iUAC^Ttn-Q^ NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1354 GENERAL THEORY OF CONICAL FLOWS AND ITS APPLICATION TO SUPERSONIC AERODYNAMICS By Paul Germain Translation of *La theorie generale des mouvements coniques et ses applications a I'aerodynamique supersonique.* Office National d'Etudes et de Recherches Aeronautiques, no. 34, 1949. Washington , iNiv^tRSlPi' OF FLOP'DA January 1955 12.- iOtLlBrVvRY P.C tUUn tjt^ ^ 5" 7(; -7 -2^ -^nn^n-- GENEEIAL THEORY OF CONICAL FLOWS AM) ITS APPLICATION TO SUPERSONIC AERODYNAMICS By Paul Germain Preface By M. J. Peres NOTICE This report deals with a method of studying the equation of cylin- drical waves particularly indicated for the solution of certain problems in aerodynamics. One of the most remarkable aspects of this method is that it reduces problems of a hyperbolic eqiiation to problems of harmonic functions. We have applied ourselves here to setting up the fundamental principles, to developing their investigation up to calculation of the pressures on the visualized obstacles, and to showing how the initial field of "conical flows" was considerably enlarged by a procedure of integral superposition. Such an undertaking entails certain dangers. In France the exist- ence of conical flows was not known before 19^6. Abroad, this question has, for a long time, given rise to niimerous reports ■vrtiich either were not published or were published only after a certain delay. Thus it must be pointed out that some of the results here obtained, original in France when found, doubtlessly were not original abroad. Nevertheless it seems possible to me to specify a certain number of points treated in this report which, even considering the lapse of time, appear as new: the parts concerning homogeneous flows, the general study of conical flows with infinitesimal cone angles, the niomerical or analogous methods for the study of flows flattened in one direction, and a certain number of the results of chapter IV. Moreover, even where the results which we found independently were already known abroad, the employed methods are not always identical. Another peculiarity should be noted. Since these questions actually are everywhere the object of numerous investigations, progress has made very rapid strides. This report edited at the beginning of 19'<5^ risks appearing, in certain aspects, slightly outmoded in 19^9- To extenuate this inconvenience we have indicated in a brief appendix placed at the end of this report the progress made in these questions during the last year. This appendix is followed by a supplementary bibliography which indicates recent reports concerning our subject, or older ones of which we had no previous knowledge . I should not have been able to successfiilly terminate this report without the advice and support of my teacher, Mr. J. Peres, and it is very important to me to express here my great respect for and gratitude to him. I should equally cite all those who directly or less directly have contributed to my intellectual development and to whom I owe so much: my teachers of special mathematics and of normal school, Mr. Bouligand who directed my first reports, Mr. Villat, promoter of the Study of the Mechanics of Fluids in France whose brilliant instruction has been of the greatest value to me. 11 I also feel obliged to thank the directors of the O.N.E.R.A. who have facilitated my task^ and especially Mr. Girerd, director of aero- dynamic research. Ill PREFACE With his research on conical flows and their application, Mr. Paul Germain has made a major contribution to the very timely study of super- sonic aerodynamics. The present volume offers a con^jrehensive expose' which had been still lacking, an expose" of elegance and solid construc- tion containing a niomber of original developments. The author has fur- thermore considered very thoroughly the applications and has shown how one may solve within the scope of linear theory, by combinations of conical flows, the general problems of the supersonic wing, taking into account dihedral and sweepback, and also fuselage and control surface effects. The analysis he develops in this respect leads him to methods which permit, either by calculation alone or with the support of electrolytic -tank experimentation, complete and accurate numerical determinations . After a few -preliminary developments (particularly on the validity of the hypothesis of linearization), chapter I is devoted to the gener- alities concerning conical flows. In such flows the velocity components depend only on two variables and their determination makes use of har- monic functions or of functions which verify the wave equation with two variables according to whether one is inside or outside of the Mach cone. Mr. Germain specifies the conditions of agreement between func- tions defined in one domain or in the other and shows that the study of conical flows amounts in general to boundary problems relative to three analytical functions connected by differential relationships. He studies, on the other hand, homogeneous flows which generalize the cone flows and are no less useful in the applications. From the viewpoint of the linear theory of supersonic flows one must maintain two principal types of conical flows, bounded respectively by an obstacle in the form of a cone with infinitesimal cone angle, and by an obstacle in the form of a cone flattened in one direction. The general investigation of the flows of the first type is entirely Mr. Germain's own and forms the object of chapter II of his book. By a subtle analysis of the approximations which may be legitimate Mr. Germain succeeds in sin^ilifying the rather complex boixndary problem he had to deal withj he replaces it by an external Hilbert problem. He shows how it is possible, after having obtained the solution for an orientation of the cone in the relative air stream, to pass, in a manner as simple as it is elegant, to the calculation of the effect of a change in inci- dence. He gives general formulas for the forces, treats completely diverse noteworthy special cases and finally applies the method of trigo- nometric operators which is also his own to the practical niomerical calculation of the flow about an arbitrary cone. The determination of movements about infinitely flattened cones has formed the object of numerous reports. The analysis which Mr. Germain develops for this question (chapter III) contributes simplifications, iv specifications, and important supplements. Thus he evolves, in the case of an obstacle inside the Mach cone, a principle of minimum singularity which enters into the determination of the solution. Mr. Germain gives two original methods for treatment of the general case: one utilizes the electrolytic -tank analogy, surmounting the difficulty arising from the experimental application of the principle of minimum singularity; the other, purely numerical, involves the trigonometric operators quoted above . In the last chapter, finally, Mr. Germain visualizes the composi- tion of conical flows with regard to aerodynamic calculation of a super- sonic aircraft. Concerning this subject he develops a complete theory which covers most of the known results and incorporates new ones. He concludes with an outline of the flows past a flat dihedral, with appli- cation to the fins and control surfaces. The creation of the National Office for Aeronautical Study and Research has already made possible the setting up of groups of investi- gators which do excellent work in several domains that are of interest to modern aviation and put us on the level of the best reseeirch centers abroad. Mr. Paiil Germain inspirits and directs one of those groups in the most efficient manner. He is one of those, and the present report will suffice to bear out this statement, on whom we can count for the development of the study of aerodynamics in France. Joseph Peres Member of the Academy of Sciences NACA TM l^^k TABLE OF CONTENTS Pages CHAPTER I - GENERALITIES ON CONICAL FLOWS 1 1.1 - Equations of Supersonic Linearized Flows 1 1.2 - Generalities on Conical Flows 10 1.5 - Homogeneous Flows 22 CHAPTER II - CONICAL FLOWS WITH INFINITESIMAL CONE ANGLES 30 2.1 - Solution of the Problem 30 2.2 - Applications kl Cone of Revolution hk Elliptic Cone ' J+T Study of a Cone With Semicircular Section 58 2.3 - Numerical Calculation of Conical Flows With Infinitesimal Cone Angles 62 Calculation of the Trigonometric Operators 68 CHAPTER III - CONICAL FLOWS INFINITELY FLATTENED IN ONE DIRECTION 79 3.1 - Cone Obstacle Entirely Inside the Mach Cone 80 Study of the Elementary Problems (Symmetrical Cone 80 Flows With Respect to Ox 1x5) 80 Nonsymmetrical Conical Flows 97 General Problem IO5 Rheo-Electric Method IO8 Purely Numerical Method II7 3.2 - Case Where the Cone Is Not Inside the Mach Cone (F) . . . . 132 Cone Totally Bisecting the Mach Cone (Fig. 28) 13l4- Cone Partially Inside and Partially Outside of the Mach Cone (r) (Fig. 30) ll^2 Cone Entirely Outside of the Cone ( r) (Fig. 29) I52 3.3 - Supplementary Remarks on the Infinitely Flattened Conical Flows 159 CHAPTER IV - THE COMPOSITION OF CONICAL FLOWS AND ITS APPLICATION TO THE AERODYNAMIC CALCULATION OF SUPERSONIC AIRCRAFT I68 U.l - Calculation of the Wings 168 Symmetrical Problems I7I Rectangular Wings I7I Sweptback Wings I86 Lifting problems 206 Rectangular Wings 2l4 Effect of Ailerons and Flaps 225 Sweptback Wing 229 The Lifting Segments 2^10 k.2 - Study of Fuselages •. . . 2^3 4.3 - Conical Flows Past a Flat Dihedral. Fins and Control Surfaces 25I vi NACA TM 135^ Pag e REFERENCES 26l APPENDIX 264 Vll Digitized by tlie Internet Archive in 2011 witli funding from University of Florida, George A. Smathers Libraries with support from LYRASIS and the Sloan Foundation http://www.archive.org/details/generaltheoryofcOOunit NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 13514- GENERAL THEORY OF CONICAL FLOWS AND ITS APPLICATION TO SUPERSONIC AERODYNAMIC S^*- By Paul Germain CHAPTER I - GENERALITIES ON CONICAL FLOWS 1.1 - Equations of Supersonic Linearized Flovs 1.1.1 - General Equation for the Velocity Potential Let us visualize the permanent irrotational flow of a compressihle perfect fluid for which the pressure p and the density p are mutual functions. The space in which the flow takes place will be fixed by three trirectangular axes Ox-^, 0x2 , "^^^^ 'the coordinates of a fluid molecule will be x-^, Xg, Xo, the projections on Ox^^ of the veloc- ity V and of the acceleration A of a molecule will be denoted by Uj^ and a^, respectively. The fundamental equations which permit determination of the flow are the Euler equations A = - — grad p P or a,.-!^ (I.l) the equation of continuity^ *"La theorie generale des mouvements coniques et ses applications a I'aerodynamique supersonique ." Office National d' Etudes et de Recherches Aeronaut iques, no. 3^j 19^9- We employ the classic convention of the silent index: t- — fp^i^ is to be read: ^(pn^) + ^(pug) + ^(P^^)- NACA TM 135l<- div pV = or ^(pu^^) = (l.2) 1 and the equation of con^jressibility P = f (P) If one notes that a. = u, ^-^ 1 k ^xv 2 and introduces the sonic velocity (1-3) c2 _ dp " dp (lA) the equation (l.l) assiimes the form a ^^ = - i ^P_ = _ 1 ^ ^£_ = _ c^ ^P_ ^ Sxj^. P Sx^ P dp Sx^ P Sx^ (1.5) We introduce the velocity potential * (x]^, Xg^ ■^^) ' defined with the exception of one constant, by — » V = grad $ u-i = OX-; i The velocity of soiind, introduced here by the symbol -^ has a dp well-known physical significance; it is the velocity of propagation of small disturbances. This significance frequently permits an intuitive interpretation of certain results which we shall encounter later on (see section l.lA). NACA TM 135i4- which is legitimate since we shall assijme the flow to be irrotational. If we make the combination ^^i _ ^ S$ ^^^ one sees, taking into accoiint equations (l.5) and (l.2), that |^|^-^^ = c2 S!^ (1.6) axi dxj^ Sxi dxj^ ^^.2 This equation is the general equation for the velocity potential. One may show, besides, that c is a function of the velocity modiilus; thus one obtains an equation with partial derivatives of the second order, linear with respect to the second derivatives, but not completely linear. The nonlinear character of the equation for the velocity potential makes the rigorous investigation of compressible flows rather difficult, at least in the three-dimensional case. In order to be able to study, at least approximately, the behavior of wings, fuselages, and other elements of aeronautical structures, at velocities due to the compressibility, one has been led to introduce simplifying hypothesis which permit "linearization" of the equation for the velocity potential. 1.1.2 - The Hypotheses of Linearization and Their Consequences For aerodynamic calculation, one may assiome that the body around which the flow occurs has a position fixed in space and that the fluid at infinity upstream is moving with a velocity U, U being a constant' vector, the modulus of which will be taken as velocity unit. We shall always assume that the axis Ox-i has the same direction as U; the hypotheses of linearization amoimt to assuming that at every point of the fluid the velocity is reasonably equivalent to U. We put in a more precise manner U-, = 1 + u u„ = V u^ = w NACA TM 13514- u, V, w are, according to definition, the components of the "pertur- bation velocity." (1) u, V, w are quantities which are very small referred to \inity; if one considers these quantities as infinitesimals of the first order, one makes it at least permissible to neglectJ in the equations all infinitesimals of the second order such as u , v , uv, etc. (2) All partial derivatives of u, v, w with respect to the coordinates are equally infinitesimals at least of the first order so Su /dv ^^ that one is justified in neglecting terms such as Sx-]_ One may deduce from these hypotheses a few immediate consequences: (a) At every point of the field, the angle of the velocity vector with the axis Ox-|_ is an infinitesimal of the first order at least. Hence there results a condition imposed on the body about which the flow is to be investigated; at every point the tangent plane must make a small angle with the direction of the nondist\irbed flow (this is what one calls the uniform motion, defined by the velocity U) . If one designates by q the velocity modulus, one has, taking the hypotheses setup into account whence (1 + u)2 + v2 + w2 = 1 + 2u q = 1 + u (b) The pressure p and the density p differ from the values p, and P-, which these magnitudes assume at infinity upstream only by an infinitesimal of the first order; the equation (l.5) is written in effect 5u _ ^1 ^P Sx^ p dx. 3 This signifies that u, v, w may very well not be infinitesimals of the same order; in this case one takes as the principal infinitesimal the perturbation velocity component which has the lowest order. NACA TM I35U with C-, denoting the sonic velocity at infinity upstream; thus c 2 u = - ^(P - P,) (1.7) On the other hand, according to equation (l.U) P - Pi = c^^Jp - pj = -p-^u If one defines the pressure coefficient C^ by P - Pn ' P,/2|"l= one has Cp = -2u (1.8) (c) Finally, an examination of what becomes of the equation for the velocity potential (equation (I.6)) under these hypotheses shows that it is reduced to Sx-|_2 ^ [bx^^ dxg^ bx^^j Let 9 (x-]_,X2,Xo^ be the "disturbance potential/' that is, the potential the gradient of which is identical with the disturbance- velocity vector; cp(x, ,x_,x^^ is the solution of the equation with partial derivatives of the second order 1 2 1 - c 1 a^qj - B^cp , a^cp (j^^^ C-]_ c3x-]_ OXg OXo a completely linear equation. 6 NACA TM 13514- The Mach number of the flow is called the dimensionless con- stant M = ^-^ which, with the velocity unit to be chosen arbitrarily^ is written here M = 1/c-j^. We put: e(M^ - l) = P , with e being equal +1 or -1 according to whether M is larger or smaller than unity. (1) If M < 1, equation (I.9) is written Sx-L^ Sxg^ Sxn^ an equation which may be easily reduced to the Laplace equation. This equation applies to flows called "subsonic" because the velocity of the nondisturbed flow is smaller than the sonic velocity at infinity upstream. These flows will not be investigated in the course of this report . (2) If M > 1, equation (I.9) is thus written p2d!c^^&.+ a^ (1. 10) 5x-]_ Sxg Sxn This equation applies to "supersonic" flows; if one interprets x-[_ as representing the time t, this equation is identical with the equa- tion for cylindrical waves, well-known in mathematical physics. Investi- gation of this equation will form the object of this report. Remarks . (1) It should be noted that, in order to write the preceding equa- tion, it was not necessary to specify the form of the equation for the state of the fluid. In particular, the formulas written above do not introduce the value of the exponent 7 of the adiabatic relation p = kp7 which is the form usually assumed by the equation of compressibility. Investigation of linear subsonic flows has formed the object of numerous reports. See references 1 and 2. NACA TM 135i<- 7 (2) The preceding analysis shows clearly the very different char- acter of subsonic flows which lead to an elliptic equation, and of supersonic flows which are represented by a hyperbolic equation. (3) When we wrote equation (l.9)^ we supposed implicitly that M^ - 1 was not infinitely small, that is, that the flow was not "tran- sonic," according to the expression of Von Karman . Thus it is impossible to make M tend toward unity in the results we shall obtain, in the hope to acquire information on the transonic case°. (k) It may happen, in agreement with the statement made in foot- note 3} that u is an infinitesimal of an order higher than first. In this case, one will take up again the analysis made in paragraph (b) of section 1.1.2, wh,ich leads to a formula yielding the Cp, more adequate than the formula (I.8) C = -2u - (v2 + w2) (1. 11) P 1.1.3 - Validity of the Hypotheses of Linearization Any simplifying hypothesis leads necessarily to resiilts different from those which one would obtain with a rigorous method. Nevertheless, it was shown in certain numerical investigations on profiles (two- dimensional flows) where the rigorous method and the method of lineari- zation were applied simultaneously that the approximation method provided a very good approximation for the calculation of forces. Besides, it is well-known that the classic Prandtl equation for the investigation of Study of the transonic flows, with simplifying hypotheses analogous to those that have been made, requires a more compact analysis of the phenomena. It leads to a nonlinear equation, described for the first time by Oswatitsch and Wieghart (ref. 3)- From it one may very easily deduce interesting relations of similitude for the transonic flows (ref. h) . One may find these relations also, in a very simple manner, by utilizing the hodograph plane. In a general manner, according to the values of M, one may be led to neglect certain terms in the final formulas found for the pressure coefficient C_. This requires an evaluation, in every particular case, of the order of magnitude of the terms occurring in the formulas when M varies. In this report, we shall never enter into such a discussion. We shall limit ourselves voluntarily to the general formulas. An inter- esting example of such a discussion may be found in the recent memorand\jm of E. Laitone (ref. 5), 8 NACA TM I35U wings of finite span in an incompressible fluid furnishes very acceptable results, and the Prandtl eq.uation results from a linearization of the rigorous problem. It happens frequently, we shall have occasion several times to point it out, that the solution found for u, v, w will not satisfy the hypotheses of section 1.1.2 in certain regions (for example in the neigh- borhood of a leading edge)j eventually certain ones among these magni- tudes could even become infinite. Under rigorous conditions such a solution should not be retained. Anyhow, if the regions where the hypotheses of linearization are not satisfied are "sufficiently small," it is permissible to assume that the expressions found for the forces (obtained by integration of the pres- sures) will still" remain valid. This constitutes a justification a posteriori for the linearization method so frequently utilized in numerous aerodynamic problems'''. Therefore, we shall not systematically discard the solutions found which will not wholly satisfy the hypotheses we set up. 1.1. U - Limiting Conditions. Existence Theorem Physically, the definition of sonic velocity leads to the rule which has been called the "rule of forbidden signals" (see footnote 2 of section 1.1. l) and which can be stated as follows: A disturbance in a uniform supersonic flow, of the velocity U produced at a point P, takes effect only inside of a half -cone of revolution of the axis U and of the apex half -angle a = Arc sin(l/M); (p cot a) a is called the Mach angle, the half -cone in question Mach after -cone at P.' Correlatively, one may state that the condition of the fluid at a point M (pressure, velocity, etc.) depends only on the character of the disturbances produced in the nondisturbed flow at points situated inside of the "Mach fore-cone at M;" the Mach fore-cone at a point is obviously the symmetrical counterpart of the Mach after-cone with respect to its apex. If one wants to Justify this rule from the mathematical viewpoint, one must start out from the formulas solving the problem of Cauchy and take into accoimt the boixndary conditions particular to the problem. Along the obstacle one must write that the velocity is tangent to the obstacle which gives the value d9/dn. Moreover, at infinity 7 For instance, in the investigation of vibratory motions of infin- itely small amplitude about slender profiles. NACA TM 13514- 9 upstream (x-^ = -00) the first derivatives of cp must be zero, since 9 is, from the aerodynamic viewpoint, only determined to within a constant, it will be assumed zero . The characteristic surfaces of the equation (l.lO) are the Mach cones. If one of the Mach cones of the point P cuts off a region (R) on a surface (E), the classic study of the problem of Cauchy" shows that the value of 9 at P is a continuous linear function of the values of 9 and of dcp/dn on R. Let us therefore consider a point M of a supersonic flow such that its fore-cone does not intersect the obstacle. We take as the surface Z a plane x-]_ = -A, with A being of arbitrary magnitude. On E, 9 and d9/dn, which are continuous functions, will be arbi- trarily small . Consequently the value of 9 at M is zero . Thus one aspect of the rule of "forbidden signal" is justified. Let us suppose that the forward-cone of M cuts off a region r(M) on the obstacle; on r(M), d9/dn is given by the boundary conditions; thus 9(M) is a linear function of the values of 9 on r(M) . One sees therefore that, if one makes M tend toward a point Mq of the obstacle, one will obtain a functional equation permitting the determination of 9 on the obstacle, at least in the case where the Q existence and uniqueness of the solution will be insured^. Consequently, 9(M) depends only on the values of d9/dn in the region r(M); this justifies the fundamental result of the rule of "forbidden signals."-'-^ 1.1.5 - General Methods for Investigation • of Linearized Supersonic Flows In a recent article-'--'- dealing with the study of linear supersonic flows. Von Karman indicates that two major general procedures exist for °For the problem of Cauchy, relative to the equation for cylindrical waves, see for instance references 6 and J. ^Such a method has been utilized by G. Temple and H. A. Jahn, in their study of a partial differential equation with two variables (ref . 8) A more exact investigation of this question may be found in appendix 1, at the end of this report. See reference h. A quick expose of the methods in question may also be found in the text, in reference 2. 10 NACA TM 135l<- the study of these flows, one called "the source method," the other "the acoustic analogy." The first is an old method and its theoretical application is fairly simple. It consists in placing on the outer surface of the obstacle a continuous distribution of singularities, called sources, the superposition of which gives at every point of the space the desired potential; the local strength of the sources may, in general, easily be determined with the aid of the boundary conditions. The second method utilizes a fundamental solution of the equation (l.lO), the composition of which permits one to obtain the desired potential; this procedure is interesting in that it permits utilization of the Fourier integrals and thus furnishes, at least in certain particular cases, rather simple expressions for the total energy. Von Karman also indicates, at the end of his report, a third general procedure, that of conical flows. We intend to investigate in this report the conical flows and the development of this third procedure which utilizes systematically the composition of the "conical flows" and, more generally, of the flows which we shall call "homogeneous flows of the order n." We shall see that this procediire permits one to find very easily, and frequently with less expenditure, a great number of the results previously obtained by other methods, and to bring to a successful end the investigation of certain problems which, to our knowledge, have not yet been solved. 1.2 - Generalities on Conical Flows 1.2.1 - History and Definition Conical flows have been introduced by A. Busemann (ref . 9) who has given the principal characteristics of these flows and has indicated briefly in what ways they could be utilized in the investigation of supersonic flows. Busemann gives as examples some results, frequently without ■ proof . Several authors have supplemented the investigation of Busemann: Stewart (ref. 10) has studied the case of the lifting wing A to which we shall come back later on; L. Beschkine (ref. 11) has fur- nished a certain number of results but generally without demonstration. We thought it of interest to attempt a summary of the entire problem. One calls "conical flows" (more precisely, "infinitesimal conical flows" )-^^ the flows in which there exists a point such that along ■'-^The adjective "infinitesimal" is remindful of the fact that the flows have been "linearized;" we shall henceforward omit this qualifica- tion since no confusion can arise in this report. NACA TM 1354 11 every straight line issuing toward one side of 0, the velocity vector remains of the same value . Let (jt) be a plane not containing 0, normal to the vector U; let us suppose only that the velocity vector at every point of (jt) is not normal to (jt); the projection of these velocity vectors on (n) determines a field of vectors, the lines of force of which we shall call (7): the cones (a) of vertex and directrix (7) are "stream cones" for the flow. More generally, let (S) be a stream surface of the flow, passing through 0; every surface deduced from (S) by homothety of the center and of k, k being an arbitrary positive niimber, is a stream surface. (S) is not necessarily a conical surface of apex 0, but having (S) given as an obstacle does not permit one to foresee the existence of such a flow. It is different if a conical obstacle of apex is given; the designation "conical flow" is thus justified. Conversely, let us consider a cone of the apex 0, situated entirely in the region x-]_ >^0, and suppose that a linearized supersonic flow exists around this cone; this flow is necessarily a conical flow such as has just been defined; in fact, if V/x-, ,Xp,x^'\ denotes this velocity field, V/X,x-]^,X,X2,X.Xo'\ (A, being any arbitrary positive number) is equally a velocity field satisfying all conditions of the problem; con- sequently, if the uniqueness of the desired flow is admitted, V must be constant along every half -straight line from -'-5. Let us also point out that according to equations (I.8) or (l.ll), the surfaces of equal pressure are also cones of the apex 0. 1.2.2 - Partial Differential Equations Satisfied by the Velocity Components According to definition, the velocity components of a conical flow depend only on two variables; on the other hand, as functions of x-j_. ■^It should be noted that this argument will no longer be valid without restriction in the case of a real supersonic flow around a cone because in this case the principle of "forbidden signals" is no longer valid in the rigorous form stated. Among other possibilities, a detached shock wave may form upstream from the cone behind which the motion is no longer irrotational. 12 NACA TM 1351^ Xo; Xo, they are naturally the solution of the equation Let us first put ^2 ^^f ^ S^f + d^f Sx-j_ ^X2 Sxo^ Xo = r cos Xo = r sin the equation then assijunes the form p2 S^fL = a!f ^ ^ Sff + 1 Sf (1.12) Sx-L^ ar2 r2 ^©2 r Sr The second term of equation (l.l2) is actually nothing else but the Laplacian of f[x, ^x^jX^A in the plane x„, x^ ^x-, being con- sidered as parameter^; naturally f (x2_,r,9\ is periodic in 9, the period being equal to 2:1. To make the conical character of the flow evident, let us put x^ = 3rx (1-13) X is a new variable; X < 1 characterizes the exterior of the Mach cone with the apex 0, X > 1 characterizes the interior of the cone. Under these conditions, the disturbance -velocity components are func- tions only of X and 9. Since f is a function of X and 9 only d2f = ^^ dx2 + 25 f dX d0 + ^-^ b9^ + ^ d2x + |£ d29 ^^2 dX c)9 ^q2 ax de but ^^ = ^{^1 - P>< 'i^) d2x = l-fd2 pr Cd2x^ - px d2r - 2 ^H-^ 1 + 2p ^ dr2\ NACA IM 135^4- 13 2_JL_ 2_£. 2^ M. are the respective coefficients of dxn^, dr^, Sxi^' Sr2 ^02' Sr ^ 1 ' ' op p d9^ d r in the expression of d f as a function of the vari- ables X-, , r, 9 . As a consequence, the equation (l.l2) becomes under these conditions (x2 - l)^ + ^ + x^ = ax2 ^02 SX (I.li^) One may try to simplify this equation fiirther by replacing the variable X by the variable |, X and I being connected by a rela- tionship X = x(|), and by making a judicious choice for the func- tion x(^). The first operation gives (x2 . l)&+ x'2 ^ + ^ XX' p X" - i;x' = with the primes denoting derivatives with respect to |. For simplifying this equation, one may make the term in 2_ disappear. This will be SI realized by putting (1) If X > 1, X = ch I (1.15) one obtains for f Laplace's equation S|2 se2 (1.16) (2) If X < 1, , X = cos Ti (1-17) in this case, one obtains the equation for waves with two variables S2f S2f _ g br]^ S02 (1.18) 11+ NACA TM I35J+ Geometrical interpretation .- X > 1 corresponds to the interior of the Mach rearward cone (T) of the point 0; every semi-infinite line, issuing from 0, inside of this cone, has as image a point Q , % . One will assume, for instance, -n < ^ n; 1=0 corresponds to the cone (r), I = 00 corresponds to the cone axis (it will always be pos- sible to assume I as positive). The image of the interior of (P) forms therefore on the region (A) of the plane (0,0 (fig> l)^ limited by the semi-infinite lines AT, A'T' and by the segment AA' . The correspondence is double valued in the sense that to a semi-infinite line issuing from there corresponds one point and one only (0,1) in the bounded region and conversely, to one point of this region there corresponds one semi-infinite line, and one only, issuing from 0, inside of (r) . Since we shall suppose, in general, that the cone investigated is entirely in the region yi.-^ >^0, only this region will be of interest (•P then being identically zero for x-|_ < 0) . The semi-infinite lines of this region issuing from 0, outside of (r), correspond to < X < 1 (fig. 2), that is, according to equation (I.I7), 0^^^ ax^ hehy^ ^n+p which finally shows that the values ^ — can be uniquely expressed Se^ax" as a function of the derivatives of 9(0) with respect to and that they, consequently, have the same value, whether calculated starting from f -, or from f p . Slimming up, one may say that it is sufficient for the establishment of the "agreement" between two solutions defined in (A) and (A'), if these solutions assume the same value on the segment AA' . l6 NACA TM 1351+ 1.2.U - Mode of Dependence of the Semi-Infinite Lines Issuing From If one puts in the plane {d ,'h) 9 + T\ = 2\ e-T] = 2n . (1.19) one sees that the characteristics of the equation (I.I8) are the parallels to the bisectrices X = c , |i = c . These characteristics are^ in the plane {t\,0), the images of the planes x-]_ = 'Pr cos(2X, - O) and x-[_ = pr cos(0 - 2|-i) which are the planes tangent to the cone (r). The characteristics passing through a point Sq^^O^'HO^ ^^^ ^^^ images of two planes tangent to the cone (r) which one may lay through the semi-infinite /Sq cor- responding to the point Sq of the plane {O,"^) (fig- 3)- "Kie gener- atrices of contact are characterized on the cone by ^he values St and ©2 °^ ^^^ angle 9. One encounters here a resiolt which seems to contradict indications of section LlA. This apparent contradiction is immediately explained if one notes that, since all points of a semi- infinite Aq issued from are equivalent, one must consider at the same time all Mach cones, the apexes of which are situated on AqJ the group of these cones admits as envelope precisely the two planes tangent to the cone (r) passing through Aq. We shall call "Mach dihedron posterior to the semi-infinite Aq that one of the dihedra formed by the two planes which contains the group of the Mach cones to the rear of the points of Aq. The region inside ot this dihedron and outside of the cone (r) has as image in the plane i9,r\) the triangle 9-^ SQ^g- A semi-infinite Aj^ will be said to be dependent on or independent of Aq according to whether the image of A-]_ will be inside or outside of the triangle 0-]_ SQ^g- This argument also explains why the equa- tion (1.1^4-) shows elliptic character inside of (r). More precisely, two semi-infinite lines Aj^ and Ag, inside of (r), are in a state of neutral dependence (ref. 9)- In fact, let M-^ be a point of A-^, Mg a point of An', let us suppose that M-, is outside of the Mach forward cone of Mgj according to the argument of section 1.1. ll the point Mo seems to be independent of M-|_; but on the other hand, if one assimies M-^' NACA TM 1354 17 to be a point of A-^, inside of the Mach forward cone of Mg, M-]_ ' and M-|_ are equivalent which explains that Mg is actually not inde- pendent of M-|_ (fig. 14-). 1.2.5 - The Conditions of Compatibility Thus one may foresee how the solution of a problem of conical flow will unfold itself. One will attempt to solve this problem in the region (A') which will generally be fairly easy since th'e general solu- tion of the equation (I.I8) is written Immediately by adjoining an arbi- trary function of the variable 9 + r\ to an arbitrary function of the variable - t) , This will have the effect of "transporting" onto the segment AA' the boundary conditions relative to the region (A'). Applying the fundamental theorem, one will be led to a problem of har- monic functions in the region (A) . But taking as \inknown functions the components u, v, w, of the disturbance velocity, we have introduced three unknown functions (while there was only one when we dealt with the function 9). One must therefore write certain relationships of compatibility which express finally that the motion is indeed irrotational, The motion will be irrotational if u dx-]_ + v dxg + w dxo is an exact differential which will be the case when, and only when X, du + x dv + X dw = r(px du + cos dv + sin G dw) is an exact differential. This can occur only if this expression is identically zero, with u, v, w being functions uniquely of 9 and of X (the total differential not containing a term. in dr must be independent of r) : In a conical flow the potential is written cp = ux-j^ + vxg + wxo = r(3uX + v cos + w sin 0) with u, V, w being the disturbance-velocity components. One will note that 9 is proportional to r. Moreover PX du + cos dv + sin dw = (l.20) 18 NACA TM 13514. This is the relationship which is to be written, and this is the point in question, on one hand in the plane {O,^), on the other in the plane (0,1). (a) Relations in the plane i9,T\). One may write U = U-l(\) + UgC^) and analogous form\ilas for v and w, X, and n being defined by the relations (l.l9)- One has in particular duT :i„, >,,, du 1 = ^ + ^ ^ = ^ _ Su (1.21) dX hT\ he dji he hT\ Besides, according to equation (l.20) p cos T] du-j_ + COS e dv-|_ + sin 9 dw-|_ = P cos T] duo + cos 9 dvo + sin (9 dwo = however: 0=A, + u, ti=X,-|-i; and consequently the first equa- tion (1.21) is written cos n p cos \ du-j_ + cos \ dv-j_ + sin X dw-]_ + sin |a p sin X du-. - sin \ dv, + cos X dv^ = since the two quantities between brackets are unique functions of X, the preceding equality causes P cos X du-|_ + cos X dv-|_ + sin X dw-|_ = 3 sin X du-[_ - sin X dv-|_ + cos X dw-]_ = or dv-, dw-| , -P dUn = i = ± (1.22) cos ZX sin 2X NACA TM I35U 19 In the same manner one will show that -P dug = dv-- dwr cos 2[x sin 2|a (1.23) (b) Relations in the plane {9,i). The calculation is perfectly analogous. The equation (I.16) causes us to introduce the complex variable ^ = 9 + i| and the func- tions U(t>), V(^), W(^), defined with the exception of an imaginary- additive constant, the real parts of which in (A) are, respectively, identical to u(0,l), v(0,l), w(0,l). The equation' (1.20) permits one to write P ch I dU + cos dV + sin dW = If one puts + i^ = ^ - il = ^ one obtains cos — 2 3 cos -^ dU + cos -^ dV + sin ^ dW 2 2 2 sm — 2 B sin -^ dU 2 sin — dV + cos -^ dW 2 2 thence one concludes as previously ■3 dU = dV dW cos ^ sin ^ ii.?-h) The formulas (l.22), (I.23), (l.2U) express the relationships of compatibility which we had in mind. 20 NACA IM 135ij- Remark. We shall utilize frequently the conforraal representation for studying the problems relative to the domain (A). If one puts, in particular Z = ei^ = e-^ei® one sees that (A) becomes in the plane Z the interior area of the circle (Cq) with the center and the radius 1 (fig. 5). i B If one puts Z = pe , the point Z is the image of a semi-infinite line, issuing from the origin of the space (x2_,X2,X2), characterized by the angle 6 and the relationship ^ = X = ^ + P^ pr 2p The origin of the plane Z corresponds to the axis of the cone (f), the circle (Cq) to the cone (r) itself. A problem of conical flow appears in a more intuitive manner in the plane Z than in the plane ^. In the plane Z, the form\ilas (l.2U) are written _3 dU = 2^^^ = 2iZ -^^— (1.25) Z^ + 1 z2 - 1 We shall moreover utilize the plane z defined by 2Z z = Z^ + 1 The domain (A) corresponds conformably to the plane z notched by the semi-infinite lines Ax, A'x' (fig. 6), the cone (F) at the edges of the cuts thus determined^ and the axis of the cone (T) at the origin lit- No confusion is possible between the point 0, origin of the sys- tem of axes x-]_, X2, Xn and the point 0, here introduced as the origin of the plane Z. NACA TM 135i|- 21 of the plane z. The relations of compatibility in the plane z then assume the form ■ P dU = z dV = iz dW {i (1.26) 1.2.6 - Boundary Conditions The Two Main Types of Conical Flows The boundary conditions are obtained by writing that the velocity vector is tangent to the cone obstacle. Let, for instance, XgCt), Xo(t) be a parametric representation of the section x-]_ = p of the cone; XoXo ' - XgXo ' , Pxo ' , -P^2' constitute a system of direction parameters of the normal to the cone obstacle, and the boundary condition reads wxc ''''3' = b(''3''2' - ^2^3') (1 + ^) (1.27) It will be possible to simplify this condition according to the cases. However, the simplification will have to be treated in a dif- ferent manner according to the conical flows investigated. As set forth in section 1.1.2, two main types of conical flows may exist. (1) The flow about cones with infinitesimal cone angles, that is, cones where every generatrix forms with the vector U an angle which remains small. Naturally, the cone section may, under these conditions, be of any arbitrary form: since the flow outside of (r) is undisturbed (velocity equivalent to U) , on the cone (r) u, v, w are zero. The problem may have to be treated in the plane Z; U(Z), V(Z), W(Z) will have real parts of zero on (Cq) . The image (C) of the obstacle, in the plane Z, is defined by a relation p = f(0); conse- quently, a parametric representation of the section x-j_ = P will be obtained by means of the formulas Xo = 2p cos 9 1 + p' ^3 = 2p 1 + p-' sin e 22 MCA TM 1354 Thus the condition (l.27) becomes w p sin - p' cos 9 + p^(p sin 9 + p' cos d)\ + vfp cos e + p' sin 9 + p^(p cos 9 - p' sin eYl = -^(l + u) (l,28) with 9 taken as parameter^ and p' denoting the derivative of p with respect to 9. The investigation of conical flows with infinitesimal cone angles will form the object of chapter II. (2) The flow about flattened cones, that is, cones, the generatrices of which deviate only little from a plane containing U. Let us remember that (section 1.1.2) the tangent plane is to form a small angle with if; consequently, rigorously speaking, the section of such a cone cannot be a regular closed c\irve, an ellipse for instance; it must present a lentic- ular profile (fig. 7) • In chapter III we shall study the flows about such cones. Remark . Actually, we have, therewith, not exhausted all types of conical flows, that is, those for which linearization is legitimate. One may, for instance, obtain flows about cones, the section of which presents the form shown in fimjre 8; the axis of such a cone has infinitely small inclination toward u. Before beginning the study of these flows we shall, in order to terminate these generalities, introduce a generalization of the flows, the possible utilization of which we shall see in a final chapter. 1.3 - Homogeneous Flows 1.3.1 - Definition and Properties The conical flows are flows for which the velocity potential is of the form 9 = rf(0,X) as we tjad seen in section 1.2.5. One may visualize flows for which cp = r'^(0,x) NACA TM 135i4- 23 We shall call them homogeneous flows of the nth order-'-^. The conical flows defined in section 1.2 are, therefore, homogeneous flows of the order I. However, we shall maintain the expression "conical flow" to designate these flows since this term has been used by numerous authors and gives a good picture . The derivatives of the velocity potential with respect to the vari- ables x-L, xg, xo all satisfy the equation (l.lO). If one then con- siders the derivatives of the nth order of the potential of an homogeneous flow of the nth order, one finds that they depend only on X- and 6 and satisfy the equation (l.lij-); the analysis made in section 1.2.2 remains entirely valid. One may make the changes in variables (l.l5) and (1.17) which lead to the equations (I.16) and (I.I8). Thus one has here a method sufficiently general to obtain solutions of the equa- tion (l.lO) which may prove useful. The simplest flows are the homogeneous flows of the order which do not give rise to any particular condition of compatibility. For the flows of nth order, in contrast, one has to write a certain number of -1 /T conditions connecting the derivatives of nth order. We shall examine-'-^ as an example the case of homogeneous flows of 2nd order. There are six second derivatives which we shall denote 9^^ (i and j may assume independently the values 1, 2, 3)j '^i^ designating o ^-^-^ — . Outside of (T) we shall put aXi ox A cp . . = cp . . + 9 . . ij ij iJ 1 2 with 9j_A being a fiuiction of \ only, 9^; of |i only (see for- m\ila 1.19). Inside of (F), ^lp2-l-L2^ ^ J 1. £_ ^1 p2 2 + 1 1 L i^(z2 - l)'(Z) (1.29) 1.3.2.2.- Let us now consider a point 0' ^x-, = e,, Xp = 0, x-, = o), €]_ being a very small quantity. Let M be a point with the coordi- nates (x-i,r,9^ with respect to 0, inside of (F), and with the para- meters (p,9) in the plane Z. For the conical flow (homogeneous of 1st order) with the vertex O', its coordinates are: (x^ - e-|_, r, 9) and its parameters in the Z-plane: ' p2 + 1 e^N ^ P 1 - 1 ""l; Since dx2^ = =1 = 3r dp 2p' P^-1 1 dp P Let US then consider two identical conical fields but with the apexes and 0', and form their difference. We shall obtain a velocity field which, due to the linear character of the equation (l.lO), will satisfy this equation. If u = r[f(z)] denotes the component u of the field with the vertex 0, one has as component u in the "difference field" u = +Rb'(z)l - R F Z p^ - 1 ""i / -t-1 ^ rTzf-Cz)! (1.30) _ 1 x-L -L J NACA TM 135i<- 27 e-|^ being considered as infinitely small. Moreover, according to the relations (l.25)^ the components v and w are written .=!lPi^R ^1 p' 1 L ^(z2 + i)f'(z) w ^1 p2 - 1 > i |(z2 - i)f'(z) (1.31) 1.3-2.3'- Let us consider the point C'^O^Sg^oV with fg being a small quantity.' Let M be a point with the coordinates (x-, ,r,0^ with respect to 0, inside of (r), with the parameters {p,d) in the plane Z. For the flow with apex 0' ', the coordinates of M are (<^ i n ^ \ X-, , r - Sp cos 9,6 + ^^ j as can be easily stated by projecting M in m on the plane XgXo (fig- 9) • But on the other hand 2x 2 dr = — ^ -*• " P dp = -eg cos 6 (i . P=)' r de = 2x 1 P 1 + p' d© = ^2 sin 9 thus dZ j^^pLp + ip dm = 1 + p^ 2xt i sin - cos 9 1 + p'^ 1 - P' .ie with Z + dZ representing the point M in the conical field with the vertex ' ' . Let us then consider two identical conical flows, but with the apexes and 0'', and form their difference. We shall obtain a velocity field which due to the linear character of the equation (l.lO) will satisfy this equation. If Vq = r[g(zJ] 28 NACA TM 1351+ denotes the component v in the field of the vertex 0, one has a com- ponent V in the "difference field" V = +nrG(z)'l - r[g(Z + dZ)] = -R[G'(z)dz] ^SP P^ + 1 2^1 p2 - 1 2x3_ p2 _ 3_ - G'(Z) cos 0(1 + p2) + i sin ^p^ - l) e^ ZG'(Z)/'z + i 10 ^2 p2 . 1 ■R ^1 p^ ^(z2 + i)g'(Z) (1.32) besides, according to equation (l.25), the components u and w are written u w = !3. 9!jl1 nlm' iz)\ ^1 p2 - 1 ^ -" = !l 9L±J. n\^(z2 - i)g'(z) 1 p - 1 L J (1.33) 1.3-2 A .- With these three lemmas established, it is easy to demon- strate the property we have in mind. Let us call "complex potential" of a homogeneous flow of zero order the function $(Z) (section 1.3. 2.1) so that cp = h[^(z)] so that the function of complex variable, the real part of which gives inside_^f (T) the projection of the disturbance velocity in the direc- tion Z, is the "complex velocity" of a conical field in the direction Z; so that, finally, the velocity field obtained by the difference of two iden- tical conical fields, the vert_ices of which are infinitely close and ranged on a line parallel^ to I, is the "field derived from a conical flow" in the direction Z; then we may state: I NACA TM 135ij- 29 Theorem; The field derived from a conical flov in the direction I is the velocity field of a homogeneous flov of zero order; the complex potential of that flov of zero order is proportional to the complex velocity of the conical field given in the direction I, since the pro - portionality factor is real. The proof follows immediately. According to sections 1.1.2 and 1.1.3 one niay be satisfied with considering, for definition of a homogeneous flow, the inside of the cone (r); comparison of the for- mulas (1.29), (1.30), _(^I.3l);. (1-32), (1.33) entails th_e^ validity of the above theorem' if" Z is parallel or orthogonal to U. Hence the general case where I is arbitrary may be deduced Immediately; if F(Z), G(Z), H(z) are the complex velocities in projection on Ox-|_, Oxo, Oxo, the expression for the component u of the field derived in the direction ^(^i^^p'^^) ^^ 1 P^ + 1 D u = — R ^1 p2 - 1 - e-LF'(Z) + egG'Cz) + e3H'(Z) Thus, with ^iF{Z) + Sg'^^^) "*" ^3H(Z) being the complex velocity in projection on I, comparison of this formula with the first formiila (I.29) completely demonstrates the theorem. Corollary: The field derived in the direction I of a conical flow, the complex velocity of which in the direction I is K(Z) , is a velocity field of a homogeneous flow dependent only on K(Z) (not on the direction Z) . The theorem just demonstrated may be extended without difficulty to the homogeneous flows of nth and (n-l)th order. A statement of this general theorem would require only specification of a few definitions; however, since we shall not have to utilize it later on, we shall not formulate this statement . 30 NACA TM 13514. * CHAPTER II - CONICAL FLOWS WITH INFINITESIMAL CONE ANGLES 2.1 - Solution of the Problem 2.1.1 - Generalities We shall now treat the first problem set up in section 1.2.6. We shall operate in the plane Z. Let us recall that the image of the cone (r) is the circle (Cq) of radius unity centered at the origin^ and that the image of the obstacle is a curve (C), defined by its polar equation p(0) . We shall denote by (D) the annular domain comprised between (C) and (Cq); we shall call (70) the circle of smallest radius centered at the origin and containing (A) in its interior, and we shall call k the radius of the circle (/q). In this entire chapter, k will be considered as the principal infinitesimal. The problem then consists in finding three functions U(z), V(z), W(Z) defined inside of (D) except for an additive imaginary constant, so that (1) ■P dU = ^Z dV = 2iZ ^y (j_25) Z^ + 1 Z^ - 1 (2) the real parts u, v, w, which are uniform become zero on (Cq); (3) on (C), one has the relation v|jD cos + p' sin e + p2(p cos - p' sin 0) + w p sin - p' cos + p2(p sin + p' cos 0) = ^H_(i + u) Put in this manner, the problem is obviously very hard to solve in its whole generality; however, an analysis of the permissible approxima- tions will simplify it considerably. 2.1.2 - Investigation of the Functions U(z), V(z), W(Z) 2.1.2.1 .- An analytical function of Z will be the said func- ^i°" (A) if its real part becomes zero on (Cq). Let us designate by NACA editor s note: Some minor inconsistencies appear in the niiraberii of equations in this chapter and subsequently in chapters III and IV, but n( attempt was made to change the numbering as given in the original text. NACA TM 135i<- 31 (7q') the circle with the radius l/k, centered at the origin, and by (D') the annulus limited by (/q) and (/q') (fig- 10)' Lemma I.- A uniform fiinction (A), defined inside the annulus limited by ^7^,^ and (^0) "^DS-y be continued over the entire domain (D') This results iimnediately from Schwartz' principle. Let M and M' be two symmetrical points with respect to (Cq)? M being inside of (Cq); 6ne defines the function (A) at the point (M') as having, respectively, an opposite real and an equal imaginary part compared to the real and the imaginary part of the function given at the point M. Lemma II.- A holomorphic function (A) inside of (P') has a Laurent development of the form- *-' ■p * Z fe - V" 1 Let H(Z) = h + ih' be such a f-unction (A) . Let us write its Laurent development in (D') provisorily in the form 00 H(Z) = ^J.Z-- . Y I 1 It is an immediate demonstration and yields the formulas defining Jj^ and Kj^ K =^ r^'' (h + ih')^ e^^^de 17 — We remember that K^ denotes the conjugate imaginary of K^^. 32 NACA TM 1354 (h + ih')^ denoting the value of H on (7o)' likewise 2rt J = iC / (h + ih')^ ,e-^^^d9 ^ (h + ih')^ ,^-in0. 2nJn ^0 Consequently, according to the lennna I: ^ = -*^n moreover P P 2jt p 27t Jo = ^- / H(Z) ^ = i / (h + ih')_^ d0 = i / h_^' d0 is purely imaginary, and the lemma II is therewith demonstrated. We shall note that, if H(Z) is limited by M on fyQ\ or (^o') ^ one has the inequality Kn < Mk'^ (ll.l) Lemma III.- A function (A) with a real and uniform part defined in (D) can be developed inside of (P') in the form B log Z + 13 + ^ (^ - K^Z^J (II. 2) Z / with B being real. Actually, the derivative of the function (A) is necessarily uni- form. Thus one knows (see for instance ref . I3) that one may consider the given function as the sum of a uniform fiinction H(Z) and a loga- rithmic term; since the critical point of the logarithm is arbitrary inside of (7q)} it is particularly indicated to choose this point at the origin; since the real part of the function is uniform, the coeffi- cient of log Z is real. Besides, since log Z has a real part zero NACA TM 13514- 33 on (^o)> H(Z) is itself a function (A). The given function may therefore be continued inside of (D') and the development (ll.2) is thus justified. Remark . If one chooses as pole of the logarithmic term a point inside of / y^\ but different from the origin, one obtains a development of the form 2.1.2.2 .- The functions U, V, W of the variable Z are all three functions (A) with a real uniform part and, consequently, can be developed in the form (II.2). We shall write henceforward ^U(Z) 2 = A log Z + ia + V(Z) = B log Z + ip +^ f^ - K^Z^j > (II. 3) W(Z) = C log Z - i7 + A, B, C are real, a, p, 7 are real and also arbitrary; but these developments are not independent since the relations (l.25) must be taken into account. For instance, Z dV/dZ must be divisible by Z + 1; otherwise we would have for U logarithmic singularities on the cone (r) which is inadmissible. Now Z^ = B dZ Z "{^ * 5^^") 3^ NACA TM 1351+ Hence one deduces the relations B =^( - 1)P2p[k2p + Kgp] 1 > 00 = ^ ( - l)P(2p + Dkp+i - Kgp J (II. 1^) obtained by putting in the preceding equality Z = i and Z = -i. Likewise J Z dW/dZ must be divisible by Z - 1 which gives C = 1 2p(L2p + Lgp) =^ (2p + 1)( ^2p+l "^ ^2p+l) (II. 5) Finally, the equalities (l.25) lead, in addition, to relationships connecting the coefficients of the developments (II.3) among themselves; thus one may write the relations B + 2K2 = -i C - 2Lc ^ Ki - % = -ifli + lJ (II. 6) nK^ - (n - 2)Kn_2 = i[(n - 2)1^.2 + rij (n ^2) and on the other hand B = -(Ji + Ji) K-L = -A + 2J2 nKji = (n - l)Jn_i + (n + l)Jn+i (II.7) (n >2) NACA TM 135^4- 35 2. 1.2. 3- Approximations for the developments (II.3) '- Moreover, the hypotheses of linearization must be taken into account which, as we shall see, will permit us to simplify the developments (II.3) consider- ably and will lead us in a very simple manner to the solution of the problem posed in section 2.1.1. The equalities (II.6) make V(Z) and W(Z) seem of the same order. We shall denote by M an upper limit of their modulus on the circle (7o)' ^ will be equally an upper limit of their modulus on (7r.'\ and hence in the entire domain (C). If one utilizes the inequality (ll.l), (11.^+) shows that-'-° B = O^Mk^) K-L - % = O^Mk^) If one assumes a, 3, y zero in what follows, which does not at all impair the generality, one may write the second formula (II.3) in the form V(Z) - R(Ki)(| - Z) - ^ ^ = B log Z - ^ K^Z^ + i^ (%)(! ^ ^) and consequently: In the annulus limited by (■yQ\ and (Cq)^ "the second term of this equality is o(Mk2log k) Likewise according to equation (II.5) C = o(Mk^) L-L + Lj^ = o(Mk3) 00 00 W(Z) - lT(Li)(| . z) - ^ -J = C log Z -H r(Li)(| - z) - ^ L^Z^ denotes Landau's symbol, " - r^im.2\ „,• — .:*.,•„„ ^.u^■^. A limited when k tends toward zero denotes Landau's symbol, A = OfMk j signifies that -^ is Mk^ 36 NACA TM 13514. In the anniilus comprised between (7r)) ^-^i^ f'-'o)' *^® second term of this equality is also o(Mk2log k) Furthermore, according to equation (11.6) Thus W(Z) - iV(Z) = O^Mk^log k) + 2iK-|_Z in the annulus fy ,C \. Finally, according to equation (II.7) A = -% + 0(Mk3) J^ = U-JLA k^^^ + o(Mk^+2^ Thus - 1 u(Z) = -R(K-[_\log Z - siCgZ + y~ S_Li ^1±1 + o/Mk3log k) 1 ^ Summing up: If one is satisfied with defining V(z) and W(z) except for o(Mk log kj and U(z) except for o(Mk-^log k), one may write in the corona (70^*^0) W(Z) = iV(Z) + 2iKiZ (ll.8) V(Z) = H(Z) - K^Z (II. 9) NACA TM 1354 37 with H (11.10) and u(z) Z ^ dZ - 2KpZ dZ ^ (11.11) The coefficient K-|_ may be supposed to be real, and the integra- tion occurring in equation (ll.ll) must be made in such a manner that r[u(Z)J will be an infinitely small quantity of the third order at least on | Z | = 1 . 2.1.2 A - Remarks . (1) The formula (II.8) which is the most important may be estab- lished immediately from the second formula (l.25). However, the method followed in the text, even though a little lengthy, seems to us more natural; also, it shows more clearly the developments of the func- tions U, V, W. (2) Strictly speaking, the hypotheses set forth in the course of this study must be verified by the solutions found in each particular case. We shall, however, omit this verification which in the usual cases is automatically satisfactory. (3) The results obtained by the preceding analysis and condensed in the formulas (II.8), (ll.9)j (ll.ll) are in all strictness valid only in the annulus (^n^'^o)' ^^^ '^°^ ^'^ ^'^^ domain (D) . However, it is very easy to extend, by analytical continuation, the definition of H to (D) . Let us now first suppose that (C) contains in its interior; since one may write V(Z) in the form V(Z) = H(Z) - V~ Kj^z" + B log Z 1 one sees that, since V(Z) is defined by hypothesis in (D), and one 00 can extend ^ K^z" and B log Z inside of (/g) up to (C), H(Z) 38 NACA TM 135i+ may itself be defined without difficulty inside of (D) . The case where (C) does not contain the origin offers no difficulty; it is then suffi- cient to utilize the development given at the end of section 2.1.2.1. As to the order of the terms neglected when one writes the equal- ity (II.9) in the domain (D), they are fo\ind to be o(Mk^log k) in (D) in the case where there exists inside of (C) a circle of the radius Xk (X, and l/X may be considered as 0(l)). Besides, if that is not the case, one may justify the validity of the results of the formiilas (II.8), (ll.9), (II.IO), (ll.ll) by making a conformal representation of the domain (D) on an annulusj the radius of the image circle of ( Cq ] may be assumed equal to linity; the image circle of (C) has a radius infinitely small of first order with respect to k and the study may be carried out in the new plane of complex variable thus introduced, without essential complication. 2.1.3 - Reduction of the Problem to a Hilbert Problem If one puts, according to the formula (II.8) V = V + iv' with v' denoting the imaginary part of V, one may write on (C) the relation w = -V Since one may, of co\xrse, with the accepted approximations, neglect u compared to 1 in the second term of the formula (I.28), one sees that this bo\indary condition (I.28) affects now only one single analytical f\inction, the function V(Z); this is a first fundamental consequence of the preceding study. Formula (II.9) shows that this condition con- sists in posing a linear relation between the real and the imaginary part of H(Z) on the obstacle. Now according to equation (II.IO) the function H(z) is a holomorphic function outside of (c), regular at infinity; the problem stated which initially referred to an annular area (D) is thus reduced to a Hilbert problem for the function H defined in a simply connected region; exactly speaking, one has to solve an exterior Hilbert problem. This is the second fundamental consequence of the results of section 2.1.2. Since we attempt to calculate V(z) and W(Z) not further than within 0(Mk2log k), and U(z) within 0(Mk5log k), the relation (I.28) NACA TM I35J+ 39 which is written (v - iw) SZp^de - i dz(l - p2) 2p^ de may be simplified and reduced to R - i dZ(v - iw) = 2p^ de On (C), K-^^Z is, according to equation (ll.l), of the order of Mk^, and therefore H=V=v+iv' =v-iw consequently, H satisfies, on (C), the Hilbert condition R - iH(z) dZ = 2p^ d0 (11.12) 2.1.^4- - Solution of the Hilbert Problem A function H(z), holomorphic outside (C), regular and zero at infinity, satisfying on (C) the relation (II.12) must be found. Let z=Z + a„+^!^ + Z (11.13) be the conformal canonical representation of the outside of (C) on the outside of a circle (7) centered at the origin of the plane z; the adjective canonical simply signifies that z and Z are equivalent at infinity. On (7) we shall put z = re i9 ko NACA TM 13514- r being constant and well determined. Let us put F(Z) = i log J (II. li^) One has on (C) or on (7) F'(Z) dZ = i ^ = -dtp = fie) d0 (11.15) with f being real; consequently R - iH^ d0 = R iH^ d0 F' (Z) d0 R lH(Z}_ F'(Z) and therefore equation (II.12) is written R H(Z) F'(Z) ^ 2 P^ dG P d9 (11.16) H(Z)/F'(Z) is a holomorphic function outside of (C) and regular at infinity. Following a classical procedure, we thus have reduced the Hilbert problem to an exterior problem of Dirichlet. Let G(Z) be the holomorphip function outside of (C), real at infinity; its real part assumes on (C) the values — . G(Z) is determined in a unique manner. According to equation (II.IS) H(Z) = -iG(Z)F'(z) + ieF'(z) (11.17) with e being a real constant . However, we have seen (section 2.1.2.3) that the coefficient of l/z in the development of H(z) around the point at infinity [coeffi- cient K, ^ was real; now, around the point at infinity iF(z) NACA m 1354 4l In order to have the development of the second term of the formula (II.I7) admit a real coefficient of l/Z, e must be zero since G(Z) is real at infinity. Thus the desired solution is H(Z) = -iG(Z)F'(Z) (11.18) With the function H(Z) thus determined, the formulas (II.8), (11. 9) > (11.11) permit calculation of the complex velocities U(Z), V(Z), W(Z) within the scope of the accepted approximations. Thus the problem posed in section 2.1.1 is solved. Remarks . (1) Uniqueness of the solution .- The preceding reasoning shows the solution of the Hilbert problem satisfying the conditions (II.I6) to be unique. This result will be valid for our problem if one shows that every function satisfying the condition (II.16) is a solution of the initially posed problem (condition (11.1^4-)) which is Immediate since it suffices to repeat the calculation. (2) Calculation of the coefficient %.- According to what has been said above, the coefficient K-|_ is equal to the (real) value assumed by G(Z) at infinity. In order to find G(Z), we may solve the Dirichlet problem in the plane z; according to a classic result of the study of harmonic functions, K-]_ is equal to the mean value of 2p =^ on the circle (7) . Hence d9 K, = i 2p^ M dCP = ^ p2d0 = 2S '1 2it Jq 3 dcp '''' ^&J (c) "" "" "P wherein S represents the area inside the contour (Cj. 2.2 - Applications 2.2.1 - General Remark Let us consider a cone of the apex in the space fOx]_,X2,Xo), the image of which in the plane Z is the curve (C), defined by its polar equation p(0). According to the definition of p (see the remark of section 1.2.5) the sections of this cone made by planes par- allel to OxgXo are homothetic to the curve k2 NACA TM 1354 sin (11.19) 2p a 2p Xo = COS 6 Xo = 1 + p2 1 + p2 -> In the case of the linear approximations, with grad u, grad v, grad w being infinitely small (it would even be sufficient that they should be limited), one sees that one may, within the scope of the approximations of section 2.1, simplify the formulas (II.I9) without inconvenience and write them Xo = 2p cos 9 Xo = 2p sin 9 hence the result,, essential for the applications. The curve (C) in the plane Z is homothetic to the sections of the cone obstacle made by planes normal to the nondisturbed velocity. Let us likewise consider a cone with variable but small incidences so that the flow about the cone should always be a flow in accordance with the hypotheses of this chapter. One sees that if the orientation of the cone varies with respect to the wind, the curve (C) in the plane Z undergoes a translation. 2.2.2 - Study of a Cone of Variable Incidence This last remark allows us to foresee that when a thorough investi- gation of a cone has been made for a certain orientation with respect to the velocity it will not be necessary to repeat all the work for any other orientation. This we shall specify after having demonstrated the following lemma. 2.2.2.1 - Lemma . - One may write on (C) that (11.20) 2p2 de P d9 . 2 R 3 - zZ and consequently i p2 de = - d

^ ^f - ^jl:°^ ^ - z2 - aSy or U(Z) = A- r2 - ^ log Z + nJz^ - kaJ" ^ \/z2~7ul2" (11.30) If one makes a = 0, one will find again the expressions already obtained for U(z) and V(Z) in the case of a cone of revolution of zero incidence (formulas (ll.2l4-) and (II.25) in which one makes a = 0) We shall denote by e and by t| the principal angles of the elliptic cone (see fig. 12). One has ep = 2 k*4 TIP = 2lr - ^ whence r =^(e + ,) a2 =gp _ ,2) 50 NACA TM 135i(- The pressure distribution on the cone circumference is easily cal- culated. It is sufficient to apply the formula (11.26); besides V(Z) II^ T)^cos^9 + e^sin^9 and .[u(z^ = ^ log 3(e + n) + 1 2 2 ri^cos^'P + e^sin^9 hence the final formiila 2CT1 log P(e + n) - 1 + zl^^^cos^P + e^sin^Cp) (11.31) The case where the velocity is not in the direction of the axis may be treated equally by utilizing the formiila (ll.2l). In this formula one must put hO(z) = 2/^2 _ A 1 ?>\ r2/ _ a2 ^ = 1 _ §". dz ,2 One then obtains „(!),.) =|(.^.g z ^ 2 z2 - a2 P a 1 2 2 or^ z^ z2 - a2/ z2 z2 - a2 P(z2 - a2) 'r2 - alU + ax2 - ^2 hence, remarking that Z = z + 2^ + a z NACA TM 13514- 51 H(Z) = ^ ,2 a (or^ - ^2)(z - g - /(Z - g)^ - 4a^) 2a2xJ(z - g)2 - l^a^ and v(z) = H(z) - ^p - arjz On the other hand, we shall calculate U "by utilizing the vari- able z and the formula (II.20). The coefficient Kg is eq,ual to Kc = ^ r^ - — |g + gr^ - ga^ and U(z) is then given by the formula U(z) = 4r r- - log z - 2a'^ + gz\ , aa - gr^ z(z2 - a2) (Zz^ + gz) Kp 3 (11.32) One will note that, if one puts g = 0, one finds again the for- mula (11.30), and that, for a = 0, one finds again the formula (II.25), except for the notations. Thus one can, without any difficulty other than the lengthy writing expenditure, calculate the pressure distribution coefficient on the elliptic cone of any arbitrary orientation with respect to the wind. 52 NACA TM 135i| 2.2.5 - Calculation of the Total Forces We have already seen in section 1.2.6 that the normal to the conical obstacle directed toward the outside has as direction parameters 7(^3^2' - ^2^3')' ^3'' "^2 Let n be the unit vector coincidental with this normal, s be the area of the section with the abscissa x-^, L the length of this section; one may make correspond to the resultant of the forces acting on a section fx-. , x-j^ + dx-^A a (dimensionless) vector "C^ = - i TCpltds (11.33) situated in the plane XpX^, and a dimensionless number Cx = - ^ rCp(nU)ds (I1.3i+) ' the vector C^ characterizes the lift, the number C^^ the drag. The integrals appearing in_^the formulas (33) a-nd (3'^) are taken along the section. Naturally C^ and C^^ are independent of this section. One may also replace C^ by a complex number C^, the real and i^ginary parts of which are equal to the components of the vec- tor C^ on Oxg and Oxo. For calculating equations (II.33) and (11.3^+) one may utilize the section x-]_ = p. If we assume I to be the length of the contour (C) in the plane Z, we may write, taking into account the habitual approximations c^ = i / Cp dz (11.35) 'c and Cx = - |- R pz - CpZ dZ (11.36) NACA TM 1354 53 with the integrals appearing in eq\iations (II.35) and (II.36) taken in the plane Z. These integrals present a certain analogy to the Blasius integrals (ref . 13) J C-p is given by the formula (ll.26)j unfortunately, it is not possible to give simple formulas for the total forces since the integrals (II.35) and (II.36) make use of all coefficients of the conformal representation-'-^. We shall apply the formulas (II.35) and (II.36) to the case of the circxilar cone; C_ is given by equation (II.27) dZ = i ^ e^^de Z dZ = 1 ^-^ d0 I = nSa 2 k One obtains C2 = -2a7 C^ = 2a3log -^ - a.^ - ay^ (II.37) Pa, In the case of the elliptic cone of zero incidence, C^ is obvi- ously zero Z = re-i^ + ai ei^ dZ = iLi^ -^e-i^1 dcp whence '^^ - wi^' - $) C '^^ '" with C_ being given by formula (II.31). Now 6T1 d*? ^ r en dt -jt 2(ti2cos^ -I- e^sin^) ^-00 t^^ + ^2^2 19 See appendix No . 7 • 5k NACA TM I35U As one can see immediately by putting t = tan cp the calculation of this last integral is immediate. Thus one obtains log p(e + Ti) 2 (11.38) with I being the length of the ellipse with the semiaxes — , — . 2.2.6 - Approximate Formula for the Calculation of Cx Let us consider the function U(z); according to formula (ll.ll) and the remark 2 of section 2.1.4 one may say that the principal term for U(z) is U(z) = k -^ log z Consequently, in first approximation Cp = - ^ log r with S being the area inside of the contour (C), and r the radius of the circle (7) on which one makes the conformal canonical repre- sentation of (C). If one now calciilates C^,, taking into account this approximate formula, one has, according to equation (II.36) Cx = 18s np3i log rR Z dZ whence C^ = + 22Sl 1 1 «p3z (11.39) NACA IM 135i^ 55 We shall state: In every first approximation the value of the drag coefficient C^ is given by the formula (11.39) • 2.2.7 - Case Where the Cone Presents an Exterior Generatrix If the contour (C) shows an exterior angular point, the various functions introduced in the course of the study (first paragraph of this chapter) present certain singularities. These singularities we shall specify. Let Zq be the designation angiilar point of (C), and &jr the angle of the two semitangents to (C) at the point Zq(0 < & < l) (see fig. 13 )j if' Zq is the image of the point Zq in the plane z, one may write, according to a well-known result, in the neighborhood of Zq with K being a complex constant and k = 1 - B; consequently k [f'(Z)]o = Ki(z - Zo)-k = K2(Z - Zq) 1+k with K-, and Kg being complex constants. F'(Z) thus becomes infinite at the point Z = Zq. In contrast, the function G(z) has, according to definition, a real part which assumes on the circle (7) the values 2 R zZ^ dz This real part thus remains finite on the circle (7) (and it satisfies there a condition of Holder) . According to a known theorem, its imaginary part likewise remains continuous on (7) (and likewise satisfies a condition of Holder). Consequently, one sees, if one refers to formula (II.I8) that 56 NACA TM 135i| H(Z) = K3(Z - Zq) k 1+k in the neighborhood of Zq; likewise, U, V, W will, in the proximity k 1 of. this point, be of the order — - — with respect to 1+k Z - Zq Thus the analysis made in section 2.1 is no longer applicable to this case. However, the formiilas (II.35) and (II.36) show that if the pressure coefficient assumes very high values ixi the neighborhood of Z = Zq, the total energy remains finite. According to what we have indicated in section 1.1. 3 ve consider the solution still valid, with the linderstanding that the values of Cp in the surroundings of Z = 2 are not reliable . 2.2.8 - Delta (a) Wing of Small Apex Angle at an Infinitely Small Incidence If one puts in the formiilas r = a^, at the end of section 2.2.U, one obtains the pressure distribution on a delta wing with small apex angle . Let us recall that a delta wing is an infinitely small angle . Its angle, according to definition, is the half -angle co at the vertex (compare fig. 1^). Thus one has cdP = ka. The formulas (II.3I) and (II.32) are applicable to a delta wing of small angle placed at an incidence also rather small. Let us moreover assume that this opening is infinitely small with respect to the incidence. Under these conditions, the formiilas yielding U(Z) and V(z) are written V(Z) = ^ CL ^ Z - \/z2 - ka^ *2 2 x/z^ - l+a^ U(Z) = - ^ 2-1-2: ^ + 8a^(a - a)Z (ll.40) NACA TM I35U 57 Actually one is justified in omitting the second-order terms with respect to a. For calculating Cp it suffices to apply the for- mula (11. 8); the second term of the second formula (ll.i^-O) may be neglected. With the incidence j , the delta ving being parallel to 0x2, °"-^ •'^^^ 7P = 2ia Finally ; one may put along the A Z = 2a cos cp = y^ cos cp 2 One then finds C^=^^ (II Al) P sin cp We remark further that cp is related to the angle ^ of figure l4 ty 2^3 = cdB cos 9 -^ = aL_cos 2 9 One may state: the pressure coefficient on a delta wing of infi- nitely small opening angle is independent of the Mach number of the flow. One has . 2^r if t . 5i if one applies formula (11.35), one finds C„ = irtoD/ This coefficient C^ has not the same significance as the one utilized in the theory of the lifting wing. Actually, it is, according 58 NACA TM 1351+ to the very manner in which it was obtained, relative to the total area of the A (pressure side and suction side); if one takes only one of these areas into account, one must write (neglecting the factor -i) 2jtai7 This form\ila has been found by other methods by R. T. Jones (ref. 1^). We shall find it again in chapter III, section 3-1.2.U, when studying the general problem of the delta wing which is here only touched on incidentally and for the particular case of a A with infinitely small opening angle . 2.2.9 - Study of a Cone With Semicircular Section As the last application, we shall treat the case of a cone with semicircular section, with the velocity U being directed along the intersection of the symmetry plane and of the face plane of the cone (fig. 15). 20 The contour (C) in the plane Z then is a semicircle, centered at the origin, of the radius a (fig. I6) . One obtains very easily the conformal canonical representation of the exterior of this contour, on the outside of a circle (7) of the radius r, centered at the origin of the plane z, by means of a par- ticiilar Karman-Trefftz transformation (ref. I3, p. 128) which is written Z - a Z + a z . .e-'tl z .5it (II. 1+2) a and r are connected by the relationship ^a = 3r^ In order to obtain the correspondence between the circle (7) the contour (C), one must distinguish two cases. Let us put .19 and re^ 20, Such a cone formed the front of supersonic models planned by German engineers . NACA TM 1354 59 (1) - 2. < cp < li, the corresponding point of (C) is on the arc 6 o of the circle. Let us put under these conditions Z = ae^ and we shall find according to form^ila (11.^4-2) tan — = 2 "^■^U ^ 12 ^^^'.2 ^ 12 (IIA3) (2) -^ < 9 < —J^, the corresponding point of (C) is on the seg- 6 6 ment AA'; let us put under these conditions Z = a cos X The formula (lI.i)-2) shows that tan - = 2 "^^'2 ^12 y ■ (^ 5jt f^H2 ^ 12 (lI.i+4) The two last formulas define completely the desired conformal representation. Figures (17) and (18) give the variations of '^ and X as functions of 9 • We shall have to utilize equally the value of dz/dZ. The simplest method for obtaining this value consists in logarithmic differentiation of the two terms of formula (II.42). One thus obtains the result dz dZ z^ + irz - r^ „2 2 Z - r (11.1^5) 60 NACA TM I35U If one has - ^ < ^'> < ^, one must put in the preceding formula 6 6 z = re-' Z = ae ill/ whence dz dZ r^ 1 + 2 sin cp ^i(cp-\tr) _ 8 1 + 2 sin 9 gi(9-i) {u.kG) 2a 2 sin t 27 sin ^ If cp is comprised between -^ and —-^, one puts z = re-'-''^, 6 D Z = a cos X. Thus one obtains dz 16 1 + 2 sm ■(1-1 ^^ 27 sin^X (11.^7) The function G(Z) has as its real part R zZ^ dz , that is 21 ar ^^^^ 8 1 + 2 sin cp if _ « < cp < II if I« < cp < llH 6 (II A8) The analytic function g^2 z dZ Z dz has a real part which, on (7), assumes these same values. This func- tion is regular at infinity, holomorphic outside of (7), but with a 2 pole z = -ir, with the corresponding residue being equal to -ia . NACA TM ISSij- 61 Let us then consider the f\inction ,2/z dZ 1 z - ir a Z dz 2 z + ir This function is holomorphic outside of (7). It is regular at infinity; its value at infinity is equal to a /s . On (7), these real and imaginary parts satisfy Holder conditions. This function is there- fore identical with the desired function G(z). Hence one deduces according to equation (II.I8) H(z) = /^ d^ - i z - irXa*^ ^ = 2:_ _ §^ z - ir dz \^Z dz " 2 z + ir/ z dZ Z ~ 2z z + ir dZ and according to equation (II.I9) V(Z) = a2fl - Z _ i z - ir dz\ \Z 2 2z z + ir dZ/ Finally, the calculation of U(Z) may be carried out with the aid of formula (II.29) 'g ^ = a2log Z - ^ r ^ - i^ d^ = a2 flog — ^ + I log z z 2/z+irz \ z+ir2 and ZH - K. = a2 fl - 1 Z dz z - ir _ iN = af A _ z - ir Z dz\ -■- V 2zdZz + ir Zj 2 \ z + irz dZ/ whence 2 log — ^i + log z z + ir / The calculation of the coefficients Ko offers no difficulty what- soever; however, as one had already opportunity to note, the term YLZ does not occur in the calculation of the pressures along the cone. 62 NACA TM 13514- This pressure distribution along the cone calculated with the aid of equation (II.26) is represented in figure I9. 2.3 - Numerical Calculation of Conical Flows With Infinitesimal Cone Angles 2.3-1 - General Remarks In the preceding paragraph, we have studied a certain number of particularly simple cases. However, if the cone (C) is arbitrary, it will be necessary to carry out various operations leading to the solu- tion by purely numerical procedures. Let us analyze the various operations necessary for the calculation: (1) The conformal canonical representation of the exterior of (C) on the outside of the circle (7) must be made; this calculation per- mits, in particular, determination of the radius r of (7), corre- spondence of the points of (C) and of (7), and calculation of the expression dZ on the contour (7). (2) The function G(z), holomorphic outside of (7), regular and real at infinity must be determined, the real part on (7) of which is known; we shall designate it by g(cp) . In fact, it suffices to know, on (7), only the imaginary part of G(z), for instance g'(?); g'(9) is the conjugate function of g{^P) and is given by the formula p2n g'(9) = ^ / g(cp')cot r_^^dCP' Zn This formiila is called "Poisson's integral." (3) With these two operations accomplished, the values of H(z) on the circle (7) (formula (II.I8)) are known which provides the values of v and w on the cone; u is obtained by the formula (II.29). The only new calculation to be made is that of the expression: G ^ g' dcp the constant of integration being determined so that u should have a mean value zero on (7). NACA TM 1354 63 All these operations always amount to the following numerical problems : (a) With a function given, to calculate its conjugate function (Poisson integral) (b) With a function prescribed, to calculate the derivative of the conjugated function (c) With a function prescribed, to calculate its derivative^-'-. We shall justify this result in the following paragraph by showing that the operation (l) may be performed by applying the calculations (a), (b), (c). We shall then indicate a general method, relatively simple and accurate, which permits solution of these problems. We shall ter- minate this chapter by giving an application. 2.3-2 - Conformal Canonical Representation of a Contour (C) on a Circle (7) The numerical problem of determination of the conformal canonical representation of a contour (C) on a circle (7) has been solved for the first time by Theodorsen^^ . We shall briefly summarize the principle of this method, simplifying, however, the initial expose of that author. Let us suppose, first of all, that the contour (C) is neighboring on a circle of the radius a, centered at the origin (fig. 20); in a more accurate manner, putting on (C) Z = ae^^i® (II. U9) with t being a function of 6, ^l' = "^(O) , we shall suppose that il^(0) and — are functions which assume small values. We shall then call de If the conformal representation of the exterior of (C) on the outside of (7) is known in explicit form, it will naturally be suffi- cient to apply operation (a). Compare references 15 and 16. One may achieve this conformal representation also by the elegant method of electrical analogies (ref. I7); the time expenditure required by the experimental method and by the purely numerical methods here described as well as the accuracy of these pro- cedures are of the same order of magnitude. 6k NACA TM 1354 (C) "quasicircular." Let 9 be the angular abscissa of the point of (7) which corresponds to the point of (c), the polar angle of which is ©; we put = cp + 7(9) 9=0- e(0) (11.50) e(9) and e(cp) representing the same function but expressed as a function of 9 or as a function of 9; we shall put likewise ?(9) = ^(0) / The desired conformal transformation may be written Z = ze^(2) with h(z) being a holomorphic function outside of (7)^ regular and zero at infinity. The equality (II.50) becomes, if one writes it on the circle (y), ae whence r(9)+ir9+e(9)"j = rei^ehCz) h(z) = I t(9) + ie(9) + log I (11.51) Finding the conformal representation of (C) on {'p) amounts to cal- culating the functions ^(9) and "(9). First of all, one knows (equa- tion of (C)) that ^(9) = ^^[9 + 7(9)] (11.52) On the other hand, according to equation (II.51), e(9) is the conju- gate function of ^(9)^ and consequently e(9) =— / \lf(9')cotri^^^^^jd9' (II.53) NACA TM I35U 65 the integral being taken at its principal value. There is no constant to add to the second term of equation (II.53), for <^(9) has a mean value zero since h(z) is zero at infinity. For the same reason, if lif^ denotes the mean value of ^(9) in an interval of the amplitude Srt r = ae^Q (II.54) an equality which will permit calculation of r if 1^(9) is known. In order to calculate 'e(9) and ^(T), one disposes therefore of the relations (II.52) and (II.53); one can solve this system by a procedure of successive approximations. We shall put first ^0(0) = eo(9) = According to equation (II.52) ^(0) = He) and according to equation (II.53) \ie) = ^ / t(e')cot ^-^d0' Thence a first approximation for "P 9^ = - ^1(0) = 9-L + \C^-l) From it on£ deduces, according to equation (II.52), a first approxima- tion for 1K9) whence a second approximation for the function e 66 NACA TM 1351+ '2rt ^2(1) =^j^ *l(.l')oot ^1 _ c^ dft^' '2(«) = ^2 €1(8) whence ^o = - ^2^^) ^2 2 (■'"2) The procedure can be followed indefinitely. The convergence of the successive approximations forms the subject of a memorandum by S. E. Warschawski (ref. I8) . We refer the reader who wants to go more deeply into that question to this meritorious report . From the practical point of view one may say that the convergence is very rapid; two approximations suffice very ajirply in the majority of cases; the different changes in variables which enc\jmber the preceding expose are very easily made by graphic method. Thus one sees that the numerical work essentially consists in calculating twice the inte- gral (11.53). This calculation is precisely the object of the prob- lem (a) stated at the end of section 2.3.1- If the contour (C) is not quasicircular,' one ma;y make, first of all, a conformal representation which transforms it into the "quasi- circular" contour (C); one will then apply the preceding analysis to the contour (C). For certain cases it will be quicker to use a direct method. Let us assiome, for instance, that (C) is a contour flattened on the axis of the X (compare fig. 21) and for simplification that X'OX is permissible as the axis of symmetry. Let us suppose that X varies along (C) from -a to +a while |y| remains bounded by ma (with m being, for instance, of the order of 1/10); it will then be indicated to operate as follows: We put along (7) ■ = |[f (^) + ±&i^')\ NACA TM 135i+ 67 One has X( n n 2 ^l_ m=l m=l (p - ki)« (p + l-^)jt cos m -^^^^ =-i_ _ cos m ' n "n (p - ^^)Jt (p + ^)it ^] with / > sm - X /-t I- \ \ (n-l) 2 •^ni^) = / <^os mx = cos -^ — - — - x n-l Z m=0 sm- Thus the coefficient of Kp is zero if -p f [i, and equal to — if p = li. Thence the desired value of h n-l ^ ^m sin mpjt (11.60) m=l Let us apply this result to the calculation of the Poisson integral, This integral defines an operator Q = A(P) of the first category for which a^ = -1. Consequently, the formula (II.60) is written n-l K- - 1 X~ o-in mpTt _ 1 /pn\ Kp - - 2_ sm ^P - - Sn(^— j if one puts n-l Sj^(x) = y sin mx = sin nx , V sin -^ [n - l)x 2 2 '. ~ sm- 72 NACA TM 135^4- Thus h ^ n 2n if p even if p odd (11.61) (2) Operators of the second category .- The considerations of parity permit one to write the general formula n-1 Qi = KoPi + X^ Kp(Pi^p + Pi_p) + K^Pi^^ (11.62) Using the same reasoning as before, one is led to determine the coeffi- cients K by the system n-1 Kq + ^ 2Kp cos m ^ + ( - l)\a = ^i p=l m (11.63) with m assuming the values 0, 1, 2, . . . n. Multiplying the first value by l/2,'the second by cos HJt/n, the third by cos ^M, the nth by cos IiL^lJiIm, and the last by ( - 1)^^/^, and adding thera^ one obtains a linear relation between the Kp, with the coefficient of IC being (p /^ 0, p / n) l.i^l_i) p+U + 1 ■'n (p + iQjH + c Rp ~ 1^)" n J '^|_ n that is, n if U = p, and if U /^ p. The coefficient of Kq is L.L^^.l ^n© ^^ni-f)-^ NACA TM 135i^- 73 The preceding conclusions remain valid, it is zero for \i f Q and equal to n if |j = 0; the same result is valid for K^. Finally, one obtains the general formula of solution s-i n-l ■^m mpjt / cos -i-— + ( m=l b 1)P ^ 2 (11.64) Let us consider, for instance, the operator transforming the func- tion- P(0) into the function d(^d0, with Q being the conjugate func- tion of P; it is an operator of the second category for which b^ = -m m Applying formula (II.6U), one obtains m % = -| s--i n-l m cos pmjt n + ( - 1)P^ ' 2 /O If one notes that n-l (x) = m cos mx = o . 2 X 2 sin — L 2 n sin (" - i)'' sm' 2 nx one sees that nfl - cos — ) if p even if p odd Ih NACA TM 1354 2.3-3 -2 - Second mode of calculation .- Examination of an important particular case will show us that in certain cases it will be advantageous to consider a second mode of calculation. The method consists in replacing the function P(0) hy a function of the form '^(0) = ^ ^n ^°^ ^^ "*" ^n ^^^ ^^ (11.66) for which the method is applied with the strictest exactness; the con- stants a and -b are such that P. = $• . One operator of the first category, one of the most important ones, is the operator of derivation which makes the fiinction dP/d0 correspond to the function P(0) . If we apply the first type of calculation, we shall replace | — ] by — ] : now, it is precisely at the points = i^ that the deriva- de)^ > ^ -^ ^ n tives — and — show the greatest deviation. In contrast, we shall d0 d0 obtain a good approximation of the desired function by replacing dP de (2i + l)jt 2n by di de (2i + l)n 2n We are thus led to the following mode of calculation: is divided into hn. equal parts; we shall put the circle fi = fi^ ^ V2n and we shall express the 2n values Qg-^ as a function of the 2n valuei ^2j+l- We shall limit ourselves to the operators of the first category. The formula expressing the Qg. as a function of Poi+i i^ written n ^2i - / S(^2i+2p-l " ^2i-2p+l) p=l MCA TM 1354 75 and we obtain for determination of the IC the system h sm (2p - l)mn ^ p=l 2n with m varying from 1 to n. Multiplying the first equation by sin(2l-i - 1) — , the second by sm (2n - l)2n 2n last by ( - 1) H-1 ^u ( ^^*h , . (2^l - l)(n - l)n ^, ., the (n - 1) by sin -^ — '—, the 2n and adding them, one obtains a linear relation in which the coefficient of s IS n-1 Sin m=l / -I \ H+p (2p - l)m. sin(2u - 1)^ + i-I-^ 2n 2n 2 n-1 IE cos(p - u)^ - cos(p + \i - 1)^ n n , ( - 1) ^-P 2 ^[cn[(P - ^)i\ -Cn[(p+ ^-1)e]^ ( -1)'"^ The coefficient is zero if p ^ 1^, and equal to 2. if p = n. Hence «P = -^ n-1 Z m=l . 2p - 1 mjt - 1 P-1 R sm ^-^ '- — + -I^ '- a^ ^ 2n 2 ^ (11.67) This procedure may be applied to the calculation of the derivative of a periodic function. In this case^ a^ = -m. Applying formula (II.67); one obtains 76 NACA TM 1354 Kp = ( - 1)P-1 — = 2n , (2p - l)n 1 - cos -^—^ — 2n (11.68) 2.3-3-^ - Remarks on the Employment of the Suggested Methods .- In order to convey some idea of the accuracy of the proposed methods we shall give first of all a few examples where the desired results are theoretically known. Let us take as the pair of functions P(0), Q(9); the functions P(0) = ^ cos 20 - 4 cos + 1 Q(g) ^ -4 sin 0(2 cos 0-1) (5-14- cos 0)2 {5 - k cos 0)2 which are the real and imaginary parts, respectively, on the circle of radius 1 of the function f(z)=^-^ (z = ei^) (2z - 1)2 One will find in figure 22 the graphic representation of the func- tions P(0), Q(0) and of the derivative Q' (0) of this function, and also the values of these functions for = y" (with p ranging between and 12) . Furthermore, one will find in figure 23 the values of Q(0), calculated from P(0) as starting point, by the method just explained (coefficients K^, defined by equation (II.6I)), and in fig- ure 2H- on one hand the values of Q'(0), calculated from P(0) as starting point (from coefficients IC defined by equation (11.65)), and, on the other, these same values calculated from Q(0) as starting point (coefficients IC defined by equation (II.68)). One will see that the accuracy obtained is excellent although the selected functions show rather rapid variations. Such calculations by means of customary- calculation methods are a delicate matter; this is particularly obvious in the case of the Poisson integral which is an integral "of principal value." Systematic comparisons of the method of trigonometric operators with those used so far have been made by M. Thwaites (ref . 23); they have shown that this method gives, in certain calculations, an accuracy largely superior to any attained before. The calculation procedure, with the aid of tables like the one represented (fig. 25) is very easy. One sees that one fills out this i NACA TM 135i+ 77 table parallel to the main diagonal of the table. With such a table, about one and a half hours suffice for a Poisson integral if one has a calculating machine at his disposal. We have had occasion to point out that the accuracy of the method obviously increases to the same degree as the functions one operates with are "regular" and present "rather slight" variations. This leads in practice to two remarks which are based on the "difference method" and reasonably Improve the resiilt in certain cases. We shall, for instance, discuss the case of the Poisson integral. (1) If the f\inction P(9) presents singularities (for instance discontinuities of the derivative for certain values of 0), it will be of interest to seek a function P-]_(8), presenting the same singularities as the function P(9), for which one knows explicitly the conjugate function Q-|_(9). One will make the calculation by means of the func- tion P(0) - P-]_(0); this function no longer presents a singularity. (2) If the function P(0) has a very extended range of variations, one will seek a function P^^CS) for which one knows explicitly the function Qn (9) so that the difference P(0) - P-, (©) remains of small value, and one will operate with this difference. Finally we note that, if the calculation of the derivative of a function P(0) as described above necessitates that P(0) be periodic, one can always return to this case, applying, precisely, the "difference method." 2.3.^ - Example: N\imerical Calculation of a Flow about a Semicircular Cone As an application, we have taken up again the case of the semicir- cular cone studied in section 2.2.9- The function g('-P) is given by the formula (lI.i<-8), and g'C^*) will be calculated by a Poisson inte- gral. Figure 26 shows the value g' i^) thus calculated compared to the theoretical value . ^e wanted to test the accuracy of the proposed method by ass\iming an extremely unfavorable case, without taking into accoiint the singu- larities presented by the fimction gC"?) • For a numerical operation of great exactness, this particular case would have required application of the lemma of Schwartz, with the contour (C) completed symmetrically with respect to OX. 78 MCA TM 135i+ It is then possible to calculate the representation of the pres- s\ires, by calculating successively the function H, ZE, and the inte- gral g'(cp). One will find the pressure distribution thus calculated in fig- ure 19; one may then compare the result obtained by the calculation method (for a very unfavorable case) with the result obtained theoretically . NACA TM 1351*. 79 CHAPTER III - CONICAL FLOWS INFINITELY FLATTENED IN ONE DIRECTION The piirpose of this chapter will be the study of conical flows of the second type (see chapter 1, section 1.2.6). Before starting this study proper, we shall make a few remarks concerning the boundary con- ditions. The conical obstacle is flattened in the direction Ox^Xg. Under these conditions, reassuming the formula (1.27) wxg' - vx^' = -^x^xg' - xgx^') (1 + u) (1.27) one may say that it reduces itself, in first approximation, to ^2' =^(^3^2' - ^2^3') (III-I) since x^, x^ ' , v, u are infinitesimals of first order, while X2 and Xp ' are not infinitesimals. Under these conditions, one may say that one knows the function w on the contour (C). On the other hand, one may write, within the scope of the approximations made, this boiindary condition on the surface (d) of the plane Ox-j^Xg, projection of the cone obstacle on the plane. Let us designate, provisionally, the value w by w'-'-m x, x^x_,\ if one operates as follows (1) w(^^[^l,X2(t),X3(tr| = w(l)[^^,X2(t),o] + X5(t) ^ |^l,X2(t),o]] With the derivatives of w being, by hypothesis, supposed to be of first order, and the boundary equation written with neglect of the terms of second order, the intended simplification is justified. Various cases may arise, according to whether the cone obstacle is entirely comprised inside the Mach cone (fig. 27), whether it entirely bisects the Mach cone (fig. 28), whether the entire obstacle is com- pletely outside the Mach cone (fig. 29), or whether it is partly inside and partly outside the Mach cone (fig. 30)- In each of these cases there are two elementary problems, the solution of which is particularly 80 NACA TM 1351+ interesting: the first, where the relation (lll.l) is reduced to w = constant = w which we shall call the elementary lifting problem (the corresponding flow is the flow about a delta wing placed at a certain incidence); the second, where the relation (lll.l) is reduced to w = wq for Xo = +0 w = -Wq for x^ = -0 I which we shall call the elementary symmetrical problem. This is the case of, for instance, the flow about a body consisting essentially of two delta wings, symmetrical with respect to Ox-|_X2 and forming an infinitely small angle with this plane . It is also the case that will be obtained, the section of which, produced by a plane parallel to 0x2X5, wovild be an infinitely flattened rhombus . The fact that one obtains the same mathematical formulation for two different cases indicates the relative character of the results which will be obtained. In the case of the symmetrical problem one may naturally assume that w is zero on the plane Ox-iXo at every point situated outside of (d) . Let us finally point out that very frequently the obtained results do not satisfy the conditions of linearized flows; in particular, the velocity components and their derivatives will frequently be infinite along the semi-infinite lines bounding the area (d) . However, we admit once more that the results obtained provide a first approximation of the problem posed above, in accordance with the remarks made in section 1.1. 3 of chapter I. 3.1 - Cone Obstacle Entirely Inside the Mach Cone 3.1.1 - Study of the Elementary Problems The case of the lifting cone has already formed the subject of a memorandum by Stewart (ref . 10); however, the demonstration we are going to give is more elementary and will permit us to treat sim\iltaneously the lifting and the symmetrical case. NACA TM 13514- 81 3.1.1.1 - Definition of the fimction F(Z) . - We shall make our study in the plane Z^ Let A'A(-a,+a) be the image of the cut of the surface (d) , i'^o)' ^^ usual, the circle of radius 1 (fig. 31) • Naturally, we shall operate with the function W(Z) . One of the conditions to be realized which we shall find again everywhere below is that dW/dZ must be divisible by (z^ - l), unless the compatibility relations show that U(Z) would admit the points Z = ±1 as singular points which is inadmissible. Thus we introduce the function F(Z) = _zi_dW (III. 2) Z2 - 1 dZ and we shall attempt to determine F(Z) for the symmetrical as well as for the lifting problem. F(Z) is a holomorphic function inside of the domain (D), bounded by the cut and the circle (Cq)^ 'the only singular points this function can present on the boundary of (D), are A and A'; on the other hand, F(z) must be divisible by 2r , unless U, V, W have singularities at the origin. On the two edges of the cut F(Z) must have a real zero part. On the circle (Cq) z2_i Z-— 2isin Z ^ = e^® — = -i — dZ dZ " d0 Consequently, F(Z) has a real zero part on (Cq) as well. The fact that F(Z) cannot be identically zero, and that its real part is zero on the boundary of (D) , admits A and A' as singular points. We shall study the nature of these singularities. 3.1.1.2 - Singularities of F(Z) .- Physically, it is clear that A and A' cannot be essential singular points. Let us therefore suppose that, in the neighborhood of Z = a, one has One assumes, as a start, that the problem permits the use of the plane Ox-]_Xt as the plane of symmetry. 82 NACA TM 1351+ F(Z) ~ K^(Z - a) niQ being arbitrary, K^jj^ / 0; let us put Z - a = re icp with cp being equal to +Tt on the upper edge of the cut, to -it on the lower edge; for sufficiently small values of r ■V^r^Oe^" and ■^r^Oe"^'' must be purely Imaginary quantities; thus the same will hold true for Kj^ cos mQjt and for ±K^ sin mQn; %^ = KmQ^cos^mjt - (iK^^ sin mjtj 2 is therefore real. On the other hand i\)^ 2 " (Sd ''°^ ^'')(i\)^^'^ ^Ort) must be real which entails sin 2m„jt = Thus there are two possibilities; let us denote by k an arbitrary integral; either mQ = k, ¥L is purely imaginary or else °^0 = ^ + p \^ is ^^^1- NACA TM 13 51*. 83 Let us now consider Fi(z) = F(Z) - Kjj^(Z - a) mo In the neighborhood of Z = a F^(Z) ~ Kjj^ (Z - af^ and the same argument shows that Smn must be an integral. Finally, one may state the • following theorem: Theorem: Inside of (Cq) "the function F(Z) may assume the form F(Z) = $(Z) + ^ i(z) (III. 3) 4 a^ - z2 with $(Z) and ''^(Z) admitting no singularities other than the poles at A and A' . The analysis we shall make will be simplified owing to certain symmetry conditions which F(Z) satisfies. Let us put W = w + iw' Obviously, X in w(X,Y) is even (when Y is constant). Consequently, F(Z) has a real part zero on OY. Applying Schwartz' principle one may write r(z) = -F(-z) (iii-^) This equation shows that knowledge of the development of F(z) around Z = a Immediately entails knowledge of F(Z) aroiind Z = -a, 84 NACA TM 1351^ 3.1.1.3 - Study of the case where F(Z) is uniform [_^Kz) = o] Let us consider the function Ap(Z) = iZ 2p E' z2)(i a^T? , (III. 5) with p an integral and >1. This function satisfies all conditions imposed on F(Z). Indeed, it satisfies equation (lll.i^-); inside of (Cq^ it does not admit singularities other than a and -a which are poles of the order p-[_. Its real part is zero on the cut as well as on (Cq); one can see when writing as Ap(Z) = Y :(z3.i)-(...^) Finally, the origin should be double zero (at least). Let us assume F(Z) to be the general solution of the problem stated; we shall then demonstrate the following theorem: Theorem: If F(Z) is uniform, one has n n ^ 2,2p F(z) .^XpAp(z) - 1 21?= — 0( Z^ 1 - a^Z' '■) (111.6) with n being an integral, and the X— being real coefficients. In case F(Z) is assumed to be a solution of the problem having a pole of the order n, one can determine a number X,^ so that F^(Z) = F(Z) - -K^k^{Z) will be a solution admitting the pole Z = a only of an order not higher than (n - l) at most. But in consequence of equation (ill. 4), 1 NACA TM 13514- 85 F-|_(Z) will allow of Z = -a as pole of, at most, the order (n - l) . Proceeding by recurrence, one finally defines a fiinction Fn(Z) = F(Z) - > \,Ap(Z) which must satisfy all conditions of the problem and be holomorphic inside of (Cq)" '^^ boundary conditions on the circle and on the cut entail F^^^) to be a constant which must be zero because ^■^i'^) J^ust become zero at the origin. 3.1-lA - Case where '^{Z) = .- We shall study the case where (ti(z) =0 in a perfectly analogous manner. Let us put _ Ni(a^-z2)(l-a2z2) ^(,^ f(Z) f(z) is a uniform function inside of (Cq) which admits as poles only the points (Z = -a, Z = a) . Actually, the origin is not a pole since, according to hypothesis, F(Z) is divisible by Z^ . The function f(Z) possesses the following properties: It is imaginary on the cut, real on (Cq)^ 3J^d. real on OY (which entails properties of symmetry if one changes Z to -Z) . Moreover, f(Z) admits the origin as zero of, at least, the order 1. All these properties appertain equally to the f unct ions p is an integral ^ 1 . . Thus one deduces, as before, the theorem: Theorem: In the case where $ (z) = 0, one may write 86 NACA TM I35J+ n F(Z) = i 2_ S 1 HZ^ - 1) / ^ ^^ ^^ Y (III. 7) (a2 - Z2) (1 - a2z2) with n being an integral, the X^ being real. 3.1.1.^ - The principle of minimum singularities .- The for- mulas (III. 6) and (III. 7) depend on an arbitrary number of coefficients . The only datum we know is the Vq, the value w assumes on the upper edge of the cut. Thus we have to introduce a principle which will guarantee the uniqueness of the solution of the problems we have set ourselves. This principle which we shall call principle of the "minimum singiilarities" may be formulated in the following manner (it is con- stantly being employed in mathematical physics): When the conditions of a problem require the introduction of func- tions presenting singularities, one will, in a case of indeterminite- ness, be satisfied with introducing the singularities of the lowest possible order permitting a complete solution of the posed problem. In the case which is of interest to us, this amounts to assuming n = 1 in the formulas (ill. 6) and (ill. 7). For the problem of interest to us, this principle has immediate significance; it amounts to stating that F(Z) and hence dW/dZ must be of an order lower than 2 in l/z - a, or W(Z) must be of an order lower than 1 with respect to that same infinity; the considerations set forth in section 2.2.7 show that these conditions entail the total energy to remain finite. 3.1.1.6 - Solution of the elementary symmetrical problem .- Let us turn again to formula (ill. 6); one deduces from it, according to equa- tion (ill. 2), that in the case where F(Z) is uniform dW _ .^^ Z2 - 1 dZ (a2 - z2)(l - a2z2, and hence W(Z) = '^^ , log (" -^)(1 -^^) 2a(l.a2) Ma + Z)(l.aZ) + Mr I NACA TM 135^4- 87 The determination of the logarithm is just that the real part of W(Z) is zero on (Cq)- Besides ^1 = 2a (l + a^jwr On the upper edge of the cut w = Wr and on the lower fedge w assumes the opposite value. This shovs us that the case investigated is that of the symmetrical problem. The value W(Z) for this problem is therefore W(Z) = IWr log (a - Z)(l - aZ) (a + Z)(l + aZ) Wr (III. 8) The calculation of the functions U(Z) and V(z) offers no diffi- culty whatsoever. It siiffices to apply the relationships of compati- bility (l.25) and to integrate; the only precaution to be taken consists in choosing the constant of integration in such a manner that the real parts of U and V on (Cq^ become zero; one then finds V(Z) = -^ _ ^0 (l + a^) "" (l - a2) log (a + Z)(l - aZ) (Z - a)(l + aZ) (III. 9) and 2wp.a U(Z) = — — ii— _ log jtp(l - a2) 1 - a^Z^ (III. 10) This last formula is the most interesting one since it permits calcula- tion of the pressure coefficient (see formula (l.8)). One finds C^ = ^^0 _ np -L log a2x2 (III. 11) 88 NACA TM 1354 In order to interpret this formula, one must connect the quanti- ties a., X, to geometrical quantities, related to the given cone. First of all Wq = a a being the constant inclination of the cone on Ox. On the other hand whence 2X = P — = P tan o) 1 + X 2 X. 1 X = cos CD - Njl - M sin^cD 3 sin CD (see fig- 32) and cos CX^r a = ^ M^sin^o), P sin odq (III. 12) In figure 33 one will find the curves giving the values of Cp as functions of u), for various Mach numbers and various values of ojq. 3.I.I.T - Solution of the elementary lifting problem .- If one starts from the formula (III.Y)^ one obtains dW dZ ^ = iXn i£ .f & ^)(l- a2z2)l i The integration which yields W(Z) introduces elliptic functions (see section 3-l-l-8)j on the other hand, it will (now) be possible to cal- culate U(Z) . We note beforehand that, according to the preceding for- mula, W(Z) assumes the same value on the two edges of the cut and that, consequently, this solution corresponds to the lifting problem. I NACA TSA 1354 89 The relationships of compatibility show that du ^ ^ z(z^ - 1) P - z2) (1 - a2z2)j 2 and hence U(Z) = - 1 z2 + 1 3(a2 + 1)' [(a2 - z2) (1 - a^Z^)\ 2 (III. 13) We still have to calculate X-, as a function of Wq. For this purpose, one may write -Wq = ^ dZ = iX. dZ J 01 (Z2 - l)2dZ [(a2 - z2) (1 - a2z22] 2,72^1 2 We put in this integral Z = iu Ol Wq = \^ (1 + u2)^du = X,-Ll(a) [(a2 +u2)(l + a2u2)] The calculation of I (a) can be made with the aid of the function E (see ref. 2U) . We shall put u + 1 = 2 u t After a few calculations one obtains Pl 1(a) = k dt \Jl - t2 l^a2 + (a2 - l)2t2 90 NACA TM 1351). Finally, the change in variable sm (a2 + l) JUa2 + (a2 - ift^ shows that if one puts 1 - a^ 1 + a2 3,2 (a2 +1)^0 \|l - k2sin2cp dP = i (a2 + l) ^Vl + a2y Hence the new formula for U(z) u(z) 2 3 a'^WQ Z2 + 1 (^^ * ^)Krti^) -, 1 (III. 14) - z2) (l - a2z We still have to connect a and Z to the geometrical quantities, One has (fig. 32) 2a 1 + a' P tan oOq 2X 1 + X^ = p tan CD One puts t = tan LP tan cjoq and obtains u = wq tan ooq 1 - p^tan^GD, f^ NACA TM 135i+ 91 and Cp = 2a tan cjoq \jl - p^-tan^coQ \|l - t^ (III. 15) if one puts J as usual Wq = a If cDq is small, E; ll - p^tan^oDQ is close to 1, and the for- mula (ill. 15), except for the notations, again gives a result found before (formula (II.33)). On the other hand, if p tan ooq — >■ 1 E J^ P^tan^cDQ and the formula (ill. 15) is written n _ ^ ^ Remark. Thus one sees that the elliptic functions need not be used in an essential manner in order to obtain the pressure coefficient. Actually they appear only in the multiplicative coefficient. (in contrast, Stevra,rt, in his demonstration (ref. 10), makes essential use of the elliptic functions.) However, these functions are indispensable in the explicit calculation of W(Z) and V(Z) . 3.1.1.8 - Calculation of W(Z) and V(z) .- There exist several simple methods for calculating W(Z); the first consists in putting^' '^For all the properties of the elliptic functions made use of in this report, see for instance reference 2k. In this paragraph, u will be a complex variable and will have no relation to the velocity compo- nent along Oxt|_. 92 NACA TM I35U Z = a snfu^a^j fk = a^j (III. 16) This transformation achieves the conformal representation of the domain (D) on a strip of the plane u (see fig. 3^)i "the values written inside of small circles indicate the values of Z taken for the corresponding value of u. One has actually sn = sn K = 1 en (i ^') = ' -if'^') = ^ 4 sn K + iK' cdfiK' V 2 1 Hf-'^' 1 _ 1 a ^ Under these conditions dW _ dW dZ iX.-L(z2 - 1) du dZ du (^2 _ 22)(i _ a2z2) i\-\ 1 1 - a^ 1 + a^ a2(a2z2 iXi 1 + p o dn u en u whence W(u) = Wp, + 1X-, '^ 2f 2 .^2 2(a2 + l)u - 2E(u) + a sn u en u ^ dn u sn u dn u en u (III. 17) For determination of X,-, , it suffices to write, for instance, that »(^) ^ ° NACA TM I35I+ 93 Now Kf-' iA,i i2(a2 + iY\l {-' l)i(l + K') - SE^i^ + Vq = However^ 2E(i^ iK' + 2i dn^^^kAscC^jk'") - 2iEf^,k' SEfS.k'^ =E(K',k') + ^^ \2 I 1 + k whence the value of X-j. Woa2 (a2 + l) A,-i — a^K' + E(k') This expression differs from the formula given for \-]_ in the course of section 3-1 -7; besides, one may, in a general manner, put the 1 - a2 formula (ill. 17 ) in another form (using a modulus ki^ = which 1 + a2 is different from the modulus k = a^ utilized so far) ty applying the Landen transformation. This transformation permits, in particular, establishment of the following formula EKl + k)u,k-L = -^ — 2E(u,k') + 2ku - k'^sn u cd u with the functions of the term at the right of the preceding equality being relative to the modulus k' = J] 11 - a^. 94 NACA TM 1351^ If one puts this formula is written u = ly (1 + k)iy,ki 1 + k 2(1 + k)y - 2E(y,k) + ^ ^ ^ "^ ^ - '^^^^ ^ en y en y dn y These last funetions are relative to the modulus k = a^ . However , 2 dn y sn y _ ^2 sn y sn y en y enydny enydny_ 2 dn^y - k'^ ^ sn y /, 2^^2 en y dn y /, 2 2 -3 2 \ i4- sn y en y dn y sn y k en y + dn y) = a ^ ^ + ^ ^ \ ' dn V en V en y If one now refers to the form\ila (III.IY)^ one sees that it may also be written W(u) = Wq + — — 1 E (1 + a2)iu,ki a-^^ia^ + ij I- and that under these conditions WfiKl) = wr Ka2 + 1) (1 + k)K' >h = However, K-i = j-= ' ' is precisely such that 2 sn ih'h) NACA TM I35I+ 95 Consequently H = ■WQa2(a2 + l) El ' 1 - a^ a + a^ which is, of course, the formula found previously. Hence W(u) = w, 1 + [jl + a2)iu,k:^ (III. 18) One may also proceed in another manner, introducing a variable other than the variable u. We put 2iZ Z2 - 1 The integration of ^=^ leads to dZ W(t) = Wq - k\^ pt dt f^ Ua2 + (a2 - l)h^ We put k. - ^-^^ ^ 1 + a2 The complementary modulus is 2a 1 + a^ 96 If one puts, therefore t = en (T",kj^) NACA TM 135i(. W(t) = wn + kx-. dT (a2 + iy^Kj_ dn^T Wq - a2(a2 + l) E(k3^) - E(T,k3_) + [^ - ^ ) sn T en T (l . a2)2 ^ If Z = i, t = 1, T = 0, one always still finds the same value for X,i ^1 = WQa'" ^a (a2 . l) El 'l_^_af a + a^ and W(t) = w. E(T,ki) ( 1 - a'^\ 1 sn T en T E(ki) [l + a2J E(ki) dn t (III. 19) The formulas (III.I8) and (III.I9) are indieated for the ealexjla- tion of W along the axis OY, whereas equation (III.I7) is more suit- able for the calculation of W along the axis OX. We now turn to the calculation of V(Z) . The calculation with the aid of the variable u is partic\jlarly simple. dV/dZ is calculated with the aid of the rela- tionships of compatibility dV dZ w_a2(a2 + 1) z'^-i E (^1) [a2 - Z2)(l - a2z2)] 1 272M 2 Let us recall that kl = 1 - a" 1 + a^ NACA TM 1354 97 and perform the change in variable (III.16). We obtain immediately ^q(^^ + 1) 1 - a^sn^u dV but V must be zero for u = 0. The integration of dV/du then gives v(u) = ""Q^f ,^ ^^ ^ — (III. 20) E(k-|_j en u dn u We verify, for instance, that for Z = i, V has a real part zero, ma iK' • Z = i corresponds to u = 2 dn^i|^") = dc(|^,k') = Njk(l + k) (k = a2) One can state that ^(■^^^) is purely imaginary. We shall not give another formula for the calculation of V(Z); the formula (ill. 20) which is particularly simple (it does not make use of the function E) permits the calc\ilation of v on the axis OX; on the other hand, v is zero on OY. 3.1.2 - Study of the Case Where the Cut is Not Symmetrical With Respect to OY 3.1.2.1 - General Principle .- The case where the cone investigated does not admit the plane Ox, ,x as the symmetry plane is easily led back to the preceding by a conformal representation, maintaining the circle (Cg)- 98 NACA OM 1354 Let us suppose, for instance, that in the plane Z the obstacle is represented by a cut along the segment (b,c) of the real axis (see fig. 35); the conformal transformation Z-i = i- (III. 21) 1 - OsyZ where a^^ is a real number (I'^il < l) maintains definitely the real \ axis and the circle (Cq)- We shall attempt to determine the numbers a-]_ and (^Y in such a manner that Z = c corresponds to Z-[_ = Q.-\_, Z = b to Z-i = -an . One must write I c - aj^ b - a;j_ an — ^^^^^^— — —an — 1 - Oq^c -^ 1 - a-ib log Z| dl (III. 30) The integrals occurring in these formiilas make sense only if Z is not on the segment be . If Z is real and comprised between b and c , one has to take the "principal value" of these integrals. Furthermore, one must demonstrate, in order to justify these formulas, that the real part of the function W(Z), defined by the first formula (III.30), actu- ally assumes the value w(X) when Z is real (Z = X) . For this purpose, one calc\ilates W(z) in a point of Z = X + it) (with T] being positive and small) by dividing the integral appearing in the first formula (III.3O) into three parts W(Z) = - i >X-e X+e nX+e X-6 After this has been done, one chooses e and t] in such a manner that the last integral is arbitrarily close to the value oX+6 I = w(X) / ^X-6 dl which is possible since this integral may be written r""^ w(i)(i - z^)^ 'x-e (^ - z)(i - ^z) dl 108 NACA TM 13514- One may then^ diminishing as necessary the upper limit fixed for choose that last number so that R >X-e 'X-e should be arbitrarily small. There is no difficulty whatsoever since the quantity under the sign / is continued in Z. Finally, I may be made arbitrarily close to i:itw(x) which shows that, if tj is sufficiently small r[w(zJ] - w(X) is arbitrarily small which had to be demonstrated. This procedure, while theoretically simple, is rather delicate in practice since the calculations to be made affect the integrals, the principal value of which has to be taken. In the lifting case, on the other hand, the application of this method would require previous solu- tion of an integral equation of a rather complicated type. For that reason we prefer to give the following calculation methods; the first utilizes the "electric analogies;" the second which is purely numerical will reduce the numerical calculation to that of a Poisson integral; in section 2.2.7 we have given a simple and accurate procedure for solving such a problem. 3 .1.3 -2 - Utilization of the electric analogies '28 The analogy consists in identifying the harmonic function w(X,Y) with an electric , potential 'Y(X,y), through a conductor constituted by a liquid occupying' a tank with horizontal bottom of half-circular shape (see fig. 39)- On the circular boundary w is constant; consequently, the semicircumference will be brought to a constant potential; it will be possible to regard that potential as the zero of the scale of potentials. This circimifer- ence will, therefore, be conducting; (this half-circle is nothing else but the part of the circle (Cq) of the plane Z for which Y > 0) . ^or all questions concerning electric analogy, see the fundamental memoranda by M. Malavard (refs. 25 and 26). i NACA TM 135i^ 109 On the cut be which represents the conical obstacle, one distributes electrodes which will be brought, by means of adjustable potentiometers, to the given potential 9. For specification of the boundary condi- tions on the segments A'b and cA, one must distinguish between the symmetrical and the lifting problem. 3.1.3-2.1 - Symmetrical problem .- w must be zero on the portions of the axis outside of the cutj consequently, the corresponding bound- aries of the tank are brought to the potential zero, that is, to the same potential as the semicircumference A'BA; w is given directly by a pure Dirichlet problem. However, the unknown of our problem is the value of the pressure along the segment be, that is, u. u is connected with w by the relationships of compatibility which permit one to- write on the axis pf the X R ^ = 2X 5w ax n y2 SY with Sw/SY being proportional to the intensity entering the tank through the electrodes; this quantity is easily measured with the aid of a convenient arrangement •^. With the value of Su/SX thus known, we must, in order to obtain the desired pressure distribution, determine, in addition, a value of u along be, for instance the one at the point o30. On the axis OY one may write -^One may, for instance, feed the electrodes of the cut .through resistances R, insiiring a drop of the potential from 'P to 9 (see fig- 39) • Under these conditions, one has a relation of the form || = k(X)(9 -cp) with k(X) being a function which depends on the chosen resistances and on the resistivity of the tank, but can always easily be obtained; the manipulation to be performed is then as follows: after the resist- ances R have been determined, one has to choose the values of 9 in order to obtain at the electrodes the values of cp prescribed by the boundary conditions. -^^e shall assume the point to lie on the cut. In the opposite case the procedure indicated here may be very easily modified. 110 NACA TM 1351+ B ^ = 2Y ^ Since u(X,Y) is zero at the point B(0,1), 3u(0) = ( -^ |^(0,t)dt = Q 1 + t^ ^^ -,1 2Y w(0,t) 1 + Y^ / w(0,t)-i -'0 ■ (1 t2 (1 . t2)2 dt Hence 3u(0) = -2 / v(0,t) ^ - "^ dt (III. 31) One will know u(0) by means of a simple integral if one knows the distribution of the w (the same as that of the cp) on the axis OY. Since this may very easily be determined, the problem is entirely solved. 3.1.3.2.2 - Lifting problem .- The boundary conditions to be realized for the lifting problem are the same as for the symmetrical problem as far as the semicircumference a'bA and the cut b^c are concerned. On the segments A'b and cA one must, of course, write Sw _ dw _ Q hY " dn that is, the corresponding walls will be insulating walls. However, this is not sufficient. If no precaution is taken, the harmonic function corresponding to the electric field thus realized will not be a solution of the aerodynamic problem posed. Actually, there is no reason whatsoever why the gradient of this potential should be zero at the points A and A', since the intensity at A and A' is, in general, not zero. Since the corresponding function dW/dZ is not zero at Z = ±1, we have already pointed out that this leads to singularities inadmissible for U(z) (see section 3. 1.1.1). I NACA TM 1354 111 The investigation of the elementary lifting problem^ admitting OY as symmetry axis, will permit us to better understand the difficulty, and to solve it . If one realizes in the tank the preceding boundary conditions by bringing the electrodes from the cut (-a,+a) to a con- stant potential, it is quite obvious that the potential thus realized in the tank will remain finite at every point of the field, even at A and A' . Thus one obtains a solution by taking for 9(X,Y) the real part of the analytic function F(Z), defined by dP ^ iX_ dZ (a2 - z2) (1 - a2z2) 17^ with X being a re'al constant, This solution does not correspond to the solution of the aerodynamic problem (see section 3-1-T) which, in contrast, gives a singularity at (a^ - z2) for the function W(z), in the neighborhood of Z = +a. As a consequence, w(X,Y) must be infinitely large at points close to +a and -a^l. This particularity must, therefore, be taken into account in the circuit . It is not the first time one encounters problems of analogy with singularities-^ . One knows that one must then realize in the neighbor- hood of the points ±a, a material model, partly conducting, partly insulating, which schematizes the arrangement of an equipotential elec- tric line and a current line. SI One encounters there an interesting example of precautions to be taken in a given problem when one applies the principle of minimum sin- gularities. This principle has led us to pose, for our aerodynamic problem, a solution for dW/dZ in ^a - Z / . But if one makes the analogy, the electric tank has no reason to "know," a priori, that realization of other conditions than those directly concerning W(Z) is desired. Thus it "applies" the principle of minimum singularities, realizing the solution for dW/dZ in (a^ - 7?) See for instance references 27 and 28. For several months, the laboratory of electric analogies of the O.N.E.R.A. has been utilizing singularities for the study of compressible subsonic flows in the hodo- graph plane . -^-^2 NACA ™ 1354 In the case of interest to us, in the neighborhood of the point X = +a, one has W(Z) = ^ \JZ - a with K being a real constant; consequently, if one puts W(Z) = w(X,Y) + iw'(X,Y) Z - a = ^ se^^ cos — - i sin and V + iw' = — the lines w = constant are determined by P „ Sq(1 + cos a) s = sn cos^ - = — ° 2 2 and the lines w' = constant by o ^ s, (1 - cos a) s = s. sin^ £ = J=2 1 -'-2 2 Sq and s-]_ being two positive constants. They are, therefore, cardi- oids; their arrangement is given by figure kO. Also, one finds in this figure the scheme of the singularity which must be placed at b and c. Thus the manipulation is as follows: after the circumference ABA' has oeen brought to the potential zero and the boundary conditions have been realized along the cut be, one brings the conductive part of the two singularities to rather high potentials which must be determined so that the intensity at the points A and A' is zero (of course, if the problem presents the axis OY as symmetry axis, the two singularities must be brought to the same potential, and the nullity of the intensity at A will insure that of the intensity at A'). This one will realize, from the practical point of view, by detaching at A (and eventually NACA TM 1354 113 at A') on the semi circumference a small electrode which will not he fed and the potential of which will be made opposite to the potential of the rest of the circumference, through a zero apparatus. It is this condition which permits determination of the potential to which the conductive part of the singularity at c (and eventually at b) must be brought. The field 9(X,Y) realized in the tank will then, in con- sequence of the principle of "minimum singularities," be proportional to the field w(X,Y) of the velocity component following Oxo . After that, the manipulation unfolds as for the symmetrical case. One measures the intensities along the cut (b,c) which f\irnishes the values du/dX. One determines the value of u at the point by restoring the field of values of w along OY and by applying the formula (III.3I) . 3.1.3-2.3 - Electric measurement of C^ in the case of the lifting problem .- In all cases, the total energy can be determined by integra- tion. In the case of the lifting problem, one will yet have a supple- mentary verification by utilizing the formula (ill. 27) which we shall write ^ 2n (1 +b2)(l + c2) ^^^ ^^ P (c - b)(l - cb) SY Actually, this last formula permits to obtain directly the C^, by a simple electric measurement which gives the intensity entering at the point B, since dw/dY(0,l) is proportional to that intensity. For this purpose, it suffices to detach, in the neighborhood of B, a small electrode (fig. 4l) and to feed it by the intermediary of an arbitrary resistance R. With all boundary conditions satisfied, it suffices to regulate 9 to make the potential at B zero as on the rest o*f the semicircle. C^ is then proportional to cp . 3 .1.3 -2.4 - Applications .- The scheme of the circuit used is given by figure 39- We do not intend to give here the details of operation, the precautions taken for increasing the accuracy, the determination of the scales, and the reduction of experiments. All this will form the subject of a later report. Here we shall give simply the results of the first experiments made following these principles'^-^. In every case studied, we have ^^There is every reason to assume that the satisfactory precision obtained could be further improved by employing a more suitable material than the one that was utilized. These tests were made frequently with utilization of chance setups with the material that happened to be at the laboratory. 111+ NACA TM 135l<. bO a 1 •H •H Td U C 0) o > p< [0 03 u w u 4-> o •iH o g 5 ^ cd ■p C (1) a a) H (U ti) ^ -p Ti a cd aj w cri o bd C •rH -P •H o P) bO a •H o -p OJ X II OJ o! X CO (U :i 03 > x: -p CJ xi -p 03 CJ (U U CO ■p ^ NACA TM 135if 115 0) 0) O O m CQ (U ft (U +3 dJ -P S 0) o H ft d o 0) > •H > O w -p H in w - M H H H O rH CVI VO H O H • • o H H H o H H ^ X X H PL, 0) ■p bO d o H 03 Ch O ^ d & O •H n -p 0} ^ xi ^ ■iH (D Vl a -p o CQ •H • •>k Tj g „,^^-^ OJ 0) ..^— s. (U X ^ X Cfl !U cd 1 > oo u C\J "^ (U ^ ft U; xi ■*■ ^ u -p o II o bD ft w II d o H •H 03 > -^ Tj O QJ ft •rH ra Td (U :3 ^ -p Fh u o o w 0} (U ^ -p OJ C a ^4 o (U > o 4h »N •H d bO -d O •H to 3 -p g O Oi t|H O ■H CO Ch X (U ■H p( !h H 0) OJ cd > xi -p l> fn 0) O O w g LO ^ o -4- cn o\ o -4- m d CT\ CTn CD ■ H CO d on CO r-\ • -4- O 15 ON H rH vo m d LO H rH o d o r-\ -4- d -4- o CVI H CO • o IT- d CVI rH d 0-) CO d o • o C3N o o d X x" CVI 116 NACA TM 135l4■ o I ■a P4 -P >> Ch •H U (U > CQ i o H d u -P oi tn CO VO oo LO ON 00 o CO 1^ f- ro -4- t- CO LO co t~- O -d- -=)- LO t- 00 m oo -:!- VO VD O OO oo oo O OO -=t- oo -* o o • • • o oo oo CO CO MD -* OO on VD VD o 00 00 PO OO o >- I>- PO m oo • • • o CVI 00 lO oo -d- 00 OO CM H H o 00 oo o 00 03 in in H o o o OJ 00 H in ■ c\a VO VO H ON ON • • o H H in in ^o o H o ON ON • • • o H H ^- ON co ON o CO CO nH H oo Pm O ft X + O OJ H X Oi OJ VD O O a H O ft tlD •H w oo in o VD OO CV] oo oo CO d VD OO in 00 in in 00 CVI d oo VD o CVI o oo o 00 VO oo o CVI -4- • -4- 00 H o oo o VO oo in H VD 00 in rH 00 o CO ON oo • o ON oo H 00 H d -4- o oo CJN CV] H CVI H o 00 -4- 00 H 00 oo CVI H VD O d CO O 00 H -4- O OO H o VO ON H H 00 H H X H -P 0) ft H cd o •H -P (U Jh O x_ oo PL, O oo ft O ft 0) o 00 c o o o ■H H o u cd ft a D •rH -P Ch •rH -4- w cd VO -4- oo in VD • • o J- CV) CO oo -4- t- o CV] VO 00 in -4- C7N • • o H O VO CVI oo in o H CO o I^ oo 00 • o H CVI -4- in 00 H o rH -4- CO ON H O o H -4- CVI ■I^- H O • • o H O VD t- O o o H 00 VO o O rH X X ■^^^ oo Ph NACA TM 1354 117 3 .1.3 -3 - PuJ^ely numerical methods. Utilization of the plane z. We have introduced this plane in section (I.2.5). Let us recall that z corresponds to Z by the conformal transformation z = - 2Z Z^ + 1 and that in this plane the relations of compatibility are written -P dU = z dV = — :i5 — dW (ill. 26) sjl - z2 One of the advantages of the plane which is of practical interest is that one has on the real axis (if z = x + iy) X = Xg Xg being the ordinate of a point of the section x-[_ = p, situated on Xo = 0, in the axis system Ox-j^XgXo. Some of the formulas established before may be written more simply. If one denotes, for instance, the image of the cut (b,c) of the plane Z in z by (A,,h), the formiila (ill. 21) is written W(z) = -i ^ log ^^-2^ (III. 32) Jt A, - Z W(z) thus appears as the complex potential due to two vortices placed at the points X, and li and of opposite intensity. Likewise, the formula (lll.2i+) may be written u(„1 - ""o \/(i + b2)(i + c^) 2^^ - z(x + ^) (III ^^) pE(ki) 1 - be r. ^ ^' \j{u - z)(z - X) If one puts X, = cos ^ \1 = cos 00 II 118 NACA TM 1351^. ^ and u) lying between and jt ■^sin •'l^ sin oj , \/(l +b)(l + c) ki = -^i i and — - ; -'- -V+m 1-bc •w+m sin '-" -^ "^ sin ^ "^ In the case where [^ = -X = k, one has, in particular r _ 2k2 ^1 \/k2 - x^ Let us recall that E'(k) = e(Ji - k^) 3.1.3 • 3 •! - Case of the symmetrical problem .- Let us now assume that the problem corresponding to the boundary conditions w = f (x) on the upper edge of the cut, w = -f(x) on the lower edge has to be solved. The fonnula (ill. 32) leads us to represent W(z) as the poten- tial of a distribution of vortices carried by the segment X.u; conse- quently W(z) = - i r^ ^^^^ du It J. u - z At a point of the upper edge of the cut, one has actually w(x) = - i r ^^^i^ + f (x) = w + Ti U\ U - X IW' with the integral taken at principal value. Let us put on the cut X = k-LJi + iL^A cos cp u = ^ + ^ + ^ - ^ cos e 2 f (u) = F(0) NACA m 1354 119 Let us assume^ to begin with, that F(0) = F(jt) = and that F(0) can be developed in a Fourier series F(0) = y A^ sin ne < 9 < jt Then I ( ^ A^ sin nejsin 6 w'(e) = - 1 LO / ^Q « cos - cos 9 '0 We shall furthermore admit that the signs \ and / are inter- changeable. According to a known result (ref . I3) n r" • a ■ a Aa i P'^ ^os(n - 1)6 - cos(n + 1)01 _ 1 / sm n6 sm 6 d0 _ _ J:_ / L . J n Jq cos - cos

- sin(n + l)3) and, consequently, that of r^. In order to set up formula (ill. 35), we have made a certain number of hypotheses. These hypotheses will be satisfied if the derivative of F with respect to 9 satisfies a condition of Cauchy-Lipschitz . In order to calculate the pressure at every point of the cone one must integrate the formula (ill. 35); for that, however, one must know the integration constant. The exact determination of the function u will be easily obtained as soon as we have studied thoroughly the character of the fiinction U(z). We suppose first -X. = p. In order to study the function U, we shall perform the conformal transformation of the plane z, provided with cuts (-oo, -l), (-|a., + |i). NACA TM 1354 121 (1,+°°) traced on the real axis, on an annular corona. This is imme- diate (see, for instance, section 3-1-7-1)- Let z-j_ first be a complex variable defined by dzi dz ^(,2-z2)(l.z2) or z = n sn/'z-,,(iV (k = ix) , then 2 The plane z provided with its cuts then is represented on a strip < T(z-]_^ < K' of the plane z-|_, and on an annular area of the plane Zg (see fig. h^) bounded by the circumferences (y^^) of the radius 1 and (72) °^ "t'^e radius q = e 2K In the plane Zg, U is of the form U(Z2) = A log zg + f(z2) with f (^2) being a uniform holomorphic function inside of the annulus (see for instance section 2.1.2.1), since U(z2) is finite, even at the image points of z = ±n, because of the hypothesis . F(0) = F(jt) = We remark that f (^g) ^^^ ^ real part zero on the circle (^i)' We assume the value of the coefficient A to be known; on the circum- ference (7i)j A. log zo maintains as constant real part 122 A log q = - 2. Kl A 2 K NACA TM I35J+ (III. 36) According to a well-known theorem of the theory of harmonic func- tions (see ref . 29) one now knows that, if a uniform harmonic func- tion H(x,y), defined inside of a circular annulus, assumes on the two limiting circles the values ^q^®) ^^'^ '^±(^) > (with 6 being the angle at the center representing the running point on each circle), one has n2n o2jt do do This theorem will allow us to demonstrate the following theorem: Theorem: If ^ = -\, the function u(9) satisfies the equality u(q))d9 fT = 2K(h)A log q ,2^„t,2cD \J± - u^cos^9 K(|a) being the elliptic function of first kind relative to the modulus n K(h) = dP Jl - li^cos^cp In fact , the mean value of the real part of f (zo ) on the circle (7±) must be zero, but the mean value of u on (^g) reads •\kK 1. JL 2n 2K u dzi = -^ dz 4^ 2 z2)(i with L designating the loop surrounding the cut (-|j.,+|i) in the posi- tive direction. However, the function u(q)) assumes the same values at points which have the same abscissa on the upper and on the lower edge NACA TM 135i| 123 of the cut; consequently, this mean value is equal to 1 u(9)d'P '0 \Jl - ji^cos^cp In order to have a mean value of f (z) on ^72) of zero, it is necessary and sufficient that the mean value of u should be equal to A log q which justifies the theorem. One utilizes this theorem in the following manner: If uq(cp) is a primitive of ^, calc\ilated by the formula (ill. 35), and if W Uo(cp)dcp = C - n^cos^ the desired value of u(9) may be written u(q5) = A log q + UQi dW Jy dZ dZ However Av = R Jy 0. But it suffices to return to the generalities 13^ NACA TM 1354 of section 1.2.2 to recognize that the obtained results will be valid in more general cases. Under these conditions, one may have in the region (a' ) (see fig. 2) domains which encroach on one another. How- ever, no difficulty arises since the relations of compatibility in the plane {t],9), formiila (l.22), show that the functions u, v, w in the plane (A') are perfectly known, owing to the boundary conditions. One will note the identity of the formulas (lII.i<-5) and (l.22). 3.2.2 - Cone Totally Bisecting the Mach Cone (Fig. 28) If one utilizes the plane Z, the problem amounts to determining the functions U(Z), V(Z), W(Z) in such a manner that u, v, w are zero on the circular arcs A-[_A2, ^±' ^z' (see fig. 46), and that w assumes prescribed values, with one part on the line A-^AA'Ag, and the other part on the line Aj^'AA'Ag'. In contrast to what happened in the preceding problem, the two half spaces, separated by the plane x^ = 0, are independent of each other. From the mathematical viewpoint, it may for instance be a matter of determining the solution in one of the semi- circles determined in (Cq^ by the cut AA' . There follows that there is no theoretical distinction between the symmetrical and the lifting problem. Naturally, one may operate in the same manner in the plane z. There will then be occasion to determine the solution in a semiplane, the upper semiplane for instance; the function w = f (x) is assumed to be known along a segment X,|-i, comprising in its interior the seg- ment -1,+1 of the real axis. The function is zero on the rest of the real axis^^. 3.2.2.1 - Elementary problem .- As before, we shall start with the study of the elementary problem, that is, the one where w = wq on the part of a cone situated in the region Xo > 0. We shall operate, for instance, in the plane Z; the f\inc- tion W(z) - Wq has a real part zero on the segment AA' and the arcs AA-|_ and A'A-]_', and equal to -Wq on the arc A-^^Ag . One can, by application of Schwartz principle, extend the definition of this function to a complete circle; its determination is then classical. (See, for instance, ref. 13, p. 162.) -^ See appendix 3 • I MCA TM 135if 135 This permits one to write immediately IWr W(Z) = wq ^ lo 1 + Z^ - 2Z cos 1 + Z^ - 2Z cos 9r (III.U7) with the logarithm being real for a real Z, and with 6-^ and ©g being the respective angular abscissas of the points A-]_ and Ag. The fiinction V(z) may be determined^ for instance, with the aid of the relations of compatibility ^0 Z^ + 1 dV dZ jt 2,2 Z-e^ Z-e ^ Z-e^ Z - e -i0c In the integration it suffices to choose the integration constant in such a manner that the real part of V(Z) becomes zero on the arc A-|_A2 • Thus one obtains IVr V(Z) = -^ cot ©2. 1°S ^ =-^ - cot 0p log ^ " ^ 1 - Ze^^l 1 - Ze^^S (III.U8) with the logarithms having an argument zero on the arc A-^Ag- One finds for V the following values V = Wq cot 02.^ o'^ "the arc A-^Ag v = Wq cot ©2 J oi^ "the arc A'Ag besides, one could have written these values directly by virtue of the relations36 (m.liS) and (111.^4-6). 3°This shows that one could have written the formula (IIIA8) directly, without writing the relations of compatibility. 136 NACA TM 1351+ In order to vrrite the value of v on the axis AA' , one must cal- culate the argument of 101 1 - Ze 101 Now Arg 101 1 - Ze 101 = Arg 101 Z 1 - Ze -10- For calculating this argument, for Z = X, one notes that the modulu / le-i \( -i0i\ / 10-1 \p of \e - Zj\l - Ze -^J is the one of \e - Xy^, under the assump- tion of 1 H- X - 2X cos 02_; on the other hand, its real part is written cos 0-i_(l -t- X^) - 2X. If one puts, therefore X = 2X 1 + X^ Arc cos cos 9-1 1 - X cos 9-1 (lII.i+9) with the arc cosine having thus, besides, its principal value. One finds likewise = -Arc cos X - cos ©2 1 - X cos 9c (III. 50) hence on the axis AA' Wr V = - cot 9-^ Arc cos cos 0-1 - X x - cos 0p -1- cot 0p Arc cos 6— 1 - X cos 9-^ ^ 1 - X cos ©2 NACA TM 135i<- 137 The calculation of U(Z) is perfectly analogous. One finds u(z) IWq sin ©c log Z - e i0n 1 - Ze ±Qo sin 9 log 1 - Ze iOi (III. 51) with the logarithms having the same value as in the formula (lll.'i^). One finds as the value of the pressure coefficient (wq = o.) C. = 2a- — 1 oj^ t^g a.rc A'Ao P 3 sin 0p ^ C_ = ^ — ^= , on the arc AAn P P sin e-L -^ (ill. 52) C^ = 2a Pit X - cos sin 9 1 cos 0-1 - X -1 Arc cos + — Arc cos 1 - X cos 0-^ sin ©2 1 - x cos 0^ on the axis AA' In the case where Ox-]_Xn is a symmetry plane 02 = Tt - 01 and the last formula (ill. 52) may also be written C^ = 2a P pjt sin 0^ cos 0-1 - X X + COS 0-1 Arc cos + Arc cos 1 - X cos 0-1 1 -*- X cos 9-1 Prt sin 0-]_ Arc sin sin 1 jl - x^cos^©! (III. 53) 138 NACA TM 1351+ In order to utilize these formulas, it is sufficient to connect the angles 0-^ and G^ with the geometrical form of the given delta wing (fig. ^7)' One has, according to definition cos ©-[_ = l/p tan u>|_ cos Qg - l/P '^^^ ^2 Let us recall also that px, X = 2 One will find in figure kQ a few applications of the for- miila (ill. 53) . 3.2.2.2 - Resultant of the nonnal forces on the upper region ( Xo > 0).- One can give, as in section 3-1.9^ a simple formula permit- ting the calculation of the resultant of the normal forces. If we des- ignate by C^"*" the dimensionless coefficient characterizing this resultant, C^''' is defined by the equality + _ <^X Cp dx 'Z X 1^ dx X Likewise we define the dimensionless number C^ , characterizing the forces normal to the lower region (xo < O), by the equality Cz" C dx X s: dx X with the integrals taken in the plane z, the first on the upper edge of the cut (X,n), the second on the lower edge. This definition entails NACA TM I35J+ 139 that the total C™ of a cone is written '-'z ~ '-'z "*" ^z Now / ix = A_ COS 0-j. i^os ^2 On the other hand X C ix = 2R U(z)d2 = -kR u(z) 1 - Z^ AgA'AAi (1 + z2)' dZ However, the integral of U(Z) 1 - Z^ alone the closed con- (1.Z2)2 tour BAgA'A-^B (fig. k6) is zero. On the other hand, with U(Z) having a real part zero on the arc A2A-]_, one has U(Z) 1 - Z^ dZ 'AgA'AAi (1 + 22)' -R 'A1A2 U(Z) ^ " ^ dZ .[i'^Ri] Rj_ denoting the residue of the function to be integrated, at the point Z = i R, = - 1 ^ = J^ dW ^ ' ' 2 dZ(^=i;) 2P dZ(2=i) Thus one obtains the general formula 2m j^i_?L^fi^ dw P COS 0-, - cos ©2 '^'^(z=i) (111.51^) ll^-O NACA TM 13514. In the case of the elementary problem, studied in section 3-2.2.1, one has dW _ ia cos ^2 - ^°^ whence dZ/r7_.\ ^ '^°^ ^^ ^'^^ ^9 C_+ = - 2a ^ 2i (III. 55) if one puts a = -L, following the notation customary in the wing theory. Thus we shall find anew a remarkable result : the value of the coefficient C^"^ is independent of the angles 0-^ and Qg* 3.2.2.3 - Study of the general case by means of the method of electric analogies .- The method set forth above (section 3'1.3.3) may be applied in superposition. The electrodes must be disposed on the arcs AA-]_, A'Ag, and on the segment AA' . These electrodes must be brought to prescribed potentials; the conductive arc A-^Ag is brought to the potential 0. Finally, one will detach a snmll electrode at the point B with the purpose of measuring the resultant of the normal forces; this resultant, given by the formula (lll.5i4-) is, in fact, pro- portional to the intensity entering at B. The value of u on the arcs AA-[_ and A'Ag is immediately known by simple integration. In fact, if for instance w-|_ designates the value of w given for = e-^_ - 6 (e positive and arbitrarily small), one has37, according to for- mula (ill. 14-6) ^7 -^'Physically, the fact that the pressure on the bounding genera- trices of the conical obstacle depends only on the inclination of the tangent plane along these generatrices is obvious . It expresses the independence (see section 'L.2.h) of these bounding generatrices with respect to the other generatrices of the conical obstacle. I NACA TM 1354 lIH ^1 U-, = - -*- 3 sin 0j_ and the formulas (lll.i+S) permit the calculation of u on the entire arc AA-i . Thus it is not necessary to measure the intensities leaving each of the electrodes except over the length of the segment AA' . As "before, this intensity, proportional to c»w/SY, furnishes immediately the value of Su/SX along the axis OX, owing to the formula P Su ^ 2X bw ax "L _ x2 ^Y Since one knows the value of u at the points A and A', one uses the superabundant data for calculation of the value of u on the axis AA' . Thus it is unnecessary to obtain the distribution of the potential, inside of the tank, as in the case described in sec- tion 3.1.3.2. 3.2.2.^ - Study of the general problem by purely numerical methods . - In order to simplify the exposition, we shall be content to examine the case where the given cone admits the plane Ox-iXo as symmetry plane. This amounts to stating that in the plane z the function w(x) is even in x on the cut (-ia,(i) representing the given cone. We assume w-j_ to be the value of w at the points x = 1 and X = -1, and put f (x) = w(x) - w^ If 1 < X < u, one will put x = — - — , |-i = — - — and F(0) = f(x). cos 9 cos 9x One notes that F(0) = 0. After this statement, it is first of all evi- dent, according to the foregoing, that one can immediately calculate the pressiire outside of the cone (r) . In order to calculate the pressure inside of (r), one will consider the flow as the superposition, 1.- of an elementary flow ^w = -w-[_, on the entire cut^ , 2.- of an infinite number of elementary flows bisecting the cone (r) and symmetrical with respect to Ox-^xt . These flows give at the li+2 NACA TM 1351+ point x(x < 1) a pressure coefficient equal to cos^e 3.- of a symmetric flow inside the Mach cone, defined by w = f(x), on the cut (-1^+1). One may apply the method described in section 3-l-3'3 for the calculation of this flow. We shall simply remark that it is not necessary to determine the integration constant since one knows that u = 0, for X = ±1. 3.2.3 - Cone Partially Inside and Partially Outside of the Mach Cone (r) (Fig. 30) 3 .2. 3-1 - Symmetrical elementary problem .- The circle bounded by (Cq) must be notched by a cut CA (see fig. ^9) j with the real part of V,'(Z) assuming the constant value w^ -• a on the upper edge of the cut; and the value -wq on the lower edge. On the circle (^0) j ^ is zero, except on the arc AA-[_ where w = Wq, and on the arc AA-[_' where w = -Wq. One will designate the point C on the circle Cq by Z = a, and the argument of A-j_ on the circle [Cq^ by 9-^. The function W(Z) can be written without difficulty „- > . Wq (Z - a)(l - aZ) W(Z) = Wq + i -!i lo^ Jt (Z - e"l)(. - e-^«l) with the argument (Z - a)(l - aZ) Z - e"l) (z - e-^«l) being chosen equal to zero at the point A on the upper edge of the cut. Since W(Z) is defined with exception of an imaginary constant only, one may also write I NACA TM I35U lJ+3 Wr W(Z) = wq + i — lo 2ZtQ - (1 + Z^) 1 + Z^ - 2Z cos 9-, (III. 56) putting tQ = l^a + ij. We shall now seek U(Z) dU ^ 2iZ dW dZ z^ - 1 ^^ 2w, Z jt ^2 Z^ - 1 Z - a 1 - aZ ±6. Z - e -^ Z - e -i0n whence U(Z) = 2wr prt 101 2 sin e-L log a log a - Z 1 - Ze '1 1 - a^ 1 - aZ (III. 57) Consequently, on the arc AA-]_ C^ = 2a 1 P P sin e3_ which is a result one could foresee immediately. One obtains easily the v to write the formula (lII.i<-9) One obtains easily the value of C^ along the axis OX; it siiff ices C^ = 2a np 1 cos 0-1 - X o„ -^ Arc cos -_ — + — ^^^— log sin 6]L 1 - X cos 02. 1 - a^ a - X 1 - aX (III. 58) Let us recall that x = 2X 1 + X^ li^u NACA TM 13514. Particular case: Let us assume that d-, = —, a = 0, under the ^ 2 following conditions 2a On the arc AA^^, C, p the segment AA', Cp = — krc cos( - x) On (III. 59) Let us recall that in all these formiilas x = 3 — , with (x-, ,r,0] being the semipolar coordinates of a point of the wing A in the system of axes (Ox-|_,X2,Xo\, and that cos ©-]_ = l/p tan cjd-|_. 3 -2. 3 -2 - Elementary lifting problem, in the case where a = .- The transformation s = \jZ transforms the circle (Cq) into a semi- circle in the plane of the complex variable s. In this plane ^ A-]_ and A-]_' have as homologues M-^ and M-^ ' (see fig. 50). The func- tion W(s) has a real part zero on the arc M-]_M-]_' and equal to wq on the arcs AM-[_, BM-^' , and on the segment AB. We shall determine directly the function U(Z) or rather the func- tion U(s). In fact, U(s) has its real part zero on M-]_M2_' and one knows, according to the relations of compatibility, that as in the pre- ceding paragraphs Wr u = p sin 0-L , on AM-j_ Wr U = 3 sin 0-|. , on BM-|_ ' I NACA TM 1354 lij-5 Moreover, the imaginary part of U(s) is constant on the real 9,xis and may, consequently, be put equal to 0. Thus one may analyti- cally continue the function U(s) across the real axis. U(s) is then determined as solution of a Dirichlet problem inside of the circle of radius unity. One has i^O (s e''l/')(s +e-''l/') U(s) = ^ log \^ - ^ /As_J: e / 3. Sin 01 (s.e''l/')(s-e-''l/') with the logarithm having the value of In for s = 1. It is then easy to calculate u on the real axis, that is, on the segment OA of the' original plane Z. Let us put 2Z 2s^ X = — 1 + Z^ 1 + s'^ The quantity under the logarithmic sign is written 2 ^1 s - 1 - 2is sin -^ 2_ 2 . .. . ^1 s"^ - 1 + 2is sin -r- Its argument is equal to that of (s^ - 1 - 2is sin — ] Now, the real part and the modulus of this expression are, respec- tively, equal to (s2 - 1) - iis^sin^ ^ = s^ + 1 - 2s2(2 - cos 9j) and 01 (s2 - 1)^ + I,s2sin2 ^ = s^ + 1 2s2cos e 1 11^6 Hence NACA TO 1354 w^ u = pn sin 0-L Arc cos 2x(l - cos 0-|.) X cos d-i C^ = 2a P 3rt sin 03_ Arc cos 2xfl 1 ^ cos 9 1 - X cos (III. 60) Particular case .- Let us suppose that 9-[_ = — C = i2i Arc cos(l - 2x) ^ Jtp 3.2.3.3 - Elementary lifting problem in the case vhere a / .- The elegant demonstration which has Just been made for a = and the principle of which is to be found in the original memorandum by Busemann, conceals one difficulty; this has caused M. Beschkine (ref . 11) to give a formula in the case where a /^ which, at least in certain cases, leads to diff ic\ilties . In working directly with the function U, one risks forgetting the supplementary conditions which, because of the relations of compatibility, must be applied if one does not want singu- larities for the functions U, V, W at points other than the ends of the cut . In fact, if U(Z) is regular inside of the circle (Cq), V(Z) and W(Z) will have a logarithmic singularity at the point Z = 0. We shall study the case where a / 0, by studying directly the function W and limiting o\irselves to not having singularities outside of the boundary generatrices of the cone. Besides, we shall again take up this important problem in section 3'3- Thus it is a matter of studying the case where w = Wq on the arc AA-|_ and on the upper and lower edges of the cut CA (see fig. ^+9) and on the arc AA-j^'; the transformation a = Z_ 1 - aZ which maintains the circle of radius unity, leads us to the case where NACA TM 1354 Ikl the cut is arranged following a radius. Finally the transformation a = s^ leads, in the plane s, to search for a function W(s) the real part of which assiaies the value wq on the arcs 'BM.-^, B'M-j_' of the semi- circle of radius unity of the positive plane and on the segment BB', and becomes zero on the arc Mj_M2_ ' by application of Schwartz' prin- ciple; one may continue the function W(s) - Wq to the lower semicircle of the plane s. This function is defined by the values of its real part on the circmnference of radius 1 of the plane s. However, since dW/dZ must become zero at the point Z = -1, dW/ds must become zero for s = ±i. In order to satisfy this condition, one decides to admit, for W(s), singular points at the points M-]_, M-|_', Mg, Mg ' , and at the point s = 0. According to the investigation of section 3'1'1'2, this point may be a pole of the order one, with the residue being necessarily purely imaginary. If ik is the residue of this pole, one may therefore write W(s) = wn + G(s) + ik ^ + ^ ^ s with G(s) being a holomorphic function inside of the circle of radius 1 . However, on the circle |s| = 1, ik ^— is purely imaginary. ' s One deduces from it immediately the function G(s); consequently, W(s) is of the form 2 "Pi iw^ 1 + s - 2s cos -— -, ^ 2 W(s) = WQ ^ log + ik 1 + ^ 1 + s + 2s cos — cp-i — ^ being the argijment of the point Mq^. 11+8 NACA TM I35I4. One calculates U(s), owing to the relations of compatibility ds 2k (s^ + a)(l + s^a) P (1 + a2)s2(s2 +1) because 2Z 2 (g + a) (1 + ga) Z2 - 1 " (g2-i)(l-a2) One verifies immediately that the points s = ±i are not poles (and that, consequently, the points Z = ±1 are not singular points), if ^0 ^1 n cos — Hence, for U(s) U(s) = 2^0 a (l - s2) ^ ^1 (1 - a2)s Pn cos -^ ^ . nu . ^1^ iwg (1 + 2a cos cpi 4- a2) ^^^ U - e^ "^Ks + e"^ 2 P« (1 - a2)sin 9i / _. cpW . fl s-e ^/\s+e '^ (III. 61) It is easy to relate the angle 3.2.2.3, and 3.2.2.i<-. Let us only indicate that, in the solution of the lifting problem by electric analogy, one must arrange a singularity at the point C like the one defined in section 3-1-3-2.3- The adjustment of the poten- tial to which the conductive part of this singularity must be brought is obtained by the condition that no intensity enters at the point A' . To verify this condition, one will use the method already indicated in J the section noted. I Naturally, the total C^ will be very easily determined by mea- surement of the intensity entering at the point B and application of the formula {lll.Gk). i Z.2.h - Cone Entirely Outside of the Cone ( T) (Fig. 29) 3-2.4.1 - Elementary symmetrical problem .- The problem consists in determining U(Z) , V(Z) , W(Z) by means of the following conditions: the real part of W(Z) assumes on the arc A]_A2 (see fig. 51) of the circle (Cq^ the constant value wq = a, and on the arc A-|_'A2' the value -Wq. On the other portions of (Cq) this real part is zero. Thus one may write immediately the value of the real part of U(Z) on the circle (Cq] (formulas (ill. 45), (III.46)). It is an even function of the argument 6 . One has NACA TM 1354 153 u = 0, on the arc A 'Ac u ^0 1 P sin ©2 on the arc A2A-]_ w, u = - 0/ 1 3 I sin ©2 ^^^ ^1 ], on the arc A^^A whence for the function U(Z), the formula U(Z) = — Q pn 1 19. sin Or log - Z i0n Ze 2 sm log 1 - Ze 1 (III. 66) the logarithms assuming the value in at the point Z = 1. The complete calculation of V(Z) and W(Z), likewise, does not offer any difficulties. One deduces from this formula the calculation of the pressures on the obstacle and outside of the obstacle. In the plane x-]_0x2 the pressure coefficient has the value Ct. = ^ ^ 3 sin 62' on the obstacle r = 2a/ 1 , in the region comprised between the obstacle P Is in 02 sin 9-^ ^nd the Mach cone of the point Let us recall that if od-, and aD„ designate the angles formed by the bounding generatrices of the obstacle with Ox-[_, one has according to definition (see fig. 52): cos 0-|_ = 1/3 tan uj-|_ cos ©2 = V? 'tan 0)2 151+ NACA TM 1351+ Inside of the Mach cone, finally, at the point Xt_, Xo, one has r - 2a 1 sin 9c Arc cos cos Qg ~ ^ 1 - X cos ©2 sin ©-]_ Arc cos cos 9-, X cos 01 if X = 3 — ^, the arc cosines having their principle values. 3.2.U.2 - General symmetrical problem .- The general symmetrical problem does not present any difficulty, since one may operate by means of superposition; let w = a(9) be the given value of velocity compo- nent following Ox^, over the length of the obstacle [9-|. < ^ < ^z)' The formulas giving the C^ may be written immediately S = s = 0O+O da , at the point of the obstacle of 3jg sin parameter 2 ^2-0 P J 9, -0 -SS — ^ behind the obstacle, outside of the ^^^ ^ cone (r) > (III. 6?) S = _ 2_ r^-o P'^^0-L-O Arc cos cos - t _da — ^ inside of the 1 - t cos sin Mach cone The integrals of the preceding formulas must be taken according to the signification of Stieljes; this is a fundamental condition for the case where a(0) presents discontinuities. In particular, one will have to take account of two discontinuities: the discontinuity +0,(^1) for = 0-|_, and the discontinuity -a- (62) ^°^ = ©2 • Not to forget these discontinuities was the reason that we wrote certain limits of the integrals 9-^-0, ©2+0. 3-2.k.3 - Elementary lifting problem .- The solution obtained for the symmetrical problem (formula (ill. 66)) is valid, since dW/dZ necessarily becomes zero at the points Z = ±1; also, dU/dZ becomes zero at the point Z = 0; thus the relations of compatibility do not entail any singularity other than the points A-j_ and Ag. We shall see NACA TM 1354 155 that in case of the lifting problem a few precautions must be taken if this condition is again to be satisfied. Let us first assume that the points A-^, A2 and A-[_', Ag ' are simple poles for —, —, and — . One may then write the values of dZ dZ dZ u, V, w on the circle (Cq) utilizing the relations (ill. 45) and (ill. 46) as well as the boundary conditions. These latter let us know that w assiimes the value Wq on the arcs A^^Ag, •^i'-^2' (fig. 51). On the other hand, the component u necessarily continues outside of the cone (since u represents the pressure except for one constant) and, being odd in Xo, must become zero in the plane OxjX^ outside of the given delta wing. Consequently, u = on the cir- cle (^0) f outside of the arcs ^j/^2' -^I'-^g'* Hence one deduces, as before, that on A-^Ag V = a cot ©2 u = - — P sin ©2 but on the arc AA-]_ sin So - sin Q-i cos On - cos St w = a V = a sin ©2 si'^ ®2 We note therefore that w assumes on the arc AA-]_ the same values as on the arc AA-^, whereas v assumes opposite values. Hence one deduces that the region of the plane Ox-^Xg, comprised between the trailing edge At and the Mach cone (see fig. 52), is thus a region of discontinuity for the velocity. One sees therefore that the hypothesis set up before ( simple poles for ^, ^, ^^ is incompatible with the fact that U, V, W do dZ dZ dZ/ not admit singularities other than the points A-|_, A2, A-]_', A2 ' . One may realize this, besides, in another manner; in order to satisfy in the simplest possible way the boundary conditions imposed on U(Z) , it siiffices to write U(Z) in the form 156 NACA TM 1351+ Ui(Z) = - la pn sin ©2 log 1 - 2Z cos 0-|_ + Z'^ 1 - 2Z cos e + z^ since this function U-|_(Z) well fulfills the boundary conditions required for the function U(Z) on the circumference ('^o)* However, dUj dz" 2ia Pn sin ©2 Z - cos 1 - 2Z cos e-]_ + Z^ cos Qr 1 - 2Z cos 02 + "^ and for Z = dU- dz" 2ia z=0 P" ^^"^ ®2 Tcos 0-]. ~ "^os ^2) If, therefore, the functions U(Z), V(Z), W(z) are not to admit singularities inside of (^0)7 "the solution Uq_(Z) cannot be retained just as it is because the corresponding functions V-[ (Z) and W-[_(Z) would have a critical logarithmic point at the origin^°. Thus we are led to modify the solution U-^Cz) by introducing a singularity at one of the points A-|_ or A2 (and, by symmetry, at A-]_' or A2 ' ) . Physically, by virtue of the rule of forbidden signals, this singularity must be placed at the pair of points ki^, A-^', because the bounding generatrix A2 (fig- 52) which takes the place of the leading edge (having as image the pair of points A2, A2 ' in the plane Z) is independent (see section 1.2. 14-) of the trailing edge (pair of points A-j_, A-|_', in the plane Z) . One then sees that, by putting U(Z) = - ig jtp sin 02 log 1 - 2Z cos 0-L + 7? 2ia {^°^ ^1 ~ ^°^ ^2)^ 2 rtp sin 02 1 - 2Z cos 6^ + Z^ 1 - 2Z cos 02 + Z (III. 68) 3°L. Beschkine (ref . 11) took the function U-]_(Z) as the value of U(Z); see further on, in section 3 -3 -2, the discussion of this question. I NACA TM 1354 157 one has, for U(Z), a function satisfying the boundary conditions on [ Cq) , holomorphic inside of (Cq)^ "the derivative of which becomes zero at the point Z = and consequently leads to fimctions W(z) and V(Z) which do not present singularities inside of (Cq) • Besides, this solu- tion is unique if one takes accoiint of the principle of minimum singu- larities . One may then calculate the functions V(Z) and W(Z) . Thus one finds for W(z) „(z) . - ia log -'"' - ^ . i<^ (" - °°° '^ °°° ^^) log ^i!i-^ It „ i©P rt sin 6-p sin 6-, ^ ±6-, 1-Ze^ "^ -L l-Ze-L a cos 93_ - cos 0^ Z^ - 1 n sin ©2 1 + Z^ - 2Z cos 0-l (III. 69) and v(z) = . ia cot a, log (z - e'^gJlz - e"^^^) ^ (z - e^«l)(z - e-^^l) o^„ cos 6-1 zfcos 01 - cos So) , 2ia _ 1 _v 1 ^ (III. 70) sin Uo 1 I ryC 2 1 + Z^ - 2Z cos 0-L Thus one finds that on the wing (arc A-^Ag) the component v has the value V = a cot 02 In the region of the plane Ox-|_X2 outside of the wing, the compo- nent V is always zero; whereas w assumes a constant value in the part comprised between the trailing edge and the cone (r) ; / 1 - cos 02^ cos 02\ 1 - cos (0-^ - 02) I sin 02 sin 0-^ j sin 0-^ sin 02 158 NACA TM 1351+ Finally^ in the part of the plane Ox-^Xg inside of (r) (seg- ment AA') V = 0, and w is given by the formula / cos 0o - X \ w = — Arc cos 3t \1 - X cos 0c ^ (1 - cos 01 cos 02) j^^ ^^J COS 01 - X N jt sin 01 sin 02 1 1 - x cos Q-^ ^ cos e-^ - cos 02 >Jl - x^ rt sin ©2 1 - X cos 0i (III. 71) 3-2.k.k - General lifting problem .- One sees immediately that, if one wants to iiniquely calculate the pressure on the obstacle, one may utilize the same formula as for the general synmietrical problem (for- mula (III.67)). Besides, the study of the values U(z), V(z), and W(Z) in the general case will also be very simple with the aid of superposi- tion. One will easily verify that, if w - a(0) is the prescribed value of the normal component along the obstacle (0i < < 02) , one has , for instance ^^2+° 1 + z2 - 2Z cos 0-1 da(0) U(Z) = -^ / log i aavy£ ^ P« J0-L 1 + z2 - 2Z cos ^^"^ ^ 212, / ^2'''° cos ©1 - cos pjt(l - 2Z cos 01 + Z^) <-'0i f. sin da(0) Analogous formulas cotild be written for V(Z) and W(Z) . Thus the electric analogy is less interesting in this case, since there is a way of solving the problem explicitly. We shall simply note that the singularity to be placed at the tank at the image point of the trailing edge is a doublet . MCA TM 1354 159 3.3 - Supplementary Remarks on the Infinitely Flattened Conical Flows 3.3-1 - Continuity of the Results At the end of this investigation, it will not be unnecessary to state briefly the continuity of the obtained results. If one takes for instance an elementary flow bisecting the Mach cone for which one makes 9-^ tend toward 0, 02 toward n, one finds, passing to the corresponding limit in the formula (ill. 52) as limiting value of the pressure coefficient r _ 2a lim, %-*o ;i^ Arc cos cos 1 1 - X cos 9-1 lima k^ Arc cos ^2 '^ sin 0o X - cos ©2 1 - X cos 0c 2a p« \Jl - X \Jl + kg, pit (III. 72) If one now makes, in an elementary flow, symmetrical or lifting (see sections 2.1.2.2 and 3-1-2.3); ti and c, respectively, tend toward -1 and 1, one again arrives at the formula (ill. 72). Besides, the formula (ill. 72) has already been written, at the end of sec- tion 3-l-l'7' One finds, finally, the same result by transferring like- wise results from section 3-2.3- If one makes, for instance, in the formula (ill. 58), 0-1 tend toward zero and a toward -1, one obtains r _ 2a lime li i™a- 1 sin 1 log 1 - a^ Arc cos e, -X ^ 1 - X cos 0-]_ - X - aX _ 2a P« _. — , a_ 1 1 -(- X 1 1 - X \1 - X 1 + X _ ha pn ^ l6o MCA TM 1351+ Likewise, starting from equation (ill. 63) and making a tend toward -1, 9-^ toward zero (^j^ tend toward zero) Q ^ ka 1 'P 23n 1 + X 2^ + 2a li^ Pit ^1 sin 6-1 Arc cos 1 2(x - Xo)(l - coscp^) \_ 2^ 1 - xXq - (x - XQ^cos '^■^1 (3jr 1 + X li "%■ 2(x - Xq) |1 + Xq (1 + Xo)(l - X)jl - Xq 2a 3« 1 - X 1 + X [ 1 + x >jl - X Likewise, one may verify the continuity of the results under the hypothesis where a single one of the generatrices of the conical obstacle is situated on the Mach cone. One thus obtains a limiting case between the flows studied in section 3'1'2 and those studied in section 3-2. 3- If one supposes, for instance, that one of the bounding generatrices has as image the point Z = 1, the second the point Z = a, -1 < a < 1, one finds, whatever the manner of making the passage to the limit, for the symmetrical problem n _ 2a ^P " 1^ VI - ^ 1 - a2 Ll X - a aX and for the lifting problem C^ = Ima P«(l -^) vj(x-a)(l -aX) "" 2(x - Xq) ikx[x(l + a)2 - 2a(x2 + l)J pn^d - x)(l - xq) 3^(1 _ x)(i . a)^(X - a)(l - aX) In the same manner one can verify the continuity between the flows studied in sections 3-2.3 a-^d 3.2.k. NACA TM 135i^- 161 3. 3 -2 - Discussion on the Possible Singularities of Lifting Problems In this entire chapter, we have limited ourselves to giving the solutions which satisfy the condition, stated frequently: To admit as singularities in the plane Z only the bounding generatrices of the A, and to choose from among all possible solutions the solution which sat- isfies the principle of minimum singularities. This is a hypothesis which is justified by its simplicity and which we have set up here with- out using the experimental results apt to guide our choice for placing the singularities-^-^. A first theoretical possibility would consist in admitting singu- larities possible on the generatrices of the Mach cone, having as image the points Z = ±1 in the plane Z. This seems to us not easily admis- sible from the physical point of view. Besides, to our knowledge, the various authors who have treated problems of infinitely flattened conical flows have always eliminated this possibility (see in particular ref s . 10 and 11). In fact, it is hard to understand how the pressure could become infinite in the neighborhood of these generatrices. In contrast, one has a means of obtaining solutions different from those obtained in the course of this investigation, in tolerating, as possible singular point, the point Z = 0. We shall first make the following general remark: Let us take the case of a cone where one of the bounding generatrices has as image the point Z = in the plane Z; in this case the pressure remains finite in the neighborhood of the corresponding bounding generatrix. This resiilts from the formulas (ill. 23) and (111.2^4-) for the case of a cone entirely inside of (F) (section 3'1'2), and from formulas (ill. 58) and (ill. 60) for the case of a cone partially outside, partially inside of (r) (section 3'2.3)' We shall show that, utilizing conformal repre- sentations and maintaining the circle (^0)^ i't will be possible, even in the case where Oy:-^ is not a bounding generatrix, to define a solu- tion of the lifting problem in such a manner that the pressure remains finite along a bounding generatrix inside of (f), under the condition of admitting the point Z = as singular point. The theoretical study of flows (movements) in incompressible fluid has been rendered possible and effective only owing to the famous hypothesis of Joukowsky which indicates the choice to be made among the singularities which are possible for the flow. The study of the prob- lems treated here shows uncertainty in the state of our actual knowledge concerning the conditions which the theoretical solution must satisfy in order to represent best the real phenomena. l62 NACA TM 135lf We retiirn to the investigation of section 3- 2.3 -3 where a. f 0: One may in fact come back to the case where a = 0, by the transforma- tions utilized, before o = Z - a 1 - aZ s2 = a The function U(s) the determination of which was the problem is then defined inside of the semicircle, and it satisfies exactly the same conditions as the function U(s) studied, in section 3 '2.3 -2 . Thus one will have U(s) = iw, Pit sin ©-]_ log s - e 9n s + e ^s + e (III. 73) 0: t 1 . On the region of the obstacle comprised between OD and Ag f\ I ^^^ MArg s| < ^^^1 one has I W = Wr V = ±w cot 0-]. a result which is quite conformal to the formulas (lll.i<-i4-) and (lII.i4-6). I 2 . On the region of the obstacle comprised between OD and A-^, s is real Ul < s < 0, for the surface Xo < oj ; one sees that w = Wq cos on every surface, whereas v ass\imes the opposite values ±wq \|ar l64 NACA TM 13514- 3. On the region of the plane Ox-, x„ comprised between A, and Ox-]^ (s is purely Imaginary and varies on the segment Qjo), v main- q>i cos -^ tains constant opposite values, equal to a ±w ^, whereas w increases infinitely in absolute value. yja^ k. On the region of the plane comprised between Oxt and OD' (s, which is purely imaginary, describes the segment coB), v is zero; w, infinite on Ox-^, becomes zero on OD'. Behind the generatrix A-^ which one may consider as the trailing edge of the wing A studied, this solution furnishes therefore a zone of discontinuity of velocity (the discontinuity being in the direction of Oxg) which occupies the region A-^, Ox-^. Moreover, the axis Ox-i is a singular straight line for the flow. Thus one encounters a scheme which seems at first rather tempting and reminds one of the study of the wing in subsonic flow; behind the wing there appears a zone of dis- continuity of velocity produced by vortices following the direction of Ox-|_, and the singularity encountered along the axis Ox-^ reminds one of the "marginal vortex" of the wing theory. As in the case of subsonic flows, this flow scheme appears linked to the condition of having a finite pressure along the trailing edge. The formulas (ill. 7^) likewise show us that the flow found does not satisfy the boixndary conditions if a is negative, that is, if the obstacle is not situated on the same side of the plane Ox-j_Xo . In fact, in this case w woixld admit on the obstacle a discontinuity in the neighborhood of the axis Ox-^; but this is incompatible with the boundary data^*^. If one wants to apply a similar method in the case of a symmetrical flow, one likewise notices immediately that the result is incompatible with the given boundary conditions since one obtains a discontinuity for w. Let us now visualize the case of a flow around a cone entirely inside of the Mach cone, with the bounding generatrices on the same ^^^Thls solution which has been suggested by Beschkine must, there- fore, certainly be rejected in the case where a is negative; the fig- ure 6 given by Beschkine (ref . 11) seems to show that this author has not seen this fundamental restriction. In this case one must certainly adopt the solution set forth in section 3-2.3'3' NACA ™ 13514- side as Ox3_ edge An. 165 (fig. 55) and the remaining finite on the trailing The function U(z) then has the form / s ■Wn , X (Z - b)(l - Zb) , , U(Z = -^ C(b,c) \) i^ i (III.75 P \J(c - Z)(l - Zc) with C(b,c) being a function of b and of c. One then sees that in calculating V(Z) and W(Z) one will find the same particularities as previously: the point Z = will be a singular point. In the region comprised between Ox-|_ and A-|_ one states a discontinuity of the component v whereas the velocity w becomes infinite along Ox2_. The following problem arises : Should one adopt in the case where the two bounding generatrices A^ and A2 are on the same side as Ox-]_ the solutions exposed in the course of this chapter, which we shall call solutions of type I (singularities on A-^ and Ag), or the solutions we just indicated, which we shall call solutions of type II (singularities along Ag and Ox-|_) ? Let us note first of all that, for reasons of continuity, it is absolutely necessary to adopt completely one or the other viewpoint; one cannot admit a solution of the type I for the flows entirely inside the Mach cone, and a solution of the type II for the flows partly inside, partly outside. Under this presupposition, the solutions of the type II are, at a first glance, rather tempting; perhaps certain authors were thinking of these solutions when they exposed the condition of the subsonic trailing edge which could be stated in the following manner: Since the tangent to the trailing edge forms with the flow an angle which is smaller than the Mach angle of the flow, one must write on the corresponding trailing edge the condition of Joukowsky in order to be sure that the velocity remains finite (see for instance ref. k) . Now the solutions corresponding to the formulas (ill. 73) and (ill. 75) seem to satisfy these conditions. And as we remarked before, these flows show, behind the trailing edge, actually a character which reminds one of subsonic flows . 166 NACA TM 135i4- We do not want to definitely reject these flows; however, we have to make three remarks . 1. As we have stated that the solutions with finite pressure along the trailing edge are not possible for the symmetrical problems, the pressure cannot remain finite in the case of a flow of the type II around a cone having thickness . 2. It would be dangerous to link the solutions of the type II to the "subsonic trailing edge" since, if the wing is entirely outside of the cone (r), there exists still another solution which yields a finite pressure on A3_ and gives rise to a surface of discontinuity between Oxn and A^^: It is the solution U2_(Z) visualized at the beginning of section 3.2.i4-.3. One has, in fact, under this hypothesis Vi(Z) = la jt sin 0. Ycos 02 - cos 9nL^log( - Z) + la n sin 02 _ cos 03_ log(l + Z^ - 2Z cos 0-l) - cos 02 logfl + Z^ - 2Z cos 02) which gives in the region comprised between Oxj^ and A^^ equal values of v a sin 0"; (COS 0o - cos 0-1 If one adopts for such a cone the lifting solution of the type II, one finds that the velocity remains finite at the trailing edge, even under the hypothesis of a cone of nonzero thickness. 3. Adopting, still by virtue of the principle of continuity, the type II for the lifting solutions in the case where the bounding genera- trices are on the same side as Ox2_ wo\ild lead us to a restriction of the range of the study of the flows with infinitely small cone angle made in chapter II; for this problem, such as it has been posed, would no longer be valid in the case where the contour (C) in the plane Z no longer contains in its interior. In contrast, we already have had occasion to state that the results of chapter III are in complete NACA TM 1354 167 agreement with those of chapter II (see section 2.2.8); this statement is valid for the case of any figure whatsoever. We may conclude that, according to the actual state of our knowl- edge, it does not seem imperative to adopt the viewpoint of the solu- tions of type II. In our opinion, only an experimental study can indi- cate where the theorist must place the singularities; the viewpoint adopted in this chapter seems to us to be the most natxiral one . It becomes required in the case where Ox-]_ is comprised in the angle A-i and Ag; in the opposite case, if in one way or another our knowledge of the physical phenomenon should widen and lead us to a change in our hypotheses on the singularities, it will still be easy to obtain the desired solutions, provided the conical character of the flow is main- tained^-'- . See Appendix No, l68 NACA TM 135i+ CHAPTER IV - THE COMPOSITION OF CONICAL FLOWS AND ITS APPLICATION TO THE AERODYNAMIC CALCULATION OF SUPERSONIC AIRCRAFT We shall show in this chapter how the conical flows studied in the previous chapter and possibly the homogeneous flows defined in sec- tion 1.3 of chapter I permit to study, at least in certain particular cases, the various elements of a supersonic airplane (fuselages, wings, controls, etc.) by "superposition" if one can apply the general method of linear approximations. Our aim is not to furnish all possible appli- cations nor to give all the formulas the constructor may need. We shall, rather, insist on the principles of such a composition; we shall give the simplest and most significant resiilts and, more specially, those which, at least to our knowledge, have a character of newness. We shall voluntarily reserve the results of technical character for a later publication. Such a superposition is justified by the linear character o£ the fundamental equation (l.lO). The simplicity of the following arguments frequently results from the rule of "forbidden signals" which we have stressed already in section 1.1.4. k.l - Application of Conical Flows to the Calculation of the Wings In his fundamental memorandum, often quoted above (ref. k) , Th. Von Karman indicates that the theory of conical flows permits the investigation of wings the profiles of which are formed by straight hp lines^^. We intend to show in this paragraph that one can investigate a wing of finite span and with a curvilinear profile by means of compo- sition of conical flows. Like the problems of conical flows (compare chapter III), a wing problem may be divided into a symmetrical and a lifting problem. We shall note 5 (x-, ,Xp ] and ^~ (x-. ,x^] , the inclinations of the top surface profiles (xo = +0\ and bottom s\irface profiles (x-d = -0\ h? The subject of a certain niimber of memoranda is the study of wings with polygonal profile. One must then superpose a finite number of conical flows. The most recent and most complete investigation of this problem is the one by A. E. Pukett and H. J. Stewart (ref. 30)- NACA TM 135i4- 169 of the wing investigated, and we shall put (compare fig. 56) 5+ = _i + a"^ &- = -i + a" with i representing the general incidence of the wing (one will define it as the incidence of the chord of one of the sections). We shall then put a = ^^ - ^- In the case of a purely symmetrical problem i = Jo = In the case of a purely lifting problem a = Let C_~ and C-j"*" be the pressure coefficients on the upper side and lower side of the wing. The local c^ and the local c^^ of a sec- tion parallel to Ox-^,x^ will be defined by (compare fig. 56) ^x = J , (Cp" - Cp^)dxi mm *-' mm ' Designating by Cp^-'-^ and Cp'^^' the pressure coefficients obtained in the study of the symmetrical and lifting problems, the superposition of which gives the general problem investigated, one has C + = c (1) + C (2) c - = C (1) - C (2) *^p ^p ^p ^p ^p *^p 170 X\CA !5t 15C4. («) C^ »C3 c,'^' ^i^ia^ ':^ IccaLI c^ of tte lifXiuais $£V3^«k 4^ki :pCD(^* ^ ^-)^a^ ♦ / sl^Kt" ^ IBB. I Sf^* *t ♦ 1 I ^^*)(-i ♦ jio)^ c, « C!,«^ :x » CxCD ♦ C3t^«) Que sees t^Kss x^srn^ ^la-^^rl^r ^msv & (tgeiiiral fcefel^ Is twMfcit iiftc £ ssjaBtftricsl sa^ 3i l:if^iJDg $i?9S£u!eK« One a«f s*;-, fl^ss^xi^icily ^miYr^r^r.., 1 *^\'v»* ->>ili Tc^t: -x'~ ~ -^ X — ' "^ -'5^- rc'izis: '""^■« NACA TM I35U 171 i+.l.i - Synmetrical Problem ^.1.1.1 - Rectangular wing wlbh symmetrical profile and zero lift U. 1.1. 1.1 - General remarks .- Thie projection of the wing is a rec- tangle (R) : ABA'B' (compare fig . 57 ) . We shall put AA' = BB' = I AB = A'B' = \l The problem is to find a flow Guch as to make the value of the normal component o) zero at every point of the plane Xo = 0, except in (R) . Fiirthermore we shall, for a start, assume that the wing cross section is constant for the entire span. This profile, symmetrical according to hypothesis, will be defined by the function a,(x-j^^ which gives the value of the inclination of the profile (supposed to be small) toward the axis of the x-^; a> will therefore assiime the value a)"*" = cl^x-l^ on the upper side (xo > 0^ of the rectangle ABB'A', and the opposite value cju" = -a^x-j^^ on the lower side (xo < 0^ . In order to solve the problem, we shall compose conical flows the vertices of which are situated on the sides AA' and BB' . In order to simplify the notation, we shall call Cg(M,a} the elementary symmetrical conical flow which has its vertex at a point M of the plane Ox-j^Xg (compare fig. 58) for which w is zero outside of the quadrant limited by the semi-infinite lines parallel to Ox^^ and Oxg issuing from M; w is equal to the constant a on the upper part of this quadrant and to -a on the lower part. Cg(M,a) will designate an analogous flow where the axis Oxg will have been replaced by its symmetrical counterpart. Such a flow has been Investigated in sec- tion 3'2.3'1' If one designates the angle x-j^MP by 9, the for- mulas (ill. 59) show that the pressure coefficient C_, is given by 2a - + - Arc sin(p tan T) 2 rt I P tan ?! < 1 (IV. 1) Cp = if p tan 9 < -1 s = 2a if p tan 9 > 1 172 NACA TM 1351+ 4.1.1.1.2 - General principle of the superposition .- Let us visu- alize^ first of all, the superposition of the following flows "^s U,a(0) and Vg B,a(0) The resultant flow gives in the plane Xo = the values of w indi- cated by the figure 59(a) . If we now subtract the two-dimensional flow about a dihedron of the angle 2a(0), it is disposed symmetrically with respect to the plane Ox-]_X2 and has Oxg as edge; the semi- infinite Ox-| is inside of the dihedron, and one obtains in the plane Xo = +0 the values of w indicated by the figure 59(b). This gives us the princij)le of the composition. One will obtain the desired flow by superposing conical flows of the type Cg the vertices M of which will be situated on AA', conical flows of the type Cg the ver- tices of -vdiich will be situated on BB', and by subtracting suitable two-dimensional flows. It will be possible to schematize the flow in a precise manner as follows I c's(M, da) + "^s^**^ ^°-) - ^Um\ (IV. 2) Jaa' Jbb' L J with E a(x-]_^ designating the two-dimensional flow about a wing of infinite span the profile of which is identical with the profile of the given rectangular wing. In fact, one verifies immediately that the flow, symbolically defined by the formula (lV.2), satisfies the given boundary conditions. We want, nevertheless, to specify that the integrals of this formula o\ight to be understood in the sense of Stieljes, in order to understand the case where the function cl(x-]_^ will represent discontinuities of the first kind. Such discontinuities exist, in general, at the leading edge AB and at the trailing edge A'B' . 14-. 1.1. 1.3 - Study of the flow Cg(M, da).- In order to make i^AA' this investigation, we introduce the axes Axy, Ax parallel to Ox-, , Ay coinciding with Oxg, and put ^ = x^ -^ = y^ a^(x^) = a(x) NACA TM 135i^ 173 The section of the Mach cone behind the point A is formed by two semi-infinites which have as equations x^ + y^ = Let (x^,y^) be the reduced coordinates of a point P of the plane Axy (fig. 60) . We shall suppose x^ < 1. If < x^ < y^, the point P is outside of the Mach cones behind all points M of the segment AA'; consequently, according to equation (iV.l) If now < y^ < x^, the point P is outside of the Mach cones of the points of the segment PiPq' ^^* inside of the Mach cones of the points situated on AP-[_, P2_ being the point of AA' of the abscissa x-'^ - y^. Besides, the conical flows, the vertex of which is on PqA', have no influence on the point P. Consequently, the pressure at the point P is written, according to equation (iV.l) P«Jo x^-y^ — + Arc sin X -y^Hi) . I iX' x^-y^ daMi) or Cp = i[2a^(x^) - P(x^,y^y P(x^,y^) = a^(x^ _ y^) - 2 r X^-y^ Arc sin y^ x^ - ^ da^(U ► (IV. 3) This formiila, set up for the case where < y-^ < x^, may be extended to the case already studied < x^ < y^ since a may be considered zero for the negative values of the abscissa. nk NACA TM 1351^ One cein now calculate the drag of the section y^ ^xM =2j Cp(l,y^)a^(l)d5 Consequently — a^(^)dl I Arc sin P'^Jy^ Jo ^ - ^ ^ da^(Tl) or, changing the order of integration in the last term and putting a^2(|)d| = ^ F(y^) = 2 a^(|)a^(| - y^)d| - ^ da^(Ti) I Arc sin / a^(l)dg "Jo Jy^+Ti ^ - n (IVA) c(y")=^-^F(yX) According to our conventions, if y-'^ ^1 F(y^) = :(y^) = ^ I NACA TM 1354 175 Such a section actually behaves like the section of a wing of infinite span which is quite ohvious according to the rule of forbidden signals. We note in addition that c^(0) = 2^ thus the drag of the section y^ = is half the drag of the same sec- tion at infinite aspect ratio. We want to point out another remarkable result r ^x(y^)'iy^ = ^ (IV. 5) Jo P that is, the mean value of the drag in the region < y-'^ < 1 where the c (y^J is not constant is equal to the value of the drag in infinite flow. In fact, first of all I dy^ a^(l)a^(| - y^)d| = / a^(e)no - n) a^(t)dt However, by definition 2 P(t,u)a^(t)dt = F(u) '0 1 I P(t,u)a^ Jo F being, besides, the function defined by equation (IVA) . Consequently "x(^) ^X'pE^'^^'^ ^F(^o-n)] (1V.7) p we remark that if t]q > 1, that is, If X > — , there is always at least one of the f\inctions F zeroj in this case the c^ of the sections V-2 close to the center is equal to ^^^^—. This is an immediate consequence of the principle of forbidden signals. Now we can finally calculate the total drag which we shall fix by the coefficient 2^oJ-Tio 178 NACA TM 1351^ If one puts nv $(v) = F(u)du Jo one sees Immediately that Cx = — *(2t1o) (IV. 8) However, the resiilt obtained by the formula (lV.5) amounts to stating that $(v) = if V ^1 Consequently, the drag of a rectangular wing has a value independent' K-2 of the aspect ratio and equal to ^^^^—, for geometrical aspect ratios X !;reater than — . Siamarizing, one may say that the complete investigation of a sym- metrical rectangular wing of zero lift amounts to calculating the func- tions P, F, which are all calculated by quadratures. ^.1.1.1.5 - Applications .- 1. The profile is a rhomb; in this case a^(t) = oq if t < i a-'^(t) = -Oq if t > - a. = OuQ We shall now calculate the function F(yx), defined by equa- tion (IVA). For this purpose we remark first that ( a^(l)a(| - y^)d| = < ao '(1 - a/") if 0^ y^^ —) — , only the discontinuity for t] = comes into play, the contribution of which is -Oq Arc sin y^ - y^ ^ + y^Arg ch — y 180 NACA TM 1354 If one assembles these partial results and puts Y(y") Y(y") -(Arc sin y^ + y^Arg ch — ^ for < y^ < 1 1 2 for y^ > 1 (IV. 9) one sees that one may write in a general manner F(y^) = 1.00^ I + Y(y^) - 2Y(2y^)1 and conseq.uently :x(n) = ka^ 2r ^ - Y(t1o + n) - Y(iio - Ti) + 2Y(2iio + 2ti) + 2Y(2tio - 2ti) (IV. 10) Figure 61 gives the variation of c^(ti) for two values of t\q. For knowing, finally, the total drag it suffices to calculate the f\inction ,! NACA TM I35U 181 Hence (u) = i+ao^ [1 + D(u)] - u D(2u) (IV. 11) Consequently, applying equation (IV.8), one obtains 'X = ^[2 D(^o) - d(2t1o)] (IV. 12) One will find in figure 62 the curve giving C^^ as a function of the reduced aspect ratio. The curves of the figures 6I and 62 have already been given by Th. Von Karman (ref. k) , but this author does not give any analytical formula. Moreover it seems as if the results Th. Von Karman' s had been obtained by application of the method of "acoustic analogy." The curve given in figure 61 may also be found in a memorandum of Lighthill (ref. 31) w'ho utilized the method of sources. 2. The profile is formed by two symmetrical parabolic arcs; in this case one must put a^(t) = 6f^(l - 2t) 6q characterizes the thickness of the profile. The problem consists in calculating the functions F (y-^j and <5(v) defined in the previous paragraph. One finds after a few integrations of elementary character he ^ ~ I n F(y^) = — ^ Arc cos y^ + y^(y^2 - 3)Arg ch i + y^ ^1 - y^^ (IV.13) and cD(v) = _0_ 3« (Zl_ _ 3y2_j^ ch ^ + y^Arc cos y^ + ^ '^'^, " y x2 (IV. lU) 182 NACA TM 135lt On the other hand 7^2 _ ^0 a = One can clearly verify that F(0) = 2 6r = 2a^ F(1) = $(l) = In figures 63 and 6U one will find the distribution of c^ over the span for a wing of reduced aspect ratio 2t] =2, and the variation of C^ (total-drag coefficient) as a function of the aspect ratio. ^.1.1.1.6 - Case where the profile is variable in span .- It is possible to calculate the symmetrical rectangular wing at zero lift in the general case where the profile is variable in span. We shall here be satisfied to examine the relatively simple case where the profiles along the span are deduced from one another by affinity; the ratio of the affinity varies with the span. We shall assume that the wing of reduced span 2r\Q has a local inclination of the form k(T))a^(t) at a point of reduced coordinates tj, t. The function k(T]) must of course satisfy the usual limitations so that the problem posed can be treated by means of linear approxima- tions. Finally, we shall assume the function k(ri) to be even in t\. Let us first of all remark that the wing of reduced span 2t\, the profile of which (which is constant along the entire span) is defined by the function a^(t), causes outside of the wing, at a point of reduced coordinates t, y-^(y^ > ''l)^ 3- pressure coefficient %i^'y^) = 7 PL P(t,y^ - n) - p(t,y^ + Ti) (IV. 15) P is the function defined by equation (IV.3) as one sees reassuming the arguments of the sections ij-. 1.1.1. 3 and U.l.l.l.i<-. One will now obtain the desired boundary conditions by superposing a succession of rectangular wings which are symmetrical with respect to 0-[_x-[_, of equal chord reduced to 1 and of variable reduced NACA TM 1354 183 span Sti^O < T] < r\ A for which the profile remains constant in span. This is justified since k(T]) had been assumed to be even. At a point (t,y^) the pressure coefficient is written s(''^) = i p '0 2a^(t)k(y^) + P(t,r, + y^)dk(ii) + Jo J I'll Cj^ ° P(t,Ti - y^)dk(Ti) - P(t,y^ - Ti)dk(Ti) V^ Jo All these integral^ are taken in the sense of Stieljes. One will obtain a simpler formula by putting l{t,y,no) = - I p[^.e(n - ^[|e dkCn) (iv.i6) G being defined by the equality e(Ti - v) = U - V In this case Cp(*>y^) = I ^""(^Hy^) - I p(t.y^.^o) + p(t.-y^,no) Plz (IV. IT) This formula is reduced to the formula (IV.6) in the case where k(ri) = 1 over the entire span. The drag of the section y^ is easily obtained -Ar') - ^ '^'(y")5' - ^|(^'io) * !:(-/^,lo)] (IV.18) 181+ NACA TM 135lf by putting l{v,^o) = 2 I P(t,v,TiQ)a^(t)dt = - I a^(t)dt / ° P[t,6(Tl - v)]€ dk(Tl) = - I ° 6 dk(Ti) J P[t,e(ii - v)]a^(t)dt whence the formiila F(v,tiq) = - I f[g(ii - v)]6 dk(Tl) F is the function defined by the equality (IV .k) (IV. 19) Thus one can see that the pressure coefficient and the local-drag coefficient are expressed by formulas analogous to those obtained in the case where the profile is constant \inder the condition that the func- tions p(t,y-^) and F(y-^) are replaced by weighted averages, P("t>v,T]g\ and F(v,t\q\, defined by the formulas (lV.l6) and (lV.19). Finally, the total -drag coefficient is obtained immediately C^ = ^^0 ^ 2t\ Od-\ e,(y-)dy- = i- P° c,(y-)dy- whence F(y^,no) + F(-y''.no) k(y^)dy^ NACA TM 1354 185 i As an example, we shall suppose k(Ti) to be defined by k(Ti) = 1 - ^ (o< Ti< no) no ^ ' One will then have F(v,no) " " f ° ^t^"^ ■ ^^ ^ ^^^'^^ " ■" / ° F[e(n - v)]^ dn If V is positive F(V,T1(.) = - — F(V - Tl)dn + i I ° F(n - v)dTl ^ "^ no Jo ^oJv = _ i. I F(u)du + ^ / ° F(u)du ^ojo ^oJo Pv $(v) = I F(u)du being the function introduced before in sec- Jo tion U. 1.1. 1.4. If V is negative: v = -v 1 r\ -^'°) ^ 4Jo '^' ' "'^"^ ^ F(u)du - ^[n^o - ^) - ^(-^)] 186 NACA TM 135if whence (IV. 20) Let us recall that $(v) = 0, if v >1. It is then easy to make applications of this formula in the case where the profile is a rhomb or lenticular formed by zero parabolic arcs. One will find the curve which gives in the first case the varia- tion of Cjj. as a function of y^, for a reduced aspect ratio tjq = 2, in figure 65. 4.1.1.2 - Study of the sweptback wing with constant profile Without investigating the sweptback wing as thoroughly as the rectangular wing, we shall show that one may, without essential diffi- culty, apply the method used for study of the rectangular wing for the sweptback wing of constant profile the plan-form of which is schematized in figure 66. We shall suppose that the plane Ox-|_Xo is a symmetry plane for the wing. With y designating the angle of sweepback, it is obvious that we shall have flows of different type according to whether the leading edge AOB will be outside or inside the Mach cone of 0. One has become accustomed to say that in the first case the leading edge is "supersonic" while it is "subsonic" in the second case, thus recalling that the velocity component normal to the leading edge is higher than sonic velocity in the first case, lower in the second. The number V, defined by: p cot 7 = — , (v < 1 characterizes the case where the leading edge is outside of the Mach cone, v > 1, in contrast, the case where it is inside) will, therefore, be an essential parameter in the investigation of sweptback wings. 4.1.1.2.1 - Case where v < 1 .- We shall put in this case V = cos Q. We shall define, as for the rectangular wing, "the reduced aspect ratio" 2t]q (compare figure 66) by the relation 2t1q = px if \ designates the span of the wing taken along Oxg. NACA TM 135ij- 187 For simplification we shall assume that the profile chord is taken as length unit^ and that the profile is defined by the function a(x-|_), with x-]_ varying from to 1. It is obvious that the desired flow will be obtained by a superposition of elementary conical flows which one may note schematically / Cg(M,da,0) - Cg(M,da,0) - I Cg(M,da/0) ' 00 ' J BB ' J AA ' Cg(M,da,9) designates a flow completely bisecting the Mach cone, admitting the plane Ox-]_Xo as symmetry plane (section 3- 2.2); Cg(M,da,6) designates a flow partially inside of the Mach cone; the sign — > indicates the direction of the bounding generatrix which forms with Oxg the angle 7; the other bounding generatrix is supposed to be parallel to the wind. Because of the symmetry it will be suffi- cient to study the region of the wing where Xo > 0. It will be convenient to put y"" = Px2 Xi = X + y^cos 6 conical flow with the vertex Mq (x-^ = 1, Xg = j , of the type C3(Mo,a,0) causes (compare formula III. 53) the following pressure field in the region y^ > 0: 188 NACA TM 13514. C^ = -^ — - — Arc sin P Pn sin t being defined by t = if < t < 1 X - I + y^cos 9 Cr. = 2a P 3 sin e if 1 < t < cos 9 Cp = if t > cos 9 At a point (x,y^) the pressure coefficient due to the flows r I Cg(M,da,0) is equal to '00' P sin a' x) - ax (x) - ax - y^(l - cos 0) + 3 c-y^(l-cos0) 2 Arc sin sin 0(x - I + y^cos 0) ^^^|^ »/(x - U(x - ^ + 2y^cos 0) 3 sin a(x) - Q(x,y^,0) i NACA TM 135^4- 189 putting Q(x,y^,0) = a X - y-^(l - cos 0) ix-y^(l-cos0) ^ . . sin dix - I + y^cos G) -, /t \ Arc sm ' '' ' da(s) = xj(x - 0(x - I + 2y^cos 0) ■ 2 x-y^(l-cos0) V . r.1 = y ^\~l ^ ^ ' sin 0(x - \ + y^cos 0) Arc COS ^x - U(x - 5 + 2y^cos 0) da(l) Let us note that Q(x,y^,0) = if y^(l - cos 0) > x and that the same holds true also in the case where the sweepback is zero ^ = Ziy For simplification, we shall henceforward assume v.^^ > ^ 1 + cos (which will always be verified if t] > l), that is, that the 'edge A/ has no influence whatsoever on the wing region Xg > 0. The contribution due to the flows I C(M,da,0) is very easily 'bb' obtained from the formula (ill. 58). The pressure coefficient due to these flows may Immediately be written Cp = P(x,,o - y",0)^ P sin 190 if one puts NACA TM 1351+ P(x,y^,0) = ax - yX(l + cos 0) 2 ix^-y^(l+cos0) Arc sin y-^sln^0 X - I cos 9 da(l) (IV. 22) If 9 = —, one falls back on the function P defined by the for- mula (IV.3); on the other hand, P^x^y^,©) is obviously zero if yx(l + cos 0) > X Finally, with the reservation that ^n > 0^1 + cos one has at a point of the wing C^ = P P sin 2a^(x) - 2Q(x,y^,0) - p(x,tio - y^,0) (IV. 23) The local-drag coefficient is immediately obtained 3 sin (3 sin P sin (IV. 24) NACA TM 135i4- 191 CD ^ CD w O o I X I X ^ CD CO O O I nI t; C •H ft OJ OJ • ^ 1 ^ ■p (U CO Tj H P" "I (U 1 CD 1 [0 CO t:|cvi 1 (U II CD 35 -p (M •1/ n (U C U H 1 1 >i 1 H l_ J Hi CO a 1 •H CO o3 CO X 0) Tj :s II 0) 1 , N 1 <: P CD CO X fl to 'S' (U Ti ■H w •H S CD CO cd + + in X >3 u H ■r\ (U '*»— ' X tH ^ 1>, >> X •rH CD CO i ' c H (U 1 X 1 X J fl + d nd CO r M d^^^ *»*-«^ ^ V- i>< 1^ (U D (U "'—^ N^-* "H ^ d d -p o—N ji"^ •«ci CD CD bO •H W CO 0) fH •d •cJ 0) + + ^ rH H 1 — i CO ^ — ■' •"• ^ cd •H —5 c X -5 -P P ^ N -=1-1 ti mj p CO II P CO ^^— ^ Ch ,, •H 05 to •\ A^ P • !k ' S ro (^4 P iH :J H cd (D (U M C ■H H H = , U 1 - cos the drag of the investigated wing is identical with that of the yawed wing of infinite span ^^ (IV. 27) 3 sin 3. If 0=0 (the leading edge is situated on the Mach cone of O), the given form\ilas present an indeterminate form. Nevertheless it is very easy to eliminate the indetermination. We shall, in particular, calculate the total drag. The value we shall obtain is very interesting NACA TM 135^ 195 because it corresponds for a given sweptback wing to the maximum of the total drag when the Mach number varies. If 9 tends toward zero, -^ — Arc sin sin_e_ sin D too Jl - f^cos^e has as a limit X - I + jr^ n|i - t2 ^x - |)(x - i + 2yx) We assume t\ > —; since our purpose is calculation of the total drag, the edge BB' may be neglected. The desired total drag which we shall denote by C„ is written nnax C^ = -8- r° dy^ r a(x)dx ^ x - | + y^ ^^^ ^ ^max RjtTi I f °Jo Jo Jo p - ^)(^ - I +2y^) ^ /'^ a(x)dx I "" da(l) n ^ - ^ ->- y^ dy^ '0 ^0 '-'0 whence, carrying out the last integration il px ^x - i)(x - I + 2y^) i^ ...^.. X - I + 2no -^max 3pnTi a(x)dx I !1^_-— ^[2(x - i) + Tijda(l) (IV. 28) One thus obtains a very sijnple formula giving the value of the total C^ when 9=0. I9i<- NACA TM 1351^ If the profile is a rhomb, one sees, writing '%a> 3«pTio da(l) '1 J x - g + 2t]q" |2(x - ^) + Tio]a(x) dx that Cv = ^ -^La) - 0(1)1 putt ing whence 0( P^ I - u) = I J^^^(2x + Tio)dx = u2(u + 2tio) Cx„ = (Cx) — ■'iiiax \ ^/oo 3n 2 3 3 (1 + Mo)^ - (1 + 2Tio)2 3n ^0 (IV. 29) max In figure 67 one will find the variation of (Cy') as a function Of r^Q. If the profile is formed by two parabolic arcs, a(x) = eo(l - 2x) and 8e, %ax 3|3nTiQ (1 - 2x) X + 2t]q (2x + tiq) - 2$(x) dx NACA TM 135i^- 195 or 8€n2 '^Snax 3P«t)q [ii(i) - 6j(i) + 6tioK(i) + 0(1^ putting ^^ 3 l(x) =1 x2(x + 2Tio)2dx = i^x(x + 2tiq) (x + 2tio)^(x - t^q) E- ^(x - 5,0) + Wl; lo X + TiQ + ^x(x + 2t1o) 1 3 J(x) = ( x2(x + 2TiQ)2dx = i ^x(x + 2tiq) .2 ^0 (x + 2Tio)'^ - ^(x + 2no) 3no' _ Ilo! log (no + ^ + \/^ + 2tiq)x 2 ^0 1 1 K(x) =1 x^^x + 2t1q^2(Jj^ |^x(x + 2t1q)(x + tIq) ^0^ Tip + X + ^/x(x + 2t]o) 2 ^ no Hence U.Q^ ■^^max pn NrT^o(no - f)(i - no) - no^iog ^ ^ ^Q ;^^^ ^ ^^Q (rv.30) One will find the corresponding curve in figure 68. 196 NACA TM 1354 4.1.1.2.2 - Case where v > 1 .- We shall begin by examining the case of an infinite half -wing inside slip (compare figure 69). It is convenient to put V = 1 -^ 2c The flow is obtained by a superposition of conical flows symbolized by c"g(M,da,c) (IV. 31) 00' with Cg(M,da,c) designating the elementary flow investigated in section 3.1.2.2. in the case where b = 0. If one puts ^ -^^ (IV. 32) x-l + vy 1 + p the pressure coefficient is given by the formula (ill. 23) which may also be written C^ = ^^ log P 3rt sjv ,2 cp c - p This formula is valid for |p|l, one has Cp = 0. One sees immediately the essential difference compared to the cases investigated before: a conical flow with the vertex (^0) can influence a point (x,y^) for which x < |. In particular, the trailing edge will play a role in the calculation of the pressure. Finally, if x = I, p = c, the C^ of the corresponding conical flow becomes infinite. If the method remains exactly the same, one must also expect a few addi- tional difficulties. The pressure coefficient at a point of the wing will therefore be written i NACA TM 135k 197 Cjx,y-) = —^ lX+(V-l)yX log cp p is of course defined by the equality (IV.32). The Cx of the section y^ is then written c - p da(l) ^x(y") = 3n^ pn Jv^ _ i*-'0 1 nt+(v-l)y^ a(t)dt / log 1 - cp c - p da(0 One will notice that, for y = 0, p = 0; and consequently _ / ^v 2a(x) , 1 2a(x) ^ Cp(x,0) = ' log - = ' log P« x/v^ - 1 3«\/ v^ - 1 Iv + Jv2 - ij As in the case of a wing of infinite span, the c^ depends only on the local inclination of the profile. Likewise (IV. 33) (v2 _ 1 The calculation of the function Cx(y^)> ^'-'^ ^^ ^ '^> presents no theoretical difficulty whatsoever. We shall now calculate the drag of the infinite half -wing, and shall show that it is finite in spite of the infinite dimensions of the wing. Assuming X to be this total drag, we shall put X = I p|u|2Cx Our purpose is the calculation of C . 198 NACA TM 135i+ The desired value of C^ will be the limit, if it exists, of the integral K) = np^^ >Yq^ nl nx+(v-l)y^ dy^ I a(x)dx log v^ - 1 1 - cp c - p da(l) when y -^ increases indefinitely. In order to calculate this triple integral, we shall replace the ensemble of the variables y-^,x,| by the variables x,|,p; the func- tional determinant —^ — - — - — ^ is equal to D(x,^,p) dyf ^ 2c^(x - l)(l - p^) ^P (p - c)2(l - pc)2 This expression one obtains from equation (lV.32) if one writes this equality in the form y^ 2pc(x - I) (c - p)(l - pc) The volume in which the triple integral must be calculated is represented in figure 70- One can write NACA TM 135l<- 199 -=1- a Ti w >.— X a N o o w 1 y — ^ w H a ■-^■^ c^a 1 y— ^ o H ^ — ^ 1 Q. — a o a 1 H o taO o H UJl 1 O a 1 I X a Q. o I H a o a H o o H X! 1 CO 0) •H -P 0) -p .a > •H ■P O a> ft CO (U cd X I -d § I AJJI A X O >5 o + I X X A O -d 1 (U a (M •iH O Ch o a 1 — ' (U Q. a -d CV] + 00 X o > I X CV) H a O a (U Ti CO ^ ^ 0) cd cd 9 ■^ o > 4J 0) ■P o o ^ a 0) 0) >» ^ ,C! H -P -P •s 0) CO 7i (fH tJ o a* CO •H g •H »^ -P ^ -p •H ,Q cd o '^ ^ fl -P -p O o ■H 0) ro CO ^ 0) rH •H +J CO > a; CU 0) u xi ^ M o B gj •H 0) fl -P ^ o • •H >j CO (U >> +J C tS CO •H Oj ?H C ^H 0) •H -P • '2 'a 0) CO d 0) 3 •H ^ ^ C +J H TJ cd -p tl 0) > fl Al cd cd 0) -p O ^ 4J CO -P cd a o a o -p •H O 0) H P -p H CD > ^ -P -P •iH Q) X to Tj >J c CO ri P t>j o •H & '^ (U .n CO 0) CU o i cd (rt CO 3 H H •H CO P^ CO OJ O -P cd ^ -P •H CI (U OJ +3 rl 43 cd 03 P -p ^ • •\ H H rH o cd Q. H ?-l •H bD > ■r) 0) t:) ^ -p rt S g C cd O o -p H ^ -rl r- Q. +J rH 200 NACA TM 13514. larger than c^ the triple integrals of equation (IV.3^) will be zero because I a{x)dx I (x - i) , the edge BB' does not influence the point 0' . '^ 1 + V In this case it may be easily shown that the contribution of the flow (equation (rV.36)) to the total drag is zero. In fact, this contribu- tion is proportional to 1 ni nx-(i+v)y'^ dy'^ a(x)dx log ^^(l+v)y'^ 00 1 - cp dcL(i) if one puts y'^ = \ - y^ NACA TM 155^ 205 One may make the change in variables used before which consists in replacing y,x,l by x,l,p; one obtains a(x)d^ r da(|)(x - I) r ^P'^ ^l - '^^^ , log 1-1 (p - c)^(l - pc)^ 1 - cp c - p dp which is evidently zero. This justifies a remark of Th. von Karman (ref. k) . For wings of high-aspect ratio, one may adopt, without large error, the formula (lV.55) for the total drag. The calciilation of the drag of an infinite sweptback wing (fig. 72), on the hypothesis that V > 1, is perfectly analogous to the one just performed. It suffices to replace, according to section 3'l-2.2, in the preceding formulas log _C£ by log 1 - cp + log 1 + cp C - p c + p Since log 1 + cp = log 1 + c2 c + p 2c 2 (1 + c2) it is sufficient to combine the expression p2rtjv2 - 1 1 + c _ = 2 2 It cos27 sin 7(1 - m2cos27) 17^ with the coefficient of the double integral of the formula (rv.55)- However, one thus attains only the drag for half the wing (x^ > O) ; one must therefore multiply by 2 in order to obtain the desired formula _ h_ cos y 1 + 2 sin 7 - M cos^y rt sm 7 (1 - m2cos27) 3/2 a(x)dx r (I x)log|x - ||da(|) (IV. 37) 206 MCA TM 1354 According to the remark just made, this formula gives, for a sweptback wing of high-aspect ratio, an approximate value of the total drag^5. We shall borrow from the memorandum Th. Von Karman's the figure 75 which illustrates the usefulness of the formulas found above for the study of the variation of the C^^ of a sweptback wing of high -aspect ratio with the Mach ntmiber (the profile is rhombic, the sweepback angle 7 = ^5°j a^id. the reduced aspect ratio rig = ^)' We obtained in the course of this investigation the value of the C„ (point A of nnax the figure) by the formula (lV.29), and the portion of the curve from B (formula IV. 27). The dotted part at the right of the abscissa M = \f2" is calculated by that same formula. One sees that it indicates also the behavior of the exact curve. Finally, for the values of M < \/2", the dotted part corresponds to the formula (lV.37)- It represents a good approximation of the rigorous values, except for the immediate surroundings of M = ^2". Here we shall stop the investigation of "symmetrical" wing prob- lems. One sees that this method leads to simple results and that the calciilations are always elementary. The field of application may easily be extended to more general cases (trapezoidal wings, leading edge cur- vature, etc . ) . i+.1.2 - Lifting Problems Study of the lifting problems is generally more delicate. In fact, the boundary conditions furnish on the wing the values of w, but out- side of the wing (in the general case) w is different from zero; on the other hand, continuity of the pressure is required which leads to supposing (in pursuance of the hypothesis of linearization as noted in chapter III) that u = in the plane 0x-|_X2 outside of the region (R) occupied by the wing. The difficulty lies in the fact that, in the general case, the boundary conditions bear up on two of the velocity components . 4.1.2.1 - Problems where the condition u = may be replaced by w = The rule of "forbidden signals" permits to define a general class of lifting problems where it will be possible to replace the Compare appendix No. 5- -NACA TM 155^ 207 condition u = by the simpler condition w = 0. This will be the case for wings ^ the projection (R) on 0xj_X2 of which will satisfy the following condition: With C designating the contoiir of the plan form (R), the tangent to (C) forms, at every point of (C), with Ox^^ an angle which is larger than the Mach angle . Naturally, such a contoiir (C) will present angular points. It is understood that, at these points, each of the semitangents must satisfy the condition stated. For the sake of abbreviation, we shall say that this contour is entirely supersonic. Let us consider a point M of (R) . As we remarked in sec- tion 1.1.^4-, the state of the fluid at M depends only on the perturba- tions inside of the Mach forecone of the point M; this forecone cuts off, in 0x-|_X2, a portion of (R) on which w is given, and a portion of the plane 0x^X2 in which the general flow is not disturbed (sec- tion 1.1. 14-) and on which u = v-w = 0. In order to calculate the press\ire at the point M, one may suppose that w = outside of (R) . One may also say that, under these conditions, the upper and the lower surface of the wing are independent. The solution of the corresponding lifting problems is therefore perfectly analogoiis to that of the sym- metrical problems visualized in the previous paragraph. Let us assume, for instance, a flat plate of the plan form indi- cated by figure 7^^ with the contour (C) being entirely supersonic; the pressure a.z every point of this plate has been calculated in chapter III. We intend to calculate the total C^. One has obviously if one puts PXp X = — ^ = p tan cp T = (M with S being the area of the region (R) . Let us put furthermore A = p tan oj-L u = p tan wq P(x) = —^ = ^^°^'^9 P(x) depends uniquely on the trailing edge B'AB. 208 MCA TM 155^^ One then obtains the formula '^ = -sJ, ^P C^(x)P(x)dx (IV. 38) Let us recall that 2i 1 P sin 9q if 1 < X < ti I , , Pi / 1 cos Ba - X T X - cos Gt C (x) = {- —\—r^ Arc cos ^:! — + _ ^ _ Arc cos -■- Pit \sin 9q 2i 1 - X cos 9q sin 9-]_ P sin 9-i_ 1 - X cos B-^ I if -1 < X < +1 if A < X < -1 with i designating the incidence counted according to the usual con- vent ions . In a recent memorandum^ M. Snow (ref. 32) has applied this method to the calculation of the total C^ of a plate in the shape of a quadri- lateral. We simply want to point out that^ in a certain number of cases, it is possible to calculate the integral (IV.38) very simply. This simplification becomes apparent when P(x) is analytic. It is then possible to use integrals in the complex field (variable z or Z) . Let us suppose to begin with that the contour B'AB is rectilinear and that its polar equation is written r = sin(cpQ - cp) OA = Z = sm ^0 Xq = p tan cpo 1^2 tan^cpQ(tan cdq - tan cjd-j_) ^2 ^ (^ - A) 2 (tan cpQ - tan mg) (tan cPq - tan co-A 2p (xq - l-'-)(xo - A) NACA TM 135^ 209 P(x) = ^^o" cos2cPq(x - Xq)! and consequently C, = - ^^0^ r^ ^p(^) ^^ ^ gP^O^ f ^ u(x)dx S cos'^cpoJa (^ - ^)' S cos^cPoJt^ (^ - ^)' 2pr. P ' — S cos cpQ '^ U(z)dz 'a (^ - ^0)' 2pro^ R S cos cpQ (iTtRo) with Rq designating the residue at the point z = xq. However, Ro dU ^ dW dz -i^ (^l - A)xc (--0) ^f^r^ '^(--0) '^^ (xo - A) (xo - ,)^;^7^ and 2rr (^ - A)xqWo 2z2xo2 (^ - A)xoWo S cos2cpo (^ - A) (xo - ^x)^xo2 . 1 P^g ^^ _ ^^ ^^^ _ ,)^,^2 _ ^ or Cz = I^Wq Xq sm cpo ^ 'xq2 - 1 ^ ^M2sin2cpQ - 1 (IV. 59) 210 NACA m 1354 The C^ is independent of odq and of a)3_; this generalizes a result already found^° in section 3.2.2.2. Let us now suppose that the arc BAB' is an arc of an ellipse with the polar equation a2b2 op op b'^cos'^cp + a'^sin^9 and let us, for simplification, assume that a^ = -a>]_. p2b2 + a^x^ e n pa2b2 p Cp(x)dx 2pa2b2 ^Z S J^ p2^2,,2,2 s -L y S U(z)d2 p2b2 + a^zS R2_ being the residue at the point z = i — • In order to calculate this residue, one must know the value of U, for z = i £^; this value is very easily obtained from the for- a mula (ill. 51). One finds "(^f) = 2w, jtp sin 9 Arc sin a sm \|a2 + b^p^cos^e In a general manner, one can obtain the C^ of a wing, the sur- face of which is a portion of a cone bisecting the Mach cone, with the vertex and a rectilinear trailing edge by measurement of the electric intensity in the tank. This result may be extended to the case where the cone is placed in any arbitrary relation to the Mach cone of provided the trailing edge is rectilinear. NACA TM 155^ . 211 On the other hand S = ab Arc tan bp cos 9 Thus, if one puts Wq = -i (incidence) n ^i n • a- sin 9 /-nr |,n^ C^ = Arc sin - (IV.40J 6 sin 9 Arc tan " L2 . ■u2o2^_^2o bp cos 9 ^a + b p cos 9 So far we have .visualized only the case where the flow on the plate was conical due to the shape of the leading edge. To terminate these few remarks about the flat plate of supersonic contour, we shall now examine the case where the leading edge is curvilinear. We shall start with the case of a polygonal leading edge (fig. 75)' The investigation is based on the following remark: if one superposes at a point A-^ two elementary lifting flows, which completely bisect the Mach cone of Aj^ and the first of which has as bounding generatrices Ai A, A2_D]_, so that w = -wq on (AA]_Di) , while the second has as bounding generatrices A-]_ A, A-^B-^, so that w = wq on (AA-]_B-^^, one obtains a resultant flow of such a type that , if ^±7^ and ^x'^'l ' ^^^ the sections of the Mach cone of A-^ in the plane 0x-|_X2, w = out- Bide of the angle (^lAiDi)' whereas w = -Wq on that angle; on the other hand, u = outside of the angle (^I'-^l^l)* Besides, one can easily verify that the resultant flow thus obtained is independent of the generatrix A (provided, however, that the latter is outside of (y'^±7±))) and that, if one puts as usual cos 9q = cos 9-1 = = P tan (Dq P tan a>i_ the pressure coefficient is equal to 2^0, Isin 9-]_ sin 9q on (riAiBi) 212 NACA TM 135^+ and to 2wn/ 1 cos 9-, - X -i cos Qq Arc cos - — Arc cos Pn \sin 01 1 - X cos Q-^ sin 9o 1 - x cos Sgy on (VAi7i) X represents as usual a semi- infinite line inside of (7i'-^i7i)* We shall note the resultant flow c (Ai^-Qq^^o - ^l) The flow about the plate schematized in figure 75 is then obtained by superimposing on the conical flow of the vertex and the bounding generatrices 0D-[_' and ODj^ the flows 'c^(Al.9o^9o - ej and t{A-^' ,Bq' ,Qq' - Q^') with Qq'^ ^l'^ ®0' ^1 characterizing the directions of the straight lines 0A-|_', ^l'^'^ '^^l' %^1' If the leading edge is curvilinear (fig. 7^), let us assume A X2^(t) ,X2(t) the point moving along this leading edge, oj(t) the angle between the tangent at the moving point and Ox3_, and let us put cos 9(t) P tan (n{t) Assuming MTx-l^x^) to be the point where one desires to calculate the pressure, one will put Xp(t) - Xp x(t) = p ^ ' ^ x-|_(t) - X3_ NACA m 155^ 213 The flow will be obtained by subtracting the flow symbolized by / c|A(t),e(t),de(t)l from the flow around a plate of infinite aspect ratio, with the leading edge 0x2 ' If MA-[_ and MA2 are, in 0x-|_X2, the two semigenerat rices of the Mach forecone at the point M, one has therefore as value of the pres- sure coefficient at M by putting de 1 Arc cos . cos e - X sin 9 X cos 9 = F(9,x) 2wn Cp(M) = -f F[e(t),x(t)] d9 (IV. 41) At a point such as M' (compare fig. 76) a slight modification of the formula will be convenient; one must write C.(M-) = '"0 P sin 9-[_ Pit 1^ [e(t),x(t]] de One thus obtains the C_ by a simple integral. We shall point out a very remarkable result for the total C^ of such a plate when the trailing edge is rectilinear. We shall show that the Cg of such a plate depends only on the trailing edgej this fact generalizes the result of the formula (lV.59)' It suffices, of course, to demonstrate the result in the case of a polygonal leading edge; thence the general case is deduced by passing to the limit (fig. 77)- According to the formula (17.39)7 "the resultant of the normal forces due to the flow C (A-|_,eQ,9Q - 9-l) 21il- NACA TM 135ij. acting on the region (R) is equal to that of the normal forces acting on the triangle A-]_B-lD-|^ in the conical flow with the vertex and the bounding generatrices OD-j_, 0D-[_' . The result stated above results from this remark. Thus one verifies that on this plate the total C^ is the same as if the direction of the flow had been reversed^T. 4.1.2.2 - Infinitely thin rectangular wing We shall now investigate the case of a rectangular wing, the pro- file of which is an arc segment (fig. 78)- In accordance with what was said before, this arc segment will be defined by the angle Jo(^l) which is formed by the tangent and the chord at the point with the abscissa x-^; if the wing has a geometric incidence defined by the angle i, we put j(xj = Jo(xi) - i (lV.1+2) w must be equal to J (x-, "^ on (R), and u must be zero outside of (R). We shall designate by C (m ,a) the lifting elementary conical flow, with the vertex M, which furnishes the value w = a on the two faces of the quadrant M, x-^, x^' With the notations of figure 58, the formula (III.60) is then written C_ = 2°[: Arc cos (1 - 2p tan cp for < p tan cp < 1 Cp = 2^ for p tan cp > 1 P (IV. 1^3) By an argument analogous to the one of section 4.1.1.1.2 we are induced to define the desired flow by the symbolic notation 'One finds here anew a remark made before by M. Snow (ref . 52) in a particular case. Besides, this result may be extended without great difficulties to any arbitrary plate of supersonic contour. NACA TM 135^ 215 Cp(M,dj) + J Cp(M,dj) - e[j(x-l)J (IV.44) 'AA' '-'BB The flow thus defined does satisfy the conditions concerning v; however, one sees Immediately that the flow gives a component u, zero outside of (R) only in the case where the aspect ratio pA is smaller than or equal to 1. The limiting case pA = 1 corresponds to the dis- position of the Mach cones given by figure 79- We shall use here^° the hypothesis where pA ^ 1, and shall then be able to calculate the flow by the formula {lY .hk) . 4.1.2.2.1. Study of the flow / C (M,dj).- We shall use the ^AA' same notations as in section 4. 1.1.1. 3. According to equation {IV .k3) , the pressure coefficient C^ at a point (x^^y^) is written ^ (0 < x^ < 1) ,xX Cp(x^,y^) = I dj(|)=|j(x^) if 0 y^ It is n6t impossible to investigate the case where pA < 1. One must then superimpose on the flow given by (equation {TV .kh}) other conical flows, the vertices of which describe the two edges of the wing, in order to establish pressure continuity without changing the w value on the wing. This investigation is clearly more complicated than the one we shall make . We shall not enter on it in order to limit ourselves to the simplest results. Further on (section i+.1.2.3.2.) one will find an application of this method in a special case. "strictly speaking, the slope of the wing shoiild be noted j^(x^) when one expresses it as a function of the reduced abscissa. We shall omit the asterisk in order to simplify the notations. 2l6 NACA TM I35I+ These two formulas may be written with R(x^,y^) = j(x^ - y^) - - I Arc cos [1 - -^^ 'dj(l) \ xx - ^^ (IVA5) stating that the function J (x^) is zero outside of the interval (O.l), It is then easy to calculate the local c^ of a section y^ with this coefficient defined by c,(y^) = -2 J Cp(x^,y^)dx^ Remarking that j(x^) = Jq(x>^) - i and putting '0 one has z - ^ ■ ^ J" ^{^'',y^y^ P Pjy^ Now NACA TM 135^ 217 / R(x^,y^)dx^ = / j(x^ - y^)dx^ dx''^ I Arc cos '^JyX Jo f - ^)^^<^' f(l -y^) - i(l - y^) Arc cos (1 — — idx-^ V x^ - W However, '1-1 'y^ Arc cos (-¥: du = 2 (1 - |)Arc sin y^ 1 - I Jy^(i - y^ - 1) ity^ Thus we put ^yM) . ^ (1 - |)Arc sin y^ 1 - I ^y^{^-y^-i) k(yx,^) =2(1-1) if y^ < 1 - I if yx > 1 - I (IV. 46) 218 NACA TM 1554 J R(x>^,y^)dx^ = f (1 - y^) - i (l - y>^) + y^[-i + Jo(l - y^J] J^i-y^ k(y^.|)dj(i) = -i + i k(y^,0) + f (1 - y^) + y^Jo(l " y"") " = -i + i k(y^,0) + pQ(y^) with (IVA7) Consequently (y-) = ^^^4^ i + ^ Po(y^) (IV A8) p p One will find in figure 80 the curve giving the variation of k(y^,Oj and of k(y^,l/2). We remark that ayx ^ k(y^,|)dJo(l) = f djo(i) r k(y^,i)dyx 1/^ (1 - l)'^Jo(^) NACA TM 155^*- 219 because 1-1 k(y^,|)dy^ = l(l - 1)2 n However (1 - D^dJQd) = 2 I (1 - I)Jq(I) = 2 / f(x)dx = 2n- putt ing r '^^ )dx = [1. On the other hand / f(l - y^)dy^ = / f(x)dx = m do Jo and i J y^Jo(i - y'')^^'' = J (1 - t)Jo(*)'^* = - I ^JQ^*) dt = (i Consequently r p p (IV A9) 4.1.2.2.2 - Study of the thin rectangular ving in the case vhere pA > 1 . - As we have said in section i|-.1.2.2, one can apply to this case a method analogous to the one employed in section 4. 1.1. 1.^4-. The pressure coefficient at a point of the wing situated on the surface x^ = +0 of reduced coordinates t, i], can inmediately be written s(*^^) = I Jn(t) 1 - R(t,TlQ + T)) + R(t,TlQ - T]) 220 NACA TM 1554 Consequently, the local c^ of the section t\ is obtained by the formula c^Cti) = -2 / C^(t,Ti)dt = ^^pp'^l[^(^o^ ^^°) ^M^o - ^^0)] -^ Po(^o-' ^) ^Po(^o - ^) or cz(^) = f\^{% + ^'°) + k(^o - ^'°) - 2] + ^|po(^o ^ ^) ■*■ Po(^o - ^)] with the functions k and Pq being defined by the equalities (lVA6) and {IV .k^) . Finally, let us calculate the total C^ Cz = ^ p c,(,)d, = - ^ -^ 1^ r''° k(t,o)dt . jl. po ^(^^ dt and since 2tio = 1 + PA - 1 one has, applying the results established at the end of the preceding paragraph, c^ = - ^ + ifii + ^ + OA - 1)^ because k(t,0) = 2 if t > 1 NACA TM 155^ 221 whence ^ p V 2pa; 02. Figure 81 gives the variation of = — =• , as a function of M, \2 di / for various values of A^*-*. One may also plan the calculation of the drag of this wing. First of all the local drag or Cx(^) = ^^^(ti) + 2 / Cp(t,Ti)Jo(t)dt Jo ^^(ti) = ic^Cri) + ^ Jo^ - ^^i\ + ^) + T(t1q - Ti)j (IV. 51) putting T(y^) = r R(t,y^)jo(t) JyX dt = if(y^) + J Jo(t - y^)jo(t)dt - '^ A. Bonney has already obtained this formula in the case where Jq = 0, n = (rectangular flat plate); compare reference 53- NACA IM I55J+ whence T(y^) = if(y^) + ii I Arc sin j^ JQ(t)dt + I Jo(t - y^)jo(t)dt Pl-yx nl Arc sin \|t - i ^^— Jo(t) the 226 MCA TM 1554 -> flow T(M,a) is identical with the flow C3(M,-a); in the region -> Xj < it is identical with tlie flow Cg(M,a). One can immediately make an interpretation of the flows T which gives account of the possible utilization in the effects of flaps and ailerons;51 the flow T(M,a) is established when, after the plane Oxjx^ has been materialized, one makes the quadrant Mx-lX2 pivot aroimd Mx2 by an angle -a (fig. 58)- Hence the investigated flow may be obtained immediately by supei~position of conical flows schematized in the following manner —> C_(A,-i) C^(B,-i) E(AB,+i) Cp(C,7i) T(D,7i) E(CD,-7i)> Cp(E,72) T(F,72) E(EF,-72) (IV. 56) If such a scheme is to be valid without further complications, the pressure coefficient outside of (R) must, of course, be zero. This will be the case if the reduced aspect ratio of the plate and the flaps is greater than, or equal to 1. Let us apply these principles to the calculation of the local C^ of a plate for which the Mach cones of the points A-|_, B, C, D, E, F are disposed as shown in figure 8^+. One may then place the origin at the point A and immediately write the local C^ as a function of yX^yX _ p,x^ ; one will put AA' = 1, CA' = c, pCD = 2, according to we have indicated this method in a note to the reports on the proceedings of the Academy of Sciences in December 19^7 (ref. 37)- The advantage of the flows T we indicate here has also been pointed out in the article of M. Snow, published at the same time as our note (ref. 52). NACA TM 135^ 227 the results obtained in sections 4.1.1.1.5 and. ij-. 1.2. 2.1: pn y\ c Arc sin — + Jy^(c - y^) i Arc sin ^ + Jy^ (l - y^) if yx < c 2 p 7-1C + ^ Arc sin Jy^ + ^y^ (l - y^) c < y^ < 1 ^z =^(i + ric) 1 < y-^ < z - c ^7 = - 7-,c 7-,c i + -=— + -=— 2 rt Arc sin ^^ + (z - y^^Arg ch I - y> Z - c < y^ < c c =i 1 + 7-^c 7-^c Arc sin ^ " ^ + (y^ - zWg ch yX _ I Z < yX < Z + c In figure 85, one will find the distribution of c^ over the span. Besides, it will be possible to write in a general manner the local c^ of any slender rectangular wing provided with flaps or ailerons. 228 NACA TM 1554 In fact, if one puts ^^f(' 'Arc sin — + u Arg ch -2- c lu f(u,c) = c f(u,c) = f(u,c) = 2c one has with the customary notations if -c < u < c if u + c < > (rv.58) if u - c > S^) = fl{% ^ v,o) . k(no - r,,o) - 2] . ij^,Q . n) ^ Pq^Io - ^)\ 27 -J^[k(TlQ+ TlA - C) - f(TlQ - Z + Tl,Cy| + 27c k(TlQ - Tl,l - C) - f (t]q - I - T1,C)I (IV. 59) The total C„ may be easily calculated. We remark for this purpose that 2t1q pc p2T)^ k(u,l - c)du = / k(u,l - c)du + ( k(u,l - c)du do Jc = lc2 + 2c(2t1q - c) The mean value of f is very easily obtained whence C, = ^Cl - ^U Jiii + 2 ih.llA(2cl - i A ^ P V 2PA; p2^ ^^-K \ 2 ; (IV. 60) One also sees that the calculation of the moments does not present any difficulties. NACA TM 155^ 229 4.1.2.3 - A few remarks regarding the study of the effect of sveepback We cannot here develop a theory of the sweptback wing. We there- fore shall content ourselves with a few remarks . 4.1. 2.3 '1 - Study of the sweptback wing with "supersonic leading edge" (p cot y > 1) , compare figure 66 .- This investigation does not present any difficulty in the case where the reduced aspect ratio. t]q is greater than . We reassume the notations of sec- 1 - cos 9 tion 4.1.1.2.1; let j(|) be the angle defining the infinitely slender profile of the wing supposed to be constant over the span fj (1 ) = JQ^^) ~ D' ^^® flow will be obtained by superimposing as before: (1) Conical flows bisecting the Mach cone, centered on 00'. (2) Lifting flows centered on AA' and BB'. (c) Finally flows about the wing of infinite span with a fin with the same profiles as the wing profile and leading edges which coincide with OA and OB. In order to simplify the investigation, we shall assume that the Mach cones of the points 0, A, B do not interfere with the wing; this will permit one to study separately the "head effect" (conical flows centered on 00') and the "end effect" (conical flows centered on AA' or BB ' ) . The "head effect" can be investigated immediately, according to the formulas of section 4.1.1.2.1. The pressure coefficient on the surface x^ = +0 is written Q being defined by the formiila (lV.2l) in which a(|) has been replaced by j(l). I 250 NACA W. 1354 -p •H CO •H ^ (U •H O •H tH tH (U o o -p I o o "-3 CD W o o X + I X CD a •H W CD m o u O O OJ + CO o o CD a •H to ca IS) o -p si to •rH V V o o •H •P c •H tsl O O H !> i I X CD •H ca 00 CD cl •H W ca to O O •H ca CO CD •H CQ ca ■ 1 l| CD 1' (^] CQ t;|OJ CJ + CD CD ca OJ -p CQ Tj -p 1 --CD - H 1 t, O ? a •H M oa a •H M ca X o o H P( 0) 1. VD OJ -P -P •H CM 1 o o bO a ■H ■H (U 1 o -P -P -P o C O o 0) -p u (U V V o cl o •H bO 0) ^1 -p bO ■p Ch O 0) oi > OJ w o o + •JUI w o o X >! cu + •1/1 I ■^ I •H CQ CO CD a CO ca CJ , CD l-n) C + •rH CO OJ •H ca CO OJ + OJ CD c •H (0 ca CD d •H W ca OJ •H CD O O I OJ 1-^ CM M ca CD a •r-l ca C! >>> •rH H t:! -P a) d hO (1) 4J i 0) > w fl cd •H ^ m 0) o o H <+H -^ (U ^^ bO -H a o §^ o -P So O -P o Id !>» OJ ,C3 ^^ U z j • g OJ 0) fl d O O •H -P o 0) w o o CM •H w ca- ft o 252 NACA TM 1354 R being the function defined by equation (lV.5^)' Consequently C7. y xN 2k(y^(l + cos e),o) . ^4- r_xf^ P sin e P sin e '0 the mean value of c„ in the interval < y^ < is written ^ 1 + cos 9 C - ^i I ^^^ ^ P sin e p sin In the same manner, one obtains without any difficulty the value of the local drag cx(y") =-z(y") + ^T^To' k P sin 9 "^ p sin 9 T[y^(l + cos 9)] and its mean value in the interval < y^ < 1 + cos 51^ k - 2 + Jo P sin 9 p sin One may summarize these results in the following manner: we con- sider a wing of an aspect ratio equal to 2i]^ (fig- 86); the total C^ of this wing is written kl P sin 9t]q 1 _ lM(i - cos 9) 5i 1 - cos 9 T]^p sin 9 1 + cos 9 Tio sin^9 - 2 2^1 Ti_P sin 9 . 2„ P sin 9ri_ 1 + cos '0 sm 9 ^ '0L_ 1.1(9) or Cz = P sin 9 L_ '0 1+Ti„\l + cos 9 n 2ki PtIq sin J+i(e) 1 + cos 9 (IV.6I1) NACA TM 135^ 253 Likewise, for the total drag -.'A -' - Jo^) 1 i4-i(e) .2 or tiqP sin 9 1 - cos 9 ^n sin 9t]q 3i^ , k 7 2 + Jo P sin 9 p sin 9 l|i2 ^(i^ + Jq2) tIq sin^e - 2 (l + cos 9)t]q P sin 9t]q sin29 P sin 9 ^Tlo 1 + cos P sin 9 (IV. 65) These formulas remain applicable as lone as Tin > = • " 1 - cos 9 4. 1.2. 3 -2 - The study of the sveptback ving with a supersonic leading edge when ^q < — 1 cos 9 , or with a subsonic leading edge, presents more serious difficulties .- A complete investigation of this kind would lead us too far. We shall content ourselves with treating a simple example which will show how to proceed in order to sunnount the difficulties. We attain this aim by introducing conical flows which we shall denote S(M,to,Uo) defined in the plane Z by the following boundary conditions (fig. 87): (1) On (Cq), u = V = w = 0. (2) On OA, w = 0. (3) On the upper edge of OC, u = Uq • On the lower edge of OC, u = -Uq . p Uq is a given constant, the point C is the image of the number Z = -a ; one puts as usual '0 2a^ 1 + a^ 25^ NACA TM 155^^ The methods of chapter III permit one to write very easily the function U(z) the real part of which gives the component u of such a flow; if one puts Z = s2 one has U(Z) = - ^ log |i-:-J^ 1 + ia^ (lV.66) 7t [s + ia 1 - iasj One verifies readily that this flow satisfies all boundary conditions, Besides, if one puts t = 2sg 1 + s^ one has on OA ^ 2to(l - t) Uf) u = ± -^ Arc cos It 1 t + tr = ± — — Arc sin n \| t + tg i tod - 1) (IV. 67) These flows will enable us to make the pressure discontinuities appearing outside of the wing disappear, without modification of the boundary conditions on the wing itself. Let us take for instance the case of a plate of the plan form indi- cated in figure 88. With y being the sweepback angle, one will put as usual P cotan 7 = cos 9 One assumes that the Mach cone A does not intersect the seg- ment 00', but that the Mach cone of does intersect the segment AA' at the point Mg. According to what was said above, one will obtain a flow which satisfies the boundary conditions on the wing portion y-^ < by super inclosing a conical flow of the vertex and bounding genera- trices OA, OB, a flow of the vertex A and bounding genera- trices AA' , AO, and by subtracting the flow about a plate of infinite span with AO as leading edge; however, the region MqPo^' then is a NACA TM 1554 i 235 zone of discontinuity for the pressure. If MqP-]_ represents the other generatrix of the Mach cone of Mq in x-^Oxq, the pressures obtained in the region MqA'P-]_ will thus be erroneous. One will obtain the desired result by superimposing on the preceding flow a flow schematized by r S(M,to,uo) 'JMqA' In this formula Uq = 2i a Pn sin e d| Arc sin sin 9(1 + t)q cos 9) ^i {i + 2tiq cos 9) dl I + TlQ COS if M is at Mq, tg = 1, i = tiq(1 - cos 9). The pressure coeff icient52 in the region MqA'P^ is given by the following formula (y^ is negative): 52 One will find in appendix No. 6 the explicit calculation of this pressure coefficient and a few important brief remarks regarding certain pecularities occurring in analogous problems. 236 NACA TM I35I+ OJl t: ca CO VD 1^ + w o o o p- OJ •H W O ^I;d 1 1 . — ^ ujl ^•^ — >» X n. >5 + + X p- p- UJ1 1 1 X 1 p- o < CD to o o I 0) OJ -p fl o -p -p O P- 1. O P* O -P ft CO OJ > OJ a o p 0) to O 0) ^ 0) CO o ^ o -P -rH o C H 0) Cd +:> ^ cd w C ■H IS S 0) -p • 0) Id Tj OJ c ^1 o O 0) w w •H ^ Ch -P OJ bO 4J -H -p -rH Ih cd X QJ C a N •rH Cd H -P S W Q, to ft (U 0) a -p ;3 ^1 o c u 0) CO -p n3 u o O Cm Ch 0) -P -p d cd ^ II -P (U 0) W 0) ^1 cd ^ o > lU OJ O cd o o t cd ft C O •rH -P Cd OJ bO a ■H ft c O cd 0) o >i 0) ,c ft •n o -p cd -p 0) 0) ■rH C! 0) M Td ■rH (D C O ID H & U 0) ■H CO o u 0) X! 4^ to OJ u l^_^i \ + ^p^ + 1 ^0 P- 1 X 1 p- Ci •H (0 /o ft o NACA TM 155^ 237 The integral of the second term represents the "end effect" of the wing AA' while the first term represents simply the pressure coeffi- cient in the conical flow with the vertex o55. As an application, we shall calculate the total C„ of this wing C. = PltT)^ '^0 dy^ 1 (x + y^)dx ^ 1 Pl Jx (x + 2y^) " u J 11 px-z dx I F(z,x,0(i| z UO 5^Had one wanted to study directly the case where 9=0 by appli- cation of the preceding method, one would have been led to write c - -iti X - y^ f (x - 2y^) .x-2 (riQ+yX) 2 Pare sin '0 ^0 X - I - 2(tio + y^) no^ + y^^ i + \ ^{^ ^2,o) di However, the integral of the second term has no meaning since the dif- ferential element is in |~^' • In order to eliminate this difficulty, one must utilize the conception: "finite part" of an integral intro- duced into the analysis by M. Hadamard (compare ref. 7)- One has in fact >x-2 (riQ+yX) Arc sin i ^0 X - ^ - 2 (iio + y^) 'J S ^ + ^0 T] X + yX| ^^ nI^ (^ - ^\) di = ^^-'(V^') 4..0 ^i (i ^ 2,0) ^^ [ \ Arc sin ^0 X - i - 2(tio + y^) t]qX + y^l di This justifies once more the interest in the motion of the "finite" part of an integral which permits a very easy performance of limiting process which may be delicate. 238 NACA TM 1554 if one puts 2(^0. y-) with F(z,x,4) designating the q^uantity under the sign / in the / formxila (lV.69). The double integral may be calculated immediately (compare the end of section i^-. 1.1. 2.1) ^tIq pi dy^ il X + y^ ^^ _ I ^^ I ^ X + yx ^rio '0 W^~^ dx = I dx ^0 ^x(x + 2yx) dy^ = Uo M ^ + 2t10 [2X+ T)^ dx - 1 = 1 3 5 (1 + 2no)^ - 1 As to the triple integral, one may write it changing the order of integrations 4 - z dz In order to calculate the last integral, one puts -t2 X - I - z It is then written t^(x - I) - (x + 2t]q)(i + t^) ^2 dt 2riQ(l . t2) ^ t2| (^ ^ ^2)' NACA TM 135^ 259 and may be calculated rapidly by residues. It has the value JtX e2 nN^^^ - 7^{' ^ ^^o) - 1 2i' Therefore, it suffices for calciilating the triple integral to utilize the following results Jo f(r^~^ ^^ ^ s^ n|2^ {i + no) (i + ^o) (i - 1^) d^ = 2 |5/2 + — I — ll0_ ^j,^ ^^j^ 5 J2^ r^o fiTT^ \1 I + 2T10 2i + Tio i 5110^ The triple integral thus has the value which leads to the following value for the desired C^ Cz = ^- Il^i^nol J5^ , (1 , ,,^).^, tan ^^ (IV. 70) One will find in figure 89 the variation of C„ as a function Of n^. 240 NACA TM 135^^ As an application J we have traced in figure 90 the variation of i- — - as a function of the Mach number for plates of the plan form 2 di defined by figure 86. The angle of sweepback is ^5°^ the geometric aspect ratios are, respectively, equal to 1, 2, and 8. The points sit- uated on the abscissa M = \J2 are obtained exactly [formula (XV-YOyi. The parts traced in solid lines are given by the formula (lV.6i+). The dotted parts are obtained by interpolation. In order to obtain them in full rigor, one would have to calculate the C^ from the formula (lV.68). i<-.1.2A - The Uniformly Lifting Segments The role played by the "horseshoe vortex" or "uniformly lifting segment" in the subsonic wing theory is well-known; the linear theory of Prandtl is based on this conception. We shall show how easy it is to obtain the corresponding supersonic flow, and shall indicate a few possible applications. According to section 5-2.3.1, the conical flow for which U(Z) = Uq + i ^ log — ^ (IV. 71) "^ 1 + Z2 represents a flow for which u has the value zero in the plane X:z = except on the quadrant Ox-^^, 0x2 where u assumes the values ±Uq. Let us then apply the results of section 1.3- The homogeneous flow of zero order, defined by the complex potential ^ (^l) -"-^ ^^^ function defining the meridian line in a plane r, x-]_. However, we shall see that Vj,^x-[_,r] is a function which is, when x-[_ is fixed, of the order — , for a small r. The boundary condition may also be written dr 1 dS rv^ = r dx-|_ 2jt dx-[_ with S ^x-| ^ = Ttr designating the area of the fuselage section of the abscissa X2^. If one makes r tend toward zero, rv^, will maintain a finite value. In a precise manner, we shall state that the investigated flow will have to verify the following boundary condition lim rv„ = i- ^^ ' (IV.7I+) r -^ 2n dxi 4.2.2 - Investigation of a Particular Case Let us consider the flow around a cone of revolution; the formu- las V(z), W(z), U(z) are functions of Z which admit inside of |Z| = 1 only the point Z = as a singularity. Thus they may be con- tinued analytically to the interior of the circle (C), image of the conical obstacle in the plane Z, under the condition of excluding the origin from this circle. After this statement we shall determine the flow around a body of revolution the meridian line of which has the simple form given by fig- ure 92- Qq naturally is an infinitely small angle. A first idea for 2kk NACA TM 1354 obtaining such a flow consists in subtracting from a flow around a cone of revolution of the vertex and the angle 9q a similar flow of vertex A. Let us put ^ ^ 1 + p^ Pr 2p Xn 1 + Pl^ pr 2Pi The radial velocity of the resultant flow is PSo' "r= 2 ^^-fe-- Let us assume that p and p-|_ are infinitely small which is the case for points M which are sufficiently distant from A Vj, ~ pe, 2r _1_ Pi ^[-1 - (-1 - -)] = a9r In order to obtain the desired flow^ it will therefore be necessary (which is^ besides, in accordance with the theorem of section 1.1. 3) to add a homogeneous flow of zero order with the vertex A defined by the complex potential $(Z-l) = -aBo^log Z-L with Z-]_ designating the complex variable Z for a flow with the ver- tex A fin parti icular Z 1 = pi) The resultant flow has for x-|_ > a the radial velocity (compare form-ula (I.29)) P9o' l-^)-t-'^ aeo2 l^Pi^/ ^^N Xi - a 1 - Pi Ply or _ peo^ 2pn - 2p - a 1 - Pl^)' 3r_ Pi (^1 - ^y NACA TM 135^ 2i+5 The obtained radial velocity is therefore not identically zero along the conical obstacle, but it is very small when x-|_ is not too close to a since p, p., and r are infinitely small quantities. For the rest, the equality (lV.7^) is satisfied for any value x-]_ > a. In first approximation, we regard the flow obtained as satisfying the conditions posed, although of course the value of v^, is not negligible if x-^ is close to a. Let us now suppose that we would want to study the flow around a body of revolution which has a meridian line schematized by figure 93' One is led to visualize the flow as a resultant of the previously defined flow and a conical flow of revolution of vertex A relative to the angle Q-^. At a point M situated on the meridian line (when the abscissa of M is distinctly larger than a), one has as the radial velocity fi 2/^1 ^1 - a\ 90 ^ V ~ Oq I \ r r 2. Q_2 ^ Qi (^1 - a) where r = (x-]_ - a') 0-|_ + aQg = r(a) + (x-^ - ^)Qj^ If one puts r(a) = a9Q r(a) designates the radius of the abscissa section x-|_ = a. Hence ei(r - a9o) _ _ r(a)9^ V ~ ^ •— ~ o-i r Since one must have v„ = 9-i, one sees that one must, moreover, add the homogeneous flow of vertex A of complex potential $(Z) = r(a)e-L log Z 2k6 NACA IM 135^^ Finally, the case investigated is obtained: (1) By adding a conical flow of the vertex relative to the angle 9q. (2) By adding a conical flow of the vertex A relative to the angle 93_. (5) By subtracting a conical flow of the vertex A relative to the angle 8q. (k) By adding a homogeneous flow of zero order of complex potential r(a)Ae(a)log Z where r(a) is the value of the radius for X]_ = a and AB(a) is the discontinuity of the angle 9 for x^ = a. i4-.2.5 - Approximate Study of a Body of Revolution of Fuselage Shape The application of the above said permits to obtain, in an approxi- mate manner, the flow about a fuselage -shaped body the meridian line of which is polygonal and, by limiting process, the flow about a body of revolution the meridian line of which possesses a continuous tangent. If one assumes first a-^, ag, • . . a^^ . . . , as the abscissas of the vertices of the polygonal line which constitutes the meridian, the desired flow will arise from the superposition: (1) Of a succession of conical flows which cause an axial velocity of the form (formula (II.23)) P^2 + 1 where 1 + p^^ xi - a^ 2Pn Pr with 9j^ being the value of 9 for a^^ < x-|_ < aj^^-|_; NACA TM 155^ 2^7 (2) Of a succession of homogeneous flows which cause an axial velocity of the form (formula (I.29)) _ ^n ^9n 1 + ^n ^1 - ^ 1 - Pn^ where ^n = ^W ^Qn =^K) = 9n " Vl However; p^^ will be very small except in the immediate neighhorhood of a.^, consequently one may expect the reduced axial velocity to be written ^ ' ^ ^1 - ^n ^- ^1 - ^n with the sums \ extending to all points k^ the abscissa a^^ of which is smaller than x-^ - Pr. The case of a meridian with a continuous tangent is obtained by performing the limiting process in the preceding expression which leads to u = - n'^' e2(|)d4 _ pi-P^ r(|)9-(|) ^^ Jo ^1 - ^ Jo ^1 - ^ However ; 6(1) = r'(0 r'2(|) + rr"(i) = ^ -^ S(0 2rt (i|2 if S(|) = nr2(|) One obtains -I -Pr p -^^ di (IV. 75) 214-8 NACA TM 135)+ I This expression is exactly the one given by Laitone (ref. 5); it is^ besides, equivalent to those suggested by the other authors named before . However, the argument just produced is somewhat summary due to the difficulties arising in the neighborhood of the points a]_, ag, • . . a . . . In the following paragraph, we shall justify the aforesaid, in particular the important formula (lV.75)' k.2.k - Justification of the Method The question is to calculate the radial and axial velocities according to the rigorous formulas, and to take the possible simplifi- cations into account only in the final result. The radial velocity con prises two terms, the first of which results from the composition of the homogeneous flows of zero order; the differential element of the corre- sponding integral is I _l_^(p . i)L±4 r(i)Ae(0 = I L±^ Hl)A9(i) 2 xi - i V P/l - p2 r 1 . p2(.) or ^[r(|)e'(i)d| + pr2(4)9'(|)dp if one assumes 9(|) different iable since r 2£!^ = pr dp 1 - p2 hence the contribution due to these flows to the radial velocity pXi-P pi r(i)e'(i)d| + r(i)e'(i)dp ^0 '-'o„ ^Pq p„ being the value of p{i) for | = 0. Likewise, the composition of the conical flow causes an integral the differential element of which is written NACA TM 135^ 2k9 y:^e2(|)A ^^— 2 d| P(l) - P(l) di 2 d| pr pe (i)dp Hence the desired integral px-j_-pr PI ^ e2(0d| + p 92(Odp '^ ^ J Po Thus the velocity is written ^ 2itr Xt -pr „ pl o 1 / -^ d2s(&) ^, ^ £. d^S(0 '0 di' 2n dp PO di^ The last integral is bounded "by the upper boundary of d^S(U and con- sequently the condition (XV-T^) is thus verified. The calculation of u is made by a quite analogous method and leads to the formula 1 1 d^S 2n ^1 - i d^2 dl '"'l"^'' r(|)e2(0 2p2(0 ^1 - ^ 1 - p2(0 d| d^S ^"^d^ -;^ 2P(0 /d^S xi - i 2« I l+p2(0Vd|2 PO dp Now it is quite obvious that this last integral is negligible com- pared to the first. Thus the formula (rv.75) is established. It fiirnishes the following approximation for the pressure coefficient ixi-pr C^ = - d!s di2 di (IV.76) 250 NACA TM 155^+ Remark . In chapter II, we had utilized the formula (l.lO) for writing the pressure coefficient. This formula would lead to write here Cp = - / ^—^^ - r'^(^) One will compare this formula with the one given in reference 36. Nevertheless, the analysis just made does not guarantee that the term r'' represents all terms of the second order; therefore, besides, in accord- ance with Laitone, we shall content ourselves with the formula (rv.76). k.2.'^ - Generalizations The method indicated above has the advantage not only of giving a new demonstration of the formulas relative to flows of revolution, but also of furnishing a more general method which lends itself to applica- tion to numerous fuselage problems. Let us take, for instance, the case of fuselages of revolution the axis of which is slightly inclined toward the wind direction. One may reassimie the preceding method, starting out from the flow about a cone of revolution inclined toward the wind (formulas (11.2^+) and (II.25)). The desired flow is obtained by suitable superposition of those conical flows and of homogeneous flows of zero order which one deduces from them by differentiating these flows in the direction of the axis of the fuse- lage (compare section l.^)- It is permissible to assume that this method will also permit the study of fuselages which are not bodies of revolution but the cross sec- tion of which remains, for instance, homothetic. Certain difficulties make their appearance, but do not seem insurmoiintable . In entering on the investigation of fuselages by the method of conical flows, we aimed only at indicating the principle of a new method. We reserve the devel- opment for a later report5o. -^Compare in appendix No. 7 the development of this idea. NACA TM 135^ 251 4.3 - First Investigation Regarding the Conical Flows Past a Flat Dihedral. Applications to the Fins and Control Surfaces. We have already indicated in the course of this chapter that there exist other conical flows than the flows with infinitesimal cone angles or the flows flattened in one direction. In this last paragraph, we shall give a few examples of flows past a flat dihedral. These flows may be utilized either for the study of the effect of dihedral on a lifting wing or for the study of the fins and control surfaces. We can here not consider developing the complete theory of these flows. We shall content ourselves with indicating a few examples. U.5.1 - Effect of Dihedral on a Wing Completely Bisecting the Mach Cone Let us consider a A wing having dihedral; this wing is infinitely flattened into two planes which intersect in Ox-[_. For simplification, we shall assume that the plane Ox-^x^ is a plane of symmetry, the wing completely bisecting the Mach cone; upper and lower sides are therefore "independents." This signifies that in the plane Z the region inside of (^o) ^^ divided into two domains (fig. 9^). The wing portion inside of the Mach cone (r) is represented by two radii OD, OD' which form with OX the angles 9q and rt - 9q. The bounding generatrices of the A have as images the points E and E' of the argument B-^ and rt - 9-]_ on the circle. One will assume, in order to better establish, the ideas, that < Gq < 9-L < .2.. The boundary conditions which permit determination of the unknown functions U(z), V(z), W(z) in the region ODEE'D'O are: f (1) On the arc EE' u=v=w=0 (2) On the arc ED and on the segment OD w cos 9q - v sin 9q = «■ (3) On the arc E'D' and on the segment OD' w cos 9q + v sin 9q = cc We shall treat here the elementary case; consequently, a will be considered constant. The condition 252 MCA TM 1554 w cos 9o - V sin 9o = a entails that on OD or also whence Z ^ cos On - Z ^ sin Qn dZ '-' dZ ^ R Z + ijsin Bq + i(Z cos 9 dU dZ = 4^i] = The normal derivative of u is zero along OD. One would have an analogous result on the segment OD' On the other hand, on ED Z ^ cos 9 dZ Z^sin dZ ^0] = which entails also dU dZ = Consequently, u maintains a constant value on ED and E'D' , Besides, it is easy to calculate this value owing to the formu- las (IIIA6); one finds Ur a P sin(9o - Qi) In order to achieve the calculation of U(z) it is then necessary to carry out the conformal transformation NACA TM 155^ 253 T = il-ni2m where m = rt - 2er The domain investigated is represented on a semicircle of the plane t (compare fig. 95)- The homologous point of E has as argument ^ ^(9l - Qq) ^1 n - 29 Now the function U(t) can be written immediately on the strength of the results of section 3 •2. 2.1 U(t) = ^iS ^ log (t t e""^^) (1 - Te"^0 (j^, ) ,psln(eo-e,) (1 , ,,-i*l) (,i*l . ,) and according to form\ila (ill. 53) one may write the value of the pressure coefficient on the wing portion inside of (P) C^ = T^S ^ Arc sin -^ '^P ^'M^l - M ^ - x2cos2cp, • putt ing X = p tan CJD In order to link 9-|_ to the angle cuq defining the bounding gen- eratrices of the cone, one will remark that 9t = ti^ + 9^ with cos ti^ = = 1 '0 'Op tan oiQ It is easy to obtain the component of the normal forces on the upper surface of each half wing; one will express this component by the dimen- sionless coefficient 25^ NACA TM 135^ 'N P tan aoQ Jo ^ ^ C^ dx xq = p tan odq In order to calculate C-^ one will use the plane t Cn = 2p tan ouq 'l ' (l . -,2)2 P tan GOo U(t). 1 - T (l + t2)= dT P tan ojQ R U(t). 1 - T^ (l ^ t2) = dT with L denoting the contour e'd'de in the plane t (fig. 95)- The calculation of this integral has already been performed in sec- tion 5.2.2.2. Hence Cn - 2a sm 9i P tan (Dq 2p cos 9-]_ sin(9-]_ - Bq) 2a cos(^o + Qq) sin cp-^ P cos cp-j^ sin(9-j^ - Sq) (IV. 78) Remarks . (1) It is obvious that the general case where a would be variable over the span can be investigated without difficulty with the aid of electric analogies. (2) The treatment of the case where the cone representing a dihedral is entirely inside the Mach cone is more difficult. The domain where the functions U(Z), V(Z), W(Z) must be studied is annular, and in contrast to what occurred in section 3.1, the conformal representation of such a domain on a circular annulus does not seem to follow immediately. NACA TM 135^ 255 (3) It is possible to study the effect of dihedral on a rectangular or on a sweptback wing by "composition" using the methods developed in section ^.1. h.3.2 - Fin at the Wing Tip Let us consider, for instance, the edge AA' of a rectangular wing of large aspect ratio; we shall assume the fin to be formed by a trian- gular plate ABB' (fig. 96) which we shall suppose, to start with, as lined up with the wind. We aim to calculate the effect of this fin on the flow. ^■3-2.1 - It is almost evident that if the semi-infinite lines AB, AB' are outside of the Mach cone of A, the fin suppresses the end effect of AA' Let us consider, for instance, the case where the wing is reduced to a lifting plate in the plane Z; the boundary conditions for the quad- rant OAB read, in fact, as follows: w = wq, on OA and AB V = 0, on OB They are the same that would be valid for a flow around a plate of infi- nite span placed at a certain incidence with respect to the wind. In contrast, the perturbation flow in the quadrant OA'B is iden- tically zero. This result applies, by the way, likewise to the "thick- ness effect." We deal, therefore, not with a new mathematical problem, but simply with a remark which can be utilized in certain technical problems . If now the fin is itself a lifting surface, that is, if v assumes on the fin a constant value different from zero, the case is particularly simple and one may conclude immediately that it is the one where the bounding generatrices of the fin are symmetrical with respect to the plane x-]_0x2. In fact, if the fin were by itself, it would give rise to a flow of such a type that the component w would be zero in the 256 NACA TM 1354- plane 0x-|_X2. Thus it suffices to add this flow to the one found in the case where the fin is lined up with the wind5T. 4.3.2.2 - The case where the bounding generatrices of the fin are inside of the Mach cone gives rise to a new problem If C and C are the images of these generatrices in the plane Z (fig. 91)) we shall suppose, for instance, that C and C are symmet- trical with respect to 0, and shall study the effect of the fin on an elementary symmetrical problem. The boundary conditions are: w = wq on the upper edge of OA and on the arc AB w = -Wq on the lower edge of OA and on the arc AB' V = on the two edges of OC and of OC ' For reasons of symmetry one also has w = on OA' . We shall discuss the function Z — (the function F(Z) introduced dZ \ in section J-l-l is proportional to Z — ). The boundary conditions inform us that Z — is real on the contour ABA'OCOA. On the other dZ hand, according to the res\ilts obtained in chapter 3, B is a simple pole for this function while C is a critical point of the order p + l/2, p being an integer. Reassuming the arguments raised in section 3-1; one sees that the simplest (in the sense of the principle of m i n i mi im singu- larities) of the functions which satisfy these conditions is written ^^=-i F =TT7^ (^-^9) ^^ ^ (Z2^l)[(z2 + c2)(l+c2z2)]^/^ We denote by the index (l) the corresponding solution. -^ 'For reasons of simplification, we have visualized the case where the bounding generatrices of the cone were normal to the wind; it is easy to treat in the same manner the case where this condition is not satisfied. NACA TM 135^ 257 Besides, u is zero on the arc A'B, and u = - — ^ on the arc AB. P Consequently, one has, if one takes as the initial determination for the radical the positive one on the upper edge of OA, according to equa- tion (lII.i+6) k = !^(X . c^) The integration of equation (lV.79) does not present any difficulty; naturally, the integration constant must be chosen in such a manner that u = for Z = -1. One finds U(1)(Z) =!^lo, pit ' i(l - Z2)(l - c2) - 2V(zg + cg)(l + c^Z^) (z2 + l) (l + c2) (IV. 80) with the logarithm having the value in for Z = 1. The explicit calculation of W^-'-'(Z) and V^-'-^(z) may be made by the elliptic functions. One must, in fact, examine whether all boundary conditions are satisfactorily verified. Now dV (1) Wo(l - c2) dZ = + Pit [(z2 + c2)(l + c2z2)] 1/2 Consequently, if one puts Z = ic sn (t,c ) the investigated region of the plane Z has as image in the plane t a rectangle (compare section 3-l'l-8 and fig. 3^) and one obtains dV (1) _ dv(l) dZ _ i^O dT dZ dT np (1 - c2) V(l) = ^(l - c2) fr - i K:.) Pit ^ \ 2 / 258 NACA TM 1351+ The integration constant is chosen in such a manner that v = on the circle (Cq) . The solution u(l)(z), v(l)(z), w(l)(z) thus does not satisfy the boundary conditions posed; it corresponds to the case where the fin itself is inclined toward the wind direction with the value of V on the fin being equal to ^(1) _ ^ (1 - c2)k' Pn 2 On the other hand, one finds for w'-'-)(z) dw(l) _ dw(l) dZ _ ^OV^ " '^ ) 1 + c^sn^i (IV. 81) dT dZ dT p« 1 _ ^23^2^ w'' ' is, therefore, expressed as a function of t by an elliptic inte- gral of the third kind. After having thus defined the solution u(^)(Z), v(l)(z), w(l)(Z) it is easy to obtain the one which is relative to the posed boundary problem; it suffices to add a solution u(2)(z), v(2)(z), w(2)(z) so that (1) u(2) = v(2) = v(2) = 0, on (Cq) (2) w(2) = 0, on OA and OA' (3) v(2) = -v(l), on the two edges of the cut CC This flow is, except for the notations, the one which has been studied in section 3.1.1.7- In particiilar, the value of the func- tion u(2)(z) is written U(2)(z) = 2 _c! vOO 1 - Z2 (IV.82) P c2 . 1 Jl - c2\ [(,2 , z2) (1 . z2c2)]l/2 One obtains thus the following general resiilt : if one must on the fin have v = Vq, the value of the function U(Z) is given by the formulJ NACA TM 1:55^ 259 u(z) iWr pn log 1(1 ■ -Z2)(l - c2) - 2n/(z2 + c :2)(l. c2z2) (l + z2)(l + c2) 2c' p(l . c2)e(^ \1 + c2/- -V0-^a-e2)K- (1 - Z2) 2pn ^(z2 + c2) (1 + c2z2) ■ (IV. 85) .0 and c2 — >1, one finds, at the limit, the result foreseen in the case where the fin bisects the Mach cone (i4-.3'2.l); and that, if c — ^0, one falls back on the solution of section 5-2. 2.1 (equation (III.57)). One may then calculate the pres- sure coefficient on the wing (Z real and positive), and finds One will see that in the case where v, 0- 2wr Pnl2 — + Arc cos J(i-x2)(: .2) PeQi - y^) 2vo-?(i-c2)k' Prt \ 1 - X x2(l - 72) ^ ^2 putt ing X = 2p 1 + p' 2c 1 + c' 4.5.5 - Crossed Wings To terminate these few remarks regarding the calculation of the effects of dihedral, we shall give a few indications regarding the case of crossed wings. Let us consider a cone flattened in two directions of the planes 0x-lX2, Ox-^Xz. The function w on the two faces of the tri- angle OAA' and the function v on the two faces of the triangle QBE' are known . Let us suppose that OB and OB' are symmetrical with respect to Ox-|_X2, and that OA and OA' are symmetrical with respect to Ox^x^; under these conditions the flow around the crossed wing is obtained in a 260 NACA TM 1354 particularly simple manner. It suffices to superimpose the flow which is infinitely flattened into the plane Ox-|_x^ and realizes the desired values for v, and the flow which is infinitely flattened into the plane 0x-|X2 and realizes the desired values for w. In fact, due to the symmetry, the first flow gives a value of zero for w in the plane OxjX^, and the second a value of zero for v in the plane Ox^x^z. The case where the crossed wing does not admit two planes of sym- metry cannot be treated as simply in the general case. Part iciilarly , the case where the bounding generatrices are all entirely inside the Mach cone leads doubtlessly to analytical solutions which can be explic- itly expressed only with difficulty, even in the elementary case. How- ever, as in all these problems concerning the effect of dihedral, the solution is facilitated by the utilization of conformal representations. Although they are hard to obtain in explicit analytical form, they may be determined accurately by judicious utilization of the general method of electric analogies . NACA TM 135^ 261 REFEHENCES 1. Sauer^ R.: Mathematische Methoden der Gasdynamick. 19^6. 2. Germain, P.: Les approximations lineaires dans 1' etude des fluides compressibles . Zeme Congres National de 1 'Aviation Francaise. 19^7. 5. Oswatitsch, K. , and Wieghardt, K. : Theorische Untersuchungen.uber stationare Potent ialstromungen und Grenzschichten bei hohen Geschwindigkeiten. Preisauschreiben der Lilienthalgesellschaft. 1943 Traduction S.D.I.T. No. 3710. k. Von Karman, Th.: Supersonic Aerodynamics Principles and Applica- tions. Journal of the Aeronautical Sciences. Juillet 19^7- 5. Laitone, E. V.: The Linearized Subsonic and Supersonic Flow About Inclined Slender Bodies of Revolution. Journal of the Aeronautical Sciences. Novembre 19^7> 6. Freda, H.: Methode des caracteristiques pour 1' integration des equa- tions aux derivees partielles, lineaires hyperboliques . Memorial des sciences mathematiques LXXXIV. 1937- 7. Hadamard, J.: Le probleme de Cauchy. Hermann. 1952. 8. Temple, G., and Jahn, H. A.: Flutter at Supersonic Speeds. R. et M. 2140. 1945. 9. Busemann, A.: Inf initesimale kegelige Uberschallstronung. Schriften der deutsch. Akad. der Luftfahrtforschung 19^3' (Available as NACA TM 1100, 1947.) 10. Stewart, H. J.: The Lift of a Delta Wing at Supersonic Speeds. Quart of Appl. Math. vol. k. 1946. 11. Beschkine, L.: Forces aerodynamiques agissant sur les surfaces portantes aux vitesses supersoniques . Les cahiers d ' aerodynamique No. 6. Janvier - Fevrier 19^7- 12. Hayes: Linearized Supersonic Flow With Axial Symetry. Quarterly of Appl. Math. IV, 3- 19^6. 13. Peres, J.: Cours de mecanique des fluides. Paris, Gauthiers Villars . 1936 . 262 NACA TM 1551^ Ik. Jones^ Robert T. : Properties of Low-Aspect -Ratio Pointed Wing at Speeds Below and Above the Speed of Sound. NACA Rep. 855, 1946. (Supersedes NACA TN 1052.) 15. Theodorsen, Theodore: Theory of Wing Sections of Arbitrary Shape. NACA Rep. iill, 1951. 16. Theodorsen^ T., and Garrick, I. E.: General Potential Theory of Arbitrary Wing Sections. NACA Rep. k'^2, 1933. 17. Malavard, L.: Sur la solution rheoelectrique de questions de representation conforme et application a la theorie des profils d'ailes. Comptes rendus Ac. des Sciences t. 218, p. IO6-IO8, 1944. 18. Warschawski, S. E.: On Theodorsen' s Method of Conformal Mapping of Nearly Circular Regions. Quarterley of Appl. Math. Ill, 19^5- 19. Germain, P.: Sur le calcul pratique de certaines fonctions inter- venant dans la theorie des profils. ler Congres de 1' Aviation Francaise, 19^5' Germain, P.: The Computation of Certain Functions Occurring in Pro- file Theory. A.R.C. Rep. 8692, 19^5. 20. Germain, P.: Sur le calcul numerique de certains operateurs lineaires. Comptes rendus Ac. des Sciences t. 220 p. 765-768. 19^5. 21. Watson, E. J.: Formulae for the Computation of the Functions Employed for Calculating the Velocity Distribution About a Giver Aerofoil. Report and Memoranda No. 2176, 19^5- 22. Naiman, I.: Numerical Evaluation by Harmonic Analysis of the e- Function of the Theodorsen Arbitrary Airfoil Potential Theory. NACA ARR L5H18, 19i<-5. 25- Thwaithes, B.: A Method by P. Germain for the Practical Evaluation of the Integral: e(e) = - ^j +($)cot ^^d$ A.R.C. Rep. No. 866O, 1914-5. 2k. Wittaker, E. T., and Watson, G. N.: A Course of Modern Analysis. Cambridge University Press. NACA TM 135^ 265 25. Malavard, L.: Application des analogies electriques a la solution de quelques problemes de I'hydrodynamique. Public, sc . et Tech. du Ministere de I'Air f asc . 57, 1934. 26. Malavard, L.: Etude de quelques problemes techniques relevant de la theorie de I'aile; application a leur solution de I'analogie- rheoelectrique . Public, sc . et Tech. du Ministere de I'Air fasc. 155. 1939- 27. Malavard, L.: L'analogie electrique comme methode auxiliaire .de la photo-elasticite. Comptes rendus Acad, des Sciences, t. 206, p. 39. 1938. 28. Peres, J., and Malavard, L.: Application du calcul experimental rheoelectrique a la solution de quelques problemes d'elasticite. Journ. math, pures et appliquees, t. XX, 19^1. 29. Villat, H.: Lecons sur I'hydrodynamique. Paris Gauthiers Villars, 1929. 30. Puckett, A. E., and Stewart, H. J.: Aerodynamic Performance of Delta Wing at Supersonic Speed. Journ. of Aeron. Sciences XIV, Octobre 19^7. 31. Lighthill, M. J.: The Supersonic Theory of Wings of Finite Span. Report and Memoranda Wo. 2001, 19^+4. 32. Snow, R. M.: Aerodynamics of Quadrilateral Wing at Supersonic Speeds. Quart, of Appl. Math., 1914-8. 33- Bonney, A.: Aerodynamic Characteristics of Rectangular Wings at Supersonic Speeds. Journal of the Aeronautical Sciences', 1947' 3^ . Schlichting: Tragflugeltheorie bei Uberschallgeschwindigkeit . Liiftfahrtforschung XIII, I936. 35- Von Karman, Th., and Moore, N. B.: The Resistance of Slender Bodies Moving With Supersonic Velocities With Special Reference to Pro- jectiles. Trans. A.S.M.E. vol. 5^+, 1932. 36. Lighthill, M. J.: Supersonic Flow Past Bodies of Revolution. Report and I^moranda No. 2003, 19^5. Note: The principal questions treated in this memorandum have been summarized in a certain number of notes to the Comptes rendus de I'Academie des Sciences: Volume 224 - 194-7 p. I83 Volume 226 - 1948 p. 311 Volume 225 - 1947 p. 487 Volume 226 - 1948 p. II26 264 NACA TM 13^k APPENDIX No. 1 - Theorem of Existence and Singularities of the Solution for a Flow Infinitely- Flattened in One Direction 1. Generalities .- The source method which should be called more exactly the "method of the fundamental solution of Hadamard" permits the general investigation of the flows about obstacles which are infinitely flattened in one direction. Several authors (compare refs. 1, 2, ^, and k of the references for the appendix) have independently investigated this problem. We ourselves have studied this question in collaboration with M. R. Bader. Since the corresponding report (ref. 5) has not been officially published, we shall give here the results which seem to us original with regard to the investigations quoted. With the same nota- tions as in the text, the problem may be formulated in the following manner (see fig. 1) : Find a solution cp ^x-]_ , X2 , X:z^ satisfying the equation L(cp) = p2 S^ _ S^ _ ^ ^ Sx-L^ Sx2 Sxv^ and the boundary conditions: (1) at infinity upstream: cp = 0, grad cp = 0; (2) on (S), projection on 0x-lX2 of the obstacle: Sep Sx^ = k'^fx-L^Xg) for Xz = +0 ^ = k-(x-L,X2) for x^ = -0 k"*" and k~ are known functions which satisfy the conditions of regu- larity (ll) relative to Scp/Sx^ which will be specified below. Figures for this appendix are found on pp. 532-535 • NACA TM 155^ 265 In order to pose the problem correctly, one must furthermore state exactly the hypothesis of regularity which one Imposes on the solution; we shall denote by (R) the portion of Ox-|_X2 which corresponds to the wake of the flattened body on (S) . (I) cp is continuous, except for, eventually, across the plane x-]_ = on (S) and (R) . (II) The first and second derivatives of cp exist and are generally continuous outside of (S); a possible exception may occur across certain characteristic surfaces where the derivatives may have either disconti- nuities of the first kind at a regular point or infinities at an excep- tional point. Nevertheless, they may have infinities on (S) in order to satisfy the hypothesis of linearization; Scp/Sx:z can become infinite only on parts of the boiindary of (S) and only when one approaches it by remaining outside of (S). Furthermore, we shall assume Scp/Sx^ and dcp/Sx-|_ to be continuous if one traverses 0x-|_X2 at a point outside of (S) . This hypothesis has an immediate physical significance for Scp/Sx:?; the same holds true for ^cp/Sx-]_ if one recalls that this quantity is proportional to the pressure. In other words, only 3cp/Sx2 can have a discontinuity of the first kind across Ox2_X2. Finally, cp can be divided (as in chapter III) into its odd and even parts with respect to x^. If cp is odd in x^ (symmetrical problem), Scp/Sx:? = outside of (S). If cp is even in x? (lifting problem), b^p/bx-^ = in 0x-[_X2 outside of (S) as it results from the hypothesis (ll). 2. Fundamental formula .- We shall utilize the generalized formula of Green In [uL(v) - vL(u)]dT = - 'XL u dv _ ^ du dv dv da y) is the surface having an element da which bounds the volume V having an element dr; the derivatives d/dv are the derivatives in the transverse direction. Thus one has, if \ is defined by F(x-[^,X2,x^) = with F(x-|_,X2,X2) > outside of V 266 NACA TM 1354 d a2 ciF a SF S BF T— = P dv hx-i Sx-[^ ^X2 Bx2 ^x^ bx-z Finally, utilizing the conception of the "finite part" of an integral originated by Hadamard, one may apply Green's formula to functions u and V which cause the employed integrals to become infinite. One then writes III ["^<^' - '^'">1 dT = u ^ - V ^ dv dv dCT Let us consider at a point ^(^2_>i2'^^ ^^3 ^ '^ ^°^ instance), the Mach forecone F and let us intersect it by the plane x-|_ = -A where A is positive and very large, and by the plane x^ = 0. We determine thus a volume V in the region x? > 0, bounded by a surface ^ . Admitting the existence of cp, we apply Green's formula to the pair u = (p(x^,X2,x^) V = H = i (^1 - -1)' - P' \ih - -2)' ^ ih - ^3)'] H is the f\indamental solution, in the sense of Hadamard, for the wave equation. We cannot discuss here all the details and all justifications but f we shall note the principal stages of the demonstration. (a) It is shown that the generalized formula of Green can be applied effectively to the pair cp, H, even if the derivatives of 9 present discontinuities of the first kind, owing to (ll) which informs us that these discontinuities occur on characteristic surfaces. (b) For the part of ^ situated on x-j_ = -A, the double integral becomes zero due to the boiindary conditions. (c) On the cone F the double integral must be taken at its finite part. Let us introduce the cone F with the equation NACA TM 135^ 267 p2[(x2-l2)'+ (X5-|3)2] = (1 -e)2(xi-^l)2 and the plane Pg x-L = 4i - 6 (& > 0) Since e and 5 are small, one will calculate the double integral on the surface adjoining ^ , formed on one hand by T^ and on the other by the circle C^g, with the section of F^ made by the plane Pg. One can easily show that the contribution due to Tg has a finite part of zero, and that the one due to C^g is -2ncp(P) . Consequently, one obtains, denoting by h the section of (F) by x^ = 0, the relation cp(p) = J^ rr cp M_ da - ^ rr H 1^ da 2jr J J ^ dxj 2jt JJ ^ dxj (d) In order to eliminate cp in the second term, one may apply the image method utilized by V. Volterra in an analogous problem. Let P' be the symmetric point of P with regard to 0x2X2; let us apply Green's formula to the volume V situated in x^ > 0, bounded by the planes x-]_ = -A, x^ = 0, and the Mach forecone of P' by putting u = cp(x2,X2,X5) V = H = H(P') One thus obtains = i rr cp ^ da - j^ rr H ^ da 2n JJ^ dxj 2it JJ^ dxj and since for x^ = H = H bx^ Bx:i 268 NACA TM 1354 cp |IL da = - r H 1^ da one has h ^^3 J^h ^^3 Combining this result with the preceding one, one obtains the desired fundamental formula q,(p) =. 1 rr H ^ da = 1 rr cp ^ da ^ dJh ^-^5 ^ uJh ^^3 3- The theorem of existence for the symmetrical problem .- In a symmetrical problem dcp/dx^ is known on every face of x^ = 0; conse- quently cp may be calculated in the entire space. The existence of the solution will be established if one verifies that this function 9 satisfies L(cp) = 0, the boundary conditions, and the conditions of regularity. (a) L(cp) = 0, for the functions k(x-L,X2) satisfying the hypothesis of regularity; one may calculate the derivatives of cp by deriving under the sum sign with respect to the coordinates of P. Since only H depends on these coordinates and H satisfies L(H) = 0, the resiilt follows from it as Hadamard has shown in a very general manner . (b) In order to verify the boundary conditions, one must show that lim (^ > 0) ^5—^0 ^^3' " ^— ^0 " JJh "^3 "^^3 This verification is easy if one puts PI 3 X-, = I-, + J- ^— ^ xp = Ip - ^^ ^o"*^ Q (1 - p.) sin in the integral and then going to the indicated limit. (c) Verification of the conditions of regularity leads to a careful study of the behavior of 9 and its derivatives. We can give here only the conclusions of this study. NACA TM 155^ 269 ^+ A. In the plane OxjX^, let P and P be two points lined up with P so that p+p = p-p = e (1) If there are only isolated points of discontinuity of Scp/Sxv on the Mach lines ahead of P, and if Scp/Sx^ is continuous at P q)(P+) = 9(P-) + 0(e) that is, cp is continuous at P, of the order e. An analogous result is valid for the first derivatives b^Ibx-^, Scp/Sx2. (2) If there is only a finite number of points of discontinuity on the Mach lines ahead of P and if P is a point of a supersonic line (compare chapter IV) of discontinuity for Swdx^, 9 is continuous of the order e, but ^/Sx-]_ and Scp/Sx2 have discontinuities of the first kind. In particular, if the tangent to the line of discontinuity at P forms with Ox-|_ the angle o), the discontinuities of ^/Sx-^ and of ^/Sx:z are connected by the well-known relation tan 00 ^pt an o) - 1 (5) If there is only a finite number of points of discontinuity on the Mach lines ahead of P, and if P is a point of a subsonic line of discontinuity, the first derivatives of cp become infinite as log e when one tends toward P. (4) If there is a discontinuity of ^/Sx^ on an entire segment of one of the Mach lines ahead of P, the first derivatives of cp become there infinite as e~ ' . B. Outside of the plane 0x-lX2 one has the following results: (1) If the boundary of h is not at any point tangent to a line of discontinuity of Scp/Sx^, and does not contain any finite part of such a line, the first derivatives of cp are continuous and of the order e . 270 NACA TM I55I+ (2) If the boundary of h is at certain points tangent to a line of discontinuity of Scp/Sx, without containing any finite part of such a line, P is situated on the characteristic surface which has this line of discontinuity as directrix, and the first derivatives of cp admit discontinuities of the first kind at P when traversing this surface . (3) If in exceptional cases the boundary of h contains a part of a line of discontinuity of ckp/Sxj, the first derivatives of cp become infinite as e ' ; besides, such a point is necessarily isolated. All these results taken together show that the conditions of regu- larity are satisfied which proves the existence of the solution fo\ind in this manner. k. The theorem of existence for the lifting problem .- We shall insist less on the calculation of the solution, which one can find in the published memoranda quoted before, particularly in reference h , than on the study of its singularities. However, in order to make this investigation, we must indicate briefly the procedure of the calcula- tion; we shall do so for the simplest case, the one where the edges of the wing are independent. (Compare fig. 2.) The fundamental formula permits the calculation of the potential when one knows Scp/dxz on the entire plane O'i^-^yj^. It is clear that this quantity is zero upstream from the line A^5M2_, with MM-|_ being the characteristic tangent to the leading edge of the wing. In order to calciilate this quantity in the regions where it remains provisionally unknown, it is advisable to make the change of variable k(x-L,X2) = K(A,|a) If |i = h-]^(A) and [I = M-2^^) ^^® ^^^ equations of the arcs AM and MNQ, one has at a point Aq, ^iq of the region M-^MNNj (since in this region cp is zero) the equation NACA TM 155^ 271 '^0 p^i2('^) / ^M nI^o - M,^(7,) nR^ '^0 dA r'° ^ d, ^ Q Am ^ J^2(A) ^ this equation entails the equality which determines ^/Sx^rAQ^^iQ^ by the inversion of an equation of Abel. One finds (ref. k) fM'^o) , '^5 '^^.o - ^2{^0) ,,. thus one knows ^/^x^ in the region M-]_MNN-l. At a point where cp^Ag^lJ-o) ^^ ^°^ zero (for instance on the wake), one has -2rtpcp(Ao,l^o) = I(^O^^^O) which gives, after a double Abel inversion 272 NACA TM l^'^k n^2 0*^2(^2) Jn2(A2) n^^2 Jn2(>v2) This equation contains two unknown functions, and in general it will be impossible to determine them both without introducing a supplementary hypothesis. But if one supposes that: ^^ is continuous in Ox2_X2 when traversing the subsonic trailing edge, Sxz it will be seen that it is easy to calculate first 9 on the wake, and then Scp/Sxz in 0x-|_X2. The preceding equation is written ll^2(M a^ _ 1 1 I K(A2,^)dn , h ■ - ^0^ n^2 xl^o - ^ d\lQ ,, dAp J An cp dAo h- ^^2 - Aq Sx 5 '^ ^2 -^2(Mj^ /. N f 2(M - ^ xf^ _itiO_ ("^^ ""^ K(A2,^)dn ^ J^^2(M J^^2(^2)-^ P^2 P^^2(M"^ / \ A. dug I ^^ ^^ K(A2,n)dn J^i2(^2) u^^l('^2) %'[^0^^2W] An 1A2 2p 1 I "0 - - ,, f 2 - ^^2(^2) J^ n|^2 - ^ 2p r ^^0 . f' %'(^o-^o) " J^i2(A2) nP^^^ °J% J^2^^ dAo NACA TM 135^ 273 If one makes [X2 tend toward M-2(^2) with e oeing a small quantity, one sees that, according to the previous hypothesis, the second term of the second member tends toward ^/Sx2^A2,M-2) whereas the third tends toward zero. Let us moreover make the provisional hypothesis that the last term tends toward zero (this hypothesis will have to be verified later on) , and we obtain ,^ . >2(^2) - ^ i \|^2 - ^0 However, since cp maintains in the wake a constant value on the lines parallel to Ox-]_, it suffices to know, for instance, the values of the potential on the straight line QI (fig. 2) in order to know them every- where . In accordance with this remark or ^l(Ap,) ^l(Ap,) f2(^P') - ^ if one defines Ap, by A - Api = l^Q - ^^2(■^P') We note that the circulation along the subsonic trailing edge is thus calculated . It remains to be verified that the provisional hypothesis adopted in the course of the calciilation is well founded which can be accomplished without difficulties. One sees thus how the solution of the lifting problem can be determined. 2'jk NACA TM 155i+ In order to establish that the calculated solution completely ful- fills the problem that is the theorem of existence, one proceeds as in the symmetrical case; thus the whole matter finally amounts to an inves- tigation of the singularities of this solution; this investigation per- mits a verification a posteriori of the conditions of regularity. In order to make this investigation, it is necessary to study first of all the behavior of ^cp/^x^ in the plane 0x-]_X2. As before, we shall indi- cate the results without demonstration. (a) Study of Scp/Sxz in 0x-lX2.- First, one sees immediately that SWdxz increases indefinitely as e ' when one tends toward the sub- sonic leading edge MN, remaining outside of (S) . On the other hand, according to hypothesis, this quantity is continuous on the subsonic trailing edge NQ. We shall now specify its behavior along the charac- teristic NW2_; a rather simple calculation which we cannot reproduce here, in order to avoid postponement of publication, permits to show that: Along the line A = Aj^, |i > ^],j, dcp/Sx^ undergoes a discontinuity of the first kind equal to ^^^(^^) K(A„.) " h - 'KM J,^(,,) \R^ d^x The manner in which Scp/Sx^ is calculated shows then readily that SqySx^ has no other discontinuities in the plane 0x-|_X2, outside of (S), of course. (b) Study of the solution in Ox-^Xq.- What has been said for the symmetrical problem remains valid by means of the following modifica- tion: First of all, b^p/^x-^ and Scp/Sx2 become infinite like e"-"-/^ along the subsonic leading edge. On the other hand, a very important fact, the derivatives Scp/c5x2^ and Bcp/Sx2 undergo discontinuities of the first kind along the characteristics issuing from the boundary points between subsonic leading edge and subsonic trailing edge. For dcp/Sx-]_, however, such a discontinuity can occur only on (S). (c) Study in space .- The only really new fact to be pointed out is that across the Mach cones behind the boundary points between subsonic leading and trailing edges, the first derivatives undergo a discontinuity of the first kind. NACA TM 135*+ 275 ^ . Final remarks . - (a) We have adhered to demonstrating the existence of the solution, but the employed procedure of demonstration shows at the same time that the solution is unique. Consequently, every solution which corresponds to the hypothesis found by other methods (particularly by the method of conical and homogeneous flows) represents the unique solution to the problem posed . (b) One will also note that the supplementary hypothesis introduced along the subsonic trailing edge in the case of a lifting problem may also be expressed by saying that the pressure remains continuous along this line. This is an immediate consequence of the investigation of the behavior of the solution. (c) We have not attempted to investigate here the most general type of surface (S) . In general, the method can be applied by means of a few precautions (compare ref. k or ref. 5)- Nevertheless, there exist cases where the application of this method actually fails, for instance, the case where the wing does not possess a supersonic leading edge, or also for certain dispositions of the trailing edge. Figure 3 shows such examples; if one traces a few Mach lines, one will understand immediately the reason for this failure. (d) One of the advantages of the method just described is the fact that it may be effectively applied to very general problems. Neverthe- less, it does, in out opinion, not minimize the advantages of the method of conical flows, since in many particular problems arising in aeronau- tics, the method of conical flows (and the method of homogeneous flows) lead in a simpler manner to the desired result. (e) The method of the fundamental solution has the great merit of permitting the study of the general conditions of the flow, particularly the study of certain pressure discontinuities which one encoionters on the surface of the wing in certain lifting problems. No. 2 - On Homogeneous Flows We developed the theory of homogeneous flows5° and gave a few applications in a recent article (ref. 7)- We shall give here a few supplements to the general study made in section l.J. If one puts -^"Simultaneously, this problem has formed the subject of an article by M. Poritzky (ref. 6). However, this author does not seem to us to have gone as far as we have in the investigation of the homogeneous flows . 276 NACA TM 1^^ ^ 125 (p + q + r = n) the '^}'^' „ „^ depend in a homogeneous flow of the order n only on (n) X and 9. Inside of the Mach cone (r) these quantities may be con- sidered as the real parts of analytic f\inctions of the variable Z defined except for an additive purely imaginary constant which we shall denote &^ ,(Z) A problem of homogeneous flows is treated for the nth derivatives, These nth derivatives are connected by the relations of compatibility which may be expressed in the following manner: All the expressions ,p+q. 1\ 7 2Z \ / 2iZ \ py U^ + 1) U2 - 1) / 2iZ \ (n-p-q,p,q) dZ are identical whatever the integers p and q may be which satisfy the inequalities < P + Q. < n In order to express the boundary conditions with the nth derivatives, and to enter the nth derivatives into the calculation of the potential or of the pressure fC = -u\ one will utilize a generalization of Euler ' s ident ity n. 1^(1,0,0) ^""(OA^O) 5^(0,0,1) a formula in which one must use the following convention concerning the cp(k) 9 (1) (1,0,0) 9 (1) (0,1,0) 9 (1) (0,0,1) = 9 (p+q+r) (p^q.;r) NACA TM 155^ 277 One will find in the quoted article an application of these general principles to the case of the flows flattened in one direction. The methods used in chapter III can be generalized without any difficulties; also, one may utilize in this investigation the analogy of the electro- lytic tank. A superposition of homogeneous flows permits, in a very simple manner, the invest igation-^9 of a rather large group of A wings: "the A wings with affine sections." No. 3 - On the Methods Utilized in Chapter III The exposition of certain problems of chapter III could be somewhat simplified not only by omitting certain intermediary calculations of wholly elementary character which we have mentioned to facilitate the reading, but also by employing slightly different methods . First of all, as we have remarked in the text, certain simplifications appear if one places oneself in the plane z. Thus the symmetrical problem may be solved by the same formulas whatever the position of the obstacle may be with respect to the Mach cone. Nevertheless one has to be very careful regarding the determinations of the solution when one passes from one case to another since the solution should be characterized by continuity. We have elected to utilize here the plane Z because the relations of compatability in Z do not cause the appearance of multi- form fimctions and the theoretical difficulties are, consequently, of distinctly lesser importance even though the calculations may sometimes be a little lengthier. Particularly, the demonstration of the theorems of sections J. 1.1. 5 and ^-l-l-^ is markedly simpler if one utilizes the plane Z. Summarizing one may say that the plane Z is simpler theo- retically while the plane z is simpler for t' j calculations Mr. Ward has stated the solution of certain elementary problems relative to obstacles flattened in one direction using a very elegant method (ref. 8). His study is based on a solution of the equation of cylindrical waves given by Whittaker. With our notations One will also refer to the article of Mr. Fenain which will appear shortly in ' La Recherche Aeronautique ; in it one will find a complete study of a certain number of these particulars. 60 In conferences at the 'Centre d' Etudes superienres de mecainque (19^9) we have made an exposition regarding conical flows flattened in one direction which is very different in form from the one given in this report. "4?he same may hold true for the electric analogies (compare on this subject the article of Mr. Fenain quoted before). 278 NACA TM 135l( 9 I fx-j_ - px2 ch u + i^Xz sh u)f(u)du Jc is the potential of a conical flow provided that the contour C joins two points u-[_ and U2 so that u-^ and U2 are roots of the equation •^1 " P^ ch u + ipx^ sh u = In contrast, the function f(u) is arbitrary. This very refined expression for 9 furnishes the relations of compatibility and permits solution of the particular problems. The homogeneous flows are given by the solutions of the wave equation of the form / fxj - px2 ch u + ipxz sh u^'^(u)du In the case of homogeneous problems of the order n, it seems neverthe- less difficult to state the boundaiy problem clearly and to solve it by this method without falling back on methods strictly equivalent to those reemployed . No. ^4- - On the Complementary Hypothesis at the Subsonic Trailing Edge The question posed in section 3.3? which we left pending, seems to admit a practically definitive answer; one must maintain the flows of the type II which give rise to a discontinuity of the potential along the wake of the wing. But as we have said before, this results from a hypothesis clearly formulated in the appendix No. 1 which may be stated as follows : The gradient of the potential is continuous across a subsonic trailing edge. All the remarks made in section 3-3 concerning the con- sequences of this hypothesis remain valid. The most decisive argument in favor of this hypothesis is that it appears to be the simplest of all one may set up that insures the con- tinuity of cp. NACA TM 135^ 279 In the case of conical flows infinitely flattened in one direction, we have seen that it entails a line of singularities following Ox-|_ along which w is infinite when the body has a trailing edge. Such an occasion does not arise in the general case (compare appendix No. 1). All methods of chapter III can be applied to the calculation of the conical flows for which this complementary hypothesis must be taken into account. In particular, we have indicated elsewhere how one must operate in this case for the analogical calculation of the solution. No . 5 - Remark on Sweptback Wings With Subsonic Leading Edge^ The fonnula (IV.JT) niay ^e written also ]+ cos 7(1 + 2 sin 7 - M cos 7} [ C^ = - ^ """ ^^" " " """ ^ "" "/" " a(x)dx a(|)logx-dd^ " Sin 7(1 -m2cos27)^/2 Jq Jo This formula lends itself well to an investigation of the optimum. We shall search, in fact, for the profile which, in delimiting a given area, provides a minimum drag; putting iX e(x) = I a(t)dt one is led to seek the minimum absolute value of the integral I de(x) I de(|)log|x - ^1 Jo Jo 6? ^ ' Communication to the Y^h Congres International de Mecanique appliquee (19^8). 63 This remark has been made by the author in the course of his communication to the 7"th Congres International de Mecanique appliquee (19^8), quoted above. 280 NACA TM 155^4- It is easily seen^ and the fact is well-known to aerodynamists, that the^ solution of the function e(x) of this problem has the form e(x) = k\^ that is, that the desired profile is an ellipse. The train of thought which leads to (lV.37) cannot be applied to the case where the profile has a tangent normal to the symmetry axis; but according to a remark already made more than once^ one may neverthe- less assume that the obtained result does not lack connection with reality. This leads to the idea that, for a wing with subsonic leading edge, it may be practical to utilize profiles with rounded leading edges. One will note that this is not the case in supersonic regime. If one takes up this problem for a wing of infinite span normal to the wind, one finds readily that the optimum profile is formed by two symmetrical parabolic arcs. No. 6 - Remarks on Lifting Sweptback Wings With Sonic and Subsonic Leading Edges (Compare Section 4.1.2.5.2) The formula (IV.69) may also be written by putting Tio£- ^ - 2(t1o + y^)] _ ^2 TlnX + y^l 1 + t^ 2 (% + y^) 1 + tr in the form i+i npjx(x - 2y^) 2 X ^^^| i. . /.- /^/ ^/^/^/:: UU uu/. / / 1 / 1 / 1 / 1 / 1 / 1 / a 8 / i / g / 1 / 1 / 1 s ■'" V / V ■^ / o / ^ / o / ■? / o / T / V / Y / _i — / / € / 3 o / R 8 / f / 1 / 1 I 1 / / p t / s « 1 / V / o / o / O / ? / ?■ / § / « / i / S / s / / 3 / 8 / 8 / i / t / / a / °, y ? / ? o ? o / ? ? =? ' ■? / s / ? ? O ° ° o o ° o " ? ? ? o ■? o ? ID o S R tr\ S p s § § 2 g s « S s fi B 1 « S g 5?t 1 •1-4 (in MCA TM 135^ 299 i lg'( /r Calculated points \ -100 -80 -60 -40 -20 20 40 60 80 100

\i- \ \ .1^ © Figure 50 Figure 51 Figure 52 512 MCA ™ 135.1+ Electric analogy 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0,9 Figure 53 NACA TM 135^ 513 Figure 54 Figure 55 U Figure 56 31^ MCA TM 15514. A' J^C Section CC' X Hc' B f a{x.) Figure 57 +a Figure 58 w=alo) (n^ W:2a (0) V — ^X A B 2 w=a(o)w=o "H w = a(o) B' A' (b) ^.Xg B W = B' Figure 59 Figure 60 MCA TM 155^ 515 /" ^^^ Cx __^^ "^ 6 / 2-^0 = 3 \ 0.6 Q4 \ / f\ •? 1 1 1 1 J_.J...J. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1.5 -1.4 -i.Z -1.0 -0.8 -0.6 -0,4 -0.2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.5 (a) 0.5 0.3 0.1 0.1 0.3 0.5 "J? (b) Figure 61 316 NA.CA TO 15514. C, ^00 0.5 Figure 62 -1 -0.8 -0.6 -0.4 -0.2 0.2 0,4 0.6 0.8 1 17 Figure 63 0.8 0.6 04 02 r^ } ' 0.2 04 0.6 0.8 1 2-70 Fig:ure 64 NACA TM 13514- 517 t i c. 1 (^^)co 0.8 ^0=2 0.6 0.4 \. 0.2 ^\ - 0.2 0.4 0,6 0.8 1 1.2 1.4 1.6 1.8 2 y' Figure 65 .^ ^ v?^ A /^ 1 \ \ A' ^^ X N \ f^ Figure 66 518 MCA m 135lf max. 0.1 0,2 0,3 0.4 0.5 0.6 07 0.8 0.9 « rj^ Figure 67 Cx 1 1 ^ r" ^x^ 00 2 t ^ ^ / X ^ / 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0,9 1 77 ^ Figure 68 NACA TM 155if 319 Figure 69 Figiare 70 520 MCA TM 1551^ \ V. y 1 \ ^ V >B^- s. ,> \" B' Figure 71 Figure 72 1 — 1 — / = 4f)'» IL 1 1 1 'y^ r 1 -IV-29 ...f 1 1 1 \^. 1- _J IV-jr _i ^v^ IV-27 r 1 . 1 vP — t \ 4(ao)2 1 V 3 \ II \ s. Jl ^i •».. / ->> ^ / *■ -H , / ' 1 ;^ ::^ 1 Vz '1.5 r e > 2 .5 3 M Figure 73 NACA ™ 155^ 321 Figure 74 B^-^l Figure 75 Figure 76 522 MCA TM 135lf "'' '-' ,. Figure 77 A B Figure 79 NACA OM 155^ 525 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0,9 1 1.1 1.2 Y Figure 80 32k NA.CA ™ 1351^ 12345678 M Figure 81 A NACA TM 155^ 525 Po(Y*) -0.4 Jo Figure 82 'x(y*) 0.8 0.6 0.4 0.2 -1 -0.8 -0.6 -0.4 -0.2 0.2 04 0.6 0.8 1 Y Figure 83 326 MCA TO 155i+ Rgure 84 4i i ■N X 1 4 / / \ 1 "^ 1 / V . / \ / 0.8 — 2 0.6 — C 04 X \ / -' 2 1 = 1/2 02 '2 ' 5 -2 * ^' c 1 - 1 Figure 85 MCA IM 155^ 527 Figure 86 Figure 87 ^ Vlx ^yv ^'^ ^^ ^ ^^v^"^^ i^ ^^ n-%-* N Figure 328 MCA TM l^^k Figxire 89 NACA TM 155^ 529 1 dCz ' I 2 di ^ p 1 1 9 . 1 1 4 c . 1 /-- *5° 1.5 \ \ ' 1 \ I X -- 8 \ V/X=2 V '^ \ - 1 ^ 1 ' V \ \, 0,D- ^ ^ 1234567 8M Figure 90 550 NA.CA TM 1554 , .X y Figure 91 Br ua M a Figure 92 MCA TM 135^ 331 552 NA.CA m 1551). u (SI/ (R) Figure 1 Figure 2 MCA ™ 155^ 553 .n^. l^ Figure 3 NACA-Langley - 1-6-55 - 1000 CM .-( CM L) ^ ^ t J C ,, -1 -H ' O) ' rvi a " • . Q) . 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