On the stability of ring modes in a trailing line vortex: the upper neutral points

The inviscid near-neutral stability of a trailing-vortex flow is investigated by using a normal-mode analysis in which all perturbation quantities exhibit a factor exp[i(nβz−nθ−ω)]. The problem is treated as a timewise-stability problem. The dependence of the eigenvalues ω on the axial wavenumber β, which has been normalized with respect to the azimuthal wavenumber n, is found both numerically and analytically for large values of n in the upper range of values of β near 1/q, where near-neutral modes are anticipated to occur. Here q, the swirl parameter of the flow, effectively compares the 'strengths’ of the swirl and axial components of motion in the undisturbed flow. Previous normal-mode analyses based on the same form of perturbation quantities have shown that for columnar vortices the unstable modes for large values of n are ring modes, and this feature is shown to persist near the upper neutral points. In fact this work on near-neutral ring modes supplements the earlier asymptotic theory for large n, which is known to fail near β= 1/q. Our numerical and asymptotic results are in excellent agreement and are also shown to be consistent with the earlier asymptotic theory through matching. It is found that ω→0 as β→(1/q)−.