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( f o  NATIONAL ADVISORY COMMITTEE FOR 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 191819 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 draglift 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 andthe 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 1T of a lifting line at supersonic speed may be derived in a simple manner from the potential PQ of a stationary source in the presence of a supersonic flow. *"Tragfligeltheorie bei Uberschallgeschwindigkeit." Jahr buch 1937 der deutschen Luftfahrtforschung, pp. I 18197. N.A.C.A. Technical Memorandum No. 897 If IQ denotes the source potential of strength 4r, the potential ET of the lifting line element dy with circulation F about the y axis is given by xl = X T = 2 / d x (1) 4 8z X' = _0 The potential of a source at the point x = y = z = 0 in the presence of a flow with velocity uo > c in the di rection of the positive x axis is Q = 1_ __ (2) 2 (u 1 (y + z2) S L c J The potential (2) is real within the double cone with half cone angle c, the axis of which cone is parallel to the direction of flow (sin a = c/uo). Outside of this cone the potential, according to the formula, is imaginary. Actually, kQ is there to be taken identically equal to .zero. The potential has physical 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 Q is the starting point for con structing the airfoil potential. We shall first derive from it the potential of a line source of finite length, then with the aid of the operation indicated in equation (1) we shall obtain the potential of a lifting line of finite length for various lift distributions. From the lifting line, there is finally obtained by the familiar method, the lifting surface. In this manner, a theory of the iairfoil of finite span for supersonic speed is ob tained that forms the counterpart of the Prandtl airfoil theory for the incompressible flow case (reference 2). N.A.C.A. Technical Memorandum No. 897 II. CONSTANr LIFT DISTRIBUTION (LIFTING LINE) 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 be the coordinates of a point in the flow and 0, y', 0, the coordinates of a source point. Then from equation (?) the potential of the line source is b t I= ,j yl b y _ g(y' ) cd y' (3) o c  1 ( y') + z 2 Se introduce nondimensional coordinates by dividing all lengths by the halflength b/2 of the line source and accordingly set 2 x 2 y 2 z 2 7' b b b b Further, we introduce the abbreviated notation u 0 1 = or (4) tanc = 1 =  1 C2 where a denotes the Mach angle. The potential of the line source then becomes ri =+ 1 in1=1 g(l') d ,1t V2 K[En *n,)2 + t2) (5) N.A.C.A. Technical Memorandum No. 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, g(il) = const. = 1. This gives 'n= +1 T, d ' : i / ..E(n T') + 3] Writing T 1' = h; t' = 1: b = 1 = 1 + 1 n' = +1ti = b3 = n 1 and (t/)j K) = a2 equation (6) becomes S=  =  arc sin arc sin (7) K, aK c a a By the operation in equation (1) there is then obtained the potential cT of the lifting line with the constant lift distribution Fo, setting 2d (a) The first step of the above operation, differentiation with respect to S, may be carried out immediately but the inte gration requires a somewhat longer computation. There is obtained a a 2 s a2 2 a 2 2 . a2 a = . /9) In integrating with respect to I, 42 and 1, are con stant. For the'first term, th6re is obtained N.A.C.A. Technical Memorandum No. 897 / d ' a2 1 2 2 2 4 2 where there has been set t* = t)2 The evaluation of the integral gives * a2 2 2 a a e K = arc tan 2 = arc tan 2 2 2 ~2 + (?2 2) 2t3 2 + It, 2 ," 2 w .~ ( 1) 2 (n i) (./u (11) where S= 2 [(n 1)2 + t2] (12) Since the integral (10) outside of the Mach cone, at the cnd point n = 1 of the lifting line with axis parallel to the x axis, t 4 [(n 1)2 + C] = o is imaginary, i.e., is to be taken equal to zero, the in tegration with respect to E' need not be extended from = * but only from the cone surface along lines paral lel to the Y axis. For the lower limit of integration, we have thus the constant arc tan m TT/2, which we may suppress. There is thus found from (8), (9), and il) the required potential of the lifting line with constant cir culation (10o) N.A.C.A. Technical Memorandum No. 897 r E r I t3 d r t2 W so t2( 1)n K =  =   arc tan 2T w asa / 2 4T 2 ( ( 1)( JL" ao (2 1) =  arc tan + similar term for the cone at S= 1 (13) This potential is different from zero only within the two Mach cones arising at the ends of the lifting line (w > 0) while in the entire remaining space it is equal to 0. For a complete circuit about each of the cone axes 1n = 1, = 0, the arc tan increases by 2 Tr. The enclosed vortex filament therefore has the circulation Fo. The lifting line assumed to extend from y = b/2 to y = +b/2 with the constant circulation Po along the span continues behind as a free vortex line in the two axes of the Mach cones. Equa tion (13) thus gives the potential of a "horseshoe vortex" at supersonic flow. As in the case of the incompressible flow, this simple horseshoe vortex becomes the starting point for more complicated lifting systems. 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 cx = ; cy = cz T ax a y z and obtain c = K  T b ( K K ) ,7 c =  (14a,b,c) TT b [ + (n 1)] ] r0 =( r 1) (w K22) TT b [2w+ (T l)2t2] See footnote on next page. N.A.C.A. Technical Memorandum No. 397 rhe field of these velocities exhibits a number of singu larities. On the cone surface all three velocity compo nents become infinite. On the cone axis cx = 0, but cy and cz become infinite as 1/r (where r is the distance from the axis). In the neighborhood of the cone axis, Cy and cz 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 r = 1, o = t = C, as may be seen immediately from (14). In the plane r 1 = 0 through the end of the lift ing line cz = 0 and > 0 : cY < 0 < : Cy > 0 In the plane 0 = 0, which contains the lifting line, c,, = 0 and i 1 > 0 : cz > 0 r 1 < 0 : cz < 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, cg and cy are infinite, although for the slope of the streamlines cz/Cy there is here obtained the sim ple value cz cy T 1 'A check for the correctness of this solution is obtained b.y substituting in the linearized continuity equation Scx 8 cy 8 c z  .+ + 8x 8y z which must be identically satisfied. N.A.C.A. 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 about 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 = 0 through the cone center, there is obtained from (14c) the simple formula 2 TT x /l 2  tan a cz = (15) o where b y  2 S= (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/a Wi= P co(y)r (y) d y (16) y = b/2 (where czg(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 the case of the supersonic flow, the relations are complicated by 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, impossible. 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 (DRAG) Before proceeding to the corresponding computations, we shall discuss briefly the supersonic flow about an in finitely long airfoil (twodimensional problem), a problem 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 angle of attack o (fig. 4). For such a plate the lift per unit span is A = 2 tan a po t P u02 (17) or A = ca = 4 tan a 0o (18) 2 2 uot On account of A = p u Fo, the relation between the an gle of attack of the wing and the circulation is 1 Fo o uo tan a (18a) 2 t From the incompressible 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.therefore may N.A.C.A. Technical Memorandum NTo. 897 *be denoted as the wave drag. For theflat plate, the wave drag per unit span is Wwave = Po A = 2 tan a Pos t p uo (19) or SWwave = 4 tan a o0 (19a) Cwave The 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 (1) and (19a), there is obtained for the polar of the wave drag ca2 wwave 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 velocity is c = Pn 0 1 (20) wave o2 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 Wave = P e Cwave (21) as may be seen by comparison with (19) and (20). In the next section it will be shown that,for a lifting surface, the velocity induced by the tip vortices like zwav is proportional to ro/t tan a. It 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 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 by y = +b / rT= +1 A = p U0 y = b/a r dy = ~ uo b t 'Y (7)) d r r = I n = +1 (d c r d y = b t z~ow 2 cz W a"d n (23) ow 1 ta Stan 3 tan a (24) is the induced wave velocity. Accordingly n = +1 wave 4 t 4 tan a , S2 d (25) By comparison of equations (22) and (25), there is found the relation between drag and lift Z b A a Wave = 2 Z (  P t tan a uo b (26) In the above equation Z is a nondimensional coef ficient that depends only on the lift distribution TI = +1 z = I Ya 2 / d \d From equation (26), it follows that: w wave (22) where (27?) czow o' N.A.C.A. Technical Memorandum No. 897 wave = tan Ca tan a (28) The numerical values of Z are given in table I for sev eral simple lift distributions. TABLE I Values of the Coefficient Z for Various Lift Distributions (Wing of Rectangular Plan Form) Number Lift distribution Z rectangular ellipse trapezoidal b' = b/2 parabola triangle 1/4 8 3 TT = 0.250. = .271 b = .370 = .300 = .667 IV. LIFTING SURFACE WITH CONSTANT LIFT DISTRIBUTION For the successful computation of the induced drag for supersonic flow, according to section II, the simul taneous assumptions must be made of a suitable drop in lift toward the edges and a surface distribution of the bound vortices. This. twofold extension means naturally a considerable swelling of the computation of the field of induced velocities as compared with the incompressible flow where the computation involves mostly a lifting line. In order to be able to recognize more clearly the effect of each of these two extensions, we proceed in two steps. N.A.C.A. Technical Memorandum No. 89? We first maintain the lift distribution constant along the span and consider only the transition from lifting line to lifting surface. The fieli of induced velocities thus obtained for a wing with constant spanwise circula tion distribution and constant chord, while it does not enable as yct the computation of the induced drag never theless furnishes many aseful results so that we orocecd first to compute this field. We assume therefore the circulation Fo constant along the span b as uniformly distributed over a rec tangular lifting surface of chord t and extending from x = 0 to x = t (fig. 5). The circulation for a strip of the lifting surface of unit width is therefore o = fo/t. It would be most convenient to make the transition from the lifting line to the lifting surface directly on the potential (13). On account of the integration diffi culties that arise, however, the transition will be made on the velocity components (equation (14)), first for the z component since the latter is the most important for the computation of the induced drag. A strip of the lifting surface of width dx' at a distance x' from the leading edge contributes to the induced z component c. at the point x, y, z, if the point lies within the Mach cone arising from the end of the strip the amount S = ', ) =  d f(  T b 2 rT where, according to equation (14) f(( ) 1)(w K t2) (29) [2W + ( 2 V If the point x, y, z lies outside the cone, the amount contributed is zero. The contributions from the plane waves starting out from the lifting surface will be sepa rately considered. Integration over the wing chord there fore gives for the downwash velocity induced by the lifting surface 2 TT z = Yo ft (',r)d U or written out in full N.A.O.A. Technical Memorandum No. 897 0 CO o II cla Q^ + CO + LO le ca I 0 L%1 i  A A I  I C.' 103 r Ia I 0.r riF II lo 11 ^J ,gf> ^_ we have 2 z 2 TT  YO T = T2 T 1 / (.T K a2 ) d T a 2 a 2 J T VT K (1) 1) T T= a 1 T= T d T VT K (T) T =T K2 2 I T= T d T T /f n2 (j _1)2 The upper integration limit C' = ri. 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 between the latter and cone I arising from the end point of the leading edge. The corresponding limits will be Ei = 2t/b = (within cone II) = K V( 1) + t (between cones I( and II) C as may be easily seen after some consideration. Introducing the new variables of integration 2 K t2 = T and writing for briefness a 2= (n i) + N.A.C.A. Technical Memorandum .No. 397 The evaluation of the integral gives 1 2 r)2 + 0 A ,) L i)0 + \ Kl arc tan L K( 1) Taking account of the different upper integration limits according to whether the point considered is within cone II or between cones I and II, equation (31), and setting for briefness Wc, = ( ) 1)2 + (32) there is obtained as tie final expression for cz: For cone II  =  1 0o (, 1)2 + t K {arc tan arc tan (3a) <(T 1) K(n 1) Between cones I and II 2 T 1 arc tan VW (33b) eo ( i)2 + t(n I) where S(T 1) 1 < < + 1 : I < arc tan < 2 2 From these formulas it follows that c on the surface of cone I is equal to zero and on account of W = 0 is con N.A.C.A. Technical Memorandum No. 897 tinuous in passing through cone II. On the common axis of cones I and II, there still occurs the same singularity as in the case of the lifting line, Fz, there becoming in finite as 1/r. In order to obtain the total downwash velocity, there is still to be added to equations (33a) and (33b) 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) Zwave 2 o wave Cxwave = (=3) The expressions for the two remaining components of the induced velocity y and x are found by similar in tegrations. We shall only indicate the results: Cone II: o ( i)3 + (n ex ( g)( 1) l) 2 = arc tan arc tan Between cones I and II: >(35) 2T  V(U 0o ( 1)2 + 3 2 TT = arc tan 1) In the above equations, the arc tan is to be taken TT/2 and +TT/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 set aside. The' singularity of Cy and cz on the cone axis (infinite as 1/r) still remains, however, and prevents the computation of the induced drag for this lift distribution. N.A.C.A. Technical Memorandum No. 897 For the downwash distribution 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 ,plane wave, there is obtained the following: 1) For x < t:  i< K < 0: = + arc tan KY 0 2 o 1 j 1 z P 1 2 0< U< + 1: = arc tan  'Yo 0 U a) (36a) where for the arc tan the same values are to be taken as in (33) and 6 is given by equation (15a). 2) For x > t: The plane waves do not contribute anything but the formulas obtained differ according as region considered is within cone II or between cones I II (fig. 5). the and t t 2cz0 I )< 5 < i t ;: o x K Y7 rr v/ x S(y7 2 ( 1 2  1 (arc tan U arc tan  TT b /  1 < t < (1 ) 1 l< < + 1 . 1 (. t 0 o\ arc tan Tn T (36b) 2cz 0 1.A.C.A. Technical Memorandum No..897 The downwash distribution..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 outerhalf 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 **! = 0. These two strips are limited by the plane waves starting out from the forward and the trailing edges of the lifting surface. Within the Mach cone the streamline picture in the outer ring is the same as for x < t and is changed only in the inner region. We shall yet consider briefly the question, what the form of the wing surface must be that corresponds to the assumed lift distribution. The wing plan form we have as sumed as rectangular. Angle of attack and twist are ob tained from the consideration that at the wing, i.e., in the plane z = 0 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(O, y) 0, i.e., straiht leadin e1te. T.en we have, d z czo(x, y) d x u where co includes the induced velocities from both the plane waves and the edge cones. There is thus obtained for the wing surface z(x, y) = czo(x', y) d x' (37) X' = 0 N.A.C.A. Technical Memorandum No. 397 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) = o x that is, a flat surface with angle of attack po. Within the Mach cone the surface bends downward more and more strongly as the edge is approached. The edge itself (y = jb) is bent infinitely downward, i.e., actually the rec tangular surface with constant spanwise and chordwise lift distribution is not possible. For this reason we may dis pense with the further computation of the wingsurface shape. V. TRAPEZOIDAL WING WITH CONSTANT LIFT DISTRIBUTION We consider now a trapezoidal wing with constant sur face density of the lift 70 (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 (7 > a), the induced drag is obviously equal to zero and only the wave drag exists (reference 4). T> a:Wi = 0 The trapezoidal wing with constant surface density of the lift 'o is plane outside the Mach cone and has the angle of attack p0 where 0o = 2 o u0 tan a The trapezoidal flat surface with constant lift distribu tion whose cutaway 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 Y d x. N.A.C.A. Technical Memorandum No. 897 Such a lifting element at x = 0 generates at a lifting element of chord d x' at x = x' y=yi(x) d2 Wiox = P = o d x' f d c (OX') d y y=yo (x) (38) (ox') where d czo denotes the downwash velocity induced by the lifting element x = 0 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 c(ox') in the plane 0 = 0, we have according to equation zo (14c) d c(ox') c= 2 _o K 2(T 1)2 zo 1 b( l) d x (39) with the aid of which equation (38) becomes d2 Wiox. =  70 d x d x' 2 TT oI 1 or with = K . 1 / J./ T)_\~ S_2(n 1)2 ((9 1) , according to equation (15a) and 8 .= tan T tan a as .the reduced angle of taper Wiox 70o2 d x d x' P 70 d x d x' 2 Ji1  Jd 4 a=6 g (e) (40) (41) N.A.C.A. Technical Memorandum No. 897 The evaluation of the definite integral gives S() = log 1 (42) e According to equation (41) the induced drag from the lift ing element x = 0 at the position x = x' is independ ent of the distance between the two elements. All ele ments lying between x = 0 and x = x' accordingly pro duce the same drag, so that the total drag induced at x = x' amounts to p 2 d Wix, 70 x' d x' g (6) 2 n The drag for the entire wing is obtained from the above by integrating over x' between the limits x' = 0 and x' = t and multiplying by two (both ends) x'=t Wi =  0 (6) d xI n .1 x'=0 = t g (6) S g (6) (43) 2 2 n 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 incompressible flow. For triangular lift distribution (lifting line) in the case of incompres sible flow, we have, for example, log 2 2 Wi =  p Fo where 1o is the circulation at the wing center. For equal total circulation Fo' iTi according to equation (43) is independent of 6, i.e., the ratio of the tangent N.A.C.A. Technical Memorandum No. 897 of the angle T to the tangent of the Mach angle (equa tion (40)). In passing to the rectangular wing, e > 0, the induced drag according to equations(42) and (43) be 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 drag alone as in the sum of the induced and wave drags. For the wave drag, according to equation (21) we have wave = F 0o Czwave where F = b t ( t tan T \ where = t 1 tne area of the wing (44) c 0 Zwave 2 tan a hence Wwave = t 02 2 tan a For the lift we have, on account of T, = 2 po uo tan a: A = P To uo F = 2 P uo2 Fo tan a (45) or A c = = 4.ao tan a (6) P F2 For the wave drag we obtain from (44) Wavee= 2 P F uo2 Ao2 tan a Wave =w 4 Po tan a = po ca (47) O wave 2 and for the induced drag from equation (43) N.A.C.A. Technical Memorandum No. 897 Ni = 2 u2 uo 2 ^oe F J^ t2 4 P 0o tan2 a g (6) 1 4 t o02 tan" a g (6) S b t tan T b cwi = 4 Po tan a g 7r 1 6 X t tan a where X =  b is the "reduced aspect ratio" of the wing. For the total drag there is thus obtained from (47) and (48) (cW)wave + ind = 4 po tan a 1 + g *n 1 6 A Ca2 X g (! ) (cw)wave + ind = a I + C) 4 tan a n 1 6 (49) It follows therefore from the above that for supersonic speed the wave plus induced drag, like the induced drag in the incompressible flow case is proportional to the square of the lift. Equation (49) is analogous to the a 2 F wellknown formula cwi _ of the elliptic lift TT b distribution for the incompressible flow. The essential difference lies in the fact that for the supersonic flow the drag parabola for small aspect ratios t/b is to a first approximation independent of the aspect ratio. The manner in which the drag increases with increasing reduced aspect ratio A = t tan a and decreasing 6 is shown in figure 10 where c/ Ca w/4 tan a is plotted against X for various values of 6. Our formulas are valid only for (48) N.A.C.A. Technical Memorandum No. 897 1 < , i.e., for the case in which the Mach cones do not overlap on the wing. In order to be able to predict what the wingshape must be 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 dcto) according to (39) gives the downwash velocity in duced at the position  by a lifting element 7o dx starting at t = 0 and ending at rT = 1. A lifting ele ment which starts at t = 1.' and ends at  , I' thus produces at the position t, T, t = 0 the downwash veloc ity d c = )2  zo 2 TT (n For the velocity induced by the entire surface there is thus obtained 0 2. (f (50) In order to evaluate this integral we introduce the new integration variables K =) 2L = K tan%)" (51) .(See fig. 9.) Since the endpoints of the lifting elements lie on the wing contour there exists the relation T' = ' tan f (52) The upper integration limit (' = in equation (50) is obtained from the condition. (See fig. 9.) N.A.C.A. Technical Memorandum No. 397 tan 0J = tan a: = 6 : &' = 1 The lifting elements whose ?' is greater than the t thus determined give no contribution at the points t, r, = 0. For the lower integration limit = 0: '' = 1: @'=u U = From equations (51) and (52) there is obtained d g' d d ' e where 0 is the abbreviation introduced in equation (40). There is then obtained from equation (50) czo (iY) 1 =1 J/  d 7' 2 F(O,6) (53) 1( 6) 2 TT In evaluating the above integral the following three cases are to be distinguished: 1. O< 6 < o ; 2. 0 < < 6e; 3. < 0 < 6 In case 1 the point P(o) 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' = 6; and in case 3, two singularities at b' = 0 and o1 = E. In cases 2 and 3 the principal values are to be taken, namely, 0 < b < e: '=9b /  l [ V I ,b F(,G) =lim { I 0' + '=o 0 ' ^ '=, S4=1 0'=e+e N.A.C.A. Technical Memorandum No. 897 and If , F( 6) =lim / . d a + c o J ^l ^. 6 *~5 1=6E / .1 ti t .. = .' : 3'=6+e The integral , (1 2 . '=. (55) may be obtained by elementary methods. We set l = t 1 where t is the new integration variable so that (55) be comes t= t t t =1 1 (1 ta)2 dt t(l + t2) t2 t 6 (1 + t2) 1 +4  ti = 'V = (56) By breaking up into partial fractions there is obtained (54b) where a < 0 < 6: E'  N.A.C.A. Technical Memoranduam o. 897 (1 t 2) t(1 + t2) t 6(1 + t2) > 2 1 1 + t2 E t 6 % 1 + t t t 6 't to t tz where 1 1 62 rerforraing the integration, there is obtained F( ,6) = I 2 L arc tan t lo t + _ lor (t t ) log ( t tI ) t % t = 1 For 0 < b < b there is therefore obtained directly F(6,6) 2 arc tarn iJ  2 6 6 +1 6 S+ i + 62 b i 6 + 1 62 / while the formation of the principal value according to equation (54a,b) gives (57) N.A.C.A. Technical Memorandum No. 897 0 < b < 6: F(6,6) = 2 arc tan log 2 6 1 6 l  log 6 S< 0< 6: log (_) F(4,6) 2 arc tan 6  2 6 '*Yi 68 + log There W 6 i +' . 92 4/ e 1 1 p8/  < arc tan V < _ 2 2 There is thus found the downwash distribution in the entire. Mach cone springing from y = b/2, x = 0. For F(.,6) we have S= :F(l; 6) 5 0 3 = 6, = 0:F(6,6) = F?(, e) = as log 3 at b = 0 (58) On the two rims of the cone (0 = xl) the induced velocity is thus zero and on the edge of the trapezoidal wing (D = 6) and on the cone axis (b = 0) it is infinite. In figures 11 and 12 for the particular case 6 = 1 (tan a = &T, tan T =  3 there is shown the induced downwash velocity in a section parallel and perpendicular, respectively, to the principal stream direction. To the above velocity field of the tip vortices there is still to be added the velocity field due to the plane wave. The latter in the plane z = 0 within the wing area is 4e1 + e6 e_ 1 l 1 6 N.A.C.A. Technical Memorandum No. 897 CZo.wave S P0 2 tan a and outside the wing area Czowave 0 From the velocity field it is now possible to compute the form of the trapezoidal wing surface that has constant lift distribution. Outside of the Mach cone we have, ac cording to equation (37) x'I x z(x, y) = wave (x', y) d x = 0 x u0 wave x, = 0 that is, a flat surface with the angle of attack 0 giv en by equation (18). The twist of this flat surface within the edge region of the trapezoidal area that is overlapped by the Mach cone is given by XI = X I 1 z(x, y) = coi d Uo .I x'=(b/2y)/tan a and according to equation (53) x'I=x x' x'=x z(x,y) _ 10o K 27 uo J x'=(b/2 F(a', 6 ) d x' = jo F(', 6 ) d x' y)/tan a x'=(b/2y)/tan a S(b y) On account of 6 = there is obtained z(x, y) = Lo y8) d ' T 2 ba S M (59) N.A.C.A. Technical Memorandum No. 897 Since the function F(6, 8) is known from equation (57), it is possible from the equation above to compute for a given 8 the profile sections of the surface at various distances (b/2 y) from the edge. The ordinate of the obliquely cutaway edge of the trapezoidal area for b/2  y < t tan T: *'=8 z(xR, y) = K Y) F y ~ d .' (60) S 2=1 The integrand becomes infinite for 'l = 8 (equation (58)). The integral exists, however, and may be evaluated by spe cial computation. There is obtained ,'=6 / F(t', 9) nd = are sin 8 1 1 (60a) d'=1 (The evaluation of the integral was performed by Dr. F. Riegels.) For 6 = 1/3, we thus have 6 = 1/3: /" d =d 4.92 The ordinate of the rear edge point x = t, b y = t tanT for 6 = 1/3 is thus z = 1.522 Po t. (Flat surface z = Po t, twist z = 0.522 po t.) For the special case 6 = 1//3 (tan a = v tan T = 1) the profile sections have been computed and are given in figure 13. If the trapezoidal wing were flat there would be a drop of the lift toward the edge down to zero. In order that full lift be maintained up to the edge, the wing must be bent downward. The twist of the N.A.C.A. Technical Memorandum No. 897 wing directly at the edge is very strong as may be seen from the "elevation contour lines" (fig. 14). VI. COMPUTATION OF THE LIFT DISTRIBUTION FOR THE UNTWISTED RECTANGULAR WING The examples thus far considered are all in connec tion with the socalled first principal problem 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 = 0 to x = t and from y = b/2 to y = +b/2 and is set at the small angle of attack 0o to the undisturbed veloc ity uo (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 czo uo Q (61) Wave o2 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/2 the lift must vanish, that is, 7 = 0 at y = b/2. There is required the lift distribution 7 = (x,y) within the region overlapped by the Mach cones. The problem is considerably N.A.C.A. Technical Memorandum.No. 897 simplified by the circumstance that, as will immediately become apparent, .7 does not depend on the two independ ent variables x, y, but only on one of the variables b = (51) x tan a (fig. 11). For the required lift distribution 'Y ) = 'Yo f () (62) of the rectangular wing there then exist the boundary con ditions 6= 0 : f(3) = 0 >. (63) i= 1 : f(.) = 1  In order to be able to set up the integral equation for V7(6) we must first compute the field of the downwash ve locities w(,) induced by a rectangular wing with the cir culation distribution 7Y() in the plane z = 0. The in tegral equation for 7Y(b) 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 Y)W =#() 1 () (64) 2 u. tan a and the induced angle of attack w.1) must be equal to the geometrical angle of attack' .o p() (t)l (65) uo The velocity field w(.) induced by the edge vortices is obtained by considering the rectangular wing with the var iable lift distribution 'Y(b) = 'Yof (W) as built up by the superposition of trapezoidal wings with various taper an gles each of which wings possesses a constant lift distri bution. Again, let 6 =  be the "reduced taper angle" tan a (equation 40), then the lift distribution Y = Y f (4) may N.A.C.A. Technical Memorandum No. 897 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 (j) = f () F (N,6) d 6 2 TT tan a and integration over 6 from 6 = 0 to 6 = 1 then gives the induced velocity field over the rectangular wing t=1 w (j) o / f'(6) F (,,6) d 6 (66) 2 n tan a / 6= o By substituting the above expression for w(t) in equation (65), there is finally obtained, taking account of (61) and (64) the required integral equation for f(0): 6=1 f() + / f'(6) F (,,6) d 6 = 1 (67) b=o to which are added the boundary conditions (53). This integral equation for the lift distribution has the same structural fornas 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 iach 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 6 = as the variable. It is neces tan a sary to solve the integral equation (67) only once to ob tain the lift distribution of the rectangular wing for all aspect ratios and all Mach numbers. N.A.C.A. Technical Memorandum go: 897 The solution of the integral equation (67) appears at first sight quite difficult, particularly on account of the complicated structure of the core F(O,6). (See equations 56 and 57.) By a simple transformation of equation (67) it is possible, however, to simplify the problem considerably.* The equation is a nonhomogeneous integrodifferential equation for f(b). Instead of it we shall consider the equivalent equation for f'(4). Taking account of the singularity of the core, equation (67) may be written 6= d / f'(#) F(4,.) d e = 1 T Differentiation with respect to b gives f,(b) + 1 { f'(b) F(6,4) + Tr f, (6) .! d6 d b 8=o 6=1  f' (6) rF(b,) + f,(6) d ed F = 0 d  j 0=b and because d Fl d i s( 6) according to equation (53): 6=0 iT .1O (68) *For this suggestion I.am indebted to Doctor Lotz and for carrying out the numerical solution of the integral equa tion to Mr. Pretsech. N.A.C.A. Technical Memorandum No. 897 The above is the equivalent integral equation for f'(C) which, however, is now homogeneous. The solution of this integral equation for f' (0) is possible by building up f' () in n steps and solving the corresponding system of linear equations f'" U 1)  1 i _b2 n 2 : X Si V+1 I f (2x+1) / d o +i 2 V ?o .j 2V+i =+ (v = 0, 1, .....n 1) (69) This is a system of n homogeneous equations for the n unknowns f'( ,mu+1) (u = 0, 1 ..., n 1). Since, as closer investigation shows, f'(0) = a, f'(2 ) is suit ably chosen not constant but equal to f'('1) = a b b There is then obtained in place of equation (69) a nonhomogeneous system of equations of the nth order for b 1 the n unknowns , f'(baV+1) (v = 1 ..., n 1). The a a further unknown a is obtained in the numerical integra tion for f(L) from the condition f(1) = a Z 2 = 1 (70) V=o a In carrying out the numerical process there were first taken five steps (Cau+1 = 0.1; 0.3; 0.5; 0.7; 0.9), then ten steps (basu+ = 0.05; 0.15; ...; 0.95). It was found that the tenstep approximation gives an improvement over the fivestep process only in the interval 0 < 2 < 0.2. In the third approximation therefore only the interval 0 < 4<0.2 was again subdivided (zsvu+ = 0.025; 0.075; 0.125; 0.175). The values obtained inthis manner for f' ( ) and f(2) are given in table IIand the function f(6) plotted in figure 15. At b = 0 the function N.A.C.A. Technical Memorandum No. 897 f(6) possesses a singularity since f'(b) there becomes infinite. The mathematical nature of this .singularity could not as yet be determined. We shall now compute the lift, wave drag, and induced drag as well as the moment about the transverse axis of the rectangular flat surface. The lift A, of that portion of the surface which lies outside the two Mach cones is t t tan a AI = P Uo o b t 1  while the lift of the two triangular portions overlapped by the Mach cones is *=1 AII = P uo t2 tana / Y7 d 6 = p uo Y0 t2 tan a E '=o where K = / f(t) d i = 0.684 (71) b=o The total lift of the rectangular plate is therefore A = P uo Y b t {l (1 K)X or, according to equation (20) A = 2 P uO2 Po F tan a {l (1 K) } (72) For the lift coefficient there is thus obtained ca = 4 3o tan a l (1 K)\} (73) For the wave drag outside of the Mach cones there is obtained simply = p b t t tan a  Wwave I = o AI = ol _t (74) tan ac b N.A.C.A. Technical Memorandum No. 897 The wave drag of the two triangular portions overlapped by the Mach cones is Wwave II = 2 P f 7 cave d f where c Zwave 2 tan a 2 tan a and 1 d f = t d (tan c) 2 Table II Lift Distribution of the Untwisted Rectangular Wing f(B) and f'(4) I f d I __ J__f 0 0.025 .075 .125 .175 .25 .35 .45 .55 .55 .75 .85 .95 CC, 4.49 1.86 1.39 1.24 1.10 .958 .850 .753 .655 .546 .417 .225 0 0.05 .1 .15 .2 .3 .4 .5 .6 .7 .8 .9 1.0 0 0.219 .312 .381 .444 .554 .349 .734 .810 .875 .930 . 971 We then have o t2 Twave II = / 2 tan a J 2B d (tan m) = t _ry 2 'Wwave II = 72 t2 K1 2 0=1 f d 4=o (75) N.A.C.A. Technical Memorandum.No. 897 where K J' f2( ) 0 d *= o Similarly there is obtained for the induced drag in the two triangular regions overlapped by the Mach cones Wi = 2 P / 7 c. d f where from equations (62) and (65) cz 1 o (1 f()) 2 tan a We then have Wi = t o f(t) [1 f(4)] d 2 i t2 Yo2 (K K) (76) 2 For the total drag W = Wwave I + wave II + Wi there is thus obtained from equations (74), (75), and (76) P b t W =  (1 K)A 2 tan a. or from equation (20) W = 2 p uo02 o0 F tan a {l (1 K), I (77) and for the drag coefficient C, = 4 po0 tan a {I (1 K) ?I (78) From equations (72) and (77) there is obtained between the lift and the drag the simple relation W = o A (79) N.A.C.A. Technical Memorandum No. 897 There is thus obtained for the plane surface of finite span the same simple result as for the infinitely long flat plate, namely, that the ratio of the total drag for a frictionless flow to the lift is 0o : 1. This may also be explained by the fact that in contrast to the in compressible flow no suction force arises at the leading edge in the supersonic case and the resultant air force is therefore at right angles to the plate. For the relation between the drag and lift coeffi cients, there is obtained finally from equations (73) and (78) ca2 1 c a 1 cw = (80) 4 tan a 1 (1 K) A 4 tan a 1 0.316 A The above formula has the same structural form as formula (49) for the trapezoidal wing with constant lift distribution. In figure 10 cw/ ca has been plotted 14 tan a against the reduced aspect ratio \ (dotted curve). It may be seen that the rectangular plane wing for the same lift has the same drag as the trapezoidal wing with con stant lift distribution with the reduced taper angle 6 = tan T ta = 0.27. For the reduced aspect ratio ;A = 0.3 the tan a rectangular plane wing has, for the same lift, about 10 percent and for A = 0.5, 19 percent more drag than the ideal trapezoidal wing whose taper angle is greater than the Mach angle. With the above results the theoretical polar and moment curves for the plane rectangular wing may be given for various aspect ratios and Mach numbers. For the mo ment MH about the transverse axis in the wing leading edge, there is obtained MH = 2 P uO2 tan a b t{ 2 (1 K) o_ 2 3 * It is interesting to note that the constant 1 K = 1  /1f(6) d e is equal to 1/T within the computational accu 0 / racy. That this is exactly so has as yet not been shown. For this it would be necessary to know the exact solution of the integral equation (67). N.A.0,A. Technical Memorandum iTo. 897 MH and for the moment coefficient Cm = P Cuo <1  mH = 4 Po tan a (1 K) 2 3 (81) 1 0.211 A cH = ca  cH 1 0.316 A 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 for the aspect ratios = 0, and I and for the Mach b 5 2 numbers = 1.2, 1.5, 2.0, and 3.0. The drag differences C between wings with var ious 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 = For the infinitely long b wing, the center of pressure lies at the midchord position and with decreasing aspect ratio it moves forward somewhat (table III). Table III t tan a 0 1/5 /2 cmH 1 0.489 0.469 Ca N.A.C.A. Technical.Memorandum No. 897 Formula (80) for the rectangular flat plate is the analogy to the familiar Cwi = cy2 F/n b2 of the incom pressible flow. Like the latter it enables the recompu tation of the drag from one reduced aspect ratio t tan a, S =  1 t2 tan a2 to another A = .From equa tions (73) and (80) there is obtained for the new angle of attack and the drag a = + Ca tan a (10.316 2) = caa 1+ 1C = 4c + (0.316 tan a (10.316X )  1 tan a (10.316Nx) S (81a) 1 tan a (l0.316XY'j .1 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 be given by P(r~') 1 n'  for n < n' I 1 o I ~ r(r') = r0 for (82)  1 11' < TIi  ~1 1 _< + :h where bn = b'/b, 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 n = r' with circulation Fo N.A...A. Technical Memorandum 17o..897 . (.71 ')[_ aK2( .)( + 2 2} ) .(rb (t . V (AA_ Ka (83) From the above there is obtained by superposition the downwash velocity cz for variable circulation Fr'): 1 . c' o (84) C2 (n,)) d Tn d i For the trapezoidal lift distribution according to equation (82) we have therefore if, on account of symmetry, we restrict ourselves to the halfwing y > 0 1 1 01 J TI cz('r') d 'n' or., according to equation (83) 2l1 (i i')){g K2[(T'i)2 +4 2t2]}d d' rb cz 1 or with K2 (I r)2 + = T b cz 1/2 d/ 1/2 1/2 /i dT T  Performing the integration there is obtained for points ., r, p within the Mach cone at the wing tip T = 1 be 1 1 +  = + log ro 1 i t, K2 t 2 (85) N.A.C.A. Technical Memorandum No. 897 and a corresponding expression with reversed sign for the Mach cone at Ti = TI. The value of w is here given by equation (12). In the cone T = 1, cz > 0 so that there is upwasi velocity. In the cone n = 1 there is a downwash velocity of the same absolute magnitude (c < 0) and outside of the two cones c. = 0, a result which is also to be expected from reasons of symmetry since, on ac dr count of = const., all separating vortices are of d r' the same strength. With K(T 1) K(TI T) =a nd = there is obtained for the downwash distribution in cone I and III respectively in the plane z = 0. b cz 1 2 1 1+ 1TTT S = + log   (86) o 1 2 1 J ~ On the cone surface according to equations (85) and (86) c, = 0 ani is thcrefore continuous in passing through the cone. On the cone axis c now becomes log arithmically infinite, whereas with the rectangular lift distribution (horseshoe vortex) cz becomes infinite on the axis as r1. The logarithmic singularity of cg 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 1 o 1 1 n ga (87) b c 1 fT T arc tan ro 1 T t(I 1) N.A.C.A. Technical Memorandum.No; 897 and corresponding expressions with reversedsigns for the cone.at T = TI For the arc tan there is to be taken the principal value 0 < arc tan < Tr. For the outer cone  (n = 1) the arc tan is zero in the upper half plane on the outer quadrants of the cone surface and equal to +rT on the inner quadrants. In the wedgeshaped space be tween the two cones cy is constant, being equal to <0: Cy= 2 ; Cx = 0 (88) b b' In passing through the plane 0 = 0, therefore there is a discontinuous increment in cy by Po 2  The region of the t plane limited by the cone b b'b b axes T = I and In = 1 distance  is thus a vor \ 2 tex surface with constant circulation density the total.circula tion of which is equal to the circulation F 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 cy and cz are equal to zero but the directions of the streamlines c/Cy 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 zoidshaped 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 = 0 to x = t. The chordwise circulation distribution is assumed to be constant of density F/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 N.A.C.A. Technical Memorandum No. 897 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. Te likewise need carry out the computation only for the cone at T = 1; for the downwash in zone T = T there is obtained the corresponding expression with re versed sizn. For tnc induced z component cz of the lifting sur face, thErc is found, according to equation (85), with 1 2 1 / ( ,>A(i ')2 [(v, l)2 + 2] 1 1 / ,)2 L  +  log   = i_ t w i: er e +i' = ? + J 1) + according, to equation (31). With the new integration var iables = *, the above equation becomes 12 2 + l2 2 o 1 ./ E*2 K 2e ^=0 1/2 1 + K2 lo (t* + .... ) d i r 1 = / 1/2 + / log (2* ....) 2 * 1 V.=j where 46 N,A.C.A. Technical MemornLdum.No. 897 ,Th =Kth ai) + ol . The three integrals are evaluated as follows: Setting *. _ K = 7, we have for J 1/2 / T d.( 1). j . d 7 J =  K ( 1) arc tan 1 ' ~(~l) With ... ( 1)2+ 2 = ais and  S* +4*Kala = there is obtained for J2 J 4 1 j1 T, 2 a 1 log 4 1 T2 Td T After a brief intermediate computation we have Ja =  l ( [) a] log O1 ) + ) log (+ Wa) + .}) and similarly S= + 1/2 ')a + s3 log ( 1 /j7 1) + 2 t log Q +) + T.A.C.A. Technical Memorandum No. 897 By adding we obtain 2 In ~1u + <( 1) arc tan o i n K(n l) 1 L U } + loo (89) 2  A corresponding expression with opposite sign is obtained for the cone T = n The arc tan in equation (39) lies within the range < arc tan < + as follows from the 2 2 fact that must be symmetrical in ( 1) since the same holds for c, according to equation (85). The induced z component thus found for the rectangu lar liftin? surface with trapezoidal lift distribution has the same singularities as the correspondin; formula (85) for the lifting line. On the cone surface c 0 ani on the cone axis logarithmically infinite. For the downwash distribution at the location of the wing in the plane G = 0, there is obtained co / r 1777 2 = 2 + b arc tan  + + log + ( 1 1  K(, 1) K(rV ) were = for coed I and = for cone III, the upper sizn 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 to obtain the field of the total downwash motion, there is still to be addel the induced downwash velocity due to the clanc wave. In the wedgeshaped space between the leading and forward edges of the wing (fig. 4), this induced veloc ity component is N.A.C.A. .Technical Memorandum.No. 897 Kr 1 r I/= = Z a 0 1/2 0 wave 2 t 2 1 ~I 1 For the total downwash velocity in the plane is thus obtained from equations (90) and (91) z.= 0, there For cone I:. 1 < ?3 < 0: S< < + 1: For cone III: 1< < 0: 0 < 7 < +1: 2(1 Tl) g () 0 WCzo = + ( Yo TT7 g ( Tr 2(1 , Ti)  Yo g (4)  ={ 1  = l Co) The downwash distribution'thus computed is plotted figure 19. Weare 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 = I does not extend beyond the wing tip and does not overlap the region of dropping circulation of'the other halfwing. 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 b b' tan a <  and 2t ab ' tan a <t t The induced drag of one halfwing lWi is composed ad ditively of the drag of half of.cone I, .Wi I1 and the (92) in (91) N.A.C.A. Technical Memorandum No. 897 drags of the two halfcones of cone III, Ii. 2 (fig. 17). lvi 'III, and i i = wi + wIII + WiI (93) Si:ce in cone I, in the plane = 0, there is upwash velocity, '7i gives a forward thrust which in absolute ii value, however, is smaller than the back thrust in cone III, since the circulation is greater here. We have x=t y=b/2 Wl = P / d x 1 ./ J7 X=0  Czo a y, ( = x tan t Y=Y, where co is known from equation (90). We thus have 0 =t/b 1 P Fob Wil / S. pn s T P t2 K =o In cone I for 1< 1 < 0: 1  and therefore l , 1 P P 2 b \2 (1 ) 8 K t =0o / d r() g () (54) i,'= i K 1 1 b 0. Ho 3=0 6=1 For briefness we set 1 j g (o) as = jg (C) d 6 = K1 0=i i0=o (95a) N.A.CA. Technical Memorandum No. 897 o +1 g ( d) d = g (0) = K2 4=_ 4='o These integrals may be exactly computed. There is obtained 3 S=8 8 7 Kg =  18 so that finally p r t Wi = 2 I (1 ,) Kb The portion WiII substituting g(b) that WIIl1 is obtained from equation (94) by for g(4) and taking r = F so 1 K Pt (98) =  n  (98) 3 1 . Finally, Wi ITin T K b is obtained by putting in equation (94) r = o 11 ) S o I  and substituting g(6) for g()). By comparison with equations (97) and (98) this gives i i + wi IIIa il llli and therefore Wi 1 S Wi = a Wii + 2 Wi 2 { + iII1 (95b) (96) (97) N.A.C.A. Technical Memorandum No. 897 Substituting the values from (97) and (98) the induced drag of the entire wing is found to be 4 K3 P 10o' t tan a 1 3 K2 t tana ( Wi  )(99) 3 1 'n b 1 2 K b 1 1 3 If, in place of Fo, there is now substituted the lift A of the entire wing A = P b ro u0 1 2 we have 16 E3 1 1 / A t tan a 3 n (1 ri) ( + Tr)2 p b uo b 1 3 K, t tan a 1 T K3 b 2 1 ( A )2 t tan a (1 )( + r2 p u b b S 1 14 t tan ac 1 l 9 TT b 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 I is 2(2 + 'r) b 1 AuA 'Twav e b (101) 3(1 + .1) 2 t tan a P o If c, denotes the coefficient of the wave plus induced drag then from equations (100) and (101) Sca2 4(2+ 1) 4 hX 1 14 Cw + 1 (102) 4 tan a 3(1+T' ) (ln )(l+r ) 1_ti 9 Tv N.A.C.A .Technical. Memorandum No. 897 The above 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 X the induced. portion of the drag is proportional to .a whereas for.the other two cases it is proportional to X. In figure 20 the coefficient c 2 cx/a i i.s plotted against the red.uc.ed aspect ratio 4 tan a t tan a = X for various trapezoid shapes b'/b. It may b 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 AX = 0.5 and b'/b ="1/2. 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 manner 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 fldw, 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 52 ... N.A.C.A. Technical Memorandum No. 897 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 problem" there is computer 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 number and may be 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 che 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. N.A.G.A. Technical Memorandum No. 897 REFERENCES 1. Prandtl, L.: Theorie des Flugzeugtragflugels im zusam mendrickbaren Medium. Luftfahrtforschung, vol. 13, no. 10, Oct: 12, 1936, pp. 31319. Prandtl, L.: General Considerations on the Flow of Compressible Fluids. T.M. No. 805, N.A.C.A., 1936. 2. Prand'l, L.: Tragflugeltheorie, 1. u. 2. Mitteilung. Nachr. von der Kgl. Gesellschaft der Wissenschaften. Math. Phys. Klasse (1918) S. 451 u. (1919) S. 107. Wider abgedruckt in Vier Ahhandlungen zur Hydrody namik und Aerodynamik. Gbttingen 1927. 3. Ackeret, J.: Air Forces on Airfoils Moving Faster, than Sound. T.M. No. 317, N.A.C.A. 1925. 4. Busemann, A.: Aerodynamischer Auftrieb bei Uberschall geschwindigkeit. Luftfahrtforschung, vol. 12, no. 6, Oct. 3, 1935, pp. 21020. 5. Betz, A.: Beitrage zur TragflUgeltheorie mit besonderer Bericksichtigung des einfachen rechteckigen Fligels. MWnchen 1919. N.A.C.A. Technical Memorandum No. 897 Figure 3. Lifting line with constant lift distribution. Downwash distrib ution in Mach cone. Potential of the lifting line. i I I I Figure 2. Lifting line with constant lift distribution (horshee vortex). Streamline picture of the y and a velocities in a plane at right angles to the axis of the Mach cone. fl I I 4 1" ,, Figu i / i I " Figure 5. Rectangular dist wing as lifting the surface with constant lift x< t distribution, for Wave of rarefaction Compression I  ,shock I I S' NWave of Compression shock rarefaction Figure 4. Plane sound waves at a flat plate. 1 s I P) I  re 6. Rectangular wing as lifting surface with constant lift ribution. Downwash distribution in wing plane. Continuous curves for (at location of wing) dotted curves x = 2t (behind the wing). Figure 1. Figs. 1,2,3,4,5,6 N.A.C.A. Technical Memorandum No. 897 I / S > '', ,", / \ / / Figure 7. Rectangular wing as Figure 8. Rectangular wing as lifting surface with lifting surface with constant lift distribution, constant lift distribution. Streamline picture of the y and 5 Streamline picture of the y and t velocities in a plane x of the Mach cone.   I I I Figure 9. Trapezoidal wing with constant lift , distribution. Figure 10. Trapesoidal wing with constant lift distribution. Coefficients of the wave plus induced drag c / a as a function of the "Oiduce aspect ratio" X = t tand/b for various trapesoid shapes 0 s4;na/tand. Figs. 7,8,9,10 N.A.C.A. Technical Memorandum No. 897 b _ Figure 11. Trapezoidal wing with constant lift distribution. Induced downwash velocity in section AB (in direction of flow) (tanT" 1/13; tan c( = 3). Figure 12. Trapezoidal wing with constant lift distribution. Induced downwash velocity in section CD (at right angles to flow direction) (tanr= 1/V3; tandV). / " .."  Figure 13. Trapezoidal wing with constant lift distribution. Profile sections. (tancf = : tan T= 1/t/). Figure 14. Trapezoidal wing with constant lift distribution. Elevation contour lines. (tand = 'FV tan = 1/V3). Figs. 11,12,13,14 N.A.C.A. Technical Memorandum No. 897 /' / r / \ Figure 17. Rectangular surface with trapezoidal lift distribution. Figure 16. Polars of plane rectangular wing for various aspect ratios. / 7fS '1 / \"  ) 1 / //// Figure 18. Lifting line with trapezoidal lift distribution. Streamline picture of the y and s velocities in a plane at right angles to the axis of the Mach cone. Figs. 15,16,17,18 N.A.C.A. Technical Memorandum No. 897 Figure 19. Rectangular wing as lifting surface with trapezoidal lift distribution. Downwash distribution for x< t. A  I dri J ~u I Figure 20. Lifting surface with trapezoidal lift distribution. Coefficient of wave plus induced drag eCw/t a s a function of the "reduced aspect ratio" A for various values of b'/b. Figs. 19,20 UNIVERSITY OF FLORIDA 3 1262 08106 286 0III 3 1262 08106 286 0 
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