VISCOUS AND INVISCID INSTABILITIES OF NONPARALLEL SELF-SIMILAR AXISYMMETRICAL 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 simila r to r(m-2) (0 < m < 2). Non-parallel effects are shown to contribute at the same order as the viscous terms in the linear governing equatio ns for the perturbations, and are consequently retained. The viscous s tability analysis for the particular case m = 1, corresponding to Long 's vortex, has recently been performed by Khorrami & Trivedi (1994). I n addition to the inviscid non-axisymmetric modes of instability found by these authors, some inviscid axisymmetric unstable modes, and pure ly 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 az imuthal wavenumber (n < 0), only the Type II Long's vortex is also uns table for axisymmetric disturbances n = 0, as well as for disturbances with n > 0. Global pictures of instabilities of Long's vortex are giv en. For m > 1, the vortex cores have the interesting property of losin g existence when the swirl number is larger than an m-dependent critic al value, in close connection with experimental results on vortex brea kdown. The instability pattern for m > 1 is similar to that found for Long's vortex, but with the important difference that the parameter ch aracterizing the different vortices, and therefore their stability, is a swirl parameter, which is precisely the one known to govern the rea l problem, while this is not the case in the highly degenerate case m = 1.