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 <= 2 x 10(4). Measurements of the Nusselt number Nu in the range 1750 <= Ra <= 2 x 10(4) are well fitted by a relationship of the form Nu D alpha Ra-3 + beta(3), for alpha(3) = 9.6 x 10(-3) and alpha(3) = 4.6. This fit indicates that the classical linear scaling Nu similar to Ra is attained, and that Nu is asymptotically approximately 40% larger than in two dimensions. The dynamical flow structure in the range 1750 <= Ra <= 2 x 10(4) is analysed, and the interior of the flow is found to be increasingly well described as Ra -> infinity 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 similar to Ra-0.52 +/- 0.05, which is distinguishably stronger than the two-dimensional scaling of k similar to Ra-0.4.