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

Asymptotic results obtained by Foster and Smith (1989) for inviscid instability modes of the Type-II Long's vortex is extended to account for the effects of finite Reynolds number. It is shown that the nonparallelism of the flow is more important than the viscous terms in determining the finite-Re behavior due to the radial velocity scales with Re exp -1 M. A critical layer of the three-layer structrue of the parallel-flow instability modes is considerably modified by radial velocity. It is found that for azimuthal wavenumber n greater than 1, the nonparallelism stabilizes the unstable inertial modes, leading to determination of neutral curves. For n less than -1, the nonparallel effects always destabilize the vortex to these helical modes.