Natural convection in a heated annulus

Natural convection in a heated vertical concentric annulus is studied. A constant heat-flux is applied to the inner cylinder and the outer cylinder is insulated. Under these conditions, the mean temperature of the fluid increases linearly while, at the same time, heat diffuses from the heated surface into the fluid, resulting in a temperature distribution on the cross-section. Subtracting this steadily increasing temperature from the total temperature results in a steady temperature stratification on the cross-section which drives fluid motion. The mathematical form of the scaled problem is shown to be identical to that of a fluid in an annulus with uniformly distributed heat sources, with the inner cylinder maintained at constant temperature and the outer cylinder insulated. At low heat addition rates, the fluid motion is steady and parallel, and heat is transferred by conduction between the fluid layers. As the rate of heating increases, the flow becomes unstable and recirculating eddies appear, which transfer heat by convection. The onset of convection is determined by linear-instability analysis of the basic-state. The results demonstrate that when the Prandtl number is small, the dominant instability obtained energy primarily from shear production. On the other hand, when the Prandtl number is large, an instability that obtains kinetic energy from buoyant production is pre-eminent. Weakly nonlinear instability theory is used to analyze finite-amplitude effects. The results show that both types of linear instabilities are supercritical.