Stability of a pair of planar counter-rotating vortices in a rectangular box

Abstract

The stability and transition of a pair of planar counter-rotating vortices in a confined region are investigated numerically. Direct numerical simulations in a rectangular box: D = [0,2kπ] × [0,π], where k is an aspect ratio, are done with symmetric external forcing. A pair of steady symmetric counter-rotating vortices are generated under weak forcing while they become unstable with stronger forcing. Special attention is paid to the effect of the aspect ratio k on the stability and the transition. It is found that a steady asymmetric pattern appears for k > k0 ≈ 0.5 while an oscillating asymmetric pattern for k < k0. In particular, the symmetric vortices become very stable around k = 0.5 (i.e. the square region). Linear stability analysis of a model flow similar to the symmetric vortices generated in the numerical simulations gives the same tendency as the above findings. The results suggest that the singular property of the stability at k = k0 ≈ 0.5 is mainly dependent on the aspect ratio k of the region but rather independent of the superficial difference of the vorticity distribution.