,HM<^HX.^n i^^pjpEiNTIAL- ^' c^f°^ 
 
 ^^«^^:^'^^'/^^?<^:>i;J;s'^SJi0 
 
 NACA 
 
 :f 
 
 RESEARCH MEMORANDUM 
 
 SOME, EXAMPLES OF THE APPLICATIONS OF THE TRANSO^C 
 
 AND SUPERSONIC AREA RULES TO THE # ^ # 
 
 PREDICTION OF WAVE DRAG ^ ^ 
 
 By Robert L. Nelson and Clement J. Welsh ^ ^ <t) ^ 
 
 Langley Aeronautical Laboratory ^^ e? ■^ *^ 
 
 Langley Field, Va. ^ *^<P c^ 
 
 UNIVERSITY OF FLORIDA ^ #/ ^' 
 
 DOCUMENTS DEPARTMENT ^ S ^ "^ 
 
 120 MARSTON SCIENCE LIBRARY ^ j^/ n^ ^ 
 
 RO. BOX 11 7011 ^ ;? J? 
 
 r-^AINPS^'lLLE. T-L 3261 1 -701 1 USA ^ «. 
 
 CLASSIFIED DOCUMENT ^ 
 
 This material contains information affecting the National Defense of the United States within the meaning 
 of the espionage laws, Title 18, U.S.C., Sees. 793 and 794, the transmission or revelation of which In any 
 manner to an unauthorized person Is prohibited by law. 
 
 NATIONAL ADVISORY COMMITTEE 
 FOR AERONAUTICS 
 
 WASHINGTON 
 
 March 20, 1957 
 
 i C<3*lFft)ENTlAL 
 
 J 
 
 '■■m^ "c.'iAf^^i^' 
 
1 3'^ O'?^'^ ^' ^^^^011 
 
 NACA RM L56DII CONFIDENTIAL 
 
 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 
 
 RESEARCH MEMORAI^IDUM 
 
 SOME EXAMPLES OF THE APPLICATIONS OF THE TRANSONIC 
 AND SUPERSONIC AREA RULES TO THE 
 PREDICTION OF WAVE DRAG 
 By Robert L. Nelson and Clement J. Welsh 
 
 SUMMARY 
 
 The experimental wave drags of bodies and wing-body combinations 
 over a wide range of Mach numbers are compared with the computed drags 
 utilizing a 24-term Fourier series application of the supersonic area 
 rule and with the results of equivalent-body tests. 
 
 The results indicate that the equivalent -body technique provides a 
 good method for predicting the wave drag of certain wing-body combina- 
 tions at and below a Mach number of 1. At Mach numbers greater than 1, 
 the equivalent -body wave drags can be misleading. The wave drags com- 
 puted using the supersonic area rule are shown to be in best agreement 
 with the experimental results for configurations employing the thinnest 
 wings. The wave drags for the bodies of revolution presented in this 
 report are predicted to a greater degree of accuracy by using the frontal 
 projections of oblique areas than by using normal areas. A rapid method 
 of computing wing area distributions and area-distribution slopes is given 
 in an appendix . 
 
 INTRODUCTION 
 
 The area rule, first advanced by Whitcomb in reference 1, has con- 
 siderably altered the methods for predicting wave drag of wing -body com- 
 binations. Studies leading to the discovery of the area rule showed that 
 interference drag between wing and body components could be very large. 
 Therefore, estimation of drag by component buildup without somehow evalu- 
 ating the interference drag could give misleading answers. However, in 
 consequence of the transonic area rule, a valuable tool was made avail- 
 able to the designers in assessing the transonic drag. This was the 
 equivalent-body concept, which states that at transonic speeds the pres- 
 sure drag of the airplane is the same as that for a body of revolution 
 
 CONFIDENTIAL 
 
CONFIDENTIAL KACA RM L56DII 
 
 having the same longitudinal distribution of cross-sectional area. As a 
 result, the drag of the configuration is obtained by either estimating 
 or experimentally determining the equivalent -body drag. Experimental 
 checks for airplane configurations presented in reference 2 generally 
 support this concept in the transonic speed range. 
 
 The supersonic area rule, given by Jones in reference 3, provided a 
 powerful method for calculating the wave drag at supersonic speeds . In 
 references k and 5 "the mechanics of the drag calculations were discussed 
 together with a number of comparisons of calculated and experimental 
 drags generally at low supersonic speeds. Jones pointed out in refer- 
 ence 5 that the method could be expected to give good results for thin 
 wings mounted on vertically symmetrical bodies. Later, Lomax in refer- 
 ence 6 gave the complete linearized theory expressions for the drag. 
 The added terms in Lomax' s result represented the limitation pointed out 
 by Jones. 
 
 The purpose of the present paper is to provide a better feel for 
 the range of applicability of both the transonic and supersonic area 
 rules. For the transonic area rule, this is done by making additional 
 comparisons between equivalent -body and wing-body experiments. For the 
 supersonic area rule, comparisons are made of calculated and experimental 
 results for both body and wing-body combinations over a wider range of Mach 
 numbers than heretofore made. The supersonic-area -rule calculations were 
 made by using a 24 -term Fourier series expression for the slope of the area 
 distribution. 
 
 SYMBOLS 
 
 A frontal projection of the area cut by a Mach plane or wing 
 
 aspect ratio 
 
 ^n 
 
 pit , 
 2 /dA 
 
 It J „ V dx 
 
 - -I -)sin nC! dC! 
 2 q; 
 
 Cj) drag coefficient, -^ 
 Cp pressure coefficient 
 
 c wing local chord 
 
 c„ root chord of particular pointed wing tip 
 
 Cj. wing root chord 
 
 CONFIDENTIAL 
 
NACA RM L56DII CONFIDENTIAL 
 
 c^ wing tip chord 
 
 D drag 
 
 d maximum body diameter, 2rjjj 
 
 f resiiltant pressure force 
 
 f(v) wing-thickness-distribution function, 
 
 G(K,Vq) wing-area-dlstributlon function 
 
 H(K, Vq") wing-area-distribution slope function 
 
 z 
 max 
 
 Yi + L^£ll\ fo^ the left-wing panel; mfl - i-£2^^ for 
 V tan A / \ tan A / 
 
 K mil + 
 
 the right-wing panel 
 
 I length of configuration 
 
 Z-t; total length of area distribution 
 
 i/d body fineness ratio 
 
 M Mach number 
 
 m = f (1 + ^j tan A 
 
 ^ (1 - A) 
 
 n ■ integer 
 
 q dynamic pressure 
 
 r local body radius 
 
 r^ maximum body radius 
 
 S reference area 
 
 S^ body frontal area 
 
 Sg wing exposed area 
 
 S^ wing total plan-form area 
 
 CONFIDENTIAL 
 
CONFIDENTIAL NACA RM L56DII 
 
 s wing semispan 
 
 Sq semispan of particular pointed-tip wing 
 
 t/c wing thickness ratio, 
 
 x,y,z Cartesian coordinates 
 
 2^max 
 
 Xq point of intersection of Mach plane with the x-axis 
 
 x' x-coordinate measured from wing leading edge 
 
 X-, dummy variable 
 
 z local maximum wing ordinate 
 max 
 
 /m2 - 
 
 ^-1 
 
 * ' ""^ Y H - ') 
 
 A wing leading-edge sweepback angle 
 
 A wing taper ratio, — 
 
 ^r 
 
 u Mach angle, sin" =^ 
 
 ' M 
 
 c 
 
 V value of v at root of particular pointed-tip wing 
 
 V value of v at root of actual wing 
 
 angle between z-axis and line of intersection of Mach plane 
 with the y,z plane 
 
 REVIEW OF THE BASIC THEORY 
 
 From reference 6, the eqiiation for the wave drag of any system of 
 bodies or wings and bodies can be written as: 
 
 CONFIDENTIAL 
 
NACA RM L56DII 
 
 CONFIDENTIAL 
 
 D ^ 
 
 4rt2. 
 
 2n 
 
 de 
 
 dx 
 
 dx-, 
 
 
 
 d A(x,9) _p_ di'(x,9) 
 " 2q 
 
 dx' 
 
 dx 
 
 d^A(xi,e) 
 
 dXn 
 
 J_d£(x^ 
 2q dxi 
 
 log(x - x^) 
 
 (1) 
 
 The equation is subject to the usual limitations of the linearized theory. 
 
 Before discussing the terms in the drag equation, it is well to review 
 the definition of Mach planes. The physical significance of equation (l) 
 is understood if the configuration is cut by Mach planes. Mach planes are 
 easily visualized by considering a Mach cone originating at a point on the 
 X-axis which is alined with the remote relative wind. A Mach plane is 
 simply a plane tangent to the Mach cone and at an angle of roll, 9 about 
 the x-axis measured from the y-axis. By moving the vertex of the Mach 
 cone along the x-axis, a series of parallel Mach planes will cut the con- 
 figuration for a fixed roll angle 9. 
 
 In the drag equation the term A(x,9) represents the frontal pro- 
 jection of the oblique area cut by a particular Mach plane, whereas f(x,9) 
 represents the net force normal to the stream direction on this section in 
 the 9 direction. These relationships are illustrated in figure 1 for 
 angle of roll 6 of the Mach plane of 0° and 90° • By neglecting the 
 
 term ^ ^ — - — -, the equation reduces to the supersonic-area-rule formula 
 
 given by Jones in reference 5« Evaluation of f(x,e) requires the pres- 
 s\ire distribution on the configuration which when integrated over the 
 configuration gives the drag directly. As a result, large values of 
 
 X 4—* — '- impose a limitation on the supersonic area rule, even within 
 
 the framework of the linearized theory. 
 
 It is not the purpose of the present paper to evaluate the drag of 
 configurations including the effect of the pressure term but to evalxxate 
 the drag of configurations using the supersonic-area-rule formula of 
 Jones. The influence of the pressure term was evaluated for one simple 
 case. 
 
 Equation (l) can be written in coefficient form as 
 
 _1_ 
 2rt 
 
 ^2Jt 
 
 CD(e)d9 
 
 (2) 
 
 CONFIDENTIAL 
 
CONFIDENTIAL NA.CA RM L56DII 
 
 where 
 
 h rh A2„ 
 
 ^ 2nSJo ^0 \<ix^ 2q dxy^<ix^2 2q dx^^ 
 
 The quantity Cj)(e) is most readily determined by solving the inte- 
 gral for C-q{Q) through a Fourier sine series expression for ^ fol- 
 lowing the method of reference 7 if 
 
 ^ = cos-l(2^-l) 
 
 a =2 r fdA P f\ i^^0^0 
 n n Jq \^dx 2 qj ^ ^ 
 
 then C-p)(0) can be written as 
 
 For the computations of this paper, only 2^1 terms were used in the 
 
 Fourier sine series expression for sii. Thus 
 
 dx 
 
 2 
 
 CD(e) =^ X "^''' 
 
 n=l 
 
 BODY DRAG RESULTS 
 
 For bodies of revolution, the calculation of the drag is simplified 
 to some extent because the area distributions are identical for all roll 
 angles. However, except for high-fineness-ratio bodies, it is not pos- 
 sible to assume that the frontal projection of the oblique area cut by 
 the Mach plane is the same as the normal area. Figure 2 shows an example 
 of this for two parabolic bodies of revolution having different fineness 
 ratios and shapes. The area-distribution curve slopes were calculated 
 from the expression 
 
 dA ^ ^ f^u (x - xo)dx 
 
 dx q2 
 
 ^1 ypV(x) - (x - xq)' 
 
 CONFIDENTIAL 
 
NACA RM L56DII COKFIDENTIAL 
 
 The derivation of this expression is given in appendix A. It has also 
 been assiimed for the calculations (and all succeeding body calculations) 
 that a cylinder can be added at the base of the body without altering 
 the drag. If this were not done, the solution would require the flow to 
 fill the area behind the base which would exceed the limitations of the 
 linearized theory. Figure 2 shows large changes in the peak slope over 
 the afterbody of the f ineness-ratio-6.04 configuration; these changes 
 would lead to a significant drag variation with Mach number. 
 
 The evaluation of the slope of the oblique area distributions is 
 extremely difficult except for simple bodies. There naturally arises 
 the question as to whether this is worth while if the pressure term is 
 Ignored. 
 
 As derived in appendix B, the local force acting on the oblique 
 area of a body of revolution is 
 
 The only assumption made in derivation of 7- is that — *- is 
 
 Q. dx 
 
 constant over the oblique area. This is a reasonable assumption except 
 
 for bodies having discontinuities, and high local slopes. Then, 
 
 dA 
 
 P f 
 
 . dA 1 P^A dCp 
 
 dx 
 
 2 q " 
 
 dx 2 dx 
 
 Thus, the error in the drag introduced by ignoring the pressure term is 
 
 dCp 
 dependent on the pressure gradient — =-. 
 
 dx 
 
 It would be expected that the drag for a conical nose with an 
 attached shock wave over which the pressure is constant at zero angle of 
 attack would be least affected by the pressure term. (The pressure term 
 takes on a value only near the juncture with the cylinder; however, the 
 pressure term was not evaluated in this region.) Figiore 3 presents a 
 comparison of the drag of various cones calculated with the supersonic- 
 area-rule formula with the exact theory drag of reference 8. The lowest 
 Mach number of the comparison corresponds to the lowest Mach number for 
 entirely supersonic flow on the cone as calculated with the exact theory. 
 The highest Mach nimiber of the comparison was arbitrarily taken as that 
 at which the slope of the Mach line equaled one-half the slope of the 
 nose. The agreement between the two theories is remarkable, within 
 5 percent except for a few points. 
 
 COKFIDMTIAL 
 
8 CONFIDENTIAL NACA RM L56DII 
 
 A better comparison may be made by plotting Cjj^Z/rm) against 
 ^T^/l , the quantity which defines the frontal projection of the oblique 
 
 area distribution. This has been done in figure ^4- to give drag in dimen- 
 sionless or collapsed form. The drags from exact theory (5° half angle 
 cone was chosen as representative), slender body theory (ref. S) , and 
 supersonic-area-rule theory are shown. The comparison shows the great 
 improvement of the area-rule theory over the slender-body theory at 
 values of P^j^/Z greater than 0.2, and the good agreement of the area- 
 rule result with the exact theory to 3rjji/i of about 0.7. At higher 
 values of P^j^/Z the area-rule theory is in error, possibly first because 
 the pressure term is neglected but finally, near Prjjj/Z = 1, because the 
 assimiptions of the linearized theory are violated. At Pr^/Z = 1, the 
 Mach line lies on the cone surface, which corresponds to the realm of 
 hypersonic flows. (See ref. 10.) 
 
 For a body with curvature, for example, a nose of parabolic profile, 
 the pressure over the nose is variable, and the influence of the pressure 
 term may be significant. Figure 5 presents the drag for noses of parabolic 
 profile in collapsed form. Here the supersonic-area-rule theory is an 
 improvement over slender-body theory but in only partial agreement with 
 the more exact second-order theory of reference 11. Inclusion of the 
 pressure term, evaluated by using second-order pressure distributions, 
 however, does give agreement with some of the second-order-theory results. 
 Since the second-order-theory drags do not collapse into one curve, agree- 
 ment should be expected only with those points for which the pressure dis- 
 tribution used in evaluating the pressure term apply. However, this was 
 not the case. For example, the pressure distribution used for the pres- 
 sure term calculation at pi'm/Z = 0.3 corresponds to the flagged symbol. 
 
 For the parabolic noses, both the area-rule theory and the area-rule 
 theory plus the pressure correction cannot be expected to apply near and 
 above Prj^/Z = O.5, where the slope of the Mach line equals the slope of 
 
 the nose tip. 
 
 Figure 6 presents a comparison of the pressure drag from supersonic- 
 area-rule theory with experiment and slender-body theory for a family of 
 parabolic bodies of revolution. The experimental drags were taken from 
 references 12 and 15 . In determining the experimental pressure drags, 
 the friction drag was assumed turbulent and evaluated by using the sub- 
 sonic drag level and the results of reference ik for the effects of Mach 
 number and Reynolds number; the fin pressure drags were assumed identical 
 and taken from reference 15; and the base drags were small and were sub- 
 tracted when available. The slender-body-theory drags were calculated 
 using the curves of reference 9- 
 
 CONFIDENTIAL 
 
NACA RM L56DII CONFIDENTIAL 
 
 As would be expected, the comparisons show the increasing ability of 
 both the area-rule theory and slender-body theory to predict the drag as 
 the body fineness ratio is increased. In most cases, the area-rule theory 
 offers a significant Improvement over slender-body theory. The area-rule 
 theory and slender-body theory are in agreement near M = 1, since at this 
 Mach number the supersonic-area-rule theory reduces to slender-body theory. 
 
 From these nose and complete-body comparisons that have been made, 
 the following conclusion can be drawn. The area-rule drag of bodies can 
 be predicted to a greater degree of accuracy by using the frontal projec- 
 tion of oblique areas at a given Mach number thaji by using normal areas, 
 if, at the Mach number under consideration, the limitations of the line- 
 arized theory are not exceeded. This is illustrated by the comparison 
 between the drag at a given Mach number and the drag near M = 1 especially 
 for the low-fineness-ratio bodies. It is not to be inferred from the above 
 statement that the supersonic -area-rule method is recommended for evalu- 
 ating the drags of bodies of revolution. However, when the drags of wing- 
 body combinations for which the body area distribution is needed are deter- 
 mined, the oblique area distribution should be used if the body is of low 
 fineness ratio or has low-fineness-ratio components. 
 
 CALCULATION OF WING-BODY DRAG 
 
 The difficulty in computing the wave drag of wing-body configurations 
 can be considerably reduced if the configuration meets the following con- 
 ditions: first, the body is of sufficiently high fineness ratio so that 
 the change in body-area distribution with Mach number is small, and sec- 
 ond, the wing is thin. These conditions imply also that the pressure 
 term is negligible. Some feel for the body fineness ratios necessary for 
 the above condition to be met can be obtained from the preceding section 
 on bodies of revolution. The assumption of a thin wing allows the Mach 
 plane intersecting the wing obliquely to be replaced by a plane perpen- 
 dicular to the wing chord plane intersecting the wing plane along the 
 same line as the Mach plane. Note that, at zero roll angle, the Mach 
 plane is normal to the wing chord plane but is not normal to the wing 
 chord plane at any other roll angle for a Mach ntmiber other than M = 1. 
 Also the angle between the Mach plane and the normal to the wing chord 
 
 plane is greatest and equal to tan" p at a roll angle of 90° • 
 
 Appendix C presents a simple analytical method for evaluating wing 
 area distributions and area-distribution-curve slopes. The curves neces- 
 sary for evaluating these quantities (figs. I6 and I7) are applicable only 
 to 65A series airfoils, but similar curves can be made up for other air- 
 foil sections. 
 
 CONFIDENTIAL 
 
10 CONFIDENTIAL MCA RM L56DII 
 
 In order to get an idea of the applicability of the thin-wing ass\mip- 
 tion, a calculation has been made of the true area-distribut ion-curve - 
 slope variation for 60° delta wing having an NACA 65AOO6 airfoil section 
 for a roll angle of 90° and a Mach number of l.i+lij-. In order to simplify 
 the calculation, the wing was approximated by a sufficient number of 
 linear-slope elements to define the airfoil section adequately. With 
 this approximation the Mach plane intersection with the wing surface was 
 made up of straight lines. The expression for the frontal projection of 
 the oblique area was then easily evaluated and differentiated to obtain 
 the slope. The results of the calculation are presented in figiire 7* 
 Altho\igh the slopes for the upper and lower half wings are significantly 
 different, the total slope agrees almost exactly with the slope obtained 
 by using the thin-wing assximption. On the basis of this result, it is 
 felt that the thin-wing solution should be adequate for wings of present- 
 day interest. 
 
 For the wing-body combinations of this paper, an additional simpli- 
 fication was allowed in the supersonic-area-rule wave-drag calculations. 
 Since the tail fins mounted on the models were thin and relatively small 
 (see ref. 16), their drags were subtracted as tares. Then, since the 
 bodies for all cases were of high fineness ratio (and identical), the 
 body-area-distribution-curve slopes were considered independent of Mach 
 number, and the changes in the area-distribution-slope curves with Mach 
 number and roll angle were due entirely to the wings. As derived in 
 appendix C, the area distribution for a given wing (m fixed) is depen- 
 
 ft r* oc; A 
 
 dent only on the value of . Thus, the area-distribution-slope 
 
 tan A 
 
 curves for the wing-body configuration are dependent only on the value 
 
 B cos 9 / V 
 
 of . Then, from equation (2) and because of the symmetry of the 
 
 tan A 
 
 configuration. 
 
 Cd(m) = f / CD(e)de 
 
 Jl Jq 
 
 In order to obtain the wave drag of the configuration, a plot of Cjj 
 against 6 is required. This can be computed if a plot of Cp against 
 
 •^T 7— is given, since the angle is known for fixed values of - — ^— - 
 
 tan A ^ ' ^ tan A 
 
 B cos 9 
 
 and . The configuration drag is simply the average drag between 
 
 tan A 
 
 9=0 and |. 
 
 For the wing-body calculations of this paper the bodies were identi- 
 cal. The body-area-distribution-slope curves are shown in figure 2(b). 
 The cvirve for M = 1.4l^ was chosen as representative for the Mach number 
 
 CONFIDENTIAL 
 
NACA RM L56DII CONFIDENTIAL 11 
 
 range of interest. The wing area distributions and area-distribution- 
 curve slopes were obtained by using methods similar to that given in 
 
 appendix C. In addition, a limiting value of ^ ^°^ =0.8 was set for 
 
 tan A 
 
 configurations having blunt leading-edge airfoils. ^Above P /^^^ ^ = 1 
 
 \ tan A ' 
 
 the Mach line lies behind the wing leading edge, and the linear theory is 
 
 no longer valid for blunt airfoils.) 
 
 / 
 
 An example of the wave drag calculation for the most extreme configu- 
 ration investigated (60° delta wing, NACA 65AOO6 airfoil) is presented in 
 
 figures 8, 9, and 10. Figure 8 shows nondimensional plots of — against 
 
 R n fl '^■^ 
 
 ^ for various values of £_2_^ — . Figure 9 shows the effect on Cn of 
 
 tan A ^ 
 
 the number of terms in the series solution. Except at ^ ^°^ ^ = and 0.£ 
 
 tan A 
 
 convergence was apparently obtained within 2k terms. Figiire 10 shows the 
 
 variation of the area distribution drag with ^r^^^r^, the variation of 
 
 tan A ' 
 
 area distribution drag with roll angle, and the variation of the config- 
 uration drag with 
 
 A 
 
 tan A 
 
 WING-BODY DRAG COMPARISONS 
 
 Figure 11 presents some wave-drag comparisons for wing-body combina- 
 tions. The experimental wing -body results were taken from references I6 
 to 19. The wing-body wave drags were obtained in the following manner. 
 The friction drags were assumed to be turbulent and were estimated by 
 using the results of reference ik. Base drags and fin pressure drags were 
 subtracted using the results of reference I6. The equivalent -body drags 
 for a Mach number of 1 were obtained experimentally by using the helium- 
 gun technique described in reference 2. These models had four scaled 
 tail fins. The friction drag was assumed to be the subsonic drag level 
 corrected at higher speeds for Reynolds number and Mach number by using 
 the results of reference ih. Base-drag rise and fin-drag rise were not 
 evaluated for the equivalent-body models. These qiiantities, however, 
 should be small in the Mach niomber range where comparison is valid. The 
 supersonic-area-rule-theory drags were evaluated by using the method of 
 the preceding section. No attempt was made to evaluate the drag with 
 the pressure term included. The drag coefficients presented in figure 11 
 are based on total wing area. 
 
 The inability of the supersonic-area-rule theory to predict the drag 
 near M = 1 is evident for nearly all cases. However, the agreement at 
 the higher Mach niimbers between the theoretical drags and the experimental 
 
 CONFIDENTIAL 
 
12 CONFIDENTIAL NACA RM L56DII 
 
 wing-body drags is excellent and within the accuracy of evaluating the 
 experimental wave drag, except for three configurations. Two of these 
 configurations (figs. ll(c) and 11(g)) had 6-percent -thick wings which 
 were the thickest wings investigated. The third configuration (fig. 11(h)) 
 
 had a ^-percent-thick airfoil but with fairly steep wedge components. 
 
 For these configurations, a significant effect of the neglected pressure 
 term may be possible. As a result, the drags calculated for configura- 
 tions having wings of these thicknesses and sections should be viewed 
 with caution. 
 
 The comparisons in figure 11 show that the equivalent -body drags 
 give a good approximation to the experimental wing-body drags up to a 
 Mach number of 1, except for the two conf ig\irations having 6-percent- 
 thick wings (figs. ll(c) and 11(g)). This res;ilt is in agreement with 
 reference 2 which shows the validity of the transonic -area rule decreases 
 with increasing wing-thickness ratio. At Mach numbers above 1, the agree- 
 ment is variable but tends to be consistent with the flatness of the cor- 
 responding theoretical c\irve. That is, as the theoretical drag variation 
 with Mach niimber becomes smaller, the equivalent body gives a better 
 approximation of the supersonic-drag level. This would be expected, since 
 a flat theoretical curve indicates that the variation in area-distribution 
 drag with roll angle or Mach number is small. Then the drag for the Mach 
 niffliber 1 or roll angle 90° area distribution (corresponding to the equiv- 
 alent body) is representative of the configuration drag. 
 
 Figure 12 shows the comparison between experimental-configuration 
 drag and equivalent-body drag for an airplane configuration. The com- 
 parison shows an extreme example, compared with the relatively good results 
 of reference 2, of the inability of the equivalent body to predict the 
 supersonic-drag level. The equivalent -body drag is approximately kO per- 
 cent low in spite of the low aspect ratio of the configuration. Apparently, 
 the configuration tail surfaces cause the area distribution to change mark- 
 edly at low supersonic speeds. Below M = 1, the equivalent body gives a 
 fair representation of the configuration drag. The drag of the configu- 
 ration minus the tail surfaces could probably be calculated to the degree 
 of acc\aracy shown in figure 11. The influence of the tail s\irfaces, how- 
 ever, may be difficult to evaliiate. If the horizontal tail supports a 
 load when the conf ig\irat ion is at zero lift, the influence of the pres- 
 sure term may be significant. Although no supersonic-area-rule drag cal- 
 culations were made for this airplane, reference 20 indicates that gen- 
 erally good predictions of complete -airplane drag can be made. 
 
 Subsequent to the preparation of this paper, NACA RM A56IO7 has 
 been prepared at the Ames Laboratory amd presents supersonic -are a -rule 
 calculations for a configuration similar to the one shown in figure 12 
 but with small differences in area distribution in addition to the 
 absence of a canopy. 
 
 CONFIDENTIAL 
 
NACA RM L56DII CONFIDENTIAL 15 
 
 EVALUATION OF COI'IPONENT AND INTERFITIENCE DRAGS 
 
 As was shown in the preceding section, the supersonic area rule can 
 be a useful tool in evaluating the supersonic drag of a wing-body con- 
 figuration. In order to assess the efficiency of the combination as a 
 whole, however, the effects of the combination on the component drags 
 and the interference drag between the components must be known. The 
 supersonic area rule provides a valuable method for evaluating these 
 effects . 
 
 The supersonic-area-rule equation can, of course, be used in evalu- 
 ating the drags of individual wing and body components. This was done 
 for a number of bodies in a previous section of this paper. The same can 
 be done for isolated wings. An example of this is shown in figure I3 
 where the drag of delta wings having 65A series sections is plotted in 
 collapsed form. The area-rule result is compared with a result obtained 
 by the method of Beane (ref, 21), The two methods are just two forms of 
 the same linearized wing theory. The agreement between the two methods 
 is good. 
 
 An example of the effect of the wing-body configuration on wing drag 
 is presented in figure 1^. The calculation is for the configuration having 
 the closest agreement between the theoretical and experimental drags 
 (fig. 11(b)). In this figure, the drag of the exposed-wing panels based 
 on total and exposed wing areas is compared with the isolated wing drag. 
 Separation of the wing panels gives approximately a lO-percent reduction 
 in wing drag coefficient at Mach numbers above 1.5« As the Mach ntmiber 
 approaches 1, this favorable effect disappears. This would be expected, 
 for at M = 1 the area distributions of the exposed wing panels and the 
 isolated wing would be identical if the body were cylindrical. 
 
 Figure I5 shows an evaluation of the interference drag for the same 
 configuration. The sum of the calculated body and wing drags is compared 
 with the calculated configuration drag. The curves show a favorable inter- 
 ference effect at Mach numbers below M = 1.3» At Mach numbers above 1.3^ 
 interference drag is, for all practical p\K"poses, zero. Thus for this 
 configuration, at Mach numbers greater than 1.^, the only beneficial effect 
 of combining the wing with the body comes from the separation of the wing 
 panels. 
 
 CONCLUSIONS 
 
 An investigation has been made of abilities of the equivalent -body 
 technique and a 24-term Foiirier series application of the supersonic - 
 area-rule method to predict wave drag at transonic and supersonic speeds, 
 From the theoretical and experimental comparisons made, the following 
 conclusions can be drawn: 
 
 CONFIDENTIAL 
 
lU CONFIDENTIAL NACA RM L56DII 
 
 1. The area-rule drag of the bodies of revolution presented in this 
 report are predicted to a greater degree of accuracy by using the frontal 
 projection of oblique areas at a given Mach number than by using normal 
 areas . 
 
 2. The supersonic wave drag of slender-wing — body configurations can 
 be predicted with the supersonic-area-rule formula. For the wing -body 
 configurations investigated, the best agreement was obtained for the con- 
 figiorations employing the thinnest wings. 
 
 3. The equivalent body technique provides a good method for predicting 
 the wave drag of certain wing-body combinations at and below Mach number 1. 
 At Mach n\imbers above 1, the equivalent body wave drags can be misleading. 
 
 Langley Aeronautical Laboratory, 
 
 National Advisory Committee for Aeronautics, 
 Langley Field, Va., April 6, I956. 
 
 CONFIDENTIAL 
 
mCA RM L56DII 
 
 CONFIDENTIAL 
 
 15 
 
 APPENDIX A 
 
 AREA DISTRIBUTION SLOPE FOR BODIES OF REVOLUTION 
 CUT BY OBLIQUE MACH PIANES 
 
 The area distributions are identical for all roll angles. For 
 simplicity a roll angle of 90° will be used in the derivation. 
 
 = Xq + 3z 
 
 The frontal projection of the oblique area cut by the Mach plane (see 
 sketch (a)) is given by the equation: 
 
 A = 2 / ^ y dz 
 
 From the equation for the Mach plane, z is related to x by 
 
 X - 
 
 z = 
 
 ^ 
 
 and 
 
 dz = i dx 
 
 The equation relating y and x is given by 
 
 = \/r^(x) 
 
 2 / 2/ s (x - xo) 
 
 z = \/h2(x) - — ^— - p 
 
 ^Jp>^^U) - (x - xof (Al) 
 
 CONFIDENTIAL 
 
l6 CONFIDENTIAL NACA RM L56DII 
 
 Then, 
 
 A = \ P^7p2r2(x) _ (x"- xo)2 dx (A2) 
 
 Differentiating the expression for A gives 
 
 dA 2 r^u (^ - ^0) 
 
 dXQ n2 
 
 P ^^^ ^pVCx) - (x - Xq)' 
 
 ^x (A3) 
 
 where x, and x^ are the roots of x = xq - pR(x) and x = xq + pR(x), 
 respectively. 
 
 CONFIDENTIAL 
 
MCA RM L56DII 
 
 CONFIDENTIAL 
 
 17 
 
 APPENDIX B 
 
 THE NET FORCE ACTING ON THE OBLIQUE AREA OF A BODY OF 
 REVOLUTION AT ZERO ANGLE OF ATTACK 
 
 For a body of revolution at zero angle of attack, the net force is 
 independent of roll angle. The derivation will be made for a roll angle 
 of 90°. 
 
 The net force in the 9 direction (the z-direction for a roll angle of 
 90°) (see sketch (b)) can be written as 
 
 f 
 
 Cp sin a dS 
 
 /■,..2 . .„2 
 
 Since dS = Ifdy + dz and sin a 
 
 dy 
 
 \/dy2 + 
 
 dz" 
 
 The pressure coefficient at zero lift is a function of x only. 
 The equation for y in terms of x is given by equation (l) of appen- 
 dix A as follows: 
 
 = Wp^'(x) - U -xo)' 
 
 CONFIDENTIAL 
 
18 CONFIDENTIAL MCA RM L56DII 
 
 and 
 
 p2R(x)g - (x - xo) 
 dy - — -—=z=^=:^:^i=^=r dx 
 
 Then, the net force can be written as 
 
 K. 
 
 I = (-2y^0 cp dy^ 4- (2 pO Cp dy 
 
 /upper surface \ -^ /lower siirface 
 
 •Xo S[p'^^-)i - (^ - -0)] ^ 2 rxo S[p'^(^)i - ^^ - ^)] 
 
 dx + ^ / — 5= =- dx 
 
 f 2 rxu ^p[p'^(^)i - (x - xo)| 
 
 — = — / dx 
 
 ^ '^^2 yp2H2(x) - (x - xo)2 
 Integrating the expression by parts gives 
 
 
 XI 
 
 dC^ 
 If — ii is essentially constant between the limits of integrations, 
 dx 
 
 Then, from equation for the frontal projection of the oblique area 
 in appendix A (eq. (A2)) 
 
 -F dCr, 
 t = _BA —2- 
 q ^ dx 
 
 CONFIDENTIAL 
 
MCA RM L56DII 
 
 CONFIDENTIAL 
 
 19 
 
 APPENDIX C 
 
 METHOD FOR DETERMINING WING-AREA DISTRIBUTION AND 
 AREA-DISTRIBUTION-CURVE SLOPE 
 
 This method assumes that the wing is thin and that the oblique Mach 
 plane can be replaced by a plane perpendicular to the wing chord plane. 
 The method also assumes the wing has straight leading and trailing edges 
 and constant thickness ratio. 
 
 The method is developed first for pointed-tip wings. Then, correc- 
 tions are made for curved-wing — body junctirres and finite wing tips. In 
 addition, the right- and left-hand wing panels are considered separately. 
 
 Pointed-tip wings , 
 lowing sketch: 
 
 Consider the right-wing panel shown in the fol- 
 
 max 
 
 X* = Xq + y(p cos 9 - tan A) 
 
 (c) 
 
 CONFIDENTIAL 
 
20 
 
 CONFIDENTIAL 
 
 NA.CA RM L56DII 
 
 The frontal projection of the area of one wing panel cut by the Mach 
 plane is given by 
 
 A = 2 Tz dy 
 
 and — can be written as 
 c 
 
 <= ' ^ -max " 2 c ^^^ 
 
 Then, for — constant. 
 
 A = ^o^ol j^^^)^^'^ 
 
 The value of t\ and v are related by the intersection line of the Mach 
 plane and the wing chord plane for the right-wing panel given by the 
 equation 
 
 x' = Xp^ + y(p cos 9 - tan A) 
 
 and for the left-wing panel by the equation 
 
 x' = Xq + y(p cos e + tan A) 
 
 Then, 
 
 V = 
 
 v„ + -^(p cos e ± tanA)T] 
 
 With 
 
 ^ = 1 - Tl 
 ^0 ' 
 
 V = — i — Vq + — (P cos e ± tanA)T] 
 1 - iL ^0 
 
 and 
 
 Let 
 
 Ti = 
 
 vq - V 
 
 (tan A ± p cos 9)-^ - v 
 ^0 
 
 m = -T^ tan A 
 ^0 
 
 CONFIDENTIAL 
 
NACA RM L56DII CONFIDENTIAL 21 
 
 In terms of tapered wing geometry, m is given by 
 
 Then, 
 
 Let 
 
 m = A LJlA tan A 
 4 1 - A 
 
 = ^0 - ^ 
 "^ " Ji ± £_co^\ _ ^ 
 \ tan A / 
 
 K = m 1 + L^2L_i 
 \ tan A ; 
 
 for the left-wing panel and 
 
 K = m 1 - P-^HlJ 
 \ tan A 
 
 for the right-wing panel. Then, 
 
 ' K - V 
 
 dT] K - Vq 
 
 '^ ~~ (K - V) 
 
 2 
 
 c , K - V 
 
 
 
 ^0 ' - K - V 
 
 ^ a. = . OLlM 
 
 dTi = — ^ ^ dv 
 
 ^0 (K - v)5 
 
 The equation for the area can then be written as 
 
 ^ ^ower (K - "y 
 
 CONFIDENTIAL 
 
22 CONFIDENTIAL NACA RM L56DII 
 
 The slope of the ELrea-distribution curve is obtained by differentiating 
 the expression for A 
 
 dA _ t 
 
 >(k - Vq) 
 
 V, 
 
 upper f(v) , _ 
 ^ ' dv + 
 
 ^o^.. (K - v)5 
 
 lower 
 
 (k - vq)2 dv^^^3^ (^ - ^of '^Vper -.. 
 
 (k-v N5 a ^^°"^^^"/K-v n5 dv, ^^PP^^' 
 \ lower j ^ upper) 
 
 or 
 
 ^= ^0^0 |h(k,vo) 
 
 Curves of G(K^vo) and H(K,vo) have been made up for a 65A series 
 airfoil and are given in figures I6 and I7 for values of K from 
 
 to 2.k and vq from to 1. In evalxiating / ^ — - — dv, f (v) was 
 
 ^ (K - v)^ 
 assumed to vary linearly between airfoil ordinate stations. Figure I8 
 gives a plot of f(v) for this ass\jmption. 
 
 For K and Vq greater than 1, G(K,vo) and H(K, vq) are given 
 by the expressions 
 
 G(K,vo) = G(K,1) il_l_!2)! 
 (K - 1)2 
 
 H(K,vo) =H(K,1) k:L-ml 
 (K - 1) 
 
 For K and Vq less than 0, G(K, vq) and H(K,vo) are given by 
 the expressions 
 
 g(k,vo) = G(K,0) (^ -J^f 
 
 h(k,Vo) = H(k,0) ^^^.^ 
 
 K 
 
 CONFIDENTIAL 
 
NACA RM L56DII 
 
 CONFIDENTIAL 
 
 25 
 
 Correction for curved-ving — body juncture .- The following sketch 
 shows a pointed-tip wing mounted on a curved body. 
 
 Line of intersection of Mach 
 plane and wing chord plane 
 
 The areas and slopes will be referred to the actual wing geometry 
 
 (cp, s, and a). The areas and slopes, however, will be for the exposed 
 
 pointed-wing tip. 
 
 In sketch (d) consider one point of intersection of the wing panel 
 
 with the body. The area of the wing cut by the Mach plane through this 
 
 point is determined only by the product of SqC^ of the exposed wing 
 
 through the point and the value of Vq for the exposed wing at the point 
 
 of intersection. As the point of intersection changes, s 
 
 0^ 
 
 '0^ 
 
 and 
 
 ^0 
 
 change and account for the intersection line. Expressed in terms of the 
 
 actual wing-body characteristics. 
 
 ^o'^o 
 
 is given by 
 
 ^0^0 
 
 I^[-(-^>iI 
 
 The quantity — is related to v^ by the expression 
 s u 
 
 ^0 = 
 
 (1 - A)m| 
 
 1 - (1 - A)| 
 
 The area of the exposed wing panel cut by the Mach plane can be written 
 as 
 
 CONFIDENTIAL 
 
2k 
 
 CONFIDENTIAL 
 
 NACA RM L56DII 
 
 c^s 4 
 
 A=^[l- (1 -A)|] G(K,vo) 
 
 The area is calculated for given value of Vq. The center-line value of 
 V is given by 
 
 Vr = V5[l - (1 - A)|] 1- K(l - A)| 
 
 The slope is obtained by differentiating the expression for A and 
 is given by 
 
 ^ crs i[l - (1 - A)|j|[l - (1 - A)|]h(k,vo) - 2(1 - A)^ g(k,vo) 
 
 dVv 
 
 If — §• = 0, 
 
 dVQ 
 
 1 - A 
 
 1 - (1 - A)^+ (k - vo)(l -A) 
 
 if 
 
 dVr 
 
 dA 
 dv. 
 
 = rTf[i- (1 - ^)flH(K,vo) 
 
 Correction for finite vlng tip .- In order to correct the pointed- 
 tip wing panel and slopes for the finite wing tip, the areas and slopes 
 outboard of the wing tip are subtracted. 
 
 Intersection line of Mach plane and wing 
 
 From sketch (e) 
 
 (e) 
 
 s„c^ = c^s 
 
 A'^ 
 
 ^ 1 _ A 
 
 CONFIDENTIAL 
 
NACA RM L56DII CONFIDENTIAL 25 
 
 Then the areas and slopes are given by 
 
 Atip = -^ _ ^ g(k,vo) 
 
 dAtip c^s|a 
 
 The center-line value of v is given by 
 
 v^ = VqA + K(l - A) 
 
 CONFIDENTIAL 
 
26 CONFIDENTIAL NACA RM L56DII 
 
 REFERENCES 
 
 1. Whitcomb, Richard T.: A Study of the Zero-Lift Drag-Rise Character- 
 
 istics of Wing-Body Combinations Near the Speed of Sound. NACA 
 RM L52H08, 1952. 
 
 2. Hall, James Rudyard: Comparison of Free-Flight Measurements of the 
 
 Zero-Lift Drag Rise of Six Airplane Configurations and Their 
 Equivalent Bodies of Revolution at Transonic Speeds. NACA 
 RM L53J21a, 195^. 
 
 3. Jones, Robert T.: Theory of Wing-Body Drag at Supersonic Speeds. 
 
 NACA RM A53Hl8a, 1953. 
 
 k. Holdaway, George H.: Comparison of Theoretical and Experimental Zero- 
 Lift Drag-Rise Characteristics of Wing-Body-Tail Combinations Near 
 the Speed of Sound. NACA RM A55H17, 1953. 
 
 5. Alksne, Alberta: A Comparison of Two Methods for Computing the Wave 
 
 Drag of Wing -Body Combinations. NACA RM A55A06a, 1955. 
 
 6. Lomax, Harvard: The Wave Drag of Arbitrary Configurations in Line- 
 
 arized Flow As Determined by Areas and Forces in Oblique Planes. 
 NACA RM A55A18, 1955. 
 
 7. Sears, William R.: On Projectiles of Minimum Wave Drag. Quarterly 
 
 Appl. Math., vol. IV, no. k, Jan. 19^7, pp. 36I-366. 
 
 8. Staff of the Computing Section, Center of Analysis (Under Direction 
 
 of Zdenek Kopal): Tables of Supersonic Flow Around Cones. Tech. 
 Rep. No. 1, M.I.T., 19^7. 
 
 9. Fraenkel, L. E.: The Theoretical Wave Drag of Some Bodies of Revolu- 
 
 tion. Rep. No. Aero. 2^4-20, British R.A.E., May 1951. 
 
 10. Van Dyke, Milton D.: Application of Hypersonic Small-Disturbance 
 
 Theory. Jour. Aero. Sci., vol. 21, no. 3, Mar. 195^4-, pp. 179-186. 
 
 11. Van Dyke, Milton D.: Practical Calculation of Second-Order Super- 
 
 sonic Flow Past Nonlifting Bodies of Revolution. NACA TN 2^hk, 
 1952. 
 
 12. Hart, Roger G., and Katz, Ellis R. : Flight Investigations at High- 
 
 Subsonic, Transonic, and Supersonic Speeds To Determine Zero-Lift 
 Drag of Fin-Stabilized Bodies of Revolution Having Fineness Ratios 
 of 12.5^ 8.91j and 6.04 and Varying Positions of Maximum Diameter. 
 NACA RM L9I3O, 19^+9. 
 
 CONFIDENTIAL 
 
MCA RM L56DII CONFIDENTIAL 27 
 
 13. Wallskog, Harvey A., and Hart, Roger G. : Investigation of the Drag 
 of Blunt -Nosed Bodies of Revolution in Free Flight at Mach Numbers 
 From 0.6 to 2.3. NACA RM L53D1W, I953. 
 
 ih. Chapman, Dean R., and Kester, Robert H.: Turbulent Boundary-Layer 
 and Skin-Friction Measurements in Axial Flow Along Cylinders at 
 Mach Numbers Between O.5 and 3.6. NACA TN 3097, 1954. 
 
 15. Stevens, Joseph E., and Purser, Paul E.: Flight Measurements of the 
 
 Transonic Drag of Models of Several Isolated External Stores and 
 Nacelles. NACA RM L5I+LO7, 1955. 
 
 16. Morrow, John D., and Nelson, Robert L. : Large-Scale Flight Measure- 
 
 ments of Zero-Lift Drag of 10 Wing-Body Configurations at Mach 
 Numbers From 0.8 to 1.6. NACA RM L52Dl8a, I953. 
 
 17. Wallskog, Harvey A., and Morrow, John D. : Large-Scale Flight Measure- 
 
 ments of Zero-Lift Drag and Low-Lift Longitudinal Characteristics of 
 a Diamond-Wing — Body Combination at Mach Numbers From O.725 to 1.5^. 
 NACA RM L53C17, 1953- 
 
 18. Welsh, Clement J., Wallskog, Harvey A., and Sandahl, Carl A.: Effects 
 
 of Some Leading-Edge Modifications, Section and Plan-Form Variations, 
 and Vertical Position on Low-Lift Wing Drag at Transonic and Super- 
 sonic Speeds. NACA RM L54K01, 1955. 
 
 19. Sandahl, Carl A., and Stoney, William E.: Effect of Some Section 
 
 Modifications and Protuberances on the Zero-Lift Drag of Delta 
 Wings at Transonic and Supersonic Speeds. NACA RM L53L2i^-a, 195^. 
 
 20. Holdaway, George H., and Mersman, William A.: Application of Tchebichef 
 
 Form of Harmonic Analysis to the Calculation of Zero-Lift Wave Drag 
 of Wing-Body-Tail Combinations. NACA RM A55J28, 1956. 
 
 21. Beane, Beverly: The Characteristics of Supersonic Wings Having 
 
 Biconvex Sections. Jour. Aero. Sci., vol. I8, no. 1, Jan. 1951? 
 pp. 7-20. 
 
 CONFIDENTIAL 
 
28 
 
 COKFIDEOTIAL 
 
 MCA RM L56DII 
 
 (a) e = 90*= 
 
 (b) e = 0°. 
 
 Flg-ure 1.- The areas and pressures which influence the drag of configura- 
 tions at supersonic speeds. 
 
 COKFIDENTIAL 
 
NACA RM L56DII 
 
 CONFIDENTIAL 
 
 29 
 
 (a) Low-flneness-ratlo body; — = 6.04. 
 
 d 
 
 /.2 
 
 (b) High-fineness-ratio body; — = 10. 
 
 d 
 
 Figvire 2.- The effect of Mach number on the area-distribution-curve slope 
 
 of bodies of revolution. 
 
 CONFIDENTIAL 
 
50 
 
 CONnDEOTIAL 
 
 NACA RM L56DII 
 
 
 <9 
 
 M 
 
 
 
 5" 
 
 /. 15- 4.68 
 
 D 
 
 1.5 
 
 /. 07- 3.94 
 
 
 
 10 
 
 1.12-3.01 
 
 A 
 
 /2.5 
 
 /. 16-2.47 
 
 t^ 
 
 /5 
 
 7.22-2. 12 
 
 G^ 
 
 17.5 
 
 728-7.88 
 
 Q 
 
 20 
 
 /. 33-7. 70 
 
 .5V 
 
 .4 - 
 
 .3 - 
 
 7ir?e of perfe>ct agreemerff — v^ /q 
 
 .2 .3 .4 
 
 Cff 7'£xact f/?eory) 
 
 Figi:ire 3.- Comparison of the drag of cones calculated with the area rule 
 with the drag from exact theory (ref . 8) . 
 
 CONFIDENTIAL 
 
CONFIDENTIAL 
 
 51 
 
 £xactt/?eory/'fi = 5''J 
 Area- rule t^eort/ 
 5/e/7der-boofi/ tJ76>ori/ 
 
 .2 .4 . .6 .d /.O 
 
 Figure \.- The drag of cones in collapsed form. 
 
 Ci)UlrJ 
 
 Slender-body t/?eory 
 Area-ru/e f/7eory 
 Area- rule ^J7eory "^ pressure term 
 O Secorfd -order theory 
 
 J I L 
 
 4 , 6 
 
 .3 
 
 J.O 
 
 Figure 5-- The drag of parabolic noses in collapsed form. 
 
 CONFIDENTIAL 
 
32 
 
 CONFIDENTIAL 
 
 NACA RM L56DII 
 
 
 I 'I 
 
 !fe » 5^ 
 
 V 
 
 N 
 
 
 QJ 
 
 
 ^ 
 
 
 -p 
 
 
 A 
 
 
 -P 
 
 
 t 
 
 
 Td 
 
 
 (D 
 
 
 +3 . 
 
 
 03 >> 
 
 
 H U 
 
 
 Ti 
 
 
 0) 
 
 
 H ^ 
 
 
 tU +3 
 
 
 
 
 
 >i 
 
 
 CO TJ 
 
 
 (U 
 
 
 •H £1 
 
 • 
 
 Ti 1 
 
 CO 
 
 ^ 
 
 • 
 
 ja Q) 
 
 
 
 Ti 
 
 
 fU 
 
 tIJ 
 
 •H 0) 
 
 § 
 
 CQ 
 
 
 XI 
 
 •^ 
 
 cd -ri 
 
 MD 
 
 ^ fi 
 
 
 cd S 
 
 6 
 
 ft 
 
 
 -P 
 
 •\ 
 
 QH £3 
 
 -4- 
 
 (U 
 
 • 
 
 6 
 
 
 
 bO -H 
 
 
 03 Jh 
 
 II 
 
 
 fl ro 
 
 X 
 
 -3 
 
 0) 0) 
 
 
 s: 
 
 
 -p ^ 
 
 * s 
 
 -p 
 
 cd 
 
 ^•^ 
 
 
 S3 (U 
 
 
 g^ 
 
 
 •H fH 
 
 
 u 
 
 
 oj oj 
 
 
 n 0) 
 
 
 e M 
 
 
 03 
 
 
 
 
 1 
 
 
 1 
 
 ■<6 
 
 
 H) 
 
 
 ^ 
 
 
 M 
 
 
 •H 
 
 
 p^ 
 
 
 CONFIDENTIAL 
 
NACA RM L56DII 
 
 CONFIDENTIAL 
 
 55 
 
 l.0r- 
 
 / 
 
 / 
 
 \ 
 
 y 
 
 <z 
 
 2 /a = 6.0 4 
 
 — /iroa-ru/e fheory 
 
 ■Sleneler'JOoety t/feori/ 
 
 '"^ M '' 
 
 /.4 
 
 J. 6 
 
 Co .2 
 
 O 
 
 2/d = 8.9/ 
 
 /.a 
 
 Af 
 
 /.2 
 
 /.4 
 
 /.6 
 
 ■4r- 
 
 <^D.2 
 
 2/cf = /2.S 
 
 .8 
 
 /.O 
 
 M 
 
 1.2 
 
 /.4 
 
 /.6 
 
 (b) ^i^2££ 
 
 = 0.2. 
 
 Figure 6.- Concluded. 
 
 CONFIDENTIAL 
 
5^ 
 
 CONFIDENTIAL 
 
 NACA RM L56DII 
 
 03 
 0) o 
 
 o 
 
 W 
 
 CO 
 
 a 
 o 
 
 •H 
 
 - -P 
 ft & 
 O S 
 H 
 m 
 
 o 
 I 
 
 a 
 o 
 
 •H 
 
 •^ 
 £1 
 •H 
 
 -P 
 M 
 •H 
 
 'd 
 I 
 
 Oj 
 0) 
 
 O 
 
 W 
 03 
 03 
 
 •H 
 I 
 
 o 
 
 •H cr\ 
 
 -p II 
 
 id CD 
 
 a 
 
 03 '^ 
 
 1 H 
 
 •H • 
 
 P 
 P 
 
 o 
 
 •H J3 
 
 03 Tj 
 
 d 
 o 
 
 ■H 
 P 
 O 
 Q) 
 W 
 
 O 
 
 o 
 
 H U 
 03 -H 
 P Oi 
 
 O VD 
 
 O 
 
 tiD O 
 
 ^•^^ 
 
 •H 
 
 vo 
 
 CONFIDENTIAL 
 
NACA RM L56DII 
 
 CONFIDEWTIAL 
 
 35 
 
 /80 
 
 Figi-ire 8.- Example of the area-distribution- curve slope for a wing-body 
 configuration for various values of p cos 9/tan A. 6o° delta wing; 
 NACA 65AOO6 airfoil section. 
 
 CONFIDENTIAL 
 
36 
 
 CONFIDENTIAL 
 
 NACA RM L56DII 
 
 tn 
 <U 
 •H 
 U 
 <u 
 w 
 
 <u 
 
 -p 
 
 o 
 
 •H 
 O 
 
 03 
 
 0) 
 O 
 
 a 
 
 bOO 
 O 
 
 O 
 
 0) 
 
 -p 
 o 
 
 CO 
 
 <: 
 
 bO 
 
 ^ ai 
 
 0) -p 
 
 -P H 
 
 (U 
 
 o 
 
 o 
 u o 
 
 (DVD 
 
 0) 
 
 -P 
 
 Ch 
 O 
 
 -P 
 O 
 
 bO 
 
 0) 
 
 -P 
 
 O 
 
 
 
 I CQ 
 
 • CO 
 
 ON (U 
 
 (U ft 
 
 u ^ 
 
 P 0) 
 M 
 •H 
 
 V? 
 
 CONFIDENTIAL 
 
NACA RM L56DII 
 
 CONFIDEOTIAL 
 
 57 
 
 .4 .6 
 
 SeoHf/tanJi. 
 
 (a) Area distribution drag. 
 
 Co 
 
 .03 
 
 a 
 
 tanA. 
 
 .02 
 
 ' _„---^ y// 
 
 
 ^ ^^ .-"^/^ 
 
 ni 
 
 ___^ — ^^Z^^i^^^ 
 
 
 
 n 
 
 1 1 1 1 1 1 
 
 JS 30 45 60 75 90 
 
 (b) Area distribution drag against roll angle. 
 
 .4 , .6 
 
 (c) Configuration drag. 
 
 Figure 10.- An example of the calculation of the configuration drag from 
 the drag of the area distributions at various values of (3 cos 9/tan A. 
 60° delta wing; NACA 65AOO6 airfoil section. 
 
 CONFIDENTIAL 
 
58 
 
 CONFIDENTIAL 
 
 NACA RM L56DII 
 
 Exper//7?e/7t 
 
 Area-raie theory 
 
 J3 
 
 -1 1 
 
 /■O /./ ^^ 12 AS /.4 AS /.6 
 
 An 
 
 (a) Basic body; model 1 (ref. 16); ^= O.O505. 
 
 Sw 
 
 .02\- 
 
 (^D .01 
 
 Arearu^e theorif 
 
 ■X- 
 
 -I L 
 
 .S .9 /.O // ^ /.^ /.3 /.4 /.S /.(. 
 
 M 
 
 (b) Model h (ref. 16); 6o° delta wing; NACA 65AOO3 airfoil section; 
 
 Sb 
 
 — = 0.0505. 
 
 Sw 
 
 .03r 
 
 .0^' 
 
 Area-ruJe thfori/ 
 ' Equii^a/ent body 
 
 (c) Model 5 (ref. 16); 6o° delta wing; NACA 65AOO6 airfoil section; 
 
 .02 r 
 
 <5) .0/ 
 
 Root airfoif 
 
 ^0003- 63 ^F/at ^0007 
 
 — = 0.0305. 
 
 Sw 
 
 I Expfrffnent 
 
 Area- rule theory 
 
 Equivalent body 
 
 (d) Model 6 (ref. 19); 6o° delta wing. Thickness ratio varies from 0.05 
 
 at root to 0.06 at 0.9 semispan. — = O.O505. 
 
 Sw 
 
 Figure 11.- Comparison of the calculated drag with experiment and 
 equivalent-body test results for wing-body combinations. 
 
 CONFIDENTIAL 
 
NACA RM L56DII 
 
 .02 
 
 .01 - 
 
 COKFIDEOTIAL 
 
 ■ Area-rale (heori/ 
 
 /.O // ,. /-^ /-? /■■f /-S 
 
 /vl 
 
 /.6 
 
 59 
 
 (e) Model of reference 17; A = 2.31; Ac/2 = 0; NACA 65AOO5 airfoil section; 
 
 Co 
 
 ^ = 0.0606. 
 Sw 
 
 Mi 
 
 - 
 
 
 
 £»per/'me/>f /^ 
 Area-ru/e /heory y^ 1 
 
 U/t 
 
 -err' — -D 
 
 
 M 
 
 .0/ 
 
 
 " 7 
 
 III III 
 
 /.O /./ ^^ /.2 /.3 /.4 /.S /.(, 
 
 (f) Model C-5 (ref. l8); A = 3; A = 0.2; Ac/J+ = ^5°; NACA 65AOO3 air- 
 
 foil section; — = O.O606. 
 Sw 
 
 Exper/ment 
 
 Area-ru/e iheory 
 
 £quiya/er>t body 
 
 (g) Model 6 (ref. 16); A = i^-; A = 0.6; A^/ij. = ^5°; NACA 65AOO6 airfoil 
 
 Sb 
 
 section: — = 0.0606. 
 
 Sw 
 
 Figure 11.- Continued. 
 
 CONFIDENTIAL 
 
1^0 
 
 CONFIDENTIAL 
 
 NACA RM L56DII 
 
 <^o 
 
 Area-ru/e t/Jeory 
 £qu/^a/enf body 
 
 (h) Model 2 (ref. 16); A = 5.0^+; A = 0.59^; Ajc/if = 0°; - = 0.0^5; 
 
 .04 
 
 ^i> .02 
 
 — = 0.0606. 
 Sw 
 
 Figure 11.- Concluded. 
 
 Exppr/znent 
 
 /.S /J> 
 
 Figure 12.- Comparison of equivalent-body drag and configuration drag 
 
 for an airplane configuration. 
 
 CONFIDENTIAL 
 
NACA RM L56DII 
 
 CONFIDENTIAL 
 
 kl 
 
 /6 
 
 14 
 
 /2 
 
 /O 
 
 (t/£)2far,f- 
 
 8 
 
 ./ 
 
 ^rea - ru/e theori^ 
 
 ■3 .4 
 
 /3 ^ane 
 
 .& 
 
 .8 
 
 Figure 13 •- Comparison of the drag of delta wings calculated with two 
 versions of the linearized theory. 
 
 CCNFIDENTIAL 
 
k2 
 
 CONFIDEFriAL 
 
 MCA RM L56DII 
 
 .a/6 
 
 .0/2 
 
 Co .008 
 
 .004 - 
 
 Figure 1^4-.- Effect of wing-panel separation on wing drag. 6o° delta 
 wing; NACA 65AOO3 airfoil section. 
 
 .0/6 
 
 .0/2 
 
 ^D .008 
 
 .004- 
 
 Figure 15-- Comparison of the sum of component drags with the configxira- 
 tion drag. 6o° delta wing; NACA 65AOO3 airfoil section. 
 
 CONFIDENTIAL 
 
NACA RM L56DII 
 
 CONFIDENTIAL 
 
 h-i 
 
 .S2r 
 
 Figure 16.- Area distribution parameter G(K, vq) for 65A series airfoil. 
 
 COI^i'IDENTIAL 
 
kk 
 
 CONFIDEETIAL 
 
 MCA RM L56DII 
 
 (a) vq from to 0.^4-5. 
 
 Figure 17-- Area-distribution- slope parameter H(K, vq) for 65A series 
 
 airfoil. 
 
 CONFIDENTIAL 
 
NACA RM L56DII 
 
 CONFIDENTIAL 
 
 h^ 
 
 (b) Vq from O.5 to 1.0. 
 Figure 17-- Concluded. 
 
 CONFIDENTIAL 
 
46 
 
 CONFIDENTIAL 
 
 NACA EM L56DII 
 
 O Airfo// ord/haie sfcrf/^/?s^ 
 
 /^v) 
 
 Figure I8.- Approximation of 65A series airfoil for the calculation of 
 
 G(K,Vo) and H(K,Vo). 
 
 CONFIDENTIAL 
 
 NACA - Lajigley Field, Va. 
 
CONFIDENTIAL 
 
 UNIVERSITY OF FLORIDA 
 
 3 1262 08106 568 1 
 
 UN!\/ERSITY OF FLORIDA 
 
 DOCUMENTS DEPARTMENT 
 
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