Bifurcation analysis of the structure of vortices in an uniform strain field.

Abstract

We have studied the stationary solutions to the two-dimensional Euler's equation. A highly accurate scheme, based on boundary integral techniques was used in investigating these steady-state configurations. Bifurcation analysis on the solution of a uniform vortex patch in an externally applied strain field, yield new non-elliptical steady-state solutions apart from the elliptical structures reported by Moore & Saffman. The elliptical solutions correspond to the points on the primary solution branch and the non-elliptical solutions correspond to points on the bifurcation branches. We also observe the presence of a turning point indicating the finite resistance of these uniform vortices. Some of these new solutions suggest the possibility of coalescence between neighboring vortices. This leads to a new problem of considering a vortex pair in a strain field and computing their steady-state solutions. Numerical computations suggest that this guess is indeed correct, as we see the solution branch corresponding to the vortex pair intersect the bifurcation branch of the single vortex at a unique strain rate. Furthermore, looking at the profiles on the other bifurcation branches, it appears that merger of neighboring vortices is a recurring phenomenon.