Velocity field within a vortex ring with a large elliptical cross section

TL;DR

This paper derives the velocity field within a steady toroidal vortex with an elliptical cross section, using coordinate transformations to solve the governing equations.

Contribution

It introduces a method to analyze vortex rings with arbitrary core radius and ellipticity by transforming to invariant coordinate sets.

Findings

Vorticity decreases monotonically with distance from the axis.

Circulation can be smaller or larger than Hill's spherical vortex for given parameters.

The method applies to steady vortex rings with elliptical cross sections.

Abstract

The velocity field within a steady toroidal vortex is found for arbitrary mean core radius and section ellipticity. The problem is solved by transforming to coordinates that define invariant sets. The method allows the properties of the coordinate system metric tensor to be exploited in the continuity equation in order to obtain the solution. The vorticity is found to decrease monotonically with distance from the symmetry axis. For a given outer radius and outer perimeter velocity, the circulation of the vortex ring can be either smaller or larger than that of Hill's spherical vortex.