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S"rk, Ur nuCopy 322 ". "RM L54A29a M k' ON SLENDERBODY THEORY AT TRANSONIC SPEEDS By Keith C. Harder and E. B. Klunker Langley Aeronautical Laboratory Langley Field, Va. C.: '.OMM TO .W. SEARCH MEMORANDUM i t; ;. ..A i ON SLIENDE ADVISORY OM MITEE SO FLORIDAFOR AERONAULTICSD TO X117011 NocslVEI. 3075 ~LLEFL 326117011 USA ,. R' CAS..IE DOC A* HTs xatria! conain. tnbrtr e affecting t Nationl Defense of the United States within te meaning T. of theu a tof laws, Title S, U.S.C., Seon. 793 and 7N4. the trasmisston or revelation of wkich Io any ... an .'..'.ar e s o I p..'"au by .. Ai:r NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WASHINGTON March 24 1954 ii., L el , S:.i'ii" i OF FLOIDAs.B] TNG~mD Z:E I I NACA RM L54A29a NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS RESEARCH MEMORANDUM ON SLENDERBODY THEORY AT TRANSONIC SPEEDS By Keith C. Harder and E. B. Klunker SUMMARY The basic ideas of the slenderbody approximation have been applied to the nonlinear transonicflow equation for the velocity potential in order to obtain some of the essential features of slenderbody theory at transonic speeds. The results of the investigation are presented from a unified point of view which demonstrates the similarity of slenderbody solutions in the various Mach number ranges. The primary difference between the results in the different flow regimes is repre sented by a certain function which is dependent upon the body area distribution and the stream Mach number. The transonic area rule and some conditions concerning its validity follow from the analysis. INTRODUCTION Slenderbody theory originated with Munk's paper (ref. 1) in 1924 in which the forces on slender airships were calculated for lowspeed flight. In 1958 Tsien (ref. 2) pointed out that Munk's airship theory also applied to the flow past inclined, pointed bodies at supersonic speeds. The subject gained new importance in 1946 with the appearance of Jones's paper (ref. 5) in which it was shown that the basic ideas of the slenderbody approximation could be used to calculate the forces on slender lifting wings at both subsonic and supersonic speeds provided that proper account was taken of trailingvortex sheets. Since Jones's paper, the subject has received wide treatment in the literature. In an important paper in 1949, Ward (ref. 4) developed a general unifying theory for the flow past smooth, slender, pointed bodies at supersonic speeds which contains as special cases the lifting planar wings of Jones and the slender nonlifting bodies treated by Von Karman (ref. 5). The corresponding problem at subsonic speeds has been examined by Adams and Sears (ref. 6) who also extended the slenderbody concepts to shapes which are "not so slender." Lighthill (ref. 7) has given a method for calculating the flow past bodies with discontinuities in slope. Keune (ref. 8) has developed solutions for slender wings with thickness and various lifting configurations have been treated by Heaslet, Spreiter, Lomax, Ribner, and others refss. 9 to 14). CONFIDENTIAL CONFIDENTIAL NACA RM L54A29a The slenderbody theory presented in references 2 to 14 has been based upon the linearized equation for the velocity potential. In the present paper, the basic ideas of the slenderbody approximation are applied to the nonlinear transonic equation for the velocity potential in order to obtain some of the essential features of slenderbody theory at transonic speeds. The attempt has been made to present the results from a unified point of view which demonstrates the similarity of the slenderbody solutions in the various Mach number ranges. The authors wish to acknowledge the invaluable aid and advice of Dr. Adolf Busemann of the Langley Laboratory during the writing of this paper. SLENDERBODY APPROXIMATION Slenderbody theory deals with that class of shapes whose length is large compared with any lateral dimension. For such shapes at both subsonic and supersonic speeds, the flow in planes normal to the stream direction can be approximated by solutions of Laplace's equation. The justification is that for very slender wings or bodies the variation of the geometrical properties in the stream direction is small and, consequently, the rate of change of the longitudinal component of the velocity in the stream direction is also small. The various slender body solutions have all been developed on the basis of the linearized potential equation. However, a similar development can be made on the basis of the nonlinear transonic equation. The simplest differential equation for the disturbance potential 0 which is generally valid at transonic speeds (ref. 15, for example) is M2 (7 )M2x xx + Orr + r + 0 (1) r r2 where x, r, and a are cylindrical coordinates, M is the stream Mach number, and 7 is the ratio of specific heats at constant pressure and constant volume. With the introduction of the dimensionless coordinates and n by x = Z and r = br, and of the dimension 2 less potential 0 by 0 = b2 0(,,), the transonic potentialflow equation becomes ( M2 (7 + 1)M2 + + =0 (2) \zI~J~ CONFIDENTIAL CONFIDENTIAL NACA RM L54A29a where Z is a characteristic length and b is a characteristic width such as the largest lateral dimension of the configuration. For suffi ciently small values of the width parameter b/1, it appears that the terms involving derivatives in the stream direction can be neglected to obtain the result that the flow satisfies Laplace's equation 1rr + r + = 0) r r2 in the crossflow plane. Equation (5) represents the slenderbody approximation to equation (1). Some conditions will be determined subsequently which are necessary in order for solutions of equation (5) to be approximate solutions of equation (1). The boundary conditions for the flow about a body in a uniform stream are the vanishing of the disturbance velocities at infinity and = dn(1 + Ox) dn on dx dx on the body where n is in the direction normal to the body contour in the crossflow plane. For flows satisfying Laplace's equation in the crossflow plane, the surface boundary condition can be integrated (ref. 4, for example) to give S dt = S'(x) (4) where dt is an element length in the direction of the tangent to any contour C in the crossflow plane enclosing the body, S(x) is the crosssectional area distribution of the body, and the prime denotes differentiation with respect to the indicated argument. The slenderbody solution of equation (1) can be represented by a solution of equation (5) plus a function of integration. Since equa tion (5) is independent of Mach number, the form of the solution is identical with the known slenderbody solutions for subsonic and super sonic flow. The slenderbody analyses of references 4 and 6 have established that the solution can be represented by a distribution of sources and higher order singularities on the axis; an equivalent form is given by a distribution of sources in the region of the crossflow plane interior to the surface boundary. The slenderbody solution is then expressed as CONFIDENTIAL CONFIDENTIAL CONFIDENTIAL + 4 lp rd I a 0r. rl 4 ca 0. CO 0 + r  0 o bl O ,t rI a C0H II or Otd d ix 0l Q) 0 0) bO O 4 CO 0d co .0 1 P OJA bD r . 4 Jrd Z po 0 r a) 12 O  *H 4 O 0 Od ai+ 4l ) cdmr S0 C M (u C *rlq 42d 0) (a 4r  Pd b 00 C H' O NACA RM L54A29a urn + rl r C. a f, .rl 'p a + aj P 0 o a rbl CaT HI~^ co ad Q) OO l ) o 0 U ,0 ., .U . 0u P t % 0 42 1 'H Od 0) *rd COd 42A co 01 rd 0 (2 A *M 42 ' 0 'Hi .0d 0)l +2  0drp uJn tQ II c0v0 II iC IF IDEI TIAL 'p Q r cQ ) 1a rl Ca 611 l *r l a + 0) 0 0 a ul a Cn II %1" _r + 'p 0. r rd 0. H in 'l 0. a 0 r4 II o NACA RM L54A29a where cp denotes the solution of Laplace's equation in the crossflow plane with i appearing as a parameter introduced by the geometry of the cross section at . The function cp, being independent of the stream Mach number, can be evaluated for incompressible flow past the shape under consideration. A necessary condition for equation (6) to be an approximate solu tion of equation (1) is that the terms neglected in equation (1) be small compared with those retained. By assuming for the moment that log p is the only singular term in 0, the ratio of the term I M2 (7 + l)M2x ]xx, which is neglected in the slenderbody approximation, to any of the remaining terms in equation (1) is of the order ()2 1 M2) log + 0(1] + 2 (log2 )+ (log + 0(1) where O( ) denotes order of and the nonsingular terms are denoted by 0(1). This ratio can be made smaller than any prescribed value E by restricting the solution to the interior of a cylinder of some radius, say R/Z. For given values of M and e, the radius of this cylinder increases with decreasing b/1 and the ratio R/b approaches infinity as b/Z approaches zero. Moreover, for given values of E and R/Z, larger values of b/Z are permitted as M approaches 1. From equation (6) it is apparent that S'(x/Z) and its deriva tives must be finite in order to satisfy the requirement of small dis turbances. Moreover, an additional restriction on the asymmetry of the body is sometimes required (ref. 4), particularly for lifting con figurations namely, the radius of curvature of the body boundary in the crossflow plane must be O(b) where the boundary is convex outward. The restrictions on body shape imposed by the function g(x/Z) will be considered subsequently after the nature of this function is established. The function g(x/1) is determined from considerations involving the complete transonic differential equation (eq. 1) and, consequently, is dependent upon the stream Mach number. Ex: the source distribution to the potential can be made smaller than any prescribed value by making b/Ro sufficiently small  To this order of approximation, then, the flow field external to Ro is axisymmetric. Moreover, there is an axisymmetric flow which approxi mately matches the pressure and flow direction of the slenderbody CONFIDENTIAL CONFIDENTIAL NACA RM L54A29a solution at r = Ro. The potential 0o for this associated axisymmetric flow satisfies equation (1) and is expressed symbolically as So/2 ) log + go(x/z,r/ (7) where the characteristic width b and length I are taken the same as for the asymmetric shape. In equation (7), so(x/1) is the area dis tribution of the associated body of revolution which has the same outer flow (r Ro) as the asymmetric shape. If this axisymmetric flow satisfies the slenderbody conditions, then go is independent of r for r < R. In order for the radial derivatives to match at r = Ro, So(x/l) must be equal to s(x/1); that is, the associated body of revolution must have the same axial distribution of crosssectional area as the asymmetric shape. In order for the pressure distributions to match at r = Ro, go(x/2) must be the same as g(x/1). Thus, g(x/Z) is the same function as that for a body of revolution having the same axial distribution of crosssectional area. In the preceding discussion the region of validity of the slender body approximation to (o was tacitly assumed to be at least as large as that for 0. This condition is certainly true since the singular terms in the two solutions are the same. In the slenderbody approximation the term 2 (7 + l)Vx.x, is required to be small compared with any of the other terms in the transonic differential equation. If this condition is to be satisfied in the neighborhood of weak shock waves where g'(x/1) would be required to have a jump proportional to the pressure rise and g"(x/Z) would be infinite, the quantities (which were included in the terms denoted as 0(1)) g"(x/Z)[ M2 (7 + l)M o] (8) and g"s a  CONFIDENTIAL CONFIDENTIAL NACA RM L54A29a must be bounded there. Thus, at shock waves the coefficients of g"(x/I) in expressions (8) and (9) must vanish. In the first of these expres sions, which is axisymmetric, the coefficient of g"(x/Z) vanishes for a local Mach number of 1. Moreover, the average value of the local Mach number ahead and behind a weak normal shock wave is 1. With this interpretation of shock waves, then, the coefficient of g"(x/1) vanishes and the solution should represent the flow in the neighborhood of weak normal shocks. Also, since g(x/1) is the same function for both a slender configuration and its associated body of revolution, their shockwave systems would have the same strength and location. In order for expression (9) to be bounded in the neighborhood of a shock wave, the double integral must vanish. Since this double integral gives rise to asymmetric shapes and is zero for chordwise locations where the body is axisymmetric, the additional restriction is obtained that the body cross section must be circular in the vicinity of shock waves. The slenderbody solutions in the various Mach number ranges are similar in that they are all represented by equation (6) although the function g(x/Z) differs for the various speed ranges. Analytic expressions for g(x/1) have been obtained for supersonic and subsonic flows by considering general solutions of the complete linearized equa tion satisfying the boundary condition of vanishirn disturbance veloc ities at infinity. These solutions were then expanded in the neighbor hood of the body to evaluate g(x/z). Ward (ref. 4) has determined this function for supersonic flows as g(x/1) = 1 log M2 1 x/l s( log  4 0 1 The corresponding result at subsonic speeds was obtained by Adams and Sears (ref. 6) as g(x/) (x/1) log s"(l)log 1)d. + 1 1 X 2 s"(l)log(1 )dJ L/ Ixl where the body extends from x = 0 to x = 1. Although an analytic expression for g(x/l) at transonic speeds is not known, it has been established that the stream Mach number enters the solution only through g(x/1) and that the only geometric property of the body influencing CONFIDENTIAL CONFIDENTIAL 8 CONFIDENTIAL NACA RM L54A29a this function is the area distribution. The transonic similarity rule for bodies of revolution (ref. 15 or 16) shows that g(x/1) can be expressed in the form g(x/z1 = ( ) log + 1)N + f(x/Z;K) where the similarity parameter is 1 M2 K = 2 ( + 1)M2( AERODYNAMIC FORCES Since the slenderbody solutions are all represented by equation (6), formal expressions for the aerodynamic forces can be determined which are valid throughout the Mach number range. Consequently, many of the essential features of slenderbody theory at transonic speeds can be obtained without resorting to detailed calculations. Lift The most significant difference between the slenderbody solutions at subsonic, transonic, and supersonic speeds is that the function g(x/Z) differs in these various speed ranges. However, the term in the pressure arising from the function g(x/Z) makes only a uniform contribution to the pressure at any value of x and, therefore, cannot affect the lift distribution or the lift. Thus, within the slenderbody approximation, the lift distribution depends only upon the function q9 and, conse quently, is independent of the stream Mach number. Several investigators (Robinson and Young (ref. 17) and Heaslet, Lomax, and Spreiter (ref. 9), for example) have previously noted that the linearized slenderbody theory gave consistent results, even at a Mach number of 1, for planar systems. According to slenderbody theory, the lift distribution can be obtained completely from solutions of Laplace's equation in the cross flow plane. Since this equation is linear, the lift is proportional to the angle of attack even at transonic speeds. Ward has obtained an especially simple form for the drag due to lift in which CONFIDENTIAL NACA RM L54A29a 1 DL = aL where a is the angle of attack measured from zero lift and L is the lift. Drag By computing the momentum change of the fluid passing through a cylinder enclosing the body, the drag is determined as D = (b2 2 s'(E)g'()dt + f d dt (10) q q O c' 6n ( where the body extends from 0 = 0 to 1 = 1, C' denotes the contour of the body at the stern which in the case of wings or wingbody com binations includes the trailingvortex wake, q is the stream dynamic pressure, and Db is the base drag. Equation (10) is valid throughout the Mach number range provided the appropriate forms of the func tion g(x/1) are employed. The line integral is zero for nonlifting configurations if the body is closed or if the body ends in a cylindrical section whose elements are parallel to the stream. The effect of Mach number (excluding the variation of base drag with Mach number) is con tained in the term involving g(x/1). When the subsonic form of g(x/Z) is used in equation (10), the correct result is obtained that the drag of nonlifting configurations is zero. By using the supersonic form of g(x/L), the drag varies with Mach number like s,'(l)]2log(M2 1). For pointed bodies, or for bodies which end in a cylindrical section, the supersonic slenderbody theory indicates that the drag is independent of Mach number. For bodies which do not satisfy these conditions, the supersonic result indicates that the drag approaches infinity as the Mach number approaches 1. These results from linear theory cannot be considered satisfactory at transonic speeds since they give a discontinuity in the drag as the Mach number is increased through 1; whereas experimental data show that the drag starts to increase rapidly at a subsonic Mach number and varies smoothly through 1. However, the few known solutions of the nonlinear transonicflow equation are in good agreement with experiment in this regard. Consequently, the drag rise of slender shapes should be correctly approximated by equation (10) when the transonic form of g(x/1) is employed in the drag equation. CONFIDENTIAL CONFIDENTIAL NACA RM L54A29a Transonic Area Rule The body shape enters into the function g(x/Z) only as a function of the crosssectional area distribution throughout the Mach number range. This property of the slenderbody solutions leads to an important result even though the analytic expression for g(x/2) is not known at transonic speeds. Examination of equation (10) shows that the body crosssectional shape enters into the slenderbody drag expression only through the contour integral evaluated at the stern of the configura tion. For a fixed geometry at the base, then, the drag of nonlifting configurations depends only on the axial distribution of the body cross sectional area and is independent of the crosssectional shape. Thus, within the slenderbody approximation, the drag of a nonlifting configu ration is the same as that of the associated body of revolution having the same streamwise distribution of crosssectional area provided the base geometry is fixed. It is in this sense that an equivalent body of revolution is associated with a wingbody combination. This result, often referred to as the area rule, is especially significant at transonic speeds where larger values of the width parameter b/Z are permitted than in other speed ranres. The property of the dependence of the drag upon the distribution of crosssectional area has previously been obtained by Ward (ref. 4) and Graham (ref. 18) for supersonic flow and has been observed experi mentally by Whitcomb (ref. 19, for example) at transonic speeds. The importance of this result was first noted by Whitcomb who demonstrated that the area rule could be used as a basis for the design of lowdrag wingbody combinations at transonic speeds. From the preceding development, the transonic area rule is subject to the restrictions of slenderbody theory with the additional condition that the base geometry be fixed. These restrictions imply that modifi caLions to the equivalent body of revolution must be performed in the region between shock waves. However, the seriousness of violating this condition is not well understood at the present time. For example, schlieren photographs seem to indicate that, even in cases where the presence of the wing affects the strength of the shock, the average strength of the shock may be close to that for the equivalent body of revolution. Langley Aeronautical Laboratory, National Advisory Committee for Aeronautics, Lr~.ilcy Field, Va., January 18, 1954. CONFIDENTIAL CONFIDENTIAL NACA RM L54A29a REFERENCES 1. Munk, Max M.: The Aerodynamic Forces on Airship Hulls. NACA Rep. 184, 1924. 2. Tsien, HsueShen: Supersonic Flow Over An Inclined Body of Revolu tion. Jour. Aero. Sci., vol. 5, no. 12, Oct. 1958, pp. 480485. 5. Jones, Robert T.: Properties of LowAspectRatio Pointed Wings at Speeds Below and Above the Speed of Sound. NACA Rep. 835, 1946. (Supersedes NACA TN 1052.) 4. Ward, G. N.: Supersonic Flow Past Slender Pointed Bodies. Quarterly Jour. Mech. and Appl. Math., vol. II, pt. 1, Mar. 1949, pp. 7597. 5. Von Ka'rman, Th.: The Problem of Resistance in Compressible Fluids. Atti del V Convegno della Fondazione Alessandro Volta, R. Accad. d'Italia, vol. XIV, 1956, pp. 559. 6. Adams, Mac C., and Sears, W. R.: SlenderBody Theory Review and Extension. Jour. Aero. Sci., vol. 20, no. 2, Feb. 1953, pp. 8598. 7. Lighthill, M. J.: Supersonic Flow Past Slender Bodies of Revolution the Slope of Whose Meridian Section Is Discontinuous. Quarterly Jour. Mech. and Appl. Math., vol. I, pt. 1, Mar. 1948, pp. 90102. 8. Keune, Friedrich: Low Aspect Ratio Wings With Small Thickness at Zero Lift in Subsonic and Supersonic Flow. KTHAero TN 21, Roy. Inst. of Tech., Div. of Aero., Stockholm, Sweden, 1952. 9. Heaslet, Max A., Lomax, Harvard, and Spreiter, John R.: Linearized CompressibleFlow Theory for Sonic Flight Speeds. NACA Rep. 956, 1950. (Supersedes NACA TN 1824.) 10. Spreiter, John R.: Aerodynamic Properties of Slender WingBody Combinations at Subsonic, Transonic and Supersonic Speeds. NACA TN 1662, 1948. 11. Lomax, Harvard, and Heaslet, Max A.: Linearized LiftingSurface Theory for SweptBack Wings With Slender Plan Forms. NACA TN 1992, 1949. 12. Spreiter, John R.: Aerodynamic Properties of CruciformWing and Body Combinations at Subsonic, Transonic, and Supersonic Speeds. NACA TN 1897, 1949. CONFIDENTIAL CONFIDENTIAL NACA RM L54A29a 15. Heaslet, Max A., and Lomax, Harvard: The Calculation of Pressure on Slender Airplanes in Subsonic and Supersonic Flow. NACA TN 2900, 1955. 14. Ribner, Herbert S.: The Stability Derivatives of LowAspectRatio Triangular Wings at Subsonic and Supersonic Speeds. NACA TN 1425, 1947. 15. Busemann, Adolf: Application of Transonic Similarity. NACA TN 2687, 1952. 16. Oswatitsch, K., and Berndt, S. B.: Aerodynamic Similarity at Axisymmetric Transonic Flow Around Slender Bodies. KTHAero TN 15, Roy. Inst. of Tech., Div. of Aero., Stockholm, Sweden, 1950. 17. Robinson, A., and Young, A. D.: Note on the Application of the Linearized Theory for Compressible Flow to Transonic Speeds. Rep. No. 2, College of Aeronautics, Cranfield, Aero. Res. Council (British), Jan. 1947. 18. Graham, Ernest W.: Pressure and Drag on Smooth Slender Bodies in Linearized Flow. Rep. No. SM13417, Douglas Aircraft Co., Inc., Apr. 20, 1949. 19. Whitcomb, Richard T.: A Study of the ZeroLift DragRise Charac teristics of WingBody Combinations Near the Speed of Sound. NACA RM L52H08, 1952. CONFIDENTIAL NACALangley 32454 325 CONFIDENTIAL C4 ci 0 Q) 0 0 y I = m , ww z U 0 s 03 a i '1 cs fi C..a .. C t 4 4 M. 0. ^ a r o r g aa "" s 14 4 c; 4; 02 z i i i< 0 i^ 0 C al4 wZ 1.00 1.2 m 2 ~ 0 a *0, 4 C', c, h ,=d 0 C 4) C: ul~ Pm ( I0 g0 Do Z po6 tTo 09' 02 u ca = C. 5 a a 4: 0 U) 'C 0' CU I ijilL22 ille I~ P 0 C! 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I' ' r,; A fl Ij c i: i ;;;i;I, CONFIDENTIAL UNIVERSITY OF FLORIDA 3 1262 08106 562 4 UNIVERSITY OF FLORIDA DOCUMENTS DEPARTMENT 120 MARSTON SCIENCE UBRARY P.O. BOX 117011 GAINESVILLE, FL 326117011 USA CONFIDENTIAL '~n~ ;''i'' ri% '"* Ir '.Ci .5 ; i ;r .;.::~ i,, ,II ,i ,;r .I ., ;.;r i ... ;; or ;?,:* L!' .I I~Ii;I : ..; 1"<. . *: *.... rl ;; ; 