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.
Swirling flow fields in combustion chambers can be determined based on swirl ratio and a velocity profile specified along some path to the vortex center. A method is presented whereby flow fields can be constructed by applying the continuity equation in a streamline coordinate system and imposing irrotationality about the symmetry axis of the vortex ring. The swirl ratio may be specified at the vortex core, along with a velocity profile along any semi-axis of the vortex cross section.
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