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 × 10 5 ; 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.