Stability of elliptical vortices from “Imperfect–Velocity–Impulse” diagrams

Abstract

In 1875, Lord Kelvin proposed an energy-based argument for determining the stability of vortical flows. While the ideas underlying Kelvin’s argument are well established, their practical use has been the subject of extensive debate. In a forthcoming paper, the authors present a methodology, based on the construction of “Imperfect–Velocity–Impulse” (IVI) diagrams, which represents a rigorous and practical implementation of Kelvin’s argument for determining the stability of inviscid flows. In this work, we describe in detail the use of the theory by considering an example involving a well-studied classical flow, namely the family of elliptical vortices discovered by Kirchhoff. By constructing the IVI diagram for this family of vortices, we detect the first three bifurcations (which are found to be associated with perturbations of azimuthal wavenumber m = 3, 4 and 5). Examination of the IVI diagram indicates that each of these bifurcations contributes an additional unstable mode to the original family; the stability properties of the bifurcated branches are also determined. By using a novel numerical approach, we proceed to explore each of the bifurcated branches in its entirety. While the locations of the changes of stability obtained from the IVI diagram approach turn out to match precisely classical results from linear analysis, the stability properties of the bifurcated branches are presented here for the first time. In addition, it appears that the m = 3, 5 branches had not been computed in their entirety before. In summary, the work presented here outlines a new approach representing a rigorous implementation of Kelvin’s argument. With reference to the Kirchhoff elliptical vortices, this method is shown to be effective and reliable.