TGl sep 1 1 1951 ke | ate Ce Thee, NAVY DEPARTMENT THE DAVID W. TAYLOR ee OT yj sy / — Me. fb Woayu WASHINGTON 7, D.C. / ‘ ( DOCUMENT | THE AXIALLY SYMMETRIC POTENTIAL FLOW ABOUT ELONGATED BODIES OF REVOLUTION by L. Landweber August I95l Report 76] NS 715-084 ls Eee Ahab sattt SEP 10 1951 INITIAL DISTRIBUTION Copies 17 Chief, BuShips, Project Records (Code 324), for distribution: 5 Project Records Research (Code 300) 2 Applied Science (Code 37/0) 2 Design (Code 410) 2 Preliminary Design (Code 420) 1 Propellers and Shafting (Code 554) 1 Technical Assistant to Chief of the Bureau (Code 106) 1 Submarines (Code 515) 5 Chief, BuOrd, Underwater Ordnance (Re6a) le raeee Aes) Meielelkere 3 Chief, BuAer, Aerodynamics and Hydrodynamics Branch (DE-3) 4 Chief of Naval Research, for distribution: 3 Fluid Mechanics (N426) 1 Undersea Warfare (466) 4 Commander, U.S. Naval Ordnance Laboratory, Mechanics Division, White Oak, Silver Spring 19, Md., 1 for Dr. Max Munk 2 Commander, Naval Ordnance Test Station, Inyokern, China Lake, Calif., 1 for Underwater Ordnance Division, Pasadena 1 Commanding Officer, Navy Underwater Sound Laboratory, Fort Trumbull, New London, Conn. 1 Commanding Officer, Naval Torpedo Station, Design Section, Newport, R.1. 6 Director, National Advisory Committee for Aeronautics, 1724 F St., N.W., Washington, D.C. 1 Mr. N. Tetervin, Langley Memorial Aeronautical Laboratory, Langley Air Force Base, Va. Dr. C. Kaplan, Langley Memorial Aeronautical Laboratory, Langley Air Force Base, Va. Commander, Portsmouth Naval Shipyard, Portsmouth, N.H., Attn: Design Superintendent Director, Woods Hole Oceanographic Institution, Woods Hole, Mass. 2 Newport News Shipbuilding and Dry Dock Co., Newport News, Va., for distribution: 1 Senior Naval Architect 1 Supervisor, Hydraulics Laboratory 3 Director, Experimental Towing Tank, SteVens Institute of Technology, 711 Hudson Street, Hoboken, N.J., 1 for Dr. Korvin-Kroukovsky 2 Dr. Hunter Rouse, Director, Iowa Institute of Hydraulic Research, State University of Iowa, lowa City, lowa 2 Director, Ordnance Research Laboratory, Pennsylvania State College, State College, Pa. 2 Dr. R.T. Knapp, Director, Hydrodynamic Laboratory, California Insti- tute of Technology, Pasadena 4, Calif. 2 Dr. L.G. Straub, Director, St. Anthony Falls Hydraulic Laboratory, University of Minnesota, Minneapolis 14, Minn. Copies 2 Director, Experimental Naval Tank, Department of Naval Architecture and Marine Engineering, University of Michigan, Ann Arbor, Mich. 2 Dr. V.L Streeter, Illinois Institute of Technology, 3300 Federal Street, Chicago 16, Ill. 2 Head, Department of Naval Architecture and Marine Engineering, Massachusetts Institute of Technology, Cambridge 59, Mass. 1 Dr. C.C. Lin, Department of Mathematics, Massachusetts Institute of Technology, Cambridge 39, Mass. 1 Director, Applied Physics Laboratory, Johns Hopkins University, 8621 Georgia Avenue, Silver Spring, Md. 1 Prof. W.S. Hamilton, Technical Institute, Northwestern University, Evanston, Ill. 1 Prof. G@. Birkhoff, Harvard University, Cambridge, Mass. ] Prof. K.E. Schoenherr, School of Engineering, Notre Dame University, South Bend, Ind. ] Prof. W. Spannhake, Armour Research Foundation, 35 West 33rd Street, Clin@zyao WO, Iu, 1 Dr. W. 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Chairman, Department of Aeronautical Engineering, New York Univer- sity, University Heights, New York 53, N.Y. Copies 1 BAR, Cornell Aeronautical Laboratory, Cornell Research Foundation, BOX 2S 5m BuUbhalo) Zils Ney 1 BAR, Pratt and Whitney Aircraft Division, United Aircraft Corp., East Hartford 8, Conn. 1 BAR, North American Aviation, Inc., 12241 Lakewood Blvd., Downey, (Capleistas 1 BAR, McDonnell Aircraft Corp., Box SiO, Se. louis 3, Mo. 1 BAR, Glenn L. Martin Co., Baltimore 3, Md. | BAR, East Akron Station, Akron 5, Ohio 1 BAR, Lockheed Aircraft Corp., 2555 North Hollywood Way, Burbank, Cauleittes 1 BAR, Consolidated Vultee Aircraft Corp., San Diego 12, Calif. 1 BAR, North American Aviation, Inc , Columbus Division, 4300 East 5th Ave., Columbus, Ohio BAR, Douglas Aircraft Cc , Inc., El Segundo, Calif. BAR, Grumann Aircraft Engineering Corp., Bethpage, Long Island, N.Y. Dr. Th. von Kaérmdn, 1051 South Marengo St., Pasadena, Calif. Prof. A.D. Hay, Princeton University, Princeton, N.J. oh ed er Oe Administrator, Webb Institute of Naval Architecture, Crescent Beach Road, Glen Cove, Long Island, N.Y. ) 1 Prof. R.C. Binder, Department of Mechanical Engineering, Purdue University, Lafayette, Ind. 1 Dr. M.S. Plesset, California Institute of Technology, Pasadena 4, Calif. 1 Dr. A.G@. Strandhagen, School of Engineering, Notre Dame University, South Bend, Ind. 1 Prof. M.H. Martin, Head of Mathematics Department, University of Maryland, College Park, Md. 1 Dr. W. Douglas Baines, Department of Civil Engineering, Michigan State College, East Lansing, Mich. 1 Dr. V.L. Schiff, Stanford University, Palo Alto, Calif. 1 Mr. C.A. Lee, Hydraulic Engineer, Research and Development Labora- tories, Kimberly-Clark Corp., Neenah, Wis. 1 Prof. A. Marcus, Physics Department, City College of New York, 137th St. and Convent Ave., New York, N.Y. 1 Dr. R.I. Wolf, Physics Department, City College of New York, 13/th St. and Convent Ave., New York, N.Y. 1 Prof. B.P. Gill, Mathematics Department, City College of New York, 13/th St. and Convent Ave., New York, N.Y. gy British Ministry of Supply, Washington, D.C. 6 British Joint Services Mission, Washington, D.C., 4 for Navy Staff and 2 for Technical Staff 1 Superintendent, Admiralty Research Laboratory, Teddington, Middlesex, England Copies Sat Australian Scientific Research Liaison Office, Washington, D.C. Hydrodynamics Laboratory, National Research Laboratory, Ottawa, Canada Dr. F. Ursell, Trinity College, Cambridge, England Dr. Vandry, Admiralty Research Laboratory, Teddington, England Prof. J.K. Lunde, Skipsmodelltanken, Tyholt Trondheim, Norway Prof. L. Troost, Superintendent, Netherlands Ship Model Basin, Haagsteeg 2, Wageningen, The Netherlands Directeur du Bassin d'Essais Des Carénes, 6, Boulevard Victor, Paris XV, France Director, Swedish State Shipbuilding Experimental Tank, G6teborg 24, Sweden Dr. G. Hughes, National Physical Laboratory, Teddington, Middlesex, England Editor, Journal of the British Shipbuilding Research Association, 5, Chesterfield Gardens, Curzon St., London W. 1, England Editor, Physics Abstracts, Institution of Electrical Engineers, Savoy Place, London W.C. 2, England Editor, Index Aeronauticus, Ministry of Supply, Millbank, London S.W. 1, England Head, Aerodynamics Department, Royal Aircraft Establishment, Farn- borough, Hants, England Head, Aerodynamics Division, National Physical Laboratory, Teddington, Middlesex, England Head, Aeronautics Department, Imperial College, London, S.W. 7, England Head, College of Aeronautics, Cranfield, Bletchley, Bucks, England Prof. J. Kampé de Fériet, Faculté des Sciences, Universite de Lille, Lille (Nord), France Directeur, Laboratoire Dauphinois d'Hydraulique des Ateliers Neyrpic, Avenue de Beauvert, Grenoble (Isere), France Office National d'Etudes et de Recherches Aéronautiques 3, rue Leon Bonnat, Paris XVI, France Prof. D.P. Riabouchinsky, Centre National de la Recherche Scientifique, 13 Quai d'Orsay, Paris VII, France Prof. J.M. Burgers, Laboratorium Voor Aero-En Hydrodynamica, Nieuwe Laan 76, Delft, The Netherlands Dr. R. Timman, National Luchtvaartlaboratorium, Sloterwag 145, Amsterdam, The Netherlands Director, Aeronautical Research Institute of Sweden, Ranhammarsvagen 12, Ulsvunda, Sweden Prof. J. Ackeret, Institut ftir Aerodynamik Der Eidgendssiche Technische Hochschule, Ztirich, Switzerland Dr. L. Malavard, Office National d'Etudes et de Recherches Aéro- nautiques, Chatillon, Paris, France TABLE OF CONTENTS Page ABSTRACT . 1 TANT OD U CAMO Nerves sce ye vue enytenh is) fepvus\ os Vert yceuc se ne 2 History Macy heer een ats 2 Formulation of the Problem . 4 RUSIOSIOID) OL AVOUNG IDIGSHNRINEOMALONNIS, 9 6 0 6 0 0 6 60 6 6 56 0 0 6 00 6 6 6 6 Sources and Sinks : ; 6 Dovlolerw; WalsiwiesblowtploMms 5 6 0 050000 00 7 Munk!s Approximate Distribution 8 igo! Woaliass Ox e\ Walsigealloicstom 5-56 o oa 6 6 615 6 0 6 oO 6 6 6 6 6 6 6 6) (NO An Improved First Approximation ... Sees ce ee eOne eran aio mlicw at 6 Atcuemcay a. *- TKS) Solwmeilom Oi Invexeal Weweteiom yy Ieee 6 5 6 6 5 0 6 0 oo ool « Velocity and Pressure Distribution on the Surface .......... If KumePICAI WWEwENeoM Git WMS 9 6 6b 6 6 0 60 Go ol «CY TOIBINUS CHANCE AMINE fu Ueieitn its h raltensicc aos oll cust cs, Peuderecu? sh ich Wei aliag Mek PRGOn tcl dongs arctan all RretorerineDe berminatO Nuon D/ Gin we esueiwes a esate on mash atin cis cunlse aera sey SAMS Compamisonmswach Karmanivand Kaplan’ Methods\js 4+ 0c) - ici ne ent SOLU TONE YEA PP IEG AMON NOR TGREEN! SiwEHEOREM ss ue) eth uenie) Gils) 0. osu ence selene nem e4 General Application to Problems in Potential Theory ......... 34 Anilnviegral iquatiion for eAxisymmetrich low sel. 060 4 - s e s e o> Kennard's Derivation of the Integral Equation ............ 38 A First Approximation ... ai eeelra fac es a esa le Ee a eS Solution of Integral en a Mer atLonics ee ay low aun! Alle Sen eene eE aa I) METAS, TRAM BANCye Cone I eeeacWS Ga Slo pid! s-6 oo 5 o oro oo oo) LO eesti ratlVeLBXAMpUey He eel han se Uebel wie. woken REIN) A lights UR) OS I ee pm ce Ne SUMMA NMR ete cts Moment OMS ak AN ass tae = Ke cee Mt NR ye a EO) ANPPINIDIDE TAD) TOINAUS Ol? JN WIGSHURIOSWHNWON =g 6 6 56 6 0 6 ll TRUST G OSS Ver ceo) rol Bains Molino Mion aictr nnn CacminuU Cia atom EOUEREMET gc Gow. ¢ voy’) THE AXIALLY SYMMETRIC POTENTIAL FLOW ABOUT ELONGATED. BODIES OF REVOLUTION by L. Landweber ABSTRACT An iteration formula for Fredholm integral equations of the first kind is ap- plied in two new methods for obtaining the steady, irrotational, axisymmetric flow of an inviscid, incompressible fluid about a body of revolution. In the first method a continuous, axial distribution of doublets is sought as a solution of an integral equa- tion of the first kind. A method of determining the end points and the initial trends of the distribution, and a first approximation to a solution of the integral equation are given. This approximation is then used to obtain a sequence of successive approxima- tions whose successive differences furnish a geometric measure of the accuracy of an approximation. When a doublet distribution has been assumed, the velocity and pres- sure can be computed by means of formulas which are also given. In the second method the velocity is given directly as the solution of an inte- gral equation of the first kind. Here also a first approximation is derived and applied to obtain a sequence of successive approximations. In contrast with the first method, which, in general, can give only an approximate solution, the integral equation of the second method has an exact solution. Both methods are illustrated in detail by an example. The results are com- pared with those obtained by other well-known methods. INTRODUCTION HISTORY > The determination of the flow about elongated bodies of revolution is of great practical and theoretical importance in aero- and hydrodynamics. Such knowledge is required in connection with bodies such as airships, tor- pedoes, projectiles, airplane fuselages, pitot tubes, etc. Since it is well known that for a streamlined body, moving in the direction of the axis of sym- metry, the actual flow is very closely approximated by the potential (inviscid) flow about the body,? numerous attempts have been made to find a convenient theoretical method for obtaining numerical solutions of the potential flow problem. At first the problem was attacked by indirect means. In 1871 Rankine* showed how one could obtain families of bodies of revolution of known potential flow, generated by placing several point sources and sinks of vari- ous strengths on the axis. This method was extended and used by D.W. Taylor® in 1894 and by G. Fuhrmann* in 1911. The latter also constructed models of the computed forms and showed that the measured distributions of the pressures over them agreed very well with the computed values except for a small region at the downstream ends. More recently, in 1944, the Rankine method was em- ployed by Munzer and Reichardt’ to obtain bodies with flat pressure distribu- tion curves, and a further refinement of the technique was published by Riegels and Brandt.” Most recently the indirect method has been employed to obtain bodies generated by axisymmetric source-sink distributions on circum- ferences, rings, disks,and cylinders. This development, which enabled bodies with much blunter noses to be generated, was initiated by Weinstein | in 1948 and continued by van Tuyl > and by Sadowsky and Sternberg” in 1950. A method of solving the direct problem, i.e., to determine the flow over a given body of revolution, appears to have been first published by von Kdrmdn?® in 1927. von Karman reduced the problem to that of solving a Fredholm integral equation of the first kind for the axial source-sink distri- bution which would generate the given body, and solved the integral equation approximately by replacing it with a set of simultaneous linear equations. Although this method has limited accuracy and becomes very laborious when, for greater refinement, a large number of linear equations are employed, neverthe- less it is the best known and most frequently used of the direct methods. A ‘modification of the von Kdrmdn method was published by Wijngaarden?? in 1948. *References are listed on page 59. An interesting attempt to solve the direct problem was made by Weinig?® in 1928. He also formulated the problem in terms of an integral equa- tion for an axial doublet distribution which would generate the given body, and employed an iteration formula to obtain successive approximations. Since the successive approximations diverged, the recommended procedure was to extra- polate one step backwards to obtain a solution. In 1935 an entirely different approach, in which a solution for the velocity potential was assumed in the form of an infinite linear sum of orthog- onal functions, was made by Kaplan?* and independently by Smith.+* The coeffi- cients of this series are given as the solution of a set of linear equations, infinite in number. In practice a finite number of these equations is solved for a finite number of coefficients, and Kaplan has shown that the approximate solution thus obtained is that due to an axial source-sink distribution which is also determined. A simplification of Kaplan's method by means of addition- al approximations was proposed by Young and Owen?® in 1943. It appears to be generally agreed, by those who have tried them, that the aforementioned methods are both laborious and approximate. Thus, ac- cording to Young and Owen:+° "In every case, however, the methods proposed are laborious to apply, and the labour and heaviness of the computations increase rapidly with the rigour and accuracy of the proc- ess. Inevitably, a compromise is necessary between the accuracy aimed at and the difficulties of computation. All the methods reduce, ultimately, to finding in one way or another the equivalent sink-source distribution, and it is this part of the process which in general involves the heaviest computing." Furthermore, a fundamental objection is that only a special class of bodies of revolution can be represented by a distribution of sources and Sinks on the axis of symmetry. According to von Karman;1?° "This (representability by an axial source-sink distribu- tion) is possible only in the exceptional case when the analytical continuation of the potential function, free from singularities in the space outside the body, can be extended to the axis of symmetry without encountering Singular spots." The dissatisfaction with these methods is reflected by the continuing attempts to devise other procedures. A new method published by Kaplan*® in 1943 is free of the assumption of axial singularities and appears to be exact in the sense that the solution ean be made as accurate as desired, but the labor required for the same ac- curacy appears to be much greater than by other methods. The application of the method requires that first the conformal transformation which transforms the given meridian profile into a circle be determined. The velocity poten- tial is then expressed as an infinite series whose terms are universal func- tions involving the coefficients of the conformal transformation. Kaplan?® has derived only the first three of these universal functions. Cummins of the David Taylor Model Basin is developing a method based on a distribution of sources and sinks on the surface of the given body. This method is also exact, but the labor involved in its application has not yet been evaluated. Another exact method, based on a distribution of vorticity over the surface of the body, is being developed by Dr. Vandry of the Admiralty Re- search Laboratory, Teddington, England. The methods of both Cummins and Vandry lead to Fredholm integral equations of the second kind, which can be solved by iteration. The present writer has developed two new methods, an approximate one in which an axial doublet distribution is assumed, and an exact one based on a general application of Green's theorem of potential theory. Both methods lead to Fredholm integral equations of the first kind for which a solution by iteration has been discussed by the author.?’ Indeed, the consideration of this iteration formula was initiated in an attempt to find more satisfactory solutions of the integral equations of von Karman?® and Weinig.22 These new methods will be presented, and, by application to a particular body, compared with other methods from the point of view of accuracy and convenience of application. FORMULATION OF THE PROBLEM We will consider the steady, irrotational, axially symmetric flow of a perfect incompressible fluid about a body of revolution. Take the x-axis as the axis of symmetry and let x, y be the coordinates in a meridian plane. Denote the equation of the body profile by 3p = 38((33) [1] Since the flow is irrotational there exists a velocity potential ¢ which, for axisymmetric flows, depends only on the cylindrical coordinates x, y and satisfies Laplace's equation in cylindrical coordinates dx (Yee) + ay (98g) 0 (21 Also, since the flow is axisymmetric, there exists a Stokes stream function y(x, y) which is related to the velocity potential by the equations f) 3 f) 3 aes ay Se oe gee [3] It is seen that Equation [2] may be interpreted as the necessary and suffi- cient condition insuring the existence of the function w. As is well known, w is constant along a streamline and, considering the surface of revolution generated by rotation of a streamline about the axis of symmetry, 2my may be considered as the flux bounded by this surface. On the surface of the given body and along the axis of symmetry outside the body we have w= 0. wy satis- fies the equation which is obtained by eliminating ¢ between Equations [3]. The velocity will be taken as the negative gradient of the velocity potential. Let u, v be the velocity components in the x, y directions, Then by [3], we have Ep Op el walked. OOK == [5] 0¢ _ 1 OW For a uniform flow of velocity U parallel to the x-axis we have @= -Ux, p= -5 Uy® [7] The boundary condition for the body to be a stream surface may be written in various ways. If the body is stationary the boundary condition is OD (2, VY =O [8a ] or, equivalently, (38), - 0 [8b] where the derivative in [8b] is evaluated on the surface of the body in the direction of the outward normal to the body. If the body is moving with veloc- ity V parallel to the x-axis the boundary condition becomes (eel = -V cos 8 [9] where B is the angle between the outward normal to the body and the x-axis. It is desired to obtain a solution of [2] or [4] which satisfies the boundary conditions [7] at infinity and [8] or [9] on the body. METHOD OF AXIAL DISTRIBUTIONS SOURCES AND SINKS The potential and stream functions for a point source of strength M situated on the x-axis at x = t are M NM x-t 0 y= M(-1 + 25) [10] where Tey at) aaet aya (a) If the sources are distributed piecewise-continuously along the x- axis between the points a and b (see Figure 1) with a strength w(x) per unit length, the potential and stream functions are Figure 1 - The Meridian Plane As is well known, Rankine bodies are obtained by superposition of these flows with a uniform stream so as to obtain a dividing streamline begin- ning at a stagnation point. Without loss of generality we may suppose this uniform stream to be of unit magnitude. This dividing streamline is the pro- file of the Rankine body for which, by [7], the stream function is abe pier ay) ees b= -ty + fate) (-1 + =*) at [14] The boundary condition, Equation [8a], then gives as the implicit equation for the body fae) (21 +%)at=gy? [15] where now y* = f(x) and r® = (x-t)® + f(x). In order to obtain a closed body the total strength of sources and sinks must be zero, i.e., [med se 0 a In this case [15] becomes [wey 28 at = 3 [15a] In general [15a] cannot be solved explicitly for f(x) when u(t) is given. A practical procedure for obtaining f(x) for a given x is to evaluate the integral numerically for various assumed values of f(x) and to determine the value which satisfies [15a] by graphical means. When f(x) is prescribed [15a] may be considered as a Fredholm inte- gral equation of the first kind for determining the unknown function u(t). This equation will not be treated. Indeed it will be shown that, when con- tinuous distributions are considered, it is a special case of the more general equation for doublet distributions which will now be derived. DOUBLET DISTRIBUTIONS Let m(x) be the strength per unit length of a continuous distribu- tion of doublets along the x-axis between the points a and b (see Figure 1). The potential and stream functions may be taken as ae [ m(t) = at [16] a TP and is Al, mE) at 017] The stream function for a Rankine flow now becomes b y= bye ey? [| MES ot [18] r Hence the boundary condition, Equation [8a], gives PInGb ames suc fae at -5 [19] Here again Equation [19] may be considered as an implicit equation for the Rankine body when m(t) is given, or as a Fredholm integral equation of the first kind when the body profile y* = f(x) is prescribed. In order to show the relation between the source and doublet distri- butions in Equations [15a] and [19], integrate by parts in [19]. We have b 2 6 6 SE ae) o-x dm x-t [mer 3G at = mt) © +48 = at Hence [19] may be written as m(t ) fe [ ° dm x-t at oils 2 [20] Teas te 5 Gig 2 The interpretation of Equation [20] is that a doublet distribution of strength m is equivalent to a source-sink distribution of strength dm/dt together with point sources of strength m(a) and -m(b) at the end points. Hence source-sink distributions are completely equivalent only tc those doublet distributions which vanish at the end points. This justifies the remark in the previous section that the integral equation for the doublet distributions is more gen- eral than that for the source-sink distributions. MUNK'S APPROXIMATE DISTRIBUTION Munk?® has given an approximate solution of Equation [19] for elon- gated bodies. His formula may be derived as follows. At a great distance from the ends of a very elongated body, the integrand of [19], m(t)/r°, will peak sharply in the neighborhood of t = x. In the range of the peak, in which the value of the integral is principally determined, m(t) will vary little from m(x). Also, only a small error will be introduced by replacing the lim- its of integration by -~ and+co. Hence, as a first approximation to a solu- tion of [19], try mix)f = 3 [21] We obtain n, (x) = 7 ¥ [22] a distribution proportional to the section-area curve of the body. This ap- “proximation was independently derived by Weinig?? who employed it as the first step in a divergent iteration procedure. It has also been rediscovered by Young and Owen?® and Laitone’® who have shown the accuracy of the approxima- tion for elongated bodies by several examples. It is apparent from its derivation that [22] also gives the asymptot- ic radius of the half-body generated by a constant axial dipole distribution extending from a point on the axis to infinity. It is readily seen that this distribution is equivalent to a point source at the initial point. As a refinement to Munk's formula, Weinblum®® has used the approxi- mation m, (x) =, Cy* [23 ] where C is a factor obtained by comparison of the distributions and section- area curves of several bodies. Weinblun's factor bears an interesting rela- A tion to the virtual mass of the body. This is seen by considering the expres- sion for the virtual mass k 4 in terms of the mass of the displaced fluid A and the totality of the AoUBIStE, fc nebe = PBS DES) b k,A = 4mp mdx - A [24 ] where k, is designated the longitudinal virtual mass coefficient, and p is the density of the fluid. But, from [23], b b 4p | m, ax = po | my2dx = 4CA a a 10 since, for elongated bodies, a and b very nearly coincide with the body ends. Hence c = +k) (25 ] In practice an approximate value of k, may be taken as that of the prolate spheroid having the same length-diameter ratio as the given body. The values of k, for a prolate spheroid may be computed from the formula®* 4 xin (a + VaF-1) - VAF-1 [26] 2 VP =T = aln (A+ VA? =1 where A is the length-diameter ratio. Hence 1 3/2 oie a [27] 2 Vr2-1 - Aln (a + Vie The values of k, versus d have also been tabulated by Lamb” and graphed by Munk .° END POINTS OF A DISTRIBUTION A difficulty in determining the doublet distribution from Equation [19] is that the limits of integration, a and b, are also unknown. In the method of von Karman?® the end points are arbitrarily chosen; Kaplan?® takes the end point of the distribution midway between the end of the body and the center of curvature at that end. Kaplan based his choice on a consideration of the prolate spheroid. Thus the equation of the spheroid of unit length and length-diameter ratio i, extending strom: xs == OM GOlexd illus 1 y° = (x - se) [28] x The radius of curvature at x = 0 is then +5. The exact doublet distribution, however, extends between the foci of the spheroid which are situated at dis- tances NING ee) OX from the end points. Hence the error in Kaplan's assumption, NN ge Silage el ( EINE 2a Xe alone 2r? diminishes rapidly with increasing 2. For the half-body generated by a constant doublet distribution (a point source), Kaplan's assumption gives a poor approximation. Let a® be the strength of the distribution. Then it can easily be shown from [19] that the source is at a distance a from the end of the body (stagnation point), and that, if the origin is chosen at the latter point, the equation of the half- body is 2 2 3 (Sar a) tae a) (291 Hence the radius of curvature at the end 1s 52, so that Kaplan's Seas ee for the start of the distribution gives za. This is in error by 32. An approximate method for determining the end points of a distribu- tion and its trends at the ends is given in Appendix 1. The given profile is assumed to extend from x = 0 to x = 1 and to have the equation = 2 3 Yr 2 Ge BLS cb AES A oon [30] The doublet distribution is assumed to extend from x = a to x = b, So that O Galen So Aehi’= «> QIMGH > Tcts))) > See, [36] - 96a? (5a? - 24a + 24) + 516a°a, is a root of the seventh-degree polynomial 2 2 2 A + a,B + a0 + aia,D + a,k + a,a,F + aa,G + a,aiH = (0) [37] A(a) = ala - 4)? (5a* - 830° + 2880 - 368a + 128) Bla) = 72(a - 4)? (50° - 250° + 40a - 16) Cla) = Yala - 4) (530° - 148a + 128) D(a) = -288(a@ - 4)(5a* - 160 + 16) E(a) = -96a(3e - 4) F(a) = 1152(2e - 3) 48a(3a - 8) Q R i H(a) = -1152(@ - 3) The solution gives, for the initial doublet strength at x = a, m(a) [la - 4) (a? - 120 + 16) + 4Bala - ¥)(a - 2) + 16a, - 96aa,)] [39] When ay, aj, 4, .-- are all small in comparison with unity, an ap- proximate solution for a@ is W / b aut S Qs Web a =c ey le a 2a [40 ] a-4+a, ifa <0 [41] and, to the same order of approximation, a a a az ax) =O +5 +52 ne \(-e ty) [42] and a a a mal) -3(1 + = i = in a VECEo | ue ey, 20 [43 ] (ay sO, i a, < © [44 ] It is seen that Kaplan's assumption that a = 4 gives the principal term of the solution in [40] or [41]. The form [42] immediately suggests a modification and refinement of the Munk-Weinblum approximation, Equation [23], which will be considered in the next section. A graphical procedure for finding the roots a@ of Equation [35] is also given in the Appendix. For this purpose the functions A(a), Bla),... Hla) are tabulated in Table 10. AN IMPROVED FIRST APPROXIMATION According to its derivation the Munk approximation could be expected to be useful only at a distance from the end points of a distribution. It was seen, however, Equation [42], that under certain circumstances a distribution which was a suitable approximation for the nose and tail of a body also ap- peared as a generalization of the Munk-Weinblum approximation, [23]. This suggests a procedure for obtaining an improved approximate distribution. It is desired to obtain a distribution m(x) which satisfies the fol- lowing conditions: (a) m(x) assumes known values m, and m, at the distribution limits a and be pent; m(a) = m m(b) = M,, [45 ] (b) m(x) is nearly equivalent to the Munk-Weinblum approximation [23] at a distance from the distribution limits, i.e., may 1Cvs for a dx - o(b-a)(f,+f,) [50] SOLUTION OF INTEGRAL EQUATION BY ITERATION Now that we have derived a good first approximation to the doublet distribution function in the integral equation [19], it would be very desir- able to apply it to obtain a second, closer approximation. This can be accom- plished by means of the iteration formula which we will now derive. Let m,(x) be a known first approximation and yp, (x) the corresponding values of the stream function won the given profile y> = f(x). Then, from Equation [18], 1 (t) Y,(x) = - f(x) + r(x) [ - dt [51] Thus wy, (x) is a measure of the error when m, (t) is tried as a solution of the integral equation [19]. If m(t) is a solution of [19], Equation [51] may be written in the form b m, (t)-m(t) 3 dt [52] p(x) = (x) | a 10 But, on the same assumptions as were used to derive Munk's approximate distri- bution, Equation [22], we obtain as an approximate solution of the integral equation [52] m, (x) - m(x) = 3 (x) [53] or, denoting the new approximation to m(x) by m, (x), m,(x) = m,(x) - 3p, (x) [54] Hence, from [51] om, (t) m(x) = m, (x) deez =|, ae | [55] ces Since the foregoing procedure can be repeated successively, we obtain the iter- ation formula 1 : om, (t) ms 44 (x) = m, (x) + + 1003 -| 2 dt [56] and m,,4(*) - m,(x) = -5 py, (x) [57] It is seen that Vs is the value of the stream function on the given profile corresponding to the th approximation m,(x) and hence serves as a measure of the error when m,(t) is tried as a Sonesta of the integral equa- tion [19]. Although successive approximations to m(x) may be computed directly from [56], an alternative form, which is both more convenient and more signif- icant, will now be derived. From [56] we may write. 16 m, (x) = m,_,(x) + + t(x) Pa | [56a J Hence, deducting [56a] from [56] and making use of [57], we get B(x) = Wy yx) see at [58] Also, from [57] we obtain i my (x) = m (x) - > Sy, (x) (591 Thus, in order to obtain ms ,4(x), we first assume an m, (x), then determine , (x) from [51]. p(x), p(X), ... can then be successively obtained from [58], and finally m, 1 (X) from [59]. It has been stated that the magnitude of yp, (x) is a measure of the proximity of m, (x). This property of W, (x) can be given a geometrical in- terpretation. Corresponding to the distribution m,(x) there is an exact stream surface on which the stream function B, (x, a) = Oy leet An, be the distance from a point (x, y) on the given body to this exact stream surface, measured along the normal to the given body, positive outwards. Let uy be the tangential component of the flow along the body. Then we have 1 Ap, (x,y) Poewes a) 8! Onan ati But Ay = -y, (x), since W(x, y) = 0 on the exact stream surface. Hence x ies 2 : [60] Ss Since, for an elongated body, UES 1, except in the neighborhood of the stag- nation points, it is seen that , (x) enables a rapid estimate to be made of the variation from the desired profile of the exact stream surface correspond- ing to m, (x). This is an important property because it can be used to monitor the successive approximations. Thus, the sequence Wp; (x) can be terminated when An, becomes uniformly less than some specified tolerance; or, since there is no assurance that the infinite sequence p, (x) converges, the sequence can conceivably give useful results even without convergence if it is continued as 7 long as An, decreases on the average, and is terminated when the error begins to increase and grows to an unacceptable magnitude at some point along the body. The strong similarity between these remarks and the discussion follow- ing Theorem 2 of Reference 1/ should be noted. There is also a strong similarity between the iteration formula of Reference 17 whose convergence was thoroughly discussed, and the present equa- tion [56]. An essential difference between the iteration formulas is that the former employs the iterated kernel of the integral equation, the latter does not, so that the convergence theorems of Reference 1/ are not applicable. Nev- ertheless, it is proposed to use the form in [56] (or the equivalent iteration formula [58]), for the following reasons: a. The labor of numerical calculations would be greatly increased by iterating the kernel, and even then only convergence in the mean would be guaranteed (Theorem 4 of Reference 17). b. The physical derivation of Equation [56] indicates that at least the first few approximations should be successively improving. ce. The successive approximations are monitored so that the sequence can be stopped when the error is as small as desired or, in the case of initial convergence and then divergence, when the errors begin to grow. VELOCITY AND PRESSURE DISTRIBUTION ON THE SURFACE When an approximate doublet distribution m, (x) has been obtained, the velocity components u, v can be computed from the corresponding stream function [18] bm, (t) y, (x,y) = y* f as dt - 1] [61] from which, in accordance with Equations [5] and [6], and -X t v = 3y ee m, (t)dt [63 ] 18 On the given surface we have, from [61], (t (x) [A een ee [64] Dig y* (x) where now > ce (ee) + ses) [65 ] Differentiating [64] with respect to x gives Wi(x) 2, (x)y"(x) jf Se nei oS [66] a or y> (x) y? (x) Hence, from [62] and [64] we obtain > m, (t) 2p, (x) = dt - [67] i I r> f(x) and, from [63], [66], and [67], see) v = uy!(x) + oes [68] y(x) where the primes denote differentiation with respect to x. Equations [67 ] and [68] are the desired expressions for u and v. If the approximation m,(t) is very good, the contributions of the error function w,(x) should be very small. It is interesting to note that the form of Equation [68] shows the deviation of the resultant velocity from the tangent to the given body. Bernoulli's equation for steady, incompressible, irrotational flow with zero pressure at infinity now gives the pressure distribution p, = 1 - (u? + v?) [69] ald where q is the stagnation pressure. 19 NUMERICAL EVALUATION OF INTEGRALS In order to perform the iterations in Equations [56] and [58] and to compute the velocity distribution it will frequently be necessary to evaluate integrals of the form 3 ti) b mt) gt ana { Se at a op B ip> where Po (c=) ieee (x) Because in this form these integrals peak sharply in the neighborhood of t = x, especially when the body is elongated, they are consequently unsuited for nu- merical evaluation. A more suitable form can be obtained by means of the following trans- formation. Let (x, y) be the coordinates of a point on the body, t the ab- scissa of a point on the axis, 6 the angle between a line joining these two points and the x-axis; see Figure 1. Then x - t = y(x) cot 6 [70] We may now transform the integrals so that 6 becomes the variable of integra- tion. Then b y? B {z nteere = | ae) ened [71] an a and b y4 B == m(t) = i m(t) sin® @ d@ [72] G12 a where = Ni 2 ae a = arctan >=, = arctan >> [73] An alternate procedure, which eliminates the peak without a trans- formation of variables, is the following. We have (Ss m(t)at = [ S[pt)-a00] aes) (3 a 20 and ie m(t)dt = [ %[n(er-moo] et + m(x) "Se at a r> la rs L r> Hence ls m(t)dt = [5 [mer -noo) at + m(x)(cos@ - cos) [71a] if SE ae ee = [ E [mct)-moxr]at r> rp? + m(x) [eos ao -cosB = (costa - cos?) | [72a ] Gauss! quadrature formula is a convenient and accurate method of evaluating these integrals. The formula may be expressed in the form 1 [Pleas - & RaF(Ena) [74] where the gy are the zeros of Legendre!'s polynomial of degree n and the Rai are weighting factors. These have been tabulated?® for values of n from 1 to 16. These numbers have the properties R eral Ga OG aa sing [75] ni > ®y nit The value of the integral given by Formula [74] is the same as could be ob- tained by fitting a polynomial of degree 2n-1 to F(x). The values of Re and are tabulated in Table 1 for n = 7, 11, and 16. When the limits of integration are w and #, as in Equations [71] and [72], Gauss' formula becomes Sai B B-a ¥ [ Fle)ae rene, RagF(8,) [76] where B- +B Oe Ga oe [77] 21 TABLE 1 ABSCISSAE AND WEIGHTING FACTORS FOR GAUSS! QUADRATURE FORMULA -0.949108 0.129485 -0.978229 0.055669 -0.989401 0.027152 741531 279705 887063 .125580 -QUU575 .062254 -0.405845 .381830 130152 . 186290 .865631 .095159 0.417959 .519096 .233194 -755404 124629 -0.269543 .262805 .617876 .149596 0 0.272925 -458017 .169157 R, =R .281604 . 182603 -0.095013 0.189451 Gry SSG aeaihy|| May = Aetaey $1 > Sn -i41 al n-it+] ILLUSTRATIVE EXAMPLE The foregoing considerations will now be applied to a body of rev- olution whose meridian profile is given, for -1< x S11, by, y? = f(x) = 0.04(1 - x*) [78] The body is symmetric fore and aft, has a length-diameter ratio A = 5, and a prismatic coefficient 1 ae | (1 - x*)ax = 0.80 [79] 0 By applying to [78] the transformation R= 26S. yoaey [80 ] We obtain the equation for the geometrically similar body of unit length, for 0oS¢@ S14, n? = 0.08(€ - 362 + 4é® - 26%) = 0.086(1 - €)(26? - 2€ + 1) [81 ] Ze We will also need the slope of the profile which, from [78], is fell (5s) ees Oe [82 ] Die J 2y (1-x4)272 The profile and f(x) are graphed in Figure 2. Figure 2 - Graphs of y(x) and y*(x) for y?(x) = 0.04(1 - x‘) First let us find the end points of the distribution. We have, from isl, 2, = Oo, A, 2S Ooty A, = 0.32. The approximate formula [40] then gives a = 3.68 or 3.84, whence a = > = 0.0217 or 0.0208. An examination of the complete polynomial [37] with the aid of Table 10 shows that its zeros oc- Ge Be OS 35655, Boel), Wale iba Wne application of Table 10 to determine these roots the regions of possible zeros should be determined by inspection, the values of the polynomial in these regions calculated from Equation [37] and Table 10, and then graphed to obtain the zeros. It is seen that in the pres- ent case the approximate formula [40] would have been sufficiently accurate for the determination of the roots near a = 4. The solution of the complete polynomial equation will always yield an additional large root, corresponding to the large root of Equation [131] of the Appendix; in general, however, this root should be rejected since as will be shown, the initial doublet distribu- tion corresponding to it is less simple than for the roots neare = 4. (45) The initial behavior of the distributions corresponding to each of the three roots, as determined from Equations [33] through [36], and [39], is shown in Table 2. It is seen from the table that the distribution for @ = 12.1 begins with practically a zero value for m(a), with a small negative slope and with upward curvature. Since the distribution curve cannot continue very far with upward curvature, there must be an inflection point nearby. In contrast, the distribution corresponding to the other two roots begin with positive slopes and downward curvatures and hence must be considered simpler. Further- more, the distribution for the first root is considered simpler than for the second since the distribution curves are practically identical except that, for the second root, the curve is extended a distance Aa = 0.0011, in the course of which m(a) changes from a positive to almost a numerically equal negative value. If we take the point of view that the positive and negative values of this extension counterbalance each other, the curve without the extension, corresponding to the first root, must be considered the simplest. TABLE 2 Characteristics of Initial Distribution Hence, for the purpose of obtaining a first approximation, we will assume @ = 3.65 and, correspondingly, a = 0.022, m(a) = 0.000022. Often, as in this case, the labor of obtaining a and m(a) can be considerably reduced by using the less exact equations [40] through [44] instead of [37] through [39]. Since, as will be seen, the iteration formulas rapidly improve upon the first approximation, great effort should not be expended to determine an initial value for m(a). The values a = 0.022 and m(a) = 0.000022 have been derived for the profile in the é, n-plane. The corresponding values in the x, y-plane are a = -0.956 and m, = 0.000088. By symmetry we also have b = -a, m, = m,. A first approximation can now be obtained from [47], [48], [49], and (50, Stinoa A S 2,9, we have k, = 0.059. Also, from [78]: it = 0.00659, [yrax = 0.0640, [ yPax = 0.0637. Hence from [50], C = 0.328. Then, from [48], 24 @ =m, - Cf, = -0.00207; from [49], e, = 0. Finally we obtain from (47 ] m,(x) = 0.328y* - 0.00207 [83] We can now apply Equation [51] and the iteration formula [58] to ob- tain the sequence of functions p, (x). Let us suppose that it is desired to obtain a distribution m, (x) whose exact stream surface deviates from the given surface by less than one percent of the maximum radius, i.e., An < 0.002. 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The first step in the determination of (x) is to read the values of w(t) from the graph, Figure 3. wR Sin 6 and Je sin 6d @are then ob- tained. w(x) is then given by [58] and graphed in Figure 3. Repeated ee | a =] eel non eae seer ae A — 0.0012 Yo : v Iteration Formula 3 v 4 ’ %k From Von Kdrman Method ¥ 0.0010 = —}|0.0008 0.0006 0.0004 —— +———-0.0004 7 eM Saal Inte —-0.0006 LL = — + -0.0008 eee -~ — + +-0.0010 + ~— ————-0.0012 — [a [ = a nv 5; J ooora -I 09 -08 -0.7 -06 -0.5 -04 -03 -0.2 -0O1 Figure 3 - Comparison of Error Functions v(x) from 4 Iteration Formula and von Karm4n Method 31 application of this procedure gives w(x) and w(x) which are also graphed in Figure 3. The sequence is stopped at p(x) since v, has increased appreciably over yp, at x = -0.956. Hence, from [59], we have the approximate distribution m, (x) = m,(x) - 3 |, (x) + (x) + v, (x)] [84 ] to which p(x) is the corresponding error function. The distance An between the stream surface for m, (x) and the given profile is seen to be very small; the largest error, v, = -0.00007 at x = -0.956, gives a An of about one per- cent of the maximum ordinate. A graph of m, (x) is given in Figure 4. For the sake of comparison the curves for m, (x) and the original Munk approximation rae) are also shown. m,(x), M(x), and y7/4 -10 -09 -08 -07 -06 -05 -04 -03 -0.2 -0.1 Q Figure 4 - Comparison of Doublet Distributions m, (x), m (x), and Munk!'s Approximation 32 Table 4 shows the calculations for obtaining the velocity components u, v from [67] and [68], in which the integrals have been evaluated in terms of the polar angle @, according to Equations FAN, (721, eae [Woe kere ails Gauss! 11-ordinate formula is used. The values of @ and t are again taken from Table 5; the values of m,(t) are given by [84], in which the y's are read from Figure 3 and m,(t) is given in Table 5. The pressure distribution can now be obtained from [69]. Graphs of p/q are shown in Figure 5. * la | 1.00 © Direct Velocity Method oO Axial Doublet Distribution Method, Gauss || Ordinate Quadrature © Axial Doublet Distribution Method, Gauss 7 Ordinate Quadrature! ——lo.90 x Kaplan Method + Karmdén Method fo) h (o) -0.20 MO 2 4h a7 LG 65 of aS 02 Ol Be Figure 5 - Comparison of Values of p/q Obtained by Various Methods 33 ERROR IN DETERMINATION OF p/q Let A(p/q), Au, Av, and Am denote errors in p/a, u, v, and m. Then, from [69], we have at = -2(u Au + v Av) from [68], A = yl Ale and from [67] and [72], except near the stagnation points, Au = 240 i sin? 6d6 = 44m 2 0 ay Hence Az = SER) 0 Bieler If now we assume u=1, y' = 0, y® = 4m (Munk's approximation), we obtain Thus an error of one percent in the determination of m would introduce an error of 0.02 in p/q. In the foregoing example the minimum value of p/q was about -0.20. Hence an error of one percent in m would have produced an error of ten percent in the minimum value of p/q. It was found, in fact, that the results with Gauss! 7-ordinate rule deviated from the values of p/q given by the 11-point rule by less than 0.003 for the entire body. The /-point rule would have sufficed if an accuracy of only 0.003 in p/q were required; see Figure 5. If greater accuracy is desired the integrals can be evaluated in the forms [71a] and [72a]. If the latter forms are used in conjunction with the Gauss quadrature formula the values of x should be chosen identical with the t's required by the Gauss formula. This enables the entire calculations, in- cluding the iterations and the velocity determinations, to be made arithmeti- cally, without resort to graphical operations, so that the method becomes suit- able for processing on an automatic-sequence computing machine. In order to obtain sufficient accuracy in the integrations and to obtain the velocities and pressures at a sufficient number of points along the body a Gauss formula 34 of high order should be used, say n = 16. For this reason the procedure that has been illustrated in detail may be less tedious for manual application. COMPARISON WITH KARMAN AND KAPLAN METHODS In order to compare the accuracy of the Kdrmdn method with the pres- ent one, the error function w(x) was computed for a distribution derived by the Karman method, employing 14 intervals extending from -0.98£ x £0.98. v(x) is graphed in Figure 3. It is seen that the errors are much greater than for the present method, especially near the ends of the body. The oscil- latory character of (x) is imposed by the condition that the stream function Should vanish at the center of each interval. It is conceivable that the amplitude of the oscillations in W(x) may remain large even when the number of intervals is greatly increased; i.e., the Kaérmdén method may give a poorer approximation when the number of source-sink segments is greatly increased. The pressure distribution obtained by the Karman method is graphed in Figure 5. Kaplan's first method?* was also applied to obtain the pressure dis- tribution. Kaplan expresses the potential function @ in the form @= FA QL(A) Plu) where A and ww are confocal elliptic coordinates, P_(u) and QA) are the nth Legendre and associated Legendre a polynomials, and the An's are coefficients to be determined from a set of linear equations which express the condition that the given profile is a stream function. In the present case it was assumed that g Was expressed in terms of the first 9 Legendre functions and the An's determined from the conditions that the stream function should vanish at 9 prescribed points (including the stagnation points) on the body. The resulting pressure distribution is also shown in Figure 5. SOLUTION BY APPLICATION OF GREEN'S THEOREM GENERAL APPLICATION TO PROBLEMS IN POTENTIAL THEORY Let @ and w be any two functions harmonic in the region exterior to a given body and vanishing at infinity. Then, a consequence of Green's second identity?’ is ox as - [{u 9? as [85] 35 where the double integrals are taken over the boundary of the body and dn denotes an element of the outwardly-directed normal to the surface S. Also derivable from Green's formulas is the well-known expression for a potential function in terms of its values and the values of its normal derivatives on the boundary* (a) = qe |[[-2 $2 + 6% Las [86] where r is the distance from the element dS on the body to a point Q exterior to the body. When a distribution of ¢or d¢/dn over the surface of the body is given then [85] may be considered as an integral equation of the first kind for finding d¢d/dn or ¢ respectively, on the surface. If the integral equation can be solved, [86] would then give the value of ¢ at any point Q of the region exterior to the body. AN INTEGRAL EQUATION FOR AXISYMMETRIC FLOW Equation [85] will now be applied to obtain an integral equation for axisymmetric flow about a body of revolution. Let y be the ordinate of a meridian section of the body and ds an element of arc length along the boundary in a meridian plane. Then we may put dS = 2my ds [87 ] It will be supposed that the body is moving with unit velocity in the negative x-direction, which is taken to coincide with the axis of symmetry. The con- dition that the body should be a solid boundary for the flow is that the com- ponent of the fluid velocity at the body normal to body is the same as the component of the velocity of the body normal to itself. This gives the bound- ary condition ‘e = -sin y» [88 ] where y is the angle of the tangent to the body with the x-axis. Substitution of Equations [87] and [88] into [85] now gives 12 [ yo 52 ds = -{ yw sin yds [89] 0 where 2P is the perimeter of a meridian section and the arc length s is meas- ured from the foremost point of the body. 36 Now let us choose for w an axisymmetric potential function and let w(x, y) be the corresponding stream function. Then dw _ ay y dn ds and Bs avy a P J v9 Gi os = ov] - J ve as Also let U be the total velocity along the body when the flow is made steady by superposing a stream of unit velocity in the positive x-direction. Then ude Ws as + cos y Furthermore, we have dx = ds cos y, dy = ds sin y. Then [89] may be written P P P ow =| w(cosy - U)ds = =|] yw dy 0 0 0 or P [90] IP P j Uwds - | (pdx - ywdy) - ow 0 0 0 But, since w and w are corresponding axisymmetric potential and stream func- tions, we have Hence wdx - ywdy is an exact differential defining a function Q(x, y) such that g2 _ y, $e = -yw [91] But since also y OH . OY Via Oox we obtain from [91] ORO) O29) a2 37 which, by comparison with [4], is seen to be the equation satisfied by the Stokes stream function. Conversely, if Q is a function satisfying [4], it can readily be verified that the functions w and w defined by [92] are correspond- ing axisymmetric potential and stream functions, i.e., that they satisfy Equa- tions [3]. Written in terms of 2, [90] now becomes oes ds = (@-4 92) [92] If we choose for 2 the stream function of a source of unit strength situated at an arbitrary point of the axis of symmetry within the body, we have, from [10], 1/2 TY eer (ecu rere | [93] Then O62 _ y* Ox r3 and, since y vanishes at both limits, (2 - 9§2)[, = 2 Hence [93] becomes jo y* (x) distal [94] It is seen that [94] is an integral equation of the first kind in which the unknown function is U(x) and the kernel is y*/2r°. In contrast with the integral equations for source-sink or doublet distributions which can be used to obtain the potential flow about bodies of revolution, the integral equation [94] has two important advantages. The first is that a solution exists, a desirable condition which is not in general the case when a solution is attempted in terms of axial source-sink or doublet distributions. The second advantage is that [94] is expressed directly in terms of the velocity along the body so that, when U is determined, the pres- sure distribution along the body is immediately given by Bernoulli's equation [69]. In the case of the aforementioned distributions, on the other hand, it would first be necessary to evaluate additional integrals, to obtain the ve- locity along the body, before the pressures could be computed. 38 KENNARD'S DERIVATION OF THE INTEGRAL EQUATION A simple, physical derivation of the integral equation [94] has been given by Dr. E.H. Kennard. This will now be presented. Imagine the body replaced by fluid at rest. Let U be the velocity on the body. Then the field of flow consists of the superposition of the uni- form (unit) flow and the flow due to a vortex sheet of density U. Now subtract the uniform flow. There remains the flow due to the vortex sheet alone, uniform inside the space originally occupied by the body, of unit magnitude. A vortex ring of strength Uds produces at an axial point distant z from its plane the velocity Soe y*Uds 2(y?+z? ye 2 where y is the radius of the ring. Let s be the distance of a point on the body measured along the generator from the forward end, in a meridian plane. The axial and radial coordinates will then be functions x(s), y(s). The ve- locity due to the sheet at a point t on the axis will then be [SLO as = 0 ers where r® = [x(s) - t]" + y*(s) and P is the total length of a generator. The equivalence of this equation with [94] is evident. A FIRST APPROXIMATION If we again make use of the polar transformation x - t = y(x) cot 6, [94] becomes 7 U(x) sin? @d@ _ o 2 sin[@->(x) ] 1 [95] When x = t, @ = 4. For an elongated body the integrand in [94] peaks sharply in the neighborhood of x = t, so that a good approximation is obtained when U(x) is replaced by U(t) for the entire range of integration. Also, »(x) will be small except near the ends of the body so that the approximation sin [6= y(x)]= sin 6 cos y(x) = sin @ cos y(t) 39 will also be introduced. We then obtain from [95] the approximation U(t) = cos y(t) [96] Just as was done in the case of Munk's approximate doublet distribu- tion we can improve upon this approximation in terms of an estimated longi- tudinal virtual mass coefficient for the body. For this purpose we will first derive a relation between this coefficient and the velocity distribution. Let T be the kinetic energy of the fluid when the body is moving with unit velocity in the negative x-direction. Then dd a eT = -p rae dS = amp | y¢@sin yds 0 by [88]. Integrating by parts and substituting for d¢é/ds from [93] now gives 2 dd 12 QW = -mp | We as ds = 7p | U(x) y*(x)ds - A 0 0 where A is the displacement of the body. But also, by definition, 2T = k 4. Hence P A(1 +k) = mp{ U(x) y?(x)ds [97] 0 This is the desired relation between k, and U(x). Now suppose, as a generalization of [96], that an approximate solu- tion of the integral equations [94] is U(x) = C cos y. If this value is sub- stituted into [97], we obtain C = 1 + kK, - Hence an improved first approxima- tion to U(x) is U, (x) (1 + k,) cos »(x) [98] Equation [98] gives an exact solution for the prolate spheroid. SOLUTION OF INTEGRAL EQUATION BY ITERATION In order to solve [94] by means of the iteration formula treated in Reference 17, it would be necessary to work with the iterated kernel of this integral equation. Since this would entail considerable computational labor it is proposed to try a similar iteration formula, but employirg the original kernel: (t) = U(t) + cos y(t) |1 - pe U (x)as] [99] 0 r Une n 40 where r= = (x - t)? + y*(x) and x = x(s). Here also it is convenient to express the iterations in terms of error functions E(t) defined by PIAUIN (xe) mays (9) us = n E(t) = 1 j aaa ds [00] or, from [99], E(t) cos y(t) = U,,(t) - u(t) [101 ] Hence n Unsq(t) = U,(t) + cos »(t) & Flt) [102] Also, from [99], 2, E (x) y* (x) Bq (t) = E(t) tae aaa [103] where X,» X, are the nose and tail abscissae. Thus, to obtain Una ft), we first obtain E(t) from U, (t) in [100], then 5 i , Be from [103], and finally Une 't) from [102]. 3’ NUMERICAL EVALUATION OF INTEGRALS In applying Equations [100] and [103] it will frequently be neces- sary to evaluate integrals of the form (See) dx, where r® = (t-x)® + y?(x) Zo r This form, however, is unsuited for numerical quadrature for elongated bodies, since y*(x) peaks sharply in the neighborhood of x = t. Here, as in the case of the integrals for the doublet distribution, two procedures are available for avoiding this difficulty. The first employs the polar transformation [70], involves several graphical operations, but in general transforms the integrand into a slowly varying function so that the integral can be evaluated by a quadrature formula using relatively few ordinates. The second retains the original variables and eliminates the peak by subtracting from the integrand an integrable function which behaves very much like the original integrand in the neighborhood of the peak. The numerical evaluation of the resulting inte- gral on the second method requires a quadrature formula with more ordinates RAR 4y than the first in order to obtain the same accuracy, but, since all graphical operations are eliminated, the second method is suitable for processing on an automatic-sequence calculating machine. The result of the polar transformation has effectively been given n [95]. We have “1 E(x) = x) E(x) sin? @ ai pe ) 4 J x= | sin[6 ->(x) [104] 0 r° where x - t = y(x) cot 6 [70] It is desired to evaluate this integral for a series of values of t. In gen- eral this can be done with sufficient accuracy by means of the Gauss 7- (or 11-) ordinate quadrature formulas. This gives 7 (or 11) values of 6 at which the integrand needs to be determined for a given t. The value of x occurring in the integrand is determined implicitly, for given values of t and 6, by the polar transformation [70]. In practice the 7 (or 11) x's can be obtained graphically from the intersections with a graph of the given profile of the / (or 11) rays from the point x = t on the axis at the angles required by the Gauss quadrature formula. If greater accuracy is desired, these graphically determined values of x can be corrected by means of the formula t-x_ + y(x_) cot 6 xe bo, See ae [105 ] & I-y! (x,) cot @ in which x_ is the graphically determined value and y!' denotes the derivative of y with respect to x. Now let us derive an alternate, completely arithmetical procedure for evaluating the integrals. Put k' (x, t) = Fale t) [Gxt Pe eles) RY where y> = f(x) is the equation of the given profile and y* = g(x, t) is the equation of the prolate spheroid whose ends coincide with the ends of the given body, and which intersects the given body at x=t. i-.e., 42 (x-X 9) (x, -x) g(x, t) = f(t) (t-x,)(x,-t) [106] The length-diameter ratio A of the spheroid is given by (t-x 4) (x, -t) ee ea ee [107] whence the longitudinal virtual mass coefficient k(t) ean be obtained from [26]. Since U(x) = (1 + k,) cos y(x) is an exact solution of [94] for the prolate spheroid, we have Jets, t) ax = Te TE] [108] We now obtain, from [98], [100], and [108] 1+k 1 Pree [109] 1+k, (t) [ tetx, t) - k'(x, t)]dx - i) Also [103] may be written in the form n+1 By i(t) = Balt) - 5] k(x, (E(x) - Eye) Jax - B,(t)[ k(x, tax 0 0 But from [98] and [100], 1-E, (t) Za Nese, ie )lebe = 2 === i: THe Hence we obtain [etx, t)[E (x) - By(t)jax [110] pean ameTegm ata ad 0 ILLUSTRATIVE EXAMPLE The present method will now be applied to the same profile [78] as before. By way of contrast with the semi-graphical procedures previously used, a completely arithmetical procedure will be employed. 43 The velocity U(t) will be determined at the 16 points along the body whose abscissae are ty = by the Gaussian values for the 16-point quadrature rule, Table 1. Since the body is symmetrical fore and aft, it is necessary to determine the velocity at only half of these points. Values of y(x), cos y(x) and k, (t) for these points are given in Table 6. In order to apply the Gauss 16-ordinate rule it is necessary to eval- uate the integrands in [109] and [110] at the 16 Gaussian abscissae x, = & for each of the 8 values of t,- Thus, there are 16 x 8 = 128 values of k(x, t) and of k'(x, t) to be determined. The matrices Ki = R k(x, t,) Lae = Rik (x5, t,) where the R,'s are the Gauss weighting factors, are given in Tables 7 and 8, and applied to evaluate E(t) from [109]. E,, Ej, and oy are then obtained from [110]. u(t) is then given by [102] and then p/q by [69], in the form p/q = 1 - De The arrangement of the calculations and the and results are given in Table 9. The graph of p/a is included in Figure 5. TABLE 6 Values of y, cos y, and k, (x) for Application of Gauss 16-Point Quadrature Formula Psa [rr |v Prove -0.9894.009]0.0408548] 1 .8965483) 1.0856 QO. Se: 9445750] .0903198]0. 7464764) 0.6412 093389 .8656312| .1324422) .3917981 .088359 27554044) .1642411] .2099651 .081862 .6178762} .1848527 4580168] .1955501| .0393076 .2816036| .1993706| .0089607 -0.0950125 |0.1999919]0.0003431}0.0003431 . 1020867 .074689 .067885 .062506 0.059509 44 TABLE 7 f(x,) Matrix of Values* kK,. = R, = ji J p 2 3/2 [(«; t,) + t(x,)] 0715 - .0098 .00505 .00341 .00208 .00121 .00062 00027 ‘4 1 Sie OOO OMe .00009 3 2 1610.00001}0.00001}0.00001| 0.00001 |0.00001| 0.00001 | 0.00002 | 0.00004 TABLE 8 Ban ea )) Matrix of Values** in = R ——__— _=_____ [Cx v7 t,) +e(x5, t,)] 0.00003 .00044 .05223| .06265| .08976] .16366} .40559| .86501| .34633 03061} .03502| .04556] .07041} .13995! .39080] .91586 01805] .01998] .02432] .03352| .05520} .11924| .37496 01058 .011 01329] .01094| .02457] .0 Nf 10217 | .03396 11| .00334] .00351| .00385 oon] .00561| .00787] .01303 3 ie .22699| .4 : 33504) 07174 I. Be 11637] .19118] .418621 .80927] .34255] .07641 9 10] .00607| .00645} .00725] -..00876] .01165] .0178 12 .00178 .0021 .00262| .00346] .0052 00081] . 15| .00007 1610.00001| 0.00001] 0.00001] 0.00001] 0.00001] 0.00001] 0.00001] 0.00002 *For i>8useK.. =K : ; Ji dP a) 5. dbt(at **For i > 8 use K'. = ail ‘ ae lag IRL 45 TABLE 9 Calculations for E(t) and U(t) Assume k; = 0.06: Put Boy = Rk 7 Si Simi 15 (ze sa (e,)) =D ji fil dj ag a” ial} nj (a) S OnSRSONs aes p & Og —— A 6. a ab . 3 v= Vo Tera (Gay) T . 2) ae eee re) 0.66458 28oee .00118 .00566 .00078 .00706 .00048 00802 .00028 .00851 -00851 .00802 .00706 00 3 0 0 Bilin : : 7 : -00171 -0520 -00062} .00078 ~00016 -03333 "4 -00027} .00029 =0. ae .01799 -00007} .00007 -0.00730 12 0.00001 |0.00001 0} Us (x, ) = 0.5450, lt+tk (2) 02S OLMIED C287? <2) as Oly T—a 0.96947 1 }0.20313]0.20017| +0.00296 . 68926} .6892 -0.00001 +0. aos .21382|_. i ‘04486 .06265 :01779 .02187| .03502 .0131 -01140|_ .01998 .008 0010 02 - 00029 .00639| .01143 0050 09357 00060 02563 00016 .00379| .00645 . 00266 .0 .00031 02312 . 00009 .00228] .00350 .00122 064.83 .00015 .01875 . 00004 .00131| .00178 0004 OU -00006 «01 344 . 00002 02603 5 .000 067 . 00081 - 0001 $0028 soueee -0. eggiee .00008] .00008 0.00001 0.00001 Be 46 TABLE 9 (Continued) l+k (en) Oe pees aeceihs Oo = Oe at Taco = 0.97391 +0.00206 } +0.0003 : +0.00013 : +0. 0000 +0.00381 0.0106 +0.0031 A +0.00096 : +0.0002 -0.00001 +0.00240 5} .00009| . +0.00001 +0. 00327 16]0.00001 {0.00001 0) +0.00634 = -0.06312 -0. ras = -0. OG 3 = 7005951 = +0.01546 = +0.00427 E sb) ra +0.11275 l+k 0.00338] 0.00271 +0.00067 +0.03333 +0.00011 +0.00004 +0. 00403 +0.00001 04802 .00712 02603 007 4 : 00044 00245 .00014 00465 01534 +0.00483 +0. +0.00152 +0.00150 +0. 00049 0) 0) ) 0) 0) 00068 One =0. 00783 =05 =0.00230 =0. 00165 -0. 00069 .02019 038 00557 : poole 00303 .01697 .05666 .00303 ; .00081 “oot 01045 06754 ; 6 0176 00041 -OOKY, -00557 20075 .00077 : .00260 ; -00035 00099 , .00014 -0.00026 ; -0.00004 -0. 0 0 0 +0.00004 onsen +0. pod +0. +0.00150 +0. poco! . 02603 00245 0.00001 {0.00001 +0.03333 +0. ae +0. 00403 -0. Ree -0.01290 = -0. aa -0. ere = +0.04u17 = +0.01079 = +0.00277 = +0.00071 = 0.09827 U(x, ) = 1.0946; + = -0.1981 See 1 47 TABLE 9 (Continued) l+k Gatsepi; (e) x, = -0-617876; cos 7 = 0.9948; +0.00026 +0.0000 j +0.00001 +0. 00568 0 ee Uy 0005 ; .00018 0047 0 +0.0000 011 : 00257 i 000 00315 0002: 00616 4 4 +0.00641 ; aE +0.0016 +0.00056 0 -0.00138 -0.0005 00234 0002 ; 0028 00012 : : 700023 283 "03792 ; 00009 -0.02006 0.00010 ; -0.00003 0 0) 0) 0 +0.00015 0187 : : +0. 00001 +0. 00165 00011 ; ; +0. goat 00315 +0. Gyo 07 344. 00410 +0.01651 +0.00568 1 E 3 i : : 10 1 12 13 14 15 16 U_(x_) = 1.0864 ; 2= -0.1803 5.5 q 1 + ky f X_, = -0.458017; cos y = 0.9992; —————— = 0.99262 (2) 36 [1 +m (x@)] 7 EB. =p me (i. 13 }) E_ -E K -E__) 2j 26 je 23 26 33 36 age eal <3 +0.00012 4 +0.00002 +0.02182 +0.00001 +0.00706 0 .00146 ; Cee aoe) . 00008 00548 +0.00002 01548 -00033 00453 .00010 01081 ata 0030 00026 +0. 00537 +0.001 +0. 0013 +0. 00048 0) 0) 0 0 =. -0.00168 -0.00096 -0.00037 0068! oe | .007 4 00015 00688 0002 00145 700005 -0.00437 -0.00007 -0.00096 -0.00002 0) 0 0 0 +0.00008 +0.00531 +0.00002 +0.00138 +0.00001 a; ; .00007 sae a 00002 700305 +0. 00001 i ; 00004 : +0.00001 00453 0) j 3 +0.00001 : 1875 0) 00548 0) i PO 0 +0.00706 0) = +0.00379 f = +0. = +0.00052 i = +0.00029 E_ = +0.00537 E_ = -0.00002 E_ = -0.00026 E = -0.00016 16 26 36 46 + E eas = = ; B=. 3 SSE = ORGY Us\(ocs) 1 06H e025 48 TABLE 9 (Continued) l+k (g) 7 = -0.281604; cos y = 1.0000; ea = 0.99764 +0.00000 -00001 . 00004 -00011 +0 .00000 00006 00006 “00002 +00001 +0.00000 7) = (h) x,= -0.095013; cos y = 1.0000; k! 0.00003 0.00003 0.00000 0.00851 0.00000 00044 00037 .00002 00693 00001 ony .00008 00 fe .00002 00 .00019 00448 0000 Scone 00283 0000! 01079 00145 .00013 00507 00049 .00917 00000 .00000 .00000 00000 00000 .00000 00420 00049 00005 00456 00145 .0000' 00321 0028 .0000 0051 00339 00179 0044 00002 .00183| .00107 00076 00548 00001 15 OE .00024 00021 : .00693 .00000 16|0.00004] 0.00002 0.00002 0.00000 0.00851 0.00000 J = +0.0432 J = +0.01219 f = +0.00063 [Bs = -0.0233 -0.00690 -0.00038 k i 0.03456 49 SUMMARY Two new methods for computing the steady, irrotational, axisymmetric flow of a perfect, incompressible fluid about a body of revolution are presented. In the first method a continuous, axial distribution of doublets which generates the prescribed body in a uniform stream is sought as a solu- tion of the integral equation where r is the distance from a point (t, 0) on the axis to a point (x, y) on the body, r= = (x - t)® + y=(x). A method of determining the end points of the distribution and the values of the distribution at the end points is given. If the equation of the body profile, with the origin of coordinates at one end, is ai 2 3 uP (8) SEL Ok te BS ar ELIS HP Goc a very good approximation for the distribution limit a at that end, when the coefficients a,, a5, .-- are small, is given by a 1 1 ae y+ a, +> Vala, if a, 2 0), 162 a, is negative, the term containing it is neglected. The cor- responding value of the doublet strength at this point is Inia a8 uD } o§ a Vaia, Formulas and tables for determining a and m(a), which may be used when the above procedure is insufficiently accurate, are also given. The values a, b, m. = m(a), m, = m(b), f= y-(a) and fe y-(b) are then used to obtain the a approximate solution of the integral equation i fy (Date a ey SEE ames — mei X) =" Cl Ven oa pend Sehr! pacacn Wid pen eat a DES 50 where 1+k, (71 ; m 2 | eax - 3(b-a) (m, +m, ) Tea Oe Re SE ee i yrdx - 5(b-a) (f,+f,) and k, is the longitudinal virtual mass coefficient for the body. This approximation is used to obtain a sequence of successive approx- imations by means of the iteration formula B i, ((te)) my (x) = (x) +3 v(x)/p - [ + at] When a doublet distribution has been assumed, the velocity components at a point (x, y) in a meridian plane are and the pressure is given by where q is the stagnation pressure. The iterations are most conveniently performed in terms of the dif- ferences between successive approximations to m(x), which also furnish, at each iteration, a geometric measure of the accuracy of an approximation. Simpler forms for the velocity components at the surface of the body are given in terms of this difference or error function. Gauss! quadrature formulas are recommended for the numerical eval- uation of the integrals. Two methods of carrying out the iterations are given. The first employs a polar transformation and a graphical operation be- tween successive iterations; the second is completely arithmetical and is suitable for processing on an automatic-sequence computing machine. All of these procedures are illustrated in detail by an example, in which the semi- graphical method is employed. The accuracy of the method is analyzed; the re- sults are compared with those obtained by the methods of Karman and Kaplan. 51 In the second method the velocity U(x) on the surface of the given body is given directly as the solution of the integral equation [Sec ieee 0 ae where s is the arc length along the profile, x is equal to x(s), and 2P is the perimeter of a meridian section. An approximate solution to this integral equation is U, (x) = (7 + k,) cos y(x) where kK is the longitudinal virtual mass coefficient and y = arctan — U, (x) is used to obtain a sequence of successive approximations by means of the iteration formula U nay (t) = U,(t) + cos y(t) | “(2 U,(x)¢s Here, also, the iterations are most conveniently carried out in terms of the differences between successive approximations to U(x) which also furnish a measure of the error in the integral equation. Two methods of carrying out the iterations are again available, of which one is semi-graphical, the other completely arithmetical. The latter technique is employed on the same example as was used to illustrate the first method. 52 APPENDIX END POINTS OF A DISTRIBUTION An approximate method for determining the end points of a distribu- tion and its trends at the ends will now be described. Let y* = f(x) be the equation of the given profile extending from x = 0 to x = 1; let m(x) be the corresponding doublet distribution, extending from x = a to x = b. It will be assumed that 0 < a << b < 1 and that a is near 0, b is near 1. Then m(x) is given by the integral equation (eit) Various conditions on m(x) may now be obtained by differentiating [111] repeatedly with respect to x. We get f B@ltex - 26 + e1(x)] at = 0 [112] a r? : 5 1 1 " , Ji m(e)[-S (2x - 2b + 8 2 Lieve )Jat = 0 [113] [ince 22 (2xa2t- tun) oe 1) (24£") (2x-2t+f!) n Aas = 0 [114] a Uy? er! r> When x = 0, r = t and, writing f(x) as a Taylor expansion (x) 2 3 Anos ap BLO ar BL Se SP oo c [115] then also f'(0) = a,, f¥(O)) s 2a,, oY (O) Ss 6a. Now, setting x = 0 in Equa- tions [111] and [113], we obtain b m(t ) ull it = at = 4 [116] 6 f mela, SUE atH= nO 117] b I a Se, - 20a,t + 4(4 - a, )t?|at = 0 [118] D2 9 . me [58s - 210a,t + 60a, (6-a,)t? + 4O(3a,-H)t? + ala,t*|at =o [119] Also assume that m(x) may be expressed as a power series = 2 m(x) = ¢, + c,x + Ce ap boc [120] Then Equation [116] gives HL -L) 4 eft-4)4 0, ms2e or, neglecting 1/b° in comparison with 1/a* and setting b = 1 in comparison walide Wily 2 al ee tp 2c a(1-a) + 2¢,a" log A oad = [121] Similarly, Equations [117], [118], and [119] give, approximately c, (3a, -8a) + 4e ala, -3a) + 6c,a"(a, -4atta*) = 0 [122 ] 2c, | 5a, -2a,a+6(H-a, )a®| + Ye a/3a -15a,ar4(4-a, )a*| [123] +0,8°| 15a -80a,a+2u(4-a, Ja? | = 0 1 -240a,a+80a, (6-a, )a?+6l (3a, -4)a9+48a,a*| 30, |35a, 240a,a4 Oa, ( -a,)a + (Ja, Ja“+48a,a +2lte, |5a,a-35a,a2+12a, (6-a, )a°+10(3a,-4)a*+8a,a>| [124] the, [35a,a?-252a,a°+90a, (6-a, )a*+80(3a,-4)a°+72a,a°] = 0 Equations [121] through [124] are sufficient in number to determine the un- knowns a, c Since the latter three equations are linear and homo- Cc Cc Oe stole geneous in Cro Gap and c,, a can be determined from the condition that the de- terminant of their coefficients must vanish. In this way the following equa- a tion of the 7th degree ina => was obtained: 54 a(a - 4)?(5a* - 830° + 28807 - 3680+ 128) - 96a or( oe - 4) + 4a ala - 4)(5307 - 148a + 128) + 1152a, a, (2c - 3) [125] + 72a, (a - 4)? (50° - 250° + 40a - 16) + 48a,a,a(3e - 8) 2 = Asc e C!S M\(Gee— NS slo) = Vis2e eo oY) = © Corresponding to a solution a of [(125], Cy» Cy» and c, can be obtained from Equations [121], [122], and [123]. The solution of the latter equations gives ¢,D = -4a?| 30° Sr tO SO ey cp Beles = Gear 10 = Ya. )] [126] e,D a|150° = ser ts Glee S oehi ae Glee ap UtRW(Ger = ais At & 6a,)) [127] e,D = -4[(a - 4)2(a - 1) + Hag] [128] where D = 2(9a° - 94a? + 272a - 192) + 8| (a - 4)?¥(e - 1) + Ya, | 1og a + 96a, -2a(15a° - 264a° + Guta - 768) - 384aa, - 96a*(5a° - 2ha + 2k) + 576a7a, [129] The initial doublet strength at x = a is = 2 m(a) = ¢, Uaaeene CLES Se lows or, from Equations [126] through [129], m(a) = -2-[(a - 4)(a? - 120 + 16) + 4Bala - 4)(a - 2) + 16a, - 96aa,] [130] ' Equations [125] through [130] determine the end points of the distri- bution and its initial trends. In general, Equation [125] will have more than one real root. In this case the initial trends corresponding to each of the roots should be examined, and that root chosen which appears to give the "simplest" trend. DD) The equations can be solved explicitly in the case of a very elon- gated body for which a,, a,, a,, ... in [115] are all very small. First let us suppose that they are so small that all the terms in [125] containing them are negligible, so that the first product term alone may be equated to zero, abo San ala - 4)?(5a* - 830° + 28807 - 3684+ 128) = 0 [131] whose real roots are a = 0, 0.547, 4.0, 4.0, and 12.429. Let us consider the solution w = 4; i.e., a= ie Since the radius of curvature at x = 0 is avV/en this solution is seen to be in accord with Kaplan's assumption for the end points of the distribution. Furthermore, sub- stituting w = 4 into Equations [129] and [130], we obtain, to the same order of approximation, a a, D = 64, Gy = Tas Os img Cie 0) whence 2 Ea Sa m(x) = Rare ae Te 8 m(a) = 0 [132] In order to obtain a second approximation it will be assumed that not only a,, 43, 4, ... but also (@ - 4) are small to the first order. Then, neglecting terms of third and higher order, Equation [125] becomes -3072(a - 4)? + 6144a (oe - 4) - 3072a, + 768a,a, = 0 [133] whence a=4+a t 1 yaa, [134] provided a 2 0 Corresponding to this value of a we obtain from Equations [126] through [129], to the same order of approximation, *The smaller of these two roots has given the preferred solution in all cases tried thus far. 56 2 ay i m(x) = c(-— +ax+a x + Lg where [135] ib) fo cad + +5 10) and m(a) == Ca" yaa, [136] The expression for m(x) in [135] may also be written as 2 n(x) = (-,2 + ¥) [1354] When a, < 0 the solution for a in [134] indicates that there would be no real roots near a = 4. In this case a graph of the complete polynomial in [125] should be examined either for the possibility that more complete cal- culations would show that there are real roots near a = 4 nevertheless, or that the maximum value of the complete polynomial in the neighborhood of o = 4 is so nearly zero, that the value of @ corresponding to this maximum may be taken as an approximate solution. On this assumption, the second order analy- sis would give Gye ob AP Bop a, < 0) [137] Since a, does not occur explicitly in Equations [135], it is seen that they would also be obtained, to the same order of approximation, if the value of o in [137] were substituted into Equations [126] through [129]. If it is determined that not even an approximate solution can be assumed near ow = 4 it would be necessary to consider solutions in the neighbor- hood of the other roots of Equation [131]. In order to facilitate the computations for graphing the polynomial in [125], the functions A(a), Bla), ... H(a), where a1 A(a) = ala - 4)?(5a* - 830% + 288a7 - 368a + 128) Bla) = 72(a@ - 4)?(50° - 250° + 40a - 16) C(a) = Yala - 4) (5307 - 148 + 128) D(a) = -288(a@ - 4)(5a® - 160 + 16) [138] E(a) = -96a(3a - 4) F(a) = 1152(2e - 3) G(a) = 48a(3a - 8) H(a) = -1152(@ - 3) have been tabulated in Table 10. In terms of these functions, Equation [125] becomes 2 2 2 A +a,B+a,C + a,a,D + aj,E + a.a,F + a,a,G + a,a,H = 0 [139] It is of interest to compare the approximate value for oa from Equa- tion [134] with the exact value for the prolate spheroid y* = sab: =e) ein this case we have 1 Seer Te2en nee 3 and the exact value of a is 1 1 1 mas 2y/1 a ape Rca Pec a NE a un* But when the length-diameter ratio A is large, Equation [134] gives the ap- proximate value a = 4 = sa which is seen to consist of the first two terms of the series expansion of the exact value of a. Table 11 shows that the approx- imate formula gives excellent agreement with the exact values even for very thick sections. Both the exact and the approximate formulas give m(a) = 0. Thus the present approximate methods work very well for the prolate spheroid. 58 TABLE 10 Functions for Determining Limits of Doublet Distributions [a [xe [ote [at [we rf ro fot Pn] (a) 3456.0 -13409 S bs 16230 B. -3225 .6| 14227 65 .3|-2995 Al 12414 89 .3|-2764 .1] 10782 107.5] -2534 0 .6 3 8 4 9324 -2304.0 2880 8029 -2073.6 6890 -1843 32 264 5898 -1612.8 253 5044 -1382 .4 2419 4320 -1152.0 2304 3716 -921.6 2188 3225 .2 2073 2838 a} 1958 2545 4 1843 MEDMO NERHDO NEDHMO NEDHHDO NERHHMO 0 0 0 7 a2 5 5 3B si an .2 5 9) off 4 33} 8 0 4A 6.6 .6 4 ou oll 4 4 5 4 6 32 8 1 al 03 0 0 0 0 .2 03 3 6 9 5 of J 6 ol 9 33} 4 2 5 9 23) 03} 9 9 9 33 .8} 2340.0] -72.0 1728 ol 9 .9] 2211.8] -122.9 4 5 5 2 2152.8] -179.5 8 32 at : 2154.2] -241.9 .2 8 4 -6| 2207.5] -310.1 6 .0 0 .O| 2304.0} -384.0 .0 8 .0 .8] 2435.0) -463.7 4 =) mil -3] 2592.0} -549.1 8 ae} 0 -1| 2766.2) -640.3 .2 .2 ot .3} 2949.1] -737.3 6 23 8 .8| 3132.0] -840.0 .0 0 29 2 3306.2] -948.5 4 ‘f 5 : 3463 .2]-1062.7 8 3 9 4) 3594.2)-1182.7 o2 29 9 .2| 3690.7)-1308.5 .6 0 0 .0] 3744.0}-1440.0] 3456.0 8 .2 4] 3745.4)-1577.3] 3686.4 a2 ol .2 .5| 3686.4)-1720.3] 3916.8 4 6 8 .0| 3558.2]-1869.1] 4147.2 6 9 3 .8] 3352.3|-2023.7| 4377.6 8 03 03 .8| 3060.0]-2184.0] 4608.0 0 o® 5 .8] 2672.6]-2350.1] 4838.4 ip 8 of .5| 2181.6]-2521.9] 5068.8 4 8 2 .0 Taubes -2699.5| 5299.2 6 0 6 8 53 .9|-2882 .9] 5529.6 8 -3072.0 0 0 5 9 .2| -3266.9 4 2 5 8 2) -3467.5 8 4 5 33} -8] -3673.9 22 6 oD 3 -5|-3886.1 6 8 8 03} -0]-4104.0 0 0 .6 B2 .0}-4327.7 4 12 2 otf -0}-4557.1 8 4 4 é 8 03 2 6 5 9 a5 6 8 0 offo NM ENaON NW EN 10 | -2790720 -615168 11 | -3417260 -897120 12 | -1966080 -1253376 13 | 4706910 3351348] -1692576 -2223360 -2854368 22052800 4728640 A+a,B+a,C+a.a,D + a,E + ajaF + a,a,G + ava A = ala - 4)?(5a* - 830° + 288° - 3680 + 128) E = -96e(3a - 44) B = +72(a - 4)? (5a° - 250° + 40a - 16) F = 1152(2a - 3) C = Hala - 4) (530 - 1480 + 128) G = 48(3a - 8)a D = -288(a - 4)(5a* - 16a + 16) He se 6 3) 29 TABLE 11 Comparison of Exact and Computed Values a of a = aa for a Prolate Spheroid A 2 Exact o Dol D2 |B Approximate a@| 3.750 (eo) On WW \N No} o re \N No} = DO REFERENCES 1. Goldstein, S., "Modern Developments in Fluid Dynamics." vol. II, Oxford: Clarendon Press, 1938, p. 523. 2. Rankine, W.J.M., "On the Mathematical Theory of Stream Lines, Especially Those with Four Foci and Upwards," Philosophical Translations, 1871. 3. Taylor, D.W., "On Ship-Shaped Stream Forms," Transactions of the Institution of Naval Architects, vol. 35, 1894. 4. Fuhrmann, G., "Theoretical and Experimental Investigations on Bal- loon Models," Dissertation, Gé6ttingen, 1912, published in Zeitschrift ftir Flugtechnik und Motorluftschiffahrt, vol. 11, 1911. 5. 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