Stability analysis of the perturbed rest state and of the finite amplitude steady double-diffusive convection in a shallow porous enclosure

Abstract

Transition from rest and steady convective states to oscillatory flows is investigated in a shallow porous enclosure subject to vertical thermal and solutal gradients. Various combinations of Dirichlet and Neumann thermal and solutal boundary conditions is considered. The unsteady form of Hazen–Darcy law with the Boussinesq approximation is used to model the convective flow through the porous medium. The governing and perturbation equations are solved numerically using finite element method. The threshold of transition, which characterizes the transition from steady to oscillatory finite amplitude flows, and the threshold of overstability (Hopf bifurcation), which characterizes the transition from the rest to oscillatory state, are obtained for a wide range of the governing parameters. The porosity and the acceleration parameter of the porous medium have a strong effect on the thresholds of transition and overstability. An increase in the acceleration parameter and the normalized porosity delays the onset of overstability and the transition to oscillatory finite amplitude flows. For Neumann boundary conditions type, the wavenumber is zero at the onset of overstabilities and finite at the transition threshold.