Similarity Solution of Two-Dimensional Point Vortices

Abstract

Theory of the similarity solution of two-dimensional point vortices in an unbounded region is presented. To get the solution, the time dependence of the solution for all vortex positions are assumed to be the same. It is shown that the system either rotates rigidly or collapses according to the initial position of vortices. Then numerical calculations are made especially for the system of three vortices. It is found that the rigid rotation corresponds to the regular triangle solution or the straight line solution, and that irrespective of the strengths of three vortices, the regular triangle solution always exists while the straight line solution is obtained by solving a certain cubic equation.