Dynamic transitions and bifurcations for thermal convection in the superposed free flow and porous media

We study the stability and dynamic transitions of thermal convection in a fluid layer overlying a saturated porous media based on the Navier–Stokes–Darcy–Boussinesq model. We take a hybrid approach that combines analysis with numerical computation. The center manifold reduction theory is applied to reduce the infinite dynamical system to a finite dimensional one together with a non-dimensional transition number that determines the types of dynamical transitions. Careful numerical computations are performed to determine the transition number as well as related temporal and flow patterns etc. Our result indicates that the system favors a continuous transition in which the steady state solution bifurcates to a local attractor at the critical Rayleigh number. Unlike the one layer case, jump transitions can occur at certain parameter regime. We also discover that the transition between shallow and deep convection associated with the variation of the ratio of free-flow to porous media depth is accompanied by the change of the most unstable mode from the lowest possible horizontal wave number to higher wave numbers which could occur with variation of the height ratio as well as the Darcy number and the ratio of thermal diffusivity among others.