Instability and transition of disturbed flow over a rotating disk

Abstract

Disturbed flow over a rotating disk can lead to transition of the von Kármán boundary layer at a much lower Reynolds number, Re , (i.e. smaller radius) than that due to the well-known Type 1 stationary mode of instability. This early transition is due to the excitation of the Type 2 instability, similar to that found in the Ekman layer. Detailed numerical values of the growth rates, phase speeds, group velocities, neutral curves, and other characteristics of these two instabilities have been calculated over a wide range of parameters. Neutral curves for the Ekman and Bödewadt boundary layers also are presented. The minimum critical Reynolds numbers for the von Kármán, Ekman and Bödewadt layers are Re c (2) = 69.4, 54.3, and 15.1 with wavelengths L = 22.5, 20.1, and 16.6 and at angles ε = −19.0°, −23.1°, and −33.2°, respectively. These minimum critical values frequently do not well describe laboratory observations, however, because at larger Re other modes grow more rapidly and dominate the flow. The computed results are in excellent agreement with laboratory observations wherever comparison is possible. The growth of representative Type 1 instabilities with radius is shown to lead to N -factors greater than 9 at Re = 520 as appears to be necessary for transition to turbulence by the interaction of Type 1 with the basic flow. The growth of Type 2 instabilities with radius can lead to three additional mechanisms of transition. The necessary levels of excitation of Type 2 for these different mechanisms are estimated. A sequence of photographs from a ciné film illustrate one of the transition mechanisms discussed: the interaction of Type 2 instabilities and a secondary instability that is nearly perpendicular to the Type 2 vortices.