Local linear stability of laminar axisymmetric plumes

Abstract

The temporal and spatiotemporal stability of thermal plumes is investigated for laminar velocity and temperature profiles, under the Boussinesq approximation, in the far self-similar region as well as in the region close to a finite-size inlet. In the self-similar case, Prandtl and Grashof numbers are systematically varied, and azimuthal wavenumbers $m=0$, 1 and 2 are considered. In the temporal analysis, helical modes of $m=1$ are found to be dominant throughout the unstable parameter space, with few exceptions. Axisymmetric modes typically present smaller growth rates, but they may dominate at very low Prandtl and Grashof numbers. Double-helical modes of $m=2$ are unstable over a very restricted range of parameters. Only the helical $m=1$ mode is found to ever become absolutely unstable, whereas $m=0$ and $m=2$ modes are at most convectively unstable. In a temporal setting, an analysis of the perturbation energy growth identifies buoyancy- and shear-related mechanisms as the two potentially destabilizing flow ingredients. Buoyancy is demonstrated to be important at low Grashof numbers and long wavelengths, whereas classical shear mechanisms are dominant at high Grashof numbers and shorter wavelengths. The physical mechanism of destabilization through the effect of buoyancy is investigated, and an interpretation is proposed. In the near-source region, both axisymmetric and helical modes may be unstable in a temporal sense over a significant range of wavenumbers. However, absolute instability is again only found for helical $m=1$ modes.