Natural Convection between Concentric Vertical Cylinders

Abstract

The motion of a fluid in the closed annular cavity formed by two concentric vertical cylinders and two horizontal planes has been analyzed by a numerical solution of the equations of motion and energy using a high-speed digital computer. The motion is generated by a radial density gradient caused by the thermal boundary conditions which are, typically: inner cylinder at a (dimensionless) temperature of unity; outer cylinder at a temperature of zero; horizontal boundaries adiabatic. The fluid is assumed to have constant thermodynamic and transport properties except for the density, which is temperature-dependent in the buoyancy term of the equation of vertical motion (the Boussinesq approximation); the flow is assumed to be axisymmetric. The equations of time-dependent motion have been solved, so that both transient and steady-state solutions are obtained. The parameters of the problem, and the respective ranges of values which have been considered, are: Rayleigh number (based on gap width) up to 2 × 105; Prandtl number 0.5 to 5; radius ratio 1 to 4; aspect ratio (cavity height/gap) 1 to 20. At moderate Rayleigh numbers the motion consists of a single cell (i.e., torus), while at higher Rayleigh numbers the onset of a multicellular motion is observed. The local and average Nusselt numbers, of interest in determining the insulating value of an annular cavity, have been obtained.