Generalization of Applications of Free Molecule Flow by I. M. Garfunkel Project MX-794 USAF Contract W33-038-ac-14222 Willow Run Research Center Engineering Research Institute University of Michigan UMM-55 July 1950

WILLOWV RUN RESEARCH CENTER UNIVERSITY OF MICHIGAN UMM-55 TABLE OF CONTENTS SECTION PAGE Abstract ii List of Symbols iii I Introduction 1 II Pressure on an Element of Surface 4 III Bodies at Arbitrary Angle of Attack 11 A. Introduction 11 B. General Method 11 C. General Bodies of Revolution 13 D. Application to Specific Bodies of Revolution 14 1. Cylinder 14 2. Cone 16 3. Double Ogive 18 4. Prolate Ellipsoid 21 5. Composite Body 24 IV Typical Numerical Calculations 26 A. Normal and Shear Stresses 26 B. Numerical Integration 28 C. Special Case: Double Ogive 30 D. Composite Body: M 35 References 34 Appendix A 36 Appendix B 37 Figures and Tables 37 Distribution 58 L. i

WVILLOW RUN RESEARCH CENTER- UNIVERSITY OF MICHIGAN UMM-55 ABSTRACT The derivation of the free molecule expressions for shear and normal pressure on an element of surface inclined at angle e with the free streamn is shown. These expressions are then used to develop a general procedure for finding lift and drag on any of a restricted class of bodies at all angles of attack. This general theory is then applied to four specific bodies of revolution: semi-infinite right circular cone, infinite right circular cylinder, prolate ellipsoid, and double ogive. A fifth application is made in obtaining drag and lift curves for a body composed of an ogive plus half a prolate ellipsoid at angles of attack of 00 and 45~. ii

WILLOW RUN U E SEATRCH CENTER -UNIVERSITY OF MICHIGAN UMM-55 -LIST OF SYMBOLS 1a - Angle of attack A - Planform area at zero angle of attack pffiB ~- Angle between the free stream vector and the normal to a general surface element C = 4 2RT c =4 2RT r r D D qA L C = L qA - Most probable velocity of the molecules in the impinging stream - Most probable velocity of the molecules in thermal equilibrium at a temperature T of the re-emitted gas - Coefficient of drag - Coefficient of lift - Drag dS - General element of surface area 2 (t -2 erf(t) = — e ds0 Error function of t F - Total force on a body F, F,F x y z c P y = v G (x,y,z) - x, y, z components of force on a body - Ratio of specific heats - Mathematical expression for the shape of an arbitrary body iii

WILLOW RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN _________ UMM- 55 LIST OF SYMBOLS (CONTINUED) L G G G x y z - Direction numbers of the normal to a general surface element r G r= x X - - - G2 + G2 + G2 x y z - Cosine of the angle between the normal to a surface element and the x-axis - Lift - Direction cosines of the free stream velocity vector U L x,.L,VY U M = 0o C i P (0 ) q = 1. PU2 2 - Molecular speed ratio - Normal pressure on an element of surface inclined at angle e with the free stream. - Dynamic pressure I R 1 - Universal gas constant, related to "ordinary" gas constant R by: R = R 1 - Temperature of the impinging stream - Temperature of the re-emitted stream T r T,T,T x y z - Direction cosines of the tangent to a general element of surface (in the plane of the velocity vector U and the normal r ) tr( e) 0 - Shear stress on an element of surface inclined at angle 8 with the free stream - Angle between a surface element and the free stream velocity iv

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-55 LIST OF SYMBOLS (CONTINUED) U - Free stream velocity W - Molecular weight x, y, z - Space variables I v

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN 16 UMM-55 I. INTRODUCTION Free molecule flow is defined as the flow (past a solid surface) of a fluid of very low density, such that the mean free path of the free stream molecules is large compared with the linear dimensions of the surface being considered. As a consequence, the collisions of the molecules of the free stream with each other have no effect whatever on the free stream velocity, and the force opposing the motion of the fluid past the surface is due solely to the action of the molecules of the fluid upon the surface. If we now also assume that the medium in which a body is being considered is composed of small spherical molecules, the concepts of kinetic theory apply. In free molecule flow, the action of the molecules may be considered to be characterized by three different reflection phenomena: A. Specular Reflection -'in which the component of molecular velocity tangent to the element of surface from which the molecule is reflected remains unchanged, while the normal component reverses its direction; there is no adjustment of gas temperature to that of the surface of the body. B. Diffuse Reflection - in which the molecules of the gas are momentarily adsorbed, (thus, one of the boundary conditions is in terms of a derivative with respect to time), so that the direction of the motion of the reflected molecules is completely unrelated to the direction of the impinging stream. This type of reflection is accompanied by an exchange of evergy between the molecules of the stream and the body surface expressible in terms of an accomodation coefficient 8, a measure of the extent to which reflected or re-emitted molecules have their energy adjusted toward that of an equivalent mass of gas traveling in a stream at the temperature of the surface. This "accommodation" of the temperature of the impinging stream toward that of the wall is due to the momentary adsorption of the gas molecules by the surface, hence is dependent on the average length of time elapsing before re-emission. Weidman determined the value of e for air on metals, and concluded that its value is independent of the nature of the metal surface (Ref. 9). 1

WILLOWP RUN RESEARCH CENTER UNIVERSITY OF MICHIGAN U —M-55 Along with the energy exchange, diffuse reflection is accompanied 'by a momentum exchange expressible in terms of f, Maxwell's coefficient, which is the fraction of tangential momentum of the oncoming molecules transferred to the wall upon collision. A table showing that values of f are usually very nearly unity for all materials at present deemed suitable for rocket "skins" is given by Millikan (Ref. 10). Therefore, it might be expected that diffuse reflection may account for practically all the action of the molecules on the body. C. Uniform Adsorbed Layer - an "in between" case in which the impinging molecules form an adsorbed layer, and later "evaporate" in such a manner that no contribution to resultant forces or moments arises when the molecules are reemitted. There is a pressure due to the re-emission from the layer, but it is distributed so that the contribution of any one part of a body is just cancelled by the contribution of the rest of the body, so that all the forces we need deal with are those needed to bring the molecules to rest with respect to the missile surface. Thus, temperature adjustment is not a factor to be considered in the calculation of these forces. Although it might prove difficult to realize this mechanism physically, use of the concept can lead to simplification of results without greatly impairing the accuracy of the calculations involved. The "cosine law"-of reflection, that the number of molecules leaving a surface element in any direction is proportional to the cosine of the angle between the direction and the normal to the surface element, expresses the result of numerous molecular ray experiments. As long as the de Broglie wave length of the incident molecules is large compared with the average roughness height times the sine of the angle of incidence, specular reflection is not observed, nor expected by theory; thus the possibility of occurrence of specular reflection is limited to angles of incidence very near grazing. Although the tendency to grazing angles of incidence increases at high Mach number, it can be shown that, at high Mach number, the increase in tendency to grazing incidence is almost exactly offset by the decrease in de Broglie wave length. For all ranges of Mach number, 2

WILLOW RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-55 ignoring the effect of specular reflection will introduce an error of less than half a percent into the calculation of restoring force (Ref. 3). 3

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-55 II. PRESSURE ON AN ELEMENT OF SURFACE As mentioned above, the collisions of the fluid particles with each other have no effect whatever on the free stream velocity; so the total effect of a stream of molecules might be found by adding the effect of collision by each molecule separately for all the molecules. In subsequent discussion, all quantities are assumed to be averages. The following closely parallels the development presented by Tsien (Ref. 4) and Ashley (Ref. 5). Letting u', v', and w' be the velocity components of a molecule in directions x', y', and z' fixed relative to U (the free stream motion of the fluid), and if collisions between molecules and other mutual forces are neglected, then the kinetic theory states -that the molecular velocity distribution is Maxwellian, i.e., the number of molecules per unit volume with velocities in the range u' to (u' + du'), v' to (v' + dv'), w' to (wl + dwT), is Wu.vw dci L Uw' & VM W=N^ dLe &v'tw ' where N = number of molecules per unit volume. Nuv_ -ki( +V V +W 7-) ~~I I h = 2RT7 - L Ci = most probable velocity of the molecules in the free stream If unit area dS of a plane surface S moves with velocity components U, '! U,V/'U in directions x', y', z', then, in the relative coordinate system fixed in the plane surface such that x is the outward normal to the surface, the velocities u, v, w in directions x, y, z are 4

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-55 u = u' - k'U where ', l, ' are the direction cosines of U v = v' - 'LU ) w = w' -V'U ) in the x' y' z' system. Then the number of molecules with velocity components in the range u to u + du, v to v + dv, w to w + dw, is - h [( +Au)1(v+'u (Wv d);'] N,LwY,. ALS d v k w = ()e dtu d cL w (2) (2) In unit time, the molecules (with velocity components in the specified range) that strike the surface dS, are contained in a cylinder with dS as base, slant height u2 + v2 + w2, and altitude -u (-u because only molecules with x velocities in the negative range - -o < u< 0 can strike the surface dS). Since the volume of the cylinder is -udS, the number of these molecules striking unit area of S is then -uNuvwdudvdw = -udN. The total number n of molecules striking this unit area is then the integrated sum of -udN over all possible velocities of impinging molecules. 7 o no X -hgwEU~+X'y)(vt'v)'+(w-tvu1h = 0 co (tL ut vWe 00 -a J-c _ a (+ iu[I+ erf(OCufi)]J where erf (t) represents the conventional error function of statistical theory, defined by erf(t)- e - e Multiplying equation (3) by the average mass, m, per molecule (mN =p, free stream density), the mass mi of the stream striking unit area per second of the plate is: I 5

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UM -55 m. P{e | e + iu terf (Wu vh)]} The total momentum of the stream of mass m; in an arbitrary direction specified by direction cosines \l, 1,UV 1, is found by integrating the momenta of molecules in each velocity range: O0 /C /C D - I8t VU tC(Vt)U (^W )Wj Ml^tl=T^ -A^-Lv));pAzaJu &v (x. ^+Z\W)B (ilz Po? c (Wv L(,L+A(vvtz-IwLee )vtu( twv. w i+w)Jtc -,Pu1 (;tClV'z4Oe) + ____________________f______ ( (5) For calculating the pressure pi due to impact of molecules on a plane inclined at angle e to the free stream velocity U, A' = Sin O = -1 1 - ' = Cos ~,L = 0 V'=o zV =o Substituting these into equation (5): -i@)=n 6e ^ e ( S ) rF( Si) (6) When U = 0 (for a gas at rest), Pi = P ci2, which is half that yielded by the kinetic theory, the other half being due to reflection. For U>> ci and e = 90~, Pi = Pci2 + PU2, which checks with the result of Zahm (Ref. 1), except for a factor of 2, since we have found only pressure due to impact and not the total pressure. 6

WILLOWN RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UJMM-55 Substituting M,= U4 = 2R U =Substituting Mo = U,= equation (6) becomes: m u L M a e1- + + A )[erf M so SLr) (6a) ^jii/32J MIV~c^T t0 Q If the stream of molecules is first absorbed by the surface before reflection, all the tangential momentum is transferred to the surface; to calculate the shearing stress Ti thus produced, the direction cosines are: X' = Sin 0e = 0 1' = Cos e [L = -1 V' =0, l= Substituting into equation (5): -- -5 7(e) ce -Mas 1e LmEi i e ite 5 (7) If the reflection of molecules is specular, the component of motion of the incoming molecules normal to the reflecting surface is simply reversed in direction, while the tangential component remains unchanged. Then the pressure due to re-emission is Pr = Pi, and the shearing stress due to re-emission is T =-T.. If, however, reflection is diffuse, no r 1 preferred direction of re-emission exists, so Tr = 0, and total tangential stress. T = 'i + Tr = Ti. To determine the pressure Pr due to diffuse re-emission, we must use the fact (as shown in texts (Ref. 11) on kinetic theory) that 13vr TCrYMr where r = 2Tr = most probable velocity of the re-emitted molecules m = mass emitted per second per unit area. r For steady state reflection, mr must equal m1 given by equation (4). I 7

WILLOWVV RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM -55 Then "r(e) - Cr -M,, Si v T S Cr F cr L1 Pr -- C-e r 4- -- - fpPuZ Ml Ci Z Mv C Mo L. - ". sin" T- L. M - 4 -2, 5 L t e (r,. si Mao L (8) since Cr CL' z ITr '2 RTT -r/ J Ti The total pressure due to both impinging and re-emission, with f fraction of the molecules reflected diffusely and (l-f) specularly, is: k- (24f) iP^ ''iPU^ l-X'ru I (9) Similarly, the shearing stress is: /f i I I. — f ",r J7. L) (10) If f is assumed to be 1 (reflection is all diffuse), -r,/.LpP j 3 = -P; -fr s~ PU I,- ~,,.,Pu I (9a) So far as is known at present, f is independent of angle of attack e, so it does not complicate any integrations of p or T over 0. 8

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-55 A table maynow be drawn up, showing the pressures and stresses that must be accounted for for each reflection mechanism mentioned above: Table of Pressures on an Element of Unit Area Reflection Specular Uniform Adsorbed Layer Diffuse Pressure Normal + Neormal __= 2Pi Pi Pi + Pr q q q q q q q Shear - 0 — i -Ta a Since, as pointed out earlier, specular reflection has a negligible influence upon calculation of drag and lift, we shall henceforward direct our attention to the other two mechanisms. Experimental data indicate that according to certain specific criteria, a body in flight may (depending on its velocity) pass into the free molecule flow regime at altitudes as low as five hundred thousand feet. (Ref. 2, 7 and 8). For bodies at those altitudes and above, such evidence as exists leads to the belief that the energy of the molecules of the incident stream generally exceeds that of the re-emitted stream. If we take unity as the value of the upper limit of the ratio r, then presumably ci the maximum value of restoring force is calculated 'by the mechanism of diffuse reflection: this we call "extreme" diffuse reflection. Conveniently enough, determination of force in the case of uniform adsorbed layer is identically that for diffuse reflection, when the ratio c r c, is arbitrarily assigned the value zero. This presumably is the i minimum value that restoring force may be. (N.B.: The physical quantities represented by cr and Tr can never be identically zero in any physical case; they are merely arbitrarily assigned that value in the case of uniform adsorbed layer to aid in simplifying the mathematical formulation of the maximum range of restoring force encountered in free molecule flow, i.e., as a simple means of bracketing the total range expected by theory.) Then the pressures (shear and normal) arising in diffuse reflection I 9

WILLOW RUN RESEARCH CENTER UNIVERSITY OF MICHIGAN UMM-55 - can be made to bracket the whole range of pressures expected by theory 'by C c letting the ratio - take on all values from 0 to 1. However, when we say C C. r 1 0,= we really mean r = 0 as used above in describing the mechanism Ci of uniform adsorbed layer. L I 10

t. WILLOW' RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-55 III. BODIES AT ARBITRARY ANGLE OF ATTACK A. INTRODUCTION In general, except for simple configurations at zero angle of attack, force components on a body can be evaluated only by numerical integration. Exposition and illustration of one of the methods involved in integrating numerically may be found in Section IV of this paper. Three of the bodies considered here have been dealt with in the case of zero angle of attack by Ashley (Ref. 5 and 6). When the treatment given here is specialized to a = 0, the results reduce identically to those obtained by Ashley. The following analysis is exactly valid only when the flow is uniform, assumed of infinite extent, and if the body under consideration is such that a tangent to the surface at any point does not intersect the body surface. B. GENERAL METHOD Given a body subject to the above conditions, orient it such that a convenient point is the origin, and the space coordinate axes coincide with axes of symmetry (if such exist) of the body. Express the shape of the body mathematically in terms of a function G(x,y,z) of the space variables, and find the partial derivatives Gx1 Gy, Gz which are the direction numbers of the normal to a general surface element of the body. The direction cosines of the normal are Fx, ry, r z, where G x x r G ' = Y y r G r -= z r G2 +G2 + G (1) x y z - i 11

I WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN,._______________ 11MM-55 If the flow is such that the free stream has direction cosines (I, A, Z ), the angleP between the free stream vector and the normal is given by: Cosp= xr x+ pr +I r (2) x y z and the surface element may be considered to be at angle of attack (2 - ) with the free stream, so that _P_ and 2 are the free molecule normal and shear pressures, respectively, on the surface elenient in question. Then the force components are to be found by integrating these pressures over the surface of the body in free molecule flight as follows: Fx (X compolet of force), FK ~ '( r)Tj LS ir Ir Fe, [rlT3, (3) where T, Ty T are the direction cosines of that tangent to a general z element of surface which is in the plane of the velocity vector U and the normal F, and are themselves determined by means of the following equations: rxTx + rT t T = ~ xTx +T y Z' T, - Si (4) The total restoring force on the body may then be found by summing vectorially: F = 2 +F F2 (5) x y z 12 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I L 12 1

WILLOW'V RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN Ir_________ I 55UMM-55 L along a line with Then the lift F and. U. F F F x y z direction cosines F ' F, L and d-rag D act in the plane d.etermined by the vectors D = F cos (F, U) L = F sin (F, U) D C _ D D qA (6) C _L L qA (7) where A is the planform area at zero angle of attack, although the reference area may be chosen arbitrarily, so long as it is treated consistently. C. GENERAL BODIES OF REVOLUTION If the body we are dealing with is a body of revolution, the axis of revolution is an axis of symmetry. If we orient the body so that its axis of symmetry lies along the x-axis, we may, without loss of generality, restrict the free stream vector to, say, the x- y- plane. Its direction cosines then are cos a, sin a, 0 where a is the angle between the free stream velocity U and the x-axis. Then the pressure components in the z-direction when summed over the surface of the body just cancel each other, so that the resultant force F acts in the plane determined by U and the x-axis; i.e., as mentioned above, in the x- y- plane. Then (3) becomes: I S r ~r, *s1 'k ( (C - d) 1 t - -- TtJ d-o v1 (3a) by symmetry, because of the choice of coordinate axes. 13

WILLOW' RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM- 55 Then (6) becomes: D = F cos ( ~-a) L = F sin (,-a) (6a) where, is the angle between F and the x-axis. D. APPLICATION TO SPECIFIC BODIES OF REVOLUTION 1. Infinite right circular cylinder of radius R at angle of attack a:* G (x, y, z) = x2 + z2 - 2 = 0 _ Parametric representation: x = - R cos z = R sin 0 y= y X (1) G = 2x G = 0 x y G = 2z, z r= - 2R, r = cos, x r = 0 Y r = - sine, z (2) i U is parallel to x- y- plane; direction cosines are: cos a, sin c, 0. *For convenience, we set up the coordinate system as follows: at zero angle of attack, the free stream vector is normal to the axis of the cylinder. Therefore, the x- y- plane is rotated through 900, so that the axis of symmetry of the body lies on the y-axis, instead of along x, as in III C. 14

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN r_______________ UMM- O5 Then: Co. A = C0o( Ccoe (35) A.RLe &e 2Rf~e 2 — (4) L ItO = sLit4 5Ly VJ1 (5) Then: F,. -fr.) F~ %. ~A Ct Ce ( t ) s.c/3 od sc~/ (6) F(J = A siW 0 YQ -/3 )cLo ~~-f5l L 13 (7) i F=FX2+F1. F:= F+ F at 7= arctan Fy -- in the x- y- plane Fx D ACL L CkA F Xc (+ 4- ) cA F Sih ( —4 4A (8) At a = 0, then: cos p = cos e, dg= de, p = e (9) TX =SLk/, Tz C.4e, Ty =o (10) then: Cc C F(( Is-P(-e) cue L =. %A 0~ L. ( + '('-e) SlQe]a (11) 15

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-55 and C = (11) This checks exactly with the results obtained by Ashley (Ref. 5 and 6). Figure 1 is the graph of CD versus M. for a = 0~ and a = 45o. Figure 1 (a) is a plot of CL versus Mm for a = 45~. 2. Semi-infinite right circular cone, semi-vertex angle e, at angle of attack a. Reaction at origin not included in this discussion. G(xy,z= -x tar^e3 +ettiz O Parametric representation: X = X y = x tan cos 7 z = x tan e sin 7 p =! y2 + x2 =xtan e (1) Gx=- 2tL e, Gy= Zy, r, -=Si I, ry=- GiE9CL, rl-CoE5LKb r (2) U is parallel to x- y- plane; direction cosines are: cos a, sin a, 0. C, 3 = S4he c:ao( - SLo( Coe c.r, A 2 7i ' z7 Stin T-C_ Co(- Sihe C,/3 (3) (4) 'A - Kg Co /3 SLrBOCeovC- c(*4 C.,sS&o( 1,XC.C J I9vY, C (5) S <^^r* J/

WILLOVW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-55 Then: - x 7?FX A 'L _t __(_ - Cot/) lb7 (6) Afl 2 s f, 1 - ( -/3 )c, e cr-,r + (-) L Cr t (st.iWit CiS SioVG Co(- C.(d s Sid sC si/ a j r (7) in the x- y-plane f V — x + 2 at + = arctlat F Fx D CD - -- JD A (8) L C, -W - %A F WAA S (t + -( ) i at a = 0, then: cos P = sin e, it 0 = - 2 (9) Tx - Is c"o=,T CsQ~~~ a L 1 Cose + 5 i O -I r'A /A n (10) Then: Co -1 JA F - -(e) (e) cM e A ' t. sit i ~o (11) 17

WILLOTW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-55 and CL =0 iL (11) This is in perfect agreement with the results obtained by Ashley (Ref. 5 and 6). Figure 2 shows a graph of CD versus Mo for a = 0~ and a = 45~ when e = 10~. Figure 2 (a) is a graph of CL versus Mo for a = 450. It should be noted here that although the treatment has been developed for a semi-infinite cone; the results apply also in the case of a finite cone, if one assumes the contribution of the base of the cone to be negligible. 3. Double-ogive, Appendix A). semi-vertex angle e, at angle of attack a (see G(6.;)- X1^ + Z e R X +RCLtz~-R + colea)- o sIt.. Parametric representation: x = - R cos 5 y = R (sin - cose) cos z = R (sinS - cos ) sin* p = R (sins - cose) = y2 + z2 (1) ZX( /JgXG- R.-(Go) Gx / — - - GaZy, G=Z (2) 1kCMV 5, - S~m b C&4.+ 18

WILLOW RUN RESEARCH CENTER- UNIVERSITY OF MICHIGAN UMM-55 U is parallel to x- y- plane; direction cosines are: cos a, sin a, 0. Cof,16' S=S imSi m 4 C&# (5) dS A R& cy d4' mni~d.<t (Sin - ce) Cod S +4 S Tr (-ce)2I (4) C.&o(- C34/13 Cot SLnr^/j + CAa Co,3 Co S - Cf:k 1 (5) Stin/ Sit<< T=- StG(+ Stn -7 1 -Sm',R Then: F n =- C -) I-t< --— i S- 4(s[W-/s) Z- A (6) Similarly, 2~A T g - cX ")I (-, SL Co + t (7) g- A -/OSL3 I - 8- si~Sno( sihlk 19

WILLOW RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN UMM-55 IF = / a = rca+k t,y plane F, (8) D F jA '.A LC F C, -- it.-d JA ^A at a = 0, C/ Co S,/3-5, d/Swd I, dS Z(sin' -CAd)c. A ~0- s)1 (9) T - si L+ 4 os43 (10) Then: (11) z~~~~~~~~~~~(1 i and C = 0 L which checks exactly with the results obtained by Ashley (Ref. 5 and 6). Figure 3 shows a graph of CD versus Moo for a = 0 and for a = 450, when = 100. Figure 3 (a) is a graph of CL versus Moo for a = 45~. A special case of the double ogive, for 0 =, is the sphere. Then G(x, y, z) = x + z2 R = 0. Parametric representation: x = - R cos 5 y = R sin 5 cosI 6x 6 2 Y-, Gy ft Y, G 2,l r-2.R 20

WILLO0V RUN RESEARCH CENTER UNIVERSITY OF MICHIGAN _____________ UMM-55 z = R sin 5 sinf4 P = R sin 8 = /? + z r)CM(S,2 —slns)+r; —s^^sl+ (2) (1) Since the drag on a sphere is independent of a, let us take a = 0. Then the direction cosines of U are * 1, 0, 0 oCai S sA A dL2 h *^ ^i~~~~~~~~~~~~~~~~~~~~ (4a) TK = SLh, I (3a) Ty - Then: Fx IS (spere c64 i WU I T/~]4~p i F Y since = qcA at for the (5a) 0 by symmetry. This is identical with the result arrived sphere by Ashley ( Ref. 5 and 6 ). 4. Prolate ellipsoid of revolution, at angle of attack X G(x,Y,Z)-., -I 0 b x Parametric representation: x = - a cos 0 y = b sin e cos * z = b sin e sin 4 p = b sin E = j/ y2 + z2 (1) 21

WILLOW0 RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN,______________ UMM-55 I 2X G,=.,xiI 2y Gi - b, b;2Z ^'z =b; b c6 rA Kcn car + K 'thSm- KLr = 4 * ----- K= /-C(e+"i> K a (2) U is parallel to x- y- plane; direction cosines are: cos a, sin a,, 0. R C"O( Cd - S4= s c) (3) c< =(-<<<^ $n(<~iC<' 5 ds A J/asin's<+b'-Cos s6dsd4+' 7rb (4) K Sinedod + X v, a. b s CA w CO19CO- IO-OCo Ty (5) SLnh sin + cs&4 K Si Then: ZiA f -o0,T -O 3iz@X ~ no;s~ ( b6 (6) 22

WILLOW RUN RESEARCH CENTER- UNIVERSITY OF MICHIGAN _________________ 0UMM- 55 - 2*A a A b &Ln2^c-...- z,Q-8) Smh' sio(s - b 4-K o (7) F= f Z- A <t = acCman -f in, +hke Fx - C (4+-) Pt X, y- pla ne D F (8) cv = Cp= CL= 9A qi L F J at o(=o = —= -A A b K Si.Lnod 0 i (9) CO 4 =O^ aK LYsn, /J=- Coe2 KUL K ~ivL' e TX S=, K (10) Tq = o D Then Co- = -I2 9i Le co e + r(,-/) s~ C o (11) This result may, if desired, be expressed solely in terms of P, 23

I WILLOXWX RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM- 55 by replacing the e terms by their equivalent p terms: L example: S b Sine — /. I -- SL c1' / Iq where: 2 2 2 C = a - b (12) A graph of CD versus M, for a = 00 and 4~5 where the ratio of minor axis to major axis is: b _ appears in figure 4. 0.09475 a Figure 4 (a) is a graph of CL versus Mo for a = 45~. A special case of the ellipsoid, for a = b ( = R), is the sphere. Then G(x, y, z) = x2 + y2 + z2 _ R = O0 K= 1 and (11) becomes C P _ _ X= t-X = 2 o) s ---cSin-a7e "t o, ID t ~ (6a) I which is identical ogive. 5. Body formed by ellopsoid with b a with that obtained by similar means with the double joining an ogive of vertex angle 10 and a semi 0.09475 at a angle of attack. y bX

I WILLOWC RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-55 At angle of attack a, the restoring force on this body (let us call it M ) may be determined by adding the contributions of each piece separately: =9o~ e — = Fx 'FY F — (Mr)I -oive+ - (ellipsoid) qA VA A=80o jA 07, L ( MI)= —^ ogie) JA <(A c =goo _ =-7E Fq t- (ellipsoid) CS8o~ sA en" where each of the pressure components is based on the area of the circle of intersection of the ellipsoid with the ogive. i Then, as before, -( 7) ( qY ci A qA F at angle 4 = arctan y. F x F C = cos (,- a) D qA F =- sin ( - a) L qA F at a = 0, CL = 0. and C is simply x L D qA Figure 5 shows a graph of CD versus MO for a = 0 and for a = 45, for MI. Figure 5 (a) is a graph of CL versus Mo for a = 45~. 25

I WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UM.M- 55 IV. TYPICAL NUMERICAL CALCULATIONS As mentioned earlier, the calculations performed to obtain the results plotted on the graphs of this report generally involve numerical integration. A typical group of such calculations is included here. Since the drag "law" changes as M, decreases through and beyond unity, there is some doubt as to the validity of the results listed for M, < 1. However, since such experimental data as do exist agree qualitatively, at least down to Mo = 0.5, with these results, we have included them in this paper. A. NORMAL AND SHEAR STRESSES As cited above, the method used in obtaining aerodynamic coefficients at angle of attack was developed on the basis of the expressions for pressure (normal and tangential) on the body in flight. These expressions, derived in Section II are: -M si s1e p SL((9) e e I +zMoSi,s-ne ^. I ~ eIrlr((M.S-Bt" (1) L^~ Moo/Ti L - 2M'o ' 'i(.) Coa,@ M oA;;7 e (2) i er (e) t Cr Ci 2M{e Mo/ S[tr(MOOS- [)]J 2 M M,~ v/~ (3) 26

WILLOWS RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-55 where the total normal pressure is: P (e)_ rP(A) fr () and the total tangential stress is: r (e) i (e) _ (4) (5) In every application so far considered (requiring numerical integration), the tangential stress term contained a factor of: 2 which is equivalent to: E (e ) q cos 0 So, instead of (2), tabulation was made of: I z (e) -^c- ^-e - m^ sine e + Sene[lerf (Ma,5SI )] Aco= /, (2a) It must be remembered that - 0, for e = + - However, values of q -2 T (e ) q cos 0 it have been tabulated forE = + - for purposes of interpolation. 2 In these calculations we were concerned with bracketing all values of lift and drag coefficients believed possible. Thus, the lower bound is that for the reflection mechanism of uniform adsorbed layer, in which case: 27

WILLOW RUN RESEARCH CENTER UNIVERSITY OF MICHIGAN UMM-55 pr (e) = O. The upper bound is that for "extreme" diffuse reflection, in q c c r r which case: = 1. To prepare for all other cases, 0<-< 1, tabulai c tion was made of: I.J- pr(4)= $; a/i +Suie[ierf(Moo56t)10 (3a) Cr 2Mo M 4i0 Ci I A typical calculation of: PI (e) T1(E) 1 p 0 P q (cos ' c /cr() at = 22~30' and the q q cose i tabulation of the results of all other such calculations performed to date by the author are presented in Tables 1, 2, 3, and 4. B. METHOD OF NU3MERICAL INTEGRATION The method used for numerically integrating was Simpson's parabolic rule: 2 If the integral we evaluate is y = F (Me ) de (1) O we divide the interval from 0 to 2 into an even number ( = n) of parts, each equal to A. Applying Simpson's rule, taking n equal to 4, Aer y -y 3- F (MO.) 4F(M,e0o+AO) F(Meo+L7a)+4F(M,*o+3Ae)+F(Ml,,3 (2) 28

I WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-55 =-*1F (M,O) +4F( M.8F,24F(M,+ '8 ) F(M) 3 8 48 9 i 8, 2 (2a) Similarly, for evaluating a double integral, such as Z = I do ( ' F (M,q,0 ) ) the process is merely repeated so that: z = y.d.4 (3) (3a) Jo 2 where y = o F (M,e,a <) de (la) is evaluated as before for each value of *, taken as the endpoints of the equal number of parts into which the interval from 0 to i has been divided. Then for Ae =, 8 ' 4 23 -3 {0F(M4~)'4F(M, "i' A4"s)+lF (Mo~ZA4~) F(M,~3A^)+F(M ) +4-.F (M,, t,)+. 4 F(M,-s,-, to,).. a.F (M,.. z,,.,+,) + 4 F (M,,+3A,,,A+) f f(M,t.,+A + + i 29

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-55 +2 z F(M, 0SIe+ZA+4)+4F(M1Ae O+e,4l^+A4)+2 F (M, BOtlABos, 42+) 4F(M, O+3tA B,4'+x.a)F(M(,o^,+s4)J +4[p(M,{,,,#+.3+)t4 F(M,OOA o,#3A4JZF)fO tAc,'+ 3A+)4F(M,0+3A^O,4+3A, )+FCM3,) 4 [F(Me,++) 4F(M1eo-Ae )+(XP'ftoAA.) A t4F(M,t3+3A,+F(M?,+f] F (4) C. DETERNNATION OF C AND C FOR DOUBLE OGIVE, HALF ANGLE 0 = 10~ AT L - D ANGLE OF ATTACK 45~ As derived in Section III C, the force components on a double ogive, 0 of half angle 10, at angle of attack of 450 are given by: I Fx A 7r(1 - 21. C?) L. R o a o I +t ) - c( (1) I F S. — r p \ 0- Scir(Sli S9e OcCS-6^A JA 7r(i-(M09)'1- Jo L I SL Jk, 8?r t(1 Xd) t V * ( S5 -c ) * C4A (s5i$- Ce4B)cGcc (2)

WILLOW RUN RESEARCH CENTER- UNIVERSITY OF MICHIGAN I UMM-55 L where cos p = cos cos ac - sin6 sin a cos 4 (3) Then 5 is to be integrated from 80 to 100, over a range of 20. Choosing A5 = 10~, we must evaluate (1) and (2) for 6 = 80~, 90~, 100~. F F x Y But x 0- - for qA qA sin 100~. 5 = 80~ and 100~, since cos 10~0 sin 80~0 Then our evaluation is limited to 5 = 90~, when (1) and (2) become: 8Coto( 7r(o.o015q) (la) 8 r (0.0 1s59), Ct;4iR) ( lSY-C Y1) I l (M&I- I- -- Co+I q L, it Ls w - ~ I (2a) Taking A4 = 30~, we must evaluate (la) and (2a) for 4 = 0~, 30~, 60~, 90~, 120~, 150~, and 180~ Since cos 90~ 0, we need not evaluate for t = 900. First we must determine f for 4 = 0~, 300, 60~, 120~, 1500, and 180. To do this, we complete the following table: r A - n '7nT 711 ^ +4m IF- - V _r )IrO JU-U P - - VUL f w - L2 U.LILS U _ *+-i _^ _ 0~ 30~ 600 900 120~ 150 180~ cos < 1 0.86603 0.5 0 - 0.5 - 0.8660.3 - 1 cos o - 0.70711 - 0.61238 - 0.35 56 0 + 0.35356 + 0.61238 + 0.70711 ___ T 1355 127.760 110.7~ 900 69.3~ 52.24~ 45~ - - 45~ - 37.76 - 2.77 0~ + 20.7 + 37.76 + 45 31

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-55 Then we evaluate (la) as follows: Fx 8.o01o11 tI?r (4) (c') r(2.r7) T(-ZO,1)o T(-3'76) '(. 45.) - -_ _-_ ---- I-+346411 — + -3,46t4 ------ a r(oo059\) 18 3 6 3 Ct45W C0(31J0 %CO 1O.7~ ~ Wl?1 to4r 3'1.1 Cl40J 0.7071o1 I t2 t 50) - '4 5) r(s-77 -T) r(-3.7n ) r'(oo.i-1t(-zo7Il _? --- ~ ' — ^ --- —^ + 3.964~Z - - -t --- —7243 0.0151qL ~Cot45 A (376~4) t CO40.7j (lb) Ordinarily a table must be set up in which the values of each of the stresses occurring in such an equation is listed for each M0o desired (see below, as was done for (2b)). However, in the case of (lb), the stresses are independent of M,, i.e., they are constant. When the values are read or interpolated from Table 1 it is a simple matter to calculate: F x - 7.66 qA (Ic) for this configuration for all M. If a better approximation had been determined, e.g., by taking A6 = 5~, it is likely that F would not have been exactly constant. x qA This is purely a result of the fact that the approximation first determined was found by computation for a single 8, namely, 8 = 90~. However, when better approximation was calculated, the variation in value of F with MN was so small that the additional work entailed x qA produced no appreciable difference in the final evaluation of the drag and lift coefficients. 32

WILLOW RUN RESEARCH CENTER- UNIVERSITY OF MICHIGAN UM_____________ M-55 Evaluation of (2a) is as follows:. A..(. 8 A 7r (o.ol51.. w I op(45*)+ p(-453) p( )t(31.). pt((o, )-p(l(8363 tozS +0o.'olil 0.8(oo03 r (1 -o.) -Z'(-3.l,o) c~io&r) -10c") t o5 -(-71 ^C^C'^T0 8U(o~' Cea (37 r7~) J (2b) The steps and tabulation involved in evaluating (2b) and then determining CD and CL by proper manipulation of (lc) and (2b) are shown in Tables 5 and 6. In like manner, the coefficients for the semi-infinite right circular cone, infinite right circular cylinder, and prolate ellipsoid at an angle of attack of 45~ were obtained. The results are presented in Table 7. D. MI AT ANGILE OF ATTACK OF 450 As discussed earlier, the M consists of half of a double ogive plus half of a prolate ellipsoid. The restoring force on the MI, at any angle of attack a, may be determined by adding the contributions of each piece separately: I (MZ)= Oa A (A HI %A $A Fx "6A 6 Fq & IT (al! $18 A a =? By the methods of part IV B above, the following table containing the contribution of each piece was completed and the coefficients obtained (see Table 8). 33

L WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-55 REFERENCES I. Free Molecule Flow 1. Zahm, A. F., "Superaerodynamics", Journal of the Franklin Institute, Vol. 217, pp. 153-166, 1934. 2. Sanger, E., "Gaskinetik Sehr Hoher Fluggeschwindigkeiten", Deutsche Luftfahrtforschung, Bericht 972, Berlin 1938. Translated by Liebkold, 1946, for Douglas Aircraft. 3. Snow, R. M., "Aerodynamics of Ultra-High Altitude Missiles", APL/ JHU-CM-498, September 1948. 4. Tsien, H. S., "Superaerodynamics, Mechanics of Rarefied Gases", Journal of the Aeronautical Sciences, Vol. 13, No. 12, pp. 653-664, December 1946. 5. Ashely, H., "Applications of the Theory of Free Molecule Flow to Aeronautics", Journal of the Aeronautical Sciences, Vol. 16, No. 2, pp. 95-104, February 1949. 6. Ashley, H., "Applications of the Theory of Free Molecule Flow to Aeronautics", IAS Preprint No. 164. II. General 7. Schultz, F. V., Spencer, N. W., and Reifman, A., "Upper Air Research Program", Report No. 2, ERI, University of Michigan, 1 July 1948. 8. Grimminger, G.., "Analysis of Temperature, Pressure and Density of the Atmosphere Extending to Extreme Altitudes", Rand R-105, 1 November 1948. 9. Weidman, M. L., "Thermal Accommodation Coefficient", Transections of ASME, Vol. 68, pp. 57-64, 1946. 34

WILLO W RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-55 -REFERENCES (CONTINUED) 10. Millikan, R. A., "Coefficients of Slip in Gases and the Law of Reflection of Molecules from the Surfaces of Solids and Liquids", Physical Review, Vol. 21, pp. 217-238, 1923. 11. Kennard, E. H., "Kinetic Theory of Gases", McGraw-Hill Book Co., Inc., New York, 1936. 35

WXILLOWN RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-55 APPENDIX A Note on the Double Ogive A double ogive is formed by rotating the arc of a circle about its chord (see below). The vertex angle (2 ) of the ogive is then equal to the central angle of the generating arc (generatrix). 5 is defined as the angle between the negative x-axis and a normal to the generatrix. y i x 36

I WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN UMM-55 APPENDIX B TABLES No. Title 1 Determination of i(e) 1 Pr(e), i(e) fore = q Cr/Ci ' q cq cos 22 30' 2 Pi(e) q Page 39 41 3 1 pr( ) Cr/Ci " q t (e) 4 q cos a 45 5 Determination of 3 I CD qA 6 Determination of+F-, CD and CL (Pr = 0) and CL (cr/ci = 1) 47 48 49 i 7 Uniform Adsorbed Layer and "Extreme" Diffuse Reflection for Infinite Right Circular Cylinder, Semi-infinite Right Circular Cone, and Prolate Ellipsoid Ratio of Axes =+ 0.09475 = b a 8 Contributions of Each Part of MI: Determination of CD and CL Fx F x = A y= -qA qA.? qA 51 FIGURES 1 Drag Coefficients and Lift Coefficients for Infinite Right Circular Cylinder 2 Drag Coefficients and Lift Coefficients for Semi-infinite Cone 53 55 37

WILLOW RUN RESEARCH CENTER -UNIVERSITY OF MICHIGAN UMM-55 FIGURES (CONT'D) No. Title Page 3 Drag Coefficients and Lift Coefficients for Double Ogive 57 4 Drag Coefficients and Lift Coefficients for Prolate Ellipsoid 59 5 Drag Coefficients and Lift Coefficients for MI 61 38

TABLE 1 p (E) 1 pT(e) T.(E) Determination of i () c Pr (0) i () q f r \ qg q cos e cI For 0 = 22~ 30' ', k4 Step Operati on 4.5 o06 o.8.0 1 1.5 2. 4,0 7.0 1o.o ( |Moosin 0.11 0.1913 0.2296 0.3061 0.5827 0.4592 0.5740 0.7654 1.5307 2.6788 3.8268 (2) M sin2e 0.0234 0.0366 0.0527 0.0937 0.1464 0.2109 0.295 0.5858 2.5451 7.1758 14.6444 0 |e-O 0.9867 0.9641 0.9487 0.9105 0.868 0.809 9 0.7195 0.5567 0.0960 0.0009 0 -1 | Moo- 1.4105 1.1284 0.9403 070520.562 02 0.761 0.2821 0.1410 0.o806 0.0564 O Ox~ i 1.917 1.0878 o.8920 0.6421 0.4875 0.3808 0.2705 0.1570 0.0135 0.0001 0 — 0) O )X sin 0 0.5326 0.4163 0.3414 0.2457.865 0.1457 0.1035 o0.06 0.10052 0- 0 -Q l+erf 1.1714 1.2132 1.2546 1.5292 1.4116 1.4840 1.5851 1.7209 1.9696 1.9998 2 -~(8) |1-erfQ 0.8286 0.7868 0.7454 0.6708 0.5884 o.516o 0.4169 0.2791 0.0304 0.0002 o_@_ _5 Ox O 3.8320 2.6041 1.9262 1.233551 0.9125 0.7326 o.5856 0.4671 0.500 0.5155 0.029 11 (x 2) 2.7106 1.6888 1.14444 0.6223 0.5804 0.2547 0.1537 0.0758 0.0054 0- -- i o )+~ 4.3646 3.0204 2.2676 1.4788 1.0990.8785.6872 5272.552 0.1029 i i @ ) o - 2.1780 1.2725 0.8031 0.5766 0.1939 0.1090 0.0502 0.0157 0.0002 - - ( 1 Q)x sin 0 0.4485 0.4645 0.4701 0.4987 0.5302 0.5579 0.5958 0.6486 0.7557 0.7655 0.7654 1) Q()x sin ) 0.171 0.5011 0.2852 0.2567 0.2252 0.1975.0.1595 o.o68 o.oi16 o - 0 ~r /A \ _. _ _ _ _ t _ _ _, > 7 < tt A t A t 7 A A t A 0 z 0 C z 0 z tp I I-I I - 01 c Z; P: q coS (D+:... ~ 1. 8400 1. 552111.36211 1.1408 1.0175 0.9367 O.b664 0. d0Uo 0. 7b73 10.7654: U. '(54 I I I.1 I I I A~ I..i i

TABLE 1 Determination of Pi( Pr( 'r (e) For. = 22 30' (c (Continued) Step 0'p 4 O, 4 0.5 0.6 0,8 1.20 I 1.5 2.0 4.0 7.0 10.0 T (-e).o -7{ti) 1.0746 0.7867 o.6o68 0.3854 0.2622 0.183 0.1110 0.0502 0.0019 0- 0 -(i6) I 2 1 2.2156 1.7725 1.4770 1.1078 0.8862 0.7385 0.5908 0.4431 0.2216 0.1266 0.0886 ___ ~2M~ c; Q9| (g) x(g) 4.0766 2.7510 2.0119 1.2638 0.9017 0.6955 0.5119 0.5570 0.1700 0.0969 0.0678 c a t ( @ | ( x ) | 2. 8 9 1 3 4 |09_____ __6 ___7 |. 3 3 0 1 4 0 06 6 0 0 3 0 0 4 |-,_______, *f, $i(O) (is"x~ 2.5809 1.5944 0.8962 0.4270 0.2525 0.1554 o0.0656 0.0225 0.0042 0 — 0 -________ r 0 C Z n o -z t C1 xn VM 0

TABLE 2 Pi(e) q H '0.4 0.5 0.6 0.8 1.0 11. 15 2.0 4.o 7.0 10. -900 1.1559 0.5597 0.2902 0.o875 0.0284 0.0095 o0.0018 0. 00010. — 0.- 0. -60~ 1.3355 0.6764 0.3674 0.1231 0.0451 0.0172 0.0041 0.0003 0. 0.- 0.-450 1.5719 0.8386 0.4807 0.1808 0.0753 0.0331 0.0102 0.0014 0.- 0. - 0. -40.80 1.6672 0.9005 0.5252 0.2050 0.0890 0.0410 0.0145 0.0022 0. — 0.- -37.76~ 1.7384 0.9502 0.5611 0.2250 0.1007 0.0480 0.0167 0.0031 0.- 0. 0. — -30~ 1.9458 1.0983 0.6707 0.2890 0.1400 0.0726 0.0292 0.0071 0. 0. 0.-22.5~ 2.1781 1.2725 0.8031 0.3766 0.1939 0.1090 0.0502 0.0157 0.0002 0. —.-20.7~ 2.2461 1.3189 0.8391 0.3945 0.2096 0.1202 0.0570 0.0188 0.0004 0.- 0. -15.7~ 2.4320 1.4590 0.9488 0.4670 0.2620 0.1572 0.0810 0.0310 0.0013 0. 0.-10~ 2.6644 1.6373 1.0912 0.5650 0.3324 0.2117 0.1188 0.0533 0.0049 0.0003 0. -50 2.8867 1.8110 1.2324 0.6658 0.4090 0.2726 0.1640 0.0830 0.0133 0.0021 0.0004 -30 2.9801 1.8846 1.2932 0.7101 0.4436 0.3007 0.1855 0.0981 0.0190 0.0041 0.0013 0~ 3.1250 2.0000 1.3889 0.7813 0.5000 0.3472 0.2222 0.1250 0.0313 0.0102 0.0050 30 3.2753 2.1208 1.4900 0.8578 0.5620 0.4000 0.2644 0.15740.0490 0.0218 0.0142 50 3.3785 2.2043 1.5606 0.9120 0.6062 0.4371 0.2958 0.1823 0.0644 0.0335 0.0248 100 3.6460 2.4230 1.7470 1.0580 0.7280 0.5430 0.3860 0.2570 0.1180 0.0805 0.0703 15.7~ 3.9644 2.6875 1.9755 1.2421 0.8844 0.6840 0.5100 0.3654 0.2076 0.1668 0.1582 20.70 4.2540 2.9310 2.1890 1.4180 1.0400 0.8240 o.680o 0.4810 0.3120 0.2700 0.2600 ~ C I r r 0 o 0 z rl C H 0 0 C l — I I cn i ^

TABLE 2 i(e) ( Continued) q e04 0. 5 0.6 o.8 1..2 1.5 2.0 4.0 7.0 10.o 22.5~ 4.3646 3.0204 2.2676 1.4788 1.0990 0.8783 0.6872 0.5272 0.3552 0.3133 0.3500 30~ 4.8042 3.4017 2.6071 1.7735 1.3600 1.1220 0.9152 0.7430 0.5625 0.5204 0.5100 37.76~ 5.2616 3.8000 2.9670 2.0875 1.6500 1.4000 1.1777 0.9970 0.8125 0.7704 0.7600 40.8~ 5.4364 3.9530 3.1062 2.2112 1.7645 1.5070 1.2835 1.1014 0.9161 0.8740 0.8636 45~ 5.6780 4.1614 3.2970 2.3820 1.9247 1.6613 1.4342 1.2486 1.0625 1.0204 1.0100 60~ 6.4246 4.8236 3.9104 2.9394 2.4550 2.1773 1.9403 1.7500 1.5625 1.5204 1.5100 90~ 7.0941 5.4403 4.4876 3.4750 2.9716 2.6850 2.4427 2.2500 2.0625 2.0204 2.0100 c t1. \, R) Cr r 0 Z tO c z Z 0 |S O~

TABLE 3 1 Pr(9) c q r Ci i C r 0 >.. 0,4 0.5 0.6 0.8 1.0 j. 2. 1.5 |2..0 4.0 7.0 10.0 -900 1.3965 0.7077 0.3839 0.1263 0.0446 o.060 o.0034 0.0002 0..- o.- 0.-600 1.5739 0.8272 0.4688 0.1695 o0.0668 0.0273 0.0072 0.0007 0. — 0.- 0.-45~ 1.8051 0.9916 0.5872 0.2353 0.1044 0.0488 0.0163 0.0027 0.- 0.- 0. -40.8~ 1.8886 1.0518 0.6320 0.2618 0.1204 0.0587 0.0202 0.0040 0.- 0.- 0.-37.76~ 1.9538 1.0992 0.6678 0.2832 0.1330 0.0673 0.0254 0.0053 0.- 0.- 0. -30~ 2.1414 1.2375 0.7735 0.3490 0.1770 0.0960 0.0413 0.0111 0.- 0. 0. -22.5~ 2.3809 1.3944 0.8962 0.4270 0.2323 0.1354 0.0656 0.0222 0.0004 0.- 0. -20.7~ 2.4040 1.4355 0.9287 0.4513 0.2480 0.1476 0.0730 0.0260 0.0007 0. - 0. -15.7~ 2.5620 1.5570 1.0260 0.5180 0.2914 0.1834 0.0980 0.0400 0.0021 0. -. -100 2.7554 1.7073 1.1475 0.60o 0.3605 0.2340 0.1345 0.0628 0.0068 0.0005 0. -5~ 2.9357 1.8493 1.2640 0.6885 0.4266 0.2866 0.1745 0.0902 0.0157 0.0015 0.0007 -3~ 3.0104 1.9086 1.3130 0.7247 0.4550 0.3100 0.1927 0.1032 0.0210 0.0024 0.0017 0~ 3.1250 2.0000 1.3889 0.7813 0.5000 0.3472 0.2222 0.1250 0.0313 0.0102 0.0050 30 3.2423 2.0941 1.4676 0.8373 0.5478 0.3873 0.2545 0.1500 0.0442 0.0182 0.0110 50 5.3220 2.1583 1.5214 0.8771 0.5788 0.4154 0.2775 0.1674 0.0543 0.0248 0.0161 10~ 3.5250 2.3230 1.6605 0.9886 0.6682 0.4905 0.3400 0.2167 0.0837 0.0444 0.008 15.7~ 3.7610 2.5161 1.8250 1.1174 0.7710 0.5830 0.4177 0.2800 0.1220 0.0690 0.0480;0 c z 0, Z " pi et Z C O r O t t4

I - TABLE 3 1 P. r(e) (Continued) q Ci ci- o0.1 0.5 0.6 0.8 1.0 1.2 1.5 2. 4.0 10.0 20.7~ 3.9706 2.6890 1.970 1.2346 0.8746 0.6690 0.4910 0.3394 0.1573 0.0895 0.0627 22.5~ 4.0766 2.7510 2.0120 1.2640 0.9020 o.6933 0.5120 0.3570 0.1700 0.0970 0.0678 30~ 4.3570 3.0100 2.2505 1.4570 1.0632 0.8345 0.6320 0.4542 0.2216 0.1266 0.0886 37.76 4.6673.2700 2.4768 1.6400 1.2193 0.9718 0.7490 0.5480 0.2714 0.1550 0.1085 40.80 4.7834 3.3677 2.5619 1.7092 1.2784 1.0236 0.7922 0.5829 0.2895 0.1654 0.1158 45~0 4.9387 3.5000 2.6762 1.8020 1.3578 1.0933 0.8520 0.6294 0.3133 0.1790 0.1253 60~ 5.4114 3.8972 3.0271 2.0883 1.6018 1.3065 1.0306 0.7682 0.3838 0.2193 0.1535 90~ 5.8276 4.2526 3.3381 2.3418 1.8170 1.4930 1.1850 0.8864 0.4431 0.2552 0.1772 r r 0 Z C z z Irl (0 t g mH nc M

TABLE 4 r (e) q cos 0 \J1 e\MO| 0.4 05 0.6 0.8 1,0 1.2L 1.5 2.0 4.o 7.0 10.0 -900 0.6303 0.3993 0.2600 0.1140 0.0503 0.0217 0.0057 0.0005 0.-.- 0.-600 0.7104 0.4667 0.3174 0.1530 0.0754 0.070 0.0122 0.001670.- 0. 0.-450 0.8147 0.5595 0.3975 0.2124 0.1178 o.o660 0.0276 0.0061 o. --- 0.- 0.-40.80 0.8524 0.5934 0.4279 0.2363 0.1359 0.0795 0.0342 0.0090 0. 0. 0.-37.76~ 0.8818 0.6202 0.4521 0.2557 0.1511 0.0911 0.0430 0.0119 0. 0.-. 0.-30 0.9665 0.6982 0.5237 0.3152 0.1996 0.1300 0.0700 0.0251 0.0002.-.-. -22030' 1.0746 0.7867 0.6o68 0.854 0.2622 0.1833 0.1110 0.0502 0.0019 0. 0.-20.70 1.0850 0.8100 0.6288 0.4074 0.2797 0.2000 0.1237 0.0590 0.0030 0.- 0.-15.7~ 1.1564 0.8784 0.6944 0.4674 0.3288 0.2483 0.1659 0.0903 0.0096 0.0002 0. — -100 1.2437 0.9632 0.7769 0.5452 0.4067 0.3168 0.2277 0.1418 0.0305 0.0035 0.0003 -5~ 1.3250 1.0434 0.8557 0.6215 0.4813 0.3881 0.2954 0.2035 0.0707 0.0217 0.0074 -3~ 1.3588 0.0768 0.8889 0.6541 0.5134 0.4197 0.3261 0.2328 0.0948 0.0388 0.0189 00 1.4105 1.1284 0.9403 0.7052 0.5642 0.4702 0.3761 0.2821 0.1410 0.0806 0.0564 3~ 1.464 1.1815 0.9934 0.7558 0.6181 0.5244 0.4308 0.3375 0.1995 0.1435 0.1236 50 1.4994 1.2177 1.0301 0.7948 0.6556 0.5625 0.4697 0.3778 0.2450 0.1960 0.1817 10~ 1.5910 1.3105 1.1242 0.8925 0.7540 0.6641 0.5750 0.4891 0.3778 0.3508 0.3476 15.7~ 1.6976 1.4196 1.2356 1.0086 0.8700 0.7895 0.7071 0.6315 0.5508 0.5414 0.5412 i~ ~~~~~.010 o5o - r 0 C o rl > C t n O I; Ol 2

TABLE 4 -L co (Continued) q cos e M\M 0.1 0.5 0.6 o.8 1.0 1.2 1.5 2.0o 4.0 7.0 10.0 20.7~ 1.7921 1.5170 1.3359 1.1145 0.9868 0.9059 0.8308 0.7660 0.7101 0.7071 0.7071 22~30' 1.8400 1.5521 1.3621 1.1408 1.0175 0.9387 0.8664 o.8056 0.7673 0.7654 0.7654 30~ 1.9665 1.6982 1.5237 1.3152 1.1996 1.1300 1.0700 11.0251 1.0002 1.- 1.27.76~ 2.1066 1.8450 1.6769 1.4804 1.3758 1.158 1.2678 1.2367 1.2248 1.2248 1.2248 40.8~ 2.1590 1.9000 1.7345 1.5429 1.4425 1.3861 1.3408 1.3155 1.3066 1.3066 1.3066 45~ 2.2291 1.9738 1.8119 1.6268 1.5321 1.4804 11.4419 1.4203 1.4142 1.4142 1.4142 60~ 2.4425 2.1988 2.0494 1.8851 1.8074 1.7690 1.7443 1.7337 1.7321 1.7321 1.7321 90~ 2.6303 2.3993 2.2600 2.1140 2.0503 2.0217 2.0057 2.0005 2.- 2. 2. 0\ r 0;o a C z 0 0 1 I= nt M O 0 ^1 0* f&

O 0 -- TABLE 5 Determination of, C and CL ( = 0) Step 0.4 0.5 0.6 0.8 1.0 1.2 1.5 2.0 4.0 7.0 10.0 I Fx 7.6600 7.6600 7.6600 7.6600 7.6600 7.6600 7.6600 7.6600 7.6600 7.6600 7.6600 qA ZT terms in (2b) 1.1250 1.1250 1.1250 1.1250 1.1250 1.1250 1.1250 1.1250 1.1250 1.1250 1.1250 Pi(45)+Pi(-45) 17.2500 5.0000 3.7777 2.5625 2.0000 1.6944 1.4444 1.2500 1.0625 1.0204 1.0100 Pi(2.7)+Pi(-20.7) 3.2500 2.1250 1.5139 0.9062 0.6250 0.4722 0.3472 0.2500 0.1562 0.1352 0.1300 I 3Pi(57.76).+p(-57.76) 1.0000 14.2500 10.5834 6.9376 5.2501 4.5334 3.5834 3.0000 2.4376 2.5113 2.2801 " _________________ 1 0+ +(0 +(0 32.6250 22.5000 17.0000 11.5514 9.0000 7.6250 6.5000 5.6250 4.7814 4.5919 4.5451 F AI ~ ci =Ix 1.856 555.5551 58.5000 28.9578 19.6290 15.5200 12.9795 11.0646 9.5750 8.1590 7.8164 7.7368 ( tan 7o = )/ 7.2500 5.0000 3.7778 2.5625 2.0000 1.6944 1.4444 1.2500 1.0625 1.0204 1.0100 0G I 182.15~ 78.70 75.2~ 68.70 65.4~ 59.450 5535~ 51. 50 46.75~ 45.60 45.50 D 7o - a 57.150 33570 50.20 23.70 18.40 14.450 10.50 6.55~ 1.57~ 0.60 0.30 (@1 cos 10 0.7971 0.8520 0.8645 0.9158 0.9487 0.9685 0.9859 0.9959 0.9995 0.9999 1.0000 ) sin 10 0.6039 0.5547 0.5026 0.4016 0.3162 0.2496 0.1789 0.1104 0.0503 0.0101 0.0050 IA = 56.0609 39.0585 29.9345 21.0707 17.1285 15.0715 1534575 12.620 11.1767 10.9440 10.8875 |D lOxX~ 44.686 52.499 25.879 19.296 16.249 14.594 13.240 12.187 11.172 10.9435 10.887 CL Ix 1355.852 21.666 15.046 8.463 5.416 5.762 2.408 1.54 0.339 0.110 0.054 Z z C, k\n o z n4 C1 -C 0 O t

TABLE 6 F Determination of y,, CD and C qA D L qA C r= 1 C i Co Step Opera t0.4 0.5 0.6 0.8 O 1.0 1.2.5 2..0 4.0 7.0 10.0 Fx ~ -F- 7.6600 7.6600 7.6600 7.66oo 7.6600 7.6600 7.6600 7.6600 7.6600 7.6600 7.6600 ~ ZT terms in (2b) 1.1250 1.1250 1.1250 1.1250 1.1250 1.1250 1.1250 1.1250 1.1250 1.1250 1.1250 Pi(45)+Pi (-45) p -45+Pi(-45) -.13.9938 9.4901 7.0412 4.6000 3.4622 2.8365 2.3126 1.8821 1.3758 1.1994 1.1353 0. 5Pi (20.7) +Pi (-20.7)..5pi( 0.7)+p(- 2.7) |6.4373 4.1872 2.9648 1.7492 1.1862 0.8805 0.6292 0.4328 0.2353 0.1800 0.1613 3Pi(37.76)+p (-37.76) 36.7029 24.6125 18.0414 11.4988 8.4594 6.7976 5.4202 4.3122 3.0812 2.6790 2.5375 ( +@+ @ 58.2590 39.4148 29.1724 18.9730 14.2328 11.6396 9.4870 7.7521 5.8173 5.1837 4.9591 () = (-1.8 456 99.1700 67.092849.6580 32.2963 24.2274 19.8132 16.1490 13.1958 9.9024 8.8238 8.4415 (a)| tan y1 = + _12.9465 8.7589 6.4828 4.2162 3.1628 2.5866 2.1082 1.7227 1.2930 1.1529 1.1020 O(9) 71 85.6~ 83.5~ 81.2~ 76.65~ 72.45 ~ 68.9~ 64.60 59.90 52.30 49.05~ 47.8~ (lo0a 71 -aC a 06 40.60 38.5~ 36.2~ 31.65~ 27.450 23.9~ 19.6~ 14.9~ 7.3~ 4.050 2.8~ laI cos 10 0.7595 0.7827 0.8066 0.8512 0.8874 0.9145 0.9419 0.9665 0.9919 0.9975 0.9988 2a) | sin 10 0.6505 0.6223 0.5910 0.5248 0.4610 0.4046 0.3358 0.2566 0.1268 0.0704 0.0485 A |= ~ +(32 99.4653 67.5287 50.2453 33.1923 25.4095 21.2424 17.8736 15.2579 12.5193 11.6848 11.3989 [C D (H^ 3x(75.5400 52.8581 40.5300 28.2536 22.5479 19.4264 16.8355 14.7472 12.4183 11.6557 11.3854 C0L ( x 64.7072 42.0251 29.6975 17.4203 11.7147 8.5938 6.0027 3.9147 1.5872 0.8230 0.5525 r r 0 z > o m C 0 0 pi z 0 0 t I1 O: C t 01 i K; 0) 3: C) Mp

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN I --- —-------— I UMM-55 I TABLE 7 Uniform Adsorbed Layer "Extreme" Diffuse Reflection cr/Ci = 0 Cr/Ci = 1 |Mo CD CL CD CL o0.- 5.49 -1.29 7.24 -3.03 0.5 4.46 -1.02 5.85 -2.42 o.6 3.78 -0.85 4.94 -2.01 0.8 2.95 -0.63 3.82 -1.50 Infinite 1.0 2..47 -0.50 3.17 -1.20 Right |1.2 2.18 -0.41 2.76 -1.00 Circular Cylinder 1.5 1.90 -0.33 2.37 -0.79 2.0 1.66 -0.24 2.01 -0.59 4.0 1.41 -0.14 1.58 -0.31 7.0 1.36 -0.11 1.46 -0.21 10.0 1.33 -0.09 1.40 -o.16 0.4 13.46 -0.59 19.19 0.28 0.5 10.63 0.01 1 14.91 0.99 o.6 8.91 0.19 12.32 1.21 0.8 6.91 0.30 9.34 1.27 Semi-infinite 1.0 5.85 0.34 7.75 1.21 Right 1.2 5.1 0.28 6.72 1.06 Circular Cone 1.5 4.55 0.23 5.80 0.90 2.0 4.05 0.17 4.98 0.70 4.0 3.52 0.05 3.99 0.34 7.0 3.41 0.02 3.68 0.18 10.0 3.39 0.01 3.58 0.12 m,

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN JUMM-55 1 I~~~~~~~~~~~~~~~~~~~~~ I I TABLE 7 (Continued) Uniform Adsorbed Layer "Extreme" Diffuse Reflection cr/ci =0 Cr/i = 1 Moo CD CL C C 0.4 63.94 14.84 85.17 35.66 0.5 51.88 11.77 68.86 28.40 0.6 43.94 9.86 58.07 23.70 0.8 34.38 7.19 44.98 17.58 Prolate 1.0 28.97 5.67 37.41 13.92 Ellipsoid 1.2 25.55 4.63 32.64 11.57 Ratio of Axes a= l 1.5 22.42 5.58 28.08 9.12 0.09475 = a 2.0 19.72 2.55 23.95 6.79 4.0 16.95 1.19 19.08 3.27 7.0 16.27 0.83 17.47 2.00 10.0 16.08 0.74 16.92 1.57 0 1 I 50

TABLE 8 r 0 CONTRIBUTIONS OF EACH PART OF M: I F F = x, y= y qA qA J1 H Item Piece 0.4 0.5 o.6 0.8 1.0 1.2 1.5 2.0 4.0 7.0 10.0 1) x 3 3.83 3.83 3.8.83 5.83 3.8 53.83 '3.83 3.83 3.83 3.83 Ogive -- 2) y 28.60 19.75 14.87 10.44 8.16.76 5.78 5.05 4.34 4.17 4.13 p =o r |5) 3x 135.56 11.54 10.03 8.26 7.18 6.50 5.90 5.40 5.02 4.93 4.87 Ellipsoid 4) y 235.1 18.69.15.68 12.02 9.9 8.58 7.52 6.20 4.96 4.65 4.57 la) x 58 3.83 583 3.83 3.83 3.83 3.83 3.83 3.83 3.8.83 3.83 Ogive Cr 2a) y 48.58 32.94 24.51 16.39 12.52 10.00 8.21 6.82 5.20 4.65 4.48 Ci 3a) x 10.50 9.44 8.55 7.40 6.60 6.05 5.59 5.20 4.92 4.87 4.85 Ellipsoid - 4a) y 53479 27.82 23.22 17.63 14.51 12.51 10.24 8.36 6.053 5.24 4.98 DETERMINATION OF C AND C D L Item Piece 0.4 0.5 0.6 0.8 1.0 1.2 5 2.0 4.0 7.0 10.0 Z x 1)+3) 17.9 15.57 15.86 12.09 11.01 10.55 9.75 9.25 8.85 8.76 8.70 Z y 2)+4) 51.91 38.44 50.55 22.46 18.09 15.34 135.10 11.25 9.50 8.82 8.70 F tan ~ y 2.9850 2.5010 2.2042 1.8577 1.6431 1.4850 1.3464 1.2189 1.0508 1.0068 1.0000 F p = __ 71 29' 68~12' 65 56' 61 42' 5840' 56 3' 553 4' 50 581 46 25' 45 12' 45 7 f - a 2629' 25~12' 200536t 1642' 1 ~4o' 11~' 8024 5038' 1~25' 0~12' 0 8024, '.50 o I~~~~ 1: t et C) 0 C O C) 0 0 -

TABLE 8 (CONITINUED) DETERNMINATION OF C AND C D L ~ I I I I I I I I I 1 1I.. Item Piece 0.4 0.5 0.6 0.8 1.0 1.2 1.5 2.0 4.0 7.0 10.0 cos y 0.8951 o.90.9190 0.961 578 0.9716 0.9815 0.9893 0.9952 0.9997 1.0000 1 sin y 10.4459 0.3939 0.3518 0.2874 0.2364 0.1917 0.1461 0.0982 0.0247 0.0034 0 F sF n l |54.75 41.40 33.55 25.51 21.18 18.49 16.32 14.55 12.84 12.43 12.30 F C - cos 7 49.00 58.05 51.40 24.45 20.58 18.15 i6.14 14.48 12.85 12.45 12.50 D qA _3 p =0 r CL F - sin y sA 24.41 16.31 11.80 7.33 5.01 3.55 2.38 1.43 0.32 0.04 0 r r z 2; 0 Z PI 0 n ^ I g t 0-4 \nJ rO Similarly for c C r = 1 C. CD 68.94 52.35 42.51 32.00 26.21 22.76 19.71 17.12 14.13 13.15 12.83 CL 48.96 35.58 25.00 16.11 11.46 8.79 6.38 4.55 1.75 0.84 0.55

26 I 22 20 18 16 I I I I I I I I DRAG COEFFICIENTS FOR _ --- — ----- _INFINITE RIGHT CIRCULAR CYLINDER _ — --- ----- WITH AXIS.L THE FREE STREAM --- --- --- = 00....X t,,.. 14 I I I I k.l 0 u I 12 10 8 6 4 2 0 I a U! E~o __ - - 1. A A X:0FX `4% I S _- _ - Of —m wnwm tE tee~~~~~~~~~~~~~~~~~~~~~~~~ DRAG COEFFICIENTS FOR INFINITE RIGHT CIRCULAR CYLINDER 1.0 - - -- WITH AXIS AT 45~ TO THE FREE STREAM 9 - 8 - 6 2 1t~~~~~~~~ t04 r r 0 o c z r 0 tI m 0 zd Cfl _: 0 1 2 3 4 5 6 7 8 9 Mco 10 0 1 2 3 4 5 6 7 8 9 10 FIG. 1

-3 -2 \J-p U u O z LIFT COEFFICIENTS FOR ] INFINITE RIGHT > CIRCULAR CYLINDER D WITH AXIS AT 45~ __ TO THE FREE STREAM ( = 450 ALL CL VALUES ARE NEGATIVE g 5 6 7 8 9 10 o| FIG. 1 a - - - - - - - - - - - - _ ^~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~,=, -1 0

_ _ 14 13 12 11 10 9 8 14 \J1 \Jn J 7 I — -- -- - ( _13 - --- --- --- -— _ 12 OEFFICIENTS FOR INFINITE CONE 11 8:_4 = 4_ u 7 3 2 6 7 8 9 10 0 1 r r 0 z C z n Ci) (C) 0 2 ct ^ 1=4 OO r V-4 6 5 4 3 2 1 0 0 1 2 3 4 5 FIG. 2

I 1.3 1.2 1.1 1.0 0.9 0.8 0.7 0n 0\ u 0 C Z I )..T bi t c C, 0 0 0.11 0.6 0.5 0.4 0.3 0.2 0.1 0 FIG. 2a

- - - - - ------- --- 70 -- -- ----- ------------ --- --- 65 --- 48- 60 - -- - - - - - - - 44 --- -- 55 --- - - - - -- DRAG COEFFICIENTS FOR DRAG COEFFICIENTS FOR 40 - - DOUBLE OGIVE 50 - DOUBLE OGIVE -. 0~ -- 450 36 -.. 45 - DIFFERENCE BETWEEN 32 W +_ ___ ___ |__ CURVES IS NEGLIGIBLE 40 \ - 28 -- - --- - - I ---- 135 - ------ - u 24 - -_ _ -_- -_ 30- - J____F 3 20 ---- 25 ------- - 16-, 20 --- _ 12 -- 15 -I_ 8 --- — i, - ----------- -10 --- -----— _'"'- --- 10 0 -- ---- 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 Moo M(o FIG. 3 r r o c C) 0 C) z z z O3 <.t1 IH 0 C) t I-4 0

48 44 40 36 32 u u 28 LIFT COEFFICIENTS FOR DOUBLE OGIVE d= 45~ ------— ____ ___ ___ ~ ~ —.,.,...,,,,...,mmmm ~ ~ eer Io C: CI (3I I,= i2: 24 20 16 12 8 4 0 0 1 2 3 4 5 6 7 8 9 10 FIG. 3 a

70 65 An a U 55 50 45 40 35 30 25 20 15 10 5 DRAG COEFFICIENTS FOR PROLATE ELLIPSOID It I___ -45' C~ X Xt I r r C z ti C3 C) I H '= 0 i 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 FIG. 4

28 26 24 22 20 18 16 u 14 Co 0 12 10 8 6 4 2 0 r r 0 o z > (3 n 0 Iz 1 =4 \ is 5 Moo FIG. 4 a

28 26 24 22 20 18 16 u 14 H r r 0 z 0 C n an I 1~ I 12 10 8 6 4 2 0 0 1 2 3 4 5 6 7 8 9 10 0 FIG. 5

WILLOW RUN RESEARCH CENTER - UNIVERSITY OF MICHIGAN -- UMM-55 1 0 U — II-L I z L _, LL. LU --- _Jm I I // ~'/~ 0 Os s 0 8o i CV, (N 0 -t C.) C. - 0 w~ r 0 00c N %O te tw CrV C( r- 0 O 62

WILLOW RUN RESEARCH CENTER-UNIVERSITY OF MICHIGAN, --- —------ -~ ~~~UMM-55 L DISTRIBUTION Distribution of this report is made in accordance with ANAF-GM Mailing List No. 14, dated 15 January 1951, to include Part A, Part B and Part C. 63