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At moolI L Copy RM L56D11 cR MEMORANDUM 0LEED T IOTIONS OF THE. TDAN7G . .r:. .. h .' ; . .'.m Jn ar T W elsh o. . ......O..E....I..O RO T I. H H . x.: :L. elme t. Wels0h,:., 0. .. ,e o." v..a..6. 'f ''T': t;..:"i:. ..ts;$.. OmM 4 ? ; I  * NACA RM L56D11 CONFIDENTIAL NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS RESEARCH 1 NEMORAIDUl4 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 wingbody combinations over a wide range of Mach numbers are compared with the computed drags utilizing a 24term Fourier series application of the supersonic area rule and with the results of equivalentbody tests. The results indicate that the equivalentbody technique provides a good method for predicting the wave drag of certain wingbody combina tions at and below a Mach number of 1. At Mach numbers greater than 1, the equivalentbody 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 evolution 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 areadistribution 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 wingbody 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 equivalentbody 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 2 CONFIDENTIAL NACA RM L56D1 having the same longitudinal distribution of crosssectional area. As a result, the drag of the configuration is obtained by either estimating or experimentally determining the equivalentbody 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 5, provided a powerful method for calculating the wave drag at supersonic speeds. In references 4 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 3 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 equivalentbody and wingbody experiments. For the supersonic area rule, comparisons are made of calculated and experimental results for both body and wingbody combinations over a wider range of Mach numbers than heretofore made. The supersonicarearule calculations were made by using a 24term 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 an = dx r A sin no do an =J \dx 2 q CD drag coefficient,  qS Cp pressure coefficient c wing local chord cO root chord of particular pointed wing tip cr wing root chord CONFIDENTIAL NACA RM L56D11 Ct wing tip chord D drag d maximum body diameter, 2rm f resultant pressure force f(v) wingthicknessdistribution function, z max G(K,v0) wingareadistribution function H(K,Vo) wingareadistribution slope function K m(l + cos ) for the leftwing panel; ml cos e for tan A / tan A / the rightwing panel 1 length of configuration lt total length of area distribution l/d body fineness ratio M Mach number m= A (1 + tan A 4 (1  n integer q dynamic pressure r local body radius rm maximum body radius S reference area Sb body frontal area Se wing exposed area S, wing total planform area CONFIDENTIAL CONFIDENTIAL NACA RM L56Dl1 So s t/c x,y,z x0 xt xl max P = yM2 1 0 = cosl(2 A S= c V0 Vr wing semispan semispan of particular pointedtip wing 2Zmax wing thickness ratio, 2z Cartesian coordinates point of intersection of Mach plane with the xaxis xcoordinate measured from wing leading edge dummy variable local maximum wing ordinate x It wing wing Mach 1) leadingedge sweepback angle taper ratio, t cr angle, sin1 R value of v at root of particular pointedtip wing value of v at root of actual wing 6 angle between zaxis and line of intersection of Mach plane with the y,z plane REVIEW OF THE BASIC THEORY From reference 6, the equation for the wave drag of any system of bodies or wings and bodies can be written as: CONFIDENTIAL CONFIDENTIAL NACA RM L56D1 D 21 d t It d d2A(x,e) t df(x,9) d2A(xi ) q 42 0 0 l x2 2q dx dX2 Sdf(xl ) log(x Xl) (1) 2q dxl J 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 (1) 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 xaxis 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, 8 about the xaxis measured from the yaxis. By moving the vertex of the Mach cone along the xaxis, a series of parallel Mach planes will cut the con figuration for a fixed roll angle 0. In the drag equation the term A(x,8) 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 8 direction. These relationships are illustrated in figure 1 for angle of roll 8 of the Mach plane of 0 and 900. By neglecting the term _2 df(x,) the equation reduces to the supersonicarearule formula given by Jones in reference 5. Evaluation of f(x,8) requires the pres sure distribution on the configuration which when integrated over the configuration gives the drag directly. As a result, large values of Sdf x 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 evaluate the drag of configurations using the supersonicarearule formula of Jones. The influence of the pressure term was evaluated for one simple case. Equation (1) can be written in coefficient form as 2 1 CD = f CD(8)de (2) C I=B% CONFIDENTIAL CONFIDENTIAL NACA RM L56D11 where c(e) =2 t St Lt dA d df og(x xl)dx dx (5) O) 1qdx tD 2n/ox0 V 2 The quantity CD(e) is most readily determined by solving the inte gral for CD(6) through a Fourier sine series expression for d fol lowing the method of reference 7 if = cos l) an J dsin no d then CD(() can be written as CD(8) = Ena2 For the computations of this paper, only 24 terms were used in the Fourier sine series expression for dA. Thus, dx n=24 CD(e) i__ 2 CD() = 4 nan2 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 highfinenessratio 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 areadistribution curve slopes were calculated from the expression dA = 2 fu (x xo)dx ,dx J 2R2(x) (x xo)2 CONFIDENTIAL CONFIDENTIAL UACA RM L56D11 The derivation of this expression is given in appendix A. It has also been assumed 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 finenessratio6.04 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 f dCp q dx The only assumption made in derivation of f is that d is q dx constant over the oblique area. This is a reasonable assumption except for bodies having discontinuities, and high local slopes. Then, iA f dA 2A 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 his region.) Figure 5 presents a comparison of the drag of various cones calculated with the supersonic arearule 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 number of the comparison was arbitrarily taken as that at which the slope of the Mach line equaled onehalf the slope of the nose. The agreement between the two theories is remarkable, within 5 percent except for a few points. CONFIDENTIAL COJFIDENTI AL NACA RM L56D11 A better comparison may be made by plotting CD(I/rm)2 against Prm/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 (50 half angle cone was chosen as representative), slender body theory (ref. 9), and supersonicarearule theory are shown. The comparison shows the great improvement of the arearule theory over the slenderbody theory at values of Prm/l greater than 0.2, and the good agreement of the area rule result with the exact theory to prmjl of about 0.7. At higher values of prm/1 the arearule theory is in error, possibly first because the pressure term is neglected but finally, near prm/1 = 1, because the assumptions of the linearized theory are violated. At prm/1 = 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 supersonicarearule theory is an improvement over slenderbody theory but in only partial agreement with the more exact secondorder theory of reference 11. Inclusion of the pressure term, evaluated by using secondorder pressure distributions, however, does give agreement with some of the secondordertheory results. Since the secondordertheory 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 prm/l = 0.5 corresponds to the flagged symbol. For the parabolic noses, both the arearule theory and the arearule theory plus the pressure correction cannot be expected to apply near and above Prm/l = 0.5, 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 arearule theory with experiment and slenderbody theory for a family of parabolic bodies of revolution. The experimental drags were taken from references 12 and 13. 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 14 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 slenderbodytheory drags were calculated using the curves of reference 9. CONFIDENTIAL CONFIDENTIAL NACA RM L56D11 As would be expected, the comparisons show the increasing ability of both the arearule theory and slenderbody theory to predict the drag as the body fineness ratio is increased. In most cases, the arearule theory offers a significant improvement over slenderbody theory. The arearule theory and slenderbody theory are in agreement near M = 1, since at this Mach number the supersonicarearule theory reduces to slenderbody theory. From these nose and completebody comparisons that have been made, the following conclusion can be drawn. The arearule 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 than 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 lowfinenessratio bodies. It is not to be inferred from the above statement that the supersonicarearule 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 lowfinenessratio components. CALCULATION OF WINGBODY DRAG The difficulty in computing the wave drag of wingbody 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 bodyarea 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 number 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'l at a roll angle of 900. Appendix C presents a simple analytical method for evaluating wing area distributions and areadistributioncurve slopes. The curves neces sary for evaluating these quantities (figs. 16 and 17) are applicable only to 65A series airfoils, but similar curves can be made up for other air foil sections. CONFIDENTIAL CONFIDENTIAL NACA RM L56D11 In order to get an idea of the applicability of the thinwing assump tion, a calculation has been made of the true areadistributioncurve slope variation for 600 delta wing having an NACA 65A006 airfoil section for a roll angle of 900 and a Mach number of 1.414. In order to simplify the calculation, the wing was approximated by a sufficient number of linearslope 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 figure 7. Although 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 thinwing assumption. On the basis of this result, it is felt that the thinwing solution should be adequate for wings of present day interest. For the wingbody combinations of this paper, an additional simpli fication was allowed in the supersonicarearule wavedrag 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 bodyareadistributioncurve slopes were considered independent of Mach number, and the changes in the areadistributionslope 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 0 cos 0 dent only on the value of ta Thus, the areadistributionslope tan A curves for the wingbody configuration are dependent only on the value B cos 8 of Cos Then, from equation (2) and because of the symmetry of the tan A configuration, CD(M) 2 / CD(9)d9 In order to obtain the wave drag of the configuration, a plot of CD against B is required. This can be computed if a plot of CD against B cos B B Ctos is given, since the angle 8 is known for fixed values of tan A tan A and C s. The configuration drag is simply the average drag between tan A S= 0 and For the wingbody calculations of this paper the bodies were identi cal. The bodyareadistributionslope curves are shown in figure 2(b). The curve for M = 1.414 was chosen as representative for the Mach number COIlFID NTAL CONFIDENTIAL NACA RM L56D11 range of interest. The wing area distributions and areadistribution curve slopes were obtained by using methods similar to that given in appendix C. In addition, a limiting value of P cos = 0.8 was set for tan A configurations having blunt leadingedge airfoils. (Above t cs A = 1, 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 (600 delta wing, IACA 65A006 airfoil) is presented in figures 8, 9, and 10. Figure 8 shows nondimensional plots of dA against dx 0 cos d for various values of Figure 9 shows the effect on CD of tan A the number of terms in the series solution. Except at P Cos 8 = 0 and 0.8 tan A convergence was apparently obtained within 24 terms. Figure 10 shows the variation of the area distribution drag with P0 s the variation of tan A ' area distribution drag with roll angle, and the variation of the config uration drag with P tan A* WINGBODY DRAG COMPARISONS Figure 11 presents some wavedrag comparisons for wingbody combina tions. The experimental wingbody results were taken from references 16 to 19. The wingbody 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 14. Base drags and fin pressure drags were subtracted using the results of reference 16. The equivalentbody 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 14. Basedrag rise and findrag rise were not evaluated for the equivalentbody models. These quantities, however, should be small in the Mach number range where comparison is valid. The supersonicarearuletheory 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 supersonicarearule theory to predict the drag near M = 1 is evident for nearly all cases. However, the agreement at the higher Mach numbers between the theoretical drags and the experimental CONFIDENTIAL CONF I DEMTIAL 12 CONFIDE TRIAL NACA RM L56Dll wingbody drags is excellent and within the accuracy of evaluating the experimental wave drag, except for three configurations. Two of these configurations (figs. 11(c) and 11(g)) had 6percentthick wings which were the thickest wings investigated. The third configuration (fig. 11(h)) had a 4 percentthick airfoil but with fairly steep wedge components. 2 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 equivalentbody drags give a good approximation to the experimental wingbody drags up to a Mach number of 1, except for the two configurations having 6percent thick wings (figs. 11(c) and 11(g)). This result is in agreement with reference 2 which shows the validity of the transonicarea rule decreases with increasing wingthickness ratio. At Mach numbers above 1, the agree ment is variable but tends to be consistent with the flatness of the cor responding theoretical curve. That is, as the theoretical drag variation with Mach number becomes smaller, the equivalent body gives a better approximation of the supersonicdrag level. This would be expected, since a flat theoretical curve indicates that the variation in areadistribution drag with roll angle or Mach number is small. Then the drag for the Mach number 1 or roll angle 900 area distribution (corresponding to the equiv alent body) is representative of the configuration drag. Figure 12 shows the comparison between experimentalconfiguration drag and equivalentbody 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 supersonicdrag level. The equivalentbody drag is approximately 40 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 accuracy shown in figure 11. The influence of the tail surfaces, how ever, may be difficult to evaluate. If the horizontal tail supports a load when the configuration is at zero lift, the influence of the pres sure term may be significant. Although no supersonicarearule drag cal culations were made for this airplane, reference 20 indicates that gen erally good predictions of completeairplane drag can be made.1 1Subsequent bo the preparation of this paper, NACA RM A56107 has been prepared at the Ames Laboratory and presents supersonicarearule 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 __~____~__~_ _ I NACA RM L56Dll EVALUATION OF CuMI.POOPEIIr AND IrJPT1FEFETICE DRAGS As was shown in the preceding section, the supersonic area rule can be a useful tool in evaluating the supersonic drag of a wingbody 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 supersonicarearule 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 15 where the drag of delta wings having 65A series sections is plotted in collapsed form. The arearule 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 wingbody configuration on wing drag is presented in figure 14. 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 exposedwing panels based on total and exposed wing areas is compared with the isolated wing drag. Separation of the wing panels gives approximately a 10percent reduction in wing drag coefficient at Mach numbers above 1.5. As the Mach number 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 15 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 1.5, interference drag is, for all practical purposes, zero. Thus for this configuration, at Mach numbers greater than 1.3, 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 equivalentbody technique and a 24term Fourier series application of the supersonic arearule method to predict wave drag at transonic and supersonic speeds. From the theoretical and experimental comparisons made, the following conclusions can be drawn: CONFIDENTIAL CONFIDENTIAL IACA RM L56D11 1. The arearule 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 slenderwingbody configurations can be predicted with the supersonicarearule formula. For the wingbody configurations investigated, the best agreement was obtained for the con figurations employing the thinnest wings. 5. The equivalent body technique provides a good method for predicting the wave drag of certain wingbody combinations at and below Mach number 1. At Mach numbers above 1, the equivalent body wave drags can be misleading. Langley Aeronautical laboratory, National Advisory Committee for Aeronautics, Langley Field, Va., April 6, 1956. CONFIDENTIAL CONFIDENrTIAL nACA RM L56D11 APPENDIX A AREA DISTRIBUTION SLOPE FOR BODIES OF REVOLUTION CUT BY OBLIQUE MACH PLANES The area distributions are identical for all roll angles. simplicity a roll angle of 900 will be used in the derivation. For The frontal projection of the oblique area cut by the Mach plane (see sketch (a)) is given by the equation: A = 2 f uy dz z1 From the equation for the Mach plane, z is related to x by Z = X and dz =1 dx The equation relating y and x is given by y = R2(x) 2 R2() (xxo) 1 P2 P 2R2x) (x xo)2 CONFIDENTIAL (Al) COIIFIDElITIAL NACA RM L56D11 Then, A =X 2R2(x (x 0)2 dx Differentiating the expression for A gives (A2) dA 2 Fxu dx0 2 Jx (x XO) /2R2(x) (x xo)2 where x, and xu are the roots of x = x0 PR(x) and x = x0 + pR(x), respectively. CONFIDENTIAL (A3) CONFIDENTIAL NACA RM L56D11 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. xu r = R(x) x = x0 + Pz NdSY dy dy The net force in the e direction (the zdirection for a roll angle of 900) (see sketch (b)) can be written as = f sin a dS f C C Since dS = dy2 + dz2 and ay sin a = dy2 + dz2 = Cp dy The pressure coefficient at zero lift is a function of x only. The equation for y in terms of x is given by equation (1) of appen dix A as follows: y pR2 (x) (x 2 y P CONFIDENTIAL CONFIDENTIAL NACA RM L56Dll and 02R(x) (x O) dy = dx p pR2(x) (x xo)2 Then, the net force can be written as SCpO C 0 ) upper surface Cp2R(x) (x xO] 2R,]2(x) (x xo)2 + i(2 Cp 0 dx + f x Px, Slower surface C 02R(x) (x x0 Sdxdx O2R2(x) (x x)2 f 2f xu x 1 Cp[P2R(x). (x x0j JpR2(x) (x xo)2 Integrating the expression by parts gives f 2 Xu q 0' x X22 dCp x) (x x0 2 dx d RP2R2(x)_ (x d xO) dOn If dc is essentially constant between the limits of integration, dx f= 2 C xu q TdxJ xl 22(x) (x x)2 dx Then, from equation for the frontal projection of the oblique area in appendix A (eq. (A2)) = C q dx CONFIDENTIAL = 2 /X0 CONFIDENTIAL NACA RM L56Dll APPENDIX C METHOD FOR DETERMINING WINGAREA DISTRIBUTION AND AREADISTRIBUTIONCURVE 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 pointedtip wings. Then, correc tions are made for curvedwingbody junctures and finite wing tips. In addition, the right and lefthand wing panels are considered separately. Pointedtip wings. Consider the rightwing panel shown in the fol lowing sketch: Zmax Sx' = x0 + y(p cos e tan A) \0 (c) CONFIDENTIAL CONFIDENTIAL NACA RM L56D11 The frontal projection of the area of one wing panel cut by the Mach plane is given by A = 2 z dy and z can be written as c z_ Zax z I_ t f(1 ) c c zmax 2 c t Then, for L constant, A = s c0 L ) dT The value of q and v are related by the intersection line of the Mach plane and the wing chord plane for the rightwing panel given by the equation x' = x + y(p cos 0 tan A) and for the leftwing panel by the equation x' = x0 + y(p cos 6 + tan A) Then, v = [0 O + ( cos e tanA) With cO v = i + CO(P cos e tan A)q V0 V VO V (tan A cos 9). v CO sO m = tan A CONFIDENTIAL CONFIDENTIAL and Let CONFIDENTIAL NACA RM L56D11 In terms of tapered wing geometry, m is given by m = A 1 + tan A 41  Then, T = V V 0t ml t P Cos  tan A / K= m( for the leftwing panel and K = ml for the rightwing panel. + cos o tan A p cos tan A / Then, SV0 V Kv dT K v dH K v0 dv (K v)2 K v0 Kv c (K vo)2 dv dq = ^ dv do (K V) The equation for the area can then be written as tS (K A = S0C0 K  2 f upper lower f(v)3 dv = s0c0 G (K, 0 (K v)3 CONFIDENTIAL CONFIDENTIAL NACA RM L56D11 The slope of the areadistribution curve is obtained by differentiating the expression for A dA t upper f() d + dvo O dv + O v0lower (K v) K vO)2 (K ~lower) dvlower do f (lower) (K v0)2 (K *upper)5 dvupper dv f(Vupper) dA sct H(Ko) do = sOcO v) Curves of G(K,vo) airfoil and are given in to 2.4 and and H(K,v0) have been made up for figures 16 and 17 for values of K v0 from 0 to 1. In evaluating / f(v) dv, (K v) assumed to vary linearly between airfoil ordinate stations. gives a plot of f(v) for this assumption. a 65A series from 0 f(v) was Figure 18 For K and vo by the expressions greater than 1, G(K,V0) and H(K,v0) G(K,vo) = G(K,1) For K and vg the expressions H(K,vo) = less than H(K,1) (K ) (K 1) 0, G(K,vO) G(K,v0) = G(K,0) (K v0)2 K2 H(K,Vo) = H(k,O) (K K0) K and H(K,vo) are given by CONFIDENTIAL are given (K v)2 (K )2 CONFIDENTIAL NACA RM L56D11 Correction for curvedwingbody juncture. The following sketch shows a pointedtip wing mounted on a curved body. Line of intersection of Mach plane and wing chord plane The areas and slopes will be (cr, s, and 7). The areas pointedwing tip. (d) referred to and slopes, the actual wing geometry however, will be for the exposed 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 SOC0 of the exposed wing through the point and the value of v0 for the exposed wing at the point of intersection. As the point of intersection changes, sO, cO, and vO change and account for the intersection line. Expressed in terms of the actual wingbody characteristics, SoC0 is given by S00 = c r (  The quantity r is related to v0 by the expression v (1 A)m ] s 1 (1 x) The area of the exposed wing panel cut by the Mach plane can be written as CONFIDENT AL CONFIDENTIAL I NACA RM L56D11 Crs t 2 A =  1 (1 )r G(K,v0) The area is calculated for given value of vO. The centerline value of v is given by Vr = v[l (1 )s] + K(l A)r The slope is given by is obtained by differentiating the expression for dA dvr A and 1 (1 )F+ (K vo)(1  If d: O, dvO dA dvr = 1 (1 A)H(Kvo) 1. A A Correction for finite wing 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 cr   s  sO From sketch (e): SoCO = ers N2 1 A CONFIDENTIAL CONFIDENTIAL NACA RM L56D11 CONFIDENTIAL 25 Then the areas and slopes are given by Crs 2 2 Atip = 1 G(K,vo) t dvr i A The centerline value of v is given by Vr = + K(l ) CONFIDENTIAL NACA RM L56D11 REFERENCES 1. Whitcomb, Richard T.: A Study of the ZeroLift DragRise Character istics of WingBody Combinations Near the Speed of Sound. NACA RM L52H08, 1952. 2. Hall, James Rudyard: Comparison of FreeFlight Measurements of the ZeroLift Drag Rise of Six Airplane Configurations and Their Equivalent Bodies of Revolution at Transonic Speeds. NACA RM L55J21a, 1954. 5. Jones, Robert T.: Theory of WingBody Drag at Supersonic Speeds. NACA RM A55H18a, 1955. 4. Holdaway, George H.: Comparison of Theoretical and Experimental Zero Lift DragRise Characteristics of WingBodyTail Combinations Near the Speed of Sound. NACA RM A55H17, 1955. 5. Alksne, Alberta: A Comparison of Two Methods for Computing the Wave Drag of WingBody Combinations. NACA RM A55AO6a, 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. L, Jan. 1947, pp. 561566. 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., 1947. 9. Fraenkel, L. E.: The Theoretical Wave Drag of Some Bodies of Revolu tion. Rep. No. Aero. 2420, British R.A.E., May 1951. 10. Van Dyke, Milton D.: Application of Hypersonic SmallDisturbance Theory. Jour. Aero. Sci., vol. 21, no. 5, Mar. 1954, pp. 179186. 11. Van Dyke, Milton D.: Practical Calculation of SecondOrder Super sonic Flow Past Nonlifting Bodies of Revolution. NACA TN 2744, 1952. 12. Hart, Roger G., and Katz, Ellis R.: Flight Investigations at High Subsonic, Transonic, and Supersonic Speeds To Determine ZeroLift Drag of FinStabilized Bodies of Revolution Having Fineness Ratios of 12.5, 8.91, and 6.04 and Varying Positions of Maximum Diameter. NACA RM L9I50, 1949. CONFIDENTIAL CONFIDENTIAL NACA RM L)6D11 13. Wallskog, Harvey A., and Hart, Roger G.: Investigation of the Drag of BluntNosed Bodies of Revolution in Free Flight at Mach Numbers From 0.6 to 2.3. NACA RM L55Dl4a, 1955. 14. Chapman, Dean R., and Kester, Robert H.: Turbulent BoundaryLayer and SkinFriction Measurements in Axial Flow Along Cylinders at Mach Numbers Between 0.5 and 3.6. NACA TN 5097, 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 L54L07, 1955. 16. Morrow, John D., and Nelson, Robert L.: LargeScale Flight Measure ments of ZeroLift Drag of 10 WingBody Configurations at Mach Numbers From 0.8 to 1.6. NACA RM L52D18a, 1953. 17. Wallskog, Harvey A., and Morrow, John D.: LargeScale Flight Measure ments of ZeroLift Drag and LowLift Longitudinal Characteristics of a DiamondWingBody Combination at Mach Numbers From 0.725 to 1.54. NACA RM L55C17, 1953. 18. Welsh, Clement J., Wallskog, Harvey A., and Sandahl, Carl A.: Effects of Some LeadingEdge Modifications, Section and PlanForm Variations, and Vertical Position on LowLift Wing Drag at Transonic and Super sonic Speeds. NACA RM L54KOl, 1955. 19. Sandahl, Carl A., and Stoney, William E.: Effect of Some Section Modifications and Protuberances on the ZeroLift Drag of Delta Wings at Transonic and Supersonic Speeds. NACA RM L53L24a, 1954. 20. Holdaway, George H., and Mersman, William A.: Application of Tchebichef Form of Harmonic Analysis to the Calculation of ZeroLift Wave Drag of WingBodyTail Combinations. NACA RM A55J28, 1956. 21. Beane, Beverly: The Characteristics of Supersonic Wings Having Biconvex Sections. Jour. Aero. Sci., vol. 18, no. 1, Jan. 1951, pp. 720. CONFIDENT IAL CONFIDENTIAL IIACA RM L56Dll (a) e = 90. (b) e = 00. Figure 1. The areas and pressures which influence the drag of configura tions at supersonic speeds. CONFIDENTIAL CONFIDENTIAL NACA RM L56D1l 2  0. ,e/x/ CON FIDENTIAL .6 x/I (a) Lowfinenessratio body; 1 = 6.04. d / ,4 l^A.r/ H.4z* M 0  2 0 (b) Highfinenessratio body; = 10. d Figure 2. The effect of Mach number on the areadistributioncurve slope of bodies of revolution. CONFIDENTIAL rIACA RN L56Dll Cb (Area 50 7.5 /0 /2.5 /5 /75 20 M /./54.68 . 073.94 . 123.01 A /62.47 /.222.12 / 281.89 /33/. 70 /Lie of perifet agreement ' 0 ,2 .3 .4 .5 C, /Exact theory) Figure 3. Comparison of the drag of cones calculated with the area rule with the drag from exact theory (ref. 8). CONFIDENTIAL CONFIDENTIAL NACA RM L56D11 4 Ea/ theory /( = 5/ Arearule theory S/endaerhody fthry 2 .2 .4 r, .6 . Figure 4. The drag of cones in collapsed form. SS/enderbody theory Arearule /heory Arearukl /theory pressure erm 0 Secondorder /thry 0 .2 .4 .8 LO Figure 5. The drag of parabolic noses in collapsed form. CONFIDENTIAL CONFIDENTIAL CONFIDENTIAL NACA RM L56D11 "0 i s u u a :s 5 r F ? I s P 0 r c Sci rda .0 o0 a,4 o o 0r LO (d $ [, 0 w cd *9 ' 49 CONFIDENTIAL lit t Ii r1 I , I 1I '1 " N '0'9 N N '0hr ^ ll I , I I I I '0 ' '* ' 4. N I 4o NACA RM L56D11 _______ _______ / .6 Z /a = 6.04 .4 .2 Arear/e hevy Slendero4dy theory .8 1.0 /2 /.4 , M 2/ = 8.91 Co, .2 .8 /.0 .2 1.4 . .4 2/d = 12. .a /V, 0 1.2 1.4 L (b) nose = 0.2. 1 Figure 6. Concluded. CONFIDENTIAL CONFIDENTIAL CONFIDENTIAL %at NACA RM L56Dll o 0* 0 * *r4 0 I .:4 0 00 .l 4. I f1 o I H .4 rd ii l0 or4 rd 43 o wI *H .0 0 aH Cd r *H u 0 D I i 0 CONFIDENTIAL NACA RM L56D11 24d /Scos a tan .A 0 ,2 7 .4  .6 .8 Figure 8. Example of the areadistributioncurve slope for a wingbody configuration for various values of P cos 9/tan A. 600 delta wing; NACA 65A006 airfoil section. CONFIDENTIAL CONFIDENTIAL 36 CONFIDENTIAL NACA RM L56D11 U mo izi cno (u ori 0 1 NO 0 0 ao z bo W 0O w ON ;q 1Q I 0 CONFIDENTIAL NACA RM L56Dll 0 .2 .4 geos&a .8 (a) Area distribution drag. 8 taeA cffA i i i i i i 60 75 90 (b) Area distribution drag against roll angle. .4 O/.tanA .8 /.0 (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 p cos 9/tan A. 600 delta wing; NACA 65A006 airfoil section. CONFIDENT LAL .03 .02 Ca .0/ 0 0 /5 30 45 *, dae IVl l l i CONFIDENTIAL NACA RM L56D11 .0"  Experiment CD  /rrArvr* hery .8 .9 .0 I /./ ..2 3 .4 /.5 /.6 (a) Basic body; model 1 (ref. 16); Sb = 0.0305. Sw  Arwa tuA. thoojy Co .o0/ N .9 1.0 i /M Ml /.3 /4 .S .6 (b) Model 4 (ref. 16); 600 delta wing; NACA 65A005 airfoil section; Sb = 0.0305. Sw .03  A fir /entr  Equval/ent mody I 0 l I I I I S 9 1.0 /./ .2 /3 /.4 .S 1.6 (c) Model 5 (ref. 16); 600 delta wing; IJACA 65AO06 airfoil section; Sb = 0.005. Sw Root ailo/l Cot  o i A~p,iu~m'm1por L  Eq..ao/e'nf booy I ru Ih~r /3 /.4 /.5 .6 (d) Model 6 (ref. 19); 600 delta wing. Thickness ratio varies from 0.05 at root to 0.06 at 0.9 semispan. = 0.0305. Sw Figure 11. Comparison of the calculated drag with experiment and equivalentbody test results for wingbody combinations. COIIFIDEITIAL .8 9 /.0 /./ /. M CONFIDENTIAL NACA RM L56D11 .03 .02  CONFIDENTIAL Arirrule thMry  E4uiYa/eLt b.vy o/ I SJ/1 .9 .O /./ /.2 M I I I /3 /.4 /5 /.6 (e) Model of reference 17; A = 2.51; Ac/2 = 0; NACA 65A005 airfoil section; b = 0.0606. Sw Aa, erime n'  Arearule /~kuw   Equval/,t kody / I I  I .9 /0 /./ /.2 M I I /3 /4 /.5 /6 (f) Model C3 (ref. 18); A = 3; A = 0.2; Ac/4 = 450; NACA 65A003 air Sb foil section; = 0.0606. Sw section; I mey 6. .02 quivaent body   Sb section; 1= on.066. Figure 11. Continued. COIIFIDEINTIAL . .. I m m m I ) NACA RM L56D11 Ex, 'er/.4ent Arearule AIOry  EqwDva/eff body .9 /. 0 / /.2 M /.4 /5 /.6 (h) Model 2 (ref. 16); A = 3.04; A = 0.594; A5c/4 = 00; = 0.045; c b = 0.0606. Sw Figure 11. Concluded. Expermenl Equid/ae'n Co'y  .9 /.0 /./ /2 114 1.3 /.4 1/. Figure 12. Comparison of equivalentbody drag and configuration drag for an airplane configuration. COIIFIDENTIAL 4irhil CONFIDENTIAL NACA RM L56D11 /6 14  Arearule theory Beane theory .3 .4 /9 ianE .5 .6 .7 . .7 .8 Figure 15. Comparison of the drag of delta wings calculated with two versions of the linearized theory. C CFIDENTIAL Co ft/iftan# 9  6 4 CONFIDENTIAL NACA RM L56Dll .0/6 .0/2 (a .008 .004 /./ 1/. /3 /. /S /16 / 7 /W Figure 14. Effect of wingpanel separation on wing drag. wing; NACA 65A005 airfoil section. 600 delta /. /2 /.3 / 4 /.S /.6 /.7 /n Figure 15. Comparison of the sum of component drags with the configura tion drag. 600 delta wing; NACA 65A003 airfoil section. COIFI DETIAL Based on4, iC1j Baswd onSu  .01/2 Co .o00 CONFIDENTIAL NACA RM L56D11 0 .2 .4 .4 .8 A.O /.2 /16 / 20 22 2A Figure 16. Area distribution parameter G(K,VO) for 65A series airfoil. CONFIDENTIAL CONFIDENTIAL NACA RM L56D11 6 9 O '0 #1 j.6 /8 20 22 N (a) v0 from 0 to 0.45. Figure 17. Areadistributionslope parameter H(K, v) for 65A series airfoil. COINFIDEITIAL CONFIDENTIAL NACA RM L56D11 (b) v0 from 0.5 to 1.0. Figure 17. Concluded. COIIFIDENTIAL CONFIDENTIAL NACA RM L56D11 0 A i/old orda/le sfr/onsa 0 .2 .6 .8 /.0 Figure 18. Approximation of 65A series airfoil for the calculation of G(K,VO) and H(K,VO). d! CONFIDENTIAL NACA Langley Field. Va. CONFIDENTIAL I." i ; ii. .: . : '':. .. i i ." .. . i!,, : ,,: .r .... .E .. .. * "l :E Il: ;. .:. : . I"'"" "''"' ii "i "" "" ..,.. ii : :i ;ic : i, ; ;.ii" UIvCnJ III UT rLwnYIun *: : *. : ...:. :; :. ::;:. *:. .;.: :. .1:..'' ^ : 3 1262 08106 568 1 UNIVERSflY OFFLORIDA ".'.i DOCUMENTS:DEPARTMN 120 MARSTONl SCIENCE UBtA P.OBOX17011v,1 GAINESVILLE, FL 32611101 S% 4 "l : "'; "." A ": " Y: % .. .; i. .. .. ,. FT .: .":. i'. Ni. ::'' ..: ! 4N, .f,1 ' ... .'...: ,. ;ii , . .. .., .. ..: : S. .**.. * : ". .ii li .t.: ..! . : .. .. :" ~I r : .r~; :ii' "~;? 