1、NATIONAL ADVISORY COMMIEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1375 Translation of UUkr eine dreidimensionale Instabilitat laminarer Gzttingen, Nachr. a. d. Math., Bd. 2, Nr. 1, 1940. Grenzschichten an konkaven nijnden.” Ges. d. Wiss. Washington June 1954 Provided by IHSNot for ResaleNo reproduction
2、 or networking permitted without license from IHS-,-,-NATIONAL ADVISORY COMMITIEE FOR AFBONAUTICS TECHNICAL MEMORANDUM 1375 ON THE THREE-DIMENSIONAL INSTRBILITY OF LAMINAR BOUNDARY LAYERS ON CONCAVE WALLS* By H. GCrtler SUMMARY “he present report is a study of the stability of laminar boundary- laye
3、r 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. -undisturbed flow at a prescribed wail, the amplification or decay is com- puted for each Reynolds number and each vortex thickness. disturbance
4、s (amplification null) r_ critical Reynolds number is detemined fied disturbances on concave walls only. The result in an eigenvalue problem by which, for a given For neutral . for each vortex distribution. The n-merical calculation produces ampli- The variation of the dimension- 7 (i less - 09 Ji w
5、ith respect to a9 is only slightly dependent on the VR shape of the boundary-layer profile. 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. amplified vortices the
6、 flow still is entirely regular; transition to tur- bulent flow may not be expected until the Reynolds numbers are higher. The numerical results yield informa- At the very first appearance of 1. INTRODUCTION Until now the stability calculations of laminar two-dimensional fluid flows on straight wall
7、s 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 Noethers comprehensive report, 1921 (ref. 2), the researches by Prandtl, Tietjens, Tollmien, md Schlichting *%her eine dr eidimens ionale Ins tab
8、ili tst laminarer Grenzsc hie ht en an konkaven ! 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
9、hydrodynamic equations with regard to the components of the disturb- ance. the stream function of the disturbance in the form y “his way the problem could be narrowed down to an expression for . A particular disturbance can then be built up by the Fourier method as a disturbance of a general kind by
10、 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. b The more general expression of three-dimensional disturbances in the form of traveling waves, which are
11、 parallel to the flat wall but oblique to the base flow direction, hence, for which the velocity compo- nents ui(i = 1,2,3) are given by (z coordinate parallel to wall and perpendicular to principal flow direc- tion), was analyzed by H. B. Squire (ref. 6). aforementioned special case (1.1) to be tre
12、ated independently, he was able to show that, in the case of the disturbances (1.2) with cy # 0, amplification always occurrs at higher Reynolds numbers than in the case of the disturbances (1.1) with Therefore, the investi- gation can be limited to two-dimensional disturbances of the form (1.1). Th
13、e 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). allowance for friction were applied by Schlichting (ref. 5) to the case By comparison with the a1 # 0, a? = u12 + 92. I Tollmiens claculations
14、for the flat plate with . Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TM 1375 3 a of flow within a rotating circular cylinder. The stabilizing effect of the wall curvature is such that the critical Reynolds number, formed with the displaceme
15、 t thickness 6* of the boundary layer, increases witoh increasing 6,jlR (R = radius of circular cylinder). This stabilizing effect corresponds likewise with the concepts associated with the action of the centrifugal force (compare Prandt.1, ref. 4). b Boundary-layer flows on slightly curved stationa
16、ry walls were inves- tigated by the writer (ref. 10) for stability against two-dimensional dis- turbances of the form (1.1). Tollmiens result for flat walls, with fric- tion neglected and, hence, with the evaluation of the critical Reynolds number disregarded, was the well-known stability criterion
17、which states that boundary-layer profiles with ififlection point are ustable. 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 R
18、 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. Howeve
19、r, the effect of the wall curvature is extremely small. sign of U“(Y), a change of sign at Ut + - 1 Ut is necessary (curvature . n 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 appea
20、r. A confirmation of the criterion fol- lows from the fact that the same can be applied also to Schlichtings Case of a rotating cylinder, as explained in detail in the aforementioned report. There the criterion, in accord with Schlichtings results, yields a stabilizing effect of the rotating concave
21、 wall. In unpublished calcu- lations, Schlichting investigated the case of the stationary curved wall in analogy to his own and Tollmiens calculations for the flat wall, with allowance for friction, for the purpose of observing the wall-curvature effect on the critical Reynolds number. Mr. Schlichti
22、ng told me that these calculations also proved the stabi- lizing effect of concave walls and amplifying effect of convex walls. In a personal conversation, In the present report, it will be shown that boundary-layer profiles on concave walls can become unstable relative to certain three-dimensional
23、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,
24、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 rec
25、tangular zones (compare fig. 1) taken from Taylors report. .I Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-4 NACA TM 1375 The expression of corresponding equidistant vortices in the boundary layer at a curved wall, in which the axes of the vortice
26、s are parallel to the principal flow direction (see representation in fig. 2), leads through the Navier-Stokes 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 vort
27、ex 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
28、, 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. 8 At the present. state of the experimental investigations, only the order of magni
29、tude 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. DEVELO
30、PMENT OF THE DIFFERENTIAL EQUATIONS OF DISTURBANCE Consider the case of two-dimensional 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 boundary-layer t
31、hickness 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. 6 on the wall The basic flow is along the x-direction (x = arc length along the wa
32、ll), y(2 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. Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-In these coordinates, the
33、 first Navier-Stokes equation, for example, reads in full rigor and generality n where and z-directions, p the pressure, P the density, and v the kinematic viscosity of the flowing medim. disregarded as customary, and R is assmed great with respect to 6 by 1 l The Navier-Stokes equa- binomial develo
34、pment of - tions and the continuity equation, up to the terms of the order $, read then u, v, w are the velocity components of the total flow in x-, y-, PA1 flc?w variatims ir, x-direction are ana n9 R-Y A J . Provided by IHSNot for ResaleNo reproduction or networking permitted without license from
35、IHS-,-,-G NACA m 1375 * TIi- undist,urbed flow u = uo(y,t), v = 0, w = 0, p = po, which itself is to be a solution of the hydrodynamic equations, for which, therefore, au, d2U0 1 duo at hY2 R 3Y - = v - - are applicable, is to change very little during the interval in which the duo disturbances are
36、to be observed. Therefore, - and, hence, its equiva- at lent viscosity term is deliberately disregarded hereafter 2nd 0 is Put = UO(Y) This basic flow uo(y) is a laniinar boundary-layer flow formed by Use -is made occa- some previous history based on the viscosity effect. sionally of the conventiona
37、l idealization of such a bourdary layer, which consists in assuming instead of the asymptotic transition in the outer flow value uo(6) = Uo at a certain point y = 6 = “boundary-layer thickness,“ while putthg uo = Uo for y 2 6. The minor effect of the assumedly slight wall curvature on the outside fl
38、ow is iaored, since it; plays no part within the framework 04 our theory of a first approximation. On the assumption that R 6, the term 3 relative to $ and can be disregarded in the equa- b 0 = !Jo = const., an increase of uo(0) = 0 at the wall up to the 0 R the term - - l - with respect to - a2u Ra
39、y ay2 two other velocity components. becomes evident in the term - of the second equation (2.1). Moreover, R no systematic difficulties are encountered if 1;he cited small terms are carried along in the subsequent calculation. But, since they only hainper the -;ask and contribute nothing to the effe
40、ct involved, they are discounted. The essential effect of the wall curvature U2 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA 1M 1375 7 So, in conformity with the arrangements at the beginning, the fol- lowing disturbance equation is used: u =
41、 uo(y) + ul(y) cos azePt v = v (y) cos azep t w = w (y) sin azep t p = Po(Y) + P,(Y) cos adt 1 1 a is to be real and the calculation for p itself is to result in real values; a = a, where h is the wave length of the disturbance. The quantity p governs the amplification or damping of the flow, depend
42、ing 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 3 represents tne streamline pattern in a section normal to the principal flow dirtctiori. h I
43、ntroduction of equation (2.2) in the equations (2.1) following the omissions arising from R 6 results in the linearized equations with respect to the disturbance They apply as long as the disturbance velocities are small with respect to the basic-flow velocity. To treat this system of ordinary diffe
44、rential equations for the Unknown functions ul, vl, wl, and pl, we insert w1 from (2.3.4) in (2.3.3). The result is p as a differential expression of the third 1 Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-8 NACA TI4 1375 3 order in vl. On substi
45、tuting this expression for p1 in (2.3.2), u1 appears as differential expression of the fourth order in vl. Combined w with (2.3 .l), the following system of coupled differential equations is obtained for u1 and vl: (2.4.1) 2 R 4 d2vl 2u uo v - - (p + 2va2) - + u*(p + vuqv, = - dY 4 dY2 u1 (2.4.2) v1
46、 When u1 and v1 are known, w1 and p1 are computed from (2.3.4) and (2.3.3). It is not recommended to set up a differential equation of thc. sixth order for u1 or for v1 alone by further elimination. The subsequent calculations are rather based direct on the systems (2.4.1) and (2.4.2) and merely pro
47、duce a simplified mode of writing. With 6 denoting a suitably chosen measure for the boundary-layer thickness, the following dimension- less factors are utilized: 4 1 u=- uO uO For neutral disturbances, that is, that state of transition in which the disturbances are neither amplified nor damped, p =
48、 0, hence T = 0. . Provided by IHSNot for ResaleNo reproduction or networking permitted without license from IHS-,-,-NACA TM 1375 9 It further is appropriate to use the quantities ? v = v1 instead of u1 and vl. The prime is also omitted in the following with- out running a chance of causing a mixup
49、with u and v defined by (2.2). The differential equations (2.4.1) and (2.4.2) can be written briefly as differen+!sl equations for u and v, as follows: d U Lu=-v d71 L Lv = - ENT SYSTEM OF INTEGRAL EQUATIONS Greens function G q; qo (1) G(q; v0) in 0 5 q 5 w at q # qo is twice differentiable with is identified by the following postulates: resp
