Ultimate Regime of High Rayleigh Number Convection in a Porous Medium

Abstract

Well-resolved direct numerical simulations of 2D Rayleigh-B\'enard convection in a porous medium are presented for Rayleigh numbers $\mathrm{Ra}\ensuremath{\le}4\ifmmode\times\else\texttimes\fi{}{10}^{4}$ which reveal that, contrary to previous indications, the linear classical scaling for the Nusselt number, $\mathrm{Nu}\ensuremath{\sim}\mathrm{Ra}$, is attained asymptotically. The flow dynamics are analyzed, and the interior of the vigorously convecting system is shown to be increasingly well described as $\mathrm{Ra}\ensuremath{\rightarrow}\ensuremath{\infty}$ by a simple columnar ``heat-exchanger'' model with a single horizontal wave number $k$ and a linear background temperature field. The numerical results are approximately fitted by $k\ensuremath{\sim}{\mathrm{Ra}}^{0.4}$.