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fA TM 2 T7b .29 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1265 AMPLITUDE DISTRIBUTION AND ENERGY BALANCE OF SMALL DISTURBANCES IN PLATE FLOW* By H. Schlichting SUMMARY In a previous report by W. Tollmlen, the stability of laminar flow past a flat plate was investigated by the method of small vibrations, and the wave length X = 2x/a, the phase velocity cr, and the Reynolds number R of the neutral disturbances established. In connection with this, the present report deals with the average disturbance veloc ities and \ V72 and the correlation coefficient u u72 l2 u v as function of the wall distance y for two special neutral disturbances (one at the lower and one at the upper branch of the equilibrium curve in the aR plane). The maximum value of the last two quantities lies in the vicinity of the critical layer where the velocity of the basic flow and the phase velocity of the disturbance motion are equal. The energy balance of the disturbance motion is investigated. The transfer of energy from primary to secondary motion occurs chiefly in the neigh borhood of the critical layer, while the dissipation is almost completely confined to a small layer next to the wall. The energy conversion in the two explored disturbances is as follows: In one oscillation period, half of the total kinetic energy of the disturbance motion on the lower branch of the equilibrium curve is destroyed by dissipation and replaced by the energy transferred from the primary to the secondary motion. For the disturbance on the upper branch of the equilibrium curve, about a fourth of the kinetic energy of the disturbance motion is dissipated and replaced in one oscillation period. The requirement that the total energy balance for the neutral disturbances be equal to zero is fulfilled with close approximation and affords a welcome check on the previous solution of the characteristic value problem. *"Amplitudenverteilung und Energiebilanz der kleinen Storungen bei der Plattenstromung." Nachrichten von der Gesellschaft der Wissenschaften zu Gbttingen, Neue Folge, Band 1, No. 4, 1935, PP. 4778. NACA TM 1265 1. Introduction The numerous efforts, within the last decade, to solve the problem of turbulence (reference 1) have at least produced some satisfactory results for a certain class of boundarylayer profiles when their stability was investigated by the method of small vibrations, with due consideration to the friction of the fluid and the profile curvature (reference 2). Referring to Tollmien (reference 3), who treated the example of laminar flow past a flat plate, the writer investigated several other cases: the Couette flow (reference 4), the amplification of the unstable disturbances in plate flow (reference 5), and the stabilizing effect of a stratification by centrifugal forces (reference 6), and temperature gradients (reference 7). Every one of the investigations was restricted to the solution of the corresponding characteristicvalue problems, without calculating the characteristic function itself. In that manner, the wave lengths of the unstable, hence "dangerous", disturbances were identified as function of the Reynolds number. In most cases, only the disturbances situated right at the boundary, between amplification and damping, were determined. For these, just as much energy is transferred from the primary to the secondary flow, as secondarymotion energy is dissipated by the friction so that the total energy balance is zero. All the stability studies made up to now were, for reasons of mathematical simplicity, based upon an assumedly plane fundamental flow, which depends only on the coordinate transverse to the direction of the flow, and a plane superposed disturbance motion which propagates in form of a wave motion in the primaryflow direction. While there is no objection to the limitation to the plane fundamental flow, since it is frequently realized experimentally, objection may be raised to the plane disturbance motion because the disturbances accidentally produced in practice are almost always threedimensional. Accordingly, it might appear as if the limitation to twodimensional disturbances was all too special. However, H. B. Squire (reference 8) recently demonstrated on the Couette flow this theorem is equally applicable to boundary layer profiles that precisely the specific case of the twodimensional disturbance motion is particularly suitable for the stability study in the following sense: According to Squire, a twodimensional flow, which is unstable against threedimensional disturbances at a certain Reynolds number, is unstable against twodimensional disturbances even at a lower Reynolds number. The twodimensional disturbances are therefore "more dangerous" for a flow than the threedimensional. The critical Reynolds number, which is defined as lowest stability limit, is thus obtained precisely from the twodimensional, not the threedimensional, disturbances. To gain a deeper insight into the mechanism of the turbulence phenomena from small unstable disturbances, a more detailed knowledge of the properties of these small disturbances is necessary. The present NACA TM 1265 report, therefore, deals first, with the distribution of the amplitude of the disturbance over the flow section, that is, the calculation of the characteristic functions and second, with the study of the energy distribution and energy balance of the disturbance motion. The investi gations are based upon the disturbances of the laminar flow past a flat plate which are situated exactly at the boundary between amplification and damping (neutral oscillations). Chapter I AMPLITUDE DISTRIBUTION 2. Discussion of the differential equation of disturbance. Let U(y) be the velocity distribution of the fundamental flow (fig. 1) and the flow function of the superposed disturbance motion, which is assume as a wave motion moving in the x direction (direction of primary flow), whose amplitude p is solely dependent on y, hence y(x,y,t) = ((y)el(at) = ((y)eia(xct) a is real and X = 2n/a is the wave length of the disturbance; P = ,r + iPi and c = P/a are, in general, complex; T = 29/Pr is the period of oscillation; Pi indicates the amplification or the lamping, depending upon whether positive or negative; cr = Pr/C is the phase velocity of the disturbance. For the disturbance amplitude (p, after introduction of dimensionless variables from the NavierStokes differential equations, it results in a lineardifferential equation of the fourth order, the differential equation of the disturbance (U c)(q" a29) U"= (p"" 2a2p" + aqp) () (R = Umb/V = Reynolds number, Um = constant velocity outside of the boundary layer, 8 = characteristic length of the boundarylayer profile = boundarylayer thickness, v = kinematic viscosity.) The general solution (p of the disturbance equation is built up from four particular solutions (pl, *..' 4 p Cl= 1 + C2P2 + C3q(3 + C(P4 NACA TM 1265 The boundary conditions (p = qP' = 0 for y = 0 and y = o give (as explained in the report cited in reference (5)) C4 = 0 (3a) and for CI, C2, C3 the system of equations CllO + C220 + C 330 = 0 ClP10' + C2cP20' + C3(30' = 0 (4) Clala + C202a = 0 (va = +va' + cPva; v = 1, 2) The subscript 0 indicates the values at the wall y = 0, subscript a the values in the connecting point y = a to the region of constant velocity. From (4) the equation of the characteristic value problem follows as Pl0 q20 qC30 Po' ^20' 30 = 0 (5) o 0 30 0 la 2a 0 This equation is discussed in the earlier reports for several cases. It contains, aside from the constants of the basic profile, the parameters a, R, cr, and ci. The complex equation (5) is equivalent to two real equations, and, if limited to the case of neutral disturbances Ici = 0), these two equations give, after elimination of cr, one equation between a and R. This is the equation of the neutral curve in the aR plane, which separates the unstable from the stable disturbance attitudes, and was originally computed by Tollmien for the plate flow. They are assumed to be known for the subsequent study (fig. 2).1 Lihe writer computed the neutral curve for the plate flow cited under reference 5 again and found some differences with respect to Tollmien's report. The newly obtained values are used in this study. NACA TM 1265 5 In the following, the amplitude and energy distribution is computed for two neutral oZcillations, one of which lies on the lower, the other on the upper branch of the equilibrium curve (fig. 2). The parameters of these two neutral oscillations, obtained from the earlier calculation, are indicated in table 1. For the calculation of the integration constants Cl, C2, C3 from (4), we put Cl = 1 (3b) because the amplitude of disturbance remains inleterminate up to a constant factor, the intensity factor of the disturbance motion. Thus, for the other two constants Sla 2 2a (6) SI la 1 la (j j =P 0 (0 10 B P20 10 The particular solutions qpl(y) and qP2(y) are readily obtainable by expansion in series from the socalled frictloaless differential equation of disturbance of the second order, which follows from the general equation (1) by omission of the terms on the righthand side afflicted with the small factor 1/aR. The point U = cr, where phase velocity of disturbance motion and basic flow velocity agree, and which is termed the critical point y = yk, is a singular point of the frictionless differential equation of disturbance, which plays a prominent part in the investigations. NACA TM 1265 Putting Y k *Y k =  = v2 ; k= a; a(a y ) = a2 S I a k 1 k 2 where subscript k denotes the values at the critical points, the frictionless solutions for linear2 velocity distribution read p = a.1 sinh(ay1); q2 = cosh(Caly) and for parabolic velocity distribution 1 2 2 = al2 + a2 2 + a2 + ...3 a1 2 2k P2 =\ b' + bl + b 2 + "2 =2 O + b 2 + 2'2 + + q)2 bo blY2 + 1 P Uk" +  ( log 2 for 2 > 0 k U "  1(logly2 in) for y2 < 0 Uk' TThe Blasius profile of the plate flow is approximated by a linear and a quadLratic function (fig. i), namely 0 5 y/= 0.175: U/Um = 1.68 y/l 0.175 5 y/b 1.015: U/U, = 1 (1.015 y/&)2 (7a) y//6 1.015: U/Um = 1 For the connection between 6 and the displacement thickne3s b* used in figure 2, b* = 0.3415 Is applicable. NACA TM 1265 According to earlier data, the coefficients are given by the equations 1 2 aL al = 1; a2 = ; a = ; a 2 4 0004 a2 a5 = 0.0013 a22 + 0.0083 24; a6 = 0.0024 a22 + 0.000 a2 Op bO = 1; b1 = O; b2 = 1 + 2 (10) b = 0.125 + 0.056 22: b4 = 0.021 0.141 a22 + 0.042 m4 b = 0.005 + 0.005 a.2 + 0'004 a4) b6 = 0.0015 + 0.0012 a22 0.0038 a24 + 0.0014 a26 The particular solution p1, with its derivatives, is regular throughout the entire range of flow (I <1 < 0, O y2 +i), and can be numerically computed with these data. But the particular solution (2 has a singularity, in which (2' becomes logarithmically infinite in the critical layer y = yk. The more detailed discussion has shown that the friction at the wall, and in a restricted vicinity of the critical layer, must be taken into consideration. The first gives the friction solution p3, the other, the friction correction for (p2. Introducing the new variable (uk)1/3 y Yk T = y yk) (aRUk) (11) gives (only the greatest terms from (1) for p(n) being taken into account) the differential equation ip" + T" = U (12) UkI from which follows the correction for p2 near the critical layer, as well as the third frictionaffected solution .3 The calculation of these two solutions merits a little closer study. 8 NACA TM 1265 3. The friction solution p, J The friction solution %cp, is obtained from the differential equation (12) when the inhomogeneous terms encumbered with the small factor e are omitted; hence, from the differential equation i "" + j(P3" = 0 A Unusual in this equation is that, in contrast to the complete disturbance equation (1) ard to the frictionless.equtation of disturbance, the depenience of the parameters a, R, and U' by (ll) enters only as scale factor for y, and that it is not at all affected by 'U. As a result, P93(Tl) can be computer once for all entirely independent from the velocity profile. In this instance, Tietjens' report (reference 9) constitutes a valuable support. A fundamental system 1" = F " and 32" = F of equation (13) is given by 1/2 J 3/2 4it1 and Ij/2 1 3/2 ei/4 (14) The expansion in series of the Bessel functions Jp(Z) (z)P (iz/2)2V p 2, V=0 V!(p + V + 1) NACA TM 1265 gives, when the constant factors (1/3 (1) /6ir (4 and ()/ (1) /r (2) are omitted 7 _13 19 2!3 4*7 413 *471013 6136.4710131619 S 4 10 16 +i  + + .o + 1314 3,33.4*7.10 5!35 *47*10*13*16 6 12 2!3 2.25 4134.2.5.811 + 1 54 + *** 1,31.2 3332.58 5135*2*5*8*111*4 Owing to the boundary condition r3 = 0 for T = + o only the solution aggregate PIF1 + 02F2" approaching zero for great positive real n (01, 02 = integration constants) comes into question. For great 1, this can be represented by the Hankel function of the second kind with the subscript and the argument q3/2 ei/4, hence by QP" F (n) = n1/2 H/(2) 3/2 ei/4] 4 : 1/3 1 31 Herewith, the lookedfor solution of (13) the constant put as cp'(n ) = 3 for the sake of simplicity3 is expansion in power series near r = 0 II S311 F.1 1 30 = F+ 2F2) cP30 NACA TM 1265 factor being given as (16a) and as asynnptotic expansion for great Tj ScP3 = BF 30 4 (B30' = 1) Integrating between the limits To and n gives cp0' 9301 = P1 Fl "d 2 + 02 Jo o0 F2'"dn + 1  '1 f To l0 F "dTidn + 2 F 2"dndTn + n n D 0 o0 where B and D are additional integration constants dependent on Toj; o is the i coordinate at the wall, y = 0; hence by (1) 1/3 1o ="k/= k (aRTUk' (lla) The boundary condition j1/ 0' = 1 at y = 0, that is, T = io, is fulfilled by adding the term 1 in equation (16b). SC3C30' is the gliding speed of the frictionless solution at the wall. 30 (16b) (16c) NACA TM 1265 Tietjens computed the integration constants Pi, 02, B, D from the fact that at a point q = 1, which lies in the range of validity of both expansions, asymptotic expansion and power series expansion of the solution of (13) up to and including the third differential quotient must agree. The resultant equation system, set up and solved by Tietjens, reads B F1111 + P2F2"' = BF41' 0P1F1 + 2F2 = BFq" P1 Fl"dTI + P 2 F2"dTi + 1 = BF4' STo oq 0 F2"'dndy + T D = BF4 (c) (17) (d) 0 F1 dnd + P2 ou o o 0 The integration constants 01, P2, B, and D can also be computed by a simpler method than Tietjens', with the aid of the transition formula from Hankel's to vessel's functions, which reads4 H /(2)(z) = e Ji/3 J /(z) J (Z) 1/3 sin R/3 [ 1/3 1/3 This obviates the joining of the two expansions, makes (17) superfluous, and (17b) gies the exact values immediately: 2 = ei/6 32/3 and = 1.o + .687r = 1.190 + 0.6871 3 3 S~.7 313 + t sin = 0.789(1 + i) (18) Tollmien, who pointed this out to the writer, had this representation as far back as 1929, but summarily took over the data by Tietjens for the sake of simplicity. NACA TM 1265 The factors P1 and D as function of no can then also be indicated explicitly, namely B = D =  1 ei/3 1 /3/ r() + Fld + + + % o o1 TI Jo  Bq3(O) (19) with Bp3(0) = i0i Table 2 contains the results of a new calculation of Tietjens' value carried out by these formulas. The differences from Tietjens' figures are insignificant. The values of F1", F2", Fl"d1 = FI', 0 10 F2"dn = F2', J F1'dT = FlI, F2'dn = F2 as function of q are indicated separately as real and imaginary part in table 3. Since, according to (15), these quantities are either symmetrical or antisymmetric functions of n, this table can be continued immediately according to the negative values of q. For the two neutral oscillations, whose amplitude and energy distri bution is to be computed, it is TI = 2.63 and To = 4.05 The corresponding values of E, according to (lla), are given in table I. along with the integration constants P1, 2', D obtained by interpolation for these no values. This takes care of all the data necessary for computing the friction solution cp with its first and second derivative as function of I by the equations (16a, b, and c). Table 4 gives the thus obtained values NACA TM 1265 (P" (P"1 of  30 30 is, according  as function of n. The connection between i and y 30 to equation (ll), Y = Vk + E In all cases, the friction solution p., from the wall toward the inside of the flow is very quickly damped out; but it still extends a little beyond the critical layer for both oscillations. h. The friction correction of 9c2 in the intermediate layer The second frictionless solution 92 behaves singularly at the critical Uk layer y = yk' namely through equation (9) as j(y7 yk)log(y yk), iC so that q2' behaves as Uk" I as  Uk ( k) U II Uk From the differential equation (12), in which only the greatest friction terms are taken into account, follows a solution q2 modified by the friction, which joins the frictionless solution at some distance from the critical point. For this purpose cp2 is expanded in powers of the previously introduced small quantity e = (aRUk)1/3 (20) P2 = P20+ E'21 + ** p20 being chosen equal to unity. From (12) follows the inhomogeneous differential equation (21) 21 ir21" = i Uk' for (21 with reference to n. NACA TM 1265 On account of the very small value of C, i can assume great values even at small y yk values. An attempt is made to find such a solution P21" of this equation, which for large j, but small (y yk) joins on to the frictionless solution d C2 Uk 1 dY2 Uk' Yk For large n there shall be: 2 d '21 Uk" 2=T (22)) dy2 Uk Ti The corresponding homogeneous equation appeared earlier in the calculation of cp.. (equation (13)). It has the fundamental system FI"(i) and F2 "() (equation (15)). A particular solution of (21) is Uk n flT S Uk' F1i" F2"d F2" F d which can be verified easily by substitution, and the general solution of (21) is P21() = iF2" T F1"dl iF1" L F2"d + clFl" + c2F2" (23) The integration constants c1 and c2, which can be complex, are evaluated from the boundary conditions. The quantity 921" is complex and shall join the real frictionless value (equation (22)) for large NACA TM 1265 5 values of 1. By decomposition in real and imaejinary parts, the four equations defining the integration constants read U '" "Lr U,'21r = G"(r) = F21"iF' F2r "Fli' + li"F2r' + Flr"F2i Uk ' F F 1 F Cr C li" 2rFr 2iF21 k (24) U, 21i = H"( ) = F21"li + F2r"Flr' + FlF FrF2r (2 +" c iF" c F + c = 0 +clrFli+t +cliF n+ C2rF21 + c2iF2r for n = +A From (21), with the boundary condition (22), it follows that P21r" is an antisymmetrical, and 1in a symmetrical function of T1. Moreover, since Flr', F21', F 2, Fli are symmetrical and Fli, 2r', F21i F r antisymmetrical functions of ri, the following must be true li = c2r = 0 (25a) for reasons of symmetry. The other two constants clr, c21 are obtained by solving the above equation system for i = n,. For the present calculation, il = 4 was chosen. The series for the Bessel functions are still fairly convergent for n = 4; but since differences of very large numbers occur, the separate terms in (24) must be computed to five digits (table 3). For c1r and c2i clr = 1.2852; c21 = 0.9373 (25b) Uk' U' so that P21r" = G"(1) and P2 = () can be calculated. The k k values are given in table 5. NACA TM 1265 d)2 The values of Q2 and  in the intermediate layer are obtained immediately by quadratures, namely dp2r Uk_ _ SUk(2a) d r= (1 + log e) + G'() (26a) and d.22 Uk" Uk" n Uk = U H )d = U ( = ) (26b) dy~ Uk' UkuJ4 Uk A check on this numerical calculation is given by the fact that 1921 for at transition from large positive to large negative j the transition substitution for q2 deduced by Tollmien (reference 3) must result again (compare equation (9)), which he obtained by discussion of the asymptotic representation of the Hankel functions. Tollmien's transition substitution gives (p21 9 2i Uk d) /y=+ dy y=m Uk the present numerical calculation gives 2i) 21) Uk Is = P +  H"(')dr d, ,/y=+ \ dy /y=oo Uk1  and the graphical integration gives +0 T"(l)dl ,;T 4 H"C(Tl)drl = 3.14 (27) that is, complete agreement within the scope of mathematical accuracy. NACA TM 1265 The intermediate layer near the critical point, which by the present calculation reaches from about q = 4 to I = +4, is already so wide at the first neutral oscillation that it reaches up to the wall (vall ro = 2.63); at the second oscillation with To = 4.05, the boundary of the intermediate layer is reached exactly at the wall. With this, all data needed for the numerical calculation of the solution (p2' corrected by the friction with p2' and cp ", are available. 5. The numerical values of the integration constants All three particular solutions Clp' P2' (3 are numerically known. To build up the required solution ( from it, the numerical values of the Integration constants C and C, must be accertained (equation (6)). First of all, equation (2) is rewritten In a more suitable form, namely 3 9 = C1 + C '2 + C,' 9 (2a) P30 where equations (3a) and (3b) were reported to and. 9 was replaced by the quantity ?(P 0 which follows immediately from the numerical calculations. Comparison with (6) gives 3' ' la O ( C920'+ 1') (6a) 2a; L 2 20 This method of writing has the advantage that the two integration constants C2 and C,' in (2a) are dependent only on the values of the frictionless solutions Q1 and 92, hence are relatively simple to compute. The values of p la' 92a' la' q2a' ,la' 2a and the values of C2 and C thus computed by (6) and (6a) for both neutral oscillations are given irn ttble 1. Table 6 and figures 3 and 4 give the values of pr' i p r I' computed with it, hence the desired amplitude distribution as function of y/b. NACA TM 1265 Outside of the boundary layer, at y/6 > 1.015 the simple formula Sr = C*eaC r = aC*eaY (pi = 0 is valid for the amplitude distribution. The constant C* is so chosen that the value of (pr' joins the already found value in y/6 = 1.015. (Table i.) 6. The average fluctuation velocities and the correction factor (compare reference 10) Changing to the real method of writing u' = = K p' cos(ax Prt) p' sin(cx Ort) UM (30) = = Kca. sin(ax Prt) + PI cos(cx Prt Um K is a freely available intensity factor. According to figures 3 and 4, the one phase (qr or pr') predominates in both neutral oscillations. Tne amplitude distribution of u' and v' can be represented most appropriately by forming, in analogy with the turbulent fluctuation velocity, the dimensionless quantities and where the dash U U m Um denotes the time average value formation over a period T at a fixed point x, y, or in other words uT u12 = ; u'2dt (T = vibration period) T t=o NACA TM 1265 19 The result is S r2 K 2 +2. v2 Ka ,2 +uM = M r2 + (pr 2 (31) and u'2 +' K2f ,2 ,2 22 2( u 2 2 TLr + Pi' + C (2Pr + ,2) (32) The last quantity gives the mean kinetic energy of the motion disturbance. (See eq. 36.) These averages, which are independent of x, are represented in figures 5, 6, and 7 and table 7 for both neutral vibrations as functions of y/6. The intensity factor itself was so chosen that the average value of 'u2 in the boundary layer is equal to 0.05Um ( '2 y = 0.05 (table 1). The maximum amplitude k 10 5 for both neutral vibrations lies near the critical layer. The correlation factor between u' and v', which is completely independent from the intensity of the motion disturbance, can then also be calculated. It is 1 1 T___ k(u', v') = vt =1 _u,2 ,2 u'v'd \u2 v,2 k(u', v') = (33) \ (r'2 + c,'2) (Pr2 + 12) The correlation factor is likewise dependent on y/6 only; its variation is indicated in figure 8 and table 7. It is negative almost throughout the entire range of the flow, for both neutral vibrations, as is to be expected, since, owing to the positive dU/dy, positive u' is usually coupled with negative v' and negative u' with positive v'. The maximum value of k is 0.17 and 0.19, respectively. It is inter esting to compare the theoretically established correlation coefficient with Townend's data in a developed turbulent flow (reference 11). The k values of 0.16 to 0.18, obtained for the flow in a channel of square NACA TM 1265 cross section at various distances from the axis, are of the same order of magnitude as those obtained by the present calculation for the incipient turbulence. Chapter II ENERGY DISTRIBUTION 7. The kinetic energy of the disturbance motion. Having established the amplitude distribution for the two neutral vibrations, the energy of the disturbance motion can be computed. The total kinetic energy of the disturbance motion in a layer of unit height, which, in x direction, extend over a wave length X and in y direc tion from the wall to infinity is E = (u'2 + v'2)dxdy 2 x=0 J=0 P= X r2U 2 2 + Ti 12 + 2(q)p2 + + 2 d(y/6) (34) The energy dE of the secondary motion in a strip of width dy and length X is accordingly E pX 2K 2 1 '2 i2 + 2 r2 + ) (35) = 22 Pr + ai + Besides, 5 dE u2 + .12 0.533 = (36) Eo dy Um2 E is the basicflow energy in a layer of unit height, length X, and width 6 (compare equation (32)). Figure 7 shows the dimensionless energy distribution by equations (35) and (36). The energy is strongly concentrated near the critical layer. To obtain the total energy E, the integral (34) must be evaluated. Dividing it in two parts with the limits 0 = y/56 1.015 and 1.015 y/b < m, the first portion is NACA TM 1265 obtained by graphical integration based on the computed amplitude distribution. The second portion is obtained analytically by (29), namely S 2 2 (pr2 + i2) d(y/b) = abCe2.03a Jy/6=1.01i r +1 The results of the evaluation are given in table 1. Now the energy of the disturbance motion is compared with the basic flow energy Eo in the space of unit height and surface area X X . It is, by equation (7a) E = U(y)cdy = 0.533 2X6 (37) x=0 y=0 Hence, for the ratio of energy of the secondary motion to the energy of the basic flow E/E, the values presented in table 1 are obtained. 8. The energy balance of the disturbance motion. Consider the time variation of the secondarymotion energy of a particle that moves with the basic flow, hence 2 ,2 o : 'u 2 + v'2(38) u'2 + = + U ) 2)} (8) For stable disturbances, the total change of energy of the secondary motion is 'J / D(u12 + v'2) dv < 0, for unstable disturbances > 0, m2 s j J Dt and for neutral disturbances = 0, the integration extending over the entire range of the particular flow. Participating on the variation of the secondarymotion energy are: first, the transfer of kinetic energy from the primary to the secondary flow, or vice versa; second, the pressure variation; and, third, dissipation. For neutral vibrations, the total energy balance is not only equal to zero for the entire space in question, but for every point y of the cross section, the energy NACA TM 1265 increase per vibration period T = 2ir/r is also equal to zero. This is easily confirmed in the following manner: It is Pf Dr(u'2 + v,2)dt 2 ftO=0 = (u2 +v'2)dt + uf T + v' 2)dt 2 It=0 6t 2 t=O 6x = P u2 + + pU u' PU v' ( dt = 0 2 0 0 x 6x The first term disappears by reason of the periodicity of u' and v'. The same holds true when the last term for u' and v' is entered according to equation (30). Thus, the energy increase per vibration period T is equal to zero at every point x, y for a neutral vibration. It is interesting to see how the several factors enumerated above participate on the energy conversion in a specific case. For both specific cases of neutral disturbance the calculation of the energy is carried out for a plane basic flow and a plane disturbance motion according to Lorentz (reference 12) Dt Cu2 + vt2)= Pu'v' p fu 22 dy 6x C) where iv' bu' = = coefficient of viscosity ox dy NACA TM 1265 The first term gives the transfer of energy from the primary to the secondary flow, the second gives the contribution resulting from the pressure variations, and the third and fourth terms, the loss of energy by dissipation. After integration of this term with respect to y over the total width of the laminar flow from y = 0 to y = and with respect to x over a wave length X of the disturbance, the second and fourth terms disappear, since u' and v' disappear for y = 0 and y = and with respect to x have the period X. Thus, the growth of the energy per unit time in a layer of unit height and base area 0 < y <, O < x < is: 'v' dxdy  dy y=0 (39) = x dxdy~ (/=0 \ax by The first integral gives the total energy passing from the primary to the secondary motion; the second, the total dissipation. The portion of the energy due to pressure variation is removed by the integration. The two energy portions for the two neutral disturbances are evaluated. Through substitution of (30), followed by integration with respect to x, we find ((rI,l rcpi,) dy IUm2 T=0 dy 2 DE = QaL2U 2 X Dt m DE = 2nK2(e1 + e2) X5 2 Um Dt 1 2 2) 2 tm eI and e2 denoting the dimensionless energy integral (40) (41a) = p X Ix=0 y=0 + (C~p a2)2 d (Pr'i Pr') d(/) d(y/b)= Y el'd(y/6) di(y/6) Uo el * o 0 Un=0 I NACA TM 1265 )o 00 e2 = 1 Pr" 2r)2 + (pi" a2i)2 d(y/6) = e2'd(y/6) (41b) To find the energy change of the disturbance motion in the vibration period T(T = X/cr, or = phase velocity), this energy change is referred to the total kinetic energy E of the secondary motion which is given by equation (34). From (40) follows then the specific energy change of the disturbance motion as T DE m 2 (Cel' + e2')d(y/b) (42) Eo Dt cr 0.533Z J (2) where Z = 0.432 and Z = 0.810 for the first and second neutral vibration, respectively, while 'Jm/cr = 2.86 for both neutral vibrations. The local energy transfer from primary to secondary motion (1) and the local dissipation (2) for the two neutral vibrations is then 12 = 78.0e l, or = 4i. 12' (43a, b) when T DE = Dt The values of e, (equations 4la, b) can be obtained (table 8) on 1,2 the basis of the computed amplitude distribution for both neutral vibrations. Figure 9 represents the local energy conversion. The dissipation in vall proximity is seen to be extremely great, while the critical layer is of no particular importance for the dissipation. But the energy transfer from the primary to the secondary motion is greatest in the neighborhood of the critical layer, while at the wall and farther outside it is very small. The curve is similar to that of the correla tion (fig. 8), as anticipated. The graphical integration of e' ani e2' gives the values DE indicated in table 1. The energy balance D = 0, or e + e2 = 0 I'ACA Tl1 1265 for the neutral vibrations is therefore fulfilled with satisfactory approximation5, and constitutes a very welcome check on the rather complicated solution of the characteristic value problem. The total energy transferred in vibration period primary to the secondary motion is T from the (LE)1/E = 78.0el or 41.6e1 and the total energy dissipated (1E)2/E = 78.0e2 or b1.6e2 These figures are also shown in table 1. Thus, at the first neutral *vibration, about half of the secondarymotion energy is destroyed by dissipation during one vibration; at the second neutral vibration, the energy conversion is only about half as great. At the second neutral vibration, the vibration period is a little greater than at the first, that is, as is readily obtainable .from the data of tale 1, is T (2) 1 = 'O) = 10.1 x 104 V; U2 m T 2n = 14.8 x 104 2 r2 Um m To illustrate; For a plate flow in water with v = 0.01 cm2sec1; Um = 20 cm sec1 5According to the present vibrations is somewhat greater primary to the secondary flow. stability calculation only the calculation, the dissipation for both than the transfer of energy from the This is due to the fact that in the dissipation of the friction solution P3 was taken into account, while the dissipation of the frictionless vibra tion ((p, (2) was ignored. But, in the energy equation, the dissipation of frictionless and frictional vibration was computed and is therefore a little greater. Thus, the "neutral vibrations" have, exactly computed, still a little damping, and the indifference curve (fig. 2) is, as a result, shifted a little toward the inside. 26 NACA TM 1265 the periods of vibration are T1 = 2.50 sec; T2 = 3.70 sec Thus, vibrations of comparatively great periods are involved. Translated by J. Vanier National Advisory Committee for Aeronautics IlACA Th 12'65 REFERENCES 1. Prandtl, L.: Bemerkungen uber ile Entstehung der Turbulenz. Ztschr. f. angew. Math. u. Mech. Bd. 1, p. 431, 1921. Uber die Entstehung der Turbulenz. Zeitschr. f. angew. Math. u. Mech. Bd. 11, p. 407, 1931. 2. Schlichting, H.: Netiere Untersuchunrgen 'iber lie TurbuLlerLzentztehung. Die Naturwissen.rchaften Bd. 22, p. 37., 1934. 3. Tollmien, W.: Uber die Entstehung der Turbulenz. Naclr. d. Ges. d. Wiss. zu Gottingen, Math.Phys. Klasse 1929, p. 21 and Verhdig. d. III. Intern. Kongr. f. techn. Mech. Stockholm 1930, p. 105. (Available as NACA TM 609.) 4. Schlichting, H.: Uber die Stabilitat der CouettestrimunLg. Ann. d. Phvysik, V. Folge, Bd. 14, p. 905, 1932. And Verhdlg. d. Intern. Mathematikerkongresses Ziurich 1932, p. 283. 5. Schlichting, H.: Zur Entstehung der Turbulenz bel der Plattenstrimunzg. Nachr. d. Ges. d. Wiss. zu Gottingen, Math.Phys. Klasse 1933, p. 181 and Ztschr. f. angew. Math. u. Mech. Bd. 13, p. 171, 1933. 6. Schlichting, H.: Uber die Entstehung der Turbulenz in eimen rotlerenden Zylinder. Nachr. d. Ges. d. Wiss. zu G.ittingen, Math. Phys. Klasse 1932, p. 160. 7. Schlichting, H.: Turbulenz bei Warmeschichtung. Summarj of results in report cited under reference 2 in Verhilg. d. IV. Intern. Kongr. fe. techn. Mechanik Cambridge 1934, p. 245. 8. Squire, H. B.: Stability for ThreeDimensional Disturbances of Viscous Fluid Flow between Parallel Walls. Proc. Roy. Soc. (London), ser. A. Vol. 142. 1933, pp. 621628. 9. Tletjens, 0.: Beitrage zum Turbulenzproblem. Diss. Gottingen 1922, and Ztschr. d. angew. Math. u. Mech. Bd. 5, p. 200, 1925. 10. Tollmien, W.: Uber die Korrelation der Geschwiniigkeitokomponenten in perlodlsch schwankenden Wirbelverteilunpen. Zeitschlr. f. angew. Math. u. Mech. Bd. 13, p. 95, 1935. 11. Townend, H. C. H.: Statistical Measurements of Turbulence in the flow of air through a pipe. Proc. Roy. Soc. A. Vol. 14.5 p. 180, 1934. 12. Lorentz, H. A.: Abhandlungen uber thoretis.he Physik. Bd. I, p. 43, Leipzig, 1907. NACA TM 1265 TABLE 1 THE PARAMETERS OF THE TWO NEUTRAL VIBRATIONS OF THE PLATE FLOW First neutral Second neutral vibration vibration cr/Um 1/E B2 D (la ('la CP2a la I/.. cla q 2a C2 10, C20 11.r015r2 + p1 Pr2 + ' o0 + a2((pr2 + (i2) S0015 S1 ,:i 0.466 2.62 x .163 .350 .209 1.625 .494 2.63 12.6 .0695 +.1021 f.1526 L. 0736i (1.374 +.2001 .416 .040 .211 2.425 .234 2.327 .101 1.005 1 .046 +1.563i ri1 O0 S.157i .706 .1454 .371 .090 0.737 6.08 x 103 .258 .350 .209 1.625 .494 4.05 19.4 t .0470 +.02761 S.0368 .06501 .395 +1.241 .435 .097 S.306 2.240 .417 2.014 .207 1.011 .114 +1.563i j .988 L .3251 1.075 .1166 .681 .183 NACA TM 1265 TABLE 1 THE PARAMETERS OF THE TWO NEUTRAL VIBRATIONS OF THE PLATE FLOW Concluded First neutral vibration Second neutral vibration E 0.432K2 0.810K2 Eo E 0.00913 0.0110 E o e x 103 5.75 6.39 ex 103 6.16 7.10 (6E)1 0.447 0.265 E ( E)2 0.479 0.294 E NACA TM 1265 TABLE 2 NEWLY CALCULATED VALUES OF AND D %o 1l D o 0.387 + 0.6721 0.672 0.3871 0.5 0.341 + 0.3661 0.770 0.3801 1.0 0.262 + 0.2131 0.892 0.3501 1.5 0.192 + 0.1421 1.023 0.2611 2.0 0.132 + 0.1131 1.202 0.1351 2.5 0.0822 + 0.10311 1.358 + 0.1241 3.0 0.0332 + 0.09721 1.397 + 0.5191 3.5 0.0165 + 0.07821 1.139 + 1.0231 4.0 0.0465 + 0.03231 0.493 + 1.2141 NACA TM 1265 TABLE 3 THE PARTICULAR SOLUTIONS OF p3 AS FUNCTION OF [Compare equation (15)] SFr FI F2 F2 F F F F ' Ir 11 2r 21 Ir ii 2r 21 0 0 0 0 0 0 0 0 0 0.5 0.021 0.000 0.125 0.000 0.1250 0.0005 0.5000 0.0026 1.0 0.167 0.003 0.500 0.008 0.4998 0.0167 0.9992 0.0417 1.5 0.563 0.032 1.122 0.063 1.1186 0.1264 1.4865 0.2065 2.0 1.318 0.177 1.974 0.266 1.9368 0.5292 1.8988 0.6588 2.5 2.59 0.678 2.97 0.797 2.7503 1.5799 2.0223 1.5544 2.63 2.87 0.901 3.24 1.022 2.90 2.02 1.95 1.88 3.0 3.97 1.94 3.86 1.91 2.9208 3.6995 1.3354 2.9269 3.5 5.07 4.55 4.02 3.72 0.9512 6.8796 1.0850 4.2113 4.0 4.20 8.70 2.35 5.77 5.6379 9.2445 6.1155 3.3059 4.05 3.84 9.13 2.00 5.89 6.62 9.20 6.76 2.90 antisy. sy. Sy. antisy. ay. antisy. anti y. sy. F Fr F F 2r F 21 F Fli" F 2r"' F 21 r ^ ^ 21 ^Ir "11 ^ 21 0 0 0 1 0 1 0 0 0 0.5 0.5000 0.0052 0.9999 0.0208 0.9998 0.0417 0.0010 0.1250 1.0 0.9980 0.0833 0.9944 0.1666 0.9861 0.3331 0.0333 0.4999 1.5 1.4661 0.4206 0.9368 0.5595 0.8442 1.1165 0.2525 1.1072 2.0 1.7472 1.3108 0.6469 1.2939 0.1187 2.5541 1.0533 1.8229 2.5 1.3100 3.0463 0.3215 2.3124 2.2818 4.3761 3.0889 2.0809 2.63 0.941 3.64 0.776 2.58 3.0 1.1161 5.4735 2.7429 3.0209 8.1612 4.7949 6.8785 0.1417 3.5 7.6314 6.7189 7.2934 1.4618 18.5245 1.7400 11.0222 7.8189 4.0 19.4902 0.6805 12.5621 6.6661 27.2703 26.6992 7.4580 26.7961 4.05 20.94 0.806 12.92 8.10 antsy. sy. sy. antisy. sy. antisy. antisy. sy. NACA TM 1265 TABLE 4 THE FRICTION SOLUTION 9, AS FUNCTION OF y/6 AND 71 First Neutrkl Vibration nl mt m' I ^ 1 /1 3ru 31i 3r 31 3r 31 ___o' _0' _30o' 3o 13 3o' 30 30 30 93 930 2.63 o 6.40 7.65 1 0 0.111 0.0159 2 .050 5.67 .794 .665 .188 .069 .0099 1 .130 3.04 1.801 .329 .106 .029 .0029 0 .209 1.92 .932 .136 .012 .009 .0061 1 .288 .995 .113 .019 .040 .001 .0035 2 .368 .202 .290 .025 .018 0 .0010 3 .447 .063 .076 .027 .003 0 .0005 4 .577 .076 .012 .020 .012 0 .0006 Second Neutral Vibration 4.05 0 17.7 34 1 0 0.0204 0.0639 3.5 .029 17.6 8.22 .460 .519 .0001 .0554 3 .054 9.71 3.08 .106 .534 .0070 .0415 2.63 .074 4.91 4.67 .030 .455 .0076 .0322 2 .105 .272 3.88 .103 .309 .0051 .0202 1 .157 1.378 2 .058 .166 .0005 .0091 0 .209 .717 1.260 0 .084 .0008 .0035 1 .261 .019 .679 .016 .035 .0002 .0012 2 .313 .194 .155 .007 .013 0 0 3 .361 .058 .019 0 .012 0 0 4 .415 .039 0 .005 .014 0 0 NACA TM 1265 TABLE 5 THE FRICTION CORRECTION p3 IN THE INTERMEDIATE LAYER AS FUNCTION OF i T1 G (TI) 1"(1n) G'(TO) S'(T) 0 0 1.285 0.774 1.570 .5 .443 1.165 .659 .953 1 .746 .857 .354 .448 1.5 .839 .483 .051 .118 2 .767 .160 .458 .035 2.5 .589 .018 .798 .073 3 .438 .072 1.056 .048 3.5 .325 .055 1.243 .013 4 .250 0 1.386 0 G"(n) = G"(TI); H =(~) = H"(n) G'(Tj) = G'(n); H'(i) = it H'(,) NACA TM 1265 TABLE 6 DISTRIBUTION OF AMPLITUDES (Pr3 ,Pi 'i P (r i AS FUNCTION OF y/b First Neutral Vibration y/6 Pr 9 i r' ci' (Prn TI 0 0 0 0 0 7.885 6.615 .050 .011 .0040 .411 .135 6.235 .003 .090 .032 .0080 .620 .115 4.580 .770 .130 .059 .0112 .782 .025 3.200 1.717 .170 .090 .0111 .896 .042 2.355 1.240 .209 .127 .0088 .976 .069 1.778 .434 .250 .166 .0063 .986 .070 .112 .220 .290 .203 .0041 .975 .058 .699 .576 .370 .276 .0010 .831 .018 1.543 .347 .451 .335 0 .670 .005 1.778 .015 .531 .380 0 .510 .003 1.641 .016 .612 .414 0 .361 0 1.606 0 .693 .438 0 .237 0 1.514 0 .774 .453 0 .118 0 1.431 0 .854 .458 0 .007 0 1.364 0 .935 .455 0 .101 0 1.303 0 1.015 .445 0 .205 0 1.252 0 NACA TM 1265 TABLE 6 DISTRIBUTION OF AMPLITUDES rp ~' (r ', i' rM, ( i", AS FUNCTION OF y/5 Concluded Second Neutral Vibration y/5 CPr 0 0 0 0 0 28.54 27.89 .029 .010 .0038 .720 .337 20.27 2.30 .054 .033 .0114 1.104 .237 9.16 6.35 .074 .057 .0140 1.245 .113 4.25 6.28 .105 .096 .0148 1.333 .052 .478 3.60 .157 .168 .0086 1.350 .129 .558 .27 .209 .236 .0041 1.306 .076 1.218 1.54 .250 .287 .0005 1.203 .014 2.967 1.31 .290 .333 .0002 1.091 .016 2.961 .334 .370 .409 0 .827 .016 2.86 .144 .451 .469 0 .607 0 2.58 0 .531 .507 0 .409 0 2.21 0 .612 .532 0 .247 0 1.93 0 .693 .547 0 .105 0 1.72 0 .774 .550 0 .026 0 1.56 0 .854 .544 0 .148 0 1.44 0 .935 .527 0 .260 0 1.32 0 1.015 .508 0 .367 0 1.23 0 NACA TM 1265 TABLE 7 THE MEAN FLUCTUATION VELOCITIES u'2, \42, THE KINETIC ENERGY OF TEE DISTURBANCE MOTION u'2 + v'2, AND THE CORRELATION COEFFICIENT k AS FUNCTION OF y/5. [EQUATIONs (31), (32), (33fl First Neutral Vibration y/b 10 L 102 o 1022 + v2' k UM UM UM _________ __ ___ _______m__ m______ 0 0 0 0 0 .050 .445 .0546 .198 .032 .090 .647 .158 .420 .061 .130 .804 .287 .648 .155 .170 .922 .434 .854 .169 .o09 1.005 .608 1.019 .140 .250 1.016 .795 1.040 .109 .290 1.003 .973 1.019 .064 .370 .854 1.322 .748 .022 .451 .689 1.605 .501 .0075 .531 .524 1.820 .309 0 .612 .371 1.984 .177 0 .693 .244 2.098 .104 0 .774 .121 2.170 .062 0 .854 .007 2.193 .048 0 .935 .103 2.180 .059 0 1.015 .211 2.108 .090 0 1.1 .203 2.027 .082 0 1.2 .193 1.935 .075 0 1.3 .185 1.849 .068 0 1.4 .176 1.763 .062 0 1.5 .168 1.681 .057 0 NACA TM 1265 37 TABLE 7 THE MEAN FLUCTUATING VELOCITIES u'\ v THE KINETIC ENERGY OF THE DISTURBANCE MOTION u'2 + v'2 AND THE CORRELATION COEFFICIENT k AS FUNCTION OF y/5. E&UATIONS (31), (32), (33) Concluded Second Neutral Vibration y/6 10 102 102 2 +  U U U2 m m Um2 0 0 0 0 0 .029 .661 .0655 .433 o .054 .937 .214 .869 .121 .074 1.038 .359 1.068 .170 .105 1.108 .595 1.218 .191 .157 1.127 1.028 1.262 .146 .209 1.086 1.445 1.190 .076 .250 .998 1.755 1.019 .010 .290 .906 2.04 .854 .015 .370 .686 2.50 .528 .019 .451 .504 2.87 .332 0 .531 .340 3.10 .208 0 .612 .205 3.26 .146 0 .693 .087 3.35 .118 0 .774 .022 3.37 .117 0 .854 .123 3.33 .124 0 .935 .216 3.22 .149 0 1.015 .305 3.11 .187 0 1.1 .292 2.925 .171 0 1.2 .272 2.72 .146 0 1.3 .252 2.53 .126 0 1.4 .234 2.35 .109 0 1.5 .218 2.18 .093 0 NACA TM 1265 TABLE 8 TEE LOCAL ENERGY CONVERSION 1) = TRANSFER FROM PRIMARY TO SECONDARY MOTION, 2) = DISSIPATION EQUATIONS (41) AND (43) First Neutral Vibration y/ el x 103 e2' x 103 E dE)1 (E)2 0 0 86.9 0 6.78 .050 .268 31.8 .021 2.48 .090 2.150 17.6 .168 1.37 .130 12.25 10.8 .955 .84 .170 23.25 5.73 1.814 .45 .209 28 2.64 2.18 .21 .250 27.30 .06 2.13 0 .290 22.85 .72 1.78 .05 .370 7.49 2.20 .584 .17 .451 1.885 2.80 .147 .22 .531 1.102 2.43 .086 .19 .612 0 2.36 0 .18 .(93 0 2.12 0 .17 .774 0 1.92 0 .15 ., 0 1.75 0 .14 35 0 1.61 0 .13 i.01: 0 1.50 0 .12 NACA TM 2L65 TABLE 8 THE LOCAL ENERGY CONVERSION 1) = TRANSFER FROM FRTMARY TO SECONDARY MOTION, 2) = DISSIPATION [EQUATIONS (41) AID (43)] Concluded Second Neutral Vibration y/6 eI x 103 e x 103 1 E)I ) 1 2E dy 15 E Edy 2 0 0 320 0 13.31 .029 1 83.6 .044 3.48 .054 7.84 24.3 .326 1.01 .074 18.45 11.15 .768 .464 .105 41.5 2.61 1.726 .109 .157 55.6 .04 2.310 .002 .209 37.5 .840 1.560 .035 .250 5.22 2.31 .217 .096 .290 7.97 2.01 .331 .084 .370 8.44 1.91 .351 .079 .451 0 1.8o 0 .075 .531 0 1.38 0 .057 .612 0 1.01 0 .042 .693 0 .819 0 .034 .774 0 .695 0 .029 .854 0 .608 0 .025 .935 0 .520 0 .022 1.01 0 .458 0 .019 NACA TM 1265 Figure 1. Laminar flow past the plate. Figure 2. The zone of the stable and unstable disturbances of plate flow. I = first neutral vibration. U = second neutral vibration. aS I NACA TM 1265 Figure 3. Real and imaginary part pr, (P, fpr' qi' of the amplitude of disturbance motion plotted against wall distance for the first neutral vibration. Figure 4. Real and imaginary part )r, q, ', y ir" of the amplitude of disturbance motion plotted against wall distance for the second neutral vibration. NACA TM 1265 0.125 I aw Figure 5. The mean fluctuating velocity in the x direction U plotted against the wall distance for both neutral vibrations. 0.04 i 0.09 " 0.0/ 61 0.4 0s oa 1.0 2 4 1i. Figure 6. The mean fluctuating velocity in the y direction rU m plotted against the wall distance for both neutral vibrations. NACA TM 1265 Figure 7. The mean kinetic energy of the disturbance motion u'2 + v2/ m2 plotted against the wall distance for both neutral vibrations. 0.20 0.15 k 0.06 a05 I /,I a' 0.2 s5 ofl u'v1 Figure 8. The correlation coefficient k = plotted against Y/6 for both neutral vibrations. NACA TM 1265 15.0   10.0 I (2) 12 024 '^ '^ a6 a0 \ 25 II Figure 9. The local energy conversion of the secondary motion for the first and second neutral vibration. I (1), U (1) = energy transfer from primary to secondary flow; I (2), II (2) = dissipation. NACALangley 41850 950 N' ~41 17 I rt to O 4a, al 0 P4 P. 14 ai ,, m o u ,1 rt a, kd4 Sfa a t Ln i\9 o\ it < T \o N I ( C.) Li a) i p $p E4 pu H o r. p u (V I M (, c > P cU r iHC 0 )I m ;A 0 +2 .0 Cd (L) LU d h04'k CH ar 0 i C ri O0)0 2CO Z 1 0 p N (d r > ) k Ca 4 +) m i 0 Sa (L H 0 0  0 H O H OOS ) 2 0 a) 0 i O 4 H ( a u (i U +) + S $ i CCi 4 C ml Sl 1 .4 mP rI o4c CU I .9CH 0)P0*) 0 A CH ) ) P aCrd 4 0 0 UP M0A v +r H +) O Sa C.) r 4 9 d+ + 1 w + c a H 41 B rd o m0 4 l 4' 0C 00 O m C cS cioa ra S k k Cr a) WU a)  41 3 p 'A ( c ca ia 0) >InP or a 0 :, ) iz3 rd +)C C 0 ) U Sr4 4 a) cri AI ob Sn 0 4' i ri Pl a) i 0 1 H Oa M ) r ) 0 40 +)'PO P w o +' d u i a a c STl .0 P C2 c +C k0 O a C > .4 O r U r C a P0 10 O2.1 + 01 PCo d P CCg0 Od m +4 + k ( Nl o i < o H Hoo H m I rUCM d c(' H Co P 0 + 0 rI c a c iia 0o 93 + 0 r. H r. 01 U C W r S 0P 0 UC r C A 4 r U n r o +)m +) 4 g MP 4 X +0 0i > + M u^p o 0 0 C 4 l 4 u a) 3 3 9ca *H < PI i rd " UNIVERSITY OF FLORIDA 3 1262 08105 023 8 