High Rayleigh number convection in a three-dimensional porous medium

High-resolution numerical simulations of statistically steady convection in a threedimensional porous medium are presented for Rayleigh numbers Ra 6 2 × 104. Measurements of the Nusselt number Nu in the range 1750 6 Ra 6 2× 104 are well fitted by a relationship of the form Nu= α3Ra+ β3, for α3= 9.6× 10−3 and β3= 4.6. This fit indicates that the classical linear scaling Nu∼ Ra is attained, and that Nu is asymptotically approximately 40 % larger than in two dimensions. The dynamical flow structure in the range 1750 6 Ra 6 2 × 104 is analysed, and the interior of the flow is found to be increasingly well described as Ra→∞ 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∼ Ra0.52±0.05, which is distinguishably stronger than the two-dimensional scaling of k∼ Ra0.4.