Local linear stability of plumes generated along vertical heated cylinders in stratified environments

Abstract

The linear temporal and absolute/convective stability characteristics of a thermal plume generated along a heated vertical cylinder are investigated theoretically under the Boussinesq approximation. Special focus is given to the uniform-wall-buoyancy-flux case whereby the cylinder surface sustains the same linear temperature gradient as the environment. A competition between the axisymmetric and helical modes is a remarkable feature of the instability, distinguishing these ‘annular plumes’ from free plumes/jets for which the helical mode is generally dominant. It is found that higher surface curvature stabilises the temporal axisymmetric mode significantly, but only has moderate effects on the helical mode. The most temporally unstable perturbation mode switches from a helical into an axisymmetric mode when the Prandtl number increases beyond a critical value. Both the roles of shear and buoyancy during the destabilisation are identified through an energy analysis which indicates that, while the shear work is usually a major source of perturbation energy, the buoyancy work manifests for long-wave axisymmetric perturbation modes, and for thin cylinders and high Prandtl numbers. For the specific temperature configuration considered herein, an annular plume is always convectively unstable whereas decreasing the cylinder radius from the planar limiting case first decreases and then increases the tendency of the flow towards being absolutely unstable. The helical mode is especially susceptible to being absolutely unstable on very thin cylinders. Several conditions for the onset of cellular thermal convection and plume detrainment are proposed based on our results and a hypothesis which connects the absolute instability to the detrainment phenomenon.