Linear stability of cylindrical Couette flow in the convection regime

Abstract

The instability of steady circular Couette flow with radial heating across a vertically oriented annulus with a rotating inner cylinder and a stationary outer cylinder is investigated using a linear stability analysis. The convection regime base flow is developed for an infinite aspect ratio geometry and constant fluid properties with buoyancy included through the Boussinesq approximation. The base flow is characterized by a dimensionless stratification parameter γ that is proportional to the vertical temperature gradient. Critical stability boundaries are calculated for this assumed base flow with respect to both toroidal and helical disturbances. The numerical investigation is primarily restricted to a radius ratio of 0.6 at a Prandtl number of 100. Critical stability boundaries in Taylor–Grashof number space are presented for two values of the stratification parameter γ (4 and 13). The results follow the development of critical stability from Taylor cells at small Grashof numbers up to a maximum Grashof number used in this calculation of 20 000 and 80 000 for γ=4 and 13, respectively. Results show that increasing the stratification parameter stabilizes the isothermal Taylor vortices, followed by a destabilization at higher azimuthal mode numbers (n>0). The results also show that for γ=4 (close to the conduction regime), two modes are obtained: one is axisymmetric and the other is nonaxisymmetric. However, for the convection regime (large γ) six asymmetric modes are obtained. Finally, the disturbance wavelength, phase speed, and spiral inclination angle are presented as a function of the critical Grashof number for the stratification parameters considered in this work.