Natural convection fluid flow and heat transfer in porous media

Abstract

Natural convection and heat transfer of fluid flow are studied numerically inside a rectangular cavity with inclination filled with a porous medium. The mass and momentum equations are given by the Darcy equations coupled with the thermal energy equation through the unsteady Boussinesq approximation. The two-dimensional restriction in terms of the stream function and vorticity variables is considered. The study is analyzed in terms of several values of the parameters that determine the evolution of the flow: the Rayleigh Number, the aspect ratio of the cavity and the angle of inclination. The mass and momentum equations in natural convection fluid flow in a porous medium are given by the Darcy equations coupled with the thermal energy equation through the unsteady Boussinesq approximation to deal with an incompressible structure. In this work the dimensionless problem is formulated in terms of the stream function and vorticity variables; then, the computation of the pressure is avoided and the incompressibility condition is satisfied automatically. Regarding the numerical method, once a convenient second order time discretization is performed, a nonlinear elliptic system is obtained which is solved through a fixed point iterative process. The iterative process leads to the solution of uncoupled, well-conditioned, symmetric linear elliptic problems for which very efficient solvers exist regardless of the space discretization. The study of natural convection and heat transfer of fluid flow in a porous medium has important technological applications: storage and preservation of grains and cereals; solar energy collectors; filter systems; transport of radioactive wastes through the soil; and postaccident heat removal in nuclear reactors. Our numerical study is carried out on tilted rectangular cavities. The study is realized through the parameters that influence directly the behavior and evolution of the flow: the Rayleigh Number Ra, the aspect ratio of the cavity A, and the angle of inclination . We mention below two categories of research in connection with natural convection problems that arise when opposing walls of a cavity are subjected to a temperature gradient and where the other set of walls is insulated—the subject of the present work.