u^/^i>h;o^ 
 
 
 < 
 
 NATIONAL ADVISORY COMMITl^EE 
 FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 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 
 
 Douglas Aircraft Company, Inc. 
 
 UNIVERSlPr' OF FLORIDA 
 DOCUMENTS DEPARTMENT 
 120 MARSTON SCIENCE LIBRARY 
 RO. BOX 11 7011 
 GAINESVILLE. FL 32611-7011 USA 
 
 I 
 
 Washington 
 January 1957 
 
TABLE OF CONTENTS 
 
 I. SUMMARY 1 
 
 II. INTRODUCTION 2 
 
 III. SINGULARITIES USED IN THE "LINEARIZED" 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 35 
 
 D. LEADING EDGE SUCTION i+9 
 
 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 NUMBER ... 55 
 
 H. SUPERPOSITION PROCEDURES AND INTERFERENCE DRAG .... 55 
 
 I. ORTHOGONAL DISTRIBUTIONS AND DRAG REDUCTION 
 
 PROCEDURES % 
 
 J. THE PHYSICAL SIGNIFICANCE OF INTERFERENCE DRAG .... 56 
 
 K. INTERFERENCE Af^ONG 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 OPTHdM 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 
 
 OPTIMUM VOLUME DISTRIBUTIONS 6l 
 
 D. UNIFORM DOWNWASH CRITERION FOR MINIMUM VORTEX DRAG . . 6l 
 
 E. ELLIPTICAL LOADING CRITERION FOR MINIMUM 
 
 WAVE DRAG DUE TO LIFT 6l 
 
 F. "ELLIPTICAL LOADING CUBED" CRITERION FOR MINIMUM 
 
 WAVE DRAG DUE TO A FIXED TOTAL VOLUME 62 
 
 G. COMPATIBILITY OF MINIMUM WAVE PLUS VORTEX DRAG 
 
 WITH MINIMUM WAVE OR MINIMUM VORTEX DRAG 62 
 
 H. ORTHOGONAL LOADING CRITERIA 63 
 
 APPENDIX V. DISTRIBUTION OF LIFT IN A TRANSVERSE 
 
 PLANE FOR MINIMUM VORTEX DRAG 6k 
 
 VI. THE OPTIMUM DISTRIBUTION OF LIFTING ELEMENTS ALONE .... 68 
 
 A. THE OPTIMUM DISTRIBUTION OF LIFT THROUGH 
 
 A SPHERICAL SPACE 68 
 
 B. THE 0PTIMUI4 DISTRIBUTION OF LIFT THROUGH 
 
 AN ELLIPSOIDAL SPACE 70 
 
 C. THE OPTIMUM DISTRIBUTION OF LIFT THROUGH 
 
 A DOUBLE I^CH CONE SPACE 72 
 
 APPENDIX VI. DERIVATION OF OPTIMUM DISTRIBUTION 
 
 OF LIFT THROUGH A SPHERICAL SPACE 75 
 
 VII. THE OPTBIUM DISTRIBUTION OF VOLUME ELEMENTS ALONE .... 78 
 
 A. THE SINGULARITY REPRESENTING M ELEMENT OF VOLUME . . 78 
 
 B. THE DISTRIBUTION OF VOLUT^E ELEfffiNTS 79 
 
 ii 
 
C. THE DRAG OF VOLUME DISTRIBUTIONS ON A STREAMWISE 
 
 LINE AND THE SEARS-HAACK BODY 80 
 
 D. THE SEARS-HAACK BODY AS AN OPTIMUM 
 
 VOLUME DISTRIBUTION IN SPACE 8l 
 
 E. RING WING AND CENTRAL BODY OF REVOLUTION COMBINATION 
 HAVING THE SMIE DRAG AS A SEARS-HAACK BODY 85 
 
 F. OPTIMUM THICKNESS DISTRIBUTION FOR A PLANAR WING 
 
 OF ELLIPTICAL PLANFORM 8^4- 
 
 VIII. UNIQUENESS PROBLEMS FOR OPTIMUM DISTRIBUTIONS IN SPACE . . 89 
 
 A. THE NON -UNIQUENESS OF OPTIMUM DISTRIBUTIONS 
 
 IN SPACE - "ZERO LOADINGS" 89 
 
 B. UNIQUENESS OF THE DISTANT FLOW FIELD PRODUCED 
 
 BY AN OPTIMUM FAMILY 90 
 
 C. UNIQUENESS OF THE ENTIRE "EXTERNAL" FLOW FIELD 
 
 PRODUCED BY AN OPTIMUM FAMILY 91 
 
 D. EXISTENCE OF SYMMETRICAL OPTIMUl-1 DISTRIBUTIONS 
 
 IN SYMMETRICAL SPACES • 92 
 
 IX. INVESTIGATION OF SEPARABILITY OF LIFT, THICKNESS, 
 
 AND SIDEFORCE PROBLEMS 9k 
 
 A. THE SEPARABILITY OF OPTIMUM DISTRIBUTIONS 
 
 PROVIDING LIFT AND VOLWE 914- 
 
 B. THE NON-INTERFERENCE OF SOURCES WITH OPTIMUM 
 DISTRIBUTIONS OF LIFTING ELEMENTS IN A SPHERICAL 
 
 SPACE 97 
 
 C. THE NON-INTERFERENCE OF SIDEFORCE ELEMENTS WITH 
 OPTIMUM DISTRIBUTIONS OF LIFTING ELE^1ENTS IN A 
 SPHERICAL SPACE 98 
 
 D. INTERFERENCE PROBLEMS IN CERTAIN SPACES BOUNDED 
 
 BY MACH ENVELOPES 101 
 
 111 
 
E. THE INTERFERENCE BETWEEN LIFT AND SIDEFORCE ELE^'[ENTS 
 
 AND AN OPTII'IUI^ DISTRIBUTION OF VOLUME ELEMENTS .... 102 
 
 F. THE RING V/ING MB CENTRAL BODY OF REVOLUTION 
 
 HAVING ZERO DRAG ■. 105 
 
 X. RESULTS AND CONCLUSIONS ■. . . . 106 
 
 XI. REFERENCES 107 
 
 IV 
 
 • 
 
NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 1^+21 
 
 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 three-dimensional 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 voliane elements are discussed, and some of the useful inter-relations 
 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 non-zero wave drag of Sears-Haack bodies and the zero wave 
 drag of Ferrari's ring wing plus central body. 
 
 Another new feature of the spatial systems is that optimism 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. 
 
 Unedited by the NACA (the Corrariittee takes no responsibility for the 
 correctness of the author's statements). 
 
NACA IM 1421 
 
 CHAPTER II. INTRODUCTION 
 
 The primary purpose of this report is to consider the problems 
 involved in exploring a broader class of aircraft conf igiirations than 
 is ordinarily studied for supersonic flight. It is necessary to deter- 
 mine whether any ion conventional arrangements of wings 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 conf igixrations 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 three-dimensional distributions of different 
 types of singxilarities 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 . 3 are included 
 in this report for convenience. The latter has also been published in 
 The Aeronautical Quarterly, May 1955:, under the title, "The Drag of 
 Non-Planar Thickness Distributions in Supersonic Flow." Permission to 
 reproduce this material has been granted by The Royal Aeronautical 
 Society. 
 
NACA TM lJ+21 3 
 
 CHAPTER III. SINGULARITIES UTILIZED IN THE "LINEARIZED" 
 DESCRIPTION OF THE FLOW ABOUT AIRCRAFT 
 
 A^ BASIC SINGULARITIES 
 
 The Source 
 
 For incompressible, non-viscous fluids the equation governing the 
 flow is the Laplace equation, 
 
 — i- + — i- + — !- = (3a-l) 
 
 ax2 ay2 Sz2 
 
 where ^ is the perturbation velocity potential. A basic solution, which 
 exhibits spherical symmetry, is the source. 
 
 9!sj = — r (3a-2) 
 
 hniix - if + (y - Ti)2 + (z - O^ 
 
 This solution can be interpreted as representing the emanation of unit 
 volume of fluid per \init of time from the point i, t), ^. Because of 
 the linearity of Eq. (ja-l), 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, 
 
 .p2 af^ , Sf^ , ^ » (5a-5) 
 
 bx^ by^ dz2 
 
 where x is the coordinate in the stream direction and p = VM - 1. Equa- 
 tion (5a-3) can also be considered as the two-dimensional wave equation 
 where the x coordinate is thought of as the "time" variable. 
 
 If the y and z coordinates in the Laplace equation (Eq. 3a-l) are 
 multiplied by i|3, then that equation is transformed into the wave equa- 
 tion; a similar transformation of the source potential from Eq. (3a-2) 
 results in 
 
NACA TM li]-21 
 
 ^(x - 1)2 - p^j^y _ ^)2^ (, _ ^2] 
 
 (5a-l^) 
 
 which can easily be shown to be a solution of Eq. (ja-j). Equation (ja-if) 
 
 n)' + 
 
 P P I 
 
 is real inside the forward and rear Mach cones, (x - O > 3 Ky 
 
 (z - O^ , and imaginary elsewhere; however, due to the nature of super- 
 sonic flow only the solution in the rear I4ach cone is used to represent 
 a source. Since half of the real solution is discarded, the constant 
 
 ffSA/> MACf^ COf/£ 
 
 *- X 
 
 ffea/oM o^ /A/F^LUEA/ce of sl/p£/?soa//c souses 
 
 associated with the incompressible must be doubled to represent a unit 
 supersonic source. Thus the supersonic source at | , t], ^ has the potential 
 
 "^ x-|>p\/(y-Ti)2+(z-02 
 
 =<24(x-02-|32[(y-'l)2+(z-02] (5a-5) 
 
 
 
 Elsewhere 
 
 where the x axis is in the free stream direction. It can be shown that 
 Eq. (5a-5) represents unit volume flow from the point |, t), ^; however, 
 care must be taken in the proof because of the singularities on the Mach 
 
 cone (cf. Ref. k) . In the proof given by Robinson ''^■' he made use of the 
 concept of the finite part of an infinite integral, an idea originally 
 
NACA TNI 1^+21 
 
 introduced by Hadamard^^'. As in incompressible flow, other solutions 
 of Eq. (5a--3) 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 going 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 = to x = x^ with strength Kx, it represents a 
 
 cone of semi -vertex angle K/2U with a semi-infinite cylindrical after- 
 body (Fig. 3a— la). Tlie velocities are constant along conical surfaces 
 
 f/A/r/^/re A T REAR /VIACH 
 \/eLOC/r/£S j ^Q^£^ Z£RO A T FORtVA RD 
 
 2 L MACH COA/e 
 
 U 
 
 SOURCE i 
 STRE^/GTH 
 
 HX„ - 
 
 V£LOC/TIES INF/NIT£ * 
 A T AiACH CONE \ 
 
 U 
 
 Fig. 3a-la: Cone-cylinder and 
 source distribution repre- 
 senting it 
 
 Fig. 3a-lb: Perturbation flow 
 lines in x-z plane for super- 
 sonic source, M = ^ 
 
NACA TM 11+21 
 
 from the origin; but on the Mach cone from x = x^ the velocities become 
 infinite due to the discontinuity in soiirce strength. The total inte- 
 grated source strength C is equal to l/2 Kxq . If Xq 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 plajie for 
 the source at M = f2^ is shown in Fig. 3a-lb. (See also Ref. 6.) For a 
 source of finite strength the velocities are infinite on the Mach cone. 
 
 The Three -Dimensional Doublet 
 
 The three-dimensional 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 soiorce 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, 
 
 2itfx2 - p2j.2j 
 
 Q Q Q 
 
 where r = y + z'^. Equation (3a-6) 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 semi-infinite 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 TM 1421 
 
 ^HSV ~ Finite part< 
 
 x-pr 
 
 0D ^i 
 
 
 
 ,x-pr 
 
 bz. 
 
 (^a d| 
 
 xz 
 
 2jtr2)/x2 - p2r2 
 
 X > pr 
 
 (5a-7) 
 
 The flow pattern for a horseshoe vortex in planes normal to the 
 free stream axis is shown in Fig. 3a-2. Far behind the bound vortex, 
 
 Fig. 3a-2: Flow pattern for supersonic horseshoe vortex 
 
8 
 
 NACA TiA 1^4-21 
 
 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 (5a-7) 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 eleqients which have as their 
 velocity potentials 
 
 ^i 
 
 xz 
 
 2npUr2^x2 - pSpS 
 
 X > pr 
 
 (5a-8) 
 
 Similarly, the potential for a unit side force element is 
 
 ^SF 
 
 ± A. 
 pU by. 
 
 iX-(3r 
 
 0a d^ 
 
 xy 
 
 2«pUr2/x2 
 
 p2r2 
 
 X > pr 
 
 (3a-9) 
 
 The force associated with Eq. (3a-9) 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. Ja-o) 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 = (Fig. 3a-3) 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. 
 
 ^ + 
 
 ^^ ^(: 
 
 -^x 
 
 Fig. 5a-3: Formation of doublet or lift transfer element from 
 
 horseshoe vortices 
 
NACA TM 1^21 
 
 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 Trefftz plane flow represents a 
 two-dimensional doublet or dipole and its potential is obtained by 
 letting x-^oo in Eq. (ja-Y) . Thus 
 
 ^2D = /p' pN (5a-10; 
 
 2n(y^ + z^j 
 
 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. 3a--2. 
 
 The Volume Element 
 
 Another useful solution is the doublet with its axis in the stream- 
 wise direction; it has as a potential 
 
 a^c 
 
 "S -X 
 
 ^^ / o o o\5/2 
 
 X > pr (3a-ll) 
 
 2n(x2 - p2r2)- 
 
 Equation (ja-H) can be shown to represent the potential for a unit of 
 volume equal to l/u (see Chapter VII) at the origin. A distribution of 
 volume elements along the x axis with strength f (x) has as a potential 
 
 ^ = 
 
 px-pr 
 '-'0 
 
 
 f(l) 
 
 x-Pr ^ ^x-pr ^,(^^^^ 
 
 2n((x - if - p2r2 
 
 "'^^O /(x-O^-pV 
 
 (5a-12) 
 
 The first term in Eq. (3a-12) is infinite; but if only the finite part 
 
 of the integral is considered (as defined by Hadamard^-^' ), then Eq. (3a-12] 
 
10 
 
 NACA TM li|21 
 
 represents the potential for a source distribution of strength f '(|). 
 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 (ja-ll) can be considered not only as a volume element but 
 
 also as a closed vortex line of circulation strength 1/3 in the yz plane 
 (Fig. 5a— ^a). The line carries a constant intensity of forces directed 
 inward so that the total vector force is zero. The negative of Eq. (ja-ll^ 
 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 l/p^ one with its axis in the negative z and the 
 other in the negative y direction (Fig. 5a— ^b); when added together the 
 trailing vortices cancel leaving the closed vortex line. 
 
 / 
 
 ■r- 
 
 *^ % 
 
 ■c- 
 
 z 
 
 <- 
 
 r- 
 
 (a) 
 
 + 
 
 (k) 
 
 -f- 
 
 C- 
 
 ■Q- 
 
 Fig, 
 
 5a-^: Formation of closed vortex line from horseshoe vortices 
 
 Two-Dimensional Singularities 
 
 In subsonic flow two-dimensional sources, obtained by integrating 
 a line of three-dimensional 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 two-dimensional source 
 potential is 
 
NACA TM 11+21 
 
 11 
 
 
 > 
 
 P- 
 
 (3a-13) 
 
 All of the disturbance created by the two-dimensional so'jrce is 
 concentrated on the Mach planes from it, thus creating a potential jump 
 across these planes. The two-dimensional vortex potential is 
 
 fcv . f ^ 
 
 
 4 
 
 1 
 
 " 2 
 
 z > 
 z < 
 
 •X > pz 
 
 (5a-ll+) 
 
 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 = 
 plane, the latter due to the discontinuity in the past history of the 
 fluid particles above and below the plane. 
 
12 
 
 NACA TM li<-21 
 
 B^ SOME EQUIVALENT SINGULARITY DISTRIBUTIONS 
 
 Statement of the Problem 
 
 The first section of this chapter reviewed the basic singulaj-itles 
 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 eind 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 A-|_ then the drag of (A2_,b) is the same as that of (A,B) 
 
 This follows from the fact that the substitution of A-[_ 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. 5b-l. 
 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. 
 
 A^ 
 
 VORTEX 
 LINES 
 
 SOURCES 
 (^STRENOTH A) 
 
 S/NHS 
 
 SOURCES 
 
 (CIRCULA T/ON tS. ; — 1 (f S TRENG TR-Jk ) (STRE/VG TH /? ^J^ \ 
 
 CASE A CASE B 
 
 p- i/ORTEX LINES 
 
 QC/RCULAT/O/V -k) 
 
 S/A/HS 
 (STRENGTH -^^^ ^ 
 
 Fig. 3b-l 
 
NACA ™ 1^4-21 15 
 
 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 vortex-line 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 axid 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. 513-1- 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 const eint strength. 
 
 To begin with we shall assume the whole infinite plajie containing 
 the front face to be covered by lifting elements. This may be physically 
 realized by a cascade of doubly infinite (two-dimensional) wings of con- 
 stant angle of attack a and such that the vertical dista nce betw een two 
 
 neighboring wings is equal to the wing chord divided by yM - 1, 
 (Fig. 3b-2). 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. 5^-1- The value of the constant k is then k = 2aU. 
 
 The lift per unit area in the plane x = -x is then 2apU . 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 > -X may be described as follows (Fig. 3ti-2). The point P receives a 
 unit of downwash (-w = aU) from the wings A and B each. It also receives 
 
 a positive unit of pressure I -u = aU/yM - 1 J from A and a negative unit 
 
lli. NACA TM lij-21 
 
 A/I ^ ~y. / p ^ ^ 
 
 Fig. 5b-2 
 
 of pressure from B. The net pressure (referred to p„,) received at P is 
 then zero and the net dovnwash is -w = 2aLJ = k. 
 
 The cascade may now be terminated from above by a wedge of opening 
 angle 2a located in the plane z = 'z with its exterior surface parallel 
 to the free stream direction (Fig. 3^-3). Actually this wedge corre- 
 sponds to a source distribution of constant source strength k = 2aJJ . 
 If the cascade is removed for z > z" 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 < z" 
 the flow field is unaffected by the introduction of the wedge. To see 
 this consider a point P with z < z" (Fig. 3^-3)- The wing at B acts as 
 before to produce a downwash of -oU at P. Only the point C on the wedge 
 
 z = z 
 
 Fig. 3b-3 
 
 affects the point P and this point C is already in the downwash field -otU 
 of wing A2. Thus the wedge tiorns the flow downward only by an angle a 
 
NACA W. 114-21 15 
 
 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 vedge 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 = iy 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, 
 I y I < y ^ 1^1 *^ ^ ^^^ downwash is -w = 2aU = 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 -x < x < x, -y < y < y > -'z < z < z". Outside this 
 cube the perturbation velocity is zero. Thus the front face is a cascade 
 of lifting wings at an angle of attack <x, 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 2a 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 = 2cdJ on the inside and 
 w = 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 resiilts. 
 
 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. J'b-^) of half -angle a = 3k/2U. 
 
 M 
 
 Z= 2 
 
 3 
 
 i^ 
 
 ^x ^ 
 
 . P 
 
 J 
 
 z - -z 
 
 Fig. yo-h 
 
16 NACA TM 1421 
 
 At a point P then the downwash is zero and the pressure is given by 
 -u = 2au/y'M2 - 1 = k. 
 
 By inserting planes of zero thickness at z = ±z, y = ±y and removing 
 the wedges outside these planes the infinite conf igiiration is cut down 
 to a conf igioration with a finite cross section. Outside these planes the 
 flow is undisturbed. Inside these planes -u maintains its value 
 
 2au/vM - 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 = +x. 
 
 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 soiirces 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. (3a-5), (3a-8), 
 and (3a-9). The strengths of the distributions considered in this case 
 are indicated in Fig. 3b-5. 
 
NACA TM 1^21 
 
 17 
 
 souses (4)-,/^ 
 
 ► 
 
 pos/r/ve 
 
 LIFT/NG 
 BLEAKS NTS 
 
 
 ^^ 
 
 y'^ 
 
 r^.c/,?; 
 
 NEG/\T/[/e 
 
 L/FT/NG 
 
 £L EMeNrS(-f>l/Jk) 
 
 X 
 
 Fig. 5b-5 
 
 The potential for the entire shell is then 
 
 ^ 2rt 
 
 (x + x) 
 
 z py 
 
 (z- ZQ)dyo dzQ 
 
 -z -> -y 
 
 (y-yo)^+ (z -Zq)' 
 
 (x + x)2- p2 
 
 (y-yof-(-^o)^] 
 
 (x-x) 
 
 z py 
 
 (z-z^jdy^ dz^ 
 
 -z-^ -y 
 
 
 21 
 
 (x-x)2.32 
 
 (y-yo)'+(--of] 
 
 2jt 
 
 X „y 
 
 -X V -y: 
 
 d^o <iyo 
 
 (x-Xo)2-p2 ^y-y„)2+(z-z)2 
 
 X ^y 
 -X ^-y. 
 
 ci^ <iyo 
 
 TrN2 
 
 (x-Xo)2_ p2 ^y_y^j2_^(^^^) 
 
 (3b-l) 
 
 This, after evaluating the integrals, equals zero. 
 
l8 NACA m lij-21 
 
 A third region of the flow field contains points slightly behind 
 and far to the side of the cuhe, where forward Mach cones from the points 
 
 include part, but not all, of the cube. For this region, Hayes' method^ ' 
 can be used to show that the potential again is zero. This method is 
 described in Ch. JVC. It requires that the distance from the cube to any 
 point P where the potential is to be computed must be large compared to 
 the dimensions of the cube . In addition, P must lie near the Mach cones 
 emanating from the singularities on the cube. P is then a point at some 
 angle (measured from the horizontal plane) on a distant cylindrical 
 control surface surrounding the cubic shell. An "equivalent lineal dis- 
 tribution" of singularities is formed by finding the singularity strength 
 intercepted from the cube by a set of parallel planes originating at 
 angle on the control cylinder and inclined at the Mach angle to the 
 free stream direction. The singularities intercepted by a given Mach 
 plane are lumped together at the intersection of the Mach plane and the 
 ELxis 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 9, the effect of all those 
 singiilarities which influence the flow field at P can be summed. Hayes 
 writes the expression for h as 
 
 h = +f - g^ sin 9 - gy cos 9 (3b-2) 
 
 where f is the source strength (per unit length), g^/P "the circulation 
 strength (per unit length) of the lifting elements, and gy/p the circu- 
 lation strength of the side force elements. 
 
 Figiire 3b-6 indicates the notation to be used in describing the 
 geometry of the intersections of the Mach planes with planes containing 
 the x,y,z axes. The Mach plane is inclined to the axis of the control 
 
 cylinder at the Mach angle \i = sin~-'-(l/M) and it is tangent to a cross - 
 section of the cylinder at angle 9.- The trace of the Mach plane in a 
 horizontal (x-y) plane is inclined to a normal to the flow direction at 
 angle 5, where tan 5 = cot |i cos 9 = p cos 9 . The trace of the Mach 
 plane in a vertical (x-z) plane forms an angle a with a line parallel 
 to the z-ajcis, where tan a = 3 sin 9. 
 
NACA TM 1421 
 
 19 
 
 r^ACS OF /\/1ACH PLANE 
 /N THE y.-Z PLANE- 
 
 TRACE OF A/1 A CH 
 PLANE /N THE 
 X-y PLANE 
 
 Fig. 3b-6 
 
 With this brief description of Hayes' procedure in mind, aji equiva- 
 lent lineal distribution of singularities is now to be computed for the 
 specific case of the cubic shell described previously. Figure yo-"{B. 
 shows the intersections of two parallel Mach plajies with the shell; the 
 Mach planes are assvimed 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. Jb-Tb 
 shows the cubic shell as though it were cut along the corner edges and 
 flattened out in one plane . 
 
20 
 
 NACA TM 1421 
 
 . % 
 
 Fig. 5b-7a 
 
 (Sinks) 
 BOTTOM 
 
 Fig. yo-To 
 
 The net singularity strength cut out by these Mach planes must be 
 lumped along a length ix 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: 
 
 ^1 
 
 cos(6 - - 
 
 dui 
 
 ^1 
 
 tan 6 
 
 dx = -k 
 
 ^1 
 
 p cos e 
 
 dx 
 
 (5b-5a) 
 
 (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: 
 
 I = 
 
 pUk -1 ;, -YT djiz = pUkz-i dz tan 
 
 jcos(jt - e)J -^ ^ ^ ' 
 
 x-| dx . 
 = pUk ^ |tan e] = -pUk 
 tan'^CT 
 
 x-]_ dx 
 
 P^sm cos 9 
 
 (5b-5b) 
 
NACA TM 11+21 21 
 
 Again, a negative sign is inserted tecause cos is negative while Z 
 should be positive. There are no forces on the side faces. In computing 
 the strength of the equivalent lineal distribution from Eq. (3b-2) it 
 must be remembered that g , g^ from that formula are circulation strengths 
 
 multiplied by P; i.e., 
 
 ^z pU ^y pU 
 
 Then 
 
 h = +f - g sin 6 - g^ cos 9 
 
 -kx2_ dx n / -plJkX2_ ±x 
 
 3 cos e pUl 32s3_j^ cos 
 
 sin =0 {yo-k) 
 
 That is, the net singularity strength is zero. This is true for all 
 angles 9, 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 
 
22 
 
 NACA ™ 114-21 
 
 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 Normal 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 p k 
 on the front face and -(3 k on the rear face. 
 
 SOURCES 
 
 LIFT/NG eL£Ai£/vrS 
 
 Sinks 
 
 C-x-<jr^) 
 
 ■SIDE FORCE 
 EL EA1£A/ TSfp U-f^) 
 
 Fig. 3b-8 
 
 IS, 
 
 First, the potential ahead of the foremost Mach waves of the cube 
 of course, zero. At a downstream point whose forward Mach cone 
 
 includes all of the sources and lifting elements the potential is 
 
MCA ™ 11+21 
 
 23 
 
 ^ 2jt 
 
 (z-z) 
 
 ,x py 
 
 (x - Xq) dXo dy^ 
 
 -x^ -y 
 
 2 o2 
 
 (y-y„)^+(z-z)^ l^(x-Xo)^-P^ (y-yo)^ + 
 
 (z-z)' 
 
 (z+ z) 
 
 .X oy 
 
 -x'-^ -y 
 
 ,x o z 
 
 (x-xo)dxo dye 
 
 (y- yo)^+ (^ + ^)" 
 
 (y-y) 
 
 (x - Xq) dxo dzo 
 
 -X"^ -z 
 
 y -y) + fz - Zq^ 
 
 (x - Xo)2 - 32 
 
 -n2 
 
 (y-y) + (z - Zo\ 
 
 X r z 
 
 (y + y) 
 
 (x- Xo)dXo dzo 
 
 ^ -^j(y + y)2+(z-Zo)2]^(x-Xo)2-p2[(y^y)2^^,.,^y 
 
 23t 
 
 -y pz 
 
 dyo d-z 
 
 O ^'"O 
 
 -y--./(x-x)2-32 
 
 (y-yo) +(^-Zo)' 
 
 ■ y pz 
 
 d-yp d-z 
 
 o ^"o 
 
 -^^-^|(x.x)2-p2 
 
 (y-yo)^+(z-zJ 
 
 (3b-5) 
 
 Carrying out the Integration, it is found that 
 
 (ji = -2kx for -y < y < y and -z < z < 
 
 ^ = Elsewhere 
 
21+ 
 
 NACA TM lij-21 
 
 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. 3^-7^ one finds that 
 
 f = -k 
 
 x-]_ dx 
 
 sm 
 
 cos 
 
 S = --pUk 
 
 Xn dx 
 P sin 9^ 
 
 I = -pUk 
 
 x-i dx 
 P cos 9 
 
 (5b-6) 
 
 and so 
 
 
 
 x-i dx 
 h - -k ^ 
 
 sin 9 cos 9 
 
 P 
 pU 
 
 ^ X-, dx ^ 
 
 -pUk — 
 
 ^ p sin 9y 
 
 cos 9 - -2-(- 
 
 pU \ ■ p cos e 
 
 X-, dx \ 
 pUk — Isin 9 
 
 = 
 
 (5b-7) 
 
 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 
 Mach cone from 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 (^ = ex. 
 
NACA TM li^21 25 
 
 CHAPTER IV, THE EVALUATION OF DRAG 
 
 A^ THE "CLOSE" AND THE "PISTAm" VIEWPOINTS 
 
 The non-viscous drag for a wing and body moving at supersonic speeds 
 
 may be obtained from two different points of view*'-'-^, 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 hali" 
 
 potential^ ' . 
 
 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 ^ ' uses a circular 
 cylinder with sixis 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 woiild 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 
 throiigh momentum and pressure considerations. 
 
26 NACA TM 1U2I 
 
 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. ^b-53^5^)- 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. 4b-l). A surface element 
 
 on S of area dS will be represented by its outward normal dn 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 
 
 u 
 
 Fig. 1+b-l 
 
 tensor be denoted by a, and let I be the region inside S and II the 
 region outside S. Then 
 
 f = a dn = Force exerted by II on I across surface element (^b-l) 
 
 If a system of coordinates x-|_, X2, x^ is chosen dn may be repre- 
 sented by its three components (dn)-^ and a by a 3x3 matrix a,- •. The 
 above equation may then be written 
 
 3 
 
 fi = (a dn)i = \~ °ij('in)j (i^b-2) 
 
 j=l 
 
 ,th 
 
 where (a dn)^ is the i"^" component of the force, 
 
NACA TI-1 li|21 27 
 
 For a non-viscous fluid the only hydrodynamical force is the pres- 
 sure p and the stress tensor is 
 
 a = -pi = -(p&ij) (^b-3) 
 
 where I is the identity tensor whose matrix is the Kronecker delta &j_ ^ . 
 In this case the force across the element is 
 
 -^ 
 
 f = -p(l d5) = -p d5 (4b-4) 
 
 or 
 
 fi = -p(dn)i 
 
 The hydrodynamical momentum equation states that the stress tensor 
 is balanced by flow-of -momentum tensor. (This is actually a restatement 
 of Newton's law that force = (mass) times (acceleration).) To define 
 the flow-of -momentum tensor we first introduce the concept of a dyadic 
 
 product of two vectors. Let a and b be two vectors with components la^j 
 
 and (bij- The dyadic product is then the tensor whose ij component 
 
 (a . b)j_. is aj_bj, i.e. 
 
 1 . b = (a. . bj) (4b-5) 
 
 Note that if c is any vector then 
 
 (a . b)?= (2:(aibj)cj'j = (aiZbjCj) = a(b . c) (4b-6) 
 
 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: 
 
 Flow-of -momentum tensor = pq . q (4b-7) 
 
28 MCA TM IU2I 
 
 Its physical interpretation may be seen by applying this tensor to the 
 normal dn 
 
 (pq . q)dn = pq( q . dn) =^Moment\:m transported throiigh dS per unit time, 
 
 (4b-8) 
 
 The basic momentum equation for stationary flow for a surface S2_ 
 which does not enclose a body is then 
 
 -* ^\ ,-* / -,-* 
 
 (pq . Ddit = a d^ (4b-9) 
 
 This is analogous to the law of conservation of mass which states that 
 
 / pq . dn = (i^■b-10) 
 
 S-, 
 
 Consider now the composite surface consisting of the surface S in 
 
 Fig. 4b-l and the body surface L. 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 (4b-ll) 
 
 ^E 
 
 Since the flow through E is zero one obtains by applying Eq. (^b-9) to 
 the composite surface S-^ = S + E 
 
 / (pq . q)dn = / a dn - / a dn (lj-b-12) 
 
 J c J a J y 
 
 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 . (4b-ll) and (J+b-12) one obtains 
 
NACA TM 1421 29 
 
 F = - / (pq . q)dn + / a dn (4b-13) 
 
 -^ S ^ S 
 
 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. (i+b-ll), 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. (i+b-lj) shows how the same force may be evalu- 
 ated from the distant point of view. 
 
 A slight modification of Eq. (kh-lj)) will be needed later. Denote 
 the flow quantities at infinity as follows 
 
 q, p, p, a at infinity = U, p^, p^, a^, respectively (ifb-l4) 
 
 The difference between a flow quantity and its value at infinity will 
 be denoted by a "prime." Thus 
 
 q' = q - U, p' = p - Pq, p' = p - Pq, a' = o - Oq (4b-15) 
 
 From the continuity equation (Eq. 4b-10) it follows that 
 
 U / pq . dn = / (pU . q)dn = (l|b-l6a) 
 
 ^-^ S ^S 
 
 Fiirthermore , since a^ = Constant 
 
 CTq dJ = (l4-b-l6b) 
 
 S 
 
 Subtracting Eqs. (4b-l6a, b) from Eq. (4b-13) one obtains 
 
 F = - / (pq' . q)dn + / a' dn (ij-b-iya) 
 
30 NACA TO li+21 
 
 where, for a non-viscous fluid, 
 
 f a' dn = - f p' dn (i^b-lTb) 
 
 -' s ■^ S 
 
 This is the fundamental momentum formula in terms of perturbation 
 
 quantities. Note 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 x-direction and use the following 
 notation. 
 
 U=Ui, q= (u,v,w), q' = (u',v',w') (l^b-l8) 
 
 where 
 
 u = U + u', v' = V, w' = w 
 From Eq. (4b-17a) then follows the fundamental momentum formula for drag ; 
 Drag = F . i = - / pu'q . dn + i . / a' dn (^^-19) 
 
 The momentijm integrals may be fiu^ther 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'>-^''j . 
 
 Inviscid Second-Order 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 rememberec 
 that the values of the pert\arbation velocities are computed from first- 
 order (linearized) theory. The result is a formula for drag according 
 to first-order theory. The term "second-order drag" refers to the fact 
 
NACA TM 1421 • 31 
 
 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 second- 
 order expression for drag. The final formula is given by Eqs . (4b-33,3i4-) 
 and the reader interested only in the final result may skip the deriva- 
 tion now presented below. 
 
 We shall first assume non-viscous flow, so that the stress tensor 
 is given by Eq. (ij-b-3). 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 + eu-|_ + e^up + . . . (4b-20) 
 
 V = evn + e^Vo + . . . 
 
 2_ + e^2 + 
 
 W = €W 
 
 P = Pq + €p-|^ + eSg + 
 
 P = Pq + 6p3_ + G Pg + 
 
 Such an expansion is valid at a distance from the body. It should 
 be remembered, however, that in slender body theory, terms involving 
 log € 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 £^ are the second order terms, 
 etc. The lowest order term in the expression for the drag will now be 
 found using the isentropic pressure-density relation and Bernoulli's 
 law. 
 
 From isentropy it follows that density is a function of pressure 
 alone. One defines 
 
 = a2 
 'constant entropy 
 
 where a is the isentropic speed of sound. Then 
 
 = Po + ^(P - Po) + • • • (^^-21) 
 
52 
 
 NACA TM lJ+21 
 
 from which then follows 
 
 Pi 
 
 1 a 2 
 
 Bernoulli ' s law may be written 
 
 d(u2 + v2+w2) 
 
 + dP = 
 
 where 
 
 P = 
 
 dp 
 
 P 
 
 or 
 
 (u' + U)^ + v^ + v^ + 2P = U^ 
 
 (^b-23; 
 
 Using Eq. (4h-2l) P may be expanded to second order 
 
 P i 
 
 dp' 
 
 i r 1 - ^ 
 
 p p 
 
 D + O 
 
 o a 2 
 
 3-0 
 
 Po^o' 
 
 |dp' = — 
 
 P' - 
 
 (P') 
 
 .^2 
 
 2Poac 2 
 
 -"o^-o 
 
 Expanding the terms in Eq. ('+b-23) to second order one obtains 
 
 eUui + 62uu2 + e2 _^ 1 ^ + _ U^ + g2p^ :L_ = 
 
 2Po= 
 
 'O 
 
 Collecting the terms of order e one finds the linearized Bernoulli's 
 law 
 
 Pi + pQ^iU = 
 
 (l^b-25] 
 
NACA TM li4-21 
 
 33 
 
 Comparing with Eq. (ij-'b-22) one sees that 
 
 UUn 
 
 P]_ = -P 
 
 ° R 2 
 
 {k-h-26) 
 
 The terms of order e yield the following expression for p^ 
 
 2 2 2 
 
 poU^2 + Po :; + p^ 
 
 p^u^%2 
 
 (l+b-27) 
 
 where Eq. (4b-25) has been used and 
 
 M = — 
 ^o 
 
 In the momentum formula, Eq. (i|b-17), "the stress and momentum flow 
 tensors may be combined to form a tensor A 
 
 A = -pq' . q - p'l 
 Using Eqs. (Ub-20, 25, 26) one may evaluate A]_-]_ 
 
 All = -pu'(U + u') - p- 
 
 = -e (pqU-iU + p-l) - e2f p^UgU + p^u-lU + p^u^S + ■p\ 
 
 ( u-,2 + v-,2 + w 2 u-,%l2 
 = -e^Po UU2 - M^^l^ + ^1^ - UU2 ; + —^, 
 
 Finally 
 
 A 
 
 11 
 
 -w 
 
 Cm2 - l)u-|_2 + v^2 ^. ^_2. 
 
 (i|b-28) 
 
 Similarly 
 
 k-^2 = -PU'V = -€'=^PoUlVi 
 
 (ifb-29) 
 
^k MCA m li^■21 
 
 A^j = -pu'w- = -e^p^u^w^ (4b-30) 
 
 Since only the first row (a-^-^, A3_2^ ^13/ ^^'^^^^ ^^ '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 
 pertiorbations of second order, or pressure and density perturbations of 
 second order, do not enter into this expression. 
 
 Thus while drag is of second order, it may be computed on the basis 
 of first order theory (linearized theory). On the other 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 Uo, 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 for the linearized 
 velocity perturbation; in other words 
 
 6U-]_, ev-]_, ew-|_ are replaced by u, v, w (4b-3l) 
 Furthermore a velocity potential will be Introduced such that 
 
 Grad = u, v, w (l+b-32) 
 
 The above results may then be summarized as follows. The drag to 
 second order is given by the formula 
 
 D = / An . dn (^b-33) 
 
 where S = Control surface enclosing the body 
 
NACA IM 1421 35 
 
 A = (All, Ai2, A15) 
 
 All = +Po |(P^^^ + v2 + w2), p2 = m2 - 1 
 
 A12 = -Po^v 
 
 ^15 = -Po^^ 
 
 and u, V and w are the components of the perturbation velocity given by 
 linearized theory. 
 
 C_^ HAYES METHOD FOR DEAG EVALUATION 
 
 The method developed by Hayes in Ref . 1 consists in applying the 
 drag formula Eq.. (4b-53) "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. Ill, 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 Karman'^'^/ 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 three-dimensional distribution of sources, lifting elements 
 and side-force 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 x-axis between 
 X = and x = L. The corresponding solid is then a body of revolution. 
 The source strength will be denoted by f , It will be ass\:imed that 
 
 f(0) = 0, f(L) = (4c-l) 
 
36 NACA TM 11+21 
 
 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 fCx) = U S'(x), f(0) means that r(0) . r'(0) = 0. 
 
 This is fulfilled if r ~ x"^, n > l/2 near the origin. In particular, 
 f (o) 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) = 
 In addition, f (L) = if the body ends smoothly in a cylinder with con- 
 stant radius, i.e. if S(x) = Constant for x > 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 
 
 (V) 2^ Jo ^{x - 1)2 - |32r2 
 
 where r^ = y2 + z,2_ pQp ^ - 3r ^ L the upper limit may be replaced by L. 
 
 Using the condition f (o) = one finds by partial integration of 
 Eq. (4c-2) and differentiation that the perturbation velocities are 
 
 0, = - i r'" , ^''^f^ ^^ (4c-5a) 
 
 ^'^^■^O ^(x - 1)2 _ p2r2 
 
 In Eqs. (lj-c-3), the upper limit is replaced by L for x - pr > L. 
 We shall introduce the notation 
 
 Then t^ = 1 on the downstream Mach cone from x = | and < t < 1 inside 
 
 this Mach cone. For x - pr > L one may also write the velocity components 
 as 
 
NACA TM 11+21 
 
 37 
 
 *^^tJ, ^'^' 
 
 ^L 
 
 (x - O^ - 3^r2j (x - ^)di 
 
 = :^ f(0(x - I 
 
 2n 
 
 )"(^ - t.^) 
 
 -5/2 
 
 d| 
 
 (4c-5a) 
 
 ^^ = S/o ^<^' 
 
 (x - O - [3 r 
 
 -5/2 
 
 p^r d| 
 
 = i!X'^(^)^^-^)"%(^-^i') 
 
 -5/2 
 
 d| 
 
 (4c-5b) 
 
 Hayes' Control Surface 
 
 Following Hayes we now Introduce the control surface shown in 
 Fig. 4c-l. It consists of a circular cylinder of radius r-]_, truncated 
 
 by a front disc x = Constant < and a rear disc x = X2_ > L. The drag 
 
 DISC 
 
 Fig. 4c-l 
 
,Q NACA m ikZL 
 
 integral (Eq. 4b-53) will be evaluated for this control surface as r-L 
 and X2_ tend to infinity. The ratio between x-l and r;^ ^i^ ^^ 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 , (4b-35,3i^-) the contribution of the rear disc to 
 the drag is 
 
 D = ^ r""^ (p2^x^ + 0r^)2nr dr (Uc-6) 
 
 The velocity components may be evaluated as follows. Write f(|) 
 as a difference of two positive functions 
 
 fii) =UU) - f-U), f+(l), f-(l)^o (^=-7) 
 
 Then by the mean value theorem and Eqs, (i4-c-5a,b) 
 
 L rL 
 
 J 
 
 ^' f+(^)d| / f-(nd| 
 
 ^0 
 
 'x 
 
 2.(x - i^f(l - t,^f 2 ^^^^ _ ^^^2^^ . ,^^2^5 2 
 
 (l+c-8) 
 
 where < £ , ^ < L. A similar expression is valid for (^^. Note that 
 
 5 2 
 in Eq. (4c-8) the continuous source distribution is replaced by a posi- 
 tive source at ^^ and a sink at Ig. However, i^ and ^2 (depend on x 
 
 and r. 
 
 As is easily seen 
 
 < X - L < X - li, tL > t^ ., i = 2, 5 
 
 (J+c-9) 
 l<l + tc, 1 + t. <2 
 
NACA ™ li+21 39 
 
 Hence, replacing i-z and ^2 ^Y L increases the absolute magnitude of 
 
 both terms in Eq. (kc-d) , Hence, on the rear disc x = x-,, < r < r-|^, 
 
 4/ i ?^ (4C-10) 
 
 where A is independent of X2_ and r-^, and 
 
 r^ rp^^S 2^ < A ri ^r 1 2pdr (^^_^^^ 
 
 '0 
 
 If one puts y = 1 - t-^^, then dy = -2tL dt-^ = -t^ • 2p dr/^x-]_ - l") . 
 
 (l+c-12) 
 
 Hence 
 
 r^l o , o . A r^ Hv fl _ Tl 1^1 
 
 Jo (x^ - L)2-/y^ y5 2(x^ - L) 
 
 y2 
 
 1 
 
 where y, = 1 - ^1 - e-i) = 2e-]^ - e-^ and e-j^ is explained in Fig. Uc-1. 
 Equation (i|c-12) may be written 
 
 "^1 
 
 Jo (xi - L)2ci2 r^2,2 
 
 where C is independent of r-^^ and x-^ for r-^ and x-^ sufficiently large; 
 6 is explained in Fig. 4c-l. The fact that X2_/[x-|_ - L^-^1, e/e^^^l 
 as r^oo has been used above. It is then clear that if e is constant 
 or if e = r"^, a < 1, then the integrand in Eq. (ij-c-lj) tends to zero 
 
 as r— »oo. A similar estimate may be shown for /i'0-p ^^- ^ comparis 
 
 on 
 
 with Eq. (4c -6) shows that; 
 
kO NACA TM 1421 
 
 Contribution of rear disc to drag is zero even if € decreases 
 as r increases. Hovever, 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 r-,, x-]_ — >«> the entire drag 
 
 contribution comes from the cylindrical part, provided e varies as pre- 
 scribed above . Thus 
 
 D = Limit Dg (i+c-lij-) 
 
 where Dg, the contribution of the cylindrical part, is 
 
 px+pr-^e 
 Dg = -Po2«r-L / (^J^ dx' (i+c-15) 
 
 '-^pr-L 
 
 ^Note that the radial component A-^-,, of the vector A^^ in the drag formiila 
 Eq. {kh-^k) is -PoS^x^r-) ^^ ^^^ above equation l/l - e has been replaced 
 by 1 + e which may be done without loss of generality. 
 
 To evaluate Dg, we write Eq. (ifc-5a,b) in the following form 
 
 0x 
 
 -1 r^''L f'(^i)^e 
 
 1 
 
 2nJo /x' - t-^ p' - i^+ 23r-L 
 
 {kc-l6) 
 ^'^L f'(^2)P ^' + P^l - ^2 
 
 ^r = — / , •, • == = ^ dip 
 
 2jtJo /x' - ^2 ^x' - I2 + 2pr-L P^l 
 
 where x' = x - pr-, . 
 
NACA TM 1421 
 
 41 
 
 The upper limit is x' for x' < L and L for x' > L. Hence 
 
 Dp = — 
 ^ 2jt 
 
 o fP^^i r^i'L r^i'L ^ 
 
 ll(iiKi2) 
 
 l/x- - ^1 ^x' - |2 
 
 /23^ 
 
 ^x' - 1^ + 2pr^ 
 
 yspr^ 
 
 x' + pr-, - |, 
 
 ^x' - I2 + 2prj_ 
 
 P^i 
 
 d|-L dig dx' 
 
 (^c-17; 
 
 Limiting Case for Infinitely Distant Control Surface 
 
 We shall now evaluate Dg as r-, — »<». The three ratios within the 
 second bracket all tend to unity as r-]_ — ?»oo and may hence be neglected 
 in the limit. Note that this approximation implies 
 
 ^(x - |-l)2 _ p2p2 ^Jx - 1^) - pr (2^ /x' - |^ /ip^ 
 
 (l+c-l8a) 
 
 Furthermore, applying the same approximation to Eq. (4c-l6) one obtains 
 that 
 
 ^r P^'x 
 
 (i|c-l8b) 
 
 ^j, and ^^ both vanish as l/^2pr^. Their ratio, however, is given by the 
 
 above equation. The corresponding relation with 0^, replaced by ^ is 
 
 exact for two-dimensional flow. Thus the flow is approximately two- 
 dimensional at large distances near the Mach cone from the leading edge 
 {^ small, i.e. tt almost unity for ^ | ^ lV 
 
 Hence 
 
 D = Limit Do = 
 
 1+rt 
 
 (i^c-19a; 
 
 where 
 
 I = 
 
 .per^ oX',L ox',L f^(^\^i^[\^ 
 ) -'0 Jo fx' - li ^x' - I2 
 
 d|-|_ d|2 tix' 
 
k2 
 
 NACk TM li)-21 
 
 The domain of integration is a region in x', |-j^, I2 space whose cross- 
 section for x' = Constant is the square < i-^ < x' , < Ig ^ x' for 
 < x' < L and the square < 1^^, I2 - ^ ^°^ L < x' < Per-^. Let 1~^ 
 be the integral where < y < L and I2 the integral over the domain 
 L < y < Per-,. Since the integrand is symmetric in i-^ and I2, half its 
 value is obtained by integrating only over the triangle ABC in the i-^, 
 io plane (i.e. i^ '^ I2) as shown in Fig. ij-c-2. In evaluating I-|_ over 
 
 f. 
 
 Fig. l+c-2 
 
 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 y-axis. 
 For ^2.^ ^2 fi^®<3. this line is inside the pyramid only when ^t ^ x' < L. 
 
 |-|_ may vary inside the triangle between and ^2^ ^^^ ^°^ ^2 ^^^ value 
 between and L may be chosen. Hence 1-^ may be written 
 
 L p^P pL f 
 
 II = 2 
 
 'N'^N 
 
 0^0 ^l2 f^' - ^1 ^^' - ^2 
 The integral I2 is (domain is triangular cylinder) 
 
 dx' d^-L d^2 
 
 (i^c-20) 
 
 operi pL ni2 f'(li)f'(l2) 
 ^L ^0 ^0 p' - li ^x' - ^2 
 
 (Uc-21) 
 
NACA TM lij-21 
 
 h3 
 
 Interchanging the order as ahove one obtains 
 
 I = Il-Hl2 = 2|^Y,''^'(^l)^'(^2) 
 
 ;2 f -hf-^2 
 
 -log C 
 
 d^l ci^2 
 (J+c-22) 
 
 Here C is a constant and it has been introduced under the assumption that 
 
 / f ii)dh, = 0, which, since f(0) = means f(L) = 
 J n 
 
 ( cf . Eq. i+c-l). (Note that the limit of integration for |, may he 
 replaced by L if the factor 2 is omitted.) 
 
 Now 
 
 dx' 
 
 52 ^x' -li ^x'- |2 
 
 - log C = 
 
 log(2x'-e^-52 + 2^(x'-|^)(x'-|2; 
 
 Per]^ 
 
 log C 
 
 J 52 
 
 -log(,3- I,) . 10. ^^-l-^l-^2^^f(^--l-k)(^--l-g 
 
 (l^c-23) 
 
 Hence if one chooses C = ^Per-|_, the second term will tend to log 1 = 
 
 as r-, — ^00. Note that for this it is essential that er2^ — > « as r-^ — >«> 
 
 (cf. pg.IVl6). In other words the simplicity of the proof depends on 
 the fact that e — >0 as r2_ — ><" (cf. Eqs. i<-c-l8). On the other hand e 
 may not tend to zero so fast that er-]_ 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. i+c-19a^ 22, 23 one obtains the final drag 
 formula 
 
 L p |2 
 
 2n 
 
 0^0 
 
 ^'(^l)^'(^2)^°s(^2 - ^O'^^l ^^2 (i+c-24; 
 
kk- NACA TM 114-21 
 
 This is von Kannan's drag formula for a lineal source distribution 
 such that f(0)=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 Three-Dimensional Soixrce 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 assijmed that f = outside a certain finite region V. A 
 special case is a planar distribution, say in the plane z = in which 
 case f(x,y,z) = f2(x,y)S(z). Another special case is the lineal distri- 
 bution on the X-axis which was discussed above. In this case 
 f(x,y, z) = f-|_(x)S(r). It will be shown below how in a certain sense the 
 
 drag evaluation for the general three-dimensional 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 x-axis 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 x-axis will be its axis 
 of symmetry. On the x-axis 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 = and the upstream 
 Mach cone from x = L touch but do not penetrate V. We now introduce a 
 control surface and define e and e-j_ as in the lineal case (cf. Fig. 4c-l). 
 
 This is shown in Fig. ifc-J. It will be assumed that r-^ and x-^ tend to 
 
 infinity as described in discussion of the lineal case. 
 
NACA TM li<-21 
 
 45 
 
 Fig. 4c-3: Hayes control surface in three-dimensional space 
 
 That is as x-|_ and r-|^ tend to infinity e sind e-^ will tend to zero. In 
 
 that sense the line AC will come arbitrarily near the Mach cone from the 
 origin. On the other hand e and €]_ will tend to zero slower than l/r3_ 
 
 so that the line AC becomes infinitely long as r-]_ — ><». 
 
 By the same methods that were used in the lineal case, it may be 
 easily seen that the contribution of the rear disc, x = x-^, 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 
 
 r cos 
 
 z = r sm 
 
 (l+c-25) 
 
kS NACA TM lij-21 
 
 Let the drag contribution of a strip on the cylinder between 9 = 0c 
 and 9 = Go + Ae be Z^. We define 
 
 — = Drae contribution per lonit angle = lim ^ as A9— >0 (Uc-26a) 
 d9 AG 
 
 d9 
 Then 
 
 n2lt 
 
 D = Total drag = / — d9 (ij-c-26b) 
 
 J Q d9 
 
 Consider now a fixed meridian plane 9 = 9o, and a point P = (^o^-'^l-'^o) 
 on the cylinder between A and C (Fig. ifc-3). The potential 0(P) depends 
 on the contribution from all sources inside the upstream Mach cone from P. 
 The contribution from a soiorce at Q = (|,ti,0 i^ proportional to the 
 source strength f (q) and inversely proportional to the hyperbolic dis- 
 tance rj^(P,Q) between P and ^ where 
 
 ^h^ 
 
 2 = (xq - |)2 - p2 (r^cos 9o - Ti)2 + (ri sin 9^ - t,)' 
 
 (^c-27) 
 
 This hyperbolic distance is constant on hyperboloids of revolution with 
 r = r-1, G = Gq as axis. Consider now the sources between two such h;yper- 
 
 boloids which intersect the x-axis at x = 5 and x - | + d^ . To evaluate 
 the contribution to ^(P) of these sources one may transfer their total 
 soiorce 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 x^ and r-i as 
 
 well as Gq. 
 
 Consider now, still for fixed 9 = 9o, the limit as r-, — >oo. Then 
 
 the hyperboloids may be replaced by Mach planes which intersect the 
 meridian plane 9 = Gq orthogonally along Mach lines. Note that for this 
 
 it is necessary that as r-^ — >oo 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 IM li^-21 
 
 hi 
 
 Fig. kc-k-: Evaluation of 0(P) 
 
 may then be transferred to the x-axis as above. However, in this limiting 
 case the resulting equivalent body of revolution depends at most on Qq. 
 It becomes independent of r-|^ and Xq. The corresponding lineal source 
 distribution will be denoted by f (x;9o) • A consequence of the independ- 
 ence of Xq and r-j_ is that f(x;9) may be used for computing 0^, and 0^^ as 
 well as ^ at P. In general it may not be used for computing ^q . Clearly 
 (/)q is zero for a lineal distribution, whereas the 0g resulting from the 
 original volume distribution is not. On the other hand 0g is not needed 
 for drag evaluation on the cylindrical surface . 
 
 Since 0^. and 0^ may be computed from the equivalent body of revolu- 
 tion for fixed 9 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. (4-0-2^4-) only by a factor 2it. Hence we have proved 
 the following: The drag D of a volume distribution of sources of 
 strength f(|,Ti,^) is given by the formulas 
 
k8 NACA TM 1421 
 
 p2rt 
 D = / ^ de (Uc-28a' 
 
 Jo ^9 
 
 ^^J^Iq / ^f'(^i;e)f'(^2^e)i°g(52 - ^i)<i^i'i^2 (^=-28b: 
 
 f(|;e)d| = j j j f(Q)<iQ (4c-28c; 
 
 vhere V(5,9o) is the region contained between two Mach planes perpen- 
 diciolar to 9 = 9o and intersecting the x-ajcis at x = | and x = i + d| . 
 
 This result was obtained by Hayes in Ref • 1. It is thus seen how 
 Hayes' derivation of von Karmdin's drag formula for bodies of revolution 
 admits an easy generalization to the general three-dimensional 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 ) = 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. (^4-0-28) 
 reduce to the previously established formiola (Eq. 4c-2ij-) . 
 
 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^ sin 9 + gy cos 9^ i^c-29] 
 
NACA ™ 1^4-21 
 
 ^9 
 
 where 
 
 f = f(^;e) = Source strength 
 
 pUg^ ^P = l{i;Q) = Lifting element strength 
 pUg /p = s(|;0) = Side force strength 
 
 The term f g sin 9 + g^ cos Gj 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 (4c-28h), as extended to include 
 lift and side force elements, is 
 
 \dQJ 
 
 ^t2 
 
 wave 
 
 i,„2 Jq Jq h'(|^;e)h'(l2;8)log(|2 - li)cili d^g 
 
 8«^^0 
 
 h'(i^;e)h'(i2;e)iog 
 
 dl^L ^^^2 
 
 (4c-30) 
 
 where h(|;9) is the equivalent lineal distribution (for a given station 9) 
 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 .theoretically complete until leading edge suction is accounted for. 
 
50 
 
 NACA TM lil-21 
 
 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 distajit viewpoint, leading edge suction cannot 
 be isolated. This is true because there is no point-to-point corre- 
 spondence between the close and the distant control surfaces. At the 
 distant control s\irface 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. 
 
 DISCONTINUITIES 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 soi^rces 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 x may be written 
 
 4 = -^ 
 
 1^ 
 
 2n 
 
 f(l)d^ 
 
 nx-pr 
 
 ^(x - 5)2 - p2r2 
 
 ^^-Pr f2(l)d? 
 
 |/(x - 1)2- p2^2 
 
 2n 
 
 ■X f^(|)d| 
 
 ^(x - i)^ - p2i.2 
 
 (^-1) 
 
 /Cf) -. 
 
 - f>^ 
 
NACA TO IU2I 
 
 51 
 
 The u-component of velocity at the point (x,r) is found by differentiating 
 Eq. (4e-l) axially. (in order to avoid indetenninant forms in the differ- 
 entiation, the equation is first transformed by means of the relation 
 I = X - pr cosh u.) This process gives the result (assuming f(0) = 0): 
 
 u 
 
 a0 
 
 2rt 
 
 X f^'(|)d5 ^x-pr f2'(|)d? 
 
 Af(x) 
 
 l/(x-02-p2r2 ^x (/(x-^^-pSrS ^'(x - x)2 _ p2r2 
 
 (^-2) 
 
 where Af(x) = f^ix.) - f2(x). 
 
 At the distant control siirface it previously was shown (Ch. IV-C ) 
 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 - |)/pr ^ 1), Eq. (4e-2) can be expressed 
 
 2jt^ 
 
 ^x f^'(|)dg 
 h' - ^ 
 
 '^' fp'(l)d| 
 
 Af(x) 
 
 'x /^^"^^ 1^ 
 
 (i^e-3) 
 
 where x' = x - pr and x' - | « 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. 4c-l8b) 
 
 pu 
 
 (k^-k) 
 
 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. (4e-3) and (4e-4) it is 
 readily seen that the drag includes a term of the form 
 
 -C{2 
 
 P" [Af(x)] 
 X (x- - x) 
 
 dx' 
 
 The integral is non-convergent . An infinite drag contribution therefore 
 results from a discontinuity in the strength of the source distribution. 
 
52 NACA m U2I 
 
 F\ THE USE OF SLENDER BODY THEORY WITH THE DISTANT VIEWPOINT 
 
 If slender body theory is applied, then the soiorce strength is 
 assumed proportional to the rate of change of cross-sectional 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 
 I'evolution having discontinuities in dS/dx. Such a prediction is, of 
 course, incorrect, and the error is caused by the application of slender 
 body theory 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 )+k 
 should be of the form R ~ d where k > 0. This does not elimi- 
 nate blunt noses or tails entirely, but excludes "excessive" bluntness. 
 (Note that the Sears-Haack optimum shape is blunt.) 
 
 The linearized theory requirement that all velocity perturbations 
 be small theoretically excludes all bluntness, but this is imimportant 
 if very small regions of the flow field are affected. 
 
 Bodies which begin or end in cylinders also may satisfy the smooth- 
 ness re qui reme nt s . 
 
 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 cross-sectional 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 yM - 1.) 
 
 Figure kt-1 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 Karmaii^'^^. 
 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 two-dimensional 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. iff -2 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 li+21 
 
 53 
 
 COMPARISON OF THEORETICAL CALCULATIONS 
 FOR DRAG OF CORRUGATED CYLINDER 
 
 {A) RATIO OF THE DRAG COMPUTED BY SLENDER BODY THEORY 
 TO THS DRAG COA/IPUTED BY Z /NEA R THEORY 
 
 (B.) RATIO OE THE DRAG CO/V1PUTED BY TI^O-D/A/1ENSIONAL 
 L/A/EAR THEORY (ASSUMING Tt^O-DI/l^ENS/O/VAL FLOl^ /N 
 EACH MER ID/AN PLANE) TO THE DRAG COA/tPUTED 
 BY THREE-DIMENSIONAL LINEAR THEORY 
 
 cs" 5r 20 
 
 X 
 
 Q: 
 
 i 
 
 Q 
 X 
 
 X 
 
 Q 
 O 
 
 I 
 
 X 
 
 i 
 
 i 
 
 X 
 
 ! 
 I 
 
 16 
 
 12 
 
 8 
 
 
 
 
 M 
 
 k- 
 
 ^ 
 
 
 
 
 
 
 
 
 
 
 te- 
 
 ^ 
 
 d 
 
 £~- ~-*\~ 
 
 PROFILE OF CYLINDER 
 CCROSS -SEC TIONSA RE CIRCULAR) 
 
 
 
 
 \ 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 V 
 
 rfA) D^^ jD^ 
 
 
 
 
 
 
 
 
 \ 
 
 \ 
 
 
 
 
 
 /5=n/ 
 
 
 
 Vl^-i 
 
 (-5)4^ 
 
 vA^l^r 
 
 
 
 
 
 
 
 
 
 
 
 \ 
 
 
 
 
 
 ■ 
 
 
 - 
 
 
 
 
 
 
 
 
 
 
 12 3 4 5 
 
 PATIO OF REDUCED WA\/E LENGTH TO CYLINDER D/AMETEP , 
 
 Ak 
 
 (FOR LINEAR THEORY DRAG OF CORRUGATED CYLINDER^ SEE PEP: 7) 
 
 Fig. 4f-l 
 
^h 
 
 NACA TM 1421 
 
 Q 
 
 
 
 o 
 
 « 
 
 S 
 
 5 
 
 
 5 
 
 k 
 o 
 
 I 
 
 Q: 
 
 8 
 
 ki 
 8 
 
 5 
 
 
 o 
 
 
 
 kj lij 
 
 Co SI 
 
 % 
 
 h 
 
 0] to 
 
 OJ 
 I 
 
 fcD 
 •H 
 
NACA TM li|21 55 
 
 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 Eleviss in Ref . 22. (This suggests that "volume elements" may retain 
 their significance even when slender body theory does not apply. ) 
 
 G^ THE DEPENDENCE OF DRAG COEFFICIENT ON MACH NUIvlBER 
 
 Hayes^ '' 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 Mach number. If the singularities are soiorces, 
 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 = 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 = 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. 
 
 }L SUPERPOSITION PROCEDURES AND INTERFERENCE 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, p-j^, correspond to a downwash 
 field, a-i , and a second pressure field, P2, correspond to a second down- 
 wash field, ao, then the pressure field p-[_ + po corresponds to the down- 
 wash field a-^ + o^- 
 
 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 D-]_ = / P2_a-L dS, where the integration extends 
 
56 NACA TM 1421 
 
 over the wing and body surfaces, and similarly the drag of the second 
 field is D2 = / p2<^2 '^^* However, the drag of the combination is 
 
 D-,,0 = / f Pi + P2)f°'l "^ <X2)dS. The terms involving cross products give 
 
 the interference drag, Dj_ = (v2S'^2 "^ P2°'l)<3-S. 
 
 1^ ORTHOGONAL DISTRIBUTIONS AND DRAG REDUCTION PROCEDUEES 
 
 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 (non-optimum) lift distribution can 
 be improved by adding the proper amount of a non- orthogonal 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 (non-optimum) lift distribution. 
 
 J^ THE PHYSICAL SIGNIFICANCE OF INTERFERENCE DRAG 
 
 It has been stated that the interference drag, D^^, is (Vi^2 "*" P2'^l') 
 
 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 cross-sectional area for the body. The V2_'^2 ^^ 
 
 dS 
 
NACA Tt-; li|21 57 
 
 gives the drag produced by the pressure field of the first body acting 
 
 on the cross-sectional area distribution of the second. The term Vncn-, dS 
 
 has a similar interpretation. 
 
 Assume that both flow fields are produced by lift distributions. 
 
 Then / P^cxg 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 p-[_ must be inclined 
 
 further because of the downwash due to po. 
 
 Let the first field be produced by a lift distribution and the second 
 
 by a thickness distribution (a body). Then / Pi ao 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 cross-sectional area distribution of the body. The 
 
 / P2^1 ^^ 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 / P]_a2 dS is drag corre- 
 sponding to the downwash field of the side force distribution acting on 
 the lift elements, while the / Poa-i dS is produced by the sidewash field 
 of the lift elements acting on the side force distribution. 
 
 K^ INTERFERENCE AMONG LIFT, THICKKESS, AND SIDE FORCE DISTRIBUTIONS 
 
 For planax 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 drag has been discussed in the 
 preceding sections. 
 
58 
 
 NACA TM li<-21 
 
 Suppose that a soiirce ajid a 
 lifting element are located as shown 
 in Fig. 4k-l, 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 y-axis (corre- 
 sponding to a planar wing problem) 
 there would be no interference . 
 
 sou/?ce 
 
 3>^ 
 
 / 
 
 / 
 
 
 Fig. J+k-1 
 
 SLBMSNT 
 
 For lift and side force ele- 
 ments^ as shown in Fig. ^k-2, 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 y-axis or on the z-axis 
 (as shown in Fig. Uk-5a 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 . 
 
 S/DE FORCE 
 
 ELEMENT 
 
 / 
 
 ^ 
 -"v^ 
 
 Fig. 4k-2 
 
 =/ 
 
 Fig. l4-k-3a 
 
 2 
 
 Fig. ij-k-5"b 
 
NACA TO 142: 
 
 59 
 
 L^ REDUCTION OF DRAG DUE TO LIFT BY ADDITION OF A THICKNESS DISTRIBUTION 
 
 Consider the two-dimensional system sketched in Fig. kl-1. The 
 cross-hatched area is a thickness distribution lying partly in the pres- 
 sure field of a flat-plate wing. The relative geometry of the thickness 
 distribution and the lifting surface are indicated in the figure . Also, 
 the pressure distributions, relative to the two-dimensional pressure 2aq/3, 
 are shown in parentheses. 
 
 Fig. i+I-l 
 
 As long as the pressure field of the thickness distribution does 
 not intersect the flat-plate, the lift of the system is the same as for 
 the flat-plate by itself . On the other hand, the interference between 
 the pressure field of the flat-plate and the thickness distribution pro- 
 duces a negative drag contribution, so that the total drag of the system 
 (omitting friction) is 12-1/2 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 (non-planar systems), 
 sources and lifting elements have an interference drag. 
 
60 - NACA TM lil-21 
 
 CHAPTER V. THE CRITERIA FOR DETERMINING OPTIMUM DISTEIBUTIONS 
 OF LIFT OR VOLUME ELETffiNTS ALONE 
 
 A^ THE "COMBINED FLOW FIELD" CONCEPT 
 
 The idea of the "combined flow field" was introduced by Munk^ '^ 
 
 and extended by R. T. Jones^ -'' ' . Consider a distribution of lifting 
 elements in a free stream of given velocity. A certain downvrash 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 axe produced at 
 each point in the field. 
 
 One-half 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. One -half 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 super-position 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 CRITERION FOR IDENTIFYING 
 OPTEvlUIvl LIFT DISTRIBUTIONS 
 
 A necessary and sufficient condition for minimum wave plus vortex 
 
 drag was given by R. T. Jones^ -'i 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 planform. This result depends on the fact 
 that a pair of lifting elements has the same drag in forward and 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 11+21 61 
 
 C_^ THE COMBINED FLOW FIELD CRITEEION FOR 
 
 IDENTIFYING OPTIMUM VOLUME DISTRIBUTIONS 
 
 A necessary and sufficient condition for minimum wave drag due to 
 
 thickness was given by R. T. Jones^ -' ' 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 planform. 
 
 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 DOWNWASH CRITERION FOR MINIMUM VORTEX DRAG 
 
 A necessary and sufficient condition for vortex drag alone to be 
 
 a minimum is that the downwash velocity throughout the wake of the wing 
 
 system shall be constant in the Trefftz plane. (The wake cross-section 
 
 is the projection of the wing system on the Trefftz plane.) This condi- 
 
 (1^5) 
 tion was given by Munk^ ' . 
 
 If the v/ake of the wing system has an elliptical cross-section then 
 a constant intensity of lift over the cross-section satisfies the above 
 condition and gives the minimum possible vortex drag. (See Appendix V-l] 
 In particular when the cross -section 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 MINIMUM WAVE DRAG DUE TO LIFT 
 
 In special cases elliptic loadings identify minimijm drag configura- 
 tions, as has been shown by Jones^ '. 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^^ 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. 
 
62 NACA TM 114-21 
 
 Such minima cannot be attained in general since the condition is 
 siifficient 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. 
 
 I\ THE "ELLIPTICAL LOADING CUBED" CRITERION FOR 
 MINEvIUM WAVE DRAG DUE TO A FIXED TOTAL VOLUME 
 
 Sears*- ' and Haack^ ''' 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 9-[_ 
 
 on the distant control surface (see Ch. IV) conforms to the Sears-Haack 
 optimum shape then the wave drag contribution at 9-i 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 Sears-Haack 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^ COI'g'ATAEILITY OF MINII€JI4 WAVE PLUS VORTEX DRAG 
 WITH MINMmi WAVE OR MINIMUM 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 planform 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 1^4-21 
 
 65 
 
 the minimum drag due to lift for the planform 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 planform 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'-"^-^'. 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 optimijm 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 symmetric wing plus central 
 body having zero wave drag is orthogonal to all sing\ilarity 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. 
 
 p. 103 f f . 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. 
 
6h 
 
 NACA TM 1421 
 
 APPENDIX V 
 
 DISTRIBUTION OF LIFT IN A TRAWSA^^SE 
 PLANE FOR MINIMUM VORTEX DRAG 
 
 As stated by Munk's Stagger Theorem' ^^, 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. A5-1). Furthermore, if there are no side force elements, 
 
 
 D/srff/BC/r/ON 
 
 i/FT /A/ SPACe 
 
 p/?OU£CT/OA/ OF' LIFT 
 OA/rO YS PLANe 
 
 -X 
 
 Fig. A5-1 
 
 then Munk's criterion for minimimi vortex drag is that in the Trefftz 
 plane, the downvash 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 = -Wq and that a uniform field w = +Wo is superimposed 
 on the original field in the Trefftz plane; then the resulting two- 
 dimensional 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 1^4-21 
 
 65 
 
 certain todies symmetrical with respect to the x-z plane; for example, 
 if is the two-dimensional potential flow around an elliptic cylinder, 
 then 
 
 'opt = ^p<i) 
 
 \°-V boundary 
 
 ^ \^K 
 
 and 
 
 <ivortexj]i 
 
 in V U/ 
 
 opt 
 
 > 
 
 Fig. A5-2 
 
 where I and d are the lift and drag intensities per unit area in the 
 transverse plane. For an ellipse oriented as in Fig. A5-2, the 
 
 potential is^^^^ 
 
 = WQ(a + b)cosh(^ - l^^sin t] 
 
 where 
 
 y + iz = fa^ - 1)2 cosh(5 + It]) 
 
 The curve ^ = ^q corresponds to the boiindary of the lift distribution 
 in the transverse plane. From the above equations one obtains 
 
 2pu(^ ^' 
 VSt] dzy 
 
 2pUwQ(a + b) 
 
 l=lr 
 
 so that the lift intensity in the transverse plane must be constant to 
 obtain minimum vortex drag. With S = rtab, the drag is 
 
 Dvortexjnin 
 
 w, 
 
 _o L ^ " 
 
 uy 4qS(l + 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 ajcis vertical, the lift should be distributed so as to 
 give a constant intensity when projected on the Trefftz plane . 
 
G6 
 
 NACA TO lij-21 
 
 This proof can 'he expended to cases in which the projected lift 
 distribution covers a rolled ellipse^ as shovm in Fig. A5-5. If only- 
 lift (and no sideforce) elements are 
 allowed, Munk's criterion of con- 
 stant downwash still holds, but the 
 lack of symmetry precludes use of the 
 formulas given above. However, the 
 optimum lift distribution can be 
 determined by a superposition of two 
 symmetrical optimum distributions, 
 as shown in Fig. k^-h. L-]_ and l,^ 
 
 Fig. A5-3 
 
 Z, 
 
 w - W.-h tv 
 
 Cc) 
 
 are constant intensity lift distributions over the elliptic areas which 
 produce constant downwashes w-|_ and Wg over those areas. Because the 
 
 governing equation is the Laplace equation, which is linear, the lift 
 distributions Li ajid L2 and the flow fields they produce can be super- 
 imposed. If L-|_ = L cos and Lp = L sin and Fig. A5-^c is rotated 
 
 through the angle 0, then Fig. A5-^c corresponds to Fig. A5-3- 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 Lo is zero so that the drag of L is obtained 
 
 simply by adding the drags of L-, and L^; that is 
 
NACA TM li|21 67 
 
 _ ^1 ^2 _ L^(a sin^0 + b cos^) 
 
 vortexjjiin " i+qs(l + a/b) "*" i^qS(l + b/a) " lfqS(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 ELEI^ENTS 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- 
 
 z, r being the radial distance from the 
 
 sity "Z." If Z = - 
 
 ;t2R2^R2 _ r2 
 
 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 cross-section 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 ) . 
 
 The lift distribution Z = 
 
 n2R2|f 
 
 then gives the minimum pos- 
 
 ^R^ - r^ 
 sible wave and vortex drag. By Hayes' procedure it can be found that 
 
 the minimum wave drag is D^j^j^ wave "= 
 
 L2p2 
 
 2itq(2R)2M2 
 
 ; the minimum vortex 
 
 drag IS D[j^in vortex 
 
 2jrq(2R)^ 
 
 and the minimum total drag is 
 
 ■'min 
 
 l2 
 
 m^ - 1 
 
 Ttq(2R)2 
 
 . m2 
 
 The largest planar wing of circular planform contained in the sphere 
 
 has a minimum drag 
 
 {Ih) 
 
 which is greater by the ratio 
 
 2M 
 
 3 
 
 This is 
 
 2M'^ 
 
 a factor of I.885 at M = f2^. However, the drag comparison is, of coiorse, 
 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. 6a-l. At 
 
NACA ™ lij-21 
 
 69 
 
 /NT£NSITy) 
 
 O^r/AAUA/t £(fUli/ALE/^T i.lN£AL D/sr/7/BUTIoN 
 
 Fig. 6a-l: Cross-sectional view of an optimum set of finite area 
 lifting surfaces in a spherical space 
 
 M = lathis 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 
 
 requirements; thus, the outer region ■= < r < R of this cascade must 
 
70 NACA TM 1^4-21 
 
 carry a constant intensity of lift. The cascades of radius K/fZ are 
 used to give the equivalent linear distribution the required elliptic 
 shape for 'R/\[2-<- i ^ R|[2. 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 l/|/2 times the radius 
 
 of the next larger one and the total wing area is S = 2.172rtR (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 0PTIMUI4 DISTRIBUTION OF LIFT THROUGH AW 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 transf oimation of one of the cartesian coordinates. This 
 transformation transforms planes into planes so that elliptical loadings 
 are preserved for the ellipsoid ajid minimum wave drag is obtained. 
 
 Also a constant intensity of lift over the wake cross-section 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 
 semi -major axis B and semi-minor axis R about the free stream (major) 
 axis, the optimum distribution of lift is 
 
 ^opt 
 
 rt^R^B 
 
 1 - (x/B)2 _ (y/R)2 _ (^/R)2 
 
 1/2 
 
 The wave drag, computed by Hayes ' method, is 
 
 p2l2 
 
 ^in wave = 
 
 8nqR' 
 
 (B/R)2 + p2] 
 
NACA IM 1421 
 
 71 
 
 and the vortex drag is also a minimum, 
 
 Dmin vortex 
 
 8nqR2 
 
 so that the total drag is 
 
 Anin ~ 
 
 8nqR^ 
 
 [(B/R)2 + p2] 
 
 + 1 
 
 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 plamform carrying 
 constant pressure. Optimum cases of this type were first discussed by 
 
 R, T. Jones^ ■'. .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 eind gap) within the elliptical 
 cross-section. The entire cascade can be analyzed as a two-dimensional 
 system. If the chord is chosen to be p times the gap then the airfoils 
 in the cascade are non-interfering but the lift distribution is suffi- 
 ciently continuous (Fig. 6b-l). 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. 
 
 U 
 
 / 
 
 / 
 
 / 
 / 
 
 A. X /y 
 
 \ 
 
 \/ \; 
 
 i^ 
 
 "^ ^ 
 
 
 / / 
 
 / 
 / 
 
 x/ 
 
 
 AIRFO/LS 
 
 AlRf^O/LS \A//rHSUFFIC/ENTL'r 
 CON T//VC/OUS t /f^TD/STff/BU T/O N 
 
 Fig. 6b-l: Examples of airfoil spacing in cascades 
 
72 
 
 NACA ™ 1421 
 
 A third limiting case is the slender body obtained when b/R^oo^ 
 
 then 
 
 I\a: 
 
 b2l2 
 
 in p ' 2 
 
 2jtq(2R) 2nq(2B) 
 
 ^vortex "*" ^wave 
 
 The wave drag portion is the same as that obtained by Jones for a planar 
 
 slender wing^ ■^', while the vortex drag for the spatial distribution is 
 one-half that obtained by Jones for the planar distribution. 
 
 C^ THE OPTIMUM DISTRIBUTION OF LIFT THROUGH A "DOUBLE MACH CONE" 
 
 Consider a space consisting of two Mach cones placed base to base 
 (Fig. 6c-l). If a uniformly loaded cascade of airfoils (with zero gap 
 and chord) is placed at the maximum cross-section of this space then 
 
 U 
 
 RADIUS - R 
 
 Fig. 6c -1: Double Mach cone space 
 with optimum cascade 
 
 elliptic loadings will be obtained when the space is cut by planes 
 inclined at the Mach angle. This airfoil cascade consequently produces 
 the minimum possible wave drag for wing systems contained within the 
 space and carrying a specified lift. The uniform distribution of load 
 over the circular cross-section 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 
 
 Dw 
 
 ave 
 
 case . 
 
 1 l2 
 2 
 
 rtq(2R)' 
 
 and the vortex drag has the same magnitude in this 
 
NACA TI! 11+21 
 
 75 
 
 The wave plus vortex drag is then D 
 
 This is equal to 
 
 «q(2R) 
 the minimiun 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 
 
 (?) 
 greater than for the cascade^ ''. 
 
 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. 6c-2). Since the downwash 
 is zero in the fore Mach cone, the downwash velocity in the combined 
 field is constant and equal to aU throughout the double Mach cone. 
 
 ^ 
 
 STP£AML/NE 
 
 p^o 
 
 Ey,PANSfON W^l/f 
 
 PfeAR A/IACNCONE O^ CASCADE 
 
 co/yrp/^£ss/oN i^avs 
 
 Fig. 6c-2: T\io-dimensional analysis of downwash in rear Mach cone 
 
 of an optimum cascade 
 
■Jk NACA TM li^■21 
 
 Far behind the cascade in the wake of the wing system e = a; this 
 can be shown by equating lift to rate of change of vertical momentum. 
 The individual wings of the cascade are non-interfering and, in the 
 limit as gap and chord go to zero, have two-dimensional wing character- 
 istics. The wing area for a sufficiently continuous lift distribution 
 (Ch. VIB) is equal to the cascade cross-sectional area A times p. Con- 
 sequently L = Cj^qS = (^/3)q(pA). By Munk's criterion (see Ch. V and 
 
 Ref . 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) (eU). The vertical momen- 
 tum of the surrounding fluid can be evaluated from the known "virtual 
 mass" of a solid circular cylinder of cross-sectional 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(£U) 
 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. 
 
NACA TM li<-21 
 
 75 
 
 APPENDIX VI 
 
 DERIVATION OF OPTIMUM DISTRIBUTION 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 90°; then the Mach planes will be parallel to 
 the y axis. The notation to be used is illustrated in Fig. a6-1; cylin- 
 drical coordinates (^,S,0) and the radial coordinate r will be used. 
 
 If the spatial lift distribution is Z(r) = I 
 lent lineal distribution along the ^ aixis will be 
 
 i^^T^ 
 
 then the equiva- 
 
 F(U 
 
 S=0 J (^=0 ^ ' ^0 ^ 
 
 u 
 
 f.rf^-R^-S' 
 
 y,^ 
 
 However, 
 
 Fig. A6-1 
 
 F(l) 
 
 opt 
 
 ^1 - (^/R)' 
 
 Kyi 
 
76 NACA TM li4-21 
 
 where K depends on the total lift of the sphere . Introducing the radial 
 coordinate r^ the integral equation to be solved is 
 
 K^l - (^/R)^ = 2rt r rZ(r)dr 
 
 The solution to this equation, found by differentiation with respect 
 to t,, is 
 
 Ur) = - ^ 
 
 2nRl/R^ - r^ 
 
 The total lift of the sphere is 
 
 L = 2 r k|/i - (;/R)2 d^ 
 
 rtRK 
 
 so that the distribution of lift for minimum wave drag is 
 
 Z(r) 
 
 .^% 
 
 R2-r2 
 
 For application of Hayes' method, the equivalent lineal distribution 
 along the x axis is needed. A plane ^ = ^' intersects the x ajcis at 
 X = -M^ ' ; since the distribution is spread out over a larger distance 
 along the x axis, its maximum intensity will be less; thus, 
 
 F(-)opt = 1^ - (x/MR)^ = ^^ - (^/MR)2 
 
 Hayes defines two functions such that for the lifting case (Ch. IV) 
 
 pUg, 
 
 F = 
 
 =z 
 
 -2BL sin 9,1 : ; 7^^ 
 h = -g„ sm e = — ^ h - x/MR ' 
 
NACA 1M li4-21 
 
 77 
 
 The expression for the wave drag contribution at each angiilar station 
 is, from Eq. (^4-0-50), 
 
 "i - S5//'^'(='^)^'W- 
 
 dx, dXp 
 
 and the total wave dra^ is 
 
 >2jt 
 
 Dw; 
 
 ave 
 
 
 
 ^de 
 de 
 
 The integration for dD/d0 has been carried out by Sears ^ '' 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- 
 
 mums in incompressible flow.) If h = cfl - (x/MR)^ then dD/d9 = pC^/l6. 
 Substituting in the equations above leads to the final result. 
 
 Dw; 
 
 ave 
 
 p2L2 
 8jtqR%2 
 
78 NACA TM II4-2I 
 
 CHAPTER VII. THE OPTIMUM DISTRIBUTION OF VOLUME ELEMENTS ALONE * 
 
 A^ THE SINGULARITY 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 
 
 Uq = Free stream velocity 
 Then the volume occupied by the fluid is 
 y(PoUo) 
 
 vol = md ; 
 
 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 "Tlie Drag of 
 Non-Planar Thickness Distributions in Supersonic Flow," published in 
 the Aeronautical Quarterly, Vol. VI, May 1955- 
 
NACA TM 11+21 79 
 
 and sink are combined in a dipole with axis in the free stream direction. 
 This singularity shoiild represent an element of volume, although the 
 fineness ratio of the element is zero. 
 
 The potential for a unit source at (|,0) in supersonic flow is 
 
 ri _ -1 
 
 2«^(x - O^ - p2r2 
 
 where 3 = ^M - 1; x and % are coordinates in the streamwise direction 
 and r is radial distance from the x axis. 
 
 Differentiating with respect to x gives 
 
 ^s. = ^^^^ ^. = 't. 
 
 -•x 
 
 2jt[(x - 1)2 - p2^2J 
 
 5/2 
 
 where 0^ is the potential for the unit dipole or an element of volume 
 equal to i/Uq. 
 
 B^ THE DISTRIBUTION OF V0LUI4E ELEMENTS 
 
 For a distribution of volume elements along the ^ axis with inten- 
 sity f(|), starting at ^ = 0, the potential is 
 
 2:t Jo T; ; o ol5/2 
 
 U-|)2_p2,2j 
 
 Integration by parts gives 
 
 f(^) 
 
 2rt^(x - 1)2 - 32r2 
 
 x-pr 
 
 1 r"^-^^ f'(^)d| 
 
 2it Jq ^(x - 02 - 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). 
 
80 NACA ™ lii-21 
 
 The smoothly faired body (indicated hy 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 +Z/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 
 
 .z/2 
 
 U 
 
 which agrees exactly with the sum of the volume elements, 
 
 o 
 
 An example of the second case is shown in Fig. 4f-2 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 VOLUME DISTRIBUTIONS ON A 
 
 STREAMWISE LINE AND THE SEARS-HAACK BODY 
 
 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 OM lij-21 
 
 81 
 
 If f(x) is the intensity of the volume element distribution for a 
 body of revolution of length "Z" then the drag is given by^-'-°^ 
 
 ^+Z/2 p + Z/2 
 
 D 
 
 
 ■ll2 ^ -1/2 
 
 f" (x-L)f'7x2')Zn x-L - Xg 
 
 dx-| dXo 
 
 To maintain constant total volume according to linearized theory 
 ^+Z/2 
 it is necessary that / f(x)dx = Constant. The body shape giving 
 
 ^-Z/2 
 
 minimum drag for a given length and volume has been determined by Sears 
 
 and Haack^-^i' independently. The corresponding f(x) (which is propor- 
 tional to the cross-sectional area) is given by 
 
 (16) 
 
 opt 
 
 (x) = 
 
 16 
 3« 
 
 1 - [^ 
 
 p+Z/2 
 5/2 / f(x)dx 
 
 J -1/2 ^ 8U 
 
 Z " 3n Z/2 
 
 o volume 
 
 f)^ 
 
 3/2 
 
 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 
 
 ^in 
 
 -% 
 
 p+l/2 
 
 / f(x)dx 
 
 J _7/p 
 
 -1/2 
 
 Uo(z/2)5 
 
 8qA 
 n\2 
 
 volume 
 
 _(z/2)5_ 
 
 D^ THE SEARS-HAACK BODY AS AN OPTIMUM 
 VOLUME DISTRIBUTION IN 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 
 
 . P+l/2 p +z/2 
 ^D ^ -P / / 
 
 ^e Qn^J-l/2 J -1/2 
 
 f'Yx-]_,9]f"[x2,9^ Znlx-L - Xj dx-[_ dxg 
 
82 
 
 NACA TM 1^4-21 
 
 Here fCx, 9) is determined by the use of "Mach, planes" for the 
 angle Q. 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 aji elliptical dis- 
 tribution for every 9, then the total drag is a minimum, and the drag 
 contribution at each 9 is a minimum and corresponds to that of an equiv- 
 alent Sears-Haack body. 
 
 U 
 
 It is not always possible to simultaneously minimize the drag 
 
 contributions at all angles 9. However if we consider the optimum 
 
 distribution of thickness 
 
 within a space which has 
 
 rotational symmetry about 
 
 a streamwise axis, then 
 
 it may be possible that 
 
 all the equivalent bodies 
 
 are Sears-Haack 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 Sears-Haack 
 
 body placed on the axis 
 
 is an optimum for this 
 
 space . It has the same 
 
 drag contribution at 
 
 every angle on the cylin- 
 drical control surface, 
 
 and of co\irse, the 
 
 "equivalent" body of 
 
 revolution for any angle 9 is identical with the real body. However, 
 
 a "ring" wing (which carried no radial forces) plus a central body of 
 
 revolution can be designed 
 
 to have exactly the same 
 
 drag as the Seeirs-Haack 
 
 body. The equivalent 
 
 bodies of revolution are 
 
 all identical with the C(, 
 
 Sears-Haack body. This * 
 
 is discussed in the next 
 
 section. (For the case 
 
 in which radial forces 
 
 are carried on the ring 
 
 wing see Ch. IX.) ^^^S t^/NG PLUS CENTRAL 
 
 BODY HA V/NG SAMB 0/?A G 
 AS SEAflS-/¥AACK BODY 
 
 SEARS-HAACK BODY BOUNDED 
 BY DOUBLE /t^ACH CO/VJE SPACE 
 
NACA TM lif21 85 
 
 E^ RING WING AND CENTRAL BODY OF REVOLUTION 
 
 COMBINATION HAVING THE SAME DRAG AS A SEARS-HAACK BODY 
 
 Consider a ring-wing 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 siirf ace . 
 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 Sears-Haack body if the equivalent 
 body of revolution is a Sears-Haack body. 
 
 To design such a system, we may select any smooth, slender profile 
 for the ring-wing and compute the cross-sectional areas cut from this 
 wing by a set of parallel Mach planes. These areas must then be sub- 
 tracted from the cross-sectional areas which would be cut from a central 
 Sears-Haack 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 cross-sectional area of the central body defined 
 in the usual way.) This body, together with the ring-wing originally 
 selected, is an optimum distribution of thickness within the double Mach 
 cone space . 
 
 As an example, consider a ring-wing with thickness distribution 
 corresponding to a bi-parabolic 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 half-way between the axis and the apex of the space . If the 
 central body of revolution is designed so that the equivalent Sears-Haack 
 body is of fineness ratio 5^ the resulting shape of the central body of 
 revolution is as shown in Fig. Je-1. 
 
81+ 
 
 NACA TM lif21 
 
 RING PV//VC?, '^/c-0.0 6 
 
 M^VW 
 
 Bq>UJ\/ALENr BODY OF 
 REi^OLUTION 
 
 CENTER BODY OF 
 RESOLUTION 
 
 Fig. 7e-l: Cross-sectional view of ring-wing and central body 
 (an optimal distribution of thickness within the double Mach 
 cone space ) 
 
 OPTIMUt^ THICKNESS DISTRIBUTION FOE A 
 PLANAE WING OF ELLIPTICAL PLANFOPiM 
 
 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'- ^. 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^ '' 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 planform 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 volimie V, then 
 
NACA TM 114-21 
 
 85 
 
 V = 2rt / t(r)r dr 
 ^0 
 
 (7f-l) 
 
 where R is the radius of the circular wing and r, 6 are polar coordinates 
 from the wing center (Fig. 7f-l)- The area cut out at each point along 
 
 the t axis by planes normal 
 to that axis is 
 
 u- 
 
 1 
 
 'V 
 
 /I 
 
 / /^ 
 
 // 
 
 ^\// 
 
 A / / 
 
 Z-^eX/ ^ 
 
 / V^R^'y^ 
 
 ^y^yC y^ ^^^/^ 
 
 % 
 
 W'^' 
 
 s(0 = 
 
 
 t dTi 
 
 = 2 r ^^^^^ ^^ (7f-2) 
 
 I TrS _ ^2 
 
 The equivalent lineal distribution along the x axis is 
 
 >R 
 
 S(x) = 2 cos \x / 
 
 rt(r )dr 
 
 X cos [X |fr2 - x2cos2|j. 
 
 (7f-3) 
 
 with 
 
 ^+R sec |i 
 -R sec [i. 
 
 S(x)dx = V 
 
 For minimum drag, this distribution should be (Ch. VF) 
 
 S(x) 
 
 1 - (x/r sec n)^ 
 
 3/2 
 
 Thus the integral equation to be solved for t(r) is 
 
 (7f-^) 
 
 K 
 
 1 - (x/r sec \x) =2 cos n / 
 
 t(r)r dr 
 "J X cos ^t ^r^ - x'^cos ^ 
 
 (7f-5) 
 
86 
 
 NACA ™ lil21 
 
 where K is a constant dependent upon the given wing volume. By a suit- 
 able transformation of coordinates, Eq. (7f-5) nis-y be written in the form 
 
 YSf) 
 
 ■3/2 
 
 
 
 t(a)da 
 
 I^ sec n J Q ^V^ 
 
 (7f-6) 
 
 where 
 
 = 1- (x/r sec |i)' 
 >2 
 
 a = 1 
 
 Vr)' 
 
 Eq. (7f-6) is called Abel's equation and its solution is well known, 
 c.f., Ref. 19. The solution to Eq. (7f-6) is 
 
 t(r) = 
 
 3K 
 
 i+R cos \x L- 
 
 1 - (r 
 
 /R)"] 
 
 and substitution of this in Eq. (7f-l) determines K; then 
 
 t(r) 
 
 2V 
 
 nR2L 
 
 (r/R)' 
 
 (7f-7) 
 
 Equation (7f-7) thus gives the distribution of 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 unyawed elliptic planform, make 
 the following change of coordinates: 
 
 U 
 
 X = °££ 
 
 Y = 5Z 
 
 R 
 
 (7f-8) 
 
 Fig. 7f-2 
 
NACA Tt4 lii-21 
 
 87 
 
 The circular wing is then transformed into an elliptic wing whose equa- 
 tion is 
 
 if ^ '' 
 
 If- 
 
 It can be verified that the thickness distribution 
 
 2V_ 
 nab 
 
 1-1^ 
 
 (7f-9) 
 
 obtained from Eq. (7f-7) through the transf onaation Eq. (7f-8) is the 
 optimum for this more general case; that is, the equivalent lineajr dis- 
 tribution for Eq. (7f-9) with a set of Mach planes inclined at the 
 angle p. as shown in Fig. 'Jf-2 is 
 
 S(X) 
 
 8v 
 
 3itZ 
 
 3/2 
 
 (7f-10) 
 
 where 
 
 I = ^a2 + b^tan^^ 
 
 Since Eq. (7f-10) represents a Sears-Haack body, the thickness given by 
 Eq. (7f-9) 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 Sears-Haack bodies as 
 seen at each angular reference station. If the reference station is 
 at an angle from the horizontal, then the Mach planes cut the elliptic 
 planform at an angle \i defined as (Ch. IVC). 
 
 tan 
 
 f^ 
 
 1 cos 9 
 
 (7f-ll) 
 
 and the total drag is 
 
 ^2n 
 
 dD 
 de 
 
 de 
 
88 
 
 NACA TM 1421 
 
 The increment of drag at each reference station is (Ch. IVC ) 
 
 — = - ^ / / S"(x)S"(|)Zn|x - ||dx d| 
 
 de 4n J ^ 
 
 (7f-l2) 
 
 1^ 
 
 and the total drag for the optimum thickness distribution Eq. (7f-9) is 
 
 D, 
 
 'opt 
 
 _ 4qV^ 
 
 Tta 
 
 \ 
 
 M^ - 1 + 
 
 2a^ 
 
 M' 
 
 -?i 
 
 >3/2 
 
 (7f-l3) 
 
 Defining 
 
 t = tr 
 
 .2 / \2 
 
 and 
 
 then 
 
 D = C£|q.nab 
 
 M^ 
 
 2a2\ 2 
 1 + Ito^ 
 
 a 
 
 2 
 
 -D 
 
 opt 
 
 
 Nf 
 
 (7f-l^) 
 
 This result agrees with that given by Jones 
 
 :iK 
 
NACA TM IkZL 
 
 CHAPTER VIII. imiQUENESS PROBLEMS FOR OPTIl-IUM DISTRIBUTIONS IN SPACE 
 
 A^ THE NON-UNIQUENESS OF OPTHvIUM 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 wing is obtained when the spanwise lift distribution is elliptical. 
 
 According to Munk's stagger theorem^ -^'' 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 lonique 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 VI-C.) The first is a constant intensity over the 
 circular disc located at the maximi^i cross-section of the space. The 
 second is an elliptical intensity concentrated on the axis of the double 
 Mach 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. 
 
90 
 
 NACA Tlvl 11+21 
 
 u 
 
 A " ZERO LOA DING ' ^ PL A C£D >V/ T/^/N A^ EL L IPSO/ DAL S^A C£- 
 
 Such a "zero loading" placed within any space alters neither the 
 lift nor the drag of the original lift distribution. For this reason 
 optimiffii 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 non -uniqueness in such cases see Ch. VII. 
 
 B^ UNIQUENESS OF THE DISTANT FLOW FIELD 
 PRODUCED BY AN OPTIMUM 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 f2_ . (?:,il,0 ^'^'^ ^2 -(-(^jT^O ^.re members of the opti- 
 mum family not included in the original group (whose members were related 
 through "zero loadings"). Assume also that f-]_ , and f2 . do not pro- 
 duce identical perturbation velocity fields far from the singularity 
 
NACA ™ 1^21 91 
 
 distribution. Then the drag of f-, equals the drag of fp (or 
 
 -^opt "^opt \ 
 
 D-[_ 4- = -^2 -t- ) ^y definition of the optimum family. Also f2 may be 
 
 set equal to f-, ^ + Af, where Af carries zero net lift (or volume), 
 -"-opt ' 
 
 but has a velocity perturbation field which is not identically zero far 
 
 from the singularities. 
 
 The distribution Z^ is orthogonal to (does not interfere with) fn 
 
 -"-opt 
 
 Tliis follows because any 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 f-]_ , , by definition, cannot be 
 
 improved, and must, therefore, be orthogonal to Af . 
 
 Since Af is orthogonal to fn . , Do , = Dt , + DAf, but we also 
 
 -Lopt' ^opt J-opt ^ ' 
 
 know that Dp = D-. and, therefore, Da^ must equal zero. Here we 
 '^opt -^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 non-negative. If Af produces any disturbances 
 
 far from the lifting system it must have positive drag, and so Af 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 non-negative at every point on the s\irf ace . 
 
 C_^ IMIQUENEGS OF THE ENTIRE "EXTERNAL" FLOW FIELD 
 PRODUCED BY AN OPTIt'IUI^ FAMILY 
 
 It has been shown that any two members of an optimum family produce 
 identical velocity perturbations on the distant control surface. 
 
 If f-i (l.T.O and fp {iy^,^) are two members of an optimum 
 -^opt ^opt 
 
 family, then f-i - fp 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 f-i - fp exists, and let "E" represent the external flow field 
 -■-opt "^opt 
 
92 
 
 NACA ™ 11+21 
 
 u 
 
 consisting of points whose aft Mach cones 
 do not intersect "S." Assume that at 
 some point in the external field "E" the 
 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 
 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-, - fp 
 
 opt ^opt 
 
 is a singularity distribution causing zero 
 
 drag, so f-]_ , - fp ^ 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 
 
 produced by f 
 
 1, 
 
 - fo 
 
 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^ EXISTENCE OF SYI-FiETRICAL OPTBOI DISTEIBUTIONS 
 IN SYMMETRICAL SPACES 
 
 It can be shown that, if the boundary of a space has a horizontal 
 plane of symmetry, 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(^?y>z) represent an optimum lift distribution in the space. 
 The distribution ^opt^^jy)-^) has the same drag and lift (the drag of 
 
NACA m li^-21 
 
 95 
 
 the individual lifting elements 
 is unaltered by the change of 
 position, and the interference 
 drag of any element pair is 
 unaltered also). 
 
 Since Z^ -^(x,y,-z) has the 
 
 same lift and drag as Z^ -^(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, 
 
 |^opt(x,y,-z) + |Zopt(x,y,z) pro- 
 duces the same external flow field 
 
 3-s Zopt(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 plajies of 
 symmetry can be used. 
 
9k NACA TM 1421 
 
 CHAPTER IX. INVESTIGATION OF SEPAEABILITY OF LIFT, 
 THICKNESS AND SIDEFORCE PROBLEMS * 
 
 A^ THE SEPARABILITY OF OPTIMUM 
 
 DISTRIBUTIONS 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 optimmn distributions of singulari- 
 ties for simultaneously providing the net lift and volume? 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 i_s 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 non-planar 
 systems, and consequently the above problems must be re -investigated 
 for these more general configurations. 
 
 * Portions of this chapter have appeared in the paper "The Drag of 
 Non-Planar Thickness Distributions in Supersonic Flow," published in 
 the Aeronautical Quarterly, Vol. VI, May 1955. 
 
NACA TM 11+21 95 
 
 Optimum Distributions Providing Lift and Volume 
 
 In non-planar distributions of lift, sideforce and thickness there 
 is generally interference among the different singularities. This means 
 that the drag for a given net lift may in some cases be decreased by 
 adding thickness or sideforce elements and taJcing advantage of negative 
 interference drag. 
 
 In order to study such cases let l{x,-y,z), t(x,y,z), and s(x,y,z) 
 represent respectively distributions of lift, thickness and sideforce 
 in x,y,z space within some boundary. Here we will exclude, without loss 
 of generality, those distributions of I and s which are completely equiva- 
 lent to elements of voTume or thickness (see III -A, the closed vortex 
 
 line). Let l-^ (x,y,z) + t-|_°(x,y,z) + s-]_°(x,y,z) give the minimum pos- 
 sible drag for one unit of net lift and zero net thickness and sideforce. 
 (The superscript simply indicates the net lift or thickness or sideforce 
 
 of the distribution.) Also let Z2°(x,y,z) + t2 (x,y,z) + S2°(x,y,z) give 
 
 the minimum possible drag for one unit of net volume and zero net lift 
 and side force. We ask what distribution gives the minimum drag when 
 both the net lift and net volume are simultaneously prescribed and equal 
 to Lq and Vq respectively? 
 
 Consider the distribution A(x,y,z) = L, 
 
 o 
 
 Z-L-'- + tjL° + s-L^ 
 
 + vJz2°+t2^+S2° 
 
 which gives the prescribed net lift and volume. For this to be the opti- 
 mum it is necessary and sufficient that it be orthogonal to every distri- 
 bution Z° + t° + s°, which contains zero net lift, zero net volume and 
 zero net sideforce. For example, any loading Z° + t° + s° multiplied by 
 an arbitrary constant C csin be superimposed on A without altering the 
 net lift, Lq and net voli;tme, Vq . If this distribution Z° + t° + s° were 
 
 not orthogonal to A, then C could be adjusted to give a negative inter- 
 ference drag with A greater than the drag of C(Z° + t° + s°) by itself. 
 Hence the distribution A could be improved and therefore wo;ild not be 
 an optimum. It is also true that any possible improvement of A must be 
 obtainable by superposition of a loading of the type C(Z° + t° + s°) on A. 
 So for A to be an optimum it is both necessary and sufficient that A be 
 orthogonal to any loading Z° + t° + s°. 
 
 However, l-^-^ + t-^^ + s-|_° and Z2° + t2 + S2° are each orthogonal to 
 
 any Z° + t° + s° since each one is an optimal distribution in its own 
 restricted class. Therefore because of the linearity of the interference 
 
 terms LqU-]_-'- + t-^° + s-|_°j + V^^Zg" + t2 + S2°)is orthogonal to any 
 
 Z° + t° + s° and A(x,y,z) is the optimum distribution having lift = Lq 
 
 and volume = Vn . 
 
96 . NACA TM 1^4-21 
 
 The Drag of the Optlmiun Distribution Providing Lift and 
 Volume in a Region Having a Horizontal Plane of Symmetry 
 
 Tlie drag of the optimum distribution A(x,y,z) must next be deter- 
 mined. If Z-|_-'- + t-|_° + s-j_° is orthogonal to Z2° + ^2 "*" ^2° "then the 
 drag of A(x,y,z) is just the sum of the drags of Lq(^Z-]_ + t-]_° + s-^") 
 and of Vq(z2° + t2'^ + S2°). We know that (12° + tg"^ + Sg") -L ftj° + Sj° 
 
 so the question arises is (Z2 + tp + ^2 ) also -LZ]_ ? (Here the sym- 
 bol "_L" indicates orthogonality.) 
 
 In order to answer this question it is convenient to represent 
 Z-1 (x,y,z) by a concentrated lift of one unit I-^q -plus a distribution 
 containing zero net lift Z2_°(x,y, z) . The concentrated unit of lift can 
 be placed anywhere in the space and then Z-|^ (x,y,z) is simply the dif- 
 ference between l-^ (x,y,z) and Z-[_g . Similarly it is convenient to 
 replace t 2 tiy t 25 + ^°2^-^'^''^'^ ' '^^'^ optimum distribution is then 
 
 A(x,y,z) = L^(l^Q^ + lj° + tj° + 5^°) + Vo(t25^ + Z2° + t2° + S2°) 
 
 The distributions in brackets are orthogonal If 
 
 1 1|+ 1,1°, +0, o 
 t' 1 c -i— ^of\ "T" ^p > ^p f sp 
 
 or if 
 
 t ^l7 I4-7O4.+ O4.QO 
 ^25 -"- 'I6 ^ '1 + ^1 + ^1 
 
 The concentrated unit of lift Z-|_g may be located at any point in 
 
 the space and has the same interference drag with (t2R + Z2° + t2° + Spy 
 for all locations. Thus if there is any point in the space where a unit 
 of lift has no interference with (t2§ + Z2° + t2° + Sg^j the orthogo- 
 nality of the two components of A(x,y,z) is assured. (This does not 
 depend on the connectivity or the convexity of the space.) 
 
NACA Tt4 1^21 
 
 97 
 
 For example, if the boundary of a space has a horizontal plane of 
 symmetry, then there are optimum distributions in the space having sym- 
 metry properties. (The proof is similar to that given in Ch. VIII for 
 a lift distribution.) If some portion of the plane of symmetry is con- 
 tained inside the space then the concentrated unit of lift can be located 
 in this plane and orthogonality demonstrated. 
 
 B^ THE NON-INTKHJ^'KKENCE OF SOURCES WITH OPTBIUM 
 
 DISTRIBUTIONS OF LIFTING ELEMENTS IN A SPHERICAL SPACE 
 
 In general there is interference between non-planar distributions 
 
 of sources and lifting elements, as shown by Hayes^-'-''. This means that 
 in general the optimimi distribution of singularities which provides one 
 unit of lift may contain volume elements or sources as well as lifting 
 elements. However for certain spaces it can be proved that there is no 
 interference between a source and the optimum distribution of lifting 
 elements alone. So for these spaces the optimum way of providing lift 
 requires no sources. 
 
 Following is a proof that a single soiirce placed at any point within 
 a sphere has no interference drag with the optimum distributions of 
 lifting elements alone in the sphere . 
 
 An optimum distribution of the total lift, L, within a sphere of 
 radius "R" (center at the origin) is given by (see Appendix VI-1) 
 
 ^opt 
 
 jt-R^^' 
 
 ,2^21/1,2 
 
 where r = Spherical radius to 
 any point. Let a source loca- 
 ted at an arbitrary point, P, 
 within the sphere be denoted 
 by S, and let P' be the pro- 
 jection of P on the horizon- 
 tal (x-y) plane. The poten- 
 tial of S is identical with 
 that caused by some lifting 
 
 element distribution, Z-"-, 
 
 on the line between P and P' plus a source S' at P'. (See Ch. Ill 
 Section B. The shells which have sources and sinks on the top and bot- 
 tom faces respectively are arranged to form a vertical column of infini- 
 tesimal cross section.) The distribution I has zero net lift. 
 
98 NACA TO lif21 
 
 The interference between ^Q-p-j^ and S is equal to the interference 
 
 between Zopt ^^^ S' plus the interference between Zopt sind I . Tlie first 
 component is zero because of the symmetry of ^Q^t s-^o^t the x-y plane, 
 
 (see the discussions of interference in Ch. IV). If the second compo- 
 nent were not zero it would be possible to obtain a distribution of 
 lifting elements alone with lower drag than ^Qp-f Since ^Qp^ has the 
 
 minimum drag by definition, the second interference component is also 
 zero. This completes the proof for a particular ^Qp-j^. 
 
 This proof can be extended to the entire family of optimum lift 
 distributions in the sphere as follows. As previously mentioned, all 
 of the optimum distributions produce identical effects far out on the 
 Mach cone and far behind the wing system. Interference drag terms can 
 be computed from these distant effects alone. Hence a source has the 
 same interference drag with each of the optimum distributions, and this 
 is zero for all cases since it has been proved zero for one case. 
 
 This proves that source distributions in a spherical volume cannot 
 reduce the drag attained with any of the optimum distributions of lifting 
 elements alone in that volime . 
 
 Similar methods may be applied to ellipsoids having one principal 
 axis vertical, to double Mach cones, and to many other volumes. It is 
 sufficient that the volume have a horizontal plane of symmetry, and that 
 the vertical lines connecting all points in the volume with this plane 
 are entirely contained within the volume. 
 
 C_^ THE NON-INTERFEEENCE OF SIDEFORCE ELEMENTS WITH OPTIMUM 
 DISTRIBUTIONS OF LIFTING ELEMENTS IN A SPHERICAL SPACE 
 
 As shown by Hayes '''' there is, in general, interference between 
 non-planar distributions of lifting elements, sideforce elements and 
 sources. It has been proven in Ch. IX-B that there is no interference 
 between a source and the optimimi distribution of lifting elements alone 
 in a spherical space. It remains to show a similar result for the inter- 
 ference of a sideforce element with the same optimum lift distributions. 
 The proof will be carried out in a manner similar to that of the previous 
 proof . 
 
 Consider an optimum distribution of total lift L in a sphere of 
 radius R; the lift distribution is given by 
 
 7 - L 
 
 "■opt 
 
 I 
 
 «2r2(^r2 - r2 
 
NACA TM li|21 
 
 99 
 
 where r is the radius to any point from the center of the spherical 
 space. Let S be a sideforce element at a point P within the sphere, 
 and let P' be the projection of P on the xy plane. 
 
 ^ 
 
 
 U ^' , 
 
 I 
 SIO£ ^^ ■ 
 
 ( 
 
 - T ^ 
 
 ^. 
 
 Fig. 9c-l 
 
 As part of the proof it is necessary to show how a sideforce ele- 
 ment can be transferred from one point to another along a line parallel 
 to the y axis. The procedure is shown in Fig. 9c-2. First a vortex 
 ring of infinitesimal height and finite width, d, is superimposed on the 
 original sideforce element; the strength and placement of the former is 
 
 S/OE FOPCE 
 
 R/AJG 
 
 + 
 
 DOUBLET 
 
 /=>(>, y-d,^; 
 
 5 rOB FO/?C£ 
 ELEMEf^T 
 
 Fig. 9c-2 
 
 to be such that the sideforce at P(x,y,z) is just canceled. The poten- 
 tial for the vortex ring can be found by integrating the potentials for 
 constant -strength infinitesimal vortex rings (Ch. IIIA) distributed along 
 
100 
 
 NACA TM lil21 
 
 the line x, z = Constants. The second step is to superimpose on the 
 vortex ring a finite width lifting line "doublet." This latter singu- 
 larity is formed by taking the limit as two equal and opposite strength 
 finite width lifting elements are brought together keeping the product 
 of lifting element strength and distance apart constant. The potential 
 of the original sideforce element at P(x,y,z) plus the two added ele- 
 ments, 0g(x,y,z) + (^ + 0-Q, is identical to that for a sideforce element 
 
 at P(x,y-d,z). 
 
 Thus the potential of a sideforce element "S" inside the spherical 
 space is the same as that for a finite vortex ring "V," plus a lifting 
 line doublet "D, " plus a sideforce element "S'" in the vertical plane of 
 symmetry (at P' in Fig. 9c-l). The interference drag between the opti- 
 mum lift distribution Iq-^^^ and S is equal to the interference drag 
 
 between Zopt ^^'^ S' plus that between Zopt and V plus that between Zopt 
 and D. The last of these drags must be zero since D is a lift distri- 
 bution having zero net lift; if this were not zero D co\ald be combined 
 with Zopt "to fonn another distribution having less drag than Zopt 
 (Section kE) , in contradiction of original assumptions. Since V can be 
 thought of as built up from distributions of infinitesimal vortex rings, 
 which in turn are made up of source-sink doublets with axes aligned with 
 the stream direction, the interference drag between V and Zopt is zero 
 by the proof given in Ch. IX-B. 
 
 The only possible interference drag with Zopt could be that of the 
 sideforce element S' in the vertical plane of symmetry, and this can be 
 shown to be zero because of the symmetry. Consider the interference 
 drag of S' with lifting elements in the rear Mach cone of S' as shown 
 in Fig. 9c-3a-. The interference drag will be due to the downwash of S' 
 
 a 
 
 / 
 
 
 / G r^^— 
 
 k 
 
 \ 
 
 (a; 
 
 \ ®[-i! 
 
 •^^h 
 
 \ 
 \ 
 
 i,~ 
 
 \ 
 
 — ^- 
 
 1 
 
 4^-^ ,^ 
 
 V /^^-^ 
 
 JK 
 
 \^r^ 
 
 ■^ i. 
 
 ii>) 
 
 Fig. 9c-3 
 
NACA TM 1^21 
 
 101 
 
 acting on the lifting elements (see Ch. IV-J); for each lift element Z-^^ 
 which receives a downwash from S' there is another lift element Z„ of 
 
 the same strength which receives an upwash of equal magnitude. Hence 
 the interference drag of S' with each pair of lifting elements in its 
 rear Mach cone is zero. Consider now Fig. 9c-3b representing S' and a 
 pair of lift elements in its fore Mach cone . The interference drag 
 here is due to sidewash fields from the lift elements acting on S' . But 
 for every lift element Za producing a sidewash Va on S ' there is a sym- 
 metrically placed l-^ producing a sidewash v-j^ = -v^^ on S'. Again the 
 
 interference drag is zero. Thus there is no interference drag between S' 
 and Zopt and hence none between S and Zopt for this particular Zopt- 
 Following the same type of reasoning as is given in Ch. IX-B, this proof 
 can be extended to all optimum distributions within the sphere; this is 
 so because of the uniqueness of the optimum external flow field. 
 
 Thus it is proven that sideforce distributions, as well as source 
 distributions, in a spherical space cannot reduce the drag attained with 
 any of the optimimi distributions of lifting elements alone in that space. 
 
 Similar methods may be applied to other spaces if those spaces have 
 both a horizontal and a vertical plane of symmetry containing the free 
 stream direction and meet certain convexity requirements. The latter 
 can be stated as requirements that straight lines from each point within 
 the space which extend to the planes of symmetry and are perpendicular 
 to them must lie entirely within the space. 
 
 D^ INTERFERENCE PROBLEMS IN CERTAIN SPACES 
 BOUNDED BY MACH ENVELOPES 
 
 Let some region "R" be chosen in the y-z plane, which is perpen- 
 
 dicular to the flow direction. Consider the space S consisting of 
 points such as "P" whose fore or ai't Mach cones intersect areas in the 
 
 A 
 
 BOUNDARY OF 
 fi?£GION "/? " 
 IN y-Z, PLANE 
 
 •X 
 
 CASCADE 
 OP A/PFO/LS 
 /A/ y-Z ^^.AA/E 
 
 A set of parallel Mach planes cutting this source distribution determines 
 an equivalent lineal source distribution according to the method of Hayes. 
 For convenience, this equivalent lineal source distribution will be denoted 
 
102 
 
 NACA TM 11+21 
 
 y-z plane which axe completely contained in the region "R." An optimum 
 distribution of lift in this space is given by a uniformly loaded cascade 
 of airfoils (of infinitesimal chord and gap) covering the region "R," 
 since this gives constant downwash in the combined flow field. 
 
 The resulting flow pattern is two-dimensional within the space "S." 
 It then follows that the sidewash and press\ire are zero in this space 
 and sideforce elements or sources introduced in "S" have no interference 
 with the optimum lift distribution. 
 
 THE INTERFERENCE BEWEEN LIFT AJTO SIDEFORCE ELEMENTS 
 AND Ml OPTIMUM DISTRIBUTION OF VOLUME ELEMENTS 
 
 l//^r E LB MS NT-, 
 
 Consider a Sears-Haack body placed on the axis of a double Mach 
 cone, and place a lifting element as shown in the illustration. The 
 interference drag between body 
 and lifting element is composed 
 of two parts, the effect of the 
 body nose on the lifting element 
 and the effect of the lifting 
 element on the tail of the body. 
 
 TRAILING 
 VOfiT/CES 
 
 The nose of the body corre- 
 sponds to a source distribution 
 and produces an up wash velocity at 
 the lifting element. This causes 
 
 negative drag. The lifting element produces a positive pressure at the 
 tail of the body. This also causes negative drag so the total inter- 
 ference drag is negative. (This argument, of course, applies not only 
 to the Sears-Haack shape but to other shapes also.) 
 
 The total drag of the combination is equal to the drag of the 
 Sears-Haack body alone plus the drag of the lifting element alone plus 
 the interference drag. The drag of the lifting element alone is pro- 
 portional to the square of the lift it carries. However, the interfer- 
 ence drag is proportional to the first power of the lift on the element 
 and to the first power of the strength of those sources amd sinks in 
 the body which are affected by interference. The lift carried by the 
 element can, therefore, always be made small enough so that the drag of 
 the element alone is less (in absolute magnitude) than the interference 
 drag. Thus, the total drag of the combination can be made less than 
 the drag of the Sears-Haack body alone . 
 
 This suggests placing elements of lift and sideforce in a ring 
 about the Sears-Haack body, and so arranged that the force on each ele- 
 ment is directed radially outward from the body. This process may be 
 used to construct a central body plus cylindrical shell which has zero 
 drag (see Ch. IX-F) . 
 
NACA TM 11+21 
 
 103 
 
 Such a system was investigated first by Ferrari^ -^ and later by 
 Ferri^ ■', and its two-dimensional analogue is the Busemann biplane. 
 
 It, therefore, appears that the optimum distribution in space of 
 volume elements alone yields minimum drag values consistent with the 
 Sears-Haack values. However, the optimum distribution of volume elements 
 plus lifting and sideforce elements should give zero drag for any total 
 volume . 
 
 I\ RING WING AND CENTRAL BODY OF REVOLUTION HAVING ZERO DRAG 
 
 The theoretical minimum drag value for a distribution of thickness 
 elements that has no interference with lift or sideforce elements is the 
 drag of a Sears-Haack body. It has been stated in Ch. IX-E that inter- 
 ferences between thickness distributions and distributions of lift or 
 sideforce may provide negative drag contributions which reduce the theo- 
 retical minimum wave drag of a system to zero. This Section illustrates, 
 for the double Mach cone volume, a central body of revolution which, 
 together with a certain distribution of radial forces on a cylindrical 
 shell, has zero wave drag. The method employed here to design such a 
 system makes use of certain equivalences between sources and line distri- 
 butions of elementary vortex shells. (These equivalences are discussed 
 in Ch. III-B.) 
 
 Consider a radially symmetric, continuous distribution of sources 
 filling a cylindrical space contained within the double Mach cone volume. 
 
 CYl /NDER 
 
 IA//TH 
 SOUf?CES 
 
 OOUBLE A/iACH 
 
104 
 
 NACA TM 1^4-21 
 
 by F(x). Because of the radial symmetry of this case, F(x) is inde- 
 pendent of the angle 9 on the distant control surface . A central body 
 of revolution which is represented by the negative of F(x) will just 
 cancel the velocities induced at the distant control surface by the 
 original sources. The drag of the combined system is then zero. The 
 remaining step is to relate the original source distribution to a dis- 
 tribution of radial forces around the boundary of the cylindrical space. 
 It can be shown (see Ch. III-B) that a source and a sihk of equal 
 strength, lying on the same line parallel to the flow direction, have 
 exactly the same effect at the distant control surface as a line of con- 
 stant strength elementary vortex shells connecting the source and the 
 sink. If such vortex shells are considered to replace a source distri- 
 bution whose strength is independent of the radial distance, the forces 
 on adjoining shells inside the cylinder cancel one another, while the 
 forces on the outer sides of the shells next to the boundary of the 
 cylindrical space determine the radial force. A cylindrical shell having 
 this radial load distribution plus a central body of revolution which 
 corresponds to the source distribution -F(x) constitute a system having 
 zero drag . 
 
 As an example, suppose that a cylinder within a double Mach cone 
 volume is considered to contain a source distribution which varies lin- 
 early with axial distance but is independent of radial distance. That 
 is, the source strength per unit area inside the cylinder is 
 
 f = 
 
 -^oll - ? 
 
 where x is measured from the leading edge of the cylinder, c is the 
 cylinder length, and fo is the strength of the sources at the rear face 
 
 of the cylinder. The equivalent linear source strength corresponding 
 to this original distribution is given by 
 
 ^ F(x) 
 
 3R 
 
 w 
 
 x\2 , _i/-x 
 — 1 + cos I 
 
 PR/ 
 
 \m 
 
 -PR < X < +pR 
 
 i^W=(#oUl-f) 
 
 PR < X < C - PR 
 
NACA TO lij-21 
 
 10^ 
 
 where R is the cylinder radius. Now, if the negative of this source 
 distribution is assumed to represent a body of revolution, then within 
 the accuracy of slender body theory the area distribution of the central 
 body is 
 
 S(x) = - i r F(x)dx 
 
 For illustration, the dimensions of the cylinder are assumed to be 
 such that 3R/C = 1/2; that is, the radius is half the distance between 
 the axis and the apex of the double Mach cone volume. The shape of the 
 central body of revolution which cancels the effect of the original 
 source distribution for this case is shown in the accompanying figure. 
 The distribution of radial force which can replace the original linearly 
 varying source distribution is 
 
 p2 
 
 1 - f )dx - 
 
 pVf^c 
 
 {!){- - f) 
 
 JM ■ ^ i 
 
 r /Uo\ 
 
 c (cTJ 1.0 
 
 
 RADIAL FORCE DISTRIBUTION 
 ON CYIINDRICAU SHELL 
 
 DOUBLE MACH 
 CONE \^OLaA/IE 
 
 CROSS- SECTIONAL 
 SHAPE OF CENTRAL BODY 
 OE REVOLUTION WM/CH 
 CANC£LS(AT DISTANT CONTROL 
 SURFACE) DISTURBANCE DUE 
 TO RADIAL FOFfCE 
 
 lO 
 
 1.5 
 
 X/C 
 
 CENTRAL BODY OF REVOLUT/ON AND RAD/AL 
 rORCE DISTRIBUTION HAVING .ZERO y[/AVE DRAG 
 
106 NACA TO 11+21 
 
 CHAPTER X. RESULTS AKD CONCLUSIONS 
 
 It appears that certain idealized spatial distributions of lift 
 and thickness may produce materially less wave drag and vortex drag thaji 
 comparable planar systems . It is by no means certain that such advan- 
 tages can be realized in practical aircraft designs, but further inves- 
 tigation of specific configurations is warranted. 
 
 One of the interesting features of spatial lift and thickness dis- 
 tributions is that optimum arrangements are generally not unique. This 
 may raise the problem of determining which member of an optimum family 
 has the least siirface area or is best adapted for structure. 
 
 Another interesting property of spatial distributions is the inter- 
 ference which may arise between lift and thickness distributions. This 
 interference can be used to account for the zero wave drag of a Busemann 
 biplane or of Ferrari's ring wing plus central body. However it is 
 shown that in some cases thickness distributions have no interference 
 with an optimum spatial distribution of lifting elements, and so cannot 
 be used to reduce the drag due to lift in such cases. 
 
 A number of other results are obtained in this report and detailed 
 discussions of the basic singularities and Hayes' method of drag evalu- 
 ation are included. However it is clear that the scope of the field is 
 such that this investigation must be regarded as a preliminary exploration. 
 
NACA ™ lif21 107 
 
 CHAPTER XI. REFERENCES 
 
 1. Hayes, Wallace D., "Linearized Supersonic Flow," North American 
 
 Aviation, Inc., Los Angeles, Report No. 
 A.L. 222, June, 19^?. 
 
 2. Graham, E. W.; Lagerstrom, P. A.; Beane, B. J.; Licher, R. M.; 
 
 Rodriguez, A. M.; "Reduction of Drag Due to 
 Lift at Supersonic Speeds," WADC Technical 
 Report 5^-524, April, 195^- 
 
 3. Graham, E, W.; Beane, B. J.; Licher, R. M.; "The Drag of Non- 
 
 Planar Thickness Distributions in Supersonic 
 Flow," Aeronautical Quarterly, Vol. VI, No. 2, 
 pg. 99, May 1955- 
 
 h. Rohinson, A., "On Source and Vortex Distributions in the Linear- 
 ized Theory of Steady Supersonic Flow," 
 Quarterly Journal of Mechanics and Applied 
 Mathematics, Vol. I, 19^8. 
 
 5. Hadamard, J., "Lectures in Cauchy's Problem in Linear Partial 
 
 Differential Equations," Yale University, 
 New Haven, 1923- 
 
 6. von Karman, T., "Supersonic Aerodynamics - Principles and Appli- 
 
 cations," Journal of the Aeronautical Sciences, 
 Vol. Ik, No. 7, 19^7. 
 
 7. von Karman, T., "The Problem of Resistance in Compressible Fluids," 
 
 Proc . 5'fch Volta Congress, R. Accad. D' Italia 
 (Rome), pp. 222 - 277 (1936). 
 
 8. Rodriguez, A. M.; Lagerstrom, P. A.; and Graham, E. W.; "Theorems 
 
 Concerning the Drag Reduction of Wings of 
 Fixed Planf orm, " Journal of the Aeronautical 
 Sciences, Vol. 21, No. 1, pg. 1, January, 1954. 
 
 9. GraJiam, E. W., "A Drag Reduction Method for Wings of Fixed Plan- 
 
 form," Journal of the Aeronautical Sciences, 
 Vol. 19, No, 12, pg. 823, December, 1952. 
 
 10. Beane, Beverly J., "Examples of Drag Reduction for Delta Wings," 
 
 Douglas Aircraft Company, Inc. Report SM-lkkk'J, 
 January 12, 1953. 
 
108 NACA ™ 11+21 
 
 11. Walker, K., "Examples of Drag Reduction for Rectangular Wings/' 
 
 Douglas Aircraft Company, Inc. Report SM-lMi46, 
 January 15, 1955- 
 
 12. Munk, Max M., "The Reversal Theorem of Linearized Supersonic 
 
 Airfoil Theory," Journal of Applied Physics, 
 Vol. 21, No. 2, pp. 159 - l6l, February, 1950. 
 
 13. Jones, Robert T., "The Minimum Drag of Thin Wings in Frictionless 
 
 Flow," Journal of the Aeronautical Sciences, 
 Vol. 18, No. 2, pg. 75, February, 1951- 
 
 ik. Jones, Robert T., "Theoretical Determination of the Minimum Drag 
 
 of Airfoils at Supersonic Speeds," Journal of 
 the Aeronautical Sciences, Vol, 19, No. 12, 
 December, 1952. 
 
 15. Mimk, Max M., "The Minimum Induced Drag of Airfoils," NACA T.R. 
 
 No. 121, 1921. 
 
 16. Sears, William R., "On Projectiles of Minimum Wave Drag," 
 
 Quarterly of Applied Mathematics, Vol. IV, 
 No. 4, January 19l<-7. 
 
 17. Haack, W., "Geschossf ormen Kleinsten Wellenwiderstandes," Bericht 
 
 158 der Lilienthal-Gesellschaft fur Luftfahrt. 
 
 18. Lamb, Sir Horace, "Hydrodynamics," 6th Revised Edition, 19^5, 
 
 Dover Publications, New York. 
 
 19. Hlldebrand, F. B., "Methods of Applied Mathematics," Prentice- 
 
 Hall, 1952. 
 
 20. Ferri, A., "Application of the Method of Characteristics to Super- 
 
 sonic Rotational Flow," NACA T.R. 8i^l, 191+6. 
 
 21. Ferrari, C, "Campi di Corrente Ipersonora Attorno a Solidi di 
 
 Rivoluzione," L'Aerotecnica, Vol. XVII, No. 6, 
 1957, pp. 507 - 518. Also available as "Super- 
 sonic Flow Fields About Bodies of Revolution," 
 Brown University, Graduate Division of Applied 
 Mathematics, Translation No. A9-T-29, 19^8. 
 
 22. Bleviss, Z. 0., "Some Integrated Volimie Properties in Linearized 
 
 Flow and Their Connection With Drag Reduction 
 at Supersonic Speeds," Do\:iglas Aircraft Company, 
 Inc. Report SM-I9I+69, February 1956. 
 
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