Collapse interactions of finite-sized two-dimensional vortices

P.W.C. Vosbeek, J.H.G.M. van Geffen, V.V. Meleshko and G.J.F. van Heijst
Physics of Fluids 9, 3315--3322 (1997)

I. Introduction

In his remarkable dissertation, Gröbli [1] established the existence of a special self-similar motion of three point vortices of chosen intensity located initially at specific positions on an infinite plane (see also Aref et al. [2] for historical and scientific background). In this special case the vortex trajectories have the form of logarithmic spirals with a common origin.

Depending on the signs of the intensities of the vortices, they can either escape to infinity (as is the case for which Gröbli constructed a solution) or move inward and collapse in the origin in a finite time. Later studies Novikov and Sedov [3], Aref [4], Kimura [5,6]) have revealed more general conditions for point vortex collapse to occur, namely some specific relations between the intensities and the initial positions of the vortices.

Although the collapse is in itself an interesting phenomenon, one may question the physical significance of this particular type of highly idealized point vortex interaction. The point vortex model has been proven to be very powerful in describing the interaction of (finite sized) vortices (see e.g. Meleshko and van Heijst [7]) and even the behaviour of dipolar and tripolar vortices in the presence of nonuniform background vorticity (see e.g. Velasco Fuentes et al. [8,9]). Yet, it is a priori not clear whether real vortices (with finite-sized, continuous vorticity distributions) show a collapse into a single vortex, as predicted by the point vortex model.

The present paper reports a numerical study of the effects of a finite vortex size on the collapse interaction. In the first set of simulations, the point vortices are replaced by initially circular vorticity patches (Rankine vortices) with corresponding circulation values. The evolution of these patches has been simulated using contour-dynamics, with the initial patch size (relative to the initial distances between the patch centres) as the main parameter of interest. In the second set of numerical simulations, the point vortices are replaced by vortices with a smooth vorticity distribution. The evolution of these vortices has been calculated by solving the two-dimensional vorticity equation (including the viscous terms) using a finite-difference method.

The main questions that will be addressed are: how are the trajectories and the shapes of the vortices affected by the finite size of the vortices? What is the influence of viscosity on the process of vortex interaction? Finally, what is the effect of solid domain boundaries on the vortex collapse for the case of point vortices?

The remainder of the paper is organized as follows. A brief description of the classical point-vortex collapse is given in Section II. The contour-dynamics simulations of the interaction of initially circular patches are discussed in Section III, while the results of the finite-difference simulations of real vortices with continuous vorticity distributions are presented in Section IV. Finally, some general conclusions are formulated in Section V.

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