High Rayleigh number convection in a three-dimensional porous medium

High-resolution numerical simulations of statistically steady convection in a three-dimensional porous medium are presented for Rayleigh numbers $Ra \leqslant 2 \times 10^4$. Measurements of the Nusselt number $Nu$ in the range $1750 \leqslant Ra \leqslant 2 \times 10^4$ are well fitted by a relationship of the form $Nu = \alpha _3 Ra + \beta _3$, for $\alpha _3 = 9.6 \times 10^{-3}$ and $\beta _3 = 4.6$. This fit indicates that the classical linear scaling $Nu \sim Ra$ is attained, and that $Nu$ is asymptotically approximately $40\, \%$ larger than in two dimensions. The dynamical flow structure in the range $1750 \leqslant Ra \leqslant 2\times 10^4$ is analysed, and the interior of the flow is found to be increasingly well described as $Ra \to \infty $ by a heat-exchanger model, which describes steady interleaving columnar flow with horizontal wavenumber $k$ and a linear background temperature field. Measurements of the interior wavenumber are approximately fitted by $k\sim Ra^{0.52 \pm 0.05}$, which is distinguishably stronger than the two-dimensional scaling of $k\sim Ra^{0.4}$.