Non-parallel effects in the instability of Long's vortex

As shown in Foster & Smith (1989), at large flow force M, Long's self-similar vortex is in the form of a swirling ring-jet, whose axial velocity profile is of sech2 form. At azimuthal wavenumber n of comparable order to the axial wavenumber, linear helical modes of instability are essentially those of the Bickley jet varicose and sinuous modes. However, at small axial wavenumbers, the three-dimensionality of the vortex is important, and the instabilities depend heavily on the effects of the swirl. We explore here the effects of finite Reynolds number Re on these long-wave inertial modes. It is shown that, because the radial velocity scales with Re−1M, the non-parallelism of the flow is more important than the viscous terms in determining the finite-Re behaviour. The three-layer structure of the parallel-flow instability modes remains, but with a critical layer considerably modified by radial velocity. In investigating the critical range Re = O(M3), we find the following: for n > 1, the non-parallelism stabilizes the unstable inertial modes, leading to determination of neutral curves; for n < − 1, the non-parallel effects always destabilize the vortex to these helical modes. Determination of the unstable modes and neutral curves for the n > 1 case requires a computational scheme that accounts for the presence of viscosity. It turns out that the n < 1 (n > − 1) modes are prograde (retrograde) with respect to the rotation of the main vortex.