The Proceedings: Fifth International Conference on Numerical Ship Hydrodynamics (1990)

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Theoretical Prediction of Midchord and Face Unsteady Propeller Sheet Cavitation S. A. Kinnas and N. E. Fine Massachusetts Institute of Technology Cambridge, USA Abstract In Ells worl;, first the linearized hydrofoil problem with arbitrary ca.vit~ deta.cl~ment points is formulated in terms of unl;no~`n source and vorticity distributions Tl~e corresponding integral equations are inverted analyt- ica.lly and the results are expressed in terms of integrals of qua.r~tities which depend only on the l~yclrofoil shape. Then, the cavitating hydrofoil problem is solved nu- merically bar cliscretizing Else problem into point source and vortex distributions and lay applying the boundary conditions at appropriately selected collocation points. I;`inally, the disrete vortex and source method is ex- tendecl to predict unsteady propeller sleet cavitation with arbitrary miclchord a,ncl/or face deta,cl~ment. ~ Introcluction Helmholtz and I~irchoff t:3] more tl~a.n a. century ago. The analysis of cavitating flows at non-zero cavitation numbers created a lot of dixer.sitv on the cavity termina,- tion models, i.e. Else Riabo~chinsl;y model t_S], the reen- tra.nt jet model t7], t94], the spiral vortex models t394~ etc. A complete description of the different cavity termination models can be found in t394 a,ncl t3S]. The difficulty of the hodogra,ph technique to treat general shaped bound- aries necessitates the introduction of tl~e li~,ea.rized cavity theory. Linear theory was first applied by rI'uliI1 t304 to the problem of a. superca.vita,ting symmetric section at zero incidence and zero cavitation number. It was then a.p- plied to general camber mea,~lines at zero cavitation num- ber t33], and to a. sl~pelca.vita.ting flat plate at incidence and a.rbitraly cavitation numbers (314. Linear theory Divas slll~seq~lentl:! ext.elldecl to supercav- itating hydrofoils of general shape at non-zero cavitation numbers [39], (11], t9~. [S], [~>9~. ~ . . . .. . Cavitation has always been a great concern in the de- sign of marine propellers. A successful propeller design ~ he partially cavitating hydrofoil problem has also is one which precludes cavitation at design conditions. been addressed in linear theory and analytical results In recent times, however, with an increasing demand for have been produced for some s~ec.i,~.l h ~ cl, Sail ~~m~i ~ ins higher propeller loadings and higher c~ciencies. the r~ro- peller cavitation is fiery often unavoiclable. The tasI; of the hyd~odyna.micist is, tl~ercfore, to predict,and con- trol the propeller cavitation and its undesirable side ef- fects. An analysis method for the prediction of unsteady propeller cavitation is, therefore, an indispensable de- si,gn tool. Furthermore. tlli.~; 1~ro~ll~l rnlrit~timn ~1 [1], [13] 7 [19], [14], [37]. The problem of a superca.~ril;atiIlg he dlofoil with arl~i- trary cavity detachment was first formulated by Fab~lla [S], who also gave results for ~ flat plate with different detachment points. Hanaol;a [15] formulated the linearized partial and su- ~ ~ _ ~ via Amp 1~~ An l~lUl1 all<(,l- percavita.ting hydrofoil problem faith al bitI ply cavit ~ de- ysis method should lee able to treat cavities ~ hich start tachment. He also gave series representations for the cav- on the suction side behind the leading edge towards the ita,tion number and the h~drody!na~nic coefficients when blade midchord and/or on the pressure side, "face", of the hydrofoil slope could lee express d in therms of poly- the propeller in front of the blade trailing e<lge, since nomials in the chord~`ise <oo~clinate. Nishivama End these types of cavitation are very Ijl;rl~r to arm at tl~^ design conditions. Cavitating or free-st.realulille flows have been studied extensively in the last century. First, the flow around flat plates or pole gonal bodies at zero cavitation n~ml~er was An alternative `` ad of form~la.ting the linearized pa.r- analyzed. The analysis of these problems was achieved tially and supercaviiating problem has been given in t91] bar applying the hodogra.ph technique as introduced lay and t204. The linearized boundary conditions hare been trary detachment. Vita leg] also gave integral expressions in terms of l;no~`rn quantities, for the cavitation n~n-~l~< r and the hydrody- na,mic coefficients for general shape l~yd~ofoils with arl~i- 685

expressed in terms of singular integral equations of un- l;nown source and vorticity distributions. Those integral equations are inverted analytically and expressions for the cavitation number, the source and vorticity distribu- tions are given in terns of integrals of functions which depend only on the geometry of the hydrofoil. Those integrals are then computed numerically and the cavity shapes are finallly computed t21], t904. The same tech- nique has also been extended for pa.rtia.1 cavities with arbitrary deta.chn~ent t93~. The leading edge correction has also been implemented in the formulation of the cav- itating hydrofoil problem to account for the non-linear foil thickness effects t2.34. In the present work, the technique used in t214 for supercavitating hydrofoils is extended to treat superca`- and ities with arbitrary detachment on either Else suction side and/or the pressure side of the l~drofoil. The effect of the detachment point on the cavity shapes and foil pres- sure distributions is investigated. In the case where the supercavity detaches on the pressure side in front of the trailing edge, an equation for the chordwise location of the cavity detachment point is given. The cavitating hydrofoil problem with arbitrary suc- tion and/or pressure cavity detachment is then solved by employing a disrcete vortex and source method. A numerical vortex and source lattice method has been developed at WIT for the prediction of the unsteady propeller sheet cavitation in spatially non-uniform wakes [25i, t4], [IS]. The computer program which implements this method is called PUF-3. Finally, PUF-3 is modified to predict unsteady pro- peller sheet cavitation with arbitrary detachment on ei- ther the pressure or the suction side of the propeller. The effect of the location of the cavity detachment on the time history of the cavity voluble and the car itv shapes is in- vestiga.ted. 2 The Cavitating Hyclrofoi! -The Analytical Method In this section, the linearized cavitating hydrofoil prol'- lem is formulated in terms of unl;no~vn vorticity and source distributions. For given cavity length and spec- ified cavity detachment points, the involved singular in- tegra.1 equations are inverted analytically. Expressions are then found for the corresponding cavitation launder, vorticity and source distributions in terms of integrals of quantities which depend only on the foil geometry. First, the superca.vitating hydrofoil problem for three different cavity detachment situations is considered. 2.1 Leading Eclge Detachment Consider a hydrofoil of chord length one, subject to a uniform flow UOO and superca`;itating at a length x = I, as shown in Figure 1. The cavity starts at the leading edge x = 0 on the suction side and at the trailing x = 1 on the pressure side. 686 The corresponding cavitation nuttier ~ is defined as: ~ eU2 (1) where pOO is the ambient pressure and PI the vapor pres- sure inside the cavity. In the context of the linearized cavity theory the cor- responding Hilbert prol~lem can be formulated t91] in terms of vorticit.y and source distril~tions Eye) and qtx) respectively, located on the x axis as shown in Figure 1. With the use of the definitions: _ Add i= tT 4(x) = At ), (3) the complete boundary value problem becomes t914: v Ye ' Ax) LU Ian ~ , ~ x o ll A I us = O2 up I 1 - ~ ~ ~ ~.L~ u-=O2UOO 1 v-=u-0 ~ 1 l\~7(x) 1 i ! 1 \ 1 1 1 1 +1 1 1 1 l 1, x 1 ~ 1 ~~t 1 ~ ibex i \ 1 Figure 1: Supercavitating hvUl ofoil

1. Cinematic Boullda.ry Condition for z > t, where: -2+2lI)~ig=~(X)O<X<l;~=o- 2. Dyna.mic Bowlda.ry Conditio~ 2 27r ~ it- ~ = 2 0 < ~ < 1 y = a+ 3. Iota condition 4. Cavity Closure Conclition o where z = )/:, t = I, r4 = 1 + t2 (13) `4~ The integrals in equations 9, 10, 11 and 19 are com- puted numerically with special care talon at the singu- larities of the integrands t214. The cavity thickness her), which also includes the foi] thickness as shown in Figure 1, is determined by integrat- ing the equation: (5) :~1) = 0 (6) Ids = 0 (7) c)! = - (~' · (~) and, y'~x) is the ordinate of the lower hydrofoil surface, as shown in Figure 1. The singular integral equations of Cauchy type, 4 and 5, can be inserted to produce expressions for the un- l;nown a, Eye) and q(2 ~ in terms of the cavity length I and Oh's), as follows [91~: 4\/ ~ ~ Jot ~' ~ ~ + ale )2 [_ Alp] do (9) _~ ~ ({ + _2~t ~ ~—0i (r7) do (10) 4~) = -~l (~) + ~ (~ - 7~) 1+_2 /~t ~ OlL ~ (11) 7r ~ ~ it- +~2~+~) for z < t, and: qua) = I- 2v~7'2 + Z2 :+ ~ {t / ~ Ol(~)dw ~ V ~ Jo N/t-~.(l+~,2~(z+~,j- /~ . (~~ + ~) _ 47.2 +~2 ~ ;t ~ Oi(~1w t19y ~ ~ ~ t - W (1 + ~,2~7 _ wy _ 687 TOO C1~ 1 = qtx) (14) 2.2 Face Detachment For thicl; symmetric foils at small angles of attack and for many foils at negative angles of attack, it is found that the supercavity detaches forward of the trailing edge on the pressure side of the foil, as shorten in Figure 2. The point of separation, s, may be found by consid- ering the following two conditions: v Figure 2: Face detachment on a supercavitating hydrofoil 1. the pressure on the wetted foil surface must be greater than the cavity pressure 2. the cavity and foil surface must not intersect aft of the separation point. In linear theory, condition l is equivalent to )(<x) > 0 for O < X < s (lS) The corresponding boundary value problem may be solved by using the analysis in the previous section and by considering as foil the part of the original foil between = 0 and x = s, cavitating at a cavity length 1/s. The vorticity distribution between x = 0 and x = s is given by equation lO, with the cavity length, however, being scaled to l/s. Tl~e behavior of the vorticity distri- bution at x = s can lee found, lay using eq~a.tion lO, to be as follows: y(<r) ~ A(s, 1) ~ (16) where A, for a. given foil geometry, depends only on the point of separation and the length of the cavity: A(s, 1) = - 2~7,2 · ~ + z2 [I + ~

1~ · ( ~ ( S)~1 + 92)d~ 0250~ ~ cry where + (3l(2S) ~~ · [A -'`/;~1 (17) zs = A; t = Am; r2 = ,~ (IS) By observing the behavior of the vorticity distribu- tion by varying s, we conclude that the correct detach- ment point is the one for which the vortic.ity distribution goes to zero at ~ with zero slope. This can be seen in Figures 3 to 5, where the cavity shapes and the vortic- ity distributions as predicted by the presented analytical method, are shown for different detachment points. In Figure 3, the circulation distribution is negative at the trailing edge ~ A(s,l) ~ O ), thus violating the condi- tion 15. In Figure 4 the vorticity distribution is positive everywhere on the foil ~ A(s, 1) ~ O ), but the cavity inter- sects the foil. The correct detachment point is somewhere between these two, and the one which satisfies both con- ditions is the one for which the vorticity has a zero slope at s, which is shown in Figure 5. At this point, we should have: A(~, 1) = 0 (19) A different approach of deriving equation 19, is given in Appendix A. The detachment point is determined lair solving equa- tion 19 with respect to ~ numerically, utilizing a Newton R.a.phson (secant) method A. A typical case requires about five iterations, depending on the accuracy of the initial guesses. The effect of the location of the detachment point on the cavitation number and the lift and drag coefficients is shown in Table 1. The importance of the correct face s . ~ .200 .0136 .542 .0247 .700 .0138 CD . ~ 0010 . .0018 .0017 SIGH .1224 .1329 .~248 Table 1: Lift and drag coefficient and cavitation number for the foils shown in Figures 3, 4 and 5. 688 0. 050 0.050~ 0. 150 _ o.2s8. 0OO 0.200 0.400 0.600 0.800 1.000 1.200 1.4001.~5C Figure 3: Cavity shape and vorticity distribution - A(s, 1) < 0 . 0.150 O. 050 O. 050 o. 2s8. 000 ' 0.'200 ' 0.'400 ' 0.'600 0.'800 ' t.'000 · 1.-200 · 1. 4001. ~t Figure 4: Cavity shape and vorticity distribution - A(s, 1) > 0 0.250L . . . . . . . . . . . . . . . 0.150 n non 0.050 0. 150 Vor tights do - Ail om ~ = . 5~2 0 258. 000 0. 200 0. 400 0. 600 0. 800 1. 000 1. 200 ~ . 4001c Figure 5: Car ity shape and vorticity distribution A(8, 1) = 0 detachment point in the prediction of the cavity extent and the forces on the foil is apparent. Some further discussion on the determination of the correct cavity detachment point is given in the Section 6.

y up ~ U+(~)= 2 _~1 ~ _ _ ~ ~ ~ l O lo ~ l :* = ~—9~l~+w—2) ; 0 < x < lo A* = By; 10 < x < 1 (26) 02 — 01 ~ F (97) Figure 6: LIidehorcl detachment on asupereavilatingfoil. Fall d f _ 1 ~ ~6 (28) 2.3 Midchord Detachment For the ease where the superea.vity detaches aft of the leading edge on the suction side of the foil, as shown in Figure 6, the linearized boundary value problem can be formulated as follows: The dynamic boundary condition on the cavity: 9~) 1 ~ 'I = tt+~2 ~ = 1 10 < X < 1, y = 0 side: where: (90) The kinematic boundary condition off the pressure 2 27r ./ ~—r = (3i 0 < x < 1, y = 0~ `~1y The kinema.tie boundary condition on the suction side: q + 1 1 judo = (~)* a < x < 10, y = 0+ (92) E)u= or A (23) with qu being the ordinate of the upper hydrofoil surface as shown in Figure 6. Equations 20 and 21 can lee reduced to the following form: 2 27r ~ ~—~ = 2 ° < .~ < 1 t2~y 2 ~ 2~ 1 6 _ ~ = 02~') 0 < x < 1 (95) with the use of the definitions: where up+ is the horizontal perturbation velocity on the wetted part on the suction side of the foil. Equations 25 and 24 are in the same form as equa- tions 4 and 5. Therefore, to invert these equations the same methodology can lee followed as described in see- tion 2.1. The perturbation velocity u+w, however, is still an unknown. To determine u+w for O < x < lo the l;inen~atie bound- ary condition, equation 22, is applied. The solution for U+W is described in Appendix B. The analysis described in this section has been ap- plied for a VLR section t16] and the results are shown in Figure 7. The top of Figure 7 shows the predicted cavity shape for a ~nidehord detachment at lo = 0.2. At the lower part of Figure 7, the corresponding total source distribution is shown together with the thickness source distribution. Notice that the two source distributions are identical for O < x < lo, as recluired by equation 56. The described theory is applied for a VLR foil t16] for a fixed cavity length I = 1.5 and for different values of the detachment point lo. The predicted cavity shapes and pressure distributions on the suction side are shown in Figures ~ to 11. The cavities in Figures 8 and 9 are unacceptable, because they intersect the foil surface. The cavity in Figure 11 is also unaeeeptal~le because it pro- duces pressures in front of the detael~nent point which are smaller than the cavity pressure. rl he correct detaeh- ment point seems to be the one corresponding to Figure 10. It appears to be the point for which the pressure distribution in front of the detachment point has a zero slope. No attempt has been made by the authors, how- ever, to generalize this condition, since the detachment point on the suction side should be determined by the viscous flow in front of the cavity [10], rather than lay any other potential flow criterion. Some more discussion on ea.vityr detachment is given in section 6. 689

0.20* 0.10. -~.100 -~.24)* '0* O.ZOO tame I ..6~6 ..Z.. _~.Z')a _~.S,.0 ~ end t .~. t ZOO - Cavity Source Distribution · Chicly ess Source Distribut ion o. 2SO _ on. - -cp o. oso I -0.050 -O. 150 -O. 2 pOO 0.200 0.400 0.600 0. 800 1.000 1.200 1. 4001. X X Figure 10: Cavity shape and pressure distribution on the suction side of a supercavitating foil with cavity detach- ment at lo = 0.07. Sable foil as in Figure ~ Figure 7: Cavity shape, total and thickness source dis- tributions for a VLR thickness profile with NACA a=.S meanline sUpercavitating `` ith n~idchord detachment lo = Figure 11: Car ity shape and pressure distribution on the .20. suction side of a. supercavitating foil with cavity detach- ment at lo = 0.10. Same foil as in leisure ~ - . 258-t too ' o.zoo o.'.oo ' 0.'600 o.aoo l.'coo ~ zoo 1.'4COl x 0.2so 0.150 "Cp 0. OSC -.OSC . -O. 150 . -O. 25n . . . . . . . . . . . . . . 0. 000 0. 200 0. 400 0. 600 0. 800 1. 000 1. 200 1. 400 1. 5C Figure 8: Cavity shape and pressure distribution on the suction side of a supercavitating foil with cavity de- tachment at lo = 0. \'LR thickness fond and NACA a=0.8 mea.nline, maximum thicl;ness/chord=0.04, max- imum camber/chord=0.03, pi; = 0.001613, car = 2°, 1= 1.5 0.250 o.~= up o. 050 - . OSC ~ BY -.258 f- ~ A ~ m 0.200 0.400 0.600 0.800 1.000 1.200 1. 4001. 5( X Figure 9: Cavity shape and pressure distribution on the suction side of a supercavitating foil Title cavity detach- ment. at lo = 0.01. Same foil as in Figure ~ 690 2.4 Partial Cavities In the case where the cavity is smaller than the chord of the foil, as shown in Figure 19, the linearized cavity problem can be formulated in a similar way as in the case of the sllpercavitating foil, in terms of vorticity and cavity source distributions t70], [1S], Aft. For given cavity end, l, and cavity deta.chn~ent, lo, the corresponding cavitation Mueller, the vo~ticity and cav- ity source distributions can be given in terms of integrals of u+, the horizontal perturbation velocity of the fully wetted foil, between lo and l t93~. 2.5 The Leading Edge Correction The linearized partial cavity theory is known to predict that, for given flow conditions, increasing the foil thicl:- ness results in an increase in the cavity extent and vol- ume. This is contrary to experimental evidence, the non- linear theory [3.5], and the short cavity theory t34~. An alternative `va.y of including the non-linear thicli- ness effects in the linear cavity theory can be achieved via the leading edge correction t93], Em>. It essentially con- sists of including Lighthill's correction t96] in the formu- lation of the linearized ca.~'it~' problem. It can be proven [2.3] that this can be ac]lieved bar modifying the linearized dynamic boundary, condition on the ca.`ritv from uc = 2 {ix; on the CCl~,ity (~)9)

- c, Figure 12: Partially cavitating hydrofoil .,.~0~, to 2. 000 . . · · · ~ ~ · . - - P=d Method L~ =~~ without I`E corn 0 ' 0.200 ' o..oo U peon z/c I Figure 13: Pressure distributions on an NACA 16-009 section with a 50% cavity at or = 3° frown panel method L /x + pr/' and linear theory without leading edge corrections. Cp = uc = ( ~ \/ —~ ~ COO; on the cavity (30) p—p=, /pU2 /2 where x is Else distance from the foil leading edge and pa is the leading edge radius. The modified boundary value problem with the intro- -C, auction of equation 30 has been solved and the solution has been expressed in terms of integrals of known quan- titles t234. A direct comparison of the linear cavity theory, with or without the leading edge correction, and the non-linear theory is shown in Figures 13 and 14. The cavity shapes O ax, as predicted by the linear theory, with or without th leading edge correction, are added normal to the foil and the produced foil geometry is analyzed with a poten- tia.l based panel method t17i, where the exact l;inema.tic boundary condition is applied on the exact foil or c< ity surface. The pressure clistril~ut.ions produced from the panel method are shown in Figures 1:3 arid 14, together with the linearized pressure dist~il~.~tio~.~.s from linear the- ory with or without tl-~e leading edge correction. In these Figures the pressure distribution from the linear theory with or without the leading edge correction is constant on the cavity, since this has l~een required via the dynamic boundary condition. The pressure distribution from the panel method, however, is not exactly constant on the cavity and this is a measure of the accuracy of the lin- ear cavity theory. Comparing Figures 13 and 14, the substantial improvement of the linear theory, when the leading edge correction is included, becomes apparent. 691 . / Palm Method Shear Theory with At: corn Figure 14: Pressure distributions on an NACA 16-009 section with a So-so cat ity at ~ = 3° from panel method and linear theory with leading edge corrections. Cp = P p~/pu2 /2

3 The Cavitating Hydrofoil- The Numerical Method The nun~erical method consists of discretizing the chord end the cavity into a. finite Hunter of segments on which the vorticity and source distributions are a.pproxi~:na.ted with point Vortices a.nc! sources respecting ely. The spacing of the panels is l~a.lf cosine on the foil and constant in the wa.l;e. The a.rrangeme~t of the vortex and source panels is shown in Figure 1.5. The following notation is used: VPsi = boundaries of source panels nisi = positions of point so~`rce.s -\'Pi = boundaries of vortex panels Ads = position of pout vortices Xki = position of ki'~.ematic boundary condition collocat'.o~. point Aid = position of dy~.amic bou~dc~ry condition collocation po'.~2.ts The arrangement of the vortex and source panels is such that the expected source and vorticity singularities at the leading edge of tl~e foil, as well as the square root singularity of the source distribution at the trailing edge of the cavity, are modeled accurately. The collocation points for the application of the l;inematic and dynamic boundary conditions are chosen such that Else Caucl~y principal value of the involved singular integrals is com- puted accurately. The detailed analysis for the selection of Else panels and control points is given in (1S], t6] and t94. The presented nu~nerica.l Netted ``as developed orig- ina.lly for pa.rtia,ll~r and supercaxita.ti~g l~>drofoils with the cavities starting at the leveling edge ore the suction side and at the tailing edge on the pressure side of the foil id], t18], t9~. To extend the numerical method to also predict face and/or midc.hord supercavities, we assume that the de- tachment points are Xs on the suction side and ,X'p on the pressure side. The points Vs and X'p coincide with any of the source panel boundaries ,X'psi on the foil. By separating the total source q into the thickness source qC and the calcite source I, the corresponding boundary integral equations become: The kinematic boundary conditions 2 + 27r j; ~ - ~ ~ ~ do ~ = 0+ 0 < x < His c = vortex pOSitions I. . = source positions 0 = dynamic c. pi ~ = kinematic c. p. X = 0 X. ~ to tot I _ . ~ ''1~ ~ 1~ ~ 1 ~ ~ i~ ~ j I 1 1 1 X, 1 1 I 1 31 Figure 15: Discrete singularities method for superca.`i- ( ) tasting foil `` ith arbitrary detachment points qc 1 f1 7(~)d: = t: dl1 ?J = 0- 0 < X ~ -~v where 11(~) is the foil mean ca,ml~er surface. The dynamic boundary conditions —UOO 2 + ~~—27r j; , —x (3:~) — Uth ~ = 0 AS < X < I (3.1) 2 2 27; 1; ~ — T lath ~ = 0 ~ < X < 1 (3,5) where uth is the horizontal perturl~a,tion velocity due to foil thickness, given as: lath = 1 jIqw(~)df (36) 27r 0 ~—~ To discretize the above integral equations, we mal;e the following definitions: · N = number of discrete vortices · M = number of discrete cavity sources Qi NS = number of fully wetted panels upstream of Xs (39) · NP Alp 692 = number of fully wetted panels upstream Or

· M discrete cavity sources Qi The strengths of the discrete vortices and cavity sources are related to the corresponding vortieity and source dis- . . .1 . ~ ll · 1 cavitation number a trll~utlons as IOllOWS: pi = )(-\V' ~ ('~Pi+~ - 'HEY') (3 Qi = 4: ('~Pi+~ - 'APE Allele If is the mean goalie of the cavity source at the corresponding source panel. At this point, we will assuage, without loss of gener- a.lity, that UOO = 1. The discretizecl boundary conditions become: The kinematic boundary conditions a) On the suction sidle: Qi 1 N _ IS _,~S are 9~ ~ V~ - i = 1, , NS (39) - ·Yv' ( d~ ) i b) On the pressure side: _ Qi _ 1 it, Fj _ {dr1~1 2(`Xpi+~—Vpi ~c' Or I= -Yki—Xvj \y dX J i i = 1, ..., lVP (~40) where of is defined as: of = qc/qc(\si) and is approx i~na.ted Fitly its value for a flat plate cavitating at the same cavity length [6] and t94. Tl~e dynamic boundary conditions a) On the suction side: al _ IS + rz + ~ ~ Qi ~ i = NS+1, , IlI (41) b) On the pressure side: a Pi 2 2( ~pi+:—gyp. ~ The cavity closure condition: ~1 ~Qi=0 i=1 There are N+~+1 unI;nov~ns: · N discrete vortices lTi 27r ~£ i, X ttth i = I\rP + 1, ? N (~4~>) There are also N+LI+1 equations: · NP + NS kinematic boundary conditions . M - NP + N - NS - 1 dynamic boundary conditions · 1 cavity closure condition · 1 equation relating F~ to Qua The last equation, which relates the discrete singu- la.rities Qua and Hi, replaces the Fist d~nan~ic boundary condition [1Si. However, in the case of midchord detach- ment, there is no dynamic l~ounda.ry condition to be sat- isfied on the first source panel and this relation is not applied. The convergence of the described numerical method is shown in Table ~ for different numbers of elements on the foil. The ana.lytica.l results shown in Table 1 have been found by using the analysis described in Section 2. Finally, the predicted cavity shapes front the analyt- ical and the numerical method are shown in Figure 16. .. ~ of ElemntS 5 0 20 do 80 100 .- N~rerical Sit .3057 .2082 ~ . .2111 .2131 .2095 _ .2095 Analytical S.: - .2357 .2070 .2110 .2131 .2100 .2099 Volure .0988 .0883 .0852 .0843 .0858 .0859 Analytical Volure .0771 .0872 .0853 .0844 . 0858 .0858 Table 9: Convergence of the numerical method. Su- (43) percavitating Joul;owsl;i thiclir~ess form with parabolic meanline, maximum tl~icl;ness/chord=0.04, maximum camber/cl~ord=0.09, cat = 3°,1 = 1.5 693

0.1SO~ O. OSO - . OSO, -.~ - HULUlC ~ ~~ W" Em-, DISCARD] "Stall ~7//~/m7~ ; ~ V.l..~ UNLESS P - EMU Ace. - ~-2 DEtACH~J 061 raft IS,Y£R SIDE ~ X-. ~ DESACRNW 011 rue UPPER SIDE ~ Xe. 1 0.2S8 300 0.200 0.;00 0.600 0.~ - I.~ t.~ t.4~1. , Figure 16: Cavity shapes from numerical and analytical method 4 The Unsteacly Cavitating Pro- peller Numerical Method. A numerical method has been developed for the unsteady sheet cavitation of marine propellers in spatially non- uniform wakes t95i, t4i, t1Si. The corresponding com- puter code is called PUF-3. The complete three di~nen- sional linearized unsteady cavity problem is solved for given propeller geometry, inflow wake and cavitation num- ber. The propeller cavitation nuder is defined as: where: In = e f2D2 (~44) · Pshaj~ = pressure at the axis of the propeller shaft · Pv = Vapor pressure · n = propeller revolutions · D = propeller diameter The flow around the blades and the cavities is rnod- eled by a lattice of vortices and line sources located on the mean camber surface of the blades and their trail- ing vortex wal;es, as shown for one blade in Figure 17. The chordwise arrangement of the vortices and sources is the same as in the hydrofoil case, described in Section 3. The spanwise spacing is constant with quarter inset at the tip. Details of the numerical grid can be found in t254 and t18~. The time.history of the cavity shapes is determined for each blade strip by applying the three-dimensional linearized unsteady cavity boundary conditions t25~. The extent of the cavity on each strip is detern~ined iteratively until the pressure on the cavity becomes equal to the vapor pressure Pa The effect of the other strips on tl~e same blade as shell as on the other I'lades is accounted for in an iterative sense. \ Figure 17: Numerical grid on one propeller blade and its wake. The leading edge correction, described in section 2.5, hats also been implemented in the numerical method for the propeller, in order to account for the non-linear blade thickness effects [1Si. In the present work, the numerical method for the unsteady propeller cavitation is extended to predict cav- ities with prescribed midchord and/or face detachment. This has been accomplished by a direct application in the propeller problem of the numerical method for mid- chord and face hydrofoil cavitation, which eras described in Section 3. The modified PUF-3 has been applied for the DTRC N4497 propeller t194. The advance coefficient is J = l7SHIP/n/D = .S and the cavitation number 0.n = 1.5. The time history of the cavity volume is shown in Figure 18 for different detachment points on the suction side of the blades. The three-dimensional Respective plots for some blade sections and their cavities are also shown in Figures 19 to 21 for diFererlt deta.cl~ment points. Tl~ose figures show the effect of the detachment point on the cavity extent and shape to be substantial. 694

~ - nln Is 8 .035 .030 Suction side detachment in ~ of local chord length 0. 7~ _ - 1 it, a\ Blade Angle in degrees :120 Figure 18: Cavity volume per blade fol different de- tacl~ment points on the suction side of the N4497 pro- peller as predicted by the modified PUF-3, An = 1.5, J= VSHIP/n/D = 0.~. W=7 pressure side suction side Figure 19: Cavity shapes for propeller N4497 at blade sections No. 3, 5 and 7 as predicted by the modified PUF-3. Detachment on the suction side, at 0.7% of the local chord; Blade angle = 12° from the top, On = 1.5, ~ = VSHIP/n/D = 0.8 Figure 20: Cavity shapes for propeller N4497 at blade sections No. 3, 5 and 7 as predicted by the modified PUF-3. Detachment on the suction side, at 3.2% of the local chord; Blade angle = 12° from the top, An = 1.5, J = VSHIP/n/D = 0.8 f;7 Figure 21: Cavity shapes for propeller N4497 at blade sections No. 3, 5 and 7 as predicted by the modified PUF-3. Detachment old the suction side, at 8.4% of the local chord; Blade angle = 12° fiom tile top, an = 1.5, J = VSHIP/n/D = 0.8 695

5 Conclusions The following has been accon~plisl~ed in the presented work: The cavitating genera,! shape h drofoil problems with arbitrary detachments, has been formulated in terms of singular integral equations of unknown source and vorticity distributions. Those equations are in- verted analytically and the cavitation number, cav- ity shapes and pressure distributions, are expressed in terms of integrals of known quantities. The effect of the detachment point on the cavity solution has been investigated. In the case where a supercavity detaches for~ravrd of the trailing on the pressure side of a. hydrofoil, an equation for the location of the cavity detachment point has been found. · A numerical discrete vortex and source method hats been developed to predict the cavitation on h~dro- foils with a.rbit~a.ry~ cavity detachment points. The numerical method has been extended to pre- dict unsteady propeller sheet cavitation with ar- bitrary midchord and/or face cavity detachment. The effect of the location of the cavity detachment on the cavity volume and the cavity shapes has been investigated. 6 Future Research . Perform experiments on car ita,ting propellers which show midcl~ord and Ol' face cavity detachment. De- termine the detachment lilies from the experiment and run the modified PUF-3 `` ith those detachment lines as input. Compare the predicted car ity shapes and propeller forces from pITE-3 with else expe~i- ment. · Employ the cavity detachment criteria. in PUF-3. 7 Acknowledgements Support of this research was provided by the AB Volvo- Penta Coorporation. At this point the authors wish to thanl; Professor Justin E. I(erwin of WIT, Mr. Lennart Brandt and LIr. Ted R.osendal of Volvo-Penta for Blair valuable comments and discussions during the course of this work. References [14 A.J. Acosta. A Note on Partial Cavitation of Fiat Plate Hy~l~ofoils. Technical R.epolt No. E-19.9, Cali- fornia Institute of Technology, I-Ix cl~odynamics La.l'- oratory, October 1955. t94 H. Aral;eri. Viscous effects on tulle position of ca~r- ita.tion separation from smooth l~odie.s. Journal of Fluid Alech(loics, sol CS(No. 43:1'P 779-799, 1975. The following items need to be addressed in order to improve on the presented worl;: Perform experiments on cavitating hydrofoils with cavities showing midcl~ord and/or face detachment, in order to validate the presented theoretical re- sults. Improve on the prediction of the cavity detachment points by coupling the presented method with a boundary layer calculation on the wetted part of the foil in front of the cavity. It is known from ex- periments All, t10] that the cavity detachment point has to be determined in relation with the bound- ary layer separation point, rather than from any potential flow criterion. The potential flow cavity detachment criterion, however, should still be valid if no boundary laster separation occurs in front of the predicted detachment point. 696 G. Birt;hoff and E.lI. Za~antonello. Jets, T47a1`es and Cavities. Academic Pi ess Iliac., New Trot k, 195 ~ . J.P. Breslin, R.J. Van IIouten, J.E. Irwin, and C-A Johnsson. Tl~eoretica,l and experimental propeller- induced hull pressures al ising from intermittent blade cavitation, loading, and thicl;ness. T'ans. SIVAIlIE, 90, 1 USA. M. Brillouin. Les surfaces de glissement de helmholtz et la resistance des fluides. Ann. Claim. Phys., vol. 23:pp. 145-230, 1911. C.N. Corrado. Investigations of lVume~ical Schemes for the Evaluation of the Cavitatiny Flow Around Hydrofoils. NIaste~'s thesis, MIT, June 19S6. [7] A. G. Efros. Hydrodynamic theory of tic o- dimensional flour Fitly cavita.tiorl. Dow. Akad. I\lauk^. S.SSR, vol 51:pp 267-970, 1946. iS] A.G. Fal~ula,. Tl~ill-ai~foil Theo applied to hydro- foils with a single finite ca,~it~ anal arbitrary free streamline detachment. Journ(ll of Fluid Alechan,- ics, viol l9:pp 797--~40, 196'7. t94 N.E. Fine. Con7l'~'tationo1 odd E.rl~e~`imental In- vestiga,tions of the Flow Around ~~!itUtin.§ HyLl'~o- foils. Technical Report No. US- I\lIT, Department of Ocean Engineering, Septen~bet 19SS.

t10] J.P. Franc and J.M. LIichel. Attached cavitation and t24] the boundary layer: experimental investigation and numerical treatment. Journal of Fluid AIechanics, vol. 154:pp 6.~--90, 1985. t11] J.A. Geurst. Linearized theory for fully cavitated hydrofoils. I''te, nal tonal .SI2.ipbu'1ding Progress, vol 7(No. 65), January 1960. i] 24 J.A. Geurst. Linearized theory for partially cavitated hydrofoils. International Shipbuilding [96] Progress' vol 6(No. 60) :pp 369--~384, August 1959. t13] J.A. Geurst and R. Ti~nman. Linearized theory of two-dimensional cavitational floss, around a wing sec- t97] tion. IX International Congress of Applied AIechan- ics, 1956. t14; J.A. Geurst and P. J. Verbrugh. A note on camber t984 effects of a partially cavitated hydrofoil. Interna- tional Shipbuilding Progress, vol 6(No. 61~:pp 409- 414, September 1959. [154 T. Hanaoka. Linearized theory of cavity flow past a hydrofoil of arbitrary shape. Journal of the Society of Naval Architects, Japan, vol. 115:pp. 56-74, June 1964. t164 J.E. Kerwin. Prig ate communication. [17] J.E. Erwin, S.A. Icings, J-T Lee, and W-Z Shih. A surface panel method for the hydrodynamic analysis of ducted propellers. Trans. SATA3IE, 95, 1987. [IS] J.E. I~er~`rin, S.A. Ixinnas, I\f.B. Wilson, and McHugh J. Experimental and analytical techniques for the study of unsteady propeller sheet cavitation. In Proceedings of the Sixteenth Symposium on Naval Hyd r odynam?cs, Ber1;ele~r, California, Jelly 1986. t194 Ixer``in.J.E. anal C-S Lee. Preelection of steady and unsteady marine propeller performance by numer- ical lifting-surface theory. Trans. Sl\rAAIE, vol 86, 1978. t204 S.A. Kinnas. Cavity Shape Characteristics for Par- tially Cavitating Hydrofoils. Technical Report 85-1, MIT, Department of Ocean Engineering, February 1985. G. Kreisel. Cavitation with Finite Cavitation Num- bers. Technical Report No. R1/lI/3G, Admiralty Res. Lab., 1946. Chung-Sup Lee. Prediction of Steady and Unsteady Performance of AIarine Propellers with or with- out Cavitation by Numerical Lifting Surface Theory. PhD thesis, M.I.T., Department of Ocean Engineer- ing, May 1979. M.J. Lighthill. A new approach to thin aerofoil the- ory. The Aeronautical Quarterly, sol 3(no Knapp. 193-210, November 19.51. B.R. Parliin. JIu''L^ I''tcyrals for Fully Cavitated IIy- drofoils. Technical Report P-2350-1, RAND, Novem- ber 1961. ~ D. Riabouchinsl;y. On steady fluid motions with free surfaces. Proceedings of London flIath. Soc., Vol. l9:pp 906-915, 1991. t294 Nishiyama T. and Ota T. Linearized potential flow models for hydrofoils in Supercavitating flows. Transactions of the A.S1lIE, December 1971. [30] M.P. Tulin. Steady Two-Dimensional Cavity Flows About Slender Bodies. Technical Report 834, DTMB, May 1953. [314 M.P. Tulin. Supercavitating flow past foils and struts. In Symposium on Cavitation in Hydrodynam- ics, NPL, Tenddington, England, September 1955. t32] M. P. Tulin. Supercavitating flows - small pertur- bation theory. Journal of Ship Research, vol 7(No. Knapp. 16-37, 1964. [33] M.P. Tulin and M.P. Burkart. Linearized Theory for Flows About Lifting Foils at Zero Cavitation Num- ber. Technical Report C-63S, DTMB, February 1955. [34] M.P. Tulin and C.C. Hsu. New applications of cavity flow theory. In 13th Symposium on Naval Hydrody- namics, Tol;yo, Japan, 1980. t35] J.S. Uhlman. The surface singularity method ap- plied to partially cavitating hydrofoils. Journal of Ship Research, sol 31(No. 2~:pp. 107-124, June 19S7. t214 S.A. l:(innas. Cavity Shape Characie7istics for Su- t36i H. Villas;. Sur ia validit;e des solutions de certain percavitating Hyd r ofoils. Technical Report 84-13, MIT, Department of Ocean Engineering, October 1984. t22] S.A. Ixinnas. Leading edge corrections to the linear theory of partially cavitating hydrofoils. Submitted for publication, March 19S9. t23] S.A. I~innas. Non-linear Corrections to the Linear Theory for the Prediction of the Cavitating Flow Around Hydrofoils. Technical Report 85-10, MIT, Department of Ocean Engineering, June l9S5. 697 problem d' hydrodvnamique. J. de AIath., vol 6(No. 10~:pp 231-290, 1914. t37] R. B. Wade. Linearized theory of a partially cav- itating piano-convex hydrofoil including the effects of camber and tl~icl;ness. Journal of Ship Research, vol 11 (No. l):pp 20-27, 1967. t384 T.Y. tofu. Cavity flows and numerical methods. Ill First International Conference on Numerical Shi Hydrodynamics, 1975.

[39] T.Y. Wu. A demote on tile Linear andl1Vonlinear T1le- ories for Fully Cavitated 1lIydl of oils. Teclluical Re- port No. 21-92, California Institute of Technology, Hydrodynamics Laboratory, August 1956. Appendix A Details of Face Detachment The Villat-Brillouin condition t36i, [53 at the cavity Appendix B detachment point requires the cavity to have the same slope and curvature with the foil. This condition satis- fies, locally, the requirements that the cavity does not intersect the foil and that the pressures on the foil are larger than the cavity pressure. If tic and Of are the or- dinates of the cavity and the foil at the vicinity of the separation point s, as shown in Figure 2, then we should have: = dyc( +) dYl( -) o d2yc( +) 42yc( ~) O (46) Equations 45 and 46, via the lcinen~atic boundary con- dition on either the cavity or the foil, become: Finally, by using equations 50, 52 and 53, we get: [do] 2 ~ (54) Thus, in order for the condition 4S to be valid we should have: A(s, 1) = 0 (55) Details of Midchord Detachment The solution By and q to the system of equations 24 and 25, is given by the expressions 10, 11 and 12 where 43` has to be replaced by 02, as defined in equation 27. To determine the unknown UC+w for O ~ ~ ~ lo, the kinematic boundary condition 22 must be applied on the upper wetted part of the hydrofoil. Combining equations 22 with 21 it can be proven that `45' the condition 22 is equivalent to: qfx)=qu,( ~ for O<~<lo (56) where qua) is the foil tl~icL;ness source defined as follows: qW(x) = TOO Cad 9~) (57) By using the coordinate transformation defined in [v] = vile+)—vim) = 0 (47) equation 13 and lay introducing to: [d ] = d (s+)—d (s~) = 0 (4S) ~~ (5S) where vim) is the vertical perturbation velocity on the cavity or foil. In the context of linearized theory: where: that: it can be proven t9] that equation 56 is equivalent to: t!(~) =—q(2 ~ + I (*) qua ~ _ H(z) = + ~ it. (Ucw2~2) 77 ~ (my) '50) where Hazy is defined as: By using equations 1O, 11 and 12 it can be shown is+—days ~ ~ 2 ~ d2(s+)—dims ~ ~ 4 ~ (53) where s+—s = s—s- = c, and A(s,l) is defined in cw 2 equation 16. 698 H( )—a°q (Z) + ~ —a°)R(z) (60) it ~) = ~ .l; —I (5~) with it(z) defined as. The total source distribution qua) is continuous at s and so is I(~) because Gypsy = 0. This malces the first condition, equation 47, always valid. it(x) = ~ 2~/2r2 (a at), (G1) and with JO and JO being the cavitation Bunker aIld tl~e cavity source solution, respective!,, corresponding to a supercavity Title the same extent x = I but starting at the leading edge, i.e. for lo = 0. The values for JO and qO are given from the equations 9 and 11. Equation 59 can finally be inverted to obtain A: 1 + 92 j(9 ~ t)(to - 9) . . 7r [ - ,lI(11) - - (1 -—) 1V(71 )] (69)

where to ~1(~) d—f If and (qw—~o[J~4O)dy (1+712)(71—Z) --- (63) N(Z) d f | \/~/7 · R(r1)dI1 (64) Equation 62 is the solution of the integral equation 59 which satisfies the condition that up+, - 2 = 0 at lo. The cavitation number, a, is obtained by applying the cavity closure condition 7. It can be shown that t9~: 31 = a0— with the following definitions: (65) 1`l def 84r. 1~/~ ~+17~/7i~ 3]F(~)d71 and N def ~ Mar where: and (66) r3(r2 + 1) ,/ i. V; Hi+ 9~-NF(~)d~ (67) ]/IF (z) d f /. ~/2 ~2 o HI (id (6S) l N ( ) def I )/(~ + t)(to—no) N( )d (69) o The integrations in equations 66, 67, 68 and 69, are performed numerically with special care tal;en at the sin- gularities of the involved integrands t9;. 699

DISCUSSION by H. Kato I appreciate the authors' effort in calculating sheet type cavitation. The authors did not compare their results with experiments. Therefore I am afraid that some of the assumptions and conditions are different from experimental observations. Firstly the sheet cavitation is closely related with boundary layer separation. The leading edge of sheet cavitation coincides with the separation point of boundary layer according to the observation by Franc and Michel[10] and Yamaguchi and Kato [A1,A2]. We can not choose the location of the cavity leading edge arbitrarily as the authors did in the paper. The authors also mention that the pressure distribution shown as Fig.11 is not realistic. However, we usually observe a negative pressure peak in the front of the cavity where the pressure is lower than the cavity pressure. The third point I would like to point out is the cavity closure condition; Eq.(7). A sheet cavitation is followed by wake flow which can not be neglected in many cases. The calculation under the assumption of Eq.(7) does not agree with the experiment especially when the sheet cavity length approaches to the foil length. [Al] H. Yamaguchi and H. Kato: A Study on a Supercavitating Hydrofoil with Rounded Nose, Naval Architecture and Ocean Engineering, Soc. Naval Arch. Japan, Vol.20 (1982). [A2] H. Yamaguchi and H. Kato: On Application of Nonlinear Cavity Flow Theory to Thick Foil Sections, 2nd Int. Conf. Cavitation, I Mech E, Edinburgh, (1983) pp.l67-174. Author's Reply First, we want to thank Prof.Kato taking the time to read our paper and for making comments on it. It is correct that we did not compare the results of our method with experimental results, and we are actually planning, as stated in Section 6 of our paper, a systematic series of experiments in the future. The objective of this paper was to produce a consistent and convergent numerical method for the midchord and face unsteady propeller cavitation. Concerning Prof. Kato's comment on the pressure distribution of Fig.ll, we do not state that the pressure distribution "is not realistic". We rather say that it is "unacceptable" according to the conditions imposed in the beginning of Section 2.2. In addition, at the end of Section 2.3 as well in Section 6, we also state, as does he, that the midchord detachment point in front of the cavity[10]. Finally, in eq.(7), we assumed cavity closure at the trailing edge of the cavity. this assumption is in accordance with a linearized Riaboushinsky or reentrant jet an cavity model. We agree, however, that more physical model, is an open cavity model with the "openness" supplied from further knowledge of the cavity viscous wake. This "openness" does not affect much the predicted cavity shape and the cavitation number, in the case of supercavitating flows. Thus, in the presented analysis of the supercavitating hydrofoils, we decided for simplicity, to take the cavity " openness" equal to zero. For partially cavitating hydrofoils, however, the cavity wake is important and should be included. An experimental analysis of the cavity wake in the case of partially cavitating hydrofoils is included in [9]. 700