y'^ 
 
 RM L57Glla 
 
 NACA 
 
 RESEARCH MEMORANDUM 
 
 THEORETICAL DETERMINATION OF LOW-DRAG SUPERCAVITATING 
 
 HYDROFOILS AND THEIR TWO- DIMENSIONAL CHARACTERISTICS 
 
 AT ZERO CAVITATION NUMBER 
 
 By Virgil E. Johnson, Jr. 
 
 Langley Aeronautical Laboratory 
 Langley Field, Va. 
 
 UNIVERSITY OF FLORIDA 
 DOCUMENTS DEPARTMENT 
 120 MARSTON SCIENCE LIBRARY 
 RO. BOX 117011 
 GAINESVILLE. FL 32611-7011 USA 
 
 NATIONAL ADVISORY COMMITTEE 
 FOR AERONAUTICS 
 
 WASHINGTON 
 
 September 50, 1957 
 Declassified May l6, I958 
 
MCA RM L5TGlla 
 
 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 
 
 RESEARCH MEMORANDUM 
 
 THEORETICAL DETERMINATION OF LOW-DRAG SUPERCAVITATING 
 HYDROFOILS AND THEIR TWO-DIMENSIONAL CHARACTERISTICS 
 
 AT ZERO CAVITATION NUMBER 
 
 By Virgil E. Johnson^ Jr. 
 
 SUMMARY 
 
 The linearized theory of Tulin and Burkart for two-dimensional super- 
 cavitating hydrofoils operating at zero cavitation number is applied to 
 the derivation of two new low -drag 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 five-term airfoils are shown 
 to be superior to the Tulin-Burkart configuration. For example, the two- 
 dimensional lift-drag ratios of these new sections operating at their 
 design lift coefficient are theoretically about ^4-5 percent and 80 percent 
 greater than the Tulin-Burkart 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 two-dimensional 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 well-known thin-airfoil theory. 
 The theory shows that hydrofoils with high lift-drag 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 low-drag section was chosen by specifying only two 
 sine terms in the pressure-distribution expansion (or equivalently, the 
 vorticity distribution) and then these two coefficients were adjusted 
 so that the necessary conditions for high lift -drag ratios were satisfied. 
 
MCA RM L57Glla 
 
 The purpose of the present report is to derive hydrofoils whose air- 
 foil center of pressure is further aft than the Tulin-Burkart configura- 
 tion and thus even higher lift -drag 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 closed-form 
 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 lift-drag ratio exactly as was 
 done by Tulin and Burkart. In this manner, the results woiild be in a 
 closed form. The nixmber 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 streainline 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 
 
 Ajj coefficients of sine-series expansion of airfoil vorticity 
 distribution; that is, 
 
 a(x) = 2V(Ao cot ^ + Ai sin 9 + A2 sin 29 . . . A^^ sin n9J 
 
 1 r^ 
 
 value of Aq when a = 0; that is, Aq ' = - — / 
 
 J n 
 
 *o 
 
 
 
 dx 
 
 distance from airfoil leading edge to center of pressure in 
 chords 
 
 ao = 
 
 Ag 
 Al 
 
 aT = 
 
 ^5 
 
 A 
 
 1 
 
MCA RM L5TGlla 
 
 c chord 
 
 Cj) drag coefficient, D/qc 
 
 C^ lift coefficient, L/qc 
 
 ^L d design lift coefficient at a = 
 
 Cjji pitching -moment coefficient about leading edge, M/qc2 
 
 Cjjj J third-moment coefficient about leading edge, lA^/qc 
 
 p - p 
 
 C-n pressure coefficient, 22 
 
 P ' q 
 
 k 
 
 drag force 
 
 k number of terms in summation ^ A^ sin n9 
 
 n=l 
 
 L lift force 
 
 M pitching moment about leading edge 
 
 M^ third moment about leading edge, 2 / p(x)x'^dx 
 
 p local press\are 
 
 p ambient or free-stream pressure 
 
 1 P 
 
 q dynamic pressure, — pV 
 
 I distance from section reference line to upper cavity streamline 
 
 V speed of advance, fps 
 
 u pertiirbation velocity in x-direction 
 
 V perturbation velocity in y-direction 
 
4 mCA RM L57Glla 
 
 x' diraensionless distance parameter along X-axis, x/c 
 
 X distance along X-axis 
 
 y' dimensionless distance parameter along Y-axis, y/c 
 
 y distance along Y-axis 
 
 a geometric angle of attack, radians 
 
 r circiilation 
 
 ft vorticity 
 
 9 variable related to distance along equivalent airfoil chord 
 by equation x = i c(l - cos B) 
 
 Subscripts : 
 
 a due to Aq or if Aq ' = 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 TULIN-BURKAET 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 suramary 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 Z-plane may be transformed into an air- 
 foil problem in the Z-plane by the relationship Z = - \jz. By denoting 
 properties of the equivalent airfoil with barred symbols and those of the 
 hydrofoil with unbarred symbols, the following relationships are derived: 
 
 dx dx ^ ' 
 u(x) = u(x2) (2) 
 
MCA RM L57Glla 
 
 ■^L = ^m = 2l^o "^ "^1 " T 
 
 Ct, = C^ = §A„ + A. 2) (3) 
 
 Cm = Cj^,3 = ^Po + TAi - TAg + 5A3 - -^j (5) 
 
 The coefficients A^ are the thin-airfoil coefficients in the sine- 
 series expansion of the velocity perturbations u(x) where 
 
 u(x) = v/aq cot I + y A^ sin n9 (6) 
 
 n=l 
 where 
 
 X = i c(l - COS 9) (0 S e S It) (7) 
 
 Since u{x) = 20, (x) , equation (6) defines the vorticity distribution 
 on the equivalent airfoil as 
 
 a{x) = 2V Aq cot ^ + \ A^ sin nG 
 
 (8) 
 
 The values of the A coefficients can be found for a given configuration 
 from the following equations : 
 
 Ao=-i / ^de + a=a + A^' (9a) 
 
 ° « Jn dx ° 
 
 A^ = I I ^ cos n9 de (9b; 
 
 J <ix 
 
KACA RM L5TGlla 
 
 The first term in equation (8); that is, the Aq 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, Aq will be con- 
 sidered zero. Any section profile derived on this basis will also, for 
 convenience, be oriented with respect to the X-axis in such a manner 
 that Aq' = 0. From equation (9a) these conditions reqiiire that a 
 also be equal to zero . Thus , the derived orientation is defined as the 
 zero -angle -of -attack case. 
 
 When Aq is set equal to zero, the hydrofoil lift-drag ratio for 
 a given lift coefficient is obtained from equations (5) and (k) as 
 follows : 
 
 T 
 
 ^L - V ^ ^' « - llfl _i2^\ (10) 
 
 CD Aj_Jk 2C3 
 
 A2 
 
 Obviously, for maximum lift-drag ratio, must be as large as possi- 
 
 ^1 
 ble. However, if the assumed condition that a cavity exists only on the 
 upper STjirface is to be real, the vorticity distribution given by equa- 
 tion (8) must be positive in the interval = 9 ^ jt; that is, the pres- 
 sure on the hydrofoil lower surface must be positive over the entire 
 chord, otherwise a cavity will exist on the lower surface. Thus, for 
 
 Ap 
 maximum hydrofoil lift-drag ratio, =■ must be as large as possible 
 
 Al 
 and still satisfy the condition that 
 
 00 
 a(x) = 2V y An sin n9 > (o ^ 9 ^ rt) (ll) 
 
 n=l 
 
 With the stipulation that the vorticity distribution is defined by 
 
 only two terms in equation (ll) , reference 1 finds the optimum relation- 
 
 Ap -1 
 ship between A^ and A2 as ~ ^ = ^' This results in a hydrofoil 
 
 A]_ 2 
 
 configuration given by the equation 
 
 y = ^ 
 
 c 2 
 
 x 
 
 *l(^f-i^f 
 
 (12) 
 
MCA RM L57Glla 
 
 From equation (5) the design lift coefficient (that is, for a = 0) for 
 this section is 
 
 ^L,a - ^ (13) 
 
 and the lift-drag ratio for this condition as obtained from equation (lO) 
 is 
 
 Since 11/20^ represents the lift-drag ratio of a flat plate, the config- 
 uration given by equation (l2) has a lift-drag ratio 2^/k times as great 
 as that of the flat plate. When the hydrofoil given in equation (12) is 
 operated at an angle of attack, the lift-drag ratio becomes 
 
 2 
 
 °^ + t CL,d 
 ^ = 1 ^ ^ 
 
 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 (ll) is defined by: (l) three terms and (2) five 
 terms . 
 
 DERIVATION OF LOW-DRAG HYDROFOILS AND THEIR CHARACTERISTICS 
 
 Statement of Problem 
 
 The problem under consideration is (l) to find the values of the 
 coefficients in the vorticity equation 
 
 n(x) = 2V y Aj^ sin ne > (o ^ 9 5 «) (16) 
 
 n=l 
 
 Ap 
 such that =• 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 (l6) . 
 
 Three -Term Solution (k = 3) 
 
 For the case k = 3, the vorticity distribution given in equa- 
 tion (l6) becomes 
 
 fi(x) = 2v(a2^ sin 9 + Ag sin 29 + A^ sin 39) > (17) 
 
 The solution of equation (17) is obtained in the following manner. Let 
 
 a2 = - ^ (18) 
 
 Al 
 
 a. = ^ (19) 
 
 ^ Ai 
 
 The problem is now to find a2 and a^ so that a2 is a maximum and 
 
 sin 9 - a2 sin 29 + a^ sin 39 > (o < 9 ^ jtj (20) 
 
 Substituting trigonometric identities for the functions of the multi- 
 ples of 9, equation (20) may be written as 
 
 1 - 2a2 cos 9 + 3a, - ka^ sin^g > (21) 
 
 The minimum of equation (21) occiors when 
 
 e = cos-1 ^- 
 5 
 
 Substituting this value of 9 into equation (21) gives 
 
 2ao^ / ap^ , 
 
MCA RM L57Glla 
 
 or 
 
 Ua^ - Ua^^ - ag^ ^ = b (b ^ 0) (25; 
 
 Therefore, 
 
 a2 = t y^l^a^ - lla^S _ b (24) 
 
 and the term under the radical has a maximum at ax = — • Thus, 
 
 J 2 ' 
 
 ag = ± Vl - t (25) 
 
 and the maximum possible value of a2 is 1 which occurs when b = 
 and 3.-Z = — . Since these values are obtained by considering the mini- 
 mum value of the vorticity or pressure on the airfoil, the condition 
 n(x) > is satisfied for all values of 9 (O S 9 ^ jt) . Thus the 
 solution for the vorticity distribution for the case k = 3 is 
 
 n(x) = 2VA-j_fsin 9 - sin 29 + | 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: 
 
 — = A-l(cos 9 - cos 29 + i cos JQ) (2?) 
 
 dx \ 2 / 
 
 Substituting trigonometric identities for the functions of the multiples 
 of 9, equation (27) becomes 
 
 1-^^ 
 
 2 cos59 - 2 cos29 - i cos 9 + l) (28) 
 
 and since cos 9=1-2— 
 
 c 
 
10 
 
 NACA RM L5TGlla 
 
 dx 
 
 2(1 - 2 ^ 
 
 2 1 
 
 2§J _i (1-2^1+1 
 
 (29) 
 
 The slope of the equivalent hydrofoil is obtained from reference 1 and 
 is given as follows : 
 
 ?(x) = ^(^) 
 dx ijc^ ' 
 
 (50) 
 
 Equation (jO) states that the slope of the hydrofoil can be obtained 
 from equation (29) by replacing x with Vx". 
 
 Thus, since c = ^, 
 
 dx -' 
 
 2 1 - 2 
 
 ^?-4-a^)^-|(x-.^) 
 
 + 1 
 
 (31) 
 
 Integrating from to x and dividing both sides by c gives the desired 
 no ndimensional hydrofoil shape; that is, 
 
 i = l[<t)--(r^^-(|f-^Hl)''' 
 
 (52) 
 
 By using equation (3), the lift coefficient of this hydrofoil becomes 
 
 ^ 3aA 
 
 i. 2\ 2 
 
 (55) 
 
 or for a = the desisn lift coefficient is 
 
 5rtA. 
 
 •'L,d 
 
 (5^) 
 
 The following drag coefficient may be obtained by using equation (k) : 
 
 2 
 
 -f(-^r=i-^; 
 
 (55) 
 
NACA RM L57Glla 11 
 
 For a = 0, the lift-drag ratio is 
 
 This value is nine times as large as that for a flat plate and l.kk times 
 as large as the value for the hydrofoil of reference 1 where ^ = -r^l— ^ 
 
 The following lift -drag ratio may be obtained for finite angles of attack 
 (eqs. (32) and (jU)) : 
 
 (37) 
 
 Five -Term Solution (k = 5) 
 
 For the case k = 5 the problem is to find the coefficients in the 
 following equation: 
 
 fi(x) = 2v(a-|_ sin e + Ag sin 29 + Aj sin 39 + Ai^. sin kQ + A^ sin 59 j (38) 
 
 so that n(x) ^ and - — is a maximum. 
 
 Al 
 
 First attempts at a solution were made on a Foiorier 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 ajnplitude 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 
 roughly equal to 1.6 was apparently possible. Unfortionately 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 encoirraging, 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 
 
12 NA.CA RM L57Glla 
 
 relative magnitude of each term. The most helpful method for obtaining 
 the best results was that used in obtaining the three -term solution. 
 This was to find first the miniraums of equation (58) in terms of the 
 
 Ap 
 coefficients. The term - — =• was then assigned a value and the other 
 
 ^1 
 coefficients were determined analytically so that three of these control 
 
 points (possible miniraums) were zero and the values of the others were 
 
 Ap 
 examined . By varying the value of - — and the choice of control 
 
 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 
 
 fi(x) = 2VA3_(sin 9 - - sin 29 + - sin 39 - ^ sin ^9 + i sin 59] (59) 
 
 In the course of deriving the solution it was proven that the value 
 
 Ap I- 
 
 of = must be less than v2. Since in the solution given by equa- 
 
 Al 
 
 tion (59) the term has a value of U/j (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 three-term solution, the shape 
 of the hydrofoil corresponding to equation (59) is obtained as follows: 
 
 y _ Ai 
 c 515 
 
 210(1) - 2,2U0(|)5/2 + 12,600(1)^ - 50,912(1)5/2 + 
 
 35,81^0(1)^ - 15,560(1)'^/^ (llO) 
 
 By using equation (5), the lift coefficient of this hydrofoil may 
 be given as 
 
 or for a = the design lift coefficient is 
 
MCA RM L57Glla I5 
 
 The following drag coefficient is obtained by using equation (U) : 
 
 n x2 
 
 Cd = §(- + ^) =$U-^4^] (^3) 
 
 and for a = the lift-drag ratio is 
 
 C 9 (^2ClJ ^ ^ 
 
 This lift-drag 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. 
 
 D 
 
 (U5) 
 
 a + -^ C 
 5Jt 
 
 L,d 
 
 Comparison of the Low -Drag Configurations 
 
 Shape . - The shapes of the two-, three-, and five-term configurations 
 given by equations (12), (52), and (UO), respectively, are compared in 
 figure 1. It is apparent in figixre 1 that the location of maximum camber 
 
 Ap 
 moves toward the trailing edge as - — =■ is increased. This movement 
 
 ^1 
 
 corresponds to moving the center of pressure of the equivalent airfoil 
 
 toward the trailing edge. It is shown in reference 1 that the limiting 
 
 Ap 
 value of is 2 and that for this value all the lift is concentrated 
 
 Al 
 at the trailing edge. 
 
 An important point to note in figure 1 is the appreciable deviation 
 of the three- and five-term hydrofoils from the X-axis. A similar devia- 
 tion from the X-axis exists in the airfoil plane where_it was originally 
 assumed that the vorticity was concentrated along the X-axis. Evidently 
 the assiimption is not as good for the higher term hydrofoils as it is 
 for the two-term 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. 
 
lif MCA RM L5TGlla 
 
 Pressure distrib-ution .- From equations (2) and (6) and the linearized 
 Bernoulli equation, it can be shown that the pressure distribution over 
 the hydrofoil chord for Aq ' = is 
 
 Cp = ^ ^ ^°° = ^(a cot I + y A^ sin nG | (1+6) 
 
 n=l 
 
 or separating the two components into contributions of angle of attack 
 Cp 1^ and camber, C_ ^ gives 
 
 ^ = 2 cot I (kl) 
 
 a 
 
 and 
 
 k 
 Cp,c = 2Ai y ^ sin ne (48) 
 
 n=l 
 
 In equations (hj) and (48) the location on the hydrofoil corresponding 
 to a given value of 9 can be found from the relationship 
 
 I = i(i - cos e)' 
 
 since ^ = I — 
 
 The coefficient A-|_ defines a particular value of the 
 ^c ■ 
 
 hydrofoil lift coefficient at a = 0; that is, the design lift coeffi- 
 cient C^ (3^ given in equations (ij), (5^), and (42). Therefore, with 
 
 the aid of these equations, equation (48) can also be written in terms 
 of Cl^^ as 
 
 ^ - 2 -^ y ^ Sin ne (49) 
 
 n=l 
 
MCA EM L57Glla I5 
 
 Thus, the total pressure distribution on the hydrofoils can be obtained 
 from 
 
 (50) 
 
 Equations (^7) and {k9) are plotted in figure 2 for the three hydrofoils 
 
 under consideration. It is apparent in figure 2(a) that the location of 
 
 Ap 
 the maximum pressure moves aft as =• is increased. It may also be 
 
 Al 
 seen that the adverse pressure gradient to the left of the pressure maxi- 
 
 ^2 . 
 mum also increases as - -^ increases. Thus the five-term hydrofoil is 
 
 more susceptible to boundary-layer 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 
 
 Ap 
 increases so rapidly with increase in -, it is believed that further 
 
 Ap ^1 
 
 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 
 five-term 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. 
 
 Lift -drag ratio .- The lift-drag ratio and lift coefficient given by 
 equations (I5) , (37) } and (^5) are plotted for the three low-drag hydro- 
 foils in figure J- The relationship k- = — S— for a flat plate is also 
 
 D 2Cl 
 
 included. The solid lines show the lift-drag ratios of the three low- 
 drag hydrofoils when operated at a = but for various magnitudes of 
 camber; that is, C^ ^- The broken cvirves are for the particular magni- 
 tude of camber for which Cl d = 0-2 and O.k, but the angle of attack 
 is varied. 
 
 In figure 5 it may be noted that the lift-drag ratios of the three - 
 and five -term solutions when operating at their design lift coefficients 
 are considerably higher than the two-term solution of reference 1. It 
 is also evident in figure 5 that the relative magnitude of the lift-drag 
 ratios of the three sections decreases with increase in angle of attack. 
 However, figure J shows that the reduction in l/D with increasing angle 
 of attack is lessened by using higher values of C-j^ i±' Only the shaded 
 
l6 NACA RM L57Glla 
 
 portion of figure 3 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 BOUTTOARY STREA^ILINE 
 
 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 can 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 two-dimensional 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 -x, 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 Aq = 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 low-drag 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 k. The symbols used in figure k are those used in reference 1 
 where u and v are the velocity perturbations in the x- and y-directions 
 in the hydrofoil plane, and u 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 
 
 fia = 2VAo cot I (51) 
 
NACA RM L5TGlla 17 
 
 and 
 
 k 
 fie = 2V \ A^ sin n9 (52) 
 
 n=l 
 
 The first of these, Q^, is taken to be distributed over the chord 
 
 exactly as given by equation (5I) • However, to simplify the problem, 
 the second component of vorticity is assumed to be concentrated at one 
 point on the chord - the center of pressixre when Aq = 0. This point 
 is given in figure il- as a distance ac aft of the leading edge . The 
 magnitude of the concentrated vorticity is denoted as F (circulation 
 
 due to camber) and is given by the following equation which is obtained 
 from thin-airfoil theory (for example, see ref . 2) : 
 
 r, = rtcV ^ (55) 
 
 The velocities induced by the circulation contributed by Aq are denoted 
 as v^, and those due to T^ are denoted as v^,. 
 
 The slope of the cavity upper surface in the hydrofoil plane is 
 
 dy ^ vW = v(-^) ^ ^a(-V^) + ^c(-^) (5^) 
 
 dx V V V V 
 
 and therefore 
 
 X v„(-^) r^v(-^) 
 
 a 
 
 ix + / — dx (55) 
 
 Jo V WQ V 
 
 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 k the induced velocity at any point along the X-axis due 
 to the circulation T^ is 
 
18 
 
 NACA RM L57Glla 
 
 - H 
 
 v,(x) = 
 
 2rtc I a - ^ 
 
 1 _j 
 
 c 
 
 V 
 
 (56) 
 
 At a point forward of the airfoil -x, 
 
 Vc(-x) = Vc(->^) = f 
 
 1 Ai 
 
 a + 
 
 ^ 
 
 since c = ^. 
 
 From equations (55) and (57) 
 
 (57) 
 
 
 c JO 
 
 a + 
 
 Al 
 dx = ^ 
 2 
 
 /t 
 
 ^ ^°^eFx^ 
 
 (58) 
 
 where the center of pressure of the airfoil a is found from the thin- 
 airfoil theory, for Aq = 0, as 
 
 = ll-l^ 
 
 2 An 
 
 (59) 
 
 By combining equation (58) with the linearized flat-plate solution of 
 reference 1, the complete solution for the shape of the cavity upper 
 streamline on arbitrary configurations is 
 
 y' = Ao 
 
 -x' + i(l + 2\^) (/x' + ^) + i logg (1 + 2^' - 2jx' + >/x 
 
 jx^ - a loggU 
 
 a + Jx' 
 
 (60) 
 
 where x' and y' denote the dimensionless parameters, x/c and y/c. 
 In this equation y is the distance from the X-axis to the cavity upper 
 surface. When the hydrofoil reference line is at an angle of attack, 
 
NACA RM L57Glla 
 
 19 
 
 the actual distance I from the reference line to the cavity streamline 
 is 
 
 or 
 
 2 = y + (XX 
 
 - = y' + ax' 
 c "^ 
 
 (61) 
 
 as can be determined from figure k. When equation (6o) is substituted 
 into equation (6l) and Aq is replaced by its equivalent a + Aq ' from 
 equation (8), the following equation is obtained: 
 
 1 = Aq'x' + (a + Aq') |(i + 2v'^) yx' + ^) + i log^ (l + 2)/x^ 
 
 ijx' + ^) 
 
 An 
 
 rr T (a + \/x' 
 
 ^x' - a loggl i — 
 
 (62) 
 
 where z/c is the dimensionless-distance 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 
 
 I = (}l * (l)< 
 
 (^). 
 
 a 
 
 a + 
 
 }). 
 
 An 
 
 (63) 
 
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 under consideration. 
 
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 + Vx' 
 
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20 
 
 MCA RM L57Glla 
 
 For each low -drag hydrofoil, A-^ may be replaced by its equivalent 
 in terms of Cl d ^^^ equation (62) may be written as 
 
 c 
 
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 CL,d 
 
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 K 
 
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 — which is — for the Tulin-Burkart design, — for the 
 
 2CL,d 5rt 3« 
 
 three-term design, ajid — for the five-term design. 
 
 The value of a may be determined from equation (59) and is '^/Q, 
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 ii) 
 
 respectively. In figure 5(a), — is plotted against x/c and in 
 
 figure 5(b), 
 hydrofoils . 
 
 coefficients 
 
 is plotted against x/c for each of the low-drag 
 
 CL,d 
 It is important to note the relative magnitudes of the 
 
 a 
 
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 for a given value of x/c. At the trailing 
 
 CL,d 
 
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 HI 
 
 This means that 
 
 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 3. The tabulated 
 coordLnates of reference J were computed for the Tulin-Burkart section by 
 considering the vorticity to be distributed as given in equation (52) and 
 perfonning the necessary complicated integration. 
 
MCA RM L57Glla 21 
 
 In figure 6 the cavity shape derived from equation (6^+) for the lov- 
 drag hydrofoils is shown for C^ ^ = 0.2. Also shown in figure 6 is the 
 
 lower surface of each design for the value of C^ ^ = 0.2. An inter- 
 esting point (first noted in reference 5) is revealed in figure 6. The 
 calciilated cavity shape at the design angle of attack falls beneath the 
 lower surface of the configuration. This result was not expected for 
 these low-drag hydrofoils because the cajnber 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 X-axis. 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 X-axis. 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 x 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 X-axis. 
 It appears, then, that low-drag 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: (l) an upper surface cavity 
 will not foiTTi 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 k . Even at 
 an angle of attack of 2°, cavitation was found to occur near the leading 
 edge on the lower surface of the Tulin-Burkart 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 lift-drag 
 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 2°. 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 five -term hydrofoil permits a thicker leading edge and a 
 more uniform section. These features are desirable structurally. 
 
22 NACA RM LpTGlla 
 
 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 two-dimensional lift-drag ratios of the two new sections 
 operating at their design lift coefficient are theoretically about k^ 
 and 80 percent greater than the Tulin-Burkart configuration. 
 
 2. The relative magnitude of the lift-drag ratios of these new con- 
 figurations as compared with those of the Tulin-Burkart design decrease 
 with increase in angle of attack. 
 
 5. 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 Tulin-Burkart Section and should be adequate for all low- 
 drag sections. 
 
 k. Low -drag 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 Committee for Aeronautics, 
 Langley Field, Va., J\ily 2, 195?. 
 
MCA RM L57Glla 25 
 
 REFERENCES 
 
 1. Talin, M. P., and Burkart, M. P.: Linearized Theory for Flows About 
 
 Lifting Foils at Zero Cavitation Number. Rep. C-658, David W. Taylor 
 Model Basin, Navy Dept., Feb. 1955. 
 
 2. Glauert, H. : The Elements of Aerofoil and Airscrew Theory. Second 
 
 ed., Cambridge Univ. Press, 19h'J . (Reprinted 19^1-8.) 
 
 5. Tachmindji, A. J., Morgan, W. B., Miller, M. L., and Hecker, R. : The 
 Design and Performance of Supercavitating Propellers. Rep. C-807, 
 David Taylor Model Basin, Navy Dept., Feb. 1957- 
 
 k. Ripken, John F.: Experimental Studies of a Hydrofoil Designed for 
 Supercavitation. Project Rep. No. 52 (Contract N6onr-2U6 Task 
 Order VI), Univ. of Minnesota, St. Anthony Falls Hydraulic Lab., 
 Sept. 1956. 
 
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 NACA RM L57Glla 
 
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NACA RM L57Glla 
 
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