]^Hh^(AA^^'^ 
 
 J, 
 
 NATIONAL ADVISORY COMMITTEE 
 FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 1265 
 
 AMPLITUDE DISTRIBUTION AND ENERGY BALANCE OF SMALL 
 DISTURBANCES IN PLATE FLOW 
 By H. Schlichting 
 
 Translation of *Amplitudenverteilung und Energiebilanz der kleinen 
 Storungen bei der Plattenstromung." Nachrichten von der 
 Gesellschaft der Wissenschaften zu Gottingen, Neue Folge, 
 
 Band l,No. 4, 1935 
 
 Washington 
 April 1950 
 
 DOCUMENTS DEPARTMfc-NT 
 
3 ^r ^^"^^ 
 
 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM I265 
 
 AMPLITUDE DISTRIBUTION AND ENERGHT BALANCE OF SMALL 
 
 DISTURBANCES IN PLATE FLOW* 
 
 By H. Schllchting 
 
 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 \ = 2.n/a., the phase velocity c , and the Reynolds 
 
 number R of the neutral disturbances established. In connection with 
 this J the present report deals with the average disturbance veloc— 
 
 \| 
 
 ities \ u' and \JT and the correlation coefficient 
 
 \|u'2 X v'2 
 
 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 oR plane). The maximum value of the last two q.uantlties 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 a 1 most 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. 
 
 *"Amplltudenverteilung und Energiebilajiz der kleinen Storungen bel 
 der Plattenstrbmung." Nachrichten von der Gesellschai't der Vis sens cheif ten 
 zu Gottingen, Neue Folge, Band 1, No. k, 1935, PP. ^7-78- 
 
NACA TM 1265 
 
 1 . Intr oduct i an 
 
 The numerous efforts, within the last decade, to solve the problem 
 of turhulence (reference l) have at least produced some satisfactory 
 results for a certain class of houndary— layer profiles when their 
 stability was Investigated "by the method of gmal 1 vibrations, with due 
 consideration to the friction of the fluid and the profile curvature 
 (reference 2). Eef erring to Toll mi en (reference 3), who treated the 
 example of laminar flow past a flat plate, the writer investigated 
 several other cases: the Couette flow (reference k) , 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 characteristic— value 
 problemB, without calculating the characteristic function itself. In 
 that manner, the wave lengths of the imstable, hence "dangerous", 
 disturbances were Identified as function of the Reynolds number. In 
 most cases, only the disturbances situated right at the boimdary, 
 between amplification and damping, were determined. For these, just 
 as much energy is transferred from the primary to the secondary flow, 
 as secondary^notion 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 primary— flow direotiono Vhile 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 three— dimensional. Accordingly, it might 
 appear aa if the limitation to two— dimensional 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 two-dimensional 
 disturbance motion is particularly suitable for the stability study in 
 the following sense: According to Squire, a two— dimensional flow, which 
 is unstable against three-dimensional disturbances at a certain Reynolds 
 number, is unstable against two-dimensional disturbances even at a lower 
 Reynolds number. The two-dimensional disturbances are therefore "more 
 dangerous" for a flow than the three— dimensional o The critical Reynolds 
 number, which is defined as lowest stability limit, is thus obtained 
 precisely from the two— dimensional, not the three— dimensional, 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 disturhance over the flow section, that is, the calculation of 
 the characteristic fimctions and second, with the study of the energy- 
 distribution and energy balance of the disturbance Jiotion. 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 DISTEIBUTION 
 2. Discussion of the differential equation of disturbance. 
 
 Let U(y) be the velocity distribution of the fundamental flow 
 (fig. 1) and )|f the flow function of the superposed disturbance motion, 
 which is assumed as a wave motion moving in the x direction (direction 
 of primary flow), whose amplitude (p is solely dependent on y, hence 
 
 *(x,y,t) = cp(y)ei(°«^^^) = q)(y) 6^°-^^"^*^ 
 
 a is real and X, = 2jr/a is the wave length of the disturbance; 
 
 P = Pp + iPj^ and c = P/a are, in general, complex; T = 2Jt/p is 
 
 the period of oscillation; P^ indicates the amplification or the damping, 
 depending upon whether positive or negative; c = Pp/a. is the phase 
 
 velocity of the disturbance. For the distiirbance amplitude cp, after 
 introduction of dimenslonless variables from the Navier-Stokes differential 
 equations, it results in a linear-differential equation of the fourth 
 order, the differential equation of the disturbance 
 
 (U - c)(cp" - a^cp) - U"cp = - ^(cp"" - 2aV + aV) (D 
 
 (R = Ujjj6/v = Reynolds number, U^j^ = constant velocity outside of the 
 
 boundary layer, 5 = characteristic length of the boundary— layer 
 profile = boundary— layer thickness, v = kinematic viscosity.) The 
 general solution cp of the disturbance equation is built up from four 
 particular solutions cp^, ..., cpj^ 
 
 cp = Cjtp^ + C2CP2 + Cf>2 "^ ^^^h ^2) 
 
MCA TM 1265 
 
 The "boundary conditions cp = qj' =0 for y = and y = 
 (as explained in the report cited in reference (5)) 
 
 give 
 
 Cj, = 
 
 (3a) 
 
 and for C^, 02^ Co the system of eq.uations 
 
 C^cpio + C2CP20 + C3CP30 = 
 
 C-,cp-,„' + C^cp_' + C,cp,^' = 
 
 ^1^10 
 
 '2^20 
 
 3^30 
 
 Cl%a + C202a = 
 
 >va = 'Pva' + °^ve.' ^ = 1. 2) 
 
 W 
 
 J 
 
 The subscript indicates the values at the wall y = 0, autecript a 
 the values in the connecting point y = a to the region of constant 
 velocity. From (i+) the equation of the characteristic value problem 
 follows as 
 
 9-Lo ^ 
 
 20 ^30 
 
 ^10' ^20' ^30' 
 
 la 2a 
 
 = 
 
 (5) 
 
 This equation is discussed in the earlier reports for several cases. 
 It contains^ aside from the constants of the hasic profile, the 
 parameters a, E^ c j and c. . The complex equation (5) is equivalent 
 
 to two real equations, and, if limited to the case of neutral disturhances 
 ic- = 0), these two equations give, after elimination of c^., one 
 
 equation "between a and E. This is the equation of the neutral curve 
 in the oR plane, which separates the unsta"ble from the stable disturbance 
 attitudes, and was originaily computed by Tollmien for the plate flow. 
 They ajr-e assumed to be known for the subsequent study (fig. 2). 
 
 -4?he 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 
 
 In the following, the amplitude and energy distribution ia computed 
 for two neutral oscillations, one of which lies on the lower, the other 
 on the upper hranch of the eq.uilibrium cijrve (fig. 2) o The parameters 
 of these two neutral oscillations, obtained from the earlier calculation, 
 are indicated in table 1. 
 
 For the calculation of the integration constants 
 from {h) , we put 
 
 a^ 
 
 ^2' ^3 
 
 Ci = l 
 
 (3b) 
 
 because the amplitude of 'disturbance remains indeterminate up to a 
 constant factor, the intensity factor of the disturbajice motion. Thus, 
 for the other two constants 
 
 Co = - 
 2 $ 
 
 la 
 2a 
 
 9 
 
 30 
 
 la 
 
 
 9 
 
 30 
 
 la ^ I 
 
 - 9 
 
 10 
 
 (6) 
 
 The particular solutions <^-,ij) and cpo^^) ^^^ readily obtainable 
 
 by expansion in series from the so— called frictionless differential 
 eq.uation of disturbance of the second order, which follows from the 
 general equation (l) by omission of the terms on the right-hand side 
 afflicted with the small factor l/oR. The point U = c , where phase 
 
 velocity of disturbance motion and basic flow velocity agree, and which 
 is termed the critical point y = y,, is a singular point of the 
 
 frictionless differential equation of disturbance, which plays a 
 prominent part in the investigations. 
 
NACA TM 1265 
 
 Putting 
 
 = y ; = y ; ay = a ; a(a — y ) = a^ (7) 
 
 y^ "^^l a - y^ ^2 ^]£ 1 ^Y 2 ^'^ 
 
 where subscript k denotes the values at the critical points, the 
 frlctionless solutions for linear velocity distribution read 
 
 cp-L = a~l sinh(ar]_y-j_) ; cp = cosh(a-^y-|_) (8) 
 
 and for parabolic velocity distribution 
 
 ^-^= a^y^ + ^^^^ + a^j^^ + ... 
 
 U " 
 
 ^2 " ^0 "^ ^1 ^2 ^ ^2 ^2 + • • • + ^ ^1 ^°S yg ^°^ ^2 ^ ° ^9^ 
 
 k 
 
 92 = tiQ + b^y^ + "b^yg^ + ... + — |- cp^^logjy^l - l^j for yg < 
 
 U?he Blasius profile of the plate flow la approximated by a linear 
 and a quadratic function (fig. l) , namely 
 
 ^ y/6 ^ 0.175: U/Ujj, = 1.68 y/6 
 0.175 ^ y/6 S 1.015: U/U^ = 1 - (1.015 - y/&)^ (7a) 
 
 y/5 > 1.015: U/U = 1 
 
 m 
 
 For the connection between 6 and the displacement thickness 5* used 
 in figiure 2, 5* = 0.3^15 is applicable. 
 
NACA TM 1265 
 
 According to earlier data, the coefficients are given by the eq.uations 
 
 « 2 2 
 
 1 °'2 °'2 
 
 ai = 1; ag = - -; a3 = — ; ai, = - — , 
 
 a = - 0.0013 ccg^ + 0.0083 ttg^; a^ = 0.002U ql^ + O.OOO6 ag 
 
 r, 2 
 
 bo =1; b^ = 0; bg = -1 + -|- (10) 
 
 b = 0.125 + 0.056 a^; bi^ = 0.021 - O.li+1 a,^ + 0.0)+2 a,^ 
 b^ = 0,005 + 0.005 °"^ + 0".00U QL^ 
 
 bg = 0.0015 + 0.0012 o^ - 0.0038 ci^ + 0.0011+ a.^ 
 
 The particular solution 9 , with its derivatives, is regular 
 
 throughout the entire range of flow (—1 ■= /-, ^ 0, '^ jp '^ -t-lj , and can 
 
 be numerically computed with these data. But the particvilar solution cpp 
 
 has a singularity, in which cpp' becomes logarithmically infinite in 
 
 the critical layer y = y, . 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 CP;,, the other, the friction correction for cp-. Introducing 
 
 the new variable 
 
 ,.(.-.,) (=BU,.)^/3 -'-4^ 
 
 (11) 
 
 gives (only the greatest terma from (l) for 9(1)) being taken into 
 account) the differential equation 
 
 i9"" + W" = e^ (12) 
 
 from which follows the correction for cpo near the critical layer, as 
 well as the third friction— affected solution cp . The calcvilation of 
 these two solutions merits a little closer study. 
 
8 
 
 NACA TM 1265 
 
 3. The friction solution cp 
 
 3 
 
 The friction solution 9^ is obtained from the differential 
 
 eq.uation (l2) when the inhomogeneous terms encum'bered with the small 
 factor 6 are omitted; hence^ from the differential equation 
 
 
 + TjCp 
 
 3 
 
 = 
 
 Unusual In this equation is that. In contrast to the complete disturbance 
 equation (l) and to the frictlonleas , equation of disturbance^ the 
 dependence of the parameters a^ K, and U' by (U) enters only as 
 scale factor for j, and that it is not at all affected by U^' . As a 
 result J 9 (tj) can be computed once for all entirely independent from the 
 
 velocity profile^ In this Instance, Tletjens' report (reference 9) 
 
 constitutes a valuable support. A fundamental system cp 
 
 and cp 
 
 32 
 
 = F, 
 
 of equation (13) is given by 
 
 31 
 
 = F, 
 
 r,V2 J 
 
 1/3 
 
 2^3/2e-Wl^' 
 
 and T\ 
 
 1/2 
 
 -1/3 
 
 2,3/2 3-lnA 
 
 (li^) 
 
 The expansion In series of the Bessel functions 
 
 V^)=(i 
 
 (iz/2: 
 
 2V 
 
 2/ vTo ^'-T^P + V + 1) 
 
NACA TM 1265 
 
 /^vl/3 1/6 .^- 
 gives, when the constant factors 1—1 (— i) / ^ [ T 
 
 audi — (—1) \^[ ] ^^® omitted 
 
 >> 
 
 F;'(ri) = T, - 
 
 13 
 
 19 
 
 2'.3^«U'7 k'.y-k'^-lO'lZ 6l3^'4-7-10-l3-l6-19 
 
 + — ° 
 
 + i 
 
 \ .^ 
 
 10 
 
 .16 
 
 F2"(ti) = 1 - 
 
 l'.3''"-ii 3133.1+. 7. 10 5l35.l+.7.iO'13.l6 
 .6 
 
 J 
 
 > (15) 
 
 .12 
 
 2'.32'2.5 lil3^.2'5-8.ll 
 
 + i-^ 
 
 15 
 
 l'.3'^-2 3'.3^'2-5.8 5I35.2.5.8.11.14 
 
 — + • 
 
 / 
 
 Owing to the boimdajy condition cpo" = for i^ = + 00 only the solution 
 aggregate P-i^-," + ^r^r^^ approaching zero for great positive real t^ 
 (Pj, Pn ~ integration constants) comes into question. For great t], this 
 can be represented by the Hankel function of the second kind with the 
 
 subscript — and the argument r^ e ^"/^ 
 
 , hence by 
 
 9," =F ''(ti) = 
 
 1/2 ^ (2) 
 
 H, 
 
 1/3 
 
 1.3/2 ^-m/k 
 
10 NACA TM 1265 
 
 Herewith, the looked— for solution of (13) — the constant factor lieing 
 put as cp '(t) ) = cp _' for the sake of simplicity 3 — is given as 
 
 expansion in power series near t) = 
 
 9 
 30 
 
 and as asymptotic expansion for great t| 
 
 Integrating between the limits r\ and tj gives 
 
 ^30 J Tl^ d\ 
 
 — = 3l / / F^"dTldTl + P2 / I* F2"dTldn + T) - T)^ -D (I6c) 
 
 ^30 ti\(/V U\U'\o 
 
 where B and D are additional integration constants dependent 
 
 on T)^; T\^ is the t\ coordinate at the wall, y = 0; hence hy (ll) 
 
 1/3 
 
 ^o = -^k/^ = -yk(°«U^') (lla) 
 
 ■o 'o 
 
 k 
 
 The boundary condition 9^'/cp^^' =1 at y = 0, that is, t) = t| , is 
 fulfilled by adding the term 1 in equation (l6b) » 
 
 •3 
 
 -CnCPon' ^s "'^■'^® gliding speed of the frictionless solution at the 
 wall. 
 
NACA TM 1265 
 
 11 
 
 Tietjens computed the integration constants ^-., Po^ ^^ ^ from 
 the fact that at a point t^ = tj, ^ which lies in the range of validity 
 
 of hoth expansions, asymptotic expansion and power series expansion of 
 
 the solution of (13) up to and including the third differential quotient 
 
 must agree. The resixltant equation system, set up and solved hy Tietjens, 
 reads 
 
 P^F^"' + ^2^2' 
 
 •^l^l" ^ ^2^2" = ^^1^" 
 
 r 
 
 Pl / Fi_"dTi + P 
 
 Al^ 
 
 2 / ^2 
 
 F2"dT) + 1 = BF, ' 
 
 o u O 
 
 (a) 
 
 (b) 
 
 (c) > (IT) 
 
 3l ' / F^"dTidTi + P2 / / Fg-dT^dn + n - Ti^ - D = BFi^ (d) 
 
 The integration constants Pj^, P2, B, and D can also be computed 
 hy a simpler method than Tietjens', with the aid of the transition 
 formula from Hankel's to Bessel's functions, which reads 
 
 H 
 
 1/3 sin ir/3 [^ 1/3 -1/3 
 
 This obviates the Joining of the two expansions, makes (I7) superfluous, 
 and {ijh) gives the exact values immediately: 
 
 !2 = ^-in/6 32/3 ll 
 Pl ^ 3 
 
 r(2 
 
 and ^=ei'^A3l/3r(^|J3i^| (18) 
 
 = -1.190 + 0.687i 
 
 = 0.789(1 + i) 
 
 h 
 Tollmien, who pointed this out to the writer, had this representation 
 
 as far back as I929, but summarily took over the data by Tietjens for the 
 
 sake of simplicity. 
 
12 
 
 NACA TM 1265 
 
 The factors P and D as fimctlon of r\ can then also he indicated 
 explicitly, namely 
 
 ,,.|(.-eW3)3V3,(i)./;o,^„,^A/; 
 
 ^ -1 
 
 ° V^n 
 
 D = -n + p, i 
 
 'o 1 
 
 > 
 
 (19) 
 
 F/'dTldTl + -^ 
 
 Po po m 
 
 1 
 
 lli^oli^o 
 
 F2"dTidTi)- - Bq) (0) 
 
 with 
 
 Bcp (0) = ip^ 
 
 Tahle 2 contains the resiilts of a new calculation of Tiet Jens' value 
 carried out hy these formulas. The differences from Tiet Jens' figures are 
 
 f>Tl f]r\ 
 
 insignificant. The values of F^", Fg", / ^^"d.^ = '^i , / ^2"^^^ = ^2*^ 
 
 / F 'di) = F-,, / Fg'dT) = F2 as function of tj are indicated separateily 
 
 uo uo 
 
 as real and imaginary part In tahle 3« Since, according to (15), these 
 quantities are either symmetrical or antisymmetric functions of r\, this 
 tahle can he continued Immediately according to the negative values of r\. 
 
 For the two neutral oscillations, whose amplitude and energT^ distri— 
 hution is to he computed, it is 
 
 y\^ = -2.63 and tj^ = -4.05 
 
 The corresponding values of e, according to (lla), are given in tahle I. 
 along with the Integration constants P-,, Po, D ohtained hy Interpolation 
 
 for these t^^ values. 
 
 This takes care of all the data necessary for computing the friction 
 solution cp^ with its first and second derivative as function of t] hy 
 the equations (l6a, t, and c) . Tahle h gives the thus ohtained values 
 
NACA TM 1265 13 
 
 ^3 ^3 ^3 
 of : : as function of ti. The connection between ti and y 
 
 ^30 ^30 ^30 
 is, according to eq.uatlon (ll), 
 
 y = yjj. + eTj 
 
 In all cases, the friction solution cp^ 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 cpg in the intermediate layer 
 
 The second frlctionless solution 9^ behaves singularly at the critical 
 layer y = y^, namely through equation (9) as jfriy - y]5-)log(y - J-^) , 
 
 so that cpg' behaves as -— ■< 1 + log(y — yj^-) [• and cpg" behaves 
 
 a V 1 
 
 as . 
 
 Uk' (y - J^) 
 
 From the differential equation (l2), in which only the greatest 
 friction terms are taken into account, follows a solution cp^ modified 
 
 by the friction, which joins the frlctionless solution at some distance 
 from the critical point. For this purpose cpp is expanded In powers 
 
 of the previously Introduced small quantity e = (oRU, ')~-^/3 
 
 ^2 " ^20 + ^^21 "^ ■"• ^^°) 
 
 ^oQ being chosen equal to unity. From (12) follows the inhomogeneous 
 differential equation 
 
 for 9^1 with reference to t). 
 
11^ 
 
 NACA TM 1265 
 
 On accoimt of the very small value of e, tj can assume great values 
 even at small y — Jy values. An attempt is made to find such a 
 
 solution 9o-i " of this equation, which for large t), hut small (y — y^.) 
 joins on to the frictionless solution 
 
 ^\ V 
 
 dy 
 
 \' y - yj. 
 
 For large t^ there shall he: 
 
 d Cp^, TT " 
 
 ^21 ^k 
 
 dy^ 
 
 V^ 
 
 (22) ) 
 
 The corresponding homogeneous equation appeared earlier in the 
 calculation of cp (equation (13)). It has the fundamental system F-,"(r|) 
 
 and F "(t)) (equation (l5))- A particular solution of (2l) is 
 
 -'k 
 
 V,/„'Vin-F,"/^%,"d, 
 
 which can he verified easily hy suhstitution, and the general solution 
 of (21) is 
 
 
 21 
 
 U,. 
 
 IF2" r F^"dT, - iF^" r F^-dTi + c^Fi" + C2F2" I (23) 
 
 The integration constants c and Cp, which can be complex, are 
 evaluated from the boundary conditions. The quantity ^p-i" is complex 
 and shall join the real frictionless value (equation (22)) for large 
 
NACA TM 1265 
 
 values of t). By decomposition in real and imaginazy parts, the four 
 equations defining the integration constants read 
 
 ^21r" = ^"(^) = -^2i"Flr' - ^2r"^ll' ^ Fii-'F^r' + F^-Fg,' 
 
 ■^k 
 
 ■^ ^ir^lr" - ^li^li" - -2r^2r" " ^2i^: 
 
 1 
 
 21 T, 
 F^21i" = S"(^) = - F2l"^li' - ^2r"^lr' - Fi.-F^i' - F^-'F^,' (^U) 
 
 ^ ^ir^li" - -li^lr" ^ ^2r^2i" ^ ^2i^2r" = ° 
 
 for T| = ±TJ^ 
 
 From (21), with the "boundary condition (22), it follows that 922j-" is 
 an antisymmetrical, and ^pil" ^ symmetrical function of t\. Moreover, 
 since F^', F^^', F^^", F^." are symmetrical and F^^', F^^' , F^.", 7^" 
 antisymmetrical functions of tj, the following must be true 
 
 c-Li = C2p = (25a) 
 
 for reasons of symmetry. The other two constants c , Cp. are obtained 
 
 by solving the above equation system for t) = tj, . For the present 
 
 calculation, tj = U was chosen. The series for the Bessel functions 
 
 are still fairly convergent for t] = ii; but since differences of very 
 large numbers occur, the separate terms in i^k) must be computed to 
 five digits (table 3). For c and Cp. 
 
 ci^ = 1.2852; C2i = 0.9373 (25b) 
 
 so that ^^21r" = ^"(^) ^^ "^;;^21i" = ^"(ti) can be calculated. The 
 
 k k 
 
 values are given in table 5. 
 
l6 NACA TM 1265 
 
 The values of cp and - — in the Intermediate layer are obtained 
 immediately by quadratures, namely 
 
 dcp, 
 
 '^2r 
 
 dy 
 
 = ^|(1 + log €) + a'CTl)| (26a) 
 
 and 
 
 
 ^--v"'^>-v,L.'"'^''^-v«''^' <^«'' 
 
 A check on this numerical calculation is given "by the fact that 
 
 dcp2i 
 for — — at transition from large positive to large negative tj the 
 
 transition substitution for cp_ 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 
 
 \ dy /y=+oo \^ dy /y=-~ U^^' 
 the present numerical calculation gives 
 
 'dcp_.\ /d.9oA U, " n+00 
 
 2i> I 2il + JL. / H"(Ti)dTi 
 
 dy /y=+~ \ dy /y=-«> U^^' U-o 
 
 and the graphical integration gives 
 
 / H"(Tl)dTl Z / H"(Tl)dTl = 3.li^ (27) 
 
 J -co U-h 
 
 that is J complete agreement within the scope of mathematical accuracy. 
 
NACA TM 1265 17 
 
 The intermediate layer near the critical point, which "by the 
 present calculation reaches from atout tj = — ij- to t) = +k, is already 
 so wide at the first neutral oscillation that it reaches up to the wall 
 (wall T) = —2.63); at the second oscillation with t) = —14-005, the 
 
 "boundaj:y of the intermediate layer is reached exactly at the wall. With 
 this, all data needed for the numerical calculation of the solution cp_, 
 
 corrected "by the friction with cp ' and 9p"> ^^® available. 
 
 5. The numerical values of the integration constants 
 All three particular solutions cp , cp , cp are numerically known. 
 
 To build up the required solution 9 from it, the numerical values of 
 the integration constants C and C-^ must be accertained (equation (i 
 
 First of all, equation (2) is rewritten in a more suitable form, namely 
 
 cpo 
 9 = cp^ + 0^92 + C^' -^ (2a) 
 
 where equations (3a) and (3^) were resorted to and 9 was replaced by 
 the quantity 9 /9 _' which follows immediately from the numerical 
 calculations. Comparicon with (6) gives 
 
 S' = I-^20' - 'lo' - -f 2'20' * '10') '«-' 
 
 This method of writing has the advantage that the two integration 
 constants Cp and C-,' in (2a) are dependent only on the values of 
 
 the frictionless solutions 9 and 9 , hence are relatively simple 
 to compute . 
 
 The values of 9^^^, 9^^, ^la'^ ''2a'' "^la' ^ 2a ^^ "^^® values 
 of C^ and C^ ' thus computed by (6) and (6a) for both neutral 
 
 oscillations are given in table 1. Table 6 and figures 3 and k give 
 the values of 9^, 9^, 9^', 9^' computed with it, hence the desired 
 
 amplitude distribution as function of y/&. 
 
18 
 
 NACA TM 1265 
 
 Outside of the "boundary layer, at y/& > I.OI5 the simple formula 
 
 'V 
 
 9^=9^' = 
 
 (29) 
 
 is valid for the amplitude distrihution. The constant C* is so chosen 
 that the value of cp ' joins the already found value in y/6 = 1 = 015. 
 (Tahle 1.) 
 
 6. The average fluctuation velocities and the correction factor 
 
 (compare reference 10) 
 Changing to the real method of writing 
 
 M r 
 
 u' = — = K Jcp^' coa(ax - 3^t) - cp^^' sin(ax - P^t) I U^ 
 
 (30) 
 
 V = — 
 
 s^ r 
 
 — = KaJ cp^ sin(aix - p^t) + cp^ cos(ca - P^t) 
 
 U, 
 
 m 
 
 K is a freely available intensity factor. According to figures 3 ajid k, 
 the one phase (cp^, or cpp' ) 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 
 
 u 
 
 ,2 
 
 U 
 
 and 
 
 ^^^•2 
 
 U„ 
 
 ; where the dash 
 
 velocity, the dlmensionless q^uantities 
 
 '"'m "m 
 
 denotes the time average value formation over a period T at a fixed 
 point X, y, or in other words 
 
 ,.2 _ 1 
 
 I'^dt (T = vibration period) 
 
 t=o 
 
NACA TM 1265 
 
 19 
 
 The result is 
 
 \u' 
 
 U 
 
 m 
 
 f,W'*^i"-- 
 
 ,2. C^ 
 
 u 
 
 m 
 
 f2 
 
 \|^i 
 
 CPi 
 
 (31) 
 
 and 
 
 CPl^) 
 
 (32) 
 
 The last quantity gives the mean kinetic ener®r 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/5. The intensity factor itself was so 
 
 to O.O5U ^ 
 m \ o 
 
 in the boundary layer is equal 
 
 dy = 0.05Uj^| (table l) . The majcimum amplitude 
 
 chosen that the average value of 
 
 
 
 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 
 
 k(u', v') = 
 
 u'v'dt = 
 
 r^ 
 
 ,2 ^,2 
 
 k(u', v') = 
 
 cpr'^i - ^r^±' 
 
 \|(9r'^ + cp^'^) (9/ + cpi^) 
 
 (33) 
 
 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 -O.I7 and -O.I9, respectively. It is inter- 
 esting to compare the theoretically established correlation coefficient 
 with Townend's data in a developed turbulent flow (reference ll) . The k 
 values of -O.I6 to -O.18, obtained for the flow in a channel of square 
 
20 - WACA 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 turhulence. 
 
 Chapter II 
 EEERGf DISTRrBUTION 
 7. The kinetic energT" of the disturbance motion. 
 
 Having established the amplitude distribution for the two neutral 
 vibrations J 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, extends over a wave length X and in y direc- 
 tion from the wall to infinity is 
 
 P>X f)oa 
 E = £ / / (u'2 + v'2)dxdy 
 2 (/x=0 iiy=0 
 
 i^v/; 
 
 cpj.'^ + 9i'^ + a2(cp^2 ^ (p^2) 
 
 d(y/6) (34) 
 
 The energy dE of the secondary motion In a strip of width dy 
 and length X is accordingly 
 
 ^ = f|a,V V"-%'=--"(<fr=-0 (35) 
 
 dy 2 2 m 
 
 Besides, 
 
 0.533 I- f = iili^ (36) 
 
 E Is the basic— flow energy in a layer of unit height, length X, and 
 width 6 (compare equation (32)). Figure 7 shows the dimenslonless 
 energy distribution by equations (35) and (36). The energy is strongly 
 concentrated near the critical layer. To obtain the total energy E, 
 the Integral (3^) must be evaluated. Dividing It In two parts with the 
 
 limits ^ y/6 % I.OI5 and I.OI5 "S y/6 < ", the first portion is 
 
WACA TM 1265 21 
 
 obtained by graphicaJ. integration tased on the computed amplitude 
 distribution. The second portion is obtained analytically by (29), 
 namely 
 
 J 
 
 P I „ ,2 , ,, ,2 , P/,, 2,^2 
 
 y/6=1.015 ' 
 
 d(y/6) = a6C*2g-^-03<:^5 
 
 The results of the evaluation are given in table 1. 
 
 Now the energy of the disturbance motion is compared with the basic- 
 flow energy E^ in the space of unit height and surface area X, X 6. 
 
 It is, by equation (7a-) 
 
 / / U(y)dxdy = 0.533 &V2x,5 (37) 
 
 Ux=Oiyy=0 
 
 E = ^ 
 o 2 
 
 Hence, for the ratio of energr of the secondary motion to the energy 
 of the basic flow E/Eq the values presented in table 1 are obtained. 
 
 8. The energy balance of the disturbance motion. 
 
 Consider the time variation of the secondary-ciotion energy of a 
 particle that moves with the basic flow, hence 
 
 r)fe^- 
 
 ^ [P(,.2 , ,.2)1 = (^ , u :^) j§(u'2 -H v.2) ^ (38) 
 
 For stable disturbances, the total change of energr of the secondary- 
 motion is - / / /— (u'2 + v'2) dv < 0, for unstable distiirbances > 0, 
 
 and for neutral disturbances =0, the integration extending over the 
 entire range of the particular flow. Participating on the variation 
 of the secondary-motion 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 
 
22 
 
 MCA TM 1265 
 
 increase per vibration period T = 23t/p is also equal to zero. This 
 is easily confinned in the following manner: It is 
 
 ^ ^(u'2 . V.2) 
 
 2jt=0^* 
 
 dt 
 
 
 2)dt 
 
 |Lu-2,,,2 
 
 + PU 
 1/0 
 
 u' -^^^ + v' I dt = 
 
 ^x Sx 
 
 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 (3O) . 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 
 
 |(u'2 . v-2) 
 
 -pu'v' ^ 
 
 dy 
 
 ^(u'p') ^ a(v'p') 
 
 ^x Sy 
 
 -(s^-rr^^{i<-^')-i<-4 
 
 where 
 
 ^' = — - — , \i = coefficient of viscosity 
 Sx By 
 
NACA TM 1265 23 
 
 The first term gives the transfer of energy from the primary to the 
 secondary flow, the second gives the contribution resixlting 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 = to y = " and with 
 respect to x over a wave length X. of the disturbance, the second and 
 foiorth terms disappear, since u' and v' disappear for y = 
 and y = 00 and with respect to x have the period X. Thus, the 
 growth of the energ/ per unit time in a layer of unit height and base 
 area 0<y<oo, 0<x<X is: 
 
 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 
 
 2 m = ^oK% ^ I (cp -q)^ _ cp 9. • ) 4U ^ _ ^U 2 / |(cp " - a2cp )2 
 
 ^ Dt ^ j^^ '^r ^i ^r^^ ) ^<^ ^ :n j^^Q l^^r ^r^ 
 
 (9/ -a2cp^)2|dy 
 
 or 
 
 t bI- ■ ^'^'''l * "2^ ^^ I "m' («" 
 
 en and eg denoting the dimensionless energr integrals 
 
 , = - (cp^'cp -9r9i') , ,'^ cL(y/b) = / e.'d(y/5) (Ula) 
 Jo d(y/6) do 
 
2k NACA TM 1265 
 
 = - ^ / |(CP/ - a29^)2 + (cp^" - a2(pi)2| d(y/6) = / eg'cKy/b) (kl^) 
 
 eo = 
 
 To find the energy change of the disturbance motion In the Tlbration 
 period T(T = X./c , c^ = phase velocity) , this energy change is referred 
 
 to the total kinetic energy E of the secondary motion which Is given 
 by equation (3^)- From {kO) follows then the specific energy change of 
 the disturbance motion as 
 
 where Z = 0.i^•32 and Z = O.81O for the first and second neutral 
 vibration, respectively, while Uj^/cj, = 2.86 for both neutral vibrations. 
 
 The local energy transfer from primary to secondary motion (l) and 
 the local dissipation (2) for the two neutral vibrations is then 
 
 i^'^h,2=^^-'^%2 °- =^l-^%2' (^3a, b) 
 
 when 
 
 Dt 
 
 The values of e ' (equations i+la, b) can be obtained (table 8) on 
 
 the basis of the computed amplitude distribution for both neutral 
 vibrations. Figure 9 represents the local energy conversion. The 
 dissipation In wall proximity is seen to be extremely great, while the 
 crlticELl layer is of no particular importance for the dissipation. But 
 the energr 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— 
 bion (fig. 8), as anticipated. 
 
 The graphical integration of e-, ' and 62' gives the values 
 indicated in table 1. The energy balance =—7 = 0, or e-, + 02 = 
 
NACA TM. 1265 
 
 25 
 
 for the neutral vitrations is therefore fiilfilled with satisfactory 
 approximation^, and constitutes a very welcome check on the rather 
 complicated solution of the characteristic value problem. 
 
 The total energy transferred in vibration period T from the 
 primary to the secondary motion is 
 
 (AE)^/E = 78. Oe^ or i^l.6e-,_ 
 
 and the total energy dissipated 
 
 (AE)2/E = yS.Oeg or i+1.6e2 
 
 These figures are also shown in table 1. Thus, at the first neutral 
 ■ vibration, about half of the aecondary-inotion 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 table 1, is 
 
 T.^g 
 
 10.1 X 10^ -^; To = (^\ = IU.8 X 10^ -^ 
 
 1 U ^ ^"^^2 
 
 -^ m 
 
 U 
 
 m 
 
 To illustrate; For a plate flow in water with 
 
 V = 0.01 cm^sec "'-; V^ = 20 cm sec "'- 
 
 ^According to the present calculation, the dissipation for both 
 vibrations is somewhat greater than the transfer of energr from the 
 primajry to the secondary flow. This is due to the fact that in the 
 stability calculation only the dissipation of the friction solution cp^ 
 
 was taken into account, while the dissipation of the frictionless vibra- 
 tion (9-1, 9p) was ignored. But, in the energy equation, the dissipation 
 
 of frictionless and frictionaJ. 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 vitratlon are 
 
 T = 2.50 sec; Tg = 3.7O sec 
 
 Thus J vibrations of comparatively great periods are Involved. 
 
 Translated by J. Vanier 
 National Advisory Committee 
 for Aeronautics 
 
IhXCA TA 1265 27 
 
 EEFERENCES 
 
 1. Prandtl, L.: Bemerkiangen uber die Entatehung der Turtulenz. Ztsclir. 
 
 f. angew. Math. u„ Mech. Bd. 1, p. U3I, 1921. 
 
 Uber die Entstehimg der Turbiilenz. Zeitschro fo angew. Math, u, 
 
 Mech. Bd. 11, p. U07, 1931. 
 
 2. Schlichting, H.: Neuere Untersuchungen uber die Turbulenzentstehiuig. 
 
 Die Naturwissennchaften Bd. 22, p. 376, 193^4-. 
 
 3. Tollmien, W.: Uber die Entatehung der Turbulenz. Nachr. d. Ges. d. 
 
 Wiss. zu Gottingen, Math.— Phys. Klasse 1929, p. 21 and Verhdlg. 
 d. III. Intern. Kongr. f. techn. Mech. Stockholm 1930, p. IO5. 
 (Available as NACA TM 609.) 
 
 h. Schlichting, H.: Uber die Stabilitat der Couettestromung. Ann. d. 
 Physlk, V. Folge, Bd. Ik, p. 905, 1932, 
 And Verhdlg. d. Intern. Mathematikerkongresses Zurich 1932, p. 283. 
 
 5. Schlichting, H.: Zur Entstehving der Turbulenz bei der Piatt enstromung. 
 
 Nachr. d. Ges. d. Wiss. zu Gottingen, Math.— Phys. Klasse 1933, 
 p. 181 and Ztschr. f. angew. Math, u, Mech. Bd. 13, p. I7I, 1933. 
 
 6. Schlichting, H.: Uber die Entstehung der Turbulenz in eimen 
 
 rotlerenden Zylinder. Nachr. d. Ges. d. Viss. zu Gottingen, 
 Math.- Phys. Klasse 1932, p. I60. 
 
 7. Schlichting, H. : Turbulenz bei Varmeschichtung. Suramar/ of results 
 
 in report cited under reference 2 in Verhdlg. d. IV. Intern. 
 Kongr. ft. techn. Mechanik Cambridge 193^^ P- 2k^. 
 
 8. Squire, H. B.: Stability for Three-Dimensional Disturbances of 
 
 Viscous Fluid Flow between Parallel Walls. Proc. Roy. Soc. 
 (London), ser. A. Vol. 1^2. 1933, PP- 62I-628. 
 
 9. Tietjens, 0.: Beitrage zum Turbulenzproblem. Diss. Gottingen 1922, 
 
 and Ztschr. d. angew. Math, u. Mech. Bd. 5, p. 200, I925. 
 
 10. Tollmien, W. : Uber die Korrelation der Geschwindigkeitskomponenten in 
 
 periodisch schwankenden Wirbelverteilungen. Zeitsclir. f . angew. 
 Math. u. Mech. Bd. I3, p. 96, 1935- 
 
 11. Townend, H. C. H.: Statistical Measiurements of Turbulence in the 
 
 flow of air through a pipe. Proc. Roy. Soc. A. Vol. li+5, p. 180, 
 193i^. 
 
 12. Lorentz, H. A.: Abhandlungen uber thoretische Physik. Bd. I, v. k3, 
 
 Leipzig, 1907. 
 
28 
 
 NACA TM 1265 
 
 TABLE 1 
 
 THE PARAMETERS OF THE TWO NEUTRAL VIBRATIONS OF THE PLATE FLOW 
 
 
 
 First neutral 
 
 Second neutral 
 
 
 
 vltratlon 
 
 vibration 
 
 
 06 
 
 (U^6/v) 
 
 0.466 
 2.62 X io3 
 
 0»737 
 6.08 X 10^ 
 
 
 Pr^/Uj 
 
 .163 
 
 .258 
 
 
 -rK 
 
 o350 
 
 .350 
 
 
 ^k 
 
 .209 
 
 .209 
 
 
 ^' 
 
 1.625 
 
 1.625 
 
 
 V/Uj,' 
 
 -.494 
 
 -.494 
 
 
 ^ 
 
 -e.63 
 
 -4.05 
 
 
 1/e 
 
 12.6 
 
 r .0695 
 
 L+.102i 
 
 19.4 
 
 r -.0470 
 
 L +.02761 
 
 
 ^2 
 
 / -.1526 
 L-.O736I 
 
 r .0368 
 I -.06501 
 
 
 D 
 
 ri.37i^ 
 
 \^ + .2001 
 
 r .395 
 
 1+1.241 
 
 
 ^la 
 
 .416 
 
 ^ .^35 
 
 
 ^la' 
 
 .040 
 
 .097 
 
 
 ^2a 
 
 .211 
 
 . .306 
 
 
 ^2a' 
 
 -2.425 
 
 -2o240 
 
 
 *la 
 
 .234 
 
 .417 
 
 
 ^2a 
 
 -2.327 
 
 -2.014 
 
 
 ^2 
 
 .101 
 
 .207 
 
 
 ^10; 
 ^20 
 
 1.005 
 
 r-»o46 
 1+1.5631 
 
 loOll 
 
 f -.114 
 1+1.5631 
 
 
 ^3* 
 
 j'-loOO 
 \ -.1571 
 
 r -.988 
 
 "L--325i 
 
 
 c* 
 
 K 
 
 .706 
 
 .1454 
 
 1.075 
 .1166 
 
 r\ 
 
 2 + cp.'? + a2(cp/ + cpi2)|^/6 
 
 •371 
 
 .681 
 
 
 iV 1.015 
 
 .090 
 
 .183 
 
NACA TM 1265 
 
 29 
 
 TABLE 1 
 
 THE PARAMETERS OF THE TWO NEUTRAL VIBEATIONS OF THE PLATE FLOW - Concluded 
 
 
 First neutral vitration 
 
 Second neutral vibration 
 
 £ 
 
 0A32K^ 
 
 O.SlOK^ 
 
 £ 
 
 0.00913 
 
 0.0110 
 
 
 
 e^ X 10^ 
 
 5.75 
 
 6.39 
 
 62 X 103 
 
 -6.16 
 
 -7.10 
 
 (AE)i 
 E 
 
 OAJ+7 
 
 0.265 
 
 (AE)^ 
 E 
 
 -0.ii79 
 
 -0,294 
 
30 
 
 NACA TM 1265 
 
 TABLE 2 
 
 NEWET CALCULATED VALUES OF P-^ AKD D 
 
 -^0 
 
 ^ 
 
 D 
 
 
 
 0.387 + 0.6721 
 
 0.672 - 0.3871 
 
 0.5 
 
 0.3U1 + 0.3661 
 
 0.770 - 0.3801 
 
 1.0 
 
 0.262 + 0.2131 
 
 0.892 - 0.3501 
 
 1.5 
 
 0.192 + 0.1U21 
 
 1.023 - 0.2611 
 
 2.0 
 
 0.132 + 0.1131 
 
 1.202 - 0.135i 
 
 2.5 
 
 0.0822 + 0.10311 
 
 1.358 + O.I2UI 
 
 3.0 
 
 0.0332 + 0.09721 
 
 1.397 + 0.519i 
 
 3.5 
 
 -0.0165 + 0.07821 
 
 1.139 + 1.0231 
 
 ii.O 
 
 -0.01^65 + 0.03231 
 
 0.493 + 1.2lJ+i 
 
NACA TM 1265 
 
 31 
 
 TABLE 3 
 
 THE PARTICULAR SOLUTIONS OF 9 AS FUNCTION OF t) 
 
 Compare equation (I5) 
 
 n 
 
 F 
 
 F 
 
 F 
 
 F 
 
 F ' 
 
 F • 
 
 F • 
 
 F • 
 
 Ir 
 
 11 
 
 2r 
 
 21 
 
 It 
 
 11 
 
 2r 
 
 21 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 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 
 
 0.1+998 
 
 0.0167 
 
 0.9992 
 
 0.01+17 
 
 1.5 
 
 0.563 
 
 0.032 
 
 1.122 
 
 0.063 
 
 1.1186 
 
 0.1261+ 
 
 1.1+865 
 
 0.2065 
 
 2.0 
 
 1.318 
 
 0.177 
 
 1.97^^ 
 
 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 
 
 I.55M+ 
 
 2.63 
 
 2.87 
 
 0.901 
 
 3. 21+ 
 
 1.022 
 
 2.90 
 
 2.02 
 
 1.95 
 
 1.88 
 
 3-0 
 
 3.97 
 
 1.94 
 
 3.86 
 
 1.91 
 
 2.9208 
 
 3.6995 
 
 1.335^^ 
 
 2.9269 
 
 3.5 
 
 5.07 
 
 i^.55 
 
 1^.02 
 
 3.72 
 
 0.9512 
 
 6.8796 
 
 -1.0850 
 
 1+.2113 
 
 k.O 
 
 i^.20 
 
 8.70 
 
 2.35 
 
 5.77 
 
 ^.6379 
 
 9.2I+I+5 
 
 -6.1155 
 
 3.3059 
 
 I1.O5 
 
 3.84 
 
 9.13 
 
 2.00 
 
 5.89 
 
 -6.62 
 
 9.20 
 
 -6.76 
 
 2.90 
 
 
 antisy. 
 
 sy. 
 
 sy- 
 
 antisy. 
 
 37- 
 
 antisy. 
 
 antisy. 
 
 sy. 
 
 n 
 
 F, " 
 Ir 
 
 ^11" 
 
 ^2r" 
 
 ^2i" 
 
 F •" 
 
 Ir 
 
 F •" 
 
 11 
 
 F '" 
 
 ^2r 
 
 F "' 
 
 ^21 
 
 
 
 
 
 
 
 1 
 
 
 
 1 
 
 
 
 
 
 
 
 0.5 
 
 0.5000 
 
 0.0052 
 
 0.9999 
 
 0.0208 
 
 0.9998 
 
 0.0417 
 
 -0.0010 
 
 0.1250 
 
 1.0 
 
 0.9980 
 
 0.0833 
 
 0.9941+ 
 
 0.1666 
 
 0.9861 
 
 0.3331 
 
 -0.0333 
 
 0.4999 
 
 1.5 
 
 1.1+661 
 
 0.1+206 
 
 0.9368 
 
 0.5595 
 
 0.8442 
 
 1.1165 
 
 -0.2525 
 
 1.1072 
 
 2.0 
 
 I.7I+72 
 
 1.3108 
 
 O.6I+69 
 
 1.2939 
 
 0.1187 
 
 2.5541 
 
 -1.0533 
 
 1.8229 
 
 2.5 
 
 1.3100 
 
 3.01+63 
 
 -0.3215 
 
 2.3124 
 
 -2.2818 
 
 4.3761 
 
 -3.0889 
 
 2.0809 
 
 2.63 
 
 0.91+1 
 
 3.61+ 
 
 -0.776 
 
 2.58 
 
 
 
 
 
 3.0 
 
 -1.1161 
 
 5.4735 
 
 -2.7I+29 
 
 3.0209 
 
 -8.1612 
 
 4.7949 
 
 -6.8785 
 
 0.1417 
 
 3.5 
 
 -7. 6311+ 
 
 6.7189 
 
 -7.2934 
 
 1.4618 
 
 -18.5245 
 
 -1.7400 
 
 -11.0222 
 
 -7.8189 
 
 l+.O 
 
 -19.1+902 
 
 0.6805 
 
 -12.5621 
 
 -6.6661 
 
 -27.2703 
 
 -26.6992 
 
 -7.4580 
 
 -26.7961 
 
 I+.05 
 
 -20.91+ 
 
 -0.806 
 
 -12.92 
 
 -8.10 
 
 
 
 
 
 
 Eintisy. 
 
 sy. 
 
 sy. 
 
 antisy. 
 
 sy. 
 
 emtlsy. 
 
 antisy. 
 
 sy. 
 
32 
 
 NACA TM 1265 
 
 TABLE k 
 
 THE FRICTION SOLUTION cp^ ^ FUNCTION 
 
 OF y/6 AND t) 
 First Neutral Vibration 
 
 Tl 
 
 y/6 
 
 V" 
 
 '31" 
 
 V 
 
 ^31' 
 
 ^3r 
 
 ^31 
 
 
 
 ^30' 
 
 ^30' 
 
 ^30' 
 
 ^30' 
 
 ^30' 
 
 ^30' 
 
 -2.63 
 
 
 
 -6.ko 
 
 7.65 
 
 1 
 
 
 
 -0.111 
 
 -0.0159 
 
 -2 
 
 .050 
 
 -5.67 
 
 .79^ 
 
 .665 
 
 .188 
 
 -.069 
 
 -.0099 
 
 -1 
 
 .130 
 
 -3.0i+ 
 
 -1.801 
 
 .329 
 
 .106 
 
 -.029 
 
 .0029 
 
 
 
 .209 
 
 -1.92 
 
 -.932 
 
 .136 
 
 -.012 
 
 -.009 
 
 .0061 
 
 1 
 
 .288 
 
 -.995 
 
 .113 
 
 .019 
 
 -.040 
 
 -.001 
 
 .0035 
 
 2 
 
 .368 
 
 -.202 
 
 .290 
 
 -.025 
 
 -.018 
 
 
 
 .0010 
 
 3 
 
 .kh7 
 
 .063 
 
 .076 
 
 -.027 
 
 -.003 
 
 
 
 .0005 
 
 h 
 
 .577 
 
 -.076 
 
 -.012 
 
 -.020 
 
 .012 
 
 
 
 .0006 
 
 Second Neutral Vibration 
 
 --1+.05 
 
 
 
 -17.7 
 
 31^ 
 
 1 
 
 
 
 -0.0204 
 
 -0.0639 
 
 -3.5 
 
 .029 
 
 -17.6 
 
 8.22 
 
 .460 
 
 .519 
 
 .0001 
 
 -.055^+ 
 
 -3 
 
 .05^^ 
 
 -9.n 
 
 -3.08 
 
 .106 
 
 .534 
 
 .0070 
 
 -.0415 
 
 -2.63 
 
 .071+ 
 
 -^.91 
 
 -1+.67 
 
 -.030 
 
 .i^55 
 
 .0076 
 
 -.0322 
 
 -2 
 
 .105 
 
 -.272 
 
 -3.88 
 
 -.103 
 
 .309 
 
 .0051 
 
 -.0202 
 
 -1 
 
 .157 
 
 1.378 
 
 -2 
 
 -.058 
 
 .166 
 
 .0005 
 
 -.0091 
 
 
 
 .209 
 
 .717 
 
 -1.260 
 
 
 
 .084 
 
 -.0008 
 
 -.0035 
 
 1 
 
 .261 
 
 -.019 
 
 -.679 
 
 .016 
 
 .035 
 
 -.0002 
 
 -.0012 
 
 2 
 
 .313 
 
 -.194 
 
 -.155 
 
 .007 
 
 .013 
 
 
 
 
 
 3 
 
 .361 
 
 -.058 
 
 .019 
 
 
 
 .012 
 
 
 
 
 
 1^ 
 
 .I1I5 
 
 .039 
 
 
 
 -.005 
 
 .014 
 
 
 
 
 
NACA TM 1265 
 
 33 
 
 TABLE 5 
 
 THE FRICTION COEEECTIOK qp IN THE 
 INTERMEDIATE LATER AS FUNCTION OF t\ 
 
 T| 
 
 G"(ti) 
 
 H"(ti) 
 
 G'(t)) 
 
 H'(tj) 
 
 
 
 
 
 1.285 
 
 -O.77I1 
 
 -1.570 
 
 .5 
 
 .kh3 
 
 1.165 
 
 -.659 
 
 -.953 
 
 1 
 
 .lk6 
 
 .857 
 
 -.35^ 
 
 -.1+1+8 
 
 1.5 
 
 .839 
 
 .i+83 
 
 .051 
 
 -.118 
 
 2 
 
 .767 
 
 .160 
 
 .1+58 
 
 .035 
 
 2.5 
 
 .589 
 
 -.018 
 
 .798 
 
 .073 
 
 3 
 
 .i^38 
 
 -.072 
 
 1.056 
 
 .01+8 
 
 3.5 
 
 .325 
 
 -.055 
 
 I.2J+3 
 
 .013 
 
 1^ 
 
 .250 
 
 
 
 1.386 
 
 
 
 G"(^) = -0"(t,); H"(^) = H"(ti) 
 G'(ti) = G'(t)); H'(-^) = -n -H'(ti) 
 
3h 
 
 NACA TM 1265 
 
 TABLE 6 
 
 DISTRIBUTION OF AMPLITUDES 9 , cp^, cp^' , cp^', cp ", cp." 
 
 AS FUNCTION OF y/6 
 First Neutral Vltratlon 
 
 y/5 
 
 ^r 
 
 9i 
 
 ^r' 
 
 ^l' 
 
 
 ^i" 
 
 
 
 
 
 
 
 
 
 
 
 7.885 
 
 -6.615 
 
 .050 
 
 .011 
 
 -.OOUO 
 
 .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 
 
 -.OOJ+1 
 
 .975 
 
 .058 
 
 -.699 
 
 -.576 
 
 .370 
 
 .276 
 
 -.0010 
 
 .831 
 
 .018 
 
 -1.543 
 
 -.347 
 
 .1+51 
 
 .335 
 
 
 
 .670 
 
 .005 
 
 -1.778 
 
 -.015 
 
 .531 
 
 .380 
 
 
 
 .510 
 
 -.003 
 
 -1.641 
 
 -.016 
 
 .612 
 
 .klk 
 
 
 
 .361 
 
 
 
 -1.606 
 
 
 
 .693 
 
 MQ 
 
 
 
 .237 
 
 
 
 -1.514 
 
 
 
 .77^ 
 
 .k^3 
 
 
 
 .118 
 
 
 
 -1.431 
 
 
 
 .Q^k 
 
 .1+58 
 
 
 
 .007 
 
 
 
 -1.364 
 
 
 
 .935 
 
 .i^55 
 
 
 
 -.101 
 
 
 
 -1.303 
 
 
 
 1.015 
 
 .kk^ 
 
 
 
 -.205 
 
 
 
 -1.252 
 
 
 
NACA TM 1265 
 
 35 
 
 TABLE 6 
 
 DISTRIBUnON OF AMPLITUDES cp , cp^, cp^' , cp^', cp^", qp^" 
 AS FUNCTION OF y/6 - Concluded 
 
 
 
 Second 
 
 Neutral Vibration 
 
 
 
 y/6 
 
 ^r 
 
 CPi 
 
 ^r" 
 
 ^i" 
 
 V' 
 
 ^i" 
 
 
 
 
 
 
 
 
 
 
 
 28.54 
 
 -27.89 
 
 .029 
 
 .010 
 
 -.0038 
 
 .720 
 
 -.337 
 
 20.27 
 
 -^.30 
 
 .051+ 
 
 .033 
 
 -.OlU 
 
 1.101+ 
 
 -.237 
 
 9.16 
 
 6.35 
 
 .071+ 
 
 •057 
 
 -.0140 
 
 1.21+5 
 
 -.113 
 
 4.25 
 
 6.28 
 
 .105 
 
 .096 
 
 -.011^ 
 
 1.333 
 
 .052 
 
 .478 
 
 3.60 
 
 .157 
 
 .168 
 
 -.0086 
 
 1.350 
 
 .129 
 
 .558 
 
 -.27 
 
 .209 
 
 .236 
 
 -.OOi+1 
 
 1.306 
 
 .076 
 
 -1.218 
 
 -1.51^ 
 
 .250 
 
 .287 
 
 .0005 
 
 1.203 
 
 .Olli 
 
 ^.967 
 
 -1.31 
 
 .290 
 
 .333 
 
 .0002 
 
 1.091 
 
 -.016 
 
 -2.961 
 
 -.33i^ 
 
 .370 
 
 .1+09 
 
 
 
 .827 
 
 -.016 
 
 -2.86 
 
 .144 
 
 A51 
 
 .1+69 
 
 
 
 .607 
 
 
 
 -2.58 
 
 
 
 .531 
 
 .507 
 
 
 
 .i+09 
 
 
 
 -2.21 
 
 
 
 .612 
 
 .532 
 
 
 
 .21+7 
 
 
 
 -1.93 
 
 
 
 .693 
 
 .5^+7 
 
 
 
 .105 
 
 
 
 -1.72 
 
 
 
 .77^ 
 
 .550 
 
 
 
 -.026 
 
 
 
 -1.56 
 
 
 
 .851+ 
 
 .^hk 
 
 
 
 -.148 
 
 
 
 -1.44 
 
 
 
 .935 
 
 .527 
 
 
 
 -.260 
 
 
 
 -1.32 
 
 
 
 1.015 
 
 .508 
 
 
 
 -.367 
 
 
 
 -1.23 
 
 
 
36 
 
 NACA TM 1265 
 
 TABLE 7 
 
 THE MEAK FLUCTUATION VELOCITIES 
 
 P", 
 
 \|v'^, THE KIHETIC 
 
 ENERGY OF THE DISTXKBANCE MOTION u' ^ + v'^, AND THE 
 COEEIELATION COEFFICIENT k AS FUNCTION 
 OF j/b. [EQUATIONS (3I), (32), (33)] 
 
 First Neutral Vibration 
 
 
 
 
 
 
 y/& 
 
 10 - 
 
 u'2 
 
 , 
 
 102^ 
 
 y'2 
 
 Urn 
 
 io2-'' V' 
 
 m 
 
 -k 
 
 
 
 
 
 
 
 
 
 
 
 .050 
 
 .1+1+5 
 
 .051+6 
 
 .198 
 
 .032 
 
 .090 
 
 .61+7 
 
 .158 
 
 .1+20 
 
 .061 
 
 .130 
 
 .801+ 
 
 .287 
 
 .61+8 
 
 .155 
 
 .170 
 
 .922 
 
 .1+31+ 
 
 .851+ 
 
 .169 
 
 .i.09 
 
 1.005 
 
 .608 
 
 1.019 
 
 .11+0 
 
 .250 
 
 1.016 
 
 .795 
 
 l.oi+o 
 
 .109 
 
 .290 
 
 1.003 
 
 .973 
 
 1.019 
 
 .061+ 
 
 .370 
 
 .851+ 
 
 1.322 
 
 .71+8 
 
 .022 
 
 .1+51 
 
 .689 
 
 1.605 
 
 .501 
 
 .0075 
 
 .531 
 
 .521+ 
 
 1.820 
 
 .309 
 
 
 
 .612 
 
 .371 
 
 1.981+ 
 
 .177 
 
 
 
 .693 
 
 .21+1+ 
 
 2.098 
 
 .101+ 
 
 
 
 .771+ 
 
 .121 
 
 2.170 
 
 .062 
 
 
 
 .851+ 
 
 .007 
 
 2.193 
 
 .01+8 
 
 
 
 .935 
 
 .103 
 
 2.180 
 
 .059 
 
 . 
 
 1.015 
 
 .211 
 
 2.108 
 
 .090 
 
 
 
 1.1 
 
 .203 
 
 2.027 
 
 .082 
 
 
 
 1.2 
 
 .193 
 
 1.935 
 
 .075 
 
 
 
 1.3 
 
 .185 
 
 1.81+9 
 
 .068 
 
 
 
 l.U 
 
 .176 
 
 1.763 
 
 .062 
 
 
 
 1.5 
 
 .168 
 
 1.681 
 
 .057 
 
 
 
NACA TM 1265 
 
 37 
 
 TABLE 7 
 
 THE MEAN FLUCTUATING VELOCITLES \ 
 
 u>2, {^, 
 
 THE KINETIC ENERGY OF THE DISTUKBANCE MOTION u'^ + v'^, 
 AND THE CORRELATION COEFFICIENT k AS FUNCTION OF y/5. 
 EQUATIONS (31), (32), (33) - Concluded 
 
 
 Second Neutral Vitratlon 
 
 
 y/6 
 
 \iu'2 
 10 ^ 
 
 .0= V' 
 
 
 -^ 
 
 
 m 
 
 m 
 
 
 
 
 
 
 
 
 
 
 
 
 
 .029 
 
 .661 
 
 .0655 
 
 .1+33 
 
 
 
 .054 
 
 .937 
 
 .211+ 
 
 .869 
 
 .121 
 
 .071+ 
 
 1.038 
 
 .359 
 
 1.068 
 
 .170 
 
 .105 
 
 1.108 
 
 .595 
 
 1.218 
 
 .191 
 
 .157 
 
 1.127 
 
 1.028 
 
 1.262 
 
 .11+6 
 
 .209 
 
 1.086 
 
 l.i+U5 
 
 1.190 
 
 .076 
 
 .250 
 
 .998 
 
 1.755 
 
 1.019 
 
 .010 
 
 .290 
 
 .906 
 
 2.04 
 
 .854 
 
 -.015 
 
 .370 
 
 .686 
 
 2.50 
 
 .528 
 
 -.019 
 
 .1^51 
 
 .^Ok 
 
 2.87 
 
 .332 
 
 
 
 .531 
 
 .3^0 
 
 3.10 
 
 .208 
 
 
 
 .612 
 
 .205 
 
 3.26 
 
 .11+6 
 
 
 
 .693 
 
 .087 
 
 3.35 
 
 .118 
 
 
 
 .77^ 
 
 .022 
 
 3.37 
 
 .117 
 
 
 
 .85U 
 
 .123 
 
 3.33 
 
 .121+ 
 
 
 
 .935 
 
 .216 
 
 3.22 
 
 .11+9 
 
 
 
 1.015 
 
 .305 
 
 . 3.11 
 
 .187 
 
 
 
 1.1 
 
 .292 
 
 2.925 
 
 .171 
 
 
 
 1.2 
 
 .272 
 
 2.72 
 
 .11+6 
 
 
 
 1.3 
 
 .252 
 
 2.53 
 
 .126 
 
 
 
 l.U 
 
 .23i+ 
 
 2.35 
 
 .109 
 
 
 
 1.5 
 
 .218 
 
 2.18 
 
 .093 
 
 
 
38 
 
 NACA TM 1265 
 
 TABLE 8 
 
 THE LOCAL ENERGY" CONVERSION l) = TRANSFER FROM 
 ERIMART TO SECONDARY MOTION, 2) = DISSIPATION 
 EQUATIONS (J+l) AND (I+3) 
 First Neutral Vibration 
 
 y/& 
 
 e-^' X lo3 
 
 e^' X 103 
 
 E d.j '1 
 
 E .^^^)2 
 
 
 
 
 
 86.9 
 
 
 
 ^.78 
 
 .050 
 
 .268 
 
 31.8 
 
 .021 
 
 -2.1+8 
 
 .090 
 
 2.150 
 
 17.6 
 
 .168 
 
 -1.37 
 
 .130 
 
 12.25 
 
 10.8 
 
 .955 
 
 -.81+ 
 
 .170 
 
 23.25 
 
 5.73 
 
 1.81)+ 
 
 -.i^5 
 
 .209 
 
 28 
 
 2.6I1 
 
 2.18 
 
 -.21 
 
 .250 
 
 27.30 
 
 .06 
 
 2.13 
 
 
 
 .290 
 
 22.85 
 
 .72 
 
 1.78 
 
 -.05 
 
 .370 
 
 7.^9 
 
 2.20 
 
 .581+ 
 
 -.17 
 
 .I151 
 
 1.885 
 
 2.80 
 
 .li+7 
 
 -.22 
 
 .531 
 
 -1.102 
 
 2.1+3 
 
 -.086 
 
 -.19 
 
 .612 
 
 
 
 2.36 
 
 
 
 -.18 
 
 .693 
 
 
 
 2.12 
 
 
 
 -.17 
 
 .77*+ 
 
 
 
 1.92 
 
 
 
 -.15 
 
 .851+ 
 
 
 
 1.75 
 
 
 
 -.11+ 
 
 .935 
 
 
 
 1.61 
 
 
 
 -.13 
 
 1.015 
 
 
 
 1.50 
 
 
 
 -.12 
 
NACA TM 1265 
 
 39 
 
 TABLE 8 
 
 THE LOCAL ENERGY CONVERSION l) = TRANSFER FROM 
 ERJMARY TO SECONDARY MOTION, 2) = DISSIPATION 
 [equations (Ul) A1\1D (43)] - Concluded 
 Second Neutral Vibration 
 
 y/& 
 
 e^' X 10^ 
 
 e^' X lo3 
 
 l¥<^>i 
 
 l^(-). 
 
 
 
 
 
 320 
 
 
 
 -13.31 
 
 .029 
 
 1 
 
 83.6 
 
 -.044 
 
 -3.48 
 
 .054 
 
 7. 84 
 
 24.3 
 
 .326 
 
 -1.01 
 
 .otV 
 
 18.1+5 
 
 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 
 
 
 
 1.80 
 
 
 
 -.075 
 
 .531 
 
 
 
 1.38 
 
 
 
 -.057 
 
 .612 
 
 
 
 1.01 
 
 
 
 -.042 
 
 •693 
 
 
 
 .819 
 
 
 
 -.034 
 
 .774 
 
 
 
 .695 
 
 
 
 -.029 
 
 .851+ 
 
 
 
 .608 
 
 
 
 -.025 
 
 .935 
 
 
 
 .520 
 
 
 
 -.022 
 
 1.015 
 
 
 
 .458 
 
 
 
 -.019 
 
ko 
 
 NACA TM 1265 
 
 Figure 1.- Laminar flow past the plate. 
 
 0.3 
 
 1 
 
 0.2 
 
 0.1 
 
 
 r 
 
 
 ._ 
 
 E 
 
 Sfobk 
 
 ' 
 
 
 
 \ 
 
 
 
 
 
 
 
 
 
 
 U, 
 
 nsfable 
 
 
 
 
 
 
 1 
 
 ■^ 
 
 ^ 
 
 
 
 
 
 
 1 S^a/>/e 
 ^Theoretical 
 
 stabili 
 
 fy limii 
 
 
 
 
 6 
 
 
 
 
 
 
 
 
 Um^* ^W^^ 
 
 Figure 2.- The zone of the stable and unstable disturbances of plate flow. 
 I = first neutral vibration, n - second neutral vibration. 
 
NACA TM 1265 
 
 kl 
 
 
 
 
 1 
 
 1 
 
 
 
 
 ^ 
 
 
 
 
 txi 
 
 
 'Ifr 
 
 
 
 fr 
 
 / 
 
 + — ■ 
 
 1 
 1 
 
 \ 
 
 X 
 
 ^' 
 
 
 
 
 
 y,-' 
 
 ao2 
 
 t/1 
 
 r 
 
 1 
 1 
 
 1 > 
 
 / 
 
 \ 
 
 N 
 
 
 
 
 
 - Q2 
 
 
 y 
 
 / 
 
 
 
 X 
 
 <. 
 
 
 
 
 
 '. ^ 
 
 ^/ 
 
 1 
 
 ^ 
 
 
 
 
 
 X 
 
 
 s 
 
 
 ' 
 
 
 
 r- 
 
 2 
 
 ^ 
 
 X4 
 
 1 
 
 £ 
 
 0.8 ^ 
 
 ^^ 
 
 'OOP 
 
 0.2- 
 
 
 1 
 
 
 
 
 
 
 
 
 --0.7 
 
 1.0 
 I 
 
 0.5 
 
 -0.6 
 
 Figures.- Real and imaginary part <Pp, cpj, cp_', 9^' of the amplitude of 
 
 disturbance motion plotted against wall distance for the first neutral 
 vibration. 
 
 o.iz 
 
 ■0.04W.2 
 
 -0.4 '1.0 
 
 Figure 4.- Real and imaginary part 9p,9;,9_',^j^' of the amplitude of 
 
 disturbance motion plotted against wall distance for the second neutral 
 vibration. 
 
h2 
 
 YikCk TM 1265 
 
 0.125 
 
 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1-6 
 
 Figure 5.- The mean fluctuating velocity in the x direction yu* y^-ax 
 plotted against the wall distance for both neutral vibrations. 
 
 0.0* 
 0.03 
 0.02 
 0.01 
 
 
 r - ~^ 
 
 1 
 
 1 
 
 ,^ 
 
 
 
 
 ■^ 
 
 
 
 
 ;/ 
 
 4 
 
 
 
 
 
 ^^ 
 
 ^ 
 
 ^/ 
 
 ^ 
 
 
 
 
 
 
 
 P 
 
 1/ 
 1 
 
 
 
 
 
 — ► 
 
 s 
 
 0.2 
 
 1.6 
 
 Figure 6.- The mean fluctuating velocity in the y direction yv ^/U 
 plotted against the wall distance for both neutral vibrations. 
 
 {^\x. 
 
 m 
 
NACA TM 1265 
 
 h3 
 
 0/2 
 
 r\ 
 
 
 
 
 
 
 
 r ■- 
 
 
 r 
 
 \ 
 
 
 
 
 
 
 
 0.08 
 
 / 
 
 \\ 
 
 
 
 
 
 
 
 / 
 
 l\\ 
 
 k 
 
 
 
 
 
 
 0.04 
 
 / \ 
 
 \ 
 
 
 
 
 
 
 / 
 
 ^ 
 
 
 
 
 
 
 l 
 
 / 
 
 >v 
 
 ^ 
 
 
 '^ ' 
 
 
 
 
 
 
 
 
 
 0.4t 
 
 0.8 
 
 ,.2 -_ ^ ,6 
 
 Figure 7.- The mean kinetic energy of the disturbance motion u'*^ + v''^/u 
 plotted against the wall distance for both neutral vibrations. 
 
 m 
 
 0.20 
 
 0.15 
 
 0.10 
 
 0.06 
 
 '005 
 
 
 
 1 
 1 
 
 
 
 
 
 \ ^\ 
 
 V 
 
 1 \ 
 
 
 
 
 
 
 1 \ 
 
 
 
 
 
 
 i\ 
 
 V 
 
 
 y 
 
 ^^^ 
 
 / 
 
 0.1 OS \0^ 
 
 3 
 
 t-"c 
 
 5 
 
 
 
 1 
 
 
 Figure 8,- The correlation coefficient k - 
 
 ^u'2.v'2 
 for both neutral vibrations. 
 
 plotted against y/e 
 
kh 
 
 NACA TM 1265 
 
 15.0 
 
 10.0 
 
 5.0 
 
 -2.5 
 
 iim.t 
 
 r" " ' 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 
 
 \ll(2) 
 
 
 
 
 
 
 
 
 
 
 
 \| 
 
 
 
 
 
 
 
 
 
 
 
 V 
 
 
 
 
 
 
 
 
 
 
 
 
 1(2) 
 
 
 
 
 
 
 
 - 
 
 
 y 
 
 
 >^ 
 
 ^) 
 
 
 0.6 
 
 0.8 
 1 
 
 LO 
 
 Figure 9.- The local energy conversion of the secondary motion for the 
 first and second neutral vibration. I (1), n (1) = energy transfer from 
 primary to secondary flow; I (2), U (2) = dissipation. 
 
 NACA-Langley - 4-18-50 -950 
 
•p 
 u 
 cd 
 U 
 +^ 
 m 
 
 4J ^ XJ 
 
 S 
 
UNIVERSITY OF FLORIDA 
 
 3 1262 08105 023 8 
 
 ^,^,,„ ..•FLORIDA 
 
 BoOUMiNTS DEPARTMENT 
 iaO N/.ARSTON SCIENCE LIBRARY 
 P.O. BOX 117011 
 GAlNiSVILLE, FL 32611-7011 USA