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.
Obtaining Expressions for Time-Dependent Functions That Describe the Unsteady Properties of Swirling Jets of Viscous Fluid
