensional frequency dropped slowly asdepth was decreased. At d/d0 = 1.34 oscillations disappeared.Therefore, somewhere between 1.34<d/50 < 2.87 value of (dl<50)min below which no cavity oscillations occur. Due to thelimitations of the model, this minimum depth could not bedetermined precisely.

Cavity Oscillations at Different Reynolds NumbersFigure 5 shows the results of the effect of Reynolds number

Red0 on nondimensional frequency fbl Ue. During this ex-periment b/d0 was kept constant. Since the nondimensionalfrequency is independent of depth for d/60'>69 depth d/50was not constant though care was taken not to decrease thedepth below the above limit. Experimental results for threeogive nose shapes with £/£> = 0.6, 1.12, and 2.12 are plotted inFig. 5. For any particular model, shear-layer thickness atseparation decreases as edge velocity increases. Therefore, tokeep the nondimensional width b/50 constant, the cavitywidth b was decreased accordingly when the edge velocity Uewas changed. Width b/50 was 12.76 throughout the ex-periment.

Within the range of Reynolds numbers Ued0/v tested(0.6xl03 to l .SxlO 3 ) , cavity oscillations occurred in thesecond mode of oscillation with a nondimensional frequencyin the range of 0.8-0,9. Data points for different finenessratios O7//}) fall on a single curve. It was stated in thebeginning of this section that nondimensional frequency fbl(4, is a function of U^t/D, blD, d' ID, etc. Since fb/Ue is in-dependent of depth and, for a fixed width b/d0, is also in-dependent of £/£>, it appears that the most appropriate par-ameters defining cavity oscillations are Ue and d0 instead ofU^f D, and L This reduces the important nondimensionalparameters of cavity oscillations by one. Hence one canconclude that nondimensional frequency fbl Ue is only a func-tion of Reynolds number Ueb0lv, depth d/d0, and width b/d0.

Mean Velocity Profile in Shear LayerThe transverse coordinate y (measured positive upward

from the edge of the cavity) is nondimensionalized by the

{ 00-.where Ue is the velocity at the edge of the shear layer. Incarrying out the above integration as one approaches insidethe cavity, a cut-off was made where the mean velocitybecame less than 5% of the edge velocity Ue. The accuracy ofmeasuring mean velocity by hot-wire anemometry in thevicinity of this region is very doubtful. Edge velocity Ue wasalmost constant along the cavity width. The mean velocityprofiles at various downstream locations in nondimensionalform are plotted in Fig. 6 against (y—yj/2)/B9 where y1/2corresponds to U/Ue = 0.5. The cavity was oscillating in thefirst mode of oscillation with/&/t/e = 0.67. The downstreamcorner was located at b/B0 = 60. The depth and the Reynoldsnumber at separation were d/60 = 100 and ReB0 - 2.42 x 102.

As is clear from Fig. 6, in the early stages of shear-layergrowth, the velocity profile changes from a boundary-layerprofile to a shear-layer profile. A similarity has beenestablished at x/60 — 15. No measurements of mean-velocityprofile could be made beyond x/B = 50 because of the presenceof the downstream corner. Velocity fluctuations (u' 2 ) 1/21 Ueas high as 0.15 were noticed in the shear layer close to thedownstream corner. Results of cavity flow visualization bysmoke injection, discussed later, show that these large velocityfluctuations are attributed to large lateral motion of the cavityshear layer close to the downstream corner.

To study the effect of cavity oscillations on the growth ofthe shear layer, mean-velocity profiles were measured forvarious widths b/00 when edge Reynolds number ReB0= 2.42x 102 and depth d/60 = 100 were kept constant. Fromthese results, the momentum thickness as defined above wascomputed at various downstream locations x/60. Figure 7indicates the growth of the shear layer B(x)/B0 as a functionof x/B 0. Results for four cavity widths b/B0 equal to 52.5, 60,85, and 105.2 were analyzed. These correspond to cavityconfigurations when oscillations just appeared, oscillations inthe first mode, cavity flow switching between the first andsecond modes of oscillation, and finally the cavity in itssecond mode of oscillation, respectively.

It is clear from these results that the growth of the shearlayer was almost linear with x/B0 in all modes of cavityoscillation. Growth rate dB/dx increased in magnitude withincreasing cavity widths, but no sudden jump in growth rateoccurred when cavity flow switched from one mode ofoscillation to another. The growth rate dB/dx, which is ameasure of rate of fluid mass entrained by the growing shearlayer from inside the cavity, can be studied as a function ofcavity width from results shown in Fig. 7. An entrainmentrate dB/dx as low as 0.006 was observed when the cavity beganto oscillate. This increased to a value as high as 0.022 for large

1.6

1.4

1.2

1.0

Q'.Q

0.6

0.4

J2/D-0.6• = 1.12

0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.41.5

Re * x 10"3

6oFig. 5 Effect of Reynolds number on nondimensional frequency atb/d0 = 12.76.

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0.8 -N

0.7 "i

,*o

SAROHIA

a *

l'**i* *i°'°^ o AD V £* FIRS

^ b/*(

AIAA JOURNAL

FIRST MODE OF OPERATIONb/0 = 60, d/0 = 100, Ren = 2.42 x 102

O O • {/O

ANDfb/U =0.67

* * .«&a ̂ ^

<V0

0 o o c °,°,!̂_ o a* x jew "*v ^X"*^ •^ ^ A A ou. o

0.4

0.3

0.2

0.1

- 3 - 2 - 1 0 1

2

f

o

A

V

X

a

2

o10

1520304050

3 4 5 6

Fig. 6 Nondimensional mean velocity profiles in shear layer at different downstream locations. Cavity oscillations in first mode of oscillation withfb/Ue = 0.67, b/00 = 60, d/90 = 100, and Re00 = 2.42 x 102.

cavity, widths when oscillations were in the second mode.Cavity oscillations are responsible for these large entrainmentrates in the laminar shear layer. It should be noted that theseentrainment rates are quite large for laminar shear layers andare comparable to the entrainment rate of the turbulentmixing layer, which is dB/d/x^0.035.20

Phase MeasurementsPhase measurements of the shear-layer velocity fluctuations

were made by cross-correlating the hot-wire signals at dif-ferent space locations. Typical constant phase lines of thepropagating disturbances relative to their phase at x=0 and y= 0 at Reynolds number Red0 = 2.S6x 103, width b/d0, anddepth d/d0 = 10 are shown in Fig. 8. The flow was oscillatingat a frequency/=625 Hz, corresponding to a nondimensionalfrequency/&/£/<, = 0.67. The numbers in Fig. 8 represent thephase \l//2ir (measured in terms of wavelength) at variousspace locations by which the propagating disturbances laggedbehind the upstream cavity corner.

It is clear from Fig. 8 that the phase at a fixed location x/50varies a great deal across the cavity flow. As one movestoward the cavity from outside (x/d0- constant), themeasurements show that the phase of the disturbancedecreases until one reaches the region across which a sharpdrop in phase occurs. As one moves further inside the cavity,the phase of the disturbance increases. Far outside the cavity,the phase shows a linear decrease with y/d0, whereas farinside the measurements show a linear increase of phase.

Phase measurements of propagating disturbances throughthe shear layer were obtained for various cavity widths. Fromthese measurements, the wavelengths of the disturbances werecalculated. For these cavity configurations, the oscillatingfrequency / was known, hence the phase speedUc = \f was computed. Present studies indicate that thedisturbance in the shear layer propagates at a constant phasevelocity, if one moves along the line where U/Ue = constant.From the mean velocity and phase measurements, the phasevelocity of the disturbances was computed as a function ofwidth b/60.

Results of the phase velocity Uc/Ue as a function of widthb/d0 when the cavity was oscillating in its first and secondmodes of oscillation are indicated in Fig. 9. These results

indicate that for a fixed depth d/d0 and Reynolds number Ued0/v> as width b/d0 was increased, the wavelength X increasedand the frequency of the disturbances dropped but theproduct J\=UC slowly increased in magnitude. This trendwas maintained until a critical value of b/d0 was reachedwhen the oscillation jumped to a higher mode. This caused asudden drop in wavelength X of the disturbances and a suddenincrease in the frequency of the oscillations. But the phasevelocity of the disturbance Uc = \f increased steadily as thewidth b/d0 was further increased without any discontinuity asthe oscillation switched modes.

10 20 30 40 50 60 70 80

Fig. 7 Effect of cavity width on shear layer growth at Ree= 2.42 xlO2 and d/0a = 100.

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JULY 1977 EXPERIMENTAL FLOWS OVER SHALLOW CAVITIES 989

-1.00 1 2 3 4 5 6

Fig. 8 Lines of constant phase in first mode of cavity oscillation atRe00 = 2.86 X 103, b/b0 = 6, d/d0 = 10, andy&/ Ue = 0.67.

0.6

o

Q

0.5-

0.4-

0.3h

0.2

FIRST MODE OF OSCILLATIONSECOND MODE OF OSCILLATION

7 8b/S

10 11

Fig. 9 Effect of width on propagation speed in the first and secondmodes of cavity oscillations at Red0 = 2.86 x 103 and d/b0 = 10.

0.7

0.6

0.5

0.4

0.3

0.2

0.1

5 6 7 8 9 1 0 1 1

Fig. 10 Effect of cavity width on wavelength of disturbance at Reb0= 2.86xl03 and d/bn = 10.

Figure 10 shows the ratio of \/b as a function of cavitywidth b/50 where Reynolds number Red0 =2.86x 103 anddepth d/d0 = 10 were fixed. Results for both first and secondmodes are shown. It should be noted that the wavelengths ofthe disturbances bear a nearly constant ratio to width b in anyparticular mode. This ratio jumps to a lower value as thecavity goes to a higher mode. This is an important ex-perimental result, indicating that the wavelengths of thepropagating disturbances have a definite integral relation tothe cavity width, for each mode of cavity oscillation.

0) ' (2)Fig. 11 Motion pictures of cavity oscillations at 500 frames/sec, /= 300 Hz, b = 0.7 in. and d=0.425 in.

Flow VisualizationFigure 1.1 shows two sets of four consecutive frames of

motion pictures taken on a l-in.-diam model. The cavity shearlayer was visualized by a continuous injection of white smokefrom inside the cavity. The flow is from left to right. Thefreestream velocity was 30 fps with cavity width Z? = 0.70 in.and depth d- 0.425 in. Due to the lack of a very intense steadylight source, one could only go as high as 500 frames/sec. Theoscillations occurred at a frequency/= 300 Hz. Results of themotion pictures are summarized below:

1) Strong interaction with the downstream corner occurredwhen the cavity oscillations began at b = bm[n.

2) When the cavity was oscillating, the mean streakline didnot oscillate much until it was very close to the downstreamcorner. Strong oscillations of the shear layer occurred in thevicinity of the downstream corner.

3) At the downstream corner, the mean streaklineoscillated in and out at cavity oscillation frequency. As thestreakline entered the cavity, the shear layer rolled up into avortex which was shed as it deflected out of the cavity. Figure11 a(i) shows the most inward position of the shear layer witha vortex, shed a little earlier, at the downstream corner. InFig. 11 a(ii), the shear layer is in its most outward positionand is ready to shed a vortex. The process continues at thefrequency of cavity oscillations. Figure l ib shows a similarsequence of cavity oscillations.

Some Miscellaneous Observations About Cavity Flow1) To measure the frequency of cavity oscillations, the

fluctuating component of the mean velocity was analyzed. On

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surveying the shear layer with the hot-wire probe, it wasfound that the u' fluctuations were almost sinusoidal in theshear layer except close to the downstream corner. In thisregion, the wave form of the u' fluctuations was quitecomplex. Due to rolling up of vortices in the vicinity of thedownstream corner, the nonlinear effects becamepredominant. Higher harmonics of the fundamentalfrequency of cavity oscillations were noticed because ofsuperimposed nonlinear velocity fluctuations. These higherharmonics should not be confused with the higher modes ofthe cavity oscillations.

2) Experiments were performed to investigate thesignificance of the details of the flowfield inside the cavity.The volume of the cavity was modified by inserting a thin diskinside the cavity and by adding a lip at the downstream of thecavity. These results indicate that the cavity volume does notaffect the phenomenon of oscillations in flows over shallowcavitites. Therefore, the cavity oscillations based on the in-teraction between the shear-layer deflection and internalpressures is inadequate to explain the mechanism ofoscillations in cavity flows. For more details of these ex-periments, the reader is advised to refer to the author'sthesis.21

IV. Conclusion and Discussion

Minimum Cavity WidthThe experimental results for laminar flow over a cavity

indicate that the nondimensional width (bmln/d0) (Red0) l/2 isindependent of depth d/50 for d/50>2. For d/d0>2, thevalue of the width (bmin/d0) (Red0) t/2 below which no cavityoscillations occur is 0.29xl03 . This result can be used indesigning nonoscillating cavities with laminar separation. Itshould be noted that for a fixed edge velocity Ue andkinematic viscosity *>, minimum width bmin ~ ( d 0 ) l/2. Fur-thermore, for laminar boundary layers without a pressuregradient, the edge velocity Ue and boundary-layer thickness atseparation (for a fixed forebody shape) are related as d0 ~ I/( U e ) t/2. Then the minimum width bm[n - d0

3/2. It is concludedthat increasing the shear-layer thickness at the upstreamcavity corner tends to delay the onset of cavity oscillations.

There is a sharp increase in the nondimensional minimumwidth (bmin/d0) (Red0)I/2 for d/d0<2, below which nooscillations occur. Present investigation shows that the effectof decreasing cavity depth d/50 is to stabilize the laminarshear flow of the cavity. It is concluded that one requireslonger cavities for the onset of cavity oscillations for cavitieswith a smaller depth d/d0, compared to those with a largedepth. This effect is very pronounced in very shallow cavities.In these, a strong lateral constraint on the cavity shear flowmay avert the growth of three-dimensional disturbances thatcontribute to the transition from a laminar flow to a turbulentone.22

Oscillation FrequencyThe overall features of cavity oscillations are given by the

effect of the width b/50, the depth d/d0 and the Reynoldsnumber Re50 = Ued0/v, on the nondimensional frequencyfb/Ue. East2 studied oscillations in rectangular cavities with aturbulent boundary-layer separation. His results fall into twobands of frequencies fb/Ue of about 0.3 to 0.4 and othersbetween 0.6 to 0.9 with a few results around 1.3. Similarbands of frequencies are noted in edge-tone generation.17

Present results fall into three bands of frequencies of 0.5 to0.6, 0.8 to 0.95, and 1.3 to 1.5. It is concluded that one gets alower nondimensional frequency for a given mode ofoscillation for a turbulent boundary-layer separation,compared to one with a laminar boundary-layer separation.

The present study shows that wavelength X of thedisturbances bears an approximate integral relation with thewidth b = \(N+ 1/2) in any particular mode of oscillation.Therefore, the nondimensional frequency can be written as

fb/Ue = (Uc/Ue) (AH-1/2). Thus, an increase in the pro-pagation speed of disturbances, with an increase of width (cf.Fig. 9) for a fixed Reynolds number, results in an increase innondimensional frequency fb/Ue, in each mode of cavityoscillation.

Free Shear-Layer RegionsOn the basis of flow visualization and hot-wire

measurements of laminar cavity shear flows, one can dividecavity shear flows into the following main regions:

1) Close to the upstream cavity corner, flow transformationfrom a boundary layer profile to a shear layer profile occurs.Present studies indicate that this region extends as far as 10 to15 momentum thicknesses 00 downstream from the point ofseparation.

2) The second region occupies the greater part of the cavityflow. Here pure sinusoidal velocity fluctuations of frequency/of cavity oscillations occur. This disturbance, at frequency/,propagates at a constant phase speed.

3) In this region, which lies very close to the downstreamcavity corner, the shear layer deflects in and out of the cavityat the frequency / of cavity oscillations. This gross lateralmotion of the shear layer causes large velocity fluctuationsand results in a periodic shedding of vortices at a frequency/from the downstream corner.

Mean and fluctuating velocity measurements further showthat the effect of the downstream cavity corner on cavityshear flow is to postpone the separated laminar layer toturbulence. The experiments show that the cavity shear layerremains laminar until a maximum width bmax is reached. Atbmax, the periodic signal could not be measured due to in-creased turbulence over a range of Reynolds number atseparation Red0 = 5xlQ2 to 2x l0 3 . A maximum widthbmax/60 > 100 was observed. This width is nearly twice as largeas the distance of transition from separation of laminar shearlayer.23 The stabilizing effect may be attributed to these largeself-sustained oscillations induced in the cavity shear layer bythe presence of the downstream corner. Presence of theseoscillations seems to delay the rolling up of the laminar shearlayer into vortices.

The experiments show that the presence of cavityoscillations in the flow induces a large increase in the shear-layer growth rate. These large growth rates may be caused byincreased "Reynolds stress" u*v' due to the presence of large "amplitude oscillations in the cavity shear layer.

Summary and Conclusions

1) The phenomenon of oscillations in low-speed flows overcavities is not an acoustic resonance phenomenon in thelongitudinal direction. These oscillations result frompropagating disturbances which get amplified along the cavityshear layer.

2) The present experiments show that the onset of cavityoscillations is accompanied by a large lateral motion of thecavity shear layer close to the downstream corner. This alsoresults in periodic shedding of the vortices from the down-stream corner at the frequency of cavity oscillations.

3) It is further observed that the transition of the laminarcavity shear flow to turbulence is postponed by the presenceof large amplitude oscillations in cavity flow until a maximumwidth bmax is reached. No rolling up of the laminar cavityshear layer into vortices occurs for cavity widths as large asb/60>\QQ. The transition phenomenon which occurs forwidths b>bmax is quite complex and needs further ex-perimental investigation.

4) It is observed that the effect of depth for d/50 - O( 1) isto delay the transition of the free laminar shear-layer flow to aturbulent one.

5) The presence of strong cavity oscillations contributes toa large growth of the shear layer. Growth rates d6/dx—0.022in laminar cavity flows are noticed.

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6) The results show that for shallow cavities, an ap-proximate integral relation between the wavelength of thepropagating disturbance X of the cavity oscillations and widthb given by b/\— (n+ 1/2) exists, where n can be 0, 1, 2, 3, ...etc., depending upon the mode of cavity oscillation.

7) Cavity volume does not affect the phenomenon ofoscillations in flows over shallow cavities.

AcknowledgmentsThe financial support provided by the U.S. Army Research

Office through Contract No. DAHC-04-72-C-0028, whichmade these experiments possible, is gratefully acknowledged.Gratitude is extended to Professors T. Kubota and A. Roshkofor their valuable guidance throughout this work. The authoralso gives his sincere appreciation to Dr. W. Behrens for hishelpful suggestions.

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