Stability of stationary endwall boundary layers during spin-down

Abstract

Since Bödewadt's (1940) seminal work on the boundary layer flow produced by a fluid in solid-body rotation over a stationary disk of infinite radius there has been much interest in determining the stability of such flows. To date, it appears that there is no theoretical study of the stability of Bödewadt's self-similar solution to perturbations that are not self-similar. Experimental studies have been compromised due to the difficulty in establishing these steady flows in the laboratory. Savaç (1983, 1987) has studied the endwall boundary layers of flow in a circular cylinder following impulsive spin-down. During the first few radians of rotation, the endwall boundary layers have a structure very similar to Bödewadt layers. For certain conditions, SavaÇ has observed a series of axisymmetric waves travelling radially inwards in the endwall boundary layers. The conjecture is that these waves represent a mode of instability of the Bödewadt layer. Within a few radians of rotation however, the centrifugal instability of the sidewall layer dominates the spin-down process and the endwall waves are difficult to examine further.Here, the impulsive spin-down problem is examined numerically for Savaç’ (1983, 1987) conditions and good agreement with his experiments is achieved. New experimental results are also presented, which include quantitative space-time information regarding the axisymmetric waves. These agree well with both the numerics and the earlier experimental work. Further, a related problem is considered numerically. This flow is also initially in solid-body rotation, but only the endwalls are impulsively stopped, keeping the sidewall rotating. This results in a flow virtually identical to the usual spin-down flow for the first few radians of rotation, except in the immediate vicinity of the sidewall. The sidewall layer is no longer centrifugally unstable and the circular waves on the endwalls are observed without the influence of the sidewall instability.