Stability of high-density trailing vortices

Abstract

The three dimensional modal linear stability of the radially stratified q-vortex is investigated. The presence of a radial density gradient in the vortex core biases the vortex stability features over the whole parameter space, i.e. varying the swirl number q, the axial k and azimuthal m wavenumbers and the density-to-vorticity radius ratio \(\epsilon \). The high swirl vortex, known to be stable in the constant-density situation becomes unstable to the Rayleigh–Taylor instability (RTI) with high amplification rates for vortex cores denser than the ambient. Hence we carry out a comprehensive stability analysis to measure the consequences of the onset of the RTI on the q-vortex linear stability. The damping effect of viscosity saturates beyond a threshold Reynolds number and we mean to address high Reynolds number situations such as those found in aircraft trailing vortices. Hence we place ourselves in the high Reynolds numbers regime for which vortices with a dense core exhibit a significant increase of the global maximum of the amplification rate. The effect of the radius ratio \(\epsilon \) is twofold. In the high swirl number regime where the homogeneous modes are stable or weakly amplified, the concentration of denser fluid at the vortex core promotes instabilities. In regions of the (k, q)-plane favouring both the homogeneous instability and the RTI mechanism, the amplification rate peaks for a radius ratio around \(\epsilon \approx 2\).