Non-Darcian Bénard convection in eccentric annuli containing spherical particles

Abstract

In this paper, a numerical investigation of natural convection in a porous medium confined by two horizontal eccentric cylinders is presented. The cylinders are impermeable to fluid motion and retained at uniform different temperatures. While, the annular porous layer is packed with glass spheres and fully-saturated with air, and the cylindrical packed bed is under the condition of local thermal non-equilibrium. The mathematical model describing the thermal and hydrodynamic phenomena consists of the two-phase energy model coupled by the Brinkman-Forchheimer-extended Darcy model under the Boussinesq approximation. The non-dimensional derived system of formulations is numerically discretised and solved using the spectral-element method. The investigation is conducted for a constant cylinder/particle diameter ratio (Di/d) = 30, porosity (ε) = 0.5, and solid/fluid thermal conductivity ratio (kr) = 38.6. The effects of the vertical, horizontal and diagonal heat source eccentricity (−0.8 ⩽e⩽0.8) and the annulus radius ratio (1.5 ⩽RR⩽ 5.0) on the temperature and velocity distributions as well as the overall heat dissipation within both the fluid and solid phases, for a broad range of Rayleigh number (104 ⩽ Ra ⩽ 8 ×107). The results show that uni-cellular, bi-cellular and tri-cellular flow regimes appear in the vertical eccentric annulus at the higher positive eccentricity e = 0.8 as Rayleigh number increases. However, in the diagonal eccentric annulus, the multi-cellular flow regimes are shown to be deformed and the isotherms are particularly distorted when Rayleigh number increases. In contrast, in the horizontal eccentric annulus, it is found that whatever the Rayleigh number is only an uni-cellular flow regime is seen. In addition, it is shown that the fluid flow is always unstable in the diagonal eccentric geometry at e = 0.8 for moderate and higher Rayleigh numbers. However, it loses its stability in the vertical eccentric geometry only at two particular cases, while it is always stable in the horizontal eccentric geometry, for all eccentricities and Rayleigh numbers.