Viscous and inviscid instabilities of non-parallel self-similar axisymmetric vortex cores

A spectral collocation method is used to analyse the linear stability, both viscous and inviscid, of a family of self-similar vortex viscous cores matching external inviscid vortices with velocity u varying as a negative power of the distance r to their axis of symmetry, u ∼ rm−2 (0 < m < 2). Non-parallel effects are shown to contribute at the same order as the viscous terms in the linear governing equations for the perturbations, and are consequently retained. The viscous stability analysis for the particular case m = 1, corresponding to Long's vortex, has recently been performed by Khorrami & Trivedi (1994). In addition to the inviscid non-axisymmetric modes of instability found by these authors, some inviscid axisymmetric unstable modes, and purely viscous unstable modes, both axisymmetric and non-axisymmetric, are also found. It is shown that, while both solution branches (I and II) of Long's vortex are destabilized by perturbations having negative azimuthal wavenumber (n < 0), only the Type II Long's vortex is also unstable for axisymmetric disturbances n = 0, as well as for disturbances with n > 0. Global pictures of instabilities of Long's vortex are given. For m > 1, the vortex cores have the interesting property of losing existence when the swirl number is larger than an m-dependent critical value, in close connection with experimental results on vortex breakdown. The instability pattern for m > 1 is similar to that found for Long's vortex, but with the important difference that the parameter characterizing the different vortices, and therefore their stability, is a swirl parameter, which is precisely the one known to govern the real problem, while this is not the case in the highly degenerate case m = 1.