l/vrA-Trv^-^Tl 
 
 B. M. LEADON 
 
 T2CHNICAL MEMOEAUDUMS 
 'lONAL ADVISORY COMMITTEE FOR AERONAUTICS 
 
 ITo. 971 
 
 THE ELLIPTIC WING BASED ON THE POTENTIAL THEORY 
 By Klaus Krienes 
 
 Zeitschrift fur angewandte Mathematik und Mechanik 
 Vol 20, No. 2, April 1940 
 
 Washington 
 March 1941 
 
 UNIVERSITY OF FLORIDA 
 DOCUMENTS DEPARTMENT 
 
 ''.O. BOX 117011 
 '^■'^cSVlli£,fi 32611-701105^ 
 
} 7 ^ ! & 
 
 NATIONAL ADYISOHY COMMITTEE FOR ASHOSTAUTICS 
 
 TECHKICAL MEMOSANDUM IIO. 971 
 
 THE ELLIPTIC WING BASED ON THE POTENTIAL THEORY* 
 By Klaus Krienes 
 
 SUMMARY 
 
 The present report deals with the elliptical wing in 
 straight and angular flow on the "basis of the potential 
 theory. Coniormahly to the theory of first approximation 
 upon v/hich the calculation rests, the ]':novv'n requirements 
 regarding the shape of the surface and its angle of attack 
 must he met. A further condition is that the slope of the 
 surface toward the streamlines must he a continuously dif- 
 ferent iahle function of the points of the surface. If 
 this is not the case, in a given example (for instance, 
 hy aileron deflection or wing dihedral - the latter heing 
 of importance in sideslips), the discontinuities must he 
 replaced hy suitahle rounding off. In general, the cal- 
 culation of a given elliptic surface requires a series of 
 -infinitely many potential functions, the coefficients of 
 which'3.re afforded from linear infinite systems of equa- 
 tions. The expansion is stopped with a certain term, de- 
 pending upon the degree of accuracy desired. Its effect 
 on the integral quantities, lift and lift moment, is prac- 
 tically negligihle. An immediate prediction of the in- 
 duced drag is ruled out, since it would involve all the 
 coefficients of the infinite numher of potential functions. 
 Otherwise, the lift distrihution at the wing tips does not 
 approach sero or the downwash becomes infinite, which is 
 due to the fact that the load distrihution of the lifting 
 line is developed here hy spherical functions (equation 
 (so)) which do not approach zero at the wing tips as do 
 the trigonometric functions employed elsewhere. On the 
 wing in sideslip, vj-hich can he summarils' replaced hy a 
 lifting line, the so-called parasite drag (reference 2) 
 
 w 
 
 ^ij- = - J /(pu - Poh) -y d X d y 
 
 *"Die eliiptische ?^ragflache auf potent ialtheoretischer 
 Grundlage." Z.f.a.M.M., vol. 20-, no. 2, April 1940, 
 pp. 65-88. 
 
ITACA Technical Memorandum Ho. 971 
 
 would have to "be defined first and the suction force on 
 the leading ed£:e subtracted therefrom, v/here , however, 
 extrapolations are recommended because of the finite num- 
 "ber of computed coefficients. Even the, resulting pres- 
 sure distribution is only conditionally valid "by fev; ex- 
 pansion terms, especially near the v;ing tips. 
 
 It may he mentioned that the computed potential and 
 
 dov;nv7ash functions change on transition to K^ S> 
 
 into Xiuner's functions for the circular wing. 
 
 A large portion of the computations were m.ade on the 
 calculating machine, the accuracy of the slide rule heing 
 insufficient in olie calculation of the elliptic integral 
 for h.i<?:her n. 
 
 INTEODUCITIOil 
 
 This article is intended as a contribution to the 
 theory of the lifting surface. The aerodynamics of the 
 elliptic wing in straight and oblique flow are explored 
 on the basis of the potential theory. The foundation of 
 the calculation is the linearized theory of the acceler~ 
 ation potential (references 1 and 2) in which all small 
 quantities of higher order are disregarded. This affords 
 the following, simplifications: 
 
 1. The z coordinate of every wing point is neglected, 
 
 i.e., the variation of the potential function 
 . ■ corresponding to- the pressure pump on the sur- 
 face is situated on the base ellipsoid (fig. 1) 
 
 2. The streamlines, along v.-hich the convective inte- 
 
 gration of the acceleration is effected, are 
 straight lines parallel to the direction of the 
 . strea.m. 
 
 In the case of the elliptic boundary, solutions of 
 Laplace's differential equation A \^ = 0, as products of 
 Lame's function, are known. 
 
 The acceleration potential and Lame's functions. - 
 Suppose that the elliptic vring is in a stationar3^ parallel 
 flov7 with the velocity V. The fluid is homogeneous, in- 
 compressible, f rictionless, and not subjected to gravity 
 and non-vortical outside the lifting surface and the shed- 
 ding vortex band. Then: 
 
iJACA Technical Memorandum No. S71 
 
 dt 
 
 grad p 
 
 (1) 
 
 The pressure 
 tion )]/ 
 
 is expressed "by the potential func- 
 
 P - Pc 
 
 p V^ ^• 
 
 (2) 
 
 where p ^^ is the static pressure at infinity. Function 
 \li satisfies Laplace's differential equation 
 
 (3) 
 
 A i' = ^^-^ + ^-^ + ^ ^' 
 
 ox 
 
 oy^ 
 
 oz 
 
 and ma3^ he visualized a.s heing the result of a superpo- 
 sition of soiirces and sinks of intensity ctIxtt,, y-, , z-^) 
 
 x j! i! 
 
 on every point of the surface. 
 
 (4) 
 
 ■vi/(x,y,z) = I I (j{x^,j z ) -—^ - ] aF 
 
 j i ^ -^ on V R / 
 
 where n denotes the direction of the normals of the sur- 
 face in (x_,,y^,z^) and 
 
 -/(>: - x-^) 
 
 ^ + (y 
 
 \ 3 ,- \ 
 
 yji / + V 2 - z -p ; 
 
 the distance of the starting point (x.y.z) from (x-., 
 ^p'^-p'* ~ integral can De exchanged for one taken over 
 the surface of the ground ellipse. 
 
 \i/(-,y.z) = // CTCx^,y^)-£_f i^ 
 
 z^ V E J 
 
 1 a x^ a 7-, 
 
 (4a) 
 
 The mathematical treatment of the present case he- 
 comes possible "by the introduction of ell ipso idal-hyper- 
 holoidal coordinates (reiererxce s). The semi-axes of th e 
 hase ellipsoid being c in direction y and cVl - K^ 
 in the direction of x (so that £ k c becomes the distance 
 of the aerodynamic centers on axis y) the nev; system of 
 coordinates is: 
 
 -sr^ ir2 „a 
 p= - K^ p^ p- - 1 
 
 = c 
 
 ir2 
 , + iL_ 
 
 2 2 
 
 1 - p.' 
 
 = C 
 
 =>P>1>|x>k>u>-k: (5) 
 
 2 2 
 
 K - V 
 
 2 
 
 1 - P' 
 
 = c 
 
4 
 
 ilACA Technical iilemorandum ITo . 971 
 
 Tlie solution of the equations leads to 
 
 2 VfC^- P^ 
 
 X = 
 
 */l - yC 
 
 J p^ - K^vi 
 
 ■\i.^ - k; ' 
 
 K 
 
 J, 
 
 yp^ - 1 yi - M-^ yi - u^ 
 
 y = cpM,- 
 
 O 
 
 V 
 
 (6) 
 
 The surfaces p, i^ , and v = const., respectively, 
 are confocal surfaces of t:ie second degree; p = 1 j'^ields 
 lliptic hase surface, p, = 1 the plane z = out- 
 the elliptic disk. The surface element of the 
 liosoid is 
 
 •<-> 
 
 ne e 
 
 side of 
 base el 
 
 d X d y = c ^ 
 
 ..^ - v^ 
 
 V 
 
 ji.^ - K^VI 
 
 d (J- d ij) 
 
 K" - U' 
 
 "besides which 
 
 Vl 
 
 =^/l - u^ -Jl - V 
 
 'c- - y- - 
 
 K. 
 
 1 - K' 
 
 (7) 
 
 (8) 
 
 is valid for p = 1. 
 
 The introduction of new coordinates u, v, w 
 
 •Y(u) = p^ - i(l + K^). V(v) = ^.^ - i(l + ^^), 
 3 o 
 
 7(w) = u^ - 1(1 +k2) 
 3 
 
 (9) 
 
 for p, lo. , 1? Dy IT. eans of /' e i ers tras s ' Y function (ref- 
 erence 4), appearing in equation (lO) 
 
 ^^^^-^ = 2 y(V(u) - ei)(7(u) - e3)(V(u) - e^ ) (lO) 
 d u 
 
 where the Quantities 
 
 ei - 1(2 - K^) ; 62 = 1(2K" - 1) ; e 3 = - i( 1 + K^) ; 
 
 ^1 + ©2 + e3 = (11) 
 
 1 ^ 
 
IvTACA Technical Memora-adum "So. 971 
 
 a/si - 63 = vi - k^ ; ve3 - 63 = K ; Vei - 63 = 1 
 
 are posted, gives 
 7(u) 
 
 a V 
 
 n 
 
 d 
 
 u = 
 
 I '' / 
 
 8VP - Si Vp - e 3v P - 
 
 83. J vp2 - lvp2 - K3 
 
 (t2) 
 
 and Lapls.ce's eq.uation reads 
 
 [V(v) --YC,,)]!^^ + [7(k) --YCu)] -|!i' + [V(u) - Y(v)]-2!i'=0 (13) 
 an ov 3w 
 
 Posting the solution in the form 
 
 -4/(u,v,w) = S(u)E(v)3(w) 
 
 (14) 
 
 gives, for each of the three functions 3, Laplace's differ- 
 ential eauation 
 
 d^3(u) 
 
 a u 
 
 = [A + B V(u)] E(u) 
 
 (15) 
 
 with the separation constants A and B. After posting 
 
 B 
 
 b = n<>n + IJ and v^ = - 
 
 1 + K 
 
 - + 
 2 3 
 
 and_,again introducing ij- , v, by means of equation (9), 
 Lame's differential equation (reference 3, Tol. I, p. 559) 
 reads 
 
 P-r-. m / 
 
 (jx^-K=)(;u,^-l)^_fin_Aiil+ (a(2i.= - 
 
 m 
 
 1} 
 
 ^.±iiL_i- 
 
 C.jJ, 
 
 + 
 
 TM ^,.2^ 
 
 lv 
 
 s 2-1 ^ in y ^ 
 
 + K^)vjjj- n(n+l).u""j3,,"(K.) =0 j (l5a) 
 
 ^or B = n(n + l) there are precisely tv/o 2n + 1 
 values Vjn, for v/hich S.,™(m,) has the form ( reference 3 , p. 350) 
 
 ^ m / \ / 2 e , /Z 
 
 
 ^a = ^ 
 
 (15) 
 
 ^1 
 
jAGA Techaical Memorandum 2To. 971 
 
 Oq^°(|j,) = aoM-"^"^! ^2~^3 + ... is an even polynoinial in jj- 
 of degree n - e -,_ - e g -£3,. j^iie values Vj^ are dissimi- 
 lar solutions of an algsliraic equation, resulting from 
 the condition that Cn (i^) is a polynomial. '/■'ith a vievr 
 to calculations later on, Sn'^d') is defined as follov/s: 
 
 e/(t;) = Vl - v^^i i^ZELZ 
 
 1; = 
 
 "G. 
 
 m 
 
 ^i o^b) 
 
 i K 
 
 £3 
 
 = /I CD ^ (-cos c) ^(sincp) ^0_ (cp ) 
 
 (15a) 
 
 by putting 
 
 — = s 1 n a ; ^^s— ^^ — 
 
 K K 
 
 = -cose:' 
 
 A-^ 
 
 = vi^ 
 
 2 2 
 k; sine; 
 
 (17) 
 
 Tlie solution 
 
 •^/•/(^x.v.p) = E^''(^OE^^(i;)E^°(p) 
 
 (18) 
 
 achieved with equation (ic) can also "be represented in 
 different form (reference 5). Tor, on denoting the zero 
 places of the polynomial 0n"(i4) v;ith Pg^. i* is readily 
 
 seen from equations (5) and (6) that 
 
 ID C. N 17 i^ c ^^ 
 
 
 const 
 
 r X sy 
 ^ 1 y yz xyz > 
 
 T T 
 
 / X' 
 
 
 zf_ ^.__lL 
 
 2 ZX 
 
 Ps ^p.s""- k; ps' 
 
 2 _ T 
 
 - c 
 
 ^Ml9) 
 
 One of the factors contained in the parentheses 
 
 is selected and the product II, is formed over all zero 
 
 Ps" 
 places heonging to 0-^ (iJ-). ?he zero places p^^ can 
 
 also oe determined direct by appl;,-ing Laplace's operator 
 q3 %a g2 
 ' '"" — + r:; — - to the right side of equation (l9) and 
 
 3x^ CY'^ oz"^ 
 
 making the result equal zero. The system of equations for 
 
 p„^ is then as folio \vs: 
 
 3^2 3^3 3^1 
 
 S — J. , e , 
 
 PS^- K^ PS^ Pg^-1 ■ ^1^'S PS^- P( 
 
 = 
 
 m 
 2 
 
 (20) 
 
 m=n- €j_- G^- £3 
 
SAGA Technical Memorandum Uo . 971 
 
 (18) 
 
 tenti 
 
 re qui 
 
 inf in 
 
 of th 
 
 tion 
 
 const 
 na + 1 
 
 The potential functions in the respective form of 
 and (19) are so-called "inner" solutions of the po- 
 al equation, not suitable for our purposes, since vre 
 re potential functions v,rhich ordinarily disappear at 
 ity. These are secured by taking the Lame ' function 
 e second type in the variable p. This is the solu- 
 of Lane's equation (l5a), which, for p — > oo as 
 
 >0. It has the form 
 
 ^n"(p) = 2/(p) 
 
 r 
 
 p 
 
 t^n" 
 
 '(p) 
 
 1 
 J 
 
 
 
 
 
 
 Vp2 - 
 
 • 1 
 
 VpS 
 
 — 
 
 K^ 
 
 a'^'s 
 
 be 
 
 red 
 
 .\iced 
 
 to 
 
 ell 
 
 i~ot 
 
 ic 
 
 (21) 
 
 m- 
 
 "Ph, 
 
 The integral can alw _ 
 tegrals of the first and second categories only, 
 aspect of the outer solution of the potential equation 
 is then as follows: 
 
 Vn (P.l^.^0 = E^ {^)E^ {v)l^ (p) 
 
 = 3/(^.)E/(v)s/(p) 
 
 .H» 
 
 do 
 
 Jp [Sn^^P^^^-^P^-l ^/P"- 
 
 (22) 
 
 and equation (19) yields 
 
 I X xy 
 
 "♦■n 
 
 .n 
 
 , m / V cons t J ^ V - T ' 
 ''' (3:,y,z) = ^ — ■( 1 y yz xyz> I I \ 
 
 C3 
 
 r 
 
 ^s 
 
 Ps-^-l 
 
 - c 
 
 2 \ 
 
 d D 
 
 y 
 
 ^ .L LSn''(p)]^ vp^- 1 ^^- k2 
 
 (23) 
 
 i'rom the representation of the potential function bjr 
 
 (4a) as source-sink superposition on the elliptic disk, it 
 
 is a.pjperent that the potential functions in the plane 
 
 z = in the outside zone of the disk must be zero. Hence 
 ! m . , . 
 
 '^ containing factor z must be taken according to (23) 
 
 I.e., only such 
 and (16)); 
 
 Lame functions as are of the form scf. (s) 
 
 i^°^(i.) = vi - 
 
 Mn'^C^^) 
 
 (24) 
 
 Lame's functions have orthogonality characteristics 
 similar to those of the sijherical functions (reference 3, 
 vol. I, pp. 369 and 379):'" 
 
KACA TecHnifial Hemorandum No. 971 
 
 + K • I ■ 
 
 Jh< 
 
 Vl - u^ A" - u^ . Ill ' if m = t 
 
 r ,/" s/rulE^^dOS^^ulE.^i;)- --— " d^.dp 
 
 f , if n ^ s 
 V_ and m = t 
 
 -1 p. ^^onnection v/ith eqxxation (4a) is established "by 
 hs.ving recGXil-seto the follov.-ing representation of 
 
 h (H =y(x - x^)2 + (;.r - y^)2 + (z - z^)^) (reference 3, v. I.I, 
 
 I\ ^ * a. 
 
 p. 172) : 
 
 p - — • E 2.. in (m-)^!! ^^>-n ^P^^n ('^F^^n ^"^F^^n ^ P^ => > ^p 
 '^t- n=o m = 1 
 
 (27) ■ 
 
 v/heiice 
 3 /'I'', ■ •■ 
 
 9zt^ V R/ ZTr = 
 
 
 2c^ -^ ^ ^- yi_^,^2Vl-Up^ 
 
 leaving the summation over 3q^'''((j,) in the form (24) to 
 iDe effected. Then "by assuming that the source-sink distri- 
 "bution on the ellijitic disk approaches zero o n the e dge with 
 the root from the edge distance, i.e., v/ith '^ i _ jj, ^ 
 
 ^i^jtJv) can he developad conf orTiahly to products of Lam.e's 
 function of the type (24): 
 
 a(x^,y^) = Z__ S .^s* Zg'^ (iav)i:s*(-^r) (23) 
 
 after which the formulation of the integral (4a) gives, 
 based on the orthogonality of Lame's products (cf. (26)), 
 
KACA 'Technical Memorandum IJo. 971 
 
 ^Hy^,V,p)=^/T-K^ „5, 2 A^"=I,/M/(l)E^^(^:)s/(i,)?/(p) (29) 
 
 n= 1 m 
 
 Scuation (29) proves the potential function (4a) to 
 "be a certain linear comhination of the functions V^^'" . 
 which are analyzed next. 
 
 Lepresentat ion of the potentig-1 function ^^^ as 
 
 definite integral . - If 
 
 X = 
 
 a-i (3^_ep) (e^-ey) 
 
 i- ^ '/'Y(t)-ea yv(u)-ec, 'A(v)-ec, v'v(w)-ec, 
 
 a 
 
 (30) 
 
 ( e^. defined "bv equation (ll)) or if (S) and (9) are taken 
 into R.C count 
 
 'v/'Y( t) - 6 3 X v' "'■ ( t )_ -e_3^ 
 
 /•YCt) -e. 
 
 yea-ej^Veg-es ° Veg -eiVes -ea "^ Vex-e^ 61-83 
 it can he shov/n that 
 
 (31) 
 
 V^^(x,y,2) =-1- / ^^(X) E^=^(t)dt 
 2tt 1 
 
 , m., s /7T7T-T- / v'V(t)-e s V^ /'-v^(0- 
 
 VBs-es 
 
 i v 63-63 
 
 e3 N^ 
 
 I 
 
 /' 
 
 (32) 
 
 _ J 
 
 polynomial [Y(t)] ^ =^ (33) 
 
 if a. loop about the points of the complex t plane cor- 
 responding to X = ±1 is taken as integration path from 
 TTj_ toward iTg. Qj^(X) satisfies Le^endre's differential 
 
 eauat ion 
 
 (1- X^) 
 
 3) ll 
 
 ra 
 
 %n 
 
 (X) 
 
 dX" 
 
 - 2X UilLlil + n(n+ 1) ^„(X) = 
 dX 
 
 ^n 
 
 (34) 
 
 and Ej^ (t) complies with Lame's differential equation 
 (15), Then 
 
10 JACA Technical Meinorand-um iSFo . S71 
 
 a^^^OO. _ £%U). ^\,-^ ^ l)[7(u) - Y(t)] ^^(X) (35) 
 
 5 u^ 3t^ - . 
 
 for any tv/o variables ea.ch of t, u, v, w; i.e., 
 
 E^^^Ct) i i-i4^ -{a^ + n(n+ l)V(u)}^^(X)l 
 
 L o u ^ J 
 
 = E/(t) ^ ^^-^ - ^^(X) ^ ^--^ ^^^ (36) 
 
 ana 
 
 -^3 
 
 -N 2 
 
 O U 
 
 \!'/-(u,v,;-;)- [Ah + n ( n + 1 )-Y ( u)] >lCi°^ (u , v , w) 
 
 .r„ m, . S w (X) d S. 
 IS (t) - — ^^^^ ^„(^'^) = 
 
 (t) 1^s 
 
 I 
 
 2Tri L ° "t " ^ ^ J TT, 
 
 (57) 
 
 The same holds true if u is replaced hy v or w. 
 The integration path is nest so chosen that 
 
 j 5^i^(t: i:^^!^- ^;,-(x) • -n ^'^ ! =0 
 
 L 3 t dt - TT 1 
 
 i.e., ''I' Yi^ ^^ every one of the three variahles u, v, w, 
 
 satisfies Lame's equation (15) and is accordingly a third 
 
 representation of the potential function defined hy (22). 
 The points X = ±1 in plane t are given "by 
 
 t = v + w + u (X = -1) ; t = V + v.r - u (X = +1) (33) 
 
 On the elliptic disk (p = l) , ^(u) = ei; i.e., 
 1 
 u= r , ^ P =-m. (39) 
 
 / ^/o■' 
 
 1 VO^ - K^ 
 
 X 
 
 The potential function V-l-*- is cited as an example. 
 The sole Lame function of t he fir st type and first degree 
 equipped with the factor ^p^ - 1 is: 
 
 2-1 (o) =vp'^ - 1 
 
iJ-iCA Technical Memorandum So. 971 
 
 11 
 
 According to (22) and (23), respectively, the poten- 
 tial function then reads, respectively: 
 
 'i^^^ (p,ij,,-i?)=vp'^-i'yi--ij.vi-i'' 
 
 n 
 
 dp 
 
 (p^ - 1) Jp^ - 1 ^'p^ - k' 
 
 ma 
 
 V^^(x,y,z) = 
 
 JT^ 
 
 u- 3 
 
 a p 
 
 J (p2- 1) /o^- 1 v'p^ - 1^^ 
 - P 
 
 Lift and lift moaents. - The lift is given hy 
 
 (40) 
 
 v.'herehy 
 
 V 
 
 /I ^ = E,^^)E,-(V)F,-(1) ^zIilMz^^ 
 
 "" 'P=i m^^(i)v'i-k2 
 
 1 
 
 1 / 
 
 c^ - y^ 
 
 x3 M^'^'(p.)Mn"'(p) 
 
 1-^2 Mj^-'^(l) 
 
 J 
 
 (41) 
 
 Based on the orthogonality of Lame's functions, only 
 contributes to the total lift 
 
 A = -§.£. V^ F 
 
 3 2 
 
 ell 
 
 (42) 
 
 ,1. 1 
 
 Vs furnishes the pitching moment about the y axis 
 
 M = 3^ c yi - K^ I Y' ?3ll 
 
 (43) 
 
 -1 
 
 Vg , the rolling moment about the x axis 
 
 L =; 
 
 8 
 
 15 2 
 
 £ Y^ E 
 
 ell 
 
 (44) 
 
 (The negative index refers to the odd functions in y.) 
 
 The elliptic v/ing in strai|£:ht fXovj. - Assume the el- 
 liptic v/ing in a stream in. direction of the positive x 
 axis y/ith velocity F,, ., Eow equation (l) enables t'he cal- 
 
1.2- :TACA Technical Menoraadum Ho. 971 
 
 culation of the velocities induced "by the pressure poten- 
 tial V in space and especis-lly on the lifting surface. 
 The z component of equation (l) reads in the stationary 
 cas e 
 
 0' + u)^-+T^— +WT— =-— ^=V -rf- (45) 
 
 ox dy az p c3z dz 
 
 Small quantities of higher order are disregarded, 
 i.e., 
 
 ^ = 7 22 (45) 
 
 ex c z 
 
 The z component \f of the velocity vector w. is 
 hereafter called "dov/nv;ash" for short. The dov;nwash on 
 the elliptic surface is chtained by integration of equa- 
 tion of equation (46) for z = and y <. c over x: 
 
 X 
 
 w = , ■ £i d X (47) 
 
 Y / 6 2 
 
 — CO 
 
 The calculation of the integral is readily secured "by hav- 
 ing recourse to the representation (32) of the potential 
 
 - ■■. n 
 
 function \i>' „"^ • After formulating Z'^ hy differentia- 
 " cz 
 
 tion below uhe integral sign, equation (47) gives 
 
 n 
 
 m 
 
 ^ 
 
 / / 
 
 iSTT 1 : / G Z 
 
 '^ -- . I ../TotYT 
 
 w.,'" 1 r r ;■ . i -/v t - e, x 
 
 -■K^ I 
 
 / G Z ^ ^ /' 
 
 J y v'e.s - e-L v^g - e. 
 
 11 
 
 -«A(t) - 63 y «A(t) - 
 
 .c.^ 
 
 r 
 
 '^^^3~®i'^'®3~®2° vex-egv'ei-e 
 
 e 1 z \ _ m - ^ 
 
 En ( t) dtdx 
 
 (48)J 
 
 Uow 
 
 3 
 
 /v 
 
 ^Jx) d^J::) 3X d«in(x) v^'(t) - ei 1 
 
 dX ^e, - es Ve, 
 
 c)z dX oz dX ./^ „../.. _ e c 
 
 ^ -^n^-"^) _ d-^^n(X) oZ dQn(X) v^ ( t ) - eg 1 
 
 5x dx dx dx ^/~^z~ir^ y^ 
 
 - e.n 
 
 c 
 
IJACA Technical iylemdrandum iTo . 971 
 
 13 
 
 that is 
 
 £qn(X) . B^n(X) VvCt) - e^ 
 
 3 z 
 
 -IK 
 
 3 X y -YCt) 
 
 (49) 
 
 which, Tsrhea inserted in eq.uat ion - (48) and fallowed ty in- 
 tegration over X, affords - since' lim Q^(X) = for 
 
 m 
 
 n 
 
 r^ iK / q_(x).-— rrr ^ 5^°^(t)dt 
 
 Y 2iTi 
 
 ■Q,^(X) is given oy I'eumann's representation 
 
 (50) 
 
 ^ 
 
 1 r ? (Y) 
 
 (X) = i / ^ ■ dY 
 
 2 J X - Y 
 - 1 
 
 (51) 
 
 Similarly X(t) is expressed v:ith 
 
 (52) 
 
 w 
 
 hence (50) "becomes 
 
 w 
 
 m 
 
 V 2TTi / 2 
 
 r n r: P^(l(s)) dY Jy ( t ) - 
 
 (t) - Y(s) ds VV(t)-e 
 
 2^En™(t)dsdt (53) 
 
 The integrand has poles at t = s^ hecause X (t) - 
 Y(s) = and at t = a^i + ioUg, where -/yTT) - es has a 
 simple zero place. The behaTior of the denominator near 
 the zero place is defined by Taylor expansion. It affords 
 
 a/p ( t ) 
 
 ~\ 
 
 ea = [t - (LU, -.1^3)] (|^yp(t)-e3)^^^^^.^^ + ... I 
 = -[t - ((jUj^ + iWs)] Ve 2-61 '/e 2-63 + 
 
 . (54) 
 
 J 
 
 taking into account equation (l), as well as 
 
 :(t).- Y(s) = (t- s)^ ^^ + 
 
 ^ St/'^^g 
 
 / J. ^ oY 
 
 = ( t - s ) -z— + . • . 
 
 OS 
 
 (55) 
 
^14 ilACA TechiliGal MemofandtLm ■ Ho . 971 
 
 according to equation '( 52) . Then the integration of 
 
 s = V + \ir + u as far a.s s =^ v + vi -u- corresponding to 
 
 Y = -1 to Y = +1 ('eq.uation (53)) gives 
 
 . . 2 / X(wi + i ujg) - Y(s) ds ^ ^ 
 
 By 
 
 r — ; 
 
 / P^'vY(s)) . E (s) d s 
 
 / 
 
 (57) 
 
 . . ^ s ; - eo 
 
 it is to be noted that the integrand for d = 1, i.e. , 
 u = -uj-|_ has the period 2 a'^, so that the integration 
 path can be shifted until s proceeds fro-n i 0)3 - ■:)^_ 
 
 to i u<2 + (-:j]_ (fig- 2), which corresponds to ^ ~ ~ 'o 
 to +-r. v/hen '/Y(s)-e^ = -Ksin £ , that is, ds = 
 
 1=^ (Ac = vi -k2 sin^ e). i-Ioreover, let 
 
 A 3 c 
 
 cvl - K*^ 
 
 I, - = n (58) 
 
 so that, because of 
 
 X(w^ + i.02) = - -p=^^^r-^===== J= n (V(a'i+ icJs) = es) (59) 
 
 ^/e 3-6^^63- = 2 
 
 and, according to (33): 
 
 the downwash function on the elliptic disk becomes 
 
 -^= E^^(K)q^(Ti)- i / P^^C^cose + r,3ine)Mn^(e)Ae-^^ (61) 
 V ■ '^ I ' cose 
 
 The coefficients . obtained' in the polynoraial of i 
 and n are complete elliptic integrals of the first and 
 second categories. The calcula-.ion of the downvrash function 
 in the case of n = 1 is expressed as: 
 
wACA Technical Memorandum llo, 971 
 
 15 
 
 E^MX) = Vl - A^ . Mi^(e) = 1 
 .,(X) = J 1 ^ ^ ^ 
 
 Pi(Y) = Y, ii. 
 
 X - 1 
 
 - 1 
 
 From (61) follows 
 
 -f 
 
 ""^ ^^ ''^■' = yi - K^ Cii(Ti) - ^ /'( e cose + n sin e) 
 V '^ J 
 
 A£ 
 
 d£ 
 
 cost 
 
 2 
 
 that is , 
 
 :iilSh:±-.JTT^^^,M-i^(f): 
 
 TT 
 
 V 2 y 
 
 A e d e 
 
 '?he lifting surface .- The shape of the surface is 
 given "by 
 
 z = z(x,y) 
 
 The slope of the surface in x direction must agree vrith 
 the direction of the flow at the same point, that is, 
 
 9 z(:^,y) ^ w(x,y) 
 ox V 
 
 from which follows, for z = z(x,y) 
 
 (62) 
 
 z(x,y) = ^ / w(x,y)dx 
 
 (63) 
 
 the lov/er integration limit being arbitrary; we equate it 
 to zero and add to the value of the integral an arbitrary 
 function in y: 
 
 X 
 
 (x,y) = ^ / w(x,y)dx + g(y) 
 
 (63a) 
 
 The 1 if t lag line , the induced drag, and the suction 
 force. - ''e merely refer to the corresponding chapters of 
 
16 
 
 1T4.CA Technical Memorandum llo; 971 
 
 Kinner's report (reference 3), where these prohlems are 
 treated in detail. The results are readily applicable 
 to the elliptic disk. The lifting line, hy which the 
 lifting surface is assumed replaced, is ohtained "by cor- 
 relating the lift elements through integration parallel 
 to the X axis : 
 
 ^n'"(0 = 
 
 (Pu- Pod) ^ ^ = 2 p r 
 
 ^l'' 
 
 , m 
 
 n 
 
 -X-; 
 
 J 
 
 -X 
 
 p=l 
 
 E 
 
 d X 
 
 It affords, for instance, 
 
 aii(Ti) = 4tt ^ 7^ c vl - k2 
 
 M - r3(r) 
 
 ^ 
 
 asMTl) = 
 
 a./(,) = 4. £ V- c v^T^ "^^"^ " ^^^^'^ (k3 - p^-) 
 
 V 
 
 a 
 
 4 (ti) = 
 
 > 
 
 (64) 
 
 The •pote.itial function of the second type . - 
 
 going potential 
 linear corahinat 
 "bins, t ions for 1 
 tential functio 
 quantities of a 
 entry of flov;, 
 no flow around 
 angle of attack 
 disk with its f 
 is suction edge 
 finity. All th 
 approach zero , 
 from the edge d 
 tial functions 
 achieved by app 
 first type at c 
 boundary transi 
 
 functions lend themselves in any 
 ion and yield the corresponding li 
 if t , lift moments, and dc-.-rnv/ash w 
 ns dealt v/ith so far afford the ae 
 
 correspondingly curved v/ing by sh 
 that is, at a certain angle of at t 
 the leading edge occurs. Thearbi 
 
 is obtained by superposing a flat 
 lov/, where, as is known, the leadi 
 , that is, the lift density'' approa 
 e potentie.l functions of the first 
 however, on the disk edge with the 
 istance; hence the task of findin 
 that have these qualities. They a 
 lying on the potential functions o 
 
 ons tant x , y , z , and 
 t ion (reference 2) J 
 
 the fo 
 
 The fore- 
 way to 
 near com- 
 The po- 
 rodynamic 
 ock-f ree 
 ack v/here 
 trary 
 
 elliptic 
 ng edge 
 ches in- 
 type 
 root 
 g poten- 
 re 
 
 f the 
 llowing 
 
 
 
 m 
 
 n 
 
 ,n- 1 ac 
 
 [c^ ^ji'°(x,y,z,c)] 
 
 (65) 
 
IJACA. Technical Memo-randum No. 971 
 
 17 
 
 These potential functions <b possess the quality 
 
 of becoming infinite on the whole "border of the disk. 
 Because of the condition of smooth efflux on the trailing 
 edge, it later is necessary to combine the functions 
 linearly. 
 
 The downwash function of the second type .- In con- 
 formity vjith equation (65), the downwash functions are 
 
 obtained by applying the operator 
 
 .n- 1 dc 
 
 c^ to equat ion 
 
 (61), while observing the int errelat ionship : 
 
 r^^.-t^" ^nM> n P,(„) - r, 
 
 -f 
 
 d PnCri) 
 d Ti 
 
 d Pn-i(^) 
 d Ti 
 
 The same applies to P^(Y) and 0,^(71). , Equation 
 
 (51) then gives 
 
 Wv 
 
 m 
 
 II 
 
 2 
 
 T 
 
 - - ^n ^"•■' 
 
 d qn-i'vTl) ^ 1 / d Pj,_^( 
 
 d- n • 
 
 TT 
 
 d Y 
 
 "^ - '^(e)Ae 
 
 M. 
 
 d £ 
 cos e 
 
 (66) 
 
 In dealing '.vith the second fundamental problem, that 
 is, in the calculation of a prescribed v;ing, potential 
 functions are used, the downwash functions of which are 
 independent 'of x on the disk. According to (46) , this 
 
 implies , since 
 
 c3v/ 
 
 = 0, that 
 
 B^\ 
 
 = 
 
 (67) 
 
 1 D=l 
 
 'rt'ith coefficient b^ still to be defined, we put 
 
 3'j^(x,y,z) = L \^^' Qn™(x,y,z) ^_^(x,y,2)= Ebji"^<i>n ''"(x.y,z) 
 m -■' -m 
 
 (68) 
 
 Correspondingly it is: 
 
 w^ = E b/ w,-II, w_,. _2 b,-w,-"II ' ■ (68a) 
 Wji(x,y) is designated as downwash function of the poten- 
 
18 
 
 IIACA ?echnicai-'Metf-6raa-d'uQff Ho, 971 
 
 tial function of the second type ''^n'. According to (68a) 
 and ( 65.) , it is : - • • -. -: - ■ , 
 
 Y m dn 
 
 +E 
 
 1 T'd Pr._ -i(Y) V 
 
 d € 
 
 . 2 
 
 d Y 
 
 cose 
 
 (69) 
 
 m 
 
 w. 
 
 The coefficients "b^ are no\ir so defined that ..^ 
 
 is a funct-ion of ti only. The first term in (69) already 
 depends on ti only; hence the second term itself, which 
 usually depends on i also, must be a function of ti only, 
 ^ fn-i(Y) -^ 
 d Y 
 
 .s a polynomial in Y with terms of the form 
 
 ^ = L tcos e + Ti sin £] 
 hence is a sum of terms of the form 
 
 (cose) -^ (sine) | ^ n 
 
 (70) 
 
 Por the folio v;ing arguments it is assumed that M^ ( c ) 
 is even in e .; so that all terms with odd powers of sin e 
 
 disappear in the intes^ration from -— to +— . But' if a 
 
 ^ - 2 2 
 
 is even, (sin c) = 1 + sum of cos terms. When this is 
 
 v;ritten in (70), y-'^-sp consists of the following sum- 
 
 mands : 
 
 (cos £)n-3q .-i-2p-a ^a 
 and the condition that 
 
 n-3p-a<n-2q<n-?p 
 . . Putt ing 
 
 cr = ^ and 
 
 n + 1 
 
 , respectively (c is an 
 ■ ■ integer) 
 
 (70a) 
 
 (71) 
 
 (72) 
 
 and stipulating that 
 
 lTL 
 
 Jn 
 
 (cos £)■>- = "- s b^"°M/(£) i£-l-^ 
 
 cos £ 
 
 ■= 0; 
 
 <1 = 
 
 (73) 
 
 1 , 2 , . . . ( 0- - 1 ) 
 
iTACA Technical Memorandum Ko . 971 
 
 19 
 
 causes all terms containing the pov/ers of cos e, i.e., 
 po'vers of ^ , to disappear. The sole nondisappearing sum- 
 man d of j^--zv ^g obtained when n - 2 -i = 0, that is, 
 when first n is even and, according to (7l) n - 2p - a = j 
 this then reads, according to (70a), Tj'^"^? and depends 
 no longer on e, so that it can he put hefore the integral. 
 Combining the sirmmands hefore the integral, wh ich noy has 
 the same value for every p, there is obtained conformably 
 d Pn-i(Y) 
 
 to 
 
 d Y 
 
 in (59) for the integral 
 
 ,11 
 
 d Pn-i(Ti) 1 r 
 
 2 I 
 
 d T| 
 
 ■^/_ n 
 
 Sbn 
 
 m 
 
 M- 
 
 '^ fey 
 
 (e) Li 
 
 a e 
 
 cos £ 
 
 (The integral disappears for odd n.) 
 
 Mj^~"^ ( £ ) contains the factor sin e; hence it is odd 
 in e. Considerations corresponding to the foregoing then 
 give the condition 
 
 (cos £ ) 
 
 n-2 q-i 
 
 sii^ e -% ^n 
 
 V. >. -'^ Mn~"'(£) Ae-^-i- 
 
 = 0, 
 
 cos e 
 
 q = 1,2. . .. , (t - 1) ; T= I- and = ^ " ^ , 
 
 respectively 
 
 > 
 
 (74) 
 
 The value of the integral in (69) is other than zero 
 only if n is odd. Summed up, it affords, by attention 
 
 to Bir(K) = Ei5+j^(K) = (equation (16)) : 
 
 - W2r+i(ri) = ksr+i ^J! ; ^v;2j.(ti) = igr :; 
 
 ~ w_(2r+i)(il) = J2r+i ,^^ '^ ; ^ w_3r(ri) = I 
 
 2r 
 
 d ^!,sr-i(n) 
 
 d n 
 
 whereby 
 
20 
 
 NaCA Technical Memorandum !;To . 971 
 
 3^? r+ 1 = "■ ^ 
 
 isr = - / 
 
 l33r+ 1 E3r+ 1 v^^ ) 
 
 "' g b°r Mir (O Ac ^^ 
 
 TT 
 
 COS £ 
 
 "2 
 
 
 Jsr+i = ^ r sine E "bi^+i ^^^^^(e) A £ -— ^ 
 
 ■-^..TT 
 
 '2r - 
 
 -m 
 
 ^2r ■^2f 'i^) 
 
 V (75) 
 
 0^°^ and bn""" satisfy equations (73) and (74), re- 
 spectively. They are c - 1 horaogeneous equations for the 
 C unknown "b^™, and T - 1 equations for the T unknown 
 hjj"^ , respectively, which can he determined therefrom up 
 to a common constant factor. The latter is so chosen that 
 
 ksr+i = 'sr = - '^1 - '''-^ '- 
 whence 
 
 1 I \ I 2 d. ■':2r(Tl) 1 ' \ c ?2--'(^) i 
 
 V dTi 
 
 2r - J 2r+ 1 
 
 = 1 
 
 (75a) 
 
 V 
 
 a ri 
 
 ( 
 Y-v/-(2r+i;(T.)= -2^—^; ---2r(Ti)— vl-K2 ^ 
 
 > (75) 
 
 Lift, lift moments, and the lifting lines of the -po - 
 tential function of the second type .- These quantities are 
 
 olitained "by the ap-olication of the operator c 
 
 c"~- dc 
 
 to the corresponding quantities of the potential factor of 
 the first type (42, 43, 44, 64). It is pointed out that, 
 during the differentiation, the areal content of the 
 ellipse ^ell emoodies the factor c^. Then, hear in 
 mind that (equations (08), (75), (75a)) 
 
 (±) gives the lift A = 8 -^ v ^ ? ^ ^ 
 
 Sell 
 
 (77) 
 
HACA Technical M:eiiiorandum, No... . 971 
 
 21 
 
 ©3 the pitching moinent M = y ^ ^ \ ~ c JT~^ ^ Y^ I ^^.l ("^8) 
 
 (79) 
 
 Mk '-2 J 
 
 8 p - 
 ^-3 the rolling moment' JJ ■ = ^ c — T^ ^gll 
 
 3 2 
 
 TT 
 
 
 n' 
 
 VI - HC® sin" 
 
 £ d e 
 
 and for the lifting lines 
 
 a^Cn) = 4 TT I V"" C-/1 - k2 P^(ti) 
 agCn) =0 
 
 .3(n) = 4 TT £ ?^ cVl - K^ PsCn) 
 
 a.CTi) = 
 
 y 
 
 (80) 
 
 'The second f-undam'ental prohlem of airfoil theory . - 
 This involves the calculation of the aerodynamic quanti- 
 ties of any given elliptic wing by means of the foregoing 
 potential functions of the first and second types. The 
 boundary condition to he met is given by 
 
 (:g,y ) = w(x.y) ^ f'- ^ 
 
 O X 
 
 V 
 
 dx 
 
 (62a) 
 
 which, differentiated, leads to (45): 
 
 3w _ y 3^ 
 
 ox 
 
 32 
 
 (46) 
 
 Posting \|; as a linear com-bination of potential 
 functions of the first and second types, leaves 
 
 'ij 
 
 L, E. a,., V 
 n=i m ^ 
 
 m 
 
 + Z Cv, §^ + 2 D 
 
 n 
 
 n -n 
 
 (81) 
 
 leaves on thedisk (p = l) 
 
 21 
 
 3z 
 
 p=i 
 
 = S Z a„ 
 n m n 
 
 m. ^^l/n'° 
 
 ^^■7^ 
 
 oz 
 
 s mce 
 
 3$ 
 
 : lp=i 3z 
 
 p=i 
 
 (81a) 
 
22 
 
 ITAGA Technical Memorandum lTo.-971 
 
 where then, however, 
 
 BU'^m 
 
 Jl - r: 
 
 ^ m / ^ ^. m / ,, 
 ■^n ^^' -n -^V 
 
 p=i 
 
 Jl - M,2 Jl - V' 
 
 a F^ (p) 
 
 -d( VpS - 1) 
 
 (82) 
 
 ^ p=i 
 
 How, if -^ is developed according to the Lame 
 ox 
 
 products 
 
 c^ c n m ^ 
 
 „m/ \ _m, « 
 E^ (p,) E^ (p) ; 
 
 Bn"^(^) =yi-t.^ m/(^) 
 
 then, because of equations (46), (31a), and (82), "based 
 upon the orthogonality of Lame's product, 
 
 m 
 
 =n 
 
 L 
 
 m 
 
 (83) 
 
 d( Ji 
 
 p' - ly.^i 
 
 This defines the coefficients a ™ of the potential func- 
 tion of the first type. Cn expanding z = z( 'i,r\) in a 
 series of 'i and ri , the a^^"' 's are deterniined by a 
 comparison of coefficients in all terms affected with pow- 
 ers of i in the form of a screen method (reference 2). 
 
 From equation (46) ■ 
 
 V 
 
 w ( ^ , n ) 
 
 m 
 
 n ,m 
 
 ■•n 
 
 / a^ dx = i w(t,) 
 3z i* 
 
 (84) 
 
 is now only a function of ti . This residuary condition 
 
 is complied with through a suitable linear combination of 
 
 the downward function of the second type, which depends 
 solely on ti . The condition reads 
 
 2 Cj^ Wj^(ti)_ + S D^ w_j.^(rj = w(ii) 
 
 n 
 
 (85) 
 
 This equation is integrated from ti = to ri , and 
 then - multiplied by ^sa-i^'n) ^^^ ^37(^1)1 respectively 
 integrated again from ri = -1 to +1, and so yields 
 through the intermediary of the int er-r els.t i ons 
 
UACA Technical Memorandoim Uo.. 971 
 
 23 
 
 d 
 
 + 1 
 
 r ■ 
 
 i ^2r- 
 
 r 
 
 — 1 
 
 ■ id-l) Pscc-i^^) <i ^ = i 2(2a-l)+l 4cx-l 
 
 lo 
 
 if r = a 
 
 if r ^ a 
 
 + 1 
 
 '^3r(Ti) Pa a-i('n) d. Ti = 
 
 a 
 
 (2a- 1) - r(2r + l) 
 
 -1 
 
 two infinite systems of equations for the coefficients 
 
 Cji and Dj^ 
 
 /T- 
 
 HC" 
 
 S C2j,4.-L— ^^l-^:: '-^ -+ Gz^ 
 
 r=o a(2a-l)-r(2r+l) 
 
 + 1 Tl 
 = ^ ./ / w(n')dTi ' p3a-i(Tl)dTi 
 
 D 
 
 - V 1 - r:3 
 
 37+ 1 
 
 4a- 1 ■/ / 
 
 -^ ° +1 ri 
 
 1 r- r 
 
 '2r 
 
 47+1 r= 1 -- 7(27+l)-r(2r-l) V/ 
 
 CC ~ J- f (O y m • • 
 
 w(r, ') dn '^37 (Ti)d'n 
 lO 7=1,2,... 
 
 X86) 
 
 T he outflow condition. - Apart from 'the compliance of 
 
 n 
 
 the ecuation systems (86), the coefficients Cj^ and D 
 must' also satisfy the outflow' conditi on , that is, they 
 must 'be so chosen that the lift density disappears on the 
 trailing edge. The potential function of the first type 
 satisfies this condition without that, since they disap- 
 
 pear on the whole edge. The 'behavior of Q 
 
 m 
 
 on the 
 
 'boundary is known v;ith the application of the differentia- 
 tion process (65) to (41). 
 
 ^ n -I p= 1 
 
 V 
 
 c2 _ y2 _ 
 
 X2 
 
 m/(i;) + ((^1^,1^)) 
 
 1- 
 
 (The term ((vl-p-^)) disappears with the root from the 
 edge distance.) vife post, respectively, 
 
 S b/ ?C(u) = M^(p) and g t^"^ M^^(cp) = M^(cp) (87) 
 
 whence the potential function of the second type becomes 
 r^n]o=r^ ^n ^-n 
 
 
 P= 
 
 yc2_y2__2- 
 
 Mn(P) + ((v^l- 1^2)) (88) 
 
 1^ 
 
 Now the functions M^j^(cp) have orthogonality proper- 
 ties similar to the trigonometric functions, as may he 
 
24 IIACA Technical MemoTan.dtim- W.o. 971 
 
 proved "by (73) and (74) if it is remem'bered that M^^C^) 
 can he written in the form 
 
 Mj^(cp) = (sin cp) ^[ao(cos cp) ^~ 3~ + a2(cos (p) ^ + ...] 
 
 It is - ■ 
 
 3TT . . 
 
 r 2 M^(cp) Mj^(cp) Acp dco = 0, if n - V = ±2 , ± 4 , . . . ( 89 ) 
 
 JH 
 
 CO 
 
 he outflow condition reads, according to (88): 
 ^li Car M2r(cp) + ^.g^ Cgr+i ii2r + i(9) = . IL<rD< ^n (goa) 
 
 CO CC r? 
 
 2 j^sr li-2r(cp) + Z Dsr+i Ii-( 2r+ i) ('+") = ^<-p< ^ (90h) 
 r= 1 ■ r= i . . , '^ ^ 
 
 which, after multiplication hy MgQ^. i ( cp) A cp and M_2Y(9)Acp 
 respec tivels'-, ■ and integration from ^ = ?- to -^j give 
 
 ^sa- i-^8a- 1 » 2a- 1 + t=\ ^^^ ^2r»2a-i = a = 1,2,... (92a) 
 
 °37 I-s'^.-s-Y + 2 Csr+i --(sr+i) .-2Y = "^ = 1,2,... (92"b) 
 
 r= 1 
 wherehy ^51! 
 
 I-Y,6= / -r^ My((?) 15(0) Acp dcp (93) 
 
 The infinite equation _ systems (85) together with (92) 
 then enahle the determination of the coefficients of the 
 potentia.1 functions Cj^ and D^^ of the second type. 
 
 These coefficients in conjunction v:ith the previously com- 
 puted coefficients a^^^ of the functions of the first 
 type make the prediction of the aerodynamic quantities of 
 the given v/ing possible, as v-'ill be illustrated on a model 
 
 case, 
 
 l^Tote . - The calculation is made on the assumption that 
 the flov/ strikes the elliptic disk parallel to the minor 
 principal axis. The length of the prin cipa l axis in y 
 direction was c, in x direction cyi-K2 , The calcula- 
 tion is readily applicaple to the case of elliptic wing 
 in flov/ parallel to the major principal axis when k^ is 
 assumed negative, that is, supposing the aerodynamic cen- 
 ter on the y axis. The conversion formulas for the el- 
 liptic integrals v/ith negative k:^ are: 
 
lACA Technical Memorandtua lo. 971 
 
 25 
 
 TT 
 
 TT 
 
 d e 
 
 d e 
 
 vO+C- K2)sin2 e Vl- k^ . 
 
 ■J 
 
 — K 
 
 1 - 
 
 1- \C 
 
 sin^e 
 
 TT 
 
 TT 
 
 > (94) 
 
 J. 
 
 I a/i + (- K^ ) s in^ £ d t = y 1- K^ 
 
 a - ~'^"' ■ sin^ £ d e 
 l-K^ 
 
 - 
 
 -/ 
 
 The integrals are as far as the factor "before the 
 integral, the same as for the reciprocal axes ratio. 
 
 T he flat elliptic disk in straight flow .- Let the 
 angle of attack he a^, that is, the elliptic area is 
 given hy 
 
 z = -X tan a^ = -x a^ 
 
 Then we have, according to (62), 
 
 w(s,y) dw 
 
 a 
 
 o ' 
 
 dx 
 
 = 
 
 on the disk. According to (83) therefore, the coefficients 
 
 m 
 
 of the potential function of the first type are all 
 
 zero; those of the second type must he computed conform- 
 ably to (73, 74, 75a), as exemplified here for n = 3 and 
 n = -3. 
 
 For an axis ratio of 
 
 vTT^ = i 
 
 K^ = 0.96, that is , A = 
 
 (2c) 
 
 ell ttVi-i 
 
 ;= 6.37 
 
 the complete elliptic integral is 
 
 F =/ 
 
 TT 
 
 d e 
 
 '•J, 
 
 VI - 0. 
 
 96 sin"" e 
 
 3.01611; 
 
 TT 
 
 Z = / VI - 0.96 sin^e de = 1.05050 
 Jo 
 In the case of n = 3 and £]_=1; c^ = c^ = 
 
 E,^(i;) = yr=^^ (u^ - Ps^) 
 
 s = 1;2 
 
 M3 (U) = U^ - p^2 
 
 M3^(e) = K^ sin^e - p, 
 
26 ilACA 'fechaicar ■Bfemorandum ;To.'971 
 
 and according to (2C) it is: 
 
 + + = 0, that is, pi^ = 0.19792; p2^ = 0.97008 
 
 Ps^-^^ Ps^ Ps^-1 
 
 ConditiorB (73) and (75a) then read: 
 
 H 
 
 1d3^^ / (0.96 sin^t - 0.19792)£.£de 
 ■J_-n TT 
 
 2 (2) ,^2 ■ ■ 
 
 + 13., i -^ (0.96 sin^ £ - 0.97008) A e-de = 
 
 - Vl - K^ = - -53^^ W'l - K 2(0.96 - 0.19792) -03" -v'' 1-kc ^ ( . 9 5 
 
 -0.97008) 
 
 whence / ^ n / \ 
 
 Td3 = 1.317, "bs^^"' = 0.3095 
 
 i. e., M3(c;5) = 1.527 k^ sin^ cp - 0.561 
 
 Por n = 3 and e-^=e^=e^=l, we find 
 
 E3-\p) =yr:-^^^-tii^-^^.: Mr^(v) ^^^tilii^i, 
 
 H~"(£) = cos£ sint 
 Equation (75a) reads: 
 
 +11 E 
 
 iT" -1 de -iPj, 
 
 •1 = -r. /- - sin e "b 3 cos e sin € A e =" ''^ 3 / sin*^ e A e d e 
 
 ^ Ti ' cose / 
 
 that is , 
 
 hs" ^ = 2.554 
 Eence 
 
 li_3 (9) = -2. 654 cos cp sin cp, Jk^ - v^ =-Kcoscp (J) 
 
 -he determination of the other necessarj" functions 
 proceeds in similar manner. The integrals (93) can be com- 
 puted ty means of the functions ^^(9) , being either ellip- 
 tic or reducible to elementary. The coefficients ly . 5 of 
 
IIACA Sechnical Memorandum lo . 971 27 
 
 (92) are herevitii known* Calculation of the right side 
 of (86) yields 
 
 + 1 r\ 
 
 +1 
 
 d ^' P2a-i(Tl)dTi=-ao / T\ P2a-i(Tl)<itn = \ 3 ^ 
 
 
 
 -^o-o for a,= 1 
 
 -tto / / d ^' i^2a-lVTi;dT1 = -Uo /- Tl ^SOi-l\r]JdT\= ]S 
 
 J "0 for a >1 
 
 Tor Y = 1,2,... the' right side of (85) is al'»irays 
 zero, whence no asymmetrical potential functions occur in 
 J. In the effected calculation, the series (Sl) of the 
 potential function was stopped with n = 4; hence equa- 
 tions (92) and (85) mus t he taken for a = 1; 2. It gives 
 
 2.101 Ci + 1.5217 Cg + 0,527 C4 = (a = 1 )1 
 
 f(92a) 
 - 0.2410 G2 + 0.5132 C3 + 0.9243 C4 = (a = 2)j 
 
 - i C, + I C3 -f ^03 ■ = -fa^ (a = 1) 1 
 
 110 f ^3^5 
 
 ~30 ^1 - ti^3 + f ^4 = - (a = 2) j 
 
 The direct solution gives 
 
 Ci = C.558 ao , 63= -0.7785 ao , C3 = -0.342 (Xq , 64 = -0.0135a 
 
 and the lift, according to (77), at: 
 
 s. = 0.568 aoX8x|¥2 ^^^^ = 4,55 ^^ | y3 Fgi3_ ' 
 
 that is , 
 
 d c 
 
 ^ a = 4.55 
 
 d tto 
 
 The moment ahout the y axis is, according to (78): 
 
 M = -0.7 785 a„ X 0.9524 X # c A - K^ 2- Y^ I ,, 
 ° ■ 3 2 ell 
 
 M = -1.9 8 a^ c Jl - K.2 -| ys p^^^ 
 
 the center of pressure is at — •^ ' = -0.435, that is, 
 
 Ca/i- k^ 
 at 28.3 percent of the maximum wing chord. 
 
 Incidental to the calculation of the induced drag, it 
 is emphasized that the lift distribution of the lifting 
 
28 IIACA TechiiiGal Memoraadnm No. 971 
 
 line does not disappear when allowing only for an infinite 
 number of series terms in (8l) at the wing tips. (fig. 3), 
 and hence must be included as a substitute lift distribu- 
 tion. In this case theelliptical is roost suitable, giv- 
 ing for the induced drag the well-known ;• formula 
 
 °wi = °a^ 
 
 ^ell 
 
 TT.(2c)2' ■ 
 
 Per the axes ratio 
 
 a/i - K^ = \, K^ = 0.75, A= 2.55 
 
 the procedure is the same, the quantities b^ being as- 
 certained from (73, 74, 75a). This affords the functions 
 M^j^(cp) for the integrals I^ g, v/herewith the coefficients 
 
 of (92) are known. The solution gives 
 
 Ci = 0.3741 a^, Cg = -0.6347 a.^, C3 = -0.2347 a^ , 
 
 C4 = -0.0138 a^ 
 
 the lift being 
 
 d c 
 
 A = 2.9S a-, £. V^ Z^-n and = 2.99 
 
 ° 2 . ell- ^ ^^ 
 
 the pitching moment 
 
 M = -1.397 ao cjl - k^ -^ V^ Fgn 
 and the center of pressure at 
 
 X 
 
 J: -^ 
 
 = -0,467 or 26.7 percent of maximum wing chord 
 
 1 - k: 
 
 Tor y 1 - tc 2 = 2 k2 = - 3 A =0.537 
 i t gives 
 
 Ci = 0.124 a , Cs =-0.5245 a^ , C3 = -0.066 a^ 
 
 64= -0.011 cXq 
 
 that is , 
 
 ^ = 0.99 
 
 d ao 
 
..HACA Technical Memorandum IJo. 971 
 
 29 
 
 The center of pressure is situated at 
 
 — ■— ^ = -0.584 = ?0.8 percent of the chord 
 
 cvi - K^ 
 
 For comparison the values for the flat circular disk 
 are repeated (reference 2): 
 
 1, 
 
 1^2 = 0, A = 1.27 2, 
 
 d (Xr 
 
 = 1.82 
 
 center of pressure: — = -0.515, i.e., at 24.3 percent of 
 chord. 
 
 The calculation method used here perrait.s e ven a "bound- 
 ary tra,nsition to the lifting line (k' = ^-A ~ K,^ = O). 
 It affords, v;hen tv;o series terms are taken into account, 
 
 d Cj 
 d a. 
 
 = 2 TT 
 
 c.p. at 28.8 percent of chord. 
 
 The latter result corresponds to an elliptic spanwise 
 load distr iout ion with a center of pressure, at 1/4 chord 
 in each airfoil section (e.g. of a homogeneous semiellipse) 
 Development of all quantities appearing in the equation 
 systems vith respect to k' affords for small k: ' : 
 
 a. a. 
 
 '3- _ 
 
 2 TT 
 
 1 + 
 
 63 
 128 
 
 TT K 
 
 v^hen 
 
 <1 Cg 
 d ar 
 
 + (In - - - , k' 
 V k' 4 / 
 
 is calculated according to linear v/ing theory 
 
 where, as is known. 
 
 a 
 
 eff 
 
 = a„ + Cc, 
 
 TT 
 
 d Cj 
 
 d (Xr 
 
 2 TT A 
 
 A + 2 
 
 there appears a marked discrepancy at small A with re- 
 spect to the values computed in accord xfith the' theory of 
 the lifting surface (fig. 4). 
 
 On the other hand, the agreement with V/einig's re- 
 sults is good. (See reference 6.) 
 
 Figure 5 shov/s the' Center of pressure position plot- 
 ted against aspect ratio. 
 
 The elliT3tic wing in yaw .- This prohlem can be treated 
 
30 IIACA 'Technical Memorandum 2To . 971 
 
 under the sane assumptions as the wing in straight flow. 
 
 The so-called angle of yaw 3 is defined in figure 6. 
 
 xjovr the streamlines are straights in the plane 
 z = , defined "by 
 
 y = -(x - XT.)tan P + yp \ 
 
 f (95) 
 
 = -X tan 3 + const j 
 
 From equation (l) 
 
 (V cos P + u) H2i + (-V sin ,6 + y) ^ + w ^ = - i 1^ 
 C'X 3y 3z pdz 
 
 under simplifying assumptions, there is found 
 
 7 cos g 1^ - V sin p §-:=T2 M (95) 
 
 ox o-^ Oz 
 
 But "becciUse of (95) it "becomes 
 
 - — = ^= — - -^ — tan p , that is — cos p = H_.cos g - ^ sm 3 
 
 ,dx ox OS- dx 3x oj 
 
 Hence equation (96) can be written in the form 
 
 — ^ cos p = V — ^ (y = - X tan ,3 + const) (95a) 
 
 d s d z 
 
 The dovfnvj-ash follov/s from integration along a stream- 
 line aga.in assumed as a straight line parallel to the di- 
 rection of flow 
 
 X 
 
 ^ = — i— /" 1^ dx (97) 
 
 , V c o s ,3 / d z 
 
 " —CO 
 
 The pooential function \[f. is again assumed as a 
 linear canbination of the potential function '*^^™ t 5^, "^.v,* 
 in the form (8l), whence the same forumlas are obtained 
 for lift and moments. 
 
 The dov/nwash function is computed on the "basis of an 
 o"bl i cue-angle system of coordinates in the xy plane, 
 given "by the ellipse diameter para.llel to the stream direc- 
 tion - |rt a,xis - and the related conjugate diameter - 
 rig axis.' Posting 
 
MACA Technical Memorandum No. 971 
 
 31 
 
 tan Og = vl - -^ tan p, that as , sm cpg = 
 
 cos Cf' ( 
 
 cos g 
 A P 
 
 (98) 
 
 we have the quoted diameters given hj 
 
 4. /" -^ ^. 
 tan CDc tan!^ cpg + „ ; 
 
 y = - ' P - X and , y = - ^ ; ■ - ^^. respectively (99) 
 
 vi - K^ 
 
 Vl - K^ 
 
 On the diskj i.e., p = 1, it is according to eo^ua- 
 tions ( 6) and (17) : 
 
 r 
 
 I = 
 
 c/l - K^ 
 
 Similarly, we post 
 
 ' ■ • cos cp ; n = — = p, sin cp 
 
 /" 
 
 h" 
 
 vTT 
 
 cos 
 
 (cp - cpp) ; T,Q = p, sin (cp -. cp„) 
 
 that is , 
 
 cos 8 
 
 A S 
 
 f, - 
 
 v'^1- K^ sin 3 
 
 a1 
 
 ■ V^l -J^ sin 3 , ^ cos ^^ f-,nn\ 
 
 Ap 
 
 A P 
 
 In this instance the potential function "^j/ ™ con- 
 
 formahle to (32) is utilised. During the respective dif- 
 ferentiations and integrations along a streamline, the 
 fact that y = - X tan P + const should he home in mind. 
 
 r- 
 
 1 m / \ 1 / „ 
 i' (x,y,z) = / Q. 
 
 vV ( t ) - e2 
 
 2T^i./^ ^Ly 
 
 e^ - e ive 3-63 
 
 X 
 
 c 
 
 V e 
 
 'Jyit) - 63 y '/v(t) - ei 
 
 - e iv^e 
 vve find that 
 
 V e 1 - eg V e 1 - 
 
 63 
 
 c I 
 
 n°'(t) d t 
 
 (32) 
 
 5 '^a(X) ^ dqa(X) 1 J'^f^t) - e^ 
 9 z d X c J^^~Z~^.J^~r 
 
32 . i^ACA Technical. Mem-br-andTim -lo . 971 
 
 and 
 
 d x d X c Lv'es -ejv/ 63-63 ^63- e3_Ve3 - 63 -' 
 
 ^^^ ttus can 136 expressed again ty ; posting 
 
 3 z d X - 
 
 the result in (97) and integrating over x gives 
 
 Wn"" 1 __1^ r / /^^ V'y (t) - ei 
 
 Y 2TTi cos p / ^ r — /- — 
 
 ".^-i ^ ■ ■ ■ . 
 
 i ! 
 
 -?j,^(t)dt(lOl) 
 
 VV(t) - 33 V-V(t) - 63 ,,,,p 
 
 v^s ~ ©xVes - 63 -763 - e■^^/e- - 63 
 
 'rt'ith equations (51) and (52) 'in^^^ -^^ ^°^ replaced 
 again "by 
 
 V+'-7+(JL! 
 
 ^ 2 / X - Y(s) ds 
 
 V+W-UJj^ 
 
 The integrand has poles at t = s, where X(t) - Y(s) = 
 and at t = tc, where 
 
 -^'"'-S' - °' -^'''f^ - »3 tan P - 
 
 ^63 — e^ves - 63 vSs - e^^ves - 63 
 i.e.. 
 
 '/^(tp) - 63 = ives. - 63 sincp; */v( tg) - 63 = -'/e3 - 63 cos cp 
 The denominator at this point is as 
 
 ,' — coscpR isincco -1 
 
 (t - tg) yv(t6) - e,: --^ - ^^ tan 3] + 
 
 Live^ - 63 V e ■)_ - 63 
 The residuum at '*' - '''8 ^^ therefore 
 
FilCA Technical Mejaor andtun . No.- 971 
 
 33 
 
 2TTiQ^(X(tp)) 
 
 = 2tt i 
 
 i cos cpg + i tan cpg sin tpg 
 ____ E^ (- Kcos cpp) ^ (r,p) 
 
 Sn"(tg) 
 
 (102) 
 
 with a Tiexv to the fact that, according to (lOO) 
 
 (tfi) = 
 
 sm cp, 
 
 CV 1 - K ^ 
 
 + COS cpp ^ = n^ 
 
 and according to equation (33) 
 
 ^/(tg) = i V^^\. 
 
 K cos q? 
 
 P 
 
 ?or the residuum at t = s, the same holds true as 
 in the case of straight flov/. It affords: 
 +11 
 
 2 
 
 Hi, • ■ i A £ m 
 
 2TTi / — Pj^ (I cose +ri sine ) ' ' ■ ■ iM^^ (e ) de 
 
 ! -„ cos e - VI- K^ tang sin e 
 
 a (103) 
 
 By replacing Vl - K^ tan g hy tan 93 and posting the 
 coordinates |p and Tig hy means of. equation (lOO), the 
 
 values (102) and (103) from (lOl) lead to 
 
 t V('3' r,g) = — En"^(- i^coscpp)qn(Tip) 
 
 1 1 
 AP 2 
 
 +5 
 
 AP 
 
 Mn (O 
 
 ?j^[f.e cos(e-Kpg)+Tigsin(e+cpc)] ;^ - 
 
 . cos (e +';pc; 
 
 Ae de 
 
 (104) 
 
 y 
 
 For the terras of P , containing powers of |g, the 
 
 denominator cos(e + cp ) in the integral disappears , af- 
 
 P 
 fording complete elliptic integrals of the first and sec- 
 ond types. To compute the others, numerator and denomina- 
 tor are expanded with cos(e - cpn)A^ p, which, with 
 
 cos(£ +cpc) cos(€ - cpo)A^3 = cos^ P - A^psin^ e 
 
 gives 
 
'34 . "^ACA Technical. Memorandum ■.No. 971 
 
 + '^ 
 1 r- - ,, , , , ,^cos(e-cpR)Mn'°(e) 
 -^A^ / T i'^QCosie^:pQ)+T\Qsln{e+c,:'p)2 E ; — Ae ds 
 
 2 
 
 The existent integrals are reducible to complete ellip- 
 tic integrals of the first and second types and one integral 
 of form ^ 
 
 r* sin^c de 
 
 J cos^ p - A^ p sin^e A£ 
 
 
 
 which, as a complete elliptic integral of the third type, 
 is reducible to incomplete first and second types (refer- 
 ence 7.) . 
 
 TT . . 
 
 (l-K^)sin p A 6 r> sin^c _d£ 
 
 cos p / cos^p-A^P sin^e Ae 
 " ' 
 
 = E(g)p(|) - e(£) F(P) - A3 tanPF(^^) ' (l05) 
 
 The calculation of the dov/nw'ash function for the po- 
 tential function '^'^^ is given as a model example. 
 
 e/(A) = Jl - A2 M3_^(e) = 1 
 
 Et."''(-h:cosC]3c) =:V1 - K^ cos^ q^a = -*• ''' ^" 
 
 ^ '^ A P 
 
 Then, according to (104a): 
 
5IACA Technical Memorandum Ho. 971 
 
 35 
 
 vTT 
 
 1 1 / ft ' - N 1 
 
 TT 
 
 ^l(Tlp) 
 
 1 ''"^^ '/ s / N-i ^°s (e-qoo)Ae 
 - -A P / L^B cos(£ + qog) + Tio sinCe +90) J ^, ,a, . ^ <i ^ 
 
 v^l - rc 3 
 
 A^P 
 
 <^Jr^^) 
 
 +n 
 
 +H 
 
 i& 1 / 
 
 , / TT 
 
 T /^'^ 'Sin^Po cosCpg+ sine cos 
 — n ^ A B / 1 1 ' ' — ' — 
 
 U-Jl 
 
 cos^ 3 - . A^ P sin^e 
 I'he eYaluation of the integral gives 
 
 ) - ^zCn 
 
 P^ A3 ^ \2, 
 
 AGde 
 
 i 1 (' a ^ v'l - K 
 
 A^g 
 
 
 - T\ 
 
 ^4 - K' 
 ^ A^g L 
 
 '^^p^Ki)- nl)^^^^ 
 
 The dov/nvfash funct i ons of the potential functions of 
 the second type. - According to (68) 
 
 S K'" ^,'' 
 
 m 
 
 n 'n 
 
 wher ehy 
 
 ,tm 1 'ipnim/ \t 
 
 '^n = -^--7 — ^° ^n (-^.y.z.c)] 
 
 according to (65). The downwash function ^n'^B'^B^ ^" 
 computed from w^^^™ ( ^g .tiq) , that is, 
 
 ■^ .r -_JL-vvmT.mr ^^^om ^ '^^n-i^^P^ 
 Y Wji - - ^ I ^n Sn (- Kcoscp^) : 
 
 -JI 
 
 d rip 
 
 d Pn-i(Y) M^(£) 
 
 in the same manner. 
 
 + i_L /- ^ ^H-i^-^ i^AVW ^A£d£ (106) 
 
 2 AP /_ dY cos ( e + cpg) 
 
36 
 
 IJACA Technical Memorandum 3^To. 971 
 
 On the disk v^e have 
 
 n 
 
 = 0, that is , on a 
 
 p = x 
 
 strea.mline (t1b= const) the downwash function is constant; 
 hence w is a function of rip only, and v;e accordingly 
 
 put in = in — P^ ,(T). To find the summand 
 ^'<=\n (e + cp )J •*■ from I 
 
 LS 1] 
 
 we put T =^ ■5- and. 
 
 n - 1 
 
 , respectively, whence 
 
 [sin (£ -t- r,.p)]"-^P =[sin(e + q)^)] '"'^''E s in (e + :Pg )] ^"^ -^P 
 
 = [sin(c + cpg)]'^" (1 + sum of cos terms) 
 
 Factor cos(e + c p ) can he extracted from the sum,. 
 
 thus becoming shorter v^ith respect to the denominator. The 
 new sum, ho\;eTer, multiplied hy [sin(e + ^^q)2^~^\ yields 
 only terms which eithe: satisfy odd in e or else condi- 
 tions (73) and (74), "espect ively , and accordingly disappear. 
 A proport ion ■ other than zero is afforded only hy 
 
 [sin(e -t- Op)j ' x Ij which is, however, no longer depend- 
 
 ent on exponent p. In consequence, all t\q^^~^'^ before 
 the integral can be combined conformably to 
 
 dY '^ dn 
 
 '^_ (r,c), since the integral for 
 
 P 
 
 every p is the sarue ; i.e.. 
 
 7 
 
 1 ■■ ^■. . dO-n-ivTip) 
 -^l ^n° 3n"(- ^cos cop) '- 
 
 +? 
 
 d riq 
 1- 
 
 dPv-i-l'\^6^ 1 A o / r . • Tn-2T 
 
 + — - — '■ '=^— — L p / [sine cosog + cos e sinq;)„j 
 
 d ri g 2 _ / ■ ^ P 
 
 rr 
 
 cos £ cos Op + sin e sin cpo 
 X -E --2- M ^ ( £ ) A e d e 
 
 cos^ p - A^ p sin 2 e 
 
 (107) 
 
 Hence the downwash functions for the wing in yaw have 
 the following form 
 
'JIACA' Technical Msmorandum. Iffo. 971 
 
 37 
 
 1 
 
 V 
 
 wa(r.p) = kjj 
 
 (p, ^32Z^SI , i_^(,) ".-l(^P) (107a) 
 
 d "n 
 
 d n. 
 
 ^ -^ '^ ^ d Ti.p ^ d Up 
 
 The second f-andamental -pro'blem for the case of the 
 wing in sideslip .- The procedure is the same as in the 
 case of the xifing in horizontal flow, "by putting 
 
 yi^~Vr3^cos gM^iiZl=I S S,g„^ K'^iy^) ^n"^ (^) ■ (108) 
 
 dx c n m ° • " 
 
 where 
 
 1 / N o az(x,y) 
 
 — w(x,y) = cos P 
 
 V dx 
 
 (108a) 
 
 and _ y = -x tan p + const; V to be given again hy 
 
 m ,!, m 
 
 + 2 C„ $ 
 
 n m 
 
 n 
 
 n n 
 
 Eq^uation (96a) then gives 
 
 s^^ = Vi- K^ 
 
 ^- ^n"(p) 
 
 d(Vp^ - 1) 
 
 D Q 
 n -n 
 
 m 
 
 (109) 
 
 (110) 
 
 p=l 
 
 and with it the coefficients of the potential functions of 
 the first type. ^ot the determination of the coefficients 
 of the functions of the second type v/e take (97) in the 
 follow ins; form: 
 
 ^ / Bzcosg n ^ oz cosp 
 
 '-—no ■-'—CO 
 
 iw(x.y) - LZ .^ f ^^n- _d X 
 
 n m 
 
 ciz cos p 
 
 2^ 
 
 wdig) 
 
 -\ 
 
 
 J 
 
 (111) 
 
 The right-hand side is a' functioh of Tig only because of 
 the' determination of the , s.^^ from equation (96a). The 
 rest of the formula then reads: 
 
38 
 
 FACA' TeG-h.nic3t3Iie2ioraad.um No. 971 
 
 
 1 
 
 d Tip 
 
 
 1 
 ^ I),,l^(&) — e_ + |. D^j^O) ' = V w(Tig)J(ll2) 
 
 This equation is integrated from Tip = to tiq 
 then - multiplied hy ^sa-i^^s^ ■ ^^^ ^sVCrig) ~ again 
 
 and 
 
 from 
 
 -1 to +1. This affords on the i)asis of the 
 
 orthogonality characteristics of the spherical functions 
 
 r = 
 
 + (CsaisaCP) + ^saJsaCP)) 
 
 'a(2a-l)-r(*r+l) 
 
 2 
 
 4a- 1 
 
 V, 
 
 +1 n 
 
 ip. 
 
 ni 
 
 I I w(ii 'p)dri 'g?2a,-i (t!^. )dn^; a = 1,3, 
 -1 'o 
 
 (113a) 
 
 Z 
 r = i 
 
 (Car -^sr (p) + ^sr Ur (g);- 
 
 + 1 
 
 1 
 ? 
 
 
 / / 
 
 4V + 1 
 w(ti'^) dTi'^ PsY^T^.p) (i^ip; .^ = 1,2, .._^ (list) 
 
 -1 
 
 The outflow condition .- The formulation of the outflow 
 condition is prefaced by the following note. The dov/nwash 
 functions (107) on the wing in sideslip have the quality of 
 ■becoming infinite hy 71-;; = ±1 like £' ^(Tia) and 
 dq^_,(T,p) ^ -n-i . . 
 , respectively. But Ti- = —1 are marginal 
 
 points of the elliptic disk in which parallels to the flow 
 direction touch the ellipse (fig. 6). Accordingly, it is 
 to be assumed that the so-called "vortsx tails" in these 
 points leave the disk parallel to the direction of the 
 stream. Hence it is postulated that no flow around the 
 trailing edge occurs between the points t\s = ±1. In 
 other words : 
 
IaCA Technical Memorandum STo . 971 
 
 39 
 
 n n 
 
 IL + P < CD < ^ 
 
 + ^< 
 
 This equation, multiplied "by 
 
 — 2a-i^^^ A cp and H.gvCcp) Acp, respectively 
 
 to ^ + \[f„ gives the fol- 
 2 P 
 
 and integrated from IL + ©„ 
 lowing systems of equ.ations 
 
 *^2a-i ^2a-i 5 2a-i+ ^ ^gylgr , 2a-i+ 2 ^st'^-st , sa-i' ^'' 
 
 O) 
 
 a 
 
 X y O y • • • 
 
 (P) 
 
 O) 
 
 (s) 
 
 >, 
 
 7 = 1? 
 
 = 0: 
 
 (114) 
 
 wherehy 
 
 3TT 
 
 .O) 
 
 "Y.S 
 
 Jtt, 
 
 M.y(g)) Mg ((p) Acp dcp 
 
 (114a) 
 
 ?hese tv:o eqtiations ((113) and (114)) make it possible 
 
 „ and D„. Together with 
 
 to determine the coefficients 
 
 the previously defined a 
 
 m ic 
 
 n 
 
 the aerodynamic quantities 
 
 of an elliptic wing in yaw can "be computed. As to the 
 equation systems themselves, they form a coupling of the 
 systems (82) and (92) for the case of straight flow, as is 
 readily apparent from the similarity of the corresponding 
 coefficients and which thus affords a first simple mathe- 
 matical check. 
 
 The flat elli-ptic wing in yaw .- The calculations are 
 carried out for the axes ratio 
 
 Vi - [t2 = i 
 
 and the angles of yaw 
 
 3 = 15° and P = 30° 
 
 Again only the potential functions of the second type 
 conformable to (108) and (llO) are required. To simplify 
 the voluminous paperwork only the functions up to the de- 
 gree n = 3 are taken, i.e., 
 
40 KACA Teclanicai Memorandum So. 971 
 
 §1, '^2> ^3} ^-a> '5_3 ('--1 does not exist) 
 
 The five unknown coefficients of these potential func- 
 tions require five equations. Two are taken from the con- 
 dition -i-wCrig) = - cos p tan a^ on the disk, equation 
 
 (I13a) for a = 1, and (llSh) for 7=1. For the other 
 three the outflov/ condition (equation (114)) with a = 1.2 
 and y ~ l) is used. The -ijxi^p) are those previously de- 
 fined in the case of straight flov/. The integrals 
 
 ( 6 ) TT 3tt 
 
 J^y i obtain now only the limits ~ + C'lg and ~- + cpp«' 
 
 v.'hile ly y remain unchanged on account of the periodicity 
 of the integrand. The values of the incomplete elliptic 
 integrals of the first and second types necessary for the 
 determination of the coefficients ^n^^)* in(^)» etc.,' 
 v.'ere taken from Legendre's tables (reference 8) 
 
 For p = 15° it affords 
 
 C 1 = a.£.q04 a^ 
 
 
 
 Cg = -0.7194 a^ 1)2= 0.C144 (Xq 
 C„ = -O.o343 a„ D^ = -0.0065 a 
 
 
 
 I.e., 
 
 A.= 4.15 a„ f V^ F^,, 
 
 M = -1.83 a^ c V'l - k2 £ V2 y^^^ 
 
 L = 0.03 84 a^ c £- V2 F 
 
 op ell 
 
 For p = 30° it is 
 
 C, = 0.407 a 
 
 
 
 Cs = -0.5S2 a^ Dg = 0.0S77 a^ 
 G^ = -0.265 a„ D, = -0.0124 a ' 
 
 I.e., 
 
 3 
 
 A,= 3.26 ao f V^ ^ell 
 
 •M =.-1.43 a^ c Jl - ^^ -I ?^ F^,, 
 ci exx 
 
 L = 0,074 a. c '£- V^ j 
 
 2 eXl 
 
 
 
NACA Technical Memorandum No. 971 
 
 41 
 
 The new additive moment L is positive according to 
 the atove calculations, v/hich is synonymous with the fact 
 that the leading wing half receives greater lift. The. 
 coordinates of the centers of pressure are: 
 
 ^ = 15°: 
 
 P = 30' 
 
 :jr~ 
 
 -0.440; 
 
 X 
 
 z/l - Ki 
 
 7 
 
 J 
 
 = 0.00925 (cp = 6°02') 
 
 ■0.439 ; ^ = 0.0237 
 c 
 
 (-^p = 13°51 «) 
 
 For comparison we repeat the values obtained at 3=0 
 when the expansion is stopped vrith n = 3. The "bracketed 
 terms contain the change in percent v/ith respect to the 
 quantities computed v^-ith the foirr expansion terms, and from 
 which inferences can he made regarding the convergence. 
 
 C-L = 0.563 tto 
 Cg = -0.776 ttQ 
 C3 = -0.365 tto 
 
 (-0.9 percent ) 
 (-0,2 percent) 
 (+7.0 percent) 
 
 D2 = 
 D3 = 
 
 1 . e 
 
 A = 4.50 an £. V^ F , , 
 2 ell 
 
 •1.98 ao c Vl - K^ £ V3 Pg3_]_ 
 
 Center of pressure: 
 
 = - 0.438 (+0.7 percent) 
 
 The results are correlated in figures 7 and 8. 
 
 For great A an approximate formula for the rolling 
 moment in relation to angle of yaw p and axes ratio 
 
 K' = V 1 - k3 is again expedient: 
 
 dcj 
 
 da. 
 
 15 
 
 128 
 
 TT' 
 
 ' sin 2 P 
 
 , 4 3 
 In 
 
 k' 2 
 
 In tan ( — + — j 
 
 sin P 
 
 1 + 
 
 TT 
 
 243 
 
 >. (115) 
 
 cos p 128 
 
 y 
 
 The formula indicates that the moment on transition 
 to the lifting line (k' = O) disappears. But if the 
 
42 
 
 ifACA Technical Mem6rahdum No, 971 
 
 moment with the half v/ing chord instead of half the span 
 is made nondimens ional , thus voiding the factor k' in 
 (115) the moment coefficient "becomes logarithmically in- 
 finite on limiting transition. In the extreme case the 
 
 lift decreases with 
 flov/ velocity in x 
 
 cos^ P, for the reason that the 
 direction is V coe ^. 
 
 Horner's results on v/in^-s of different plan forms in 
 sideslip are in very close agreement viith the values given 
 here. The assumption that the rolling moment is, aside 
 from the angle of yav; , largely dependent upon the aspect 
 ratio rather than the chord dis trihut ion appears therefore 
 just if ied. 
 
 Translation ty J. Vanier, , 
 National Advisory Committee 
 for Aeronautics. 
 
HACA Technical Memorandum No. 971 43 
 
 aEPEEENCES 
 
 1. Prandtl, L.: Beitrasg zur Theorie der tragenden Plache. 
 
 Z.f .a.M.M. , vol.16, no. 6, Dec. 1936, pp. 360-61. 
 
 2. Kinner, 'li . ', Die kreisformige Tragflache auf potential- 
 
 the oret ischer G-rundlage. Ing.-Arch., vol. 8, 1937, 
 pp. 47-80. 
 
 3. Heine, E.: Handtuch der Kugelf unkt i onen . vol. I, 
 
 Berlin, 1878, p. 347 f f ; vol. II, 1881. 
 
 4. V.'hittaker-V/atson : Modern Analysis. Cambridge, 1920, 
 
 p. 549 ff. 
 
 5. Hotson, E. V< . : Spherical and Ellipsoidal Harmonics. 
 
 Camhridge, 1931, p. 476. 
 
 6. Weinig, ? . ; Beitrag zur Theorie des Tragf liigels endlicher, 
 
 instesondere kleiner Spannvreite. Luf tf ahrtf or schung , 
 vol. 13, no. 12, Dec. 20, 1936, pp. 405-09. 
 
 7. Schlornilch , D.: Gomp. d. hoh. Analysis, vol. II, 
 
 Braunschweig, 18 55, pp. 336-39. 
 
 8. Legendre , A. M.: Tafeln der ellipt. Uormalintegrale . 
 
 Stuttgart, 1931. 
 
 9. Hoerner, S.: Krafte und Momente schragangestromter 
 
 Tragflilgel. Luf tf ahrtf or schung , vol. 15, no. 4, 
 April 20, 1939, pp. 173-83. 
 
TSACA Technical Meuicrandxim Uo. 971 
 
 Figs. 1,2, 3 
 
 "isi.l Base Zllipsoid 
 
 iant 
 
 ^ 2 
 
 ?M -\-iuo 
 
 1 o 
 
 Tip-. 2 
 
 i \/T^k'=i/5 
 
 L J 
 
 Cornputei lift distribution of the 
 lifting line. 
 
 Elliptic lift distribution with 
 equal total lift. 
 
 Fig. 3 
 
NACA Technical Heraorand-om No. 971 
 
 Figs. 4,5 
 
 iCg 
 
 1 ,u 
 2Tf 
 
 \ 
 
 ' 
 
 
 
 
 
 
 
 6.0 
 
 \l 
 
 
 
 
 
 
 
 
 4.0 
 
 ^^ 
 
 \ 
 
 
 
 
 
 
 
 
 \^ 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 v. 
 
 
 
 
 
 2.0 
 
 
 1 
 
 ■V. 
 
 
 
 '-~5-5 
 
 ■~ - 
 
 
 
 
 
 
 T> 
 
 .a 
 
 0.4 0.3 1.2 
 
 Circular iisfc 
 
 l.P 
 
 Lifting; surface 
 Lifting line 
 
 Jif^ure •*. 
 
 u 
 
 n 
 O 
 
 30 
 
 n 
 
 25 
 
 'Hi 
 o 
 
 20 
 
 Pi 
 
 •H 
 
 15 
 
 Q, 
 
 
 a 
 
 
 ~->i 
 
 
 
 
 ^"7 
 
 
 
 
 
 
 
 j "~ 
 
 
 " "1 
 
 _^ 
 
 
 
 
 1 
 
 
 ' 
 
 1 
 
 A 
 
 0.4 0.8 1.2 
 
 Circular iis.sc 
 
 1.- 
 
 Figure 5. 
 
NACA Technical Menorandum No. 971 
 
 Fifcs.6,7,8 
 
 cp=0 
 
 TT 
 
 2 
 
 Figure 6. 
 
 dc 
 
 0.08 
 
 0.06 
 
 'L 
 
 da 
 
 o.o:- 
 
 dc. 
 
 ! 1 
 
 /i 
 
 12 ^ 24 30^ 
 
 Fifirure 7 plottei against angle of yaw 6 
 
 da o ./ . 
 
 for aspect ratio y\ -6.3'^ 
 
 
 
 
 4 
 
 2 
 
 1 
 
 » ' . 
 
 - ! 
 
 ! 
 
 " 
 
 
 
 
 ' '■•■^ 
 
 da 
 
 i 
 
 1 
 
 
 1 
 
 1 
 
 
 
 
 1 1 
 
 
 l^- 
 
 24 
 
 30° 
 
 Figure 8. '^a plotted against angle of yaw ,8 
 dec 
 for aspect ratio/V = 5.37. 
 
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