\ii\r/\'j'm NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1375 ON THE THREE-DIMENSIONAL INSTABILITY OF LAMINAR BOUNDARY LAYERS ON CONCAVE WALLS By H. Gortler Translation of ^Ubsr eine dreidimensionale Instabilitat laminarer Grenzschichten an konkaven \A/aiiden.'* Ges. d. Wiss. Gottingen, Nachr. a. d. Math., Bd. 2, Nr. 1, 1940. Washington June 1954 3SI ni^^i -hbo 1 1^1 7 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1375 ON THE THREE-DIMENSIONAL INSTABILITY OF LAMINAR BOUNDARY LAYERS ON CONCAVE WALLS*"^ By H. Gortler SUMMARY The present report is a study of the stability of laminar boimdary- 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 n'omerical calculation produces ampli- fied disturbances on concave walls only. The variation of the dimension- less -i=^ / — with respect to a^ is only slightly dependent on the shape of the boundary -layer 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 two-dimensional 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 Instabilitat laminarer Grenzschichten an konkaven Wanden." Ges. d. Wiss. Gottingen, Nachr. a. d. M;ith., Bd. 2, Nr. 1, 19^+0. ■'"Presents part of a thesis by H. Gortler submitted in partial ful- fillment of the requirements for the degree of Dr. Phil, in Math - Natural Sciences Faculty of the University of Gottingen. 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 distiirbances . Schlichting, 193^ (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 x-direction (= 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 x-direction 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 distiorbance in the form •(x,y,t) =(p(y)ei(°«-Pt) (1.1) A particular disturbance can then be built up by the Fourier method as a distiorbance 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 three-dimensional 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 Uj_(i = 1,2,5) are given by Ui = f^(y)ei( 0) 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 In these coordinates, the first Navier-Stokes equation, for example, reads in full rigor and generality ^+ -^— u ^ + V ^ - -^^^ + w ^ St R - y Sx ^ R - y bz -R— 1 ^P + V i—^— S^u , S^u , a^u R - y P Sx l(R - y)2 5x2 ay2 dz2 1 au 2R bv u R - y ^ " (R - y)2 Sx " (r - y)2 where u, v, w are the velocity components of the total flow in x-, y-, and z-directions, p the pressure, p the density, and v the kinematic viscosity of the flowing medium. All flow variations in x-direction are disregarded as customary, and R is assimied great with respect to 6 by 1 „„. 1 binomial development of R and (R - y)2- The Navier-Stokes equa- tions and the continuity equation, up to the terms of the order ^, read then bt \ by -R Sz \ ^ az2 R ^ ^-fv . - bt Sy R |l+^ + w^=-i|£+v/'^+^^^ 1^^ P by ^2 az2 R by dw , bw , bw :r— + V :— + W -— ot by dz bv V , bw _ — _ _ ^ _ — _ (J dy R 5z P az I ay2 az2 " R Sy ^(2.1) 6 NACA 1M 1575 The undisturbed flow u = UQ(y,t), v = 0, w = 0, p = Pq, which itself is to be a solution of the hydrodynamic equations, for which, therefore, ^Uq at 2 a Uq 1 ^^0 "^ dy2 R ay 4 1^0 R P dy are applicable, is to change very little during the interval in which the disturbances are to be observed. Therefore, ^---^ and, hence, its equiva- ot lent viscosity term is deliberately disregarded hereafter end Uq is put = UQ(y). This basic flow UQ(y) is a laminar boundary -layer flow formed by some previous history based on the viscosity effect. Use is made occa- sionally of the conventional idealization of such a boimdary layer, which consists in assuming instead of the asymptotic transition in the outer flow Uq = Uq = const., an increase of Uq(0) = at the wall up to the value Uq(B) = Uq at a certain point y = 5 = "boimdary-layer thickness," while putting Uq = Uq for y ^ &. The minor effect of the assumedly slight wall curvature on the outside flow is ignored, since it plays no part within the framework o^ our theory of a first approximation. On the assvmiption that R » 6, the term -^ relative to ^ and R oy the term - — ^n— with respect to 2_^ can be disregarded in the equa- R ^y ay"^ tions (2.1) on account of |ii - ^, ^-^ ~ ^ 21i,. The same applies to the ay & ay^ & ay two other velocity components. The essential effect of the wall curvature 2 becomes evident in the term -— of the second equation (2.1). Moreover, R 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 TO 1575 So, in conformity with the arrangements at the beginning, the fol- lowing disturbance equation is used: u = UQ(y) + U]_(y) cos azeP"*^ "v = v., (y) cos azeP^ ■w = w, (y) sin azeP*" P = Po^y) + Pi(y) COS azeP"t V (2.2) a is to be real and the calculation for p itself is to result in real values; a = ^, where A is the wave length of the disttirbance. The A 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 ctirved 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 Pu-j_ + v^ dU(- d u-[ 2„'\ a u V 2Ur -LQ 1 tiPi P^i ^ "i — ^ P dT = ' (2.3.1) (2.3.2) Pw, p Pi ^d2wi 2 \ ,d^ - " "1 (2.3.3) 1 il2 ^1 - - a dy i2.3.h) They apply as long as the disturbance velocities are small with respect to the basic-flow velocity. To treat this system of ordinary differential equations for the unknown functions u-^, v-^, w-|_, and p-j^, we insert w-]_ from (2.3.^) in (2.3.3). The result is p as a differential expression of the third 8 NACA 1M 1575 order in v, . On substituting this expression for p^ in (2.3.2), u^ appears as differential expression of the fourth order in v 1' Combined with (2.3.1), the following system of coupled differential equations is obtained for u, and v 1' 2, V _ _ (p ^ va2) a^ = VI — (2.i^.l) d Vi dy d^v U. V -| (P + 2va2) + a2(p + va2)v-,_ 2af\i. R u^ (2.1i.2) \'Ihen U]_ and vi are known, W]_ and p-, are computed from (2.3A) and (2.3.3). It is not recommended to set up a differential equation of the sixth order for u-, or for v-, alone by further elimination. The subsequent calculations are rather based direct on the systems (2.i4-.l) and (2.ii-.2) and merely produce a simplified mode of writing. With 6 denoting a suitably chosen measure for the boimdary-layer thickness, the following dimension- less factors are utilized: --i a = a£) ^ =2 ^0 Uo& T = V a262 + ^ I (2.5) For neutral disturbances, that is, that state of transition in which the disturbances are neither amplified nor damped, p = 0, hence t = a. NACA TM 1375 It further is appropriate to use the quantities (2.6) instead of u-, and v^. 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.U.1) and (2A.2) can be written briefly as differential equations for u and v, as follows: '^ Lu = — V dTl LqLv = o^nUu > (2.7) ^ by utilizing the differential operators L = - T dTl^ ^0 = dT]^ o2 In conformity with the order of this system, six boimdary conditions can be prescribed. It is especially stipulated that ui(0) = vi(0) = w-^(O) =0, i.e., that the fluid hiigs the wall. So with consideration to (2.3.^), it is required that u(0) = v(0) = v'(0) =0, With the other three conditions, the decay of the disturbance at r\ — ?» oo is attainable, or when the boundary layer at y = &, that is, T = 1 is permitted to change to the cons'tant ouLside 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 "<»" signifies "sufficiently great." The homogeneous system of differential equations (2.7) together with six homogeneous boundary conditions produce an eignevalue problem for the 10 NACA TM 1375 proposed values of U(t)) and R/6: The magnitude of amplification 3 Uo& for every given wave length A and every given Reynolds number Re = must be determined (i.e., the relationship existing between the parameters T, a, and [i, required for solving the homogeneous boundary value prob- lem, must be calculated). The neutral disturbances (3 = 0, that is, T = a) especially, call for the determination of a "critical" Reynolds number of every wave length of disturbance A, 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 object — namely, 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? Aj:-e there appreciable differences when different boundary-layer profiles U(ti) are used as basis? The question of calculating the eigen- fimction 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 Uo& FFT wall ciorvature appear only in the form of the dimensionless / — V \/R (namely, in parameter n). 3. CONVERSION OF THE DIFFERENTIAL EQUATIONS OF DISTURBANCE TO AN EQUIVALENT SYSTEM OF INTEGRAL EQUATIONS Green's function G(t]; t]„\ is identified by the following postulates: (1) Gm; t)qJ in ^ t] ^ 00 at T / ^r, is twice differentiable with respect to t\. -A 2 (2) LG - ^ - t2 G = at T ^ ri^ in ^ t] ^ dri^ (3) Gff]; T] J is continuous at the point t^ = f] , but has in its first derivative the discontinuity defined by {k) G(0; tIqi = and G also disappears at t\ — $.00. NACA TM 1375 11 Green's function HTt]; r[^\ is to have the following quality: (1) E(t\; t]^\ is four times continuously differentiable with respect to t| in ^ t) ^ oo at t^ ^ t]q. (2) LqLH = at n^^'Ho i^ O^ti^oc. (5) At the point tj = t\q, Eli]; t)q") is continuous including first and second derivatives, but the third derivative -has the discontinuity d^H lim <^^ (tio + e; no\ ^ (t^q - e; t] ) L = - 1 [dn (k) h(0; t]q\ = ^ (O; t^q) = and H disappears at r\ dTi By these requirements, G and H at ^ t] < oo are clearly identified. The calculation gives g(ti; tiq) = J i e~'''TO sinh tt] for t] ^ t] — 6""^^ sinh ttiq for t]q ^ t] (a + t) (a - t)' ■^^0 _ p-^^o e '*-• [cosh 0T\ -. cosh tt] h(ti; Tin) = < •)] > re ^0 - ae ^0 j f t sinh err] - a sinh tti| 1 for t] S tj '-'^ - e""""^ I I cosh OTi^ - cosh t (a + T)(a - t)' i Te-°^ - ae-^^ CTT OT]q - cosh TTJQ T sinh oTi^ - a sinh tt)^ for T) < TJ (5.1) 12 NACA TM 1375 In the event that p = 0, H/'t); r\Q\ becomes ka^ for T] ^ T] Wj.la) n + nn ) + 1 M O^T] - Tlf oj-^ij for no ^ ■H The differential equation system (2.7) is equivalent to the integral equation system uU) - - /; G(., no) ^ v(,„>n v(ti) = a^^j h(ti; n^) u(^T]Q)u(^TiQ^dTiQ (5.2) i+. METHOD OF DEFINING THE EIGENVALUES To begin with, the integration interval is divided into partial intervals of the same length d, and tjq^ ' and j]^^ signify points of the k-th partial interval: (k - 1) d "g ilp,^^^ < kd (k = 1, 2, ^...), and where, for simplicity sake, t\^'^) = t] ^ ^ , The subscript k added to a function symbol indicates that the particular function is to be formed at a point of the k-th partial interval, say about U-^ = UHO^ ' j; furthermore, G^^ = gM^); ^Iq^^Mj %k = ^U^^^; ^O^^/'' ^ reason of the symmetry of the Green functions, G^^ = G-^^ and E^-^ = Hj^j . NACA TM 1575 15 Patterned after the Fredholm theory the integral in (5.2) is replaced by summation Ui = - d^Gii^Ui^'vj, k=l k=l (k.la) (i = 1,2,5,...) (i^.lb) Uj^' disappears for sufficiently great arguments t^qV^). Letting, as approximation to the asymptotic transition, the boundary layer change to U = Constant at t] = 1 (that is, y = &), it can be stated more accu- rately that Uj^' = at t\q^^) ^ 1. Thus involved in the summation (U.la). If nd at finite d a finite sum is must be extended from 1 to 1, the summation along k As a result, only the v, with k ^ n appear on the right-hand side of the equation (i+.la). Correspondingly, considering only the equations with the Vj_ at which i ■^ n in (i+.lb), the infinite sum on the right-hand side can be approximately replaced by a finite sum of k - 1 up to a sufficiently great k = N, because the values H-j^j^ decrease rapidly with increasing k owing to the upwardly restricted i < n (the Uj^ themselves decay with increasing k) . The homogeneous system of the n + W equations is therefore investigated Ui + d^G.,U. k=l 'ik"k 'k = (i = 1,2, ...,N) > N (^.2) a^^dy~E^ k^k^k - Vj^ = (i = l,2,...,n) k=l for the N unknown u.(i = 1,2,...N) and the n unknown v^(i = 1,2, . . .n) . lU NACA TM 1375 The vanishing of the determinants 1 G^^U^'d G^^Ug'd ... G,^u;d 1 G U 'd 21 1 G U 'd 22 2 ... G U 'd 2n n 1 ^Nl^i'^ 'S2"2'^' ... G^- U 'd Nn n a^[xE-^-^lJ-^d a^[iE-,2^2^ . o^^H-^jjU]^ - 1 a2^iH23_U^d a^^E^^U^d . • ^^^^H2j^%d - 1 a^^E^-^Ujd a^^E^JJ^d . . a2^H^Uf^d - 1 postulated for the existance of a nontrivial solution system leaves an algebraic equation for [i at given U, cr, and t and, especially, the critical value of the dimensionless » — for t = ct. Every point t]^^' within the boundary layer contributes two non- trivial series to the determinant — ^that 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 Tiil)) produces only trivial series because there the Green functions (and U also) disappear. 5. CALCULATION OF CRITICAL REYNOLDS NUMBER VARIATION IN ITS MINIMUI4 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 boundary-layer 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 5 for the boundary- layer thickness. This is the wall y /Un distance at which the variable — \/— ^ in Blasius 's report assumes the 2 V vx value 5. NACA TM 1575 15 As a practical check on the quality of convergence of the calcula- tion method developed in section h, t]W = ti„(^) = — - — and d = — were selected and the following three approximations calciilated: for ^i^) and ^q(^), the points 0, l/2, 1; 0, l/3, 2/5, 1; 0, l/k, l/2, 3/^, 1 were taken. According to the remarks made at the conclusion of the preceding section k, the calculation of three-, five-, or seven-row determinants is involved, which result in t, linear or quadratic or cubic equation for n with respect to a and r. Evaluation for neutral disturbances (t = a) showed that, to each value of the parameter a = ctS, that is, to each wave length of disturbance there corresponds the related value of p. as smallest root. Since the equations exhibit, on the whole, coefficients with alternating prefix, only positive roots [i are obtained by this calculation, that is, positive values of the critical dimension- less (-^) p, hence an instability of the assumed type only on concave walls (R > 0) result. The results of this preliminary calculation are shown in figure h . The convergence for the parts of the curve above greater or smaller aS 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 tj (k) outside the boundary layer were assumed for individual oL values as a check that the approximations achieved in figure k 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 t] = on was very slow. Therefore, series from t) = 1 on were resorted to. Corresponding to the order of the differential-equation 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 six-row 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 tj = O.5, but on account of the unavoid- able large figures appearing in the solution of the determinants, the 16 NACA TM 1375 results could no longer be regarded as reliable. On the other hand, near the minimum on the curve of -^^-\l^ against cg5, 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('n). To this end the calculations with the T](^) places 0, l/^, l/2, 3A, 1 were repeated for the boundary- layer profiles U(ti) (sin It n + _e sin jt e '^ 1 + e ■d i + e> for < n < 1 1 for T) > 1 It sm — 2 1 + £ ^ (5.1) with ^ — S — = ±1, that is, e = 1-752, e^ = - O.389O. The first pro- file (e = 1) has negative curvature throughout, the second, (e = Gp) has an inversion point. (See fig. 5j curves (2) and (5).) To assure a physically logical comparison of the results for the several boundary-layer profiles, it was postulated that all profiles have the same momentum thickness V^O (Uq - Uo)uQdy = 5 / (1 U)UdT] (5.2) which is a measure for the loss of momentum in the boundary layer. This condition is met when between the individual boundary-layer thicknesses the relation -3 = 0.1115 = 0.132Si = O.I3762 exists. Here 6 denotes the previously defined thickness of the Blasius plate boundary layer, 5i and &2 "the boundary-layer thickness (t] = 1) for the sine profiles (5.1) with ex 3J^^ £2- The result of the comparison is shown in figure 6. It was found that, when . /— is plotted against ai3, the individual cixrves within NACA TM 1375 IT the scope of our approximation do not differ appreciably from one another. (a corresponding comparison based on the displacement thickness 6* instead of {j produces curves which differ from one another considerably.) A final calculation, as the roughest approximation to an actual boundary -layer profile, was made on the section profile JTi for "i T) < 1 U = -i 1 / (5.3) Q. for T ^ 1 (compare curve k, fig. 5)- If &, ^^ "the boundary-layer thickness of this profile (y = &:j at ti = 1), then t3 = ^ 5,. At identical momentum thickness with that of the profiles used so far, the difference is slightly greater, but, considering the rough approximation (5.3), the departure from the results so far is not very great. (Compare curve U, fig. 6.) The amplifications in the explored wave-length 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 bo\andary-layer profile. Instead of the extreme case (3.1a) of Green's function H(ti; ^q) , the more general expression is obtained from (3.1). 352 To each a& and ^t"? that is, to each pair of parameters ct, t, there corresponds a particular value of the dimensionless — ^^— / — . The curves ^^^2- = Constant are obtained by graphical interpolation after conversion gn2 of 6 to t3. (Compare fig. 8.) For greater parameter values ii.H_^ -^he V quality of the approximate calculation decreases quickly. 6. ASYMPTOTIC STATEMENTS Supplemental to these results for great and small values of ccS, 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 o 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 18 NACA TM 1375 order differential equation T2(2a2 + t2)u'2u" - 2T^{a^ + t2)u'U"u' + \r2{a2 + t2)(2U"2 - U'U'") - a^r^U^^ u = - fiUU'^u (6-1) Integration of this equation across the boundary layer gives the relation "qS 2 6 ^0 /"A'+U'S - t (a2 + t2)(2U"2 - U'U'")"}" + Sr^ia^ + t2)u'U"u' - T2(2a2 + t2)U'2u' dn V R nl 2a2 / UU'^udTi 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, r, so that the essential contributions to the integral are already included in the estimation (6.2a) (6.2) Uo^^s 2 / U'2udTi ^ . V y R / UU'^udTi -' Equation (6.2a) is evaluated by an approximation expression for u by means of a polynomial of the fourth degree in i^, taking into considera- tion the boundary conditions u(0) = 0, u"(0) = (hence v(0) = 0), u'" (0) - t2u'(0) = (hence w(0) = 0) and u' (l) + tu(1) = (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 e-'^T on account of U'(t]) = for ii > 1 and hence u" - t'^u = O). 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 ^(t + h)y^ + t2(t + U)^ _ f 1 + t + I^ + iyj^Y ^^"^^ NACA TM 1575 19 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 cr and t gives for the Blasius plate profile the asymptotic formula corrected for t3 . The same calculation for the section profile (5-5) gives the factor 2.1 instead of 2.3, hence, a slight difference only. The tie-in with the results obtained for average a values is readily accomplished with the asymptotic formula (G.k). Figure 8 represents the variation of the crit- Uqi3 r~ ical factor /^ plotted against a-3 in the double logarithmic net / Q{\'- Bi3 (curve ^^-^— = 0). The first amplification curves '^ — = c = Constant > V V are also shown. The variation of these curves at high ai3 values is obtained by addition of 2.5c to the critical values of J ^ at equal ai3, as is readily apparent from (6.^). Moreover, by (6.U) ^ = oM\K- a^ 4. oMxf^ (6.5) at great a-d , which constitutes an upper limit for the dimensionless amplification quantity ^ solely dependent on ^. Uq R An asymptotic prediction for small a is obtained also by an appro- Un^ fa priate analysis. It is found that the critical factor -^''—\h^ increases V y K proportional to (coi3)~ with decreasing 03, 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 20 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 M V 0.58- Ji (^-^ri) (T.1) It involves vortices at which ai3 == 0.1^, that is, whose wave length A is given by A « l+5^(= 5. 06) (7.2) For the Taylor vortices between stationary outside and rotating inside cylinder (ref. 3), 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 2-1/2 times the boundary-layer thickness; they even extend beyond the boundary layer. The stability limit for the Taylor vortices is given by -^^ = kl.^J—, where d is the cylinder spacing and Uq the velocity of the rotating Inside cylinder while the outside cylinder is at rest. At -a = - d, ^^ = 2.81,/^. 6 V U 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 6 turbulence can be produced imtil 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 eno\:igh to be disre- garded. When the variation in x-direction 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 boundary-layer 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 21 In the -^\/— , ai3) diagram the vortices of constant wave length V V Vr' I '^ A describe, by virtue of the identity "o** ^. (2,)-5/2 M^l (,,)3/2 (,.3. curves of the configuration with (2n:)5/2 c = —y—XJ— = Constant. These curves cross the system of curves of constant amplification and are reproduced for some values of the parameter ~^\ i^i figure 8. Tlriere are curves in this series which cross the zone of unstable disturbances - when they enter it they always cross it since the curves of constant amplification, and especially the curve ^ — =0 at great a values, vary proportional to (a-fi)2 (see equation {6.k)) - 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 ^ — = 0. This is the case for the curve with the parameter ~^\ = 50 shown in figure 8. Therefore, if the disturbance of the wave length in wandering through the thickening boundary layer ever is to reach an undamped state, w - must be ^ 5O; that is, the inequality 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 boimdary 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 ai9 ~ 1.1, where, therefore, A « 5.7^ = 0.635 (7.6) 22 NACA TM 1575 Therefore, the wave length A of the disturbance must have a certain magnitude characterized by (T.5) if it ever is to get in a critical situ- ation with the increasing boundary-layer 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 boundary-layer thickness is already reached at a certain stage, where as yet ^ > O.63, after which the disturbance is amplified until a certain second ratio ^ < 0.63 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- rience the highest amplification, or in figure 8, the curves = Constant, which prevail at the start of their amplification path before transition to turbulent flow in the range of minimums of the curves '^ — = 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 Re^ = -2— = 2.6 x lO-' -X V at point ^ = 0.75 and for Re^ = 5.1 X 10^ at point - = O.U5. Using R R the Blasius law of growth of the boundary layer at the flat plate \ ~ ^ \; t7~ ) ^^ basis, the values of the critical dimensionless factor — ^y^- 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 ^ = Constant = 7^ ^ — which, based upon the law -6 = — \ ; — , is the same] Uq I|. V ' D \|Uq' y are minimijm at about a, = 0.6, hence oS = 5-5- Assuming that the turbu- lence is caused by vortices of the type investigated here, at reversal the boundary -layer thickness would, roughly speaking, have increased up to the order of magnitude of the width of the highest amplified vortices. The /lJoR^-2/3 latter, m turn, would have a wave length A of about A ~ 50Rl-^"^) NACA TM 1575 25 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 ^ — become unreliable, is insufficient. Translated by J, Vanier National Advisory Committee for Aeronautics 2k NACA TM 1375 REFERENCES 1. Blasius, H. : Grenzschichten in Fliissigkeiten mit kleiner Reibung. Diss. Gottingen 1907> erschienen in Z. f. Math. u. Physik, Bd. 5^, 1908, pp. 1-37. (Available as NACA TM I256.) 2. Noether, F. : Das Turbulenzproblem. ZAMM 1, 1921, pp. 125-138. 3. Taylor, G. I.: Stability of Viscous Liquid Contained Between Two Rotating Cylinders. Phil. Trans. Roy. Soc . (London), vol. 223, Feb. 8, 1923, pp. 289-3^3. h. Prandtl, L.: Einfluss stabilisierender Krafte auf die Turbulenz. Vorticige aus dem Gebiet der Aerodynamik und verwandter Gebiete. Aachen 1929, pp. 1-7 . (Available as NACA TM 625.) 5. Schlichting, H. : tJber die Entstehung der Turbiilenz in einem rotie- renden Zylinder . Nachr. Ges. Wiss. Gttttingen, Math.-Phys. Kl. 1932, pp. 160-198. 6. Squire, H. B.: Stability for Three-Dimensional Disturbances of Viscous Fluid Flow Between Parallel Walls. Proc . Roy. Soc. (London), ser. A, vol. 142, Nov. 1, 1933, pp. 621-628. 7. Schlichting, H. : Neuere Uhtersuchungen 'liber die Turbulenzentstehung. Naturwissenschaften, 22. Jahrg, 193^, PP. 376-381. 8. Prandtl, L. : The Mechanics of Viscous Fluids. Vol. Ill of Aerodynamic Theory, div. G, W. F. Durand, ed., Julius Springer (Berlin), 1935, pp. 3^^-208. 9. Clauser, Milton, and Clauser, Francis: The Effect of Curvature on the Transition From Laminar to Turbulent Boundary Layer. NACA TN 6l3, 1957. 10. G'ortler, H. : ^er den Einfluss der WandkrVtmnung auf die Entstehung der Turbulenz. Z.f.a.M.M., Bd. 20, Heft 3, June 19^0, pp. 138-1^17. NACA TM 1375 25 Colored fluid s:^-^;,-^r^^^^^;^^^^^^^ r Inside cylinder Outside cylinder Figure 1.- Vortex between the walls of two coneentric rotating cylinders according to G. I. Taylor, streamline pattern following incipient instability (inside and outside cylinder rotate in same direction). 26 NACA TM 1575 Figure 2. - Vortex distiirbances in the flow of a flmd on a concave wall, axes of vortices parallel to principal flow direction. NACA TM 1575 27 y A Figure 3.- Scheme of streamline pattern in a section at right angle to the principal flow direction. 28 NACA TM 1375 Figure 4.- The critical factor Uo^ — for Blasius's flat plate bound- R ary layer plotted against a& computed by three increasing approximations. NACA TM 1575 29 Figure 5.- The boundary-layer profiles of equal momentum thick- ness i3 used as basis of the calculation. 50 NACA TM 1575 2.0 1.5 1.0 05 Uot> V^ V |/r k \ 1 1 \ \ \ \ \ (4) \ 1 \ >'// \ \ ^ /A \ \ /M/ \ \ y V / \ \^ .-"" ^y /(3) ^ ^ — ^-ai) 0.2 0.4 06 Figure 6.- The, critical factor — S- -L plotted against a-a for the V \( R boundary -layer profiles of figure 5. NACA TM 1575 51 Figure 7.- Approximate function for u (according to equation (6.3)). 52 WACA TM 1575 0.2 0.4 0.6 0.81.0 2.0 4.0 6J0 8PIO 20 40 60 80 2000 -1000 0.01 0.02 5 0,05 0.075 0.10 0.25 0.5 0.75 1.0 2.5 5.0 7.5 10 Figures.- Total variation of the critical factor — ^J— plotted against ai3 and the first amplification curves '=^77— = const. The V parallel lines in this diagram represent individual vortices (A = const.) in a slowly thickened boundary layer at constant out- side flow velocity Uq. NACA-Langley -6-29-54-1000 cd C CO ^ -5 " CI> CO 5 S HI O 2 O ^ (S S 2 - -iB a O :J3 >< H ^ S- o r;^ H > 3 5 J r^ -^ < 1 u O J CQ " ^ <.S Q ^ S tS gSg S3 m '5 05 O p H ':3 W ''! r/^ ^ c 33 £ J < aZ.<< Z Z O J? QJ £ c o XI t^ (1) c ■rt ""^ "S - •^ < o > = z w 0) C^ C - > CD JS 1 . M n — ;^ tH j:: 5 ■'=' <" ti -frt - J= " J= c -c ^' _ _ 0) « O " ^ c _ ^ -- 3 g Z I O g t, rt ii Qj' , rt ti > ^ a " « o o "S"" •^ -c ■" >.i ^ CJ ^ QJ 3 c S r K a ■a .2 ^■§ o rt^ ■° S U O • iH iH _-, ii , , --i M O O 2. •§"•«£« ^^ g. o a, Q, g j- '■ - '- -g js c " s-.a .2 2 oj g « S X .2 5 ^^ QJ O — ' ' nl ss% >. 5, 13 ■a _ 3 ^ ::5 •S OJ c« <3 g 2 M 3 ;- 'U ^ > c CO _ .*- nl rt J C "CMC ii<:^ 53 - w ozo>c5S2 *-4 ^ *"* M O ^5 2 H>3 3 « U m =s oa z = 2S8^ ^§z| O J CO " - = Z " > QJ Ji! 1 . M n ■S " .2 rt 5 ^ <" ti « -O rt j= " x; c CJ o QJ N 2 W >- C £ QJ QJ O O o QJ ii ^ ^ t! Si, IS^32 o -C £1 :;:; w o o o r- **- QJ - J= 73 "O rt T3 rt oj g ■o .2 tS QJ ;;; C 5 ^ QJ j3 -O ^ rt ^ ? S > ;S QJ OJ J= 2 -c m£ •"«?>-■=;■ o ^§ nl CO QJ g « £ c ? 0) o O M^ O ■ <" 3 ;,' OS « 5^ ^ ■M QJ o • 'C t; .^ in cj o Tj oj rt "" c ,^-C 3-^=2 a .2 .2 S S X ^ ° 03 <*-!*- a ;;^ ! n c "i* ■ 2 2 S a-. 2 QJ CO ^ — ' ^ >, o to > •S " 3 -O a « s| rt g QJ ^ _ 3 t, S ^ « s « S C t. CO 3 rt £ => S « s X ^ .S OJ O 73 ^73 ■•^ ^ u ^ C4 0) C OJ CJ J3 nl XI s s-e g — 333 < < z tg ::; « C C CO 1 s 1 5 =- - ii <; — 01 c — r^ QJ CO - tH ^ CO cn ti 5 o •- O < QJ — o rt 2, fcM ozo^OSS - « a a • >< w ^ 2h>3 3 El U CO « 03 Z c 2S8-- ^§z| O -J M 2 E Z J ^ S wg £ CO ■> b: 2 fi - -o a= CO D g < H K S -2 H s J < rt z <: < z z o J> 5 cvT co . '-' 3 _- 5 2 c o 73 t-' QJ C ■^ < o > = Z " QJ Ci c > > 4> J^ QJ - > ±i ■::; ^ 3 rt " CK nJ a u c u a a. QJ rt ■O QJ cd 3 -^ a. 2 -- QJ 5 « QJ -C 2 J= Ml£ •s § = ^ ,„ c« 2 2 -o S ii ^ ^ — 3 t- S ■" QJ c< £ c > 01 o to — QJ C 0) — cfl -P CJ X coS 2 10 3 t, QJ OJ qJ C3 - -^ O 73 ^ 73 >-. " 3 r tH o t, rt ~ c« QJ C OJ ,. '' C O J3 rt XJ i QJ « X ^ .S rt H lao ^ s — 3 a c .2 5 Q. V rt g •a « ^i; ^ rt C >>Lf5 OJ C t* c- -n QJ ■-^ " ^ it; i5 "? ii - ^ rt 2 j^* ^ *S.gf3^ o; <■ — OJ c o) t^ ozoS(5S2 - « a a < « z < < z z o J > ■o 0) s u o c t- o .2 « 5 > il rt t < 2 e p V CO ■o •a ■a S o c .2rt < 2 u o c t< o .2 rt t 2 I o c ■c ° CD -JS H 2 S am <: o <; z c 3 H 2 u ' c ,_! CO QJ r4 3 CJ c *-t ;^ u u >>m r a ;-. r^ Tl ZJ r; s rtl 0^ ?. ^ S hJ Lh H m 0) - -a" ^^ dj C a> O Gort NAC Gese Wiss 1 o O j3 • >-i w ^ 5 3 <.S 6 Z ►J « Q ;^ s " ■> 05 O a g a» ci c ^ > CD ^ CI , cfl -r^ ■? " .2 « 5 ^ « CU T-3 £ " J= c C « X Q) CD O " ^ 1= « 5 o 3 S z IV "at. c« ii § "> o S ^ S CO ct! CD -r: o o "2 o 15 rt -O -O M .£ i: 0) „ -5 -T ^ OJ 3 K I* O S rt \- ^ ■*-* ^ > tj ^ O CO , C T3 0) I- CD c^ ,n i! u, en 3 .2^ « °-- 3 ch :^ ■" CD rt " >> p IP. c ■2 S ;3 "a a « 2 I c< S ° Si ID a, C 0) 14 X! S 6 \ u, ^^ >>m ID C CO <\ fi 0) -1 £; ".'A = s^ < ^ C4 z/ J t. "^ CO / - -2<:r ^ t. y CO ■.0 •< CD (^ 020 < o •< z CD . S § 3 2 = 5-' I" 2 > I si CMC CD 3 ID " S -3 o ■- i « I O S 3 « U M « m z 3 sS8ii ID iZ) „ cd <; z z 3 o J m 2 - < K S S wt„^ >> I P CO ID ^ CO 2 o ID £ C O ■a ^ V a :ccl ^ ^ - ■^ < o > = Z " CD CI 3 ., > CD ^ ^ . CO -rj •? " .2 rt O •* "^ ^ ■*-' t. iS S 35 :5 trt 111 —I O > CO 9 -Q ^ 3 1-. >i CD O ^ ^ S5 Q p >>:3 S '^ 2 2 o '^ ^ ;-2 t-. > ;^ rf 3 X a J u -2 H g ^ <; c« z <: < z z o J & 3 o.~ — -f « - J3 " £ 3 O CJ CD S . CO •" C CD J= CD CD o a ^ „ CO ii .S- 2 -S M a. o m CJ o '^ £ 3 S ID O CO CJ O J ^ ID " ID ^"sS I o 3 - 1-5 g t- cd ii Ijl g .0 >- :o 5 .2 O M >■ cvTJ ■o — 3 t, 1=1 ■" CD c« " >. 3 ^i* ccl "^ X! -i 3 S - rt -e cfl *-* 2 m 3 a- .2 S§ 3 S 2 3 '~' c S CD 1; >< £ .5 •O S cd > I" 3 ,^-3 ■3-^3" S- -2 .2 S CD '^ CO *i: "ii 73 ■-I t- O, CD >> 2 3 5 « ^ 5 c JS t> o '-' ^ « t; ii t, 01 CD qJ O -a ^T3 ■^ 5 CO ™ 2 S — 1^ Lh CJ tH cd CD 3 U ^ CTJ XI = S-P S < < z Eg u; k-; o ^ ■ - S S ^ „ ;3 3 s 3 a 3 5 - ' >^ en CD 3 CO _ chI " £ s ^ J2 3 ho S ^ < — ID 3 CD ^ . CD CO ^ £ Q o<9g||; (^ O z a O .rt >i w 5 H > 3 J <5 w U m CQ Z ^ S8ii 5« g Z W 3 O >- .5 Z '-' CD 3<: 3 CO 3 d a 3 0" S ■n ■^ "5 - < o > = Z M CD S- 3 . > a) ^ c4 . CO -rl ■g CO M rt o wi5 S 5 t: CD ccJ 13 ^v cfl s <: H n < 3 n < rt zz K O X3 Sz" X ^ J z<: < 3 £i„ , Sf-rt - J= " J= 3 CJ CJ OJ S V-2S S CD J= CD ID O ^ ^ 3 „ " 3 6 Z 2 I "• 3 „ 3 t S S § ISh32 ^ o S ■a 2 ° 3 -3 3 ^ S C4 V -— t- CTJ CD T1 3 tn t^ S J2 ° >» ID - > ±5 3 r. " rt " g; CD £ s J= bJ5£ . CO r- cr; ■" 01 - J3 3 S X 5 1 g 3 24= 3 c X ■S ^ « £ 3 s CD O 01^ — 2^^§ 3 s oB CD 3 ?*i -.H K 01 0; jH 2 o rt ^ .^ CO CJ O \« CO ^ > XJ d, cc! 01 c ,^-3 3-^=2 ■ a .H .2 2 g£ -3 ^ t/1 ^ -^^ u 1 _. 0; ■ OJ T3 3 S _. CO -g 2 CO ^ Q.- .2 CD 3 "5 S CO 3 "< £ C cd t-i bi 3i « S ii u O ^ ■o cci S: ii L. 01 OJ a, ■l-l *^ t* I — I 3 3 a 3 5 < z a. g 5 CO (U u . 3 >, T3 — O C fc. o °-» c — •2 s 13* •S 0) H > i^ 5 a " u S If5 o t- rt " CO o c '"' r o s (U to < 1 = U 0) -Q < -c i; z H 5 i < < z irt __, < < 2 n •r^ ■o ■o a •p4 CM •>H f-l c E c 00 a u 3 >. c o a m c o 73 3 s CD 3 > O c< CJ S O 13 o o c L. o -<:::^ MAY 2 6 im ^""^ ■) 'dm Renew online @ http://www.ufiib.ufl.edu/ Select: My Accounts Loan Policy information @ http www.uflib. ufl. edu/as/circ.html UNIVERSITY OF FLORIDA 3 1262 08106 544 2 Cf ifOU LE.FL cikun -7011 USA