Nonlinear phenomena in closed flows near critical Reynolds numbers

Abstract

It is shown that small normal perturbations in a closed steady fluid flow always either decrease or increase monotonically irrespective of the form of the walls or the character of the motion. Making use of this result, steady motions near critical Reynolds number R are investigated in general form. In the general case two series of steady motions, analytically dependent on R, intersect at the critical R. The motions of one series, starting from equilibrium at R = 0, are stable for R less than a critical value R 0, and are unstable for R > R 0. The motions of the second series, on the contrary, are stable above some critical point, are unstable below it and do not exist at a certain R 1 < R 0. The motions of both series coincide at the critical point itself, and near R 0 their difference varies as ( R − R 0). A special case may occur if a problem permits a symmetric transformation for example, an arbitrary displacement along an axis) and the perturbation which disturbs the stability is invariant with respect to this transformation. Two new series of steady motions, analytically dependent on (R − R 0) 1 2 , then appear above the critical point. The situation is exactly the same in the case of a fluid moving between two rotating cylinders. Until recently in hydrodynamic stability theory, the stability of non-closed flows was studied almost exclusively, efforts being directed mainly to the calculatation of the critical Reynolds number, i.e. to the solution of the linear small perturbation equations. Only Landau [1,2] excess of the critical, has shown that there must exist (he had in mind non-closed flows) unsteady periodic motion, the amplitude of which is proportional to (R − R 0) 1 2 . As regards closed flows, apparently only the motion of a fluid between two rotating cylinders (Taylor's problem) has been investigated. Taylor [3] calculated the critical Reynolds number for this problem and the pertubation which breaks down the steady flow, and it was then shown in his experiments [3] (cf. also the work of Lewis [4]) and in the experiments of Stuart that all the theoretical conclusions are correct. In his experiments it is also seen that after the break-down of the basic flow a new steady motion is established, whose intensity differes from the intensity of the basic flow by an amount which is proportional to (R − R 0) 1 2 . Stuart applied the Landau concept to the Taylor problem and showed that, although the motion here is closed, the conclusions of Landau remain partially in effect and the theory is in excellent quantitative agreement with experiment. The impression generated is that the laws indicated by Landau must also hold for closed flows. The Taylor case, however, is not a typical case of closed flow. The length of the cylinders in the Taylor experiments was 800 times greater than the width of the space filled with fluid, and consequently it may be thought that the phenomena observed there should be like the phenomena in infinite non-closed flows. It would be very interesting to investigate experimentally some typical motion, for example the motion between two rotating spherical surfaces. In this paper, a general investigation of the nonlinear hydrodynamic equations is carried out for closed flows near critical Reynolds numbers, and it is shown that the Taylor problem is really a special case and that phenomena near critical points in typical closed flows look absolutely different, the method used here is a development of the method of [6].