00 CO < 3 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1383 FINITE SPAN WINGS IN COMPRESSIBLE FLOW By E. A. Krasilshchikova From Scientific Records of the Moscow State University, Vol. 154, Mechanics No. 4, 1951, with appendix condensed from a document "Modern Problems of Mechanics,* Govt. Pub. House of Tech. Theor. Literature, (Moscow, Leningrad) 1952. v^V''ii«»->- Washington September 1956 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM I385 FINITE SPAN WINtJS IN COMERESSIBLE FLOW^ By E. A. Kras ilshchikova This work is devoted to the study of the perturbations of an airstream by the motion of a slender wing at supersonic speeds . A siorvey of the work related to the theory of the compressible flow around slender bodies was given in reference 1^4- by F. I. Frankl and E. A. Karpovich. The first works in this direction were those of L. Prandtl (ref . U) and J, Ackeret (ref. 25) in which the simple problem of the steady motion of an infinite span wing was studied. Borbely (ref. 25) considered the two-dimensional problem of the harmonically-oscillating nondefonnable wing in supersonic flow by using integrals of special types for solutions. Schlichting (ref. 2J+) considered the particular problem of the flow over two-dimensional 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 19^5? 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 arrow-shaped and triangular wings when the flow, pertiorbed by the wing motion, is conical. The work of E. A. Karpovich and F. I. Frankl (ref. I5) was devoted entirely to the problem of the suction forces of arrow-shaped wings. In 19^2, 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 1914.6-14-7 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 oiir candidate's dissertation (ref. 5)^ in which we found the effective solution for a limited class ^Scientific Records of the Moscow State University, Vol. 15^, Mechanics No. 4, 1951, pp. I8I-259. The appendix represents a condensation made by the translator from a document "Modern Problems of Mechanics," Govt. Pub. House of Tech. Theor. Literature, (Moscow, Leningrad) 1952, pp. 9^-112. 2 MCA TM 1583 of harmonically-oscillating wings. In reference 6 we solved the problem for wing influences by "tip effect." Later works (refs. I5, 16, and I7) were devoted to the same problem. In reference G, using an idea of L. I. Sedov as a basis, we reduced the problem of the influence of the tip effect on harmonically-oscillating wings to an integral equation. The question of the flow over wings of finite span remained open for some time. At the start of 19^7^ 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 I8, M. D. Khaskind and S. V. Falkovich solved the problem, in the form of a series of special functions, for a harmonically 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 steady-motion 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 finite-span 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. 2l) and Von Karman (ref. 22) who solved the problem of the steady flow over finite-span, symmetrical' 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, 'J, 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 wing -mot ion in quadratures and for the harmonically-oscillating wing in a power series of the parameter defining the oscillation frequency. The present work is a detailed explanation and further development of oiir papers (refs. 6 to 9) which were published in the Doklady, Akad. MCA TM 1583 5 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 ISkj-kS 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 two-dimensional 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 pressvure 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 ll 1. SEOTING 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 fiindamental wing velocity u. The Ox-axis is directed opposite to the wing motion and we take the x,y-plane such that the z coordinates of points on the wing shall be small (figs. 1 and 2). We will consider the normal velocity component on both sides of the wing surface to be given by Aq + Aj^f jt + q] (1.1) Results of Part I, sections 6 and 7 were found by the author in May, 19^7 at the Mathematics Institute, Akad. Nauk, USSR. k NACA TM 1585 The first component defines the wing siirface Ao = -uPo (1-2) where Pq ^^ '^^^ angle of attack of a wing element. The second compo- nent defines the additional unsteady motion of the wing. The functions Aq and Ai 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 ex±ernal forces . The velocity potential of the perturbed stream cp(x,y,z,t) is represented in the form cp(x,y,z,t) = cpo(x,y,z) + q)i(x,y,z,t) (1.5) where the potential cpQ corresponds to the basic steady motion of the wing and the potential cp-, 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 dx dx. dy dy dz dz The functions cpQ and cp-^ and their derivatives will be considered first-order q^uantities and second-order quantities will be neglected. With these assumptions it is known that the potential cp-, satisfies the wave equation which in the moving axes is = (1.4) (a2 -u2) dx2 2 ^^9i ■ + a"^ i Sy2 2 ^^Ti + a"^ az2 at2 2u dtdx and the potential To satisfies {> a2 - a2) ''^0 , ax2 a2 ^'^0 , 2 ^% 6.^ = (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 siorface opposite to its motion. Just as on the wing surface the velocity potential undergoes a junip discontinuity on this sheet. NACA TM 1585 We represent the projection of the vortex sheet on the x,y-plane as the semi-infinite strip E]_ (fig. l) extending along the x-axis to infinity from the trailing edge of the wing. Let us establish the boundary conditions which the functions (p„ and cp]^ satisfy. Let us transfer the boundary conditions on the wing surface parallel to the z-axis onto the projection E of the wing on the x,y-plane, which is equivalent to neglecting second-order quantities in comparison with first-order ones. Therefore on the basis of equation (l.l) we obtain the streamline condition ^=Ao(x,y), OZ -^ = Ai(x,y)f[t + a(x,y)] OZ (1.6) which must be fulfilled on both the upper and lower sides of Z. The kinematic condition^ which expresses the continuity of the normal velocity components of the fluid particles, must be fulfilled 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 z-axis onto its projection E]^ on the x,y-plane which is again neglecting second- order q\iantities. Therefore we have the conditions dcpo dz Z=H-0 dCPQ dz_ z=-0 dcpi z=fO dcp-. i'z Jz=-0 (1.7) to be fulfilled on E-^. Furthermore, the dynamic condition which the potentials cpQ and qp-^ 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 dt dx 2 \dxj Vdy/ \dz, ' + r(t). ^- f MCA TM 1383 Keeping equation (I.3) in mind and neglecting second-order quantities, we obtain dcpo dx z=fO dcpQ dx z=-0 at dx z=fO ^9l ^9l — i + u — - dt dx (1.8) ^z=-0 which must also be fulfilled on E]_. After boundary conditions (I.6) and (I.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,y-plane. are odd functions in z Therefore, the functions cp^ and cp-j^ '?o(x^y^-z) = -^0^^'^'^^' 9i(x,y,-z,t) = -9-L(x,y,z,t) (I.9) Combining equations (I.8) and (I.9) ^e conclude that the functions and 9-, satisfy the respective conditions dqv, ^cpi ^9i — - = 0, — - + u — - dx dt dx = on Z- (1.10) Since the 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 which we assume to be zero. Hence we have the condition on the disturb- ance wave 9o(x,y,z) = 0, cp3_(x,y,z,t) = (1.11) The potentials cp_ and cp^^ are continuous functions everywhere outside the two dimensional region Z + Zq^ and, as was established, are odd in z, therefore, in the whole x,y-plsLne outside of the region Z + Z-j^ where the medium is pertxirbed, the following conditions are satisfied: cpQ(x,y,0) = 0, cp-L(x,y,0,t) = (1.12) RACA TM 1583 7 The region where equation (l.l2) is satisfied is denoted in figure 1 by E2 3-ncL Sg'. Thus the considered hydrodynamic problem is reduced to the following two boundary problems: I. To find the function cp. (x,y,z,t) which satisfies equation (l.i|) and boundary conditions (I.6), (l.lO)^ (l.ll), and (l,12). II. To find the function cp^Cx^y^z) which satisfies equation (I.5) and boundary conditions (I.6), (l.lO), (l.ll), and (I.I2). Since the functions cp„ and cp, are antisymmetric functions rela- tive to the z = plane, it is sufficient to solve the problem for the upper half plane. From the solution of boundary 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 Aq replaces A^^. 2. VELOCITY POTENTIAL OF A MOVING SOURCE WITH VARIABLE INTENSITY 1. Let us construct a solution of equation (l.^) 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 fi(t) . 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 = fQ(t - t]_)f]_(t) . The law of variation of the function Tq is the same for all the sources if the initial moment of each source is considered to be the moment when it came into 2 being. The function f]_ 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 ti (fig. 3) • 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 .ti" fo t - t^ -iKf-f $l*(x',y',z',t) =A / 1 dti tl' i x' + ut)2 + y'2 + z;'2 (2.1) ^Prandtl (ref . 3) considered an analogous problem with q = fgCt - tj.) 8 MCA TM 1585 where A is a constant with the dimensions of a velocity. The limits of integration t-^' and t]_" take into account those sources which affect M at time t. The origin of the fixed coordinates 0' is placed at the point at which the source started at t = 0. Introducing the new variable of integration t = a(t - t^) - 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.I) into %^'^i/(-;'r-tf--)fr--4' 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 and 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 T-[_. 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 f]^(t) . The derivation remains valid if the additive constant aj^ is added to the argument t of the function f]_. Putting the sources at the origin, we find the velocity potential from equation (2.2) by considering the interval of integration from to Ti to be vanishingly small. Then, neglecting the term (—It and putting — / fof— |dT = C where C is a constant, we obtain the desired solu- ^ Jo V^/ tion for equation (1.^4-) in the general form (p*(x,y,z,t) = C ^' ^ u2 - a2 u2 - a2 ^x2-(g.l)(y2.z2) f C (y2 + z2'' , ux a t + o-i - — + u2 - o2 u2 - a2 |/x2 - (g - l)(y2 . z2) . ^.2 _ ^1 _ ,j(,2 , .2) (2.5) MCA TM 1385 Let us note that each component of the arbitrary function f]_ as well as the constant C and a]_ in equation (2.5) is separately also a solution of equation (l.ij-). In equation (2.5) putting ax = and the velocity of motion of the source u = 0, we arrive at the well-known 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 f^ in equation (2.3) to be constant, we arrive at the Prandtl (ref . 3) solution for the retarded potential of a moving source of constant intensity * - ± ^ -(^-A{y^^-^) 90* = 2. It is possible to obtain, by the same method, the velocity potential of a source with the variable intensity f^Ct) moving arbitrarily. the motion is given by X = F-]^(t), Y = 0, Z = and when > a. For example, in the case of rectilinear motion of the source when dFx(t) dt that is, the motion of the so\irce is supersonic, the velocity potential of the source at the origin of a coordinate system moving with the source is ^ A ^ R.p. A2(x,y)e (iv.l) where A2(x,y) defines the amplitude and initial phase of the oscillations. Using the obvious relation e^^ + e-i9 = 2 cos 9 and equation (3-5)^ the basic formula for the velocity potential (3-1) is represented as cp,(x,y,z,t) = - i eP'^ J- 7l ^ -PI e ^cos A \/(x - if - k2(y - T,)2 - kV S(x,y,z) L^-lz=0 y/(x . |)2 . k^Cy . t,)^ . k' J dT)d| (1+.2) 2z2 where and X = Ofi. P = u2 - a2 Icou u2 _ a2 Keeping the second inequality of equation (5 '9) in mind, let us compute the inner integral after which we obtain a solution of the prob- lem for a wing of infinite span cp^(x,z,t) = - i eP^ -^x-kz e-P^I z=0 A ^(x - 1)2 - k2z2_ where Iq is the Bessel function of zero order. By means of equation (4.3) the velocity potential may be computed at those points of the x,z-plane for which the interval of integration on the Ox-axis does not extend beyond the wing, i.e., at those points of the NACA TM 1583 15 x,z-plane not affected by the vortices trailing from the wing because the function — - is given only on the wing. In order to compute the dz velocity potential at any point of the x,z-plane by equation {k.J>) it is dcp-, necessary to determine — =, using equation (1.8), everywhere on the dz Ox-axis outside the wing. Let us express, by equation (U.j), the velocity potential 9-, for any points lying on the Ox-axis outside the wing, which, according to equation (I.8), equals on the Ox-axis everywhere outside the wing 9j^(x,t) = R.p. q>i^{l)( '-(x-l) (k.k) where V = u and I is the abscissa of the trailing edge. Then we obtain the integral equation Scp^ (h.^) which — = satisfies on the Ox-axis outside the wing. In reference 5^ dz we solved such an integral equation. The inversion of equation (^.5) is dz ■ z=0 .-^^,^..jyun,{u..o]^ (4.6) where F* denotes the right side of equation (4.5)^ the known function, and where I-^ is the Bessel function of first order. Therefore, keeping equation {k.6) in mind, we can calculate the velocity potential at any point of the x,z-plane by equation (k.^). l6 MCA TM 1385 The problem considered in this section was solved and explained in reference 5 from another point of view. 5. INFLUENCE OF THE TIP EFFECT 1. To calculate the velocity potential according to equation (5-l) and also through equation (j-lO) or {k.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 dcpi component — - everywhere in the region of integration S from the dz boundary conditions of the problem on the z = plane. Let us consider the case when the region of integration S inter- sects the wing surface and the region Sx lying outside the wing and outside the region of the vortex system from the wing. Region 2L-, (fig. 6) is part of the region Z2 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 the leading edge is defined so that condition (5-7) 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 construct the integral equation for Cvx,y), connected to — =• dz by relation (5 -5)^ in ^5- Let us select the velocity potential cp-, at any point N(x,y,0) lying in ^^ by means of equation {'^.l) , equal to zero everywhere in ^2 according to equation (l.l2) . 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 part of the wing falling in the Mach fore-cone from N(x,y,0), and the region a(x,y) is that part of L-, lying in the same fore-cone. According to equation (3-6) C(x,y) is given in s. In a, C(x,y) is unknown. We therefore arrive at the integral equation which C(x,y) satisfies in Ztl. NACA TM 1585 17 a(x,y) C(|,Ti)K(|,T);x,y;t)diTd| = F(x,y,t) (5.1) where the kernel is ft K = {l,T\;x,y,t) = ^ a(|,n) - 4^^ - ^^ l/(x - 1)2 - k2(y . ,) 2 2 2 2 '} \/(x - 1)2 - k2(y - n)2 '(x - I)'' - k^(y - n) (5.2) and the known function F(x,y,t) = ^ jj Ai(|,Ti)K(|,Ti;x,y;t)dTid| (5.3) i(x,y) If the characteristic coordinates are introduced Xj^ = X - Xq - k(y - yo) , y^ = x - Xq + k(y - yo) , z-^ = kz (where xq ani. jq may be any numbers) then integral equation (5-l) 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 HARMDNICALLY OSCILLATING 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 (i4-.l), then equation {^.l) becomes "(x,y) \Ax - 1)2 - k2(y - ,)2 18 MCA TM 1383 (Sep where the fiinction 9(x,y) = — ^ oz function is z=0' :.~P^ in a and where the known F(x,y) = - IT A(|,Ti) s(x,y) cos [■K /(x - if - k2(y - n)2. /(x - 1)^ - k^(y - n)^ dT] d| (6.2) where A(x,y) = a7 -px z=0 in s. In order to solve this integral equation we Introduce the characteristic coordinates x^ , y^ z with origin at "O" by means of the formula X]_ 3 X - ky, yj^ = X + ky, z-l = kz (6.5) In the new coordinates the variables of integration in a will vary between the limits < _ < ^ = ^i = v K^i) = ^i = ^i (6A) where y\ = ^{'>^\) is "bhe equation of the wing tip - the arc ED of the wing contour - in the transformed coordinates, and x,^ is the abscissa of E defined in section 5 in these same coordinates (fig. 8) Equation (6.I) is transfomied to /(>=! - H)(yi - 11) (6.5) MCA TM 1585 19 where the function 6i(xi,yi) = h% Sz-j_ P (xi4yi) zj_=0 and where the known function is Fi(x^,yi) = - ^1(^1,^1) — * -. ^^ ^ '-^ cini ^h 1(^1 - ^i)(yi - 11) An = \ ■ — —V e Szi P (xi+yi) z^=0 (6.6) Sep, Let us note that the normal velocity of the perturbed flow — — Sz-j_ is related to Sq>j_/Sz-j_ by Bz Sz-j_ For brevity, the index "l" will be left off the independent variable everywhere from now on. 2. Let us look for a solution of equation (6.5) in the form of the power series n=0 2n (6.7) Into both sides of equation (6.5) let us introduce cos H/(^ - ^)(y - n) = y^ ^^ U - O'^Cy - nf a^^ (6.8) 20 NA.CA TM 15 85 Keeping the absolute convergence of equations (6.7) and (6.8) in mind, we multiply them term hy term with the result e(i,Ti)cos[A /(x - O (y - n) ] ^"^ (_j_)n-k n=0 '""S [2(n - .)]l ['" - ^' <^ - ^>]"'' ^2^<^'^' f«-9' Substituting equations (6.7)^ (6.8) and (6.9) into equation (6.5) the latter becomes X y 1 r r 00 k=n , n-k — / J H-K^^'Yl rJ-^) "1. ^2k (^^^^P^ - (y - n)] ^dri d| ^ /,, n=0 k=0 r2(n - k)]l 2k L J Xg t(U n + 1 "i ^^ fe (2n)l ^ -• (6.10) s(x,y) Taking into account the uniform convergence of the series in both sides of equation (6. 10) with respect to the variables i and t] we integrate term by term k=n , .,n-k X X n-k-i vn + l n-- H ^^""-^TTT- /T A(|,Ti)[(x - ^)(y - Ti)] %^ d^ (6.11) n=0 (2n): s(x,y) MCA TM 1585 21 In equation (6.LI) equating coefficients in identical powers of \ we obtain the integral equation which the functions Q^x,y) satisfy r^ p e2,(^,n) ., ^\^' . = Fjx,y) (6.12) '^ ^i(0 where k=0 where, in its turn, _1 fnU,y) = iril^ jj A(^,Ti)[(x - 0(y - n)] ^dr, d^ ^^^^^^ s(x,y) and ^ (6.15) from which the functions f^ are defined for k ^ and n > 0. Let us note that the right side F^(x,y) of equation (6.12) depends, for 92j^, on the coefficients 92jj "but only for k = 0,1,2, .. .,n-l. There- fore, if we find Gq, 62, 9i^,..., 92(n-l)j "then Fj^(x,y) is a known function in the equation which the coefficient 82^ in the general term of series equation (6.7) satisfies. For n=0 the right side in equa- tion (6.12) Fo(x,y) = f;3(x,y) = - ff A(4,ti) -—=^=M= (6.I6) JJ /(x - |)(y - n) s(x,y) is a known function of x and y. Let us solve equation (6.12) for 9_ (x,y). 22 NACA TM I583 The two dimensional integral equation (6.12) is equivalent to the two homogeneous integral eqxxations px e* (^ y) r f ' , d^ = F^(x,y) (6.17) and Ji(0%^^T = e2n(^^y) (6.18) Vy - n each of which reduces to an Abel equation. Using the inversion formula of the Abel integral equation and observing that for any n functions Fj^(x£,y) = hence the solution of equation (6.I7) for the function 62n(x,y) is e^U,y)=ir!^« (6.19) Let us turn to equation (6.I8). We denote the parameter | by x, and again using the inversion foimiula for the Abel equation and keeping in mind that according to equation (6.I9) the right side 62jj^[x,\|f(x)] of equation (6.I8) for y = ilf(x) is different from zero, the solution of equation (6.I8) for 6 p^ is l^lntvK^^l Py e*2nTi(x^n) "^ /y - V(x) «Ji|f(x) /y - n e^(x,y) = i ^ ^"^-^:7 . 1 / ^ L^^Z^ an (6.20) Substituting in equation (6.20) in place of 9L(x,y) its value from equation (6.I9) we obtain the solution of eq\aation (6.12) in the following form: Jt2 ,/y _ ijf(x) ^^E /FT^ , PX py Fn|T^(|,Tl) ^ ' ' ^ " dri dl (6.21) it2 Jxg J^ (x ) /(x - 0(y -Tj) KACA TM 1385 25 Thus, according to eqiaation (6.21), we can evalimte successively, the coefficients Gq, 62^ %}•"> ®2k' ^'tc. Formula (6.21) shows that all the coefficients (n=0,l,2, . . . ) for y = \|f(x), i.e., on the wing tip ED, become infinite as R~ ' where R is the distance of the point (x,y) from ED. Therefore, the velocity of the perturbed stream becomes infinite 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 e^ (x,y) = i JL r P , '■■"'^' a, « (6.3.) which can be confirmed without difficulty by direct differentiation with respect to the pajrameter. Therefore, the solutions of integral equation (6.5) are constructed in the form of the absolutely convergent series (6.7) for any value of the parameter 7\. The coefficients 92Q(x,y) are expanded in the series n=0 We find the function e«(x,y) = Scp-j_ e ^ in e! z=0 ^ (fig. 6) lying off the wing to the left, from eq-uations (6.21) or (6.22) by replacing in the latter the function i|''(x) by il'2(x) (where y = \l'2(x) is the equation of the arc E'd' of the wing contour - the left wing tip) and interchange the role of the coordinates. 2k MCA TM 1583 5. Let us consider a wing of small span. Let the characteristic cones from £]_ and E2_' intersect the wing as shown in figure 9- The points En and E2_' are defined just as are E and E^^ in section 5- Let us divide the x,y-plane where the medium is perturbed into the regions Sq, S^, S2, • • • , Sn, • • • • The region Sn is the M-shaped region lying within the character- istic aft-cones from En and E^' (or within one of them) and outside the characteristic aft-cones from Ej^+i and Ej^+i'- In its turn, we divide the part of the x,y-plane lying to the right and left of the wing into the strips a]_, 02? ■ • ■> O^i, . . . and o±' , •^2'-' • * '' a.^' , . . ., respectively. The strip a.^ lies within the characteristic aft-cone from En. Therefore, a-^ and a^^' are the parts of S^ lying respectively to the right and to the left of the wing. Let us return to the fundamental fonnula for the velocity potential, equation (J4-.2), which is in the characteristic coordinates cp]_(x,y,z,t) = S(x,y,z) ^\ _?ll^ r I -1 e 2 cos[a\/(x - 0(y - ti) - z^\ ^ 2=0 \|(x - l)(y - Ti) - z2 (6.21+) dTl d^ In order to compute the velocity potential by means of this formula 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 S^^ of the x,y-plane, we must first determine — ^ e 2 outside the az wing in the strips a^, a^, . . ., ^n, and a^', ag', • • ., o^\ . . respectively. a^\ ... by e', e'(2)^ . . .^ e»(n) NACA TM 1385 25 Let us denote e "^ in the d-i, ffp, . . . , cTj^, . . • strips by 0, e^^)^ e(3)^ . . ., e^"^) . . . and in a^^', 02', . . ., (2) Let us construct the integral equation for 9^ . Let us express the velocity potential at the point N(x,y,0) in ag by formula (6.2^+) which is equal to zero everywhere in the strips a^., 02, • • ■ (^n (correspondingly in a-]_', ffgS • • • ^^n')- Let us divide the region of integration into the three parts S=s + a+CT-|'* as shown in figiire 10. The function e ^ = A(x,y) is given in s(x,y) on the ^2 ;. n (x+y) wing. In a-|'*(x,y) of a-,', the function r^^ e '^ =6 (x,y) is -^ oz determined by the solution of equation (6.23). In cr(x,y) we denote — ^e ^ ^y Ql^;(x,y). Then we arrive at dz the integral equation satisfied by 9^2) ff e(=)(5,n) °°'=[V('- U(y-1)] a, a^ , F(2)(x,y) (6.25) 26 NACA TM 1583 where the limits of Integration are bounded by xg = I = x and \jf(|) = T] ^ y and the known f\inction f'^^ is defined as f(2) (x,y) = - ff A(|,Ti) c°b|: a/(x- U (y - n jl ,^ ^^ . JJ ^x - l)(y - n) s(x,y) Ti- e.(^^Tj) cos [A|/(x- 0(y_^J l)J ^^ ^^ (g^26) /(x - 0(y - n) We look for the solution of integral eqtiation (6.25) in the form of the power series 9^2^ (x,y) =11 e(2)u,y) A2n (6.27) S=0 2n Moreover, by reasoning slmilaxly to the preceding section we arrive 1 integral e( of series (6.27) (2) at an integral equation for the coefficient Sp^ in the general term (2) (M) ,, '^^^^ =. = 4n(-,Y) (6.28) Vi(0^2n ..vw-^^==== where If f^2)k (^^y) (6.29) fc=0 ^2k (^^^)P^ - ^^^y - ^)]''"^"2 dT) d| (6.50) Equation (6.28) differs from equation (6.12) only in the form of the f(2) function on the right side. Taking into account the condition on the boundary fJ^2} (x^^y) = for any n=0, 1, 2, . . . the solution of (6.28) for 9^22) j_3 obtained by using the solution (6.21) or (6.22) of (6.12) as a final formula if Fj^(2) replaces Fn in the latter. The A^\^,7) = Fn(x. where , in its t\irn. f(2)k( n ^ x,y) [2(n- •k+l ■ k)]l NACA TM 1585 27 function Fn^^'(x,y) depends on the coefficient 92k^^ where k=0, 1, 2j . . ., n-1. Therefore, just as in the previous section, if the 9 2k ^°^ ^^ 0>1;.2, . . ., n-1 are already found, then F^{2) in the right side of (6.28) is a known quantity. Therefore, the functions ^0 ' ^2 ' • ' *f ^2n ' • * * '^°^'^ ^^ found successively. (2) (2) Let us note that Fn , and therefore the coefficient 92n , depends only on the first n + 1 coefficients 9q , 92', . . ., 92n' of the series expansion of e'(x,y) = ^ e 2 oz in a, ' . Reasoning in the same manner, we may find the values of 9^^', 9> ', . . ., 9^ ' . . . in a^, aj^^, . . ., o-^, . . . (correspondingly W ) " ,..., ,... in On , 02^ • • • Ot\t ) • 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 nxomber of strips a-tj or cTtt' . All the results hold for the case when the wing tips are not given hy one equation y = ■>^(x) but consist of curves given by the equations y = ''l'i5^(x) k = 1, 2, . . .,m . The same observation applies to the leading edges E'E (or E^E-, ') of the wing. Therefore, in our problcn the wing contour may be piecewise smooth. If the frequency of oscillation oj of the wing be put equal to zero then the coefficients Bq, qJ^',..., ^0^^^ . . . coincide with the values of the derivatives ^q/^^ ^ '^^^ strips o-^, 02, • • •, o^, . respectively, for the steady motion of a wing when the streamline condi- tion (1.6) on the wing Is given in the form OZ 28 NACA TM 1583 7. INFLUENCE OF THE VORTEX SYSTEM FROM THE WING FOR A HARMONICALLY OSCILLATING WING 1. Let us consider the case when the region of integration S(x,y,z) in formula (^.2) for the velocity potential intersects the vortex sheet y.-< as shown in figure 26(a) (see also fig. 11). That is, let us consider the case when the trailing edge of the wing - the arc DT of the wing contour - or, so to speak, the vortex sheet, acts on the point M(x,y,z) of space. Using condition (l.lO) we determine dcp-j^/Sz in the region Q. of the x,y-plane and shown in figure 11. The region Q. is off the wing within the characteristic aft-cone from D and outside the characteristic cones from T. Therefore, Q is affected by the vortices trailing from the edge UT of the wing but not from D'T'. The region Q partially intersects the vortex sheet Let us return to the characteristic coordinates x-|_, y-^, Zj. which we introduced earlier by formula (6.5). As before, for brevity we omit the subscript 1 from the independent variables. IS Condition (l.lO) fulfilled on E In the characteristic coordinates ^ ^ u ^ + u 5^ = (7.1) St dx dy From equation (T.I) it follows that the function ^00 = cp-L(x,y,0,t)e u 2 remains constant everywhere on the vortex sheet along lines parallel to the direction of the incoming stream^ i.e., along vortex lines from the wing. MCA TM 1585 29 Since the velocity potential cp]_ = everywhere in the x,y-plane off the wing surface and the vortex sheet, then it may be verified that cp^ possesses the specified property everywhere in fi. Let us construct the equation for the function «(x,y) Scp^ ^^ Jz=0 e in a. Let us express cp^ at the arbitrary point N(x,y,0) lying in Q. by using the basic formula for the velocity potential (6.24). We divide the region of integration S into three parts, as shown in figure 12, into s(x,y), CT2_*(x,y) and a(x,y). The regions s and a-j_* are parts of the wing surface and E^, defined above, respectively, which fall within the characteristic fore-cone from N(x,y,0) . The region a is the part of Q. in the same cone. The variables of integration in a vary between xj) % ^ ^ x and X(^) ■^ tj •^ y where xp is the abscissa of D and y = X (x) is the equation of the arc DT of the wing contour. The expression obtained for cp^ is differentiated in a direction parallel to the velocity vector of the impinging stream. Therefore we arrive at the integro-differential equation which •3 satisfies in Q, hxJ^Mi) /(x - 4)(y - n) ^"f ,(^.,) cos[ x/(x- 0(y- T,)] ^^^^^ ^J^Jxi^) •"" /(x - 0(y - ^) .a2 rr Hi,n) -o^^ -^^(- - ^ny - ^ 1 d, dl = .(x,y) (7.2) \M^) /(x - n(y - Ti) 30 MCA TM 1583 where n = -i 2 2 - — ^ and the known function is UO) $(x,y) = |- . - rr A(|,Ti)K-L(|,n;x,y;X) di d| - s(x,y) JJ e(^,n)K-L(l,il;x,y;A) dn d| .- <^l(x,y) hA2 j^^ A(|,T])Ki(|,Ti;x,y;A) dn d^ - s(x,y) ^^^ J] e(^.^)Ki(^,n;x,yjA) dn di (7.5) 2n n=0 k=0 [2(n - k)] I W^^^)[(^- ^)(y - ^)]'"^ (7.5) MCA TM 1585 31 Substituting equation (7.5) j (6.8), and (6.9) into equation {'J .2) , the latter becomes ^ pp 00 p„ k=n / T ^n-k ^ ^ n-k- i op " k=n , TxH-k ^n-k- i k flTL ^'" E rj .M . W^,^)[(x - l)(y - r,)] 2 ^^ ^ ^ ff^ a2(-+i) P ^ (-^)"'' Wi,Ti)[(x - 0(y - n)]"'^"l dn d^ ^-^ f-^ t-:^ {2(n - k)1 I (J n=0. k=0 - (-1)"+^ s n=0 ^^"J • op 00 k=n n-k+1 n-k- i '^^ Oi* n=0 k=D L^^"^ - ^)J • , jjf^ ^^ a2(-i) A(|,,)[(x - 0(y - ri)]'" 2 ^, ,^ ^ s n=0 (2-)' k=n , .n-k+L , 1 a, n=0 k^O L J (7.6) Taking into account the imifomi convergence of the series with respect to | and t) in both sides of equation (7.6), we integrate it term by term. Then, keeping in mind, the uniform convergence of the obtained series with respect to x and y which Is also maintained after 52 NACA TM 1383 differentiation, we differentiate the specified series term by term with respect to x and y. After these operations on both sides of the obtained equation we equate coefficients in identical powers of A. There- fore we arrive at the integro-differential equation which the coefficients of equation (7.^) satisfy ^ r- r' ,^u,,) ^^^^ ^x Jxp ^x(0 /(x - |)(y - Ti) f r r W^.n)- ^^^ = Mx,y) (7.7) ^ ^Xd JxU) /(x - |)(y - T]) where ^ ^ _^xjj A(|,^)[(x- 0(y - n)] dri d| + [2(n-k)Jl f k=n r -,Nn-k+l k=0 ^ -■ CT^* k=n-l . -l^'^-k p. -u. 1 k=0 u^ k=0 •- J CT (7.8) NACA TM 1585 53 in which the last sum and also the terms in \i are defined for nX). Let us note that the right side, 0^, of equation (7.7) for ■a2n contains terms with coefficients ^2k ^^"t only for k = 0, 1, 2, . . ., n-1. Let us transform equation (7.7) • We integrate by parts with respect to 5 the first integral on the left side of equation (7.7) j the second by parts with respect to t\, afterward we differentiate with respect to the parameters x and y, respectively. Equation (7.7) becomes ' ' dTi d| = %*{x,y) (7.9) where ''^^■^^(^)] rdX(0 ^C ^(x - |)|y - x(|) {^..}.,..,(x,y) (7.10) Let us note that the first term in equation (7. 10) of the right side of equation (7*9) becomes infinite for x = x-p. Let us return to expression (7.8) foi" $n and separate out of it the terms corresponding to the value k = n in the first sum - the compo- nent - f // . dri dl = R ^^^^. \/{^- U(y- n) We Integrate this integral by parts with respect to ^ keeping in mind that the limits of integration in a-^* are x-g ^ ^ ^ x-q and \|r(^) ^ T) = y and that 92^(x£,y) = 0. Then we differentiate with respect to X 5i^ mcA TM 1385 R = 1 r^ 92n(xD,Tl) ■E dT] + 1 A, dTl d^ (7.11) Let us subject the desired fionction -8 in equation (7.2) to a sup- plementary condition. Let us assume that at the trailing edges - the arc DT (or D'T', respectively) of the wing contour - and on the straight line DD* (figs. 11 and 12) - the intersection of the characteristic aft-cone from D with the z=0 plane (correspondingly the line D'Di*) - the velocity of the perturbed flow, and therefore the fiuiction t3, is a continuous func- tion, then the conditions are fulfilled ^[x,x(x)] = a[x,)C(x)] (7.12) (7.13) These conditions are einalogous to the Joukowsky condition for flow around a wing by an incompressible fluid. From equation (7.13) follows , rv VV1)^^^^_ py "an'-D^") ,, (,.,,) /^ - ^D ^x(xd) {/y - -t] l/x - x-Q ^i(xj3) /y _ T] since x(xd) = i(xp) . Substituting equations (7.II) and (7.1^) in. equation (7.IO), the latter becomes ^^D /(x - 0[y - xd] mn. _ 1 di (7.15) dl "X: '""E 1 ^ y ^oJ^.n) 2n' ^^(0 [^ dT] d^ + -I-,,' (x,y) NACA TM 1385 55 where ^n' = "^n - R (7.16) For n = O, the right side in equation (7 .9) is a known function of X and y 00* = ^o[l,x(l)] [ dx(n _ ij^^ ^. ^D /(x - 0[y - x(0] ^ ^^ ^"^E 1 S /x. /TTISI ry 9o(i,Ti) ku) /y - T (ill d5 - E J7 ^^^'^^ ^^ ^ r eo(l,n) "^ /(x - l)(y - n) ^ /(x - n(y - ^) (7.17) Let us solve equation (7 •9) for ■®2nx "*■ ■^2nv y The two-dimensional integral equation (7*9) is equivalent to two homogeneous eqiiations ^XD \/(x- I) dl = On*(x,y) (7.18) and ^^ WMllV,li::il a. . .,„»a,y) Ml) |/y - n (7.19) each of which reduces to an Abel equation. Using the Abel inversion formula we find the solutions of equations (7.I8) and (7.19) as 56 MCA TM 1585 (7.20) and (7.21) Substituting equation (7 •20) into equation (7. 21)^ first replacing in the latter by x, we obtain the solution of equation (7 •9) as ^ l/x - XD^/y - x(x) 1 1 r^ ^nr&'^(^^] ,, -^ , / , d| + "" h - x(x) ^xd ^7T ^2 ^ - Xj^^X (x) f^~^ ,f2 Jxd ^x(x) /(x - 0(y - Ti) (7.22) Integrating equation (7.22) alon£ the straight line parallel to the free-stream between the limits of N(x,y,0) and N(x,y,0) we find the formula determining -a in the general form of equation i'J.k) NACA TM 1385 57 « "X ^xi - Xd[/xi + y - X -X(xi) :r2 ^x ^Xd ^x^ - | j/x^ + y - x - x(xi) 3_ ^x oxi+y-x ^nTi*(xD,n) dT) dx^ ^2 Jx Jx(xi) Jx-L - xj) |/xi + y - X - Ti 1 r^ r^'i r^i+y-^ -^nin ■x- 2j- T( X ^Xi (e,Tl) ■D ' X(x-j_) Jxj^ - I i/x-L + y - X - Ti cLt] d| dx-j_ (7.23) If in equation (7.25) the coordinates x and y are taken as solu- tions of y-x + x-y = and y - x(x) = and the value of ■32n(^^y) is determined from condition (7. 12) on the trailing edge, then we find •d py, on the vortex sheet. If in the same formiila, the coordinates x and y are set equal to X = Xj) and y = y - x + Xj^ and the value of iSgj^Cxjy) is determined from equation (7. 13) on the line x = Xj^, then we find •62^ outside the vortex sheet in the region it affects. Thus, through equation (7.23), we can compute successively the coef- ficients •^Q, 182, . . ., ■32n' .... Therefore, the solution of equation (7*2) is Qonstructed as the absolutely convergent series (7.^) for any value of A. The coefficients -62^' are expanded in the series 00 ^'(x,y;A) = ^ ^2n'(x,y)A2n ^^^24) n=0 58 NACA TM 1585 The function iS' = e in fi ' (fig. U) may be computed through equation (7-25) if the fimction X oC^) replaces x(x) in it (where y = ^2(^) ^^ "^^^ equation of D'T' of the wing contour) and we inter- change the role of the coordinates . 5. Let us consider the general case of the flow over an oscillating wing by a supersonic stream. Let the characteristic aft-cones from E-^ and E-^' and D-]_ and D-[_' intersect the wing as shown in figure I3. Then E-|_ (correspondingly E-,'), as shown above, are defined so that to the left on the leading edge equation (3.?) is satisfied and to the right it is not. The points D^ and D-^' are, respectively, the most right and left points on the wing plan form. The space of the considered wing plan form as transformed by equa- tion (5.^) is illustrated in figure ik. Let us divide the x,y-plane where the medium is perturbed into a series of regions: the regions considered in the preceding section, Sq, S-j_, . . ., Sn, . • ., % and the regions A^, Aq^, • • •, £^, .... The region S-^^ is the M-shaped region bounded downstream by the intersection of the characteristic cones from D-^ and D^' with the z = plane. In the z = plane, these lines are the upper bounds of the region of influence of the trailing vortex sheet. The region Z^i is M-shaped lying between the characteristic cones from Dn, Dn'^ ^n+1' ■'^n+l' • ^® divide, in its turn, the part of the x,y-plane lying to the right and left of the wing, respectively, into the strips cr-^.' cf2> • • • ^n' • • • > i^W defined above and into &2^, ^2, • . ., 5n, • • . and into a]_', 02^ • • • , ^^i' > ' ' ' ' %' defined above and &i'.» 52* > ■ • •> ^n'' • * * correspondingly. The strip Sn is that part of A^ to the right and 5j^' is the corresponding part of Z^ to the left of the wing. It is easy to see that the region Q defined at the beginning of this section is in 6]_. In order to solve completely the problem of the flow over the wing shown in figures 15 and Ik, the derivative Scp-j_/Sz must be determined NACA TM 1385 59 in 62_, 62, • • •, 5n, • • • and. In bj^' , 62', • • •, 5j^', . . . . ,(x+y) ^e"^~^ by ^ ^(2), ,(3), Let us denote the function ■d^^' . . . and -a', -B'^^"), . . ., -B'^^^ . . . in the b^, b2, ■ - • , bxi, . . . and 5i', 62', - • •, 6^', • • • strips, respectively. Applying equation (6.24) for the velocity potential we construct cp^ for any point N(x,y,0) in 62- We divide the region of integration S which depends on the form of the function e 2 into the following: S = s + cr* + a-,*' + S* + a, Sz as shown in figure I5. This function is given in s. It was determined in a* and a'* in the preceding section by the solutions of equa- tions (6.7), (6.25), (6.27), etc. In s* it is determined by the solu- ^1 -P^ (2) tion of equation (7-24). We denote e ^ in a by •a^ '. 5z Using the boundary conditions (l.lO) and (l.l2) we arrive at the integro- (2) differential equation which t3^ ' satisfies and which differs from equa- tion (7 -2) only in the form of the right side. On the one hand the right side depends on the solutions 9, 9 , . . ., Q^^' , 9', Q'^^\ . . . , 9'(N) and on the other hand on the solutions ■d' . We construct •a(2) in the form of a power series in the parameter X. Requiring the fulfillment of equations (7. 12) and (7. 13) for -a ^2) (2) (2) (2) we obtain for the coefficients -Bq , ^2. j • • •> ■^2n j • • • an expansion in series of t3^ ' of equations of the form (7-9) which differ from each other in the form of the right side. (2) The right side in the equation for the coefficient ■32n ^^ "the general term of the series for -d^ ' depends on the first n+1 coef- ficients of the expansion of 9^-'-' and 9'^^^ where i takes all values (2) less than or equal to N, and on the first n coefficients i3q^ \ tSo^^), . . ., •32k^^^ 0'^^} ^} ^} • • -J "^-l) °f "t'^® series expansion of jj^ NACA TM 1383 the desired function ^^^'. Therefore, it is possible to find succes- sively the coefficients ■Oq^ , ^2 ^^ • • '> '^an using the solu- tion (7.22) of (7.9) as a final formula if there is put in the latter, instead of i^-^*, right sides in the equations of the form of (7 •9) for the respective coefficients of the expansion of '^2n • By the same reasoning, values may be found of ■d^^^ , -d^^', . . ., ■d\^> , ... in 6^, 61^, ' ' ') ^k' • • • • Therefore the velocity potential may be computed by equation {6,2k) at any point of the space perturbed by the motion of the ving shown in figures 15 and Ik. 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, ui, be put equal (2) (k) to zero, then the coefficients ^q, •^o* , . . ., iSq , • • • coin- cide, respectively, with the values of Scp^~)/Sz in 6]_, &2^ • • '^ 6jj., . . . for steady motion when the streamline condition (I.6) is given on the wing as Sc(:^/dz = k-^{x,y) . We apply the proposed method of determining b(f^/hz for the oscil- lating motion of a wing by constructing an integral equation, to wings of completely arbitrary plan form. For example, the wing contoior may not be cambered but may have the shape shown in figures I8, 2k, etc. In all cases, the part of the x,y-plane where b(^/hz must be deter- mined should be divided into the corresponding characteristic regions. Then successively passing downstream from one region to another, construct the integral and integro-differential equations using the boundary condi- tions on the x,y-plane. The solution of these equations for Sep, /Sz or for functions related to Scp^/Sz 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 formula for the Abel integral equation. The form of the wing contoixr is the limits of integration. The influence on the considered region, of determining NACA TM 1385 kl 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 AROUND AN OSCILLATING WING OF NON-ZERO THICKNESS 1. Let us consider the motion of a thin wing at a small angle of attack (fig. 15a-) • I 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 iipper svirface of the wing will be considered given by Qny. = AouU,y) + ^'P- A2u(x,y)ei"^ (g^^^ and on the lower surface by Onl = ^Qi^^^^y) + K-P- A2^(x,y)e'-'"^ ^g^^) where A^. and A^-^ define the wing surfaces and ^2u = %u(^'y)^^°^^^'^'* ^^ A2J = A^.j(x,y)ei<^l(x,y) define the ampli- tude and initial phases of the additional oscillating motion of the wing. We consider the functions Aqu? A]_^ and a^ given at each point of the upper surface and Aq^, ^±i) ^^^ °,^ given on the lower surface. The x,y, z coordinates were defined in section 1. The velocity potential cp^ is 9T3(x,y,z,t) = cp(x,y,z,t) + cp (x,y,z,t) (8.5) ^2. . NACA TM 1385 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,y-plane (fig. 15^1 ) . The potential cpg is specified by the motion of a thin oscillating wing with a profile symmetric relative to the x,y-plane. 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 symmetric flow and cpg satisfies 9s(x,y,-z,t) = cps(x,y,z,t) (8.4) Each of the potentials cp and cps is represented, in its turn, by 9 = cpo + ^ (8.5) ^s = ^Os + ^s (Q-Q where cPq and 9Qg correspond to the steady motion of the wing and cp^ and cpj^g correspond to the additional motion of the wing. Let us set up the streamline condition u^ing 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 2: of the wing on the x,y-plane (fig. 1). Therefore, we obtain the streamline conditions based on equa- tions (8.1) and (8.2) NACA TM 1383 k^ Sep az = Aou(x,y) + R-P. A2u(>^,y)ei^^ (8.7) Jz=r+-0 and Sep Sz = AQ^(x,y) + R.P. A2i(x,y)e loot (8.8) . z=-0 which must be satisfied on the upper and lower sides of £, respec- tively. Using eq-uations (8.5) and (8.6) we establish boundary conditions for the desired potentials cpQ, cpi, Tos? '•^^^ 'Pis* Keeping in mind that on the z=0 plane the normal derivatives of the potentials cp^g and (^^ are specified by the symmetry of the flow over the wing satisfying the condition M Os Sz -'z=+0 S9, Os z=-0 S9. Is -'z=+0 Scpic (8.9) z=-0 We find the boundary conditions for cp^g and cp^^g which must be satis- fied on the upper surface z to be Sep, Os = rQ(x,y), - .z=+0 Bcp. Is . h-z = R.P. r2(x,y)ei^t (8.IO) Z=rfO where the functions Tq and Tg are related to quantities given on the wing surface through ro(x,y) = Aou^^'^) - ^OZ^^^^) r2(x,y) = A2u(x^y) - A (x,y) '21 (8.11) The conditions to be satisfied by cpQg and cp^g on the lower s\arface of Z are Bcp Os 1^ ro(x,y). ,z=-0 Sep. Is lost I = - R.P. r2(x,y)e^"^(8.12) z=-0 kk NACA TM 1383 Since the normal derivative of the potentials cpQ and cp^^ specified by the antisynnnetrlc flow over the wing, on the z=0 plane, satisfy Sep. hz z=fO Sep, ^^ Jz=-0 Scp-j_] - Jz=+0 Scp^ (8.13) z=-0 the boundary conditions which must "be satisfied simultaneously on the upper and lower siirfaces of E are Scpr = AQ(x,y) hz = R.P. A2(x,y)ei<^ (8.1i^) where Aq and Ag are related to quantities given on the wing through Ao = V+ Aqz A2 = A2u + ^2^ (8.15) The boundary problems for cp^(x,y, z,t) and cpQ(x,y,z) were set up in section 1 where in the case of a harmonically oscillating wing, equa- tion (8.1^+) rather than equation (I.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 <^g and ^qs* I. Find cp-]_g(x,y,z,t) satisfying equation (l.k), condition (l.ll) on the disturbance wave, condition (8. 10) on the plane region E and Sep. Is Sz = (8.16) everywhere in the x,y-plane off Z where the medium is perturbed. II. Find the function ^si^fV)^) satisfying equation (I.5), condi- tion (1.11) on the disturbance wave, condition (8. 10) in the plane region L , and Sep, Os = (8.17) everywhere off z in the x,y-plane where the mediiom is pertiorbed. NACA TM 1583 i^5 Since the potentials q>Ls ^^^ ^b ^^^ functions which are symmetric relative to the x,y-plane, it is sufficient to solve the problem for the upper half- space. The solution of boundary problem I is given by equation {k.2). By means of this formula it is possible to compute the velocity poten- tial cf>j_g everywhere since in the case of symmetric flow over a wing the derivative Sq>j_s/Sz 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 cpj^g at M according to equation (^.2) the function Sep. Is = R.P. r2(x,y)e iujt must be substituted for - — =• y and integration is over that part of ^^Jz=0 the wing within the characteristic cone from M. The solution of boundary problem II as is known (refs. 21 and 22)^ is ScPq given by formula (5.10) if the function « dz is replaced by z=0 ^^ OS = ro(>^>y) QJ^d integration is also over the region defined irame- Sz I 2^+0 diately above. If the wing is vibrating as a rigid body then the functions A2u ^^^ A21 coincide and therefore, to solve the flow problem in this case, it is siiff icient in antisymmetric streams excited by the motion of an oscillating wing with profile of zero thickness to superpose steady symmetric streams. k6 KACA TM 1585 PART Il5 To apply the integral eqioations 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 cpQ specified "by the steady motion of the wing may be computed through equation (J-IO) at those points M(x2,yi,Z2_) of the space for which the region of integration S(x-|_,y3_, z-]_) , already known from Part I, does not extend outside the limits of the wing where IS given. If Scpo/Sz-|_ 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 (6.3) the form ^o(xi.yi.zi) = - — 2jt hCPQ S(x3^,y2^,Z]_) i\ d|^ ^1=0 \l (^1 - ^i)(yi - ^i) - ^1^ (21.1) and to obtain the effective solution of the problem, it is necessary, first of all, to find dcpQ/Sz2_ everywhere in S by constructing and solving an integral equation. 1. INFLUENCE OF THE TIP EFFECT FOR STEADY WItTG MOTION 1. The integral equation (5.I) in the coordinates (6.3) is, for the steady wing motion /T ei(xi.yi) -—=^,^k==- F(xi,yi) (21.2) .(x,,y,) /(>=!- 61) (yi-ni) ^The results of Part II, sections 1, 2, and 5 were completed in April, 19^4-8 at the Math. Inst, of the Acad, of Science, USSR. NACA TM 1383 47 where 0-[_ is the value of 59q/Sz-]_ on z; , (fig. 6) and where the known function is n^l'^l) - - U *(^l'1l) -5====, (21-3) The function A given on the wing is Sz-j_ k k I 2 2k It is easy to see that the velocity of the pertixrbed flow nonnal to the x,y-plane is related to Scf^/Sz-^ through 5^ ^ _ ^ Sep, = k-0 The regions of integration in are x-]_g - ^1 = ^1 ^'^^ ^(^1) - ^1 ^ yi where, as before, y-|_ = ^(^l) ^^ "^^^ equation of the wing tip ED in the transformed coordinates and x-]_g is the abscissa of E in the same coordinates. The regions of integration for ^j_ ^^ s are the same limits x^^ < !-[_ < X]_ and ti(li) ^ ^i ^ ^^^l) where y = ij'-i(xi) is the eqixation of the leading edge e'E of the wing contour, Let us note that equation (21.2) may also be obtained from equa- tion (6.5) if the frequency ca of the wing oscillation is set equal to zero in it. Let us delete the index "l" from the independent variables. We solve the double integral equation (21.2) with respect to &, by means of a repeated application of the inversion formula for Abel's inte- gral equation. 48 NACA TM 1585 We write equation (21.2) as \ ^^^ M^) 4(1) ^^^ ^V^^ ,^~^ (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 ^y e^(x,n) .T|r(x) , dTi = - / , aX^ ^^ ^y(x) /7^ ^^i(^) ^7T7 (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 = t(x) we find the solution using the well-known inversion formula for the Abel equation ei(x,y) = i ^ /y - ^U) i(x) A(x,Ti) ^il(x) /y - Ti dTi y = iU) It ^i(x) /y - Ti Sii dTl (21.7) Let us note that the solution (21. 7) for the steady motion of a wing may be obtained from the solution (6.22) of equation (6.12) for the vibrating wing if the index n and the frequency of oscillation o) are both set 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 ei(x,y) = - 1 ^ /y - ^(x) ^ti(x) A(x,Ti) Ill±' 1 dn (21.8) y - n NACA TM 1385 U9 In a similar maimer, we find the value — — = 6i'(x,y) in L^i (fig. 6) ei'(x,y) = - i \ ^ _ ^ A(l,y) ^ ^ d^ (21.9) ^^TTl^ ^+i(y) /x - I The fimctions x = ''(fj^Cy) and x = ■(I'gCy) 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 infinity as R 2 where R is the distance of N(x,y,0) from the points ED or E'D' (see fig. 7). 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 E and the region z-5 or ^3'* The region of integration S in equation (21.1) is divided into three parts: S = s-, + s^ + Sq, as shown in figure I6 cpo(x,y,z) --^ ff A(|,n) ^^"^ 2rt SQ+S2 /(x - l)(y - n) ^1 The limits of region s-j_ are Xj- < ^ < Xy^ and i (|) = t] ^ y z2 X - I rfhere x^ is the coordinate of the point A which is the intersection of the characteristic forecone from M with the side edge ED of the zing. The equation t) = y _ 2.2 fx - | is the equation of the hyperbola in which the aforementioned cone intersects the z=0 plane. The limits of region S2 are x^ = i - ^A ^^^ ^±(^) - T - tU). 50 NACA TM 1383 Using equation (21.8), let us evalaiate the integral over s-^ in equation (21.10) I = // 9i(^,Ti) dT] di /(x - 0(y - n) 1 r^f -^. ' riU) A(i,V)\/i(i) - V ^^1 'ti(i) /t) - t(0(n - T) dTl d^ |/(x - l)(y - Ti)-z^ (21.11) we interchange the order of integration of tj ' , T if/'"' „„,,^ nV- x-l dTl ' + (5) \J^ - *(l)(i - V)\/y - 7^~ • dii' d| (21.12) The resiolt of the inner Integration is I* = -^y- x-l dTl /^(O r\ - i(l)(T] - Ti')|/y - —5 n |/t(0 - T ^ (21.13) ^' V X - I Putting the value of equation (21.13) into equation (21.12) we obtain 1 = - /A(i,n') dTl d| /(x - n(y - n) - z^ (21.11^) NACA TM 1583 51 Equating (21.114-) and (21. ll) we obtain fJ^^U^r,) "^ ^' = -Jj^^''''^ /(x _ o(y - n) - z2 S2 l(x^O(y-n) (21.15) Therefore, to find the velocity potential, on the basis of equa- tion (21.1), at a point M(x,y,z) projected onto M'(x,y,0) in the x,y-plane as shown in figure I6, it is sufficient to integrate over Sq 90(x,y,z) = - ^ f[A{i,n) ''' ^^ z= (21.16) 2rt 3Q /(x - l)(y - Ti) - z2 The limits of region Sq are i;i{i) = T = y - z^x-l and x. = I = x-p where x^ is the abscissa of the point of intersection of the Mach forecone from M with the leading edge E'E. The velocity potential on the wing surface can be calculated from equation (21. 16) by setting z=0 in it and considering the region of integration to be x^ "^ ^ 5 x and i|'2_(0 - T - y because the lines of intersection of the characteristic forecone from M with the x,y-plane, in this case, are the lines ^ = x and t^ = y. In order to conipute 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 2, and L-,* i that is, at points o 5 of space where there is felt the effect of both side edges ED and E'D', it is s-ufficient to integrate equation (21.1) over the region B = S _. + S , the cross-hatched region in figure I7. Hence the Integral over S Q in equation (21.1) imist be taken with the opposite sign, i.e., the plus sign. 52 NACA TM 1385 3. Let us consider the ving of more general form shown in figure I8, Let the forward part of the wing have the "break, the arc EGG '£-]_', in the wing 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 hy 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 figure 18. If the region of integration S in equation (21. l) intersects regions 2, 2', 3 and does not intersect k. then the velocity potential may be evaluated by using equation (21. I6) (see figs. 16 and I7) . The simple res\ilt which is expressible by equation (21. I6) does not hold in the general case. If S intersects h on the wing, in the curvilinear triangle K'0-j_K, then according to equation (21.1) Scp^/oz must first of all be foimd in the triangle. Let us express, by equation (21. l), the velocity potential at any point of K'O-lK as equal to zero everywhere outside the wing and the vortex sheet from the wing, hence in K'0-]_K. Therefore, we arrive at an integral equation of the form of (21.2) for the function 6l*(x,y) = ScPq/Sz in K'O-j^K but with a more coniplicated known function. Applying the Abel inversion formula twice, we arrive at the solution in the following final form: ei*(x,y) = - i . ^ ■ ■ / ^ dTl - "^ /y - t(x) Ji|f3_(x) y - Ti ^t2(y) A(|,y)\/i2(y) - | d| "" /x - T2^'^^i(y) X - i (21.17) where j, = ;|,(x) is the equation of EG, y = ^lix) is the equation of E'] X = ?2*^y) °^ ^l'*^' and. X = '^■^^{y) of E-^'E. NACA TM 1385 53 Substituting equations (21. I7), (21.8), and (21. 9) into equa- tion (21.1) we obtain the formula for the velocity potential at M which has the projection M* shown on figure I8, and for which the region S intersects k on the wing and, therefore, the region K'02_K outside the wing, as ifQ{x,y,z) = - ^ JJ ^^^''"^ S*(x,y,z) '^ A(5,n) dTl d| (x - l)(y - T,) 1* /(x - 0(y - 1) - z^ ten-^ /[V*(^) - ^(^)i[(- - ^)(^ - ") - -^] a. d^ . [*(!) - n]{(x - 0[y - r^ii)]- ^JJ A(|,Ti) So* /(x - l)(y - Ti) - z2 ta.-! / [^('')-^2(n)][(x-i)(y-n)-z^] ^^ ^^ [*2( 1) - 4[(y l)[x - **(il)]- 2l ^ J (21.18) where y = i*(x) and x = ^*(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 cross-hatched in fig- \jre 18. The regions Sj^* and S2* are part of S* and are marked in Sl^ and S2* are the same figure by horizontal stripes. The regions bounded downstream by lines parallel to the coordinate axes passing through G and G*. The points G and G' are respectively the points with the largest x and y coordinate on the arc EGG'Et'. 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'02_K and -3 and off the wing. ^^ NACA TM 1383 2. FLOW OVER WINGS OF SMALL SPAN 1. Let us assume that the characteristic cones from £]_ and E^' intersect the wing as shown in figure I9. This occiirs, for example, for small span wings . Let us divide the x,y-plane where the medium is disturbed into the regions Sq, S]_, . . . , S^, . . . . The region Sn is an M-shaped region lying between the character- istic cones from En and E^' (or in one of them) and E^+i and En+i'. In its turn, we divide the part of the x,y-plane to the right and left of the wing into the strips cr-]_, ffg? • • -^ ^xn • • • and cr-iS ^o' > • • •? ^n' > • • •> respectively. The strip a^ lies between the after cones from E^ and Ej^^3_. Therefore, a^^ is that part of Sn lying 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 the leading edge E]_'E-|_ be given as in part I, section 6, by the equation y = \l(3_(x) and the side edges E-]_Ej^,-|_ and E-, 'E .n ' by y = \|((x) and y = \|;2(x) , respectively, or as x = ■»|/(y) and x = ■>|/2(y) correspondingly . To compute the velocity potential at M according to equation (21.1) in that part of space (or, in particular, on the wing surface) the region of which intersects S^ of the x,y-plane but not S^+i, we must first of all determine ^q/Sz off the wing in a-, , 02, cr^, . . . , a-^ and also in <7j_ ' , ^2' , '^^' } • • • > % ' ^ • • • • We construct the integral equation for ^q/Sz in the arbitrary strip a-^. Let us express a velocity potential which is equal to zero every- where off the wing and outside the region of influence of the vortex system from the wing, at N of the cr^ strip (fig. 20) according to the fundamental formula (21.1) NACA TM 1383 55 S(x,y,0) Sq^ hz dT] di z=0 /(x - |)(y - Ti) = (22.1) The limits of integration in S are ^i < ^ '^ x and y-^ < T < y. For convenience in later writing, ve make S a rectangle, which is pos- sible since the medium ahead of the wing is not distvirbed and dcf^^/dz is zero. The region S is shown in figure 20 bounded by the line£3 LN, NL-i_, L^O, and 0,L. Let us denote dcf^mz by 6-j_, Qg, . . ., 6j^, . . . and 9-j_', '2', % > in the respective regions a-,, Op, '^2 ' ' ' ' > ^k ' ' ■ * ■ ■ In conformance with this new notation we write equation (22.1) as ^y ei^(^,Ti) 'xd ^^- ^ 't(0 /y - ^ dT] + r^^^) A(^,n) '^2(0 ^y - ^ dT) + i=k-2 ^y.+l e.'(5,Tl) t:! Jy^ {f^^ ^2^^) ej,_i (^,n) 'k-i W dTi>d| = (22.2) 56 NACA TM 1585 Applying the Abel inversion formula twice to equation (22.2) we find e^ for k > 2 ei,(x,y)=-l_i- ^■^^^^ A(x,ti)/m.(x) - Ti '+2W y - n dT] + i|S^2 ^y.+l e^'ix,n)JHx) - TI ^ ^ nt2(x) ej^_3_' (x,!]) /t(x) - T] 1=1 -^^i y - n yk-i y - T Correspondingly^ for &j^' we obtain dTi (22.3) \'(^>y)=-i _ 1 1 "|/x - t2(y) rt2(y) A(g,y)fe(y) - ^ ^^ ^ 't(y) ^ - ^ i=k-2 ..x^+i ei(^,y)/t2(y) - ^ ^>(y) e ^_.^_(^,y)/^g(y) - ^ dl 1=1 ^1 X - i ^k-1 X - i (22A) NACA ™ 1385 57 where the terms in equations (22.5) and (22.U) containing the svumnations are defined only for k ^ 3. If Bj_, 02, ' • -, ^k-1 ^"^"^ therefore, Q^^' , 82', . . ., ^k-l' are already defined in cr^^', ^2' ' ' ' '' '^k-l' "then we can conipute 9k in ak foi" any k by means of equation (22.3). The value of Scp|-)/^z in aj_ and a^^' is determined by solving equations (21.8) and (21.9). The value of ScPq/^z in 02 is found from equation (22.5) by putting k = 2: >(x) e2(x,y) - - 1 _L= f ^'^ A(x,,) IMIa dn « i/y - t(x) Jt^(x) y - T — -=^= / /_ A(^,Ti) I ^ ^ d^ dn 3t2 /y - ilf(x) Jy^ -'>|f3_(n) (y - n)(x - O/x - Ifg^^) (22.5) We find Scf^-j/Sz in 02' in the same way .. ^ 1 1 r'^2^''^ . , \l+2(y) - ^ e2'(x,y) = - i ^_ ^ _ A(^,y) L£ dl + « /x - i2(y) ^^(y) "" ~ ^ , , rT(y) pt(0 \/tp(y) - ^\/l'(0 - Ti Jt^ /x - t2(y) Jx^ ^t-L(0 (x - 0(y - Ti)/y - ^^(0 (22.6) Thus, step by step we compute Bcf^/Sz in o-^. 58 NACA TO 1385 Using the solution of equation (22.5), ve now prove the relation xq* ry-^ /x-5 n = xj_* yi c>% az dTl d| = '=^ /(x - 0(y - n) - z2 (22.7) where x, * eind. Xp* are any numbers satisfying x, < x.* = x. (x. is the coordinate of the point A shown in fig. 21), x-^ < Xj_* < x^. For the proof, we write Q. in the equivalent form n = -X2* fy~ Qkd.n) dTl d^ V ^(^) /(x - 0(y - n) - z2 "2* r^^^^ A(^,Ti) dn d^ ^1* '^^2(^) /{x- |)(y- n) _ z2 i=k-2 2* ,yi+i e^'(l,Ti)dndi ^xg* ^M^) e^_^'U,n) ^^ ^^ ^ Jx,* Jv. /(x - 0(y - n) - ^ ""i* ^^k-l /(x - i)(y - Ti) - z2 1=1 ^V ^i (22.8) NACA TM 1383 59 where 9j^ in the first of the integrals is replaced by its value according to equatiori (22.3). Then, we obtain fi = - i 3t 1 r^* r^(^) A(|,V)/^(U - V Xi* ^i2(^) /x - ^ I* dT)' d| - - L_ / ; I* dTl' dl « i=l Jx^* Jy {^^^ 1 rX2* .t2(^) ej^_^(5,Ti')/^K0 - T ^1* ^^k-l /x - I I* diT d^ + XI* ^tgCO /(x - |)(y - n) - z2 i=k^ ^X2* ^Yi+i e^'C^,.) dTl d^ i=l ^1 ^1 A X - 0(y - .) - z^ ^X2* Pt2(0 9k_i'd,Ti) dT] dg J^ * J,.. . L -. : — ^2* ^k-l /(x- |)(y - Ti) - z2 (22.9) where I* denotes the integral (21.13) 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 •^2 r ^1* ^1 y-T Sep, az d| dn ^=0 /(x - |)(y - Ti) - z2 = NACA TM 1383 60 y * and Yp"* ar^ ^^ numoers satxsiyxng, j-^ = ^ ^ V * ^ V (v is the coordinate of B shown in fig. 21) ''I "^2 " B B where y * and y^* aj-e any nunibers satisfying y-^ = Vj* < y^ and Using equations (22.5) and (22.1^) it is possible to prove equa- tions (22.11) and (22. L2) correspondingly rX2* ry* Js^l an di Jxj* JtU) [s4z=0 /(x - l){y - n) -^ Ki 2 r^* r*(^) 1 az j,^ ^^.1 /[(x- |)(y- Ti) - z^][y*- ^(1)] ^| ^^ '^ Jxi* ^71 /(x- O(y-n) - z2 f[(x- l)(y-y*) - ^^Jii) - n] where y* may depend on I and satisfies i|r(x]_*) < y* = y-;^ (22.11) 2 .^^iz=o |/(x - U(y - n) - z2 2 r^^* r*2(n) i^Jz^ ^^.x / [(x-0(y-n)-zfx*-?2(n)] ^^ ^^ '^ V ^-1 /(x-|)(y-,)-z2 f [(- -*) (^ - ^) - ^'][^2(1) - ^ ] (22.12) where x* may depend on t) and satisfies ^2^^-]*^ < x* = x — ~. The relations (22.10) and (22.12) may be obtained, respectively, from equations (22.7) and. (22.11) if the role of the coordinates is inter- changed in the latter. NACA TM 1383 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 ' i--h -^= = (22.13) Interchanging the role of the coordinates in equation (22.13) we obtain ^^ = (22.li|) It is possible to consider equations (22.13) and (22.1^4-) as rela- tions fulfilled along the characteristic lines LN and L''N' in the x,y-plane where y and x are, respectively, the coordinates of N or N' lying off the wing and off the region of influence of the trailing vortex system (fig. 20). The points N and N' lie to the right and left of the wing, respectively. These relations can be useful for coitrpu- t at ions . 2. Let us turn to the fundamental formula (2I.I). Using eq\iations (22-7) (22.10), (22.11), and (22.12) we obtain, by calculation, the formula for the velocity potential cpQ at M(x,y,z) for which S intersects Sn for any n > ib'^-y- ., 1 - A(6,n)d, di ^1 ■ A{|,,)dr|d5 1 rr *<*'''"*l d-ldi-l ' A(l,ii)n2 ^ 3-, /(x - l)(y - n) - .2 " sg itx-O(y-i) -z2 "'3-^1 /(x - |)(y - n) - .2 «'S2« /(x - I)(y - 1) - l2 1 ^"' -"^-1 ^♦^l'*' ^(5.^>"2 .. ., . i n' ''"'' ^*'<''' '^''^'"'"^ I. a- «= k=l Xi -t(U ■ (, . 5)(y . ^) . ^2 «2 fcl ,.y^ J,j(l) ^(^ _ ^,(^ . ,, . ^2 1 .^^[♦2('a!| ^♦2('a; V2(5'1"'2 ,, , 1 ('■♦P^b)] pKy^) e„.,(t,l)n^ ^_ ^^ (22 „2 „^_^ ,,(5) ^j^ _ ^jj^ . ,) . j2 «== Jy„.2 JtaCl) /(, . |)(y . ,,) . r2 62 NACA TM 1383 where the functions O, and fig are defined as fil = tan ' [(x - 0(y- V^) - z2][^tr(0 - Tl] _i /[(x - 0(y - Ti) - z2][x^- .ir^Cn)] ^2 = tan ,/ ^— — |f[(x - XA)(y - Ti) - z2][t2(Ti) - ^J and where the regions S^ and S_ are regions of the wing marked on figure 21. The region 3-^* is the vertically-striped region on the wing siorface. The region S^* is the horizontally- striped region of the wing surface. It is clear that S-|_* and S2* intersect each other and S0 on the wing. The region S-]_ lies off the wing and is vertically- striped in figure 21. This region is the sum of the regions over which are taken the integrals containing 9^^' for k=l, 2, , , ., n-2 in equation (22.15). The region S2 lies off the wing and is horizontally-striped in the figure. All the integrals are evaliiated over it which together con- tain 6-|^ for k=l, 2, . . ., n-2. If M is such that S in the basic formula intersects Sj^ falling in the characteristic cone from Eq and lying outside the cone from En', then n must be replaced by n-1 in the second sum and in the last term of equation (22.15). If S falls inside the cone from Ej^' and lies outside the cone from E^ then n-1 must be substituted for n in the first sum and the penultimate term of equation (22.15). Let us note that the simis in equation (22.15) are defined for n > 5 and the last two terms in equation (22.15) for n > 3. NACA TM 1383 63 If n=l, then the formula for the velocity potential in equa- tion (22.15) is limited to the first two tenus . This result was already obtained before . If n=2, the formula in equation (22.15) is limited to the first foiu: terms, the region of integration is shown in figure 22. Thus, to evaluate the velocity potential, by equation (22.15), a-"t a point M(x,y,z) which has the projection M'(x,y,0) shown in fig- ure 21, it is necessary, first of all, to compute Q-^ for k=l, 2, 3> . . ., n-2 by equation (22.5) for 1^2 and by equation (21.8) for k=l (Gj^' correspondingly). As an example we present the expression for the potential for n=3 in the expanded form ^ S ® /(x- O(y-Tl) - z2 2« JJ /(x - i){y - T,) _ z2 H^,r\)ilj_ rr A(|,T])"p d^] d| dn d| - i // ^ + '^^ s-L* /OT- 0(y - Tj) - z2 ^2 -^^ /(x - 0(y - ^) - ^2 1 r*2(^A) r^(^) pt(0 A(|,Ti»)y/i(5) . n'^i^ ^^"'^2 h V)^T7^)(n-v)/(I-0(y^):z2"^"^^^^ _i_ r*(yB) r^^2(^) r^2(l) A(r,Ti) fe(n) - |.o «5 ^Xp ^ Jy,(Tl) / - , , ^ -— -dg'dTld^ (22.16) 6k NACA TM 1383 The region of integration in the last two integrals over | and t] are, respectively, the regions S-]_ and S2 lying off the wing and shown striped in figure 25- Formula (22.15) for the velocity potential contains an n-iterated integral with the integrand an arbitrary given function on the wing: Scpo/Sz = A(x,y). In the general case, it is not possible to reduce the number of iterations in the computation of equation (22. I5) for arbitrary wing- tip shapes since the arbitrary functions \|f, ilfp, and A all contain the variables of integration. If the functions \|/ and \(/o are fixed then the wing to be considered has completely determined tips and it is easy to see that all the integrals in equation (22.15) a^^e reduced to double integrals taken over the wing surface with integrands containing the arbitrary given function A(x,y) which defines the form of the wing surface. Let us turn to the wing of small span which has a break in its leading edge as shown, for example, in figure 2k. The derivative ^q/^z may be evaluated in a-, and ao ^Y equa- tions (21.8) and (22.3). It is impossible to evaluate Scpg/^z in a^z using equation (22.5) and, therefore, a surface -integral equation must again be constructed which will also reduce to two Abel eqiiations but with more complex right sides than occurred for 0^ in figure 19 • Hence, we note that it is impossible to construct one formula which would determine dq)Q/Sz for all cases, but a single method of solution. may be shown to depend on the wing plan form. The formation of the surface -integral equation for ^q/Sz is explained above, for each characteristic region. Each of these equa- tions is of the same type, reducing to two Abel equations with different right sides in different cases. In particular, the right side of one of the Abel equations, in some cases, may be identically zero. NACA TM 1383 65 3. INFLUENCE OF THE VORTEX SYSTEM FROM THE WING FOR STEADY WING MOTION 1. To study the influence on the air flow of the trailing vortex system in steady motion, it is convenient to operate with the acceleration potential ^q which, in linearized theory, is related to the velocity potential derivatives in the characteristic coordinates through Let us turn to the wing shown in figure 25. Let us take a point M(x,y,0) on the wing surface, which lies between the characteristic cones from D and D'. Therefore the trailing edge DT affects M. Using equation (21.15) "the velocity potential at M according to equation (21.1) is q^(x,y,0) = - ^ S=S-j^+Sq A(|,Tl) dTl d? 1 rr Hi,r\) dn di /(x- 0(y - n) B2 (23.2) where the regions s = S3_ + Sq and S2 axe shown in figiore 25. The region S2 belongs to Q,, considered in section 7 of part I and shown in figure 11. We denoted the derivative ^q/^z in H by tS where this derivative is an \inknown. We subject Scpo/^z to an additional condition, analogous to the Kutta-Joukowsky incompressible-flow condition. We assume that the perturbation velocity potential at the trailing edge - the arcs DT and D'T' of the wing contour (figs. 11 or 25) - and therefore, the specified derivative, is a continuous function. Then the respective conditions are fulfilled: gg NACA TM 1383 ^[x,x(x)] = a[x,x(x)] (25.3) ^[x,>^(x)] = a[x,X2(x)] (25.1f) where, as above, the function y = x(x) is the equation of DT and y = Xp(x) is the equation of D'T' of the wing contour. In order to obtain the acceleration potential Oq at M on the wing surface, we must take the derivative of equation (23.2) 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 t, In the second with respect to f]. During these operations, we use equation (23.3) and the relation (22.13) which is fvilfilled along characteristic lines, and which on the line DD* (fig. 25) is ' ^ dTl = - / , ^ ^ dTl (23.5) X(xj)) /y - T] "^iI'-l(xj)) /y - ti We keep in mind, moreover, that the limits of integration of s-j_ are ^D = ^ = ^A ^^'^ X(0 ^ T ^ ^1(0 where xj) is the abscissa of D and ^A = x^(y) is "the abscissa of A, the limits of Sq are x^ < | < x and ^^^(O = T % J and finally the limits of S2 are xj) < | < x^ and x(0 ^ n CD /x - I 1 a ^x(l) A(g,Tl) il(^) /y - n .dTl d| (23.9) and ^x py ^^ ^XD ^MO /(x - 0(y - Ti) ^ ' '^ dTl d| = -^z==: -I ^= ^"^ + /x - xj^ '-^^{^j)) /y - n xd /x - I 1 a S5 ■d^ (23.10) Keeping equation (23.5) ^^ mind, which is fulfilled on the characteristic DD* we substitute equations (23.10) and (23.9) into equation (23-8) obtaining ^x Xj) /x - I 3(1, n) x(0 dT) + A(|,Tl) L dn ^ ^x(i) /y - n '^^1(0 /y - n -d| == (25.11) NACA TM 1583 69 This equatipn is equivalent to A r :fe^d,.A r'^'^^^a, = (23.12) Sy ^x(x) ^y - T] ^y ^t^(x) /y - Ti according to the inversion of the Abel integral equation. We integrate the last two integrals in equation (23.12) by parts with respect to T after which, as above, we differentiate with respect to the parameter. Using equation (23.3) we arrive at -y \(x,Tl) + ^^(x,Tl) X(x) /y - n dn = - ^x(x) A^(x,Ti) + A^(x,Ti) '^l(x) / dT] y - n A[x,i-L(x)] /y - i-^U) di^i(x) dx (23.15) Let us apply once again Abel's inversion foritiula, keeping in mind that the right side of equation (23.13)^ generally speaking, is different from zero for y = X (x) we obtain the solution for ^^^ + -Q as 1 r^^""^ ^x(^,y) + \(^.y) = - J , , , / , , ^ '^ /y - < (x) ^ti(x) Ax(x,Ti) At^(x,ti) EKZa d, . 1 ^ ALti(x)l y - T '^ /y - X(x) L "- J 1 dii(x) 1 ix /X(x) - i^(x) y - t;i^(x) (23.14) 70 NACA TM 1383 Using eq-uation (23.1^) we prove iU - l)(y - n) dn di = =1 dTi d| A[^.4r(|)] 1 - ~~di Ml ^^1 /(x - l)[y - ti(0] (25.15) where Z-]_ = RQ. The regions Sg and s-[_ are shown in figure 25. Substituting equation (25.15) into equation (25.6) we obtain the formula for the acceleration potential '5o(^^y) u rr A^(^,Ti) + A^(^,Ti) = ^x + %y = - ^ / / -T==- -in |f-]_(x) Sep, dTl L=o /y - n = (23.1?) NACA TM 1383 71 Interchanging the role of the coordinates In equation (23.17) we obtain Sx ^ti(y) d^ z=0 i^ ^ \{y) ScPr Sz .^ =0 f" = (23.18) where x = +-■ (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. (25. I8) as rela- tions which hold along characteristic lines in the x,y-plane where the vor- tex sheet has effect. Relation (23.17) is fulfilled along characteristic lines parallel to the Oy-axis (the line NQ on figure 26); the y-parameter 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. I8) is fulfilled along lines parallel to the Ox- axis; the x parameter is the abscissa of a point lying off the wing to the left. If the point N or to the left of D'H' relations (22.13) and (22.li4-) also hold. is thus located to the right of the vortex line DH , then along characteristic lines the respective If N is located to the left of DH or to the right of D'H', respectively, then relations (23.17) and (23. I8) 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 accleration potential in the simplest case of the vor- tex sheet affecting the flow. For any other case, the potential Oq is found in an analogous way. In each case an integral equation is constructed for ^^ + •fly. All the integral equations are of the same type but with different right sides in the different cases, and they are inverted by means of a double application of the Abel integral equation inversion formula. In the following paragraph we present resiilts defining the accelera- tion potential Oq at any point of a wing surface. 72 NACA TM 1585 3. Let us find the velocity potential cpo(x,y,z) at a point M lying within the characteristic aft-cone from D and outside the charac- teristic aft-cone from D' . The region of integration S in the funda- mental formula (21.1) intersects the plane region Q. (fig. 11) in this case. The projection M' of M on the x,y-plane is shown in figure 26a. Starting from condition (1.12) (of part I) we express the derivative 3cPq/Sz for any 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 fonnula (21.8). We then obtain the desired representation for the derivative Bcp x+yp-xj3 ^ _ 1 1 ^2 ^ /y - X - yj) + xp^^^l(x) S9^(x,T),z) -'z=0 ^x + yp - Xp - n y - n dT] Using equation (23.19) it is easy to prove (23.19) X- X2- y- t+y^-^ Sep. dT] d^ ^=0 /(x - 0(y - T)) - z^ X2* P^+yD-^D fScpol dTl d| 1* ^^l^id) 1 ^^Jz=0 /(x - 0(y - Ti) - z2 (23.20) by the same methods used in proving equation (21. I5). NACA TM 1383 73 The limits of integration in equation (23 .20), x-^^* and X2*, are Xj) ^ x^' < = X •F and X- ■D < * r^ < where x-p is F is the any numbers satisfying the coordinate of the point F shown in figure 26a. The point intersection of the vortex line DH, which has the equation y = X + y-Q - x-Q, with the characteristic cone from the point with the coordinates (x,y,z). In particular, there holds S2 acpr dT] d5 ^=0 /(x - |)(y - Ti) - z2 dT] d| Sz '^=0 /(x - |)(y - Ti) - z2 (23.21) where the regions S-j_ and S2 are shown in figure 26a. The region S-|_ is marked with horizontal and the region S2 with vertical crosslines. Keeping in mind equation (23.21) we obtain an expression for the velocity potential at the point M defined above 2jc A(|,T))dTl d^ _1^ 2rt ^(|,Tl)dTl di ^^ /(x - |)(y --Ti) - z2 2«JJ /(^ _ |)(y _ n) - z2 (23.22) where Sq and S' are shown on figure 26a. Therefore, the region of integration S in equation (23.22) inter- sects the wing surface only in that part of 9. which lies to the left of the vortex line DH. Before evaluating the velocity potential by equation (23.22) it is necessary to determine Scpg/Sz = -6 in the region S' of fi. We find ■d from the solution (23.1^) 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 ScPq/Sz = -3 in Q, to the left of DH, the coordinates x and y on the vortex sheet should be taken as the solution of the eq-ua- tions f-2-y^+x-Q = and y = x(x) and the value of •9(x,y) is determined from equation (23.3) at the trailing edge. 1^ NACA TM 1383 If the X and y coordinates are set equal to x = x-p, and y = y-x+Xp and the value of ^(x,y) is determined on DH from the solution of equation (21.8) then the obtained expression will correspond to the value of Bcp^^/Sz In Q. to the right of DH off the vortex sheet but in its sphere of influence. h. ERESSUEE DISTRIBUTION ON A WING SURFACE 1. Let us consider a wing of arbitrary plan form. Let the wing contour in the characteristic coordinates be given by the following equa- tions: The leading edge E'E by y= \(f(x) or x = iF]_(y), the side edges ED and E«D' by y = i|r(x) and y = t2(x) or x = ilf(y) and X = ^2^y), the trailing edges DT' and D'T' by y = x(x) and y = X2(x) or X = x(y) and x = XgCy). 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 Oq by P(x,y) = P^Cx^y) - Pu(x,y) = 2pOo(x,y) (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. Let us express the stream pressure on the wing siirface in each characteristic region by the function A(x,y) which is given on the wing, defining the 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,y-plane. It is clear that these segments are parts of the lines of Intersection of the charac- teristic cones, with vertices in the x,y-plane, and the x,y-plane itself. Region I is the region where the tip effect is not felt. This part of the wing lies ahead of the characteristic aft-cones with vertices at E' and E. NACA TM 1583 75 Region II is where the tip effect is felt but not the influence of the trailing vortex sheet. This region lies betveen the characteristic aft-cones from E' and E and D and D'. At M of region II, for which the lines M-^M^ and l^M^, intersect on the wing as shown on figure 27, the pressure difference is up p(x,y) = - — JJ D(|,Ti;x,y)dTl U +^ // D(5,Ti;x,y)dTi d^ + Si S2 ^/bp„,(i),., J'x - 2iiy_| .f\.- %l\f BL*(.),,,x,,|an - i^PJl dt2(x) ix ] !- / B|'^,i/;L(^)j^'y|^^ L2 (24.2) Inhere S]_ is the region of the wing boimded by the lines Wr^, WIq, U-^A^ and lA^i^_, S2 is the region bo\mded by M^M^, M^Mj^ and the arc L = M^Mj and where D(5,Ti;x,y) = A|(|,Tl) + A^(|,T]) /(x - |)(y - n) B(^,Tl;x,y) = A(l,Tl) /(x - l)(y - Ti) T6 KACA TM 1385 If the lines M,M, and MpM do not intersect on the wing, as shown in figure 28, then the pressiore difference is p(x,y) = - ^^'^D(|,Ti;x,y)dTi d| - ^ J B[t,1fj_U);^,j]il S2 L \ d^±U) d| -d| up J d.v(y) up dv ^^1^2^^) 'B[i(y),Ti;x,y]dTi Ll dx B[i,^^{x);x,y]d^ (24.3) L2 where S-j_ is bounded by the lines Wl-^, M-]_Mz, MM2, MglV^ and Arrows in the figures show the direction of integration in the con- tour integral and the integrals taken over the lines L-|_ = K-zU-^ and L2 = Mi|M2. In region III, which lies between the characteristic cones from E and the characteristic cones from E', D and D', the pressure differ- ence is p(x,y) = - ^JJ D(|,Ti;x,y)dTi dl - ^ / B[|,ti(U;x,yJ S]_ L 1 - di|^l(0 up 1 _ ^iklj^ y Bp(y),Ti;x,y]dTi d& fd^ - The pressure difference in region III' is expressed in the same way. NACA TM 1383 77 p(x,y) = ^// D(5,Ti;x,y)dTl d^ - up 1 - di|/2(^) dx B i,^2(^)j^>y]'i^ (2i^.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- respondingly. At M(x,y) of region IV, when M-^M^ and M2Mlj. intersect on the wing, the pressure difference is P(x,y) = - ^ /y^D(|,n;x,y)dTi di + ^ /TD(|,Ti;x,y)dTi d| + Si S2 up ^JB[|,^3_(|);x,y] jl L a* and 62 > o-^J that is, a wing surface not affected by the trailing vortex sheet. 86 NACA TM 1583 We will assume that the wing siirface is a plane inclined "by an angle Pq "^o 'the free-stream direction. Therefore^ the derivative — ii v/lll "be a constant everywhere on both sides of the wing surface and will be given in the form S9f ^ - - uPq tan a* (A1) In conformance with the method we divide the wing surface into the three characteristic regions la, Xb, and Ic, with each region having its own analytic characteristic solution ajid taking into account the angular point A of the leading edge (fig. 5^) . Let us compute the stream pressure on the wing surface in each region. Using the formula (5-9) , we find the pressure in the regions la and lb, lying outside the characteristic cone from A, to be p = ^ = 2u^pPo tan a* (A2) This formula shows that the pressure in regions la and lb is a constant. In region Ic, lying inside the characteristic cone from A, we find, by using the same formula, the pressure to be . 2u2pPq tan 6-1 Jcot2 a* tan2 S^ - 1 2 _-|^ /I + cot a* tan 5;^ ^ *^°t b2_ - X2_ yi - cot a* tan 8-^ y-]_ - Z cot 6-]_ 2 -1 P- ~ ^°t a* tan 5]_ I cot 63^ - x-]_ X — — tan I — ^^— ^— »-^— ^— ^— + rt Vl + cot a* tan 5-j_ y-, - Z cot S-^ (A3) NACA TM 1585 87 In the original coordinate system shown in figures 3^ aJ^d. 55^ (A5) becomes . , 2u2ppo tan 81 p(x,y) = - X v/cot2 a* tan2 b^ - 1 -1 /-'- ~ ^o't °''* "taJi &! ^ cot 5]_ - X + y cot a* ' 1 + cot a* tan 63_ y cot a'^ + x - Z cot h-^ 2 taji-^. /l + cot a* tan h-^ I cot 5^^ - x + y cot a"*^ 3t ^^ \ / 1 - cot a* tan 6]_ y cot a* + x - Z cot 83^ (Al^) These formulas show that the pressure is constant along each ray from A in region Ic. Shown in figures 56 and 37^ respectively, are the pressures along a section ^j^j. paJ^allel to the y-axis and aJ-ong the section A2B2 par- allel to the x-axis. The lift P of the considered wing is P = 2u2pPQz2(tan 6]_ - tan 62) tan 82ycot2 a* tan2 S-j_ - 1 2 1 /cot a* tan 81 - 1 1 + - tan-1 ,/ ■ + rt V cot a* tan 8]^ + 1 2 tan 8]_ - taji 82 _]_ /cot a* tan 8]_ + 1 - - "Can \ / — ^— ^^^^-^«— ^-^■^— ■— — ^— — + rt taxL 82_ + tan 82 k tan3 8-- cot a* tan 8]_ - 1 ^ tan 8i(tan2 81 - tan2 82) tan' 1> [cot a* tan 82 - 1 \ J cot a* tan 82 + 1 (A5) The lift coefficient C^ is J+Pq "tan 82^ Cz = Wcot2 a* tan2 82_ - 1 o -1 /cot a* tan 8i - 1 1 _ £ tan"-^ \/ i + rt Vcot a* tan 8i + 1 2 tan &! - tan 62 _l |cot a* tan 8^^ + 1 ^ tan 82_ + tan 82 Vcot a* tan 81 - 1 16 Po taxi 82 >+ tan' K(tan 81 + tan 82)\/cot2 a^ tan2 82 - 1 ^ _2_ /cot a* tan 82 - 1 (A6) 88 NACA TO 1383 As is well known, the wave drag coefficient C.^ is related to the lift coefficient thro\:igh C^^ = PqC^,. Let us consider particular cases of (a6) . In the limit as 5]_— > ^, we obtain for the triangular wing the well known result for the lift coefficient of a triangle. Comparing (a6) and (A7) we conclude that for identical wing speeds and identical angles of attack the lift coefficient of the arrow-shaped wing exceeds the lift coefficient of the triangular wing. In the particular case when 62 = 5i, we obtain the infinite span arrow-shaped wing. In the limit as 62 — ^ S^. (^^"^ yields i^-^Q tan 6]_ \/cot2 a* tan2 S3_ - 1 This restilt shows that the lift coefficient of an infinite span arrow- shaped wing eqiials the lift coefficient of an infinite span slipping wing with slip angle b±. Formula (a6) shows that with increasing 5]_ and Sg, the angles between the leading and trailing edges and the free stream, respectively, the wing lift coefficient decreases. The dependence of C^ for an arrow-shaped wing on Sj^ and 82 is shown in figures 58 aJ^<3- 39. B„ Semielliptic Wing Let us consider the wing plan form which is half an ellipse as shown in figure kO , Let the semiaxis an and b-, of the ellipse be given. Let us assimie that the wing moves, as shown in the figure, in the direc- tion of the axis of symmetry. k See the work of M. I. Giirevich: On the Lift of an Arrow -Shaped Wing in Supersonic Flow. Prik. Mate. Nekh., Vol. X, No. k, 19ij-6. NACA TM 1383 89 The equation of the leading edge, the line D'D, in characteristic coordinates with origin at is Yl = - x^ and the trailing edge equation in these same coordinates is f a-]_^ - b-]_2 cot2 a*jx-L ± 2a-]_l3-]_ cot a^^a-^S + 13^2 cot2 a-*^ - Xj^S •^1 ~ aj^S + t,j2 cot2 a* In the original x,y coordinates the trailing edge eqioation is y = 1 h. ,2 xri' - "^ (Bi) ^1 The plus sign relates to the arc C3D of the ellipse axid the minus sign to the arc CD ' . Let us assiome that the wing surface is a plane inclined at an angle Pq to the free-stream direction, therefore the normal derivative — — as given by (A1) . Let us consider the flow around the semiellipse when the character- 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. Eegion 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 cha,racteristic 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 pressiore in I is constant everywhere and expressed by (A2). In VI the pressure distribution in the x,y coordinates is given by p = u^pPq tan a* X _-L cot a* B^y + B^f^ + 2a-^b^ cot a-^^B^ - f ^2 xB^ 1 -5. sm (B2) 90 NACA TM 1583 where Bi = ai^ + 1d;l' 2 4. -h.S (,Q^2 ^* X + y cot a'** B2 = &^ - "b-^^ cot^ a Similaxly for region VI'. The pressure distribution in V is 2 2u pp^ tan a-^ p(x,y) = ^ X f cot a* &j_y' + Bgf^ + Sa-^bj cot a^h^^ - f.-^ \ - sin"l + 1 ^1 sin' ^ cot a* B-^y + ^'^'^ - Sa^b^ cot a*p3_ - fg' (B5) where f2 = X y cot a* and B2_^ Bo^ and f-L are as defined in (B2) Graphs of the pressure distributions along the respective sections A]_Bj_ and A2B2 parallel to the y-axis are given in figures ij-1 and \2 and along the corresponding segments A^B^ and Ai^B^^^ parallel to the X-axis axe shown in figures ^4-5 axid M4-. Spanwise section lines A-]_B-]_ and A B are shown in figure ^-5; whereas chordwise section lines A B_ and A|^Bl are shown in figure UO. If the semiajcis of the ellipse are given in a special way; namely, if there exists between the semiaxes the relation a-j_ = b2_ cot a*, then formula (B2) for the pressure distribution in region VI simplifies, becoming p(x,y) = u^pp^tan a* 1 - ^ sin- ■a cot a* y + y2a-j_^ - (x + cot a^y)^ (Bll) This corresponds to the case where the characteristic cones with apexes at D and D' intersect the wing trailing edge on the axis of symmetry of the wing at the point C; consequently the region V on the wing now vanishes. NACA TM 1585 91 In the general case for the flow aroiond a semi elliptical wing, it may be shown that on the siorface of the wing in region V, there exists a certain curve along which the pressure difference "between the upper and lower surfaces of the wing reduces to zero. Downstream from 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 (B5) to zero. fa^a + b^^ cot2 a*)^ + Uj2 _ b^^ cot^ a*)^ ^^-l^^^ cot^ a*j ^2 _ b]_2 cot2 a*) 2 ha-f-bj_^ cot^ a* + l6a-^-^^ cot^ a* y^ x2 + = i+a^L^^L^ cot2 a* (a^^ _ b^^ cot^ a*)^ (a^^ + b^^ cot2 a^ After obvious transformations, we represent the desired geometric locus in the following final form 2_ + Z_ = 1 (b5) 22 3-2 bg where 2aib^ cot a* a^^ _ -^2 ^^^2 ^^r ^2 = I ^ ^ ^ ^2 = ^ai2 + b-L cot^ a* cot a^/ai + b-L^ cot2 a* (B6) 92 NACA TM 1383 These results show that the zero-pressure line is the arc of an ellipse with semiaxes a2 and b2 related through (b6) to the semiaxes a^ and b2_ of the axe of the ellipse which is the wing trailing edge. The directions of the semiaxes ag and bg coincide with those of the semi- axes a and b-, . In order that the zero-pressure line should not pass through the wing surface, the elliptical arc forming the trailing edge of the wing should not have a real point of intersection with (B5), which determines the zero-press\jre line. Comparing (B1) and (B5) we obtain the following result. In order that the zero-pressure line, of a plane wing of semielliptlc plan form moving at the supersonic speed u, should not pass through the wing surface, it is necessary and sufficient that the geometric parameters of the wing satisfy the condition a^ < ^ b;L co^ °^* (E''') Constructed in figure k6 is an isometric view of the pressure on a semielliptic wing in the general case when (BY) 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 form shown in figure k'J . Let the leading edges be the lines OE-,, and OE-,', the side edges E-j_D and E]_'D' parallel to the free stream, and the trailing edges DB and D'B. In characteristic- coordinate space, the wing has plan form as shown in figure kQ. 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; I - semispan and h chord. Let us consider that wing for which a > a*, 7 > a*. The first inequality means 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 1585 93 The equations of the lines forming the wing contours are: the line OE-^ y = X tan a or in characteristic coordinates ^1 = i ^1 where _ 1 - cot g* tan a 1 + ctg a* tg a here m < 0, since 6 > a*; the line OE-j_' y = - X tan a and y, = mx-. the line E-|_D J = I and y-]_ = x-|_ + 2 cot a*l the line E]_'D' y = - Z and Yi = ^1 - 2 cot a*l the line DB y = - X tan 7 + h tan 7 and y-i = — x-, + n-i 9^ NACA TM 1383 and finally D'B y = X tan 7 - h tan 7 and y-]_ = ni-[_x-]_ + n2 where 1 + cot a* tan 7 2h cot a* tan 7 m]_ = ^ — -— n-L = "2 = cot a* tan 7 1 + cot a* tan 7 2h cot g* tan 7 1 - cot a* tan 7 In conformance with the method we divide the wing surface into the 13 characteristic regions shown in figure k8. Assuming that the 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 the wing surface as formulas already transformed back to the original coordinate system. The pressure in la and lb is constant and expressed by (A2) . In Ic the pressure is f \ ^^P^O f rt , 1 1 /x - cot a* y P(x,y) = J - i + tan-1 -^ ' - '' /x - cot a* f X + cot a* j: \/ - m. cot a* 1 ^ / - m f x + cot a* y (CI) ' |fx + tan- r-^ '" " '°* °^* ^ cot a* y Hence it follows that the pressure is constant along each ray starting from in Ic. In Ilia p(x,y) = —-± 1 tan-1 ,/- ^ ^°^ ^* ^^ " ^^ (C2) Ti /- m cot a* F (m - 1) (x + cot a* y) + 2Z cot a* I NACA TM 1585 95 In Illb p(x,y) = 2u pPqChi - 1) rt V - m cot a* ta.i-1 ^=_ /^ - "°^ ^; rr^ |f X + cot a^ tan -1 2 cot a*(z - y) (1 - m)(x + cot a* y) + 21 cot a* tan-1 [^ ra X - cot a'**" y X + cot a* y (C5) In IIIc p(x,y) = 2u^P|3q(1 - m) rt 1/- m cot a* tan -1 2m cot a*(y - l) (1 - m) (x + cot a* y) + 2ml cot a* {Ok) In Ila p(x,y) = 2u pPq(1 - m) jt 'f-m cot CL* tan 2m cot a*(y - l) (1 - m)(x + cot a* y) + 2mZ cot a tan"^ 7(1 - m)(x - cot a* y) - 21 cot a* I |f 2 cot a*(z + y) l (C5) 96 NACA TM 1583 In lib p(x,y) = 2u pPq(1 - m) jt !/■- m cot a,* tan' _1 1 /x - cot g* y /- m f X + cot a* y tan" tan' 1/ 2 cot a*(z - y) r(x- m) (x + cot a* y) - 21 cot a* -. /(I - m)(x + cot a* y) - 21 cot a* 2 cot a*(Z + y) tan -1 cot a* y x + cot a^ y (C6) In lie p(x,y) = 2u'^p3Q(m - 1) jt ^ - m cot a* tan _-| 1 /(l - ni)(x - cot g* y) + 2ml cot g^ f m 2 cot a*{l + y) 1 1 /x - cot g* y ^ -] , tan--^ , 1/ ^ ^ + tan-1 pTi „ ^ „ (/_ m » X + cot g* y |f x + cot g* y X - cot a* y tan' m 2 cot g*(z - y) (1 - m) (x + cot g* y) + 2ml cot g* (C7) Formulas for the press\ire distribution on the wing surface in regions Ilia', lllb ' , IIIc ' , and lla' may be obtained from (C2) , (C5), (C4), and (C5), respectively, if coordinates appropriate to the specific regions are chosen. The formulas for the pressure show that there is a zero-pressure line on the wing siirface, downstream of which the pressure difference below and above the wing becomes negative. This line is formed of the two segments KN and KN' the equations of which are NACA TM 1583 '^'^ y = X tan h - 21 tan a* tan a y = _ x tan & + 21 tan a* tan a (C8) and which are parallel to the leading edges E-]_0 ajid E2_'0. The zero-pressure line may easily be constructed graphically. Graphical representations of the respective pressure distributions in the sections A]_B2_, A2B2, A-xBx, A.i^.Bi^, and ArB^ parallel to the y-axis are given in figures kS, ^0, ^1, 52, and 55. An isometric press\are surface is shown in figure 5^ for the hexagonal plane wing. Translated by Morris D. Friedman 98 NACA TM 1383 REFERENCES 1. Sedov, L. I.: Theory of Plane Motion of an Ideal Fluid, 1959. (Russian) 2. Kochin, W. E.: On the Steady Oscillations of a Wing of Circular Plan Form. P.M.M., vol. VI, no. k, 19i]-2. MCA translation. 3. Prandtl, L.: Theorie des FlugzeugtragfliJgels im Zusanmendruckbaren Medium. Luftfahrtf orschung, no. 10, vol. I3, 195^. h. 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, 19^7- Also D.A.W., vol. LVI, no. 6, 19^7' Brown translation. 6. Krasilshchikova, E. A.: Tip Effect on a Vibrating Wing at Supersonic Speeds. D.A.N. , vol. LVIII, no. 5, 19^7. 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, 19i^7. 8. Krasilshchikova, E. A. : Tip Effect on a Wing Moving at Supersonic Speed. D.A.N. , vol. LVIII, no. k, ISk-J. 9. Krasilshchikova, E. A.: On the Theory of the Unsteady 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 Supersonic Flow. P.M.M., vol. XI, no. 1, 19^^-7. 11. Gixrevich, M. I.: On the Lift on an Arrow-Shaped Wing in a Supersonic Flow. P.M.M., vol. X, no. 2, 19^7. 12. Gxirevich, M. I.: Remarks on the Flow Over Triangular Wings in Super- sonic Flow. P.M.M., vol. XI, no. 2, 1914-7. 13. Karpovich, E. A., and Frankl, F. I.: Drag of an Arrow-Shaped Wing at Supersonic Speeds. P.M.M., vol. XI, no. h, 19^7. Brown translation. Ik. Frankl, F. I., and Karpovich, E. A.: Gas Dynamics of Thin Bodies. 19^8. Translation published by Interscience Publ., N. Y. 1954. NACA TO 1383 99 15. Panichkin, I. A.: On the Forces Acting on an Oscillating Wing Profile in a Supersonic Gas Flow. P.M.M. , vol. XI, no. 1, I9I+7. 16. Galin, L. A,: On a Finite-Span Wing in Supersonic Flow. P.M.M., vol. XI _, no. 3, 19ij-T. 17. Fallcovich, S. V.: On the Theory of Finite-Span Wings in Supersonic Flow. P.M.M. , vol. XI, no. J>, 19^1. 18. Khaskind, M. D., and Falkovich, S. V.: Finite-Span Oscillating Wing in S\ipersonic Flow. P.M.M. , vol. XI, no. 3, 19k'J. 19. Gurevich, M. I. : On the Question of the Thin Triangular Wing Moving at Supersonic Speeds. P.M.M., vol. XI, no. 3^ ^9^1' 20. Galin, L. A.: Wing of Triangular Plan Form in Supersonic Flow. P.M.M., vol. XI, no. k, 1947. 21. Puckett, A.: Supersonic Wave Drag of Thin Aerofoils. Jnl. Aero. Sci., vol. 13, no. 9, 19^6. 22. Von Karman, T. : Supersonic Aerodynamics. Jnl. Aero. Sci., vol. ill-, no. 7, 19hl. 23. Ackeret, J.: Luftkrafte auf Flugel, die mit grosserer, als schallge- schwindigkeit bewegt werden. Zeit. fiir Flugt. u. Mot., vol. I6, 1928. 2U. Schlichting, H. : Tragflugeltheorie bei Uberschallgeschwindigkeit. Luf t f ahr tf or s Chung., vol. I3, No. 10, 1936. 25. Borhely, N. : Uber die Luftkrafte, die auf einen harmonischschwingenden zweidimensionalen Flugel bei ifberschallstromimg wirken. ZAMM, vol. 22, no. k, 19^3. 26. Busemann, A.: Inf initesiraale kegelige Uberschallstromung. Luftfahrt- f or s Chung, no. 3, 19^+3. 27. Frenkel: Electrodynamics, vol. 1. 100 NACA TM 1583 u t / 2 / j^ M f E. 1 -y Figure 1. Figure 2. Figure 3. NACA TM 1383 101 1 77=77/0 M' (x-k2,y,0) Figure 4. Figure 5. 102 NACA TM 1585 '\ N(x,y,0) Figure 7. • NACA TM 1583 103 Figure 8. 104 NACA TM 1583 °f Figure 9. NACA TM 1385 105 Figure 10. Figure 11. 106 NACA TM 1385 Figure 12. ri 1 E Figure 13. NACA TM 1383 107 Figure 14. 108 NACA TM 1385 'I Figure 15. Figure 15(a). NACA TM 1385 109 Figure 15(b). Figure 15(c). 110 NACA TM 1583 Figiire 16. M'(x,y,0) Figure 17. NACA TM 1383 111 n M'(x,y,0) Figure 18. 112 NACA TM 1583 i E2(x2,yi)^^^l(^t'y2) Ezl^s'^z) E^lM'Vs) E3(x3,y^) n (^^n'Vn+l) En + i(^n + l .Vn+z) En+i(xn+2,yn + i) / Fig:ure 19. NACA TM 1383 115 Oi(xi.yi) X C7, Figure 20. lli^ NACA TM 1583 Figure 21. X cr Figure 22. NACA TM 1385 115 Figure 23. Figure 24. 116 NACA TM 1383 M(x,y,0) Figure 25. Figure 26. • NACA TM 1585 117 Figure 26(a). Figure 27. 118 NACA TM 1385 Figure 28. Figure 29. NACA TM 1585 119 Figure 30. Figure 31. 120 NACA TM 1383 Figure 32, Figure 33. NACA TM 1383 121 Figure 34. Figure 35, 122 NACA TM 1383 P--P(y) Figure 36. P = P (x) K. M. Figure 37. NACA TM 1385 125 §2= Const 8^ > S, c,= c,(S,) 4/3 ^^=T 1 Figure 38. S^ = Const Cz-'Cz ih) IT 2 Figure 39. 12lj- NACA TM 1383 Figure 40. P--P (y) •-y M, N, Figure 41. P=P(y) Mg Ng •-y Figure 42. NACA TM 1585 125 K3 1-3 M3 N3 Figiire 43. Figure 44. 126 NACA TM 1385 *^y Figure 45. Figure 46. NACA TM 1385 127 Figure 47. Figure 48. 128 NACA TM 1583 Figure 49. 1 2 3 P=P(y) 5 6 *-y Figure 50. NACA TM 1583 129 P--P{y) Figiire 52. 150 NACA TM 1383 Figure 53. Figiire 54. NACA - Langley Field. Vu - < . y C CO ,! « uT Ih 'tn CO •S ^ 1 .5 o ° c c « |S5 0< u ^5 «a3 OJ m •^^ XJ o ^ in O 0) >2SS CO T c J o m u tin D CO < OT °? -' p. 41 S 20 - o O M (J O ^ -o < .g < 0. 2 ^ M gj V .s o >,5 13 13 S 25 •o ::2 jq w 'I o c (V CO o «-. :5H < <: a M t« S iS .tiS CO s o O rf u ir> c CO oa t^ o ^ >» o a T3 > 0) d c = t. J2 « S3 CO Q. g "* M - o !? 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" > Zi2 2 ^ OT '^ « M zz fe ;s i H J< m O n — -c ^ ■ 3 - ^ ;m s o ' O -H "^ u m a O ^ OJ (U W ■o O M C , fii' o J= " . o o o ° S ti c >> o -3 -3 . ■a 3 u, ^ o — s y s ■ .2 -c ,S f B — s b o "^ -e -n Jr. H to U3 ^ m he — ^ (D — 03 ° OJ •s ^ S - o s e- a 3 ™ o ° '- is 3 C CT O I T3 rt I « ^ I en — . 1 o u 1 — m lU o sr ^3.2 S 0) T3 M ' a. 3 CO o < z , . :s m U oj - c cj t: c tt " w ,. 0) . 3 o — ■- T3 W Wings, Complet Theory (1 Vibration and F Wings and Ailer likova 1383 nivers Recor chanic Kxasilshc NACA TM Moscow U Scientific v.154, Me 1951 O (U »aa ^< Sw; c >J ' o to 1) M < m o? _ p, CD S Sz S" O CO U O , OO 2 -7- -H -O < .- < d. 2 __| CO ^ 5 w < §H ^ -r e 2 o >.5 s r: > « .S - T3 ' •^ rn ',- -2 s > c o c -^ Ci. ^^ T3 ^ C ^ 0) O a 2 • .s o 3 CO ,2 0) o >. S=3 0) o 3 2 T3 a- ^ S-3 in ^ a b o 03 Z M o ° "' ^ l- i3 .■= .Q £ * 2 0 CO u o O 00 0) o J^ CO > QJ 0) t-i 5 ly — ' c q to?" o C'r_aj£ 2<°r!-'!5°'" «aa ^ s o J Ui s u c ^ o n (D ai < CO UJ n K U< S " >^ OS CJ ■" '^ W 0} a> .. a> •? S.2 £3 O C CU tH p O.^ cu cu ^ CO . rt oo c«co ? > •-' o O tcH CJ Ji 2 CO l<^ " z »-( rt . — o CO 5.H ■= •o S c« C rt ^ o 5 « .2§ T3 3 c »^ u v o o n. 0) a. J5 rt (3 H o >.£ S3 S 5 tJi " ■ T3 - -a a. S §^ s- S M « s g 3' ^ 5 § 13 - <" §2-§^ « ■= « ^ ■ Q. 2 x: 2 si oj a ki (0 -, o .5 * S H CO O 3 b O C4 en > :S .Q j= S o o u. cu m e 'g T3 j:i ■a o lU — o > X a CO e > Si o TJ S CU oE rt T3 O CU is 3 c: a" o c g * cu ° u ho5 I" ° .2 I. WJ VJ —1 „ cu o CJ ii t. r; CD G a,U o cu o >. ■o hO — I 3.2 . a> ^ c o 0) TD CO t3 1. ' ■ J .. ^' j.i I - ..J BUILOrH"; 701- Ro:-: ;30TA ATT: fuUr^oJJu ii.«. L£ADO:i U-V l-i-5_{;,_7_.^_ ^_i j-ii;-i5-i7-?.0-21 T. k. 1333