Some developments in the theory of vortex breakdown

Abstract

The primary aim of the analysis presented herein is to consolidate the ideas of the ‘conjugate-flow’ theory, which proposes that vortex breakdown is fundamentally a transition from a uniform state of swirling flow to one featuring stationary waves of finite amplitude. The original flow is assumed to be supercritical (i.e. incapable of bearing infinitesimal stationary waves), and the mechanism of the transition is explained on the basis of physical principles that are well established in relation to the analogous supercritical-flow phenomenon of the hydraulic jump or bore. In previous presentations of the theory the existence of appropriately descriptive solutions to the full equations of motion has only been inferred from these general principles, but here the solutions are demonstrated explicitly by means of a perturbation analysis. This has basically much in common with the classical theory of solitary and cnoidal waves, which is known to explain well the essential properties of weak bores.In § 2 the basic equations of the problem are set out and the leading results of the original theoretical treatment are recalled. The new developments are mainly presented in § 3, where an analysis of finite-amplitude waves is completed by two different methods, each serving to illustrate points of interest. The effects of small energy losses and of small flow-force reductions (i.e. wave-resistance effects) are considered, and the analysis leads to a general classification of possible phenomena accompanying such changes of integral properties in either slightly supercritical or slightly subcritical vortex flows. The application to vortex breakdown remains the focus of attention, however, and § 3 includes a careful appraisal of some experimental observations on the phenomenon. In § 4 a summary is given of a variant on the previous methods which is required when the radial boundary of the flow is taken to infinity. The main analysis is developed without restriction to particular flow models, but in § 5 the results are applied to a specific example.