When a sheet cavity on a hydrofoil section attains a certain size, it starts violent periodical oscillation shedding a harmful cloud cavity downstream at each oscillation cycle. 
This phenomenon is due to the occurrence of the re-entrant jet. In this paper, the behavior of the re-entrant jet was observed in detail using a transparent foil section model and a high-speed video camera. Time variation of pressure distribution on the foil was measured simultaneously. 
It was found that the re-entrant jet can start at any point in sheet cavity elongating stage. Even two re-entrant jets can appear in one cycle. 
When a re-entrant jet is generated upstream, the jet velocity is lower compared to the case when a re-entrant jet is generated downstream. The jet velocity is almost constant at the value determined by the location of the generation. 
As a result, the cavity oscillation cycle becomes constant when it is normalized by the sheet cavity surface velocity and the maximum sheet cavity length. 
The jet velocity is calculated from the pressure gradient at the sheet cavity T.E., using a simple theoretical model. The calculated jet velocity agrees with the measurement, showing that the jet velocity increases as its generation point shifts downstream. It is possible that pressure gradient at the sheet cavity T.E. is the driving force of re-entrant jet

Maeda, Masatusgu

Sakoda, Motoyuki

Yakushiji, Ryo

Yamaguchi, Hajime

English

CaltechCONF

CAV2001:sessionA9.004 1MECHANISM OF CLOUD CAVITATION GENERATIONON A 2-D HYDROFOILMotoyuki SAKODA Ryo YAKUSHIJI Masatsugu MAEDAHajime YAMAGUCHIDepartment of Environmental and Ocean Engineering,Graduate School of Engineering, University of Tokyo,7-3-1 Hongo Bunkyo-ku, Tokyo, 113-8656 JAPANAbstractWhen a sheet cavity on a hydrofoil section attains a certain size, it starts violent periodical oscillationshedding a harmful cloud cavity downstream at each oscillation cycle. This phenomenon is due to theoccurrence of the re-entrant jet. In this paper, the behavior of the re-entrant jet was observed in detail usinga transparent foil section model and a high-speed video camera. Time variation of pressure distribution onthe foil was measured simultaneously. It was found that the re-entrant jet can start at any point in sheetcavity elongating stage. Even two re-entrant jets can appear in one cycle. When a re-entrant jet is generatedupstream, the jet velocity is lower compared to the case when a re-entrant jet is generated downstream. Thejet velocity is almost constant at the value determined by the location of the generation. As a result, thecavity oscillation cycle becomes constant when it is normalized by the sheet cavity surface velocity and themaximum sheet cavity length. The jet velocity is calculated from the pressure gradient at the sheet cavityT.E., using a simple theoretical model. The calculated jet velocity agrees with the measurement, showingthat the jet velocity increases as its generation point shifts downstream. It is possible that pressure gradientat the sheet cavity T.E. is the driving force of re-entrant jet.NOMENCLATUREC Chord length of hydrofoil (m) σ cavitation numberU∞ Uniform ﬂow velocity (m/s) σ = (P∞ − Pv)/ 12ρU2∞ (-)P∞ Referece static pressure (Pa) L Sheet cavity length (m)Pv Vapor pressure of water (Pa) Lmax Maximum sheet cavity length (m)x Distance from the cavity L.E. (m)ρ Density of water (kg/m3) Vj Velocity of re-entrant jet (m/s)1 IntroductionWhen a sheet cavity on a body such as hydrofoils attains a certain size, it starts violent periodical oscillationshedding a harmful cloud cavity downstream at each oscillation cycle.One of the most well-known factors of this phenomenon is re-entrant jet, which runs from downstreamto upstream along the bottom of the sheet cavity, i.e. on the hydrofoil surface covered by the sheet cavity.Concerning the re-entrant jet, various researches have been made, e.g. visualization of cavity closure usingdye injection (Le et al., 1993) and high-speed cinematography (De Lange et al., 1994). Kawanami et al.(1996) reported that cloud cavitation was not generated on the foil with obstacle which blocked the jet.Recently several investigations were conducted to know the characteristics, especially velocity, of re-entrant jet. Kawanami et al. (1996) measured re-entrant jet velocity electrically and reported that it wason the same order of magnitude as the main ﬂow velocity. Mathieu C. et al.(1998) estimated re-entrant jetCAV2001:sessionA9.004 2velocity by visualization and pointed out that the re-entrant jet showed diﬀerent behaviors depending onthe thickness of the sheet cavity. Tuyeˆt M.P.(1998) measured the local mean velocity of the re-entrant jetusing the electric probe and reported that its value increased with the distance from the L.E.These reports are very interesting, however, information about the re-entrant jet is not enough to clarifythe mechanism of cloud cavitation generation. The purposes of the present work are to understand thecharacteristics of the re-entrant jet more in detail and to discuss its generation mechanism. For thesepurposes, we carried out experiments to locate the starting point and to measure the velocity of the re-entrant jet, and ﬁnally we propose a simple theoretical model to discuss the eﬀect of the pressure gradientas the driving force of the re-entrant jet.2 Experimental Setup and ConditionsThe experiments were carried out in the Marine Propeller Cavitation Tunnel, the University of Tokyo. Thetank has a 1.0m long rectangular test section of 0.15 × 0.60 m. The proﬁle of tested foil was NACA 0015.The model foil was 0.15m in chord length and 0.15m in span length. One hydrofoil was made of acrylicresin(Figure 1-(a)), so that one could observe the cavity and the re-entrant jet from pressure side of thehydrofoil. In order to measure the time variation of the sheet cavity length and the position of re-entrantjet, the state of cavity was ﬁlmed by a high-speed video camera with a frame rate of 9,000 frames per second(Figure 1). The other hydrofoil was made of brass(Figure 1-(b)), where six pressure pick-ups were installedon its back from 20%C to 70%C at each 10%C. Time variation of pressure distribution on the hydrofoil wasmeasured simultaneously with the high-speed video records. The experimental conditions were determinedso that re-entrant jet could be observed clearly, as follows:angle of attack α = 8◦(ﬁxed)uniform flow velocity U∞=5m/s, 8m/scavitation number σ=1.4, 1.6Flowacrylic resinL.E.T.E.(a) foil model for observation and image analysis(b) foil model equipped with 7 pressure pick-ups 20%C70%C60%C50%C40%C30%Cspan centerbrasspressure pick-up (?= 6.2mm)velocity measurement point static pressuremeasurement pointhydrofoil(NACA0015)mirrorhigh-speed video camerabottom view20mm 140mm150mmcavitymirrorhigh-speed video camerain case of foil (b)in case of foil (a)top viewFigure 1: Experimental setup and foil modelsCAV2001:sessionA9.004 33 Results and Discussion3.1 High-Speed Video ObservationFigure 2 shows high-speed video images that were ﬁlmed from the pressure side of acrylic foil model. InFigure 2 white arrows represent front line of re-entrant jet.After the separation of sheet cavity (generation of cloud cavity), a new sheet cavity appears near the foilL.E. (Figure 2-(a)), and it grows with the time(Figure 2-(b), (c)). Figure 2-(d) is a image immediately aftergeneration of re-entrant jet. The re-entrant jet runs from downstream to upstream(Figure 2-(e)). In thiscase, two re-entrant jets are observed (Figure 2-(f)). The second re-entrant jet also moves from downstreamto upstream, however it seems to be faster than the ﬁrst one (Figure 2-(g)). The sheet cavity begins tocollapse when the re-entrant jets reach the leading edge of the sheet cavity (Figure 2-(h)).Uniform Flow(a) t=0.0033(sec)(b) t=0.0099(sec)(c) t=0.0165(sec) (g) t=0.0429(sec)(h) t=0.0464(sec)(d) t=0.0231(sec)(e) t=0.0297(sec)(f) t=0.0363(sec)Cloud cavitySheet cavityL.E.T.E.Sheet cavity T.E. Sheet cavity L.E.Re-entrant jetSecond re-entrant jetFigure 2: Typical pattern of cavity bottom view in one cycle(U∞=5m/s, σ=1.4)CAV2001:sessionA9.004 43.2 Behavior of Re-entrant JetIn order to know the behavior of re-entrant jet in detail, we captured video images every 10 frames(∼= 1msinterval) and measured the length of the sheet cavity and the position of the re-entrant jet from these images.A total of 131 periods were investigated, and re-entrant jets were observed in 93 periods, which are 70.2%of the investigated periods. More than one re-entrant was sometimes observed in one oscillation cycle. Asingle jet was observed in 69 periods (51.9%), while two jets were observed in 24 periods (18.3%). Someexamples of the time variation of the length of the sheet cavity and the position of the re-entrant jet in onecycle are shown in Figure 3. In Figure 3 the vertical axis shows a distance from the sheet cavity L.E., andthe horizontal axis is the time from the release of the former cavity.010203040506001020304050600102030405060x/C(%)0 0.01 0.02 0.03 0.04 0.05time (sec)Sheet cavityRe-entrant jet1Re-entrant jet20102030405060x/C(%)x/C(%)x/C(%)0 0.01 0.02 0.03 0.04 0.05time (sec)0 0.01 0.02 0.03 0.04 0.05time (sec)0 0.01 0.02 0.03 0.04 0.05time (sec)Sheet cavityRe-entrant jet1Re-entrant jet2Sheet cavityRe-entrant jet1Re-entrant jet2Sheet cavityRe-entrant jet1Re-entrant jet2re-entrant jetSheet cavityEstimeted generation pointSecond re-entrant jetOne oscillation cycle (a) (b)(c) (d)Figure 3: Examples of time variation of cavity length and re-entrant jet position in one oscillation cycle(U∞=5m/s, σ=1.6)Comparing Figure 3-(a) with (b), it appears that the re-entrant jet can start at any point in the sheetcavity elongating stage even under the same experimental condition. In the case of two jets (see Figure 3-(c),(d)), the second re-entrant jet is faster than the ﬁrst one. It is noticed that each re-entrant jet moves almostlinearly toward the upstream direction at a constant speed. This agrees with the results of Mathien C. etal. (1998).The above results suggests that the velocity of a re-entrant jet is correlated to its starting position. Weestimated the generation points of re-entrant jets by ﬁnding the cross point of the cavity end and jet trace(seeEstimated generation point in Figure 3).The measured jet velocities (Vj) are plotted against the estimated position of the generation point (x/C)in Figure 4-(a). It is noticed that re-entrant jets can appear at any point in the sheet cavity elongating stage,and that the jet speed generally increases as its generation point shifts toward the downstream direction.Jakobsen(1964) approximated the velocity at the cavity surface by U∞√1 + σ, and Kawanami(1999)showed that Strouhal number based on the cavity surface velocity, the shedding frequency of cloud cavityand the maximum sheet cavity length was almost constant. The above reports, it is assumed that velocity ofCAV2001:sessionA9.004 5012345678910 20 30 40 50 60 70Vj (m/s)x/C (%)00.20.40.60.811.20.2 0.4 0.6 0.8 1x/Lmaxnon-dimensional VU??5m/s, ??1.4U??5m/s, ??1.6U??8m/s, ??1.4U??8m/s, ??1.6(a) (b)*U??5m/s, ??1.4U??5m/s, ??1.6U??8m/s, ??1.4U??8m/s, ??1.6 Eq. (1)Figure 4: (a): Re-entrant jet velocity (Vj) VS. its generation point(x/C), (b): Non-dimensional velocity(V ∗= VjU∞√1+σ ) VS. non-dimensional generation point of re-entrant jet (normalized by maximum cavitylength: Lmax)the jet has a close relation to its generation position. Actually when the jet velocity normalized by the cavitysurface velocity is plotted against its generation point normalized by maximum cavity length(Figure 4-(b)),all points seems to fall onto a single curve as a whole. It is satisfactory to consider that the jet velocityis almost constant at the value determined by the location of its generation, so the cavity oscillation cyclebecomes constant when it is normalized as above. We propose the following empirical formula (1) for thenormalized jet speed(Vj) as a function of the normalized distance(x/Lmax) from the L.E. to the generationpoint of the jet.VjU∞√1 + σ= A tan(π2xLmax)+B sin(π2xLmax)A = 0.0721916, B = 0.200598 (1)3.3 Eﬀect of Pressure GradientThen, why does the jet velocity increase as its generation point shifts downstream? The measurement ofthe pressure distribution conducted in the present work showed that pressure gradient at the cavity T.E.became steeper as the cavity became longer. We assumed that pressure gradient at the sheet cavity T.E.was driving force of re-entrant jet.Figure 5 shows measured pressure distribution on hydrofoil at the time sheet cavity length was 25%C,35%C, 45%C and 55%C. A result of BEM calculation with boundary layer eﬀect is also shown in Figure 5.Figure 5 dipicts that Cp is almost equal to −σ on the upstream side of the sheet cavity T.E. It indicatesthat the sheet cavity is ﬁlled with vapor, because Cp is deﬁned as follows:Cp =P − P∞1/2ρU2∞(2)On the other hand, measured Cp at about 5%C downstream of the sheet cavity end is lower than thecalculated one. This indicates the existence of a stagnant pool. Thus, the pressure gradient at the sheetcavity T.E. becomes steeper as the cavity becomes longer.So, we calculated the jet velocity from the measured pressure gradient at cavity T.E. using a simplemodel. The procedure of the estimation is mentioned below. We assumed the following conditions (seeFigure 6):1. There is a stagnant pool of l in chordwise length at the sheet cavity T.E.CAV2001:sessionA9.004 6-?= -1.4-3-2-10120 10 20 30 40 50 60 70 80 90 100Cpx/C (%)Pressure gradient    at cavity T.E. BEM calculation with boundary layer effect25%C35%C45%C55%CCavity T.E. atFigure 5: Measured pressure distribution on hydrofoil in various elongating stage of sheet cavity (U∞=8m/s,σ=1.4)-?Flowx/CPressure distribution in cavitating conditionPressure distribution in non-cavitating conditionSheet cavityCp-+Stagnant poolSheet cavity growth rate : Vs = dtdLjet speed :  VjLength of StagnantydxP(x) P(x+dx)VjBSpanwiseChordwiseConstant accelationConstant velocityElement modelxlFigure 6: Calculation model2. The pressure gradient around the sheet cavity T.E. is constant.3. The ﬂuid element is accelerated constantly by the pressure gradient.4. The ﬂuid element moves at constant velocity inside the sheet cavity.5. There is no viscous eﬀect.The equation of motion of the ﬂuid element isρBydxdVjdt= yB{P (x) − P (x+ dx)} (3)CAV2001:sessionA9.004 7On the other hand, partial derivative of Cp(x) with respect to x is∂Cp(x)∂x=112ρU2∞∂P (x)∂x(4)By solving (3) and (4) as simultaneous equations, Vj is described as follows:−dVjdt= −U2∞2∂Cp(x)∂x(5)Using the above assumption 2, the pressure gradient at the cavity T.E. is approximated with:∂Cp(x)∂x=σ + Cp(x+ l)l(6)Then, (5) is rewritten as follows:−dVjdt= −U2∞2σ + Cp(x+ l)l(7)The time (τ) until the ﬂuid element reaches the cavity end is given by(∫ τ0Vjdt)+ Vsτ = l (8)If (8) is solved for τ and (7) is integrated with respect to t from 0 to τ , ﬁnally V j is described as follows:Vj = −Vs +√V 2s − U2∞(Cp(x+ l) + σ) (9)We estimated the jet veloicity by substituting the measured values to the RHS of the Equation (9) assumingl to be 5%C. The results are shown in Figure 7.Comparing the estimation with the experimental results, the estimated value of jet is on the same orderof magnitude as the experiment results. Moreover the estimated jet velocity shows the same tendency as theexperiment that the jet velocity increases with the increasing distance from the L.E. to its generation point.So, it is possible that the pressure gradient at the sheet cavity T.E. is the driving force of the re-entrant jet.The diﬀerence between the estimation and the experiment, expect for case (b), increases as the jetgeneration point shifts upstream. This might be due to the hypothetic value of l, which was given as aconstant value for all conditions. It is considered that the value of l requires some correction correspondingto the size of the sheet cavity. In case (b), measured Cp were rather scattered, which might be inﬂuence ofthe cloud cavity. So the pressure gradient at the cavity T.E. turned gentle and the estimated jet velocitybecame lower than those of other cases.4 Concluding RemarksIn the present work, experiments were performed to clarify the mechanism of the generation of cloud cavi-tation. The main reults are summarized as follows:• The re-entrant jet can start at any point in sheet cavity elongating stage. Even two re-entrant jets canappear in one cycle.• When a re-entrant jet is generated upstream, the jet velocity is lower compared to the case when are-entrant jet is generated downstream.• The jet velocity is almost constant at the value determined by the location of its generation. As aresult, the cavity oscillation cycle becomes constant when it is normalized by the sheet cavity surfacevelocity and the maximum sheet cavity length.• A simple model for the jet velocity has been proposed based on the pressure gradient. The jet velocitypredicted by the model agreed with the measurement, suggesting that the pressure gradient at theT.E. of the sheet cavity is the driving force of the re-entrant jet.CAV2001:sessionA9.004 800.20.40.60.810 0.2 0.4 0.6 0.8 1x/Lmax00.20.40.60.810 0.2 0.4 0.6 0.8 1x/Lmax00.20.40.60.810 0.2 0.4 0.6 0.8 1x/Lmax00.20.40.60.810 0.2 0.4 0.6 0.8 1x/Lmaxestimationexperimentestimationexperimentestimationexperimentestimationexperimentnondimentional V*nondimentional V*nondimentional V*nondimentional V*(a)  U?= 5m/s, ? = 1.6?(d)  U?= 8m/s, ? = 1.4?(c)  U?= 8m/s, ? = 1.6?(b)  U?= 5m/s, ? = 1.4?Figure 7: Comparison of re-entrant jet velocity between experiment and estimationReferencesLe Q., Franc J.P. and Michel J.M. (1993). ASME Journal of Basic Engineering, 115, 243-248.Kawanami Y., Kato H., Yamaguchi H., Tagaya Y. and Tanimura M. (1996). Proc. FEDSM‘96, FED-236,329-336.De Lange D.F., DeBruin G.J. and Van Wijngaarden L. (1994). Proc. 2nd Int. Symp. on cavitation, 45-49.Jakobsen. J.K. (1964). ASME Journal of Basic Engineering, Vol.86, 291-305.Mathieu C., Jean-Pierre F. and Jean-Marie M. (1998). Proc. 3rd Int. Symp. on cavitation, Vol.1, 209-241.Tuyeˆt M., Fre´de´rique L. and danie H.F. (1998). Proc. 3rd Int. Symp. on cavitation, Vol.1, 215-219.Kawanami Y. (1999). Ph.D thesis, 70-76(in Japanese).

Mechanism of Cloud Cavitation Generation on a 2-D Hydrofoil

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