The velocity of a vortex ring with a thin core of elliptical cross section

Abstract

The motion of a circular vortex ring with a thin elliptical core is considered. The core is untwisted so that the vortex ring is axisymmetric and the vorticity in the core is proportional to distance from the axis of symmetry. The core rotates with a constant angular velocity comparable to the circulation frequency, as in Kirchoff's two-dimensional solution. The velocity of the ring, suitably defined, is periodic and the average velocity is $\frac{\Gamma}{4\pi R}\big [\ln\big (\frac{16R}{a+b}\big ) - \frac{1}{4}\big ],$ where $\Gamma$ is the circulation around the core, a and b are the semi-major and semi-minor axes of the core cross section and R is the radius of the ring. This mean velocity is smaller than the velocity of translation of a ring of the same radius and circulation but with a circular core of the same-cross-sectional area.