Time-dependent solution for natural convection in a porous enclosure using the Darcy–Lapwood–Brinkman model

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Natural convection (NC) in high permeable porous media is usually investigated using the Darcy–Lapwood–Brinkman model (DLB). The problem of the porous squared cavity is widely used as a common benchmark case for NC in porous media. The solutions to this problem with the DLB model are limited to steady-state conditions. In this paper, we developed a time-dependent high accurate solution based on the Fourier–Galerkin method (FG). The solution is derived considering two configurations dealing with unsteady and transient modes. The governing equations are reformulated using the stream function. The Temperature and the stream functions are expended as unknowns in space using Fourier series which are appropriately substituted in the equations. The equations are then projected to the spectral space using Fourier trigonometric trial functions. The obtained developed equations form a nonlinear differential algebraic system of equations. An appropriate technique is used to integrate the spectral system in time and to ensure high accuracy. The results of the FG method are compared to a finite element solution for different Rayleigh and Darcy numbers values. The transient and unsteady solutions are obtained with a feasible and low computational cost. The paper provides high accurate time-dependent solutions useful for benchmarking numerical models dealing with NC in porous media. The results of the developed solutions are efficient to gain physical insight into the time-dependent NC processes.