B.}^ 
 
 ^ 
 
 NATIONAL ADVISORY COMMITTEE 
 FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 1291 
 
 ON STABILITY AND TURBULENCE OF FLUID FLOWS 
 By Werner Heisenberg 
 
 Translation of "ijber Stabilitat und Turbulenz von Fliissigkeitsstromen." 
 Annalen der Physik, Band 74, No. 15, 1924 
 
 Washington -.-jv r^r- r-. 
 
 -'JYQF FLORIDA 
 June 1951 ^^^S DEPARTMENT 
 
 GA,NESV.UE.FL3Jei,-.o„usA 
 
NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 1291 
 
 ON STABILITY AND TURBULENCE OF FLUID FLOWS* 
 
 By Werner Heisenberg 
 
 INTRODUCTION 
 
 The t-urbulence problem, which will quite generally form the subject 
 of the following investigations, has been treated in the course of time 
 in so many reports from so many different viewpoints that it is not oiir 
 intention to give, as an introduction, a survey of the results obtained 
 so far. For that purpose, we refer the reader to a report by Noether-'- 
 on the present state of the turbulence problem, where most biblio- 
 graphical data may be found as well. 
 
 For our purpose, a rough outline of the present state of the tur- 
 bulence problem will be sufficient. The investigations made so far are 
 divided into two parts; one part deals with the stability investiga- 
 tion of any laminar motion, the other with the turbulent motion itself. 
 
 The first-mentioned investigations led, at the beginning, to the 
 negative result that all laminar motions investigated are stable. 
 V. Mises2 and L. Hopf3 proved, on the basis of a formula by Sommerfeld^, 
 the stability of the linear velocity profile corresponding to Couette's 
 arrangement. Blumenthal5 reached the same result for a profile of the 
 third degree, upon which Noether invited discussion. On the other hand, 
 Noether" later succeeded in specifying an unstable profile - a profile 
 which is unstable even in the case of a frictionless fluid can never be 
 realized as a steady state of motion for actual conditions. More 
 
 *"Uber Stabilitat und Tiirbulenz von Fllissigkeitsstromen. " Annalen 
 der Physik, Band 7^, No. 15, 192U, pp. 577-627- 
 
 -"-Noether, F. : Zeitschr. f. angew. Math. u. Mech. 1, 1921, p. 125. 
 %ises, R. v.: Beitrag z. Oszillationsprobl. : Heinr. Weber- 
 Festchrift, 1912, p. 252. 
 
 ^Hopf, L.: Ann. d. Phys. kk, 19lk, p. 1. 
 
 ^Sommerfeld, A.: Atti d. IV. congr. int. dei Mathem. Rom 1909- 
 •^Blumenthal, 0.: Sitziongsber. d. bayr. Akad. d. Wiss., 1913^ P- 563. 
 %oether, F. : Nachr. d. Ges. d. Wiss., Gottingen 1917. 
 
2 NACA TM 1291 
 
 recently, however, Prandtl' has shown that indeed profiles exist which 
 possess unstable characteristics only if the friction is taken into 
 consideration. 
 
 The other group of reports which achieved great success quite 
 recently by the calculations"^ of Von Karman, Latzko, and others 
 investigates the turbulent motion itself proceeding by a semiempirical 
 method using the laws of similarity. Theoretically, the reports of 
 this group are based almost throughout on Prandtl' s boundary-layer 
 theory. Their most important result for our purpose is the so-called 
 y-*-/ '-law of turbulent velocity distribution which follows from Blasius' s 
 law of resistance (examples can be found in Schiller's report^. ) 
 
 The determination of the critical Reynolds nimiber was always one 
 of the main aims of the first-mentioned reports, the stability inves- 
 tigations. So far, a satisfactory calculation of this number has not 
 been accomplished and it must be regarded as doubtful whether it could 
 be achieved by stability investigations. The tests of EkmanlO, RuckesH 
 and Schiller-^, together with the negative results of Hopf concerning 
 the linear velocity profile, rather suggest the notion that the critical 
 Reynolds number does not indicate the point where the laminar motion 
 starts to become unstable, but the point where, for the first time, the 
 turbulent motion is possible as steady state. From the viewpoint of 
 theory, we must thus be prepared to find eventually two critical 
 Reynolds numbers, one corresponding to the beginning of turbulence, the 
 other to the breaking down of the laminar motion. 
 
 The present investigation also will be divided into two different 
 parts, the treatment of the stability problem on the one hand, that of 
 the turbulent motion on the other. 
 
 Tprandtl, L. : Physik. Zeitschr. 23, 1922, p. 19, and Tietjens, 0., 
 Dissert. Gottingen, 1922. 
 
 °See Zeitschr. f. angew. Math. u. Mech. , 1, 1921, p. 233f. 
 
 9schiller, L. : Rauhigkeit und'kritische Zahl. Physik. Zeitschr. 3^ 
 1920, p. 4l2. 
 
 -'-'^Ekman, V.: Turbulent Motions of Liquids. Arch. f. Mat. och 
 fysik 6, 1919. p. 12. 
 
 llRuckes, W. : Dissert. ¥urzburg 190?. See also Lect\ire by W. Wien, 
 Uber turbul. Bewegungen. Phys, Zeitschr. 8., 190^^ and Verh. d. deutsch. 
 phys. Gesellsch. 9, 190? . 
 
NACA TM 1291 
 
 The aim of the first part is to summarize all previous investiga- 
 tions under a unified point of view, that is, to set up as generally as 
 possible the conditions under which a profile possesses unstable or 
 stable characteristics, and to indicate the methods for solution of the 
 stability equation for any arbitrary velocity profile and for calcula- 
 tion of the critical Reynolds number for unstable profiles. This aim 
 ca,n, of course, be attained only imperfectly by the use of approxima- 
 tion methods. Nevertheless, we hope to be able to clarify by such 
 calculations the qualitatively essential viewpoints. At first, the 
 investigation of any arbitrary profile seems physically meaningless 
 since only certain profiles actually occur; however, since we may 
 interpret any profile as finite disturbance of another, as for instance 
 Noether has done elsewhere, and since we must, on the other hand, later 
 extend the investigations to the (at first unknown) basic profile of the 
 turbulent motion, the investigation of an arbitrary profile seems, after 
 all, to be of great importance. 
 
 As application of the methods, the parabola profile will be cal- 
 culated completely. 
 
 In the second part, we shall attempt to derive, under certain 
 greatly idealizing assumptions, differential equations for the tiorbulent 
 motions and to obtain from them qualitative information about several 
 properties of the turbulent velocity distribution. 
 
 PART I: THE STABILITY EQUATION 
 1. Statement of the Mathematical Problem 
 
 The most essential limitation we impose on our calculations con- 
 sists in the exclusive consideration of two-dimensional laminar motions 
 and only two-dimensional disturbances of these motions. Taking a rec- 
 tangular coordinate system X, Y, Z as basis, we therefore assume that 
 the velocity in the Z direction is zero and all remaining quantities 
 independent of Z. Furthermore, however, we shall only examine the sta- 
 bility of such laminar motions as occur between two straight parallel 
 walls. We assume the walls to be parallel to the X axisj therefore, 
 the laminar motion to be investigated also promises a, velocity com- 
 ponent only in the X direction. This velocity w in the X direction 
 will, in some way, be dependent on y. Concerning the function 
 w = w(y) we reserve for later making a few assumptions about continuity, 
 symmetry, etc.; otherwise, however, this function is to be at first 
 quite arbitrary. 
 
NACA TM 1291 
 
 If we put w = ay, our formulations become exactly identical with 
 those investigated by Hopf in the Couette case. The problem whether 
 the investigated profiles w = w(y) can be realized as steady motions 
 will not be dealt with for the present. 
 
 Before deriving once more the stability equation (already set up 
 elsewhere by Sommerfeld) briefly from Stokes's differential equations, 
 we introduce dimensionless variables in the known manner. Let h be 
 a characteristic length (for instance the distance between the two 
 walls), U a characteristic velocity of the profile, \i the viscosity, 
 
 Uhp 
 
 p the density, and 
 
 = R the Reynolds number; we introduce instead 
 
 of X, y, u, V, t, and p (u, v being the velocity in x or 
 y direction, respectively, t the time, and p the pressure) new 
 
 variables x 
 
 0' 
 
 Jr 
 
 u^ 
 
 /"q, t , and Pq, according to the relations 
 
 X y u "v,,!! Li /^K 
 
 ""0 " h' ^0 " h^ "^ ^ ^' ""0 = u^ " h^ Po = P — (1) 
 
 _h_ 
 
 If the index is subsequently omitted, Stokes's equations read 
 
 Su Su 
 
 — + u — + V 
 
 St Sx 
 
 3u ^ 1/ _^ _^ S£u _^ S£u\ 
 
 Sy R\ ax ^ Sx2 ^ Sy2/ 
 
 Sv ^ ^ 
 — + u + V 
 
 dt Bx Sy 
 
 K- 
 
 Sp d^v S'-v' 
 
 Sy Sx^ ^ By^j 
 
 Since we presuppose incompressibility, we write 
 
 (2) 
 
 u = 
 
 St 
 
 V = 
 
 S\tf 
 
 Sx 
 
 (3) 
 
 As is well known, we obtain by the elimination of p 
 
 
 w 
 
NACA TM 1291 
 
 By A one understands here the differentiation symbol 
 
 + 
 
 Sx2 Sy2 
 
 Equation {k) does not yet contain anything about our special 
 problem^ the stability investigation of a certain laminar flow. Accord- 
 ingly, equation (h) will form also the basis for the calculations of 
 part II. In order to pass over specifically to the stability investi- 
 gation, we divide the motion and therewith also the vector potential i|r 
 into a basic flow and small oscillations superimposed over it. Thus we 
 set up the formula 
 
 t= $(y) + q)(y)e^(P*-^) (5) 
 
 2- = v{y) = w (6) 
 
 oy 
 
 If we enter this formula into equation (4), omitting all terms not con- 
 taining 9 (since we regard equation {W) as satisfied for cp = O), 
 furthermore omitting all terms quadratic in cp (since we assume cp as 
 small), the corresponding differential equation for cp reads 
 
 (cp" - a2cp)fw - i") - cpw" = ^ (cp" " - 2a2cp" + a\) (7) 
 
 \ a/ oR 
 
 The fact that we regard equation {h) as satisfied for qp = signifies 
 physically that we consider only such basic flows w which either, by 
 virtue of external forces, are really steady, or show a variation with 
 time which is slow compared to that of the small oscillations. 
 
 Equation (7) is in this generality already derived elsewhere by 
 Noether. It is an ordinary differential equation for cp of the fourth 
 order. It corresponds to the fact that the function cp must fulfill 
 four boundary conditions; u a,nd v, thus also cp and cp' , must dis- 
 appear at the two walls. If we put p/a = c so that c essentially 
 signifies the wave velocity, the mathematical problem may be formulated 
 as follows: The solutions of the equation 
 
NACA TM 1291 
 
 (9" - <x^(p){v - c) - cpv" = — (cp"" - 2aV + cc\) (Ta) 
 
 aR 
 
 axe to be investigated with the secondary condition that at the bounding 
 walls (for instance, y = 1 and y = -l) cp = and cp' = 0. For each 
 value of a and R the corresponding value of c and p is to be 
 calculated; let a for rea,sons of simplicity always be positive. 
 According to whether the ima,ginary part of 3 is positive, zero, or 
 negative, we are dealing with a stable, undamped (undamped oscilla- 
 tions = neutrally stable oscillations), or unstable oscillation. The 
 conditions for profile w are to be found imder which equation (7a) 
 admits only stable oscillations or, respectively, also unstable ones. 
 
 Before turning to the methods of solution we want to point out a 
 special property of the equation (7a). In the limit of the frictionless 
 fluid, R = 00, equation (7a) is transformed into a differential equa- 
 tion of the second order for cp 
 
 1 
 1 
 1 
 
 (cp" - a2q))(w - c) - cpw" = (8) 
 
 Accordingly, only two boundary conditions must now be satisfied which 
 signify that the normal velocity component, thus v or cp but no 
 longer cp' , is to disappear at the two walls. 
 
 The conditions for the solvability of equation (8) have already 
 been investigated in detail by Rayleigh.-'-^ Introducing a simple desig- 
 nation, one may distinguish basic flows "capable of oscillation" or 
 "not capable of oscillation" according to whether or not equation (8) 
 possesses a solution with real c which satisfies the boundary con- 
 ditions. -^ If solutions with complex c exist, the stability problem 
 for these oscillations has, as will be shown later, already been 
 decided by equation (8), also in case of consideration of the friction; 
 the oscillations are then always unstahle. 
 
 One is, however, beyond this led to the conjecture that the pro- 
 file w, under influence of friction, permits unstable or undamped 
 
 ^^Lord Rayleigh, Papers I., p. 36I; III. pp. 575, 59^; IV. p. 203. 
 
 13 
 
 Here, however, it is by no means siifficient to approximate the 
 
 profile by tangents polygons; the result with respect to possible 
 
 oscillations would thereby be completely falsified. 
 
NACA TM 1291 7 
 
 oscillations only in one case: when it belongs to the "basic flows 
 capable of oscillation. 
 
 This supposition is the more obvious as it ha.s been confirmed for 
 all profiles investigated so far.l^ Nevertheless, it is by no means 
 motivated by the fact that equation (8) results from equation (7a) in 
 the limit R = 0° since it has been proved, for instance, in the reports 
 of Oseen-^5 that the limiting process R = 00 has led more than once to 
 false results in the differential equations, particularly with respect 
 to the boundary conditions of the frictionless fluid, and that one may 
 therefore apply the limiting process only to the integrals of equa- 
 tion (7a). Moreover, it can by no means be decided beforehand whether 
 the friction modifies the undamped oscillations of equation (8) in the 
 sense of a damping or an excitation. 
 
 Following, we shall attempt to prove our surmise mentioned above by 
 showing that the systems capable of oscillation are shown to possess 
 above a certain value of the Reynolds number and in general unstable 
 character, whereas all systems not capable of oscillation are shown to 
 possess a, stable character. 
 
 By this principle the problem of the stability of a profile is 
 quite considerably simplified since, as is well known, the solutions of 
 equation (8) may be directly written down for very small values of a. 
 
 2. The Methods of Solution and the General Behavior of 
 
 the Integrals of Equation (7a) 
 
 The most important property of equation (7a) which permits an 
 approximate representation of its solutions consists in the fact that 
 R ma.y be regarded as very large. It will become evident that if a 
 stability limit exists, this limit lies^ in general, at very high values 
 of R. Since it is, moreover, physically quite improbable that for 
 small values of R instability of the respective profile could occur, 
 it is sufficient for our next purpose to regard R as very large. 
 
 This assumption makes it possible to approxima,te the solutions of 
 equation (7a) by development in negative powers of R or, as will be 
 
 shown, of SJaR. Furthermore, we shall assume a as small and shall 
 develop the solutions in a given case in positive powers of a^. 
 
 -'•Compare also Pra.ndtl, Physikalische Zeitschrift. 
 
 -'compare, for instance, C. W. Oseen, Beitrage z, Hydr. Annalen 
 der Physik k6, pp. 231 and 623, 1915. 
 
8 NACA TM 1291 
 
 Both methods of development in (oR)"-'-/^ and or, respectively, 
 seem contradictory insofar as in the first case oR is assumed large, 
 in the second case a^ small; however, the contradiction is eliminated 
 "by the fact that R ir;3.y be regarded, in general, as extraordinarily 
 large so that for instance for R = 2000, a = l/lO, aR becomes eqiml 
 to 200, a? - 1/100 which is fully sufficient for a satisfactory con- 
 vergence of the two developments. However, the convergence properties 
 of these approximation methods must be considered more exactly. The 
 
 investigation shows that the series in (oR)" / are generally diver- 
 gent, yet show the well-known characteristics of the semiconvergent 
 series, that is, that the terms first decrease, then again increase, 
 and that one obtains the optimum a,pproximation if one breaks off the 
 series with the smallest term. Our approximation method has, there- 
 fore, convergence properties similar to those of the series of the 
 perturbation theory used in astronomy, the behavior of which is 
 described in detail by Poincare', Meth. nouv. d. 1. mec. eel, II. 
 
 The use of the semiconvergent developments is rendered con- 
 siderably difficult by the fact that they lose their validity in the 
 neighborhood of a certain point so that it ca.nnot be immediately 
 decided in wha,t manner the approximate solutions on both sides of the 
 point must be joined in order to approximate a certain integral of the 
 equation (7a) on both sides. This question will be discussed in detail 
 in section 3- 
 
 The development in positive power series of a^ seems, in general, 
 to be actually convergent. For special profiles this development may 
 be strictly proved (for instance, for the linear profile); however, we 
 have not carried out an investigation of the problem under what con- 
 ditions for the profile w this convergence actually occurs. 
 
 We start with the derivation of the a,pproximate solutions of 
 equation (7a) 
 
 oR^ 
 
 (9" - a2q))(w - c) - w"q) = — (cp"" - 2<x^^' + a\) {la', 
 
 For this purpose we first put 
 
 cp = e , g = \pRgn + Si + T=" go + • • • (9) 
 
NACA TM 1291 
 
 We shall limit the development to the two highest terms in ^aR. There 
 follows 
 
 aRgO^(w - c) + ^(go' + 2gQgJ(w - c) = ioRgQ^ + i JS^R (Ug^^g^ + egQ^gQ*) 
 
 a and w" are presupposed to be of the order of ma,gnitude 1 or, at 
 a,ny rate, « WaR. By mea,ns of simple calculation there now results 
 
 5 So' r„ ^„ _ 5 
 
 gQ=|-i(7T^^ g^=_2^^ |g^ dy = -| log gQ (10) 
 Thus we obtain two particular integrals of the equation (Ta) 
 
 ± 
 
 ^ ^-iaR(w-c) 
 
 9l^2 = ^^ - c)-5/^^^0 (11) 
 
 The point y^ is to be determined by w being = c for y = yQ. 
 Thus yQ may, under certain conditions, be complex. The sign of the 
 root is to be chosen so that for 
 
 w - c = -ae^ -iaR(w - c) = oRae ^ 2' 
 
 the root becomes 
 
 (aEa)l/2e ^2 k) 
 
 A remarkable fact about these two integrals is that a^ in cp 
 does not appear in this approximation (that is, only in the combina- 
 tion ciR which, a,s follows from eq\aation (7a), could in a certain 
 sense, be called the true Reynolds nimiber). 
 
 As we sha,ll see later, the two integrals (ll) determine the 
 behavior of cp in the boundary layer, and the nonoccurrence of a^ 
 in equation (ll) signifies physically that we consider only oscillations 
 
10 
 
 NACA TM 1291 
 
 the wave length of which is large compared to the boundary-layer thick- 
 ness^ which is certainly the case for the empirically observed unstable 
 oscillations. 
 
 For a complete system of solution of equation (7a), however, we 
 need two fixrther integrals; naturally, we shall choose those which 
 result from the integrals of equation (8) by the development of 
 
 (oR)" in a power series. 
 
 For this purpose we first solve equation (8) by the development 
 of a^ in a power series. Thus we put 
 
 (cp" - a2q))(w - c) - cpw" = 
 
 (8) 
 
 cp = 9(0) + a2cp(l) + aV^) + 
 
 (12) 
 
 Hence follows 
 
 cp(0)"(v _ c) - q)(0)w" = 
 
 CP^^)"( 
 
 w - c) - cp 
 
 (l)v" = cd(0)(w 
 
 9 
 
 (2)"(v - c) - cp(2)v" = q)(l)(w - c) 
 
 By the method of variation of the constants the two integrals 
 
 ^3(R=«>) 
 
 = (w - c) 1 + 
 
 a 
 
 dy 
 (w - c)' 
 
 dy(w - c)2 + 
 
 ) 
 
 ^4(R=oo) 
 
 = (v - c) 
 
 dy 
 (w - c)^ 
 
 1 + 
 
 >(13) 
 
 a2 / dy(w 
 
 dy 
 
 (w - c)^ 
 
 + . . 
 
MCA TM 1291 
 
 11 
 
 result. In addition^ these integrals have now to be corrected "by- 
 quantities of the order (oR)"^ . . . etc., if they are to satisfy 
 equation (7a). 
 
 Without writing the corresponding series development down in 
 detail;, we give as result cp with the quantities of the order (aR)-l 
 
 ,3=(w-c)|l..^ J^-^Jdy(w 
 
 ^ r ^y ^ (w 
 
 cxRJ (w - c)^ dy3 
 
 - c) + 
 
 \2. k 
 
 c) + a . . . + 
 
 cpi, = (v - c) — ^i^ 11 + ^^ <iy(^ - '^y 
 
 (w - c)2 
 
 a 
 
 i d3 , . P 
 
 -— — - (w - c) / 
 
 ^ dy2 J (w 
 
 <iy 
 
 dy 
 
 (v - c)^ 
 
 (iJ+) 
 
 With equations (ll) and (lU), a complete solution system of the equa- 
 tion (7a) has been found approximately. 
 
 Before applying this system of solution for satisfying. the boun- 
 dary conditions, it will be useful to clarify the physical significance 
 of the four integrals cp, , cp , cp^, and cp,, and to anticipate a few 
 
 results which we cannot establish until later. 
 
 The integrals cp-, , cpp are very rapidly variable for the high 
 values of R which are of interest to us, as can be seen from the 
 exponent of the order \/clR. If, therefore, for instance cp-|_ is at 
 
 one wall of the order of magnitude 1, it will va.nish exponentially at 
 some distance from the wall. (in itself, it also could become extraor- 
 dinarily large; however, this is naturally prevented by the boundary 
 conditions.) Consequently, cp is composed, except for the immediate 
 neighborhood of the wall, of cp^ and cpij. alone, that is, is 
 very similar to the behavior of cp in the frictionless fluid. 
 
12 NACA TM 1291 
 
 The fact that a? does not explicitly occur in cp]_^ ^, (com- 
 pare equation (ll))^ but does appear in cpo, qp|, (compare equation l4) ) 
 must evidently be interpreted physically to signify that^ if a2 is 
 
 assiomed to be not very large (a^ < ^oR, compare equations (7a) and (9))^ 
 the wave length may be considered as infinitely large compared to the 
 boundary- layer thickness, but not compared to the width of the channel. 
 The characteristic difference between the "boundary- layer integrals" 
 ^1^ 92-' '^'^ °^^ hand, and the integrals corresponding to the friction- 
 less fluid cp^^ 94^ °^ "the other, is therefore significantly expressed 
 in the occurrence or nonoccurrence of a^. 
 
 Concerning the convergence of the development in power series 
 of a?, we may hope that for values of o? of the order of magnitude 1 
 it is still amply siifficient to enable a good approximation, since for 
 a linear profile the series for cpo^ qp|, become series of the type of 
 
 power development of cos a which in the neighborhood a = 1 still 
 converges very rapidly. 
 
 The flow pattern to be expected after all these conclusions corre- 
 sponds to the formulations made in Prandtl' s boundary- layer theory. 
 Except for the immediate neighborhood of the walls, the motion obeys 
 very nearly the differential equation of the frictionless fluids. To 
 the walls themselves, however, adheres a boundary layer the thickness 
 
 of which is of the order of magnitude (aR)~ / . In this boundary 
 layer the velocity u decreases toward the wall rapidly toward zero 
 whereas v is almost zero even outside of the boundary layer. 
 
 3- The Connecting Substitutions 
 
 If we want to study the course of the integrals of equation (7a) 
 from one bounding wall to the other, we must take into account that, 
 at a, point y = y^ in the channel w - c = (or that at least the 
 
 the real part of w - c is zero), therefore the wave velocity there 
 
 agrees with the velocity of the basic flow. At this point, the 
 
 approximation formulas (ll) and ( 1^) for the integrals of equation (7a) 
 cease to be valid. 
 
 Thus it is necessary to know the connecting substitutions for cp]^, 
 cpp, cpT, and cp, which have to be applied for the transition from 
 
 Re(w - c) > to Re(w - c) < 0. For this purpose, we develop w 
 
NACA TM 1291 13 
 
 and cp in the neighborhood of the critical point Jq in the power 
 series cf (aR)~ /-^ a,nd put therefore 
 
 y - y^ = Ti(aR)-V3 
 
 Furthermore we assume the imaginary part of y^ to be of smaller 
 
 order of magnitude than (aR)~ ' ■^. If it is of higher order of magni- 
 tude, the connecting substititions are self evident because then 
 nowhere in the entire range of real y does a "critical point" appear.- 
 If the imaginary part is of the same order of magnitude, the behavior 
 of cp and w may be easily interpolated from the two limiting cases 
 just mentioned. At first we may even put 
 
 Im(yo) = 
 
 since cp m our case 
 
 Im(yQ) « (clR)-V3 
 
 may be developed in power series of IIIl(y„^ and at first only the 
 behavior of cp for Im^yQ'\ =0 is needed. 
 
 Thus we now put 
 
 cp = cPq + ecp^ + e^cpg + 
 
 e = (clR)-i/3 
 
 2 2 
 w - c = eaT) + e brj + 
 
ih 
 
 NACA TM 1291 
 
 Then there results from equation (Ta) 
 
 + ecpj_"" + 
 
 ..^,„2 
 
 = -i[(^o" ^ ^^1" ^ • ■ ■)^'^ -^ e^o"^!! 
 
 2ecpo'b] + 
 
 Thus in first a.pprox imat ion 
 
 q^o 
 
 
 9o V 
 
 (15) 
 
 in second approximation 
 
 'Pi"" = -i[^i"^^- + ^o"^"^^ " ^V] 
 
 (15a) 
 
 For the integrals cp-i, cpb^ equation (ll), we infer from equa- 
 tion (15) that they beha,ve in the critical range (t) order of magni- 
 tude 1) like the integrals found by Hopf for the linear profile, that 
 is, like certain cylindrical functions. Thus, we may conclude at 
 this point that the connecting substitutions for cp-, , cpp from equa,- 
 
 tion (11) except for quantities of the order (aR)-l/3 must be the 
 same as for 4;he linear profile 
 
 "Pi — >• <?! + icpg 
 q?i - 1^2 — > 9i 
 
 (16) 
 
 corresponding to a transition of 
 
 Re(w - c) < > Re(w - c) > 
 
NACA TM 1291 15 
 
 However, for the study of the integrals cpo, cpi, in the neighbor- 
 hood of Y - Yq = 0, the simple calculations ina,de so far are not suf- 
 ficient since for the latter the approximate solution (l5) would 
 read cp" = Oj however, we know from equation (l4) that, in the limit 
 R — >«r, cpK ' , in general, becomes logarithmically infinite at the 
 
 point y - yQ = 0. Equations (15) and (l5a) are therefore in this- 
 form unsuitable for expressing this singularity. 
 
 Instead, we now set a = and w'" =0 (that is, we break off 
 the development of w with the second term); otherwise, however, 
 integrate equation (7a) exactly at first. In doing so, we notice that 
 cp = w - c must be a particular integral of this simplified equa- 
 tion (Ta) and we make, therefore, for cp the statement familiar from 
 the theory of linear differential equations 
 
 cp = (w - c) / \lf dy 
 
 /' 
 
 Then there follows from 
 
 cp"" = -iaR(cp"(w - c) - w"cp) 
 
 for 
 
 9 = (w - c) r ^ dy 
 
 i|f"'(w - c) + Ww' + 6i|f'w" = -iaE(2w'(w - c)t + (w - c)^^') 
 which after repeated integration becomes 
 
 \|/"(w - c) + 3itf'w' + 3W = -iaR((w - c)2i|f - c) (--T) 
 
16 MCA TM 1291 
 
 C is an integration constant. If one now again introduces 
 
 I 
 
 n = (y - yo)(aR)l/3, e = (aR)-l/3^ 
 
 w - c = ear] + e 'br)'^, ^ = -^ + ^-^f + 
 
 there results 
 
 (iTa) 
 
 V'a^ + 3to'a = -i(a2Ti2^Q _ c) 
 
 " » " 2 ^ • ^ 
 
 i|f-[^ ai] + 3\|;-|^ a + iITq bT) + b^lfQ bi) + b^Qh = 
 
 -i^2ab\|rQTi + a^T]'^^^ 
 
 Of course, these differential equations still contain all solu- 
 tions of equation (7a). We intend to study particularly cp]^ f qp^ 
 
 shows for a = at the point y = Jq regular behavior); therefore, 
 
 we select the one solution of equation (iTa) which behaves at some 
 
 distance from y^, thus for large (w - c)af{, like 
 
 (v - c) 
 
 2' 
 
 since 
 
 we know from equation (l^) that cp|, at some distance from y„ is 
 
 given by 
 
 (w - c) 
 
 r dy 
 J (v - c)^ 
 
 Thus we obtain according to equation (iTa) 
 
 ° " a^^ 
 
 and 
 
 . /2b 2 ^ 
 
 il'l n + 3\ = -i(— Cti + ari yi/-^j 
 
 (18) 
 
MCA TM 1291 
 
 IT 
 
 f-, is again fully determined "by the fact that it is to behave "at 
 
 2b C 
 
 infinity" like 
 
 aSri 
 
 We now ask for the transformation substitutions for cp^ (and cpo) 
 
 meaning thereby the following: In the asymptotically valid repre- 
 sentations for cpo, cpi (equation (l^)), we always find the integral 
 
 — ;r- which loses its sense if it is to be extended beyond the 
 
 (w - c)2 
 
 point y = yQ (w - c = 0). Actually^ is^ near the critical 
 
 (w - c)2 
 
 point, replaced by the function ■!. Thus the behavior of t (partic- 
 ularly ^-,) in the critical neighborhood is the solely decisive factor. 
 
 If this behavior and therein the magnitude of the integral / ij/ dy 
 
 (extended beyond the critical point) be known, this knowledge is 
 equivalent to knowing the transformation substitutions for cpo, cp^,. 
 
 The solution ^-. characterized by equation (18) and the boundary 
 condition at infinity reads: 
 
 tI'i(ti) = 
 
 
 H, 
 
 2/3 
 
 (1) 
 
 H2/3^2)t^2^^ 
 
 ^/3 
 
 (2) I 22/3(1)^2^11 
 
 (19) 
 
 Therein Hankel' s cylinder functions of the index 2/3 and the 
 argument -(-iaQTl^3/2 appear ^ag = a-^'^). The sign of (-iaQT))-'' 
 
 / v3/2 re("l)/2 
 is to be taken so that |-ia_Ti) ' becomes positive for t\ = • 
 
 ^ ' / ttQ 
 
 A closer investigation of equation (19) shows that ^ behaves in the 
 entire upper semlplane, and partially, even in the lower one, namely, 
 
 for -n = rei^ within the limits -— < I < -^ at infinity like 
 ' 6 6 
 
18 
 
 NACA TM 1291 
 
 2bC 
 
 '-, if a^ or a is positive. If a is negative^ the upper and 
 lower semiplane are interchanged 
 
 a-'\\ 
 
 lim \i/-]_^rei^ ) = — is valid, if 
 
 r — >oo s T 
 
 2"bC 
 37 
 
 Hence we infer the important result: 
 
 or 
 
 5ijr ISiti 
 
 a < 
 in Titi 
 
 a > 
 
 (20) 
 
 ^1^1 ci"ri 
 
 2t)C 
 
 ijt 
 
 2bC 
 
 in 
 
 a > 
 
 a < 
 
 (21) 
 
 Thus the transformation substitutions for cpo, cp^, accurate up to the 
 
 magnitudes of the order (clR)~ ' , are now found for finite values 
 of a also; we now know - and that is sufficient - what, according 
 
 to equation (21), the integral 
 to w - c > 0, signifies. 
 
 ,w - c 
 
 ^2 
 
 , extended from w - c < 
 
 The formulas (I6) also may be derived once more from equa.tion (l7)j 
 to the asymptotic solutions (ll) of equation (7a) correspond the 
 integrals 
 
 H, 
 
 2/3 
 
 (i),(2; 
 
 j(-i-o^)^/^] 
 
 (19a: 
 
 of the homogeneous equation (18) (C = 0). The problem of finding the 
 transformation substitutions of the "asymptotic" integrals (ll) and (l4) 
 is therewith completed with the required accuracy ^except for quantities 
 
 of the order {oR)-^!\ 
 
MCA TM 1291 19 
 
 h. Fulfillment of the Boundary Conditions and the 
 
 Stability of the Oscillations Corresponding 
 
 to the Solution System I 
 
 Our considerations so far have been quite independent of the type 
 of profile except for a few limitations concerning the singular points 
 which had to be imposed on the profile. In order not to lose our- 
 selves in an excessive number of different possibilities^ we shall 
 further specialize the character of the basic flow. The considera- 
 tions, however, have much more general validity. We thus assume that 
 the bounding walls are represented by the equations y = +1 and 
 y = -1, that, furthermore, the wall y = 1 possesses, with respect to 
 the other, a relative velocity in the positive X direction (of the 
 magnitude w(+l) - w(-l)) and that the laminar flow adheres to the 
 walls (which corresponds to Couette' s test arrangement); finally, we 
 assume that in the range -1 < y < +1, that is, in the fluid, Re(w - c) 
 once and only once is zero. Moreover, we shall presuppose in the 
 entire region continuity for w and the derivatives of w and, beyond 
 this, make the additional assumption that the functions w, w' , w" , 
 etc., always are of normal magnitude, that is, that they do not, for 
 
 1/2 
 instance, at certa,in points, assume a magnitude of the order (clR) 
 
 Furthermore, for the following calculations, we at first regard 
 a as so small and aR as so large that we may put with sufficient 
 accuracy 
 
 93 = w 
 
 (l^a) 
 
 The fixing of the lower limit of the integral in cp^ obviously does 
 
 not signify a limitation of the generality of our solutions. Rather 
 we determine thereby cp. as that linear combination of cp and 9 
 
 which disappears a,t the point y = -1. In case w - c should dis- 
 appear there also, cp. obviously is replaced by the function 
 
 cp = w - c which now for y = -1 becomes zero. 
 
20 
 
 NACA TM 1291 
 
 In order to satisfy the boundary conditions first at the vail 
 y = -1, we form two aggregates f , f from cp , cp , cp , cpr for 
 
 which really cp = cp' =0 for y = -1 
 
 q>4 + 
 
 go(-l)[w(-l) - c] - f w'(-l) [^ 
 
 f 2 = ^4 
 
 gQ(-l)[;.(-l) - c] + |w'(-l) L' 
 
 > (22) 
 
 Therein we understand from now on by g the root y -iaR(w - c), not 
 
 as in equation (lO)^ M-i(w - c)^ in order to save writing down the 
 factor ^oR. 
 
 In order to satisfy the boundary conditions at the other wall as 
 well, one must attempt to determine two constants A and B so that 
 
 Af^(+1) + Bf2(+1) = 
 
 Af^'(+1) + Bf^'C+l) = 
 
 The condition for the possibility of such a determination is 
 
 f^(+l) fgC+l) 
 f^'(+l) f '(+1) 
 
 (23; 
 
WACA TM 1291 
 
 21 
 
 By this condition c or ^, respectively^ is determined if R and a 
 are given. Thus it is now a question of solving equation (23) for c 
 and of determining the sign of the imaginary part of p. Equation (23) 
 forms the perfect analogue to Sommerf eld' s turbulence equation for the 
 linear profile. 
 
 From equation (16) we infer 
 
 '-\ 
 
 f;L(+l) = CPi^(+l) + 
 
 fl'(+l) 
 
 go(-l)[w(-l) - c] - |W(-1) 
 
 cp]_(+l) + icp2(+l) 
 9i(-l) 
 
 = <(+l) + 
 
 93(4-1) 
 w(-l) - c 
 
 /-l)[w(-l) - cj - |w'(-l) 
 
 CP3'(+1) 
 
 w -1 
 
 cpi'(+l) + icF^'(+l) 
 
 ^^(-l) 
 
 > i2k) 
 
 f2(+l) = qpi^(+l) - 
 
 q:^(+l) 
 
 )(-l)[;(-l) - £1 + |w'(-l) 
 
 CP3( 
 
 +1 
 
 w(-l) - 
 
 cPgC-l) 
 
 f2'(+l) = cpi^'(+l) - 
 
 CP2'(+1) 
 Cp2(-1) 
 
 gQ(-l)[.(-l) - c] + |w'(-l) L' 
 
 cp3'(+l) 
 K-1) - c 
 
22 
 
 NACA TM 1291 
 
 We insert these values of f-. 
 
 into equation (23] 
 
 •1' ^2' ^1 ' ^2 
 
 after having made an estimate of the magnitude of the individual terms 
 in order to eliminate unnecessary complications of the calculation by 
 writing down unessential terms. For this purpose we note that there 
 
 or 
 
 will be^ in general^ either cpp( + l) « cpp(-l 
 
 This is caused by the factor \|aR in the exponent of 
 equation (ll) if there does not exist the equality 
 
 92^+1' 
 
 9i 
 
 » 
 
 CP2 in 
 
 -i(w - c)dy = Re 
 
 c)dy 
 
 which we exclude. 
 
 Which one of the two cases will occur cannot be decided before- 
 hand; generally, both are possible and yield both solutions. In the 
 case of an obliquely symmetric profile, one case gives the solutions 
 symmetrical to that of the other. At any rate, the two possibilities 
 behave principally quite analogously and it is therefore sufficient 
 to investigate one of the two. Thus we assume 
 
 92^+1) « 92(-l' 
 
 that is (compare page 9)? the point w = c is to lie nearer to 
 w = w(+l) than to w = w(-l). 
 
 Hence it follows that 
 
 9i(' 
 
 cp2(-l' 
 
 is extraordinarily small, thus 
 
 9i(-l] 
 
 is very large. Thus there 
 
 remain in fj_ and f]_' only the terms which have ^^(-l) in the 
 denominator; in f2 and f2' the terms containing ^^ ^^^ eliminated. 
 
NACA TM 1291 
 
 23 
 
 From equation (23) we thus obtain 
 
 rq)-,_(+l) + icp2(+l) 
 
 1 - c 
 
 [go(-i)B- 
 
 rq)-j_'(+l) + icpg'C+l)] 
 
 L(-i)[^(-i) -3+1 ^'(-1)] f (-1) - {] 
 
 (25: 
 
 Even in this form the equa,tion for c is still rather complicated. 
 We therefore further simplify equation (25) hy cancelling now not only 
 
 quantities of the order of magnitude e N , but also quantities of the 
 order (aR)"^/^, 
 
 For this purpose we determine that g„(+l) is of the order 
 
 1/2 
 (oR) ' , thus at first excluding the possibility of w(+l) - c being 
 
 very small;, and that furthermore 
 
 P^'(+1) + 1^2' (+1) = - I ^—4- [9i( + l) + i92(+l^ 
 
 gQ(+l) rcpi(+l) - i92( + l^ 
 
 Thus we retain only those terms of equation (25) which are multiplied 
 by the factor gQ(+l). 
 
2k 
 
 MCA TM 1291 
 
 Thus the simple result is found 
 
 U-lC+I) - icp2(+l2J'Pl^.( + l) = 
 
 or 
 
 Py=+1 
 
 2/ \/-iaR(w-c)dy 
 
 -1 
 
 Pi 
 
 dy 
 
 ■1 
 
 (w - c)' 
 
 (26) 
 
 This equation possesses two completely different solution systems 
 
 py=+l 
 
 yo 
 
 ^ 
 
 i(xR(w-c)dy 
 
 = 1 
 
 (I) 
 
 <iy 
 
 (w - c)' 
 
 (II) 
 
 System I represents the perfect ana,logue to the solutions Hopf 
 obtained for the linea.r profile and has discussed in detail elsewhere, 
 section k. Actually it is shown that the oscillations corresponding 
 to system I always are of stable character. From 
 
 2N 
 
 Uy^ 
 
 2 1 V -icLR(w-c)dy 
 ^^0 
 
 = 1 
 
 follows 
 
 2 ^-iaR(w - c)dy = «i(| + 2n\ 
 
 Jyo 
 
 (27) 
 
NACA TM 1291 
 
 25 
 
 where n is a positive (compare page 9) integer. It is easily seen 
 that this eq\iation can only be satisfied when ac = P possesses a 
 positive imaginary part. Thus the oscillations characterized "by equa- 
 tion (27) are actually damped, the amount of the damping being of the 
 order of magnitude w(+l) - c, and therefore by no means need be small. 
 
 5. The Solution System II and the Conditions for 
 
 Instability of a Profile 
 
 The solutions in the system II are identical with the solutions 
 of the Rayleigh equation (8) and satisfy the condition 
 
 dy 
 
 .w - c, 
 
 = 
 
 (23) 
 
 or (compare the remark to equation (l^), page 19) quite generally 
 
 for 
 
 and 
 
 ^y dy 
 
 cpi^ = (w - c) I ^ = 
 
 (w - c)^ 
 
 y = +1 
 
 y = -1 
 
 The latter form differs from the first in certain exceptional cases 
 which will be discussed later; moreover, equation (28) represents, of 
 coiirse, only a first approximation (a = 0). For the solutions of 
 equation (28) one must now distinguish four different possibilities: 
 Either, (l) equation (28) has solutions with complex c, then the 
 profile is always unstable since the conjugate complex value of c 
 also always represents a solution; (2) there exist solutions of equa- 
 tion (28) with real c. Then we designate, as Prandtl did elsewhere, 
 the profile as "capable of oscillations." This can, according to 
 equation (2l), occur only if, at the point (w = c), w" = 0, if, 
 therefore, the profile either possesses a point of inflection or is 
 
26 
 
 NACA TM 1291 
 
 composed of linear pieces; (3) real values of c 
 at least the real part of 
 
 exist which make 
 
 <iy 
 
 (w 
 
 zero; or, finally, (U) if none of these three cases occur, equation (28) 
 has no solution. In cases (3) and {h), we call the profile "not 
 capable of oscillations." We contend that case 1 always results in 
 instability, cases (3) and (4) always in stability, case (2) generally 
 in instability of the profile taken as a basis. For cases (l) and (4), 
 this has already been proved above. In case (3); we put c = c-p + icj^ 
 
 with Cj, signifying that real value of c for which the real part of 
 
 dy 
 
 (w - 
 
 disappears. Then we know from section 3 tha.t for c-j^ ^ 0, the imaginary 
 
 part of the integral becomes 
 
 2b 
 Ia3| 
 
 ni, for Cj^ » 
 
 (oB) 
 
 -1/3 
 
 , however, 
 
 2b 
 
 .3| 
 
 rti. Thus, for reasons of continuity (compare section 3)? a 
 
 point c-j^ > must exist where the imaginary part of the integral (28) 
 
 also disappears. The four solutions of equation (28) thus character- 
 ized yield therefore a quantity c with a positive imaginary part, 
 thus stable oscillations. 
 
 Case (2) finally requires somewhat more detailed calculations. 
 Before performing them we note that to case (2) pertain two types of 
 solution for equation (23) which cannot be represented in the form 
 
 <iy 
 
 (w - c)' 
 
 If w(+l) = w(-l) a solution of equation (28) is cp = w - w(+l); in 
 fact, here cp = for y = +1 and y = -1. Fiirthermore , it 
 
NACA TM 1291 27 
 
 may happen that, for instance, w becomes infinite for w = w(+l). 
 Then 
 
 cp = [w - w(+l)J 
 
 ^y 
 
 <iy 
 
 f_l [w - w( + l)] 
 
 also is a solution of equation (3) which satisfies the boundary con- 
 ditions. We shall not treat this case here in more detail since it 
 will be discussed more thoroughly in part II; however, it must be 
 noted as essential that the difference between cases (2) and (3) is 
 very large and that it is, for instance, by no means sufficient to 
 approximate, according to Rayleigh, a curved profile by a polygon. 
 
 For an illustration of this difference 
 
 .nj(c)| = 
 
 Re J(c) = Re 
 
 <iy 
 
 (w - c) 
 
 2 
 
 is represented qioalitatively as a function of c in figure 1 where 
 the solid curve corresponds to the curved profile, the dashed curves 
 to the one consisting of linear pieces. One sees that every break 
 causes a new root Re(j) = because J at the point c = ■w-j^v.ga.k "^°'^ 
 
 the broken profile varies like . This corresponds to 
 
 ^ - ^break 
 Rayleigh' s well-known theorem that there are as many oscillation roots 
 as breaks. Nevertheless, the ciorved profile does not possess an oscil- 
 lation root. After this comment, we revert to our contention that the 
 profiles capable of oscillation generally become unstable if the fric- 
 tion is taken into consideration. 
 
 For a proof of this instability, we return to eq-uation (25) and 
 
 to the more exact solutions in system II. Since we know that c is 
 
 -I/2 
 real, except for q\:iantities of the order of magnitude (aR)~ ' , we 
 
 may assume 
 
 P2(+l) »cp^(+] 
 
23 NACA TM 1291 
 
 If we furthermore neglect the terms of the order (aR)~ ' in equa- 
 tion (25), we obtain after slight transformations 
 
 ■^ dy 1 1 
 
 ■1 (v - c)2 gQ(-l)[w(-l) - ^2 g^( + i)[^( + i) _ g 
 
 We put further c = Cq + 5 where Cq is the zero of J, 5a small 
 
 -1/2 
 quantity of the order (oR) . We assume for reasons of simplicity 
 
 a to be positive; then we may on the right side replace c by Cq 
 
 and may develop the left side into a Taylor series in &. Thus there 
 results, if we break off the Taylor series with the second term which 
 we presuppose as sufficient approxima,tion 
 
 J(c) = j(co) + 5 M 
 
 dc / \ 
 
 and from equation (25) because of 
 
 j(co) =0; gQ = ^-iaR(w - c] 
 (concerning the sign, compare page 9) 
 
 dc 
 
 Hence follows, because of 
 
 - w(-l) > w(+l) - c. 
 
NACA TM 1291 
 
 29 
 
 (cq is to lie nearer to w(+l)) that the imaginary part of 5 and 
 thus also that of c and of p has the same sign as 
 
 dJ 
 dc 
 
 (^==0) 
 
 dJ 
 
 and that oscillations corresponding to a negative value of — have 
 
 dc 
 
 an unstable character. If therefore our partly linear profile still 
 
 has the property that — < at the point w = c, it is unstable. 
 
 dc 
 
 This condition — < 0, however, is satisfied very frequently, for 
 dc 
 
 instance, always when the point w = c lies near one wall (for 
 instance, y = +l) and the profile is linear from the point w = c to 
 the boundary. 
 
 SuTTimarizing, we conclude: The instability or the stability of a 
 profile can be decided for all profiles considered so far by their 
 behavior in the case of frictionless fluid. Profiles which are capable 
 of undamped oscillations in the latter case and where the friction is 
 taken into account become, under certain presuppositions, unstable. 
 The latter profiles must have very special properties as shown above; 
 they must, for instance, be partly composed of linear pieces or they 
 must have a point of inflection w" = 0. (Compare above.) 
 
 At the same time, however, these profiles of type 2 are the only 
 ones still to claim physical interest since they are the only ones 
 whose behavior with respect to their stability corresponds approximately 
 to Reynolds' conjectures. Following, we shall show that these profiles, 
 in general, really have a critical Reynolds number (with the exception 
 of the broken profiles). 
 
 6. The Reynolds Number of the Stability Limit; Nijmerical 
 Calculation on the Parabola Profile 
 
 If, therefore, a profile is prescribed which, for frictionless 
 fluid, permits undamped oscillations and with friction is unstable, 
 the question arises, for what minimii.m value of the Reynolds nijmber does 
 instability occur? The simplified equations (25), (26), etc., do not 
 suffice for answering this question. We must revert to equation (23) 
 and to the forms (ll) and (l^) for the integrals cp, , cp^; 9q> and 
 
 cp. ; however, it is, of course, quite impossible generally, for an 
 
30 NACA TM 1291 
 
 arbitrary profile w, to represent the critical Reynolds number as a 
 function of w and of integrals over w; it will only be our task to 
 Indicate the way by which one arrives at the critical velocity and 
 then to perform the calculation on a special example. 
 
 Since in our last calculations a and R had appeared only in 
 the combination aR (because we had assumed a? as small), these 
 calculations can yield at best a critical value for oR only, not 
 for R alone. Thus we must first investigate the behavior of the 
 roots of equation (23) for increasing a?. Of the roots of equa- 
 tion (23), only those in the solution system II which satisfy the 
 equation cp].(+l) = are of interest. 
 
 Instead of equation (23) we must therefore discuss the equation 
 
 q) = (w - c) / -^fl . a2 rdy(w - c)^ T-^^ . 
 (w - c)^\ ,1 ,1 (w - c)2 
 
 (28a) 
 
 9=0 for y = -1, y = +1 
 
 If the profile consists, as in Rayleigh' s exa.mple, of linear pieces, 
 there exists (compare page 27) always a root of equation (28a) for 
 every break and these roots remain in existence for every value of 
 a^. Thus, the broken profile yields no maximum value of a^ and 
 therefore cannot ever lead to a critical Reynolds number. 1° 
 
 This is different if (cf. pp. 26 and 27) a solution of equations (28) 
 or (28a), respectively, with real c is possible for the reason that 
 either somewhere in the profile w" =0 or that w(+l) = w(-l), 
 cp = w - w(+l) represents a solution of equation (28). These latter 
 types of solution always yield a solution of equation (28a) only for 
 a very definite value of a . For w" =0, c is determined by the 
 very fact that for w" =0, w is to be w = c; thus the equation (28a) 
 defines a quite definite value of a ; however, for the case 
 w(+l) = w(-l) a, solution of equation (28a) obviously exists only 
 for a2 = 0. 
 
 It is still presupposed that R and oR are large and a « R. 
 Thus critical Reynolds numbers will possibly appear if these presup- 
 positions are no longer valid; however, the respective Reynolds num- 
 bers R would then probably assiame values so small that they certainly 
 would be of no physical significance. 
 
MCA TM 1291 31 
 
 For this type of solution of eqioations (28) or (28a), which are 
 characterized in the limit R = 00 by a very definite value of a?-, we 
 shall expect that, with the friction taken into consideration, a also 
 may vary from its definite value only by small amounts. For these 
 profiles the appearance of a maximum value (and in the case w" = 
 also of a minimum value) for a is very understandable. Thus all 
 oscillations, the wave length of which is smaller than a certain 
 critical wave length, are in such cases damped for all values of oR. 
 
 After having found an upper limit for o?- , one will attempt to 
 determine the approximate magnitude for the lower limit of oR, A 
 simple investigation of equation (25) shows that essential variations 
 in the imaginary part of r occur only after the exponent of e in 
 the approximate representation (ll) in cp-,(+l), cp2(+l) has decreased 
 
 to values of the order of magnitude 1; however, if this is the case, 
 we very soon reach the critical value (for which the imaginary part 
 of c is changed from negative to positive values) as will be shown 
 in the numerical example. If we assume that w is essentially linear 
 between w = Cq (cq = real part of c) and w(+l) the condition for 
 the approximate ma.gnitude of clR reads 
 
 (aR)^/^[v(.l) - Co] 3/2 
 
 (29) 
 
 or 
 
 w'(+l) 
 
 W(+1) - Cq 
 
 Since in the cases of interest to us w(+l) - c_ will probably be 
 
 small, we may by assianption form a conclusion as to high critical 
 Reynolds numbers. At the same time we note that for a certain value 
 of R there will always exist not only a maximum value but also a 
 minimum value for a of the unstable oscillations. This follows from 
 the fact that we did find a minimum value of oR (not R). 
 
 « 
 As numerical example for our general calculations made so far, we 
 
 select the parabola profile because it is physically the most 
 
 interesting one. It is to be classified as "profile capable of 
 
 oscillation" of the type w(+l) = w(-l). 
 
32 MCA TM 1291 
 
 Here too we shall consider only the two-dimensional motion, that 
 is, not Poiseuille's flow in tubes "but the flow prevailing between two 
 parallel walls at rest (y = +1, y = -l) under the influence of a 
 constant pressure gradient. Thus we put 
 
 w = 1 - y2 (30; 
 
 The symmetry of w and w - c permits the deduction that cp must 
 be an even function of y.^^ Thus we single out, from among the solu- 
 tions of equation (7a), two symmetrical particular integrals and attempt 
 to satisfy the boundary conditions at one of the walls, for instance, 
 y = -1. Those at the other wall then are fulfilled automatically. 
 Obviously we may take simply cpo as one of those symmetrical integrals. 
 For the other we choose 
 
 q>i(y) <P2(y) 
 
 9^(0) q)^{0) 
 
 It follows from equation (29a) that for our profile near critical 
 
 -I/3 
 velocity c will be small of the order (oR) ' ; in the following 
 
 calculations we shall thus cancel terms of higher than first order 
 
 in c. Furthermore, we state that cpp^^) will be » cp-,(0) so that 
 
 in the neighborhood of w = and w = c the second symmetrical 
 function cp simply is reduced to cp (y). From equation (16) then 
 
 follows that we have the two integrals (p^ and cp-, - Icpp at disposal 
 
 for fulfillment of the boundary conditions for y = -1. Equation (23) 
 is therewith transformed into 
 
 17 
 
 If one divides cp into a part even m y and a part odd in y, 
 
 each part of cp by itself must satisfy the differential equation (7a) 
 and the boiondary conditions because of the symmetry of w - c and w. 
 For the general stability investigation of the profile 1 - y^ it is 
 therefore sufficient to treat the two cases "cp even" and "odd" 
 separately and only these two cases; however, it may easily be seen 
 that the assumption of symmetrical oscillations, that is, "cp odd" does 
 not lead to a solution of equation (23) and thus not to unstable 
 oscillations. The assumption "cp even" therefore suffices for the 
 stability investigation. This is noteworthy insofar as, accordingly, 
 all symmetrical oscillations are stable and only unsymmetrical dis- 
 turbances unstable. 
 
MCA TM 1291 
 
 33 
 
 cp3(-l) cpi(-l) - icp2(-l) 
 CP2'(-1) q5-|_'(-l) - icp2'(-l) 
 
 = 
 
 (31) 
 
 In cpo one must always take y = as lower limit for the occiorring 
 integrals in order to guarantee the symmetry of cp^. Furthermore, we 
 shall develop in cpo only up to magnitudes of the order a^ and in 
 
 the development in (ccR) break-off with the terms of the order 
 
 (oR)" . We now write equation (31) in the fonn 
 
 cpi'(-l) - icp2'(-l) _ cp3'(-l) 
 cp-l(-I) - icp2(-l) cp^{-l) 
 
 If one inserts equations (ll) and (ih) , there results 
 
 (32; 
 
 2 / \/-iaR(w-c)dy 
 
 uyo 
 
 + 1 
 
 \liaRc = - — 
 ' 2c 
 
 2 / \| -iaR(w-c)dy 
 
 a£ 
 
 dy(w - c) + a 
 
 1 + a2 
 
 ^-1 
 
 dy 
 
 OCR 
 
 (q (w - c)2Jo 
 
 / n2 k 
 dy(w - cj + a 
 
 oR 
 
 (33) 
 
 Since c becomes very sm9.ll, we assume in first approximation w 
 from to c as linear; w~ 2(y + l). Then we obtain 
 
3^ NACA TM 1291 
 
 ^pTVdy = - i c3/2 (31,) 
 
 If we put 
 
 z 
 
 1 c3/2(2aR)l/2 
 3 
 
 there arises from equation (33) 
 e-(l+i)^ + i 3z(l + i) 
 
 ^-(l.i)z _ . 
 
 dy(w - c)"^ + 
 
 -'S 
 
 (35) 
 
 This equation is perfectly analogous to equation (26). We are 
 interested, above all, in the limiting value R or z, respectively, 
 for which the unstable oscillations are transformed into stable ones; 
 thus the imaginary part of c is exactly zero. This limiting value z 
 will, of course, also be a function of a. Thus we now assume c as 
 real and thereby obtain the limiting value of z or R, respectively, 
 as a function of a. The minimum value of R on this R(a) curve 
 will be denoted as the characteristic Reynolds number for the parabola 
 profile. Detailed calculation shows that one obtains by means of the 
 form (35) of the stability equation only the upper part of the curve 
 R = R(a) (solid line in fig. 2) which was to be expected according to 
 the deliberations of section 6; however, one can calculate the lower 
 part of the curve R = R(a) only by using for cp, , cpo; approximations 
 
 other (compare equation (l9a)) than the asymptotic formulas (ll). The 
 critical Reynolds number denotes just the range where the asymptotic 
 formulas cease to be valid. Since this circumstance would lead to very 
 complicated numerical calculations and since we cannot attach, in 
 general, (compare section 7) any essential physical significance to the 
 type of instability here characterized, we used rough estimates for 
 calculation of the lower part (dashed line in fig. l) of the curve 
 R = R(a) which, of course, cannot yield quantitative results; however, 
 the qualitative behavior of the curve is surely reproduced correctly. 
 Thus, we conclude from figure 2: 
 
NACA TM 1291 35 
 
 1. There exist both a maxim-um value of a and a minimum value 
 of R for instability. 
 
 2. For a definite value of R there exists a maximum as well as 
 a minimum value of a; instability prevails within these values^ sta- 
 bility outside. 
 
 3. The maximum value of a lies approximately at a = 0.7 
 [a^ = iV the magnitude of the minimum value of R is of the order 
 
 of lo3. A calculation of this mini mum value with some degree of 
 exactness is not possible from the figure. 
 
 7. Physical Discussion of the Results of Part 1 
 
 Let us summarize once more in detail all results found concerning 
 the stability problem. Above all, it became clear in the course of 
 the calculation that the problem of the stability of a profile for a 
 viscous fluid can generally be decided by treating like Lord Rayleigh 
 the limiting case of frictionless fluid (equation (8)). Profiles which 
 are unstable then (that is, for R = 00) remain so for sufficiently 
 large finite values of R (section k) as was to be expected beforehand. 
 Likewise profiles which, in the frictionless case, are not capable of 
 oscillations are found to be stable (section k) and profiles, which 
 a,ccording to the investigation by equation (8) permit \andamped oscil- 
 lations, generally to be lonstable (section 5). This latter case 
 obviously is the only one which, physically, signifies something new 
 compared to frictionless hydrodynamics; however, it should be emphasized 
 that this case, contrary to what one might conclude at first from 
 Rayleigh' s reports, represents an exceptional case. If one disregards 
 the possibilities w(+l) = w(-l), w'^Q^^^^^g^ = 00 (section 5);. it is 
 
 a necessary condition for the occurrence of this exceptional case that 
 somewhere in the fluid w" = 0. The broken profiles consisting of 
 linear pieces introduced by Rayleigh belong to those exceptional pro- 
 files; however, the only permissible conclusion is that curved profiles 
 for the purpose of stability investigation may not be approximated by 
 polygons according to Rayleigh (page 27). It is true that one may 
 find also for profiles ciirved everywhere (w" / O) (section 5) when 
 using the differential equations with friction terms oscillations 
 which are damped for every value of aR for which, however, in the 
 limit R = 00 the amount of the damping like (aR)-l/3 tends toward 
 zero; thus one has here also \indamped oscillations for R = «=. However, 
 these oscillation roots are lost if one takes the simplified differ- 
 ential equation (8) without friction terms as a basis. Insofar, there- 
 fore, this also is not a case of exception to the rule according to 
 
36 NACA TM 1291 
 
 which consideration of friction, in cases where the frictionless equa- 
 tion (8) permits undamped oscillations, results in an excitation; 
 however, as said above, the possibility of undamped oscillations for 
 equation (8) must be regarded as an exceptional case. The parabola 
 profile belongs to these exceptional profiles (section 6). If we 
 investigate further for the linstable profiles concerning the range of 
 values for R within which instability occurs, it is found that 
 generally, too, only profiles of the last class may lead to critical 
 Reynolds numbers, that is, only profiles which permit without friction 
 undamped oscillations; however, among the latter, Rayleigh' s broken 
 profiles nevertheless did not yield (compare also page 30^ footnote I6) 
 a critical Reynolds number. For Rayleigh' s profiles a minimum value 
 of oR does exist but no maximum value of a; therefore, no minimum 
 value of R for the neutral stability either (section 6). Only those 
 profiles of the last class for which in the frictionless case only a, 
 definite value of a leads to undamped oscillations (for instance, 
 the types w" = for w = c, w(+l) = w(-l)) result in a maximum 
 value of a and a minimum value of oR, thus also in a minimum value 
 for R. For a definite value of R there exists therefore for those 
 profiles a maximum as well as a minim\jm value of a for the unstable 
 oscillations. All these results are in agreement with the stability 
 investigations of hydrodynamic profiles made so far. 
 
 The question is now how these mathematical results will manifest 
 themselves experimentally. It seems surprising that the stable pro- 
 files (for instance, Couette' s-*-") and the unstable ones (for instance, 
 Poiseuille' s) empirically show exactly the same behavior. Above a 
 certain Reynolds number turbulence occurs in case of sufficient dis- 
 turbances; if the distiirbances are made as small as possible, the 
 laminar profile may be obtained for arbitrarily high Reynolds numbers. 
 Especially the last fact, which has been tested by Ekman (footnote 10, 
 p. 2), on the parabola profile seems to contradict the theory for 
 unstable profiles; however, it can easily be seen that this contradic- 
 tion is only illusory: The smaller the external disturbances, the 
 
 1 R 
 
 Compare the reports quoted in the introduction. 
 
 -'However, compare the interesting investigations of Couette's 
 motion concerning its stability against three-dimensional disturbances. 
 G. J. Taylor, Stability of a Viscous Liquid Contained between Two 
 Rotating Cylinders. Phil. Transact, of the Royal Society London 223., 
 pages 289-3^3, 1922. 
 
 This possibility of interpreting Ekman' s tests as a sort of 
 starting effect has been pointed out to me by Professor Prandtl. I 
 should like here to express my deepest gratitude to him for this and 
 many other valuable suggestions. 
 
MCA TM 1291 37 
 
 longer it takes (particularly for high Reynolds niimbers since the 
 
 1 /p 
 excitation there is of the order (oB) ' ; compare section 5) until 
 
 they noticeably influence the motion. Thus it will always be possible 
 to postpone, for flow in tubes, this moment so long that the quantity 
 of fluid, the stability of which is dealt with, has already left the 
 tube when its instability becomes apparent. The Reynolds number we 
 
 calculated could therefore be tested only on a closed system of tubes 
 
 21 
 where the same qua,ntity of fluid always flows. On the other hand, 
 
 the tests by Schiller (footnote S, p. 2) which show that below a 
 certain Reynolds number only laminar motion exists cannot be included 
 at all in stability investigations. The original motion here is not 
 laminar; one rather deals with existence or nonexistence of a turbulent 
 form of motion. At any rate, one may conclude from all these con- 
 siderations only that a solution of the turbulence problem by stability 
 considerations alone is absolutely not possible. 
 
 Still, the previous investigations may yield important qualitative ■ 
 results for our real purpose, the calculation of the turbulent motion. 
 If we interpret the turbulent motion as a certain basic flow with 
 superimposed undamped oscillations, we may conclude from our calcula,- 
 tions that the minimum value R for which this type of motion is 
 possible probably also lies at values of the order of magnitude 10^; that 
 the wave length of the undamped oscillations lies at 2jth/2, namely a 
 at values of the order 1, and that a for a prescribed R is confined 
 to certain occasionally very narrow limits; that furthermore these 
 oscillations have the character of a wall disturbance as may be con- 
 cluded from the smallness of w(+l) - c. These qualitative results 
 are quite independent of the special form of the basic flow; however, 
 beyond such qualitative criteria the calculations made so far do not 
 contribute anything toward the actual solution of the turbulence 
 problem. 
 
 PART II: THE TURBULENT MOTION 
 1. Statement of the Mathematical Problem 
 
 The Reynolds number usually denoted as critical (which is, for 
 instance, measured in Schiller's tests and indicates the appearance 
 
 21 
 
 However, the disturbances in stability observed by Ruckes 
 
 (footnote 10, p. 2) for rather small Reynolds numbers are perhaps 
 
 caused by instability according to section 7- This would be quite 
 
 conceivable when the critical Reynolds number according to section 7 
 
 lies below the one for which turbulence (compare part II) is possible 
 
 for the first time. 
 
38 NACA TM 1291 
 
 of tiirbulence in case of siiff iciently large disturbances) has no con- 
 nection with stability problems and with the laminar flow; it is 
 absolutely a characteristic constant of the turbulent motion. Like- 
 wise, the Blasius drag law and the well-known conclusion derived from 
 it (that the turbulent velocity in the proximity of a wall increases 
 with the 1/7 power of the distance from the wall) show clearly that 
 the so-called turbulent motion has its own well-defined regularities 
 and that it represents a second possible form of motion of the viscous 
 fluids. Thus the only way to a solution of the turbulence problem is 
 to attempt to eliminate the indef initeness of the turbulent motion 
 and to idealize it until it permits mathematical analysis by Stokes's 
 equations. 
 
 The turbulence problem of hydrodynamics is a problem of energetic, 
 not of dynamic stability. There exist two different forms of motion 
 of the viscous fluid, each of which has a certain range of values of 
 Reynolds numbers within which it is possible. Laminar flow is possible 
 from R = to R = » but becomes, however, under certain conditions, 
 above a certain value of R dynamically neutrally stable. The turbulent 
 
 motion on the other hand exists only above a certain critical value 
 
 22 
 of R and is always energetically more stable than the laminar 
 
 motion. Thus one may in the range of R in which both forms of motion 
 are possible always let the fluid drop from the laminar to the tur- 
 bulent state by means of sufficiently large disturbances. 
 
 In order to make approximate mathematical treatment of the tur- 
 bulent motion possible, we again consider the flow between two parallel 
 walls and make, first, the following assumptions:"^^ The flow is to 
 be (a) symmetrical about the X axis with the boiniding walls at rest, 
 
 (b) periodic in the X direction with the period^^ 2it/a, and 
 
 (c) periodic with time with the period 2jt/3, and (d) all disturbances 
 are to propagate with a speed p/a relative to the X axis, that is 
 
 if the motion is developed into a Fourier series, only products of 
 i( pt - ax) are to appear in the exponents of^^ e. 
 
 22compare F. Noether, elsewhere. 
 
 ^his statement, too, which represents a simple generalization 
 of Soramerf eld' s stability theorem was indicated, for investigation of 
 the turbulent motion itself, by F. Noether without further conclusions 
 in his paper entitled "Zur Theorie der Turbulenz" (Concerning the 
 theory of turbulence), Jahresberichte des deutschen I^th. Vereins 23, 
 page 138, 191^. 
 
 The assumption of a definite a is justified by the result of 
 part I that a is confined between certain limits which are, partic- 
 ularly in the proximity of the minimum value of R, very narrow. 
 
 25we need not emphasize the part that the actual motions are doubt- 
 lessly much more complicated; nevertheless, one may well expect these 
 statements to permit qualitative statements concerning the turbulence. 
 
NACA TM 1291 
 
 39 
 
 The Fourier development of the stream function should therefore 
 
 I 
 
 read 
 
 1^ = cpQ(y) + cp-L(y)e^ 
 
 (pt-ax) — 7—7 -i(|3t-ax) , ^ 2i(pt-cxx) 
 
 (36) 
 The mathematical problem then consists in the determination of the 
 odd (according to a) functions cpQ, cp-]_j and cp2 (^i^ ^ . . . con- 
 jugate to cp2_, 92^' -^^ first, the degree of convergence of the 
 
 series (36) is completely iinknown; the question of convergence can be 
 decided only after calculation of cpQ, cp-|_ . . . If we want to carry 
 
 accuracy so far as to the n approximation, that is, if we want to 
 calculate q3„ . . , cp^^, we obviously obtain (n + l) simultaneous 
 
 differential equations for the (n + l) unknown functions cp . . . cp^^. 
 
 Following, we shall need partly the first, partly the second approxi- 
 mation. Thus we enter equation (h) with the statement eqiiation (36), 
 compare the coefficients of the periodic functions on both sides, and 
 break off with the term Q'21{P>t-ajx) ^^^^ thus with cp2, Cp2. For cpQ* 
 
 we write w (as in equation (6) for $ ). Thus three simultaneous 
 differential equations are produced (the simultaneously obtained con- 
 jugate eqiiations need not be written down) 
 
 dy 
 
 (q^i'q^i - 9i'9i) + 2(92*92 - ^2 ^2 
 
 ^-h^" 
 
 (cpi" - a^cp^Uw - ^j - 9iw" - cp2'(9i" - cc^^i) - 2^2 (<?i"'- ^^^i') 
 
 
 2^ .. . h, 
 
 > (37) 
 
 2a cp-i " + 
 
 aSi) 
 
 
kO ■ NACA TM 1291 
 
 The first of these equations may be integrated twice and yields 
 
 C and C]_ signify arbitrary integration constants. 
 
 Since the left side of eqioation (37a) and w are, according to 
 
 re 
 
 quirement (a), odd in y, C must disappear in our case. 
 
 If we go back from the second to the first approxiiiia,tion, our 
 system of equations is reduced to two simultaneous differential equa- 
 tions for w and cp-. 
 
 cpi'9^ - cp^'cp^ = -^(w' - Cj^y) 
 
 (cp," - a2pj(w - i) - w>, = ^(q,;- - 2a2cp," . a^J 
 
 I (33) 
 
 By way of a,n interpolation we shall now reflect what replaces eqiia- 
 tion (33) if we do not consider a flow symmetrical about the X axis 
 (requirement a), (that is, the flow of a fluid under a pressure 
 gradient between two walls at rest), but instead a flow antisymmetrical 
 about the X axis (that is, a flow between two walls moved relative to 
 each other without pressure gradient as in the Couette case). Require- 
 ments (b) and (c) axe to be maintained. The statement (36) will then 
 nq longer be satisfactory since cp-i, cpp, etc., for arbitrary p/a 
 
 are no longer even functions; in order to obtain the entire flow 
 pa.ttern in terms of odd functions, we must also include the symmetrical 
 
 oscillations of the form e^' "'^"^""'■'^^ in the formulation for \|f, that 
 is, -^ must start with the terms 
 
 „ / V i(Bt-ax) / -i(Bt-ax) / \ -i((3t-ax) 
 cPq + qPi(y)e + <Pi(-ye ^^ + ^-^{jje '^ ' + 
 
MCA TM 1291 
 
 kl 
 
 As a consequence finally all the terms of the form 
 appear in ilf (elimination of requirement (d)). 
 
 In place of equation (33) there results 
 
 a(mpt-ncLx) 
 
 9i(y)9i(y)' - cpj_(y)'cp-|_(y) + cpi(-y)9-L(-y) ' 
 
 cPi(-y)'<Pi(-y) = ^(v* - c) 
 
 clR 
 
 (V--S)f -!)-->! =^(V" -^-\ 
 
 + a\j 
 
 I (39) 
 
 The two equations of the system (38) and (39! 
 simple illustrative significance. 
 
 respectively, are of 
 
 The second equation is none other than our former stability equa- 
 tion (7) which determines the amplitude of the oscillation superimposed 
 on a hasic flow w and which formed the basis for our investigations 
 in part I. The first equation, however, represents the theorem of 
 momentum. The left side of this equation essentially indicates the 
 momentum transferred on the average by the turbulent vorticity^ , the 
 term with w' on the right represents the laminar momentum transfer, 
 and the constant C or C-|_y, respectively, is the constant of the 
 
 momentum theorem. 
 
 Due to the boundary conditions at the walls cp-, = cp-[_ =0. There- 
 fore there w' = C or C-iy^g^-^i^ respectively; thus at the walls the lami- 
 nar momentiim transport surpasses the turbulent one, '"''wall ^iH gen- 
 erally be very large. (Compare the next section.) At the channel 
 center, however, that is, in the entire tunnel outside of the immediate 
 
 ^e are referring here to the mean momentum in the X direction 
 which, for omt problem, is transferred in the Y direction. The 
 momentum in the X direction equals, on the whole u, the velocity of 
 the particle transporting the momentum in the Y direction is v; thus 
 the momentum transferred during the \init time uv, on the average uv 
 which for the case (36) results in 
 
 uv = -ia(cp-Lcp-[_' - 9i'<Pi) 
 
h2 MCA TM 1291 
 
 wall proximity, w is of the order of magnitude 1, thus very small 
 compared to C or Cjj, respectively. The turbulent momentum trans- 
 port will therefore here completely overbalance the laminar one. 
 
 It corresponds to the structure of the systems (33) and (39) 
 that we are able to give immediately a trivial solution of them, 
 namely cp-j^ = 0, w' = C or, respectively, w' = Cjj, that is, we thus 
 
 revert to the laminar motion. 
 
 Our problem now is to obtain definite results concerning the non- 
 trivial solutions of equations (38) and (39)- 
 
 2. The Turbulent Motion in Wall Proximity and 
 
 the Law of Resistance 
 
 The most important result concerning the behavior of w in the 
 immediate proximity of the walls is the law derived by V. Karman 
 (elsewhere) from the Blasius drag law by means of considerations of 
 
 similitude that w in the proximity of a wall increases with t\ ' ' 
 (ti representing the distance from the wall). We repeat briefly 
 Von Karman' s train of thought since we are thereby enabled to a general 
 visualization of what to expect, even without knowing the Blasius law, 
 concerning the behavior of w of the wall. 
 
 As can easily be seen from considerations of similitude, it must 
 be possible to represent the shearing stress t acting at a wall 
 (that is, the drag) in the form 
 
 T = Kp. H f(R) 
 ^ h 
 
 where k signifies a certain dimensionless constant. 
 If we specially assume a power law there is 
 
 U „| 
 T = k;(i - R = K[i 
 
 U/Uhp\ 
 
NACA TM 1291 1+3 
 
 From equation {ho) follovs inversely 
 
 U 
 
 fe)"-3 
 
 1 
 1+1 
 
 The velocity distribution in wall proximity must then be represented 
 by an equation of the form 
 
 1 
 
 r-Hr!l"^Ka) 
 
 (Here again (compare equations (l) and (6)) w has been selected 
 dimensionless and therefore contains U in the denominator; t] denotes 
 the distance from the wall. ) 
 
 We again assume in first approximation a power law (let a be a 
 dimensionless constant) 
 
 " = §E^-^r T"(sr 
 
 If one now requires the velocity distribution in innnediate wall proxi- 
 mity to be a function only of t, ^^ p, "but not of h which is 
 physically very plausible, there follows 
 
 - = 6 (1^1) 
 
 1 + I 
 
 3 1 
 
 For ^ - T 3-s corresponds to Blasius' law there results e = — . 
 
 In order to understand the physical significance of this result 
 
 we note: w ^ -n ' signifies that w' = -r^ is infinite at the 
 ' dr| 
 
 boundary, that therefore w Infinitely clings to the wall; however, 
 it is clear that actually w' at the wall cannot be infinite since 
 w' , on the contrary, essentially denotes the shearing stress at the 
 
hk NACA TM 1291 
 
 wall and is therefore, according to equation (hO) , to be equated 
 to R^ except for a numerical factor independent of R. 
 
 ^'edge~R^ (^2) 
 
 w' at the wall is therefore very large for the large values of R 
 
 which are of interest to us. Corresponding to its derivation the 
 
 1/7 
 velocity distribution w ~ j]-^' ' will be strictly valid only in the 
 
 limiting case of infinitely large distance from the wall or of friction- 
 less fluid (R = 00). These facts can be still more easily comprehended 
 
 if the law w ~ t\^/ ' is written in the form -q ~ w'. From the fact 
 that the shearing stress is finite we know that the first term of the 
 power development ii(w) must be of the form j-^v. This term, however, 
 
 is very small, essentially equalling the reciprocal value of w' and 
 thus being of the order R-5 (compare equation (^2)). 
 
 The meaning of the derived law w ~ t\^i ' is thus obviously that 
 the series development of ti(w) is to stg,rt with the terms 
 
 7^w + 7^w + . . . (ij-3) 
 
 where j, is extraordinarily small and that therefore the first 
 term y-^ for somewhat large values of w may be cancelled compared 
 with the second J-f^- . 
 
 According to the explanations above we expect independent of the 
 validity of the l/7-law for the basic flow of the turbulent motion 
 small curvatures at the centers, and in the wall proximity, clinging 
 of the basic flow to the walls. 
 
 For such a profile the investigations of part I do not directly 
 apply since there w' , w" , etc., had been presupposed as finite; 
 however, these investigations can easily be generalized to include 
 profiles like the one considered here, (Compare section 5j Page 26.) 
 Particularly, the solution of the reduced equation (3) (thus lim R = <») 
 with satisfaction of the boundary conditions becomes especially simple 
 here; the profile characterized just now belongs, according to sec- 
 tion 5; page 26, to those capable of oscillation; a solution of equa- 
 tion (28) with real c exists. This is extremely important because 
 
MCA TM 1291 " 45 
 
 it shows that the turbulent profiles are always unstable according to 
 section 5^ or, in other words, that it is just the deviation {h2) from 
 the laminar resistance law which makes an unstable profile and thus 
 makes turbulence possible. 
 
 The solution of equation (28) is for a = 
 
 ^ = t - -(-ill / " r ^'^ n2 ^^^) 
 
 [w - w( + l)J^ 
 
 By selection of the lower limit of the integral we made cp become 
 zero for y = -1; that it becomes zero also for y = +1 due to 
 selection c = w(+l) can be seen easily from the following 
 transformation 
 
 = w 
 
 t 
 
 dw 
 
 w(-i) [^ - -(+i3 -' 
 
 The integral of the right side becomes at the point w = w(+l) 
 
 infinite of lower order than since w' there ( in the 
 
 w - w(+l) 
 
 limit R = 00) becomes infinite. Thus cp-, = for y = +1. By equa- 
 tion {kh) we have represented in the limit of frictionless fluid the 
 amplitude of the turbulent oscillations and derived from the boundary 
 conditions the va.lue for p/o., namely p/a = w(+l). It is, however, 
 self-evident that the solution symmetrical to equation {hh) 
 
 9 
 
 = fw - w(-l)l I ^L.^ ikka) 
 
 also satisfies the boiuidary conditions; thus c = w(-l) is valid 
 here. In case of the Couette arrangement we therefore conclude from 
 equations {hk) and (^^) that two mutually symmetrical oscillation 
 systems exist, the wave velocities of which agree respectively with 
 the velocities of the two walls (w(+l) and w(-l)). 
 
k6 ' . NACA TM 1291 
 
 For the symmetrical flow "between two walls at rest, on the other 
 hand, w(+l) = w(-l) = 0. From equations (kk) and (i^-i^a) we then con- 
 clude that every integral of the form 
 
 Pdy 
 
 satisfies the boixndary conditions; however, from the requirement (a) 
 that cp is to be odd there results that we must select as lower limit 
 
 Tdy 
 J w2 
 
 of the integral / — y = 0. Thus 
 
 (kkh) 
 
 In the case of symmetrical flow there is therefore, particularly 
 for 3/a 
 
 w(+l) = w(-l) = £ = (1+l^c) 
 
 a 
 
 In the turbulent basic flow, the type of its singularity at the 
 walls is of foremost interest to us; thus for the assumption w ~ -q^, 
 the exponent e. ¥e shall attempt to show that from the differential 
 equations (38) and (39) respectively in the limit R = «> at least in 
 Immediate proximity of the wall such a power law with the exponent 
 6 = 1/7 actually follows. It is true that the domain of convergence 
 of the power series used is not established so that the conclusions, 
 as far as they apply to the shape of the profile at some distance from 
 the wall, are uncertain. We develop cp-i and w in the neighborhood 
 
 of T] = in integral and positive powers of t] - this is possible for 
 any finite value - and then inversely t\ in integral powers .of w. 
 Thus we are led directly to the formula (^3) for ti(w). 
 
 We contend, and this is the most important result we shall need 
 later, that cp-]_ in first approxima,tion may be represented by a series 
 of the form 
 
 2 5 3 
 
 + 
 
NACA TM 1291 kl 
 
 where a^, ag , . , are real, at-, a-,-, . . . purely imaginary con- 
 stants; furthermore, w is of the form 
 
 w = p^T] + p^T]'^ + . . , (U5) 
 
 This contention may "be proved for the differential equations (38) 
 directly by expressing cp and w in undetermined coefficients if 
 the terms a^, a-,, clo, o.^, and 3^, 3-, are prescrihed. Thus we 
 
 will, above all, attempt to determine these terms. First, cp-j^ and 
 9 ' for T) = must be zero beca.use of the boundary conditions; thus 
 
 the series for cp-, starts with apT] (oq = a]_ = O). We can verify 
 
 afterward that, furthermore, the following term cn.^r]-' is eliminated, 
 
 that is, becomes very small compared tc the other terms. By way of an 
 interpolation we shall prove here for this purpose by a single approxi- 
 mate integration of the second equation (38) that a-, assumes the 
 
 order of magnitude ccR. For a^ = equation (33) reads 
 
 „ It/, PI _ ,,ii i _ im 
 
 whence follows 
 
 The constant A is here of the same order of magnitude as the left 
 side of equation (U6) at the center of the tunnel, thus almost of the 
 order of magnitude 1. (Compare part I, section 2.) The term cp-,'" 
 
 at the edge is therefore of the order cxR due to the boundary condi- 
 tions. The same is valid for ao. 
 
 Thus we shall meanwhile assume ao as small and later attempt to 
 justify that assumption. Of the constants Pq, 3i; the first, Pq, 
 equals zero because of the requirement (a), section 1. 
 
h3 NACA TM 1291 
 
 The constants Op and p^ are, at first, arbitrary ' and there 
 
 is no possibility of deriving them from the solution of the differential 
 equations (38) and (39) in the proximity of the wall. This possibility 
 would. arise only if we should succeed in continuing the solution (^5) 
 analytically up to the other wall; however, this is an extremely com- 
 pj.icated mathematical problem if only for the reason that, as will be 
 seen, the simplified equations (38) and (39) si's not sufficient for 
 determining qp-, and w at the center of the tunnel. Although we 
 
 must therefore forego the solution of this problem, we may still expect 
 to obtain, by merely developing cp^ and w in the proximity of one 
 
 wall with undetermined coefficients cx2, 3-]_, those qualitative char- 
 acteristics of w and qpQ_ in wall proximity which are, according to 
 experience, quite independent of the behavior of the fluid at the 
 tunnel center as, for instance, the law w ~ tj / ' . 
 
 We enter equation (38) with the statement 
 
 ^1 " °'2^^ + o-^n + o-^^^ + • 
 w = 3-|_Tl + PgTl^ + ^3^^ + • . 
 
 replacing the second equation by {k6). We therefore again assume a 
 as very small which here only signifies (compare part I, section 2) 
 that the wave length of the oscillations is to be large compared to 
 the boundary- layer thickness; moreover, we put, according to equa- 
 tion (hkc) 
 
 P = o 
 
 a 
 
 For the first equation (38) we write furthermore 
 
 -ioR^^Cp^' cp-j_ - cp^'cpi) = 2aR(cp^.'cp^^ - qPiiCp^j.') = w' - C^y 
 Therein cp-^j, denotes the real, (p^ • the imaginary part of cp . 
 
 27 
 
 We shall assiMie a^ as real. This does not imply a limitation 
 
 of generality since cp-^ is determined only up to a factor of the form 
 
 e^^ as the initial point of the time coordinate in equation (36) may 
 be chosen arbitrarily. 
 
MCA TM 1291 49 
 
 From .equations (46) and (4?) now follow the recursion formulas 
 
 n-2 
 n(n - l)(n - 2)0^ = -loR Y~ s(n - 2s) a^_i_sPs-l (^) 
 
 nPj^ = 2aR 2~~ s(n - 2s)a^_3-^Ps^ 
 
 5=2 
 
 {h9) 
 
 in addition 
 
 Pi = 'l^ed 
 
 ge 
 
 2p2 = C^ 
 
 Therein ag''^ denotes the real, a ■'- the imaginary part of a^ 
 From equation (48) there follows first 
 
 0-4 = 
 
 From eqioatlon (4-9) there then results 
 
 P3 = p^ = 3^ = Pg = 
 
 The term P2 ™^y also he approximately equated to zero. 
 
 From equation (49) there follows 
 
 Pi 
 P2 = 
 
 "^^edge 
 
50 MCA TM 1291 
 
 p 
 For very small t] the term PgT ■'■^ therefore to be neglected com- 
 pared to the first term P>jj\; for larger •(], however, the higher terms 
 P'tT]'^ etc.j are completely predominant. 
 
 Let us thus assume also ^2 - '^ ^^^ thus calculate the higher 
 terms of the series for p-|_ and w. 
 
 There follows 
 
 2 
 agPi 2 °'2Pl 
 
 cxcr = -ioR ; ; tt/r = av = 0; ao = -(oRj ^r— 
 
 ^ 3x4x5 ° ' ° 3x5x6x7x8 
 
 f o^3^ "-a^^S °^Pl^ \ 
 
 2x7x9x10x11 3x5x6x8x9x10x11, 
 
 (Xto = ^1^ ~ ^ 
 
 P7 = -(•^f %^; Pa " P9 ■ ''lo = I'll - »12 ■ °' 
 
 ■^ V5 X 5 X 6 X 8 X 11 X 13 X l4 7 x 10 x 11 x 1: 
 
 The representation (^5) for w we contended is therefore proved and 
 it may easily be shown too that of the further terms only in every 
 sixth term has 3 a finite value. 
 
 Hence follows for the representation of r\ as power series in w 
 
 7 13 
 ^ = y^v + y^v' + 7-L2^ -^ + . . . 
 
 1 9 °2^ 
 
 ' H' ^ 703j 
 
 X50) 
 
 13 \7xlOx 10x11 xl3x p-^lS 5x5x6x8 xllx 13 xl^xpj^ll- 
 
NACA TM 1291 '51 
 
 zero 
 
 The terms j^ to 7g, 7q to J y^.' ^lU ^"^^-^ ^H equal 
 28 
 
 The development (50) now actually completely agrees with equa- 
 tion (^3) and we seem thus to arrive, even without knowledge of the 
 
 constants a2 and ^-^, to the law t) .^ w' semiempirically derived 
 
 "by von Karman. The coefficients 7-, and j^, however, cannot he 
 
 calculated. Inversely, we may perhaps conclude from the empirical 
 findings for the coefficients P-j. ^^^ ^2 that jn is of the order 
 
 o /Ji 1 "3 /R 
 
 of magnitude 1, 7]_ of the order (olR) , thus 02 ~ (oR) . 
 
 Subsequently, we thus also confirm our former assertion aoT] « ap. 
 
 Raising the question of what order of magnitude are the values of w 
 for which the third term in equation (50) is small compared to the 
 
 second for which therefore the w' profile actually is valid, one 
 finds w -v ^-^ ' } thus '^"-'-/°. Accordingly, the profile w ~ t\}-I^ 
 
 follows from the differential equations (38) only qualitatively ?.t 
 first. No information about the fact that the 1/7 profile has been 
 observed almost up to the tunnel center is given in our calculations; 
 however, this was not to be expected since the other constants entering 
 the law also depend on the behavior of the fluid at the opposite wall. 
 
 As an interpolation, we shall once more briefly s-ummarize what 
 factors we have neglected in deriving equation (50) from equations (48) 
 and (^9) and attempt thereby to determine within what accuracy the 
 conclusions drawn from (50) are correct. First, we used system (38) 
 instead of (37)^ thus cancelled magnitudes of the order cpo/'w* Further- 
 more, we equated a-^ = 0, 3/a = 0, 32 = and therewith neglected 
 
 aoT] o 32^ Pi 
 
 magnitudes of the order -^^—, -^—, , and -:, respectively. 
 
 a.2 aw Pi p^TiD 
 
 The acciiracy of our calculations will be determined by the largest of 
 the terms here neglected. Simple considerations of the order of magni- 
 tude, not executed here, make it probable that of these terms 92/^ 
 
 is the largest but that this term, too, goes toward zero with R — ^00. 
 
 ?8 
 
 This power series ti(w) may, of course, also be derived directly 
 
 from equations (46) and {hj) without the detour over the series of w(t|) 
 if w is introduced as independent variable; however, the calculations 
 required for this purpose are somewhat more complicated. 
 
52 NACA TM 1291 
 
 Selection of a sufficiently large value for R will therefore make 
 it possible to carry the accuracy of the results derived from equa- 
 tions {h3) , {hS) , and (50) arbitrarily far. 
 
 As to Blasius' law of resistance, it can, of course, be derived 
 inversely according to the method described above from the law t] ~ wT^ 
 by means of consideration of similitude if one assumes, as we did, that 
 the behavior of w in the proximity of the wall is independent of the 
 tunnel width; however, for the reasons stated above (impossibility of 
 analytical continuation) we must leave the question unanswered whether 
 this latter - physically very plausible - assiomption also follows from 
 the differential equations (38) and (39)> respectively. 
 
 We are, however, able to draw a noteworthy direct conclusion con- 
 cerning the law of resistance from equations (38) and (39) t)y means of 
 consideration of similitude. In the tunnel, except for immediate wall 
 proximity and the point y = (compare below equation (66)) one may 
 write instead of the first equation (38) because of the magnitude of 
 C-L (compare pages kl and k2) 
 
 iaE(cp-L'qp-i_ " 'Pl''^l) " ^1^ ^^l) 
 
 Since the amplitude cp-, cannot go toward infinity with R — >« - this 
 
 would render all our calculations devoid of physical sense - there 
 follows that Cj^ is at most of the order of magnitude oR, that there- 
 fore the exponent | of eqiJiation (hO) must be <1 (which in a certain 
 ma,nner also can be seen from equation (4l)), Hence follows (compare 
 equations (^2) and (^O)) that the law of resistance t = const. U^ 
 usually assumed in hydraulics represents an upper limit for all imagi- 
 nable laws of resistance of turbulence which is independent of the wall 
 characteristics. One may conclude as an assumption that the law 
 
 T ~ U ' is valid only for smooth walls - it was for those only that 
 
 we obtained r) -^ w' - that the law of resistance for rough walls, 
 
 » 2Q 
 
 however, more and more approaches the quadratic law. ^ For rough walls 
 
 the amplitude cpj will be independent of R and of the magnitude of 
 the wall disturbances; moreover, for rough walls the boimdary conditions 
 will no longer cause cp-, to be real in first approximation as corre- 
 sponds to equation {hk). 
 
 29 / / 
 
 Compare the more exact investigations by Von Karman, elsewhere, 
 
 and the experimental investigations by Schiller, same periodical 3> 
 page 2, 1923. 
 
NA.CA TM 1291 53 
 
 Nothing is changed in the conclusions of this paragraph if the 
 equations (39) are taken as a basis instead of the differential equa- 
 tions (38). 
 
 3. The T\arbulent Motion Outside of the Ijcamediate 
 Proximity of the Wall 
 
 It is essential for the motion at the tunnel center that cp-i here 
 
 is composed of those two integrals of (7a) which appear in case of 
 frictionless fluid, thus for eqxoation (8), (Compare part I, section 2.) 
 The most important characteristic of cp^^ following from this fact is 
 that it satisfies - except for magnitudes of the order cpg and 
 (aR)"l - the condition 
 
 cp-j'qp-, - cp-,'cp-, = Const. (52) 
 
 This results, according to Abel's theorem, from the fact that, 
 except for magnitudes of the order 92 ^.nd (oB)"-'-, 'Pi-rt 8.ndL cp-, j 
 
 (the real and imaginary part of (p-^) are solutions of the differential 
 eq\;ra,tion (8). 
 
 Hence it can be concluded that the equations (38) and (39) are not 
 sufficient for establishment of the motion over the entire tunnel width 
 but that we have to go back to eqioation (37) and to the system of eqxxa- 
 tlons which corresponds to it for Couette's case. 
 
 This, in general, involves a complication of the mathematical 
 problem. Only in Couette's case may 'the problem be solved comparatively 
 easily because the first equation (39)^ except for magnitudes of the 
 
 order w'/c, thus (oE)"-^' and <?n , compare equation (37)^ agrees 
 
 with equation (52). Whereas, therefore, eqxxation (52) in consequence 
 of its derivation from Abel' s theorem is correct only up to magnitudes 
 of order (p2> ^^ Couette's case equation (39) shoiild still be valid up to 
 
 magnitudes of the order cp2^. This requirement is satisfied if we put 
 
 (P2 = (53) 
 
5^ MCA TM 1291 
 
 This equation is therefore to be regarded as solution for Couette' s 
 case, of the differential equation we took as a hasis. 
 
 According to equation (53) it would follow for cp-, from (37) 
 
 q)i"'cpi - 9i"cPi = (54) 
 
 The system applying to Couette' s case is not equation (37) hut a more 
 complicated one which we are not going to write down. We do, however, 
 state about it that it leads, like (37), for (p2 = to the solu- 
 tion (5^) and thus to the result 
 
 cp-L = ae^'y + he-yy (55) 
 
 Here, a, b, and 7 are any complex constants. For w then follows 
 from the second equation (39) or, respectively, from its reduced 
 form (8) 
 
 w - c = a-j^e ^ + b-j^e ■'• (56) 
 
 Since at one of the walls there should be w - c = 0, and since, on 
 the other hand, w should be odd in the neighborhood of y = 0, it 
 follows that w, simply by the vanishing of 7-|^ and a suitable 
 
 Increase of a-, and b-, , must degenerate to a linear profile. 
 
 Thus, we obtain the important result that for Couette' s case the 
 basic profile w of the turbulent motion takes an essentially linear 
 course over the entire tunnel width - however, strongly deviating from 
 the laminar profile, it will be much flatter than the laminar one - 
 
 [that, however, (compare II, section 2) at the edge it clings again 
 
 like Ti^T -to the wall^ . 
 
 We shall now turn to the more complicated case of a flow between 
 two walls at rest, thus exactly to the system (37). For a solution 
 we must naturally be content with rough approximations. First, we 
 
MCA TM 1291 55 
 
 can cancel in equation (37) the right sides of all three equations, 
 namely the friction terms; this is fully justified by the considera- 
 tions of Part I, section 2. Then we equate p/a = (compare 
 equation ( ij-^c) ) . 
 
 We thus obtain for cp2_ in the place of the second equation (37) 
 cp3_"w - w"cp-L - a2wcp-j_ - 92' (^1" - "■^^i) - 2CP2(^-1_"' " °'^^l') "^ ^9^' (^2" " 
 i|a2cp2) + ^^(cp^'" - ^\') = (57) 
 
 If we develop cp-]_ as solution of the eqviation (57) in powers of a^ 
 on one hand and powers of cp2 on the other, and if we further note 
 that cp]_ is to be odd (compare (44b)) and write cp-, = cp, „ + cp, , there 
 results with only the linear terms taken into consideration 
 
 cp = aw 
 
 Jo ^ Jo ''^(^^''^10 ^ ^2>lo" " ^cPgCPio' 
 
 2^10'^2" - ^10^2'") (59) 
 
 Naturally cp-, is herein not fully determined - the constant factor a 
 assumed as real which does not signify a limitation remains undetermined. 
 
 If we substitute this value of cp]_ into the simplified first 
 equation (37)^ namely 
 
 cp-.^"-," - ^'-.cp-, = Const. (60) 
 
56 MCA TM 1291 
 
 we obtain, with cpp. denoting the imaginary part of cPp, the result 
 
 Now, however, as follows from the third eqijation (37) and from the 
 fact that cp]_ is real in first approximation, cppi satisfies the 
 equation 
 
 cp2i"w - (p2iw" =0 (62) 
 
 thus 
 
 ^ - -no (63) 
 
 If we substitute this value of qp2i into equation (63) and if we 
 
 further consider that for y = the left side of equation (61) and 
 therewith the constant on the right side is zero (this signifies for 
 
 the constant of the right side of equation (60) only that it is in 
 
 p 
 first approximation zero, that is, small of the order Cp2.'''2 ^■'^ 
 
 ? h 
 
 au'^(f>jP2> respectively, or a cp-j^ , we obtain 
 
 ^10^10'" - ^io"^io' = ° (6^) 
 
 which fully agrees with (5^). 
 
 This equation, it is true, becomes, like eqijation (5^), trivial 
 in the neighborhood of the point y = 0; it is there fulfilled 
 identically since qp is an odd function of y. Thus it cannot permit 
 there a determination of w. This leads for the symmetrical profile 
 (6^) to a remarkable discontinuity at the point y = 0. (For the odd 
 profile such a discontinuity cannot be seen from the differential 
 equations.) If one integrates (37a) one obtains, as shown above, 
 after a single integration the equation 
 
 2aR|^-^^"cp-,_ - cpi"?! + 2(^2"^2 " 'P2"^2) "^ * ' • 1 = w" - C (65) 
 
NACA TM 1291 57 
 
 where C (compare pp. i^-1 and h2) , Blaslus' law of resistance being valid, 
 is of the order of magnitude (dR)-'' , thus at any rate very large . 
 The left side of equation (65) disappears, however, with cp]_ and cpo 
 
 (which, as we know, are odd functions of y) at the point y = 0. Thus 
 
 Vo" = c (66) 
 
 must be valid there. This signifies that ^-^=0" "^^ very large 
 
 {~(aB) ) and that therefore w at the point y = shows a sharp 
 break^ ^radius of curvature ~(aR)~-^/ ). At a small distance from 
 this point the course of w must, according to equation {6k), again 
 be essentially linear. 
 
 We obtain the result: For the flow between two walls at rest as 
 well - and surely this may be applied also to the flow in the tube - 
 the profile is linear approximately over the entire tunnel width; at 
 the center, however, it shows a sharp break (it clings to the walls 
 with the y '' law). (Compare figure 3.) 
 
 The physical cause of the sharp break is the fact that the gradient 
 of the turbulent momentum transfer for y = disappears for reasons 
 of symmetry and that therefore, because the gradient of the entire 
 momentum transfer over the tunnel width is constant, the gradient of 
 the laminar momentum transfer, that is w", must be very large there. 
 
 k. Final Remarks and Summary of the Physical Results 
 
 Our investigations still show two important gaps. First, they do 
 not yield the transition from the t]-"-/! profile to the linear profile 
 valid in the center part. Second, they are limited to large values 
 of R and thus do not yield the minimum value of R, either, if such 
 a minimum value exists for which the turbulent motion is still possible. 
 The first of these two gaps is most difficult to fill in (compare 
 page kS) ; we cannot even indicate a method which would satisfactorily 
 
 ^Professor Prandtl was so kind as to point out this break to me 
 on the basis of empirical material. The break seems less sharp empiri- 
 cally than according to calculation results, which is easily explained by 
 the fact that the assumption (a), page 38, concerning the symmetry of the 
 vortices and distiirbances also does not exactly correspond to actual 
 conditions. 
 
53 MCA TM 1291 
 
 solve this particular problem. One may attempt to piece the two 
 approximations together that come from the wall and from the tunnel 
 center. This would have to be done by means of the condition that 
 at the respective junction cp, , cp ' , cp-,"; 9-,"'? ^i and w' are to 
 
 be continuous; however, the convergence of the developments (^5) and 
 (50) is hardly sufficient thus to guarantee a somewhat defined approxi- 
 mation. At any rate the ultimate result, the profile w, is still to 
 a great deal dependent on the type of joining the two approximations. 
 Finally, it must be regarded as dubious whether such an exact carrying 
 out of the formulation (page 39) would yield essentially new physical 
 results in agreement with experience since these statements certainly 
 represent a very strong idealization of actual conditions. 
 
 In contrast, filling in of the second gap does not offer any 
 basic difficulties whatsoever; all necessary expedients are contained 
 in Part I and once the profile w is completely known, the methods 
 described in Part I are, on principle, sufficient to calculate according 
 to Part I, section 6, the minimum value of R for which the turbulent 
 motion is possible. One could, for instance, calculate the critical 
 Reynolds number for a profile obtained, according to the method 
 mentioned above, by piecing together the two approximations, or one 
 could base this investigation on the empirically observed profile and 
 thus calculate the Reynolds number in a semiempirical manner. In any 
 case one will - the investigations in Part I made this probable and 
 direct calculations, here not reproduced, confirmed it - arrive at 
 the same order of magnitude of the critical Reynolds number, namely 
 
 R ~ 103. The exact value of R will, it is true, still be too 
 dependent on the manner in which the profile was obtained to permit 
 comparisons with experience. For that reason we did not perform here 
 such a calculation of R. 
 
 Let us finally summarize what may be concluded as physical result 
 from our investigations concerning the turbulence problem. In Part I 
 we recognize that the laminar motion and its stability condition are 
 not of essential significance for the turbulence problem and the 
 critical Reynolds number. In Part II, however, we investigated the 
 turbulent motion itself and may hence give a few data on the turbulent 
 state of motion. In general, the velocity distribution over the 
 entire tunnel is of the simplest type; it is - according to the test 
 conditions - linear or constant (section 3)- At the center there is, 
 for symmetrical flow between two walls at rest, a sharp break; at 
 the walls the flow clings, for the t\^i ' profile, to the walls 
 (section 2). The calculations do not disclose anything about the fact 
 that the 1/7 profile is valid until far into the tunnel interior. The 
 turbulent oscillations are for Couette's case almost harmonic in the 
 interior of the tunnel (section 3; equation (53)); in the proximity 
 
MCA TM 1291 59 
 
 of the walls all oscillations will occur. The velocity of the waves 
 agrees with the wall velocity (section 2, equations {kk) - (hkc)) ', for 
 Couette's case there exist two groups of turbulent oscillations, one 
 of which agrees, with respect to its velocity of propagation, with one 
 of the walls, whereas the other group possesses the velocity of the 
 other wall. Thus the turbulent disturbances show, superficially, the 
 character of a wall disturbance. It must, however, be emphasized that 
 these disturbances are capable of existence as free oscillations, inde- 
 pendently of wall roughness and similar influences. The amplitude of 
 the turbulent waves considerably increases toward the walls (this 
 follows from eqiiation (4^), section 2) and goes toward zero only 
 directly at the wall. 
 
 The wave length of the occurring oscillations (Part I, section 3) 
 is, with respect to order of magnitude, equal to (rather somewhat 
 larger than) the tunnel width. The minimim value of the Reynolds num- 
 ber (Part I, section 8) for which turbulence is still possible, lies - 
 with respect to order of magnitude - near 10^. From the profile t) ' ', 
 
 Blasius' T ~ u'/ seems to result, under certain presuppositions, as 
 the law of resistance for smooth walls. For rough walls it probably 
 approaches (section 2) the hydraulic law f ~ u^. The piorpose of the 
 present report was not so much to establish these regularities, to a 
 great part known before, as it was to prove that all results obtained 
 so far (seemingly partly contradicting each other) can be uniformly 
 described mathematically with the aid of simple basic assumptions. 
 
 I wish to express here my deepest gratitude to my revered teacher. 
 Professor Sommerfeld, for suggesting this report and for frequent 
 assistance. 
 
 Translated by Mary L. Mahler 
 National Advisory Committee 
 for Aeronautics 
 
60 
 
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