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;' 5/ . NATIONAL ADVISORY COrMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1575 ON THE THREEDIIMESIONAL INSTABILITY OF LAMINAR BOUNDARY LAYERS ON CONCAVE WALLS* By H. Cbrtler SUMM ARY The present report is a study of the stability of laminar boundary layer profiles on slightly curved walls relative to small disturbances, in the shape of vortices, whose axes are parallel to the principal direc tion of flow. The result in an eigenvalue problem by which, for a given undisturbed flow at a prescribed wall, the amplification or decay is com puted for each Reynolds number and each vortex thickness. For neutral disturbances (amplification null) a critical Reynolds number is determined for each vortex distribution. The nfrimerical calculation produces ampli fied disturbances on concave walls only. The variation of the dimension less U0 with respect to at is only slightly dependent on the V HR shape of the boundarylayer profile. The numerical results yield informa tion about stability limit, range of wave length of vortices that can be amplified, and about the most dangerous vortices with regard to the tran sition from laminar to turbulent flow. At the very first appearance of amplified vortices the flow still is entirely regular; transition to tur bulent flow may not be expected until the Reynolds numbers are higher. 1. INTRODUCTION Until now the stability calculations of laminar twodimensional fluid flows on straight walls had usually been based upon disturbances in the shape of plane wave motions which travel in the direction of the flow. After some initial failures (see Noether's comprehensive report, 1921 (ref. 2)), the researches by Prandtl, Tietjens, Tollmien, and Schlichting *"Uber eine dreidimensionale Instabilit'it liminarer Grenzschichten an konkaven Wnden." Ges. d. Wiss. Gottingen, Nachr. a. d. Math., Bd. 2, Nr. 1, 1940. presents part of a thesis by H. Gortler submitted in partial ful fillment of the requirements for the degree of Dr. Phil. in Math :Naturtl Sciences Faculty of the University of Gdttingen. 2 NACA TM 1575 have, since 1921, produced results which compared well with observations and to a certain extent yielded information about the important question of the origin of turbulence from small disturbances. Schlichting, 195h (ref. 7), gave a report on the results of these investigations. A brief glance at the method is indicated. In these calculations, a velocity distribution U(y) that depends only on the coordinate y at right angles to the plane of the wall is assumed as the basic flow. The omission of variations of the laminar basic flow in the xdirection (= principal flow direction parallel to wall) was dictated by mathematical reasons; and the results of the calcu lations enabled valuable deductions to be made as long as the variations in xdirection were not excessive. To the basic flow U(y) were added disturbances of assumedly sufficient smallness to permit linearization of the hydrodynamic equations with regard to the components of the disturb ance. This way the problem could be narrowed down to an expression for the stream function of the disturbance in the form j(x,y,t) = p(y)ei(axt) (1.1) A particular disturbance can then be built up by the Fourier method as a disturbance of a general kind by a linear combination of such partial oscillations. While a is assumed as real, the prefix of the imaginary part of p determines whether there is amplification or damping with increasing time t. The more general expression of threedimensional disturbances in the form of traveling waves, which are parallel to the flat wall but oblique to the base flow direction, hence, for which the velocity compo nents ui(i = 1,2,5) are given by ui =fi(y)ei(alX+a2zt) (1.2) (z coordinate parallel to wall and perpendicular to principal flow direc tion), was analyzed by H. B. Squire (ref. 6). By comparison with the aforementioned special case (1.1) to be treated independently, he was able to show that, in the case of the disturbances (1.2) with al 0, 2 0, amplification always occurrs at higher Reynolds numbers than in the case of the disturbances (1.1) with a2 = a12 + a22. Therefore, the investi gation can be limited to twodimensional disturbances of the form (1.1). The stability investigation of laminar boundary layers relative to these disturbances was also applied to curved walls (x is then the arc length of the wall). Tollmien's claculations for the flat plate with allowance for friction were applied by Schlichting (ref. 5) to the case MNACA TM 15375 of flow within a rotating circular cylinder. The stabilizing effect of the wall curvature is such that the critical Reynolds number, formed with the displacement thickness 6* of the boundary layer, increases with increasing 6*/R (R = radius of circular cylinder). This stabilizing effect corresponds likewise with the concepts associated with the action of the centrifugal force (compare Prandtl, ref. 4). Boundarylayer flows on slightly curved stationary walls were inves tigated by the writer (ref. 10) for stability against twodimensional dis turbances of the form (1.1). Tollmien's result for flat walls, with fric tion neglected and, hence, with the evaluation of the critical Reynolds number disregarded, was the wellknown stability criterion which states that boundarylayer profiles with inflection point are unstable. Such profiles are characterized by a pressure rise from the outside of the boundary layer in the direction of flow. For curved walls, this criterion is modified to the extent that, instead of the stipulated change of the sign of U"(y), a change of sign at U" + 1 U' is necessary (curvature radius R of wall positive on walls convex to flow, negative on walls concave to flow). This Tollmien instability occurs, therefore, on concave walls only after the minimum of the pressure impressed on the boundary layer from without, and on convex walls already before the pressure mini mum. However, the effect of the wall curvature is extremely small. It is surprising that convex stationary walls in this sense act amplifying, but concave walls stabilizing, hence, that the effect of the centrifugal force does not appear. A confirmation of the criterion fol lows from the fact that the same can be applied also to Schlichting's 'case of a rotating cylinder, as explained in detail in the aforementioned report. There the criterion, in accord with Schlichting's results, yields a stabilizing effect of the rotating concave wall. In unpublished calcu lations, Schlichting investigated the case of the stationary curved wall in analogy to his own and Tollmien's calculations for the flat wall, with allowance for friction, for the purpose of observing the wallcurvature effect on the critical Reynolds number. In a personal conversation, Mr. Schlichting told me that these calculations also proved the stabi lizing effect of concave walls and amplifying effect of convex walls. In the present report, it will be shown that boundarylayer profiles on concave walls can become unstable relative to certain threedimensional disturbances. It involves an instability that does not occur on flat or convex walls. The friction is duly allowed for in the calculations and even the impediment due to the now more complicated type of disturbance can be overcome in a relatively simple manner. As to the type of these disturbances, they are similar to those investigated by Taylor in 1923 (ref. 3) in the flow between rotating cylinders and which led to the well known instability (excellently confirmed by experiments Taylor made at the same time) in the form of appearance of sharply defined vortices distrib uted boxlike in rectangular zones (compare fig. 1) taken from Taylor's report. IIACA TM 1375 The expression of corresponding equidistant vortices in the boundary layer at a curved wall,'in which the axes of the vortices are parallel to the principal flow direction (see representation in fig. 2), leads through the lavierStokes equations and the equation of continuity to an eigen value problem: The amplification of the vortex disturbance for a pre scribed basic flow on a given wall must be computed for each vortex dis turbance and each Reynolds number of the basic flow; in particular for the neutral disturbances (zero amplification) a "critical Reynolds number" for each vortex distance must be determined. It is interesting to know how these results tie in with the type of basic flow and the wall curvature, and also the size of the vortices which are amplified first at increasing Reynolds number, as well as the question of the most dangerous vortices from the point of view of transition from laminar to turbulent flow. At the present state of the experimental investigations, only the order of magnitude of the effect is of interest. All the calculations are centered on these claims, hence do not aim at an exhaustive mathemat ical treatment of the present disturbance problem but rather to a reply to the questions of interest in practice with an expenditure justifiable to the claim. 2. DEVELOPMENT OF THE DIFFEELI1TIAL EU,'ATIONS OF DISTURBANCE Consider the case of twodimensional flow of a viscous fluid on a slightly curved stationary wall. The finite curvature radius R of the wall is, for the sake of simplicity, assumed as constant, and R is assumed great compared to the boundarylayer thickness & on the wall formed under the influence of the viscosity; R is chosen positive for walls concave to the flow since the instability to be explored occurs only on concave walls and negative for walls convex to the flow. The basic flow is along the xdirection (x = arc length along the wall), y( O) is the vertical wall distance, and z is the coordinate at right angle to both in the direction of the cylinder axis out of whose surface portion the wall is formed. NACA TM 1375 5 In these coordinates, the first NavierStokes equation, for example, reads in full rigor and generality aU + R u u + v uv 8_+ 6t R y 6x 6y R y 6z R 1 P R2 2u 82u 2u +v + + _ R y P0 x (R y)2x2 6x 2 z2 1 6u 2R av u R yy (R y)2x (R y)2 where u, v, w are the velocity components of the total flow in x, y, and zdirections, p the pressure, p the density, and v the kinematic viscosity of the flowing medium. All flow variations in xdirection are disregarded as customary, and R is assumed great with respect to 5 by binomial development of 1 and R The NavierStokes equa R y (E y)2 tions and the continuity equation, up to the terms of the order read then 5+ v( "R"+) 6z= 2" ;3 z2 R ly + v + + w z 1 + v + 2 ot by R 6z P2 62 R y y ;3w + 1 6p +2 v 1 Ew2 + ( 11w +v+W +v + 6t 6y oz P 7z 6y2 az2 R oy Sv w ay R 6z 6 NACA TM 1575 Thi undisturbed flow u = u0(y,t), v = O, w = O, p = PO, which itself is to be a solution of the hydrodynamic equations, for which, therefore, 6t 6 y2 R cy u 1 0o R P y are applicable, is to change very little during the interval in which the aUo disturbances are to be observed. Therefore, uo and, hence, its equiva ot lent viscosity term is deliberately disregarded hereafter and uO is put = uO(y). This basic flow uO(y) is a laminar boundarylayer flow formed by some previous history based on the viscosity effect. Use is made occa sionally of the conventional idealization of such a boundary layer, which consists in assuming instead of the asymptotic transition in the outer flow uO = U0 = const., an increase of uo(O) = 0 at the wall up to the value u0(5) = UO at a certain point y = 6 = "boundarylayer thickness," while putting u0 = UO for y 5. The minor effect of the assumedly slight wall curvature on the outside flow is ignored, since it plays no part within the framework o4 our theory of a first approximation. On the assumption that R >> the term R relative to u and the term with respect to 2u can be disregarded in the equa R ay y2 tions (2.1) on account of u uu a2 1 _u. h The same applies to the by 6' y2 6 sy two other velocity components. The essential effect of the wall curvature becomes evident in the term of the second equation (2.1). Moreover, no systematic difficulties are encountered if the cited small terms are carried along in the subsequent calculation. But, since they only hamper the task and contribute nothing to the effect involved, they are discounted. NACA IM 1575 So, in conformity with the arrangements at the beginning, the fol lowing disturbance equation is used: u = u0(y) + ul(y) cos azePt v = vl(y) cos azePt w = wl(y) sin azePt p = pO(y) + pl(y) cos azet (2.2) a is to be real and the calculation for p itself is to result in real values; a = where 7 is the wave length of the disturbance. The quantity p governs the amplification or damping of the flow, depending upon whether it is greater or smaller than zero. The equation (2.2) cor responds to a vortex distribution at the curved wall, the axes of which coincide with the direction of the principal flow. Figure ) represents the streamline pattern in a section normal to the principal flow direction. Introduction of equation (2.2) in the equations (2.1) following the omissions arising from R >> results in the linearized equations with respect to the disturbance pu1 + v dy S 2u0 Pv1 + u1  wl 1 P  dUl a2u) p1 dPl v P iy ( dy 2  a2) S(l 2 a.2w W 1 dvl 1 = a dy They apply as long as the disturbance velocities are small with respect to the basicflow velocity. To treat this system of unknown functions ul, vl, (2.5.5). The result is pl ordinary differential equations for the Wl, and pl, we insert wl from (2.5.4) in as a differential expression of the third (2.3.1) (2.5.2) (2.5.5) (2.5.4) 8 NACA TM 1575 order in v1. On substituting this expression for pl in (2.5.2), ul appears as differential expression of the fourth order in v1. Combined with (2.3.1), the following system of coupled differential equations is obtained for u1 and vl: d2ul v (P + va2) dy2 duo .1 ( 2dy d2vl (p + 2va2) + a2(p + va2)v1 dv' 2C2uo R u1 (2.4.2) When u1 and v1 are known, wl and pl are computed from (2.5.4) and (2.3.3). It is not recommended to set up a differential equation of the sixth order for ul or for vI alone by further elimination. The subsequent calculations are rather based direct on the systems (2.4.1) and (2.4.2) and merely produce a simplified mode of writing. With 5 denoting a suitably chosen measure for the boundarylayer thickness, the following dimension less factors are utilized: T*g a = T2 % T : V/2 2 + 052 (2.5) For neutral disturbances, that is, that state of the disturbances are neither amplified nor damped, p transition in which = 0, hence T = a. d4v dvy (2.4.1) NACA TM 1575 It further is appropriate to use the quantities UO 1 u' = u/ S1 ) (2.6) v' = instead of ul and v1. The prime is also omitted in the following with out running a chance of causing a mixup with u and v defined by (2.2). The differential equations (2.4.1) and (2.4.2) can be written briefly as differential equations for u and v, as follows: dU Lu = v dl. LOLv = 02Uu (2.7) by utilizing the differential operators d2 L d2 2 O d2 0 d2 In conformity with the order of this system, six boundary conditions can be prescribed. It is especially stipulated that ul(0) = v1(0) = wl(0) = 0, i.e., that the fluid hugs the wall. So with consideration to (2.5.4), it is required that u(O) = v(O) = v'(O) = 0. With the other three conditions, the decay of the disturbance at n  oo is attainable, or when the boundary layer at y = 5, that is, q = 1 is permitted to change to the constant outside flow. Thus the smooth junction of three disturbance components with the respective values decaying with T > oo outside the boundary layer is assured. The symbol "c" signifies "sufficiently great." The homogeneous system of differential equations (2.7) together with six homogeneous boundary conditions produce an eignevalue problem for the NACA TM 1575 proposed values of U(n) and R/5: The magnitude of amplification P UO, for every given wave length A and every given Reynolds number Re  V must be determined (i.e., the relationship existing between the parameters T, a, and 4, required for solving the homogeneous boundary value prob lem, must be calculated). The neutral disturbances (p = O, that is, T = a) especially, call for the determination of a "critical" Reynolds number of every wave length of disturbance h, at which the particular disturbance is exactly maintained without amplifying or decaying. In the subsequent analysis of the eigenvalue, the practical aspect is the primary objectnamely, at what Reynolds number does amplification appear (stability limit)? What is the range of the wave lengths of dis turbances that can be amplified at all? At what wave lengths does ampli fication appear first when Re increases? What disturbances are ampli fied most and are therefore most dangerous from the point of view of turbulence? What effect has the amount of the wall curvature on these data? Are there appreciable differences when different boundarylayer profiles U(j) are used as basis? The question of calculating the eigen function is disregarded in the present report, although it may be stated that the method developed enables an approximate representation of it. It is readily apparent from (2.7) that the Reynolds number and the wall curvature appear only in the form of the dimensionless Uo/j v RE (namely, in parameter p). 5. CONVERSION OF THE DIFFERENTIAL EQUATIONS OF DISTURBANJCE TO AN EQUIVALENT SYSTEM OF INTE.CRAL EQUATIONS Green's function G(TI; 10) is identified by the following postulates: (1) G (r; 10) in 0 < !5 o at n J 10 is twice differentiable with respect to n. (2) LG d T2 G = O at n / nO in 0 5 T . dn2 (3) G(r; nO) is continuous at the point n = no, but has in its first derivative the discontinuity defined by lim (0O + (0) )o 1E; ) = 1 (4) G(O; n) = 0 and G also disappears at n , (0 D NACA TM 1375 Green's function H(T; n0) is to have the following quality: (1) H(q; T0) is four times continuously differentiable with respect to n in 0 14 oo at T / To. (2) LOLH = 0 at q io0 in 0 1 5= o. (5) At the point T = o, H(TI; T0) is continuous including first and second derivatives, but the third derivative has the discontinuity im 3 (dIO E 0 LdTi + E; BO) dH3 drI (4) H(0; T0)= (0; 0) = 0 and H By these requirements, G and H at identified. The calculation gives disappears at n> . 0 < I oo are clearly G(n; lo) 1 H(T; ) = L eTIO sinh TTI SeT sinh TTI for T) T0 for rTO TI 1 2 eaOO eTO )(cosh (oa + T)(o T) 2 o , cosh TTI)  Te~" 0 ae TI (T sinh crq a sinh TTI) for TI TT = 0 1 2 e_Ol (a + T)(a T)2  eTTI (cosh aOQ cosh  I Te' aeT sinh cOTIQ a sinh TTIO 7TO) for T0 = TI (3.1) NACA TM 1375 In the event that 0 = 0, H(T; 10) becomes Se O + TI) +.L2,, + aG(O + TI) +1 e( T),(TI 0 n) + 1 Sea + o) {2T2no + (I + ;7  for T < 10) + e TIO)a(TI T10) + for The differential equation system (2.7) is equivalent equation system u(n) = J v) = 02p G(n; 'o) d v(no)dTb0 dTIo H(T; TIO U00) U()U()dTIO (35.1a) to the to the integral (3.2) 4. METHOD OF DEFINING THE EIGENVALUES To begin with, the integration interval is divided into partial intervals of the same length d, and 110(k) and q(k) signify points of the kth partial interval: (k 1) d < _(k) kd (k = 1, 2, 3...), and where, for simplicity sake, I(k) = n (k). The subscript k added to a function symbol indicates that the particular function is to be formed at a point of the kth partial interval, say about Uk = U 1O (k) furthermore, Gik = G(n(i); I\(k)), ik = Hk (i); TIO(k)), By reason of the symmetry of the Green functions, Gik = Gki and Hik = Hki. H(1; o0)= NACA TM 1575 Patterned after the Fredholm theory the integral in (3.2) is replaced by summation ui = dZGikUk Vk (4.1a) k=l (i = 1,2,5,...) 00 vi= c2LdZ HikUkuk (4.lb) k=l Uk' disappears for sufficiently great arguments TI(k). Letting, as approximation to the asymptotic transition, the boundary layer change to U = Constant at Tr = 1 (that is, y = 6), it can be stated more accu rately that Uk' = 0 at qO(k) > 1. Thus at finite d a finite sum is involved in the summation (4.1a). If nd = 1, the summation along k must be extended from 1 to n. As a result, only the vk with k : n appear on the righthand side of the equation (4.1a). Correspondingly, considering only the equations with the vi at which i < n in (4.1b), the infinite sum on the righthand side can be approximately replaced by a finite sum of k 1 up to a sufficiently great k = N, because the values Hik decrease rapidly with increasing Sk owing to the upwardly restricted i < n (the uk themselves decay with increasing k). The homogeneous system of the n + N equations is therefore investigated n ui + d GikUk 'vk (i = 1,2,...,N) k=l (4.2) N a2Pd~IHikUkuk vi = 0 (i = 1,2,...,n) k=l for the N unknown ui(i = 1,2,...N) and the n unknown vi(i = 1,2,...n). 14 NACA TM 1575 The vanishing of the determinants 1 0 ... 0 GllU'd G12U2'd ... GnU 'd 0 1 ... 0 G U 'd G U 'd ... G U 'd 21 1 22 2 2n n 0 0 ... 1 GNIUl'd GN2Ud ... GnU'd C2pH11Uld oa2H12U2d ... a2CHIUNd 1 0 0 21iH2H1Uld a2.H22U2d ... j~pH21,P1jd 0 1 ... 0 G2IiH~1U1d a2p.H2U2d ... 2pnNUNd 0 0 ... 1 (4.3) postulated for the existence of a nontrivial solution system leaves an algebraic equation for i at given U, a, and T and, especially, the critical value of the dimensionless 05 for T = a. Every point q(k) within the boundary layer contributes two non trivial series to the determinantthat is, two series (or gaps) in which not only the terms of the principal diagonal are different from zero; a point at the border or outside of the boundary layer supplies only a non trivial series; the wall point (say, chosen as n(1)) produces only trivial series because there the Green functions (and U also) disappear. 5. CALCULATION OF CRITICAL REYNOLDS NUMBER VARIATION IN ITS MINIMUM WAVE LENGTH RANGE A few words concerning the choice of basic flow for the proposed numerical calculations are indicated. Theoretically, the basic flow U represents any boundarylayer flow formed at a wall due to friction and some earlier history. The present calculations are based on the data of the Blasius boundary layer of the flat plate (ref. 8). Several other profile forms are included for comparison. As regards the profile U of the plate boundary layer, the wall distance at which the boundary layer in its asymptotic transition to the outside flow diverges only 1 percent from this flow (curve 1, fig. 5) serves as measure 6 for the boundarylayer thickness. This is the wall distance at which the variable E U in Blasius's report assumes the 2 vx value 3. NACA T1 1375 As a practical check on the quality of convergence of the calcula tion method developed in section 4, 1(k) = T (k) = k1 and d = were selected and the following three approximations calculated: for q(k) and 1O(k), the points 0, 1/2, 1; O, 1/5, 2/5, 1; 0, 1/4, 1/2, 5/4, 1 were taken. According to the remarks made at the conclusion of the preceding section 4, the calculation of three, five, or sevenrow determinants is involved, which result in & linear or quadratic or cubic equation for jp with respect to a and T. Evaluation for neutral disturbances (T = a) showed that, to each value of the paranmeter a = ab, that is, to each wave length of disturbance there corresponds the related value of [i as smallest root. Since the equations exhibit, on the whole, coefficients with alternating prefix, only positive roots 4 are obtained by this calculation, that is, positive values of the critical dimension less (U 2vo hence an instability of the assumed type only on concave walls (R > 0) result. The results of this preliminary calculation are shown in figure 4. The convergence for the parts of the curve above greater or smaller a5 values, where the curves continue to rise, was not quite satisfactory. But the range of the minimum, which is of chief interest here, emerges sufficiently accurate. Beyond these approximations, other points q(k) outside the boundary layer were assumed for individual ca values as a check that the approximations achieved in figure 4 are not subjected to appreciable changes. The minimum becomes a few percent less and shifts slightly toward smaller ab values. Incidentally, it should be noted that the order of magnitude of these numerical values had been checked by special calculations. Origi nally it had been attempted to solve (2.7) by expanding T in power series. The convergence for u and v from n = 0 on was very slow. Therefore, series from I = 1 on were resorted to. Corresponding to the order of the differentialequation system and the number of boundary conditions, three coefficients each had to be determined. In consequence, u and v had to be joined continuously with continuous first and second derivatives within the boundary layer. This gave six linear homogeneous equations for the six still indeterminate coefficients and the stipulated disappearance of the sixrow determinant of this equation system produced the conditional equation between p, a, and T. But these calculations failed at the evaluation of the determinants. The values of u and v to be gained from the series and their derivatives could still be determined with an accuracy of 1 percent at T = 0.5, but on account of the unavoid able large figures appearing in the solution of the determinants, the NACA TM 1375 results could no longer be regarded as reliable. On the other hand, near the minimum on the curve of U0 against a, they yielded results which in order of magnitude agreed with the previous calculations. Because of the surprisingly small values of the critical Reynolds number, the new calculations explained above were carried out. The next step was to find the extent of the change in the results by a different choice of basic flow U(i). To this end the calculations with the q(k) places 0, 1/4, 1/2, 5/4, 1 were repeated for the boundary layer profiles TT. \TA + it E it _ U() = (sin inn s in 1 sinn E 1 1 2 1 + E 21+ E 1 s + E for 0 < T 1 (5.1) 1 for N 1 1 with _ = l, that is, E = 1.752, E = 0.3890. The first pro 21+ 1 1 file (e = 1) has negative curvature throughout, the second, (E = E2) has an inversion point. (See fig. 5, curves (2) and (3).) To assure a physically logical comparison of the results for the several boundarylayer profiles, it was postulated that all profiles have the same momentum thickness S=1 (UO uO)uody = 5 (1 U)Ud (5.2) UO 0 0 which is a measure for the loss of momentum in the boundary layer. This condition is met when between the individual boundarylayer thicknesses the relation 4 = 0.1116 = 0.15261 = 0.13752 exists. Here 6 denotes the previously defined thickness of the Blasius plate boundary layer, 61 and 62 the boundarylayer thickness (n = 1) for the sine profiles (5.1) with El and E2. The result of the comparison is shown in figure 6. It was found that, when r is plotted against ac, the individual curves within v Rli NACA TM 1575 the scope of our approximation do not differ appreciably from one another. (A corresponding comparison based on the displacement thickness 5* instead of a produces curves which differ from one another considerably.) A final calculation, as the roughest approximation to an actual boundarylayer profile, was made on the section profile fl for 0 1 u = (5.3) Sfor n 1 (compare curve 4, fig. 5). If 83 is the boundarylayer thickness of this profile (y = 63 at T = 1), then '8 = 1 5. At identical momentum 6 ? thickness with that of the profiles used so far, the difference is slightly greater, but, considering the rough approximation (5.5), the departure from the results so far is not very great. (Compare curve 4, fig. 6.) The amplifications in the explored wavelength range, at least in vicinity of the critical Reynolds numbers, can be determined by'the same approximate method. These calculations were made on the Blasius plate boundarylayer profile. Instead of the extreme case (3.la) of Green's function H(T; qO), the more general expression is obtained from (5.1). To each ab and L, that is, to each pair of parameters a, T, there corresponds a particular value of the dimensionless U6 /. The curves v YR 8d = Constant are obtained by graphical interpolation after conversion V of 6 to 3. (Compare fig. 8.) For greater parameter values 92, the quality of the approximate calculation decreases quickly. 6. ASYMPTOTIC STATEMENTS Supplemental to these results for great and small values of ab, a few statements are indicated. A differential equation of the sixth order for u alone can be obtained from (2.7) by elimination of v. Its form is disagreeable for the general calculation, but it enables a pre diction to be made for the extreme cases of great and small values of a and T. In this differential equation the coefficients relative to a and T represent polynomials up to the sixth degree. Considering only the two highest powers on the assumption of sufficiently great values of a and T inside the boundary layer, the problem reduces to the second NACA TM 1575 order differential equation T2(2a2 + T2)U'2u" 2T2(a2 + T2)U'U"u' + {T2(o2 + T2)(2U",2 U'U'') a2T4U2 u = iUU'u (6.1) Integration of this equation across the boundary layer gives the relation S2 1 T4U'2 + T2)(2U"2 U'U' )' + 2T2(c2 +T 2)U'U" r2(2a2 + T2) 2u 1 ..:, 2 5 0(6.2 v R R1 2a2 / UU'3udi "0 The integrals still contain the unknown function u and its first and second derivative, but the derivatives multiplied by polynomials of lower degree in a, T, so that the essential contributions to the integral are already included in the estimation 0 2 5ur2udT S) u 2 (6.2a) UU'5udT 0 Equation (6.2a) is evaluated by an approximation expression for u by means of a polynomial of the fourth degree in T, taking into considera tion the boundary conditions u(0) = 0, u"(0) = 0 (hence v(0) = 0), u"'(0) T2u'(0) = 0 (hence w(O) = 0) and u'(l) + Tu(l) = 0 (constant connection with constant tangent to the solution for u outside the bound ary layer, which according to (2.7) is given by u = Constant eTn on account of U'(1) = 0 for il > 1 and hence u" v2u = 0). As a result u is closely approximated in wall proximity and the postulated decay toward the outside is attained. Minor errors in u near the outer edge of the boundary layer are of no consequence in view of the rapidly decreasing U'; errors of u in the numerator and denominator act in the same direction (positive integrands throughout), thus affecting the result very little. Again the asymptotic relation (6.2a) manifests the existence of instability at concave walls only (R > 0). The defined polynomial for u reads rigorously u = constant + 4) + T2(T + 4) 1 + T + + !) (6 + 3! 2 3 NACA TM 1375 but in practice only the highest powers of T are effective for great T. Figure 7 represents this approximate function u for several values of T. The evaluation of the above appraisal for very great a and T gives for the Blasius plate profile the asymptotic formula S2, 32 + (6.4) corrected for 3. The same calculation for the section profile (5.5) gives the factor 2.1 instead of 2.3, hence, a slight difference only. The tiein with the results obtained for average a values is readily accomplished with the asymptotic formula (6.4). Figure 8 represents the variation of the crit ical factor kJ plotted against ac in the double logarithmic net v \J p 2 (curve =0). The first amplification curves c = Constant > 0 V V are also shown. The variation of these curves at high a~ values is obtained by addition of 2.3c to the critical values of at equal ma, as is readily apparent from (6.4). Moreover, by (6.4) S0.43 2 043 (6.5) 70 at great am, which constitutes an upper limit for the dimensionless amplification quantity a solely dependent on . UO R An asymptotic prediction for small a is obtained also by an appro priate analysis. It is found that the critical factor u increases proportional to (c~)l with decreasing ca, as expressed in figure 8. (For a more accurate prediction, data about the sixth derivative of u and v are necessary.) 7. DISCUSSION OF THE RESULTS On the basis of the data collected in the foregoing, the questions formulated above can now be answered in some detail. As regards the NACA TM 1375 stability limit, that is, the Reynolds number at which vortices of the particular type can exist for the first time without decaying again, is 0 5 16 (71) It involves vortices at which am 0.14, that is, whose wave length \ is given by h 45(= 5.06) (7.2) For the Taylor vortices between stationary outside and rotating inside cylinder (ref. 5), the vortex appearing at the stability limit has a wave length of about double the distance of the two cylinders, the vor tices thus filling quadratic cells (fig. 1). In the present case they fill cells with a width of about 21/2 times the boundarylayer thickness; they even extend beyond the boundary layer. The stability limit for the Taylor vortices is given by = 41.5 where is the cylinder spacing and UO the velocity of the rotating inside cylinder while the outside cylinder is at rest. At 8 = d, U = 2.81 6 v1  The appearance of the first vortices in the boundary layer does in no way indicate incipient turbulence of the flow. On the contrary, it should be emphasized that the flow will be regular in every way, just the same as before. (Naturally, it does not include the case in which ordi nary plate turbulence already occurs at very great ..) No incipient turbulence can be produced until the Reynolds numbers become considerably higher so that the disturbances of an entire range of wave lengths expe rience sufficient amplification. The same holds true for the Taylor vor tices between fixed outside and rotating inside cylinder; the vortices first appear as predicted, but the flow does not become turbulent until the velocities are higher. The theory developed in the present report postulates that the vari ation of the flow in principal flow direction is small enough to be disre garded. When the variation in xdirection is small, the results obtained retain their validity as good approximations. In consequence it is justi fied, under this hypothesis, to inquire into the fate of a vortex of given wave length in its wandering in flow direction through a boundarylayer thickening up at constant outside velocity. The momentary shape of the boundary layer has no appreciable effect on the results, as already seen, when it is referred to the momentum thickness as characteristic length. NACA TM 1375 In the ( ad) diagram the vortices of constant wave length A describe, by virtue of the identity U.w (a=)/2 U0 ( (7.3) curves of the configuration UO()/2 UV = C (aR)5/2 (7.4) with (2i)3/2 C =  = Constant. These curves cross the system of curves of constant amplification and are reproduced for some values of the parameter 6 in figure 8. Tierec are curves in this series which v VR cross the zone of unstable disturbances when they enter it they always cross it since the curves of constant amplification, and especially the curve 2 = 0 at great a values, vary proportional to (a5)2 (see V equation (6.4)) and there are curves in the series that never reach the instability range. Thus, the vortices corresponding to the latter are never amplified but always swallowed by the viscosity effect. In the extreme case there is a curve which is tangent to the neutral curve "2 = 0. This is the case for the curve with the parameter UN =50 V V I P shown in figure 8. Therefore, if the disturbance of the wave length in wandering through the thickening boundary layer ever is to reach an uinLdired state, VR must be 50; that is, the inequality v 1R 12/3 > 13.6t 2 (7.5) R = vY) must be fulfilled, which affords a measure for the smallest vortices which are able to experience amplification at all. At the instant where it reaches its solitary neutral state, the particular boundary disturbance has a specific wave length referred to the momentum thickness prevailing at that point. According to figure 8, the contact of the aforementioned curves occurs at about ab 1.1, where, therefore, (7.6) X 5.78 = 0.635 NACA TM 1575 Therefore, the wave length A of the disturbance must have a certain magnitude characterized by (7.5) if it ever is to get in a critical situ ation with the increasing boundarylayer thickness. If equality exists in (7.5), this instant is given by the fulfillment of (7.6); damping occurs before and after. If inequality exists in (7.5), then the critical ratio of wave length to boundarylayer thickness is already reached at a certain stage, where as yet > 0.63, after which the disturbance is amplified until a certain second ratio < 0.65 is reached; from then on the disturbance is damped again. However, the prediction about the second critical ratio is applicable only when the disturbance on the pre viously transversed path of amplification does not exceed the theoreti cally specified range of "small" disturbances. The last question to be answered concerns the most dangerous disturb ances, that is, disturbances in the whole range of wave lengths which in traveling through the boundary layer at equal Reynolds number  expe v rience the highest amplification, or in figure 8, the curves UH = Constant, which prevail at the start of their amplification path v yR before transition to turbulent flow in the range of minimums of the curves 2 = Constant. v In an article by M. and F. Clauser (ref. 9), the appearance of tur bulent flow at the concave wall was observed for Rex = Ux =2.6 x 10 Sv at point X = 0.75 and for Rex = 3.1 x 105 at point x = 0.45. Using R R the Blasius law of growth of the boundary layer at the flat plate 8 = 7 ) as basis, the values of the critical dimensionless factor fl~ are 10.6 and 8.6. They are indicated by the dashed markings in figure 8. A rough extrapolation indicates that, in the vicinity of these values, the amplification curves = Constant (or the curves V x= Constant = which, based upon the law = 2 /, is the same) UO 4 v 5 5Uo are minimum at about a = 0.6, hence aS = 5.5. Assuming that the turbu lence is caused by vortices of the type investigated here, at reversal the boundarylayer thickness would, roughly speaking, have increased up to the order of magnitude of the width of the highest amplified vortices. The latter, in turn, would have a wave length A of about h 50R21. NACA TM 1375 23 For a more accurate prediction about the most dangerous vortices, the par ticular total amplification throughout the unstable range would have to be determined for different vortices a = Constant by integration; but for this, the few amplification curves, which at greater values of v become unreliable, is insufficient. Translated by J. Vanier National Advisory Committee for Aeronautics NACA TM 1575 REFERENCES 1. Blasius, H.: 3renzschichten in FlUssigkeiten mit kleiner Reibung. Diss. Gottingen 1907, erschienen in Z. f. Math. u. Physik, Bd. 56, 1908, pp. 157. (Available as NACA TM 12.6.) 2. Noether, F.: Das Turbulenzproblem. ZAMM 1, 1921, pp. 125158. 5. Taylor, G. I.: Stability of Viscous Liquid Contained Between Two Rotating Cylinders. Phil. Trans. Roy. Soc. (London), vol. 225, Feb. 8, 1925, pp. 289545. 4. Prandtl, L.: Einfluss stabilisierender Krafte auf die Turbulenz. Vortrage aus dem Gebiet der Aerodynamik und verwandter Gebiete. Aachen 1929, pp. 17. (Available as NACA TM 625.) 5. Schlichting, H.: iber die Entstehung der Turbulenz in einem rotie renden Zylinder. Nachr. Ges. Wiss. Gottingen, Math.Phys. Kl. 1952, pp. 160198. 6. Squire, H. B.: Stability for ThreeDimensional Disturbances of Viscous Fluid Flow Between Parallel Walls. Proc. Roy. Soc., (London), ser. A, vol. 142, Nov. 1, 1955, pp. 621628. 7. Schlichting, H.: Neuere Untersuchungen Uber die Turbulenzentstehung. Naturwissenschaften, 22. Jahrg, 1954, pp. 376581. 8. Prandtl, L.: The Mechanics of Viscous Fluids. Vol. III of Aerodynamic Theory, div. G, W. F. Durand, ed., Julius Springer (Berlin), 1935, pp. 34208. 9. Clauser, Milton, and Clauser, Francis: The Effect of Curvature on the Transition From Laminar to Turbulent Boundary Layer. NACA TN 615, 1957. 10. Gortler, H.: Ober den Einfluss der WandkrUmmung auf die Entstehung der Turbulenz. Z.f.a.M.M., Bd. 20, Heft 3, June 1940, pp. 158147. NACA TM 1575 Colored fluid Inside cylinder cylinder Figure 1. Vortex between the walls of two concentric rotating cylinders according to G. I. Taylor, streamline pattern following incipient instability (inside and outside cylinder rotate in same direction). NACA TM 1575 Figure 2. Vortex disturbances in the flow of a fluid on a concave wall, axes of vortices parallel to principal flow direction. NACA TM 1375 y I~ x  Figure 3. Scheme of streamline pattern in a section at right angle to the principal flow direction. NACA TM 1375 25 Uo0 8 ujs 20 15 I 2 3 ,5 ~ ~ ~ ~ ~ 0 (F^ Figure 4. The critical factor  for Blasius's flat plate bound ary layer plotted against as computed by three increasing approximations. NACA TM 1575 (2) (3) (4) a S81 2 83 Figure 5. The boundarylayer profiles of equal momentum thick ness b used as basis of the calculation. NACA TM 1375 1.5 (4) 1.0 \  .. a 0 02 0.4 0.6 Figure 6. The critical factor  4T plotted against a% for the boundarylayer profiles of figure 5. 2.0 NACA TM 1575 1.25 77 1.00 0.75 0.50 0.25 Figure 7. Approximate function for u (according to equation (6.3)). NACA TM 1375 0.1 / I I V I I I I I I I I 0.01 0.025 0.050,075 0.10 0.25 0.5 0.75 1.0 2.5 5.0 7.5 10 Figure 8. Total variation of the critical factor U0 plotted against as and the first amplification curves = const. The parallel lines in this diagram represent individual vortices (h = const.) in a slowly thickened boundary layer at constant out side flow velocity U0. NACALangley 62954 1000  ozo J cr oJ ! a r~D o cUnZ~O o n SAW CuE o W aj n m> G .5 0 0 *ra  In c EW Z A w C E r8Z U r~s r.T Q 9 Z c Z a o 1 M "o ,S  Ui z C.zo szzolCI 53P e S z' S'S S 3 "Lg 5 5 sEjs >e ^CP icgS s;^^ W4.,u~' s<s s L S zzo>3sj(So o r~ cu n ac  o i to il C > . o z~oL C a SIE P 4 e R l~.  U . < iz zzo C. 58 11 *3 " *g 8 o . li a s s. o *  S 0s I . .0 cs 0 0 I0 BS 0b * u.c < ( U c "~ e o < 3 U2(SS(SS 7 _*, .= = i .. w j L. E o o ^ ,igs^ .g i > F 0 u i m .M. " 0 0  0C y lcd Cd Cud 0Cu Cr CuU ru m e u C.. 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