Effects of a vertical fracture on the stability of thermal convection in a porous cavity

The hydrodynamic stability of a fluid-saturated cavity with a vertical narrow fracture is analyzed by using a tangential simplification mode that only retains the tangential components of dominating the fluid flow and heat transfer within a fracture. The effects of the fracture aperture, fracture permeability, fracture thermal conductivity, and flow properties (including the Darcy's, Darcy-Brinkman, and Navier-Stokes modes) within it on the stability of convection are investigated by re-deriving the mass and energy conservation forms in the fracture. The results in this study are summarized below. The tangential simplification can be used in a narrow fracture with a wide range of apertures. A fracture with high-permeability or a relatively wide aperture can effectively promote the development of convection near itself. As the Darcy number lowers, the Darcy-Brinkman mode approaches to the Darcy's mode, and the Navier-Stokes mode represents the supreme form of fracture flow. In contrast, at high Darcy number, the fracture tends to approach to an adiabatic and impermeable wall. Additionally, the cavity with a narrow vertical fracture (e.g., dimensionless aperture d(f) < 10(-3)) has a minimum asymptotic critical Rayleigh number Ra-c of 19.74 at a relatively large cavity aspect ratio (e.g., aspect ratio L > 2) but has a global minimum Ra-c of 19.24 at cavity aspect ratio L of 0.89.