Theoretical Model of Buoyancy-Induced Flow in Rotating Cavities

The Ekman-layer equations, which have previously been solved for isothermal source-sink flow in a rotating cavity, are derived for buoyancy-induced flow. Although the flow in the inviscid core is three-dimensional and unsteady, it is assumed that the flow in the Ekman layers is axisymmetric and steady; and, as for source-sink flow, the average mass flow rate in the Ekman layers is assumed to be invariant with radius. In addition, it is assumed that the flow in the core is adiabatic, and consequently the core temperature increases with radius and with rotational speed. Approximate solutions are obtained for laminar flow, and it is shown that the Nusselt numbers for the rotating disks and the mass flow rate in the Ekman layers are proportional to Gr(c)(1/4), where Gr(c) is a Grashof number based on the rotational Reynolds number and the temperature difference between the disk and the core. The equation for the Nusselt numbers, which includes two empirical constants, depends strongly on the radial distribution of the temperature of the disks.