ENGINEERING RESEARCH INSTITUTE UNIVERSITY OF MICHIGAN ANN ARBOR FINAL REPORT WING-BODY INTERFERENCE PART I. THEORETICAL INVESTIGATION By H. E. BAILEY R. E. PHINNEY Projects M937 and M957-1 WRIGHT AIR DEVELOPMENT CENTER, U.S. AIR FORCE CONTRACT AF 533(038)-19747, E.O. NO. 460-31-12-11 SR-lg January, 1954

LITHOPRINTED IN THE UNITED STATES OF AMERICA

TABLE OF CONTENTS Page LIST OF FIGURES iv LIST OF SYMBOLS v I. INTRODUCTION1 II. APPLICATION OF NIELSEN'S METHOD TO THE CASE aB # 01 1. Decomposition of Wing-Body Interference Problems 1 2. Outline of Solution for Problem (c) 3 3. Free-Stream Velocity Potential 3 4. Body-Alone Velocity Potential 3 5. Body-Upwash Field 5 6. Fictitious-Wing Potential 6 7. Interference Potential 9 8. Fourier Coefficients of )c3/jr r=l 10 9. Computation of Cp4 12 10. Computation of Cp 14 11. Computation of Cp3 14 12. Curves of Cp due to Interference of Wing on the Body 15 13. Corrections Applied to the Theoretical Cp Curves 17 III. CONCLUSIONS 17 REFERENCES 18 iii

LIST OF FIGURES Page Fig. 1. Decomposition of the Wing-Body Interference Problem 2 Fig. 2. Coordinate System 4 Fig. 5. Fictitious Warped Wing 7 Fig. 4. Fourier Coefficients of the Velocity Induced Normal to the Body by the Warped Wing 13 Fig. 5. Fourier Coefficients of Cp4, the Pressure Coefficient Induced on the Body by the Warped Wing 15 Fig. 6. Theoretical Value of Cp on a Cylindrical Body at Angle of Attack OCB in the Presence of a Flat Surface Wing at Zero Angle of Attack 16 iv

LIST OF SYMBOLS x,y,z Rectangular Cartesian coordinates x,G,r Cylindrical coordinates aB Angle of attack of the body aw Angle of attack of the wing cu Angle of attack induced by the body-upwash field 1 Velocity potential of the free stream 2 Velocity potential of a doublet 3 Velocity potential of the fictitious wing 04 Interference velocity potential Vo Free-stream velocity Mo Free-stream Mach number Cp Pressure coefficient 1,029, 3 Velocity potentials of the fictitious wing f 2nth Fourier coefficient of 83/dr rl 2n ir=l 11,I2,I3,14 Integrals defined by Equs. (18) and (19) Cp2 Pressure coefficient due to 02 Cp3 Pressure coefficient due to 03 Cp4 Pressure coefficient due to 04 Cp4 2nth Fourier coefficient of Cp p42n W2n Characteristic function defined and tabulated in Reference 2. v

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN FINAL REPORT WING-BODY INTERFERENCE PART I. THEORETICAL INVESTIGATION I. INTRODUCTION This is the first of three parts of the final report for AF Contract AF 33(038)-19747 and contains the results of the theoretical work accomplished. Part II will contain the experimental results obtained with a model composed of a cylindrical body with wings, while Part III will contain the experimental results obtained from the body simulator plate and half-wing. The general aim of this program is to study the effect of viscosity on wing-body interference at supersonic speeds. The theoretical solution of wing-body interference problems is impossible without the aid of certain simplifying assumptions, the two most important of which are (1) an inviscid flow and (2) a linearization of the differential equations of motion. The effect which viscosity has on wing-body interference may then be found by comparing the results of experiment with the results given by a theory which neglects viscosity. II. APPLICATION OF NIELSEN'S METHOD TO THE CASE CB Z 0 1. Decomposition of Wing-Body Interference Problems Any wing-body interference problem may be decomposed into a combination of simpler wing-body problems. This decomposition, which is presented in References 2 and 3, is illustrated diagramatically in Fig. 1. If the body is at some angle of attack QB and the wing is at some angle of attack cw, then 11

^ ^ ^~~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~ ~ ~ ^ - - -- - - ~ i -- --- i f ^ O? ~w / aCI )~~~~~~~~~~~~I ^~~~~~~~~~~~~~~~ + C ^}+1. ^C / (a) (b) (C) Fig.l. Decomposition of the Wing-Body Interference Problem.

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN the problem may be decomposed into three separate problems: (a) body at zero angle of attack and wing at angle of attack aw; (b) body alone at angle of attack cB; and (c) body at zero angle of attack and wing at angle of attack -au, where cu is the angle of attack induced in the wing plane due to the body upwash field generated by the body alone at angle of attack aB' Problem (a) is solved by Nielsen in References 2 and 3. Solutions to problem (b) are relatively easy to obtain using the method of Reference 4. Furthermore, the method of Nielsen may be used to obtain the solution of problem (c). Since in some of the configurations tested on this project the body was at an angle of attack, it was necessary to solve problem (c). 2. Outline of Solution for Problem (c) In order to solve problem (c) the following procedure is used: (1) the body-alone potential is found; (2) a fictitious wing potential is found which cancels the velocities induced in the wing plane by the body-alone potential; and (3) an interference potential is found which cancels the velocities induced on the cylindrical body by the wing potential but which does not induce any velocities in the wing plane. Physically the problem is that the doublet flow about an infinite circular cylinder at angle of attack aB with respect to the free-stream direction is suddenly arrested due to the presence of a flat-plate wing with leading edge at x = 0. 3. Free-Stream Velocity Potential The coordinate system used henceforth is shown in Fig. 2. The velocity potential in a uniform stream inclined at an angle OB with respect to the x-axis is $1 = V [x cos cB + z sin aB] = Vo x + Vo cB z. (1) 4. Body-Alone Velocity Potential Now in order to make the cylinder of unit radius whose axis coincides with the x-axis a stream surface, it is necessary to add to $1 the potential 02 for a doublet of strength Vg4B: Voo CIB =, sin. (2) r In the linearized theory the pressure coefficient is generally taken to be a function only of the axial perturbation component. The axial component of ~1 + O2 is 5

me o C C\ \ a. * ~~~~~N C,) a) \ V~ ~,/,^" \ \ \I\ r /'ei\ (O~~~~~~~~~

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN A [1 + 2] = Vo COS aB = Vo. (3) (x Thus, if only the axial component of the perturbation velocity is used to determine the pressure coefficient for the body alone at an angle of attack, the pressure coefficient will be zero. However, this is not in agreement with the actual pressure distribution for a body at an angle of attack, which must vary with e. If the second and higher powers of the axial and tangential components of the perturbation velocities are included in computing the pressure coefficient, then the pressure distribution in 9 for the body alone at an angle of attack will be that for a cylinder in a uniform stream of velocity Voocg. Finding the pressure distribution in this fashion is merely an application of the local-sweepback principle. This procedure is valid only because the body under consideration is cylindrical; i.e., the contour does not vary with x. If the body contour varied with x, it would be necessary to include first- and second-order terms according to the formula of Reference 5. Physically, this means that while the linearized form of the Bernoulli equation may be used in computing the pressure coefficient for problems (a) and (c), since in those cases the chief contribution will be from u', the linearized form of the Bernoulli equation cannot be used to compute the pressure coefficient for problem (b), since in this case the first-order perturbation terms are zero, so that the second-order terms become of paramount importance for computing the pressure coefficient on the body alone. The component of the velocity normal to the cylinder of unit radius is readily seen to be zero, since ^,= VVsiV CnB sin 9 A- [(D + (2] = V. (cB sin 9 -..... -.i (4) r2 Hence, 9- [i + 2] = 0 (5) r=l and the cylinder is a stream surface for the velocity potential <1 + 02. 5. Body-Upwash Field The tangential velocity component on the cylinder is v d = [~ [ + 2]r= = 2VocB cos *. (6) r=l

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN It might be pointed out here that the pressure coefficient for the body alone at an angle of attack could be computed from this expression using the equation from Reference 5. Cp = 2u - 2 (7) VO V002 The upwash velocity in the xy plane is given by vu = I [ + 2] _ [1 + <2] = Vo OB [ + ] - (8) 8 z z=0 r 9=0 y2 This is the value of the upwash velocity in the plane 9 = 0, which is inclined at an angle oB with respect to the free-stream direction. The plane of the wing contains the free-stream velocity vector and therefore makes an angle o% with the plane e = 0. Now if z is perpendicular to the plane 9 = 0 and if z' is the direction perpendicular to the wing plane, then Z' = z - aB. (9) Hence the upwash velocity in the wing plane is I- [(i+ 0t = Voo UB (10) dz' z=0 y2 6. Fictitious-Wing Potential If the wing plane is to be a stream surface, it is necessary to find the velocity potential for a wing which will just cancel the upwash velocity by inducing an equal and opposite velocity -vu = -(V0aB)/y2 in the xy plane. Thus the desired potential must be that for a wing whose angle of attack varies spanwise as CB/y2. Since, as pointed out in Reference 2 the wing may be extended through the body in any arbitrary manner, it will be extended through the body at a constant angle of attack aB in order to avoid infinite upwash velocities at y = 0. Figure 3 is a drawing of the warped wing which will give the necessary angle-of-attack variation. It will be noticed that the wing is divided into three panels: panel I is at a constant angle of attack CXB, and panels II and III are at an angle of attack CB/ y2. 6

/ / ^^< / /' / / y Fig. 3. Fictitious Warped Wing

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The velocity potential 01 for panel I is obtained by integrating sources of constant strength -VOB over the surface of panel I. If x' and y' are the coordinates of the source point, then V0C +1 X l-2(y-yT)2 dy'_ d =01 = Id; (11) t(x - x')2- z2- (y - y')2 -l 0 01: OB y _ osh x (y + 1) cosh - (y 1)2+ z2 x + x sin-1 1 - x sin-'1 + 1 (12) 4(y + 1)2 + z2 VX2 - Z2 VX2 - z2 - z tan- x (y - 1) + tan- (+ 1) z Jx2 - (y - 1)2 z +x2 z2 - (y + 1)2 The velocity potential for panel II is obtained by integrating sources whose strength varies as -cgBVy/y2 over panel II; thus, yV4ZB y Y xZ 4+(y yT )2 dy' dx.02 _ d (13) t J (y')2 4(x-x)2 -z2 _ (y _ -y )2 1 0 The integral is easiest to evaluate if formula 161 of Reference 6 is first used. to perform the integration in x' and then, after an integration by parts together with a decomposition into partial fractions, formulae 195, 229, and 230 are used to perform the integrations in y'. The final result is Voo COB -1 _ x xy 1 02 =- B cosh- + y 1 it 4IZ2 + (y - 1)2 y2 + Z2 Vy2 + z2 2 X2- Z2_ v(y-l\ v x cos cosh-1 COS-l z y(y-)_,y cosh-l x xo -z Z y2 z2 + (y 1)2 ( _ Z - z tan-1 x (Y - 1) 1 2 y2 + z2 y2 + z2 z x 2Z _ (y _ )

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN The velocity potential for panel III may be found immediately from that for panel II simply by replacing y by -y in 02. The total velocity potential for the entire warped wing is given by 3 = 01 + 02 + 03= [1- - (y- t [ Py2 + z2 *cosh"1 x + + + (y + 1) Nz2 + (y 1)2 y2 + z2 cosh-1 x + xy 1 -/Z2 + (y + 1)2 y2 + z2 4/y2 + z2 2 (15) cosX2 - z2 - Y( -1) CO1 X2 _- 2 - y(y + 1) 1. yos-1 COs — -X2_ Z2 yX2 - Z2 -x sin1 y -l + x sinl y+ 1 z z 1 -x sin- x sin 2 (yl2 Z _2 -+ z y2 + z2. tan x(y - l2) + ta x (y+) 1) 3_ Interference Potential If the velocity potential 03 is now added to the velocity potential 1 + 02, the resulting flow will be parallel to the wing plane, but the addition of 03 will lead to a violation of the boundary conditions on the circular cylinder. In other words, 03 will induce a velocity normal to the cylinder which is given by....... —............ —- 9

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN __3: VooW C/B 2(1 - cos ) + x /x2 _ 2(1 + cos c) )r r=l 1 - x 1 - x2 + K sin e + 2 sin 0 [tan-i1 x(cos 0 - 1) aL sin 9 Nx2 - 2(1 cos Q) (16) _ tan-1 x(cos Q + 1) x cos (x2 - 2) _tan- ++ sin 9 7x2 -2(1 + cos 4) (1 - x2)3/2 cos-1 x2- (1- cos ) -1 x - (1 + cos Q) Lx2 - sin2 2 - sin2 9 The following discussion concerns the region in which the various terms in the above equation give a contribution to c)03/ar Ir=l The first and sixth terms give a contribution only inside the Mach cone whose apex is the point x = O, y = +1, z = O. Furthermore, even though both of these terms appear to be unbounded at x = 1, it can be shown that together they give a finite contribution at x = 1. Similarly, the second and seventh terms give a contribution only inside the Mach cone whose apex is the point x = 0, y = -1, z = O. The sum of these two terms gives a finite contribution at x = 1. The third term contributes to 8)3/r Jr=l everywhere behind the Mach plane from the leading edge. The fourth term will give a constant contribution outside the Mach cone with apex at x = O, y = +1, z = 0 and for the values of y in the range -1 - y z +1. The contribution will be variable inside this Mach cone and behind the Mach plane outside the range -1 z y z +1. The fifth term is similar to the fourth term except that the Mach cone in which the contribution is variable has its apex at the point x = O, y = -1, z = 0. 8. Fourier Coefficients of s3/9r Ir=l Now, as mentioned previously, it is necessary to find an interference potential 04 in order to satisfy the condition of no flow normal to the cylindrical body. This 04 is exactly the potential which is given by the Nielsen method. 4 will have the following properties: 10

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN (1) it will induce no velocities normal to the wing plane, and (2) it will just cancel the velocities normal to the cylinder r = 1 induced by the velocity potential 03. In order to find the interference potential 04 using the Nielsen method, it is necessary to obtain the Fourier expansion of 9/3/r Jr=l in a cosine series of even multiples of 9; i.e., 00 3r1 = fo + f2 cos2n (17) Jr r=l 2 n=l where the f2n are functions of x only, sin-1 x f2n = cos 2nG sin 9dG + 4 x i/ 1- x2 cos-' [(2-x2)/2] d9 ~* /5 cos 2n X2 - 2(1 - cos 9) sin-1 x + - cos 2nG sin 9 tan-1 x(cos- d (18) J sin e 2 - 2(1 - cos ) 0 cos-1 [(2-x2)/2] cos 2nG + 4 x(x2 - 2) cos 2n9 t (1- x2) 3/2 0 cos 9 cos- x2 - (1 - cos 9) d, 2 2 f2n = I + I + +4 (19) f 2n =I + 12 + Is + I4 ~ (19) The integral I1 is easily evaluated by means of formula 560 in Reference 6. The integrals I2, I3, and 14 are more complicated, but may easily be shown to be expressible in terms of complete elliptic integrals of the first, second, and third kinds as long as x lies in the range 0 - x _ 2. For values of x > 2, the elliptic integrals involved in the evaluation of I2, I3, and 14 become incomplete. By means of the substitution cos 9 = 1 -(x2w2)/2 it is readily seen that 12 may be written in terms of complete elliptic integrals of the first and 11

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN second kinds with modulus k = x/2 and with coefficients depending only on x. Integrals I3 and I4 are to be integrated by parts before making use of the substitution cos 9 = (1 - x2w2)/2. When this is done, 13 and I4 may be expressed in terms of complete elliptic integrals of the first, second, and third kinds with modulus k = x/2 and with coefficients depending only on x. The first four Fourier coefficients have been computed as functions of x for intervals of x = 0.2 over the range 0' x 4 2 with the aid of the tables in References 7 and 8. These values of the f2n are plotted as functions of x in Fig. 4. 9. Computation of Cp4 After the Fourier coefficients f2n of 3s/dr r-=l have been found, the Fourier coefficients of the pressure coefficient Cp4n due to the interference potential 04 may be obtained from 2f2n (x) cos 2ng 2 cos 2nG CP4 (1, -)42n VcV VO, ~~~~~~~~~~x ~~~(20) f2n (. ) W2n (x -. ) ~ - 0 Equation (20) is the interference pressure coefficient as given in Reference 2. The convolution integral which appears in this equation was evaluated numerically using the values of W2n tabulated in Reference 2. The integration was performed using Simpson's Rule with an interval between successive points of x = 0.2. Care must be exercised in performing this integration due to the discontinuities in the derivatives of the f2n(x) at x = 1.0. This difficulty was avoided by performing the integration in two parts for all values of x > 1. Several values of the convolution integral were checked by plotting the integrand as a function of x and integrating the resulting curve with a planimeter. The values obtained in this manner were in good agreement with those obtained using Simpson's Rule. After the values of Cp42, which are plotted as functions of x in Fig. 5, are known, the value of Cp4 for any value of 0 - 9 - 90 and 0 L x L 2 may be obtained as the sum of the Fourier series, 4 Cp4 = Cp cos 2n9. (21) n=0 12

. —----- --------------------— 6 f6 r 4 \ j -2.4 20 0.2 0.4 0.6 0.8 1.0 1.2 1.4 l.6 1.8 2.0 X/8R Fig. 4- Fourier Coefficients of the Velocity Induced Normal to the Body by the Warped Wing. 1.2 ---— 0.2 OA 0. O.2 1.4 I —-------—.0 -0.41 X/R Fig.5- Fourier Coefficients of Cp4, the PressureCoefficient Induced on the Body by the Warped Wing. o.___^v-_:\_^<i__~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Induced on the Body by the Worped Wing.

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN Since only the first four terms in the Fourier series for Cp4 are retained, the value given by the series will not be exact. The approximation of this series to the true value of Cp4 will be good for large values of x, but for values of x in the range 0' x - 1 the approximation is not very good. The cause of this inaccurate representation of Cp4 for small values of x is a direct consequence of the inability of a Fourier series of only a few terms to represent accurately a function with a jump discontinuity in the immediate vicinity of the discontinuity. 10. Computation of The pressure coefficient Cp4 is the pressure coefficient on the body due to the interference potential 4-. In order to find the total pressure coefficient Cp on the body due to the presence of the wing attached to the body at an angle of attack, it is necessary to include the effect of Cp3, which is the pressure coefficient induced on the body by the fictitious-wing potential 03, and the effect of Cp2, the pressure coefficient on the body alone at an angle of attack. If the pressure coefficients Cp, Cp3, and Cp4 were computed from the usual linearized formula, i.e., Cp 2f/dx)/VcxJ, then the total pressure coefficient would be the sum of Cp2, Cp3, and Cp4. However, if Cp2 is obtained from the linearized formula, then Cp- = 0, as pointed out in Section II, 4 above. Therefore, it is essential in computing Cp2 for the body alone that second-order terms in the perturbation velocities be retained. As a result the total value of C has been computed as Cp = CP2, ahead of leading-edge Mach helix (22) Cp = Cp3 + Cp4, behind leading-edge Mach helix. Thus second-order terms are used in the computation of Cp ahead of the Mach helix from the juncture of the wing leading edge and the body, since in this region the first-order terms in the perturbation velocities are zero. Behind the Mach helix from the wing leading edge juncture, only first-order terms are used in the computation of Cp, since in this region the contribution of the first-order terms outweighs the contribution of the second-order terms. 11. Computation of Ci As seen above, it will be necessary to obtain a Cp3 based on the velocity potential C3 before the total Cp may be found. Cp3 may be found readily 14

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN from the derivative of 03 with respect to x when evaluated at r = 1, which is ^)3 ___Voo XB {1x2 - 2(1 - cos Q) 1x2 - 2(1 + cos @) a r=l K x2 - 1 x2 - 1 + cos- 2 1 - cos ) cos (23) 4- ------------ COS-Z,,, 11. - os- -- (23} (1 - X2)3/2 2 - sin2 (1 - X2)3/2 x2- (l+cos s) sinl- cos e - 1- + sin-' cos + lxe2 sin29 - sin2 29 x i2 - sin 2 - s2 The first and third terms contribute to 03/dx 1 only inside the Mach cone with apex at x = 0, y = +1, and x = 0, while the second and fourth terms contribute only inside the Mach cone with apex at x = 0, y = -1, and z = 0. The fifth and sixth terms give a variable contribution inside their respective Mach cones and a constant contribution outside their Mach cones. Once )03/x jr 1 is known, it is a simple matter to obtain CI3 as long as only first-order terms are retained in the expression for the pressure coefficient, since then )x r=l P33 C = -'2 c) Or 1, (24) V00 12. Curves of Cp due to Interference of Wing on the Body The various values of Cp and Cp have been combined according to Equation (22) and are plotted in Fig. 6. These curves have been nondimensionalized so that it is possible to find Cp on a cylindrical body at any angle of attack due to the presence of a flat-plate wing at zero angle of attack for any value of the free-stream Mach number for which the linearized approximations are valid. This nondimensionalization is accomplished by plotting PCp/IB versus x/p, where P = Mo2 - 1 for various values of 9. The more practical problem of a flat-plate wing and a cylindrical body inclined at the same angle of attack with respect to the free stream may, of course, be solved by combining the solution presented here with the solution presented in Reference 2 as indicated at the beginning of this section. However, it is important to remember that this solution is applicable only forward of the Mach helix originating at the juncture of the body and the wing trailing edge. 15 -..

')OD440D Jo eIuo oJez tD 6u!M eoolins 4Dl D o o eouessed 9eq u! SD 0ODlID;)0 e6UD D0 Apoq IOO!JPU!IAO D UO do o 0nlDA IDo!00 JOSe l'9'6!as/x' 0'z 9'1'1 8'0' 0 0 r'001+=0 9'1og+=8-, I0 S ~= IN _____ 06+=___ —---------- | ----- - - _ -------- 98'0v06e0 0 006W=8 006-=9_ _ 8,0 ~09-9 ~'1 ----—: —- 9 1.01-3,I.... -I (3'7,

ENGINEERING RESEARCH INSTITUTE * UNIVERSITY OF MICHIGAN 13. Corrections Applied to the Theoretical Cp Curves As mentioned previously, the approximation of the pressure coefficient near a jump discontinuity by a Fourier series of only four terms will give a rather crude result. If the value of the jump and its position were known, it is probable that a considerably improved approximation to the actual value of the pressure coefficient curves could be obtained near the jump discontinuity. As a consequence, the following procedure was used to obtain the value of the jumps in the pressure coefficient curves for the present problem. The axial position at which the jump discontinuity must occur is known to be on the Mach helix originating at the intersection of the wing leading edge and the cylindrical body. At intervals of x = 0.2 for x in the range 0 - x 1.4 the values of G lying on this helix were computed. These values of 9 were then substituted into the Fourier-series representation of Cp to obtain the value of Cp at the jump. Then, according to Reference 9, this value of Cp is approximately one-half of the actual jump in Cp. If, therefore, the values of 2Cp, where Cp is the value obtained from the Fourier series, are plotted versus x, then the resulting curve should approximate the variation in the jump discontinuity along the Mach helix. Actually, in both cases this curve was found to be linear within the accuracy of the method. From this curve the value of the jump discontinuity for any desired meridional angle may be found if use is made of the equation defining the Mach helix, 9 = x. Once the value of the curve at the jump is known as well as the fourterm Fourier-series representation of the curve, a smooth curve may be faired through these data. This faired curve should give a fairly good approximation to the actual value of the pressure coefficient curve. III. CONCLUSIONS The solution presented here may be used in combination with the solution of References 2 and 3 to obtain the pressure distribution on a cylindrical body due to the presence of a flat surface wing for any combination of body and wing angles of attack. This solution, of course, is valid only forward of the trailing edge of the wing. 17

UNIVERSITY OF MICHIGAN fIU 6II;l 13.11 REFERENCES 3 9015 03695 6525 1. Phinney, R. E., "Wing-Body Interference," Progress Report No. 4. Univ. of Mich. Eng. Res. Inst. Project M937, April, 1952. 2. Nielsen, J. N., "Supersonic Wing-Body Interference." Ph.D. Thesis, California Institute of Technology, 1951. 3. Nielsen, J. N., and Pitts, W. C., "Wing-Body Interference at Supersonic Speeds with an Application to Combinations with Rectangular Wings," NACAXTN2677, April, 1952. 4. Tsien, H. S., "Supersonic Flow over an Inclined Body of Revolution," Journal of the Aeronautical Sciences, 5, No. 12 (1938). 5. Dye, F. E., "A Comparison of Pressures Predicted by Exact and Approximate Theories with Some Experimental Results on an Ogival-Nosed Body at a Mach Number of 2.00," Cornell Report CAL/CF-1723, December, 1951. 6. Pierce, B. 0., A Short Table of Integrals, Ginn and Company, 1929. 7. Milne-Thomson, L. M., Jacobian Elliptic Function Tables, Dover Publications, Inc., 1950. 8. Spenceley, G. W., and Spenceley, R. M., Smithsonian Elliptic Functions Tables, Smithsonian Institution, November 1, 1947. 9. Carslaw, H. S. Fourier Series and Integrals, Dover Publications, Inc., 1930. 18