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NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMOjRANDUM 1383 FINITE SPAN WIlfGS IN COMPRE~SSIRLE FLOW* By E. A. Krasilshchikova This work is devoted to the study of the perturbations of an airstream by the motion of a slender wing at supersonic speeds. A survey of the work related to the theory of the compressible flow around slender bodies was given in reference 14 by F. I. Frank and E. A. Karpovich. The first works in this direction were those of L. Prandtl (ref. 4) and J. Ackeret (ref. 23) in which the simple problem of the steady motion of an infinite span wing was studied. Borbely (ref. 25) considered the twodimensional problem of the harmonicallyojscillating norndeformable wing in supersonic flow by using integrals of special types for solutions. Schlichting (ref. 24) considered the particular problem of the flow over twodimensional rectangular and trapezoidal wings. To solve this problem, he applied Prandtl's method of the acceleration potential which he looked for in the form of a potential of a double layer. However, as shown later, Schlichting made an error and arrived at an incorrect result. In 194), Busemann (ref. 26) proposed the method of solving the prob lem of the conical flow over a body by starting from the homogeneous solution of the wave equation. This method was modified by M. I. Gurevich who, in references 11 and 12, solved a series of problems for arrowshaped and triangular wings when the flow, perturbed by the wing motion, is conical. The work of E. A. Karpovich and F. I. Frankl (ref. 15) was devoted entirely to the problem of the suction forces of arrowshaped wings. In 1942, at a hydrodynamics seminar in Moscow University, Prof. L. I. Sedov proposed the problem of the supersonic flow over slender wings of finite span of arbitrary plan form. In response to this proposal of L. I. Sedov, there appeared in 194647 a series of works by Soviet authors on the question of the super sonic flow over wings of finite span. The first work in this direction was our candidate's dissertation (ref. 5), in which we found the effective solution for a limited class ~Scientific Records of the Moscow State University, Vol. 154, Mechanics No. 4, 1951, PP. 181239. The appendix represents a condensation made by the translator from a document I'Modern Problems of Mechanics," Covt. Pub. House of Tech. Theor. Literature, (Moscow, Leningrad) 1952, pp. 94112. NACA TM 1585 of harmonicallyoscillating wings. In reference 6 we solved the problem for wing influences by "tip effect." Later works refss. 15, 16, and 17) were devoted to the same problem. In reference 6, using an idea of L. I. Bedov as a basis, we reduced the problem of the influence of the tip effect on harmonicallyoscillating wings to an integral equation. The question of the flow over wings of finite span remained open for some time. At the start of 1947, there appeared works in which different methods were proposed for solving the tip effect problem which would be applicable to any particular wing plan forms. In reference 18, M. D. Khaskind and S. V. Falkovich solved the problem, in the form of a series of special functions, for a hannonically oscillating triangular wing. Later, M. I. Gurevich generalized this method (ref. 19). In reference 20, L. A. Galin reduced the problem of determining the velocity potential of an oscillating wing to the problem of finding the steadymotion velocity potential and gave a solution, in series, for the velocity potential of a rectangular, oscillating wing cambered in the direction of the oncoming stream. The methods, proposed by different authors, for solving the problem of the flow over wings of finite span do not permit the solution of the problem for any finitespan wing and may only be applied to a limited class of wings. Parallel developments in this direction were made by the foreign authors Puckett (ref. 21) and Von Karmin (ref. 22) who solved the problem of the steady flow over finitespan, symmetricall wings at zero angle of attack. As is known, such wings produce no "tip effect" and the study of the perturbation of the airstream by their motion presents no mathe matical difficulties . In references 6, 7, and 8 we proposed a method of solving the finite span wing problem by constructing and solving an integral equation which considered the wing plan form in both steady motion and oscillating harmonically. In reference 9 we generalized the problem to more general forms of unsteady wing motion by the method of retarded source potentials. Introducing characteristic coordinates we solved the integral equa tion for wings of arbitrary plan form and represented the solution for steady wingmotion in quadratures and for the harmonicallyoscillating wing in a power series of the parameter defining the oscillation frequency. The present work is a detailed explanation and further development of ur apes (efs 6 o 9 whchwere published in the Doklady, Akad. Nauk, USSR. In this work we propose an effective method of solving aerodynamic problems of slender wings in supersonic flow. All the results and problems explained in this paper were reported by the author in 194748 to the USSR Mechanics Institute, V. A. Steklov Mathematics Institute, Moscow University, etc. In the first part of the work we find a class of solutions of the wave equation, starting from which we obtain the solution to the problem of determining the velocity potential of some wing plan form in unsteady deforming motion. The obtained solution contains the solution of the twodimensional problem as a special case. In the same part of the work, we solve in quadratures the problem of steady supersonic flow over a wing of arbitrary surface and plan form. The effective solution for wings of small span is similarly given. We obtain formulas determining the pressure on the wing surface in the form of contour integrals and integrals over the wing surface. The author thanks L. I. Sedov for reading the manuscript. PART Il 1. SETTING UP THE PROBLEM 1. Let us consider the motion of a thin slightly cambered wing at a small angle of attack. We will consider the basic motion of the wing to consist of an advancing, rectilinear motion at the constant supersonic speed u. Let be superposed on the basic motion, a small additional unsteady motion in which the wing surface may be deformed. Let us take the system of rectangular rectilinear coordinates Oxyz moving forward with the fundamental wing velocity u. The Oxaxis is directed opposite to the wing motion and we take the x,yplane such that the z coordinates of points on the wing shall be small (figs. I and 2). We will consider the normal velocity component on both sides of the wing surface to be given by vn = AO + Alf~t + al (1.1) LResults of Part I, sections 6 and 7 were found by the author in May, 1947 at the Mathematics Institute, Akad. Nauk, USSR. NACA TM 138j NACA TM 1585 The first component defines the wing surface (1.2) AO = up0 where B0 is the angle of attack of a wing element. The second compo nent defines the additional unsteady motion of the wing. The functions AO and AL and a are considered given at each point of the wing surface. We will assume that the fluid motion is irrotational and that there are no external forces. The velocity potential of the perturbed stream cp(x,y,z,t) is represented in the form q(x,y,z,t) = c90(x,yiz) + 91(xiyiz,t) (1.5) where the potential (p0 corresponds to the basic steady motion of the win and the potential cl corresponds to the additional unsteady motion. Thus the projections of the velocity v of the fluid particles on the moving Oxyz coordinates are determined by "(0 1 Vy  + , dx ox ) VY = + , by ey z z b The functions c0 firstorder quantities With these assumptions and q? and their derivatives will be considered and secondorder quantities will be neglected. it is known that the potential (l satisfies the wave equation which in the moving axes is a291 ax2 + ,2a ~ y2 2 a429 + a az2 4291 at2 a2r1 S2u = 0 atdx (a' u2) (1.4) and the potential cp0 satisfies (,2 2) a2rp0 ax2 4 2 a2p0 a2 ,2 42 0 0 dz2 (1.5) where a is the speed of sound in the undisturbed stream. A vortex surface, called the vortex sheet, trails from the side of the wing surface opposite to its motf~on. Just as on the wing surface the velocity potential undergoes a jump discontinuity on this sheet. NACA T1M 138) We represent the projection of the vortex sheet on the x,yplane as the semiinfinite strip Z1 (fig. 1) extending along the~ xaxis to infinity from the trailing edge of the wing. Let us establish the boundary conditions which the functions c90 and 91 satisfy. Let us transfer the boundary conditions on the ving surface parallel to the zaxis onto the projection E of the wing on. thie x,yplane, which is equivalent to neglecting secondorder quantities in comarison with firstorder ones. Therefore on the basis of equation (1.1) we~ obtain the streamline condition = AO(x~y), dz d1=A (x'y)f t + a(x,y) dz (1.6) which must be fulfilled on both the upper and lower sides of C. The kinematic condition, which expresses the continuity of the normal velocity components of the fluid particles, must be fufilled on the dis continuous surface of the velocity potential and on the vortex sheet. We transfer the condition on the vortex sheet parallel to the zaxcis onto its projection ZL on the x,yplane which is again neglecting second order quantities. Therefore we have the conditions (1.7) to be fulfilled on El" Furthermore, the dynamic condition which the potentials cp0 and qg satisfy must be fulfilled on the vortex sheet. Since the pressure remains continuous on crossing from one side of the vortex sheet to the other, then from the Lagrange integral P = u 12 2 390 90  =  Oz z=+0 dz z=0 91 91  =  bz z=+0 SE z=0 2~(, IPSE MACA TM 138J Keeping equation (1.5) in mind and neglecting secondorder quantities, we obtain zt ,=0 1 +Oa u + u (1.8) which must also be fulfilled on C1 After boundary conditions (1.6~) and (1.7) are established, we correctly consider that, to the same degree of approximation, the surface of discontinuity of the velocity potential the vortex surface lies entirely within the x,yplane. Therefore, the functions 9O and cq are odd functions in z (P0(x,yIz) = 0(Fx'y~z), 91(x'y,zrt) = (Pl(x',yz,t) (1.9) Combining equations (1.8) and (1.9) we conclude that the functions CP0 and cp1 satisfy the respective conditions =0, + u on El (1.10) ox it ax Since th motion of the wing is supersonic, the medium is disturbed only in the region bounded by the respective disturbance waves represent able by a surface enveloping the characteristic cones with vertices at points of the wing contour. Ahead of this surface in front of the wing  the medium is at rest, therefore, the velocity potential is a constant whih we assume~ to be zero. Hence we have the condition on the disturb ance wave D(POxIy~Z) = O, rp1(xiyiz,t) = 0 (1.11) Th potentials g0 and (l are continuous functions everywhere outside the tw dimensional region C + El and, as was established, are odd in z, therefore, in the whole x,yplane outside of the region E + 51 where the medium is perturbed, the following conditions are satisfied: PO(x'y,0) = 0, 9 (x',y,,t) = 0 (1.12) The region where equation (1.12) is satisfied is denoted in figure 1 by C2 and C~ Thus the considered hydrodynamic problem is reduced to the following two boundary problems: I. To find the function cPl(x,y,z,t) which satisfies equation (1.4) and boundary conditions (1.6), (1.10), (1.11), and (1.12). II. To find the function rp0(x,y,z) which satisfies equation (1.5) and boundary conditions (1.6~), (1.10), (1.11), and (1.12). Since the functions cp0 and 91are antisymmetric functions rela tive to the z = 0 plane, it is sufficient to solve the problem for the upper half plane. From the solution of boun~dary problem I it is possible to obtain the solution of II if the function f in the first be considered a constant equal to unity, and AO replaces Al* 2. VELOCITY POTENTIAL OF A MOVING SOURCE WITH VARIABLE INTENSITY 1. Let us construct a solution of equation (1.4) as the retarded potential of a source moving in a straight line with the constant velocity u and having an intensity which varies with time according to fl i)* Let us consider the infinite line along which, at each point from Left to right, sources with velocity u start to function one after the other with the variable intensity q = fO~ tl)fl(t). The law of variation of the function f0 is the same for all the sources if the initial moment of each source is considered to be the moment when it came into being.2 The function fl has the same value for all the sources at each instant. Let a source at an arbitrary point of the O'x'axis be acting at time tl fg ) The retarded potential of the velocity at the point M as a result of such a system of sources is represented in the fixed coordinates by 01 7 >y'z',t) = A r dtl r = \(x' + ut)2 + ,2 + z.2 (2.1) 2Prandt1 (ref. j) considered an analogous problem with q = fO L t13 MAC.A TM 158) 8 NACA TM 1583 where A is a constant with the dimensions of a velocity. The limits of integration tl' and tl" take into account those sources which affect M at time t. The origin of the fixed coordinates O' is placed at the point at which the source started at t = 0. Introducing the new variable of integration T = a(t tl) r and transforming to the coordinate system x = x' + ut, y = y', z = z' which is moving forward in a straight line with the velocity u, we transform equation (2.1) into If it is assumed that u > a then the velocity potential at M(x,y,z) is the sum of the expressions (2.2), with the minus sign in front of the radical taking into account the effect of the sources in the strip AC on M a~nd with the plus sign taking into account the sources on CB. The smaller root of the radicand is taken as the upper limit of integration 7l. It is easy to see that in this case both roots are real, positive quantities (fig. 5). On the basis of expression (2.2) we now construct a velocity potential at M from the sources moving with speed u > a which have an intensity .which varies with time as fl(t). The derivation remains valid if the additive constant al is added to the argument t of the function fl* Putting the sources at the origin, we find the velocity potential from equation (2.2) bDy considering the interval of integration from 0 to rL to be vanishingly small. Then, neglecting the term ( I) and putting A o,1. 0 &7 = C where C is a constant, we obtain the desired solu tion for equation (1.4) in the general form 1 1 "u2 uxa,2 u2 a2 x2_u1)y 2 9 ~(x,y,z,t) = C I x2212 422 flt a ux a x2 u2a 1) y2 7 .iit a 'u2' a2 u2 a2 g (2.5) x2 (a 1 y2 + z2 MACA TM 1585 Let us note that each component of the arbitrary function fl as well as the constant C and all in equation (2.3) is separately also a solution of equation (1.4). In equation (2.)) putting al = 0 and the velocity of motion of the source u = O, we arrive at the wellknown solution for a spherical wave. If the velocity of motion of the source is u < a then to obtain the retarded potential of a moving source the right side of equation (2.3) must be limited to the first component. Considering the function fl in equation (2.5) to be constant, we arrive at the Prandt1 (ref. 3) solution for the retarded potential of a moving source of constant intensity Cl 2. It is possible to obtain, by the same method, the velocity potential of a source with the variable intensity fl(t) moving arbitrarily. For example, in the case of rectilinear motion of the source when the motion is given by X = Fl(t), Y = O, Z = 0an whn a at that is, the motion of the source is supersonic, the velocity potential of the source at the origin of a coordinate system moving with the source is [x + F ( t) F~llt * + y2 + z2 [x + F1(Z 1 Flt citldt1 a~t tl) lx+ Fl(t) Fl t1)2+ y2+ 2 = (2.5) 10 MACA TM 1585 If dbFl(t)/dtl < a, i.e., the source velocity is subsonic, then to obtain th~e velocity potential one must be limited to the one component in equa tion (2.4) which corresponds to the smaller of the values of the parameters tl and ty*. Th function expressed by equation (2.4) satisfies the linear equa tion with variable coefficients 2 2 dFt a2p 2 dt axat 42F,( ) b d2 ax ax2 + a2 I a2 +a2 a ,2 (2.6) If the source moves with constant acceleration as Fl(t) = ut bt2 (where b is a constant) then equation (2.0) is an algebraic equation of the fourth degree in ti with two real roots. Formula (2.4) contains the LienardWeigert (ref. 27) formula as a special case when the source intensity is constant. j. DERIVATION OF TEE BASIC VELOCITY POTENTIAL FORMULA 1. We apply a solution of the form (2.3) of the wave equation (1.4) to the abovementioned boundary problem I. At each point of the x,yplane let us place sources with the poten tial 9*. Hence, we will consider C and al in equation (2.3) functions of points of the x,yplane and we will replace al by a and fl by f. As a consequence of the linearity of equation (1.4), its solution is a flunction cpl expressed by u(x CI h X ~12 kC(3 91 IC.2 f ;t c aiSr'll 111_ ~2 UI _.1 I U~S~i x drpC + plix,~,zri I / c(x.~;il f t+ ((n) uBx h1 V x )2 k2(y I 2.,2; 1 ' C(S,91 x uB 2 u g n( (j.1) where k = 1. NACA TM 158) The region of integration S(x,y,z) is that part of the x,yplane which lies within the characteristic forecone of equation (1.4) from the point with coordinates x,y,z (fig. 4). The solution of equation (3.1) will give the velocity potential arising from the additional motion of the wing if C(x,y) is determined from the boundary conditions of the problem on the x,yplane. Let us introduce the new variable of integration 9 into equa tion (3.1) in place of (x F)2 k~z2cos 8 (3.5) 9 y  Then equation (j.1) becomes 9 x  1 x 2 k2z2 cos pl(xiyIz~t) = sin 6 ddj + (x )2 k2z u(x 5) .2 _2 S(x,yIz) a .2 2 cos 9j X c scy  (x I)2 k~z2 cos 9  u2X a2 u 2 a2 V'~aa (3.4) ~~(;C1 iy f~~~~ t x[) `2cs f. t+ a ,y  NACA TM: 1585 Let us note that for any point M(x,y,z) to isolate from the region S(x,ys,z) a region able of integration has the limits of space it is possible S' in which the vari x kz 6 5 1 C', O 4 $ ,1 = ( )2k22 9 y x ) kz where C' remaining on z or is a constant satisfying the inequality C' < x region S S' the limits of integration either depend on z only in the combination kz2. kz. In the do not depend Differentiating equation (3.4) with respect to a we find the rela tion between C(x,y) and a(x,y) and the normal derivative of the velocity potential 891 d~z at any point of the x,yplane (35) Corpa~ring equation (3.5) with equation (1.0) we conclude that on the wing C(x,y) =1 Al(x~y) (3.6) i.e., the function C(x,y) is given. Therefore, the velocity potential pl may be computed from equa tion (3.1) by taking equation (3.6) into account for those points M(x,y,z) of space for which the region of integration S(x,y,z) does not extend beyond the limits of the wing. If the leading and trailing edges of the wing are given by x = Jr(y) and x= X1(y'), respectively, and if, therefore, JI and X1 satisfy C(x,y) =R ft +afxiy  z=0 NACA TM 1383 (3.7) (3.8) (where a* is the semivertex angle of the characteristic cone) on the leading and trailing edges of the wing, respectively, then in particular, equation (5.1) yields the effective solution of the problem of finding the velocity potential pl everywhere on the wing surface because in this case the region of integration S does not extend beyond the wing for any point M(x,y,0) on it (fig. 5). Also, in particular, equation (5.1) gives a solution of the plane problems if C and a are considered as functions of one variable  C = C(x) and a = a(x) and the variables of integration in the region S are considered to vary between 0 [ x km 12 = Y 1 x ()2 k2z2 < Y+ 1 (xg2 k,2 1'I (39) where ql and 12 are as defined previously. Considering f in equation (3.1) a constant equation (3.5), we obtain the fundamental formula tial cp0 specified by the basic steady motion of and taking into account for the velocity poten thre wing PO(x,y,Z) =  (3.10) Formula (3.10) contains, as special cases, the results of Prandt1 (ref. 3), Ackeret (ref. 23), Schlichting (ref. 4) when thne wing surface is a plane and when the leading edge is a straight line perpendicular to the free stream. <1 cot a* dy d;il 7) < cot 2* dy S(x,y,1) z0 x )2 22 kz n d(g 2 k2 2 2z2 14 NACA TM 1383 4. HAIRMONIC OSCILLATIONS OF A WING 1. Let us turn. to the case when the additional motions of the wing are harmonic oscillations, i.e., on the wing equation (1.6) is given as acq,1, i a~),Ut + "(xtuy =RP. lxye = R.P. A2(x,y)e (4.1) where A2(x,y) defines the amplitude and initial phase of the oscillations. Using the obvious relation ei9 + ei8 = 2 cos 6 and equation (55), the basic formla for the velocity potential (3.1) is represented as a lcosl x(2 k( k2_r o2 kz (q(x,y,s,t) = epx S bz s~3 dqde (4.2) where u2 a2 u2 a2 Keeping the second inequality of equation (3.9) in mind, let us compute the inner integral after which we obtain a solution of the prob lan for a wing of infinite span cl(x,z,t) = epx xkz z=0 e4 I~'O~ A L)2 k2z2 d where IO is the Bessel function of zero order. By means of eqaion. (4.5) the velocity potential may be computed at those points of the x,zplane for which the interval of integration on the Oxeaxis does not extend beyond the wing, i.e., at those points of the NACA TM 1858 x,zplane not affected by the vertices trailing from the wing because the function 01is given onl on the wing. In. order to copue the oz velocity potential at any point of the x,zplane by equation (4.3) it is necessary to determine 1, using eqution. (1.8), everywhere on the Oxaxis outside the wing. Let us express, by equation (4.3), the velocity potential 1j for any points lying on the Oxaxis outside the wing, which, according to equation (1.8), equals on the Oxaxis everywhere outside the wing rp (x,t) = R.P. 91(E)ev(x3) where iu and 1 is the abscissa of the trailing edge. Thn we obtain the integral equation 3x z=e IOJ 8)d = k ~lep 3 1z=0*1 ~ ) bzp we solved such an integral equation. The inversion of equation (4.5) is ar,1 z0PX= dF*(x)+ x dg.6 dz dx where F" denotes the right side of equation (4.5)J the know function, and where Il is the Bessel function of first order. Therefore, keeping equation (4.6) in mind, we can calculate the velocity potential at any point of the x,z;plane by equation (4.3). NACA TM 138) The problem considered in this section was solved and explained in refe~rence 5 from another Ipoint of view. 5. INLUN~CE OF THE TIP EEFECrT 1. To calculate the velocity potential according to equation (3.1) and also through equation (3.10) or (4.2) for those points M(x,y,z) of space for which the :region of integration S extends outside the limits of the wing surface, it is necessary to determine the normal velocity component everywhere in the region of integration Sfo h boundry conditions of the problem on the z = O plane. Let us consider the~ case when the region of integration S inter sects the wing surface and the region E3 lying outside the wing and outside the region of the vortex system from the wing. Region ZE (fig. 6) is part of the region 12 defined above. That is, let us con sider the case when the wing tips the arcs ED and E'D' of the wing contour act on the point M(x,y,z) or so to speak, the influence of the "tip effect" and not the influence of the vortex sheet trailing from the wing surface. The point E on th leading edge is defined so that condition (3.7) is fulfilled to its left and violated to its right. The point E' is similarly defined. The points D and D' are, respectively, the right most and leftmost points on the wing contour as shown in figure 6. Let us conzstruct the integral equation for C(x,y), connected to1 by relation (j.5), in 2 . Let us select the vel~ocity potential rl at any point N(x,y,0) lying in C3 by means of equation (3.1), equal to zero everywhere in 2 according to equation (1.12). The region of integration S(x,y,0) is divided into two parts, as shown in figure 7; the region s (x,y) is that pat of the wing falling in the Mach forecone from N(x,y,0), and the region o(x,y) is that part of Cj lying in the same forecone. According to equation (3.6) C(x,y) is given La s. In aC(x,y) is unknown. We therefore arrive at the integral equation which C(x,y) satisfies in r" . NACA TM 1385 a(x,y) C!g,g)K(g,9;x~yyttddqdj = F(x,y,t) (5.1) where the kernel is C~~u + (~l a( u2 a 2I~ l2y)j (x ~2k2 g)2k2_2 K =(E,?;x,ylt) = t+ a(C,r) u(x 9) a (x 2 1) 2 d Lu2 a2 u2 2 a2  and the known function F(x,y;t) = A~,)~~~~~~qj 2~stx,y) If the characteristic coordinates are introduced (5.2) (5.5) xl = x x k(y yO)s Y1 = x x0 + k(y yO), Z1 = kz (where x0 an~d yO may be any numbers) then integral equation (5.1) is simplified and in some cases this integral equation is easily inverted as will be shown below. 6. SOLUTION OF THE INTEGRAL EQUATION FOR A HARNDNICALLY OSCILL~ATING WING 1. If the additional motions of the wing are harmonic oscillations, i.e., the condition on the wing is given in the form of (4.1), then equation (5.1) becomes C06[ \k~TY~dldC = F(x,y) (6.1) IS 9(E,ll) o(x,y) NACA 554 1585 where the function 8(x,y) = . p Laaa hrete nw function is F(x,y) = SSA(S,1)co (x)222d d(.2 (x 5) k2( 2 L s(x y) wher A(xy) = epx.B in s. In order to solve this integral equation we introduce the characteristic coordinates x1, 1, El with origin at "'O" by means of the formula (6.5) X~l' = x y, = x + kyJ El = km In thte ne~w coordinates the variables of integration in a will vary between the limits xE 1 = ~xl'/ I 1 (6* ) where yl = ~(x) is the equation of the wing tip the wing contour in the transformed coordinates, abscissa of E defined in section 5 in these same Equation (6.1) is transformed to  the are ED of and xE is the coordinates (fig. 8). cos A xl 1 1 1)S'~;~T xl 1 1 d(1d = FI(1 xl 1) (6.5) 'xl_ Il eL5 Tl xE il, 1171 cosh A xl 1 71 il) NACA TM 1383 where the furnctionn P(x1 71) e2 61(xlrYl) = and where the known function is S " xldl) FL(xlY1y) = Al 1791) d11 dS1 (c.) s (xlfl) e2 Al= sl=0 Let us note that the normal velocity of the perturbed flow a' /8z1 by is related to abz1 For brevity, the index "1" will be left everywhere from now on. off the independent variable 2. Let us look for a solution of equation (6.5) ia the form of the power series e(x,y;h) = n=0 92n(xry) h2n (~.7) Into both sides of equation (6.5) let us introduce cos A (x )(y rl) = (1)n (x ))ny In 2n (6.8) NACA TM 1585 Keeping the absolute convergence of equations (6.7) and (6.8) in mind, we multiply them term by term with the result e(S,rl)cos[A J(x e)7 ) ] h2n 1(_)nk 1(x 9) (y 4) nk 82k(Sn) k=0 2(n k)! = nw0 (6.9) Substituting equations (6.7), (6.8) and (6.9) into equation (6.5) the latter becomes n=0 1 nk 2dyd k=n h2n k=,0 1 nk 6 k ( ) ( ) 2(naj k)l~x i 1 n 2 ai dS Sn+1 S(1) h2nCG (x) ( 9) LG0 (2n)1 = JlA(5,rl) s(x,y) (6.10) Taking into account the uniform convergence of the sides of equation (6.10) with respect to the variables integrate tern by term series in both 5 and we nk1, 2 a k=n nk xx fX 2n"t (1) Id=0 k=0 2~(n k)l S xE r(~ 2k, (e,r) ~x i) (Y 91 n+1 n 12n (1(2)./ A(i,) [(x i) (y 1)] 2 d t s(x,y) = (6.11) (y 1) N~ACA TM 1385 In equation (6.11) equating coefficients in identical powers of h we obtain the integral equation which the functions 02n(x~y) satisfy x r 2n~g,9) = Fn(x,Y) (6.12) where n1 k F (x,y) = fn(x,y) + f_" f (x,y) k=0 (6.15) where, in its turn, 1 s(x,y) 1 nk+1 nk fk(x,y) = 2(1) x 'u 2k(ll (xE(y1 2y E (6.15) from which the functions fr; are defined for k $ 0 and n > 0. Let us note that the right side En(x,y) of equation (6.12) depends, for 92n,~ on the coefficients 02k but only for k ,,,.,.Tee fore, if we find 90, 82 84,*** 82(n1), then F,(x,y) is a knzown function in the equation which the coefficient 92n in the general term of series equation (6.7) satisfies. For n=0 the right side in equn tion (6.12) F (x,y); = (x,y) = s(x,y) is a knnown function of x and y. A(S,rl) d? (6.,16) Let us solve equation (6.12) for 92n(x,y). 22 NACA TM 1383 The two dimensional tutegral equation (6.12) is equivalent to the two homogeneous integral equations x 92n at = F,(xy) x E (6.17) and rY I 82n(5,'1) dsl JY ?  B~(Sy) (6.18) each of which reduces to an Abel equation. Using the inversion forrmula of the Abel integral equation and observing that for any n functions Fn(xEJy) = 0 hence the solution of equation (6.17) for the function 9 (x,y) is d5 92n(x~y) = 1 Let us turn to equation (6.18). We denote the parameter 5 by x, and again using the inversion formula for the Abel equation and kee ing in mind that according to equation (6.19) the right side 92n[x,3(xl of equation ( 6.18) for y = #(x) is different from zero, the solution of equation (6.18) for 62n is 1 a2n x,9(x)~ 1 92n(x'y) + 82n?(x,4) dB r (6.20) Substituting in equation (6.20) in place of 02n(x,y) its vaue from equation (6.19) we obtain the solution of equation (6.12) in the following form: 92n(xIy) % ~,~j ag + x2 .~? xE ~ FnSt? S,) x i)(y 4 x2 JxE JS(x) NACA TM 1585 Thus, according to equation (6.21), we can evalae successively, the coefficients 90, 82, 84>"** 82k, etc. Formula (6.21) shows that all the coefficients (n=0,1,2,...) for y = f(x), i.e., on the wing tip ED, become infinite as R1/2 where R is the distance of the point (x,y) from ED. Therefore, the velocity of the perturbed stream becomes in~finite as the specified order on the wing tips, approaching from outside the wing. It is possible to represent the inversion (6.21) of (6.12) as drl di a2 92n (x,y)=  lr2 axay  xE (r(x) (6.22) which can be confirmed without difficulty by direct differentiation with respect to the parameter. Therefore, the solutions of integral equation 16.5) are construted in the form of the absolutely convergent series (6.7) for any val~ue of the parameter h. 02n(x,y) are expanded in the series The coefficients 8' (x,y;h)) = 2nxy)An n=0 (6.23) (x~y) e in. I: (fig. 6) lying off the wing to the left, from equations (6.21) or (6.22) by replacing in the latter the function Jr(x) by e2(x) (where y = 92(x) is the equation of the are E'D' of the wing contour  the left wing tip) and interchang~e the role of the coordinates. We find the function 9'(x,y) = ~' z= NACA TM 1585 5. Let cones from points El us consider a El and EI' and EI' are wing of small span. Let the characteristic intersect the wing as shown in figure 9. The defined just as are E and El in section 5. Let us divide the x,yplane where the medium is perturbed into the regions SO, S1, S2, .., Sn>* The regionn Sn istic aftcones from is the Msh~aped region lying: within the character En and E (or within one of them) and outside the characteristic aftcones from En+1 and En+1'. In its turn, we divide the part of the x,yplane lying to the right and left of the wing into the strips al, o2, *.. > n ***"d #1 * ag', .., respectively. The strip on lies within the characteristic aftcone from n. Therefore, an and ag' are the parts of Sn lying respectively to the right and to the left of the wing. Let us return to the fundamental formula for the velocity potential, equation (4.2), which is in the characteristic coordinates e2 e 2 (x g)(y ) z2  In order to comrpute the velocity potential by means of this fornnula in those parts of the space (or, in particular, on the wing surface) for which the region of integration S(x,y,z) intersects the region Sn of the x,ylplane, we must first determine 1 e 2 outside the wing in the strips a 02'.. U, and arl', @2>* *nl * respectively. S(x,y,) z=0 NACA ?TM 1585 (x+y)i 891 Let us denote e 2 az strips by 9, a(2) ,() in the a 0'""" B(n). and in al', CI (121 * by 8', er(2) "n' . Let us construct the integral equation for 6(2)~ Let us express the velocity potential at the point N(x,y,0) in by formula (6.24) which is equal to zero everywhere in the strips l 0r2, an correspondinglyy in al 42 ,.' "n Let us divide the region of integration into the three parts S = s + (T + a1'* as shown in figure 10. The fuinction e 2 =A(x(,y) is given in six,y) on the az (x+y) wing;. In ol'+(x,y) of ol', the function azi e 2 = '(xgy) determined by the solution of equation (6.23). (x+yr) we denote e az by e(2)(x,y). Th~en we arive~ a~t In a(x,y) the integral equation satisfied by a(2) e ")(5,1) cos[x/(x a(y )] d d= F( 2) (X~Y) SS ~(xy> (6;.25) NACA TM 1585 where the limits of integration are bounded by xE = x and ~(5) 6 9 y and the known function F(2) is defined as F 2) (x,Y) = ASi(Z,1) co hIcry ~ ids d5  s(x,yr) / 9(%8)cos dCx ~~ E)y ) aTn at (6.26) We look for the solution of integral equation (6.25) in the form of the power series 9(2)(x,y) =~ 6(2)(,y 2n (6.27) n=0O 2n MIoreover, by reasoning similarly to the preceding section we arrive at an integral equation for the coefficient eg, in the general term of series (6.27) eg ) (tn) dq dE n2)xy) (6.28) where Fn 2(x,y) = Fn(x,y) + f(2k(,) where, in its tur, (6.29) nk+1 f(2)k(x,y) = (1 (9 x n C2(n k) I 2 1 (6.j0) Equation (6.28) differs from equation (6.12) o:Ely in the form of the y(2) fuctiojn on the right side. Taking into account the condition on nh onaJF~I ouino the ounary 2)EIy) = O for any n=0, 1, 2,. thesouino (6.28i) for 6~1 is obtained by using the solution (6j.21) or (6.22) of (6.12) as a final formula if F,(2) replaces Fn in the latter. The C)(y rl) n k NACA TM 1583 function Fn(2)(x,y) depends on the coefficient 92k(2) where k=0, 1, 2 ., 1.Therefore, just as in the previous section, if the 92k(2) for k = 0,1,2, .. ., n1 are already found, then Fn(2) in the right side of (6.28) is a known quantity. Therefore, the functions e0(2), 82(2)1 .., 8 2n(2), may be found successively. Let us note that Fn 2, and therefore the coefficient 92n(2 depends only on the first n + 1 coefficients 80 r, 62' > *, *>2n' of the series expansion of (x~y) e'(x,y) =  e coz in 0J1 Reasoning in the same manner, we may find the values of 6 5), 9 ., 9(N). .. in Ir ag, ..,UN .. (correspondingly 8 (', 64, ., 9,(N), in "lr > 2: M7 Therefore, the velocity potential can be computed by equation (6.24) at every point M(x,y,z) of the space for which the region S(x,y,z) intersects any number of strips aN or UrN All the results hold for the case when the wing tips are not given by one equation y = #(x) but consist of curves given by the equations y = ek(x) k = 1, 2, .,m The same observation applies to the leading edges E'E (or E1E ') of the wing. Therefore, in our problem the wing contour may be piecewise smooth. If the frequency of oscillation a> of the wing be put equal to zero then the coefficients a0, 80 g), **, 80()... coincide with t values of the derivatives as0/az in the strips als Q2, .., aN,. respectively, for the steady motion of a wing when the streamline conldi tion (1.6) on the wing is given in the form = Al(Xry) NACA SS( 1585 7. INFLUENCE OF THIE VORTEX SYSTEM FRON.THE WING FOR A KARMIONIC.MGWf OSCILATTIVG WING 1. Let us consider the case wh~en the region of integration S(x,y,s) in formula (4.2~) for the velocity potential intersects the vortex sheet I1 as shown in figure 26(a) (see also fig. 11). That is, let us consider the case when ,he trailing edge of the wing the are DTr of the wing contour or, so to speak., the vortex sheet, acts on the point M(x,y,z) of space. Using condition (1.10) we determine b~l/az in the region R of thie x,yplane and shown in figure 11. Thle region 0I is off the wing within the characteristic aftcone from D and outside the characteristic cones from T. Therefore, R is affected by the vortices trailing from the edge DTof the wing but not :from Dr'T'. The region :i partially intersects the vortex sheet C1' Let us return to the characteristic coordinates xl, y1, El which we introduced earlier by formula (6.5). As before, for brevity we omit the subscript 1 from the independent variables . Condition (1.10) fulfilled on in the characteristic coordinates '1 is 319 + ~ u +u 1 (7 .1) at ax ay From equation (7.1) it follows that the function .m x+y 9.m 91l(x',y,,t)e u 2 remains constat everywhere on the vortex sheet along lines parallel to the direction of the incoming stream, i.e., along vortex lines from the wing. NACA TM 1585 Since the velocity potential r1 = 0 everywhere in the x,yplane off the wing surface and the vortex sheet, then it may be verified that ap possesses the specified property everywhere in SZ. Let us construct the equation for the fiction 6(x,y) 0e in R. Let us express cpm at the arbitrary point N(x,y,0) lying in by using the basic fonrmla for the velocity potential (6.24). We divide the region of integration S into thrz~ee pars, a~s shown in. figure 12, into s(x,y), a *(x,y) and cr(x,y). The regions s anda a~re parts of the wing surface and E defined above, respectively, which fall within theP characteristic forecone from. N(x,y,0). The region a is the part of R in the same cone. The variables of integration in ar vary' between xD 5 ( x and XE where x is the abscissa of D and y = X(x) is the equation of the arc DT of the wing contour. The expression obtained for cpe is differentiated in a direction parallel to the velocity vector of the impinging streamn. Therefore we arrive at the integrodi fferential equation, which 4 satisfies in x_h~ x co[ x 5)(y ?) a x 4 (1,1) C5h )( lldtl dS + 2 I~ x "y r cos(A/(x 5)(Y '1)1 r SEPXY 7 NACA TM 1585 where 1 2a and the known function is we A(S,?)Kl(S,8;x,y;X) dp dS  Qb(x,y) = L. 9(5,4)Kl(t,9;xy;A) di di  A(S,9)Kl(S,9;x~y;A) d? dS  y) al(x,y) 012 s(x, r" S al(x~y) (7.5) 9(S,a)Kl(S,9;x,y;A) da di cos A) ~ 5)yi 1) x E)(y ) wJhere K1(5,rl; x,y;A) = and the operator  = a+b, aL ax. ay The function 6 is determined from equation (6.7) of the preceding section. 2. We will look for a solution of equation (7.2) in the form of the power series 4(x,y;A) = 42nB(x,y) h~n (7.4) Keeping in mind the absolute convergence of equation (7.4) and using the expansion (6.8) for the cosine we obtain 6(5,9ih) cos 3/(x 4)(y 8 k=n k=0 (ln k 1) 2k(e,9) (x e)(y rl) nk C2(n k)f 1 e (7.5) S Js(x,y) NACA TM 1385 51 Substituting equation (7.5), (6.8), and (6;.9) into equation (7.2), the latter becomes nk nk = k~n a ax s n=0 at~ aS + lo=n nk nk 1 an (1)) 1 2(c,l) Cx f)(y r) 2a dyd + nk nk 1: (1) d2k(S,9) (x 6)(Y 9)] n dtl at [2(n k)l 1 k~n h2(n+1) r (1)n+ 2n A(,)( 4( )n d? di + (2n)! r 0 s n=0 t)(y 9nk  s9 n2ad ag + (x~~~~c )y9nk k= n nk+1 h2n ZZ(1) 92k(E,?) [(X k=) [2(n k) I +l n=0 (2) 1 ,, n=0 2n. 2 h(n+1) n0 s '1 h2(n+1) A(S,9) (x e) (y k=n nk+1 = I( In k) :2kt,) (7.6) Taking into account the uniform convergence of the series with respect to 5 and ? in both sides of equation (7.6)J we integrate it term by term. Then, keeping in mind, the uniform convergence of the3 obtained series with respect to xv and y which is also maintained after (7.8) 32 NACA TM 1583 differentiation, we3 differentiate the specified series term by term with respect to x andy. After these operations on both sides of the obtained equation we equate coefficients in identical powers of A. There fore we3 arriv at thae in~tegrodifferential equation which the coefficients of equation (7.4) satisfy IL rx bx xD 3r gr XI) 2}' dg at /(x )(y )) 2# >9~) =~ dSn(xiy) L(x 5)(y 'I) (7.7) where (1)+ a (2n)I BL art dS SA(r,8) + 8 x 6)y 1 Pn(x,y) = (1) n5/2 '~ AI') ~(r,l) (x 6)(y 1) C2(n k) 1 d? dS + khn nk+1 [~2(n k) 1 k=0 2d >4) (xr[ i)(y n) nk 92k~,9)(x I)y _4 2dTn dS + 1l~ kc=0 LI Ir/ a nk (1) I [2(n k)J E nk+1 1= nk MACA TM 1385 in which the last sum and also the terms in CI are defined for n>O. Let us note that the right side, en, of equation (7.7) for 92n contains terms with coefficients 42k but only fork= ,12 ... n1. Let us transform equation (7.7). We integrate by parts with respect to the first integral on the left side of equation (7.7), the second by parts with respect to ?, afterward we differentiate with respect to the parameters x and y, respectively. Equation (7.7) becomes Ix Y 92n5(E,Tn) + 62n (5 1) dn 15 = *n*(x,y) (7.9) where *~ 1 2(xDa'I) n ,(x,y) = d'l + 1) at + 4(x,y) (x ) y X(i ) ( Let us note that the first term in equation (7.10) of the right side of equation (7.9) becomes infinite for x = x:D. Let us return to expression (7.8) for egand separate out of it the terms corresponding to the value k = n in the first sum the compo nent a e~~n~g? ~dr dS = R We integrate this integral by parts with respect to 5 keeping in mind that the limits of integration in ol* are xE AR x(D and e(5) 5. 9 y and that 62n(xE,Y) = 0. Then we differentiate with respect to x NACA TM 158.5 JxE 1 r +. xD e J~s 2#%>8rT ) y. 1I 6 2n( ~D,1) r /xxD (xD) ~Y _9 (7.11) Let us subject the desired function 4 in equation (7.2) to a sup plemtentar condition. Let us assume that at the trailing edges the are DT (or D'T', respectively) of the wing contour and on the straight line DD)* (figs. 11 and 12) the intersection of the characteristic aftcone from D with the z=0 plane correspondinglyy the line D'Dlx) the velocity of the perturbed flow, and therefore the function 4, is a continuous fune tion, then the conditions are fulfilled 6 [x,x(x)] = A x,i(x) B~xD,y] = a [xD,Y (7.12) (7.15) These conditions are analogous to the Joukowsky~ condition for flow around a wing: by an incompressible fluid. From eqluatiojn (7.13) follows 1 r'' d2n(xD ~ FD~~~ J(x y ~2n(xDI1 1 j dTn = Jx xD J(xD (7.14) since X(xD) = 9(xD)* Substituting equations (7.11) and (7.1 ) in equation (7.10), the latter becomes 92n /~.:(g) x 8)y X(S (7.LS) .x " xD o i 62n 51 xE 1 , Id .p i i:; NACA TM 1583 where (7.16) on' n hR For n = O, the right side in equation (7.9) is a known function of x aind y bOO= xD (cx S) y X(5)] Ij dt  a 0i9 1 3 T d? dS (x )(y ) A(5,11) d? at 014 (7.17) Let us solve equation (7.9) for d2nx + 92ny* The twodimensional integral equation (7.9) is equivalent to two homogeneous equations xx 42n 7di = On*(x,y) D x ) (7.18) and 9 n(t,9) + 92i Y daT = B2n*(S~y) (7.19) each of which reduces to an Abel equation. Using the Abel inversion formula we find the solutions of equations (7.18) and (7.19) as i d 'X(5)d5 HIACA TM 1585 at xD ni '(~y di n*"(xD Y y xD 4s2n (x~y) =  X (7.20) 42n@(,Y 7 B2ny(@,Y) =I~+S j() (7.21) rY I Jx(S) 1 + Substituting equation (7.20) Into equation (7.21), first replacing in the latter by x, we obtain the solution of equation (7.9) as e n* xDX(x) 92nx(x,y) + 42ny(x~y) 2 gx~Tj 1 x enS* EX(x) dS + 1 r n{i(x, ?DT> ~xXDdx + %c2 iD Xn5 (x) ( Cx 5)(y q)5( t d r (7.22) Integrating equation (7.22) along the straight line parallel to the freestream betwJeen the limits of ;N(x,y,0) and N(x,y,0) we find the formula determining 42n in the gBneral forn of equation (7.4) ntl*(xD,rl) aT) dX1 xl x l!  MACA TM 1585 x xXx1 dxl up xlxD xJ yxXixl 92n(x,Y) = 42n(3T,g) +  x2 x" x, en( g+C,X(l)a di xl JXD l 5 I xl + Y x X(xl) Sx rx1+yrx 2I VX( xl ont/ ^l(Sr 11 x xl yx+yx xx D'(xl) an at dx1 (723) y are taken as solu the value of 42n(x,9i~:) edge, then we find If in equation (7.25) the coordinates tions of y x + x y = 0 and 21() is determined from condition (7.12) on the 92n on the vortex sheet. x and = O and trailing If in the same fornaLa, the coordinates and y are set equal to jI = xD and y7 = y x + xD and the value of 92n(jT,Y) is determined from equation (7.15) on the line x = xD, then we find 32n outside the vortex sheet in the region it affects. Thus, through equation (7.23), we can compute successively th coef ficients 40, 92, *.. 2ns** Therefore, the solution of equation (7.2) is constructed as the absolutely convergent series (7.4) for any value of A. The coefficients 42n' are expanded in the series J'(x,y;h) = 42~n'(x,y)A2n n=0, (7.24) NACA TM 1585 x~ p+y The funtion 9' e 2 in (r' ig. 11) may~ be computed through equation (7.23) if the function X2(x) replaces X(x) in it (where y = X2(x) is the equation of D'T' of the wing contour) and we inter change the role of the coordinates. 3. Let us consider thie general case of the flow over an oscillating wing by a supersonic stream. Let the characteristic aftcones from El and El' and Dl and Dqr intersect the winlg as shown in figure 13. Then El (correspondingly El'), as shown above, are defined so that to the left on the leading edge equation (3.7) is satisfied and to the right it is not. The points D1 and DI' are, respectively, the most right and left points on, the wing plan form. Tlhe space of the considered wing plan form as transformed by equa tion (5.4) is illustrated. in figure 14. Let us divide the x,yplane where the medium is perturbed into a series of regions: the :regions considered in thle preceding section, SO, Sl>***>S>* *S and the regions nl, a2' " Ch, The region Spj is the M~shaped. region bounded downstream by the intersection of the chlaracteristic cones from DI and D1' with the a = O plane. In the z = 0 plane, these lines are the upper bounds of the region, of influence of thle trailing vortex sheet. The region n~ is Mlshaped lying between the characteristic cones from Da, Dn', D,+1, Dn+11. We divide, in its turn, the part of the x,y:plane lying to thne right and left of the wing, respectively, into thne ;trips al, a12> n *" defined above and into 81, 82 .. ,6n, .. and into al' a n** *" defined above and 81" 621 *, ** n', correspondingly. The strip 6n is that partt of Lh to the right and in' is the corresponding part of Lh to the left of the wing. It is easy to see that the region a defined at the beginnling of this section is in bl* In order to solve completely the problem of the flow over the wing shown in figures 13 and 14, the der,(~vative a891/az must be determined N~ACA ~TM 183 in 81J s2, *> ;6n, and in 81' 621 ** 6n >** ( x+y) Let us denote the function e 2 by4 2) g) ti(n). .. and B', es(2), s~g(n), in the 817 62> ** Sn, and 811 6 21 s **En,. strips, respectively. Applying equation (6.24) for the velocity potential we construct cp, for any point N(x,y,0) in 82* We: divid the? region of integration S which depends on the form of the Function e 2 into the following: S = s + QE + al*' + S" + a, as shown in figure 15. This functiLon is given in s. It was determline~d in e*and o'lt in the~ preceding section by the soluions of equ, tions (6.7l), (6.23), (6j.27), etc. In e4 i~t is determiined by the solu tion of equation (7.24). We denote  e 2 in a by 4(2). Using the boundary conditions (1.10) and (1.12) we~ arrive at the~ integro differential equation which g(2 satisfies and which differs from equa tion (7.2) only in the form of the right side. On thet one hand the righ side depends on the solutions 9, 9(2), (N), 6,I et(2)1.. e'(N) and on the other hand on the solutions 9'. We~ construct 9(2) in the form of a power series in the parameter h. Requiring the fulfillment of equations (7.12) and: (7.13) for 6(2) we obtain for the coefficients 0() 2),... n(2) an expansion in series of 6 2) of equations of the form (7.9) which differ from each other in the form of the right sid~e. The right side in the equation for the coefficient 9;tn(2) in the general term of the series for 9(2) depends on the first n+1 coef ficients of the expansion of 6~i and 6' 1) where i takes all values less than or equal to Na, and on the first n coefficients 4,2), 92(2), .. 2k(2) (k=0, 1, 2, .., n1) of the series expansion of NACA TM 1585 the desired function 9(2). Therefore, it is possible to find succes siel te oefiiets4(2)J 2(2) 42(2) using the solu tion (7.22) of (7.9) as a final formula if there is put in the latter,, instead of cI,*~, right sides in the equations of the form of (7.9) for the respective coefficients of the expansion of 42n(2), By the same reasoning, values may be found of 4 5), 4 ,... 9(k), .. in 85> 64, kr Therefore the velocity potential may be computed by equation (6.24) at any point of the space perturbed by the motion of the wing shown in figures 15 and 14., In particular, the velocity potential may be eval uated at any point of the wing surface. All the results are valid when the contour of the wing is piecewise smooth. If the frequency of the oscillations of the wing, m, be put equal to zero, then the coefficients 901 902) 900() o cide, respectively, with the values of a90/8 in bl> 62> * sk, for steady motion when the streamline condition (1.6) is given on the wing as ac90/az = Al(x,y). We apply the proposed method of determining ac1 he for the oscil Lating mo~ction, of a wing by constructing an integral equation, to wings of competely arbitrary plan form. For example, the wing contour mayr not be camibered bu may have the shape shown in figures 18, 24, etc. In all cases, the part of the x,yplane where acq1$z must be deter mined should be divided into the corresponding characteristic regions. Thren successively passing downstream from one region to another, construct the integral and integ;rodifferential equations using the boundary condi tions on the x,yplane. The solution of these equations for al1/32 or for functions related to a(1/az is obtained as a series in even powers of the :parameter A, which defines the frequency of oscillation. The whole problem of determining the coefficients of the expansion reduces to a double integral equation in each characteristic region. Each of the equations after transformation appears to be an equation of the same type which is solved by means of a double application of the inversion fonrula for the Abel integral equation. The form of the wing contour is the limits of integration. The influence on the considered region, of determining NACA TM 1a38 the desired function in the preceding upstream characteristic region, is reflected in the form of the function in the right side of the integral equations. 8. FLOW AROjUND AN OSCIL;LATINGC WING OF NONZERO THICKH~ESS 1. Let us consider the motion of a thin wing at a small angle of attack (fig. 10a). Let the wing be moving forward in a straight line with the constant supersonic velocity u. Let an additional small oscillating motion be superposed on the basic motion of the wing so that the wing surface may be deformed. The normal velocity component on the upper surface of the wing will be considered given by Png = i~u(x,) + R.P. A2u(x~y)ei*t (8.1) and on the lower surface by on = AOl(x~y) + R.P. A2u(x~y)e ir (8.2) where Ab~u and AO 1 define the wing surfaces and A~u = Alu(xIy)eiagl~xty) and A21 = Al3(xly)e ia(x,y) define the ampli tude and initial phases of the additional oscillating motion of the wing. We consider the functions A~,Alu and amu given at each point of the upper surface and AOZ> All, and al given on the lower surface. The x,y,z coordinates were defined in section 1. The velocity potential pp is 9Pp(x,y,zrt) = cF(xryIZ,t) + (Ps(x,y,z,t) (8,5) NACA TM 1383 The potential cp is specified by the motion of an oscillating wing of zero thickness, which creates at each moment an antisymmetric flow with respect to the x,yplane (fig. 15b). The potential 9,is specified by the motion of a. thin oscillating wing with a profile symmetric relative to the x,yplane. Therefore the motion proceeds in such a manner that at each moment the wing surface wiLl be symmetric relative to a designated plane (fig. 15c). Such a wing creates a symmnetric flow and cp, satisfies Ps(x~y,z,t) = q,(x,y,z,t) (8.4) Each of the potentials Sand Wa is represented, in its turn, by (85) 'P = P0 91 C (8*6) Q's ''* +0 1s where cp0 and 0 correspond to the steady motion of the wing and rq and cY1s correspond to the additional motion of the wing. Let us set up the streamline condition using the representation (8.5) for the velocity potential. We transfer the boundary conditions on the wing surface parallel to the Oz axis onto the projection C of the wing on the x,yplane Therefore, we obtain the streamline conditions based on equa tions (8.1) and (8.2) NACA IN1 1)S3 ==AOu(x,y) + R.P. A2u(x~y)eiwt (8.7) and lz= which mut be satisfied on the upper and lower sides of C, respec tively. Using equations (8.5) an (8.6) we establish boudr conditions for the desired potentials cP0s (1, 'P0s, and 4)1s* Keeping in mind that on the z=0 plane the normal derivatives of the potentials 91bs and (1ls are specified by the syrmmetry of thre flow over the wing satisfying the condition Os Os1 1 (~', = Ui 4s (8.9) z "J=+0 z= %=+ a = We find the boundary conditions for c90s and r1s which rmust be satis fied on the upper surface to be where the functions rO and P2, are related to quantities given on the wing surface through A~u(x,y) A02(x'y) A2u(X,Y) A2 (X'y) TO(x,y) = P2(x,y) = 2 2 (8.11) on the lower surface of R.P. r2(xty)e'ist(8.12) The conditions to be satisfied by j re I dI a c = 0 z=+ cP0s and Q1s 1g~s  az= 44 NACA TM 1838 Since the normal derivative of the potentials c'b and cq specified by the antisymmetric flow over the wing, on the z=0 plane, satisfy (8.13) the boundary conditions which muI~st be satisfied simultaneously on the upper and lower surfaces of I are 90 AO(xIy) 3z 1=R.P. A2(x~y)eiid az (8.14) where AO and A2are related to quantities given on the wing through A~u + AOL AO = 2 A~u + A22 2 (8.15) The boundary problems for c1(x,y,z,t) and c90(x,y,z) were set up In section 1 where in the case of a harmonically oscillating wing, equa tion (8.14) rather than equation (1.6) should be taken on the wing. The solution of these boundary problems is contained in the present work. Let us formulate the boundary problems for c1s and r90s* I. Find r91s(x, y~z,t) satisfying equation (1.4), condition (1.11) on the disturbance wave, condition (8.10) on the plane region C and 1~s = bz (8.16) everywhere in the x,;'plane off r where the medium is perturbed. II. Find the function cpg(x,y,z) satisfying equation (1.5), condi tion (1.11) on the disturbance wave, condition (8.10) in the plane region Z, and Os =o (8.17) everywhere off Z in the x,yplane where the median is perturbed. =~  a z s z=+0 NIACA ?TM 1385 Since thre potentials 01s and q are functions which are symmetric relative to the x,yplane, it is sufficient to solve the problem for the upper halfspace. The solution of boundary problem I is given by equation (4.2). By means of this formula it is possible to comp~ute the velocity poten tial 91s everywhere since in the case of symmetric flow over a wing the derivative ac1s//8z is a given quantity for any point M(x,y,z) of the space in the region of integration S(x,y,z). To compute c1s at M according to equation (4.2) the function SR.P. 02(x,y)eirnt and integration is over that part of the wing within the characteristic cone from M. The solution of boundary problem II as is known refss. 21 and 22), is is replaced by = EO(IDxy) and integration is also over the region defined imme diately above. If the ving is vibrating as a rigid body then the functions Auand A~2 coincide and therefore, to solve the flow problem in this case, it is sufficient in antisymmetric streams excited by the motion of an oscillating wing with profile of zero thickness to superpose steady symmetric streams. must be substituted for q given by fonrmla (3.10) if the function 9b ap azz0 r PART IIS To apply the integral equations method explained in Part I of the present work, let us consider the problem of the flow over thin wings of finite span in steady supersonic flow. The velocity potential rp0 specified by the steady motion of the wing may be computed through equation (3.10) at those points M(x141l,zl) of the space for which the region of integration S(xldli,zl), already known from Part I, does not extend outside the limits of the wing where  is given. If a90/3zl appears to be unknown at any part of S, then, to use equation (3.10) in these cases, where it has in the characteristic coordinates 6t.3) the form 90(x ,yl~Zzl) li Sq d= (21.1) and to obtain the effective solution of the problem, it is necessary, first of all, to find 890/az1 everywhere in S by constructing and solving an integral equation. 1. INFLUENCE OF THE TIP EFFECT FOR STEADY WINIG MOTION 1. The integral equation (5.1) in the coordinates (6.5) is, for the steady ving motion 1(xldQdqldi F(xlrYl) (21.2) a(x'1 1 3The results of Part II, sections 1, 2, and 5 were completed in April, 1948 at the Math. Inst. of the Acad. of Science, USSR. NACA TM 1838 NACA TM 158) where 81 is the value of a99/zl on E3 (fig. 6) and where the known function is F~14) A(Elsil) (21.3) s xl 7Y1) The function A given on the wing is a'P0 up0(x,y) _u xl +71 ;v1 xl (14 A x1,Y1) ; (21.4 azl k kl 2 2 It is easy to see that the velocity of the pertured flow  normal to the x,yplane is related to arPO/aZ4 through The regions of integration in ar are x1E: 'i 51 x l and ~(El) 1 81 6 Y1 where, as before, yl = I#(xl) is the equation of the wing tip ED in the transformed coordinates and X1E is the abscissa of E in the same coordinates. The regions of integration for 91 in s are the same limits x1E r 51 < Xl and ltl 1 91~ ~( 9 ) where Yl1 = 1xl) is the equation of the leading edge E1E of the wing contour. Let us note that equation (21.2) may also be obtained fronn equa tion (6.5) if the frequency a of the wing oscillation is set equl to zero in it. Let us delete the index "1" from the independent variables. We solve the double integral equation (21.2) with respect to 65, by means of a repeated application of the inversion formula for Abel's inte gral equation. NACA TM 1838 We write equation (21.2) as dll +\I) dy dI = Jyrl S~ J," 1 Jx~ (21.5) This is an Abel equation with right side identically zero, therefore, the brace equals zero for ( = x. Hence, equation (21.5) is equivalent to dT) = A~x,4) dn (x), 1/y ?x) 1 (x) (21.6) which is also an Abel equation. Noting that the right side of' equa tion '21.6) is, generally speaking, different from zero for Y = #(x) we find the~ solution using the wellknown inversion formula for the Abel equation 19J(x) 01(x,Y) =1 iy 1 8 (x)/y ~ i (x) Aix,.) (21.7) I 21.7) for the steady motion of a wing may (6.22) of equation (6.12) for the vibrating frequency of oscillation as are both set Let us note that the solution be obtained from the solution wing if the index n and the equal to zero. Carrying out the operations specified on the right side of equa tion (21.7) we find the solution of equation (21.2) to be 91(x~) = _1 1 (x) 8L('Y)= f y /I(x) 9() A(x,l)' W~x dr Y . (21.8) x 3: d y = (x) dn' dq NACA TM 1383 a90 in a smlar manner, wJe find the value  1'(x,y) in Zjr OZ x (Y r RL'(x,y) = A((Y Jx c~Y 2 1(y) ( (21.9) The functions x = 1(rly) and x = 92(y) are, respectively, the equa tions of the arcs ED and E'D' of the wing contour solved for x. The solutions (21.8) and (21.9) show that the velocity of the perturbed stream, when the arcs ED and E'D' are approached from off the wing, goes to 1 R 2 where R is the distance of N(x,y,0) from the points (see fig. 7). infinity as ED or E'D' 2. Let us find the velocity potential according to equation (21.1) at the point M(x,y,z) of space for which the region of integration S intersects the wing surface T, and the region Ly or Zp' The region of integration S in equation (21.1) is divided intor three parts: S = sl + 2 + sO, as shown in figure 16 1 Aay dn d BO+s2 rr dq at rp0(x,y,z) = (21.10) The limits of region s1 are xE 5 ( xA and t (5)'1 & 22 here xA is the coordinate of the point A which is the intersection of the characteristic forecone from M with the side edge ED of the .ring. The equation rl = Y z2 x 5 is the equation of the hyperbola in which the aforementioned cone intersects the z=0 plane. The limits of region s2 are xE 5 E & xA and Jrl( *5? l() x  NACA TM 1585 Using equation (21.8), let us evaluate the integral over equation (21.10) s1 to S1 drl l (21.11) .  we interchange the order of integration of 1, ,2 t(g)? Yx6 ~c i, (5) I 1 ~1) Ag4 dq dq' dt (21.12) Tlhe result of the inner integration is 2 z x5 S2 T x  (21.15) Putting the valu of equation (21.13) into equation (21.12) we obtain I ~ ~ ~ = d5q) (x 5)y 4) z (21.14) 22 x6 (x l )y ) NACA TM~ 138) Equating (21.116) and (21.11) we obtain s1 "2 y(x 5)(y ?) z2 (21.1~5) Therefore, to find the velocity potential, on the tion (21.1), at a point M(x,y,z) projected onto x,yplane as shown in figure 16, it is sufficient basis of equa M' (x,y,0) in the to integrate over A~~t,9) dn a (21.16) "0O )y )z 'P0(x,y,z) = 1 2J The limits of region sO are ~1(() 1'1 y z2,/x and xA < Bweex is the abscissa of the point of intersection of the Mach forecone from M with the leading edge ErE. The velocity potential equation (21.16) by setting integration to be xA E( on the wing surface can be calculated fromn =0in it and considering the region of x and S (C) 6 1 y because the lines of intersection of the characteristic forecone from M with the x,yplane, in this case, are the lines 5 = x and ? = y. In order to compute the velocity potential at points of space, or in particular, on the surface of the wing for which the region of inte gration S intersects simultaneously I~ and Z I; that is, at points of space where there is felt the effect of both side edges ED and ErD', it is sufficient to integrate equation (21.1) over the region a = S + S e, the crosshatched region in figure 17. Hence the integral over S e in equation (21.1) must be taken with the opposite sign, i.e., the plus sign. 52 NACA TM 1383 3. L~et us consider the wing of more general forn shown in figure 18. Let the forward part of the wing have the break, the are EGG'E1', in the wig contour which affects the flow just as do the side edges. Let us show how to compute the velocity potential at all points M(x,y,z) of the space disturbed by the motion of the wing, which is not affected by the trailing vortex sheet, in particular, on all points of the wing surface. We divide the wing surface into the characteristic regions shown in figue 18. If the region of integration S in equation (21.1) intersects regions 2, 2', 3 and does not intersect 4 then the velocity potential may be evaluated by using equation (21.161 (see figs. 16 and 1'1). The simple result which is expressible by equation (21.16) does not hold in the general case. If S intersects 4 on the wing, in the curvilinear triangle K'4KL then according to equation (21.1) ar0/os must first of all be found in the triangle. Le us express, by equation (21.1), the velocity potential at any point of Kr'OlK as equal to zero everywhere outside the wing and the vortex sheet from the wing, hence in K('O1K. Therefore, we arrive at an integral equation of the form of (21.2) for the function 91*(x,y) = acFypz in K'01K but with a more complicated known function. Applying the Abel inversion formula twice, we arrive at the solution in the following final form: 01*(x,y) = 1 1 1 1 I k (~) 27 _ dS n /x #2 1k)x 5 (21.17) where = 1x i the equation of EG, y = 1l(x) is the equation of E'1 x = 7(y) of El'O' and x = gl2(Y) of E1IE. NACA TMr 158) Substituting equaittion~s (21.17), 2.) n 3 )it qa tion (21.1) we obtain the formula orthe velocity potential at M which has the projection M' shown on figure 18, and for which the region S intersects 4 on the wing and, therefore, the region K'O1K outside the wing, as I s*(x,y,z) A~g~l)drl dE A x E(y, 9) z qq(x,y,z) = tem g ySlx )  $(c) 1 (x () y *(5)  drl de + z22  ] ar2 (x ( )(y1 2 A(Srl) tsn'i I[Fcl~ hx tl(r 1) 22 [Y2(1) dE drl ~]l~y I)[x r(?J] z~~ (21.18) where y = Y*(x) and x = rl*(y) are the equations of GG' contour in terms of x and y, respectively. of the wing The region S" is the part of the wing shown crosshatched in fig ure 18. The regions Sl*C and S2* are part of S* and are marked in the same figure by horizontal stripes. The regions Sl* and S2* are bounded downstream by lines parallel to the coordinate axes passing through C and G'. The points G and C' are respectively the points with the largest x and y coordinate on the arc EGG'El By combining the results of equations (21.1) and (21.18) there is found in the form of integrals taken over the wing surface, an effective expression for the velocity potential at points of space for which S in equation (21.1) intersects 5 or 6 on the wing and therefore a K'01K( and and c31 off the wing. NACA TM 1383 2. FLOW OVER WINGS OF SMALL SEAN 1. Let us assume that the characteristic cones from El and El' intersect the wing as shown in figure 19. This occurs, for example, for small span wings. Let usdivide the x,yplan~e where the mediumm1 is disturbed into the regions SO S,.. ., Sn',, The region Sn is an Mshaped region lying between the cha~racter istic cones from En and E'(or in one of them) and En+1 and En+1'. In its turn, we divide the part of the x,yrplane to the right and left of the wing into the strips ol, a2, .., ag, .. and al', 02= ', ., respectively. The strip an lies between the after cones from E, and En+1. Therefore, an is that part of S, lyinlg to the right of the wing. The coordinates of E and E' with their indices are shown in figure 19. The strip a,' is similarly defined. Let thle leading edge El'El be given as in part I, section 6, by the equation y ()and the side edges E1En+1 and El n+1' by y = Jr(x) and y = 2(!x), respectively, or as x = l(y) and x = JI2 7) correspondingly. To compute the velocity potential at M according to equation (21.1) in that part of' space (ojr, in particular, on the wing surface) the region of which intersects Sn of the x,yplane but not Sn+1, we must first of all determine agO/az off the wing in al, cr2, ,... n and also in crl', 52 / .. n a** We construct the integral equation for hp0/a5 in the arbitrary strip ok. Let us express a velocity potential which is equal to zero every where off the wing an~d outside the region of influence of the vortex system from the winlg, at U~ ofsithe ok strip (fig. 20) according to th~e fundamental formula (21.1) NrACA TM 1383 if by dr( di 0 S(x,y,0) (22.1) The limits of integration in 8 are xl < 5 5 x and y1 i tl I * For convenience in. later writing, we make B a rectangle, which is pos sible since the medianm ahead of the wing is ntot distured and abla b is zero. The region S is shown in figure 20 bouned by the lined ~LN, rL1, L101 and OpL. Let us denote aby 32 by 91 21 *> kJ . and 9311 2" '' 92 s Sk, in the respective regions al' ... and al1 2 '> k ** In confojrmance with this new notation we write equ~ation (22.1_) as /,x dyL + drl + 1 Jx 5 k1 (22.2) i=k2 rYi+1 eir S? dqt + i=1 Ci  r7 56 NACA TM 138) Applyi~ng the Abel inversion fomumLa twice to equation (22.2) we find efor k 2 4(x) A(x,1)/9(x) ? anl + 2(x) 7  Bk(XI1 1 syi+1 ~ ~ ds +1(,)()4 Fi r i=1 92(x) ekIr(x,9) Vx) d Yk1 T (22.3) Correspondingly, for Gk'(x,y)= fi~ we obtain [ 2 7 ) F) di + k() k i=1 ~Xi+ da 2() (22.4) NACA TM 138) jT where the terms in equations (22.5) and (22.4) containing the summations are defined only for k 5 . If 61, 82r 8k1 and therefore, 8l', 621 *~ ** k/ are already defined in ol, 21, ., crk1' then we can compute 8k in ok for any k by means of equation (22.5). The value of a(Fg/az in al and crl' is determined by solving equations (21.8) and (21.9). The value of ac90/az in a2 is found from equation (22.5) by putting k = 2: 1 1 9(x) e,(x~y) = [ x gy (x )J 2 x) A(x,rl) d'l + R,11) at d'l (y ?)(x 5)/x J2 1) (22.5) We find acyg/az in a2' in the same way 1 12 e,'(xJy) = _xT Y A(5,y) x d5 + t~(y) ,9(5)2 ()  1. 1 ~ J, A(ST,) d? dS (22.6) Thus, step by step we compute arpg/as in ok* NACA TM 1583 Usihrg the soluion, of equation (22.i), we now prove the relation = xl:2s _z a'xt da at (22.7) where xl*F and x2" are any numbers satisfying x1 < x2* 2 the coordinate of the point A shown in fig. 21), xl xl For the proof, we write n in the equivalent form xA. (xA is < A* 22 [ x t(s) /(x 8)(y 4) z2 a' = F JX * xl ~ 5 2(x g)( r 4) ai' (5,1) dl at x2 r2(5 $L1 ,9) drl di i=k2 x 71+1 (22.8) NACA TM 1383 where 8k in the first of the integrals is replaced by its value according to equatio3 (22.5). Then, we obtain 0 = 1 2+ r~ A(S,g') 4) .q I+ dnl di  ei I) 1'TTrl i=l 1 X i=1 x y x1 1 I" drl' d5  x2" JX~L" JYkl C/x5 I" d?' dS t ( * x, t ) xl 2 A( ) dq dS i=k2 i=1 x2 i+1 x~l 1 ei'(S,5) dq di /(x S)(y 8) z2 x2 "2 x2 Ik1 ekl'(t,9) dq di /(:x )y ) 2 (22.9) where I* denotes the integral (21.15) evaluated before. It is easy to see that all the terms in the right side of equation (22.9) cancel in pairs. Hence, equation (22.7) is proved. It is also clear that the following holds z2 ya Y2Jx 71* xl Zjz=0 J(x g)(y rl) z2 O at art NACA TM 1583 where y and y2" are any ntanbers satisfying Y1 1 JT1 <2 Bg BY is the coordinate of B shown in fig. 21) . Using equations (22.3) and (22.1+) it is possible to prove equa tions (22.11) and (22.12) correspondingly Tx,* xt* drl at kx S)(y ?) "2 132!zC tan1 (x 5)(y ?) z2]y ( 1 dC d? 2 x2 5 and satisfies 1(x1r~ y =Yxl whe re y* maLy depend on Jy2: S* drl dS lten t 2 .'`12 nit Y1 ~2('1) ~ Xi (22.12) where x* may depend on TI and satisfies #2 Y 1*) < x*C = x ;L The relations (22.10) from equations (22.7) and changed in the latter. and (22.12) may be obtained, respectively, (22.11) if the role of the coordinates is inter , y* 8 4(() bz 0 Sag a1, NACA TM 1385 61 Let us note that the result of a single application of the Abel inversion formula to equation (22.2) or directly to equation (22.1) yields Interchianging the role of the coordinates in equation (22.15) we obtain jI' I J O (22.14) It is possible to consider equations (22.15) and (22.14) as rela tions fulLfilled long the characteristic lines LII and LN in the x,yplane where y and x are, respectively, the coordinates of NT or N' Lying off the wing and off the region of influence of the trailing vortex system (fig. 20j). The points NI and Ii' lie to the right and left of the wing, respectively. These relations can be usefull for compu tations. 2. Let us turn to ther fundamental formula (21.1). Using equations (22.7r) (22.10), (22.11), and (22.12) we obtain, by calculation, the formula for the velocity potential cpg at M(x(,y,z) for which S intersects S, for any n > 0 NACA TM 1838 where th functions 01 and R2 are defined as 1 l(x 5)(y rl) z2 .B "(C) 01L = tan 1(x S)(y Jr) z2 xA" 2rl 0L2 = tar [(x xA)(y 4) z2 2]()1 and where the regions S~ and Se are regions of the wing marked on figure 21. Th region Sl* is the verticallystriped region on the wing surface. The region S2* is the horizontally striped region of the wing surface. It is clear that Sl* and S2* intersect each other and Sqp on the wing., The region S1 lies off the wing and is verticallystriped in figure 21. This region is the sum of the regions over which are taken the integrals containing 8k' for k=1, 2, .., n2 in equation (22.15). The region S2 lies off the wing and is hori zontally striped in the figure. All the integrals are evaluated over it which together con tain 8k for 1,2 .. n. If M is such that S in the basic formula intersects S, falling La the characteristic cone from En and lying outside the cone from E' then n must be replaced by nr1 in the second sum and in the last term of equation (22.15). If S falls inside the cone from Ene and lies outside the cone from E, then n1 must be substituted for n in the first sun and the penultimate term of equation (22.15). Let us note that the sums in equation (22.15) are defined for n > 3 and the last two terms in equation (22.15) for ny 3. NACA TM 138) If n=1, then the formula for the velocity potential in equa tion (22.19) is limited to the first two terms. This result was already obtained before. If n=2, the formula in equation (22.15) is limited to the first four terms, the region of integration is shown in figure 22. Thus, to evaluate the velocity potential, by equation (22.15), at a point M(x,y,z) which has the projection Ml'(x,y,0) shown in fig ure 21, it is necessary, first of all, to compute Bk for k1 ,5 n2 by equation (22.5) for k>2 and by equation (21.8) for k=1 (8k1 correspondingly). As an example we present the expression for the potential for n=3 in the expanded form 1 AE9d i eb(x,y, z) =1 JJ (~iaid 2xB so/x )(y 9) 2 A(C,?)drl d5 8 (x )(y ) 22 1 A(C,?)n2 da dS Sl* A(S,ri)na1 1 I (x 4)(y 4) 2 ,2 82  an'd5 drl + I 1 1 5dd n 7r2 xl (y (5 3 x2 71 (22.16) 64 NACA TM~ 1383 The~ region of integration in the last two integrals over 5 and ? are, respectively, thle regions S1 and S2, lying off the wing and shown striped in figure 23. Formula (22.1)) for the velocity potential contains an niterated integral with the integrand an arbitrary given function on the wing: #0/lbs = A(x,y). In the general case, it is not possible to reduce the number of iterations in the compu~utationur of equationU L (22.15) for arbIitray vinj tip shapes since the arbitrary functions y4, 2 and A all contain the variables of integration. If the functions and $2 are fixed t~hen th~e wing to be considered has completely determined tips and it is easy to see that all the integrals in equation (22~.15) are reduced to double integrals taken over the wing surface with integrands containing thte arbitrary given function A(x,y) which defines the form of the wing surface. Let us turn to the wing of small span which has a break in its leading edge as shown, for example, in figure 24. The derivative a00/az may be evaluated in Gi andl up by equa tions (21.8) and (22.j). It is impossible to evaluate a O/az in 05 using equation (22.3) anrd, therefore, a surfaceintegral equation must again be constructed which will also reduce to two Abel equations but with more complex right sides than occurred for a3 in figure 19. Hence, we note that it is impossible to construct one formula which would determine a670/az for all cases, but a single method of solution may be shown to depend on the wing plan form. The formation of the surfaceintegral equation for 870/az is explained above, for each characteristic region. Each of these equa tions is of the same tyrpe, reducing to two Abel equations with different right sides in different cases. In particular, the right side of one of the Abel equations, in some cases, may be identically zero. NACA TM 138) 5. INFLUENCE OF THE VORTEX SYSTEM FROM THE WING FOR STEADY WING MIDTION 1. To study the influence on the air flow of the trailing vortex system in steady motion, it is convenient to operate with the acceleration potential 40 which, in linearized theory, is related to the velocity potential derivatives in the characteristic coordinates through 0 = u 9x 90 (2).1) Let us turn to the wing shown in figure 25. Let M(x,y,0) on the wing surface, which lies between the cones from D and D'. Therefore the trailing edge us take a point characteristic D]T affects M. Using equation (21.15) the velocity potential at M~ according to equation (21.1) is ssis0 A(S,rl) in at 1 e(S,?) dn dS 82rr~Lr)Y 90(xly,0) = (2~.2) where the regions s = s1 + sO and s2 are shown in figure 2$. The region s2 belongs to R, considered in section 7 of part I and shown in figure 11. We denoted the derivative 890/az in R by 9 where this derivative is an unknown. We subject tf90/az to an additional condition, analogous to the KuttaJoukowsky incompressibleflow condition. perturbation velocity potential at the trailing and D'T' of the wing contour (figs. 11 or 25) specified derivative, is a continuous function. conditions are fulfilled: We assume that the edge the arcs D  and therefore, the Then the respective NACA TM 1585 (25.5) (23.4) D~T and In order to obtain the acceleration potential OO at M on the wJing surface, we must take the derivative of equation (25.2) in a direc tion parallel to the oncoming stream. Before differentiating the double integral with respect to x and y we integrate by parts in the first case with respect to 5, in the second with respect to ?. During these operations, we use equation (25.j) and the relation (22.15) which is fulfilled along characteristic lines, and which on the line DD" (fig. 25) is X~xD) A D fl(xD) ;XxD) Y I (23.5) We keep in mind, moreover, that the limits of x:D 5 ( xA and X(5E)( '1 fl(5) where xD XA. xA(y) is the~ abscissa of A, the limits and t() yand finally the limits of integration of s1 are is the abscissa of D and of sO are xA After the specified operations, the results of differentiation are 1 ff A (S,1)fr(1),) + An(S,1) d sl+sO s d'l dS  P~x(x,y) + 44 (x,y) = 25.6) AZ 8, I 5 21, (23.6) 9[x,xj~x)] = A x,X(x) B[x,:,(x)] = A[x,x2(x)] where, as above, the function y = XIc(x) is the equation of y = 2(x)is the equation of D'T' of the wing contour. dS NACA TMI 1358 where the arc 2 = RP is shown in figure 25. In order to evaluate the acceleration potential 00 at M according to equation (2j.6) it is first of all necessary to determine 63 + 4y in s2' 2, Let us construct the integral equation for 9, + d Let us express the acceleration potential through equation (21.1) at an arbitrary point N(x,y,0) outside the wing in n affected by the vortex sheet trailing from the wing 44(x~yo) 1A(EB 2x d' di s(xy)x S)(y .4) a(x,y) (23.7) for which the limits of integration in a are xD ( 5 3 x and 7(( q y and in s, 5 varies between the same limits but rl between 01(5) I rl X(S) (fig. 26). Let us differentiate this expression in the freestream direction. Since, according to the condition((1.10) of part I) the velocity poten tial (p0 off the wing in the x,yplane remains constant along lines in the specified direction, then the left side of equation (2).7) goes to zero as a result of differentiation an~d therefore we obtain x xy ag)? axaII d'l = O 2)8 by ~xD 1 ) /(x S)(y ?) N~ACA TMI 1383 (25.6) by parts respect to x. .1 xD) A xD We integrate the first two integrals in equation with respect to 5, after which we differentiate with result is The X(~) Sd'l + x xD A(E,tl) /(x~c' 5 g)y ) 1 Ilx xD x(g) A(g,q) dlyl dg x 1 ai (259) Sy 9 xD') .I X xD) J rl x bxgD rY JX(5) ' d7 d5 = x D ix 5 yy~YSSr) rx 1 a ~x 5 a~ (25.10) Keeping equation (2).)) in mind, which is fulfilled on the characteristic DD*n we substitue equtions (23.10) and (2J.9) into equation (25.8) obtaining d? +a I Ex rY ') y A(5,1) d x~)A(S,rl) 1() y 1 IT ,i) drl + (25.11) dt 9  NACA TM 1383 This equatipn is equivalent to ,X(x) 1i(x) J1(x)  9(x,1) dr 9(x,l) Jy 1 A(x,rl) dll + A(x,rl) =0 dy O (23.12) according to the inversion of the Abel integral equation. We integrate the last two integrals in equation (23.12) by parts with respect to tl after which, as above, we differentiate with respect to the parameter. Using equation (23.3) we arrive at Bx(x,q) + 4 (x,q) yt = rX(x) 1~(x) Ax(x,rl) + A l(x,9) y X.x) (23.13) Let uts aply once again Abel's inversion formula, he~eping in. mind that the right side of equation (23.13), generally speaking, is different from zero for y = )((x) we obtain the solution for 4x, +y as yx(x) F O,(x,y) + b (x"y) = 1 1 Az. x,9) + <~T(x) l d 1l 1~ 1~ d 1l(x) j~)1x A 4 x @ Xx) dx 7 x) (25.14) A x, 1(x Sl(x) 1   7 1(x)j dx NJACA TMu 1585 Using equation (25.14) we prove d? de = s2 6(x,9 d)(y ) d9 dS (x 4)(y ?) s1 (x S) y 9 (SE  [ 21 (23.15) where 21 = HQ. The regions s2 and sl are shown in figure 25. Substituting equation (2).15) into equation (23.6) we obtain the forml fo~r the acceleration potential 0g(x~Y) n = 0Px + by = A (5,9) + A (5,1) (Ix i)(y 9) 1 s1 sO dtl dS  A E,el1~ i (25.16) where L = QP, the direction of the integration is shown by the arrows in figure 25S. Thus to evaluate the acceleration potential at M on a wing sur face two integrals, the surface integral over sO and the contour inte gral ove L of the leading edge are to be comIputed. Let us turn to equation (25.12) and write it in the form J,1(x) 1(x) d 9 7   1 1 )3 d be z=0 bz z=0 NACA TM 1583 Interchanging the role of the coordinates in equation (29.17) we! obtain a+ 0 ax 1y sz z=0, B (P)b = (23.18) where x = el(y) is the equation of E'E of the wing leading edge solved for x in terms of y. It is possible to consider equations (23.17) and (23.18) as rela tions which hold along characteristic lines in the x,yplane where the vor tex sheet has effect. Relation (23.17) is fulfilled along characteristic lines parallel to the 0y;axis (the line NQ on figure 26); the yparameter is the ordinate of a point lying off the wing to the right, in the effective range of the vortex sheet (point N in fig. 26). Relation (23.18) is fulfilled along lines parallel to the Oxaxis; the x parameter is the abscissa of a point lying off the wing to the left. If the point N is thus located to the right of the vortex line DH or to the left of D'H', then along characteristic lines the respective relations (22.13) and (22.14) also hold. If N is located to the left of DH or to the right of D'H', respectively, then relations (23.17) and (23.18) hold along characteristic lines. In this case, equations (22.13) and (22.14) are not fulfilled, In this section, we wrote down the transformation and obtained the formula for the acceleration potential in the simplest case of the vor tex sheet affecting the flow. For any other case, the potential 0~ is found in an analogous way. In each case an integral equation is constructed for 9x + y. All the integral equations are of the same type but with different right sides La the different cases, and they are inverted by means of a double application of the A~bel integral equation inversion formula. In the following paragraph we present results defining the accelera tion potential OO at any point of a wing surface. NACA TM 1585 3. Let us find the velocity potential 90O(x,y,z) at a point M lying within the characteristic aftcone from D and outside the charrac teristic aftcone from D'. The region of integration S in the funda mental formula (21.11 intersects the plane region ii (fig. 11) in this case. The projection M4' of M on the x,yplane is shown in figure 26a. Starting from condition (1.12) (of part 1) we express the derivative ifr0/az for 3ny point where the velocity potential equals zero and where, simultaneously, the effect of the vortex sheet is felt through the same derivative at points located upstream on the same characteristic line with the point studied. To do this we reason just as we did to obtain formula (21.8). We then obtain the desired representation for the derivative 390 1 1 .x'yDxD OQ0(x'rl~z)x+yDD bz xn~ _l D+xD(x) b z=0 j (25.19) Using equation (23.19) it is easy to prove x2" ,E+YDxD 80da d5, (23.20) by the same methods used in proving equation~ (21.15). TIACA TM 138) The limits firgaini equation (25.20), xl*~sto and x2*, are any numbers satisfying xD xl* xF and xDj x2 X' xF where xF is the coordinate of the point F shown in figure 2ba. Th~e point F is the intersection of the vortex line DH, wh~ich~ has the equation y =x + YD xD, with the characteristic cone fr~om the point with the coordinates (x,y,z). In particular, there holds /J 3r00 dndi_ drl dS (25.21) where the regions S1 and. S2 are shown in figure 26a. The region S1 is marked with horizontal and the region S2 with vertical crosslines. Keeping in miind equation (25.21) we obtain an expression for the velocity potential at the point M1 defined above 1 / A(5,1)dy dS 1 (,d d 2x0/x8(y1 2 2 S' (~x S)(y 1) z (25.22) where SO and S' are spown on figure 20a. Therefore, the region of integration S in equation (23.22) inter sects the wing surface only in that part of which lies to the left of the vortex line DH. Before evaluating the velocity potential by equation (23.22) it is necessary to determine a'F0/a2 = in the region S' of R. We find 4 from the solution (25.14) if the latter is integrated in a free stream direction between N(x,y) and N(x,y). Hence in order that the obtained expression correspond to the value of the deriva tive ac90/az = in .' to the left of DH, the coordinates r. and on the vortex sheet should be taken as the solution of the equa tions yf 32 yD + xD = 0 and y' = :r(x) and the value of 9(ii,) is determined from equation (25.5) at the trailing edge. NACA T1M 1838 If the E and coordinates are set equal to 2 = xD and fi = yx+xD and the value of 4(.3,f) is determined on DHI from the solution of equation (21.8) then the obtained expression will correspond to the value of acpg/az in R to the right of DH off the vortex sheet but in its sphere of influence. 4. PRESSURE DISTRIBUTION ON A WING SURFACE 1. L~et us consider a wing of arbitrary plan form. Let the wing contour in the characteristic coordinates be given by the following equn tions: The leading edge E'E by y= J(x) or x = ~1(y), the side edges ED and E'D' by y = #(x) and y = 92(x) or x = #(y) and x = Z(2y), the trailing edges DT' and D'T' by y = Xr(x) and Y = X2Z(x) or x = ji(y) and x = 22 7)* Let us find the pressure of the flow on the wing surface. According to the Bernoulli integral, the pressure difference of the flow above and below the wing is related to the acceleration potential 90 p(x,y) = pZ(xIY) pu(x,y) = 2peO(xiy) (24.1) where p is the density of the undisturbed flow. We divide the wing surface into the ten characteristic regions shown in figures 27 and 28. Ltet us express the stream pressure on the wing surface in each characteristic region by the function A(x,y) which is given on the wing, defining th shape of the surface. We denote by M and M with a subscript the ends of line segments parallel to the coordinate axes and lying in the x,yplane. It is clear that these segments are parts of the lines of intersection of the charae teristic cones, with vertices in the x,yplane, and the x,yplane itself. Region I is the region where the tip effect is not felt. This part of the wing lies ahead of the characteristic aftcones with vertices at E' and E. NACA TM 1583 Region II is where the tip effect is felt but not the influence of the trailing vortex sheet. This region lies between the characteristic aftcones from E' and E and D and D'. At M of region II, for which the lines M1MrI and M2Mq intersect on the wing as shown on figure 27, the pressure difference is p(x,y)=up D(S,?;x,y)d? dS + up D(S,?;x,y)d? d5 + d yS) Y L1 L  1 B 9(y),q;x,y dn 1 B5 (x; dV'rI~i (24.2) where S1 is the region of the wing bounded by NISand M12Mq S2 is the region bounded by the lines MI~I, M1yM M2Mq and the arc L = ~ZMk and where Ag(~,?) + A?(~,?) A(51T~) B(S,?;xy) = D(S,~;xy) = NACA TM 1583 If the lines M1N and 2M2 do not intersect on the in figure 28, then the pressure difference is wing, as shown d i15~,  p(x,y) = u 82 D(S,9;x,y)d9 dS 1 u. 1 d r(y) B y)8xyd L2 (24.5) wJhere S1 is bounded by the lines MMI> I1 L = MM .~ IrZM M~g M2MA 4 and Arrows in the figures show the direction of integration in the con tour integral and the integrals taken over the lines L1 = M5"I and L2 = M4 2* In region III, which lies between the characteristic cones from E anrd the characteristic cones from E', D and D', the pressure differ ence is P(x~Y) = US 81 D(g,9jx,y)drl dS  B ~(y),?jx,y d' (24.4) The pressure difference in region III' is expressed in the same way. .B ,tlE);~y  B 8 1E);j 1 d5 p 1 a~yd; J) L2 NIACA TM 158) p(x,y) = up D(g, ix~y)dl dC  B 8,9 (S);xiy~l 1~(5 d5  dC~dx 1()~B 8, 2g(x) ;x, y di L2 up (24.5) Region IV lies in the characteristic cones from E and E' and D and outside the characteristic cone from D'. Region IV' is defined cor respondingl. At M(x,y) of region TV, when 1MIM and M2 14 intersect on the wing, the pressure difference is p(x,y) =p D(t,?;x,y)dy di Sl S2 u B E,t1(S);xIyI 1 d ll() dS  up id92(x) j~(, ~ X dx (24.6;) For the MI, for which M1M~ and M2Nq do not intersect on the wing, the pressure difference is expressed by equation (24.5). Similarly, the pressure difference for region I~V' is u J D(E,9;x1)dy1 di + ufr(e,l;x~y~dy di + SL S2 p(x,y) =  d~l() d5dS  L 3 . B E,9 (5) (24.7) ;x,y] 1 B y,1xy G L1 NACA TM 1385 if My@and M2 4 intersect on the wing. If these lines do not inter sect on the wing the pressure difference can be expressed by equa tion (24.4). In region V, which lies within the characteristic cones from E, E', D) and D)' where the influence of the trailing vortex sheet is felt, the pressure difference is p(x,y) = ]Dj(i,1;x~yldn di + DI(t,9;x.y~dy dr + S1 S2 Bp I 8, 1(E);,y V1 dd RE(5 (24.8) if MME and ML2Mq intersect on the wing, and p(x,y) = p SD(i,9;xly)dl dS ,9()xp drl( 51 L (24.9) if they do not intersect. In region VI, lying in the characteristic cones from E and D and ouside the characteristic cones from E' and D' (also in region VI' ) the pressure difference is expressed by equation (24.9). TPhe pressure difference for rg~ Lon I has the same form. Ths, if M, at which the pressure is desired, is in one of the regions II, IV (IV' correspondingly), or V, as shown in the figures, then to set up the regions and contours of integration in the pressure formulas it is necessary to proceed as follows: Draw two lines MM1 and MM2 upstream from M to intersect with the side (or trailing) edges of the ving. From these points of intersection M1 and Mr2 again draw lines M1M3 and M2 4q upstream to intersect the leading edge E'E at M3 and If M4 is in region III or VI (III' or VI' correspondingly) then from M draw the lines N~y and MM1'i upstream; the line MMimmediately intersects the leading edge E'E at MgZ; MMI1 intersects the side edge NACA TM 138) 7 ED in the case of region III or the trailing edge DT' in the case of region VI. From the point of intersection MIagain draw the line I01 53 to intersect the leading edge E'E. Let us consider particular cases. (I) Let the side edges of the wing ED and E'D' be straight lines parallel to the free stream. In this case by x and, therefore, formulas (24.2) and (24.3) are simplified substantially;, because the Last two terms in them become zero. A particular wing of this class is the rectangular wing. (II) Let the ving surface be such that D(S,1;x,y) O This holds, firstly, when the wing surface is a plane, i.e., the function A = uPO/k is given on the wing, where 80 is the angle of attack, as a constant. Secondly, this holds when the wing surface is linear, generally speaking, uncambered, with generators lying in planes parallel to the y = xplane (x~zplane in the original coordinates), then the derivative of the function A(x,y) given on the wing satisfies the rela tion Ax = A In particular this is a wing with a cylindrical surface formed in the manner described. In these cases, only the contour integrals and the integrals over the line segments L1 and L2 remain in the formulas for the pressure. (III) The pressure formulas take an especially simple form when the wing surface is such that the function D(S,rl;x,y) 50 on the wing, at the same time as the side edges ED and E'D' are straight lines parallel to the stream (combination of cases I and II). In this case, the pressure difference above and below the wing in any region can be repre sented by p(x,y) = ,1()xy 1 d di~ (24.10) L 5 NACA TM 1585 where the~ plus sign, is taken if the lines MIM3 and M2M4 intersect on the wing and the minus sign if these lines do not intersect on the wing. B~ence, the pressure on the wing surface is expressed by the curvi linear integral. taken over the arc L of the wing leading edge. (IV) Let t~he wing plan form be such that the points D and E and E' and D' coincide. In this case, calculation of the pressure on the wing surface is also simplified because there are no regions II, III and III' on the wing. In particular, the trapezoidal wing belongs to this case. 2. The pressure forrmulas show that there can exist a geometrical locus F*(x,y) = O where the pressure on the wing p(x,y) = 0. Down stream of this geometrical locus, the pressure difference p = p2 pu is negative. For example, if D(t,?;x,y) 5 0 locus F* = O is found in the region aeteristic cones with vertices E and regions II and IV or through TV an The first case occurs only when K;, th on the wing then the geometrical of the wing l\ying inside the char E' and passing through either Id V or or lying entirely in V. re intersections of' the lines 01K and 02K parallel to the coordinate axes, appears to be outside the region of influence of the vortex sheet, as shown in figure 27, for example.. In all these cases, the points T and T' are on the geo metrical locus of F* = 0. The curve F* = O may also be shaped convex downstream and not as shown on the figures. Let us write t~he equation for the geometrical locus. Fw = 0. In region II: d 1E, d5 F*(x,y) = = 0 p Y) / 1 2 1 idibiL 1~~) 2 1 dt2(x) lx fl 02(x) dy \I x (y) dx y Y \/ 2(x) (24.11) N~ACA TMl 138) In region IV: JC1(5)1  F*(xy)  'j;i J xe[~= O (24.12) In region V: F*(x,y) = 1 ~(x) Y.(y) = 0 (24.13) If the side edges of the u~ing are direction or the wing is such that E inglyr) coincide, then F*F = O takes a not changed, blut in regions II aLnd of equations (24.11) aind (24.12) lines paraLlel to the free stream and D (E' and D' correspond Ssimplle form. In region V it is IV, we haive, respectively, in place (~24.1,4) and (24.1_5) In all cases when the pressure difference on the wing, according to equations (24.2) to (24.9), is expressed onily by means of curvilinear integrals taken over L of the wing contour, it is easy to constrct the zeropressure curve graphically, keeping in mind that the zeropressuree curve in these cases is the geometrical locus of such points M on the wing surface for which the points M1 and Mg on the wing contour coin cide. That is, the are on the leading edge over which the cuirvilinear integral is taken shrinlks to a point. We construct the zeropressure curve as follows: From each pojint NIO on the leading edge we draw the lines NfyI and 110 I2 parallel. to the coordinate axes intersecting the side edges ED and E'D' as shown 1 F* = $1 2(x) r(y) = 0 F* = [2(x) ~Y) = O NACA TM 1585 in figures 29 and 3O, or the trailing edges as shown in figures 31 and 32. From the~ points of intersection N1 and N2 within the wing again we draw lines N1N and 2NW" parallel to the coordinate axes. The geo metrical locus of NS*, where these lines intersect, is the desired zero pressure lin3e. For exampn~le, for a asymmetric wing, if the side edges ED and E'D' are parallel to the~ stream, the zeropressure curve passes through G and G' and is the line equidistant from the leading edge (fig. SO). The? points G an are shown on figures 29 to 32. If E and D, E' and D'", correspondinglyr, coincide and the trailing edges are straight lines then F*n = O passes through C and 0' and. is the curve obtained by inverting the leading edge 'Erelative to the center of inver sion. O*. The center 0* is the point of intersection of the trailing edges (fig. 31). If the wing is asynmmmetric and if the side edges ED and E'D' are parallel to the free stream then the zeropressure curve is the reflection of the curve equidistant to thne leading edge and passing through 0 and G', relative to the line equidistant from the side edges (fig. 29). If the points E: and D, and also E' and. D', coincide and the trailing edges are straight lines making identical angles with the stream then the geometrical locus F*n = O is the reflection of the curve obtained by an inversion, with center 0*, of the leading edge and passing through the points G and G;' relative to the line equidistant from the side edges (fifg. 32). 3. All the obtained results are generalized to the case when the leading edge E'E is given not by one equation y = gl(x) but consists of segments of smooth curves given. by y = $1k(x), where k=1 ,... with n anyr integer. In such cases the surface and contour inteFr~als in the formulas for the pressure should be divided into component parts for the actual evaluations. The side, ED and E'D', and trailing, DT' and D'T, edges may also be piecewise smooth. The same generalization holds for the previous three sections. LC. All the results are generalized in the case of the asymmetric flow over a ving which occurs, for example, in the motion of a yawed wing. Let us consider a wing of arbitrary plan form with an angle of yaw r as shown in figure 55 NACA TN 158) The pressure on the wing can be comp~uted by the same formulas if the equation of the are EO'EO, in the coordinates transformed to the origin O, is taken as the function y = fl(x). The equation of EODO (corre spojnd i ngly EO'DO') is y = 4(x). In this case EODO acts as the~ wing tip. Finally~, for the trailing edge, DO 0, we have the equation y = X(x) (correspondingly for DO'TOI) ). As is known, knowing the acceleration potential or the velocity potential on the wing surface, we can easily compute the aerodynamic forces on the wing. In order, we represent the aerodynamicforce foruas using the adi ginal coordinate system shown in figures 1 and 2. The lift P on the wing is P = 2pS O0(x.y) dx d (24.16;) where the region of integration in C is defined by Jr0(y) fx X X1 (y) and yD' J Y 6 YD where x E Or(y) is the equation of D'E'ED atnd x = X1(y) is the equation of the trailing edge D'TT'D (figs. 217 and 28). The limits yDL and yD are respectively the coordinates of D' and D) of the ving. Since according to linearized theory B0(x,y) = u by, bx then integrating (24.16~) over x and keeping in mind that the velocity poten tial is zero on D'E'ED fran conditions (1.11) and (1.12) of part I, the lift is ~P = 2pu W 1 b~ldy D1. If the trailing edge is piecevise smaooth~, then in actual computa tions the contour integral must be divided into its component parts. NACA TM 1383 The expression for the moment M~by due to lift relative to the Oy,axis is Mby = 2p jl O (x,y)x dx dy 2.) The moments relative to the other axes have the same form. 6. The explained theory can be generalized to the case of the flow over a tail or over a biplane in tandem. We proceed as follows to obtain formulas to compute the pressure on the tail taking into account the influence of the wing. Express P0% + (P0y at M(x,yr) on the tail using the basic formula (21.1). In the expression for q0x t 0y under the integral sign insert 4x y j on the vortex sheet. The function 9x y $ is found from the Abel integral equation which is constructed by the method of section 5 Ijn the case of flow over the tail the different characteristic regions on the tail must be separated just as was done in figures 27 and 28 for the uniform motion over a wing. Only in this case, to divide the tail surface into regions, there must be taken into account, on the one hand, the wing effect and on the other hand, the tip effect and also the effect of the vortex sheet of the tail itself. NACA TM 1J83 APPElfDIX EXAMPLES The following examples, solved by N. S. Burrow and M. M. Priluk, will serve to illustrate the methods explained before. A. ArrowShaped (or Svallowtail) Wing Let us consider the arrowshaped (or swallowtail) wing plan form where the leading edges are formed by the segments AD and AD' and the trailing edges by the segments DB and D'B as shown in figure 34, Let the following geometric parameters be given: 61 the angle between the leading edge and the freestream direction; 62 the angle between the trailing edge and the freestream direction and 2 the wing semispan. The equations of the wing leading coordinates with origin at 0 are edges in the x,y characteristic line AD yl + o a+ 1a (1 cot a" tan 61)xl + 21 cot line AD ' and the trailing edge equations are line DB yl =1 + o "tnB(1 cot aWe tan line D'B YL=1 cot ate tan 82 c a n 82)xl + 21 cot a*l 82 x 2 ota where the angle am* is the semiapex angle of the characteristic cone. Let us consider the wing for which 81 > a+ and 52 > at; that is, a wing surface not affected by the trailing vortex sheet. NA.3CA TM 1383 We will assume that the wing surface is a plane inclined by an angle B0 to the freestream direction. Thrfoe tedriaiv  will be a constant everywhere on both sides of the wing. surface and will bre given in the form 2 p0tn2 (Al) In conformance with the method we divide the wing surface into the three characteristic regions la, Ib, and Ic, with each region having its own analytic characteristic solution and taking into account the angular point A of the leading edge (fig. 34). Let us canpute the stream pressure on the wing surface in each region. Using the formula ($1.3), we find the pressure in the regions la and Ib, lying outside the characteristic cone from A, to be 2u2pp0 p 2 u2pp0 tan a* (A2~) lu2 1 1 Sa2 This formula shows that the pressure in regions la and Ib, is a constant. In region Ic, lying inside the characteristic cone from A, we find, by using the same formula, the pressure to be 2u2FPpO tan 81 2 o *tn5 o lx p(x,y) = coi tn 1 b2 tanlOlt LC4t; + ~ 1X cot2a* an2 l n1 + cot a* tan 61 y1 I cot 61 p. tan~1 1 + cot a* tan 61 y 1 cot 61 x (Ag) 1 cot a* tan 61 71 cot 81 x x + y cot an x '1 cot 81j 2u12pp0 tan 61 x Jco;t2 tan2 51  1 1I cot a" tan 81 1 cit 1 tan n 1 + cot (z" tan 61 y cot I NACA TM 1585 In the original coordinate system shotin in figures 34 and ~5, (AS) becomes p(x,y) = 61 x + y cojt rz4 a" + x 1 cot 61 2 1 1 + cot a" tan 81 co 1 tan i x V1 cot a" tan 61 y cot a* + (A4) These formulas show that the pressure is constant along each ray from A in region Ic. figures 56 and j7, respectively, are the pressures along a parallel to the yaxtis and along the section A2B2 pa~r Shown in section AlBy allel to the xaxis. The lift P of the considered wing is 2u2 02tan 62cta(;lbll tan 82 2 1 ot a*tan 81 1+ tan 82 ot2 n 61 tan 61 1 cot a* a 8 2.ta 51 tan82 tn1cot a" tan s1 + 1 xr tan 81 + tan 62 Vcot a+ tan 61 1 1 lcot a*" tan 62 1 tan cot a'A tanz 82 + 1 4 tan5 62 " tan 81 1802 61 ta2 62) (A5) The lift coefficient Cz is 480 tan 81 2 1 o an C, = 1acn~2 tan1 Icta z K cot a" tan co2 tan 1 tan 2 gl n' 1 ~ 4 ~ 2 tan 61 tan 62 1cot a+ tan 51 + 1 16B0 tan2 62 tan1 rc(tan 61 + tan 62) ct~2 a tn g1 61 1 81 + 1 cot a* tan 821 Vcot a+ tan 82 + 1 (A6) See the work of M. I. Gureyic'h: On the Lift of an ArrowSha~ped Wing in Surpersonic Flow. Prik. Mate. Nekb., Vol. X, No. 4, 1946. NACA TM 1383 As is well known, the wave drag coefficient C, is related to the lift coefficient through Cx = POC,. Let us consider particular cases of (A6). In the limit as 61 y>~ we obtain for the triangular iin7g C, = 4P0 tan a" (A7) the well known result for the lift coefficient of a triangle. Comparing (A6i) and (A7) we conclude that for identical wing speeds and identical angles of attack the lift coefficient of the arrowshaped wing exceeds the lift coefficient of the triangular wing. In the particular case when 62 = 61; we obtain the infinite span arrowshaped wing. In the limit as 52' 1 (A6;) yields 4P0 tanl 81 C, z cot2 a* tan2 61 1 This result shows that the lift coefficient of an infinite span arrow shaped wing equals the lift coefficient of an infinite span slipping wing with slip angle 6L. Formula (A6) shows that with increasing 61 and 62, the angles between the leading and trailing edges and the free stream, respectively, the wing lift coefficient decreases. The dependence of Cz for an a~rrowshaped wing on 61 and 62 is shown in figures j8 and 59. B. Semielliptic Wing Let us consider the wing plan form which is half an ellipse as shown in figure 40. Let the semiaxis al and bl of the ellipse be given. Let us assume that the wing moves, as shown in the figure, in the direc tion of the axis of symmetry. NACA ?TM 1838 The equation of the leading edge, the line D'D, in characteristic coordinates with origin at 0 is yl = 1 and the trailing edge equation in these same coordinates is 71 a2 bl2 cot2 aw~xl r 2alb1 cot aX a 2 + bl2 cot2 aX x12 al l cot2 ase In the original x,y coordinates the trailing edge equation is b \al2 x2 y = + I (Bl) a The plus sign relates to the are CD of the ellipse and the minus sign to the aeC CD'. Let us assume that the wing surface is a plane inclined at an angle "i90 BO to the freestream direction, therefore the normal derivative  oz1 as given by (Al). Let us consider the flow around the semietlipise when the cha~racter istic cones from D and D' intersect on the wing surface. In con formance with the method we divide the wing surface into the four regions I, VI, VI', and V. Region I is outside the characteristic cones from D and D', hence the vortex sheet trailing from the wing exerts no effect here. Region VI is within the characteristic cone from D but outside the cone from D'. Conversely, VI' is within the cone from D' and outside the cone from D. Region V, however, falls within both the characteristic cones from D and D'. Using the formulas, we compute the pressure in each region on the wing surface. The pressure in I is constant everywhere and expressed by (A2). In VI the pressure distribution in the x,y coordinates is given by p = u2 pO tan a" x 12 sin1 cot aX B1y + B2 1 + 2albl cot a* 31 ~12 B2 NACA TM 138) where B1 = al2 + bl2 cot2 a+s f1 = x + y cot a" B2 = al2 b12 cot2 a* Similarly for region VIr. The pressure distribution in V is 2u2 0g tan a+ p(x,y) = x 1 cot a? Bly' + B2 1 + 2alb1 cot a? IBI 12 1i cot at Bly + B2 2 2albl cot a* IBI f22 sin (B3) where f2 = x y cot a* and Bl, B2, and fl are as defined in (B2). Graphs of the pressure distributions along the respective sections AlBI and A2B2 parallel to the yaxils are given in figures 41 and 42 and along the corresponding segments A3B and A4B4 parallel to the xaxis are shown in figures 45 an~d 44. Spanwise section lines A BL and A2B2 a~re shown in figure 45; whereas chordwise section lines 3B3 and A4B4 are shown in figure 4O. If the seTmiaxis of the ellipse are given in a special way; namely, if there exists between the semiaxes the relation al = bl cot a", then forml (~B2) for the pressure distribution in region VI simplifies, becoming p(x,y) = u2 Otan a* 2_ sin1 .,, 2Tx ctx~ (B4) This corresponds to the case where the characteristic cones with apexes at D) and D' intersect the wingstrailing edge on the axis of symmetry of the wing at; the point C; consequently the region V on the wing now vanishes. al2 b cot2 a cot a* l+ bl2 cot2 a, NACA TM 1585 In the general case for the flow around a sermielliptical wing, it may be shown that on the surface of the wing in region V, there exists a certain curve along which the pressure difference beaten the upper and lower surfaces of the wing reduces to zero. Downstream, fran this curve on the surface of the wing the pressure difference becomes nega tive. We find the equation for this line of zero pressure by equating the right side of (BS) to zero. l2+b2cot 2 ,i t 1 2 bl cot 2 2 4al2bl12 cot2 ,*x]2 cot2 a*2 4al1bl2 cojt4 + 16a bl32 cot6 a* y2 = 4al~bl2 cot2 a' al2 bl2 cot2 dx2 (al2 + b 2 cot2 a*) After obvious transformations, we in the following final form represent the desired geometric locus y2 + 1 b22 (BS) where 2alb1 cot G* a = 2 a2 + bl cot2 ~A* b2  (B6') al82 bl2 NACA TM 1585 These results show. that the3 zeropressure line is the are of an ellipse with semiaxes a2 ndb2 related through (B6) to the semiaxes al and b1 of the are of the ellipse which is the wing trailing edge. The directions of the semiaxes a2 and b2 coincide with those of the semi axes al and bl. In order that the zeropressure line should not pass through the wing surface, the elliptical are forming the trailing edge of the wing should not have a real point of intersection with (BS), which determines the zeropressure line. Comparing (Bl) and (B5) we obtain the following result. In order that the zeropressure line, of a plane wing of se~mielliptic plan formn moving at the supersonic speed u, should not pass through the~ wig surface, it is necessary and sufficient that the geometric parameters of the wing satisfy the condition al 5 35 b cot a* (BT) Constructed in figure 46 is an isometric view of the pressure on a semielliptic wing in the general case when (B7) is not fulfilled and there exist the regions I, VI, VI', V on the wing. C. Hexagonal Wing Let us consider the wing of hexagonal plan forml shown in figure 47. Let the leading edges be the lines 081, and 081', the side edges ElD and ED'parallel to the free stream, and the trailing edges DB and D'B3. In characteristiccoordinate space, the wing has plan fornn as shown in figue 48., Let us assign the following geometric parameters: a the angle the leading edge makes with the free stream; 7 the angle the trailing edge makes with the free stream; 2 semispan and h chard. Let us consider that wing for which a > a*, 7 > a*. The first :inequality me~ans that the wing surface extends outside of the character istic cone from 0. The second inequality means that the wing surface is outside the sphere of influence of the trailing vortex sheet. NACA TM 15385 The equaEtions of the lines forming the uing contours are: the line~ 0El y = x ta~n a or in characteristic coordinates 7T1 = E x~l where 1 cot 0.* tan a m 1 + cet a.* tg g b > a,*; the line OE1' here m < O, since y = x tan a and yl = mxl the line E1D y = 1 and yl = xl + 2 cot a+2 the line E1'D' y = I and yl = xl 2 cot a'c3 the line DB y = xtan 7 + h tan r and yg= xl+n tan .1 NACA TM 1385 and finally D)'B y = x tan r h tan r and yl mlxl + n2 :the re 2h cot a* tan r nl = 1 + cot a" tan r 1 + co"V a" tan 7 1 cot a" tan~ 7 2h cot a* tan r n2 = 1 cot a* to' 7 In conformance with the method we divide the wing surface into the 13 characteristic regions shown in figure 48. Assuming that thie surface of the wing is a plane, we give the stream line condition in the form (Al) and we compute the pressure in each characteristic region. We produce below the results of computing the pressure on th wing surface as formulas already transformed back to the original coordinate system. Th pressure in la and Ib is constant and expressed by (A2). In le the pressure is 2upp 1 xcta* y p(x,y) = c\rmco t2LP + tan1 __ . MV ITcota* 2 I nm x + cot a* y 1 cot a* yl tan rn f x m o X(1 Yx + cot a* Hence it follows that the pressure is constant along each ray starting from 0 in Ic. In IIIa 2u2po (1 m) p x,y) = n ( m cot a* . 2 cot a* (1 Y) (C2) (m 1)(x + cot a" y) + 22 cot a* ~ 2m cot a*(Y 1) Y(1 m)(x + cot OL* Y) + 2ml cot a*t 2u 2pp0(m ) i a1 1 xco *y p(x,y)=tn  Jrf\fm cot a* JC x + cot a y 2 cot a*(l y) (1 m)(x + cot af y) + 2E cot a2* x cot a* y tan1 (IE x + cot a* y (03) In IIIc 2u2 O(1 m) It {cTE cot ar* 1 2m cot a"(y 2) ttan (1 )(x + cot a*" y) + 2mZ cot a* (c4) In IIa. tan1 2 2u pPO0 m) p(x,y) = r Jm cot a* tan 1 (1 m)(x cot ar* y) 21 cot at 1 r ~2 cot cr*(l + y) (CS) NACA TM 138) In IIIb  ta1 i I .I S(1 m)(x + cot a* y) 21 cot a* 1 1(1 m)(x + cot at y) 21 cot a+ tan 2 cot a+(l + y) tn1 xcot a* y tan~ mx + cojt a* y (1 m)(x cot a*y)~ + 2 cot (z* 1 2 cot a*(l y) tan' ; (1 m)(x + cot a* y) + 2m? cot a* 96 In It NACA TM 1585 2u p~~y=2u pBO(1 m) 1n~ 1 Ix cot 2* y n {1~ E cot a* ( mYx + cot a2* y 2 cot a*C(2 y) tan1 (C6) In IIe 2 2u FpB(m 1) 1 p(x,y) =I tan i 1Jmcot a* 1 1 ' 2 cot a"(2 + y) 1 1 x" cot a* y tan" (7 m Vx + cot a* y x cot a" y x + cot a* y + tani II m (C7) Formulas for the pressure distribution on the wing surface in regions IIIa', IIIb', IIIc', and IIa' may be obtained from (C2), (C3), and (CSf), :respectively, if coordinates appropriate to the specific regions are chosen, The formulas for the pressure show that there is a zeropressure line on the wing surface, downstream of which the pressure difference below and above the wing becomes negative. This line is formed of the two segments KN and K'the equations of which are (C4 ); NACA TM 1J83 y = x tan 8 22 tan aA tan a y = x tan 6 + 23 tan aiw tan~ a. (CS) and which are parallel to the~ leading edges E10 and E1'O. The zeropressure line mayr easily be constructed graphically. Graphical representations of the respective pressure distributions in the sections A1Bl, A2B2, A~B A4B6, and AjB5 parallel to the yaxis are given in figures 49, jO, 51, 52, and 53. An isometric pressure surface is shown in figure Sk for the hexagonal plane wing. Translated by Morris D. Friedman NACA TM 1583 REFERENCES 1. Sedov, L. I.: Theory of Plane Motion of an Ideal Fluid, 1939, (Russian) 2. Kochin, N. E.: On the Steady Oscillations of a Wing of Circular Plan Form. P.M.M., vol. VI, no. 4, 1942. NACA translation. 3. Prandtl, L.: Theorie des Flugzeugtragfluigels in Zusammnendruickbaren Medium. Lufifahrtforschung, no. 10, vol. 13, 1936. 4. Ackeret, J.: Gasdynamik. Handbuch der Physik, vol. VII. 5. Krasilshchikova, E. A.: Disturbed Motion of Air for a Vibrating Wing Moving at Supersonic Speeds. P.M.M., vol. XI, 1947. Also D.A.N., vol. LVI, no. 6, 1947. Brown translation. 6. Krasilshchikova, E. A.: Tip Effect on a Vibrating Wing at Supersonic Speeds. D.A.N., vol, LVIII, no. 5, 1947. 7, Krasilshchikova, E. A.: Effect of the Vortex Sheet on the Steady Motion of a Wing at Supersonic Speeds. D.A.N., vol. LVIII, no. 6, 8. Krasilshchikova, E. A.: Tip Effect on a Wing Moving at Supersonic Speed. D.A.N., vol. LVIII, no. 4, 1947. 9. Krasilshchikova, E. A.: On the Theory of the Uhsteady Motion of a Compressible Fluid. D.A.N., vol. LXXII, no. 1, 1950. 10. Falkovich, S. V.: On the Lift of a Finite Span Wing in a Supersanic Flow. P.M.M., vol. XI, no. 1, 1947. 11. Gurevich, M. I.: On the Lift on an ArrowShaped Wing in a Supersonic Flow. P.M.M., vol. X, no, 2, 1947. 12. Curevich, M. I.: Remarks on the Flow Over Triangular Wings in Super sonic Flow. P.M..M., vol. XI, no. 2, 1947. 13. Karpovich, E. A., and Frankl, F. I.: Drag of an ArrowShaped Wing at Supersonic Speeds. P.M.M., vol. XI, no. 4, 1947. Brown translation. 14. Frankly, F. I., and Karpovich, E. A.: Gas Dynamics of Thin Bodies. 1948. Translation published by Interscience Publ., N. Y. 1954. 
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