On the relation between the viscous and inviscid absolute instabilities of the rotating-disk boundary layer

Abstract

In this paper we consider the stability of the flow produced by an infinite rotating disk. A large-Reynolds-number asymptotic theory is developed to obtain the non-parallel correction to the local absolute instability (AI) found for this flow by Lingwood (1995), who used the parallel-flow approximation. Our asymptotic theory is based on the inviscid AI underlying the viscous AI and so is expected to give the non-parallel correction to the upper branch of Lingwood's neutral curve for the AI. It is found that non-parallel terms have a destabilizing effect on the AI. Also, it is shown that, although the position of the neutral curve for convective instability is known to depend on choice of measurement quantity, for AI it does not. However, in relating the asymptotic non-parallel results to the numerical parallel results at large Reynolds numbers, it is found that Lingwood's viscous AI does not, after all, asymptote towards the inviscid results. Instead, Lingwood's family of branch points is distinct from a second family of branch points that do asymptote towards the inviscid limit. We show that these two families of branch points are related by a ‘super branch point’ at which three spatial branches connect simultaneously. Lingwood's branch points, in fact, have a viscous long-wave origin, and will therefore be subjected to non-parallel effects that are some power of the Reynolds number larger than if they had been of inviscid origin.