Continuous parametric families of stationary and translating periodic point vortex configurations

Abstract

The number of periodic arrangements of point vortices – point vortex streets – in two-dimensional fluid flow that are stationary is known to be finite for a generic choice of vortex circulations. When all circulations are the same in absolute value, however, stationary vortex street configurations have been associated with the zeros of certain trigonometric polynomials containing free complex parameters. The presence of these parameters may prove useful in constructing point vortex models of shear layers and wakes. In this paper it is shown that such a continuum of stationary configurations exists in a much wider class of point vortex street systems. The circulations may take on many values, not just two, providing increased flexibility in the modelling context. A simple method for computing these configurations is derived. The effects of symmetries on the solution polynomials are described, and illustrated with examples. In addition, novel translating vortex street configurations are found having arbitrary translation velocity and containing free parameters for vortex circulations ±1 and also for vortex circulations +1, −2.