Spectral time-dependent solutions for Darcy model of natural convection in a porous enclosure

Abstract

According to his high accuracy in the simply connected domains, the Galerkin spectral method having Fourier series as trial functions is chosen to develop a space-time dependent solution for the Darcy model coupled with the convection-diffusion heat transfer equation in a square porous cavity. Applying the Fourier-Galerkin (FG) procedure for the stream function formulation of the governing equations, we obtain a system of nonlinear algebraic differential equations that we simplify in ordinary differential equations and integrate in time using the variable order Runge-Kutta method. Two configurations dealing with transient and unsteady regimes are considered where the solution is derived for a wide range of Rayleigh numbers. The developed FG solution shows high accuracy with a reduced computational cost and excellent agreement with a finite element solution. The FG solution is used to provide physical insight into the effect of time variation on the mechanisms of natural convection (NC). The results provide a set of high-accurate data that can be used for testing numerical codes of heat transfer in time-dependent configurations.