Flow structure and modeling issues in the closure region of attached cavitation · 2002-09-06 · Flow structure and modeling issues in the closure region of attached cavitation Shridhar

PHYSICS OF FLUIDS VOLUME 12, NUMBER 4 APRIL 2000

Flow structure and modeling issues in the closure regionof attached cavitation

Shridhar Gopalan and Joseph KatzDepartment of Mechanical Engineering, The Johns Hopkins University, Baltimore, Maryland 21218

~Received 20 May 1999; accepted 13 December 1999!

Particle image velocimetry~PIV! and high-speed photography are used to measure the flowstructure at the closure region and downstream of sheet cavitation. The experiments are performedin a water tunnel of cross section 6.3535.08 cm2 whose test area contains transparent nozzles witha prescribed pressure distribution. This study presents data on instantaneous and averaged velocity,vorticity and turbulence when the ambient pressure is reduced slightly below the cavitationinception level. The results demonstrate that the collapse of the vapor cavities in the closure regionis the primary mechanism of vorticity production. When the cavity is thin there is no reverse flowdownstream and below the cavitation, i.e., a reentrant flow does not occur. Instead, the cavitiescollapse as the vapor condenses, creating in the process hairpin-like vortices with microscopicbubbles in their cores. These hairpin vortices, some of which have sizes as much as three times theheight of the stable cavity, dominate the flow downstream of the cavitating region. The averagedvelocity distributions show that the unsteady collapse of the cavities in the closure region involvessubstantial increase in turbulence, momentum, and displacement thickness. Two series of testsperformed at the same velocity and pressure, i.e., at the same hydrodynamic conditions, but atdifferent water temperatures, 35 °C and 45 °C, show the effect of small changes in the cavitationindex ~s54.69 vs.s54.41!. This small decrease causes only a slight increase in the size of thecavity, but has a significant impact on the turbulence level and momentum deficit in the boundarylayer downstream. Ensemble averaging of the measured instantaneous velocity distributions is usedfor estimating the liquid void fraction, average velocities, Reynolds stresses, turbulent kineticenergy and pressure distributions. The results are used to examine the mass and momentum balancedownstream of the cavitating region. It is shown that in dealing with the ensemble-averaged flow inthe closure region of attached cavitation, one should account for the sharp~but still finite! gradientsin the liquid void fraction. The 2-D continuity equation can only be satisfied when the gradients invoid fraction are included in the analysis. Using the momentum equation it is possible to estimatethe magnitude of the ‘‘interaction term,’’ i.e., the impact of the vapor phase on the liquidmomentum. It is demonstrated that, at least for the present test conditions, the interaction term canbe estimated as the local pressure multiplied by the gradient in void fraction. ©2000 AmericanInstitute of Physics.@S1070-6631~00!00804-7#

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I. INTRODUCTION

Sheet cavitation appears on lifting surfaces as the psure on parts of the body, typically near the leading edgereduced below the vapor pressure. At early phases, whencavitation index is just below the inception level, the cavis typically thin and has a glossy leading edge with eitheblunt front or a series of sharp thin ‘‘fingers.’’ Their occurence depends on the surface roughness~a discussion fol-lows!. The liquid–vapor interface becomes wavy, unstaand eventually breaks up at the trailing edge~closure region!of the vapor sheet. The flow downstream of the cavitydominated by bubble clusters and contains, even in mcases, large-scale eddies~for example, Ref. 1! that are morepowerful ~i.e., strength and size are larger! than any bound-ary layer structure that exists prior to the cavitation. At tpresent time we have no substantiated quantitative or qutative explanation for these phenomena and their relatships to the body geometry, boundary layer structure, et

The development of attached cavitation from nuclei

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flows with laminar separation was demonstrated first byakeri and Acosta2 and later confirmed by Gates and Acost3

and Katz.4 They showed that ‘‘band type cavitation’’ occurred as free stream bubbles were entrained into the srated region through the reattachment zone, where they wpushed upstream by the reverse flow. Within the relatquiescent flow of the separated region the bubbles gslowly. These observations provided for the first time a con the process of sheet formation. However, they were apuzzling, since sheet cavitation occurs on surfaces withlaminar separation. Thus, the basic mechanism of sheetmation in attached flows remains unanswered.

If the cavitation nucleus is a free stream bubble, ittypically separated from the solid boundary by some liquThe attachment of this bubble to the surface and the menism that prevents the bubbles from being swept awaydemonstrated by Li and Ceccio,5 are unresolved issues. Thprocess must involve favorable conditions that may incluthe local pressure distribution, boundary layer thicknesuper-saturation~supply of gas! and local surface imperfec

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896 Phys. Fluids, Vol. 12, No. 4, April 2000 S. Gopalan and J. Katz

tion. The latter creates localized flow separation withassociated quiescent region, where bubbles may grow wout being swept away. Note also that the presence of a ctation bubble near the surface changes the local flow stture, and may cause local boundary layer separation.well established that the size and shape of surface roughaffect the conditions for inception.6 The origin of cavitationcan also be a surface nucleus, such as air pockets in cron the model.7 Acosta and Hamaguchi8 demonstrated theeffect of surface nuclei. They coated a model with a thin fiof silicon oil ~which dissolves large amounts of air!, anddemonstrated that, as long as the film was new, it hadnificant impact on the inception indices. Peterson9 showedsimilar effects by degassing the surface of a body.

Predictions of the length of the cavity have been mosbased on potential flow analysis and empirical data~Refs.10–14 and a summary by Wu15!. Recently, Laberteaux anCeccio16 compared measured results to theoretical pretions and showed major discrepancies. Of the past effortmodel the flow one should mention the singularity methoof Furness and Hutton17 and the bubble two-phase flowmodel developed by Kubotaet al.18 The latter followed ear-lier experiments~Ref. 1! that had demonstrated that the florolled up to large vortex structures as the cavity surfacecame unstable. Their model predicted the unsteady shedprocess in the closure region of the cavity. However,origin of the vorticity causing the rollup was still in thboundary layer upstream of the cavitation and the rollupcurred as the shear layer~interface! became unstable. As wilbe shown in this paper, although some vorticity is generain the boundary layer upstream, the dominant origin ofvorticity downstream of the cavitation is the collapse of caties in the closure region. It is also worthwhile to mentithat not all sheet cavities necessarily shed cloud cavitatSome cavities are ‘‘closed,’’ especially those formingthree-dimensional test objects.

As the pressure is reduced the cavitating region groand becomes increasingly unstable. Portions of the cavitybeing shed in larger sections and at lower frequenciesform large bubbly clouds. This shedding process involvsubstantial changes to the length of the attached sheet.development, motion and collapse of the cloud is the mdestructive form of cavitation. We have very little dataeven qualitative understanding of the breakdown of the ctation and the bubble size distribution within the cloud. Sface pressure fluctuations measured under the cloud aretremely high.19,20 The flow mechanism causing these higpressure fluctuations is also unclear, and the current moare based on the assumption that they are caused by swaves that develop within the bubble cloud~for example,Refs. 21–25!.

The shedding process of large sections of the cavitypuzzled researchers for quite a while. If potential flow anasis is used, there must be an unsteady reverse flow inclosure area and below the cavity. Thus, it has been argthat the shedding is caused by a ‘‘reentrant jet,’’ a reveflow that forms below the cavity and ‘‘pinches’’ it intermittently ~Refs. 1, 17, 18, 20, and 26–32 and many othe!.Velocity measurements performed by Kawanamiet al.20 pro-

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vided evidence of reverse flow with magnitude of the saorder as the free stream velocity. They also showedsmall fences installed on the wall, close to the point whthe sheet cavitation ended, prevented the shedding of ccavitation. The noise levels and drag force also decreaCallenaereet al.32 demonstrated the upstream motion of tcontact point of the cavity with the wall prior to cloud sheding. However, they mention that when the cavity is ththere can be a situation where adverse pressure gradientoo weak for a reentrant jet. This situation is consistent wthe present observations that at early stages of cavitathere is no reverse flow below or downstream of the attaccavitation.

The present paper describes part of our on-going efto resolve the flow structure around and downstream of shcavitation. We use PIV to map the flow~instantaneous andmean!, vorticity and turbulence as the cavitation indexreduced below the inception level. We provide evidence tthe collapse of the cavities is a primary source of vorticand that small change in the size of the cavity cause substial increase in the turbulence level and momentum thicknin the boundary layer downstream. We then address moing issues of the ensemble-averaged flow.

II. EXPERIMENTAL APPARATUSAND INSTRUMENTATION

A. Test facility

To provide detailed answers on the flow structure, itconvenient to perform the experiments in an apparatusallows careful observations on the flow structure and bubdistribution. Some of the measurements must be performat a very high magnification under controlled conditions thinclude boundary layer thickness and characteristics, lopressure gradients, surface roughness and properties, potion of nuclei, etc. However, the facility must also allooperations at relevant Reynolds numbers (.106). Theseconsiderations have led to the design of the present expmental apparatus. A schematic description of the test facis provided in Fig. 1 and the test section is described in mdetail in Fig. 2. The pumps are located 5 m below the testsection, reducing the likelihood of pump cavitation, and t1000 l tank is used for separating undesired free strebubbles. The settling chamber contains screens and hocombs to reduce the turbulence level~along with the 9:1contraction! and the vertical tank above is used for controling the pressure in the facility. The 6.3535.08 cm2 testsection has a minimum length of 41 cm and maximumtrance velocity of 13 m/sec. Thus, it enables generationboundary layers with Reynolds numbers~based on axial dis-tance! well into the 106 range. Windows on all sides andtransparent contoured nozzle~which is actually part of thebottom window! enable unobstructed observations fromdirections.

The contoured surfaces in the test section were desigusing a commercial CFD code~Fluent Inc.!. We chose pres-sure distributions that resemble a typical suction side of ling surfaces at incidence. This typical pattern includes agion with low speed~stagnation!, sharp decrease in pressu

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897Phys. Fluids, Vol. 12, No. 4, April 2000 Flow structure and modeling issues in closure region of attached . . .

to a minimum, recovery to a pressure lower than the ambpressure, a region of fairly constant low pressure and furincrease to the ambient conditions. The computational gwas made sufficiently fine that further refinement did naffect the results and we used renormalization group~RNG!for turbulence modeling. We assumed a free stream tulence level of 1%~in reality it is significantly lower—about0.1%! and a uniform inflow to the test section. A turbuleboundary layer was used since the beginning of the nowas located 10 in. from the origin of the test section. Slitsthe top and bottom walls of the tunnel, located upstreamthe nozzle, were used for boundary layer suction. This wused for determining the effect of the boundary layer oncavitation ~qualitative observations only!. In reality, theboundary layer turned out to be laminar. If we had to repthe design, we would use a laminar boundary layer. Systatic evaluation of a series of designs eventually led togeometries and pressure distributions shown in Fig. 3~theyare all fifth-order polynomials!. They have different pressurgradients and minimum levels, but similar recovery attrailing edge. Four nozzles identified as shapes 13, 14and 22 were manufactured using computer numerical con~CNC! at an accuracy of 0.03 mm. As noted before, t

FIG. 1. Schematic description of the experimental facility.

FIG. 2. Details of the test section.

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nozzle and the bottom window were machined as one unallow unobstructed visual access and to prevent inaccuraassociated with matching them. For one of these cases~shape13! we compared the computed velocity to measured dand that will be discussed shortly.

B. Measurement techniques

Planar velocity measurements are performed using PThe data acquisition and analysis procedures have beenveloped in our laboratory for several years.33–41 In thepresent experiments the light source is a dual head, 300pulse, Nd:YAG laser whose beam is expanded to a 1-mwide sheet. Images are recorded using a 204832048 pixels,2

4 frames/sec, digital camera manufactured by Silicon Motain Design. This camera has a custom, hardware-basedage shifter that allows us to record two images on the saframe with a prescribed, fixed displacement~image shift!between them!.42 This feature solves the directional ambigity problem. We use in-house-developed, correlation-basoftware for computing the velocity distributions from thimages. The vorticity is determined using a second-ordenite difference scheme. Calibrations and uncertainties ofprocedures are discussed in detail in Refs. 33, 39, andExtended discussion on uncertainty in PIV measuremecan be found also in Refs. 44 and 45. The uncertainty cankept at about 1% provided there are a sufficient numbeparticles per interrogation window~;4–5 according to ouranalysis! and the displacement between exposures exce20 pixels. In the PIV analysis of the present flow the sizean interrogation window is 1.35 mm and the distancetween vectors is 0.68 mm, i.e., 50% overlap between wdows.

As discussed by Sridhar and Katz,39,40 by using fluores-cent particles as velocity tracers we can easily distingubetween bubbles and particles within the illuminated plaBubbles reflect light, and as a result, maintain the origilaser color~green—532 nm!, whereas the particles fluorescat 560–570 nm. During liquid velocity measurements we ua filter that removes the green light, which eliminates mosthe bubble traces. When the bubble motion is measured

FIG. 3. Geometry of the nozzle surface and the corresponding presdistributions.


FIG. 4. ~a! Computed velocity field using Fluent.~b! Instantaneous measured velocity using PIV.~c! Computed velocity field downstream.~d! Instantaneousmeasured velocity field downstream.

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III. CAVITATION INCEPTION INDICES AND GENERALFEATURES

Although detailed validation of the computed pressudistributions is beyond the present scope, it is still usefucompare the measured minimum pressure to the compdata in order to verify the pressure field. The computedlocity distribution and sample instantaneous PIV data, bfor shape 13, are presented in Fig. 4. Near the minimpressure point the ratio of the~maximum! instantaneous velocity to the inlet velocity is 2.4, whereas this ratio for thcomputed velocity averaged over the area covered byinterrogation window is 2.23. Even this 7% differencecaused in part by the thicker~turbulent! computationalboundary layer, whereas the experimental boundary layelaminar. Consistent with the numerical results, boundlayer separation does not occur in the high adverse presgradients downstream of the minimum pressure point. Fther downstream, atx/L.0.58 @Fig. 4~d!, data shown fromx/L50.66], the experimental results show the effectboundary layer transition in a region with adverse pressgradients, with the typical rollup of large-scale eddies cloto the inflection point in the boundary layer mean velocprofile. The mean velocity in this region is still positiv

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namely there is no boundary layer separation. Note that tsition occurs also on the upper wall at almost the same alocation, but with a thinner layer. The numerical results, bing steady and with an initially turbulent boundary layer,not show these trends@Fig. 4~c!#. Since all the cavitationphenomena that we focus on occur atx/L,0.5, these flowphenomena associated with transition are beyond the sof the present paper. For the present purpose, the compaseems to indicate that the computed minimum pressurreliable.

The cavitation inception indices of sheet cavitation (s i)for the three nozzles are shown in Fig. 5. They are define

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wherePinlet is the pressure measured at the inlet to thesection,Pv is the vapor pressure,r is the density of theliquid and Vinlet is the velocity at the inlet. The inceptioindices are measured using visual observations by keethe velocity constant and gradually reducing the pressuretil cavitation appears. In the case of sheet cavitation, visobservation is straightforward. Each point is an averageseveral measurements that do not differ significantly. Incasess i is lower than the computed2Cpmin and there islittle dependence on velocity. However, the difference btweens i and2Cpmin increases with increasing pressure g

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dients near the minimum pressure point and can be as mas 10%. The fact thats i52Cp min does not hold for sheecavitation has been discussed in Refs. 46–48. Howeverfact that the differences inCp min ands i depend on the pressure gradients~induced by the geometry of the test object! isnew. Thus, the model geometry seems to have some~but notsubstantial! effect on the inception indices beyond the manitude of2Cp min . The reasons for these differences areclear, and may be related to the length of the region wpressure below the vapor pressure, which may affect bothavailability of surface nuclei and the duration of exposurefree stream nuclei~in the order of 100msec in the presenexperiments!. We performed tests with deaerated water~3–5ppm! and saturated water (;15 ppm! and could not see major impact on the inception indices of ‘‘sheet cavitation~which is not the case for bubble cavitation!. Consistent withprevious results on sheet cavitation,49 the dissolved air con-tent and free stream bubble distributions have insignificeffects on the inception indices. However, when the testcility contains a high concentration of ‘‘large’’ bubbles(.1 mm! they destroy the ‘‘stable,’’ upstream portion of thattached sheet, as reported before by Katz4 and Ceccio andBrennen.5

Sample photographs of the cavitation in nozzle 13presented in Fig. 6. Figures 6~a! and 6~b!, respectively, showthe same flow conditions before and after polishing the sface of the very same nozzle. The small roughness elemof the unpolished surface are sufficient for fixing the origof the cavitation on specific points near the minimum prsure point. Polishing the surface changes the shape ofleading edge of the cavitation to a broad and ‘‘blunt’’ pattewith a glossy leading edge. When the pressure is redufurther ~to s,3!, as shown in Fig. 6~c!, the glossy leadingedge becomes aligned along the same axial location.interface is still initially laminar~glossy! and is followed bya region with distinct, orderly, two-dimensional interfaci

FIG. 5. Measured cavitation inception indices at different inlet velocitcompared to computed minimum pressure coefficients.

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waves. These waves have been observed beforeBrennen50 on the surface of large-scale cavitation arouspheres. According to Brennen, this phenomenon is a reof organized, boundary layer instability at the interface, i.Tollmien–Schlichting waves. After a short distance theterface becomes increasingly unstable and three-dimensand rolls up into a series of bubbly eddies that are being sintermittently behind the attached sheet. The relationshiptween the instabilities on the surface close to the leadedge and the large-scale vapor-filled cavities/structudownstreamseemsto be a process of growth of interfaciainstabilities. The size and shedding frequency of these edon the same test object depend on the cavitation index.already discussed in many studies, at sufficiently low cavtion indices the shedding process may involve formationcloud cavitation~e.g., Refs. 1, 18, 20, 26, 28, and 32 aothers mentioned in the Introduction!. Different phases information of cloud cavitation in the present facility are illutrated in Fig. 6~d!.

In the present paper, however, we focus on the flstructure in the closure region when the cavitation indexhigh enough that massive cloud shedding does not ocTwo top views are shown in Figs. 6~b! and 6~e!. They bothdemonstrate clearly that the flow downstream of the closregion contains what appears to be hairpin-like structucontaining bubbles. The transition between the continusheet to these hairpin structures is demonstrated in Figs.~f!and 6~g! in a sequence of successive images~side view! re-corded at 1000 frames/sec using a high-speed digital cam~Kodak Ektapro EM Motion Analyzer, Model 1012!. Theseimages show the evolution of the same cavity~indicated byarrows! as it progresses downstream. The initia‘‘rounded’’ mostly vapor cavity shrinks as the vapor codenses due to the local increase in pressure, tilts forw~presumably due to differences in magnitudes of the lovelocities! and becomes a slender hairpin vortex with a bubly core. This vortex is considerably taller than the vapcavity from which it evolves and, in some cases, it reacheheight of as much as three times the height of the cavitaupstream~height of stable sheet cavity'5 mm; refer toTable I!. Examinations of the high-speed movies show athat the process involves an increase in the angular veloi.e., as the cavity shrinks, the bubbles also start spinninincreasing speeds. Measured vorticity will be introduclater. The process repeats itself and, in fact, Fig. 6~f! alsoshows in the lowest frame a thicker cavity in the processrolling up in addition to the thin hairpin vortices. In Tablewe present some quantitative information obtained fromframes.

IV. FLOW STRUCTURE

A. Instantaneous flow

All the data presented in this section have been obtaiwith shape 13. Sample velocity and vorticity distributionsdifferent sections along the nozzle are presented in F7~a!–7~h!. As the pressure is reduced below the inceptlevel (s54.69 vss i56), there are no signs of new vorticitalong the stable glossy region of the interface. The vel

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FIG. 6. Appearance of the leading edge from the top during low levels of cavitation and when the surface is~a! ‘‘rough’’ and ~b! ‘‘smooth.’’ Also, note thehairpin like structures in~b!, where the cavity collapses.~c! The leading edge of the sheet cavity~top view! during advanced stages of cavitation. Glasfinger-like structures are visible, which immediately become wavy.~d! Photographs~side view! showing different stages of the unsteady~advanced! cavitationphenomenon.~e! The trailing edge of the sheet cavity, as seen from the top. At least one hairpin~or horseshoe! -like structure with a bubbly core is clearlyvisible. ~f! and~g! High-speed series of a side view. Time interval between frames is 1 msec. Note the slender structure generated after the cavityFlow is from right to left.

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ity is shown in Fig. 7~a! and the vorticity is not presentedsince besides two very small peaks near the interfacex/L.0.25 the map does not show any other phenomen~itcan be found in Ref. 51!. Note that the PIV interrogationwindow size, 1.36 mm, and the distance between windo0.68 mm, do not resolve the laminar boundary layer

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stream of the cavitation. Signs of high vorticity peaks stappearing as the interface becomes wavy@velocity and vor-ticity in Figs. 7~b! and 7~c!, respectively#, but become quitepronounced only at the trailing edge of the [email protected]~d!–7~g!#, i.e., atx/L50.35. This region is characterized bunsteady, intermittent patches of cavities at various stage


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collapse as demonstrated by the three characteristic examof the instantaneous flow. In one of the samples@Fig. 7~e!#,one can even identify two inclined cavities that resembleshapes of the vapor cavities seen in the high-speed phgraphs as hairpin vortices are being formed. In all casescollapse of the cavities involves formation of high vorticipeaks of both signs, but mostly negative. Thus, the collaof the vapor cavities involves substantial vorticity productiin the closure region of the sheet cavity. The large eddgenerated in this region are convected downstream withevidence of additional significant level of vorticity prodution. In Fig. 7~h! that covers the region upstream of the fuwetted boundary layer transition@see Fig. 4~d!#. The eddiesreach elevation of almost three times the height of the shcavity—consistent with the photographs shown in Figs. 6~f!and 6~g!.

TABLE I. Data from high speed frames@Figs. 6~f! and 6~g!#.

Height of cavity/structure~mm!

Convection speed ofcavity/structure~m/sec!

Frame No. Figure 6~f! Figure 6~g! Figure 6~f! Figure 6~g!

1 8.6 82 9 10.3 8.33 10.4 10.9 10.6 10.64 13.2 11.8 12 10.65 14.3 12.9 13 11

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To summarize the process, the PIV data and the phgraphs show that as the water vapor interface becomes wand condensation starts, portions of the cavitation becodetached. These odd-shaped, mostly vapor, cavities arevected briefly downstream but condense quickly~withinabout 2 msec!, creating in the process powerful vortices. Athe sample velocity distributions in the closure region deonstrate@Figs. 7~a!, 7~b!, and 7~d!#, under the present flowconditions, i.e., just below inception with cavity heightabout 3 mm, there is no reverse flow even very close towall at the trailing edge of the cavitation. This conclusionbased on careful examination of all the PIV images offlow in the closure region~a total of 120 instantaneous vectomaps, as will be discussed shortly!. To confirm this conclu-sion we have also checked the traces of individual particvery close to the wall in a few images to examine wheththere is reverse flow at scales smaller than the interrogawindow. We have seen no evidence of reverse flow.There isno reentrant flow under the present flow conditions. The‘‘large’’ vapor cavities simply shrink as the vapor condensand the horizontal velocity around them remains positiThis observation is not consistent with the typical reentrflow model for the closure region of attached cavitation.agrees, however, with a comment made in a recent papeCallenaereet al.32 that when the cavities are thin, the adverpressure gradients may be too weak for creating a reenjet. Note also that we do not claim here that there isreverse flow under more developed states of cavitation

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FIG. 7. ~a! Sample instantaneous velocity map at the leading edge of thcavity for s54.69. ~b! and ~c! showvelocity and vorticity slightly down-stream.~d!–~h! all show instantaneousvorticity at various axial locations. In-crement in contour lines is 500 l/secDotted lines represent negative vorticity, i.e., clockwise rotation.

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FIG. 8. Running average of~a! horizontal velocity,~b! vertical velocity, and~c! and~d! normalized stresses at random sample points to illustrate convergof data. Turbulence level is maximum atx/L50.31 and significantly drops atx/L50.34.

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fact, intermittent reverse flow occurs under the giant bubvortices that are being shed behind the closure region, ein the present facility, but only during advanced stagescloud cavitation~much lowers!. From observations so farwe say that an adverse pressure gradient is a necessarydition for the formation of a reentrant flow. The questioremains as to the magnitude of the adverse pressure grathat would cause the reentrant flow. There can be twosons for an adverse pressure gradient—~a! induced by thegeometry of test object and~b! caused by the cavitation@pressure downstream of cavitation is higher thanPv , Fig.19~a!#. In our experiments in the region 0.2,x/L,0.3, ~a! isvery mild and ~b! is clearly dominant. If the cavitation isextended~by lowering s! to x/L50.4, we see reverse flowbecause of the strong geometry-induced adverse presgradient~Fig. 3! in addition to the one already caused by tcavitation. We have not made any systematic study of thtrends. However, increased likelihood of reverse flow un

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the cavitation with increasing adverse pressure gradiealong the wall prior to the onset of cavitation is consistewith the observations of Callenaereet al.32 This advancedlevel of cavitation is not in the scope of the present stud

B. Mean flow and void fraction distributions

Two data series, both at the same velocity and pressbut at different water temperatures, 35 °C and 45 °C, hbeen recorded. They provide essentially the same hydronamic conditions withVinlet55.2 m/sec~the effect on viscos-ity is negligible!, but the resulting cavitation indices, 4.6and 4.41, respectively, vary slightly due to the different vpor pressure. This slight change has a strong impact onextent of cavitation, flow structure and turbulence. Trendscavity length as a function of the cavitation indexs ~usuallyscales as 1/s) are examined in Ref. 16. We analyzed 7images ats54.69 and 48 PIV images ats54.41. Running

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averages of velocity components and Reynolds stresss54.69 are shown in Fig. 8. The data show that convergeis reasonably achieved. Due to the presence of bubbles athe walls, each image had to be examined carefully tomove the bubble traces before computing the velocity disbutions~although the laser wavelength was filtered, theresome secondary reflection of fluorescent light from the sface of the bubbles!. Otherwise, the bubbles contaminatthe liquid velocity measurements. As discussed before,length of the cavities varied from one image to the next aodd-shaped detached patches appeared frequently. Stwas evident that the height and length of the cavitationcrease substantially with the small change ins.

When one attempts to determine the ensemble aveof the flow in the closure region, the averaging proceshould take into account the existence of two phases wnonuniform spatial distributions and the phase change.ensemble-averaged continuity equation for the liquid is

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]xi5FDavr v

DtG , ~2!

where over-bar indicates ensemble average, the subscrland v refer to liquid and vapor, respectively, anda i is thevoid fraction, i.e. the fraction of observations~images! forwhich a certain interrogation window contains the phasi.Note thata l51 corresponds toav50. A ‘‘phasic indicator’’was used on each image and vapor/voids were given a vof 0, and liquid were given a value of 1. Then an ensemaverage was obtained~from 72 images fors54.69 and 48for s54.41; thus, data ats54.69 is more reliable!, providingthe distribution ofa l . In principle it is possible that ‘‘holes’’could show up on PIV images from bubbles that are nothe light sheet plane but between the sheet and camera.is not the case in the present data since the cavities are scompared to the size of the camera lens that is located qclose to the light sheet. Thus, out-of-plane cavities doblock details in the light sheet plane. Equation~2! brings intoaccount that the only source of mass for the liquid is cdensation of the vapor phase. Assuming that the mean flosteady and that the liquid density is constant,

]~a lui !

]xi52

]~av ~rv /r l !uv i !

]xi. ~3!

Due to the substantial differences in density, the source ton the right-hand side is negligible and, assuming thataveraged flow is two-dimensional, the continuity equatbecomes

]~a l ul !

]x1

]~a l v l !

]y50. ~4!

Distributions ofa l , ul anda lul for s54.69,ul anda lul fors54.41,v l anda lv l for s54.69 andv l anda lv l for s54.41are presented in Figs. 9–13, respectively. As Fig. 9 showboth casesa l gradually increases from zero to one, but tdimensions of the cavitating region increase substantiwith the slight decrease ins. This trend is evident for all thevalues of void fraction, but especially for thea l50.1 and 0.2

orceng-

i-sr-

ed, it-

gesthe

ts

uee

nhisallitet

-is

me

n

in

ly

lines. Also, the vertical and horizontal gradients ina l in-crease with decreasings. Thus, while attempting to modethe void fraction distributions, the expressions that comemind are in the form of

a l~s2s i ,Cp!512expF2KxS x2xref

hcavD nx

2KyS y2y0

hcavD nyG ,

~5!

where Ki and ni are coefficients,hcav is the height of thecavity,xref is a reference location~such as the location of themaximum height of the cavitation! andy0 is the wall eleva-tion. All these coefficients are functions of (s2s i) and thepressure coefficient in the closure region. In the presentper we present only examples of the effect ofs2s i , butother studies, such as Ref. 32, demonstrate also the subtial effect of the pressure gradients on the flow structurecavitation in the closure region~including the formation of areentrant jet!. In the present flow conditions]Cp /]x;0.xref can be chosen as the location of the transition betwstable to intermittent shedding/collapse of the cavitation. Bsides being a function ofs2s i and pressure gradients, sincthe interface stability is affected by the Reynolds numbe50

a l is probably also a function of the Reynolds number. Sinwe did not determine the distribution ofa l for many condi-tions, we cannot report on trends. However, for the presdata at s54.69, nx52.07,ny52.6, Kx50.027 and Kx

51.24. This distribution is shown in Fig. 14. The relativrms difference between this curve and the experimental d@Fig. 9~a!# in the region between the wall andy/L50.14,0.215,x/L,0.36, is 4%.

Another feature of the closure region can be observethe distributions of vertical velocity~Figs. 12 and 13!. Abovethe immediate vicinity of the wall~1–2 mm! and all aroundthe cavity in the closure region, the vertical velocity is negtive, a trend associated with the flow surrounding the cav

FIG. 9. Measured distributions of void fraction (a l) for ~a! s54.69 and~b!s54.41. Increment in contour lines is 0.1.

vem

tn

15ehewol

h

.

-ibledi-s of

viz.

bu-2,

ed

meents,


tion. For both the cavitation indicesv l anda lv l are positivevery close to the wall. Examination of the instantaneouslocity distributions in this region shows that this phenoenon is caused by the collapse of the vapor cavities andgeneration of large vortices. Sample close-up velocity avorticity maps demonstrating this are presented in Fig.They show that the vertical velocity is positive on both sidof the collapsing vapor cavities. The upward motion of tlower cavity wall contributes to this positive motion belothe cavity. The vortical motion that develops as a resultthe cavity collapse@Fig. 15~b!# causes the positive verticavelocity above the cavity.

V. MODELING ISSUES

A. Mass balance

Due to the sharp gradients ina l , as expected, the higgradients in vertical velocity (] v l /]y,0, Fig. 12! are notbalanced by]ul /]x.0 ~Fig. 10! of the same magnitude

FIG. 10. Distributions of~a! u and ~b! a l u for the cases54.69. Hatchedzone represents the region where points have less than 20 measurei.e., liquid velocity.

--hed.

s

f

Clearly, the average liquid velocity distributions, by themselves, do not satisfy the two-dimensional incompressflow continuity equation. In fact, in some cases both graents have the same sign. On the other hand, the value]a lv l /]y,0 are matched by the proper values of]a lul /]x.0, satisfying Eq.~4!. To illustrate this conclusion, Fig. 16shows the distribution ofh andha , defined as

h5~]ul /]x1]v l /]y!2

~]ul /]x!21~] v l /]y!2;

~6!

ha5~]a l ul /]x1]a lv l /]y!2

~]a l ul /]x!21~]a l v l /]y!2 .

These distributions are obtained in the two-phase region,0.23,x/L,0.3, 0.118,y/L,0.14, for s54.69, hence ob-tained from 72 instantaneous realizations. Note that distritions of ul , v l , a lul , a lv l are presented in Figs. 10 and 1respectively. We did not computeh andha for s54.41. Thequantityh equals 0 when the continuity equation is satisfi

nts,

FIG. 11. Distributions of~a! u and ~b! a l u for the cases54.41. Hatchedzone represents the region where points have less than 20 measuremi.e., liquid velocity.

th

ttha

0.a

liq


and is larger than 1 when both velocity gradients havesame sign. The average value ofh is 1 whenu and v arerandom, unrelated numbers.52 The results clearly show thaintroducing the void fraction causes a substantial drop innumber of cases with values equal to or greater than 1,an increase in the number of cases with values less thanClearly, the closure region must be treated as a two-phflow.

B. Reynolds stresses, momentum balance andinteraction terms

The Reynolds averaged momentum equations for theuid phase, neglecting the viscous terms, can be written~from Ref. 53!

]~a l ul ul !

]x1

]~a l ul v l !

]y52

1

r

]a l p

]x2

]~a lul8ul8!

]x

2]~a lul8v l8!

]y1Mx , ~7!

FIG. 12. Distributions of~a! v and ~b! a l v for the cases54.69. Dashedlines represent negative quantity.

e

end25.se

-as

]~a l ul v l !

]x1

]~a l v l v l !

]y52

1

r

]a l p

]y2

]~a lul8v l8!

]x

2]~a lv l8v l8!

]y1M y2g, ~8!

FIG. 13. Distributions of~a! v and ~b! a l v for the cases54.41. Dashedlines represent negative quantity.

FIG. 14. Empirical distribution of the void fraction fors54.69, using Eq.~5!. The coefficients arenx52.07,ny52.6, Kx50.027andKy51.24.

oro

m

fues

ns17a-

e

rentns.ity

ple,

r-

e


whereMi is the interaction term, i.e., the effect of the vapphase on the momentum of the liquid phase. Distributionsnormal stressesa lul8ul8, a lv l8v l8 and shear stressa lul8v l8 andthe estimated turbulent kinetic energya lk* 50.75(a lul8ul81a lv l8v l8) ~it is based on the assumption that the third coponent is the average of the two measured components! fors54.69 are presented in Fig. 17~note the multiplicative con-stants!. Although the number of samples fors54.41 issmaller~48 as compared to 72 fors54.69!, we present theturbulent kinetic energy~Fig. 18! to demonstrate the effect oa lowers. As seen in the figures, along the wall the turblence level increases considerably with the slight decreass. The peaks ina lk* more than double and their location

FIG. 15. Sample close-up of~a! velocity and ~b! vorticity during cavitycollapse. First contour level is6500 l/sec~dashed lines represent negativvorticity! and increments are of 500 l/sec.

FIG. 16. Distributions ofh andha @Eq. ~6!#, illustrating the effect of voidfraction on the continuity equation.

f

-

-in

shift downstream. Note that the highest velocity fluctuatioactually occur upstream of the regions indicated in Figs.and 18, but their impacts are diminished by the multipliction with the void fraction~Fig. 9!. Interestingly, above thecavitation, namely atx/L,0.27 and away from the wall, theeffect of s is relatively small, i.e., in both casesa lk* havesimilar magnitudes. Also the data ats54.69 show thata lul8u8 l and a lv l8v8 l have comparable magnitudes. Thpeaks in2a lul8v l8 are about 25%–40% of thea lk* peaks, aratio that characterizes separated flows with large cohestructures, consistent with the instantaneous distributioNote that these levels are significantly higher than velocfluctuations in turbulent boundary layers,54 but are compa-rable to peak levels within a separated region, for exambehind backward-facing steps.55

It is also important to determine the effect of the inte

FIG. 17. Distributions of~a! and ~b! normal Reynolds stresses,~c! shearstress, and~d! turbulent kinetic energy fors54.69.

ethtionnothi.

umdc

fe

e

dfa

seid-ero

e

a

aplu

ri-

reac

w

ms.es,A-

umre-

ents

uree.t

dthe


action terms. Since we cannot determine the magnitudMi directly, one needs to evaluate the rest of the terms inmomentum equation and determine whether the interacterms contribute substantially to the overall liquid mometum. A troublesome term is the pressure gradient. It isclear to us that the streamwise pressure distribution inclosure region satisfies the boundary layer assumption,that the pressure is predominantly a function ofx only. Topartially answer this question, one can look at the pressdistribution above the cavitation, where the turbulent terare negligible anda l51. Consequently, the horizontal anvertical pressure gradients can be determined from the aceration terms only. Pressure gradient distributions at difent elevations, estimated from the values of2(ul]ul /]x

1 v l]ul /]y) and 2(ul] v l /]x1 v l] v l /]y)2g, are pre-sented in Fig. 19. Here the pressure is scaled withU2/L,whereU510.5 m/sec is the velocity in the undisturbed rgion above the cavitation (U2/L5441 m/sec2 or 45 g!. It isclear that]p/]x varies significantly with y, as one woulexpect, considering that the pressure near the cavity intershould be close to vapor pressure. Thus, thebestthat we cando with the present data is to choose the pressure as clopossible to the cavitation interface, where the liquid vofraction is still one, i.e.,y/L50.14. Note that near the closure region of the cavitation there are substantial advpressure gradients that do not exist in the noncavitating fl~compare to Fig. 3!.

Interestingly, since we measure the pressure at thetrance to the test section~to determines!, where theul

55.2 m/sec, we can use the Bernoulli equation to estimthe pressure at any point in the test section as longviscous/turbulence effects are not important. Using thisproach one can determine, for example, that the absopressure atx/L50.215, y/L50.215, where the velocity isalmost uniform and]p/]y'0, is 1.643104 Pa over the va-por pressure. Then, vertical integration using the measuvalues of]p/]y ~Fig. 19! leads to a pressure of approxmately 5000 Pa over vapor pressure aty/L50.132, x/L50.215. This result also shows the validity of our measuments since this point is located near the cavitation interfand hence the pressure should be close toPv . Once the localpressure can be evaluated,](a l p)/]x and](a l p)/]y can becomputed at every point alongy/L50.132.

Thus, using the pressure gradients aty/L50.14, theterms in the momentum equations are now available and

FIG. 18. Distribution of turbulent kinetic energy fors54.41.

ofen

-te

e.,

res

el-r-

-

ce

as

sew

n-

teas-te

ed

-e

e

can substitute them in order to estimate the interaction terIn order to reduce the jitter associated with the derivativthe vectors are spatially filtered prior to differentiation.simple ‘‘box filter’’ is used, i.e., each velocity vector is replaced with an average of the 434 neighboring vectors.Again, all the results are scaled withU2/L ~545 g! and aresummarized in Figs. 20 and 21. In the horizontal momentbalance the turbulent stresses have little impact on thesults. The dominant terms are clearly the pressure gradiand a lul]ul /]x, but a lv l]ul /]y is not negligible.Mx issignificantly smaller than the convective terms and pressgradients except forx/L,0.23, where it becomes negativIn this region, due to the proximity toa50, we suspect thathere are significant differences between]p/]xuy /L50.132and]p/]xuy /L50.14 @evident in Fig. 19~a!, where the gradientsbegin to vary considerably as the cavity is approache!,which may cause this deviation from zero. Conversely, invertical pressure gradients@Fig. 19~b!# the changes with el-

FIG. 19. Distributions of~a! (1/r)]p/]x and ~b! (1/r)]p/]y at severalelevations. The quantities are normalized byU2/L~545 g!.

er

refoe

ure-s arebal-adi-

thethena-w-

ey-ther

the

ok-

de

stan-p-

ingexe

nd

ainnd-cit

ape


evation are small, yielding more reliable results for the vtical momentum equation.

All the terms in the vertical momentum equation asmall compared to the horizontal terms except](a l p)/]y. This term is large for a significant fraction of thclosure region due to the large vertical gradients ina l ~Fig.9!, leading to a high interaction term in they direction. Sincea l]p/]y is small,rM y is essentially equal top]a l /]y, veryclearly seen in Fig. 21.

In the equations derived by Drew and Lahey53 for flowswithout phase change, the expression forMi is

rMi5p]a l

]xi2t i j

]a l

]xj1Mi8 , ~9!

FIG. 20. Terms of the horizonal momentum balance@Eq. ~7!#. Mx is theinteraction term. Quantities are normalized byU2/L~545 g!.

FIG. 21. Terms of the vertical momentum balance@Eq. ~8!#. M y is theinteraction term. Quantities are normalized byU2/L~545 g!.

-

r

wheret i j is the viscous stress andMi8 is a fluctuating termresulting from ensemble averaging. In the present measments, both the viscous stresses and the fluctuating termsmall ~e.g., see the effect of Reynolds stresses on theance! compared to the convective terms and pressure grents, resulting in

rMi'p]a l

axi. ~10!

Thus, at least for the present low levels of cavitation,interaction term is equal to the pressure multiplied bygradients of the void fraction. Consequently, in order to alyze the flow in the closure region one can solve the folloing momentum equation,

a l ul j

]ul i

]xj52

a l

r

] p

]xi2

]~a luli8ul j8 !

]xj. ~11!

Since, for the existing conditions, the gradients of the Rnolds stresses are still substantially smaller than the oterms,a l may be canceled and Eq.~11! can be reduced to asimple ‘‘Euler equation’’ for the liquid velocity. Note thathis conclusion is appropriate only for conditions where tstress gradients~viscous and Reynolds! are small. However,a l still remains in the continuity equation.

C. Momentum and displacement thickness

Finally, for boundary layer modeling, it is of interest tdetermine the effect of cavitation on the momentum thicnessu, a lu and the displacement thickness,d*, a ld*. Theyare defined as

u5 (wall

y/L50.19u

uy/L50.19S 12

u

uy/L50.19DDy,

d* 5 (wall

y/L50.19 S 12u

uy/L50.19DDy,

~12!

a lu5 (wall

y/L50.19a lu

uy/L50.19S 12

u

uy/L50.19DDy,

a ld* 5 (wall

y/L50.19

a l S 12u

uy/L50.19DDy.

Distributions ofu, a lu, d* and a ld* are presented in Fig.22. Note that the integral momentum equation would incluthe terms multiplied bya l . It is clear that the turbulencegenerated by the collapse of the cavitation causes a subtial increase in the momentum deficit near the boundary. Ustream of~or in the absence of! the cavitation, the entireboundary layer thickness is smaller than our vector spacof 0.675 mm. Also, the slight change in the cavitation indincreases~almost doubles! the magnitude and extends thhorizontal length of the region with high momentum adisplacement thickness. Botha lu anda ld* decrease as thevoid fraction recovers to 1.0, but their magnitudes remsubstantially higher than any value that exists in the bouary layer upstream of the cavitation. The momentum defiis created predominantly in the closure region. The sh

s.

owheth

ita

m°Cth

v-.enad

thisof

clo-entseureientwethe

hlyigh-pse

ialhatt ofu-ub-ent

avi-ighttheyeresture%–

har-c-ike

le-ita-uidnitebeeudeseri-the

mul-

o.od.n,u-

ream-

n

s onval


factor d*/ u;1.2, which presently occurs atx/L50.25– 0.35, is characteristic to turbulent boundary layer56

VI. SUMMARY AND CONCLUSION

Particle image velocimetry is used to resolve the flstructure in the closure region and downstream of attaccavitation. The present paper focuses on the flow whenambient pressure is reduced only slightly below the cavtion inception level, i.e., when the cavity is thin~3–5 mm!.Two sets of vector maps recorded at the same hydrodynaconditions but at slightly different water temperatures, 35and 45 °C, provide data on the effect of small changes tocavitation index~4.69 vs 4.41! on the instantaneous and aeraged flow structure, turbulence and vorticity production

The instantaneous velocity distributions show that whthe cavity is thin there is no reverse flow downstream abelow the cavitation, i.e., there is no reentrant flow. Inste

FIG. 22. Momentum and displacement thickness for~a! s54.69 and~b!s54.41.

de-

ic

e

nd,

the cavities collapse as the vapor condenses. Note thatconclusion applies to conditions where, in the absencecavitation, there are no adverse pressure gradients in thesure region~there are substantial adverse pressure gradiwhen cavitation occurs!. As discussed in Ref. 32, reversflow and a reentrant jet start developing when the closregion is located in a region with adverse pressure gradeven without cavitation. Consistent with this conclusion,do see reverse flow when the cavitation is extended todiverging part of the present nozzle.

The shape of the cavities in the closure region is higirregular and unsteady. The flow measurements and hspeed photographs show that the process of cavity collainvolves rollup of large hairpin-like vortices and substantvorticity production. These vortices extend to elevations tare considerably higher than the height of the stable parthe cavitation. Ensemble averaging of the velocity distribtions shows that the unsteady cavity collapse involves sstantial increases in turbulence, momentum and displacemthickness in the boundary layer. The small decrease in ctation index between the two data sets increases the heand length of the cavity, and has a strong impact onturbulence level and momentum deficit in the boundary ladownstream of the cavitation. The region with the highturbulence level is located just downstream of the closregion. The Reynolds shear stresses reach levels of 2540% of the normal stresses. Such high correlations are cacteristic of flows containing large coherent vortex strutures, consistent with the observations. However, unltypical turbulent shear flow, the large eddies~i.e., turbulenceproduction! originate as the vapor cavities collapse.

It is demonstrated that in dealing with the ensembaveraged flow in the closure region of the attached cavtion, one should account for the sharp gradients in the liqvoid fraction. Since the collapse process occurs over a firegion, the incompressible 2D continuity equation cansatisfied only whena l is included in the analysis. Using thmomentum equation it is possible to estimate the magnitof the ‘‘interaction term,’’ i.e., the impact of the vapor phaon the liquid momentum. It is demonstrated from the expemental data that, at least for the present test conditions,interaction term can be estimated as the local pressuretiplied by the gradient in void fraction. Equation~11! is theresulting ensemble-averaged momentum equation.

ACKNOWLEDGMENTS

This project has been funded by ONR, under Grant NN00014-95-1-0329. The program manager is Dr. E. RoThe authors would like to thank Ye Zhang, Mancang TiaSteven King and Manish Sinha for their valuable contribtions.

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Flow structure and modeling issues in the closure region of attached cavitation · 2002-09-06 · Flow structure and modeling issues in the closure region of attached cavitation Shridhar

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