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 × 10⁴ . Measurements of the Nusselt number $Nu$ in the range 1750 Ra 2 × 10⁴ are well fitted by a relationship of the form Nu = α ₃ Ra + β ₃ , for α ₃ = 9.6 × 10⁻³ and β ₃ = 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 Ra 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 .