Convection in porous media with dispersion

We investigate the effect of dispersion on convection in porous media by performing direct numerical simulations (DNS) in a two-dimensional Rayleigh-Darcy domain. Scaling analysis of the governing equations shows that the dynamics of this system are not only controlled by the classical Rayleigh-Darcy number based on molecular diffusion, $Ra_m$, and the domain aspect ratio, but also controlled by two other dimensionless parameters: the dispersive Rayleigh number $Ra_d = H/\alpha_t$ and the dispersivity ratio $r = \alpha_l/\alpha_t$, where $H$ is the domain height, $\alpha_t$ and $\alpha_l$ are the transverse and longitudinal dispersivities, respectively. For $\Delta = Ra_d/Ra_m > O(1)$, the influence from the mechanical dispersion is minor; for $\Delta \ll 1$, however, the flow pattern is controlled by $Ra_d$ while the convective flux is $F\sim Ra_m$ for large $Ra_m$, but with a prefactor that has a non-monotonic dependence on $Ra_d$. Our DNS results also show that the increase of mechanical dispersion, i.e. decreasing $Ra_d$, will coarsen the convective pattern by increasing the plume spacing. Moreover, the inherent anisotropy of mechanical dispersion breaks the columnar structure of the mega-plumes at large $Ra_m$, if $Ra_d < 5000$. This results in a fan-flow geometry that reduces the convective flux.