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, $R{a}_m$, and the domain aspect ratio, but also controlled by two other dimensionless parameters: the dispersive Rayleigh number $R{a}_d=H\mathrm{/}{\alpha }_t$ and the dispersivity ratio $r = \alpha_l/\alpha_t$,where$H$isthedomainheight,$\alpha_t$and$\alpha_l$arethetransverseandlongitudinaldispersivities,respectively\mathrm{.}For$\Delta = Ra_d/Ra_m > O(1)$,theinfluencefromthemechanicaldispersionisminor;for$\Delta \ll 1$,however,theflowpatterniscontrolledby$Ra_d$ while the convective flux is $F\sim R{a}_m$ for large $R{a}_m$, but with a prefactor that has a non-monotonic dependence on $R{a}_d$. Our DNS results also show that the increase of mechanical dispersion, i.e. decreasing $R{a}_d$, 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 $R{a}_m$, if $R{a}_d<5000$. This results in a fan-flow geometry that reduces the convective flux.