Stability of Vortices in Equilibrium with a Cylinder

The stability of steady, inviscid vortex pairs in equilibrium with a circular cylinder are studied by discretizing equations derived from contour dynamics. These flows were found previously using different methods in Elcrat, Fornberg, Horn & Miller (2000). There are two families of vortices, one with a pair of counter-rotating vortices standing behind the cylinder, which may be thought of as desingularizing the Föppl point vortices, and the other with the vortices standing directly above and below the cylinder. Vortices in the first family are found to be neutrally stable with respect to symmetric perturbations. When asymmetric perturbations are included, there is a single unstable mode and a single asymptotically stable mode. Vortices above and below the cylinder have two modes of instability, one symmetric and the other asymmetric, and likewise two asymptotically stable modes.