The stability of a trailing line vortex. Part 2. Viscous theory

Abstract

In a previous paper, the inviscid stability of a swirling far wake was investigated, and the superposition of a swirling flow on the axisymmetric wake was shown to be initially destabilizing, although all modes investigated eventually become more stable at sufficiently large swirl. The most unstable disturbances were non-axisymmetric modes with negative azimuthal wavenumber n representing helical wave paths opposite in sense to the wake rotation. The disturbance growth rate appeared to increase continuously with |n|, while all modes with |n| > 1 represented disturbances which are completely stable for the non-swirling wake. In the present analysis, both timewise and spacewise growth rates are calculated for the lowest three negative non-axisymmetric modes (n = −1, −2 and −3). Vortex intensity is characterized by a swirl parameter q proportional to the ratio of the maximum swirling velocity to the maximum axial velocity defect. The large wavenumbers associated with the disturbances at large |n| allow the n = −1 mode to have the minimum critical Reynolds number of 16 (q ≃ 0·40). The other two modes investigated have minimum Reynolds numbers on the neutral curve of 31 (n = −2, q = 0·60) and 57 (n = −3, q = 0·80). For each mode, the neutralstability curve is shown to shift rapidly towards infinite Reynolds numbers once the swirl becomes sufficiently large. Some of the most unstable swirling flows are shown to possess spacewise amplification factors almost ten times that for the most unstable wavenumber for the non-swirling wake at moderate Reynolds numbers.