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EMEMMMMMMMMMMMM9 IVA"TM1193 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1293 APPROXIMATE METHOD OF INTEGRATION OF LAMINAR BOUNDARY LAYER IN INCOMPRESSIBLE FLUID* By L. G. Loitsianskii Among all existing methods of the approximate integration of the differential equations of the laminar boundary layer, the most widely used is the method based on the application of the momentumI. equation (reference 1). The accuracy of this method depends on the more or less successful choice of a oneparameter family of velocity profiles. Thus, for example, the polynomial of the fourth degree proposed by Pohihausen (reference 1) does not give velocity distributions closely agreeing with actual values in the neighbor hood of the separation point, so that in the computations a strong retardation of the separation is obtained as compared with experi mental results (reference 2). The moreaccurate methods employed in recent times (references 2 to 4) assume as a singleparameter family of profiles the exact solutions of some special class of flows with given simple velocity distributions on the edge of the boundary layer (single term raised to a power, linear function). The transition to the more complicated two and moreparameter families of profiles would require, in addition to the momentum equation, the employment of other possible equations (for example, the equations of energy (reference 5) and others (reference 6)). A greater accuracy might also then be expected fcr relatively simple velocity profiles that satisfy only the fundamental boundary con ditions on the surface of the body and on the edge of the boundary layer. This second approach, however, as far as is known, has not been considered except for very simple solution for the case of axial flow past a plate (reference 7). In the present paper, a solution is given of the problem of the plane laminar boundary layer in an incompressible gas; the method is based on the use of a system of equations of successive moments (including that of zero moment, the momentum equation) of the equation of the boundary layer. Such statement of the problem "Priblizhennyi Metod Integrirovania Uravnenii Laminarnogo Pogra nichnogo Sloia v Neszhimaemom Gaze." Prikladnaya Matematika i Mek hanika, USSR, Vol. 13, no. 5, Oct. 1949, p. 513525. NACA TM 1293 leads to a complex system of equations, which, however, is easily solved for simple supplementary assumptions. The solution obtained is given in closed form by very simple formulas and is noless accurate than the previously mentioned complicated solutions that are based on the use of special classes of accurate solutions of the boundarylayer equations. 1. Derivation of Fundamental System of Successive Moments of BoundaryLayer Equation. The wellknown equations of the stationary plane laminar boundary layer in the absence of compressibility have the form 2 Ou ou 0 U u + v  = jUU' + u x ~(1.1) ou ov  + E 0 where u(x,y) and v(x,y) are the projections of the velocity at a section of the boundary layer on the axial and transverse axes of coordinates x and y, U(x) is a given longitudinal velocity on the outer boundary U' = dU/dx, and U is the kineratic coeffi cient of viscosity. When the equation of continuity is applied, the first of equations (1.1) may b! given the more convenient form I0 L(u,v) = [u(Uu)1 + [v(Uu)] + U'(Uu) 0 (Uu) = 0 ox oy 2 cy (1.2) The left side of equation (1.2) is multiplied by 9' and integrated with respect to y from zero to infinity in the case of an asymptotically infinite layer or from zero to the outer limit of the layer y = 5(x) for the assumption of a layer of finite thickness. In either case, the following expression is obtained: L(u,v)yk dy = Uu) dy + fy [v(Uu)] dy + fo dx 0 0o oy U5 U' y (Uu) dy U o I k q2(Uu) y k 2 dy = 0 y y2 (1.3) FTACA TM 1295 7 It is assumed in this equation and in what follows that, in view of the very rapid approach of the velocity difference Uu to zero as y4 all integrals with the infinite upper limit have a finite value. For k = 0, d f u(Uu) dy U' (Uu) dy (1.4) ax 0 0 .p where the magnitude "V = ^(^)}= (1.5) Ou Ty 1=0 represents the friction stress on the surface of the body. Equation (1.4), the wellknown impulse or momentum equation, is readily transformed into its usual form db** UIB** (  + (2+H) = (1.6) dx U p2 where ',5 j= ( dy \ U} ,5 > (1.7) S = 1 u dy 0 U U H 5* For k = 1, a new equation of the 'first moment' is obtained from equation (1.3) NACA TM 1293 d yu(Uu) dy  dx Jo v(Uu) dy + U' T y(Uu) dy = VU and, in general, for k > 2, the equations of successively increas ing moments are obtained S yku(Uu) dy k yv(Uu) dy + U' yk(Uu) dy U0 0 0 = k(kl) OT yk2(Uu) dy In all these equations, the transverse velocity assumed expressed in terms of the axial u(x,y) from of continuity. It is now assumed that the family of functions u = I? (x, y;, 2 X k) v(x,y) is the equation (1.10) satisfies the boundary conditions of the problem with k param eters X, k, which are functions of x, such that the k successive moments of equation (1.3) GD, ykL(uo, v0) dy (1.11) become zero. On the assumption that it is permissible to pass to the limit k *, it would then be possible to state that the function u(x,y) = lim k.K" uo x ; (x), X2(x), ,X k(XA (1.12) with parameters Xl(x), K2(x), Xk(x) satisfying the infinite system of equations (1.8) (1.9) NACA TM 1293 yakL(uo, v) dy 0 (k = 0, 1, 2, ) or, what is equivalent, systems (1.4), (1.83), and (1.9) will be an exact solution of the fundamental system (1.1) for the assumed boundary conditions. For this solution, it is i.;?relV necessary to recall the known theorem that a continuous functir.n, call successive derivatives of which are eoual to zero, is identically e0ual to zero (reference 8). The question of the proof ocf the validity of this theorem is not considered in the case of an infinite interval or of an inter val the boundaries of which are fuiinctions of a certain variable with respect to which the differentiation is effected. A certain construction, not based it is true on a rigorous proof, of the solution cof the problem will be emplc.:.ed with the aid of the suc cessive euations of the moments of the basic boundar.,la.er aviation. 2. Choice of 1ara:eters of FarJi.l, of Velocity Profiles at Sec tions of Boundary La:cr. Special Form of Equations of Moments. As is seen from the previously discussed considerations, the funda mental difficulty lies in the choice of a family; of vclocity pro files (1.10) and the determination cf the parameters hfk of the family. One of the simplest :rethods of the solution of this prob lem is indicated herein. In the converging part of the toundar; layer, the velocity profiles at various sections of the layer are known to be almost similar; the velocity profile is deformed mainly in the diffuser part of the boundary layer downstream of the point of minimum pres sure. The deformation of the profile consists of the appearance of a point of flexure that arises near the surface of the body and then moves awa. from it as tho separation point is approached. The presence of this deformiation of the profile near the sur face should greatly affect the magnitude T prporticnal to the normal derivative of the velocit& on the surface of the body:,; it will therefore diminish to zero as the point of separation is approached. The deformation of the profile will have a smaller effect cr such Integral magnitudes as Q* Mni 5** And ver, little esfec+. cn magnItudes that contain under the !rtCilcr.l s!;n 'runcthns t.h'it. raridll;' de5reas?9 as the srrfaoe of 'r.e 1 :iy s orprach?'. NACA TM 1293 For the parameters characterizing the effect of the deforma tion of the velocity profile, it is natural to assume those magni tudes that depend relatively strongly on the deformation of the velocity profile. With regard to the magnitudes that vary little with the deformation of the velocity profile, however, it is natural to assume that they do not depend on the chosen parameters. For the fundamental parameters determining a change in the shape of the velocity profiles, which may be called form parameters, the nondimensional combination of the magnitudes Tv, B* and 5** will be employed with the given functions U(x) and U'(x) and physical constants, namely, the parameters 1)1 f U'8o**2 r= Ts** (2.1) k(y/b**) }y= 1U H  For the computation of the remaining magnitudes in the equa tion of moments according to the assumption, the velocity profile will be assumed in a section of the boundary layer in a form that does not depend on the parameters f, t, and H: U U ,).,(,) (2.2) This assumption permits, as will be subsequently seen, obtain ing on the basis of very simple computations a sufficiently accurate solution of the boundarylayer equations for arbitrary distribution of the velocity on the edge of the layer. The transformation of equation (1.6) will now be considered. If the parameter ( is introduced, then by equation (2.1), dL** U'8** P ,1, Ufb** +(2+H) = dx U U 8** or I2 ^ o U _d (6 *4f2 + (2+H) f= NACA TM 1293 It is not difficult to obtain finally 1 U df + H 1 H U "f O ( T x 2+ ~ f = 2 (2 (2 U'dx 2 U,2 For the transformation of the left side of equation (1.8) the first integral can be written by equation (2.2) (i = y/l*) yu(Uu) dy = U23**2 1p(lp) dj = H1U2**2 (2 .0 0 where the magnitude H1, equal to er ( H1 = 9p(14) dTi g0 (2.5) represents a constant computed by the given function T(j). In order to compute the following integral, the transverse velocity v is first expressed by the formula v = j dy = U% J 9 d) ox ox Jo\J0 0 qdrq U dx 0 0 d JU 5** <9 dx dx or, when it is noted that in d ( ) ") dx dx 5* Y dB** g *2 dx 1 d5** 1 ** dx the following expression is obtained: dj U d dx / f I 1 0 1 di / 0J .3) .4) v = U'F** j (2.6) = U'3** NACA TM 1293 There is thus obtained v(Uu) dy = U25** ) dTI o S UU'**2 ( jp 9 d)(14 ) dti  J0^~0 uUY'**2Js7J0' Pdn(19 dq or v(Uu) dy = HU2 5** H5 JU' 2 (2.7) ~dx 0 where H2 and H. denote the constants H2 = ( dT (1p) dy( J0 \ 0 \ b = f q~dy](1p) dy (2.8) H3 d) 1 dq Finally, the last Integral in equation (1.8) is transformed into f1 0( y(Uu) dy = H4U 8**2 (2.9) where the constant H4 is equal to H4 = Tl(lp) dyl (2.10) CJ NACA TM 1293 9 By substituting the integrals obtained in equation (1.8), d (HA KU 2) HU2 d + UU 2 + LUU'8*2 = u dx L / 2 dx 4 (2.11) or by replacing 8**2/v = f/U' by equation (2.1) aind carrying out the transformation, ( H 1 ) = [1 (2~1+3+)f +37 )f (2.12) When the nev constants are introduced, 1 Hi 1 Hz BlJ 2 b 2 H, + H+(2.13) df U1(abf U (2.14) The third equation is obtained from the system (1.9) "by setting bk = 2. d ( y~u(Uu)dyv 2 /yv'Uu~dy + U1 7 2y^U_dy 2v (Uu)d:y dxo Jof Jo Jo (2.15) the equation of the first moment is reduced to the form df U' U" dx u(abf) +  (2.14) The third equation is obtained from the system (1.9) by Betting k = 2. y~uUudy 2Jyv(Uu~dy + U'fY2(Uu)dy = 2uJ (Uu)dy (2.15) There is obtained, as before, 2 u(Uu)dy = HgU2**3 (2.16) 10 NACA TM 1293 where the constant Hc is equal to H5 H5 = 2p(19)da (2.17) Further, by analogy with equation (2.7) OD  yv(Uvu)dy 6U2r**2 d* H7UU'r**3 (2.18) Jo dx where H6 f T( P f T dt d (19p)dyl ) (2.19) H7 = pdj (1cp)di 0 0C The last integral on the left side of equation (2.15) is equal to f2(Uu)dy = HgU 5**3 (H8 f 2(l9)d) (2.20) The integral in equation (2.15) on the right reduces to the unknown prameter H11 (Uu)y = ) dy = U ** = U ?**H (2.21) Jo Jb~d = Ur 8U 3* By substituting the expressions obtained for the integrals in the secondmoment equation (2.15), there is obtained, after simple transformations, S(3H52H6) = ][H (H+H7+ ) f + (3B52H6) Z f (2.22) NACA TM 1293 The system cf three equations (2.3), (2.14), and (2.22) has thus been established for determining there three unknown magnitudes , and H. The solution of this system is now considered. 3. Determination of the Constants H.. Approximate Formulas for Paranieters f, r, and H. For the determination of the numeri cal values of the constants H1, H2, ., H8, the form of the function (c (1) must be known. The simplest velocity profile in the theory of the asymptotic boundary layer is the velocity profile in the sections of the boundary layer of the flow past a plate. The function (pi) for this case can easily be determined from the generally known table of values of the velocity ratio u/U as a function of ; = Ux/2. Superfluous computations may be avoided by noting that the con =*.s to be crin'uted are connected with one another by certain sim le reltAtions. First of all, frcm equations (2.3) and (2.14), = a + + U f (3.1) 5y setting f = C, there is obtained a = 2Co, where t0 denotes the magnitude 6 ccmpuLed for the plate (U = 0, f = 0). From' the definition of C and frcm the known relations for the plate, S= 2 , 0.664 .32 = .664 = 0.4408 U i U 11 VU (3.2) Further, by comparing with one another the magnitudes H1, H2, H_, and 84, H3 = Bl H2 (3.3) H4 H3 = I( IdTI (1.;p)dy = (1p)dl 2 ( 02 (3.4) NACA TM 1293 where H0 is the value of H for f = 0, that for a plate is equal, as is known, to H0 = 12 = 121= 2.59 6** 0.664 is, the ratio It is (3.3), and, then easy to obtain the value of b (3.4). by equations (2.13), b = 2H1+H3+4 = a (2H1+H1H2+H1H2+ "H62) = a (4H12H2+ 1H2) = 4 + a H02 = 5.48 5.5 (3.5 When df/dx is eliminated from equations (2.22) and (2.14), i ~3H52H6 H = (5+H7+ 8) f + (2H1+H3+H4) f 4(H1 2H2) = (H5+H7+ H) f + (3H52H6) 1 f (3.6 By setting f = 0, S(3H52H6) = = 5.89 (3.7 = a =0.44Z58 The only magnitude that must be computed again of values 4(i) is the magnitude H5 + H7 + H8/2,. gration gives H5 + H7 + H8 = 24.73 after which there is immediately obtained from the table Numerical inte (3.8) H = 2.59 7.55 f ) ) ) (3.9) NACA TM 1293 Substituting this expression for R in equation (3.1) gives C = 0.22 + 1.85 f 7.55 f2 (3.10) Finally, integrating the simple linear equation (2.14) gives x aU' b1 0.44U' U4.5 f ub Uo )u 5 Jo (3.11) U 0 ou.5 Fo Equations (3.9), (3.10) and (3.11) give the required solution. The simple, approximate solution just obtained is now compared with the actual values. The almost complete agreement of the val ues of f obtained with the first approximation (which is practic ally the only one that is applied) of the preceding works (refer ences 2 and 3) will be noted. The closedform relation between ( and f likewise differs little from the corresponding tabulated functions in the references cited. For comparison, the curves C(f) and H(f) obtained accord ing to the formulas of reference 2 and by the formulas (3.10) and (3.9) are shown in figure 1. The results obtained will also be compared with the formulas of Wright and Bailey (reference 9). An approximate method of computation of the laminar boundary layer is proposed therein in which the equation of momentum (1.6) is employed with T. and 6** substituted by the formulas for the flow past a plate. By expressing the results of Vright and Bailey in the parameters of the present report, the analogs of equations (3.9), (3.10), and (3.11) are obtained. H = 2.59 C = 0.22 + 4.09 f (3.12) f = 0.44 UJ U It is easily seen that this formula for f corresponds to equation (3.11) for b = 1. The straight lines for ( and H shown dotted in figure 1 indicate the considerable deviation of the formulas of Wright and Bailey from more accurate formulas presented herein. NACA TM 1293 For confirmation, the particular case of the laminar boundary layer corresponding to the socalled singleelope velocity distri bution at the outer boundary of the layer U = 1 x will be con sidered. This case has been theoretically solved and an exact solution in a tabulated form (reference 10) is available. The results of the recomputation of these accurate solutions in the form assumed by the parameters are given in figure 2. Also shown for comparison are the corresponding curves obtained by the pro posed approximate method and by the method of Wright and Bailey. 4. Possible Methods of Rendering the Foregoing Solution More Accurate. The method described in the preceding sections was based on the assumption of a slight dependence of Hi on the form param eters f, C, and H. This assumption may be eliminated and the method rendered more accurate, although it thereby becomes con siderably more complicated. In order to discuss the possible generalizations of the method, the complete system of equations, for example, for the three parameter case is written out; that is, a threeparameter family of velocity profiles is assumed in place of equation (2.2). f, H) (4.1) By substituting this velocity profile in the system of the three equations of successive moments (1.6), (1.8) and (2.15), there is obtained a system of three ordinary nonlinear differential equations that determine the magnitudes of the parameters f, (, and H: 1 U df 1 UU" ZI Uf+ 2 f + Ef (4.2) F U'dx 2 U' r [1 ( ++4 + (x1 )H2 ) f (4.3) H2 \+ o+ )f +dx +2+ dx (5 + / dx U [1 (2H,+H3+H4)f] + '1(]l H2) f (4.3) MACA TM 1293 ["1 ~ ~ M5 f H. It (352E6) + K4 ) 2 (K6 "+2H dx U f (4 dx +K) (+s+H7 f + dx + R8f] 2  (3H52H6) f in which, in addition to the previous notations, the following definitions are chosen: OD K1 = K2 *o K3 L K4 T Jf K5 = 0 0 K6 = f9 (1fq)d) 6 0 0 (4.4) (4.5) di)j (1 )dTq 7f d ) (1 p)dTi a d: 1 ( 1 rp ) d y l L d) (1cp)dl 6f 1) NACA TM 1293 It is noted that, in the system of equations (4.2), (4.3), and (4.4), Hi and Ki are not constant magnitudes as pre viously, but known functions of the form parameters f, C, and H; the form of these functions depends on the chosen family of profiles (4.1). The equations (2.3), (2.14), and (2.22) earlier employed evi dently represent a particular case of the system (4.2), (4.3), and (4.4) on the assumption that the family of velocity profiles at the different sections of the boundary layer has the form of equa tion (2.2); in other words, these profiles are similar to one another. All values of Ki are of course then equal to zero and Hi is constant. The proposed method may be rendered considerably by assuming, for example, the singleparameter family profiles = h(e; f) Then =K3 = =K 3 OH 1 more accurate of velocity (4.6) S= K6 = = 0 ac 2EO and the system as follows: of equations (4.2), (4.3) and (4.4) is transformed 1 5df 1UU SU + 2 f + 2 U,'dx ( 2 U12) (4.7) S 2 U1+ f f L1  (2H1+H3+H4)f] + (3H25H6 z = H56 + f V 5 dx UH (R5+H7+i 1: ])fl + IU"352H6 T \2 4j 411 2 C1 (4.8) (4.9) i l H 2 f NACA TM 1293 Equation (4.8) can be given the form U' 1 (2H1+H3+H4)f U ul (K++H/f U U 1 H}2 + (Kl+6H1/f~ U' H1 1 S2 HK (4.10) which represents a generalization of equation (2.12) where equa tion (4.10) approximates equation (2.12) because of the small change in Hi with change in the parameter f and the smallness of the magnitude (Kl+3H1/of)f in comparison with H1 H2/2. This gen eralization permits obtaining the integral of equation (4.10) by introducing a correction to the solution of equation (2.12). By dividing both sides of equation (4.9) by the corresponding sides of equation (4.8) and thus eliminating df/dx, there is obtained H = (H5+H7+: H8)f+ (3H52H6) + H 1 H2 + 1K + f 681 Kl* F 1 (2Hl+H3+H4)f] .+ E(1 (H12 H 2) 1 (3H52H6) + + 1 Z) 1f ( 1 f  1 ( 3 H5 2 H6 ) 4 1 H2 + (K1 f(. (4.11) By similar considerations on the smallness of the magnitudes (K4+1/2 oH5/6f)f in comparison with (3H52H6)/4 and of (Ki+oH1/6f)f in comparison with H, H2/2 and on the slight variability of Hi, it may be concluded that the value of H determined by equation (4.11) is an improvement in the accuracy of the approximate value of H according to equation (3.6). It may be remarked that in this more accurate approximation there is no longer that universal relation between the parameters H and f, independent of the form of the function U(x), character izing the given particular problem. The presence in equation (4.11) UU" U'2 NACA TM 1293 of a second term with the factor JU't/U'2 shows that in the more accurate approximation the magnitude H in a given section of the layer depends not only on the value of the form parameter f in this section, as was the case in the rougher approximation of equa tion (3.6) or (3.9), but also on the value of the magnitude UU"/U' in the section considered, that is, on the values of the func tion U(x) and its first two derivatives. It is readily observed that the second term on the right side of equation (4.11) will give a small correction to the solution (3.6) for relatively small values of the magnitude UU"/U'2. The same considerations hold for the expression for t, which may be obtained by substituting df/dx from equation (4.10) and H from equation (4.11) into equation (4.7): S1 ~ (21+8*4 1l 1 /, H^JC.1 ^ f8 1 (2E+H3+H4)f F+ 1 (3H52H6)f +('K4 1 f 2] + 1 1 H2 L2f 4 2 Zif 51"2 2 + Si Kl)g~ 1 2 2f + (H5+7+ H8)f2 /f[ +H, or1 f,3/ i H i \ U K (+ f2 + (3H526) K4 + 2 2) U12 1 (3H HJ l 2 2 + 1+ 2 \ (4.12) As is seen, in this new approximation, in contrast to the pre ceding one, there is no universal relation between t and f. The presence of a term with the factor UU'"/U,2 makes the magnitude t depend not only on the value of the parameter f but also on the form of the function U(x) and its first two derivatives in the given section of the boundary layer. It is of interest to remark that in this approximation the position of the point of separation of the boundary layer, that is, the value of x = x. for which C is equal to zero, will no longer be determined by some universal value of the form param eter f., but in each individual case the value of x = X8 must be determined for which the right side of equation (4.12) become zero. NACA TM 1293 By assuming a particular form of a family of velocity profiles (4.6), employing, for example, the sets of velocity profiles applied in the previous investigations (references 2 to 4), the values of the functions Hi and Ki are determined; the form parameters f, (, and H, that is, the thickness of the momentum loss **, the friction stress Tw and the displacement thick ness 5* may then be found without difficulty. The solution of equation (4.10) and the determination of H and t by equa tions (4.11) and (4.12) offers no particular difficulty. Further improvement in the accuracy requiring the solution of a system of the type of equations (4.2), (4.3) and (4.4) is hardly of practical interest. In the previous discussion, the scheme of the asymptotically infinite boundary layer was used, but similar equations may be obtained also for the case where the boundary layer is assumed to be of finite thickness. The method here proposed may evidently also be applied to the case of the thermal boundary layer. The characteristic feature of the method for the cases of both the dynamic and the thermal bound ary layer lies in the fact that the friction stress and the quantity of beat given off by a unit area of the body are expressed in inte gral form and not in terms of the derivatives of functions that represent the approximate velocity and temperature distributions in the sections of the boundary layer. Translated by S. Reiss National Advisory Committee for Aeronautics. REFERENCES 1. Loitsianskii, L. G.: Aerodynamics of the Boundary Layer. GTTI, 1941, pp. 170, 187. 2. Loitsianskii, L. G.: Approximate Method for Calculating the Laminar Boundary Layer on the Airfoil. Comptes Rendus (Doklady) de l'Acad. des Sci. de L'URSS, vol. XXXV, no. 8, 1942, pp. 227232. 3. Kochin, N. E., and Loitsianskii, L. G.: An Approximate Method of Computation of the Boundary Layer. Doklady AN SSSR, T. XXXVI, No. 9, 1942. NACA TM 1293 4. Melnikov, A. P.: On Certain Problems in the Theory of a Wing in a Nonideal Medium. Doctoral dissertation, L. Voenno vozdushnaia inzhenernaia akademia, 1942. 5. Leibenson, L. S.: Energetic Form of the Integral Condition in the Theory of the Boundary Layer. Rep. No. 240, CAHI, 1935. 6. Kochin, N. E., Kibel, I. A., and Roze, N. V.: Theoretical Hydrodynamics, pt. II. GTTI, 3d ed., 1948, p. 450. 7. Sutton, W. G. L.: An Approximate Solution of the Boundary Layer Equations for a Flat Plate. Phil. Mag. and Jour. Sci., vol. 23, ser. 7, 1937, pp. 11461152. 8. Carslaw, H., and Jaeger, J.t Operational Methods in Applied Mathematics. 1948. 9. Wright, E. A., and Bailey, G. W.: Laminar Frictional Resistance with Pressure Gradient. Jour. Aero. Sci., vol. 6, no. 12, Oct. 1939, pp. 485488. 10. Howarth, L.: On the Solution of the Laminar Boundary Layer Equations. Proc. Roy. Soc. (London), vol. 164, no. A919, Feb. 1938, pp. 547579. NAjA TM 1293 1 0 1 .10 0.08 0 f Figure 1. x Figure 2. NACALangley 72651 1000 4 H 3 2 L. C: 0 ca u 3 s!" w" Bd .5 jkW5. ci2 o 4 o'v :tC S oa b.E J 0 ^ a a. o a "'i1 I 0) E o C .o . s a.' 2b R:. 5 =a'.~ .mi~ ..a CjS a Ei in m t i CU U? IUL E ,g 0J V'1 U " Q M 0 i 5 s v, *S^w) r .2 * an c Gocif lL' 0) 2 g 0c IV. C r 3 ci a5! aE Sa m Imu , ba 26" n r_ 3a. 02a; 'oN 0 c i0 Q. ~ ~ > 0 1 S U; z5 I 4;n . o E w oo ci 2 u Mr ii 5? i A a 20 r Lhfw Z ,.'D LT ", :ai ' t ca IV< a Sw 1w N o a n cZ 0 . C 0 I z s J; <[ IZA) 4 o CL. .<. Iz E4S> CLq Y~~ 2,:i~ o o rsrc4!2 r4 o q AA I0 S~o Ic z2 S l 0102 0)w F < C) zz F, L. F cn? z z 3 Ea 1 cd'il0 S. 'n toM , ~ F .u tsK S o r. I..~ c2U .. I's3 kp\ I~ ~ u l5. = g 2 .2 ta 0.. 0 P, z 0 0 i ZL 0l >g 0? 01Jl CLg~ 6 i5 '0 0a t^ .S t j a se 40 E a 4E NO z: Go0" S S aMI.0f 00 ?0 g RH ^ e Sf Oll8r~ 014 300 cc 0 7l a I E 2j~j^ ull~l 00OS ll~i~ll f.. 5, Q kga~ l a '8 ' % U,0 .g^ ~ g S 5 s gu Lu.C2 v S 5;. : sa o *~ 'S > jSgF *a sa S *S a I "SCuOa 44 ~v W .6E ) 4 . 0. U OE~~3 '0 .uj~ ~0 C S.cdO0 WW ,w V 0Cu 11 (,oo SO *Cu0 m000m0 a s 0..C Cu O C; E PO C4 Vf i W i wuVg '9 * .. 5.2 D . .Q .g .... ..... ....... F.0 .4o Cd %J a A co Q' z CUl o 0 toa~l 0 Cu 2 _ .o V Z > 0 0 4) 0i .i bl,S S 0=' t 'dB **~~~~~ ~ z" z *w, g o t S 'S  =1 t., Ig. 0  li i'l l E D4 F4l^ 2 ". !Rs a , g!l^ i s ; W llq 0 :g S I a IEl 0 ~go S o. W S ~ t53'^ Safc ta^ 13 0 * 4:z~~l  W l0 .R.. 0 0 > 0  j UNIVERSITY OF FLORIDA 11 11 0811111 64 7 3 1262 08106 645 7 
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