KLEADOi^ TECHNICAL MEMORANDUMS NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS No. 897 AIRPOIL THEORY AT SUPERSONIC SPEED By H. Schlicliting Deutsche Luf tf ahrtf orschung Jahrbuch 1937 Verlag von R. Oldentourg, Miinchen und Berlin ■RSPTYOFFLORDA "NTSDEPARTf Washington Juno 1939 (10 NATIONAL ADVISOfiY COMMITTEE FOH AERONAUTICS TECHNICAL MEMORANDUM NO. 897 AIRFOIL THEORY AT SUPERSONIC SPEED* By H. Schlichting A theory is developed for the airfoil of finite span at supersonic speed analogous to the Prandtl airfoil theory of 1918-19 for incompressible flow. In addition to the profile and induced drags, account must be taken at super- sonic flow of still another drag, namely, the wave drag, which is independent of the wing aspect ratio. Both wave and induced drags are proportional to the square of the lift and depend on the Mach number, that is, the ratio of the flight to sound speed. In general, in the case of supersonic flow, the drag-lift ratio is considerably less favorable than is the case for incompressible flow. Among others, the following examples are considered; 1. Lifting line with constant lift distribution (horseshoe vortex). 2. Computation of wave and induced drag and the twist of a trapezoidal wing of constant lift density. 3. Computation of the lift distribution and drag of an untwisted rectangular wing. I. INTRODUCTION The basic principles for the following computation of airfoil flow at supersonic speed are presented in the paper of Professor Prandtl (reference 1) , and a detailed expla- nation of the method may therefore be dispensed with here. The potential $- of a lifting line at supersonic speed may be derived in a simple manner from the potential flow. $Q of a stationary source in the presence of a supersonic * "Tragf lugeltheorie bei Uberschallgeschwindigkeit . " Jahr- buch 1937 der deutschen Luf tf ahrtf or schung, pp. I 181-97. N.A.C.A. Teclanical Iviemoranduin No. 897 If $Q denotes the source potential of strength 4tt , the potential $_ of the lifting line element dy with circulation F about the y axis is given by t — $, 4 TT d x' 3 0, ^ a z (1) X' = - oo The potential of a source at the point x=y=z=0 in the presence of a flow with velocity u^ > c in the di- rection of the Dositive x axis is $ Q '-r - 1 ! (7 J (S) + z2) The potential (2) is real within the double cone with half cone angle a, the axis of which cone is parallel to the direction of flovi (sin a = c /uq ) . Outside of this cone the potential, according to the formula, is imaginary. Actually, $, is there to be taken identically eaual to zero. The potential has phjrsical reality only in the "af- ter cone" of the point x = y = z = 0. In the "forward cone" it is similarly to be taken identically equal to zero . The potential 3)^^ is the starting point for con- structing the airf from it the potent then with the aid (1) we shall obtai finite length for lifting line, ther method, the liftin of the airfoil of tained that forms theory for the inc Q oil potential ial of a line of the operat n the potenti various lift e i s finally g surface. I finite span f the count erpa ompressible f We shall source of ion indicat al of a lif di stribut i o obtained by n this mann or super son rt of the P low case (r first derive finite length, ed in equati on ting line of ns. From the the familiar er , a theory ic speed is ob- randtl airfoil eference 2). N.A.C.A, Technical Memorandum No. 897 II, CONSTANT LIFT DISTRIBUTION ■ (LIFTUTO LIUl) We shall now assume a line source of length "b (later - span of wing) which lies in the direction of the y axis and extends from y' = -b/2 to y' = +'b/2 (fig. 1) . Let g(y') "be the initially given local source intensity (later = the lift distribution). Further, let x, y, z he the coordinates of a point in the flow and 0, y' , 0, the coordinates of a source point. Then from equation (2) the potential of the line source is y' = + % = J g(y' ) i y' (3) X - "/ ^0\^ L ^ c {(y - yO' + z^] ie introduce nondimensional coordinates hy dividing all lengths hy the half-length b/2 of the line source and accordingly set 2 X h b » 2 y 2 z 2 T]! Tl> Further, we introduce the abbreviated notation 2 u. _ 1 := K' or tan a = 1 /u. > where a denotes the Mach angle. The potential of the line source then becomes (4) ^'qd.'n.U oT^l'=+l g('n') d TT 'Tl': •^ /e"- ^^'^[(Ti - ^')^ + i^i (5) N.A.C.A. Technical Memorandum Ho. 897 We shall now carry out the integration in equation (5) for the simplest case of the line source with constant source density, that is, gCTi') - const. = 1, This gives ri'= +1 $n = / = (5) |2 _ ^a^(^ _ ^.)2 ^ ^23 Writing t] - Ti« = ■^j ri' = -Ij ^ = ^^ = n + 1 1^1 = +lt ^ = •63 =. r, - 1 ■ / c / s 2 4.2 2 and ( t/"^) - t = a equation (6) becomes \ - --. 1 /• ^ d ^ 1 / . ■'^a - 7i-=. ■ . ■'■: = -— arc sm — - arc sm — 1 (7) /r^ _ ^2 "^ ^ ^ a y ^ ^ By the operation in equation (1) there is then obtained the potential $j, of the lifting line with the constant lift distribution T^ , setting ^T = — / — r ^ ^' (3) ^ 2 TT / at The first step of the above operation, differentiation with respect to t,, may be carried put immediately but the inte- gration requires a somewhat longer computation. There is obtained 3 % 16^ Ux ^^ a^ya^ - •dg^ a2 yi^ - ^^ (9) In integrating with respect to i, ^g ^^'^ ^1 ^^^ con- stant. For the first term, there is obtained K.A.G.A. Technical Memorandum No. 897 .2 ^ / _ 2 ^ (^^^ .2 Va^ _ ^^2 2 J ^* y|*2 + ^2^*(^2j^^2)_^4^3^^: where there has he en set The evaluation of the integral gives ^' t ^2 i r 2 / a (.2 =— arctan — 2 iU. v{*^ + ^^ e*(t^ - ^p^) - K^ t^ d 2 2 6 2 > 2 / • . ; 1^ I w - i [r] - 1) = — arc tan 2 2 e (ti - 1) t^^^ where (11) O) ^ e - ^ [(ri - 1)^ + t^] (12) Since the integral (10) outside of the Mach cone, at the end point T) = 1 of the lifting line with axis parallel to the x axis, r - ^' [(ti - 1)^ + rj = ■ is imaginary, i.e., is to he taken equal to zero, the in- tegration with respect to |' need not he extended from i' = -^. but only from the cone surface along lines paral- lel to the X axis. For the lower limit of integration, we have thus the constant arc tan oo tt/2, which we may suppress. There is thus found from (8), (9), and (11) the required potential of the lifting line with constant cir- culat ion N.A.C.A. Technical Memorandum No. 897 C = V = i e'=-c t ^2 d e' J^ 2 r^ ^' u)- e'Cn - if arc tan 4iT 2 e (Tl - l)t TTu" Kti - 1) = — arc tan 2tt I / cu + similar term for the cone at Tl = -1 ri3) This potential is different from z iAach cones arising at the ends of while in the entire remaining spac a complete circuit about each of t t = 0, the arc tan increases by 2 fila.r.ent therefore has the circula line assumed to extend from y = - constant circulation Fq along the a free vortex line in the two axes tion (13) thus gives the potential at supersonic flow. As in the cas flow, this simple horseshoe vortex for more complicated lifting syste ero only within the two the lifting line (ai > 0) e it is equal to 0. For he cone axes T\ - ±1, TT. The enclosed vortex tion To. The lifting ti/2 to y = +b/2 with the span continues behind as of the Mach cones. Equa- of a "horseshoe vortex" e of the incompressible becomes the starting point ms . In order to obtain an idea as to the appearance of the supersonic flow in the neighborhood of a horseshoe vortex, we differentiate the potential (13) to find the induced ve- locities Cy = 8 On y Cz = 9 $T and obtain c^ Cz = (r; - 1) .t TT b (e^ - K i^) r^ a TT b [ ^2+ (Tl - 1)^ /;r LSL ^^^ - 1) (ou - ^^f) TT t [t^a.+ (-n - 1)'^^] JZ / / (14a. b.c) ' ^See footnote on next page. U.A.C.A. Technical Memorandum No. 397 7 The field of these velocities exhibits a number of singu- larities. On the cone surface all three velocity compo-- nentg become infinite. On the cone axis c-^ - 0, but and c„ become infinite as l/r (where r is the Cy ana (Jjj distance from the axis). In the neighborhood of the cone axis, c„ and c^ thus behave exactly as in the neigh- borhood of a vortex filament in the incompressible flow. The field of the induced velocities gives a motion which encircles the vortex filament traveling downstream from the end of the lifting line T| = 1 , | = ^ = 0, as may be seen immediately from (14). In the plane T) - 1 = through the end of the lift- ing line Cg = and t > : Cy < I < Q '. Cy > In the plane t = 0, which contains the lifting line, c^ = and 'n-l>0 : C2>0 ri-l<0: C2<0 The flow picture in the cone, however, in its detail is essentially different from that in the neighborhood of a vortex filament in the incompressible case. Figure 2 shows the flow picture of the y and z velocities in a plane perpendicular to the cone axis downstream of the lifting line. The figure was obtained by computing the isocline field Cz/cy = const. On the cone surface, as has been said, Cj. and Cy are infinite, although for the slope of the streamlines c^/cy. there is here obtained the sim- ple value t Cy Tl - 1 ^A check for the correctness of this solution is obtained by substituting in the linearized continuity equation „ 3 c B Cv 9 ^z - K^ £ 4 Z + 5. z= 3 X by 8 z which must be identically satisfied. N.A.CA. Technical Memorandum No. 897 The direction of the streamlines is therefore radial to the center. The flow consists partly of the closed stream- lines which circulate atout the vortex filament and partly of the streamlines that enter on one side of the cone and leave it again on the other side. In addition to the two Mach cones that arise from each of its ends, the lifting line generates two plane waves, which enclose a "wedge space" and which appear in the streamline picture as the common tangents of the two cones . For the downwash distribution in the plane t = through the cone center, there is obtained from (14c) the simple formula 2 TT X J \ - ^^ where ^0 tan a Cg^ = (15) y 2 ^ = (15a) X tan a This downwash distribution is shown in figure 3. In order to study the processes on an airfoil of fi- nite length at supersonic speed, particularly the induced drag, the replacement of the wing by a lifting line with constant circulation as in the case of the incompressible flow, appears inadmissible since on account of the infi- nite velocity at the end of the lifting line an infinite induced drag would be obtained. This difficulty in the case of the incompressible flow is avoided, as is known, by allowing the circulation to drop to zero in a suitable manner toward the wing tips. The induced drag is then computed by the formula. y = +b / e Wi = P f =Zo^^^ r (y) d y (16) y = -b/s (where c^ (y) is the induced downwash velocity at the place of the lifting line, and P the density). N.A.C.A. Technical Memorandum No. 897 In tlie case of the supersonic flow, the relations are complicated ty the fact that in spite of the assumption of a lift distribution decreasing to zero toward the wing tips, there are obtained singularities at the lifting line posi- tion of such a character as to make the computation of the induced drag by formula (16), which maintains its validity for supersonic flow, impossihle. As closer investigation shows, this is due to the fact that the lifting line is the geometric locus of the vertices of all the' Mach cones that pass down "behind. This difficulty may "be overcome "by pass- ing from the lifting line to the lifting surface. III. WAVE RESISTANCE (ORAG) Before proceeding to the corresponding computations, we shall discuss "briefly the supersonic flow a"bout an in- finitely long airfoil (two-dimensional problem), a pro"blem that had "been considered "by J. Ackeret in 1925 (reference 3). The simplest and at the same time the ideal supersonic profile is that of the infinitely thin flat plate of chord t set to a small a^ngle of attack p (fig. 4). For such a plate the lift per unit span is A = 2 tan a fio t p u^^ (17) or P 2 = 4 tan a pQ (18) 2 . "u.„ t On account of A = p u T the relation "between the an- gle of attack of the wing and the circulation is 1 r B^ u tan a = 2. (18a) ~ o 2 t From the incompressi"ble flow, the supersonic flow about the airfoil differs in that, for the latter case, even if the fluid friction is neglected, there is always associated a drag that originates from the plane waves which start out from the lifting surface and are inclined to the latter by the Mach angle and which, therefor e may 10 IT.A^C.A. Technical Memorandum TTo . 897 be denoted as the wave drag. Tor the flat plate, the wave drag per unit span is Wwave = - Po -^ = - 2 tan a Po^ t p u^^ (I9) or "wave a = c^ =4 tan a B^ (l9a) D p wave ° 2 Ihe resultant of the lift and the drag is here at right angles to the plate. This comes from the fact that at su- personic flows there is no suction force at the leading edge of the plate. From equations (18) and (l?a) , there is obtained for the polar of the wave drag °w "wave 4 tan a which is thus a parabola as in the case of the incompres- sible flow. Plane waves start out from the leading and trailing edges of the inclined flat plate (fig. 4) and in the space between them the induced downwash v eloci ty is 1 r Cz = -P. ^-^n = - - (20) '^a^e ^° ° 2 t tan a The wave drag, on the other hand, can also be computed from this downwash velocity induced by the plane waves, accord- ing to the formula 'to 'f = p r C7 (21) "wave ^ ^Tisive ^ ' as may be seen by comparison with (19) and (20). In the nest section it will be shown that, for a lifting surface, the velocity induced by the tip vortices like c„ is proportional to '^r.i'^ ^^^ ^' ^^ then follows from equa- tions (21) and (16) that the. wave drag behaves in exactly the same way as the induced drag from the tip vortices. For practical applications it is therefore of no inter- est to consider the induced drag alone, but it is the sum of the induced and wave drags that must be considered. N.A.C.A. Technical Memorandum No. 897 11 For an airfoil of finite span and constant chord with circulation that is constant along the chord and variable along the span F (y) = t 7 (y) the total lift and wave drag are given hy y = +'b/2 r|= +1 A^PUq/ rdy=:ipUoht/ 7(ri)dri (22) y - -'b fs r\- -i +b/2 T] = +1 ^wave = P /' ^^ow ^ d y = £ b t j^ c,^ J d r, (23) y = -b/a n = -1 where 1 7 (24) "w 2 tan a is the induced wave velocity. Accordingly T) = +1 ^wave = - - -^-^ / 'Y^ ^ :u ■ - + — ^- ra 0] -VJl rH. + 1 ra ^ r-t I 1 1 n r 5^ 1 0] 03 ^—^ — - .vv_f. ^, 1 I 1 ~»_o I + ro I *" — ^ r-1 I •^ t The upper integration limit t' = 1 1- is different according to whether the point lies within the Mach cone II arising from the end point of the trailing edge (fig. 5) or hetween the latter and cone I arising from the end point of the leading edge. The corresponding limits will be 1^ = 2t/li - £ (within cone II) (ri, - 1)" + {^ ("between cones I and II) V, (31 (31) as may he easily seen after some consideration, Introducing the new variables of integration (e - e')^ - K t^ = T and writing for briefness ai-= (r, --1)- + t^ we have r, _ 1 /"' (t - k2 a^^ ) d T- 2 TT 7, T = T T = T, -n -1 T Vt - K^ (ri - 1 )''^ d T /t - k2 (r, - iT' T = T. T = T. . ^ (r, - 1) d T I t/T T = T k2 (r, - 1)' o 1 '^ IT.A.C.A. Technical Memoranduni .No . 897 15 The evaluation of the integral gives Jii -r)'- «"{(ti -1)' + C] 2 TT r\ - 1 S *2 (Tl - 1) Vt - 1^1 arc tan L /T- e' = K(r, - 1) e^= Taking account of the different upper integra,tion limits according to whether the point considered is within cone II or between cones I and II, equation (31), and setting for "briefness a,c = (e - £)^ - i^^[(n - 1)% f^ there is obtained as the final expression for c" • For cone II Z -n -^ Tl - 1 (Tl-- 1)^ + ■\ '^/uj - ^/u)7 r •2 k. (32) -.{ arc tan v'a) K(Ti - 1) arc tan 'u)e K(T1 - 1) I (33a) Between cones I and II 2 TT -1 = 'Yo n. _ 1 where - 1 < (Tl - 1)^ + i^ K (Tl - 1) -/(JU - K arc tan y O) K(n - 1) < + 1 : - 5 < arc tan < ^ (33b) From these formulas it follows that c on the surface of z cone I is equal to zero and on account of tu = is con- N.A.C.A. Technical Memorandum No. 397 tinuous in passing through cone II. On the common axis of cones I and II, there still occurs the same singularity as 3*2, there hecoming in- in the case of the lifting line, finite as l/r. In order to ohtain the total downwash velocity, there is still to be added to equations (33a) and (SSh) the por- tion contributed by the plane wave. This contribution is different from zero only between the plane waves starting from the leading and trailing edges (fig. 4), According to equation (20) "wave __ K ry ' ywav e = c ^wave (34) The expressions for the two remaining components of the induced velocity "Cy and c"^- are found by similar in- t egrat ions Cone I I : c. 2 n -^ = - 7, We shall only indicate the results: ( a/uj - (ri - D" + V d TT — = arc tan •Yo t^Ti - arc tan IJU, ii'f] - 1) Between cones I and II: "c„ t >(35) 2 TT — - - ^, (Tl - 1)' + ^' -/uT 2 n 7, = - arc tan t{T,- 1) J In the above equations, the arc tan is to be taken -tt/2 and +Tr/2. As may be seen from equations (33) and (35) by comparison with equation (14) in passing from the lifting line to the lifting surface, the difficulty of the infi- nite velocity at the cone surface has been sot aside. The' singularity of c"y and c^ on the cone axis (infinite as l/r) still remains, however, and prevents the computation of the induced drag for this lift distribution-. N.A.C.A. Technical Memorandum No . 897 17 Tor the downwash di stritut ion in the plane z =0, at 1) the location of the wing x < t, and at 2) behind the wing x > t, taking account of the iplane wave, there is obtained the following: 1) For X < t : 2c„ - 1< •^< 0; K7, - - 1 + 1 jVl - ^^ — s arc. tan TT ^ A :__ J \ - -a^ \ 2c, 0< i>< + !:• 1 j" V 1 - k7. = #1 arc tan ■J^. - ■} S (35a) J where for the arc tan the same values are to be taken as in (33) and •d is given by equation (15a). -, 2) For X > t: The plane waves do not contribute anything but the formulas obtained differ according as the region considered is within cone II or between cones I and II (fig. 5). - 1 I) < < 1 - i X 2C: K7 = -t(/^~^-/(-7)^-*=) TT ^ - ■=■ ( arc tan TT t ^ 2 X Jl'- -!<■§< 1 - -^ < X < + 1 > 2c, J ^y. yr - - arc tan - /V-FF^ ) — - arc tan ■) (36b) 18" IT.A.C.A. Technical Memorandum. No . 897 The downwash di strihut ion for x < t and for x = 2 t computed by the above equations is shown in figure 6. Further, we have in the same manner as for the lift- ing line determined for the lifting surface the stream- line field of the y and z velocities in a plane at right angles to the cone axis. At the location of the wing (x < t, fig. 7) there is obtained outside of the cones springing from the wing tips a constant downwash, due to the plane waves, along the span. The streamline picture within the Mach cone in the outer half is similar to that of the lifting line (fig. 2); the inner half how- ever is entirely changed by the additional downwash ve- locity from the plane wave. The streamline picture behind the wing (x = 2t , fig. 8) has, outside the Mach cones springing from the wing tips, a constant downwash velocity due to the plane waves in two strips symmetrical to the plane -z = 0. These two strips are limited by the plane waves starting out from the forward and the trailing edges of the lifting surface, flthin the Mach cone the streamline picture in the outer ring is the same as for x < t and is cha,nged only in the inner region. We shall yet consider briefly the question, what the form of the wing sxirface must be that corresponds to the assumed lift distribution. The wing plan form we have as- suraed as rectangular. Angle of attack and twist are ob- tained from the consideration that at the wing, i.e., in the plane z = in each section parallel to the flow di- rection, the direction of the streamlines must be parallel to the wing tangent. Let z = z(x, y) be the equation of the wing surface and z(0, y) = 0, i.e., straight leadin- 3 I Te . Then we have^ (i z _ c z ( X , y ) d X u where c„ includes the induced velocities from both the Zo plane waves and the edge cones. There is thus obtained for the wing surface ■ X ' = X z(x, y) = — /' Cz„(x', y) d x« (37) ^° J ■0 X' = K.A.C. A. Technical Memorandum Ho. 897 19 so that a further quadrature is required to compute the form of surface wing. For the case considered of constant lift distribution there is obtained for the region outside of the two Mach cones at the wing tips, from equations (37) and (20): z (x, y) = -pQ X that is, a flat surface with angle of attack p . Within the Mach cone the surface "bends downward more and more strongly as the edge is approached. The edge itself (y = i;"b) is hent infinitely downward, i.e., actually the rec- tangular surface with constant spanwise and chordwise lift distr ihut ion is not possible. For this reason we may dis- pense with the further computation of the wing-surface shape . V. TRAPEZOIDAL WING WITH CONSTANT LIFT DISTRIBUTION We consider now a trapezoidal wing with constant sur- face density of the lift 7 (fig. 9). If the wing is cut away behind (taper angle t, fig. 9) in such a manner that the Mach cone at the tip of the leading edge does not overlap the wing (t > a) , the induced drag is obviously equal to zero and only the wave drag exists (reference 4) T > a: Wj_ = The trapezoidal wing with constant surface density of the lift 7q is plane outside the Mach cone and has the angle of attack Pq where ■ '^0 = 2 0Q u^ tan a The trapezoidal flat surface with constant lift distribu- tion whose cut-away angle t is greater than the Mach cone angle may be looked upon as the "ideal supersonic wing with finite span" since for it the ratio of drag to lift is no greater than for the wing of infinite span. The computation of the induced drag for t < a is possible in a simple manner from the above results. By a lifting element we shall mean a strip of the lifting sur- face of chord d x and therefore with circulation 7 d x. 30 U.A.C.A. Technical Memorandum No. 897 Such a lifting element at x = generates at a lifting element of chord d -x' at x = x' y=yi(x) d^ Wj^^, = P 7o d X' / d c^^°^'^ d y (38) '' ox ' ) where d c^^ denotes the downwash velocity induced ty the lifting element x = at the position x = x'. The integration limits are the surface of the Mach cone aris- ing from the tip of the wing leading edge and the side edge of the plate. For the downwash velocity in the plane ^ = 0, we have according to equation c zo t a4c) with the aid of which equation (38) becomes I J A lOX' p „ ° / ' Tl - 1 or with t5 =K , according to equation (15a) and e ■ . tan T , > 6 = (40) tan a as .the reduced angle of taper ^=1 ^ ^iox' = '^0^ d X d X' / dr^ 2 TT J ^ ^=6 _£- 7/ d X d X' g (G) (41) 2 TT N.A.C.A. Technical Memorandum No. 897 21 The evaluation of the definite integral gives (6) = - 7] 1 - 62 log 1 - (42) According to equation (41) the induced drag from the lift- ing element x = ent of the distance ments lying "between duce the same drag, s' amounts to at the position X = X ' is independ- tetween the two elements. All ele- X = and X = X ' accordingly pro- so that the total drag induced at x d Wi: The drag for the entire wing is ohtained from the above "by integrating over x' "between the limits x ' = and x' = t and multiplying "by two ("both ends) W^ g (6) / ,/ x'=0 d x' 7, IT (6) = - g (6) (43) The minus sign is explained "by the fact that with our choice of coordinate system the drag component of a force is in the direction of the negative z axis. Formula (43) for the induced drag of a trapezoidal wing with constant surface density of the lift is of the same structural form that is found for the incompressi"ble flow. For triangular lift di str i"bution (lifting line) in the case of incompres- si"ble flow, we have, for example. Wa i2£-£ p r ' •^ TT where T is the circulation at the wing center. For equal total circulation F^, W^ according to equation (43) is independent of 6, i.e., the ratio of the tangent 22 N.A.C.A. Technical Memorandum Ko . 397 of the angle t to the tangent of the Mach angle (equa- tion (40)). In passing to the rectangular wing, 6 > 0, the induced drag according to equati ons (42) and (43) he- comes logarithmically infinite, in agreement with our re- sults of the previous section. Actually, we are not interested so much in the value of the induced dra^ alone as in the sum of the induced and wave drags. For the wave drag, according to equation (21) we have Vr„„„„ = p F 7^ C2 wav e wave where ? = t K-- t tan T \ ;■ txie area of the win.c- 7t > (44) 'wave hence W, wave 2 tan a 2 tan a y For the lift we have, on account of '^q ~ ^ ^o ^o "^^ ^' A=P7nUnF=2P u^^ F p^ tan a '0 "-0 Ho (45) or A 2 ° Ca - 4.^0 tan a For the wave drag we ohtain from (44) ^wave.= 2 P F Uq^ ,3o^ tan a (46) W. wave -Uo"F 'w. = 4 wave '0^ tan a = pQ c. (47) and for the induced drag from equation (45) K.A.C.A, Teclinical Memorandum No. 897 33 r^ = _£_ t^ 4 p/- u/ tan^ a g (6) 2 TT W. i _ 1 4 t ^ - tan- a g (6) °W-i f u,- I ^ t Ti ' '^" ' / t tan T \ A g (6) Cwi = 4 ^^-^ tan a - -— ~ . (48) t tan a where X = -= is the "reduced asoect ratio" of the wing. For t?ae total drag there is thus obtained from (47) and (48) (^w)wave + ind^= ^ Po^ *^^ ^ l^ + ": — " , TT 1 - b X v^w^wave + ind = c 2 . X g (e) 'a J 1 + I (49) >w TT 1 _ ft -X -^ 4 tan a TT 1 - d A it follows therefore from the ahovfe that for supersonic speed the wave plus induced drag, like the induced drag in the incompressihle flow case is proportional to the sq,uare of the lift. Equation (49) is analogous to the well-known formula c„. = —2— — «■ of the elliptic lift " 1 v^ TT D distrihution for the incompressihle flow. The essential difference lies in the fact that for the supersonic flow the drag parabola for small aspect ratios t /h is to a first approximation independent of the aspect ratio. The manner in which the drag increases with increasing reduced , . . . t tan a , , • fi • -u aspect ratio X - and decreasing o is shown m h . ^ / ^a' figure 10 where c^/ is plotted against X for / 4 tan a ~ • various values of 6. Our formulas are valid only for 24 H.A.C.A. Technical Memorandum No. 897 ?v < — , i.e., for the case in which the Mach cones do not overlap on the wing. In order to be ahle to predict what the wing' shape must he sc that our assumed lift distribution may be pos- sible, we must first compute the field of the induced velocities. For this purpose equation (39) is to be in- tegrated over the trapezoidal area. The value dc^°^' according to (39) gives the downwash velocity in- Z duced at the position | by a lifting element 7q dx starting at f = and ending at T] = 1 . A lifting ele- ment which starts at ^ = t' and ends at 1^ = 1^' thus producos at the position ^, T), ^ = the downwash veloc- ity ' ' d c^l'O = _ lo. d ^. y (p. -ry- - y^'- {r - r,')3 2 TT (e - f ') (r)< - Tl) For the velocity induced by the entire surface there is thus obtained 2 ^ J (^ . |.)(n' - Ti) d^' (50) In order to evaluate this integral we introduce the new integration variables ■Q = K TJ — ZJH = K tan^'' (51) e - ?' (See fig, 9.) Since the end points of the lifting elements lie on the wing contour there exists the relation 1 _ n' = t ' tan T (52) The upper integration limit E' = L, in equation (50) is obtained from the condition. (See fig. 9.) K.A.C.A. Technical Memorandum No. 897 25 tan cp' = tan a:^' = ^:^'=l The lifting elements whose f' is greater than the I thus determined give no contribution at the points t, r], t = 0. lor the lower integration limit e' = : Ti' = 1 : ^' = K-}.^ZJ1 From equations (51) and (52) there is obtained d ^ _ e where 6 is the abbreviation introduced in equation (40). There is then obtained from equation (50) = 1 ;o (^^ = - / ^ 2 TT ^ . d ^' = -^ K J(^.6) (53) •(^' - e) 2tt In evaluating the above integral the following three cases are to be distinguished: 1. < 6 < 2 . < ^ < e ; ^ < < 6 In case 1 the point P (■§ ) lies within, in cases 2 and 3 without the trapezoidal wing. In case 1 the integrand is regular over the entire range of integration; in case 2 it possesses a singularity at ^' = 6; and in case 3, two singularities at ^' = and .^ ' = £ . In cases 2 and 3 the principal values are to be taken, namely. < ^ < P(^. 6) : lim i / ; d ^' + / J d ^ ) (54a) 26 N.A.C.A. Technical Memorandum Ko, 897 and ^ < < 6: ■S =-€ p(^s^6) = lim| / J\- ^'^ i'U' - 6) ci ^' + / ^i =1 ./ d ^ = tS ■=+€ ^ '=t+e (54d) Tiie integral ■• a/i - ^'2' F(^),e) = / -;^— -_ d ^' U^' - 6) i'=^ (55) may be obtained by elementary methods. ^e set yi-^'2 = t^i'-l where t is the new integration variable so that (55) be- comes ' (1 - t^)2 dt . F(^,6) = ; _ J t(i + t-) s 2 t - e (1 + t-) ^ t =rl S '' wnere *I = ^ - 1 + V 1 - d' (56) By breaking up into partial fractions there is obtained N.A.C.A. Technical Hemorandum Ko, 897 27 2\ 3 (1 - t^) t (1 + t^) |2 t - G(l + t^) j 1 + t2 e t 1 - 6- + t - t „ t - t ■; wnere 1 ± -7 1 / e^ i.n Q performing the integration, there is obtained 2" (^,6) log t Vl - £2 - 2 arc tan t - + t = V ^ log (t - tj - log ( t - t. ) }. Tor < 6 < d there is therefore obtained directly TT FC^, 6) = - - 2 arc tan ^ - log \|/ 6 + yr: log / . 'i +^1 - e^ V 6 - 1 •+yi - (57) 1 - \J; 6 + J^ 6^ / while the formation of the principal value according to equation (54a,b) gives 28 N.A.C.A. Technical Memorandum No. 897 < ^ < 6: loa; ^ lii),b) = H - 2 arc tan \|/ - i^ 2 ^ ^/l - fe^ fl +71-63 ^^ e - 1 +Jl'- 62 log 4^6-1-^1- 62/ ^^ < < 6: F(i5,6) = - H - 2 arc tan ^ 2 log (-4') VT - 6^ /i +yi - e's v^ e - 1 +./l~- + log V(57) vt/ 6 - 1 -/i - 62 / y wner e - H < arc tan ^ < - 2 2 There is thus found the downwash distribution in the entire Mach cone springing; from y = b/S, x = . ?or ^(■d.t') we have ■d =±1:F(±1; 6) = ^ =: e , ^ = 0:F(6,e) = ?(0, 6) = " as log ^ d ^^ (60) TT ^2 // Qia The integrand becomes infinite for {) ' = 6 (eqaation (58)). The integral exists, however, and may be evaluated by spe- cial computation. There is obtained J ^^^ 2 6 6 6^ ^ + XT~i- (The evaluation of the integral was performed by Dr. I. Riegels , ) For 6 = 1/3, we thus have 6 = 1/3: y Ilill-ild.' =- 4.92 ^' = ^' = 1 The ordinate of the rear edge point x=t, i.-y=t tan t for 6 = 1/3 is thus z = - 1.522 6 t. (Flat surface z = -pQ t, twist z = -0.522 pQ t.) For the special case 6 = I/3 (^tan a = v 3 ;_ tan t = — =— ) the profile sections have been computed and are given in figure 13. If the trapezoidal wing were flat there would be a drop of tne lift toward the edge dcTvn to zero. In order that full lift be main.tained up to the edge, the wing must be bent downward. The twist of the N.A.C.A. Technical Memorandum No. 397 31 wing directly at the edge is very strong as may he seen from the "elevation contour lines" (fig. 14). VI. COMPUTATION OF THE LIFT DISTRIBUTION FOR THE UNTWISTED RECTANG-ULAR WING The examples thus far considered are all in connec- tion with the so-called first principal prohleni of the airfoil theory where the lift distribution is given and it is required to find the drag and the wing shape. Of greater practical importance is the second principal prob- lem where the wing shape being given it is required to find the lift distribution and the drag. As in the case of the incompressible flow, so also in the case of the compressible flow the first problem, which leads only to quadratures, is considerably more simple than the second, which requires the solution of an integral equation. In what follows there will now be given an example of the second principal problem, namely, the computation of the lift distribution for a plane rectangular wing (span = b, chord = t), that is to say, the same problem that was first considered by A. Betz (reference 5) for the case of incompressible flow. In the treatment of this problem we can utilize to a large extent the results we had obtained in the previous section for the trapezoidal wing with constant surface density of the lift. We con- sider a rectangular flat plate which extends from x = to X = t and from y = -b/2 to y = +b/2 and is set at the small angle of attack ^ to the undisturbed veloc- ity Uq (fig. 5). Within the region bounded by the plane waves starting out from the leading and trailing edges and the two Mach cones there is the constant downwash velocity due to the plane waves °20wave = -f^o ^0 = - I ~-^- (SD wave 2 tan a Outside the region of the flat plate overlapped by the Mach cones at the tips there thus exists the constant lift distribution 7o . At the tips y = ^b/S the lift must vanish, that is, 7=0 at y = ±b/2. There is required the lift distribution 7=7 (x,y) within the region overlapped by the Mach cones. The problem is considerably 32 N.A.C.A. Technical Memorandum .No . 397 simplified by the circumstance that, as will immediately "become apparent, .7 does not depend on the t':"o independ- ent variables x, y, but only on one of the variables b "2 - y ^= _f (51) X tan a ' ' (fig. 11). i'or the required lift distribution ^ii)) = y^ f(^) (52) of the rectangular wing there then exist the boundary con- ditions ^ = : f ( tO = (53) ^ = 1 : f ( O - 1 I In order to be able to set up the integral equation for 7(^) we must first compute the field of the downwash ve- locities w(-i)) induced oj a rectangular wing with the cir- culation distribution 7(i5) in the plane z = 0. The in- tegral equation for 7('d) is then obtained in the known manner from the consideration that for each position of the wing the sum of the effective angle of attack 1 7(^) ■ ■ ■ p(^) = — (64) . ' , 2 Ug tan a and the induced angle of attack - 1 L^ must be equal ^0 to the geometrical angle of attack ^^ pU) -liil-^^, ' (65) ■^0 The velocity field w('d) induced by the edge vortices is obtained by considering the rectangular wing with the var- iable lift distribution y {■&) = 7q f (■&) as built up by the superposition of trapezoidal wings with various taper an- gles each of which winsrs possesses a constant lift distri- tan T bution. Again, let d = be the "reduced' taper angle" tan a (equation 40), then the lift distribution 7 = 7 f (•&) may N.A.C.A. Technical Memorandum No. 897 33 be obtained by the superposition of trapezoids with angles 6 and lift densities 7of'(6) d 6. Each of these trape- zoids produces, according to equation (53) the velocity- field d w (^) = - -^^-^ !^ F (^,6) d 6 2 rr tan a and integration over 6 from 6=0 to 6=1 then gives the induced velocity field over the rectangular wing 6=1 w (^) = 1° /' f ' (6) F (^,6) d e (66) 2 rr tan a «/ 6=0 By substituting the above expression for w('d) in equation (65), there is finally obtained, taking account of (51) and (54) the required integral equation for f (■&) ; 6=1 f (^) + i / f ' (6) F (,^6) d 6 = 1 (67) ^ J 6=0 to which are added the boundary conditions (33). This integral equation for the lift distribution has the same structural formas that for the incompressible flow. It differs from the latter, however, by the different core F(^,6), which is given by equation (57), and for the super- sonic flow is of a much more complicated form that for the incompressible flow. Equation (67) also exhibits the nota- ble property that neither the aspect ratio of the wing nor the Mach number appears explicitly, whereas in the incom- pressible case the characteristic value of the integral equation depends on the aspect ratio. The dependence of the lift distribution on the Mach number appears in the introduction instead of the geometric angle cp (fig. 9) tan cp the reduced angle ■d = as the variable. It is neces- tan a sary to solve the integral equation (57) only once to ob- tain the lift distribution of the rectangular wing for all aspect ratios and all Mach numbers. 34 N.A.C.A. Technical Memorandum ITo.' 897 The solution of the integral equation (57) appears at first sight quite difficult, particularly on account of the complicated structure of the core F(-g,G). (See equations 56 and 57.) By a simple transformation of equation (57) it is possible, however, to simplify the problem considerably.* The equation is a nonhomogeneous int egr oiif f er ent ial equation for f ( -d) . Instead of it we shall consider the equivalent equation for f ' ( ^) . Taking account of the singularity of the core, equation (57) may be written 6=, 6 = f (^) + -| /' f (e) r(t',.6) d G,+ /' i'(*) F(i.6) d 6 j= 1 6=0 6 = ^ Differentiation with respect to •8 gives f^{^)+}i\f^{i)i(s,i)+ I f ' ( 6) i_I d e TT d ^ 6=] f (,3) p(*.*) + / f (6) i-I d e} = d i) e=^ and because. d ? Jl - ^^ d ^ ^s( t (81) Through equations (73), (80), and (81), the polar and mo- ment curves not considering the frictional drag, are com- pletely determined. In figure 16, the polars are given til for the aspect ratios — = 0, — , and — and for the Mach z ^ ^ ^ numbers -2. = 1.2, 1.5, 2.0, and 3.0. The drag differences c "between wings with various aspect ratios are considerably- smaller in the case of the supersonic flow than for the incompressible flow since in the first case the greatest part of the drag is contributed by the wave resistance, which is independent of the aspect ratio. The plane rectangular wing at supersonic flow is one with constant center of pressure position, if the fric- tional drag is disregarded. The position of the center of pressure depends only to a slight extent on the re- t tan a duced aspect ratio X = . ?or the infinitely long b wing, the center of pressure lies at the midchord position and with decreasing aspect ratio it moves forirard somewhat (table III). Table III t tan a b ■ 1/5 1/2 Cmg °a 1_ 2 0.489 0.469 IT.A.C.A. Technical Memoranduin No. 897 41 Formula (80) for the rectangular flat plate is the analogy to the familiar . a^. = Cy^ ■$ Jx[ "b^ of the incom- pressible flow. Like the latter it enahles the recompu- tation of the drag from one reduced aspect ratio t tan a, ^ = -^ i ta tan aa to another Xg = . Prom equa- tions (73) and (80) there is ohtained for the new angle of attack and the drag 1 1 = ^^-f i CWg - °W^+4 a w^ - ^w, "^~r \ tan ag (1-0.316 Xg) tan a^ (1-0 . 316 Xj 1 1 } tan a^(l-0.316xJ tan a, (1-0 . 316 X^ 2 2 1 ••"' (81a) VII. TRAPEZOIDAL LIFT DISTRIBUTION a) Lifting Line As a further example we now compute the induced drag and the velocity field for trapezoidal lift distribution . for "both the lifting line and the lifting surface (fig. 17). Let the lift distribution therefore he given by r('n') Tl' ^1 r(Ti-) = r^ for for Ti^ < Tj' < 1 - 1 < T < - 111 - Til < 'H < + Til •\ (82) ,.wh.ere T,^ = b ' /t , according to figure 17. The field of the induced velocities and induced drag for variable lift distribution may be obtained in the familiar manner from the lift distribution by superposition. On account of integration difficulties, however, this computation can not directly be made on the potential but must be carried out separately for the three velocity components. From equation (14c) we have for the induced downwash velocity of a lifting line ending at r\ = Ti' with circulation r 0" 42 N.A.C.A. Technical Memorandum llo. .897 Jri') = -^ r^ e(^i '-'ri'){e' - ^"[(^ - ^.•)^ + 3 in} TT-b ( |3_K3.^2)[(r,_Tl>)a+^2jy^3_^2 [(tT -Tl) ^ + ^ ^ ] (83) From the ahove there is ohtained by superposition the downwash velocity c^ for variable circulation r(rj'): c. — - ri '=+1 Ti =-1 Cz(tT) d. tT dri' (84) For the traToezoidal lift distribution according to equation (82) we have therefore if, on account of symmetry, we restrict ourselves to the half-wing y > tT=i = z = c„(r,') d r^,« 1 - 'n, J or, accordini^ to equation (33) TT b C; T7 1 '^'=/ ?(n - ri'){a^- Kn(Ti'-.i)^ + 2t']} d ri' l-Ti, J (?^-Kn')[(Ti'-^)%t"]y^-Kn(ri'-n) = + t = ] or with k2 (r)i _ r,)2 + l^ = -^ b c, 1/2 d T 1/2 d T Performing the integration there is obtained for points ^,11. t within the Iviach cone at the wing tip T, = 1 b c, TT — {- Vd) ■'■ . f + V(.U 1 _ n^ ^ |2 _ K2 ^' , i log i±^^ } (85) f -V (U N.A.C.A. Technical Memorandum Ho. 897 43 and a corresponding expression with reversed sign for the Mach cone at "H = 'Hi* The value of o) is here given by- equation (12). In the cone T\ = 1 , c^ > so that there is upwash velocity. In the cone r\ = r\^ there is a downwash velocity of the same absolute magnitude (c^ < O) and outside of the two cones = 0, a result which is also to he expected from reasons of symmetry since, on ac- d r count of = const., all separating vortices are of d ri' the same strength. With ^(T) - 1) i i,nd = i there is obtained for the downwash distribution in cone I and III respectively in the plane z = 0. TT b cz - ± ■^1 {-Jl^ •s 1 > + - log 2 1 + / 1 - ^' 1- / 1 - } (86) On the cone surface according to equations (85) and (85) Cg = and is therefore continuous in passing through the cone. On the cone axis c^. now becomes log- arithmically infinite, whereas with the rectangular lift distribution (horseshoe vortex) c„ becomes infinite on the axis as r '"■'•. The logarithmic singularity of e^ is no longer a disturbing factor for the computation of the induced drag. For the sake of completeness there will also be given the remaining two components of the induced velocity. There is found for the cone at T\ = 1: b c. TT TT ■■- ' '■• r t ^AD 1 - T^^ e"" - t^ t' - 1 t V (.U — arc tan 1 - -n^ |(ri - 1) (87) 44' N.A.C.A. Technical Memorandum. Kio . 897 and corresponding expressions with reversed signs for the cone at 11 = T] . For the arc tan there is to he taken the principal value < arc tan < tt. For the outer cone (t] - l) the arc tan is zero in the upper half plane on the outer quadrants of the cone surface and equal to +tt on the inner quadrants. In the wedge-shaped space "be- tween the two cones Cy is constant, being equal to r i ^ : c =:;, 2 ; . cx = (88) h - h' In passing through the plane ^ = , therefore there is a discontinuous increment in c„ by 2 . The region of the t, plane limited oy the cone b - b« is t hu s a V r - 2 tex surface with constant circulation density the total circula- tion of which is equal to the circulation T^ of the bound vortex in the region of the constant lift. A streamline picture of the y and z velocity components for a plane x = constant that intersects both cones is drawn in figure 18. Like the streamline picture for the constant lift distribution (fig. 2) it was ob- tained by computing the field of isoclines. On the outer halves of the cone surfaces c„ and c„ are equal to y ^ -^ zero but the directions of the streamlines c„/c,r have a value different from zero. In this case, too, not all streamlines are closed, part of the streamlines entering from the undisturbed region into the one cone and coming out from the other again into the undisturbed region. b) Lifting Surface In order to compute the induced drag for the trape- zoid-shaped lift distribution, we must, as in section IV, make the transition from the lifting line to the lifting surface. A rectangular lifting surface will therefore now be assumed of span b and extending from x = to X = t. The chordwise circulation distribution is assumed to be constant of density T/t, while along the span the distribution is that given by equation (82). For the com- putation we may here restrict ourselves to the region be- IT, A. C. A. Technical. Memorandum JFo . 897 45 tween the cones springing from the leading and trailing edges of the lifting surface, since only this region en- ters into the question of the computation of the induced drag. We likewise need carry out the computation only for the cone at 1^=1; for the downwash in cone 'H = 11 -, there is obtained the corresponding expression with re- versed sign. For the induced z component c^ of the lifting sur- face, there is found, according to equation (85), with 'Yo = ^o/t: t=i. 2 TT = / — : i I ' 7 1 - 'Hi 7 (e - ? ')^- K I' ^'=^' 1/2 /* ^ t - r + /( e - e")"- ^" [...J ,, + _ / log ^ 7p= ^ ====r d e 1 - Ti, J e - e' - yr e - e')^- ^^ [...J e'=o Wiier e ( t^' = t - ^ Jir\ - 1)^ + V according to equation (31). With the new integration var- iables I - l^ - I*, the above equation becomes 3 „ i. . _JL_ )' iHi^-^iiii^iin , ,. \J2 -P * ^log (I* + yi*2 - K^i — ]) d e* 1 - ^1 4*=s 1 1/2 .'^ ^ 1 J / log ii* . Ji*^ - K^C....]) d !■• 1 - \J^*^^ wh ere 45 K.A.C.A. Teclinical Memorandtun.llo; 897 I The three integrals are evaluated as follpws Setting l*^ - K^ i^ = T, we have for J^ Jx- Ji = Tg / 1/2 ,. y T - K'"( ri - 1)2 1 - ^a .. d T V O) 1 - Tl 1 I , , V ^ V O) ") i«/uj - 1^ ("^ - 1; arc tan > r> ^ K(r, - 1) -J With and (Tl - 1)^ + t" = a. ^* +y^*s_K2ai^ = T there is ohtained for J. ^3 = 1 1 A2„2 ^2^3 'i-^^T - K^ a, 4 1 - r, ■l^T, T2 — los T d T After a "brief intermediate computation we have 1/2 ~^^ |k[(i - Ti)^ + ^"3 log f 7(1 - n)^ + ^s) 1 - -Hi ■ ^ ■ ^ - f log (t + VuL) ) + Vuj and similarly 1/2 1 - Tl, ^ ■ - ? log (e - ^/a^) + Vt^j , A.C.A. Technical Memorandiim No. 897 47 By adding we ottain 2 T7 -^ = \- Vu) + 1^(11 - l) arc tan 1 - 'Hi K(n - 1) + - i log \ (89) r. - V o) A corresponding expression with, opposite sign is oDtained for the cone T) = T) . The arc tan in equation (89) lies TT TT within the range < arc tan < + — as follows from the _ 2 2 fact that Cg must he symmetrical in (ri - l) since the same holds for c^ according to equation (85). The induced z component thus found for the rectangu- lar lifting sxirface with trapezoidal lift distribution has the same singularities as the corresponding formula (85) for the lifting line. On the cone surface "c,^ "= and on the cone axis logarithmically infinite. S'or the downwash dis tr ihut i on at the location of the wing in the plane t = 0. there is ohtained o '^zo ^ f j / 1 Vl - ^2 2 TT = ± ' ^ - V 1 - d + ^ arc tan + -i"" 1 - TH^ ^ % 1 1 +Vl - d^ + ^ lyg 1 ^ -/l - ^2 ± 1 - 11, g (^) (so) 6 K(in - 1) K(ri - ri, ) wnere ^ - — for co.'^e I and ^ = — lor i I cone III, the upper sign holding for cone I and the lower for cone III. Equations (89) and (90) include only the downwash velocity induced by the edge vortices. In order tn obtain the field of the total downwash motion, there is still to be added the induced downwash velocity due to the plane wave. In the wedge-shaped space between the leading and forY/ard edges of the wing (fig. 4), this induced veloc- ity component is 48 K.A.C.A. Technical Memorandum . No . 8 97 ■zo wave ^ jc r _ « 1 - Ti 2 t 2 ° 1 - 'P. = 7, 1/2 1 - ^, e t") (91) For the total downwash velocity in the plane z = 0, there is thus ohtained from equations (90) and (91) For cone I - 1 < -S < 0: < ^ < + 1; For cone III 2(1 - 111) - o-i{ n 2(1 - Til) - 1 < -6 < 0: =— c zo < -d < + 1: ^ + g (^) TT > (M I TT V (^) TT (92) The dcYnwash distribution thus computed is plotted in figure 19.. We' are now in a position to compute, for the wing with trapezoidal lift distribution, the induced drag. In order to avoid special complications, we shall assume that the Mach cone springing from the leading edge at T) = r] ^ does not extend beyond the wing tip and does not overlap the region of dropping circulation of the other half-wing. The first is identical with the condition that the cone springing from T] = 1 does not extend into the region of the wing where the circulation is constant. This gives for the Mach angle the two conditions tan a < b - b> 3t and tan a < b' The induced drag of one half-wing — Yf^ is composed ad- ditively of the drag of half of- cone I, and the U.A.C.A. Technical Memorand-om Ho, 897 49 drags of the two half-cones of cone III, Wj and ^IIIi I W. = Wij^ + Wijjj^ + Wijjj^ (93) Since in cone I, in the plane t = 0. there is upwash velocity, W^ gives a forward thrust which in ahsolute value, however, is smaller than the tack thrust in cone III, since the circulation is greater here. We have x=t 5^=1) /2 r _ i_ =P / dx / -Cgod-y. (^1=7--^ ^an a J W where "^o ^^ known from equation (90). We thus have zo 3 1= ot/h ^=0 'hr '^ -:; — / i''^ i r (^) g (^) d ^ (94) P V ■^1 l-Tl^STTPt^K In cone I for -1 < •Q < 0; 1 - Tl^ and therefore h 1 P r 2 ^ h s. 2 r- / ' w. = i°_( — ) /e=de / ^ g(^) d ^ -^ (1 - -nj^ ^.8 TT ^ t^ t^ J J ^=0 ^=-1 For hriefness we set 1 g (^) d ^ = / g (^) d ^ =. K3 (95a) P ^=-1 ^=0 50 N.A.CmA. Technical MemorandTim No. 897- -/ ^g (^) d ^ = / ^ g (^) = Kg . (951)) These integrals may be exactly computed. There is obtained 1.3 = i-^2 - — (96) 8 18 so that finally ■'*i 1 ~( ) , (97) (1 - T,l) 2 TT t^ b Th e portion Wj^^^_' is obtained froia equation (94) by substituting -g(ii) for g(^) and taking F = F so t ha t ^ 3 1 - Tl TT 1^ b 1' Finally, W^ is obtained by putting in equation (94) ^ 1 - -n, ^ and substituting -g(^) for g(^). By comparison with equations (97) and (98) this gives ' ' ■ ir. = I. ■■ + W. and therefore ^i i Wi = 2 Wijjj^ + 2 fij^ ^ 2 Wij^j^ {1 + i-} ''nil N.A.C.A. Teclinical Memorandum No.' 897 51 Substituting the values from (97) and (98) the induced drag of the entire wing is found to be 4 E^ p Tq^ t tan a f 1 Z Iq t tan a \ , , 31-Ti^ TT b 1-t1^2K:^ b If, in place of T^, there is now substituted the lift A oft heentire wing w e ha V e 15 Kg 1 _ 1 ^ A_ ^^^ t tan a Ur 3 TT . (1 - ri^) (1 + Tl^) ^ p ^b 3 Kp t tan a Jl i _j ^ ■■ 1 - Ti, 2 K^ b ■' Til 2 K3 "^ -. -2 1 / A \^ t tan a if .' = — 1 (1 - Tl,)(l + r,,)^ P ^0 b' ^ 1 ^— ^1^ l_li^^ (100) L l-noQTT b J Thus the formula has been found for the induced drag with trapezoidal lift distribution. To this must be added the wave drag. The latter according to equation (26) and table lis - £l£-±J2il _L__ i (.L^)' (101) ^^^^e " , ,3 Vu^ b^ ^ ^ 3(1 + Ti^) ^ t tan a P ° W. If c„ denotes the coefficient of the wave plus induced w drag then from equations (lOO) and (lOl) c a 4(2+Tl,) ■ 4 X" . 1 14 + A_ ^) (102) 4 tan a 3(l+'n^)'' (l-T] ^) (l+Tl ^) ^ ^ 1-Tli 9tt 52-.. N.A.C.A', -Technical. Memorandum ITo." 397 The alDove formula differs from the corresponding formulas for the rectangular flat plate (equation (80)) and the trape- zoidal wing with constant lift distribution (equation (49)) in that for small A the induced, portion of the drag is proportional to ^s whereas for- the- other two cases it is proportional to x« ^^ figure 20 the coefficiont .Cxf/——— — is plotted against the reduced aspect ratio ' 4 tan a — tan a _ ^ f^j, various traioezoid shaioes "b ' /"b . It may be seen that by far the greatest portion of the drag is contributed by the wave resistance. The portion contrib- uted by the induced drag, within the range of validity of our formulas, amounts to a maximum of 11 percent of the wave resistance for X - 0.5 and b ' /b =l/2. It is therefore smaller than for the rectangular flat plate where for the same aspect ratio it amounts to 19 percent (fig. 10). VIII. SUMMARY With the aid of the expressions given by L. Prandtl (reference 2) a theory is developed of the airfoil of fi- nite span at supersonic speed. As in the case of the Prandtl airfoil theory for the incompressible flow, it is a first order approximation theory. The airfoil is first replaced by a "horseshoe vortex" and the induced velocity field of the latter .computed. This field is considerably different from that of the incompressible flow. From the horseshoe vortex there are obtained in the familiar mazinor by superposition more complicated lifting systems. The computation of the induced drag., in contrast to the incom- pressible case, is for the compressible flow possible only if there is first assumed a surface vortex distribution and secondly a suitable dropping off of the lift toward the wing tips. . ■ ■ As an example of the "first principal problem" there are computed the induced drag and the wing surface shape for a wing of trapezoidal plan form with constant surface density of the lift. The induced drag, as in the case of the incompressible flow, is found to be proportional to the square of the lift and depends on the Mach number as well as on, the aspect ratio. In addition to the frictional and' induced drag there is present in the- supersonic case also the wave drag, produced by the sound waves, which N.A.G.A. Technical Memorandum No. 897 53 varies as the induced drag. It is therefore only the sum of the wave and induced drags that is of practical inter- est. As an example of- the "second principal protlom" there is computed the lift distribution and induced drag for the rectangular flat plate (untwisted rectangular wing). Out- side the two Mach cones springing from the leading edges of the wing tips the lift density is constant; within these cones the lift drops from the full value at the cone rim to the value zero at the lateral wing edge. The inte- gral equation that arises is independent of the aspect ratio and of the Mach numter and may he solved numerically by approximate methods. In general for airfoils of normal aspect ratios at supersonic flows the greatest portion of the total drag is contributed by the wave resistance while the induced drag contributes only a small proportional part . Finally, there is considered the lifting line with trapezoidal lift distribution and the lifting surface of rectangular plan form whose lift is constant along the chord and trapezoidal along the span. For these cases the downwash distribution and induced drag are computed. Translation by S. Reiss, National Advisory Committee for Aeronautics. 54 N.A.C.A, Technical Memorandum No. 897 REFERENCES 1. Prandtl, L.: Theorie des Flugze'ugtragf liigels im zusam- inendriicktaren idedium. Luf t falirtf or schung, vol. 13, no. 10, Oct; 12, 1936, pp. 313-19. Prandtl, L,: G-eneral Cons iderat ions on the Flow of Compressible Fluids. T.M. Ho. 805, N.A.G.A., 1935. 2. Prandtl, L.: Tragf liigeltheor ie , 1. u. 2. ilitt eilung. Nachr. von der Kgl . Gesell schaf t der Wi ssenschaf t en. Math. Phys. Zlasse (1918) S. 451 u. (191S) S. 107. Wicder atgedruckt in Vier Ahhandlungen zur Hydrody- nanik und Aerodynamik. G-bttingen 1927. 3. Ackeret, J.: Air Forces on Airfoils luoving Faster- than Sound. T.M. llo. 317, H.A.C.A. 1925. 4. Busemann, A.: Aerodynamischer Auftriet tei U"b©rschall- geschwindigkeit . Luf tf ahr tf orschung, \'ol. 12, no. 6, Oct. 3, 1935, pp. 210-20. 5. Betz, A.: Beitrage zur Tragf lugeltheori e mit besonderer Beriicksicht igung des einfachen rechteckigen Fliigels. MUnchen 1919. N.AoC.A. Technical Memorandum No. 897 Pigs. 1,8,3,4,5,6 Figure 3.^ Lifting line with constant lift distribution. Donnwash distrib- -w r. ution in Mach cone. Figure 1.- Potential of the lifting line. 4L, V xiaa. Wave of rarefaction^.' ^ — H^ Compression 2 shock _ ^— < |£| ^0-^ N.— N;Wave of Compression shoclc" rarefacti Figure 4.- Plane sound waves at a flat plate. on \ ^> ; t ^'' I X / ^-'•'" '^A . -^ v. Figure 2.- Lifting line with constant lift distribution (horshee vortex). Streamline picture of the y- and 2- velocities in a plane at right angles to the axis of the Mach cone. — I I I j l I I - ■ ■ ■ i , ^ .^0- \ V \ \ f ''' t -6, 5 T~ "^1 — _^ ^ =^ ^ s \ \ \ -p — — -Ttja -^ \ \ -ift -» ° Figure 5.- Rectangular wing as lifting Bxurface with constant lift distribution. Figure 6,- Rectangular wing as lifting surface with constant lift distribution. Downwash distribution in the wing plane. Continuous curves for x< t (at location of wing) dotted curves for X = 2t (behind the wing). N.A.C.A. Technical UemoraQduvi No. 897 Fig8. 7,8,9,10 Figure 7.- Hectangialar wing as lifting Burface with constant lift distribution. Streamline pictvire of the y- and a- velocities in a plane x < t at right angles to the axis of the Sdach cone. Figure 8.- Rectauigular wing as lifting surface with constant lift distribution. Streamline picture of the y- and z- velocity components in the plane X = St at right angles to the axis of the Mach cone. /■P-- ' ' ' Figure 9.- Trapezoidal wing with constant lift distribution. Figure 10.- Trapesoidal wing with constant lift distribution. c Coefficients of the wave plus induced drag c^/ ^^^^^ as a function of the "reduced aspect ratio" X s t tan«V^ for various trapezoid shapes 9 » tanr/tanof. N.A.C.A. Technical Memoranduiii No. 897 Figs. 11,12,13,14 Figure 11.- Trapezoidal wing with constant lift distribution. Induced downwash velocity in section AB (in direction of flow) (tanT« l/yZ; tanoCnV^). Figure 18.- Trapeeoidal wing with constant lift distribution. Induced downwash velocity in section CD (at right angles to flow direction) (tanrs l/VS; tand-Vs). Figure 14.- Trapezoidal wing with constant lift distribution. Elevation contour lines. (tano^sYS; tanT = l/Vs) . Figure 13.- Trapezoidal wing with constant lift distribution. Profile sections. (tanoC = V3; tan T =1/^3). N.A.C.A. Technical Ueisorandum No. 897 ip, , ^ — -, , , , , , ■ I w Figs. 15,16,17,18 /a ^ Inc( ^ ^ ^ ^ A ( Incompressible^^, fi 0,1 Ijl X tgo- Figiire 15.- Rectangular plane wing. Lift at wing edge. = t!! C, Cm ^^ Figure 17.- Rectangular surface with trapezoidal lift distribution. Figure 16.- Polars of plane rectangular wing for various aspect ratios. ' I ' ' ■ I ^ ' ' I I ' I -I ' I I ' I ' ' I ' ' I I I I I I I I 1 —1 — r Figure 18,- Lifting line with trapezoidal lift distribution. Streamline picture of the y- and z- velocities in a plane at right angles to the axis of the Idach cone. N.A.C.A, Technical Memorandum No. 897 Pigs. 19,20 Figure 19.- Rectangular wing as lifting stirface with trapecoidal lift distribution. Downwash distribution for x