Constant-flux discrete heating in a unit aspect-ratio annulus

Natural convection in an annulus with a discrete heat source on the inner cylinder is studied numerically. The outer cylinder is isothermally cooled at a fixed low temperature, and the top wall, the bottom wall and unheated portions of the inner cylinder are thermally insulated. For low applied heat flux through the heater, as measured non-dimensionally by a Grashof number, Gr, the flow in the annular gap consists of a single-cell overturning meridional flow driven by the radial temperature gradient between the heater on the inner cylinder and the cold outer cylinder. In this regime, the flow is very weak and heat is transported primarily via conduction. The flow structure does not change until Gr ∼ 10⁴, although the flow strength steadily increases with Gr. As the nonlinear convection terms become more important, the meridional circulation sweeps the isotherms from being almost vertical near the outer cylinder to almost horizontal near the bottom wall. By the end of the transition from the conduction-dominated regime (Gr < 10⁴) to the convection-dominated regime (Gr ∼ 10⁶), the flow becomes segregated into three distinct regions: (i) for vertical levels below that of the bottom of the heater, an essentially cold stagnant pool develops, with the heat flux through the outer cylinder dropping to zero. (ii) At vertical levels between the bottom and the top of the heater, most of the region in between the two cylinders is stably stratified with a relatively weak radial flow from the cold to the heated cylinder. The horizontal isotherms adjust to the temperatures on the cylinders in thin buoyancy boundary layers which drive fluid down the cold cylinder and up the heated cylinder segment. The boundary layer on the heater is about half as thick as that on the cold cylinder, but about twice as intense. (iii) The third reion is above the heater top. The boundary layer flow from the heater continues upward where it meets the top endwall and bounces off of it. A wavy jet bounces on its way radially outward between the top insulated wall and the stably stratified region below. Flow separation on the top wall leads to the formation of a recirculation zone there. The vast majority of the heat flux through the outer cylinder occurs at this upper level, and is heavily concentrated near the very top. Geometric factors, such as the radius ratio of the cylinders and the heater placement, have quantitative effects, which are described, but the overall qualitative picture remains unchanged.