Sixth International Symposium on CavitationCAV2006, Wageningen, The Netherlands, September 2006

The structure of unsteady cavitation. Part I : Observations ofan attached cavity on a three-dimensional hydrofoil

Evert-Jan FoethTechnical University of Delft

Laboratory of Ship Hydromechanics+31-152786603

e.j.foeth@wbmt.tudelft.nl

Tom van TerwisgaTechnical University of Delft &

Maritime Research Institute of the Netherlands

Abstract

Fully developed sheet cavitation on ship propellers and rudders is a major cause of noise, vibrationand erosion. A 3D twisted hydrofoil with a cavitation pattern closely related to propellers was observedwith a high-speed camera at the University of Delft Cavitation Tunnel. Re-entrant flow coming from thesides of the cavity aimed at the center plane, termed side-entrant flow, collided in the closure region ofthe cavity, pinching off a part of the sheet resulting in a periodic shedding. The collapse of the remainderof the sheet appears to be a mixing layer, originating in the closure region moving upstream up to theleading edge. Collision of side-entrant jets in the closure region of a cavity is identified as a secondshedding mechanism, in addition to re-entrant flow impinging the sheet interface at the leading edge.

1 Introduction

Fully developed sheet cavitation on ship propellers is a major cause of noise, vibration and erosion. Asavoiding cavitation is usually impossible it must be controlled. Although the final evaluation of a pro-peller design is based on model experiments, numerical methods for cavitation simulation are becomingincreasingly more important. Potential flow solvers are now the industry standards (e.g., Young & Kinnas[27]), but in the last decades an increase in both Euler (e.g., Choi & Kinnas , Sauer & Schnerr [8]) andRANS (e.g., Kunz et.at. [20], Senocak & Shyy [25]) codes has been observed. However, up to now thesesimulations are not able to capture the pressures radiated by cavitation or to predict erosion location andseverity on propellers [17]. To improve the description of the cavity behavior and especially the unsteadyshedding in the form of cloud cavitation and in support of the rapidly expanding field of numerical simu-lation this experimental research was started with a threefold goal. 1 st analyze the physical mechanismsof the instability of the cavity 2nd: build a data set of simple cavitating flows to be used as benchmarkmaterial for computational fluid dynamics validation, and 3 rd, extend the insights gained to guidelinesfor propeller design. Here we focus on the description of the flow field around an attached cavity and itsshedding mechanism.

Cavitation is the evaporation of liquid when the pressure falls below the vapor pressure. For anoverview, see Rood (inception [24]), Brennen (bubble cavitation [5]), Arndt (vortical cavitation [2]) andFranc (sheet cavitation [14]). When this low pressure region forms near the leading edge of a hydrofoil, theflow detaches and reattaches downstream; an attached pocket of vapor is formed called sheet cavitation.In the closure region of this sheet cavity, the pressure gradient is positive, forcing a thin stream of liquidupstream into the cavity, denoted as the re-entrant jet. When this re-entrant jet impinges on the fluid-vaporinterface, the resulting disturbance will force a region of the sheet to be pinched off from the main sheet

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cavity and is advected with the flow. This structure is often both highly turbulent and bubbly in nature andhence called cloud cavitation. Depending on the pressure condition, this cloud may collapse violently withensuing erosion and noise. The remainder of the sheet cavity grows again and the process repeats itself.This behavior has been reported by several authors (i.e. de Lange [12]). The importance of the re-entrantjet is further demonstrated by Kawanami by blocking the re-entrant jet consequently altering the behaviorof the cavity [18].

Cavitation has been extensively tested in the past on two-dimensional hydrofoils (i.e. Franc [15]. How-ever, cavitation on ship propellers is distinctly three-dimensional due to the propellers three-dimensionalgeometry, the radially increasing velocity and change in blade loading, and a periodic change of inflowconditions due to the wake behind a ship’s hull. Three-dimensional effects were taken into account by deLange & de Bruin [11] and Laberteaux & Ceccio [21] using swept hydrofoils; the sweep (or skew) hada significant effect on the cavity topology and direction of the re-entrant jet. Near the sides of a cavitythe re-entrant jet is directed inward and sideways at a near 90-degree angle with the incoming flow [10];colliding re-entrant jets can form disturbances leading to shedding.

2D sheet cavitation on 2D hydrofoils is often shedding intermittently and at random locations makingthe flow field far more complex and three-dimensional, as shown by Callenaere et.at. using a backward fac-ing step to create a two-dimensional sheet cavity [6]. The presence of the cavitation tunnel walls suppressescavitation on 2D hydrofoils near the wall, leading to a re-entrant flow from the sides directed inward, mak-ing the cavity locally 3D and sometimes simplifying the shedding.As studying cavitation on a rotating object is inherently more difficult, three-dimensional hydrofoils weredesigned resulting in a cavitation topology closely related to propellers with periodically changing inflowconditions. A three-dimensional hydrofoil with its shedding mechanisms depending on its spanwise load-ing and reduced wall influence has a more structured and repeatable sheet than a 2D hydrofoil at a constantangle of attack. This allows for observations of the influence of controlled three-dimensional effects. Forthis study a span-symmetric 3D hydrofoil is chosen, creating an isolated sheet cavity around the plane ofsymmetry. The 3D hydrofoil is lightly loaded at the tunnel walls to deny any interaction of the cavity withthe tunnel boundary layer. Visualization is an indispensable tool for studying cavitating flows. In this paperhigh speed recordings are presented with our interpretation for the shedding behavior for two distinct cases,a cavity of roughly half the chord length and a (near) super cavity. The shedding mechanism for both casesdiffered from 2D shedding, where the re-entrant jet reaches the leading edge, but was governed by the 3Dtoplogy of both hydrofoil and sheet cavity.

A brief description of the setup of the experiment is given in section 2. The observations are describedand interpreted in section 3 and conclusions are in section 4.

2 Setup

The experiments were performed in the University of Delft Cavitation Tunnel (see figures 1 & 2), with aneffective measuring channel 0.60 m in length with a 0.3 m x 0.3 m cross section with optical access fromall sides; velocities up to 10 m/s per second can be attained and the local pressure can be reduced to 5,000Pa. The non-dimensional cavitation number is defined as

σ =p0 − pV

12ρV 2

0

(1)

or, the ratio of the pressure head to the vaporization pressure (p V ) and the dynamic pressure located at thetest section entrance. The pressure was measured with 10 Keller PAA-15 pressure transducers, at each wallat both the inlet and the outlet of the test section, as well as on two locations before the contraction upstreamof the test section These sensors were calibrated in situ to within 1% error with Keller PAA-33 pressuretransmitters in the test section. Uncertainty analysis and error propagation indicated an uncertainty in σnear7.5% within a 95% confidence interval for the examples shown. All data were acquired by a measurement& control computer fitted with three synchronized National Instruments DAQ’s (48 channels).

The test object is a three-dimensional hydrofoil, previously used by Dang [9], with a chord length ofC = 150 mm, a span of S = 300 mm (spanning the entire test section), and a span wise varying angle of

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attack. For the sectional profile, a simple NACA0009 thickness distribution (z) is chosen [1].

z (x, y) =t

0.20(0.29690

√x − 0.12600x− 0.35160x2 + 0.28430x3 − 0.10150x4

)(2)

For the actual manufacturing of the foil, the trailing edge must be sufficiently thick to prevent damageduring the milling process and handling. A minimum trailing edge thickness of t min = 0.4mm waschosen. To increase the thickness of the foil, a correction function was chosen with a zero first orderderivative at the point of initialization. A 2nd order polynomial satisfies this condition.

tTE (y) =(

x − xsp

1 − xxp

)2 (tmin

2C− z|x=1

)H(xsp) (3)

with x = [0, ..., 1], xsp = 0.35 the starting point of the above thickness correction function, and H theHeaviside function. This function effectively increases the symmetrical offset of the NACA0009 hydrofoil(continuously) smoothly from 35% to 100% chord.

To avoid cavitation near the walls of the tunnel and to generate a span wise varying cavitation pattern,the angle of attack increases toward the center of the hydrofoil as given by

α (y, β) = αT

(16

∣∣y − 12

∣∣3 − 12∣∣y − 1

2

∣∣2 + 1)

+ β (4)

with y = [0..1] the dimensionless span wise coordinate. This is a simple polynomial allowing for the angleof attack to rise (continuously) smoothly from zero at the walls to αT , with an αT of 8 degrees, creating alow-pressure region in the center of the hydrofoil. The angle β is the rotation angle of the entire hydrofoiland is always equal to the local angle of attack at the wall. The hydrofoil and angle of attack distribution ofthe sections are sketched in figures 3 & 4, respectively. The center of rotation is located halfway down thechord. The hydrofoil was manufactured in both anodized aluminum and perspex. The perspex hydrofoilwas filmed from the pressure side so that the cavity could be viewed from the inside.

At low Reynolds numbers the boundary layer near the minimum pressure region will be laminar andno cavity sheet will appear. When the boundary layer separates and a separation region is formed, asmooth and glassy cavity appears. With the limited Reynolds numbers typically present at small scales,transition to turbulence does not occur unless the boundary layer is locally disturbed. When it does occurnatural transition to turbulence can temporarily suppress leading edge detachment [15]. Experiments withaxisymmetric headforms showed that for laminar detachment, local indentations in the detachment lineoccur, called ’divots’, locally disturbing the interface (Tassin Leger et al. [26]. As a 3D hydrofoil isused at moderate velocities, the sides of the attached cavity can be locally suppressed by laminar effects,so roughness elements of 120 μm were applied at the leading edge (4% chord length) as a turbulencetripping mechanism. The roughness elements can lead to local streaks of cavitation appearing next tothe main cavity. At too low speeds, the entire detachment region near the leading edge may resemble anagglomeration of such streaks which was observed at 5 m/s. The gas content was not measured, but theroughness will supply the degassed flow with ample nuclei for sheet cavitation to develop [19]; incipientcavitation on roughness elements is typically observed when σ equals the minimum pressure coefficient[7] and the nuclei content of the flow is no longer critical.

The camera used for the high-speed imaging is a Flowmaster HighSpeedStar 4 (Photron Ultima APX)with a 10 bit dynamic range, 1 Mpix resolution at 2 kHz with a maximum acquisition frequency of 120kHz (0.4% full resolution) with 2.6 GB memory. The lens is a Nikon AF Nikkor 50 mm, used with af-stop of 2.8. Illumination experiments with stroboscopes, continuous light, and pulsed lasers indicated thatmotion blur is significantly reduced with a short illumination time of both the stroboscopes and the pulsedlasers, compared to continuous lighting, even with a shutter time as small as 10 μs. The stroboscopes usedwere not designed for frequencies of 1 kHz and ran prohibitively hot after a few seconds and were not used,even though they are more flexible to aim than the laser. A New Wave Pegasus dual-head, high repetition,diode pumped Nd: YLF laser, with a 180 ns duration with a 10 mJ/pulse power is used at 1,000 Hz. Thelaser is expanded for volumetric illumination with two cylindrical lenses and aimed through a side-windowat the hydrofoil. The system is controlled by LaVision’s Davis 7.0 and triggered by the measurement &control computer.

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3 Observations

The shedding process of the attached cavity is classified into three regimes. At high cavitation numbers theattached cavity is very small and present over a wide range of the leading edge. This cavity is sheddingvortices intermittently. No large structures are identified. At moderate cavitation numbers large structuresare shed regularly. Lowering the pressure further creates an attached cavity reaching a length comparableto the chord length of the hydrofoil. Shedding is then intermittent and irregular.

Simple analysis of the high speed video indicated that the Strouhal number

St =fl

V(5)

based on cavity shedding frequency f and cavity length l is around St = 0.185 for moderate cavitationlengths when 0.8 < σ < 1.1. For 2D hydrofoils Strouhal numbers of St = 0.25 − 0.40 are reported orspecified by Arndt [3] as:

St = 1/4√

1 + σ (6)

from which we concluded that the resulting topology of the sheet differs significantly from a 2D cavity dueto the 3D geometry of the foil. In figure 5 the Strouhal numbers are plotted versus the cavitation number,σ as well as eq. 6. There does not seem to be an indication that the Strouhal number is dependent on thecavitation number, σ, nor that for the limited number of points there is a transition from one shedding cycletime to the next.

3.1 Side entrant flow

A small cavity refers to an attached cavity with its maximum stable length (prior to shedding) is roughly20 to 70 % of its chord length. In figure 6 a full shedding cycle at 5 m/s is shown with the hydrofoilfilmed from above, flow top to bottom. The shedding was repeatable, constant in its shedding frequency,and always followed the same macro structural collapse. This makes the 3D geometry and subsequentcavitation a viable study object for the investigation of its mechanism.

Due to the crescent -or convex- shape of the closure, the re-entrant jet was distinctly different frommerely upstream. First, the closure region in a 2D flow would normally be succeeded by a stagnation line(orthogonal to the 2D plane), here it was a stagnation point at the mid plane. Second, at the sides of thesheet the pressure gradient forces the flow over the sheet into the cavity roughly ”mirroring” the streamlinesat the interface contour as sketched in figure 7.1. De lange & de Brujn concluded that the re-entrant jetof the 2D hydrofoil was directly upstream, but in the 3D case the re-entrant jet component normal to theclosure line was reflected inward. As the (instationary) pressure gradient is perpendicular to the closureline, it is to be expected that the component tangential to that closure line remains unchanged. (Of course,the same is true to the 2D re-entrant jet which has no transverse velocity component).

To distinguish between various directions of the re-entrant flow, the term side-entrant is introduced.This term refers to the that part of the re-entrant jet that has a strong span wise velocity component directedinto the cavity originating from the sides (and may have a small up- or down stream component). Theterm re-entrant is reserved for the flow originating from that part of the cavity where the closure is more orless perpendicular to the flow and is thus mainly directed upstream (but may have a small lateral velocitycomponent). The re-entrant jet is thus split up in re-entrant and side entrant components, even though atcertain points of the flow both terms may apply. The term side-entrant is introduced to emphasize the three-dimensional character of the flow. Side-entrant components of the re-entrant jet do not reach the leadingedge but may form an equally important source for the shedding, as explained in the following section.

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3.2 description of the shedding mechanismThe shedding cycle of the cavity in figure 6 is broken up into 4 parts;

1 6.1-6.4 Interface becomes turbulent or frothy2 6.5-6.12 Primary Shedding (cavity center)3 6.9-6.16 Secondary Shedding (cavity sides)4 6.17-6.20 Growth

Note that there is a short overlap between primary and secondary shedding (and growth). The primaryshedding is located at the mid plane of the hydrofoil, the secondary shedding is visible at the sides of theprimary shedding as two distinct smaller vortices. After the secondary shedding the cavity grows and atsome point the interface was visibly turbulent. Sketches of the shedding are presented in figure 7 to aid ourinterpretation.

At the sides of the cavity, re-entrant flow has a very small span wise component and is directed down-stream. The span wise component is largest when the cavity closure contour is roughly at 45 ◦ with theincoming flow where the velocity component in downstream direction is zero. At larger angles, re-entrantjet will have an upstream component (fig. 7.1). Side-entrant flow was present during the entire sheddingperiod as the sides of the cavity were present. Flow from the sides was not obstructed, nor was it directedat the leading edge. When the sheet cavity was growing, it flowed unobstructed to the center. The side-entrant jets from both sides were focused into the closure region of the sheet where they collided; it is in theclosure region where the cavity became turbulent first, not near the leading edge, as will be described below.

The primary shedding originated at this location in the center of the sheet. However, only a smallerportion was broken off and advected with the flow while most of the cavity remained attached. The velocityof a simple streamline at the cavity surface can be estimated [13] at VV = V0

√1 + σ . Although the exact

velocity of the re-entrant flow is notoriously difficult to measure, the velocity of the jets is unlikely tobe a order of magnitude lower. If we assume that during the shedding cycle the two side-entrant jets areconverging for about a third of the shedding cycle (15 Hz), the amount of fluid through a square millimeter(taking a homogeneous velocity distribution) at this velocity of 6.4m/s was about (6400 · 2)/(15 · 3) ∼=285 mm3/shedding per mm2 of jet . At this rate the cavity closure could be collecting fluid quicklyeven if the jet is thin. Any fluid ejected upward through the interface is likely to have created a significantdisturbance, isolating a small portion of vapor and creating a bubbly flow as the jet entrains vapor. As theside-entrant jets were aimed at the closure, it is here that the fluid impinges on the interface first. Thisstructure can be seen to roll up quickly in figures 6.5 -6.8 by self-induction into a hairpin. This structuregrows significantly in height, of the order of the cavity length. The cavity closure after the initial cut-off(figure 7.6.a) is now turbulent. The radially diverging re-entrant jet (fig. 7.5) was observed early in thegrowth of the sheet cavity. The vapor interface at the leading edge was not visibly disturbed upon contact;its weakness apparently did not lead to immediate shedding. The re-entrant jet did not seem to be a causefor the detachment of the main structure.

The remaining topology of the sheet in fig.’s 6.5 to 6.12 is concave. The re-entrant flow direction in thecenter is still directed radially outward. The main side-entrant jets and radially diverging re-entrant jet arenow converging in both downstream lobes of the remaining cavity shape (fig. 7.3 & 7.4). A new collapseis created, here denoted by the secondary shedding, by the collision of these two flows. However, themechanism does not seem to be any different from the primary shedding. Basically, the main shedding asvisible on fig’s 6.5-12 is repeated next to the center plane as visible in fig. 6.9-18. The secondary sheddingwill disappear when the closure is no longer concave, as visible from fig. 6.15 to 6.20.

After the secondary shedding the remaining cavity has a reasonably convex shape with two concavehollows (fig. 7.5). Side-entrant flow will appear at either side of the hollows. On the outer sides this flowwill collide with the side entrant flows on the outer-most section of the cavity, and on the inner sides the jetsare directed inward as sketched. At this point, side entrant and re-entrant flow are at a near 90 ◦ angle.. Ina brief period of converging side-entrant jets, the cavity grows into its crescent shape again but it is merelya period of respite before side-entrant jets at the center plane collide once again. The cavity never reachesa constant length.

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3.3 Re-entrant flow directions

In order to visualize the re-entrant jet more clearly, a series of frames of the transparent hydrofoil is pre-sented in figs 8 & 9. The cavitation is filmed through the hydrofoil showing the internal structure. Theradially diverging re-entrant flow is visible in fig. 8(denoted as A) as waves on the jet surface occasionallyreflected the laser light. The re-entrant flow directed upstream in a 2D situation would be constrained inits lateral movement. As the re-entrant flow is only directed upstream at the plane of symmetry, it doesnot experience this constraint and diverges. If the re-entrant jet is radially dispersed, it may loose some ofboth its velocity and momentum delaying the shedding until side-entrant jets collide in the cavity’s closure.This may explain the systematically lower Strouhal number for this particular experiment. Figure 9 showsthe cavity after the secondary shedding, corresponding to fig. 7.5. The movement of the front of the side-entrant jets (A) can be seen as re-entrant flow forced into the cavity collides with re-entrant flow from theplane of symmetry. The re-entrant jet is also present (B) and a turbulent region is created upon impact atthe lower corners of the cavity closure. Both situations clearly show the presence and influence of a lateralvelocity component at the cavity closure.

3.4 Cloud cavitation

An example of a (super) cavity is given in figure 10 showing the shedding in more detail. The sheddingstarts in the closure region although converging side-entrant jets cannot be identified. In fig. 10.1-6 thelocation of the front of the re-entrant jet is given by the arrow on the top-center location of each frame. Al-though difficult to identify on photographs, it is clearly seen to slowly move forward on film. The collapseof the sheet in fig. 10.1-10 starts from the end, moving upstream. Although the origin is not well visible,it seems to result from the impact of side-entrant jets as the frames before or during the initial collapse aresimilar to the frames of figure 6.

An initial structure is pinched off as illustrated in figure 7.6 after which there may no longer be re-attachment between the cavity and newly formed vortex. If one takes the boundary integral of the isolatedvapor structure with surface S and boundary ∂S as in figure 7.6a,

Γ =∮∂S

V ds ∼= 2πhVo

√1 + σ (7)

with h the estimated cavity thickness of at the closure, it is immediately apparent that circulation ispresent (taking the region circular for simplicity’s sake), and must lead to a reduction of the bound circu-lation (or lift) of the hydrofoil. As the mechanism of the generation of the first shed vortex is by isolatingit from the main cavity by means of re-entrant flow, its mechanism is inertial and not viscous in nature.It is therefore expected that a non-viscous numerical model should be able to capture the generation andadvection of this structure.

3.5 Remaining cloud structure

The shed flow structure in fig. 10.7-8 consists of primary span wise and secondary stream wise vortices,similar to the turbulent cavitating shear flow structure observed behind steps and other mixing layers [23].Figure 12 shows the formation of a span wise vortex from figure 10.5 (showing intermediate frames aswell). Such a span wise vortex system can be a result of a Kelvin-Helmholtz instability with a street ofpositive vortices (where the vorticity has the same sign as the hydrofoil’s net circulation). A close-up offig. 10.6-9 is given in fig. 13, including all intermediate frames. Bernal & Roskho describe this observedstructure of span wise and stream wise vortices in more detail using a helium-nitrogen mixing layer [4].The observed cloud cavitation is similarly structured and greatly resembles that of a mixing layer.The stream wise vortices originated as a single span wise vortex warped around the primary span wisevortices. The smaller scale vortices can be seen to be stretched around the periphery of the span wise

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structures with an increase of their vaporous cores. Normally in shear layers, cavitation inception is firstobserved in these streamwise vortices [23], but in the case of a sheet cavity break-up vapor is trapped inthe initial formation. As the height of the shear layer increases, the distance between vortices increases asits frequency decreases due to conservation of vorticity.Gopalan & Katz already concluded [16] that vorticity is produced in the closure region of a sheet cavitybut no production of vorticity was measured during advection using PIV; vorticity is produced during sheetcollapse. If the closure of the cavity is indeed a shear layer, its modeling will require viscous numericalmodels able to capture the observed dynamics such as unsteady RANS or LES. Three mechanisms forvorticity generation can be identified: a) change in bound circulation b) shear and c) baroclinic torque.Vorticity production in a shear layers is naturally due to shear, but baroclinic torque due to the densitydifferences may play a part during this complex cavity collapse, the extent of which is not known.

3.6 Closure of the attached cavity

From figure 10.4-8 it is visible that the origin of the mixing moves forward. The direction of the collapseof the sheet is radially divergent, accelerating up to the mean stream velocity when reaching the leadingedge, as determined from frame-by-frame analysis. The approximated location of the front at the centerplane is identified and plotted in fig. 11. It appears that the collapse front accelerates (roughly) linearly.The collapse is thought to be pressure-driven as it follows the same route as the re-entrant jet.At the start of the collapse cycle the cavity is a well-defined structure and due to its 3D geometry with asymmetry plane, only a stagnation point is present in the closure region (fig 14.1). After the first pinch-offthe closure region of the cavity has changed from a convex into a concave or flat shape and a short highpressure line element is present (fig 14.2), widening with each further pinch off, gradually changing from apoint source into a line source (fig 14.3) as the cavity loses its three dimensionality. This can be a possibleexplanation for the acceleration of the collapse front. If both the fluid is collected at the center from side-entrant jets and the pressure at the cavity closure increases on average, the re-entrant flow can be expectedto be much thicker, perhaps as thick as the cavity interface itself.

When a cavity is very thin, the side-entrant jet does not reach the center plane. The closure of thesheet cavity remains turbulent at all times, shedding small vortices, but no large structure is shed. If noappreciable change in pressure is generated at the closure, no enhanced re-entrant flow as visualized infig. 10 is present and the disturbance does not migrate toward the leading edge.

4 Conclusions

From this experiment follows that re-entrant flow from the sides dictates the behavior of the shedding cycleand the flow from the sides depends on the cavity shape. Thus the cavity topology largely dictates there-entrant flow directions and focal points of this flow. Its shape and motion governs its behavior and theconvex cavity shapes seem to be intrinsically unstable. Re-entrant flow reaching the leading edge doesnot seem to be the only cause for shedding. The re-entrant flow can be both moving upstream but alsoin span wise direction denoted as a side-entrant component. For any convex cavity shapes, side-entrantcomponents of the re-entrant jet focus in the closure region of the sheet, creating a disturbance causinga local break-off of the main sheet structure. Side-entrant components may collide before the re-entrantflow reaches the leading edge or the upstream directed re-entrant flow is too weak to cause shedding at theleading, changing into a different shedding mechanism. Side entrant component focusing is suggested as asecond shedding mechanism for attached sheet cavitation in addition to the re-entrant components reachingthe leading edge.The alternating shedding seen on the presented hydrofoil results in two shedding cycles, but the two-dimensional hydrofoil lacks the symmetry plane in the center, resulting in the seemingly random localshedding along its cavity closure. Any disturbance at its closure will re-direct the re-entrant flow into side-entrant flow resulting in focal points and subsequently into local shedding. As a result, the two-dimensionalcavity has a highly three-dimensional structure making it a more difficult study object, either numerical orexperimental.The shedding mechanism observed after side-entrant jet collision at the center plane is a pinch-off of a part

7

of the attached cavity. The observed (cavitating) vortices after the shedding lead to the conclusion that amixing layer is present with its characteristic span wise and stream wise vortices which are clearly visibleon the images presented.Pressure measurements by Le et al. [22] indicated that thin cavities have a smooth pressure recovery at thecavity closure generating re-entrant flow with a minimum of momentum. Also, a thin cavity does not allowfor flow reaching the leading edge. In these experiments continuous mixing can be present. The cavities ofthis experiment are thicker, but some cavities of significant length (over 50% chord) do not show sheddingbut have an open closure, although intermittent shedding is sometimes observed. In the absence of shed-ding of large structures, the closure region is relatively constant with side-entrant flow continuously aimedat the same location of the sheet leading to continuous vortex shedding; a cavity does not need to be thinto have an open closure if its closure is continuously supplied with fluid.

For future experiments the effects of an unsteady inflow will be investigated. We intend to use a flowoscillator to stimulate the cavity with frequencies both higher and lower than the natural shedding frequencyand expand the range of hydrofoils. We intend to use hydrofoils with a steeper angle of attack distributionto enhance the 3D effects further and with different sectional profiles to compare the influence of cavitythickness.

5 Acknowledgments

This research is funded by the Dutch Technology Foundation STW project TSF.6170 and the Royal Nether-lands Navy. See www.stw.nl for more details.

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[21] K.R. Laberteaux and S.L. Ceccio. Partial cavity flows pt 2. cavities forming on test objects withspanwise variation. Journal of Fluid Mechanics, 431:43–63, 2001.

[22] Q. Le, J.P. Franc, and J.M. Michel. Partial cavities: Global behavior and mean pressure distribution.Journal of Fluids Engineering, 115:243–248, 1993.

[23] T.J. OHern. An experimental investigation of turbulent shear flow cavitation. Journal of Fluid Me-chanics, 215:365–391, 1990.

[24] E.P. Rood. Review - mechanisms of cavitation inception. Journal of Fluids Engineering, 113:163–175, 1991.

[25] I. Senocak and W. Shyy. Numerical simulation of turbulent flows with sheet cavitation. In 4 th

International Symposium on Cavitation, Pasadena, Ca, USA, 2004.

[26] A.L. Tassin Leger, L.P. Bernal, and S.L. Ceccio. Examination of the flow near the leading edgeof attached cavitation, part 2: Incipient breakdown of two-dimensional and axysymmetric cavities.Journal of Fluid Mechanics, 376:91–113, 1998.

[27] Y.L. Young and S.A. Kinnas. A bem for the prediction of unsteady midchord face, and/or backpropeller cavitation. Journal of Fluids Engineering, 123(2):311–319, 2001.

9

Figure 1 A sketch of the Delft Cavitation Tun-nel consisting of two cylindrical and twosquare channels

Figure 2 A close-up of the test section showing thehydrofoil (suction side up, transparent hy-drofoil mounted upsdie down), the cameralocation and effective viewing area

Figure 3 Top, side, and front view of the hydrofoil.The black outline indicates the viewing areaof figure 6 & 10

Geometric angle of attack

0

1

2

3

4

5

6

7

8

100 150 200 250 300Span [mm]

Ang

le[d

eg]

500

Figure 4 Distribution of the geomtetric angle of at-tack of the hydrofoil. The angle at the sidesis taken as the reference angle for the wholegeometry

10

0.0

0.1

0.2

0.3

0.4

0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40σ

St

5 m/s β=17 m/s β=07 m/s β=17 m/s β=2Arndt

Figure 5 Strouhal numbers at two different velocities based on maximum cavity length with an average ofSt = 0.185, significantly lower than for a two-dimensional foil. Several points at St = 0 arevisible indicating either irregular shedding (σ < 0.80) or the absence of large structure sheddingresulting in an ’open’ cavity (σ > 1.1). The continuous line is the relation by Arndt [3] for 2Dcavity shedding.

11

6.1 6.2 6.3 6.4

6.5 6.6 6.7 6.8

6.9 6.10 6.11 6.12

6.13 6.14 6.15 6.16

6.17 6.18 6.19 6.20

Figure 6 Visualization at 4.96ms ± 6.4%, α = 1, σ = 0.66 ± 7.94%, f = 285.7Hz (every 7th frame ofthe 2,000 Hz recording). Viewing area as outlined in figure 3, flow from top to bottom.

12

7.1 Streamlines over the cavity directed inward 7.2 Estimate of the direction of re-entrant flow fo-cusing in the lobes causing a second pinch-off

7.3 The streamlines at the side planes in the con-cave part are partly directed away from the cen-ter plane

7.4 Estimate of the direction of re-entrant flow fo-cusing in the lobes causing a second pinch-off

7.5 The streamlines at the side planes in the con-cave part are partly directed away from the cen-ter plane.

7.6 Side-entrant jets converge in the closure regionand cut off the first vortical structure. The re-maining cavity closure is now ”open”.

Figure 7 Sketches of the re-entrant flow at two instances, during its maximum growth position (top) andafter the primary shedding (bottom)

13

Figure 8 The re-entrant flow was filmed through a transparent hydrofoil. The images show the re-entrant jetafter cleaning up the pictures (despecle, color & histogram enhancement) . These figures showsthe radially diverging re-entrant (A) emanating from the center of the foil at two different sheddingcycles as sketched in fig. 7.4. The two horizontal lines are holes for ink injection (not presented)

Figure 9 This series shows the cavity at the end of its secondary shedding cycle. The side-entrant jet is seento develop at both corners of the sheet (A) as visualized in fig. 7.5. The re-entrant jet is visiblenear the leading edge (B)

14

10.1 10.2 10.3 10.4

10.5 10.6 10.7 10.8

10.9 10.10 10.11 10.12

10.13 10.14 10.15 10.16

10.17 10.18 10.19 10.20

Figure 10 Visualization at 6.89 m/s± 7.70%, α=1, σ=0.49± 28.4%, f=400 Hz. Viewing area as outlinedin figure 3, flow from top to bottom. Every 5th frame is show. White outlines indicate area’senhanced in fig. 12 & 13

15

Estimate Velocity Collapse Front Midplane

0

10

20

30

40

50

60

70

0 1 2 3 4 5 6 7 8 9 10 11Time [ms]

Dist

ance

from

lead

ing

edge

[mm

]

0

1

2

3

4

5

6

7

8

9

Vel

ocity

[m/s

]

VelocityLocation

Figure 11 The location and velocity of the visible collapse front as visible in figs 10.5-10.10 determined byframe to frame analysis. Error bars indicate a 10 pixel error in the location of the collapse front.The front seems to accelerate linearly.

16

Figure 12 A close-up of figure 10 shows the formation of a large span wise vortex at 2,000 Hz. As themain sheet collapses, a trail of very small vortices is created, merging in several distinct largerstructures

17

Figure 13 Close-ups of fig. 10.6-9 including intermediate frames. The stream wise cavitating vorticesoriginate from a small vortex generated at the cavity collapse and are stretched around the spanwise vortices.

Figure 14 The stagnation point behind the cavity may grow into a stagnation line element during its collapseas it increasingly more resembles a two-dimensional sheet

18