A fourth order compact scheme for heat transfer problem in porous media

In this paper, we have extended the recently proposed fourth order compact scheme by Pandit et al. (2007) which was used earlier only for the convection–diffusion equations without considering reaction and nonhomogeneous source terms, and designed to compute flow in a two sided lid-driven differentially heated square cavity filled with a fluid saturated porous medium. The governing Navier–Stokes (N–S) equations in streamfunction–vorticity (ψ−ζψ−ζ) form of Brinkman-extended Darcy model including the energy transport equation are all solved as a coupled system of equations for the five field variables consisting of streamfunction, vorticity, two velocities and temperature. Moreover, local entropy generation distributions are determined based on the obtained dimensionless velocity and temperature values. The derivative source term present in vorticity equation has been treated as fourth order compact using Padé scheme. The details for the derivation of difference relations at boundaries to generate accurate and stable solutions are also given. We have computed the results for three different cases depending on the direction of moving walls. To assess the numerical accuracy of our proposed scheme, one pertinent test problem with known exact solution is used. The higher order compact scheme adopted in the present study yields consistent performance for variation of key parameters e.g. Richardson number (RiRi), Darcy number (DaDa), Grashof number (GrGr) and fixed Prandtl number (PrPr)=0.7. The present results are compared with numerical results available in the literature and excellent match is observed in all the cases, establishing the efficiency and accuracy of the extended combined formulations of our fourth order compact scheme and Padé schemes.