The Steady Motion of a Symmetric, Finite Core Size, Counterrotating Vortex Pair

Abstract

The steady motion of a symmetric, finite core size, counterrotating vortex pair is characterized by circulation $\Gamma $, a velocity V, and a spacing $2x_\infty $. In the classical limit of a point vortex, the normalized velocity,$Vx_\infty \Gamma /\Gamma $, is $1/( 4\pi )$. The effect of finite core size is to reduce the normalized velocity below the value for a point vortex. The flow is governed by a single geometrical parameter$R/ x_\infty$, the ratio of effective vortex size to vortex half-spacing. Perturbation analysis is used to derive general, closed-form analytical solutions for the complete velocity field, the vortex pair velocity, and the boundary shape for a continuum of values of $R/x_\infty $. Both uniform and piecewise constant density cases are treated. These solutions illustrate the different orders at which the solution deviates from the point vortex pair. For example, the vortex shape becomes noncircular at order $(R/x_\infty)^2$, but the normalized velocity does not change until order $( R/x_\infty )^5 $. For the uniform density case, calculation of specific values of vortex pair velocity, aspect ratio, and gap ratio shows good agreement with previous numerical results.