VISCOUS AND INVISCID INSTABILITIES OF A TRAILING VORTEX

A spectral collocation and matrix eigenvalue method is used to study the linear stability of the trailing line vortex model of Batchelor. For both the inviscid and viscous stability problem, the entire unstable region in the swirl/axial wavenumber parameter space is mapped out for various azimuthal wavenumbers m. In the inviscid case, the non-axisymmetric perturbation with azimuthal wavenumber m = 1 has an unstable region of larger extent than any other, with an unusual two-lobed structure; also, the location and numerical value of the maximum disturbance growth rate previously reported for this case are shown to be incorrect. Exploiting the increasingly localized structure of perturbation eigenfunctions allows accurate results to be obtained up to values of m more than 3 orders of magnitude larger than previously, and the results for the most unstable mode are in excellent agreement with the asymptotic theory of Leibovich & Stewartson. A viscous analysis of these fundamentally inviscid modes reveals that the critical Reynolds number at which instability first occurs increases as O(m2) for m much greater than 1, and finds the critical values of swirl and wavenumber, which approach limiting values as m --> infinity. In the viscous case, the instabilities for m = 0 and 1 recently reported by Khorrami are found via a simplified numerical approach and the entire unstable region for each of these modes is mapped out over a wide range of Reynolds numbers. The critical Reynolds numbers for these modes are found to be 322.42 and 17.527, respectively, the latter having been unreported previously. The instabilities persist in the limit of large Reynolds number, with corresponding disturbance growth rates decreasing roughly as 1/Re. In addition to the primary mode, a new family of long-wave viscous instabilities is found for the m = 1 case.