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1 O r 1 0 f I .. 1 5 y / 1 f TABLE OF CONTENTS I. SUMMARY . . . 1 II. INTRODUCTION . . 2 III. SINGULARITIES USED IN THE LINEARIZEDD" DESCRIPTION OF THE FLOW ABOUT AIRCRAFT . . 5 A. BASIC SINGULARITIES . . 5 B. SOME EQUIVALENT SINGULARITY DISTRIBUTIONS 12 IV. THE EVALUATION OF DRAG . . 25 A. THE "CLOSE" AND THE "DISTANT" VIEWPOINTS 25 B. GENERAL MOMENTUM THEOREM FOR THE EVALUATION OF DRAG 26 C. HAYES' METHOD FOR DRAG EVALUATION . 355 D. LEADING EDGE SUCTION . . 49 E. DISCONTINUITIES IN LOADINGS . 50 F. THE USE OF SLENDER BODY THEORY WITH THE DISTANT VIEWPOINT . . 52 G. THE DEPENDENCE OF DRAG COEFFICIENT ON MACH NUMEBEF 55 H. SUPERPOSITION PROCEDURES AND INTERFERENCE DRAG 55 I. ORTHOGONAL DISTRIBUTIONS AND DRAG REDUCTION PROCEDURES . . 5$6 J. THE PHYSICAL SIGNIFICANCE OF INTERFERENCE DRAG 56 K. INTERFERENCE AMONG LIFT, THICKNESS AND SIDEFORCE DISTRIBUTIONS . . 57 L. REDUCTION OF DRAG DUE TO LIFT BY ADDITION OF A THICKNESS DISTRIBUTION . . 59 V. THE CRITERIA FOR DETERMINING OPTIMUM DISTRIBUTIONS OF LIFT OR VOLUME ELEMENTS ALONE . 60 A. COMBINED FLOW FIELD CONCEPT . 60 B. COMBINED FLOW FIELD CRITERION FOR IDENTIFYING OPTIMUM LIFT DISTRIBUTIONS . 60 C. COMBINED FLOW FIELD CRITERION FOR IDENTIFYING OPTIJMUM VOLUME DISTRIBUTIONS . 6i D. UIIFOPJR DOWNWASH CRITERION FOR MITNDIU4 VORTEX DRAG 61 E. ELLIPTICAL LOADING CRITERION FOR MINIMUM WAVE DRAG DUE TO LIFT . . 61 F. "ELLIPTICAL LOADING CUBED" CRITERION FOR MfI14UM ..!AVE DRAG DUE TO A FIXED TOTAL VOLUME . 62 3. COMPATIBILITY OF MINIMUM WAVE PLUS VORTEX DRAG WITH MINMUM WAVE OR MINflIDTUlI VORTEX DRAG 62 H. ORTHOGONAL LOADDIG CRITERIA . 65 APPENDIX V. DISTRIBUTION OF LIFT III A TRANSVERSE PLANE FOE MINIMUM VORTEX DRAG . 64 VI. THE OPTIMUM DISTRIBUTION OF LIFTING ELEMENTS ALONE 68 A. THE OPTDIMUMll DISTRIBUTION OF LIIT' THROUGH A SPHERICAL SPACE . . 68 B. THE OPTEIMUM DISTRIBUTION OF LIFT THROUGH All ELLIPSOIDAL SPACE . . 70 C. THE OPTIMUM DISTRIBUTION OF LIFT THROUGH A DOUBLE MACH CONE SPACE . 72 APPENDIX VI. DERIVATION OF OPTDIUMI DISTRIBUTION OF LIFT THROUGH A SPHERICAL SPACE . 75 VII. THE OPTDIMUM DISTRIBUTION OF VOLUME ELEMENTS ALONE 78 A. THE SINGULARITY REPRESENTING AN ELEMENT OF VOLUME 78 B. THE DISTRIBUTIONi OF VOLUME ELEMENTS . 79 * ii C. THE DRAG OF VOLUME DISTRIBUTIONS ON A STREAMWISE LINE AND THE SEARSHAACK BODY . . D. THE SEAFSHAACK BODY AS AN OPT.IMUM VOLUME DISTRIBUTIONI IN SPACE . . E. RING '.JirIG AND CENTRAL BODY OF REVOLUTI01[ COMBINATION HAVING THE SAME DRAG AS A SEARSHAACK BODY . F. OPTIMUM THICKNESS DISTRIBUTION FOR A PLANAR WING OF ELLIPTICAL PLAfJFORM . . VIII. UNIQUENESS PROBLEMS FOR OPTIMUM DISTRIBUTIONS IN SPACE . A. THE IIONUNIQUENESS OF OPTEI.UM DISTRIBUTIONS IN SPACE "ZERO LOADINjGS" . . B. UNIIUEJhESS OF THE DISTANT FLOW FIELD PRODUCED BY AN OPT.fL1 FAMILY . . C. ENIQUJEirESS OF THE ENTIRE "EXTERNAL" FLOW FIELD PRODUCED BY AN OPTIMUM FAMILY . . D. EXISTENCE OF SYMMETRICAL OPTIMUM DISTRIBUTIONS IN SYMIETRICAL SPACES I. . . IX. IliVESTIGATION OF SEPARABILITY OF LIFT, THICKNESS, AND SIDEFORFCE PROBLEMS . . A. THE SEPARABILITY OF OPTIR1MU DISTFJRIBUTIONS PROVIDING LIFT AND VOLUblE . . B. THE HONINTERFERENCE OF SOURCES WITH OPTIMUM DISTRIBUTIONS OF LIFTING ELEMENTS IN A SPHERICAL SPACE . . . C. THE JONiIUJTERFERENCE OF SIDEFORCE ELEMrENTS WITH OPTRIM1 DISTRIBUlTIONS OF LIFTING ELEMENTS IN A SPHERICAL SPACE . . . D. IlITERFERENCE PROBLEMS IN CERTAIN SPACES BOUNDED BY MACH ENVELOPES . . . 94 . 94 * 97 . 98 iii E. THE I.'ITEPFEREIICE BETWEEN LIFT AI1D SIDEFORCE ELEMENTS AND AN OPTII.IUT i DISTRIBUTION OF VOLIJUlE EL1EMETS 102 F. THE RING WING AND CENTRAL BODY OF REVOLUTION HAVING ZERO DRAG . . .. 105 X. RESULTS Arm COnCLUSIONS . . 106 XI. REFEEENrCES . . .. 107 NATIONAL ADVISORY COMMITTEE FOB AERONAUTICS TECHrICAL M1EMORAUDUM 1421 A THEORETICAL INVESTIGATION OF THE DRAG OF GENERALIZED AIRCRAFT CONFIGURATIONS IN SUPERSONIC FLOW* By E. W. Graham, P. A. Lagerstrom, R. M. Licher, and B. J. Beane CHAPTER I. SUMMARY It seems possible that, in supersonic flight, unconventional arrange ments of wings and bodies may offer advantages in the form of drag reduc tion. It is the purpose of this report to consider the methods for deter mining the pressure drag for such unconventional configurations, and to consider a few of the possibilities for drag reduction in highly idealized aircraft. The idealized aircraft are defined by distributions of lift and volume in threedimensional space, and Hayes' method of drag evaluation, which is well adapted to such problems, is the fundamental tool employed. Other methods of drag evaluation are considered also wherever they appear to offer simplifications. The basic singularities such as sources, dipoles, lifting elements and volume elements are discussed, and some of the useful interrelations between these elements are presented. Hayes' method of drag evaluation is derived in detail starting with the general momentum theorem. In going from planar systems to spatial systems certain new problems arise. For example, interference between lift and thickness distributions generally appears, and such effects are used to explain the difference between the nonzero wave drag of SearsHaack bodies and the zero wave drag of Ferrari's ring wing plus central body. Another new feature of the spatial systems is that optimum configu rations generally are not unique, there being an infinite family of lift or thickness distributions producing the same minimum drag. However it is shown that all members of an optimum family produce the same flow field in a certain region external to the singularity distribution. Other results of this study indicate that certain spatial distri butions may produce materially less wave drag and vortex drag than com parable planar systems. It is not at all certain that such advantages can be realized in practical aircraft designs, but further investigation seems to be warranted. uneditedd by the HIACA (the Corrmittee takes no responsibility for the correctness of the author's statements). correctness of the author's statements). NACA T!4 1421 CHAPTER II. INTRODUCTION The primary purpose of this report is to consider the problems involved in exploring a broader class of aircraft configurations than is ordinarily studied for supersonic flight. It is necessary to deter mine whether any unconventional arrangements cf dings and bodies offer sufficient aerodynamic advantages in the form of drag reduction to merit more detailed study. As a first step in this direction attention is directed to optimum configurations, even though they are highly idealized in form and do not necessarily represent practical aircraft. In the preliminary exploration of such ccnfigurations it is not necessary to know their detailed shapes. It is sufficient to define the aircraft as a distribution of lift and volume in space, without knowing the camber and twist of the wing surfaces supporting the lift distri bution, and knowing only approximately the shapes of the bodies con taining the volume. Hayes' method of drag evaluation is well adapted to this type of analysis and is one of the primary tools used. However other methods and points of view are employed wherever they appear to offer further understanding of the problems. The properties of sources, dipoles, etc., are reviewed, and a sin gularity corresponding to an element of volume is introduced. Some useful relations between threedimensional distributions of different types of singularities are developed and later applied. Also Hayes' method for drag evaluation is developed in detail. Since this report is exploratory in nature the investigations made are frequently incomplete and somewhat isolated from each other. Some of the material of Ref. 2 and most of the material of Ref. 5 are included in this report for convenience. The latter has also been published in The Aeronautical Quarterly, [lay 1955, under the title, "The Drag of NonPlanar Thickness Distributions in Supersonic Flow." Permission to reproduce this material has been granted by The Royal Aeronautical Society. tical Society. NACA TM 1421 CHAPTER III. SINGULARITIES UTILIZED IN THE LINEARIZEDD" DESCRIPTION OF THE FLOW ABOUT AIRCRAFT A. BASIC SINGULARITIES The Source For incompressible, nonviscous fluids the flow is the Laplace equation, )2 +2 ;x2 ,y2 2 + ) = _ 3Z2 where 0 is the perturbation velocity potential. exhibits spherical symmetry, is the source, equation governing the (5al) A basic solution, which S(x 4)2 + (y )2 + (z 2 (5a2) This solution can be interpreted as representing the emanation of unit volume of fluid per unit of time from the point t, 1, t. Because of the linearity of Eq. (j5al), other solutions of it can be built up by a superposition of sources through the use of certain limiting proce dures; such resulting solutions are the horseshoe vortex, doublet, line vortex, etc. Much is known about these solutions and with them the flow over wings and bodies can be described mathematically. In supersonic flow the governing differential equation is the linearized potential equation, 2 62o  x2 8x2 2 ,= 0 6z2 (3a3) where x is the coordinate in the stream direction and 0 = M2 1. Equa tion (5a5) can also be considered as the twodimensional wave equation where the x coordinate is thought of as the "time" variable. If the y and z coordinates in the Laplace equation (Eq. 5al) are multiplied by io, then that equation is transformed into the wave equa tion; a similar transformation of the source potential from Eq. (3a2) results in [JACA TM 1421 (5a4) (x )2 P2 [ n)2+ ( )2] which can easily be shown to be a solution of Eq. (5a3). Equation (5a4) is real inside the forward and rear Mach cones, (x 0)2> 2 y T)2 + (z )2] and imaginary elsewhere; however, due to the nature of super sonic flow only the solution in the rear Mach cone is used to represent a source. Since half of the real solution is discarded, the constant RE6S/ON 0c /i'ZUEA/CE OF SUPERSON/C SOURCE associated with the incompressible 2 must be doubled to represent a unit supersonic source. Thus the supersonic source at 1, T1, ( has the potential (5a5) Elsewhere where the x axis is in the free stream direction. It can be shown that Eq. (Ba5) represents unit volume flow from the point E, T, t; however, care must be taken in the proof because of the singularities on the Mach cone (cf. Ref. 4). In the proof given by Robinson(k) he made use of the concept of the finite part of an infinite integral, an idea originally Os~ x t 'P (y i)2+ (z_ )2 S NACA TM 1421 introduced by Hadamard"5. As in incompressible flow, other solutions of Eq. (5a5) can be built up by superposition of the basic source solu tions; some of the solutions can also be obtained, as was the source, by analogy with the incompressible solutions. Before icine on to other solutions let us examine the supersonic source in more detail. Since the velocities are infinite on the Mach cone from a finite source, care must be taken in using such sources to describe real flows. It is instructive to examine the isolated source in terms of the limit of a finite line of sources in the free stream direction as the length tends to zero while the total strength remains constant. Under the assumptions of slender body theory, if the line of sources extends from x = 0 to x = x with strength Kx, it represents a cone of semivertex angle K/2U with a semiinfinite cylindrical after body (Fig. 5ala). The velocities are constant along conical surfaces ///N/TE AT REAR MACH VELOCITIES1 CONE, ZERO AT FORWA RD MACH CONE SOURCE A STRE THO pvX0 U X Fig. 5ala: Conecylinder and source distribution repre senting it Fig. 5alb: Perturbation flow lines in xz plane for super sonic source, M = F2 VELOCiTrIES INFINITE AT MACH CONE b IACA TM 1421 from the origin; but on the Mach cone from x = xo the velocities become infinite due to the discontinuity in source strength. The total inte grated source strength C is equal to 1/2 Kxo2. If xo is allowed to approach zero while C remains constant then in the limit a concentrated source of strength C is obtained. The flow pattern in the xz plane for the source at M = f2 is shown in Fig. 5alb. (See also Ref. 6.) For a source of finite strength the velocities are infinite on the Mach cone. The ThreeDimensional Doublet The threedimensional doublet (or dipole) is a second basic solution of the wave equation; it is obtained by allowing a source and sink of equal strength to approach one another while the product of source strength and distance between source and sink remains constant (and equal to unity for a unit doublet). The axis of the doublet is defined here as the vector extending from the center of the sink to the center of the source; positive values are taken to be those along the positive directions of the coordinate system. For a doublet with its axis vertical, the above method of derivation is equivalent to taking the negative partial deriva tive of the source potential in the z direction; that is, S 2_ 2z x 3r (5a6) 2 (x2 22 where r2 = y2 + z2. Equation (5a6) represents a positive doublet at the origin, i.e., one with the source above the sink. The Horseshoe Vortex In supersonic theory, as well as in subsonic, the flow around a wing of finite span can be described by certain solutions of the wave equation called horseshoe vortices. In the subsonic case this singularity is derived by integrating in the streamwise direction a semiinfinite line of negative doublets with axes vertical. The supersonic horseshoe vortex can be derived in the same way as the subsonic one if only the finite part of the integral, as defined by Hadamard) is taken as the solution. This solution can also be obtained without the use of the Hadamard finite part if a streamwise line of sources is differentiated in the vertical direction; thus, NACA TM4 1421 7 0 pxpr 1 xpr HSV = Finite part {xr O d} x O3 d9 f. 0I 6z 0 _____xz x > pr (5a7) 2itr2 x2 22 The flow pattern for a horseshoe vortex in planes normal to the free stream axis is shown in Fig. 5a2. Far behind the bound vortex, Fig. 5a2: Flow pattern for supersonic horseshoe vortex 8 NACA TM 1421 the flow near the x axis is similar to the flow far downstream around a subsonic horseshoe vortex and it is this part which gives rise to the "vortex" drag. The drag associated with the flow near the Mach cone is called "wave" drag. Equation (5a7) represents a supersonic horseshoe vortex of unit strength, i.e., unit circulation around the bound vortex. Since a force pUr is associated with a bound vortex of strength r, we shall, for convenience, discuss unit lifting elegients which have as their velocity potentials = xz 21tpUr2 fx2 22 x prr (5a8) Similarly, the potential for a unit side force element is = 1 0 f o SF = U px1r pU byJO S dE = xy 2itpUr2 x2 P2r2 The force associated with Eq. (3a9) is directed in the positive y direc tion; a force in any direction normal to the flow direction may be repre sented by a combination of lift and side force elements. In the light of the discussion of the horseshoe vortex, the three dimensional doublet (Eq. 3a6) may be given added significance as a lift transfer element or element of moment. That is, the doublet potential can be formed by subtracting the potential for a horseshoe vortex at x = Ax from one at x = 0 (Fig. 5a5) and applying the proper limiting processes (equivalent to differentiating the horseshoe vortex potential); in this process the trailing vortices from the negative or rear element are can celed out by those from the positive one, and the remaining part forms the doublet or lift transfer element. U it> z z Fij. 5a5: Formation of doublet or lift transfer element from horseshoe vortices x > pr (5a9) HACA TM 14i21 The horseshoe vortex consists of a bound vortex of infinitesimal length plus two free vortices trailing back to infinity. Since the vortex drag and the lift associated with a finite wing can be evaluated by considering the flow velocities far behind the wing, it is useful to consider the trailing vortices as they appear in the Trefftz plane far downstream from the bound vortex. The Trefftj plane flow represents a twodimensional doublet or dipole and its potential is obtained by letting x> in Eq. (5a7). Thus (5a10) 02D z + 2 2n y2 + z2) It should be noted that the potential for this doublet is independent of Mach number, and thus the vortex drag calculations for a given lift distribution are the same for supersonic and incompressible flows. The flow pattern about the doublet will be similar to the planar flow inside the dashed circle in Fig. 5a2. The Volume Element Another useful solution is the doublet with its axis in the stream wise direction; it has as a potential d v OV 60S Sx 21t(x2 02r2)3/2 x ? or (5all) Equation (5aUl) can be shown tc represent the potential for a unit of volume equal to 1/U (see Chapter VII) at the origin. A distribution of volume elements along the x axis with strength f(x) has as a potential rxpr = 0f Jo0 f(t )v dt f (U x3r 24( (x *)2 p~r220 1 fxr 2 Ao The first term in Eq. (5a12) is infinite; but if only the finite part of the integral is considered (as defined by Hadamard(5) ), then Eq. (5a12) (x )2 p2r2 (5a12) NJACA TI 1421 represents the potential for a source distribution of strength f'(0). Thus a body can be built up from a series of volume elements as well as from a series of sources and sinks. The Closed Vortex Line Equation (5a11) can be considered not only as a volume element but also as a closed vortex line of circulation strength 1/ 2 in the yz plane (Fig. 3a4a). The line carries a constant intensity of forces directed inward so that the total vector force is zero. The negative of Eq. (3a11U) would represent an element with the forces directed outward from it. The potential for the closed vortex line can also be obtained by applying the standard limiting process to an element composed of two pairs of horseshoe vortices of strength 1/P 2, one with its axis in the negative z and the other in the negative y direction (Fig. 5a4b); when added together the trailing vortices cancel leaving the closed vortex line. r 4,S. + C (a) Fi,. 5a4: Formation of closed vortex line from horseshoe vortices TwoDimensional Singularities In subsonic flow twodimensional sources, obtained by integrating a line of threedimensional sources in the lateral direction, have proven useful in many problems; so also has the infinite bound vortex obtained by a lateral integration of horseshoe vortices. The same types of solu tions can be derived for supersonic flow and these provide more insight into the nature of the supersonic solutions. The twodimensional source potential is NACA TMt 1421 + 1 x2_2z2 28 J x2p2z2 S = 20 x > M ( 1 02sf12 All of the disturbance created by the twodimensional source is concentrated on the Mach planes from it, thus creating a potential jump across these planes. The twodimensional vortex potential is +1 x2 2Z2 +1 z> 0 2V 0 &V dT = 2 x > pZ (3a14) 1 x2p_23z2 2 z < 0 Again all of the disturbance is concentrated on the Mach planes. There is a potential jump across the Mach planes and also across the z = 0 plane, the latter due to the discontinuity in the past history of the fluid particles above and below the plane. NACA T1 1421 B. SOME EQUIVALENT SINGULARITY DISTRIBUTIOIUS Statement of the Problem The first section of this chapter reviewed the basic singularities which represent elements of lift, side force and volume in linearized supersonic flow theory. It was noted that these singularities may all be obtained from the source singularity with the aid of the simple proc esses of integration and differentiation. The fact that the basic singularities are so related will be shown to imply that certain dis tributions of singularities are equivalent, i.e. they produce the same flow field, at least outside of a finite region. In the present section an equivalence theorem will be proved regarding constant strength dis tributions of sources, lifting and side force elements and vortex sheets. Such a theorem will later prove useful in the study of interference between distributions (Ch. IX B,C,F). Note that if the distribution A is part of a larger distribution (A,B) and if A is replaced by an equiva lent distribution Al then the drag of (A1,B) is the same as that of (A,B). This follows from the fact that the substitution of Al for A does not change the flow field at infinity (Ch. IV). The distributions to be studied will be located on a cubic shell which has two faces perpendicular to the free stream direction. One face of the cube will be covered by sources of constant strength and the opposite face by sinks of constant strength. The remaining four faces will be encircled by vortex lines of constant strength. Two cases may then be distinguished: (A) The source distribution is on a face parallel to the free stream; (B) The source distribution is on a face normal to the free stream. These two cases are illustrated in Fig. 5bl. The source, sink and vortex distributions are uniform and of constant intensity as indicated. The vortex lines are continuous around the cube with the circulation directed so as to induce in the interior of the cube downwash velocities in case A and upstream velocities in case B. J Z SOURCES VORTEX LINES (IsrENGTrH ) (cCIRCULATION 4) ^ . ^y x ^ AC MOTEX iNES 6i 'f S/ SOUES 57 (/ ) (ciRCUIAr/oN &) (S TREPNGTHw) (STREfNGTH CASE A CASE B Fig; 5b1 NACA TM 1421 We shall now prove the following theorem. Theorem In both cases A and B the perturbation velocities are zero every where outside the cubic shell. Inside the shell the downwash is constant in case A (w = k) and the pressure is constant in case B (u = k). This theorem implies in particular that the source sink distribution, say in case A, is equivalent to the negative of the vortexline distribu tion in case A, in the sense that the associated flow fields are identical outside the cube. Note that in case A the vortex distribution on the front and rear faces gives rise to a lifting force, whereas the vorticity on the side faces produces no force. In case B the vorticity on the top and bottom produces lift and the vorticity on the side faces produces side force. The theorem will first be proved by a geometrical argument and then an alternative proof by analytical methods will be outlined. Geometrical Proof of Theorem Consider first case A. We shall construct a geometrical configu ration which corresponds to the distribution of singularities indicated in Fig. 5b1. This construction will proceed in several steps by succes sively cutting down configurations of infinite extent. The vortex dis tribution on the front face is equivalent to a distribution of lifting elements of constant strength. To begin with we shall assume the whole infinite plane containing the front face to be covered by lifting elements. This may be physically realized by a cascade of doubly infinite (twodimensional) wings of con stant angle of attack a and such that the vertical distance between two neighboring wings is equal to the wing chord divided by M2 1, (Fig. 5b2). In the limiting case of zero chord length the plane x = x is then covered by vortex lines with the circulation (of strength k) oriented as in Fig. 5b1. The value of the constant k is then k = 2aU. The lift per unit area in the plane x = 3 is then 2apU2. Since the wings are spaced so as not to interfere with each other but still influence every point downstream of the cascade, the flow field at any point P with x > 3 may be described as follows (Fig. 5b2). The point P receives a unit of downwash (w = cU) from the wings A and B each. It also receives a positive unit of pressure (u = aU /M2 from A and a negative unit m A and a negative unit NACA TM 1421 ^P )7 P x Fig. 5b2 of pressure from B. The net pressure (referred to p.) received at P is then zero and the net downwash is w = 2aP = k. The cascade may now be terminated from above by a wedge of opening angle 2a located in the plane z = T with its exterior surface parallel to the free stream direction (Fig. 5b5). Actually this wedge corre sponds to a source distribution of constant source strength k = 2aU. If the cascade is removed for z > T the flow field is zero there since the wedge isolates this region from the rest of the cascade and since the exterior surface of the wedge is at zero angle of attack. For z < T the flow field is unaffected by the introduction of the wedge. To see this consider a point P with z < T (Fig. 5b5). The wing at B acts as before to produce a downwash of adU at P. Only the point C on the wedge A, C 2C  A2' >P / Fig. 5b5 affects the point P and this point C is already in the downwash field al of wing A2. Thus the wedge turns the flow downward only by an angle a A4I MACA TM 1421 so that the total downwash at P is again 2aU. Conditions at P are the same as in the infinite cascade. Similarly the cascade may be terminated from below at z = Z by placing a wedge there of opening angle 2a.. This corresponds to a dis tribution of sinks of strength k. The cascade may then be cut down to finite width by placing planes of zero thickness parallel to the plane at y = *y and removing the part of the wings for y > Y. Since no sidewash is present the flow field is undisturbed by the introduction of these planes. Thus for x > x, y < Y, z < Y the downwash is w = 2UJ = k and the pressure is zero. Outside this region all perturbation quantities are zero. Finally one may restrict the flow field to the inside of a cube by taking the negative of the above configuration and placing it at x = x. Thus the resulting flow field has constant downwash and zero pres sure inside the cube x3 < x < x, y < y < y, z < z < z. Outside thiis cube the perturbation velocity is zero. Thus the front face is a cascade of lifting wings at an angle of attack a, which bends the flow down. The rear face is a cascade of wings of angle of attack a which straighten the flow out again. The top and bottom faces consist of wedges whose inside surfaces follow the direction of the flow which has been bent by the cascade. These outside surfaces are parallel to the free stream. (Note that for the wedge of negative angle the "interior" top surface is directed downward at an angle 2am and the "exterior" bottom surface is parallel to the flow.) Finally the side faces are planes that carry no forces. For each such plane the downwash is w = 2aU on the inside and w = 0 on the outside. These planes are then surfaces of constant vortic ity. However, the vorticity vector is parallel to the free stream and hence no force results. Thus a geometric configuration (using a wedge of negative opening angle) corresponding to case A has been constructed and the theorem has been proved for this case. The corresponding construction for case B will only be indicated. The source distribution on the front face is obtained by placing wedges there (Fig. 5b4) of halfangle a = pk/2U. A4 N, x 0.1 \k I N f ^ [^ Fig. 5b4 NACA TM 1421 At a point P then the downwash is zero and the pressure is given by u = 2aUJ/M2 1 = k. By inserting planes of zero thickness at z = z, y = y and removing the wedges outside these planes the infinite configuration is cut down to a configuration with a finite cross section. Outside these planes the flow is undisturbed. Inside these planes u maintains its value 2aU/M2 1 = k. These planes are then pressure discontinuities and hence carry lift and side force respectively. They are also vortex sheets. Finally the configuration may be terminated by placing its negative at x = +3. A geometric configuration (again using wedges of negative opening angles) corresponding to case B has thus been constructed and the theorem has been proved for case B. Analytical Proof of Theorem (Outline) Case A. Source Distribution Face Parallel to Free Stream Consider a cube with sources of strength k on the top and k on the bottom, and with lifting elements of strength pUk on the front face and pUk on the front face and pUk on the rear face. On the side faces of the cube there are no forces associated with the vortex lines parallel to the flow direction; these are the trailing vortex system of the ele ments on the rear face. In computing the potential due to the singularities on the cubic shell, various regions of the flow field are considered separately. For the region ahead of the foremost Mach waves from the cube no disturbance is possible in supersonic flow. Behind the cube, if the forward Mach cone from a point includes all of the shell, the potential at that point may be found simply by integrating the total effect of the singularities covering the shell. The potential due to individual unit source elements, lifting elements, and side force elements are given in Eqs. (5a5), (5a8), and (3a9). The strengths of the distributions considered in this case are indicated in Fig. 5b5. n Fig. 5b5. NACA TMn 1421 / fy' SOURC~s (i4) (i~9, 44 ~ Pos/;r/v EI/ /A____V/j IFT/ ; ^ 7( ,S) ELEMENTS f/(' / (x, y, z) //(x,y,2) Ls/wHsp.4 Fig. 5b5 The potential for the entire shell is then k (x y) (z zo)dyo dzo "^^ly2y y)2 + (2]^)2j ?^ ^ ^2 2A (x+E) _ [y _7 ) +2(Z zo) 2] Xx ) 2 [(yyo) + (zzo) 2 (X z zo)dyo dzo 0 )2x ( 3 ) y yo 2+ (z 2 (x ) (Y_ Y" 702+ (Z zo) 2] k x y do do f rJ dxo dyo (3b1) (x x2 2 YYo )2+ (z+.)2 This, after evaluating the integrals, equals zero. NACA TM 1421 A third region of the flow field contains points slightlyr behind and far to the! side of the3 cube, where forward N~ach cones from the points (1) include part, but not all, of the cube. For this region, Hayes' method can be used to sho that the? potential again is zero. Th~is method is described in, Ch. IVC. It requires that the distance from the cube to any point P whre the potential is to be compulted must be Large compared to thne dimecnsions of the cube~. In addition, P must lie near the Mach cones emnanatingi from the singuilarities on the cube. P is then a point at some angle 8 (measured from the horizontal plane) on a distant cylindrical control surface surrounding the cubic she~ll. An "equivalent lineal dis tribution" of singularities is formd by finding the singularity strength intercepted from the cube by a set of parallel_ planes originating at angle 6 on the control cylinder and inclined at the Mach angle to the free streak direction. The sinigularities intercepted by a given Mach plan are lumped together at the intersection of the Mlach plane and the axis of the cylinder, such that the total strength of the equivalent distribution is equal to the total. strength of the original distribution. After determining the strength (h) of the equivalent lineal distribution which represents the cubic shell for a fixed 6, the effect of' all those singularities which influence the flow field att P can be summed. Hayes writes the expression for h as h = +f gz sin 9 gy cos 6 (jb2) where f is the source strength. (per unit length), gzI p the circulation strength (per unit length) of the lifting elements, and g 0ll the circu lation strengh of the side force elements. Figure 5b6 indicates the notation to be used in describing the geomretry of the intersections of the Mach planes with planes containing thie x,y,z axes. The Mlach plane is inclined to the axis of the control cylinder at the Mach angle CL = sinl(1/M) and it is tangtent to a cross section of the cylinder at angle 6.* The trace of the Masch plane in a horizontal (xy) plane~ is inclined to a normal to the flow direction at angle 6, where tan 6 = cot 4 cos 9 = 8 cos 8. Th trace of the M~ach plane in a vertical (xz) plane f'onns an angle a with a line parallel to the zaxis, where tan a = p sin 8. 1IACA TM 1421 TrACE OF A ACH PLANE /N THE XZ PLAVE p I /^> TRACE OF AA CH / PLA'E PV THE Xy PLANE / Fig. 5b6 With this brief description of Hayes' procedure in mind, an equiva lent lineal distribution of singularities is now to be computed for the specific case of the cubic Shell described previously. Figure 5b7a shows the intersections of two parallel Mach planes with the shell; the Hach planes are assumed to be separated by an infinitesimal distance. The case illustrated shows only three faces of the cube intersected by the Mach planes since the procedure would be the same if four faces were affected. In order to better define the geometry and notation, Fig. 5b7b shows the cubic shell as though it were cut along the corner edges and flattened out in one plane. Tj NACA TM 1421 /9 Fig. b7a b LIe di' FRONT SIDE (5/NVKS) BOT TOM Fig. 5b7b The net singularity strength cut out by these Mach planes must be lmnped along a length dx of the axis. The total source strength is the product of the strength per unit area (k) and the area intercepted from the top surface of the cube by the Mach planes: x_ xli xl f = k x1 dn, = k dx = k 1 dx cos( = tan 8 cos e [ 2; (5b5a) (The negative sign is inserted because 9 is in the second quadrant for the example shown, but f is positive.) The total lifting element strength is pUk multiplied by the area intercepted from the front face: Zi I = pUk 1 dn5 = pUkzi dz tan 6 p cos(n 67) Ata dx x, dx = pUk t a 1= pUk 2 1 6 cos tan20o p2sin 8 cos 9 (5b3b) NACA TM 1421 Again, a negative sign is inserted because cos 9 is negative while I should be positive. There are no forces on the side faces. In computing the strength of the equivalent lineal distribution from Eq. (5b2) it must be remembered that gp, gz from that formula are circulation strengths multiplied by 0; i.e., g = PS z pu y =pu Then h = +f g. sin 9 gy cos 9 kx1 dx pUkx Cl dx =1 Li   sin 9 =0 (5b4) P cos 9 pU p2sin 0 cos That is, the net singularity strength is zero. This is true for all angles 8, and similar calculations show that it is also true for every station x along the cylinder axis. Therefore, the velocity potential is zero at all distant points for which Hayes' method is applicable. There remains to find the velocity potential in the neighborhood of the shell. The cube may be subdivided into smaller cubic shells, each similar to the original. Singularities on interfaces of adjoining shells then cancel so the net singularity distribution is unchanged. Those shells which lie behind and outside the forward Mach cone from any point cannot influence the velocity potential at that point. It was shown earlier that those shells which lie completely inside the for ward Mach cone from the point also do not influence the potential there. Therefore, only those shells lying along the forward Mach cone need be considered. However, these may be further subdivided into cubic shells of elementary proportions so that the distance from the point to any one of the shells is very large compared to the dimensions of that shell. Then the analysis based on Hayes' procedure shows that these shells do not contribute to the velocity potential at the point either. This indi cates that the velocity potential is zero everywhere outside the cubic shell. To find the velocity perturbations inside the shell, again consider it divided into smaller shells. None of these except the one containing the point P can influence the potential at P according to the preceding analysis. Therefore, all of the small shells located more than a dis tance E ahead of P can be removed without affecting the potential at P. The forward Mach cone from P then intersects only the front face of the NACA ITM 1421 remaining part of the original cube, so that, effectively, P is aware only of an infinite distribution of lifting elements. Since this result is independent of the location of P inside the original cubic shell, the downwash inside the shell must be constant. Case B. Source Distribution Face formal to the Free Stream Consider now a cube with lifting elements of strength pUk on the top face and pUk on the bottom'face, with side force elements of strength pUk, pUk on the side faces, and with sources of strength 02k on the front face and p2k on the rear face. 2 eLmeEvrsfpUk) Fig. 5b8 First, the potential ahead of the foremost Mach waves of the cube is, of course, zero. At a downstream point whose forward Mach cone includes all of the sources and lifting elements the potential is NACA m 1421 k (z (xx)dx, dyo E (Y YYo)2+(z )2(x x) 2 2 [( Yo)2+(zT)2] (x + , r_(y xxo)dxo d0z+ x (y(y yy))2 _y y) + zo_ xz)2o 0 (y )2 + (Z zo) (Y + x xo) dxo, dzo, 3E~y +' ( )2 +( z _z) 2 x z )2+(z)o 2(xxo)2_ 2(y+y) 2+ z zo)2] 2k dyy d 4 zo y xo)2 .x X), z d & I SO (5b5) ( +  [ + 3 (z zJ 12 Carrying out the integration, it is found that S= 2k for < J < y andz < z < z Elsewhere ( = 0 NACA TM 1421 By Hayes' procedure, when forward Mach cones from distant points include only part of the singularities, the potential at those points is the same as would be contributed by a lineal distribution whose strength, h, can be computed in the manner described previously. For Mach planes intersecting the cube in the same location illustrated for another case in Fig. 3b7, one finds that X1 dx xli dx Xdxl f = k S = pk 1S, 1 = pUk x (5b6) sin 6 cos 0 p sin e' cos 6 and so Xh dx px dx ex d h = k 1 pUk 1 cos LpUk sin 6=0 sin 9 cos 9 pU p sin / pU cos (5b7) Thus, the potential due to the cubic shell is zero at all distant points of the flow field which lie near the Mach cone of the shell. In the neighborhood of the cube, the same arguments used for the first cubic shell show that the perturbation velocities for this case are zero there also. Therefore, the perturbation velocities are proved to be zero in every region of the flow field external to the cubic shell. To find the potential at a point inside the shell, the shell is sub divided as before into smaller shells, each similar to the original. The analysis just completed shows that the velocity perturbations at P cannot be influenced by any of these shells except the one containing P. There fore, all of the small shells located more than a distance e ahead of P can be removed. The net singularity, strength intersected by, the forward 'Tach cone frrom P then includes only sources on the front face of the remaining group of cubes. Effectively, then, conditions at P are the same as behind an infinite distribution of sources of constant intensity. This result is independent of the location of P inside the cubic shell, so the pressure must be constant inside the shell and the potential is of the form 0 = ex. NJACA m 1421 CHAPTER IV. THE EVALUATION OF DRAG A. THE "CLOSE" AND THE "DISTANT" VIEWPOINTS The nonviscous drag for a wing and body moving at supersonic speeds may be obtained from two different points of view(1), using linearized theory. First, the drag can be evaluated by integrating the local pres sure times frontal area over the wing and body surfaces. Second, the drag can be evaluated from momentum or energy considerations involving the flow field at a great distance from the aircraft. These two pro cedures are actually variations of the same basic method. In the latter case part of the drag due to lift is associated with the production of kinetic energy in the trailing vortex system, and is called "vortex drag." This drag is Identical with that produced by the same spanwise lift distribution in an incompressible flow, (frequently called "induced drag"). The remainder of the drag due to lift and all of the drag due to thickness is associated with the production of energy near the surface of a downstream Mach cone whose vertex is in or near the aircraft. This is called wave drag, and the associated energy is half kinetic and half potentially). The wave drag plus the vortex drag is equal to the drag evaluated at the wing and body surfaces by the first method. (It may be necessary to retain nonlinear terms in the expression for pressure coefficient to get this agreement.) The momentum theorem is utilized in both of the above methods but different "control surfaces" are used. In the first case the control surface is close to the aircraft surface, but in the second case the control surface is a distant one. For example Hayes(1) uses a circular cylinder with axis passing through the aircraft and parallel to the free stream direction. The radius of the cylinder is chosen to be very large compared to the aircraft dimensions since this simplifies the calculations. The wave drag. of the aircraft is then evaluated from the rate at which momentum (in the free stream direction) is carried across the sur face of the cylinder. (If the control surface had been chosen as a sur face containing streamlines instead of a perfect cylinder, then the wave drag would have appeared as pressure on the streamline surface.) The cylindrical control surface is closed far downstream by a plane normal to the flow direction. The vortex drag is then determined, as in incompressible flow, by the rate at which the kinetic energy of the trailing vortex system passes through this plane, or alternatively through momentum and pressure considerations. ure considerations. NACA ITM 1421 B. GENERAL MOMENTUM THEOREM FOR EVALUATION OF DRAG In the present section a momentum integral for the drag, as given by linearized theory, will be derived (Eqs. 4b53,53). The drag will be given as an integral over an arbitrary control surface enclosing the solid. The integrand is a quadratic expression in the velocity compo nents as given by linearized theory. First a more general momentum integral will be considered. Consider a control surface S enclosing a solid (Fig. 4bl). A surface element on S of area dS will be represented by its outward normal din where the length of dn is equal to the area of the surface element. Thus dn = (dS)n if n is the outward normal of unit length. Let the hydrodynamical stress U  0 Fig. 4bl tensor be denoted by a, and region outside S. Then f = a dn = Force exerted let I be the region inside S and II the by II on I across surface element (4bl) If a system of coordinates xl, X2, x3 is chosen dn may be repre sented by its three components (dn)i and a by a 35x3 matrix oaij. The above equation may then be written 5 fi = (a dn)i => .ij(dn) j j=l (4b2) where (a dn)i is the ith component of the force. NACA TM 1421 For a nonviscous fluid the only hydrodynamical force is the pres sure p and the stress tensor is a = pl = (Pij) (4b5) where I is the identity tensor whose matrix is the Kronecker delta bij. In this case the force across the element is f = p(I dn) = p dh (4b4) or fi = p(dn)i The bydrodynamical momentum equation states that the stress tensor is balanced by flowofmomentum tensor. (This is actually a restatement of Newton's law that force = (mass) times (acceleration).) To define the flowofmomentum tensor we first introduce the concept of a dyadic _> I> product of two vectors. Let a and b be two vectors with components (ai) and bi). The dyadic product is then the tensor whose ij component (a b is aib i.e. a b = ai bj (4b5) Note that if c is any vector then ( b) = ((aibq)c = (aibjCj) = ( ) (b6) where b c is the ordinary dot product. The flow of momentum tensor is then the dyadic product of pq (momentum per unit volume) and q velocity: Flowofmomentum tensor = pq q ) (4b7) NACA TM 1421 Its physical interpretation may be seen by applying this tensor to the normal dn > = P> (pq q)dn = p(q dn) =kMomentum transported through dS per unit time. (4b8) The basic momentum equation for stationary flow for a surface Sl which does not enclose a body is then S(pq q)dn = f a dn (4b9) S1 JS1 This is analogous to the law of conservation of mass which states that Spq dn = 0 (4b10) S1 Consider now the composite surface consisting of the surface S in Fig. 4bl and the body surface E. Let dn denote normals on E which point outwards with respect to the body (i.e. into region I). From the defini tion of the stress tensor a F = Total force exerted by fluid on body = a dn (kbll) Since the flow through E is zero one obtains by applying Eq. (4b9) to the composite surface 81 = S + E S(p )dq. =Lf an adn (4b12) S S fE The minus sign in the last term is due to the convention that on the surface E the quantity dn denotes the inward normal with respect to the region I. Comparing Eqs. (4b11) and (4b12) one obtains NACA Tm 1421 F = (pq )d + a dn S S (4b15) This is the fundamental momentum formula which gives the total hydro dynamical force on the solid as an integral over a control surface enclosing the solid. Note that in Eq. (4bll), the force is given by an integral of the stresses on the body surface. This is the "close" point of view for evaluating the force. Eq. (4b15) shows how the same force may be evalu ated from the distant point of view. A slight modification of Eq. (4b15) will be needed later. Denote the flow quantities at infinity as follows q, p, p, a at infinity = U, po, p0, O0, respectively (4bl4) The difference between a flow quantity and its value at infinity will be denoted by a "prime." Thus q =q U, p' =p p, po = p p0, a' = a o From the continuity equation (Eq. 4b10) it follows that (kbl5) (4bl6a) / (pU q)dn = 0 JS Furthermore, since a. = Constant aCT dn = 0 Subtracting Eqs. (4b16a, b) from Eq. (4b15) one obtains F = (pq' q)dn + a' Cn S B (4bl6b) (4b17a) U < pq dn = S IJACA TI1 1421 where, for a nonviscous fluid, 0' dn = p' dn (4b17b) This is the fundamental momentLum formula in terms of perturbation quantities. iHote that the latter are not assumed to be small. The drag is the component of F in the free stream direction. We shall take this direction as the xdirection and use the following notation. L' = i, q = (uvw), q = (u',v',w') (4b18) where u = U + u', v' = V, w' = w From Eq. (4b17a) then follows the fundamental momentum formula for drag: Drag = F i = pu'q in + i f a' din (4b19) The momentum integrals may be further simplified for special choices of the control surface S, in particular by letting S recede to infinity. However, we shall first derive an approximate form of the drag formula, valid within linearized theory. In the following section this linearized formula will then be specialized to a special infinitely distant control surface (method of Hayes(l)). Inviscid SecondOrder Drag It will be shown below that for a thin or slender body the largest contribution to the drag may be evaluated by an integral of a quadratic expression of the linearized perturbation velocities. It is usually stated that the drag is of second order. However, it should be remembered that the values of the perturbation velocities are computed from first order (linearized) theory. The result is a formula for drag according to firstorder theory. The tdrm "secondorder drag" refers to the fact [ACA TM 1421 the integrand is quadratic in u', v' and w' and hence of second order if u', v' and w' are themselves of first order. Furthermore, the second order correction to u', v' and w' will contribute nothing to the secorid order expression for drag. The final formula is given by Eqs. (4b55,34) and the reader interested only in the final result may skip the deriva tion now presented below. We shall first assume nonviscous flow, so that the stress tensor is given by Eq. (4b5). Furthermore, we shall assume that the solid is characterized by a parameter E, which is small, e.g. the fineness or thickness ratio. We shall furthermore assume that the flow quantities may be expressed by power series in e: u = U + EUl + Eu2 + (4b20) v = V1 + E2v2 + . w = Ew + E2p +2 . p = p + Ep + Ep + . 0 = po + epl + E 2 P2 + . Such an expansion is valid at a distance from the body. It should be remembered, however, that in slender body theory, terms involving log e are of importance very near the body. The coefficients of e are the first order terms and are given by linearized theory. The coefficients of E2 are the second order terms, etc. The lowest order term in the expression for the drag will now be found using the isentropic pressuredensity relation and Bernoulli's law. From isentropy it follows that density is a function of pressure alone. One defines (p) = a2 S)constant entropy where a is the isentropic speed of sound. Then p = p+ p p + (4b21) ao HACA TM 1421 from which then follows P1 p^ 1 a 2 Bernoulli's law may be written d(u2 + v2 + w2) + dP = 2 where p P = p T^o (u' + U)2 + 2 + w2 + 2P = U2 Using Eq. (4b21) P may be expanded to second order 1 p '  O 02o p0^ \Poao (4b24) P 2poa)2 PO 2poao 2 Expanding the terms in Eq. (4b25) to second order one obtains S2 u+ 2 + v+ 2 + wl2 1 EUui + 2 + E + p o + E2P2  = 0 2poa3 Collecting the terms of order e one finds the linearized Bernoulli's law PI + PoulU = 0 (4b22) P dp' p + 0 a02 (4b25) (4b25) NACA TM 1421 Comparing with Eq. (kb22) one sees that Uu1 P = PO 2 O* The terms of order ield the following expression for p2 The terms of order yield, the following ex'pressioni for p2 ul2 + v2 l+ w2 p0ulM2 Po0Uu2 + Po 22 + + 2 2 = 0 (4b27) where Eq. (4b25) has been used and M=U ao In the momentum formula, Eq. (4b17), the stress and momentum flow tensors may be combined to form a tensor A A = pq' q p'I Using Eqs. (4b20, 25, 26) one may evaluate All All = pu'(U + u') p' = E (PoulU + p E 2(pou2U + plu1U + pu2 + p2 p p ul2 12 + w2 = 2P Uu2 M2u1 + U12 Uu2 +Vi2 1 Finally, A E11 = oP [M2 lu2 + V12 + W2 Ul2M2 2) (kb28) Simrnilarly A12 = pu'v' u C2PoulV1 (kb26) (4b29) NACA Thi 1421 A13 = pu'w' I 2 2pou1w (4b50) Since only the first row (All, A12, A15) enters in the drag computation we have proved the following: 1. The dominant term in the drag formula is of second order in E 2. The integrand in the drag formula is, to second order, a second degree polynomial in the first order velocity perturbations. The velocity perturbations of second order, or pressure and density perturbations of second order, do not t ter into this expression. Thus while drag is of second order, it may be computed on the basis of first order theory linearizedd theory). On the cther hand, one may easily check from the above expressions that in general the lift has a first order term. Furthermore to compute lift to second order one needs to know u2, that is, u to second order. In the remainder of this report we shall only be concerned with the drag as given by linearized theory. It is then convenient to introduce a change of notation. We shall let u, v, w stand fcr the linearized velocity perturbation; in other words Eul, EVJ, Ew1 are replaced by u, v, w (4b1l) Furthermore a velocity potential 0 will be introduced such that Grad = u, v, w (4b52) The above results may then be summarized as follows. The drag to second order is given by the formula D = Al dn (4b55) S where S = Control surface enclosing the body NACA TM 1421 A = (All, A12, A1) A = +Po 0u2 + v2 + W2) p2 = M2 1 Ali + Pou (4b34) A12 = POuv A5 = Pouw and u, v and w are the components of the perturbation velocity given by linearized theory. C. HAYES METHOD FOR DRAG EVALUATION The method developed by Hayes in Ref. 1 consists in applying the drag formula Eq. (4b55) to a special control surface, a truncated cir cular cylinder, surrounding the body and in considering the limiting case when the control surface recedes to infinity. The general momentum integral for the drag then assumes a simplified form. (This results in certain simplifications in the integrand.) Furthermore, if the body is represented by singularities (sources, lifting elements, etc.) as dis cussed in Ch. III, the velocities at large distances may be represented very simply in terms of the strength of the singularities. As a result the drag may also be represented as an integral over the singularities (distribution of source strength, etc.). This result of Hayes' gener alizes a previous result by von Kirmain(7) for a body of revolution. First a somewhat detailed demonstration of the method of Hayes will be given for the case of a lineal source distribution. This part may be skipped by a reader not interested in mathematical details. Then the results of Hayes and related results will be stated in intuitive terms for general threedimensional distribution of sources, lifting elements and sideforce elements. Detailed proofs will not be given. However, the results may be proved by methods closely analogous to the method exhibited for the case of a lineal source distribution. Hayes Method for Lineal Source Distribution We shall consider a distribution of sources along the xaxis between x = 0C and x = L. The corresponding solid is then a body of revolution. The source strength will be denoted by f. It will be assumed that f(0) = 0, f(L) = 0 (4c1) NACA TM 1421 These assumptions lead to certain restrictions on the body shape. Let the radius of the body be r(x). The cross sectional area S(x) is then nr2(x). Since f(x) = U S'(x), f(0) means that r(0) r'(0) = 0. This is fulfilled if r ~ xn, n > 1/2 near the origin. In particular, f(0) is equal to zero if the body starts in a point with finite slope, i.e. r ~ x near x = 0. The analogous condition at x = L insures f(L) = 0. In addition, f(L) = 0 if the body ends smoothly in a cylinder with con stant radius, i.e. if S(x) = Constant for x l L and S'(x) is continuous and hence zero at x = L. It will be indicated in the proof below why the restrictions on f are necessary. Expression for Velocities The potential due to the source distribution is then l xpr ^xQ1r) = 2 f (4c2) f(E)d (x t)2 02r2 where r2 = y2 + z2. For x pr L the upper limit may be replaced by L. Using the condition f(0) = 0 one finds by partial integration of Eq. (4c2) and differentiation that the perturbation velocities are xPr ,, 1 ^ x  21 0 r 2~tr xfpr f'()dE (x t)2 p2r2 S(x g)f(g)d x 2r2 V(x ) r In Eqs. (4c3), the upper limit is replaced by L for x pr >j L. We shall introduce the notation = x ) Then t = 1 on the downstream Mach cone from x =  and 0 < t <. 1 inside this Mach cone. For x "r > L one may also write the velocity components as (4c5a) (4c5b) NACA TM 1421 x 1= L f() [(x )2 P2r2] 3/2(x S)dt L 2(/ p5/2 L f()(x ) 21 t 2 3/2d = pL r 1 r 2 d 21t fr ^xs2?? p1d( 12t~ L / 0 f L _2 t & =IJFLf(W)X E) _tti  (4c5a) (k4c5b) Hayes' Control Surface Following Hayes we now introduce the control surface shown in 4cl. It consists of a circular cylinder of radius rl, truncated front disc x = Constant < 0 and a rear disc x = xl > L. The drag FORWlARD o/SC  y REAR DISC Fig. 4c1 Fig. by a NACA TM 1421 integral (Eq. 4b55) will be evaluated for this control surface as rl and x, tend to infinity. The ratio between xl and rl will be determined later in such a way that the contribution of the rear disc to the drag will vanish in the limit. Contribution of Rear Disc According to Eqs. (4b55,54) the contribution of the rear disc to the drag is D = rl (p2 x2 + r 2)2r dr 0 (4c6) The velocity components may be evaluated as follows. Write f(g) as a difference of two positive functions f(0) = f+(0) f_(g), f+(S), f(6) o 0 (4c7) Then by the mean value theorem and Eqs. (4c5a,b) L 0 f+(S)dt L 0 2x 3)2(l tt5 2) 2mx ) (l t 5/2 2% 2 21 te22) where 0 : EY, 2 L. A similar expression is valid for .r Note that in Eq. (4c8) the continuous source distribution is replaced by a posi tive source at t and a sink at g2. However, 93 and t2 depend on x and r. As is easily seen 0 x L tL i i = 2, 5 (4c8) (4c 9) ff()dS 1 : 1 + tt 1 + tt2 < 2 NACA Tm 1421 Hence, replacing E and E2 by L increases the absolute magnitude of both terms in Eq. (4c8). Hence, on the rear disc x = xl, 0 r rl, X2 < 22A (4c10) (xl L) 1 tL) where A is independent of x, and rl, and 2r, rp2x2 dr A fro1 x, r L 1 20 dr (4cil) O (x1 L)20 xi L ( 2)5 xl L If one puts y = 1 tL 2, then dy = 2tL dtL = t^ 2p dr/(x L). Hence Sr_20ar A 12 dy A I I (4c12) o (xl L)2 y1 y5 2(x/ L)217A where yl = 1 (1 e )2 = 2E1 E12 and el is explained in Fig. 4c1. Equation (4c12) may be written rB rpx2dr< B C (4c15) o (xi L) 212 r122 where C is independent of ri and x1 for rl and x1 sufficiently large; e is explained in Fig. 4c1. The fact that x/(x1 L) l, e/e1l as r :. has been used above. It is then clear that if e is constant or if E = r", a < 1, then the integrand in Eq. (4c15) tends to zero as rc. A similar estimate may be shown for fyr2dx. A comparison with Eq. (4c6) shows that: NACA TM 1421 Contribution of rear disc to drag is zero even if E decreases as r increases. However, E should decrease more slowly than r. Since the distance BC is of the order er it follows that this distance becomes infinite in the limit. Contribution of Cylindrical Part Since the contribution of the forward disc to the drag integral is identically zero it follows then in the limit rl, x1m the entire drag contribution comes from the cylindrical part, provided E varies as pre scribed above. Thus D = Limit D2 (4c14) where D2, the contribution of the cylindrical part, is D2 = po2irlI x+pr1E x r dx' (4c15) o3r, Note that the radial component Ai. of the vector A1 in the drag formula Eq. (kb4) is pdx r.) In the above equation 1/1 E has been replaced by 1 + e which may be done without loss of generality. To evaluate D2, we write Eq. (4c5a,b) in the following form 1 x_,L r1) 1 x 2n 021 1 V l + 20rl (4c16) S= x',L f'(f2) xf + or 2 2Ko x' g2 + 2pr, Pr1 where x' = x pr,. fNACA TM 1421 41 The upper limit is x' for x' L and L for x' L. Hence p o erl x1'L xl'L 1iF) 20pr, 2 = 0 0 2 x' x' 2 x' + 2prl x' + prl t2 r  dl 2 dx' (4c17) x' 2 + 2pr, Pr1 j Limiting Case for Infinitely Distant Control Surface We shall now evaluate D2 as rl. The three ratios within the second bracket all tend to unity as rl:o? and may hence be neglected in the limit. Note that this approximation implies 1 1 1 ,1 1 (4c18a) (x 12 p2r2 x o) pr f2pr x' 1 20r Furthermore, applying the same approximation to Eq. (4c16) one obtains that Or ~ 0Ox (4c18b) r and x both vanish as l/2f3rl. Their ratio, however, is given by the above equation. The corresponding relation with or replaced by Oy is exact for twodimensional flow. Thus the flow is approximately two dimensional at large distances near the Mach cone from the leading edge (E small, i.e. tt almost unity for 0 S L). Hence D = Limit D2 I (4c19a) 41r where prl x',L x',L f'f l)f'( 2) d d2dx' I = /$ d/ dx' \j_ J0 JQ J ^x 1 ^x, 2 NACA TM 1421 The domain of integration is a region in x', l, t2 space whose cross section for x' = Constant is the square 0 1 & x', 0 E2 :! x' for 0 5 x' L and the square 0 < l, E2 L for L x' E per,. Let I, be the integral where 0 y L and 12 the integral over the domain L y per,. Since the integrand is symmetric in 1 and t2, half its value is obtained by integrating only over the triangle ABC in the k1, 2 plane (i.e. l < t2) as shown in Fig. 4c2. In evaluating I over B C ^______________ V A X/ Fig. 4c2 its domain (a truncated triangular cylinder with base at x = L and vertex at x' = 0) we shall first integrate along a line parallel to the yaxis. For tl' S2 fixed this line is inside the pyramid only when ki, x' < L. Sl may vary inside the triangle between 0 and E2, and for t2 any value between 0 and L may be chosen. Hence Il may be written pL n2 PL f gl 2a II = 2 2L ft N fIj2 dX1 dt1 dk2 (4c20) 0 0 S2 F xt ^ ~2 The integral 12 is (domain is triangular cylinder) 0Er 0 L t2 f'(1 2f'S l2 = 2 a/ / l 2did dx' (4c21) "L J0 0 ^x' El ^x' (2 NACA TM 1421 Interchanging the order as above one obtains I = Il+12 = 2 L t2 ft 2 rlf'_x) x_ log C dt, d2 fo (o L f t t t2 id (4c22) Here C is a constant and it has been introduced under the assumption that fL f f'(E)dt = 0, which, since f(0) = 0 means f(L) = 0 J0 (cf. Eq. 4c1). (lcute that the limit of integration for t1 may be replaced by L if the factor 2 is omitted.) Now ,ppe r l C + 21p e r l p ,2) Er log C = [0x' E) E2 +( 1) (x!;2)] t2 = log(t2 91) + log2r1E2+2 (1 1)(w Q (4.25) C Hence if one chooses C = 43Erl, the second term will tend to log 1 = 0 as r74. Note that for this it is essential that er1 as ri)> (cf. pg.TV16). In other words the simplicity of the proof depends on the fact that e0 as rl> (cf. Eqs. 4cl8). On the other hand e may not tend to zero so fast that erl remains bounded. In this case the above proof would be invalidated. Actually a drag contribution would come from the rear disc in that case. By combining the Eqs. 4c19a, 22, 25 one obtains the final drag formula D = pOfLf 2 1 Qd)a d9 42 (c24) NACA TM 1421 This is von Karman's drag formula for a lineal source distribution such that f(D) = f(L) = 0. It has been derived above by the method of Hayes. This derivation has the advantage that it may be extended immediately to cases of a more general distribution of singularities. Such generalizations will now be discussed. General ThreeDimensional Source Distributions We shall now consider a more general case of a spatial distribution of sources. It will still be assumed that no lifting or side force ele ments are present. The source strength will be denoted by f(x,y,z). It will be assumed that f = 0 outside a certain finite region V. A special case is a planar distribution, say in the plane z = 0 in which case f(x,y,z) = f2(x,y)8(z). Another special case is the lineal distri bution on the xaxis which was discussed above. In this case f(x,y,z) = fl(x)S(r). It will be shown below how in a certain sense the drag evaluation for the general threedimensional case may be reduced to a consideration of certain equivalent lineal distributions. In the course of this discussion certain restrictions on f(x,y,z) will be made in addition to the requirement that it vanish outside a finite region. Consider a line in the streamwise direction passing through V. The position of the line which will be taken as the xaxis is actually arbi trary, but for practical purposes it will be assumed that it is "well centered." This is, of course, a somewhat vague requirement. However, if for example f has rotational symmetry, the xaxis will be its axis of symmetry. On the xaxis choose as origin a point, 0, whose downstream Mach cone contains V. For convenience choose this point as far down stream as possible. Also choose the point, L, for convenience as far upstream as possible, whose upstream Mach cone contains V. An equivalent requirement is that the downstream Mach cone from L is contained within the downstream Mach cones from every point in V. Let the value of x at point L be L. Thus the downstream Mach cone from x = 0 and the upstream Mach cone from x = L touch but do not penetrate V. We now introduce a control surface and define e and el as in the lineal case (cf. Fig. 4c1). This is shown in Fig. 4c5. It will be assumed that rl and x, tend to infinity as described in discussion of the lineal case. IJACA TM 1421 45 4X Fig. 4c5: Hayes control surface in threedimensional space That is as x and r1 tend to infinity e and el will tend to zero. In that sense the line AC will come arbitrarily near the Mach cone from the origin. On the other hand c and E Will tend to zero slower than 1/r1 so that the line AC becomes infinitely long as rlm. By the same methods that were used in the lineal case, it may be easily seen that the contribution of the rear disc, x = xl, becomes zero in the limit. All the drag thus comes from a portion on the cylindrical surface arbitrarily near the Mach cone from the origin and is hence pure wave drag. To evaluate the drag contribution from the cylindrical surface we introduce cylindrical coordinates x, r, 9 where x = r cos 6, z = r sin (4c25) NACA TM 1421 Let the drag contribution of a strip on the cylinder between 9 = Bo and 9 = o00 + AO be AD. We define dD = Drag contribution per unit angle = lim r as AO ;0 (4c26a) To9 AO Then r2jx D = Total drag = d d (4c26b) J 0 dO Consider now a fixed meridian plane 6 = 09, and a point P = (x,,rlo) on the cylinder between A and C (Fig. 4c5). The potential O(P) depends on the contribution from all sources inside the upstream Mach cone from P. The contribution from a source at Q = (t,rjt) is proportional to the source strength f(Q) and inversely proportional to the hyperbolic dis tance rh(P,Q) between P and Q where rh2 = (xo )2 p2 (r1cos 00o TI)2 + (rl sin 9 t)2] (4c27) This hyperbolic distance is constant on hyperboloids of revolution with r = rl, 0 = o00 as axis. Consider now the sources between two such hyper boloids which intersect the xaxis at x = t and x = + d. To evaluate the contribution to O(P) of these sources one may transfer their total source strength to the axis. In this way the distribution in V is replaced by an equivalent lineal distribution i.e. by an equivalent body of revolution. So far this lineal distribution depends on xo and r, as well as o00. Consider now, still for fixed 0 = o00, the limit as rl>co. Then the hyperboloids may be replaced by Mach planes which intersect the meridian plane 0 = o00 orthogonally along Mach lines. Note that for this it is necessary that as rlw> any point between A and C comes arbi trarily near the downstream Mach cone from the origin in the sense described above. The source strength between two such neighboring planes NACA T4 1421 e0=q X=o Fig. k4c4: Evaluation of O(P) may then be transferred to the xaxis as above. However, in this limiting case the resulting equivalent body of revolution depends at most on 0o. It becomes independent of r1 and x,,. The corresponding lineal source distribution will be denoted by f(x;9o). A consequence of the independ ence of x. and ri is that f(x;9) may be used for computing 0, and as well as 0 at P. In general it may not be used for computing e9. Clearly 00 is zero for a lineal distribution, whereas the 00 resulting from the original volume distribution is not. On the other hand 9 is not needed for drag evaluation on the cylindrical surface. Since or and x may be computed from the equivalent body of revolu tion for fixed 6 it follows that dD/d9 may be computed in exactly the same way as the drag of a body of revolution was computed. The result will differ from Eq. (4c24) only by a factor 2it. Hence we have proved the following: The drag D of a volume distribution of sources of strength f(t,q,() is given by the formulas )(= L. 48 MACA TM 1421 21r D = de (4c28a Jo de dL = L t2 1 ; 2;)og (2 1 2 (4c28b f(S;9)dt = f f f(Q)dQ (4c28c v(6,e) where V(g,9o) is the region contained between two Mach planes perpen dicular to 9 = 0o and intersecting the xaxis at x = t and x = g + de. This result was obtained by Hayes in Ref. 1. It is thus seen how Hayes' derivation of von Karman's drag formula for bodies of revolution admits an easy generalization to the general threedimensional case. This proof obviously presupposes the following requirement on the strength distribution f(Q) in addition to the requirement that it vanish outside a finite volume: f(Q) must be such that for each 9 f(x;9) sat isfies the same requirements as f(x) in the lineal case. In particular for each 9: f(0;9) = f(L;9) = 0 and f(x;9) must be differentiable with respect to x. If f(Q) has rotational symmetry, i.e. depends on r and x only then it may obviously be replaced by one equivalent lineal distribution, independent of 9, for computing the distant flow field and the drag. In the special case when f(Q) is lineal to begin with, Eqs. (4c28) reduce to the previously established formula (Eq. 4c24). Extension to Include Lift and Side Force Elements For simplicity only sources have been considered in the preceding development. However lift and side force elements can be included and were included by Hayes in his original report. We will not go into the details here, but merely indicate the final results, since the funda mental ideas of the method have been illustrated in the discussion of source distributions. Following Hayes we define a function h such that h = f g ggsin 9 + gy cos 0) (4c29 NACA Tm 1421 where f = f(t;9) = Source strength pUgz/P = Z(;9) = Lifting element strength pUgy/P = s(;9) = Side force strength The term (g sin a + g cos e) is proportional to the component of force in the direction 0, and is the only component contributing to the wave drag in the Hayes calculation. Equation (4c28b), as extended to include lift and side force elements, is pL fL TO) e 2 OL, (h' gl;0)h' E2;O)10g (2 1 1d~ dE2 = L L h' (El;9)h'(2;9)log J2 d11 d2 (4c50) where h(t;9) is the equivalent lineal distribution (for a given station 0) of the original spatial distribution of singularities. This equation makes it possible to determine the wave drag of an arbitrary spatial system containing thickness and carrying both lift and side forces. In order to determine the total pressure drag of the system it is necessary to evaluate the vortex drag produced by the lift and side force. In Hayes method the vortex drag appears as a momentum outflow through and a pressure on the end of the cylindrical control surface. It can be evaluated by calculating this momentum and pressure or by determining the kinetic energy associated with the vortex system in the Trefftz plane. Since this is identical with the induced drag problem of incompressible flow, we will not discuss it further. D. LEADING EDGE SUCTION The evaluation of the drag of a lifting wing of zero thickness by integrating local pressure times frontal area over the wing surface is not theoreticallycomplete until leading edge suction is accounted for. 50 NACA TM 1421 This means that the infinite negative pressures acting on subsonic leading edges should be included. In practical applications this leading edge suction is sometimes discarded since in many cases only a fraction of the theoretical value is actually realized. However, from the distant viewpoint, leading edge suction cannot be isolated. This is true because there is no pointtopoint corre spondence between the close and the distant control surfaces. At the distant control surface the velocity field created by the wing leading edges is merged with the fields created by other areas on the wing and body. From the distant point of view leading edge suction is automatically assumed to be fully effective, and therefore it must be so assumed from the close viewpoint to get correspondence in the drag values. E. DISCO'TIIJUITIES IN LOADINGS For a planar wing, vortex drag is dependent only on the spanwise lift distribution. A discontinuity in the ordinates of this lift dis tribution produces a concentrated vortex of finite strength and infinite energy, which corresponds to infinite drag. Wave drag is similarly affected by discontinuities in loadings. For example, consider a distribution of sources on a streamwise line. If there is a discontinuity in source strength, then the drag evaluated on the distant control surface is infinite. To prove this, assume a source distribution with a discontinuity at the point x = X (see sketch). The velocity potential at a point (x,r) downstream of the rearward Mach cone from Y may be written I f xr f(t)d 1 E f ( ()d r = + 2 0 V(x _)2 p2r2 2x O (x _)2 2r2 xr f(dE(4e1) 3i r(K E p2r:J f2 (5) NACA TM 1421 51 The ucomponent of velocity at the point (x,r) is found by differentiating Eq. (4el) axially. (In order to avoid indeterminant forms in the differ entiation, the equation is first transformed by means of the relation = x f3r cosh u.) This process gives the result (assuming f(0) = 0): U j f 1'()d + xPr f2()d Af (x ) ax 21t o(x g)2 _p2r2 J (x E)2 p2r2 (x X)2 2r2 (4e2) where Af(x) = fl) f 2). At the distant control surface it previously was shown (Ch. IVC) that one need consider only conditions very near the Mach cones from the source distribution. Introducing the approximations used in Hayes' method (i.e., (x t)/pr t 1), Eq. (4e2) can be expressed Tf I( f e5)4 2r 2pr 0o + X fx' ( ) x' (x [ where x' = x pr and x' O pr. Since the radius of the control sur face is large compared to the length of the source distribution, the Mach cones originating at the sources are essentially plane waves when they intersect the control surface, so that the radial component of velocity (at the control surface) is (Eq. k4c18b) v = pu (ke4) The drag, being equal to the transport of horizontal momentum across the control surface, is proportional to the product of u and v integrated axially along the control surface. From Eqs. (4e5) and (4e4) it is readily seen that the drag includes a term of the form dxt J^ (x' ) The integral is nonconvergent. An infinite drag contribution therefore results from a discontinuity in the strength of the source distribution. NACA TM 1421 F. THE USE OF SLENiDER BODY THEORY WITH THE DISTANT VIEWPOINT If slender body theory is applied, then the source strength is assumed proportional to the rate of change of crosssectional area, dS/dx, for a corresponding body of revolution. This means that infinite drag will be predicted (by the distant procedure) for all bodies of revolution having discontinuities in dS/dx. Such a prediction is, of course, incorrect, and the error is caused by the application of slender body theord to bodies which are not sufficiently smooth. The use of slender body theory requires that smoothness should be maintained at the nose and tail of the body and therefore dS/dx should be zero at these locations. In order that dS/dx should be zero at the nose or tail of a closed body of revolution it is necessary that the variation of body radius, R, with distance, d, from the nose or tail (1 (/2)+kc should be of the form R ~ d(2)+k where k > 0. This does not elimi nate blunt noses or tails entirely, but excludes "excessive" bluntness. (Note that the SearsHaack optimum shape is blunt.) The linearized theory requirement that all velocity perturbations be small theoretically excludes all bluntness, but this is unimportant if very small regions of the flow field are affected. Bodies which begin or end in cylinders also may satisfy the smooth ness requirements. For a body to be sufficiently smooth to permit the use of slender body theory, it is necessary to restrict the "short" wave length fluctu ations in the plot of crosssectional area versus length. The word "short" cannot be defined exactly here, but should probably apply to all wave lengths less than the body diameter times F2 1) Figure 4fl illustrates the effect of wave length on the accuracy of the slender body theory. The drag for an infinitely long corrugated cylinder according to strict linear theory was found by von Karman (7). Slender body theory is in good agreement with these results only where the reduced wave lengths are large compared to the cylinder radius. At the other extreme twodimensional theory is approached. It should be remembered that when the distant viewpoint is used the drag of a singularity distribution is evaluated. The body shape corresponding to the singularities may, be determined either by "exact" linear theory or approximated by slender body theory. For example in Fig. kf2 a specific source distribution is considered, and is inter preted as a "bump" on a cylinder by "exact" linear theory and by the slender body approximation. For this ratio of wave length to cylinder NACA TM 1421 COMPARISON OF THEORETICAL CALCULATIONS FOR DRAG OF CORRUGATED CYLINDER (A)PRATIO OF THE DRAG COMPUTED BY SLENDER BODY THEORY TO THE DRAG COA4PU/TED BY L /NEAR THEORY (B.) RATIO OF THE DRAG COMPUTED BY TWOD/MENSIONAL L/NEAR THEORY (ASS(UM/AlNG TWOD/MENS/IONAL FLOW/N EACH MERIDIAN PLANE) TO THE DRAG COMPUTED BY T/REED/MENS/ONAL LINEAR THEORY c 40 'K d S16 . I I o  Z\ PROFILE OF CYLI/NDER (Q: K ( CCROSSSEC7IONSA RE CIRCULAR) Cr 12  4 IN_________ __ __ __ k i RATIO OF REDUCED WAVE Z ENGTH TO CYL /IDER DIAMETER,, d (FOR LINEAR THEORY DRAG OF CORRUGATED CYLINDER, SEE PEF:7) Fig. if1 54 NACA TM 1421 Qt (0 ~ n. Zt N RI r kII c~co 41. Ell cz^ i ^$ M4P? ^^ 0^ fe ^ ? ~0 0 NACA TM 1421 diameter the bump shapes and locations are quite different. It is of interest however that the net volumes contained in the bumps are iden tical. This has been proved by Lagerstrom and Bleviss and generalized by Bleviss in Ref. 22. (This suggests that "volume elements" may retain their significance even when slender body theory does not apply.) G. THE DEPENDETJCE OF DRAG COEFFICIENT ON MACH RJMLBER Hayes(1) has pointed out that, for a distribution of singularities on a single streamwise line, the drag, evaluated from the distant view point, is independent of Mlach nuriber. If the singularities are sources, and slender body theory is applied, this indicates that the drag of a given body of revolution is independent of Mach number. However the application of slender body theory in conjunction with the distant view point requires that dS/dx = 0 at the tail of the body. Hayes' result is.therefore consistent with a fact previously deter mined, that the drag coefficient of a slender body satisfying the "closure" condition (dS/dx = 0 at the tail) is independent of Mach number. If the singularities are not confined to a single streamwise line, then the distant viewpoint gives a drag coefficient which varies with Mach number. This can be seen from the fact that the projection of the singularity distribution onto a single streamwise line varies with the inclination of the Mach planes used for the projection. H. SUPERPOSITION P1RO'CEDURES AND INTEPFERENCE DRAG In all the developments discussed in this report the linearized. supersonic flow equation is used. This means that one flow field and the lift (or volume) distribution which causes it can be superimposed on a second flow field with its corresponding lift (or volume) distri bution. If the individual flow fields satisfy the linearized flow equation, then their sum does also. For example, let a pressure field, pl, correspond to a downwash field, a.,, and a second pressure field, p2, ?orrespond to a second down wash field, p2, then the pressure field pl + p2 corresponds to the down wash field al + a2. However, the drag of the sum of the two fields is not in general the sum of the drags of the individual fields. For example, the drag of the first field would be D1 = J 1 dS, where the integration extends NACA TM 1421 over the wing and body surfaces, and similarly the drag of the second field is D2 = Na2 dS. However, the drag of the combination is D1+2 = fP1 + P2 g + a2)dS. The terms involving cross products give the interference drag, Di = 1(pia2 + p2al)dS. I. CiPTHOGlIAL DISTRIBUTIONS A1D DRAG REDUCTION PROCEDURES If the interference drag is zero then the two distributions are said to be orthogonal. The use of orthogonal distributions for reducing drag has been studied in Refs. 8, 9, 10, and 11. For example consider two types of lift distributions which are orthogonal and assume that each one carries a net lift. It has been shown (see for example Ref. 9) that some combination of the two will carry a given total lift with less drag than would be produced if either one of the individual types of distribution carried all of the lift. On the other hand, any given (nonoptimum) lift distribution can be improved by adding the proper amount of a nonorthogonal type of dis tribution which carries zero net lift. The improvement is obtained by utilizing negative interference drag. This can be seen as follows. The total drag of the combination is the sum of the individual drags plus the interference drag. The interference drag can always be made negative by proper choice of the sign of the distribution that carries zero net lift. Also, since the strength of the zero lift distribution enters linearly into the interference drag, but enters quadratically into its individual drag, the magnitude can be so chosen that the interference drag dominates. Thus the total drag of the combination can be made less than the drag of the given (nonoptimum) lift distribution. J. THE PHYSICAL SIGNIFICANJCE OF INTERFERENCE DRAG It has been stated that the interference drag, Di, is pl2 + P2ca, dSE where the subscripts designate the two flow fields which have been super imposed, and the integration is to be carried over all surfaces. Assume that both flow fields are produced by thickness distributions. Then the a values are the body surface inclinations which correspond to dS/dx, the rate of change of crosEectional area for the body. The fpila2 dS NACA Ti 1421 gives the drag produced by the pressure field of the first body acting on the crosssectional area distribution of the second. The term p2al dS has a similar interpretation. Assume that both flow fields are produced by lift distributions. Then pla2 dS is the drag created by the downwash field of the second distribution acting on the lifting elements of the first distribution. (The surface which supports the lift corresponding to p1 must be inclined further because of the downwash due to p2) Let the first field be produced by a lift distribution and the second by a thickness distribution (a body). Then plax2 dS is the drag produced by the downwash field of the thickness distribution acting on the lift elements plus the drag caused by the pressure field of the lift distri bution acting on the crosssectional area distribution of the body. The f p2l, dS gives no contribution to the drag in this case. Assume that the first field is produced by a lift distribution and the second by a side force distribution. The p1a,2 dS is drag corre sponding to the downwash field of the side force distribution acting on the lift elements, while the p2a1 dS is produced by the sidewash field of the lift elements acting on the side force distribution. K. INITERFEREICE AMOUG LIFT, THICKIJESSI, AID SIDE FORCE DISTRIBUTIONS For planar distributions of lift and thickness (the lift being normal to the plane) there are no interference drag terms, and the two problems can be studied independently. However, for spatial distributions, inter ference generally exists. This has been discussed by Hayes, and the physical meaning of the interference irag has been discussed in the preceding sections. NACA TI 1421 Suppose that a source and a lifting element are located as shown in Fig. 4kl, the direction of flow being perpendicular to the page. Then the component of the lift which lies in the line connecting the two singularities causes all of the interference. If the lift element were located on the yaxis (corre sponding to a planar wing problem) there would be no interference. For lift and side force ele ments, as shown in Fig. 4k2, there is interference between the force components which lie in the line connecting the singularities, and also interference between the com ponents normal to the connecting line. If the side force element lies either on the yaxis or on the zaxis (as shown in Fig. 4k5a and b), then there is no interference. This can also be seen from symmetry considera tions, which show that the lift ele ment produces no sidewash at the side force element and similarly the side force element produces no downwash at the lift element. SOURCE, Fig. 4k1 L IF r LEMEN/T* _ _ S/DE FORCE EL E44EN 7';a oS Fl. aT. Fig. 4k2 Fig. 4k3a Fig. 4k5b E LEMENr  P1W y NACA TL 1421 L. REDUCTION OF DRAG DUE TO LIFT BY ADDITION OF A THICKNTESS DISTRIBUTION Consider the twodimensional system sketched in Fig. 411. The crosshatched area is a thickness distribution lying partly in the pres sure field of a flatplate wing. The relative geometry of the thickness distribution and the lifting surface are indicated in the figure. Also, the pressure distributions, relative to the twodimensional pressure 2amq/p, are shown in parentheses. Fig. 421 As long as the pressure field of the thickness distribution does not intersect the flatplate, the lift of the system is the same as for the flatplate by itself. On the other hand, the interference between the pressure field of the flatplate and the thickness distribution pro duces a negative drag contribution, so that the total drag of the system (omitting friction) is 121/2 percent less than the drag of the flat plate alone. Thus, the total lift in this case is unaffected by intro duction of the thickness distribution and a drag reduction is obtained. This example is related to the Busemann biplane. The result obtained illustrates the fact that, in the general case (nonplanar systems), sources and lifting elements have an interference drag. terference drag. NACA T1 1421 CHAPTER V. THE CRITERIA FOR DETERJIIJG OPTII.MUM DISTRIBUTIONS OF LIFT OR VOLUME ELEIIETTS ALOIE A. THE "C'MBlJED FLOW FIELD" CONCEPT The idea of the "combined flow field" was introduced by Munk'(12) and extended by R. T. Jones(15,14). Consider a distribution of lifting elements in a free stream of given velocity. A certain downwash velocity and pressure are produced at each point in the field. If the direction of the free stream is now reversed without moving the lift elements or altering the directions and magnitude of these lift contributions, then in general different downwash velocities and pressures are produced at each point in the field. Onehalf the sum of the downwash velocities produced at a given point in the forward and reverse flows is called the downwash velocity of the combined flow field at that point. A similar definition applies to sidewash velocity. Onehalf the difference of the pressures in the forward and reverse flows is called the pressure in the combined flow field. These definitions follow from the superposition of the perturbation velocity fields for forward and reverse flow. It should be remembered that in the flow reversal the lift distribution (not the wing geometry) is fixed. The same ideas may be applied if other singularities such as sources, side force elements and volume elements are considered. When sources are used the signs must be reversed when the flow direction is reversed. A source in forward flow becomes a sink in reverse flow. B. COMBINED FLOW FIELD CRITEP.IOII FOR IDEfTIFYIN1G OPTIMUM LIFT DISTRIBUTIONS A necessary and sufficient condition for ndminimum wave plus vortex drag was given by T. Jones (15) in connection with planar systems. The condition is that the downwash in the combined flow field shall be con stant at all points of the platform. This result depends on the fact that a pair of lifting elements has the same drag in forward arid reverse flow, which is also true when the lifting.elements are not in the same horizontal plane. Hence the above criterion can be extended immediately to lift distributions in space by requiring constant downwash (in the combined flow field) throughout the space. NACA TM 1421 C. THE COMBIrNED FLOW FIELD CRITERION FOR IDENTIFYING OPTDIUl VOLUME DISTRIBUTIONS A necessary and sufficient condition for minimum wave drag due to thickness was given by R. T. Jones(13) in connection with planar systems. If total volume is fixed then the optimum distribution of volume gives a pressure gradient in the combined flow field which is constant over the platform. As in the case of lifting elements this criterion can be extended to cover thickness distributions in space. It is then necessary for the pressure gradient in the combined flow field to be constant throughout the space. D. UNIFORM DOWJWASH CRITERION FOR [1iIIJIiT1 VORTEX DRAG A necessary and sufficient condition for vortex drag alone to be a minimum is hat the downwash velocity throughout the wake of the win: system shall be constant in the Trefftz plane. (The wake crosssection is the projection of the wing system on the Trefftz plane.) This condi tion was given by Munk(15). If the wake of the wing system has an elliptical crosssection then a constant intensity of lift over the crosssection satisfies the above condition and gives the minimum possible vortex drag. (See Appendix Vl). In particular when the crosssection of the wing wake degenerates into a horizontal line, (corresponding to a planar.wing) the familiar require ment of elliptic spanwise load distribution is obtained. E. ELLIPTICAL LOADING CRITERION FOR I1IUTUNIRl WAVE DRAG DUE TO LIFT In special cases ellipti' loadings identify minimum drag configura tions, as has been shown by Jones(1 Let the space containing the lifting elements be cut by a series of parallel planes each inclined at the Mach angle to the flow axis. Consider all the lift intensity cut by any one plane to be located at the intersection of the plane with the flow axis. If the resulting load distribution on the axis is elliptical, and if this is true for all possible sets of parallel planes (inclined at the Mach angle)., then the wave drag is a minimum. In Hayes(1) procedure for calculating drag (see Ch. IV) this con dition corresponds to obtaining the minimum possible drag contribution at every angular position on the cylindrical control surface. NACA TM 1421 Such minima cannot be attained in general since the condition is sufficient but not necessary. However if they are attained and if the vortex drag is also a minimum then the more general criterion (constant downwash in the combined flow field) is satisfied. F. THE "ELLIPTICAL LOADING CUBED" CRITERION FOR MIIJDLRJl WAVE DRAG DUE TO A FIXED TOTAL VOLUME Sears(16) and Haack(17) in determining optimum shapes for bodies of revolution in supersonic flow have also determined sufficient condi tions for identifying optimum distributions of volume elements within a prescribed space. We consider a distribution of volume elements within a prescribed space and ask how these elements should be arranged in order that they should cause the least wave drag while providing a fixed total volume. If the equivalent body of revolution for a given angular position 01 on the distant control surface (see Ch. IV) conforms to the SearsHaack :.ptinium shape then the wave drag contribution at 01 is a minimum. There fore if the equivalent bodies of revolution for all values of 9 are optimum shapes the total wave drag is a minimum. The density of the lineal distribution of volume elements repre senting the SearsHaack optimum shape corresponds to the cube of an elliptical distribution over the length of the line. Hence if all the equivalent lineal distributions have this form an optimum is ensured. Such minima cannot be attained in general since the "Elliptical Loading Cubed" criterion is a sufficient, but not a necessary condition for minimum drag. When such minima are attained the more general cri terion (constant pressure gradient in the combined flow field) is also satisfied. G. COMPATABILITY OF 1IIIIflUM WAVE PLUS VOPTEX DRAG WITH 11IlIDilA WAVE OR M.1IrFnIJ1 VORTEX DRAG It is possible for minimum wave plus vortex drag to be obtained when neither the wave nor the vortex drag is individually a minimum. For example consider that the "space" within which lifting elements may be distributed is the platform shown in the figure. For the vortex drag to be a minimum it is necessary to maintain an elliptic spanwise loading over b. This requires a finite load on "a" which in turn pro duces infinite wave drag if the chord for "a" goes to zero. However NACA TM 1i+21 b the minimum drag due to lift for the platform is certainly finite (load the end pieces only and consider them as isolated wings) hence minimum vortex drag is not consistent with minimum total drag in this case. On the other hand, for a planar wing of elliptical platform minimum wave drag and minimum vortex drag are obtained with the same (constant intensity) lift distribution. H. ORTHOGONAL LOADING CRITERIA Optimum distributions can be identified also through orthogonality considerations(8,9). The optimum distribution of lifting elements in a space is orthogonal to every distribution carrying zero net lift and is not orthogonal to any other distributions. A similar statement can be made for the optimum distribution of volume elements alone (assuming for the moment that negative local vol umes are not excluded). However if lifting (and side force) elements are introduced in addition to volume elements, then the criterion must be modified. For example the rotationally symmnietric wing plus central body having zero wave drag is orthogonal to all singularity distributions although it contains a net volume.* The criteria discussed in preceding sections of this chapter have not been thoroughly investigated for cases involving lift and volume elements simultaneously. However, some material on interference between lift and volume distributions is given in Ch. IX. See p. 105 ff. Since the wave drag is zero the disturbances on a distant control cylinder are identically zero. Hence its interference with any other singularity distribution is zero. NACA iTM 1421 APPENDIX V DISTRIBUTION OF LIFT IN A TRANSVERSE PLANfE FOR I4IIMTUM VORTEX DRAG As stated by Mank's Stagger Theorem(15), the vortex drag of a spa tial wing system is not changed if all lift and side force elements in the system are projected onto a single plane normal to the flight direc tion (see Fig. A'l). Furthermore, if there are no side force elements, DO/SrTQ/87T/ON OF y Z IFTIN SPACE PROJECT/ON OFLIFT ONrO )Ye PLANE U ^ / ____ Fig. A51 then Munk's criterion for minimum vortex drag is that in the Trefftz plane, the downwash in the wake must be constant. (The wake cross section is defined as the projection of the wing system on the Trefftz plane.) Assume that the downwash field associated with the optimum lift distribution is w = wo and that a uniform field w = +wo is superimposed on the original field in the Trefftz plane; then the resulting two dimnensional flow pattern is equivalent to a uniform flow around a solid body. Munk gives the expression for the lift distribution in the trans verse plane in terms of the velocity potential of this new flow for NACA TM 1421 certain bodies symmetrical with respect to the xz plane; for example, if 0 is the twodimensional potential flow around an elliptic cylinder, then zopt = 2pU"() Sdzboundary dvortexmin where I and d are transverse plane. potential is(18) Fig. A52 the lift and drag intensities per unit area in the For an ellipse oriented as in Fig. A52, the = wo(a + b)cosh@ go)sin I where y + iz = a2 b2 cosh(S + in) The curve t = to corresponds to the boundary of the lift distribution in the transverse plane. From the above equations one obtains I = 2pU ( T1 dz _t $=So 2pUwo(a + b) b so that the lift intensity in the transverse plane must be constant to obtain minimum vortex drag. With S = nab, the drag is (w,\ L2 Dvortexmin L gS(+ a/b) where L is the total lift generated. Thus to obtain minimum vortex drag for a spatial distribution of lift whose Trefftz plane projection is an ellipse with one axis vertical, the lift should be distributed so as to give a constant intensity when projected on the Trefftz plane. \u) Iopt NACA TM 1421 This proof can be extended to cases in which the projected lift distribution covers a rolled ellipse, as shown in Fig. A55. If only lift (and no sideforce) elements are allowed, Munk's criterion of con L stant downwash still holds, but the lack of symmetry precludes use of the formulas given above. However, the (P optimum lift distribution can be determined by a superposition of two  / symmetrical optimum distributions, ( / as shown in Fig. A54. L1 and L2 Fig. A55 4 W W 1w = W, f W (a) (b) (c) Fig. A54 are constant intensity lift distributions over the elliptic areas which produce constant downwashes wl and w2 over those areas. Because the governing equation is the Laplace equation, which is linear, the lift distributions L1 and L2 and the flow fields they produce can be super imposed. If L1 = L cos 0 and L2 = L sin 0 and Fig. A4c is rotated through the angle 0, then Fig. A54c corresponds to Fig. A53. There is a uniform downwash w corresponding to the uniform lift L. Thus Munk's criterion is satisfied and the drag is a minimum. It can be shown by symmetry that the total interference drag between the lift distributions L and L2 is zero so that the drag of L is obtained simply by adding the drags of L1 and L2; that is NACA TM 1421 L 2 L2 2 I'ortexin 4qS(l + a/b) + qS(l + b/a) L2(a sin2o + b cos2) 4qS(a + b) It should be noted that for this optimum rolled ellipse case there is also a uniform sidewash generated. If a distribution of side force elements were available, it would be possible to utilize the uniform sidewash to reduce the vortex drag below the value given above. 68 NACA TM 1421 CHAPTER VI. THE OPTIMUM DISTRIBUTION OF LIFTING ELEMENTS ALONE A. THE OPTIMUM DISTRIBUTION OF LIFT THROUGH A SPERICAL SPACE Consider a sphere of radius "R" with its center at the origin, and let a total lift "L" be distributed through the sphere with local inten sity "Z." If I = L r being the radial distance from the 2RR2 2 r72 origin, then elliptic loadings are obtained when the sphere is cut by any set of parallel planes (see Appendix VI for derivation). The fact that elliptic loadings are produced when the planes are inclined at the Mach angle (to the free stream direction) insures that the wave drag is a minimum (Ch. V). The crosssection of the wake is circular, and if the lift intensity is projected onto a plane normal to the free stream direction it can be shown that the lift is uniformly distributed over this circular cross section. This insures that the vortex drag is also a minimum (Ch. V). L The lift distribution 7 = then gives the minimum pos I(P2R 2 2 sible wave and vortex drag. By Hayes' procedure it can be found that LB22 the minimum wave drag is Dmin wave = 2 ; the minimum vortex 2nq(2R)dM2 _L2 drag(l) is Dmin vortex and the minimum total drag is 2nrq(2R)2 1> L' 212 l Dmin I LI.L 2 vq(.2R)2 1.2 The largest planar wing of circular planform contained in the sphere has a minimum drag(14) which is greater by the ratio 21 This is 2M2 1 a factor of 1.885 at M = {2. However, the drag comparison is, of course, not complete without consideration of the viscous drag (and thickness drag. For the spatial lift distribution described above, the required wing area is infinite and so, then, is the viscous drag. But the same minimum of wave and vortex drag can be achieved with a number of wing systems having finite wing area. For example, consider the infinite set of cascades enclosed in a spherical space as shown in Fig. 6a1. At NACA TM 1421 O rT/tsw/M FOR MIN/IM/UM voqrec ORAG(COnsrA^r IN rTENSITY) Oor/mA4UM 1Erui/ALENT LINEAL DISTRABUTrioN PORA4IN/Mf/OI1 VAVE DRAG (Ett LPrT/C) Fig. 6al: Crosssectional view of an optimum set of finite area lifting surfaces in a spherical space M = (2 this set of cascades covers the region adequately so that the equivalent linear distribution will be continuous. Determining the lift distributions for the cascades is essentially a stepwise process in that the vortex drag criterion is satisfied over part of the space and then the wave drag criterion over part, alternating back and forth until both conditions are satisfied everywhere. In this example rotational symmetry is assumed and the center cascade is used to satisfy the vortex drag p requirements; thus, the outer region L < r P. of this cascade must V2 NACA TM 1421 carry a constant intensity of lift. The cascades of radius R/VF are used to give the equivalent linear distribution the required elliptic shape for R/V2$< t < R2. The next step is to evaluate the distribution over another section of the center cascade to give constant lift inten sity when elements are summed up in the free stream direction, then satisfy the wave drag criterion with the next cascade, etc. This proc ess is continued working inward to the center of the space; although an infinite number of cascades are required the total wing area is finite. Each of the small cascades has a radius 1/W2 times the radius of the next larger one and the total wing area is S = 2.172itR2 (Ch. VI B). It should be noted that this is not necessarily the minimum wing area that could be used, so the distribution obtained is an optimum one with respect to wave and vortex drag only and not with respect to friction drag. B. THE OPTIMUM DISTRIBUTION OF LIFT THROUGH AN ELLIPSOIDAL SPACE The spherical space with its optimum lift distribution can be changed into an ellipsoidal space with a corresponding lift distribution by a scale transformation of one of the cartesian coordinates. This transformation transforms planes into planes so that elliptical loadings are preserved for the ellipsoid and minimum wave drag is obtained. Also a constant intensity of lift over the wake crosssection is maintained for the ellipsoid so that the vortex drag is also a minimum. Although the optimum lift distribution for an ellipsoid is obtain able from the spherical case, the value of the minimum drag is not nec essarily the same. For an ellipsoid formed by revolving an ellipse of semimajor axis B and semiminor axis R about the free stream (major) axis, the optimum distribution of lift is L Zopt = xB2 112 tA2R2B 1 (x/B)2 (y/R)2 (z/R)2 The wave drag, computed by Hayes' method, is Dmin wave = rq 2L2[/ + ] 8 TtqR 2 [( B /1R,)2 + 02] 1ACA TM 1421 and the vortex drag is also a minimum, l2 Dmin vortex L 8itqR2 so that the total drag is Dmin = L2 P2 +1 81tqR2 (B/R)2 +21 For B = R the results reduce to the spherical case. Several limiting cases can be examined; in one an ellipsoid is collapsed into a horizontal planar wing of elliptic planfornm carrying constant pressure. Optimum cases of this type were first discussed by R. T. Jones(1'). Another limiting case which gives minimum drag occurs when an ellipsoid is collapsed into a plane normal to the flow direc tion (B/R>0). Then the wing system can be interpreted as a uniformly loaded airfoil cascade (of zero chord and gap) within the elliptical crosssection. The entire cascade can be analyzed as a twodimensional system. If the chord is chosen to be p times the gap then the airfoils in the cascade are noninterfering but the lift distribution is suffi ciently continuous (Fig. 6bl). In other words, when the cascade is cut by planes inclined at the Mach angle, the resulting load distribu tions used in Hayes' method will be continuous. The total wing area is then p times the area of the ellipse. /4 /X/ U_ 7'\ \* \ /A/ rRFmA/VVG HON/NTrRFER?/NG MOv,NTrERFeR1NG A /.FO/LS A/RFO/LS AIRFO/LS W1/THSU/FFICENL7Yf CONrI/N/OUS L/FrTD/SrR/BU TION Examples of airfoil spacing in cascades Fig. ob1: NACA T14 1421 A third limiting case is the slender body obtained when B/R4>; then Dmin L + A 2nq(2R) 2L2 2 = Dvortex + Dwave 2nq(2B) The wave drag portion is the samnie as that obtained by Jones for a planar slender ding while the vortex drag for the spatial distribution is onehalf that obtained by Jones for the planar distribution. C. THE OPTEIUM DISTPIBUTIO[ OF LIFT THROUGH A "DOUBLE RACH CONE" Consider (Fig. bc1). and chord) is a space consisting of two Mach cones placed base to base If a uniformly loaded cascade of airfoils (with zero gap placed at the maximum crosssection of this space then *RADIUS = R u Fig. 6r1: Double 11ach cone space with optimnuit ascade elliptic loadings will be obtained when the space is cut by planes inclined at the Ilach angle. This airfoil cascade consequently produces the minimum possible wave drag fo.r wing systems contained within the space and carrying a specified lift. The uniform distribution of load over the circular crosssection insures minimum vortex drag also, so the lift distribution is an optimum for the double Mach cone. The value of the minimum wave drag (obtained by Hayes' method) is 1 L2 Dwave = r : and the vortex drag has the same magnitude in this anql2e. case. NACA T1M 1421 r2 The wave plus vortex drag is then D = L This is equal to nq(2R) the minimum vortex drag alone for a planar wing of span 2R. If the air foil cascade is compared to the largest planar wing of diamond planform which can be contained within the double Mach cone, the minimum wave plus vortex drag of the diamond planform is approximately 1.52 times (2) greater than for the cascade(2) Again it must be emphasized that the drag comparison is not com plete without the inclusion of viscous drag and thickness drag for the wing system. Since the circular cascade is an optimum arrangement, it satisfies Jones' criterion (Ch. V). This can be checked as follows: By two dimensional analysis the downwash, e, in the aft Mach cone is 2a where a is the angle of attack of each airfoil (Fig. 6c2). Since the downwash is zero in the fore Mach cone, the downwash velocity in the combined field is constant and equal to dU throughout the double Mach cone. STREAML/NE /_5 _____ V ~EXPANSION WAVe 4R +A C CONE O CASCA DE F~ReA R MAdACHCONE Oc* CASCADE \TCO4fPRESS/ONV WAVE Fig. 6c2: Twodimensional analysis of downwash in rear Mach cone of an optimumr cascade NACA TM 1421 Far behind the cascade in the wake of the wing system c = a.; this can be shown by equating lift to rate of change of vertical momentum. The individual wings of the cascade are noninterfering and, in the limit as gap and chord go to zero, have twodimensional wing character istics. The wing area for a sufficiently continuous lift distribution (Ch. VIB) is equal to the cascade crosssectional area A times 3. Con sequently L = CLqS = (1Ix/p)q(pA). By Munk's criterion (see Ch. V and Rnf. 12) the downwash in the Trefftz plane over the area behind the cascade is constant; thus, the vertical momentum of the fluid in the downwash region behind the cascade is (pAU) (cU). The vertical momen tum of the surrounding fluid can be evaluated from the known "virtual mass" of a solid circular cylinder of crosssectional area A moving downward in the fluid; this latter momentum is equal to that of the downwash region itself. Thus, by the momentum theorem, L = 2pAU(EU) and equating the two expressions for L gives E = a. The airfoil cascade is not the only distribution of lift in the double Mach cone which has minimum wave drag. A true lineal distribu tion of lift distributed as an elliptic loading along the axis of the double Mach cone will produce the same minimum value of wave drag. So also will a lift distribution of constant intensity throughout the entire double Mach cone. However, the latter two cases will not give the mini mum value of vortex drag; in fact, the true lineal distribution will have infinite vortex drag. ortex drag. NACA o4 1421 APPEfDIX VI DERIVATION OF OPTD.IUMI DISTRIBUTIOIl OF LIFT THROUGH A SPHERICAL SPACE A sufficient condition for minimum drag is that each equivalent lineal distribution of lift should be elliptic (Ch. V). For the spherical space these equivalent lineal distributions will be the same at all angu lar stations if the optimum lift distribution is rotationally symmetric. For simplicity, examine the problem from the angular position 9 (on the control surface) equal to 900; then the Mach planes will be parallel to the y axis. The notation to be used is illustrated in Fig. A61; cylin drical coordinates (t,S,O) and the radial coordinate r will be used. If the spatial lift distribution is 2(r) = 1( 2 + S2) then the equiva lent lineal distribution along the axis will be r R2t 2 r2t/ \ t 2 F() = / 2 (2 + S 2 Sd dS = 2n t D 3=0 Jo=0 0 9.z U S1 (2 + S2dS 7^1 2~ R2 71 If K> Fig. A61 However, F() opt = K1 (/R)2 76 I1ACA TM 1421 where K depends on the total lift of the sphere. Introducing the radial coordinate r, the integral equation to be solved is K 1l (R/B)2 = 2it R rl(r)dr The solution to this equation, found by differentiation with respect to t, is K I(r) =   21rR 2 r2 The total lift of the sphere is L = 2 R K l (/R)2 d K = nRK L=22 so that the distribution of lift for minimum wave drag is 1(r) = L IC2R2 :R r2 For application of Hayes' method, the equivalent lineal distribution along the x axis is needed. A plane t = C' intersects the x axis at x = Mt'; since the distribution is spread out over a larger distance along the x axis, its maximum intensity will be less; thus, F(x)opt = 1 (x/MR)2 1 (x/MR)2 Hayes defines two functions such that for the lifting case (Ch. IV) F pUgz 23L sin u8 /_, h = g. sin =,2L sin x/ME) npULRM NACA TN 1421 The expression for the wave drag contribution at each angular station 9 is, from Eq. (4c50), dD = Pffh'(x2)h(xl)lnx2 x l dx dx2 and the total wave drag is p2i Dwave f2 dO The integration for dD/d9 has been carried out by Sears(16) in terms of a Fourier series expansion of an arbitrary function h. For the wave drag optimum the distribution h is elliptic and only the first term in the series for h appears. (Note the similarity to the vortex drag opti mumrs in Incompressible flow.) If h = C01 (x/MR)2 then dD/d9 = pC2 /16. Substituting in the equations above leads to the final result, 02L2 Dwave = o o 8nqR m 78 NACA TM 1421 CHAPTER VII. THE OPTfIUM DISTRIBUTION OF VOLUME ELEMENTS ALONE* A. THE SINrGULARITY REPRESENTING AN ELEMENT OF VOLUME The investigation of lift distributions is simplified by the use of a singularity which represents an element of lift. This singularity is the elementary horseshoe vortex. The intensity of lift corresponds to the strength of the singularity and the location of the lift force is identical with that of the bound vortex. The study of volume (or thickness) distributions is similarly simplified by identifying the sin gularity which corresponds to an element of volume. Consider a source and sink of equal strength and located on the same streamwise line. In each unit of time a certain quantity of fluid is introduced into the flow pattern by the source and the same quantity is removed by the sink. The volume occupied by the fluid flowing from source to sink depends on the strength of the source and sink and the distance between them, and also depends on the velocity and density of the fluid flowing from source to sink. However, if the volume is to be considered a linear function of the strength of the singularities, then the mean value of density times velocity must be unaffected by the perturbation velocities created by the source and sink. This means that in a line arized treatment of the problem the fluid flowing from source to sink may be considered to have free stream density and velocity. Let m = Mass of fluid introduced per unit time d = Distance between source and sink p = Free stream density Uo = Free stream velocity Then the volume occupied by the fluid is vol = md poU) Since the volume is proportional to md, doubling the intensity of source and sink and halving the distance between them should produce a shorter, but thicker volume of the same magnitude. This suggests pro ceeding to the limiting case (as in incompressible flow) where the source The contents of this chapter have appeared in the paper "The Drag of NonPlanar Thickness Distributions in Supersonic Flow," published in the Aeronautical quarterly, Vol. VI, May 1955. NACA TM 1421 and sink are combined in a dipole with axis in the free stream direction. This singularity should represent an element of volume, although the fineness ratio of the element is zero. The potential for a unit source at (E,0) in supersonic flow is OS = 24x 12 2n (x )2 r2 where = VM 1; x and S are coordinates in the streamwise direction and r is radial distance from the x axis. Differentiating with respect to x gives Sx = (x ) 2g (x _)2 2r2]5/2 where is the potential for the unit dipole or an element of volume equal to 1 /U0. B. THE DISTRIBUTION OF VOLUME ELEMENTS For a distribution of volume elements along the t axis with inten sity f(), starting at t = 0, the potential is = 1fxpr 1 0 f(t)(x 2)dt [(x )2 2r2 Integration by parts gives f() xOr 2n (x t)2 p2r2IO 1 lfxOr 2_ 0 f'(t)dt (x E)2 p2r2 The first term in the expression for the potential is infinite, and apparently corresponds to the "roughness" of the body, which is an assembly of blunt elements (see illustration). IU = O NACA TM 1421 The smoothly faired body (indicated by dash lines) is all that we are concerned with, and this creates the finite part of the potential. This finite part is also the potential for a source distribution of intensity equal to +f'(). This source distribution can be used to con struct a body of revolution extending from Z/2 to +1/2. The shape of the body of revolution created by the singularity dis tribution may be obtained approximately by slender body theory or more accurately by "exact" linear theory. In the first case the volume is +1/2 / I`d which agrees exactly with the sum of the volume elements. 1/2 U0 An example of the second case is shown in Fig. 4f2 where a singularity distribution on the axis is interpreted first by slender body theory then by "exact" linear theory as a "bump" on a cylinder. The bump shapes and locations are quite different but the volumes are identical. This has been proved by Lagerstrom and Bleviss and generalized by Bleviss in Ref. 22. A planar distribution of volume elements may be interpreted by ("exact") linear theory as a thin planar wing. The volume contained in this wing is exactly equal to the sum of the volume elements. The concept of the volume element is not necessary for the study of smooth slender bodies of revolution and planar wings, since these con figurations are relatively simple. However the use of the volume element does help to clarify problems involving more general spatial distributions of thickness. The points to be emphasized are that fixing the sum of the volume elements fixes the total volume, and fixing the distribution of volume elements determines the drag. It is therefore possible to study the drag of a distribution of volume elements without calculating the exact shape of the corresponding body. This is analogous to the fact that the drag of a distribution of lifting elements can be studied without calcu lating the twist and camber of the corresponding wing surfaces. C. THE DRAG OF VOLUTE DISTRIBUTIONS Oil A STREAlWMISE LIIE AJ1D THE SEARSHAACK EODY A body of revolution may be constructed from a distribution of vol ume elements along a streamwise line, or from the equivalent distribu tion of sources. The body constructed from volume elements is an "infinitely rough" body and has infinite drag. However, discarding the infinite part of the potential leaves a "smooth" body (with finite drag) which is equivalent in every respect to the body created by a source distribution. NACA '1 1421 If f(x) is the intensity of the volume element distribution for a body of revolution of length "1" then the drag is given by(16) p+1/2 +1/2 D = /2 41/2 f" (x1)f" (x2) In xl x2 dxl dx2 To maintain constant total volume according to linearized theory + 1/2 it is necessary that f/2 f(x)dx = Constant. The body shape giving minimum drag for a given length and volume has been determined by Sears(16) and Haack(17) independently. The corresponding f(x) (which is propor tional to the crosssectional area) is given by opt (x) = 1  +1/2 5/2 / f(x)dx . 2x 32 1 1/2 f(x)dx 8Uo volume I 5n 1/2 L  ) /2 Thus the optimum distribution of volume elements along the axis corresponds to the cube of an elliptical distribution. (For lifting elements the optimum distribution is elliptical.) The value of the minimum drag is Dmin = V2 2 2 8q volume ~ 2) (/2)5 D. THE SEARSHAACK BODY AS AN OPTIMUM VOLUME DISTRIBUTION IDJ SPACE If the volume elements are not confined to a single streamwise line, then the drag contributions at different angles, 9, on Hayes' cylindrical control surface are not necessarily the same. For any one angle, 9, the drag is given by .+1/2 +1/2 = 2 / Sir2 _1/2 1/2 f" (xl),9 f" t(x,9) In xl x2l dxl dx2 NACA TM 1421 Here f(x,9) is determined by the use of "Mach planes" for the angle 9. All the volume elements intercepted by any one "Mach plane" are transferred (in the plane) to the streamwise axis. The resulting distribution along the axis is f(x,9). The problem of finding the mini mum drag contribution at the one angle 9 is then similar to the Sears Haack problem. If f(x,9) corresponds to the cube of an elliptical dis tribution for every 9, then the total drag is a minimum, and the drag contribution at each 8 is a minimum and corresponds to that of an equiv alent SearsHaack body. It is not always possible to simultaneously mini contributions at all angles 8. However if we consider distribution of thickness within a space which has rotational symmetry about a streanwise axis, then it may be possible that all the equivalent bodies are SearsHaack bodies having the same length. For example, consider that a double Mach cone bounds the space within which  thickness is to be distrib uted. The SearsHaack body placed on the axis is an optimum for this space. It has the same drag contribution at SEARSHAACK BOD every angle on the cylin BYDOUBLE AIACH drical control sirface, and of course, the "equivalent" body of revolution for any angle 9 is identical with the real a "ring" wing (which carried no radial forces) plus a revolution can be designed to have exactly the same drag as the SearsHaack body. The equivalent  bodies of revolution are all identical with the  SearsHaack body. This is discussed in the next section. (For the case  in which radial forces are carried on the ring wing see Ch. IX.) RING WINVG PLUS C nize the drag r the optimum Y BOUNDED CONE SPACE body. However, central body of EA'7RAL BODY IA V/NG SAME ORAG AS SEARS1VAACK BODY NACA um 1421 E. RINIG WING AND CENTRAL BODY OF REVOLUTION COMBINATIOIJ HAVING THE SAME DRAG AS A SEAJRSHAACK BODY Consider a ringwing plus a central body of revolution contained within the space bounded by a double Mach cone. Because of the rota tional symmetry of this particular system, the equivalent body of revo lution is independent of the angle 9 on the cylindrical control surface. In this case, if the local radial force on the wing is everywhere zero, the drag of the equivalent body of revolution is, according to Hayes' formula, identical to the drag of the original system. Thus, a ring wing (which carries no radial force) plus a central body of revolution will have exactly the same drag as a SearsHaack body if the equivalent body of revolution is a SearsHaack body. To design such a system, we may select any smooth, slender profile for the ringwing and compute the crosssectional areas cut from this wing by a set of parallel Mach planes. These areas must then be sub tracted from the crosssectional areas which would be cut from a central SearsHaack body by the corresponding Mach planes. The resulting area difference defines the area distribution (in the Mach planes) of the correct central body. (This area must be projected normal to the flow direction to obtain the crosssectional area of the central body defined in the usual way.) This body, together with the ringwing originally selected, is an optimum distribution of thickness within the double Mach cone space. As an example, consider a ringwing with thickness distribution corresponding to a biparabolic arc profile. The camber necessary for zero local radial force need not be determined, since it does not affect the shape of the central body. Assume that the wing is six percent thick and located halfway between the axis and the apex of the'space. If the central body of revolution is designed so that the equivalent SearsHaack body is of fineness ratio 5, the resulting shape of the central body of revolution is as shown in Fig. 7el. as shown in Fig. 7e1. 84 NACA TM4 1421 RING W/V/ NG, /C = 0.06 EQUIVALENr BODY OF REVOLUTION CENTER BODY OF REVOLUrTION Fig. 7el: Crosssectional view of ringwing and central body (an optimal distribution of thickness within the double Mach cone space) F. OPTDTUJ THICKITESS DISTRIBUTION FOB A PLAIUAR WIJG OF ELLIPTICAL PLAIFORV It is desired to find the optimum thickness distribution for a planar wing of elliptic planform and given volume; this problem was first solved by R. T. Jones(14). A geometrically simpler problem, which will be examined first, is to find the optimum thickness distribution for a circular wing of given volume. The method of Hayes(l) in which the drag is evaluated by summing increments of drag at each angular sta tion around a cylindrical control surface far away from the body, will be used. For the total drag to be a minimum, the increment of drag at each angular station should also be a minimum. If the thickness distribution of the circular platform is rota tionally symmetric, then the equivalent bodies at each angular station will have the same shape (although different "fineness ratios") due to symmetry. If t(r) is the thickness distribution to be optimized for a given volume V, then NACA TM 1421 R V = 2t t(r)r dr 0 (7f1) where R is the radius of the circular wing and r, 0 are polar coordirnates from the wing center (Fig. 7fl). The area cut out at each point along the t axis by planes normal to that axis is + t s(0 = / r^ t .R =2 J t(r)r dr r2 2 (7f2) The equivalent lineal distribution along the x axis is R S(x) = 2 cos p X COS 11 rt(r)dr r2 x2cos2p +R sec . R sec c S(x)dx = V For minimum drag, this distribution should be (Ch. VF) S(x) a 1 (x/B sec )2 /2 Thus the integral equation to be solved for t(r) is K I (x/B sec )2] = 2 cos B t(r)r dr Scos I r2 2cos2 fJ xCos V yr x Cos j. U with (7f 5) (7f4) (7f5) 86 NACA TM 1421 where K is a constant dependent upon the given wing volume. By a suit able transformation of coordinates, Eq. (7f5) may be written in the form K (2 (7f6)d) R sec p o . where = 1 (x/R sec 2) a = 1 (r/R)2 Eq. (7f6) is called Abel's equation and its solution is well known, c.f., Ref. 19. The solution to Eq. (7f6) is t(r) = [K (r/R)2] 4R cos uL 1 and substitution of this in Eq. (7fl) determines K; then t(r) = 2V 1 (r/R)2] (7f7) Equation (7f7) thus gives the distribution cf thicKness which will result in minimum drag for the circular planform wing of given volume. To apply the circular planform solution to the original problem of finding the optimum thickness for an urnyawed elliptic planform, make the following change of coordinates: y X I w ib (7f8) yY/. x yAY// ^ U 00 A 4 &  Fig. 7f2 NACA TMi 1421 87 The circular wing is then transformed into an elliptic wing whose equa tion is (x)2 + ()2 1 a b It can be verified that the thickness distribution 2V XY2 /\ t = 2V1 ( (7f9) rtab[ ( a ( b2] obtained from Eq. (7f7) through the transformation Eq. (7f8) is the optimum for this more general case; that is, the equivalent linear dis tribution for Eq. (7f9) with a set of Mach planes inclined at the angle as shown in Fig. 7f2 is 13/2 s(x) = 1 (2 (7f10) where S= a2 + b2tan2. Since Eq. Tf10) represents a SearsHaack body, the thickness given by Eq. (7f') is optimum for the unyawed elliptic wing. Determination of the total drag in this optimum case involves an integration of the drag increments from these SearsHaack bodies as seen at each angular reference station. If the reference station is at an angle 6 from the horizontal, then the Mach planes cut the elliptic planforir at an angle p. defined as (Ch. IVC). tan p = M2 1 cos 9 (7fll) and the total draF is ~de D 21c dO lo d9 88 NACA TM 1421 The increment of drag at each reference station is (Ch. IVC) f S"(x)S"(6)iZnx fIdx d (7M12) 4q2 \214 and the total drag for the optimum thickness distribution Eq. (7f9' is SM2 1 + 2a2 Dopt = 4ab )/2 (7f15) +2 a2 Di 1n+i bD Defining t = to 1 (x)2 a (Y J2 D = CDqnab then M2 1 + a 2 CDopt (2 + M2 1+ 5/2 Ti l + b2 This result a,reec with that given by Jones( 14. (7f14) NACA TM1 1421 CHAPTER VIII. UIJIQUEifESS PROBLEMS FOR UPTIRr1 DISThIBUTIONS IJ SPACE A. THE IOfJUNIiUEirESS OF OCPTMlM DISTRIBUTIONS IN SPACE "ZERO LOADINGS" In the subsonic flow of a perfect fluid the only drag caused by a lifting wing is vortex drag. The minimum possible vortex drag for a planar ding is obtained when the spanwise lift distribution is elliptical. According to [unk's stagger theorem(15) the chordwise location of the lifting elements is unimportant, so there are infinitely many distribu tions of lift over a given planform which produce the minimum drag. In supersonic flow lift causes both vortex drag and wave drag. The chordwise location of lifting elements is still unimportant in deter mining vortex drag, but does affect the wave drag. For this reason the optimum lift distribution for a planar wing is generally unique in super sonic flow. However, spatial lift distributions offer more freedom in the arrangement of lifting elements and the optimum distributions in space are not generally unique even in supersonic flow. For example, the minimum wave drag due to lift in a double Mach cone space can be attained with each of three different simple lift dis tributions. (See VIC.) The first is a constant intensity over the circular disc located at the maximum crosssection of the space. The second is an elliptical intensity concentrated on the axis of the double 11ach cone. The third is a constant intensity throughout the entire double Mach cone. If the first two distributions are superimposed, one carrying a unit of positive lift and the other a unit of negative lift, the result is a net lift equal to zero. Also, the net strength of the lifting elements intercepted by any cutting plane inclined at the Mach angle is zero. This means that the combined distribution has zero wave drag. Furthermore, there are no disturbances whatsoever produced on the distant control surface near the Mach cone and no wave drag inter ference can exist with any other loading. If another such combined dis tribution with opposite sign is placed on the same streamwise line with the first one, then, by Munk's stagger theorem, the vortex drag is zero also. This is one example of a "zero loading" (see illustration), and many others can be constructed. NACA TM 1421 low1  ^ / / K / A "ZERO LOADING "PLACED WI/I//N AN ELLIPSO/DAL SPACE Such a "zero loading" placed within any space alters neither the lift nor the drag of the original lift distribution. For this reason optimum lift distributions in three dimensions are never unique (unless the space degenerates into a surface). Similar arguments can be applied to optimum distributions of volume. For an example of nonuniquieness in such cases see Ch. VII. B. UNIQUENESS OF THE DISTANT FLOW FIELD PRODUCED BY AN OPTDU4JM FAMILY It has been shown that optimum lift or volume distributions in space are not generally unique, since a group of optimum distributions can be obtained from one given optimum distribution by superposition of "zero loadings." Each member of the group produces the same (minimum) value of drag for a given total lift or volume. From the method of construction of this group (by the use of "zero loadings") it follows that each member produces the same velocity per turbation field in the Trefftz plane and on the distant control surface near the Mach cone. It can also be shown that there are no optimum dis tributions outside this group, since all possible optimum distributions are indistinguishable from the "distant" viewpoint. Assume that flopt(EYTI,) and f2pt(t,T1,) are members of the opti mum family not included in the original group (whose members were related through "zero loadings"). Assume also that f lopt and f2pt do not pro duce identical perturbation velocity fields far from the singularity NACA TM 1421 distribution. Then the drag of fl equals the drag of f2opt (or Dlopt = D2pt) by definition of the optimum family. Also f2opt may be set equal to flpt + Lf, where 2f carries zero net lift (or volume), but has a velocity, perturbation field which is not identically zero far from the singularities. The distribution Lf is orthogonal to (does not interfere with) flopt opt, This follows because an,' given lift or volume distribution can be improved through combining it with a distribution having zero net lift or volume if there is interference drag. However flt by definition, cannot be improved, and must, therefore, be orthogonal to Af. Since If is orthogonal to flo, D2,,pt = DIopt + DAf, but we also lopt' opt lopt I know that D2 = DI and, therefore, Dp must equal zero. Here we dopt opt can obtain a contradiction since both the vortex drag and the wave drag depend on the squares of velocity perturbations (in the Trefftz plane and far out on the Mach cone) and the drag contribution from each portion of the control surface is nonnegative. If t6f produces any disturbances far from the lifting system it must have positive drag, and so6f must produce identically zero disturbances to have zero drag. The above contradiction shows that all the members of the optimum family are indistinguishable from the distant viewpoint. If drag is computed from the "close" viewpoint the above argument cannot be made. Drag contributions then appear as the product of local pressure times angle of attack on the wing surfaces, and these quantities are not necessarily nonnegative at every point on the surface. C. UNIIULENESS OF THE ENTIRE "EXTlERIJAL" FLOW FTELD PRODUCED BY AN OPTEfUA FAMILY It has been shown that any two members of an optimum family produce identical velocity perturbations on the distant control surface. If fl (t,,t) and f2 (t 1 are two members of an optimum opt 2opt family, then flopt f2pt must produce identically zero velocity per 'opt 'opt turbations on the distant control surface, and the drag will be zero. Let "S" designate the space within which the singularity distribu tion flopt f2pt exists, and let "E" represent the external flow field NACA Ti1 1421 consisting of points whose aft Mach cones do not intersect "S." Assume that at some point in the external field "E" the C resultant velocity vector is inclined to \ the free stream direction. Then an ele mentary wing can be inserted at that point with the angle of attack adjusted to give negative drag on the wing. Since F the singularities in "S" are outside the aft Mach cones of all points in "E," the net drag change produced by the elementary  wing is negative. However, f f2 'opt opt c is a singularity distribution causing zero  drag, so flopt f2opt plus the elementary wing is a system having negative drag, although it is an isolated system inserted in a uniform flow field. However, the drag of this system eval uated on a distant control surface comes from a summation of positive quantities and cannot be negative. This contradiction shows that the external flow field "E" produced by flopt f2pt must consist of velocity opt opt vectors aligned with the free stream direction. These vectors must also have the magnitude of the free stream velocity; hence, the external flow field is completely undisturbed, and it can be concluded that all members of the optimum family produce the same flow pattern in the external field "E." It is of interest that a similar proof cannot be made for subsonic flows. In such cases there is no external region where an elementary airfoil can be inserted without producing interference effects at the original singularities. D. E:ISTETICE OF SYI.1ETRIC AL OPTITUI DISTRIBUTIONS IN SYMMETRICAL SPACES It can be shown that, if the boundary of a space has a horizontal plane of syrrmmetry, then there is one member of the family of optimum lift distribution within the space which is symmetrical about the plane. The proof is as follows: Let iopt(x,y,z) represent an optimum lift distribution in the space. The distribution iopt(x,y,z) has the same drag and lift (the drag of NACA Ti 1421 the individual lifting elements is unaltered by the change of position, and the interference drag of any element pair is unaltered also). Since topt(x,y,z) has the same lift and drag as opt(x,y,z)) it is also a member of the optimum family. All members of the opti mum family produce the same exter nal flow field, and any distribu tion producing that field is an optimum. The distribution, lopt(xy,z) + lopt(x,y,z) pro duces the same external flow field as lopt(x,y,z). It is, therefore, an optimum, and since it is also symmetrical about the horizontal plane the proof is completed. Similar proofs can be developed for cases where lift, thickness, and side force elements are present. Also certain other planes of symmetry can be used. NACA 24 1421 CHAPTER IX. IlfVESTIGATIOrJ OF SEPARABILITY OF LIFT, THICKNESS ANJJD SIDEFORCE PROBLEMS* A. THE SEPARABILITY OF OPTEMU DISTRIBUTIOUIS PROVIDING LIFT AND VOLUME Separability Questions For the purpose of drag evaluation a complete aircraft is repre sented by a distribution of lift elements, volume elements and possibly sideforce elements in space. A certain net lift must be provided to support the weight and a net volume must be provided to house payload, fuel, structure, etc. The drag should then be made as small as possible with the net lift and volume equal to the prescribed values. Several questions arise. Can we first study the problem of how best to provide the required lift (with no net volume), then determine the best way to provide the required volume (with no net lift), and finally by superposition obtain the optimum distributions of singulari ties for simultaneously providing the net lift and volume 7 If this procedure is possible will the drag of the combination be the sum of the drags of the two superimposed distributions? Does the optimum way of providing the lift with no net volume require only lifting elements or are volume and sideforce elements necessary? Similarly does the optimum way of providing the volume with no net lift require singulari ties other than volume elements? For horizontal planar systems the answers to these questions are comparatively simple. The lift and volume problems can be studied sepa rately and the optimum singularity distributions superimposed. The drag of the combination is the sum of the drags of the individual distribu tions. Finally, the optimum way of providing the lift requires only lifting elements and the optimum way of providing volume requires only volume elements. All of the above results follow from the fact that in horizontal planar systems there is no interference drag among lift, sideforce, and volume elements. However this is not true in general for nonplanar systems, and consequently the above problems must be reinvestigated for these more general configurations. * Portions of this chapter have appeared in the paper "The Drag of NonPlanar Thickness Distributions in Supersonic Flow," published in the Aeronautical Quarterly, Vol. VI, lay 1955. 
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