HIGH PEFORMANCE FOIL SECTIONS WITH DELAYED CAVITATION INCEPTION

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1 Proceedings of the 3rd ASME/JSME Joint Fluids Engineering Conference July 8-22, 999, San Francisco, California, FEDSM HIGH PEFORMANCE FOIL SECTIONS WITH DELAYED CAVITATION INCEPTION Hajime Yamaguchi Dept. of Environmental and Ocean Eng. University of Tokyo Hongo 7-3-, Bunkyo-ku, Tokyo Japan Phone: ext.6536 Fax: Hiroharu Kato Dept. of Environmental and Ocean Eng. University of Tokyo Hongo 7-3-, Bunkyo-ku, Tokyo Japan Phone: ext.6535 Fax: Masatsugu Maeda Dept. of Environmental and Oce an Eng. University of Tokyo Hongo 7-3-, Bunkyo-ku, Tokyo Japan Phone: ext.6539 Fax: Makoto Toyoda Dept. of Environmental and Ocean Eng. University of Tokyo Hongo 7-3-, Bunkyo-ku, Tokyo Japan Phone: ext.6525 Fax: ABSTRACT Three foil sections with different thickness ratios are theoretically designed to delay cavitation inception and give high lift/ drag ratio. The thickness to chord ratios are 3.86%, 6.48% and 9.7%, respectively. Cavitation performance as well as noncavitating performance are examined by the noncavitating and cavitating foil theories, and by cavitation tunnel tests such as hydrodynamic force measurement and cavitation observation. From these test results, the foil performance and design method are evaluated as well as the accuracy of the theoretical prediction is discussed. It is found that the present designs have been successful. INTRODUCTION New trend of marine transportation driven by modal shift has led to increasing demand for high speed vessels such as car ferry of more than 3knots. Such a vessel should often have a shallow-immersion and highly-powered propeller which requires new design under severe condition against cavitation. On this background, the Japan Shipbuilding Research Association commenced a three-year research panel, SR23 [997, 998, 999], in 996 to investigate a new propeller design method. The authors carried out a fundamental part of that research program. They are theoretically developing new foil section series which can be applied for blade sections of those propellers. This paper describes the design and experimental verification of three foil sections which have been executed as a base for the study of developing new foil section series. Three foil sections are designed to delay cavitation occurrence and give high lift/drag ratio. Two of them are designed for lift coefficients and cavitation numbers at.7r and.5r positions of a typical propeller for such a high speed vessel, respectively. The other foil section is designed for an intermediate condition between these two. The foils are designed by the Eppler code [Eppler and Somers, 98] with prescribed pressure distribution and then empirically modified. Foil performance in noncavitating condition and incipient cavitation numbers are predicted by iterative calculations between a panel method and boundary layer calculation [Yamaguchi, 999]. Cavitation performance is predicted by a nonlinear panel method theory [Yamaguchi and Kato, 983] for partial cavitation and by a linearized singular distribution method [Kikuchi et al., 994] for supercavitation. Model tests of these foils are performed at a cavitation tunnel of the University of Tokyo [Kato et al., 98; Applied Fluid Engineering Laboratory, 998]. Hydrodynamic forces are successfully measured by a load cell, in addition to usual cavitation observa- Copyright 999 by ASME

2 Table Design Conditions for Three Foil Sections, SR23-NC, NC2 and NC3 Thickness Ratio as Corresponding Radius Position of an over-3 Cavitation Number Strength Requirement knots Ship Propeller SR23-NC %.7R SR23-NC %.5R SR23-NC % Intermediate Condition between NC and NC2 tion. The effectiveness of the present design is verified and issues to be solved in future are discussed. NOMENCLATURE c = Foil chord length. Cp = Pressure coefficient defined by uniform flow velocity and pressure. Cpmin =Minimum pressure coefficient on the foil surface. Cpsep = Pressure coefficient at laminar separation point. Cptr = Pressure coefficient at natural transition point. H2 = Shape factor of boundary layer = (displacement thickness) / (momentum thickness). Re = Reynolds number defined by uniform flow velocity and foil chord length. t/c = Foil thickness ratio. (X,Y) = Cartesian coordinate system nondimensionalized by the foil chord length. The origin (,) is the leading edge of the foil, while the point (,) is the trailing edge. Yc = Camber of the foil. Yt = Half thickness of the foil. α = Angle of attack. α = Zero-lift angle of attack. σ = Cavitation number defined by the uniform flow velocity and pressure, and vapour pressure. FOIL DESIGN The SR23 research panel assumed a 8m long high speed vessel with twin screws. A controllable pitch propeller was then designed by a propeller designer as a base for the improvement. Table shows the design conditions for the present three foil sections, SR23-NC, NC2 and NC3. Here "NC" means "noncavitating". Since the propeller is a controllable pitch one, blade area ratio is restricted, resulting in severe requirements of high lift coefficients and thickness ratios against low cavitation numbers. Although new supercavitating foil sections have also been designed and investigated [Yamaguchi et a., 997], only noncavitating foil sections are described here to meet the subject of the symposium. The foil sections are designed by the Eppler code [Eppler and Somers, 98], which gives noncavitating foil shape for given pressure distribution. This code is often used for foil section development since it can control the cavitation bucket by giving pressure distributions at different angles of attack. For example, Flow Outer Potential Flow Displacement Thickness of Boundary Layer Foil Section "Displacement Body" Fig. Displacement Body Concept Shen [985] used this code to develop a new foil section with delayed cavitation inception. Yamaguchi et al. [988a, 988b] applied this code for developing propellers with new blade sections to suppress and/or control cavitation. The present study is similar to that one. The design conditions, however, have become more severe to cavitation. Also, more accurate theoretical prediction method is adopted in this study. The aims of the present design are as follows: () to delay cavitation inception, and (2) to realize high lift/drag ratio. After section shape is designed by the Eppler code, aft part of thickness distribution is modified to thicken there using a similar method to that in the previous study [Yamaguchi, et al., 988a]. The Eppler code usually gives a section shape with unpractically thin aft part particularly for such severe design conditions since it always gives zero trailing edge angle as well as zero trailing edge thickness as a property of conformal mapping procedure. Then the foil performance in noncavitating condition, i.e. pressure distribution, lift and drag coefficients and boundary layer characteristics, are predicted by interactive calculations between outer potential flow and boundary layer based on the "displacement body" concept [Cebeci and Bradshaw, 977] as shown in Fig.. Since the boundary layer characteristics are determined by the pressure distribution which is obtained from outer potential flow solution, the calculation should be done in iterative manner. The pressure distribution with viscous effect taken into account is calculated iteratively by thickening the foil as much as the displacement thickness of boundary layer. Outer potential flow is computed by a panel method similar to Hess [972]. Boundary layer calculation is performed by a so-called integral method: the Thwaites method for laminar flow, the Cebeci and Smith equation for natural transition and the Head method for turbulent flow. All of them are described in a textbook by Cebeci and Bradshaw [977]. A FOR- TRAN77 program package of this computation can be downloaded as a freeware from the authors' Internet WEB page [Yamaguchi, 999]. 2 Copyright 999 by ASME

3 Y Y SR23-NC (Thickness Ratio = 3.86%) SR23-NC2 (Thickness Ratio = 9.7%) SR23-NC3 (Thickness Ratio = 6.48%) X:Y = : X:Y = : Fig. 2 Shape of Three Foil Sections Figure 2 illustrates the shape of three foil sections. Figure 3 and Table 2 denote the thickness and camber distributions. Figure 4 shows calculated pressure distributions at design angles of attack. As shown in Fig. 3, the present foils have low cambers near the leading edges and high cambers in the aft parts, while fore parts are thickened, resulting in wide flat pressure distribution area on the upper surface as shown in Fig. 4. Similar pressure distribution on the upper surface can be realized by NACA 6 or 66 foil section series. But the NACA foils are designed with a combination of thickness distribution of flat pressure and camber distribution of flat loading. Thus, there is a limitation of controlling the cavitation bucket in the NACA foil design. The present design method has more flexibility because it can control the pressure distribution directly. Larger thickness and lower camber near the leading edge than those of the NACA foils, are the results of widening and deepening the cavitation bucket. However, the leading edge radii of the present foils are rather smaller than those of the NACA foils. This is also the result of widening and deepening the bucket. Instead, the negative pressure peak near the leading edge grows faster than the NACA foils, when the angle of attack becomes too high or too low and goes out of the bucket bottom range. This means that the present foils would stall earlier than the NACA foils. This is a sacrifice due to Camber, Yc Half Thickness, Yt SR23-NC SR23-NC2 SR23-NC3.6 Half Thickness Distribution X:Yt = : SR23-NC SR23-NC2 SR23-NC3 Camber Distribution X:Yc = : Fig. 3 Comparison of Thickness and Camber Distributions Table 2 Maximum Thickness and Camber, and Their Positions SR23-NC SR23-NC2 SR23-NC3 Thickness Ratio 3.86% 9.7% 6.48% Maximum Thickness Position X=.2844 X=.4352 X=.3975 Camber Ratio.2%.2%.2% Maximum Camber Position X=.622 X=.7543 X=.74 the wider and deeper bucket. The stall characteristics are, however, not important in the present case since stall due to cavitation would occur much earlier than the stall due to noncavitating flow separation. Figure 5 compares boundary layer characteristics, using shape factor, H2, as an indicator. Since no leading edge separation is predicted for SR23-NC2 and NC3, slight negative pressure peaks at the leading edge shown in Fig. 4 do not cause cavitation occurrence. 3 Copyright 999 by ASME

4 Pressure Coefficient, Cp Pressure Coefficient, Cp Pressure Coefficient, Cp SR23-NC Angle of Attack = -.25deg. Reynolds Number =.5 6 =.77 Drag Coefficient =.53 = SR23-NC2 Angle of Attack =.99deg. Reynolds Number =.5 6 =.2892 Drag Coefficient =.53 = SR23-NC3 Angle of Attack =.56deg. Reynolds Number =.5 6 =.224 Drag Coefficient =.49 = Fig. 4 Calculated Pressure Distribution in Noncavitating Condition at Design Angle of Attack; The solid and broken lines denote pressure distributions with and without the boundary layer on the foil surface, respectively. Shape Factor of Boundary Layer, H2 Shape Factor of Boundary Layer, H Laminar Boundary Layer 3 SR23-NC (t/c=3.86%) SR23-NC2 (t/c=9.7%) SR23-NC3 (t/c=6.48%) Laminar Boundary Layer Turbulent Boundary Layer Upper Surface Natural Transition No Turbulent Separation (NC and NC2) Turbulent Separation at X=.994 (NC3) Lower Surface Natural Transition Turbulent Boundary Layer No Turbulent Separation Fig. 5 Calculated Boundary Layer Characteristics in Noncavitating Condition at Design Angle of Attack 4 Copyright 999 by ASME

5 Table 3 Comparison of Predicted and Measured Foil Performances in Noncavitating Condition Angle of Attack (deg.) Drag Coefficient Calculation Measurement Predicted Measured Predicted Measured Predicted Measured SR23-NC SR23-NC SR23-NC Although the measurement of hydrodynamic forces will be described later, Table 3 compares the predicted and measured foil performances in noncavitating condition. As shown in this table, the thinnest foil SR23-NC has showed slightly lower lift coefficient in the experiment than that predicted by the viscous/ inviscid flow interaction computation. The difference in lift coefficient corresponds to the difference of about.25deg. in angle of attack. Since the experimental error in adjusting the angle of attack is about.deg., this difference is not a measurement error. Actually, the SR23-NC was designed and tested in the first year of the project. The other two were done in the second year. Therefore, we investigated the pressure distribution and boundary layer characteristics in detail. The reason for this discrepancy is not necessarily clear, but one possibility is the boundary layer characteristics on the foil lower side. It is seen in Fig. 5 that transition to turbulence occurs more upstream compared to the other two foils. This is due to higher adverse pressure gradient on the lower surface of the SR23-NC (Fig. 4). Early transition on the lower surface leads to thicker boundary layer, resulting in loss of camber of the displacement body. This effect is, of course, taken into account in the theoretical prediction since viscous/inviscid flow interaction is computed in an iterative manner. However, this camber reduction effect might be more in the experiment than that in the computation. Based on this consideration, the lower side pressure distribution is improved in designing the SR23-NC2 and NC3 foils. As a result, fine agreement has been realized for these two foils not only at the design angles of attack but also at the other angles. This will be shown in Figs. 8 and 9 later. EXPERIMENT The experiments were performed at the Foil Test Section of the Marine Propeller Cavitation Tunnel, University of Tokyo [Kato et al., 98; Applied Fluid Engineering Laboratory, 998]. The working section was 6mm high, 5mm wide and,mm long. Turbulence level of the main flow was.2-.5%, and uniformity of the flow was about % at most. Stainless steel foil models of 5mm chord and 5mm span were manufactured by an NC machine and hand-finished with an accuracy of.5mm. Hydrodynamic forces were measured by a load cell with the arrangement shown in Fig. 6. A cantilever type foil model with disk was fixed to a 3-component load cell which measured lift Load Cell Flow.2-.3mm Foil Model 5.mm Flat 6mm Fig. 6 Arrangement of Foil Model and Load Cell to Measure Hydrodynamic Forces at the Cavitation Tunnel and drag forces and moment. As well known, it is very difficult to measure the drag force with such an arrangement since the drag is two-orders lower than the lift. Slight mis-arrangement can lead to a significant error. After several trials, accurate measurements have been realized by paying special attention to keeping side wall and disk as flat as possible and also keeping the gap between the foil tip and another side wall as short as possible. Also, an observation window on the foil tip side wall was completely fixed not to move due to pressure difference between inside and outside of the tunnel. Since the tip correction for drag coefficient [Treaster and Gurney, 985] is almost negligible for the present arrangement, the drag coefficients with no tip correction are shown in this paper. The mirror image effect due to upper and lower walls is also negligible since the working section is high enough compared to the chord length of the foil model. The experimental conditions are as follows: SR23-NC Uniform Flow Velocity (m/s) = Water Temperature ( C) = Air Content Ratio for Saturated Condition at atm. = 7-5% (Measured by a dissolved oxygen meter, D.O. meter) Cavitation Number, σ =.2 - Noncavi. 5 Copyright 999 by ASME

6 Reynolds Number, Re = ( ) 6 = -2., -.5, -., -.5,.,.5,.,.2,.5, 2., 3., 4. SR23-NC2 Uniform Flow Velocity (m/s) = Water Temperature ( C) = Air Content Ratio for Saturated Condition at atm. = 9 - % (Measured by a dissolved oxygen meter, D.O. meter) Cavitation Number, σ =.4 - Noncavi. Reynolds Number, Re = ( ) 6 = -3., -2., -.,.,.5,.,.5, 2., 2.5, 3., 4., 4.5, 5., 5.5, 6., 7. SR23-NC3 Uniform Flow Velocity (m/s) = Water Temperature ( C) = Air Content Ratio for Saturated Condition at atm. = 3-4% (Measured by a dissolved oxygen meter, D.O. meter) Cavitation Number, σ =.4 - Noncavi. Reynolds Number, Re = ( ) 6 = -3., -2., -.,.,.5,.,.5, 2., 3., 4., 5., 6., 7. FOIL PERFORMANCE IN NONCAVITATING CONDITION Figures 7-9 show lift and drag coefficients and lift/drag ratio of the three foil sections, comparing those between calculations and experiments. As aforementioned, measured lift coefficients of SR23-NC foil are a little lower than those predicted, while the other two foils show fine agreement not only on the lift coefficient but also on the drag coefficient and lift/drag ratio. Lift coefficient can be well predicted by a simple equation from thin wing theory when the angle of attack is low, although it is overpredicted for high angles of attack. The present computation follows the decrease in lift curve inclination as the angle of attack increases. This is because the boundary layer transition point moves upstream with increasing angle of attack. No significant trailing edge turbulent separation is predicted. At very high or low angles of attack, the iterative calculation could not converge since this computation method is based on boundary layer calculation. As shown in Table 3, high lift/drag ratios have been achieved by the present foil section designs. CAVITATION INCEPTION Incipient cavitation numbers were determined by visual observation. The values near the design points were as follows: SR23-NC:.5 (α=-.5deg.),.2 (α=.deg.) SR23-NC2:.42 (α=.deg.) SR23-NC3:.23 (α=.5deg.) All were bubble cavitation. Compared to the design cavitation numbers shown in Table, the SR23-NC3 did not cavitate as Drag Coefficient SR23-NC (t/c=3.86%) Calculation Thin Wing Theory (C L = 2π(α-α )) Experiment Design Angle of Attack = -.25deg. - 2 Design Angle of Attack = -.25deg Drag Coefficient Design Angle of Attack = -.25deg. Fig. 7 Comparison of Hydrodynamic Performance of SR23-NC Foil in Noncavitating Condition; Re = (.8-.29) 6 (Experiment),.5 6 (Calculation) Copyright 999 by ASME

7 Drag Coefficient Design Angle of Attack =.99deg Drag Coefficient SR23-NC2 (t/c=9.7%) Calculation Thin Wing Theory (C L = 2π(α-α )) Experiment Design Angle of Attack =.99deg Design Angle of Attack =.99deg Fig. 8 Comparison of Hydrodynamic Performance of SR23-NC2 Foil in Noncavitating Condition; Re = (.23-.3) 6 (Experiment),.5 6 (Calculation) Drag Coefficient Design Angle of Attack =.56deg SR23-NC3 (t/c=6.48%) Calculation Thin Wing Theory (C L = 2π(α-α )) Experiment Drag Coefficient Design Angle of Attack =.56deg Design Angle of Attack =.56deg Fig. 9 Comparison of Hydrodynamic Performance of SR23-NC3 Foil in Noncavitating Condition; Re = (.3-.22) 6 (Experiment),.5 6 (Calculation) 7 Copyright 999 by ASME

8 Calculation -Cpmin = -Cp at minimum pressure point -Cpsep = -Cp at laminar separation point -Cptr = -Cp at natural transition point -Cptr(face) = -Cptr on foil lower surface Experiment Incipient Cavitation Number (Sheet Cavity) Incipient Cavitation Number (Bubble or Streak Cavity) Re =.5 6 (Calculation) Re = ( ) 6 (Experiment) Design Angle of Attack = -.25deg. SR23-NC (t/c=3.86%) σ, -Cpmin, -Cpsep, -Cptr Re =.5 6 (Calculation) Re = ( ) 6 (Experiment) σ, -Cpmin, -Cpsep, -Cptr σ, -Cpmin, -Cpsep, -Cptr 3. Design Angle of Attack =.99deg. Re =.5 6 (Calculation) Re = ( ) 6 (Experiment) SR23-NC2 (t/c=9.7%) 4. Design Angle of Attack =.56deg. SR23-NC3 (t/c=6.48%) 3.5 Fig. Comparison of Cavitation Bucket Chart (a) Top View (b) Side View Fig. Bubble Cavitation; SR23-NC3, α=.5deg., σ=.9 expected. Very weak bubble cavitation was generated on the SR23-NC3, but they were insignificant. It is a little difficult to evaluate the SR23-NC. It did not cavitate at the design angle of attack of -.25deg. But it generated very slight bubble cavitation at the angle of attack of deg., where design lift coefficient was measured. All the data are summarized in Fig. as cavitation bucket charts. As well known, bubble cavitation or streak-like cavitation occurs at the bottom (left side in the figure) of the bucket. The streak-like cavitation occurs near the edges of the bucket bottom. Sheet cavitation is generated at high or low angles of attack. Even the thinnest foil, SR23-NC, has a wide bucket of about 2.5deg. In terms of comparison to the theoretical predictions, the buckets drawn by the minimum pressure are too narrow as is well known too. The bucket width determined by the pressure at laminar separation or natural transition points agrees with that of the experiments. CAVITATION APPEARANCE AND FOIL PERFORMANCE IN CAVITATING CONDITION Appearance of developed cavitation and foil performance variation due to cavitation are described here. Since all the foil sections showed the same trend, we show here only the results of SR23-NC3 foil. Figure shows typical bubble cavitation which appears at low angles of attack and very low cavitation numbers. Figure 2 demonstrates bubble and streak mixed cavitation which appears 8 Copyright 999 by ASME

9 Cavitation Number, σ Sheet Cavitation Super Cavitation Streak + Bubble Bubble Back Bubble + Face Cav. Face Cav. Design Angle of Attack =.56deg. Cavity Oscillation (a) Top View Fig. 4 Cavitation Diagram Denoting Types of Observed Cavitation in Accordance with Combination of (α, σ) (b) Side View Fig. 2 Bubble and Streak Cavitation; SR23-NC3, α=3.deg., σ=.24 (a) Top View (b) Side View Fig. 3 Sheet Cavitation; SR23-NC3, α=5.deg., σ=.89 at medium angles of attack and low cavitation numbers. One might be afraid if the streak cavity is originated by a small irregularity on the model surface. But such irregularity was not seen actually, and the positions of streak cavities were not always fixed. Therefore, this cavitation behaviour is inherent. Figure 3 shows clear sheet cavitation which appears at high angles of attack. Figure 4 shows the types of cavitation observed at all test conditions, (α, σ). Only bubble cavitation occurs around the design angle of attack, i.e. around the bottom of the cavitation bucket. At medium angles of attack of about 2-4 deg., sheet cavitation occurs first from the leading edge of the foil. As cavitation number is decreased, the sheet cavity develops and starts oscillating when its termination position approaches the foil trailing edge. Then the sheet cavitation, partial cavitation, does not become supercavitation but turns into bubble cavitation or bubble/streak combined cavitation. This is because the effective angle of attack of the cavity/foil combined body is reduced. It is still difficult to predict this phenomenon theoretically. At high angles of attack of more than about 5deg., the partial cavitation turns into supercavitation keeping a sheet appearance. Figures 5 and 6 compare the variations of cavity termination points and lift coefficients between theory and experiment. Also shown are regions of observed cavitation types whose approximate boundaries are drawn by thin dotted lines. Computations are made by a nonlinear panel method theory [Yamaguchi and Kato, 983] for partial cavitation and by a linearized singular distribution method [Kikuchi et al., 994] for supercavitation. Since both of them are free-streamline flow theories, fairly good agreement is obtained in the regions of supercavitation and partial cavitation, although data points in the latter region are not many. It is rather normal that calculated results are far from the measured ones in the other regions, since cavitation types assumed 9 Copyright 999 by ASME

10 (Cavity Termination Point)/(Foil Chord Length) T.E σ=.2(exp.) σ=.26(exp.) σ=.3(exp.) σ=.35(exp.) σ=.4(exp.) σ=.5(exp.) σ=.7(exp.) Thick Lines = Supercavitating Foil Theory (Thin Wing Theory) Thin Lines = Partially Cavitating Foil Theory (Thick Wing Theory) σ =.26 σ =.3 Bubble σ =.35 σ =.4 σ =.5 σ =.7 Design Angle of Attack =.56deg. Bubble +Streak Super Cavitation Fig. 5 Comparison of Cavity Termination Point between Experiment and Predictions using Free Streamline Flow Theories; SR23-NC2; Thin dotted lines denote the region of each cavitation type observed. in the theories are different from the actual ones. Although it is still difficult to theoretically predict the cavitation behaviour and foil performance in these regions of bubble or bubble/streak combined cavitation, the foil performance becomes in between those in noncavitating condition and calculated by a free-streamline flow theory. Although the figures are not shown here, the maximum lift/drag ratios of all the three foils are obtained in bubble cavitation condition. They are: SR23-NC: Lift/drag ratio = 59 at α=.5deg. and σ=.2. SR23-NC2: Lift/drag ratio = 69 at α=.5deg. and σ=.36. SR23-NC3: Lift/drag ratio = 6 at α=.5deg. and σ=.3. These maximum lift/drag ratios are actually higher than those of design points shown in Table 3 and also even maximum values in noncavitating condition shown in Figs This is because the lift coefficients are larger than those at the design points, although σ =.2 σ =.26 σ =.3 σ =.35 σ =.4 Oscillation Partial Cavitation SR23-NC3 (t/c=6.48%) L.E σ=.2(exp.) σ=.26(exp.) σ=.3(exp.) σ=.35(exp.) σ=.4(exp.) σ=.5(exp.) σ=.7(exp.) Partial Cavitation Bubble σ =.26 σ =.35 σ =.3 σ =.4 Bubble +Streak Design Angle of Attack =.56deg. Thick Lines = Supercavitating Foil Theory (Thin Wing Theory) Thin Lines = Partially Cavitating Foil Theory (Thick Wing Theory) σ =.7 σ =.5 Oscillation Super Cavitation σ =.4 σ =.35 σ =.3 σ =.26 σ =.2 SR23-NC3 (t/c=6.48%) Fig. 6 Comparison of between Experiment and Predictions using Free Streamline Flow Theories; SR23-NC2; Thin dotted lines denote the region of each cavitation type observed. drag coefficients are increased a little by the occurrence of cavitation. CONCLUSIONS Three foil sections with different thickness ratios were designed to delay cavitation inception and give high lift/drag ratio. The thickness to chord ratios were 3.86%, 6.48% and 9.7%, respectively. Cavitation performance as well as noncavitating performance were examined by noncavitating and cavitating foil theo- Copyright 999 by ASME

11 ries, and by cavitation tunnel tests such as hydrodynamic force measurement and cavitation observation. From these test results, the foil performance and design method were evaluated as well as the accuracy of the theoretical prediction was discussed. The following conclusions were obtained.. A method to measure foil hydrodynamic forces accurately has been developed at the University of Tokyo's cavitation tunnel. 2. Noncavitating foil performance such as lift and drag forces is well predicted by a panel method / boundary layer iteration computations with "displacement body" concept. The thinnest foil showed slightly lower lift force than the theoretical prediction. The reason for this is not necessarily clear, but might be early boundary layer transition due to adverse pressure gradient at lower surface which is higher compared to those of the other foils. This should be taken into consideration in future design. 3. Sheet cavity inception can be predicted by the pressure at the leading edge separation point when the pressure distribution is computed with boundary layer effect taken into account. Bubble cavitation inception nearly coincides with -Cpmin, absolute value of the pressure coefficient at the minimum pressure point. Thus, cavitation buckets are wider than those drawn by -Cpmin and should be discussed with viscous effects taken into consideration. 4. Foil performance in sheet cavitation condition can be predicted fairly well by free-stream line theories. It is difficult to predict foil performance in severe bubble cavitation condition with very low cavitation numbers. ACKNOWLEDGMENTS The authors thank Mr. M. Tanimura, who conducted the experiment on the SR23-NC foil section. This study is executed as a part of the SR23 Research Panel program under Japan Shipbuilding Research Association, subsidized by The Nippon Foundation. The authors acknowledge the persons concerned. REFERENCES Applied Fluid Engineering Laboratory, 998, Marine Propeller Cavitation Tunnel, ~yama/cavtun_www/marineprop/index-e.html. Cebeci, T. and Bradshaw, P., 977, Momentum Transfer in Boundary Layers, Series in Thermal and Fluids Engineering, Hemisphere Publishing Co., McGraw-Hill Book Company. Eppler, R. and Somers, D.M., 98, A Computer Program for the Design and Analysis of Low-Speed Airfoils, NASA Technical Memorandum 82, 77p. + FORTRAN program list. Hess, J.L., 972, Calculation of Potential Flow about Arbitrary Three Dimensional Lifting Bodies, Douglas Report MDC- J5679. Kato, H., Watanabe, Y., Komura, T., Maeda, M. and Miyanaga, M., 98, New Marine Propeller Cavitation Tunnel at the University of Tokyo, J. Soc. Nav. Archi. Jpn., Vol. 5, pp (in Japanese) Kikuchi, Y., Kato, H., Yamaguchi, H. and Maeda, M., 994, Study on a Supercavitating Foil, Proc. 2nd Intern. Symp. Cavitation, Tokyo, pp Shen, Y.T., 985, Wing Sections for Hydrofoils - Part3: Experimental Verifications, J Ship Res., SNAME, Vol.29 No., pp SR23, 997, 998, 999, Studies on High-powered Propeller for Shallow Draft Ships, Ship Research Panel 23, Annual Reports of Japan Shipbuilding Research Association. (in Japanese) Treaster, A.L. and Gurney, G.B., 985, Sidewall Boundary- Layer Corrections in Subsonic, Two-Dimensional Airfoil/Hydrofoil Testing, J. Aircraft, Vol.22 No.3, pp Yamaguchi, H. and Kato, H., 983, On Application of Nonlinear Cavity Flow Theory to Thick Foil Sections, Proc. 2nd Conf. Cavitation, IMechE, Edinburgh, pp Yamaguchi, H., Kato, H., Kamijo, A. and Maeda, M., 988a, Development of Marine Propellers with Better Cavitation Performance (2nd Report: Effect of design lift coefficient for propellers with flat pressure distribution), J. Soc. Nav. Archi. Jpn., Vol. 63, pp Yamaguchi, H., Kato, H., Sugatani, A., Kamijo, A., Honda, T. and Maeda, M., 988b, Development of Marine Propellers with Better Cavitation Performance (3rd Report: Pressure distribution to stabilize cavitation), J. Soc. Nav. Archi. Jpn., Vol. 64, pp Yamaguchi, H., Kato, H., Tanimura, M. and Toyoda, M., 997, A Study on Blade Sections for a Trans-Cavitating Propeller, J. Soc. Nav. Archi. Jpn., Vol. 82, pp (in Japanese) Yamaguchi, H., 999, A Computer Program for Predicting the Hydrodynamic Characteristics of a Two-Dimensional Foil or Cascade in Steady Flow with Boundary Layer Effects Taken into Account - Outline of the "prblg.f" Program - (with freeware package), Copyright 999 by ASME