The bimodal instability of thermal convection in a tall vertical annulus

Abstract

We pursue a three-dimensional linear stability analysis to investigate the convective stability in a cylindrical annulus imposed with a horizontal temperature gradient. The investigation focuses on the stability of both axisymmetric and asymmetric modes in a wide range of Prandtl number Pr and the radii ratio η between the inner to outer cylinders. Results show that, for the axisymmetric mode, the stability strongly depends on the Prandtl number. For 0≤Pr≤1.4, the instability sets in as the shear mode that the instability enhances as η increases while it is virtually independent of Pr. The critical Grashof number can be approximated by Grc(η)≈103×4η−0.68+3. For 1.5≤Pr≤12.5, the buoyant mode appears and competes with the shear mode to predominate the stability under various Pr and η, namely, the bimodal instability occurs. The transition between the two modes occurs at the specific radii ratio η1=0.21Pr0.727−0.31. For Pr≥12.6, the shear mode vanishes and the buoyant mode prevails. The asymmetric modes are less prevailing than the axisymmetric mode for small Prandtl numbers, except that the mode of an azimuthal wavenumber equal to 1 becomes dominant in two small regions of η. When the Prandtl number is large, the axisymmetric mode predominates over the asymmetric mode except when η is very small. Otherwise, the asymmetric mode of a larger azimuthal wavenumber is less prevailing.