Viscous effects in the absolute–convective instability of the Batchelor vortex

Abstract

The effects of viscosity on the instability properties of the Batchelor vortex are investigated. The characteristics of spatially amplified branches are first documented in the convectively unstable regime for different values of the swirl parameter q and the co-flow parameter a at several Reynolds numbers Re. The absolute–convective instability transition curves, determined by the Briggs–Bers zero-group velocity criterion, are delineated in the (a, q)-parameter plane as a function of Re. The azimuthal wavenumber m of the critical transitional mode is found to depend on the magnitude of the swirl q and on the jet (a > −0.5) or wake (a < −0.5) nature of the axial flow. At large Reynolds numbers, the inviscid results of Olendraru et al. (1999) are recovered. As the Reynolds number decreases, the pocket of absolute instability in the (a, q)-plane is found to shrink gradually. At Re = 667; the critical transitional modes for swirling jets are m = −2 or m = −3 and absolute instability prevails at moderate swirl values even in the absence of counterflow. For higher swirl levels, the bending mode m = −1 becomes critical. The results are in good overall agreement with those obtained by Delbende et al. (1998) at the same Reynolds number. However, a bending (m = +1) viscous mode is found to partake in the outer absolute–convective instability transition for jets at very low positive levels of swirl. This asymmetric branch is the spatial counterpart of the temporal viscous mode isolated by Khorrami (1991) and Mayer & Powell (1992). At Re = 100, the critical transitional mode for swirling jets is m = −2 at moderate and high swirl values and, in order to trigger an absolute instability, a slight counterflow is always required. A bending (m = +1) viscous mode again becomes critical at very low swirl values. For wakes (a < −0.5) the critical transitional mode is always found to be the bending mode m = −1, whatever the Reynolds number. However, above q = 1.5, near-neutral centre modes are found to define a tongue of weak absolute instability in the (a, q)-plane. Such modes had been analytically predicted by Stewartson & Brown (1985) in a strictly temporal inviscid framework.