Natural Convection in a Vertical Porous Cavity: A Numerical Study for Brinkman-Extended Darcy Formulation

A dimensional analysis of the Brinkman-extended Darcy formulation, which includes the transport and viscous terms, leads to four governing parameters for steady-state natural convection in a vertical porous cavity. They are: Rayleigh number, Darcy number, diffusion parameter Ω, and aspect ratio. Numerical results for 0≤Da≤10⁻¹, 10≤Ra*≤5 ×10³, and A=1 and 5, indicate that the temperature and velocity fields are significantly modified, the flow regimes are delayed, and the heat transfer rate is decreased when the Darcy number is increased beyond 10⁻⁵ for fixed Ra* and A. The slope of the ln (Nu) versus ln (Ra*) curve in the boundary layer regime decreases from 0.53 at Da=0 to 0.264 at Da=10⁻¹ when A=5. The contribution of the transport term increases with Ω, Da, and Ra*, but the effect on the overall heat transfer is insignificant. However, the problem becomes ill formulated at high values of these parameters and may require the consideration of Forchheimer modifications. A scale analysis is also presented to show that the inertia term is of a low order of magnitude in comparison with the viscous term at high Prandtl numbers.