Higher order compact computations of transient natural convection in a deep cavity with porous medium

In this paper, an extension of the fourth order compact scheme on nonuniform grids (Pandit et al. (2007) [28]) is proposed for solving two dimensional (2D) unsteady natural convection flows in a rectangular cavity (with different aspect ratios) filled with a fluid saturated porous medium. The bottom wall of the cavity is uniformly and non-uniformly heated and the top wall is adiabatic while the vertical walls are cold maintained at constant temperature. We have used streamfunction (ψ)-vorticity (ζ) formulation of Navier-Stokes equations with the consideration of Brinkmann-extended Darcy model to simulate the momentum transfer in the porous medium. The streamfunction-vorticity and the energy equations are all solved as a coupled system of equations for the five field variables consisting of streamfunction, vorticity, two velocities and temperature. In this ψ-ζ formulation, the temperature gradient source term also has been treated as fourth order compact. The higher order compact scheme adopted in the present study yields consistent performance for a wide range of key parameters e.g. Rayleigh number Ra (from 103 to 108), Darcy number Da (from 10-5 to 10-3). Results are presented in the form of streamline and isotherm plots as well as the plots of Nusselt number at the heat source surface under different conditions. The present scheme is not only robust as evidenced from computations at higher Ra, but also accurate as is seen from comparisons with reliable existing results.