L ^EMB26 ENGINEERING RESEARCH INSTITUTE Page EMWO __________6 _ l __UNIVERSITY OF MICHIGAN age April 7, 1949 THIS ISA REPORT OF WORK IN PROGRESS. IT IS TENTATIVE, AND SUBJECT TO REVISION. TEE DERIVATION OF FIRST ORDER EXPRESSION FOR LIFT COEFFICIET, MOMENT COEFFICIENT AND CENTER OF PRESSURE LOCATION FOR AXIALLY SYMMETRIC 'LBODIES IN A SUPERSONIC STREAM EMB-26 I~~rlL \q-1 Prepared by.Z'/ A'< /o W. a. Dance Approved by -b. /.; F. W. Roas

ENGINEERING RESEARCH INSTITUTE Page EIB-26.UNIVERSITY OF MICHIGAN a SYMBOLS ase base area of body A m an cross-sectional area of body meanc.p.: center of pressure or point of zero aerodynemic omeant %:. lift coefficient CL llft WErrirAt MOMEhT CM: s. coefficient of moment about body vertex k^: function representing distribution of source or sink strength along x axis at pointse = (A s function reprsenting distribution of doublet strength,along x axis at points) 1 length of bod 1 c.p. distance from vertex to center of pressure of body L x lift norml to x axis M: aoment of L about body vertex p: static pressure R s radius of body at station x u,v,vw perturbation velocities in x, r, 9 directions, respectively (AO: free stream velocity x,r,0: cylindrical coordinate system used. x axis oriented along body axis of syaetry

ENGINEERING RESEARCH INSTITUTE Pa E-MB-26 UNIVERSITY OF MICHIGAN Pge EEK SYMBOLS 0( - angle of attack /3://~ T-/ - compressibility factor:: source, sink or doublet position along x axis - ifree stream deMity /W * - perturbation potential /: perturbation potential for axially syaaetrioal flov tz = perturbation potential for spatial flow Sm!ary of Conclusions Simple first order expressions for lift coefficient, pitching moment coefficient and center of pressure location, are derived, using solutions to the linearized potential equation for spatial supersonic flow. These expressions have been used as a basil for examining the trends and orders of agnitude of these aerodyamic parameters for various supersonic bodies. In all cases where these formulas are used, more rigorous solutions should be and have been used. It should be noted that these expressions simply point out the way to an optiaum missile shape and are first order solutions only in the case of very thin bodies of revolution. These relations have their counterpart in subsonic incompressible flow theories such as the thin airfoil theory and the theory of airship bodies employing apparent mass concepts. The expressions as derived are: CL - 2o a,/vanel cfi r2o( W6^5I l ^-^-^.^ 0 =R [l - 4 e 3. "M I I....

ENGINEERING RESEARCH INSTITUTE Page EMB-26 f UNIVERSITY OF MICHIGAN t 5 In the aerodynamic design of supersonic missiles some theory has to be used to determine the conventional aerodynasiC coefficients. Such theories employ solutions to the rigorous non-linear, hyperbolic, partial differential equation of flow or its linearized derivative. As yet, knovn solutions to the rigorous non-linear equation are few in number and restricted in nature. Such solutions include the TaylorMaccoll solution for the flow around a cone, the Rankine-Hugoniot relations leading to two-dimensional deflection through a shock wave, and the Prandtl-Meyer solution for expansion flow around a convex corner. The method of characteristics for problems having a twodimensional natural ocrdimte system is a solution to the rigorous equation only when the characteristic quadrangles become smaller approaching the limit. None of these rigprous solutions offers a ready tool for examining the spatial flow around supersonic bodies* A standard approach to physical phenowna, obeying laws defined by non-linear differential equations, is to attempt to examine and Justify the approximate solution to the linear differential equation derived from the more rigorous non-linear equation. In aerodynaxics, numerous solutions have appeared for problems of bodies mlarsed in the flow field of the linear potential equation for supersonic flow. The solution dealt with in this paper is one of the earliest solutions of the linear equation for flow about axial symmetrical bodies of revolution. Because the method of linearizing the rigorous non-linear potential equation was the method of small perturbations, these solutions are restricted to slender pointed bodies at small angles of attack. The expressions derived herein are first order approximations to the more rigorous solutions of the linearized equation. As such their application is more restricted than are the more lengthy solutions. However, these expressions can be used to indicate desig trends and orders of itude for preliminary analysis prior to obtaining the more lengthy solutions to the linearized equations appearing in Ref. 1. The following text contains the derivation of the first order exprest. sions for lift,. moment and center of pressure location for very slender axially symmetrical bodies in a supersonic stream. The linear potential equation for spatial supersonic flow is: (/-^72) 31- ^ 52 a ^* ^ ftr dur-o (1) Following in form the procedure outlined in Lamb (Ref. 2) the general solution to (1) when Mach number exceeds one is )w- 5; COOV.O^f f 9 Ad + s6 Use 5-,(2) 5sa

I I I I F 2B-26.ENGINEERING RESEARCH INSTITUTE Page............ UNIVERSITY OF MICHIGAN. where ) and O must satisfy the equation below obtained by substituting equation (2) into equation (1). d} t, t} ' Ld~{8 I ), 4 Z~ t ~ E' (3) A known solution tc equation (3) is 5 _ ( d / j (U) where 4 is identically the solution to equation (5) given by Karman and Moore (Ref. 3)* (/ )J ~ dt (5) Solution (2) to equation (1) was introduced by Lamb and further developed by Ferrari (Ref. 4) and Tsien (Ref. 5). The solution to equation (5) is O - 4_ where _ Solution (6) becomes more familiar when put in the form of a new variable, where using the substitution below f: -=. _E- <^-^C _ '' equation (6) becomes /-/s 0 /9tr (7) 0 Equation (7) is the form of the source-sink solution often used in subsonic source-sink flow solutions. Here f() is a function representing the distribution of strengths of the sources and sinks at points. of the x axis of the cylindrical coordinate system of equa tion (5}. x and r are the coordinates of the particular field points for which the potential due to sources or sinks at points I is desired. According to (h), differentiation of (7) with res ect to r --

I I EMB26 l....ENGINEERING RESEARCH INSTITUTE Pae 5 2...............UNIVERSITY OF MICHIGAN Page will yield a solution for (1). However, (7) blows up at the upper limit aid hence the differentiation must be performed on (7) in the form of (6). This is not the case in subsonic flow where the limits of (7) are constant. Differentiate (6) with respect to r. t(X^). /-rL ti 2?4Xo~l)=v-vft04c^)c (8) By taking the iritexS equal to one in equation (2) and (h) the corresponding solution to (1) is: wI* 4 t^/^( - </ * =d - (9) using sr _ D o (9) becomes r2t~ ~ _~ are d #! J(10) (10) represents the potential for field point (x, r, ) due to a system of doublets distributed at points P along the x axis according to the function Now, visualize a slender body of revolution placed in the linearized compressible supersonic stream of equation (1) at an angle of attack. Assume a system of sources and sinks and doublets is distributed along the x axis at points J according to the rule of functions /() and f\ respectively. These siigularties are so arranged such that the effec of them in displacing the stream is to reproduce the effect of the body displacing the str-e Thia restriction upon the singularity distribution will be ela:birated upon further in the statement of the boundary conditions of solutions to (1) and (5). It is desired now to obtain aa siple first order rule for the lift, moment about bodV vertex, and center of pressure location using solution (9). Let L I lift of a slender. A'ally saymetrical body in the direction normal to the x, a plane* Let M w moment of the slender axially syaetrical body about the vertex of the body. While determining a first order approximation to L and M, reference will be made to the sketch below. i --- - ~ -~~~ ~~~~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

1 El2 ENGINEERING RESEARCH INSTITUTE I ______ EMB-26_________ I __UNIVERSITY OF MICHIGAN 6 0 I. 4t ---- ^ —'"Y/~~~~~~~~. w W- -- -- -- Ir "~ I" — I.- - - - - I -. —l l -! 0 0. m I. -. ~ i __.040 _ I ~ -- - - - - - I 0 - Z/. I U.O __ _,I -~~ --- —-- - j

ENGINEERING RESEARCH INSTITUTE P EMB-26 UNIVERSITY OF MICHIGAN Pge 7 The lift L on the elemental surface is: i.- d R rB dt c (11) whereap s p surface - p free stream Correspondingly, the moment of this elemental area i1: a /-M ^.-^ rp Cr^. r Aic (12) Integration over the surface now yields the general expressions for lift and mgaent about thq vertex. That is, q Q;r L - | fi (13) O o and T z M <s f /- t 8R7^ y 4; (14) O Q Within the rigor of the linearization process employed to obtain equation (1) from the rigorous non-linear flow equation, the expression for pressure coefficient is C W = (15) where ^ ^ v!^a (16) and hence, f-Z-6 ^ (17) Substituting (17) into (13) and (14) yields r Ir( ~L - -8" 2 SI ~/6c f f^ r (18) 0 O r (19) I

L ENGINEERING RESEARCH INSTITUTE EM-26 UNIVERSITY OF MICHIGAN Page We nust now determine the value of d as R: r O. (i.e. for a first order, very slender, body). Here the boundary conditions cowe into play. The boundary condition on solution (10) is that the derivative with respect to r (the r component of the perturbation velocity) must be equal and opposite to the component of the free streac velocity in the r direction in order that the flow be tangent to the surface of the body. To illustrate: I, VO 4....~i1l, (r- )~2 - (20) ad d 4I 4e. Uf C.-q B (21) Where r R, (10) becomes oi....; s ' (22) We must determine the value of \-, /r Ra as R O. Noticing again that in (22) the denominator vanishes at the upper limit, return to - ---

I12 ENGINEERING RESEARCH INSTITUTE........ ~"1~:26.,, ' i:_,............ Page -_ MeB~-26 __ _jUNIVERSITY OF MICHIGAN 9 O I I... I.,.. the more convenient forn of (9) to perform the desired differentiation C0 substituting s x -/,R cosh u yields eL4^; lire? ^//S (24) Let RB~O in (24) as for very slender bodies.. z-/ (d)e $7 + /p t9-FJ" (25) eo4 -The integral appearing in (25) can be integrated by parts. Performing this integration yields 3~8,~~E) k (;4 ~; 7 k A (26) as R 0 o, f x -A R cosh u -+x. Noting that g (O) s O for pointed bodies, expression (26) becomes for slender bodies, ( -rvo /J (27) Substituting this value into (21), yields '/ie 'd C) "c^(28) o o Now ie A (x) the cross-sectional area of the body at x, ence J/ ft^ _ ~/ ' I) (29) Differentiate with respect to zx calling g (0) - 0 for a pointed body.. 6.z) - r -^ a (30)

ENGINEERING RESEARCH INSTITUTE P EM1B-26 UNIVERSITY OF MICHIGAN ge 10 This establishes a rule connecting the doublet distribution g (x) with the shape of the body. i.e.,4 ) = r';t (31) Now, exaaine From (9) 0 sas R R —0.,,,e(2) 45^44{@4 e -^^ o if j x - BR cosh u ^j -.^ (8;2 4 (335) This becomes as R -O. r2k.+ I0^ OT '//& 417 or im""o"IIek 4. "Now — ^ +-:4b + - we (35) as g (0) is taken as zero for pointed bodies. Combining (35) and (30), yields (iX)X7o + 77-e (36) This is the desired value for- which must be substituted into the equations for lift and ent, (18) and (19). Hence, f Pe&1 aoo. 6.* ~ 21 d 4:E(57) (37) and

ENGINEERING RESEARCH INSTITUTE DEMB26 _ i _ UNIVERSITY OF MICHIGAN ge 1 A4" 7r I 0 0 Performing the indicated rt~loC (?t. I.. 0101 V I a4lt (58) integrations yields (39) 6 0 ^/ ~Ae< Ze, Now (40) (41),- V f --- j m -Lar>J.+-67 <St since A (o) = 0 for a pointed body. Then L 1-00 1 /4v (42) Define lift coefficient as L /-r n the first order approximation for very slender bodies is Then the first order approxlation for very slender bodies is (43) (44) where c is in radians. Define moment coefficient as C I "'f t Then the first order approximation (45) for very slender bodies is C -7 2 L_ se!u #; 4s t d X: (46) 0

ENGINEERING RESEARCH INSTITUTE Page ____ 26_EMB-26____.....UNIVERSITY OF MICHIGAN 12 Now the Integral J/A44 can be integrated by parts. That is, Pierce (Ref. 6) (19a): /lXf& y Pe*,C/t -Ot A (47) Here let then. /^ ^- -97y -/ '(48) 0 0 or -. /<^ t _,4- ^/sto (49) 41 He'nce, C. -/ — v (50) 4-,L'- ^ J 4 This simple formula for Atsent has been independently derived by Munk (Ref. 7), Tsien (Ref. 5), and Laitofe (Ref. 8). Equations (50) and (44) can nov be coabined to yield one first order approximation for the center of pressure location. This is:;".-. -- "'-i -... _ (51) Now: Y^ 7<y^4 e - x < y (52) and hence A.L2- ^ " (53) OR et-o re I - --

ENGINEERING RESEARCH INSTITUTE _ EMB-26 _ __ UNIVERSITY OF MICHIGAN Pa 1 This equation was used primarily by the Wizard Aerodynamics Group to predict the proper trend of shape of a body of revolution to yield favorable rearward c.p. location. In all cases, calculations based on thesoe sml expreasson ee t la E. Xa rlerop Moluttos j

ENGINEERING RESEARCH INSTITUTE P I~MB-26 i Page __ EMB1-26__ 1 ____UNIVERSITV OF MICHIGAN i14, REFERENCES 1. "The Application of the Singularity Method to Finding the Linearized Pressure Distribution over a Body of Revolution". W. H. Dorrance, University of Michigan, EMB-9. December 1948. 2. "Hydrodynamics". H. Lab 3. "The Resistance of Slender Bodies Moving with Supersonic Velocities with Special Reference to Projectiles". Von Karman and Moore, Transactions of A.S.M.E., Vol. 54. 1932. 4. "Supersonic Flow Fields about Bodies of Revolution". Ferra l, Brown University Translation.A9-T-29 5. "Supersonic Flow over an Inclined Body of Revolution". Tsien, Journal of Aeronautical Sciences, Vol. 5. October 1958. 6. "Table of Integrals". Pierce 7. "The Aerodynamic Forces on Airship Hulls". unk1, NACA T.R. 184.: 1923. 8. "The Linearized Subsonic and Supersonic Flow about Inclined Slender Bodies of Revolution". Laitone, Journal of Aeronautical Sciences, Vol. 14. November 1947..1 -i