Short-wavelength stability analysis of thin vortex rings

Abstract

The linear stability of thin vortex rings are studied by short-wavelength stability analysis. The modified Hill-Schrsodinger equation for vortex rings, which incorporates curvature effect, is derived. It is used to evaluate growth rates analytically. The growth rates are also evaluated by numerical calculation and they agree well with analytical values for small ε which is the ratio of core radius to ring radius. Two types of vortex rings are considered: Kelvin's vortex ring and a Gaussian vortex ring. For Kelvin's vortex ring the maximum first-order growth rate is found to be 165/356ε. For the Gaussian vortex ring the first-order growth rate is large in the skirts of the vortex core. The first-order instability is significant for both vortex rings. © 2003 American Institute of Physics.