Inviscid long-wave theory for the absolute instability of the rotating-disc boundary layer

Abstract

This paper is concerned with the absolute instability of the boundary-layer flow produced when an infinite disc rotates in otherwise still fluid. A greater understanding of the mechanisms and properties of the absolute instability is sought through the development of an analytic theory in the inviscid long-wave limit. It is shown that the fundamental basic flow characteristic of the absolute instability is a wall-jet in the radial direction superposed with an asymptotically small cross-flow, which generates a small reverse flow outside the boundary layer in the appropriately resolved basic velocity profile. The absolute instability is produced by a modal coalescence involving the interaction of eight saddle-points of the dispersion relation. An explicit expression for the growth rate has been obtained in terms of basic flow parameters. Most curiously, the pinch-point for absolute instability is shown to become asymptotically close to a branch-cut on the imaginary axis of the complex wavenumber plane, and unstable spatial branches emanating from the pinch-point cross this imaginary axis onto a Riemann sheet of the dispersion relation composed of solutions growing exponentially with distance from the disc. The existence of such modes contradicts the expectation of monotonic exponential decay of disturbances outside a boundary layer.