Pseudosteady-state natural convection inside spherical containers partially filled with a porous medium

Abstract

A computational study of the pseudosteady-state two-dimensional natural convection within spherical containers partially filled with a porous medium is presented. The computations are based on an iterative, finite-volume numerical procedure using primitive dependent variables, whereby the time-dependent continuity, momentum and energy equations in the spherical coordinate system are solved within the composite system. The natural convection effect is modeled via the Boussinesq approximation, whereas the Darcy Law is utilized to treat the porous medium. For a reference case, flow and temperature field details during the transient evolution to the pseudosteady-state are presented. It is shown that the dominant transport mechanism at the early stages is due to heat conduction and natural convection plays no role. A parametric study was performed with the values of the Rayleigh number ( Ra) , Darcy number ( Da) and the thermal conductivity ratio varying one at a time. The dependence of the flow and thermal fields on these parameters was elucidated. For low Ra and Da numbers, the flow field is restricted within the central fluid core. Only for high Ra and Da numbers, one can observe comparable fluid motion in both the porous medium and central fluid core regions. The local Nusselt number on the surface and interface temperature exhibit nearly uniform variations for low Ra and Da numbers, signifying little deviation from the limiting pure conduction case. For high Ra and Da numbers, marked heat transfer is observed on the bottom of the sphere. The interface temperature is also seen to deviate from uniform variation for high Ra and Da numbers. Only the intensity of the recirculating flow in the central fluid core region was seen to depend on the thermal conductivity ratio. The thermal conductivity ratio modifies the time scale of the thermal transport and only the relative magnitudes of the monitored quantities are affected.