Mixing induced by buoyancy-driven flows in porous media

A theoretical model for fluid mixing in steady and transient buoyancy-driven flows induced by laminar natural convection in porous layers is presented. This problem follows a highly nonlinear dynamics and its accurate modeling poses numerical challenges. Based on the Taylor dispersion theory, a one-dimensional analytical model is developed for steady and transient velocity fields. To investigate steady-state mixing, a unicellular steady velocity field is established by maintaining a thermal gradient across a porous layer of finite thickness. A passive tracer is then introduced into the flow field and the mixing process is studied. In the case of transient flows, as the convective flow grows and decays with time the behavior of the dispersion coefficient is characterized by a four-parameter Weibull function. The simple analytical model developed here can recover scaling relations that have been reported in the literature to characterize the mixing process in steady and transient buoyancy-driven flows.