Three-Dimensional Natural Convection in a Confined Fluid Overlying a Porous Layer

This paper presents a numerical study of three-dimensional, laminar natural convection in an enclosure containing a fluid layer overlying a porous layer saturated with the same fluid. The Brinkman-extended Darcy formulation is used to model fluid flow in the porous layer as this facilitates the imposition of a no-slip boundary condition at the fluid/porous layer interface. The enclosure is heated from one side wall and cooled from an opposite wall, while the remaining walls are adiabatic. The mathematical analysis is carried out in terms of a vorticity-vector potential formulation that ensures the conservation of mass. The governing equations in non-dimensional form are transformed into parabolic equations by means of a false transient method in order to facilitate a solution procedure by an alternating direction implicit method. Accuracy of the numerical solutions with respect to uniformly and nonuniformly spaced grid points has been tested by performing extensive numerical experiments. As expected, it is found that the intensity of free convection is much more profound in the fluid layer. The numerical results indicate that penetration of the fluid into the porous region depends strongly upon the Darcy and Rayleigh numbers. The effect of the ratio of thermal conductivities (porous to fluid regions) is to intensify the convection current in the fluid layer.