Supersonic Wave Drag of Thin Airfoils Citation Puckett, Allen Emerson (1949) Supersonic Wave Drag of Thin Airfoils. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/HYJJ-3241. https://resolver.caltech.edu/CaltechETD:etd-06232004-141931 Abstract The linearization of the equations of motion for the supersonic flow of a perfect fluid is discussed, and methods of solution using elementary source sink solutions are developed. These methods are applied to the calculation of performance of several types of three-dimensional supersonic airfoils; in particular, the drag at zero lift of a family of almost triangular, symmetrical wings is calculated. The significance of the results is discussed. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: airfoils ; (Aeronautics and Mathematics) Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Aeronautics Minor Option: Mathematics Thesis Availability: Public (worldwide access) Research Advisor(s): von Kármán, Theodore (advisor) Stewart, Homer Joseph (advisor) Group: GALCIT Thesis Committee: Unknown, Unknown Defense Date: 1 January 1949 Record Number: CaltechETD:etd-06232004-141931 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-06232004-141931 DOI: 10.7907/HYJJ-3241 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 2697 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 24 Jun 2004 Last Modified: 29 Mar 2023 16:49 Thesis Files Preview PDF (Puckett_ae_1949.pdf) - Final Version See Usage Policy. 13MB Repository Staff Only: item control page

SUPERSONIC WAVE DRAG OF THIN AIRFOILS Thesis by Allen E. Puckett In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 1949 pote ABSTRACT The linearization of the equations of motion fer the supersonic flew of a perfect fluid is discussed, and methods of solution using elementary source sink solutions are developed. These methods are applied to the calculation of the performance of several types of three~dimensionel supersonic airfoils; in particular, the drag at zero lift of a family of almost triangular, symmetrical wings is calculatede The significance of the results is discussed. ii ACKNOWLEDGEMENTS The author wishes te express his appreciation to Prof. Th. von Karman for his suggestions and assistance in the initial formulation of the problem, and for the inspiration furnished by his methods and solutions in approaching other problems in this field. Sincere thanks are also due Prof. He Je Stewart for his constant help and puidence during the course of the work, and to Miss Shirley Woodbury for her assistance in the preparation of the manuscripts Part If Tit Iv vit VIIT iii TABIE OF CONTENTS Title Introduction and Sumary Linearization of Equations and Method of Solution Special Applications to Infinite Wings Application to Delta Wings; Scheme of Calculations Basic Pressure Distributions and Surface Integrals Drag Coefficients and Pressure Distributions for Delta Wings Limiting Cases Conclusions References Appendix I; Summary of Symbols Figures 24 34 38 61: 82 95 97 99 101 14 15 16 1? 18 20 21 22 28 iv - LIST OF FIGURES Title Influence Cone for Source Forecone of P Supersonic leading and Trailing Edge Subsonic Tips Blementary Doublet Line Swept Wing, k < 8 Swert Wing, k > 2 Geometry of Delta Wing Four Mach Number Ranges Basic Triengle, n> 1 Influence Zone for t > 1 Basic Triangle, n <1 Pressure Coefficients on Basic Triangle Integretion Region for Sy Integration Region for £5 G(n, ¢) for n> G(n, s) for n <1 F(n, s) Double Wedge Profile AeWing Dreg vs. Maximum Thickness Position, B = 0. A-Wing Drag vs. Maximum Thickness Position, a = 0.5 Ae“Wing Drag for a = 6 AeWing Drag for a = 0925 102 102 103 103 104 108 108 109 119 Fige Now | Title Page 240 AeWing Drag for a = 0,50 117 25 => wing Drag for & = 0075 | 118 26. ‘AsWing Drag: Effect of Trailing Edge Sweep for b = 002 119 27? - ~~» AwMing Drag; Effect of Trailing Edge Sweep for , b = 0.5 120 28 AeWing Drag; Effect of Swee pbhack 121 29 AeWing Drags; Effect of Peak Location, ¥, = 30°, a =0 1l2Z2 30 AwWing Drag for a = O08, b = 0c2 123 31 AeWing Drag for w, = 30°, a = 005 124 32 AeWing Drag for w) = 18°, b= 02 | 125 33 ThreesSlope Delta Wing 126 54 AeWing Three Slope Profile, n = 2,5 127 35 . DragsStrength Ratio, Three Slope Profile, n = 2,5 128 36 Pressure Distribution, TwoesSlepe A-Wing, n = 0.8 129 37 . Pressure Distribution, n = 2.5 130 38 Rectanguler Wing, c/b < 6 131 39 Rectangular Wing, c/b > B 131 PART I INTRODUCTION AND SUMMARY The flow of an incompressible, non=viscous fluid around a three dimensional non-lifting body can, in principle, be calculated for rather gene rel shapes, and reasonably simple closed solutions exist for a few specie] shapes. These possibilities exist, of course, because the incompressible equations of motion are linear. These solutions, however, do not answer directly one of the fundamental aeronautical problems =~ the calculation of the drag.« since in an ‘incompressible fluid this arises essentially from viscous effects. On the other hand, in & compressible fluid, at supersonic speeds, an importent part of the drag does arise from the non-viscous flow, and is known @s wave or pressure drag, Unfortunately, however, the complete compressible flow equations are non-linear, and at present can be solved only in two-dimensicnal and axially symmetric eases, and then only by an approximate numerical procedure, However the importence of the perfect fluid solutions in drag calculations (aside from their obvious importance in lift calculations) has empha= sized the usefulness of approximate solutions for engineering purposes, and perhaps stimulated the carrying out of such calculations to an even greater degree than existed in the low-speed case. . The most practical approach to an approximate sclution is the "enall perturbation" assumption = that is, the assumption that the importent features of the flow are represented by velocity fields in which the velecities de not differ much from the free stream velocity. It ean be shown that this amounts te requiring that the slopes of elements of the surface of the body are aimost parallel to the direction of the free stream velocity. This is most likely to be true for thin airfoils, or thin bodies of revelution. Even in these cases, there may be regions in which the lecal (approximately j caleulated velocities clearly do not satisfy the smell perturbation assumption. In these regions, at worst we must recognize that the approximate calculation is lecally in serious error, while at best we may hope that the general behaviour of the solution, as to the extent and order of magnitude of "pressure peaks", for instance, may still be predicted correctly, and that in the integrals of the solutions, representing total forces or total drag, the error may be small, Comparison with analogous situations in approximete incompressible flaw calculations allow some optimism regarding the letter possibility, while at the same time careful examination of the approximations made to the complete equations permit more quantitative specification of the probable limits within which the errors of the sclution lie below certein tolerances. In the last analysis, from an engineering point of view, the justification for any approximate method of solution, and the determination of its range of usefulness, lies in comparison with experimental results. In the following chapters a brief summary will be given of the methods of linearizing the general equations of motion for a compress= ible fluid, and of obtaining solutions to these equations for supersonic flows, An extended account will be given of the application of one method, the’ "source solutions" to the calculation of the flow around thin three-dimensional airfoils et zero lift, In particuler, calcula tions are carried cut and results. presented in detail for one simple but especially interesting femily cf airfoil plenforms, the "delta wings", which have an essentially triangular planform, but are also allowed to. have various trailing edge angles, This family is particue larly interesting because it is characterized by only a few geometrical parameters, but at the same time exhibits most of the peculiarities, also possesed by more complicated airfoils, in the behaviour of the drag coefficient as the leading edge angle or pesition of maximum thickness line are varied, Local behaviour of the pressure coefficient is alse typical of local conditions in more complicated cases. In this way, the calculetions allow the development of some engineering feeling for the relation of airfoil drag performance to the critical geometrical parame ters The most important results of the calculations are presented. in the curves of drag coefficient vs. Mach number for airfoils of this . family, in Figs. 20 to 32. It is seen there that the most important geometrical parameters are the ratios of the angles defining position of leading edge and trailing edge, and maximum thickness line, to the Mach angle, and that certain singularities occur in the solution when any one of these ratios is le. In general, for an airfoil of given thickness, the drag of the airfoil is decreased considerably when both the leading edge and the line of maximum thickness are swept well behind the angle of the Mach waves. it is also found that in certain cases the drag coefficients approach finite values when the Mach number approaches 1, in contrast to the results in two-dimensional cases, and in spite of the usual suspicion that the linearized approximation fails completely in this neighborhood. It seems likely that some of the details of the flew in this limiting case do fall outside the scope of the linearized theory, but it also seems quite possible that this evidence of great improvement in undesirable drag characteristics near Mach number 1 by using swept leading and trailing edges is evidence of a real phenomenon. This is borne out to a considerable extent by existing experimental data. PART IT LINBARI ZATION OF EQUATIONS AND METHOD OF SOLUTION A. Basic Equations The following symbols will be used _U, V, W, = velocities in direction of x, y, and 2 (or x,) or Uz axis respectively U, = undisturbed or free stream velocity, in direction of x-axis u, Vv, w, = perturbation velocities in direction of x, y, and ra) or us z (or x, J axis respectively D = perturbation velocity potential M * Mach number, U/a a * speed of sound p * pressure pep = density Y = yatio of specific heats, e/e, po =u? ~ 2 A= Slope of airfoil surface in xedirection Subseript ( J, will denote conditions in the free stream (i.e., at infinity ). ; Certain additional symbols will be introduced later in the discus= sion of properties of particular airfoils. Although the technique of linearizing the equations of motion is well known, it will be exhibited here in somewhat complete form in order to demonstrate clearly the order of magnitude of the terms neglected. The fundamental equations governing the steady motion of a compressible, woGee inviscid fluid are the Ruler equation, dT; / ZZ _ _ d ax, aX, : (1) and the continuity equation, 9. (ou;) =O (2) Ze é F] BL; It can also be shown from the equation for conservation of energy that, in the absence of viscosity, and heat transfer, the density will be a unique funetion of pressure, given by the isentropic law ox p = conste p * (3) In this case it can alec be shown that a flow initially irrotation will, in the absence of discontinuities (shock waves) remain always irrotational, and that equation (1) can be reduced to the Bernoulli equation TU; 7; - a 2 = lores. (4) 2 %-/ TZ, 7; ¢ xa Bo = Lost, (4a) 2 — where @ is the locel speed of sound, \/7p/p » The continuity equation ean then be written ag_ 22. Ui de Pe. », AG | ak, 7 ap PX; a2 Ob (5) or, in expanded form, as Fon a2 ()- 22 82) 4 att) WF (/- B)e Ce 4 21 =) Wav > UW AW pV AW _ oO 4277 oy ar A* AK a* OY (5a) We now suppose the net velocity to consist of small disturbances to the free stream velocity, U,, so that g=uU, tu VYRqv w=w (6) where u/U,, v/U,, w/U, are all small compared to ones The speed of sound, @, can be expressed in terms of these perturbations, using equation (4a), as f BG “2 +(e) (Z) “(Bf (7) a " that 4. és {1 +0-Mg | 1 a4) 1 } (8) where HM, = 0,/8¢ Using this expression in equation (5aJj, we obtain Ga yt dor, Our me _ 2) fa. £4 (1 - Ma) + BE + SE = SE Me 247-1) My {(E) Or 2 wf? AL Gu 2 ur The following assumptions are now made: {1) (af, J is small compared to (1 « MJ at low supersonic Mach tumberse This implies that either WI, must be extrenely small, or the Mach number cannot be allowed to approach too closely to ls (2) M on (/, ) is small compared to 1 for high Mech numbers, Mje Since it will be found from the first approximate solutions that (u/U,) is of the order (7/0, 1/My» or a/M., where @ is the local angle between a surface element and the flow direction, this assumption implies that eM is small compared to le Since M is of the order ce » where 4 is the Mach angle, we thus have ofu << 1, Thus at very high Mach numbers, with small values of Va) the second order effects become incPeasingly important. If these assumptions are made, (9) becomes the basic equation of the linearized theory: ae Our = ar ae ; The solution to this can be regarded as the first step in a series of successive approximations. Substituting the values of u, V, Ww found from (10) inte the right hand side of (9) a new linear equation is obtained which might be solved for the second approximation. This ig essentially the procedure used by Hentsche and Wendt, (Ref. 7) in the subsonic two-dimensional case; it can also be carried out explicitly in the supersonic twoudimen sional ease, but has not as yet been worked out in the general three-dimensional case, However there is some question about the convergence of this process even in the twoxdimensional CASEe Since the flow is irrotational, the velocities can be given ass: v= I/IK = B, v= Y/ay = Py w= Ip/ 32 Ls lI and equation (10) becomess This is the three»dimensional form of the perturbation equation first introduced by Ackeret (Ref. 1) and Glauert (Ref. 3 Je The boundary condition, that the velocity component normal to the surface be zero, may be expressed completely ass uZ +i¥mt+Wn = 0 > (12) where L, m, nare the direction cosines of the normal to the surface, If we deseribe the surface of a thin airfoil as + = t(x,yJ, thens ae IL ‘ , — —_— — / — — a ZL: wm 7) = DL OY 4 so that equation (12) becomes (ne gir ww =e (13) if Ff/AxX° ana It/ Jy are both small, even though possibly of the same order of magnitude, we have ca) Thus in this first approximation, the boundary conditions relates only the downwash, Ws to the x»slope of the surfaces A corresponding linearized calculation of the pressure changes in the flow is necessary. If the Bernoulli equation for the pressure in _ terms of the velocities (equation (4)) is expanded ina Taylor series about the free stream velocity, we find that Ppa AG Ck + Se) Blt VE) 20,7 3 Zs + of(é | We may def ine the pressure coefficient, oe by 2 _ aye Tp = TZ x 0 a (15) awlLl= In the first order approximation, we then have 4 = ww O \ c 2u/t, (16) Ba Methods of Solution There are two general methods of arriving at solutions to equation (11). One is to investigate immediately the existence of certain typical “conical” solutions, that is, a solution with u, v, w constant along each ray through a point. This form of solution was discussed by Busemann (Refs 6). As shown by Stewart (Ref. 8) such solutions can be represented as functions of a complex variable in a distorted plane ‘perpendicular to U. More general solutions can then be produced by combinations of special eoenical solutions. The other method of attack is to represent the solution as an integral of certain elementary solutions which can be interpreted as supersonic sources or doublets, Such elementary solutions were used by Karmen and Moore in calculating the flow srowd slender bodies of _ Pevelution (Refe 2), and were discussed by Prandtl, and used by Schlichting, in connection with liftingewing problems (Refs 4 and 9). This type of solution is particularly adaptable to calculation of flow around thin wings at zero lift, as will be seen. C. The Source Solution Tf we let. i oo £CE, @ LILY, 2) = J lr- = )* S£ly-2)" + 2° f (17) where pe = a? #1, this potential will be a solution to equation (11), “Lee 6 . If goes to zero, it does, in fact, become the potential for a subsonic source located at (= , 7 )inthe x, y plenee However for g@ > 0, the expression under the radical venishes for certein values of x, y, and z, namely, at those values representing the surface of the cone defined by _,2 2 2 2 This cone is, of course, the Mach cons, with cesntral halfeangle of 4 = cot’ 8 “which is the Mach angle, and vertex at ( © , @, Oj. This cone is shown in Fig. 1. Equation (18) defines two cones, one estending up= stream, which may be called the “"forecone", and one extending downstream, which may be called the "aftercone". Apparently Qg is defined for X, ¥, 2 lying within either of these SONS o However from physical sonsiderations we shall suppose that the disturbance q at ( & , @ j ean affect only conditions downstream, that is, in the aftercone of ga The cone is the stationary wave front produced by a disturbance at (5 , Yj) ina flow at Mach number Moe If the point P = (x, y, z) lies in the aftercone of ( F , 4 do then ( & , Y% ) aiso lies in the Mach cone extending upstream with vertex at P, ises, the forecone of P, Apparently go at Pis affected only by points whose aftercones include P, and therefore by disturbances at points lying within the forecone of P. This cone intersects the £, 7 plane in a hyperbole, given essentially by equation (18), and shown in Fige Ze aloe A general solution ean now be constructed by superpositions of elementary solutions of the type in equation (17), with qf F,7 j interpreted as a source strength per unit area. It can be noted that these solutions will produce w = Ip [a2 antisymmetric in z, but u and v symmetric. The general solution can be represented by els 2) LM (18; df. ) $a 9,2) o-Jotlge 5) [y-0) 242) where it is assumed that no disturbances exist ahead of £ =0, The curves Z, ,», and 4%, are defined by p= ¥- Vlesl 8 - 2, = ¥ + VERT B — 2* EF = x - G2 that is, the two branches of the hyperbola of Figs 2s The downwash, or wevelocity, in the xy=-plane, associated with the potential of equation (19) has a very simple expressione However the ealeulation of w = ad / 32 directly from equation (19) is somewhat awlward, sinee the limits of integration involve z, and the integrand becomes infinite at these limits, Therefore certain transformations are necessary. Integrating the second integral of (19) by parts, with a Ltn! {CG -0) aide and noting that | le | hy! SCG 4) | : ESO sn ple | ~~ Z/psa)-2650)} Z, / we obtains mp < BE M2 7 | ) oy aly p ~ Hoa.) ser} « gfas ss ad (owes: dy ra] a L, The wevelocity mey now be calculated; ag | 52 ~ 7 a a(x Be, yj) + Ty + Ty (20) where z- Ae T = 26905 a) ~ gls, 2) } bs o 7 fe 20) )4 - C= _ ‘= 5 Ste (Al - CELI and 1-f2 ’ ) y 2. 22 IE") oy 2 7g 22 [@ op" Vas} sr2® og 4, v4 hur 96. emp BGI / a Wu -5)* 872" c A 2 Ig. st ’ =z G 6 C1 (22) @15< Denoting the first of the two integrals in the last expression for T, by I, and Ip respectively, it is clear that [z,1 < ($8 i, la (9,-2,) = 0 / FP? /mal, 2 *«-$2 ip ( I¢ /Ig) is finite. We also note that, inT., M2 oP fd inn! ha A) _ IG Jer Mae ~ BC4- Zz) 72 Sp eee % - 92 gr fe 7) VK -5)* — (82 = %, _ 2%, har (LF = pt La) lina (off) 7 az Jl ¥)?— — 927 Na , 2.2(¢-7@) +f: me [le-5)- p72 fv-s)*- Blige 2} f (23) We are interested in the value of wat 2= 0, The integral in the last term of equation (21) is obviously convergent, and therefore aporoaches zero as z —~ 0, The first two terms are EB S Ea 7 CH | which also approaches zero as 2 —~0,. As seen from equation (22), it is actually necessary to prove that the integral of this expression with respect to 5 vanishes, which is easily shown as follows. If z= [rset (2p) C24] } . 2 a _ | [Z| S Trp a gezt where x = |G) - 68), So that | a |Z | S K 2 log fers) fas? | ra} Om and lim (Z| = lima Zz (og za = Z>O eo Thus qT, becomes zero, and in a similar manner, Ty approaches zero as Zz —» 0, Therefore wr = - 2,68) - — T gl, ¥) (24) This extremely simple result indicates that the velocity potential may be written immediately in terms of a known downwash distribution ‘in the xy~plane, with q =a2 | (24a) The same result was demonstrated previously (Ref. 10} in a somewhat simpler manner, relying to sone extent on a physical argument, but which unfortunately was. not rigorous matheneticelly. As noted previously this solution will produce a distribution of w which is antisymmetric about the xy=plane, which, from the boundary condition, (equation 14) corresponds to a symmetric body at zero lift. A potential of this type will thus be useful mainly for calculations of drag of thin airfolis. In this case, the local slope of the airfeil surface in the x-direc= tion, Ay » Gives the source distribution immediately, for, ,from equation (14), w= 0, Ax (25) In sertain special cases, this solution can also be used to calcue late the Lift distribution on an airfoil. Consider the airfoil of Fig. 3,- in which the angle between the tangent to any part of the leading edge or trailing edge and the xeaxis, a » is always greater than the Mach angle, “~ . In this case there is no disturbance in the field ahead of the wing. Suppose that the angle of attack distribution over the surface of the wing is @; then in the upper half-space (Z > 0) we may represent the potential in the form of equation (19), with q = “405 n)/ Tt. In the lower half-space we may define the solution as the negative of | this, thus cbteining a total solution for which w is symmetric, and u antisymetric about (x, y), iee., @ solution for a Lifting wing of zero thickness. Since p is zero everywhere ahead of the airfoil, and the Mach weves from the tips, the two solutions agree here, and since conditions behind the airfceil cannot influence conditions ef the surface, it is unnecessary to calculate conditions heres The type of solution cannot, of course, be used to caloulate directly the flow in the wake of the airfoil, when the solution. must again be symmetric inw. Leading and trailing edges of this type are called "supersonic" since the component of the free stream Mach number normal to the leading edge is greater than ls M, = M sin @ >Msinx =1le (26) Thus the "source distribution" technique ean be used to calculate the lift as well ss drag of any airfoil with entirely supersonic leading and trailing edges. Por airfoils with "tips" which are essentially subsonic leading or trailing edges, as in Fig. 4, the source solution can be applied to calculate lift on all parts of the airfoil except the regions, A, affected by the tips (the shaded zones), In principal, the same technique can be used even here, if a source distribution can be found for both regions A and B which results in the proper upwash over A, and a zero usvelocity over B, for in this case the two potentials for the upper and lower halfspaces would match in the free stream areas B, In general, this would appear to lead to an integral equation: for the source distribution; however Evaard (Ref. 11) has shown that a solution can be written immedi« ately in terms of the angle of attack distribution in region A. Thus even in this case the solution can be represented immediately by simple integrals cf the type of equation (19), This method fails if the influence zone from one of the tip regions B, or from the area ahead of a subsonic part of the leading edge, falls on part of the opposite region A. In the’ general case of airfoils with subsonic leading edges, the lift problem may be solved by super positions of special conical flow solutions of the type described in the first part of this chaptere Another technique is also suggested by the special properties of the solution in equation (19), We observe first that not only the poten» tial, os but also the perturbation velocity components, u, v, and w must satisfy the basic equation (11). Thus if ; x, L> _ 2. ulsn) dF a4R &(t,4,2#) =-7 53 ds s > a (27) Jle-5)*- 2*ftg-2) 4 2° 0 2, we see from the properties of (19) as indicated in (24) that Ulx, y, 0) = u( Za, Y ds This distribution of u will be antisymmetric in the xyeplane = i.e@.s, it represents a lifting surfaces A potential which will produce this u-field iss P= fk (t,4,2) dz If the integration with respect to x and the differentiation with respect to z are carried out, we find that Sen g--4/ ulsn)2#. (1-8) ds de (oa) USL Gg + 2 les) a fy e From equation (16), the velocity u is essentially the lift per P ' t unit area, L., for = = —-206 = # 4 = | (29) poye 7 — © TZ Thus the potential for a wing of given lift distribution may be written down immediately, but the angle of attack distribution must, in general, be calculated from equation (28). In the inverse problem, that of given. shape, the lift distribution will, according to this method, be the solution of an integral equations The above results may be arrived at in another (rather similar) manner, but which brings out more clearly its vhysical interpretations Imagine a row of supersonic "sources" of constant strength q starting at the origin and extending back along the xeaxis to oe , According to equation (19), the potential for this row of sources will be = | _ £1 4F ( (a) where ee ee ee Thus ) ax Z, (44, 2) = + B, cosh 2 (yee 2 A row of "doublets", obtained by bringing another source row up from below, will be represented by Zz: A -Z2 oA, | BY, 2)= 32 oo (yrs 2a? )(x7- Bg" +27) (31) =2l= Now by analogy with the subsonic interpretation of such a doublet line, this should represent an infinitesimal horseshoe vortex, commence ing at the origin, as in Fige 5, with finite product of circulation 7” and width Ai » but, in this limiting case, infinite circulation, The flow field associated with the potential g, might be considered a distorted version of the subsonic horseshoe vortex, and is therefore ‘not recognizable as such - but if we proceed to the Treftz plane ( x —= 0), and restrict ourselves to the neighborhood of the x-axis, the flow field is immediately recognizable as identical to the subsonic doublet. For , _ _f, 7 (32) wo. fo = (gi+ 27) which is the potential for a horseshoe vortex of width a, and eirculation I’, fora —0, with q == [ 2/2, and with circulation in the sense shown in Fige 5. Since the vortex laws and the Kutta-Joukowski law still hold in the supersonic flow, we know that the total lift on the bound part of this vortex system (i.e., at the origin) is L= pul a= pU. 21q » (33 ) Thus 1 4ne pue/2 (34) If we now suppose qy to be a doublet strength per unit area, introduce this in equation (31) and integrate over an area in the 3 , / plane the result agrees with equations (28) and (29) exactlye =oee This approach has incidentally indicated a method for calculating the downwash behind a supersonic wing of known lift distribution in a plane very far behind the wing, and in a region well inside the Mach cones, In this plane, the chordwise distribution of lift on the wing is obviously unimportant, If the total lift per unit span is L', than ‘the potential in the Treftz plane is approximately, from equations (32) and (34), | b/2. | g-2-— L(g) 2 de (385) ~ amp pUV2 (y-97)* # 27 —b/ where the wing span runs from «b/2 to b/2. Or, alternatively, this field can be calculated using any of the standard techniques for obtaine ing the flow piela far behind e@ subsonic wing of known lift distribution. The above calculation will not, of course, give the field correctly near the Mach cone from the WiNge In the remainder of this presentation, attention will be centered on the calculations which can be made using the potential of the type in equation (19), with particular attention to drag calculations. In Ref. 13, R. T. Jones makes use of a source distribution similar to the solution of equation (19), actually an integral of a uniform distribution over certain basic areas, to calculate the drag of certain swept back wings. Again, in Ref. 14, Jones makes use of the solutions of the type in equation (28) to calculate the characteristics of wings with given Lift distribution, Further application of an extended form of the solution in equatien (19) was used by Evvard in Refs, (15), (16) and (17) to calculate lift near subsonic wing tips, downwash fields, and performance of yawed wings. The doublet distributions of equation (28) are used again by Heaslet and Lomax in Ref. 16 for the calculation cas £ downwesh fields. Extensive calculations of airfoil drag for vaious specific plan= forms were made using the source distribution method by Mergolis in Refse (19), (20) and (21). The source solutions were also presented Gr by Tha von Karman in terms of Pourier integrals in Ref. 2Ze PART III SPECIAL APPLICATIONS , TO INFINITE WINGS As.a first illustration of the application of the source type of potential, a calculation will be made of the pressure distribution and drag on an infinite twoedimensional wing of small but finite thickness cand at rere angle of attack which is swept back or yawed, Two cases ere of interest: (1) the angle GO between wing leading edge and xeaxis is greater than the Mach angle, < , or (2) @ is less than 4. The first case is shown in Fig. S» In this case, the “sweepback angle", @ , is such that tan 7 > 8, The pressure esefficient at a point P, which we may take to be on tthe x axis, and in the plane of the airfoil, will be determined by the source distribution within the ferecone of P, the region A in Fig. 6. This forecone is bounded by the Mach lines g= 4 (X-)/8 and by the leading edge = II KY where Ik =tan JT 4, From equations (19) and (24), the potential at P is given by x- F)F— 4°" wheres wis the (known J downsash inthe = , “7 plane, Let A be the Plt) = - Lf wr AE ad (36) f 7 2 7¥ I glope of a surface element of the airfoil measured in the x=direction. Then A is constant along lines parallel te the leading edge of the airfoil. If we introduce — s= F- kw then s is constant along Lines parallel to the leading edge, and w= Als). U Equation (36) now becomes: “2 a = go pforas - Vlx- 5- £e)?-38°9* 71 while the limits of integration are: @, = - &-S)/p = — \ He = + lt s)/3 = Equation (39) may be rewritten ass g _ -7 +s) ais) Ww. SJ£- Z at = [Ree or} This is immediately integrated to x _t fwtsdds fk , Uk df Dp n/a 82 k? oe 3B ao a (38) (39, (40) or, - wills) ds ? = -f- (42) ad wl) _ TA OK nn) 1 coe le k* (43) The pressure coefficient can, of course, be calculated immediately from equation (16). It will, however, be more convenient to express the pressure coefficient in terms of the component of velocity normal to the airfoil leading edgee We may let Uo =U cos 7 n M =WM cos @ ne and Ce = AP (44) n v2 /2 [Vn where Ap is the pressure disturbance on the airfoil surface, We may further note that cosem = JV1ltkE end we my define a new "8" for the normal Mach number by 2 2 poe 2 rq? BAK. (f ) = Mr -/ = 274 = 2 (45) (#E JHE The slope of the airfoil measured normal to the leading edge is Ag = 4 = AV/4 &* | (46) Combining equations (43), (44), (45), and (46), we obtain finally 7 oo ye 1 This is identical with the result obtained by Ackeret (Ref. 1) for the pressure coefficient on a twoedimensionsal airfoil, referred here to & the velocity and Mach number of the flow normal to the leading edges The initial requirement that B > k ensures that MO is greater than 1, This result is sasily obtained also by the direct physical argument that the pressures in this case must be determined only by the component flow normal to the leading edge, as was pointed out in 1935 by Busemann (Ref. 5). In other words, the true free stream velocity can be supposed to consist of one component perallel to the wing leading edge, which cannct affect the pressure distribution, and a normal component, The pressure coefficient referred to the true free stream dynamic pressure is 2 A, cas * G~ Co = — (48) /7,°—/ For this twoedimensional airfoil, the normal to each surface element is also norsial to the airfoil leading edge, and therefore the horizontal component of the net force on each element is normal to the leading edges The component of this force, AF, parallel to the x-axis will give its contribution to the airfoil drag, AD, as mien AD = AP cos T (49) Thus the elementary drag coeffielent will be 2A. Cos? OT ° Jn? = 1 or 2 \* cas TT v7 where the x-direction slope, A has again been introduced, re % + AC, = If we consider an airfoil with fixed profile in the xz~plane, but variable sweepback, and thus consider only the variation in cos T / V M° » 1 with 7 , we see that the drag is actually increased with increasing Ts and approaches the usual twoedimensional sigulerity at MT =i. Even in this very simplified case it thus appears likely thet primary drag advantage from sweepback must come from sweeping the wing back so far that M, < 1, or else taking advantage of the new geometry to decrease the slopes, A » measured parallel to the free stream Plows. A similar caleulation may now be made for the second case mentioned at the beginning of this section =— the infinite wing swept at an angle such that the leading edge is behind the Mach waves, ies, tan 7 > Ba this case is illustrated in Pize 7. The forecone of P will now intersect both leading and trailing edges of the wing, so the limits of integration must be revised somewhat, It is convenient to carry out the integration in two parts, one over the area ahead of the line L, and bounded by the upper Mach wave, and the other over the region behind L. Using again s= ¥ = kW we have the complete potential oy weids a7 (52) p= tN eae kB) 9" y ’ where A is the forecone of Pa If we divide the integral as above, with f= +1, then (52) and D ay | (53) -tf> Sy (ask le, } eee (gaat 2 where 2, and Q, ere still given as in equation oy and the lower limit a will be allowed to approach minus infinity. In order to obtain the principal value of these integrals, the squared quantities under the square root must be taken in such sense that they are positive in the range considered, The expression in brackets under the square root is positive in both cases» InI,, (x-s) is greater than zero, so | x 2 f -/ (4% ft / och Le _ Lee 41 * afer | 8” big eves) ag At “= f, 4 the expression in brackets becomes mity, so that roe “/ _ Kae )a (54) 7 [Blk eee) The second integral is similar except that (s-x) is now greater than gard, Thus %. _— _ kB? 42 ; ee cosh” (-#- (B05 - ae [ Cc I, = f tr as cosh Ck De ao fs 8 ls-x) These two integrals may be combined tos Cc p a ( > fords cots” a (E78 }8 (56 J Yee gp? fs le-s) oO sines the expression in brackets in both equations (54) and (55) is positive in the range of s considered, The incremental uevelocity is thens . Cc GL | Th (8-*)/f kle-s) ~Cep2)aj ®pelu-s)* oh. Vee : a urls) as Pe kd If a is now ‘allowed to approach minus infinity, we find Cc l= ag. = — fe wrdts) as (57) oe d This disturbance velocity is of the. same form as that obtained in the two-dimensional thin wing theory for the representation of the pressures over a symmetrical airfoil of finite thickness at zero lift. In fact, for an airfoil at zero Mach number, with a chordwise slope distribution ) ol 84s in a flow at velocity Us» the perturbation velocity over the surface would be ab | - 2 f Als) ds (58) oa PL (s- xX) The pressure coefficient corresponding to the supersonis case will be, from equations (38) and (57), Cc c 2 Als) ds = -4F e T/k* 3 / (sx- x) 0 where A (s) is still the slope in the x-direction. In the incompress= (59) ible case, from equation (58), the pressure coefficient is c C2 + 2 ff AG) as (60) a. 77 J (s- 2x) In our supersonic ease, we may now rewrite equation (59) in terms of the velocity, Mach number and slope normal to the leading edge, using equations (38), (44), and (46), and referring the pressure coefficient to the dynamic pressure of the normal flow as before. We note that wn 2am 2 _2 2 lew? = 172-0 = E38 (61) : 1+ x? 1+ x? The new supersonic pressure coefficient, Cpe becomes ; Cc Cc. = é As) ads CAAT, Aa (s-x) oO — og (62 ) 7 ———3 P. Js Ne e Thus, as agein might be expected from very simple physical considera tions, the pressure distribution over the airfoil is determined only by the component of flow normal to the leading edges In this case, the normal Mach number is less then 1, and the pressure distribution is subs sonic in character, being exactly equal to that computed for the two dimensional normal profile with a Prandtl«Glauert correction based on the normal Mach number, The drag in this case is obviously zero, according to the D'Alembert paradox, or as may be easily shown by integrating the pressure distribution of equation (59) multiplied by the local slope over the chord of the airfoil. It must be noted that the integrals of equations (54) and (55), and particularly of (56) and (57) are all singular at s-= x, In equation (57) it is necessary to take the Cauchy principle value. In (54), (55) and (56) the singularity is essentially logarithmic, and causes no trouble. The above calculation suggests strongly that appreciable drag reduc= tions may be obtained with wings swept back behind the Mach angle. However, in the form presented it cannot give any quantatetive indication of this wh So in ® practical case, eince any real wing must be finite in span, and therefore no part of ut will be free from the influence of the center sections If.a real wing consists of two two-dimensional wings both swept back from a center plane of symmetry then the conditions infinites ly far from the center must approach the results of the previous calculae tione The real problem is the calculation of the flow resulting from the geometry of the center sections PART IV APPLICATION TO DELTA WINGS; SCHEME OF CALCULATIONS As @ particularly simple case of a finite three=dimensionsl wing, we shall sonsider the wing with planform shown in Pig. 8, called a delta wings For simplicity the profile is taken to be a double wedge, with variable lovation of the vertex, B. The three essential geometrical perameters are the threes sweepback angles of the leading edge, maximum thickness line, and trailing edge; Ty: Tos and Tas respectively, or their tengents, k, = tan an Ll Ky =tan TO. kp, =ten T, (68) It will also be convenient to define *y f k of, 2 = kg /tey (64) The area, S, and aspect ratio, K , are respectively Sb fe. F Se eS, és) » «© * k,- &s | (65) 4° = 2 (1-4) k, The thickness is Z times the true reot chord, Z= (1 - ak. The slopes (in the x-direction) in regions i and II respectively will thus be A, = Gas, A {i = 8) (66) 2(1 =r) 2(r = a) il oo It is desired to compute the drag of this wing at zere lift, using the source solutions of the type of equation (19) to describe the flow field. The cemputation scheme thus becomes quite simple in principle; since the slope in region I is a constant, a ys its contribution to the solution is represented by a constant source distribution in region I, and simile: rly in region II, In order to simplify still further the classification of the basic integrals required, the wing planform can be composed ino somewhat different way. The slope distribution can be supposed te result from the superposition of three triangles, with vertices at A, B and C, respectively, seach extending indefinitely in the positive x=direction. The slopes of the surfaces of these triangles will then be A i An = A, “ A, which, upon superposition will give the correct actual slopes in regions I and Il, and zero slope in the wake. The basic pressure distribution required will therefore be that associated with a single such triangle of constant slope. For the sake of order, the scheme of superposition and calculation of mutue] influences will be set up fairly completely in this section, before any of the calculations are carried out. in the calculation of the drag of the complete wing, four cases must be considered, determined by the direction of the Mach weve relative te the three lines Ag, Br , ond CT on the winge These cases are character+ ised by = Gm of #8 on BS and are illustrated in Figs le We notiee thet in cases (1) and the ure easures associated with the source distribution in the triangle A, will act upon both regions I and II, while the scuree distribution in B will act only upen region II, In case (Sj, the pressures due to B ean also act on region I, while in case (4), the pressures due to both B and C can act on both regions I and II. Let Cp, be the pressure coefficient for the Pield associated with & uniform source distribution in a triengle with vertex at j and a (positive } unit slope in the x-direction. Then let D.:. be the Ars. £ ig in region i due to the source distribution in triangle with vertex j. In general, the total drag for both sides of the delta wing will then consist of the fellowing six terms: ti) 224 2 24,Aa, [op 4A F wa (2) re 2 A, Ag | oe AA a ul a 4) "tA 2 2A, A, [ &, 44 ; ZO G) Lae _ 2A, As [av & a GZ) 20 = 2A, Ac J & aA & wr | (68 ) 08 Foe Intredue ing the values of the slopes from equations (65) and re : £ ; aq and dividing by the actual area of the wing, we heve the individual 2 contributions to the dreg coefficient as: ze foal’ 1 G Cy = = /% aA G74) ZA é (1-4)? wa Z GZ) Gp. 2 ~-26 (64 Lp, dA GF 4) tS 2 &- —+)*(r-4) SS _ 2? Ga aA Fc) GC) Cope © C-tW(r- 4) 4/4 | - 2% Gna" G9 df A) Cor, 2 (-r)(r-a) SJ i iq aA / ze l-4)?_ () Cong * & fala) LG aA (ete) 2 47-4)" () Cy 7 ~F= (4) ¢/& dA GAA) The total drag coefficient for beth sides of the airfoil will be the sum of thess tertigs as be ba In the first and second cases of Figs 9, terms (2), (3) and (6) erms (3) and (6) are zero. Term (1) and (4) will heave a singularity when ky = 8, and tern (8) will have a singularity when ky ~ 8, or kyr = PB. This essentially completes the eatalogin (a) ; , ot aa? 4.4 y 4 i ae 5 te) “ A of the necessary calculations. The only problem remaining is the caleu- dation of the various pressure distributions and the special integrals mn ben ‘4 49) ae Se 5 f of these pressure distributions as required by equetions (69). PART V BASIC PRESSURE DISTRIBUTIONS AND SURFACE INTRGRALS A. Pressures over unit-slope triangle. As coneluded in the last section, the initial problem is the calculation of the pressure field associated with one of the basic triangles, This pressure field will have a different representation depending on whether k is greater or less than BR, i.ee, whether the leading edge is ahead of or behind the Mech wave from its vertexe The first case is illustrated in Pigs 10, for a point within the triangles The potential on the airfoil surfece at point P = (x, y) will be the integral over the region between the leading edges end the Mach lines going forward from P of the uniform source distribution, This integration is most conveniently carried out in three parts, corres« ponding to the regions (1), (2) and (3) in Figs 10. Since the pressure field is to be calculated for a triangle with unit slope, the potential iss B (x, y) = Ty + Ty + Ins Bays where | 5s ; ~_ . oF L, = # fee [ ts2) ds | I, ~-ky Y ¥) o tky I, =-Z2 fly 5 7) 0 | sla was Ye) with 2 2 2 — fz. Hen)= fle-s)*p*ly-2)* f and : | F, = z= By =a) 5, = x+ Bly +f) The points Vy and Yo ore given by yg = e+ By = yp end ky, =x Bly = yy) ore = xtB _ + fe Yo. ae yy, * core (71) Iutegrating these expressions once, one obtains: | 0 ¥, Lo = FZ / dg cos 7 FE } é 77 44-7) L | J, ~@ 0 2 ow -/ Ltrke = = J ad cosht bly -2) Gr 7 L-k , CO / - I = 7 [a 0s4 ° . o ee 20g) AE ws _~ x fae cosh i aad 4 7 WT {BC4-#) adh. Go _ TZ ost - is = 7 Z fy . 3l4-2) ky F = — z fa cosh 2 ke (72) These integrals may inmediately be differentiated to find the uevelocity. Since the integrands vanish at “Vy oY Yo, we obtains AW PL 77 Vigo £4) —-£ (g-7) Gg. 2 dy _ & 7 I fame aya)* (73 j These integrations can be carried out easily if one notes that (Le ko)*- ¢%9-9)*= Ke +4 4 y lee) tleetel* & pe* (rk) tga) = ( lena 0 eR) Silk- ky) ° ke 2 8 * (74) fle Denoting the integrals of equation (73) by I, and I, respectively, these becor me ¢ Oo r or he! kx 4/84 cetera | z= - (a7) “ 7 [eB felix + ky) 4, Fe T= eZ 4 cosh” cen Ae leg" | 2 7 (k* 82 blx-kg) 2 At the limits y, and the inverse hyverbolic cosine terms Vy Vos Ni vanish, sc that one finally obtains: ee [sxie pcs AX-BY a - Tee is ae, fale- are ty) (75) It is clear, of course, that this solution must be essentially econleal, that is, the velocities and pressures must be constant on reys through the vertex of the triengliee Therefore let t= xy /x so that t = 1 on the leading edge. It is alse clear that the solution must depend essentially on the ratio k/B, se we shall let n = k/B » Thus t = non the Mach wave from the vertex of the triangle. This factor, n, might be called the "relative sweepback", since it is essentially the ratio of leading edge sweepback to the “sweepbacle" of the Mach wavese The. present caleulation has required n> le Then 2 -/ nr ue 2 - _ oF _v LOS. Sf AFC + Coste Ae 2k 773 Vr |} allée) nli-t) (76) Now if 2 + Ca +t cosh a = — im » cosh B = aot a@l + &) n(l = +) then it can be easily proved by elementary identities that ash *tB /— (€/07)* 2 y /—4? The final expression for u may therefore be written: | 27 bl | tat - COS ——_a> , ©" ~ wel) /— ¢? (77) As t approaches 1, thet is, the leading edge of the triangle, the disturbance velocity has a logarithmic eingulerity. Since in this case, with n> 1, the leading edge of the triangle is "subsonic", this behaviour might well be expected. The expression also becomes complex for 1 <t < ns thet is, for points between the leading edge and the Mach wave from the vertexe However for the case n> 1, there is also a disturbance in this region which is required in the airfoil drag calculations In erder to verify the form of the pressure distribution in this region, it is necessary to make e calouletion similar te the preceding for a point off of the triangle, but behind the Mach waves. This situation is illustrated in Fige lie The integration of the source influence at P may be carried out in two parts. Again, for the trisngle of unit slope in the x direction, 0 6 6&-Blg-@) y P= - 2 ay = 7 [(v-=)*—- Blg-@)* Ye ty 4, t-y-2) o | J As 7 2 a | SE (78) oO +ky where - x™ By _ xo B Vy kop ? 92 “ cP Q A __2 pl tthe Oy ag! HF d 2/ 2 ply) WY" alge) Y, a Ginee the integrands again vanish at ¥z 8nd Yo, the derivative with respect to x becomes: a) un 2b- - 2 / ae “ "4 Vlxe kg)? Bly -e)* - Fz. ap -2/ Re A ilg¢-#)* (78) 6 A transformetion essentially the same as that in equation (74) may.now be made. in order to obtein the proper value of the integral, it must be remembered that now x < ky, and kx > 6“y, so that the sign of the lest term in parenthesis in the second of equation (74) must be reversed, The result is then cbtained that: uzr-ZZ —— cosh’ bxeey + COSI) (Ee | a kg? Ali tk) (£S (£¢-k) or tt = _ oT a [cosh +2 - LOS: sty = “/ nae — “f 778 Un*/ nte) ea) (80) As before this expression may be transformed tos _ Zz Cosh nit (81) 20 / | > ad a Wal / ~7—/ in which form it is real for 1 << + < ne This expression again has e logarithmic singularity as + approsches i, which matches the singularity of equation (77), but it approaches zero as t approaches n, that is, es the point P approsches the Mach waves Rquations (77) and (81) complete the description of the pressure field for n> 1, that is, sweepback behind the Mach line, or subsonic leading edges It is now necessary to calculate the disturbance field associated with the triangle having k < 8, orn < 1, ises, a supersonic leading edgs. This situation is illustrated in Pige 12s Fork < 6, there igs no disturbance ahead of the leading edge of the triangle, so the calculations are eoncerned only with points on the wing. However here, again, two types of points must be distinguished, Points such as Pye Fig. 12, lying between the leading edge and the Mach wave M, (l>+t>n) have ® potential determined only by the sources lying within the cross~hatched region (1), But this potential apparently must be the same as thet fors point on an infinite wing, since the vertex does not lie in this region, The disturbance velocity at Py must thus be equal to that calculated for the infinite wing in Part III, and from equation (43) one observes imme diately om T i Yo (88 At point P, the influence of the vertex is felt. However the potential at P, must be equal to the potential at Py decreased by the influence of a uniform source distribution in the cross#hatched region (2), or, for positive unit slepe, 2 (P,) = P(Py) a AD (83) where Ag = "S V (x-) *- 8-9)" =26— Writing this integral in two parts, we have: a Ye x-B(9-7) -ke Ap = - 2 [ee Ms njds - E fx fat gids (34) | — ky , G2 ky where | = , xo by - x= fy vy “SEak * V2 "EB +t Cerrying out these es there results: -£ aste~ Ag = £/- “ At - fal cash 1 K-£@ _ cosh % +k SY-2) ACY -@) o 4g = 2 fey cosh ~ #-*% +H Je coh Cos An! KekG KCG-9) UG-2) (85) cA The uevelocity associated with this potential may now be computeda The integrands vanish for Va =Y¥ 1 OF Yo, and the uevelocity is therefore o oO . da T ae - 2_(Ag) = - T 2 (BEY 46 = SW) =F ee ety-0)° Wier ke )J* e142)" These become elementary integrels if it is noted that 2 + Dy /E—K* (BUx- ky)” ki- Sy {3° —k* MB? k (t-ky)*— 8-2)" = _ - ee sépit 2 72 i die i Gk oe Ohl “ aa ( ; 87) (rk These expressions are essentially the same as those of equations (74), except that the signs of the terms have been chosen so as to keep the quantities in brackets positive through the range of integration, in order te obtein the principal value of the integral. Introducing equations (87, inte (86) and integrating, we obtain | 4 kx~ g ay & E | Awa = stig FE BF yp sia~ kx eka | (38) -~F Wf 187k? A(k—ky) Blk + kg) If the variable t = ky/x is introduced, and this equation is reduced by means of elementary trigonometric identities, it can be shown that coe 2 2 sin [| PUWE (89) 2a Ak = > Ss ) Tel /—n? /—- €* According to equation (83), the totel uevelocity at point P,, Fig. 12, will consist of the expression in equation (82) minus the increment in equation (89). Thus Tr. —— 2 2 ’ “ul(E) =z — {/—- 450 [eae | (90) 2 2 T —_ Zz Si l-11 =A Be As t approaches n, equation (90) will, of course, approach equation (82 jo We have now calculated the x«component of the perturbation veloe cities for both the case of n> 1] and n < l, and for points both ahead of and behind the Mach weve. The pressure coefficients are obtained immediately from equation (16), Ce = es 2u/t These pressure ccefficients, all calculated for wmit slope of the besic triangle, may be summprized as follows: Case le n>] a) 0 <+t< i (Pige 16) # cosh)" 2 £* Co = 173 f 17 / /— t* (91a) b) I < c < 8 (Fige 11) Cp > (91b) Pp Bfr—~n* (91¢) (914) The pressure coefficient for a triangle of slope A is, of course, obtained by multiplying the right hand side cf equations (91) by A. For these triengles of slope A y itt is seen from equations (91) that the pressure coefficient in the form Cpf/ A, or Cp V Wael nr, is e function only of t (position on the airfoil) and n (relative sweep= pack Je Curves of the pressure coefficients in this form are plotted in Fige 1% The case n = © will correspond to the case of the two-dimensional wing with no sweepback, for which it is well known that Cpf/A = 2. Values of t > 1 correspond to points ahead of the triangles It can be shown that all of the expressions for pressure coefficient in equations (91) are equivalent to the real part of a single expression, identical to (9la)s 4 | pe Eb Tape Of (92) For purposes of the surface integrals to be carried out later, however, and certainly for ordinary calculations, it is more convenient to have the real expressions cf equations (91). Be Surfs ce Integrels of Pressures in order to carry out the drag calculations, it is now necessary to calculate the surface integrals required in equation (68) or (69 Je The two basie types of integretione required correspond te parts (1) and (2) of equations (68); #11 the other integrals may be expressed in terms of integrals of these bY peso Consider first the integral over the region show in Figs 14 of the pressures due to sources in this region - lees, of pressure coeffie cient as given by equations (Sle, c, or dj. This integral will be required in the calculation of parts (1), (4j and (5) of equations (69). Since the pressure distributions are all conical in form, the integrations can be carried out over triangular strips, as shown in Fige 14. In terms of the spanwise variable t, and the geometrical parameters showm in Fige 14, the elementary area dA becomes: b-s)* dA= A, ho wt)® AL «os, where | | A= o®/eke = 0° /2fn Now let > / 5, (n, 5) = |tp 4A = ai arse cosh EL ote (04 A é | where A ig the srea of figure 14, and Cp is the pressure distribution Ta of equation (Sle), and n = k/Be The integrel in equation (94), ‘ 4 (/f* t* / 2 - _ o wale may be evaluated by an: integration by parts. Let 2 2 So = . Lo’ a—é - a Los ae: Aw /7— st)? tft. + |} = FI sé)” l-5) (-e*) i } (1-&) ~ Ast )(1-s) and This particular form of v is used in order te avoid an infinity at one of the limits of integration. Since 2 ‘ ’ hon Ut) cosh *[2 EL binr (t-t) log (-) = /-t2 fog Eos we have 7 / oA ( -/ Vi77- . 7 = cS cosh 7 4. oe | PERE 72 (96) 6 ' Since t / ft (- st Nie €) tes (—st (+E the integral in equation (96) can be written as I = 1, +1,, with / roe te. Lo Se fees J lH st) [n@ er a / 3a _ oo — AL Uoon substituting t =n sin OY _ these integrals reduce to elementary forms , 6 ; 6, , ~ @-64. / r = fo - (og ; IFS J /- JS 7 sin 6 +s) [sn] 64s Oto p) 2 d 6, . / L, = ; oo 2_ 2 ft+s/ /+n sind - OF 6-8, (97) oO 2. Oo where -l ~~ _/ = S¢ — = 54 6, va) , 6, SH The second of these two integrals is simply —/ Lo * Vins) [ne (ag (1+ V1 — Un}? } (98) The appropriate form for the evaluation of the second integral in (97) will apparently depend on whether sn = sk/B is greater or less than 1, ise., whether the trailing edge of this region is ahead of or behind its Mach waves In the ease that sn > 1, it can be written Unt) - stan. [s*n®/ oT My [22 a oa / hb / 2/sin@/ - | +s 577% 7 4 4/nr/ ~s1 —fs*n?-/ (99) while in the case that sn < 1, it can be written — S+7 ES + Tan _ 2 tp ~/ ale Jn sn Vase G — s7r7* ) ya TF n> ne [ne — sh? (100,) When introducing I, into equation (96), one may observe that Ap /, i+ cosh = 0g (1 + V7— U/n}*) on SS vosh ny 4 log 1) (+s Vs Finally, using equations (98) and (99) or (100) in (96) and (94), the surface’ integral S, may be evaluated as FA, where, for ns > 1 jas [ s_cesh'n /o (Gln, s) = / - 4+ S56 Vr77~/ Jnr / L (0g [0 os 2 Jsn=/ / am A+ f(ri-( —~ sa—f/sne/ or, for sn < 1 Ss cosh 5 cosh 4 leg la} les + 2 (Le s* ) Ji nts? h+Jn2-/ -hAS The same type of surface integral must also be caleulated for the ease that n < 1, iste, for the pressure distributions of equations {91lc) and (91ldj. Since these pressure distributions are actually the real part of the expression in equation (9la), as mentioned previously, it is clear that the desired integral could be obtained by finding the real part of equation (102) or (103) for n < 1, It is, however, easler to calculate the iategral directly. If Sns)= [tx, 44 A / / 4(1-S)*A, / wT .-t/net* 5, (4, 5) = wa liane Lf [Zain Te fe (104) (a4 then the integral will be i yy no! a ne £7 Zz At Ef - [=P ee (108) The first integral is immediately: . wil a 2 (/—-s) =5 5a The second integral cari be evaluated by the same technique as that used in equation (95), with . 1 Jp® #2 _ At wom Te A Ost from which / wilt _ V ee |, tae “ (t- st Nitt}i n?— t? 2 4-S 7 The integration of this last expressions proceeds in much the same way as the preceding computations, except that the principal values for un < 1 must be computed. Carrying out this integration and combining terms, one finds that | / nl m_ Ver? 7 Z-als 1+ FB eee (00 S47 £)/ Combining the above expressions for I, and I, in equation (105), the value of T becomes: Ss -/ Ji-n> —_ 7 = eS +e f Va (sin “sn + ZL ) (106) /-S (-s° ~/-s717 2 If the surface integral for a < 1 is written in the same form as equation (101), with , | FA, | Sn, 5) = ma 2 s) (107) we find, using equation (104), thet _ A - «/ G, (h, 5) = ies lia 1 +—==5(2 + SU‘7 sn) | (108) The first basic surface integral, as given by equation (94), has thus been expressed in three forms, 5 (ns) = 74 Bins), for n>l, ns>i 7/3 / = —— G, (n, 5), forn>i1, m<1 7 FA ge (n, s), forn<1, ns<1 u/s (109) 4 with Gy. G's and G, given by equations (102), (103) and (108) respece tively, The three expressions are, of course, completely equivalent, and can be derived directly from each other. Their equivalence at the various limits is seen immediately, for atns = 1, ah’ n log , 2 A) Gla, zs ~ P41 ner (ee * r=) ” ple (1 (110) fs n approaches 1, the expressions in general become indeterminate, but it is observed that cosh hy oe log h7 hind = f hint = @ n»y VIET ; Ny V7} and that A ISAS con 2 Tar jos = W- sin’ /- 57 = 7 — cos)! os = oa 4 sta 5 The values of G as n approaches 1 then become lira Gln, s)= lim G, ins 5)= pg )t phe (E+ sin ls) (112) hi->/ h->/ A particularly important limit occurs as s approachss O for n < le From equation (108), linn GG, C47, 5) = + S>d that is, the surface integral becomes independent of the sweepback, me The other necessary type of surface integral is that corresponding to part (2), (3) and (6) of equations (69). Consider the area B shown in Pigs 15, bounded at the front by a Mach wave from P. The integral B of the pressure due to a uniform source distribution in the triangle with vertex at P must be calculated. In terms of t = sky/x, the elementary area will be Ptetate) Ui- s)~ dA = A, (ins f aC where A, = se® /2k = se” /2fn The pressure distribution is given by equation (91b), but with n replaced by snes Then we may write Sz", 5) = |b, AA 8 sn = 40-3)" ai _/ cosh [Sn —! sin*/ ae 7 | s37 Me-5f* sf Vv eT (118 If one designates the integral in this expression by T, and integrate by parts, as in the previous caleuletions, with -~/ 22 AL tt = 4 [Ss ¥I°~/ = —___—_ 7 2,2, (£-/ du =— EVE? ae Aw = ; (e1)[sn® 0? ? OMe~s) then 7 Vs7n 7 t AL 114) J (e+iNe- s)f/s*n*% zt? 4, This integral is again. separated by partial fractions inte qt and Los with S47 re /[ dt tts J tt) (sn €2 , / r _ Ss Jat 42> SS (es) [sb (115) These integrals are, as previously, elementary forms, and are evaluated asz a c 7g =" snr fin®i(s*n®/) / = 1 42 G+ s) fay FS MCl-Ss) ~ (116) Writing equation (113) as 4 | S; (7,5) = 73a A, Fln, Ss) (117) one then finds thet fas (og St? F(n, s) = hs / en 7 / / snl CeNen) f © Tae, F Nli-s) wGQes This expression has. physical significance only when both n > 1 and si > 1, It goes to zero, of course, for sn * ly This completes the caloulation of the fundamental pressure distri« butions and integrals of the pressure distributions required for the calculation of the drag of a delta wing, To summarize, the various expressions for the pressure distributions are given in equations (91a) through (91d); the two types of surfaces integrals, illustrated in Figse 14 and 15, ere given in equations (109) and (117), together with the expressions for the related funetions F and Ge Plots of F(n, s) and G(n, s) are given in Pigs. 17 and 18, The G function is continuous in derivative across the line sn = 1, even though the real expression changes form. However the real part of P(n, sj) vanishes abruptly, with infinite slope, at sn = 1, This is to be expected, of course, since the pressure distribution acting ehead of the triangle in this case has an infinite pressure peake This singularity will be seen later to have an essential effect on the character of the drag curves for sn * le The curves of Figs. 17 and 13 ean be used for approximate calculas tion of the drag coefficient components of equations deseribed in the next section as long as none of the parameters are near any “eritical" values, © ie@e, M, SH, are not near O or 1, As any of the discontin= uitiss are aporoached, the drag expressions involve small differences between large quantities, end more accurate calculations must be usede oaG lee PART VI DRAG COBFFICIENTS AND PRESSURE DISTRIBUTIONS FOR THE DELTA WING he calculation of Drag Coefficients, Double Wedge Profiles The scheme of calculetions set up in Part IV, as summarized in equations (68) and (69), conteined surface pressure integrals which we c@n now express in terms of the basic integrals S4(n, s) and So{n, ¢) calculated in the last section, It is necessary only to relate the geometrical parameters of the basic wing, as shown ‘in Pige 8, and equations (63) and (64), to the parameters defined for S, and Sp, as illustrated in Figs. 14 and 15. It must be noted that we will use a single relative sweepback parameter, referred to the leading edge, n = ky/f o We then have, by inspection, for the integrals required in equations (68) or (69), J G, dA = Sh Yr) zr J Chg 44 ~ 5p 7, 7) r J Ce, AA = 5, (7,4) — 5, Cr, 4) [<n 44 = S,(n,a) =~ S07) S Ca AA = 5 (rn, &) a Se dA = By (rn, = | y C (119) The expressions for S, and Sp as given in equations (109) and (117) may now be introduced, and the result substituted in equations (69) to obtain the finel expressions for the drag coefficient. It must be noted that the integrations for s) and 8, were, in effect, carried out over only one half of the airfoil, so the area S appearing in equations (69) must be interpreted as one~half the totel area. Thus in (69a )3 AS r/(1 ~ a) In (69d): A /s = y/(1 = a) ' and similarly for the ether terms. The complete expressions for the components of the drag coefficient of the twows lope or double wedge delta wing then become: . 2 (j- C — 22 /-@) Gln,r) a, = 7737 (/- r)* (1208 j c a 27° rli-a) Flr7,r) (120d) "8 37 (l-r)* (r-a) | 2c? alli-a) - O,¢ = am (ra) AE! [Vy a)— E(rn, <)f (120¢) FF - BF al [iain a)— Gln, r) | C . ?aA BU U-rflr-4 (1204 ) 2 274 r(/-a) Grr, 4) (1200) °re” en -rXr-a)* 2t* ali-a) A COrc = Ba [r—a J? Err, Pr ) (120f) The totel wave drag coefficient, based on the actual airfoil plan area, is the sum of these six contributions, The expressions for G(n, s) are given in equations (102), (103) or (108) depending on the values of n and s, while F(n, sj is defined by equation (116). We my recell again the significance of the parameters appearing in these equations:. n is the basic "relative eweepback" parameter « since n= k,/#, n gives the sweepback of the airfoil leading edge relative to a Mach wave from its vertexe The parameters a@ and r give the relative geometry of the planforms r defines the maximum thickness lecation, as a fraction of total chord, from apex to wing tips. It will also be convenient to express the location of the maximum thickness point as a fractien of the actual root chord, say b, in which case, from Figs 8, (121) The number & gives the sweepback of the treiling edge relative to the sweepback of the leading edges as a —»1 one obtains essentially a two~dimensional swept wing or arrow winge It is noticed first that the totel drag coefficient resulting from equations (120) may be represented in the form Cy P/ ra 2 in which case it is a function only of n, a, re Thus the drag for all Mach numbers and geometries may be represented in a reasonably simple family of plots, If, for instance, CyP/ 7 2 is plotted against 1/n, this parame ter , for & given sweepback, becomes proportional to // we «1, and the plot will immediately show relative variation of drag with Mach number e Before proceeding with discussions of the final calculations of drag coefficient, it will be desirable to calculate one simple limiting case, As n —» 0, we have either a swept wing at an indefinitely high Mach number, or & wing with zero sweepback ~ ise., an essentially two dimensional wings In this case, the contribution from the terms in (120b), (120c), and (1Z0f) is zeros. Furthermore, as n 0, we have from equaticn (105), a(o, s) = S(l=s) Substituting this in (120a), (120d) and (12Ge}, there results: inn COB _ Cina) ier) — 4-4) Cy, Neo FT? li-r}* | O-rXr- fa fa 4) - “rf rll-a)* A * Wr)irca)? (/- *) I Using equation (121) to express r in terms of b, the location of maximum thickness, this becomes: | / ' C, lirn Of Lg 05 n~o Cc? 4-6) (122) But this is exactly the expression for the drag coefficient of a twoedimensionel double+wedge airfoil, with crossesection as shown in Figs 19, The eptimum lccation of maximum thickness for this airfoil, as is well know, occurs with the peak at the 50% chord point = i.e., b = 0.5, and then —~ 3 = 4 (125) ~66- Thus the equations (120) reduce te the correct vaiue for the case of zero sweepback. Moreover, we have learned also that at extremely high Mach numbers, such that the Machwaves are swept well behind the leading edge, the drag coefficient again approaches the value pertain~ ing to e twoedimensional wing with the same lecation of maximum thick« ness, independently of the sweep of the delta wing trailing edge « 6 rather remarkeble result. This also suggests ‘thet a result of essential interest in the case of the delta wing will be the location of the maximum thickness point which will result in least drag for various Mach number rengese From the above calculation, it appears that b = 0.5 will be optimum for very high Mach numbers; it is certainly unlikely that this will be true at all Mach numberse In order to examine this, in Figs 20 is plotted Cy8/z 7 vse b, with nas parameter, for the case of zero trailing edge sweepback (a =O). It is noticed first that with O <n <1, for ail values of b, the drag of the delte wing is greater than that of a twoedimen= sional wing; this case corresponds to case (1) in Fig. 9a. As the sweepback is increased (n increased), the drag begins to drop, and one observes the large discontinuity in drag curve slepe at the point r=(l«b)= ifn, ieee, when the Mach waves are parallel to the maximum thickness line, This situation forms the dividing line between eases (2) and (3) in Figse (9b) and (Se). It is only with the maximum thickness point forward of this position, isees, b < 1 1/n, that the drag finally dreps below the value for a twoedimensional winge =O 7a A remarkable and apparently fortuitous result now appears; for values of n greater than approximately 123, the minimum drag occurs with the mmexci mum thickness line somewhere between the 10% chord and 20% chord positions, practically independently of the sweepback ne This means that the maximum thickness line is swept back approximately 80% as much as the leading edgee With this location of maximum thick= ness, the drag drops rapidly with increasing n; for n = 1.4, the drag has decreased to the minimum two-dimensional value, and for n = 2.0, it is approximately one«half of the minimum two-dimensional value. With increasing n, (greater sweepback er decreasing Mach number ) the location of maximum thickness point becomes less and less critical. The corresponding variation of Cyp/z with b for the wing with trailing edge swept 50% as much as the leading edge, ices, & = 005, is plotted in Fige 21. The results are again qualitatively very similar to the case of a = Oe The important drag break now occurs, of course, at somewhat more rearward locations of the maximum thick«= ness, since rn = 1 corresponds to b = (n ~ 1j/n(l «~ aj. However the minimum drag again oceurs with maximum thickness located between the 10 and 20% chord points. It is significant that the dreg drops much more abruptly with increasing n in this case; at n = 1.3 it is appreciably belew the minimum twoedimensional drag, while at n = 2 it is almost 1/4 of the twomdimensional value. Similar plets have been made for a *~ 0.25 and a = 0.75; in both these cases, & similar result concerning optimum location of maximum thickness point for high n's is observed = the minimum drag is achieved with b in the range from 10 te 20% of root chord. As the trailing edge =6 8a sweepback approaches that of the leading edge, ieee, as @ approaches 1, it is hardly to be expected that this result would still be true. The wing in this case is approaching & two~dimensional swept wing, and the optimum location of maximum. thickness should be more nearly at the 50% chord point. Bs Effects of Sweepback on Drag. The effect of variation in sweepback on drag is seen more easily in plots of Cy8/t # vs. i/n; in Figs. 22, 23, 24, and 25, these plots are given for a = 0, 0225, 0.50 and 0.75 respectively, with bas para= meter for each. The real character of the peaks occurring at the critical values of n and rn can now be seens Consider first Fig. 22, for a= 0. At low values of 1/n (high sweepback or low M) the superiority of the small value of b is evident. It might also be remembered here that the minimum drag for a twowdimensional double wedge, with maximum thickness at 50% chord, gives CBr et = 4, which furnishes an indication of the significance of the levels of the curves. As 1/n is increased, the drag curves approach a first peak at rn (in this case (1 « byn, = 1, at which point the Mach waves are parallel to the maximum thickness line, Sometime before this point, the delta wing loses its superiority over the two=dimensional wings Another essential effect becomes evident here; the smaller values of b will, of course, delay the occurrence of this peak to a higher Mach 6 Qu number, but the height of the peak itself increases. Thus the forward lecetion of maximum thickness is clearly an advantage only if the design operating condition for the wing is to the left of the peak; if for any reason the design condition must lie to the right of the peak, then a value of b near 0.5 would be advantageous, in thet the peak itself is reducedg | | A second peak occurs at "= 1 with all values of b, iee., Mach waves parallel to the leading edges Again, the height of this peak is increased by forward location of maximum thickness point. The drag decreases for greater values of 1/n, and as U/n approaches infinity (large M) the curves approach asymptotically the value given in equation (122) for a twoedimensional wing with the same maximum thickness location. In Fige 23 are shown the corresponding results for small sweepback of the trailing edge, a = 0.25. The general character of the curves, and the relative drags for various values of b are much the same, An additional discontinuity has entered at an = 1, i.e., when the Mach waves are parallel to the trailing edges The drag rise occurring at this discontinuity is, however, very small, and not particularly impor- tants. The next peak has been delayed, but also increased in magnitude by the trailing edge sweepbacks The "leading edge peak" (n = 1) is particularly increased in magnitude over the case with a =0, For large values of 1/n, the curves approach the same asymtotic values as in the a = 0 cases | The trends already exhibited in Figs. 22 and 23 are continued with the larger values of trailing edge sweep shown in Figs. 24 and 25, for & = 0.50 and 0.75. In these cases, the value of the drag coefficient woe for Low (1/2) has become so small compared to the two-dimensional wing that the location of maximum thickness point actually becomes relatively unimportant. In fact, in practical cases, for the thick ness ratios, 2 , of practical importance, the value of the wave drag coefficient in this range of sweepback will be of the same order ag oresmeller than the drag due to skin friction, On the other hand, the drag peaks, at a = 0.75, for instance, now become so violent that the design condition for the airfoil must clearly lie at low values of 1/ne As shown by von Karmen in Ref. 22, the corresponding drag peak for an infinite two-dimensional swept wing is infinite in magni» tudes To clarify further the effect of trailing edge sweep for constant location of maximum thickness point, in Figs, 26 and 27 are plotted CyP/ Zz 2 ws, ifn with b = 0.2 and 0.5 respectively, and with a as parameter, With Porward position of maximum thickness (Figs 26), the amount of trailing edge sweep has an important influence on drag for all values of n, decreasing it strongly to the left of the peak, and increasing it through the peak, With the 50% chord location of maximum thickness point (Fig. 27), this effeet becomes even more pronourceds The important conclusions from these two sets of curves are easily summarized: | a) Moving maximum thickness forward decreases drag for small values of ifn, although at very small values of 1/n, and partie eularly with large trailing edge sweep, it is not so important. On the other hand, the drag peak, although delayed, is greatly increased = 7 le in severity. b) ‘Increasing trailing edge sweep has a similar effect, but somewhat more pronounced. . ' The above representations of the drag variation, while quite general, do not allow immediate visualization of the variation of actual arag coefficient with Mach number, since this is partly included in the factor B = w= 1. In order to present a clearer physical picture of the behaviour of the drag coefficient in typical eases, the funetion C/T 2 is plotted vs. M in Fig. 28 for a family of airfoils with no trailing edge sweep, with maximum thickness at a constant 15% chord, but with varying leading edge angle. The sweep angle is there replaced by the apex halfeangle, @,, These are, of course, obtained from a single curve in Fig. 22. The shift in pressure peak location and height, and the change in low Mach number drag coefficient with increasing sweep is clearly seens The behaviour of the curves near M =~ 1 should be particularly notede In Figs 22, the curve of C,$/Z 2 is seen to approach 0 as 1/n approaches 0, so that c/r? is in this order indeterminate. The calculation of this limiting case is outlined in the next section, and it is shown there that ty Zz 2 has a logarithmic approach to infinity at M= 1 for a*=0, This is, of course, much less severe than the one half power approach to infinity for the infinite two-dimensional wing. The effect of peak location is also interesting when C,/ Tc 2 is plotted directly, In Fig. 29 the drag coefficient is plotted against M for a family of airfoils with constant leading and trailing edge sweep, (@ = 30° and a = 0), but with varying location of maximum thickness, The shift of the pressure peak to higher Mach numbers, and-its increase in height with decreasing b are easily seen. In Fig. 30 are drag coefficient curves corresponding to the airfoil family of Pig. 28, but with trailing edge sweep of a = 0.5, and b* 0s2. The general behaviour for low Mach numbers is the same as in Fige 28, with the important difference that the drag coefficient approaches a finite limit as M approaches 1. This limit is again indeterminate to the order of the curves of Figs 24, butit is obvious that the limit is finite, as the curves in Fig. 24 approach 1/n = 0 with finite slope. The evaluation of this limit is somewhat lengthy, and is carried out in the next section. In Fig. 31 the effect of maximum thickness location on o/c * is exhibited for the family of airfoils with 30° apex halfeangle, and a=0.5. The characteristics are much the same as Fig. 29a in order. to compare more clearly the effect of trailing edge sweep, Fige 32 presents curves of c/u* for wings with 30° ana 15° apex halfeangles, and values of a = 0 and O.5e The similarity between the effects of sweeping maximum thickness line and sweeping trailing edge is again evident. The character of the above curves, and in particular of Figs. 29 and 31, suggests that there are two ways in which advantage may be taken of the deltaewing drag properties, At reasonably low Mach numbers, the leading edge may be swept far enough so that the drag peak lies at a Mach number considerably higher than the design point, and with proper forward meximum thickness location, a substantial reduction in design point drag is possible. On the other hand, the design Mach number may be so high that it is impractical to sweep the wing back far enough for this. In this case, the maximum thickness point may ‘be located near the 50% chord point, and the primary effect of the delta wing will be to cause a very small drag rise through the low Mach number region, as compared to the twoedimensional wing, and to approach finally the two-dimensional value, The drag peak is then very small. This application may be the more important of the two for very high Mach numbers» Another rather obvious consideration must be pointed out, which in an engineering problem may have decisive influence on the choice of optimum configuration. The representations so far have been of 0,/t 2 where @ is the thickness-chord ratio. Ina specific design problem, the actual wing root thickness will be determined by structural considerations, and presumably will not vary greatly within a reasonable range of configurations, However the change in configuration for a wing of given ares, or given ac, Jat, may involve considerable changes in wing root chord and therefore in TZ . It is actually in this way that the low aspect ratio (large sweepback delta wings often achieve their greatest advantage, For the low aspect ratic generally implies much larger root chord (and often smaller absolute thickness) and therefore decreases TZ materially» Since the drag coefficient varies with zr, the effects may be very lar ges The comparisons of drag coefficient discussed above then becomes essentially relative in character, and it is clear that the real choice of optimum configuration must depend largely on the special design parameters or conditions in a particular engineering problems aa 4m Ce Drag Coefficients and. Drag-Gtrength Ratio for Modified Double-Wedges . A further extension of the above drag calculations which has parti cular engineering importance now suggests itself. In the case of an infinite two-dimensional wing, the profile with least drag for a given thickness is the familiar double—wedge section, with maximm thickness at the 50% chord point, as previously described. However this is not, in general, the optimum profile in a practical problem; the usual design problem is to fina a profile of minimum drag with some other section property given » either stiffness (moment of inertia) or strength (section modulus), The variation in profile may then include variation in thickness ratio. The section property must even depend, then, upon the type of structure chosen - whether, for instance, it is solid, or essentially monocoque, or a combination, it was shown by the present author that for a two-dimensional solid airfoil, the least drag for given stiffness is obtained with an airfoil having a prefile defined by an elliptic function - actually very close to a circular arc. For & monocoque structure, of given stiffness, the optimum section actually is a bieconvex circular are profile, Optimum profiles for given section modulus varied only slightly from this, Only if the structure is concentrated primarily in a single spar at the maximum thickness point will the double=wedge section be optimum, These caloulations suggest that a) the doubleewedge delta wing very probably may be the optimum profile for given thickness ratio, but that b) the optimum delta wing profile from the standpoint of a likely engineering requirement will probably have a continuous curvature and more distributed thickness. The canplete variational problem cen be set up, much as was done for the two-dimensional wings However, in order to avoid this generality and still establish the trend of the desired change in thickness distribution, a simple family of profiles will be considered which has one more parameter added. This prefile, called a "three slope" profile, is shown in Fige 33. The slope of the central part of the airfoil will be held zero, and the location of the forward and aft limits of this zone, ie@e, band a, allowed to vary. Only the case of zero trailing edge sweep will be considered, The scheme of the calculation is essentially the same as for the two slope delta wing, and will be indicated briefly. Referring to Pig. 33, we will superimpose three triangles with vertices at A, B, and C, having slopes Aad Ag = 7 4, Ae = A A / X N Furthermore, - £ _/ ~~. 2 ft A,= 3 (-r iy Az Se 2a Using the surface integral designations of equations (94) and (113), Part V, and the drag component notation introduced in equations (68), Part IV, we can immediately write = 2 A, Az SC, r) D z = 2A, Ac (200, @)-20rn, 4) D ‘2A = 2A, A, [5, tn, 0) —.5, Sn, a) Dne - | “ee - 2A,A, Slan, 0) Substituting for the slopes, A » &®S given above, and expressing S and S, in terms of F and G according to equations (109) and (118), we have finally: —_— 2 22 / Gin, r) ,, * Baw (—rj2 C - _ 27z? r — C = 277 &. Ore or — [etn a)—Firn, 2 )f C - . 27? _ aA adh Wen) (etn o)— G&n, 4) § Come = 22° [ater 0) — Grn, 4) § id Amt ater) “ Co, = 22° 1 Glan o) LIC — A A (125) 7 Too Caleulations of the drag coefficient have been carried out for one particular value of relative sweepback, n = 2.5, and for various values of a and re The variation of drag coefficient with position of this two maxim thickness points is shown in Fige 34, where Cy8/z * is. plotted against (1 « aj, with b = 1l=r as the parameter. As (1 =a) approaches b, the airfoil is identical to the two slope delta wing with zero trailing edge sweep considered previously, so that the end points of the curves for various b's will produce the drag curve for n = 2.5 in Fige 20. It is immediately clear from Fig. 34 that for any location of Py 2 that is, the double-wedge airfoil does have the least drag for given the minimum CyP/ T 2 will oecur when P. is identical with Pye thickness for this airfoil family. However , rearward motion of Po eauses only a small increase in drag « for instance, with P, at 10% chord (b = 0.1), Py can be moved aft to the 30% chord point with only a 30% increase in drag, At the same time, it is clear that the strength of the section must be increased greatly by this changes The section modulus, &, for this profile, assuming it to be solid, is given by z? SI = ef co (l+ sr 3a ) ~ fe 2F (126) In order to find the airfoil of least drag for given section modulus, we may plot the ratio CoB/T7 oR s/t S/e =78e which does not involve the thickness ratio. In Fige 35, this ratio is plotted as a function of 1 » a for b= 0.05, 0610 and Os15. For each of these curves, the improvement in drag is very significant as F, is moved aft, with an optimum for Py at approximately 40% chorde The best drag=strength ratio is obtained for the value b = 0.10 and approximately (1 = a) = 0.46 At this point the dragestrength ratio has decreased approximately 19% below the value for the two slope wing with b = 0.1 or Oe2. Thus for a solid three-siope or modified doubleswedge wing of this family, the optimum profile will have a maximum thickness region distributed between approximately the 10% chord and the 40% chord stations. The resulting design is certainly more attractive from a structural standpoint in any case, Consideration of a more general family of shapes would undoubtedly lead to better profiles, but this first step clearly indicates the trend, namely, that the thickness must increase very rapidly near the leading edge, but then may continue with out much decrease to the neighborhood of the 40 or 50% chord point. The location of this second point will, of course, depend on n, the leading edge sweepback, and will be higher for larger values of ne De Pressure Distributions Over Double-Wedge Wings. (The pressure distributions over the delta wings sre of some interest, and shed some light on their possible behaviour in a real fluid, These distributions are easily calculated, of course, by proper superposition of the basic pressure fields of equations (91), using the source strengths defined by equations (66) and (67), The two important types oceur when (ajn < 1, and (oj) n and rn > 1 (cases (a) and (cj, Fige 9). wf Gee “An example of the. first type is shown in Fige 56, where the chord= wise pressure coefficient is plotted in the form CpB/ Z for several spanwise stations, with n = 0.8, and b = 0.5, i.¢., maximum thickness at 50% chord, and no trailing edge sweep. The pressure remains constant for a short distance behind the leading edge, with the typical "supersonic leading edge” characteristic, and then drops rather rapidly. When the maxima thickness Line is reached, a large drop occurs, with no pressure pie until the Mach wave from the maximum thickness line is crosseds Thus all pressure gradients are favorable up to this Mach wave, which lies well aft on the wings In Fig. 37 are plotted similar chordwise pressure distributions for the case n = 2.5 and b = 0.2, with no trailing edge sweep. The leading edge now has the subsonic (logarithmic) positive pressure peek, . but the pressure begins to drop very rapidly as the maximum thickness line is approached. At this line ® negative peak occurs, followed by a very rapid rise, This behaviour is expected, by analogy with the subsonic flow over such a twowdimensional corner, in which case it is first accelerated and then decelerated very rapidly. The important point in the present ease is that the maximum thickness line for the twoeslope airfoil will always introduce a large adverse presswre gradient’, in cases when this line is swept behind the Mach waves. This gradient may have undesirable effects on the boundary layer. A profile with con= tinuous curvature would certainly be more desirable in this respects E. Lift for Delta Wings With Supersonic Leading Edgese One further application can be made of the calculations presented previously. In Part Il it was pointed out that the source solutions can be used to caleulate the Lift on airfoils with "supersonic" leading a30« edges, since the flow in the upper halfespace and lower halfespace may be calculated separately. Imagine a flateplate delta wing with n l, of zero thickness, at angle of attack & In the upper half-space the potential will be essentially that given by equation (83), which produces a constant downwash over the entire triangle, and the pressure distribu= tion will be | - kL 1/17. z* - )— 8 _ 0° alent CF [Fe J om This distribution is shovm in Fige 135. In the lower half-space, the potential is represented by the negative of that in the upper halfespace, which reverses the sign of all perturbation velocities, and therefore gives a solution with downwash symmetric in 2. The lift will then be twice the integral of the pressure distribution of equation (127) over the surface, But this is exactly the integral S,(n, s), carried out for a triangle with unit slope, and given by equations (109) and (108), Rewriting equation (108) slightly, and using equation (109), the lift coefficient becomes s Pa (-4) - / C 78 (1a) (Ame cos 7 * Fat cos (-an) } (128) where & is the trailing edge sweepback parameter, as in Figs 8, and n is less than le For the case of no trailing edge sweepback, (a= 0), it is observed (as shown in equation (112)) that for any n, we les C= (129 ) This is exactiy the lift coefficient on an infinite twowdimensional “ uniswe pt wings Thus we have the remarkeble result that for a delta wing with leading edges ahead of the Mach waves, and ne trailing edge sweep, the lift coefficient is independent of sweerback, and has the same value as the two-dimensional wings By sweeping the trailing edge, the lift coefficient can be nade much greater than the twoedimensionoal values Curves of this lift coefficient were presented in Ref. 23, together with the lift coefficient curves for n > 1 which were caleulated with the conical flow technique. FART VII LIMITING CASES The behaviour of the dreg coefficients previously calculsted is of interest in several limiting cases. In particuler, its behaviour as the iach number approaches 1 can be caleculeted. The case of the wing with no trailing edge sweepback will be considered first, since this does not form a simple special case of the more general problem for a not equal to zeros | The drag is given by equations (120), with CDre = Corse = and a = 0 in the other drag components, As M approaches 1, with finite sweepback factor k, then B= \/M’ = 1 approaches 0, and the parameter n = k/B approsches infinity, Therefore the limiting forms of G{n, sj) and F(n, s) for large n must be computeds Noting thet for large x, mm? ; eosh *~ x & log 2x = log x, one finds from equation (102) that G(n, s} approaches the value /- 5 s log fl lag "7 / | G (ns) — a yn 7 + Bz PTs > ((-s5) Cg” (1s0) where only terms of the order (log nj/n have been retaineds From equation (118) one also finds thet as n —» oo ? [-.S5 log -¥ / so # sr* En, 5) ae 5 «+ log Wlr-s) se SS fog n (132) s va) to the same order of (log nj/ne Substituting these in equations 120a, b, d, and 6, with a = 0, one obtains: 26 mw _, len fir (iH-r) ort Ze 2 DW bUHr)*® pGery® WiH-0) ~ Ppr) _» log 7 (182) ra ia) Now substituting n = k/B, G 2 / i | Ge GE Er? Fa (288 &s Mapproaches 1, therefore, the drag coefficient for the delta wing with no trailing edge sweepback approaches « logarithmic infinity. The character ef this weak peak is shown in the previously discussed plotse For the wing with trailing edge sweepback the situation is slightly more complicated. The terms in G and F may be arranged in the following decressing orders of magnitude in ne (og (0g 77 . ae LOG” ele, #7 2 2 4 7? 4 “od Terms of the first two orders of magnitude will be needed, It will be most convenient, however, to calculate them separately. De» noting the contributions of terms of the largest order by ont, ene finds, upon introducing equations (180) and (131) in (120)s @) “of 7 _> fan f (-@ -4)* n ali-a) ce 2 mq (-r Ut-r fl r-4) ,lr-a) —~/-42 | ((1-a)* _ (t-4) ‘"F rli-r 0r-4) rlr-a) —— —» “gr /-a) f 1-4@ (£-1) - (-& rn lr-a@)( l-r 7 r —- O Thus there is no contribution te the drag coefficient from terms of this order of magnitude. In order to obtain terms cf the next order, we must first find terms of that order in G(n, sj and F(n, s), which might be designated G2) ana B(2), Prom equation (102) we have (2) _ f-S ws (og 2 fF v/ -s J a G (1,5) ig | 2 - ae 9 } (155) and from equetion (118) . ; 2 (2) l-5 f 10g l fog 22a! S) —-~ = LEFT f£. fo a 7, ) les sn 7 A 7 172((-5 ) _ ZS se SF eg © te (09 Tas SS sw 4? (136) wi Be Denoting the contributions to the drag ecefficient in this order by. on {? ) we have, from equations (120), @ _ 20? (4) Lf ch rleg2 _/ fog (1-1) f OLA BN l-r a CDrs } Bm (F-al/-r?) A % FD Tp (2) _ 20? A C/-a) (og a ZL 2a [rc BW U-rlr-a)[ (+a avy 7 (9 [A /-alr [loa “sr +. ag 2) Hi 1+ Br ar? t+) [- 4/6 Q) 2 _ y CH = -2n Gea) [ee fala 1. fogli-al) “LA BW (l-r\Nr-a) | (+4 na F — be f $22 _ a fo lg Ge) | C2 22? rlinal? 7 aleg2 ~gg eg - 4), Zs BW U-rNr=a?ll rer oo 1 2zr* ala) (09 a/r +L log Zee Za Bul rea* ari ‘11 /-@f/r Collecting terms, the entire expression can finally be reduced tes oa s 2 26 _, 2 li-a)” [Ia Zz? T 7 ali-rYr-a) Jag ¢Ct-a) Zr b-& (137) _ (og (r- 2) | a a Te) Replacing n by k/B, it is seen that the drag coefficient, Cf = 2 approaches a finite limit as M approaches 1, This limit is shown for the curves of drag vs. Mach number pletted in Pigs. 30 and 31. If a is allowed to approach C in this expression, the first and lest terms in (157) give essentially a constent, independent of By while from the middle term one obtains Oe 2 / a 7-2 ral nr (og ce (158) The entire expression for drag coefficient was, however, valid only for the case that an 2 1, otherwise the expressions for Ca and aC Cy would have been imaginary, Therefore a can approach 0 no fester TIC ~ then I/n, ors _ fe 2 SS lag S aso) ze 7 kre 7 8 (L5S) which agrees with equation (133), although the method of approach to a= 06 is quite differents The existence of this finite limiting value of Cy, for wings with swept trailing edge raises an interesting question concerning its significance and validity. On the one hand, the present linearized approach to the solution of the compressible flow problem certainly breaks down at a Mach number of exactly 13; in fact, as brought out in the derivation of the basic equations in Part II, the approximation requires smaller and smaller disturbances as M approeches 1, Furthere more in the present case, if one considers only the area ef the delta wing ahead of the maximum thickness point, the pressure coefficient approaches infinity everywhere ‘et & rate given by the limiting form of equation (9le), 4 1 C —~_ — } anil P tk °6 i and the integral of these pressures over this forwerd surface, which is proportional then essentially to G(n, oj, approaches infinity at the rate 1 gas —>~ Const: log B The finite limit of the drag coefficient is caused by the existence of negative drag on the remainder of the wing, which approaches infinity at essentially the same rate as the positive contribution. On the other hand, for a sufficiently thin airfoil, the result may be regarded as accurate for Mach numbers which approach extremely close to 1, The practical question therefore seems tec be how far from Mach number 1 the non-linear terms in the equations have an important effect, This question can properly be answered only by experiment, including the possibility that viscous effects might also become importents =B Ben Experimental results aveilable so fer in this range indicate very strongly thet the drag curves are predicted with reasonably good accuracy almost down to Mach number 1 by the linearized results; in fact, the drag rise near or through Mach number 1 seems to be predicted very closely by the limits calculated above, The experiments do not, of course, show eny infinite drags or even large peaks near M* 1 for the delta wing with zero trailing edge sweepback; to what extent this is a function of acceleration or viscous effects, or to whet extent it could be predicted by a more accurate solution to the equations near Mach number 1, is not yet clear, Another interesting special case of the above calculation is obtained if a is allowed to approach 1 ~ which should in the limit correspond to & twomdimensional swept wing, or “arrow wing". For convenience the maximum thickness point may be taken at mid root chord, which implies 2(lerj = (1a) (140> In this case, from equation (187), the drag coefficient is given by 3 DE 2a)” [4 __ 4, atr-a) ret T + aC/-4) 7 2. _ FZ Aali-4) 4 hog LA f (- a® 4 Z. - 4li~2) °F “2 & (-@ 8 afr / > > A lz log Cota) — fog & | (ar) This drag coefficient, based upon the actual plan area, S, apparently approaches 0 as a_ approaches 1 and the span approaches infinity. In order to obtain a physical idea of the forces involved, however, a new drag coefficient, Cpt, might be calculated, based on the square of the root chord, A. From Fig.e 8, L 2 k(1 = a) and thus 2 QD = oF, = — (<z log li+a)—_". log af (142) (ta which, as @ approaches 1, gives Ezr* | ne log 2 eo “p= ge OF 4) > eo (143) Since the thickness + = ze » this result may also be interpreted 2 8&t = 72 log 2 (144) D q where q is the dynamic pressure, Thus for a sweepback of 45°, the drag is of the order of the force produced by one dynamic pressure acting over an ares equal to the square of the airfoil thickness « a surprise ingly low result for the limit at Mach number 1,. The result in equation (143) is exactly twice that computed by von Kerman (Ref. 22) for the limiting case of a strictly two-dimen= Sional swept wing as Mach number 1 is approachede Presumably the discrepancy is due to the fact that the limit in the present case is approached for a wing with tips which begin and remain as sharp points, so that there is an essential difference in the character of the tips during the limiting process, Similar results can be obtained for the case of a rectangular wing of finite aspect ratio. The classical result, for infinite aspect ratio, predicts that the drag increases near M = 1 ag (nu o vee, while for the finite aspect ratio case, it can be shown that the drag increases only as log (Me “2 ij. The finite aspect ratio drag calculation will be outlined briefly belowe Consider first the rectangular wing of Fige 38, for which the Mach wave from one tip does not reach the ether tip, ieee, Bo > Ce The pressure at P will have the value calculated for an infinite twoedimensionel wing plus a tip correction. The tip correction may be regarded as the influence at P of a source distribution in region A equal and opposite to the source distribution at corresponding chordwise stations on the remainder of the wing. If A (x) is the slope (in the x-direction) of the airfoil surface, then the correction to the twoedimensional potential, Ag » for a point influenced only by one wing tip will be a-BY a d2 ~ b= 2 [rele [ERG | _ @ 2, where 7 = F * Ck- V6 then - x bY ia | cos! ~o ds and the incremental u-velocity is 2, laf) = 7 as) g as (X-§)V(x-)*- 87g? (145) 7) The loeal pressure increment is Ap = = cud” and the local drag increment, for both sides of the airfoil is a(ap) = 2 Ax} Ap dxe The totel drag correction, consisting of the integral of this incree ment over the region OMN, oe AD= ~ ze2 fits 7 as | | J 8) la-5)— py" (146 J a-s/s Als) as | veld o tae [2 [pd OD BY ‘| ¥70 where the order of integration was reversed in the last stepe From i this, x Cc 4D= — oa [rae (20) as (147) 0. , O «Gla But if t(x) is the thickness of the airfoil, then (Ms) = Fels) = £6) and AD= 222" Li/t dv = 0 (148) 77/8 Thus the drag correction for thisc ase, Pb > C, is zero, and the drag has its two-dimensional value a7 “ Dp, = af Vaxe, rk (14g j In the case that Bb > C, illustrated in Pig. 39, the calculetion proceeds as before, except that the integration of the drag increment is carried out only over WORQs But since the integral over MNO will be gero, as before, it is necessary only to calculate minus the integral of the drag increment over MQR. Thus AD = a 26° tra foa fi "ae as A- EN (e- 5)? — 87Y? for the influence of one wing ips For both tips, and reversing the order of integration, (e-8Y/3 Ap - one Fae} ody LX-5) Vlr-5) 2 *_ ety? or, ¥-xX-26 | . 2 4D= 26 fuone/ Adz) [le-s)*— 98? dF Ck- §) (149) This increment will always be negative, and can be easily calculated in specific cases, Consider, for example, an airfoil with double-wedge profile, and maximum thickness at the 50% chord pointe Then for a< F< c/a | ACE) = 4+ Az 25 F< , ACE) =— aA, Inserting these values of A in equation (149/, and carrying out the successive integrations over the necessary intervals, one obtains finally: eord 2 ye? es “rae LEY ~ 96" + 2°b cash a _ 2 pbc cos ‘fe $/c*-¢ ~ 2 cash hee + GC LOS (2 | (151) The basic twoedimensionel drag is, of course, 2 Zz 2 bc fo oT A, 8 (152) D, = saQ Gwe As the Mach number approaches 1, (Pb = 0), the dominant terms im equation (151) ares a AD —» weoihe [8 th? a9 (8 ~ BC /2, | Combining this with equation (181), and noting that A,= Z for this wing, one finds that Cp 42 S Jo —_— Zz? Wc oD 2 ~ (153) Thus the drag coefficient has the same type of logarithmic singularity near M= 1] as was found for the delta wing with no trailing edge sweep= backs The drag reduction from the two-dimensional case, as given by equation (151) will be quite appreciable for low aspect ratios. =35= PART VIII CONC LUS IONS: In Part. IT it was shown that a reasonable approximate solution to the equations of motion for the supersonic flow of a perfect fluid is. obtained by assuming the disturbance velocities to be small. One possible method of representing solutions to the linearized equations, in terms of sources and doublets, was develeped and discussed. It was shown that this representation was particulerly suitable for calculating the flew over thin three«dimensional airfoils at sere angle of attack, and can also be used to represent the flow over airfoils at finite angles of attack. The equations were applied in Parts IV, V and VI to the calculae tion of the pressure distributions and drag forces acting on a family of wings of nearly triangular planform, with a simple thickness distrie# tution, called the delta wings, It was found that with the exception of small regions on the airfoil, the disturbances resulting from the solution do fall within the assumptions of the linearized theory, if the airfoil thickness is sufficiently smell. It was also found that the drag acting on these airfoils may depend critically upon the details of the planform and thickness distribution, and that optimum choices may be made of these various geometrical parameters for particular design problems. The essential results of the drag calcue- lations for this eirfoil family were summarized in Figs. 20 to 32. In general, the results indicated that (a) the "sweepback" of a wing is characterized not only by its planform, but the geometry of its meximom thickness lines, and (b/ the essential effect of increasing =SGe sweepback is to lower the drag coefficient in the low supersonic Speed range, with a drag rise or peak at seme higher Mach number, near the range where the Mach weves are parallel to the leading edge or maximum thickness lines. Finally, it was concluded that the possible adjustments to the shape of. the drag curve as a function of Mach number could be used to advantage in several ways, with the choice depending on the requirements of a particular engineering probleme le 4e 5 60 a=G7= REFERENCES Ackeret, J., "Luftkrafte auf Flugel, di ran s mit grossere al fa Schallgeschwindigkeit bewegt werden", ZoF.Me, Vol. 15, pDe 72-74, 1925. vou Karmen, Th,, and Moore, N. Be, "Resistance of Slender Bodies FJ 3% % 8 tt Moving with Supersonic Velocities", Transastions of A.S.ME., ’ LOS Z6 Slavert, He, "The Effect of Compressibility on the Lift of Airfoils", Proce Royal Society, (A), Vole 118, pe 113, 1927. Prandtl, Le, "Theory of the Lifting Wing in a Compressible Medium", Luftfahrtforsshung, Vole 13, ose 313, 1936. Busemann, Ae, "Aerodynamischer Auftrieb bie Uberschallzeschwindigkeit", Proce Volta Congrass, 1935.6 Busemenn, Ae, "Infinitesimale kegelige Uberschallstromung", Schriften der Dutschen Akademie der Luftfahrtforschung, 7B, pe 105, 1943. Hantasche, W., and Wendt, M., "Gasstromungen mit Ubergang von Unterschall~zu Uberschallgeschwindigkciten"™, ZA MeM., Vol. 22, pe 726 Stewart, He Je, "The Lift of a Delta Wing at Supersonic Speeds", Quarterly of Applied Mathematics, Vol. IV, No. 3, October, 1946, Schlichting, H., "Deutsche Luftfahrtforschung? Jahrbuch, 1937. Puckett, A. Fe, “Supersonic Wave Drag of Thin Airfoils", Journal of the Aeronautical Sciences, Vol. 13, No. 9, 1946. Evvard, John Ce, "Distribution of Wave Drag and Lift in the Vicinity of Wing Tips at Supersonic Speeds", NACA TN-1382, 1947. Py 16. 17. 186 19. 200 Zl. 226 256 Liepmann; H. We, and Puckett, As Ee, “Introduction to the Aero= dynamics of a Compressible Fluid", Wiley and Sons, New York 1947. Jones, R. T., "Thin Oblique Airfoils at Supersonic Speed", NACA TNel107, 1946, Jones, Ro. To, “Sstimated Lift-Drag Ratios at Supersonic Speed", NACA TN+1350, 1947. Evvard, John C., "Theoretical Lift Distribution and Upwash Velocities at Supersonic Speeds", NACA TNe1484, 1947, Evvard, John C., "Theoretical Distribution of Lift on Thin Wings at Supersonic Speeds, (An Extension)", NACA 1-158, 1948. Evvard, John C., "The Effects of Yawing Thin Pointed Wings at Supersonic Speeds", NACA TNe1429, 1947, Heaslet, Me A., and Lomax, H., "The Calculation of Downwash Behind Supersonic Wings", NACA TN=1520, 1948. Margolis, K., "Supersonic Wave Drag of Sweptback Tapered Wings at ‘Zero Lift", NACA TNel448, 1947. Margolis, Ke, "Effects of Chordwise Location of Maximum Thickness on the Supersonic Wave Drag of Sweptback Wings", NACA TN=1543, 1948. Margolis, Ke, “Supersonic Wave Drag of Nonlifting Sweptback Tapered Wings with Maximum Lines Behind the Line of Maximum Thickness", NACA TNel672, 1948, von Karman, The, "Supersonic Aerodynamics", Journal of the Aeronautical Seiences, Vol. 14, po 373, 19470 Puckett, A. E., and Stewart, He Ja, “Aerodynamic Performance of Delta Wings", Journal of the Aeronautical Sciences, Vol. 14, pe 567, 19470 ‘S APPENDIX I SUMMARY OF SYMBOLS Undisturbed or free stream velocity (in x-direction), Notes The subscript is dropped in the linearized solutions. Total velocities in directions of x, y and zg axes respectively. Perturbation velocities in directions of x, y and 2 axes. Perturbation velocity potential. Mach number pressure density dynamic pressure = eu" /20 ratio of specific heats, C S°4 pressure coefficient = (p = Po W/4s where p, is free stream pressure. ue = 1 = cotangent of Mach angle. angle between airfoil leading edge and normal to stream direction (sweepback angle); see Fig. 6 or 86 _ tan k/8, = relative sweepback parameter. slope of airfoil surface in x«directione total airfoil chord, apex to tip, (see Fige 3 je airfoil root chord xedistance from trailing edge at root to airfoil tip. x-distance from maximum thickness point at root chord to airfoil tips i] =100= maximum thickness point location, fraction of root chords airfoil drag, both sides of a symmetrical airfoil. airfoil actual planform AYe&ea drag coefficient =D/S. airfoil maximum thickness. tf = thickness ratio. functions used in drag calculations, defined by equations. -101- FIG f ee ee i) wot x L2G EE. FHKE AVAL ADAL GF ! -102- ) F/G. S Se PER SOME LEADING HIE PUPS SUB SONIC -103- FIG. 5 ee eet) ameter La pt Ed FARE LXOWLELE TF LAKE NN ‘ i { i i | | i { | 4 by -104- — -¢ ws a» . a “ ——— ee ~ a“ a“ a a LC “ tT a 3 7 "xX, EF L- a ns ae , Co ene nage 0 F/G, 7 SWE F par say & , K > ‘3 Fev Far 4, TK, fari Fp 7 Ke 2 FR, tan Fp = kp 8 BK, -105- “ vB a vag Z va Pe “ea Le “fA a oO A vod c F/G 9 Four AD AC PT M UATE ERE FOAM GES BASIC TRANS EL & fv >t a -106- FIG, S/ INFLUENCE ZONE £0 EL >/ FIG. (2 / PASIC. 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Hy ian agal R Glass Gene ats ‘ TE pt i { te at Lt nei eee ae ry “ rhein Pence nancies at oeoee i fob on Maya ygrr pares an $+ tet 1 1 H at Sean eeeen cua i | . i Ek =119< aS a 44 a at tr t.. He wd : fet ee i + better tb +t aa tty eter Pte tebe yy Hi imi ot us ; t Sa a Seana eee ae) + i eM ee meal pee -121= theatre oe } ‘ : epee Seca o Le fi f: GALOCIT* LF- a “Ff? f Ah leh bo lr ZO J IVVE ELE NACH en ae ane eee i So aes pean Bt fre meee fi 2125 L is Sa grec beware em A) AV A Cet KA PES ERE ' -126- FIG. 35 Sac uneee boone =129@ SP Og Ahk VOY OY ee pe { 1 ‘ he! Fone heee ane sored