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CL PX' 4A o '5 itq I "V I e It$ Pit A 44 4 L you All 0, v Fr lt,"t ',t I w IVA! 0 twtl4A, y*t' "Saw. I It, t 1 4 OT l Tt "''t T`nlr* Mw t,"r' 16"', mttt out S' Of me, 40 d Val NACA RM L57Glla NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS RESEARCH MEMORANDUM THEORETICAL DETERMINATION OF LOWDRAG SUPERCAVITATING HYDROFOILS AND THEIR TWODIMENSIONAL CHARACTERISTICS AT ZERO CAVITATION NUMBER By Virgil E. Johnson, Jr. SUMMARY The linearized theory of Tulin and Burkart for twodimensional super cavitating hydrofoils operating at zero cavitation number is applied to the derivation of two new lowdrag configurations. These sections were derived by assuming additional terms in the vorticity distribution of the equivalent airfoil; in particular, three and five terms were considered. The characteristics of both the three and fiveterm airfoils are shown to be superior to the TulinBurkart configuration. For example, the two dimensional liftdrag ratios of these new sections operating at their design lift coefficient are theoretically about 45 percent and 80 percent greater than the TulinBurkart configuration. A simplified calculation of the location of the cavity boundary streamline for arbitrary configurations is also presented. The method assumes that the contribution of camber to the equivalent airfoil vor ticity distribution is concentrated at the center of pressure. INTRODUCTION In reference 1 Tulin and Burkart present a linearized theory for determining the characteristics of supercavitating twodimensional hydro foils of arbitrary section operating at zero cavitation number. It is shown that the hydrofoil problem may be transformed into an equivalent airfoil problem which can be treated by wellknown thinairfoil theory. The theory shows that hydrofoils with high liftdrag ratios are those whose equivalent airfoils have their centers of pressure as far aft as possible while maintaining all positive vorticity over the chord. In reference 1 such a lowdrag section was chosen by specifying only two sine terms in the pressuredistribution expansion (or equivalently, the vorticity distribution) and then these two coefficients were adjusted so that the necessary conditions for high liftdrag ratios were satisfied. NACA RM L57Glla The purpose of the present report is to derive hydrofoils whose air foil center of pressure is further aft than the TulinBurkart configura tion and thus even higher liftdrag ratios are obtained. One obvious means of accomplishing this is to specify a given pressure distribution on the airfoil and then to determine the Fourier coefficients which describe it. This procedure usually will not lead to a closedform expression for the airfoil or hydrofoil shape; however, the method does permit adequate solutions in tabular form to be made. In the present case, it was reasoned that superior configurations could be derived merely by choosing more terms in the vorticity series expansion and adjusting the coefficients for maximum liftdrag ratio exactly as was done by Tulin and Burkart. In this manner, the results would be in a closed form. The number of terms chosen for the analysis was specified as three for the first case and five for the second case. The results of calculations based on these presumptions are presented. Since, for practical reasons the hydrofoils must have some thick ness; the shape of the cavity streamline leaving the leading edge is required. The thickness of the hydrofoil that can be permitted is then such that the hydrofoil upper surface lies below this free streamline. The linearized theory permits, in principle, this streamline location to be calculated; however, when the expression for the airfoil vorticity distribution becomes very lengthy, the calculation is very difficult. If the vorticity due to camber is assumed to be concentrated at the cen ter of pressure of the airfoil, the problem is greatly simplified. The results of an analysis based on this assumption are also presented. SYMBOLS An coefficients of sineseries expansion of airfoil vorticity distribution; that is, 0(x) = 2V (Ao cot + Al sin B + A2 sin 29 An sin n) 2 A' value of A0 when a.= 0; that is, A0' f a d9 0A 0 dR a distance from airfoil leading edge to center of pressure in chords a2 A= 2 A5 az = 3 Al NACA RM L57Glla c chord CD drag coefficient, D/qc CL lift coefficient, L/qc CL,d design lift coefficient at a = 0 Cm pitchingmoment coefficient about leading edge, M/qc2 C0m thirdmoment coefficient about leading edge, M /qc4 p p Cp pressure coefficient, qP D drag force k k number of terms in summation An sin n9 nM n=l L lift force M pitching moment about leading edge M3 third moment about leading edge, 2 p(xi)idi 0 p local pressure p ambient or freestream pressure q dynamic pressure, pV2 I distance from section reference line to upper cavity streamline V speed of advance, fps u perturbation velocity in xdirection v perturbation velocity in ydirection NACA RM L57G11a x' dimensionless distance parameter along Xaxis, x/c x distance along Xaxis y' dimensionless distance parameter along Yaxis, y/c y distance along Yaxis a geometric angle of attack, radians P circulation Q vorticity 6 variable related to distance along equivalent airfoil chord by equation x = 2 c(l cos 9) Subscripts: a due to A0 or if Ao' = 0 is due to angle of attack c due to camber Barred symbols refer to quantities in the airfoil plane and unbarred symbols to quantities in the hydrofoil plane. SUMMARY OF THE TULINBURKART LINEARIZED THEORY Since it will be necessary in the derivation of the new hydrofoils to refer frequently to the linear theory of reference 1, a summary of the principal results of that theory is useful. In reference 1 it is shown that the problem of a hydrofoil operating at zero cavitation number in the Zplane may be transformed into an air foil problem in the Zplane by the relationship Z = ZF7. By denoting properties of the equivalent airfoil with barred symbols and those of the hydrofoil with unbarred symbols, the following relationships are derived: d&R)= 2(R2) (1) di dx () = u (2) (2) NACA RM L57Gl1la CL = m = A0 + Al T CD = L2) = A A 2 Cm = C,5 = 32(o + 71 7A2 + )A z) The coefficients An are the thinairfoil coefficients in the sine series expansion of the velocity perturbations U(i) where u() = V Ao cot 2 + A sin + L \ n=l where I = 1 (1 cos 9) 2 (o 9 6 x) Since U(R) = 2n(R), equation (6) defines the vorticity on the equivalent airfoil as I w, () = 2V Ao cot + n=l distribution An sin no) The values of the A coefficients can be found for a from the following equations: given configuration Ao = 1 .n f o SdO + a = a + Ao' dx An = I d cos nO dO IC f dA (9a) (9b) NACA RM L57Glla The first term in equation (8); that is, the Ao term, is the vorticity due to angle of attack and the second term is that due to camber. In order to isolate the effects of camber, Ao will be con sidered zero. Any section profile derived on this basis will also, for convenience, be oriented with respect to the Xaxis in such a manner that Ao' = 0. From equation (9a) these conditions require that a also be equal to zero. Thus, the derived orientation is defined as the zeroangleofattack case. When A0 is set equal to zero, the hydrofoil liftdrag ratio for a given lift coefficient is obtained from equations (5) and (4) as follows: 2 CL 2(A%) = A2 (10) CD A/4 2CL 2 Al) A2 Obviously, for maximum liftdrag ratio, must be as large as possi A1 ble. However, if the assumed condition that a cavity exists only on the upper surface is to be real, the vorticity distribution given by equa tion (8) must be positive in the interval 0 S 9 5 v; that is, the pres sure on the hydrofoil lover surface must be positive over the entire chord, otherwise a cavity will exist on the lower surface. Thus, for maximum hydrofoil liftdrag ratio, A2 must be as large as possible Al and still satisfy the condition that 00 () = 2V An sin n G 0 (o0 9 5 ) (11) n=l With the stipulation that the vorticity distribution is defined by only two terms in equation (ll), reference 1 finds the optimum relation ship between Al and A2 as = 1. This results in a hydrofoil configuration given by the equation = + 5/2 ()1 (12) NACA RM L57G11a From equation (5) the design lift coefficient (that is, for a. = 0) for this section is CLd 5tA (15) and the liftdrag ratio for this condition as obtained from equation (10) is L _25/ I (14) D 7 2CL Since 7/2CL represents the liftdrag ratio of a flat plate, the config uration given by equation (12) has a liftdrag ratio 25/h times as great as that of the flat plate. When the hydrofoil given in equation (12) is operated at an angle of attack, the liftdrag ratio becomes a + 2 CLd L I 2 (15) ( + CL,d) The present analysis is concerned with the derivation of two new configurations. The problem is exactly the same as that discussed in reference 1 and summarized above except that the vorticity distribution given by equation (11) is defined by: (1) three terms and (2) five terms. DERIVATION OF LOWDRAG HYDROFOILS AND THEIR CHARACTERISTICS Statement of Problem The problem under consideration is (1) to find the values of the coefficients in the vorticity equation k n() = 2V A sin n 0 (o0 9 it) (16) n=l such that A2 is a maximum for the specific cases of k = 5 and Al k = 5, and (2) to use the method of reference 1 to find the shape of 8 NACA RM L57Glla the hydrofoil which when transformed to the airfoil plane has the vor ticity distribution given in equation (16). ThreeTerm Solution (k = 5) For the case k = 5, the vorticity distribution given in equa tion (16) becomes Q(R) = 2V(Al sin 9 + A2 sin 29 + A3 sin 30) 0 (17) The solution of equation (17) is obtained in the following manner. Let a2 = (18) Al a = A (19) SAl The problem is now to find a2 and a so that a2 is a maximum and sin a2 sin 29 + a sin 30 5 0 o 9 ) (20) Substituting trigonometric identities for the functions of the multi ples of 9, equation (20) may be written as 1 2a2 cos + 5a ha5 sin29 0 (21) The minimum of equation (21) occurs when 1 a 9 = cos 2 4a5 Substituting this value of 9 into equation (21) gives 2a22 / 2 1 + 5a a al 2 0 (22) a5 5 \ l+a 3 INACA RM L57Glla or ha 4aa2 a22 >0 = b (b 0) (25) Therefore, a2 =t +4a ha2 b (24) and the term under the radical has a maximum at a = Thus, 2 a = t 4l b (25) and the maximum possible value of a2 is 1 which occurs when b = 0 1 and a = . Since these values are obtained by considering the mini mum value of the vorticity or pressure on the airfoil, the condition (x) 0 is satisfied for all values of 6 (0 5 6 it). Thus the solution for the vorticity distribution for the case k = 5 is Q(5) = 2VAl sin 9 sin 29 + 6 sin 59) (26) The airfoil slope which has the vorticity distribution given by equation (26) is obtained from reference 2 and is given as follows: d =Al cos 6 cos 29 + I cos 5I) (27) dA (C 2 0) Substituting trigonometric identities for the functions of the multiples of 6, equation (27) becomes dy Af2 cose 2 cos2 1 cos 6 + 1 (28) and since cos 6 = 1 2 . NACA RM L57Glla d ) = A, 21 2 dx 2(l1 2 ) The slope of the equivalent hydrofoil is obtained from reference 1 and is given as follows: S(x) =  dx d ( (30) Equation (50) states that the slope of the hydrofoil can be obtained from equation (29) by replacing i with 4x. Thus, since E = Fc, S= A 1[2 (1 2  2(1 2 2 Integrating from 0 to x and dividing both sides by c gives the desired nondimensional hydrofoil shape; that is, + 8o(0)2 64()5/2 (52) )= 51 20 c 10 (2/ ^c By using equation (5), the lift coefficient of this hydrofoil becomes CL + L (35) or for a = 0 the design lift coefficient is CL,d = 4nA (34) The following drag coefficient may be obtained by using equation (4): C = (.A)2 +2L 2 33ff / (55) (29)  1 2 +1 (51) c112 + 1 NACA RM L57011Ga For a = 0, the liftdrag ratio is L = 9( (56) This value is nine times as large as that for a flat plate and 1.44 times as large as the value for the hydrofoil of reference 1 where L = J . D 4 k2CL) The following liftdrag ratio may be obtained for finite angles of attack (eqs. (52) and (34)): L a + CLd7) ( + 2CLd)2 FiveTerm Solution (k = 5) For the case k = 5 the problem is to find the coefficients in the following equation: Q(f) = 2V(Al sin 9 + A2 sin 29 + A sin 59 + A4 sin 49 + A5 sin 50 (38) A2 so that Q(0) o 0 and is a maximum. Al First attempts at a solution were made on a Fourier synthesizer. The synthesizer is an electronic device which is capable of generating 80 harmonics of a Fourier series and recording the summation of these components over any desired interval. The amplitude and phase angle of each harmonic generator is controllable. By using only the first five components and zero phase angle, it was discovered that a solution with A2 A2 roughly equal to 1.6 was apparently possible. Unfortunately the Al sensitivity of the equipment was not sufficient to assure positive values of the summation of components near the leading edge. However, the syn thesizer result was encouraging, since it showed that apparently there was a considerable advantage to using five terms, and revealed some of the characteristics of the solution; for example, the algebraic sign and NACA RM L57Glla relative magnitude of each term. The most helpful method for obtaining the best results was that used in obtaining the threeterm solution. This was to find first the minimums of equation (58) in terms of the coefficients. The term was then assigned a value and the other A1 coefficients were determined analytically so that three of these control points (possible minimums) were zero and the values of the others were A2 examined. By varying the value of and the choice of control Al points, a solution was obtained. The method is admittedly one of trial and error and, since the process is somewhat lengthy, the details are omitted. The best solution obtained was (x) = 2VAl(sin 9 sin 26 + sin 5 2 sin 6 + 1 sin 9 (59) In the course of deriving the solution it was proven that the value of must be less than V2. Since in the solution given by equa Al tion (39) the term A2 has a value of 4/5 (very close to the estab Al lished maximum), further efforts to find a better solution were not con sidered worthwhile. By following the method used for the threeterm solution, the shape of the hydrofoil corresponding to equation (59) is obtained as follows: S= 210) 2,240(5/2 + 12,600(2 30912 2 + 55,80)5 15,560( )7/2 (40) By using equation (5), the lift coefficient of this hydrofoil may be given as CL = i + (41) or for a = 0 the design lift coefficient is CL,d = 5itA (42) NACA RM L57Glla The following drag coefficient is obtained by using equation (14): CD = + 1) = 2 4 and for a = 0 the liftdrag ratio is L 100/c (4) D 9 CL This liftdrag ratio is about 11 times as large as the value for a flat plate and nearly twice as efficient as the configuration of reference 1. For finite angles of attack, 2 L a + 2 CL,d D % '2 ( + CLd Comparison of the LowDrag Configurations Shape. The shapes of the two, three, and fiveterm configurations given by equations (12), (52), and (o40), respectively, are compared in figure 1. It is apparent in figure 1 that the location of maximum camber moves toward the trailing edge as 2 is increased. This movement Al corresponds to moving the center of pressure of the equivalent airfoil toward the trailing edge. It is shown in reference 1 that the limiting A2 value of is 2 and that for this value all the lift is concentrated A1 at the trailing edge. An important point to note in figure 1 is the appreciable deviation of the three and fiveterm hydrofoils from the Xaxis. A similar devia tion from the Xaxis exists in the airfoil plane whereit was originally assumed that the vorticity was concentrated along the Xaxis. Evidently the assumption is not as good for the higher term hydrofoils as it is for the twoterm configuration, particularly for large magnitudes of camber. As a result, the linearized theory may be less accurate in pre dicting the characteristics of the new hydrofoils. NACA RM L57Gl1a Pressure distribution. From equations (2) and (6) and the linearized Bernoulli equation, it can be shown that the pressure distribution over the hydrofoil chord for Ao' = 0 is P Pa2 k cot + An sin n=1 (46) or separating the Cp ,a and camber, two components Cp gives Cp into contributions of angle of attack = 2 cot 2 (47) k Cp,c =2A n sin n6 n=l1 (48) In equations (47) and (48) the location on the hydrofoil corresponding to a given value of e can be found from the relationship x= 1(1 cos 9) E 1T since = ( )2 The coefficient A1 defines a particular value of the hydrofoil lift coefficient at a = 0; that is, the design lift coeffi cient CLd given in equations (15), (54), and (42). Therefore, with the aid of these equations, equation (48) can also be written in terms of CLd as Cp,c CL,d Al S2 A CL, d (49) k A s i n n O Ax NACA RM L57Glla Thus, the total pressure distribution on the hydrofoils can be obtained from C pp = a + 'P CLd (50) Equations (47) and (49) are plotted in figure 2 for the three hydrofoils under consideration. It is apparent in figure 2(a) that the location of the maximum pressure moves aft as A2 is increased. It may also be Al seen that the adverse pressure gradient to the left of the pressure maxi A2 mum also increases as A increases. Thus the fiveterm hydrofoil is A1 more susceptible to boundarylayer separation than the other two. If such separation occurs, the pressure distribution shown will be con siderably altered. This of course also applies to the two and three term solutions but to a lesser degree. Because the adverse gradient increases so rapidly with increase in 2, it is believed that further A2 Al increases in , attained by considering more terms in the vorticity Al expansion, will not be practical. The small pressure "humps" near the leading edge of the three and fiveterm hydrofoils are peculiar to the solutions found but could be eliminated by proper adjustment of the coefficients. However, the exist ence of these humps is probably not important in a practical configuration. Liftdrag ratio. The liftdrag ratio and lift coefficient given by equations (15), (57), and (45) are plotted for the three lowdrag hydro foils in figure 5. The relationship L=  for a flat plate is also 7 2CL included. The solid lines show the liftdrag ratios of the three low drag hydrofoils when operated at a = 0 but for various magnitudes of camber; that is, CL,d. The broken curves are for the particular magni tude of camber for which CL,d = 0.2 and 0.4, but the angle of attack is varied. In figure 5 it may be noted that the liftdrag ratios of the three and fiveterm solutions when operating at their design lift coefficients are considerably higher than the twoterm solution of reference 1. It is also evident in figure 5 that the relative magnitude of the liftdrag ratios of the three sections decreases with increase in angle of attack. However, figure 5 shows that the reduction in L/D with increasing angle of attack is lessened by using higher values of CLd. Only the shaded NACA RM L57Glla portion of figure 5 is considered of practical value because the hydro foil must operate at finite angles of attack as will be pointed out in the following section. APPROXIMATE LOCATION OF THE CAVITY BOUNDARY STREAMLINE The desirability of operating as near the design lift coefficient as possible is obvious from figure 3. Therefore, since the hydrofoil must have some thickness, the minimum angle at which a hydrofoil with finite thickness canr operate with a cavity from the leading edge is needed. The angle can be determined from the linearized theory of ref erence 1 by determining the location of the upper cavity boundary. The minimum angle at which the upper cavity streamline clears the upper sur face of a hydrofoil of finite thickness is the angle desired. An approx imate solution for the location of the cavity streamline is derived in the following analysis. It is shown in reference 1 that the slope of the cavity upper sur face formed on a twodimensional hydrofoil operating at zero cavitation number and infinite depth can be obtained by transforming the vertical velocity perturbations ahead of the equivalent airfoil to the cavity upper surface. These velocity perturbations are obtained by setting up the expression for the velocity induced at a point i, upstream of the equivalent airfoil. The procedure usually leads to very complex prob lems in integration, particularly if the series expansion of the vor ticity distribution contributed by the camber is very lengthy. This complication is avoided in the analysis to follow by assuming that the vorticity contributed by the camber is concentrated at only one location, the center of pressure of the airfoil when Ao = 0. The magnitude of the concentrated circulation is similarly prescribed. The method can be expected to give only an approximate answer, particularly if very much of the camber vorticity is located near the leading edge. However, for the new lowdrag cambered sections being considered, the vorticity due to camber is in fact concentrated away from the leading edge (as indi cated by fig. 2) and the approximation should be very good. The hydrofoil and its equivalent transformed airfoil are shown in figure 4. The symbols used in figure 4 are those used in reference 1 where u and v are the velocity perturbations in the x and ydirections in the hydrofoil plane, and a and V are the perturbations in the air foil plane induced by the airfoil circulation. In the airfoil plane the vorticity is divided into two components a = 2VAo cot (51) 0VA 2 NACA PJRM L57Gila (52) c = 2V ) An sin no n=1 The first of these, :, is taken to be distributed over the chord exactly as given by equation (51). However, to simplify the problem, the second component of vorticity is assumed to be concentrated at one point on the chord the center of pressure when Ao = 0. This point is given in figure 4 as a distance aE aft of the leading edge. The magnitude of the concentrated vorticity is denoted as c (circulation due to camber) and is given by the following equation which is obtained from thinairfoil theory (for example, see ref. 2): r. = TcV (55) The velocities induced by the circulation contributed by as 7a, and those due to rc are denoted as vc. A. are denoted The slope of the cavity upper surface in the hydrofoil plane is dyx v(x) V() V V( + (5 x V V V V and therefore fx c x) O V Ix f a ( dx +) Odx + Jo v (55) The first term of equation (55) has been evaluated in reference 1; therefore, only the contribution due to camber need be considered here. In figure 4 the induced velocity at any point along the Xaxis due to the circulation rc is MACA RM L57G11a c11) = i7V A1 2jtF(a.) 2i5a A1 V a  5 (56) (57) At a point forward of the airfoil x, e() = 'cF ) = 1 Al V 4 a + v since E = Fc. From equations (55) and (57) A1 ~4c ca + where the center of pressure of the airfoil airfoil theory, for Ao = 0, as  a loge ( ) +. (58) a is found from the thin a = 1  2 (l2 A,) (59) By combining equation (58) with the linearized flatplate solution of reference 1, the complete solution for the shape of the cavity upper streamline on arbitrary configurations is y' = Ao x' + j 1 + 2F (x' + x') + I logl + 2rx' A l a loe+ ax+ T[^~a loge  2 + ii' + (60o) where x' and y' denote the dimensionless parameters, x/c and y/c. In this equation y is the distance from the Xaxis to the cavity upper surface. When the hydrofoil reference line is at an angle of attack, Sc = T NACA RM L57Glla the actual distance I from the reference line to the cavity streamline is I = y + ax (61) S= y + ax c as can be determined from figure 4. When equation (60) is into equation (61) and Ao is replaced by its equivalent equation (8), the following equation is obtained: substituted a + Ao' from c= A'x + (a + A') + 2x x' + + + x  2x'+ +] a loge(a +aF) (62) where I/c is the dimensionlessdistance parameter from the hydrofoil reference line to the cavity upper surface. By separating the angle of attack and camber contributions, equa tion (62) may be written as = + (!c = where for the case of under consideration, (65) a Al A0' = 0, which applies to the lowdrag hydrofoils l = (I + 2%) (x' + 1) + 1 log el + 2x' 2 x' + x (ci 4 & x and c 1 r a + Ix'\  F = S a loge Al = a aa 20 NACA RM L57GUa For each lowdrag hydrofoil, A, may be replaced by its equivalent in terms of CLd and equation (62) may be written as S 4 CLd(64) where 1= K S Ka loge a + x' cL,d a and K = Al which is for the TulinBurkart design, 2 for the 2CL,d 5r 53r threeterm design, and  for the fiveterm design. 57 The value of a may be determined from equation (59) and is 5/8, 5/4, and 5/6 for the TulinBurkart, threeterm, and fiveterm hydrofoils, respectively. In figure 5(a), a is plotted against x/c and in figure 5(b), a is plotted against x/c for each of the lowdrag CL,d hydrofoils. It is important to note the relative magnitudes of the (i.) (GO coefficients E and a for a given value of x/c. At the trailing a, CL,d edge, is roughly 10 times as great as . This means that a CL,d except for small angles, the angle of attack is predominant in pre scribing the cavity shape. The adequacy of the assumption of concentrated camber vorticity is shown in figure 5(b) by comparing the solid (A) curve with the dashed one. The solid curve was computed from equation (58) and the dashed curve obtained from the coordinates given in reference 5. The tabulated coordinates of reference 5 were computed for the TulinBurkart section by considering the vorticity to be distributed as given in equation (52) and performing the necessary complicated integration. NACA RM L57Glla In figure o the cavity shape derived from equation (64) for the low drag hydrofoils is shown for CLd = 0.2. Also shown in figure 6 is the lower surface of each design for the value of CL,d = 0.2. An inter esting point (first noted in reference 5) is revealed in figure 6. The calculated cavity shape at the design angle of attack falls beneath the lower surface of the configuration. This result was not expected for these lowdrag hydrofoils because the camber was selected to have posi tive pressure everywhere on the lower surface. It is believed that the disagreement is due to the inability of the linear theory to accurately predict the pressure distribution when the airfoil vorticity is not in reality distributed along the Kaxis. However, the shape of the cavity as determined from the linear theory is much less sensitive to the devia tion of the true location of the vorticity from the Xaxis. That is, the distance from a point on the equivalent airfoil to a point forward of the leading edge is approximated very well by only the R component of the distance. Thus, it is seen that the pressure distribution predicted from the linear theory will be more nearly correct when the equivalent airfoil is at an angle of attack and more symmetrically located about the Xaxis. It appears, then, that lowdrag hydrofoils such as those derived in the present paper and reference 1 can never be operated at the design angle of attack for the following two reasons: (1) an upper surface cavity will not form even on an infinitely thin configuration and (2) some thick ness must be provided for strength. The possibility that, near the design angle of attack, the pressure distributions shown in figure 2 are incorrect has been indicated by experimental investigation in reference 4. Even at an angle of attack of 20, cavitation was found to occur near the leading edge on the lower surface of the TulinBurkart configuration used in the investigation. Because of the need for operating at finite angles of attack, the upper portion of figure 5 has been shaded to indicate that the liftdrag ratios calculated near the design lift coefficient are of academic interest only. In general, the minimum angle at which supercavitating flow from the leading edge is possible will be equal to or greater than about 20. The exact minimum angle and, thus, the practical range of operation will be determined by the type and magnitude of camber and the thickness required for strength. In figure 6 the cavity streamline shown may be considered as possible upper surfaces of practical hydrofoil configurations. For a given angle of attack the fiveterm hydrofoil permits a thicker leading edge and a more uniform section. These features are desirable structurally. NACA RM L57G11a CONCLUSIONS The principal results obtained in the application of the linearized theory to the design of new configurations may be summarized as follows: 1. The twodimensional liftdrag ratios of the two new sections operating at their design lift coefficient are theoretically about 45 and 80 percent greater than the TulinBurkart configuration. 2. The relative magnitude of the liftdrag ratios of these new con figurations as compared with those of the TulinBurkart design decrease with increase in angle of attack. 3. The simplified equation developed for the cavity boundary stream line for arbitrary shapes is in good agreement with the more exact solu tion for the TulinBurkart Section and should be adequate for all low drag sections. 4. Lowdrag hydrofoils developed from the linear theory cannot operate at the design angle of attack because an upper surface cavity will not form even for sections with zero thickness. The sections must be operated at an angle of attack slightly greater than the design angle. Langley Aeronautical Laboratory, National Advisory Commttee for Aeronautics, Lanley Fie~ld, Va., July ;2, 1957. NACA RM L57Gla REFERENCES 1. Tulin, M. P., and Burkart, M. P.: Linearized Theory for Flows About Lifting Foils at Zero Cavitation Number. Rep. C658, David W. Taylor Model Basin, Navy Dept., Feb. 1955. 2. Clauert, H.: The Elements of Aerofoil and Airscrew Theory. Second ed., Cambridge Univ. Press, 1947. (Reprinted 1948.) 5. Tachmindji, A. J., Morgan, W. B.. Miller, M. L., and Hecker, R.: The Design and Performance of Supercavitating Propellers. Rep. C807, David Taylor Model Basin, Havy Dept., Feb. 1957. 4. Ripken, John F.: Experimental Studies of a Hydrofoil Designed for Supercavitation. Project Rep. No. 52 (Contract N6onr246 Task Order VI), Univ. of Minnesota, St. Anthony Falls Hydraulic Lab., Sept. 1956. NACA RM L57G11a u m 4 U3 3 * 0 C S 11 H 0) OJ g E m bD 0) O m I' O a '4 0 0 *rI * '4 0) CH K VI C P /( a) 'umo q jo euMpzo IWuOTBuUiTPUON NACA RM L57Glla 25 1 C^^ q ______ ____ 0 H^^ / H  g' U H P4 0 u 0 + 0 40 co 4' 4) &4' 4 0 4 U,~ *r4 0e 0 4  . .00 '0/  ______ '0 K> 4'"r \4)   4') *r 0 0 E4 E U 4 0: '~~4 0  4\rs G 4', 00 d H ~~4 CC i_ _ cm I _____ __ H 4'*I __ __ \ MI0 f ^ r o  H    ____ / C0 0 0 0 0 Nr H 'aNX LI L 26 NACA RM L57Glla 100 i....i.*.. .. IL. JCONFIGURATIONS A Tulin Burkart f... 'w. o B Three term C Five term Io 80 r... i S Note Only the unshaded portion is considered practical. *1'~ (See text.) 1 . ) 50 *tLowdrag hydrofoils operating at design sCo 30 * I ~ jl ~^l ,o~ 1, \1 e'''*^ ^ ^ ^ gl 40 .. '. *"; ^ ^ 1 : ^ ;.;.i' 10 6T11"2' o o a^^^ o .1 .2 *h .5 .6 .7 .( ev 30 10. 20 10 0 1 .2 .3 .7 CL L/D and CL for lowdrag hydrofoils. Figure 3. Variation of IACA RM L57Glla HYDROFOIL PLANE (Z) AIRFOIL PLANE (Z= Vi) Figure 4. The hydrofoil and equivalent airfoil planes. NACA RM L57G11a * hil U~^ C,^" I. 31 , +0 d d P I 0. 00 CD 0 0 H id gi 42 >C 0 +2 di . *rc U2 o 0 il o b V7 Cd 4, 00 (U .b( 0 0) 0 C 0 U cI U; NACA RM L57Glla 29 Hydrofoil lower surface Cavity upper surface Angle of attack.a .2 deg..b .1  3 .1 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 x/c (a) TulinBurkart. .3 o(,deg .2 6 r I cc 01 .1 .2 .3 4 .0 .7 . (b) Three term. .3 o(, des .2 6 o .1 .2 .3 *1.4 .5 .b .7 .8 .9 1.0 K/C (c) Five term. Figure 6. Location of cavity upper surface for lowdrag supercavitating hydrofoils, CL d = 0.2. NACA Langley Field. V. I eIIl UNNERSIry OF FLORWA 31 '4 v 4 i 4 , 4' I It It I Ilk 1 41 'T k'r' iJ'l "'k Y t"Tt 
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