TM-'hof 
 
 ...a:«AJMJN 
 
 Ir' 
 
 NATIONAL ADVISORY COMMITTEE 
 FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 1308 
 
 ON MOTION OF FLUID IN BOUNDARY LAYER NEAR LINE 
 
 OF INTERSECTION OF TWO PLANES 
 
 By L. G. Loitsianskii and V. P. Bolshakov 
 
 Translation 
 
 "O Dvizhenii Zhidkosti v Pogranichnom Sloe Vblizi Linii Peresechenia 
 Dvukh Ploskostei." Rep. No. 279, CAHI, 1936. 
 
 Washington 
 
 November 195^,,,,, ,.^5,^^ ^^ p^ORIDA 
 
 ;^'^NTS DEPARTMENT 
 . ^o iViAnSTON SCIENCE LIBRARY 
 P.O. BOX 117011 
 GAINESVILLE, FL 32611-7011 USA 
 
^"^^ b-iL. ut>^ ^7"? 
 
 NACA TM 1308 
 
 NATIONAL ADyiSOEY COMMITTEE FOE AEEOWAUTICS 
 
 TECHNICAL MEMOEANDIM 1308 
 
 ON MOTION OF FLUID IN BOUNDARY LAYEE NEAR LINE 
 OF INTERSECTION OF TWO PLANES* 
 By L. Or. Loitslanskii and V. P. Bolshakov 
 
 SUMMARY . 
 
 In the paper "The Mutual Interference of Boundary Layers , " the 
 authors InTestigated the prohlem of the interference of two planes 
 intersecting at- right angles on the "boundary layers formed by the motion 
 of fluid along the line of intersection of these planes . 
 
 In the present paper, the results of the preceding one are general- 
 ized to the case of planes intersecting at any angle. The motion of a 
 fluid in an angle less than 180° is discussed and the enlargement of the 
 "boundary layers near the line of intersection of the planes, the limits 
 of the interference effects of the "boundary layers, and the corrections 
 on the drag are determined. All computations are conducted "by the 
 Earman-PohUiausen method for laminar and tur"bulent "boundary layers . The 
 results are reduced to tabulated form. 
 
 INTRODUCTION 
 
 The pro"blem of the interaction of the "boundary layers formed in the 
 dihedral angle between two thin plates parallel to the intersection of 
 the plates was apparently first proposed in reference 1 (for the case 
 of a right angle). In the discussion of reference 1, the need for a 
 generalization of results obtained for the case of the right angle to 
 the case of any angle less than two right angles (180°) was pointed out. 
 The present paper is concerned with this problem and its solution. 
 Although the limits of application for the approximate method of the 
 finite -thickness layer previously used are retained, the problem of the 
 interaction of the boundary layers near the intersection of a dihedral 
 angle of any magnitude from 0° to 180° is solved herein. For a laminar 
 layer, a first and a second approximation are given and also, for a 
 check, a sixth approximation (in the terminology of Pohlhausen). It is 
 
 *"0 Dvizhenii Zhidkosti v Pogranlchnom Sloe Vblizi Linii Peresechenia 
 DvukkL Ploskostei." Eep. No. 279, CAHI, 1936, pp. 3-18. 
 
NACA TM 1308 
 
 shown that the sixth approximation differs comparatively little from the 
 second. In concliision, the case of a turbulent boundary layer is considered 
 with the assumption of the validity of the '1/7' power law for the velocity 
 profiles. As in reference 1, all computations can be carried through to 
 the end, although the procedure is somewhat more cumbersome. The limits 
 of interference of the layers and the correction on the drag due to the 
 Interference effect are determined. 
 
 1. DERTVATIOW OF FUKDAMEIWAL IM'EGEAL CONDITION 
 
 Consider the flow of a fluid, approaching from infinity with 
 velocity T, in the dihedral oblique angle 9 between two plates of the 
 same finite length x along the flow and infinite in the transverse 
 direction (fig. 1). The Y- and Z-axes are taken along the leading edges 
 of the plates and the X-axis along the flow parallel to the line of 
 intersection of the plates. In the oblique system of coordinates thus 
 obtained, the distribution of the velocities in the boundary layer may 
 be given in the same manner as for the case of the rectangular system 
 of coordinates for the flow in a right angle. 
 
 If each plate worked Independently of the other, there would be 
 formed on it a layer of thickness 5q, which is a function only of x 
 and the profile of the longitudinal velocities 
 
 u„ = Uq(x, z, sin 6) 
 
 or 
 
 Uq = Uo(x, y, sin 0) 
 
 depending on whether the boundary layer is considered to lie in the 
 plane XOY or XOZ. 
 
 Because of the retarding effect of one of the plates on the other 
 near the line of intersection, the layers must not be considered as in 
 the plane problem. The layers become three dimensional and the thick- 
 ness will now be a function of two variables : 
 
 S = 5(x,y) on the plate containing the Y-axis 
 
 5 = 6(x,z) on the plate containing the Z-axis 
 
 The component of velocity parallel to the X-axis, which will be denoted 
 by u without the subscript 0, will be a fimction of the three 
 variables ; that is , 
 
 u = u(x,y,z) 
 
WACA TM 1308 
 
 By the conditions of the prohlem, under the "basic assumption of the 
 concept of a finite region of influence of the viscosity, this function 
 must tiecome zero on the surface of the plates, a constant at the outer 
 limit of the houndary layer, and, particularly, at a finite distance from 
 the line of intersection of the plates, must "become the velocity distri- 
 bution uo(x,z, sin G) or uo(x,y, sin 0), which corresponds to the 
 isolated plate with "boundary layer undistur'bed "by the adjacent plate. 
 The "boundary separating the region distur"bed "by the adjacent plate from 
 the undistur'bed region, which corresponds to the isolated plate, shall, 
 for "briefness, be denoted as the "interference "boundary" of the layers. 
 The eq_uation of this "boundary In the planes XOY and XOZ will "be 
 
 hQ = hgCx) 
 
 A section of the "boundary layer cut "by the plane x = ^ is shown 
 in figure 2 . Inasmuch as the coordinates of the system YOZ are 
 oblique, the equation of the boundary layer in this section for the 
 undistiirbed region for y>y2(?)j where 
 
 ygC?) = ^qU) - \U) cos e = h^{^) - Sq(C) cot e, win have the form 
 
 Equation (Ij holds true both for 9<n- and for G> :^. The non- 
 
 ^ 2 
 
 coincidence for Q f — of the interference boundary ^rSX) and the 
 
 coordinate 1q{^) should be noted, from which starting point the 
 ordinate z of the outer limit of the boundary layer becomes and remains 
 constant, independent of y. All that has been stated about the YOX 
 planes also remains true, of course, for the ZOX plane (because of 
 the symmetry y-^C?) = z-|_(^) and ygC?) = Z2(?)> which should be remembered 
 in the following development) . 
 
 For the present, the question as to the equation of the boundary 
 layer in the section x = ^ will be ignored. The fundamental integral 
 condition of the problem will now be set up. For this purpose, as in 
 the work previously cited, the momentum theorem is applied for a tube 
 of flow formed by the coordinate planes, by the surface of the lines of 
 flow passing through "tlie edge of the boundary layer at section x = x, 
 and, finally, by the surfaces of the lines of flow passing through the 
 perpendiculars to the plates located at the points y = hQ(x) and 
 
 z = hQ(x), where the part of the flow tube considered is between the 
 
 sections x = x and the section located upstream of the flow at a 
 sufficient distance from the section x = 0, that is, from the entry of 
 the fluid on the plates. Then, as is known. 
 
VJ 
 
 (a) 
 
 p(V-u) u da = W 
 
 NACA TM 1308 
 
 (2) 
 
 where W is the drag of the plates applied to the segment of the flow- 
 tube considered and a is the section of the boundary layer cut by a 
 plane perpendicular to the X-axis at distance x from the origin 0, 
 
 The double integral on the left may be expressed in the following 
 manner (fig. 2): 
 
 r 1 
 
 1 
 
 (V-u) u da = sin 
 
 dy 
 
 r\ 
 
 
 
 ^2 
 
 p(V-u) u dz + 
 
 '72 PZ 
 
 d-J I pCV-u) u dz + 
 
 .y2+z-|_ cos B 
 
 dy 
 
 cos B 
 
 p(Y-u) u dz 
 
 (3) 
 
 where Z, the ordinate of the outer edge of the boundary layer, is, as 
 yet, an unknown function of y and x if the angle B is considered 
 as a parameter maintaining a constant value for the given problem. The 
 first two integrals have a very obvious origin; the presence of the last 
 integral is due to the obliq.uenes3 of the coordinates and the differences 
 in the direction between the ordinate Z]_(x) and the thickness of the 
 layer 5o(x) , which makes it necessary to take the integration over the 
 two areas of the triangles shown hatched in figure 2 . 
 
 The drag W, as is easily seen from figure 3, is determined by the 
 integral 
 
WAG A TM 1308 
 
 r\ X 
 
 W = 2 
 
 O^inC?) 
 
 0' 
 
 00 
 
 Tdy 
 
 I»h(x) 
 
 r» x 
 
 d? = 2 
 
 py2(?)+^l(?) °°s ^ 
 
 T dy + 
 
 'y2(^)+Zl(^) °°^ ® 
 
 Jy2(^)+^i(?) °°^ ^ 
 
 Tq dy 
 
 iC 
 
 ^4) 
 
 where T denotes the friction stress in the region S disturhed "by the 
 adjacent plate and t denotes the corresponding stress in the undis- 
 
 tirrbed region Sq. The boundary "between the regions S and Sq is, 
 of course, the boundary of interference of the boundary layers. 
 
 Combining equations (3) and (4) yields, in general form, the 
 integral condition of the problem: 
 
 sin 
 
 yi 
 
 cLy 
 
 p(V-u) u dz + 
 
 o z 
 
 dy 
 
 p(Y-u) u dz + 
 
 ^Q 
 
 yg+z^ cos 
 
 yg+Zj^ cos 0-y 
 
 cos 6 
 
 iy 
 
 p(V-u) u dz 
 
 r» X 
 
 = 2 
 
 ygCO+z-j^C^) cos 
 
 py2(x)+z^(x) cos 
 
 T dy + 1 '^0 ^7 
 
 Jy2(C)+2i(?) cos 
 
 <i? 
 
 (5) 
 
 Eq^uation (5) is a generalization of the integral condition derived in 
 the previously cited paper for the case of a right-angled dihedral 
 
 (0 = — ) . The limits of integration are as yet unknown functions of the 
 
 arguments. In order to determine the forTn of these functions, it is 
 first necessary that the shape of the boundary layer be given. 
 
NAG A TM 1308 
 
 2. SHAPE OF CROSS SECTION OF BOIMDARY LAYEE; VELOCITY DIACJRAMS; 
 
 SOLUTION OF PROBLEM BY FIRST APPROXIMATION 
 
 The section of the "boundary layer on the plane x = ^ (fig- 2) ia 
 considered and it is assumed that the equation of the curve defining the 
 edge of the "boundary layer is 
 
 f(y,z) = a(?,e) 
 
 JiU) » y - yg^^) 
 
 > 
 
 (6) 
 
 The form of the function f (y,z) cannot "be determined unless the 
 additional assumption is made as to the similarity of the approximate 
 velocity diagrams in the different planes x = 4- The curve is sought 
 in the form 
 
 u 
 
 f(y,^) 
 
 (7) 
 
 (8) 
 
 (9) 
 
 where the function <p(t) is subjected to the conditions 
 
 <p(0) = 
 
 <P(1) = 1 
 
 and the function f(y,z) is su"bjected to the conditions 
 
 f(0,z) = f(y,0) = 
 
 Then the diagram of velocities of equation (7) vill evidently satisfy 
 the end conditions of the pro"blem at the walls and at the edge of the 
 "boundary layer of equation (6). Setting 
 
 y = iiK,e) y- 
 z = i{^,e) z- 
 
 where 2(^,0) is a certain length characteristic for the section x = ^ 
 and y' and z' are nondlmensional magnitudes, yields eqixation (7) in 
 the form 
 
 u 
 V 
 
 <P 
 
 
NACA TM 1308 
 
 From the condition of similarity of the velocity diagrams^ the right side 
 of this equation must not contain £,, which can be the case If f(y,z) 
 is a homogeneous function of the zi^^ degree, so that 
 
 u 
 
 ^ '<--') 
 
 vhere 
 
 ^ (€;0) = constant (lO) 
 
 the constant depending, of course, on 0. It is easy to see that the 
 degree of homogeneity should he equal to 2 (n = 2) and the function 
 f(y,z) must simply he equal to the product of the variables yz because 
 otherwise, with conditions (9) satisfied, the derivatives of the velocity 
 with respect to the coordinates would become zero at the walls and would 
 not give any friction. From equation (s), the following equation of the 
 edge of the boundary layer at the section x = ^ is obtained: 
 
 yz = a(C,0) (11) 
 
 where, from equation (lO), for example, choosing the coordinate 
 j-j_iK) = Z]^(g) of the undisturbed layer for the characteristic length 
 
 liK,e) yields 
 
 a(c,e) = k(e) z/ (^) = J£(|L 5 2(0 (12) 
 
 ■^ sin2 e 
 
 where k(0) is a certain nondimensional constant parametrically 
 dependent on the angle 6. 
 
 It is now easy to obtain the magnitudes J2{K) } ^q{K) > a^<3. the 
 function Z(y,^,0) for given angle 0, From equations (ll) and (12), 
 
 where S^CC) is a known function of ^ independent of the angle 6 
 and determined by solving the problem of the boundary layer of the 
 isolated plate. 
 
 From the definition of the interference boundary ho(?) given in 
 equation (l), 
 
NACA TM 1308 
 
 ho(C,0) = y^{K) + Zi(C) cos =[k(0) + cos eJziC?) = ^^^^.^^ g°" ^ SqC?) 
 
 (14) 
 
 Finally, from equation (ll), there directly follows that 
 
 2, 
 
 y sm^ 7 
 
 (15) 
 
 It is easily verified that, for y = J^^^) > ^ 'becomes z-,(4). The 
 Telocity diagram in the disturhed region of the "boundary layer will 
 therefore he smoothly Joined with the velocity diagram in the undistiurbed 
 region, that is, with the diagram of the isolated plate. 
 
 When the equation of the boundary layer is found, the required 
 velocity profile is obtained from equation (7) : 
 
 
 (16) 
 or from equation (15): 
 
 V ^ '^\^{J,K,Q)) 
 
 (17) 
 
 If Y denotes the variable ordinate of the edge of the boundary layer, 
 that is, the magnitude that, from equation (ll), satisfies the equation 
 
 Y-Z = a(?,0) 
 then 
 
 (17-) 
 
 } = \T{I,K,e)) 
 
 It is Important to note that equations (16), (17), and (17') are true 
 not only for those values of y and z that satisfy inequalities (6) 
 
 z^(?,e)-z=Z2(C,0) 
 but in the entire range of interest: 
 
WAG A TM 1308 9 
 
 When the velocities of the points located in the rhomh 
 
 are considered^ the houndary layer for these points is as though infinite, 
 hut the velocities are determined from computation on the hyperholic edge 
 of the houndary layer. 
 
 The velocity profile has thus been determined and the edge of the 
 houndary layer is known. ^Substituting the values of u and Z in the 
 integral condition (5) yields an ordinary differential equation with one 
 unknown a{S,^6) inasmuch as all the magnitudes Involved, including the 
 friction at the wall 
 
 are expressed in terms of this function. 
 
 As is easily seen, however, the differential equation reduces to 
 a simple equation in finite form for determining the coefficient k(0) 
 appearing in equation (12) . In order to obtain this single unknown 
 coefficient, certain boundary conditions are assigned for the function 
 (P(t) and its derivatives, as in the classical Karman-Pohlhausen method. 
 
 A consideration of the first approximation is the first step. The 
 function <p(t) is subjected only to the conditions 
 
 (p(0) = 
 
 <P(1) = 1 
 
 that is, the velocity u(x,y,z) becomes zero at the walls and 1 at the 
 outer limit of the boundary layer, which leads to the profile 
 
10 
 
 MCA TM 1308 
 
 - = yz 
 
 V a(x,0) 
 
 For substitution in the integral condition, the integrals that enter 
 this condition are first computed: 
 
 (19) 
 
 ,yi 
 
 r\ Zr 
 
 dy 
 )o o 
 
 ■yi 
 
 u dz = 
 
 
 
 l(x,0) 
 
 y iy 
 
 V ^\'-2' 1 
 
 z dz = ■ / . -V^- = T a(x,0) V 
 
 pyg /^z 
 
 dy 
 Jyi 
 
 Jn Z pyg 
 u dz = 
 Jyn 
 
 dy 
 
 71 ^ 
 
 
 
 yg 
 
 u dz 
 
 V 
 a(x,0) 
 
 'yi 
 
 i^(x,e) 
 
 2y 
 
 dy 
 
 |va(x,e) logg ^ 
 
 (x,9 ) 
 
 py^+z^ cos 
 
 yg+Zj^ cos 0-y 
 cos 
 
 dy 
 
 u dz 
 
 Al^n(x) 
 
 hn(x)-y 
 
 = 2 
 
 V 
 
 t{x,e) 
 
 0^""' 1 cos 
 
 y dy 1 z dz 
 
 yg 
 
 10 
 
 .l.,(x) 
 
 . ^. 2 fl I rho2(x)-2hQ(x) y + y'\i dy 
 a(x-[_0J cos^ ^,lv„ L 
 
 '^2 
 
 V 
 a(x,0) 
 
 \ a(x,0) V ^ cos + -^ y 2 cos^ 
 o 1 1"^ 1 _) 
 
WAG A TM 1308 
 
 11 
 
 In a similar manner, the integrals of the square of the velocity are 
 computed. The development -will be limited to computing the integrals 
 just given and the remaining ones are written out in ready form; 
 
 ^yi 
 
 dy I u" dz 
 OO 
 
 I v2 a(x,e) 
 
 a(x]^) e 
 
 2 dz= iv^ a(x,9) logg ^^2^ 
 
 ^1 
 
 ,hn(^) 
 
 ,ho(x)-y 
 cos 6 
 
 dy 
 
 u^ dz = Y^ 
 
 Jy. 
 
 1 2 1 ^1^ ^°^^ ^ 
 6 ^1 cos e + 15 ^(^^g) 
 
 1 J-^^ cos^ 
 90 a2(x,0) 
 
 Thus, the left side of the integral condition reduces to the form. 
 
 ,/ ) p(V-u) u da = pV^ sin 6^ a(x,0) + | a(x,0) log^ Sl^Ei^ 
 
 L ^1 
 
 + 
 
 12 « 1 ^l^'°"' 
 
 6 ^1 °°^ ^ ^ 60 a(x,0) 
 
 y 6 cos^ 
 90 a2(x,e) 
 
 ] 
 
 The computation of the right side of integral condition (5) reduces to 
 
12 
 
 WACA TM 1308 
 
 r> X 
 
 
 
 T ay + 
 
 L^o 
 
 ,h^(x) 
 
 lho(€) 
 
 Tq dy 
 
 n ^ 
 
 d^ = 
 
 sin 
 
 \U) 
 
 Uo 
 
 ^ ^ 5Ti1^ ^' ■" 
 
 ■\U) 
 
 )h,(C) 
 
 ^^^J 
 
 d? 
 
 sin 
 
 NCOS' 
 
 a(x,0) 1 dg o / ^ 
 
 
 
 
 According to the first approxiniation for the isolated plate (according 
 to Pohlhausen), 
 
 yi(C) = 
 
 5o(?) 1 
 
 sm y am 
 
 V^ 
 
 Ox 
 
 n^f 
 
 dg 
 
 FKcT 
 
 = sm 
 
 V 12v 
 
 — = g vSo(x) am 
 
 so that 
 
 W 
 
 ^ V 
 
 sin 
 
 n X 
 
 2 n 
 
 cos 
 
 vJ 
 
 /^ X 
 
 a(C,0) 
 
 d? 
 
 vJ 
 
 yi'(€) 
 
 l^(£^V3in2 1 5o2(x) ^ cos 
 
 O V -J ^ V 
 
NACA TM 1308 13 
 
 All these expressions are Immediately simplified as soon as eq^uation (12) 
 is considered; then 
 
 / X k(e) 2/ x 
 sin^ e ^ 
 
 After all the simplifications are made, the following equation for 
 determining k(0) is oTatained : 
 
 2 3 
 
 -, , fr,\ 2 COS 2 coa 6 1 cos 6 ,„^n 
 
 ^°«. >^(^) = 3 - TT5T - 5 J2(^ - 15 P(^ (^°) 
 
 It is readily seen that, for 6 = —, equation (20) "becomes 
 
 log k = 2. 
 e 3 
 
 k = 1.95 
 
 as given in the previously cited work on the interference of the "boundary 
 layers on mutually perpendicular plates. 
 
 The explicit dependence of k on is not given in equation (20). 
 This relation may be obtained "by the following simple device: When 
 
 22^ -y f2l) 
 
 equation (20) "becomes 
 
 cos = ^e^ ^ 1^ (22) 
 
 "When the values of ^ are given in the interval where the absolute value 
 of the right side of equation (22) will not exceed 1, the corresponding 
 value of is obtained, K is determined, and then k(0) is found 
 from equation (21). It would also be possible, of course, to proceed in 
 another manner: Equation (22) may be rewritten 
 
 k(^) = e (22-) 
 
 From a given value of ^, k(^) is obtained and then cos 0, and so forth. 
 The simplest method is to draw the function 
 
14 
 
 NAG A TM 1308 
 
 Tj = -i()C) = ^e (23) 
 
 and obtain its intersection with straight lines parallel to the ^-axis 
 
 ■r] = cos Q 
 
 and then to obtain k from X.- 
 
 The values of 0°, lc(0), and ^(0) are tabulated in table 1. 
 
 The dimensions of the region of interference of the layers are first 
 determined. According to equation (14), 
 
 hQ(x,0) = m(0) 5q(x) 
 
 (14-) 
 
 vhere 
 
 m 
 
 k + cos 
 
 sin 
 
 1+^ 
 
 cot 
 
 (24) 
 
 The values of m for different can be determined from table 1 
 The value = 180° is somewhat isolated; for this value of the 
 value of m becomes indeterminate. The value of m can easily be 
 obtained, however, by the usual device of analysis; 
 
 (.liX) M eos2 0^ = - ^' f (^) ^^^ f l 
 
 = 
 
 =1 
 
 so that, for = 180°, ^ = - 1 and m = 0. The values of m(0) are 
 given in the last column of table 1. 
 
 The correction in the drag of one side of the plate due to inter- 
 ference of the layers may be computed by the equation 
 
 AW = 
 
 O I 
 
 
 
 ■V«' 
 
 (Tq-t) dy 
 
 i? 
 
 (25) 
 
 inasmuch as the difference in the drag may show up only in the region S 
 (fig. 3). The quantity 2 is the length of the plate in the direction 
 of the flow. 
 
WACA TM 1308 
 
 15 
 
 The integral equation (25) is readily computed and yields 
 
 AW 
 
 py2(c)+zi(?) cos e 
 
 uv 
 
 (t -t) dy 
 
 dC 
 
 Finally, 
 
 f^ T- rni^2^^^'^^i^^^ °°^ ^ 
 
 sin 6 
 
 ^ V 
 
 1 , , y ^ 
 
 iTTfT a(?,0) 
 
 2 sin 
 
 ..(a) - H2|i^U 
 
 § 
 
 A¥ = p(e) ij vz 
 
 k^(e) - cos^ 
 
 p(0) = 
 
 2k(0) sin 
 
 (26) 
 
 If L denotes the width of the plate in the direction transverse 
 to the flow and the effect of its free end is not considered, the 
 relative correction due to the Interference of the laoundary layers may 
 l)e computed : 
 
 A¥ 
 
 p(0) [1 Yi ^ p(e) uf^ 
 
 0.578 L.P^Z 0-5^8 LyVZ 
 
 or, when the Reynolds numher Et of the plate, which is equal to YZ/v, 
 is Introduced, 
 
 AW /„>, Z 1 
 
 (27) 
 
 where q(0) denotes the relation 
 
 q(0) = 
 
 p(e) 
 
 0.578 
 
 Generally, the correction ohtalned is extremely Insignificant for plates 
 that are long in the transverse direction. If, however, the transverse 
 length L is comparable with the width of the region of the disturbed 
 layer at the end of the plate x = Z, the correction is not insignificant. 
 Thus, if 
 
16 
 
 WACA TM 1308 
 
 L = n-h^il) = n-m(e) 6q(Z) = n-m{e)\f^^ 
 
 the relative correction will now "be equal to 
 
 AW ^ q(0) .1 
 
 or 
 
 percent 
 
 where 
 
 s = 
 
 100 q(e) 
 Vl2 m(0) 
 
 (28) 
 
 (29) 
 
 All the magnitudes introduced in the preceding eq^uations may be expressed 
 in terms of the previously given parameter E, : 
 
 1 ^ 
 
 q(^) = /;^ ^ cot e 
 
 •\ 
 
 3(^) 
 
 1.156 K 
 
 1-^ 
 
 i.iseVis 
 
 > 
 
 100 
 
 (30) 
 
 / 
 
 In table 1, it is possible to find the values of these magnitudes for 
 different angles 6 or the corresponding values of the parameter S^. 
 The magnitude s(0) increases with an increase of the angle Q. Some- 
 what paradoxical is the value of 50 percent for s(e) at 0° = 180 
 and m = 0. It is found that the region of interference is equal to 
 zero, and that a relative effect occurs on the drag, which is due to the 
 fact that the width of the region and the absolute correction on the 
 drag slmultdneoi:isly approach zero. On the other hand, when 6 decreases 
 
 to and m-» 
 limiting cases. 
 
 decreases to 16 percent. Both these cases are 
 
 In figure 4, the curves of the relation between J^ and G are 
 drawn for the first and other approximations and also for the case of 
 turbulent layers . 
 
MCA TM 1308 17 
 
 3. SECOND AHD SIXTH APPEOXIMATIOHS 
 
 The following approximations differ from the first only in the 
 hovmdary conditions that are imposed on the function <p(t) . 
 
 The second approximation corresponds to the assumptions 
 
 9(0) = 
 
 9(1) = 1 
 
 9'(1) = 
 
 The velocity profile for these conditions will hare the form 
 
 .2 
 
 u = T 
 
 J yz \ / yz Y 
 \a(x,e);"^a(x,e)j 
 
 No new difficulties in principle, as compared with the first approximation, 
 are obtained. Inasmuch as the "basic computations have "been explained in 
 the first approx im ation, the results are presented with the same 
 notations . 
 
 The magnitudes are all given in tahle 2. 
 
 The final form of the equation for the determination of k(0) is 
 
 1 )r(Q\ _ _ii 5 cos 15 cos^ 9 _2_ cos^ 6 5 cos^ 6 
 
 % ^ ' ~ 120 -^ 4k(0) + 28" 2,^, + 21^3,^^ " 168' 4. 
 
 r(0) "" k"(0) """ k"(0) 
 
 1 cos^ 
 
 420 v4 
 
 k^(0) 
 
 (31) 
 
 The solution is effected hy the same device as in the case of the first 
 approximat ion . 
 
 All computations were also carried out for the sixth approximation, 
 TDy which is meant the results of the assumption of the following 
 boundary conditions : 
 
18 MCA TM 1308 
 
 (p(0) - 9(1) = 1 
 
 ♦"(O) i= <P'(1) - 
 
 <p'"(0) - <»"(1) = 
 
 <p'"(l) « 
 
 Thla approximation gives, for the undiaturted layer, the velocity 
 profile 
 
 Ug.V 
 
 <i^-iit^iil-%1 
 
 and the following values of the frictional force and layer thickness, 
 which are extremely close to the accurate solution of Blasius: 
 
 T„ = o.ssi'v/tipy^ 
 
 iV¥ 
 
 6 
 
 
 = 6.048^^ 
 
 It is therefore reasonable to suppose that, for investigating the proh- 
 lem of the interference of the layers, the sixth approximation is 
 considerably more satisfactory than the second. 
 
 The results are again collected in table 3. 
 
 In figure 4, the curve of K against 6 is given for the second 
 and sixth approximations. 
 
 As before, the eq^uation for the determination of k(0) is also 
 given 
 
 2 3 
 
 loge k = 0.1857 + 0.6370 ^^|-^ + 0.6107 £2£^ + 0.1043 22S^ . 
 
 ^ k'^ k^ 
 
 0.0524 224-^ ^ 0.0053 ^2^^ + 0.0025 ^^^ - 0.0001 ^-^^ 
 k^ k^ k^ k^ 
 
 (32) 
 
MCA TM 1308 
 
 19 
 
 4. INTERFERENCE OF TUEBULEHT BOUNDAEff LAYEE?S 
 
 For the solution of this concluding part of the prohlem, the -well- 
 knovm 'l/V power law is applied, which, when the houndary layer is 
 not two dimensional, assumes the form. 
 
 " = i^}' 
 
 (33) 
 
 As is known, for an infinite plate without interference of the 
 layers, this law gives excellent agreement with experiment j it may "be 
 expected that the application of the 'l/?' power law to this case will 
 he justified "by experiment. 
 
 The generalized integral condition (5) is substituted in the 
 Telocity -prof ile equation (33) . Unfortunately, the integrals will now 
 not he so easily evaluated hecause the presence of fractional exponents, 
 particularly in the integrals taken over the hatched areas of the 
 triangles shown in figure 2, strongly complicates the computations and 
 leads to the necessity of taking integrals of "binomial differentials. 
 All the computations can he made with a sufficient degree of accuracy, 
 however, hy using converging series and integrating them. The left 
 side of the integral condition reduces to the following expression: 
 
 p(T-u) u da = pV a(x,0) sin 
 
 0.0023 
 
 J-. cos e 
 
 i(x.0) 
 
 0.1607 + 0.0972 log„ ^^^!^> + 
 
 j^ cos 
 
 0.1360 -=7 — -c 0.023 
 
 a(x,0) 
 
 0.0004 
 
 yi'(x) 
 
 ry^2 cos eY 
 
 fj^^ cos ey 
 
 [a(x,0) J -^ • • 
 
 (34) 
 
 In computing the right side of the integral condition, the 
 Earman formula is used for the expression of the friction at the wall 
 in the undisturhed region of the layer: 
 
 V VA 
 
 Tq = 0.0225 pV^f^Y' = 0.0225 pV 
 
 ^Vy^ sin 9 J 
 
 A 
 
 (35) 
 
20 
 
 NACA TM 1308 
 
 In the disturbed part of the layer, the analogous formula 
 
 L/4 
 
 T = 0,0225 p 
 
 4^r 
 
 0.0225 pT 
 
 = 0.0225 p' 
 
 \VZ sin ej 
 
 J ^.y \ l/4 
 
 \V a(x,e) sin QJ 
 
 (36) 
 
 is assumed. These values are substituted, aa in the previous sections, 
 in the formula for the resistance of the walls fonning part of the 
 boundary of a tube of flow. Then, 
 
 W = 2 
 
 io LO 
 
 r\iK) 
 
 
 d£, = 0.1943 p- 
 
 ^ ^o^-{j^ ^ 
 
 1 1 
 
 4 
 
 1 ^v 
 
 r, (x) cos 01+ p 
 
 1 J 1 
 
 0.0090 
 
 4 
 sin e 
 
 \J0 
 
 a(g.e) dg ^ 
 5 
 4 
 
 C 11 
 
 n"^ 19 
 
 0.0056 
 
 Jo 
 
 y^ *(C) cos^ 9 
 a(C,0) 
 
 dC - 0.0014 
 
 y-|^ (g) cos^ 
 
 0.0006 
 
 27 
 4 
 
 
 
 a2(C,0) 
 
 d^ + 
 
 Oo 
 
 y^ (C) cos* 
 a^(?,0) 
 
 dg - 0.0003 
 
 Jo 
 
 35 
 
 4 
 y^ (?) cos5 
 
 a4(?,0) 
 
 d^ + 
 
 ^' 43 
 
 4 
 
 0.0002 
 
 Jo 
 
 y^ (g) cos*^ 
 a5(?,0) 
 
 d^ + . 
 
 (37) 
 
NACA TM 1308 21 
 
 Ihe Earman formula was then used; 
 
 S^Cx) = 0.370(^) 
 
 1/5 
 
 X 
 
 When it Is known (see equation (12)) in advance that the solution of the 
 prohlem will have the form 
 
 2 ^O^(^) 
 
 a(x,0) = k(0) j^ (x) = k(0) ^ g (38) 
 
 sin 6 
 
 this value is auhatituted in equations (34) and (37); then, after some 
 simplifications, the following solution Is ohtained: 
 
 p(Y-u) u da = p 10.1607 k + 0.0972 k log_, k + 0.1360 cos - 
 
 sin \ ^ 
 
 coa^ ' cos^ cos"^ coa^ 
 
 0.0230 — r + 0.0023 —^ — - 0.0004 — + 0.0001 — ^ — + . . . j 
 
 k k k ^ 
 
 ^^O^(^) / cos2 
 
 W = p <0.1949 k + 0.1949 cos - 0.0195 k + 0.0122 
 
 sin 
 
 0.00305 H24^ + 0.00133 ^2^ + 0.00073 2^1^ 
 
 k2 k3 k^ , y 
 
 Equating these two expressions yields the following equation for 
 de t ermining k ( ) : 
 
 n 1 n ncno n cncc COS0 , „ ^„„„ COa^ ^ „ _ COS"^ 
 
 log k = 0.1518 + 0.6056 — r: — + 0.3632 5 — - 0.0556 = — + 
 
 e K -^^ -^:> 
 
 „ ^^^^ cos"^ _ _-o_ cos^ cos° 
 
 0.0175 r 0.0082 — + 0.0051 r — + . . . 
 
 k^ k^ k^ 
 
 As in the preceding sections, the change of variables is made: 
 
 ^=^(0) ■ (39) 
 
22 NACA TM 1308 
 
 The transcendental equation then becomes 
 
 cos = ^ exp (0.1518 + 0.6056 Y, + 0.3632 ^^ _ 0.0556 ^3 + 
 
 0.0175 'd - 0.0082 'C' + . . .) (40) 
 
 This equation is easily solved "by the tabular method, where 
 00< 0< 180° 0.5540 >^>- 1,0000 
 
 The corresponding values of 9, k(0), and ^(0) are given in table 4. 
 The further investigation In no vay differs from that for laminar layers. 
 The values of the coefficients characterizing the boundary of the region 
 of interference and the corrections on the drag are given in table 4 
 for various values of 0, with all the computations ommitted, in the 
 notation previously used. 
 
 In connection with the formulas of turbulent friction, the 
 coefficients q and p are determined by the formulas : 
 
 AW = p-p V^Z^f-^r (41) 
 
 and 
 
 (41') 
 
 The dependence of p on ^ is determined by the following series : 
 p = p cot (0.00133 - 0.00083 ^'^ + 0.00021^3 _ 0.00009 ^^ + 
 
 0.00005^5 _ 0.00003^^ + . . .) (42) 
 
 ^ = oyfe (^^) 
 
 The dependence of ^ on is given for the case of turbulent layers 
 in figure 4. The boundary of the region of interference (the 
 coefficient m(0)) in the case of the turbulent layers differs little 
 from the corresponding boundary of the laminar layers, according to 
 the sixth approximation; whereas the relative correction on the drag s 
 in percent is several times less in the turbulent case. 
 
WACA TM 1308 23 
 
 CONCLUSION 
 
 The results oTatained have a readily understandable form. In general, 
 the effect of the interference of the layers on the drag of the plates 
 is insignificant. The effect assumes an appreciahle value only in the 
 case vhere the plates in the dimensions transverse to the flow "become 
 comparahle vith the width of the region of disturbed boundary layer. 
 Moreover, interference plays a large part in the motion of fluids through 
 small dihedral angles. Thus, for example, in the motion near the inter- 
 section of a dihedral angle of about 10°, the region of interference 
 exceeds by 16 times the thickness of the layer at the given section. 
 At smaller angles, the phenomenon is still more marked. 
 
 All the conclusions of the present and preceding papers require 
 experimental check. 
 
 By agreement with the Central Aero-Hydrodynamical Institute, the 
 aerodynamic laboratory of the Leningrad Industrial Institute is under- 
 taking an experimental investigation of the phenomenon of the inter- 
 ference of boundary layers . It is proposed, through use of the method 
 of microtunnels, to observe directly the distortion in the velocity 
 profiles, and so forth, of the phenomenon. 
 
 The present work was carried out at the Aerodynamic Laboratory of 
 the Leningrad Industrial Institute. 
 
 Translated by S. Eeiss, 
 National Advisory Com mi ttee 
 for Aeronautics 
 
 BKh'KHMGE 
 
 1. Loizlansky, L. G. : Interference of Boundary Layers. CAHI Eep. 
 No. 249, (Moskow), 1936. 
 
24 
 
 NACA TM 1308 
 
 TABLE 1 
 
 e 
 
 k(e) 
 
 K{e) 
 
 m(e) 
 
 P(0) 
 
 q(0) 
 
 3(0) 
 
 (deg) 
 
 
 
 
 
 
 (percent) 
 
 
 
 .2.894 
 
 0.3455 
 
 m 
 
 OD 
 
 «• 
 
 16.3 
 
 10 
 
 2.880 
 
 .3419 
 
 22.257 
 
 7.324 
 
 12.671 
 
 16.4 
 
 20 
 
 2.840 
 
 .3309 
 
 11.315 
 
 3.697 
 
 6.396 
 
 16.7 
 
 30 
 
 2.773 
 
 .3123 
 
 7.278 
 
 2.503 
 
 4.330 
 
 17.2 
 
 40 
 
 2.682 
 
 .2856 
 
 5.364 
 
 1.916 
 
 3.315 
 
 17.8 
 
 50 
 
 2.567 
 
 .2504 
 
 4.190 
 
 1.570 
 
 2.716 
 
 18.7 
 
 60 
 
 2.434 
 
 .2054 
 
 3.388 
 
 1.346 
 
 2.329 
 
 19.8 
 
 70 
 
 2.283 
 
 .1498 
 
 2.793 
 
 1.187 
 
 2.054 
 
 21.2 
 
 80 
 
 2.120 
 
 .0819 
 
 2.329 
 
 1.069 
 
 1.849 
 
 22.9 
 
 90 
 
 1.948 
 
 
 
 1.948 
 
 .974 
 
 1.685 
 
 25.0 
 
 100 
 
 1.773 
 
 -.0980 
 
 1.624 
 
 .892 
 
 1.543 
 
 27.4 
 
 110 
 
 1.601 
 
 -.2136 
 
 1.340 
 
 .813 
 
 1.407 
 
 30.3 
 
 120 
 
 1.442 
 
 -.3467 
 
 1.088 
 
 .732 
 
 1.266 
 
 33.6 
 
 130 
 
 1.300 
 
 -.4945 
 
 .858 
 
 .641 
 
 1.109 
 
 37.3 
 
 140 
 
 1.185 
 
 -.6464 
 
 .6 2 
 
 .536 
 
 .927 
 
 41.0 
 
 150 
 
 1.099 
 
 -.7880 
 
 .466 
 
 .417 
 
 .721 
 
 44.7 
 
 160 
 
 1.042 
 
 -.9018 
 
 .301 
 
 .285 
 
 .493 
 
 47.3 
 
 170 
 
 1.010 
 
 -.9750 
 
 .144 
 
 .142 
 
 .246 
 
 49.3 
 
 180 
 
 1.000 
 
 -1.0000 
 
 
 
 
 
 
 
 50.0 
 
 TABLE 2 
 
 G 
 
 k(0) 
 
 Kie) 
 
 m(0) 
 
 P(0) 
 
 q(0) 
 
 3(0) 
 
 (deg) 
 
 
 
 
 
 
 (percent) 
 
 
 
 2,217 
 
 0.4511 
 
 m 
 
 «• 
 
 0> 
 
 13.7 
 
 10 
 
 2.205 
 
 .4466 
 
 18.360 
 
 10.162 
 
 13.921 
 
 13.8 
 
 20 
 
 2,168 
 
 .4334 
 
 9.090 
 
 5.148 
 
 7.052 
 
 14.2 
 
 30 
 
 2.108 
 
 .4108 
 
 5.948 
 
 3.584 
 
 4.910 
 
 14.7 
 
 40 
 
 2.026 
 
 .3781 
 
 4.343 
 
 2.702 
 
 3.701 
 
 15.6 
 
 50 
 
 1.925 
 
 .3339 
 
 3.352 
 
 2,233 
 
 3.059 
 
 16.7 
 
 60 
 
 1.808 
 
 .2765 
 
 2.665 
 
 1.928 
 
 2.641 
 
 18.1 
 
 70 
 
 1.678 
 
 .2038 
 
 2.150 
 
 1.712 
 
 2.345 
 
 19.9 
 
 80 
 
 1.542 
 
 .1126 
 
 1.742 
 
 1.546 
 
 2.118 
 
 22.2 
 
 90 
 
 1,407 
 
 
 
 1.407 
 
 1.407 
 
 1.927 
 
 25.0 
 
 100 
 
 1.284 
 
 -0.1353 
 
 1.127 
 
 1.279 
 
 1.752 
 
 28.4 
 
 110 
 
 1.182 
 
 -0.2893 
 
 .894 
 
 1.153 
 
 1.579 
 
 32.2 
 
 120 
 
 1.108 
 
 -0.4513 
 
 .702 
 
 1.018 
 
 1.395 
 
 36.3 
 
 130 
 
 1.060 
 
 -0.6064 
 
 .544 
 
 .875 
 
 1.199 
 
 40.2 
 
 140 
 
 1.032 
 
 -0.7422 
 
 .414 
 
 .721 
 
 .988 
 
 43.6 
 
 150 
 
 1.016 
 
 -0.8524 
 
 .300 
 
 .556 
 
 .762 
 
 46.3 
 
 160 
 
 1.007 
 
 -0.9332 
 
 .196 
 
 .379 
 
 .519 
 
 48.3 
 
 170 
 
 1.002 
 
 -0.9832 
 
 .097 
 
 .192 
 
 .263 
 
 49.6 
 
 180 
 
 1.000 
 
 -1.0000 
 
 
 
 
 
 
 
 50.0 
 
NACA TM 1308 
 
 25 
 
 TABLE 3 
 
 e 
 
 He) 
 
 ^(0) 
 
 m(e) 
 
 P(0) 
 
 4(0) 
 
 s(0) 
 
 (deg) 
 
 
 
 
 
 
 (percent) 
 
 
 
 1.969 
 
 0.5079 
 
 oo 
 
 . 
 
 •• 
 
 12.3 
 
 10 
 
 1.957 
 
 .5032 
 
 16 . 940 
 
 8.416 
 
 12.713 
 
 12.4 
 
 20 
 
 1.921 
 
 .4892 
 
 8,365 
 
 4.272 
 
 6.453 
 
 12.8 
 
 30 
 
 1.863 
 
 .4648 
 
 5.458 
 
 2.921 
 
 4.412 
 
 13.4 
 
 40 
 
 1.783 
 
 .4296 
 
 3.965 
 
 2.262 
 
 3.417 
 
 14.3 
 
 50 
 
 1.685 
 
 .3815 
 
 3.039 
 
 1.879 
 
 2.838 
 
 15.5 
 
 60 
 
 1.572 
 
 .3181 
 
 2.393 
 
 1.631 
 
 2.464 
 
 17.0 
 
 70 
 
 1.450 
 
 .2359 
 
 1.907 
 
 1.457 
 
 2.201 
 
 19.1 
 
 80 
 
 1.327 
 
 .1309 
 
 1.524 
 
 1.324 
 
 2.000 
 
 21.7 
 
 90 
 
 1.204 
 
 
 
 1.204 
 
 1.204 
 
 1.819 
 
 25.0 
 
 100 
 
 1.106 
 
 -0.1571 
 
 .946 
 
 1.094 
 
 1.653 
 
 28.9 
 
 110 
 
 1.038 
 
 -0.3295 
 
 .741 
 
 .985 
 
 1.480 
 
 33.2 
 
 120 
 
 1.004 
 
 -0.4982 
 
 .582 
 
 .887 
 
 1.340 
 
 37.5 
 
 130 
 
 1.000 
 
 -0.6428 
 
 .466 
 
 .766 
 
 1.157 
 
 41.1 
 
 140 
 
 1.000 
 
 -0.7660 
 
 .364 
 
 .643 
 
 .971 
 
 44.2 
 
 150 
 
 1.000 
 
 -0.8660 
 
 .268 
 
 .500 
 
 .755 
 
 46.7 
 
 160 
 
 1.000 
 
 -0.9397 
 
 .176 
 
 .342 
 
 .517 
 
 48.5 
 
 170 
 
 1.000 
 
 -0.9848 
 
 .088 
 
 .174 
 
 .263 
 
 49.6 
 
 180 
 
 1.000 
 
 -1.0000 
 
 
 
 
 
 
 
 50.0 
 
 TABLE 4 
 
 
 
 k(0) 
 
 ^(0) 
 
 m(0) 
 
 P(0) 
 
 q(0) 
 
 s(0) 
 
 (deg) 
 
 
 
 
 
 
 (percent) 
 
 
 
 1.805 
 
 0.5540 
 
 0* 
 
 •• 
 
 •• 
 
 5.35 
 
 10 
 
 1.796 
 
 .5483 
 
 16.010 
 
 0.0115 
 
 0.3189 
 
 5.38 
 
 20 
 
 1.767 
 
 .5318 
 
 7.915 
 
 .0058 
 
 .1614 
 
 5.51 
 
 30 
 
 1.720 
 
 .5035 
 
 5.172 
 
 .0039 
 
 .1094 
 
 5.73 
 
 40 
 
 1.654 
 
 .4631 
 
 3.765 
 
 .0030 
 
 .0839 
 
 6.03 
 
 50 
 
 1.577 
 
 .4076 
 
 2.898 
 
 .0025 
 
 .0689 
 
 6.43 
 
 60 
 
 1.485 
 
 .3367 
 
 2.292 
 
 .0021 
 
 .0594 
 
 7.00 
 
 70 
 
 1.382 
 
 .2475 
 
 1.835 
 
 .0019 
 
 .0525 
 
 7.73 
 
 SO 
 
 1.273 
 
 .1364 
 
 1.469 
 
 .0017 
 
 .0472 
 
 8.68 
 
 90 
 
 1.164 
 
 
 
 1.164 
 
 .0015 
 
 .0431 
 
 10.00 
 
 100 
 
 1.065 
 
 -0.1631 
 
 .905 
 
 .0014 
 
 .0394 
 
 11.76 
 
 110 
 
 1.000 
 
 -0.3420 
 
 .700 
 
 .0013 
 
 .0364 
 
 14.05 
 
 120 
 
 1.000 
 
 -0.5000 
 
 .577 
 
 .0012 
 
 .0325 
 
 15.22 
 
 130 
 
 1.000 
 
 -0.6428 
 
 .466 
 
 .0010 
 
 .0280 
 
 16.24 
 
 140 
 
 1.000 
 
 -0.7660 
 
 .364 
 
 .0008 
 
 .0230 
 
 17.08 
 
 150 
 
 1.000 
 
 -0.8660 
 
 .268 
 
 .0006 
 
 .0178 
 
 17.95 
 
 160 
 
 1.000 
 
 -0.9397 
 
 .176 
 
 .0004 
 
 .0122 
 
 18.73 
 
 170 
 
 1.000 
 
 -0.9848 
 
 .088 
 
 .0002 
 
 .0064 
 
 19.65 
 
 180 
 
 1.000 
 
 -1 . 000 
 
 
 
 
 
 
 
 19.97 
 
26 
 
 NACA TO 1308 
 
 6o(x) 
 
 6o(0 
 
 Figure 1. 
 
 *— Y 
 
 Figure 2. 
 
tl 
 
 
 
 a 
 
 ■n 
 
 
 > 
 
 g 
 
 £1 
 
 1. 
 
 > 
 
 J 
 
 S 
 
 s 
 
 o 
 
 ^■■ 
 
 S" 
 
 w 
 
 si 
 
 CO 
 
 
 
 
 
 
 
 
 
 fafe JO) 
 
 en 
 i .^ 
 
 ooaSos 
 
 o 1 la - 
 
 " ° C " m 
 
 • a> ^ o "Y 
 S •< rt S CO 
 
 ^ "^ •- « a 
 
 I c4 H n 5 c 
 
 ^ •3 •-; M a 
 
 <; 2 a " - 
 
 rj -a cn E to 
 
 v; c c " CO 
 
 '1 « 
 
 53 
 o o 
 
 O . 
 
 ■ Oi 
 
 2 5 
 
 2 <" 
 
 « -S 
 
 o o 
 
 J CO 
 
 i .^ 
 
 •o -s s- 
 
 O > u] •« 
 
 ■ „ *> ^ .2 '7 
 
 < 2 a S ^ 
 
 G C ra CO 
 
 SJ c g .. „ 
 ZU-ffE-iS 
 
 T-< eg w H 3 G 
 
 ■< 
 o 
 < 
 z 
 
 < 
 o 
 < 
 z 
 
 — ' 
 
 ^^ 
 
 o 
 
 
 u 
 
 a) 
 
 J 
 
 D, 
 
 c 
 
 
 ^ 
 
 > 
 
 a 
 
 u 
 
 3 
 
 ^ 
 
 3 
 
 & 
 
 § 
 
 
 s 
 
 ^ 
 
 w 
 
 
 o 
 
 o 
 
 o 
 
 o 
 
 |x< 
 
 ii< J pa 
 
 i .^ 
 
 ■o 5 & 
 
 " O C " m 
 « _^ O "Y 
 
 «-3 S - 
 
 ^=o 
 
 o - 
 
 U QJ 
 
 E-i rt " " • 
 
 < 2 a I 't 
 
 jS ^ c " CO 
 
 zo-enS 
 
 .S3' 
 IS 3 
 
 o o 
 
 o o 
 
 ■o -S S" 
 - CO a 3 05 
 
 o 1 t5 - 
 
 CO o c " ,m 
 
 ^ I- a c 2 
 
 -^ 0) _ o ";* 
 S <! « S CO 
 H -, =J " • 
 
 " a " - 
 
 rt S to 
 
 ^4 ca a n 5 C 
 
 < 2 
 
 < S = S," s 
 
 Z U-ffHS 
 
 r-i c^J a 9 n c 
 
 < 
 
 < 
 z 
 
 CO o I** fe • 
 
 o .2 O o I 
 -' -S g w : 
 
 < mZH I 
 
 zzoz- 
 
 rS "^ 
 
 
 u 
 
 <: 
 z 
 

 ■rt *j a 
 
 CO aS * 
 
 P^ III iJ OQ 
 
 rH cj ■_; n B u 
 
 *— ' ' CO " 
 
 . 0) or* 
 
 ; CO 
 
 < 2 a " . 
 
 *S G c rt CO 
 
 
 £■3 
 
 M O 
 
 « 3 S O 
 
 o o 
 fob 
 
 o o 
 
 i .^ 
 
 •« -s §• 
 
 coaS« 
 
 O 1 M - 
 
 CO O c " ™, 
 
 « _ O "^ 
 
 5 <J rt in CO 
 
 H -H " " • 
 
 B •- en D, 
 
 <; 2 a " - 
 
 iS ^ c " CO 
 
 ;nS^ 
 
 
 u 
 <: 
 z 
 
 CO 
 <U O (V ^ 
 
 -■° S5 
 
 r. ^ ° 2 
 
 §2 e S. 
 
 <; 
 o 
 
 <: 
 
 <-< CM 
 
 CO CO 
 
 ^■=0 
 
 s-S 
 
 o o 
 fob 
 
 in >Z 
 
 A S 
 
 O -. . 
 
 ■o -S §• 
 
 coSS« 
 
 =^ O e to ^ 
 
 - a C » 
 
 *- -* ^ o r* 
 
 S ^ H -H " " • 
 ■S 2 -,. « fl S "^ 
 
 o o 
 
 < 2 a 52 - 
 
 O -S rt S to 
 
 zo€'h3 
 
 ^ c4 1.; n 5 c 
 
 
 < 
 o 
 ■< 
 z 
 
1 
 
UNIVERSITY OF FLORIDA 
 
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 JNIVERSITY OF FLORIDA 
 DOCUMENTS DEPARTMENT 
 
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