Short-wavelength stability analysis of Hill’s vortex with/without swirl

Abstract

The stability of Hill’s vortex with/without swirl is studied by the short-wavelength stability analysis or WKB analysis. It is shown that the classical Hill’s spherical vortex is subjected not only to the Widnall instability but also to the curvature instability found for thin vortex rings and helical vortex tubes. A new “combined” mode of instability caused by the two instabilities is discovered. The magnitude of the exponential growth rate of the combined mode is similar with the curvature instability around the stagnation point; it exceeds the Widnall instability near the boundary. The effects of swirl on the instabilities are investigated using a family of solutions obtained by Moffatt [“The degree of knottedness of tangled vortex lines,” J. Fluid Mech. 35, 117 (1969)]. As the swirl parameter α increases, a stable region appears around the stagnation point; the maxima of the growth rates decrease; the combined mode region disappears for α≥3. As α increases further, however, the region of the generalized centrifugal instability emerges from the stagnation point.