Azimuthal vorticity gradient in the formative stages of vortex breakdown

Abstract

This paper is motivated by an observation: in the nascent state of vortex breakdown before it develops into a full-grown radial expansion, an initially straight vortex core first swells, and does so even in a straight pipe for no apparent reason. Although this initial swelling may be explained in many ways according to the perspectives chosen, we offer our own interpretation framed solely within vorticity dynamics: the radial swelling as well as the subsequent growth are induced by the azimuthal vorticity gradient decreasing downstream. The negative azimuthal vorticity gradient first appears at start-up and moves eventually into the region where the circulation reaches its steady-state value. The vorticity gradient can become negative without necessarily being accompanied by a sign-switch of the azimuthal vorticity itself.The key point – that the negative azimuthal vorticity gradient induces initial radial swelling and its growth – is demonstrated in two analyses. First, a kinematic analysis results in an equation for the radial velocity where the azimuthal vorticity gradient appears as a source term. Its solution shows, in general and explicitly, that the negative azimuthal vorticity gradient does induce radially outward velocity. Two heuristic examples serve to illustrate this point further. In the second analysis, using the equation of motion in the streamline coordinates, the negative azimuthal vorticity gradient is shown to diverge the meridional streamlines radially. A numerical simulation using a modified vortex filament method not only corroborates this role of the azimuthal vorticity gradient in initiating and promoting the radial expansion, but also adds details to track the formation process. Both analyses and simulation support our interpretation that the initial radial swelling and its subsequent growth are induced by the negative azimuthal vorticity gradient.