On the stability of three-dimensional boundary layers with application to the flow due to a rotating disk

Abstract

A phenomenon of boundary-layer instability is discussed from the theoretical and experimental points of view. The china-clay evaporation technique shows streaks on the surface, denoting a vortex system generated in the region of flow upstream of transition. Experiments on a swept wing are described briefly, while experiments on the flow due to a rotating disk receive much greater attention. In the latter case, the axes of the disturbance vortices take the form of equi-angular spirals, bounded by radii of instability and of transition. A frequency analysis of the disturbances shows that there is a narrow band of disturbance components of high amplitude, some frequencies within this band corresponding to disturbances fixed relative to the surface and others corresponding to moving waves. Furthermore, the determination of velocity profiles for the rotating-disk flow is described, the agreement with the theoretical solution for laminar flow being quite satisfactory; for turbulent flow, however, the empirical theories are not very satisfactory. In order to explain the vortex phenomenon just discussed, the general equations of motion in orthogonal curvilinear co-ordinates are examined by superimposing an infinitesimal disturbance periodic in space and time on the main flow, and linearizing for small disturbances. An important result is that, within the range of certain approximations, the velocity component in the direction of propagation of the disturbance may be regarded as a two-dimensional flow for stability purposes; then the problem of stability formally resembles the well-known two-dimensional problem. However, it is important to emphasize that this result-namely, that the flow curvature has little influence on stability-is applicable only to the possible modes of instability in a local region. The nature of three-dimensional flows is discussed, and the importance of co-ordinates along and normal to the stream-lines outside the boundary layer is examined. In accord with the formal two-dimensional nature of the instability, there is a whole class of velocity distributions, corresponding to different directions, which may exhibit instability. The question of stability at infinite Reynolds number is examined in detail for these profiles. As for ordinary two-dimensional flows, the wave velocity of the disturbance must lie somewhere between the maximum and minimum of the velocity profile considered. The points where the wave velocity equals the fluid velocity are called critical points, of which most of the profiles considered have two. Then Tollmien's criterion that velocity profiles with a point of inflexion are unstable at infinite Reynolds number is extended to the case of profiles with two critical points. One particular profile-namely, that for which the point of inflexion lies at the point of zero velocity-may generate neutral disturbances of zero phase velocity, corresponding to the disturbances visualized by the china-clay technique. A variational method for the solution of certain of the eigenvalue problems associated with stability at infinite Reynolds number is derived, found by comparison with an exact solution to be very accurate, and applied to the rotating disk. The fixed vortices predicted by the theory have as their axes equi-angular spirals of angle 103 degrees, in good agreement with experiment, but the agreement between theoretical and experimental wave number is not good, the discrepancy being attributed to viscosity. Finally, the correlation between the experimentally observed and theoretically possible disturbances is discussed and certain conclusions drawn therefrom. The streamlines of the disturbed boundary layer show the existence of a double row of vortices, one row of which produces the streaks in the china clay. Application of the theory to other physical phenomena is described.