High-Rayleigh-number convection in a fluid-saturated porous layer

The Darcy–Boussinesq equations at infinite Darcy–Prandtl number are used to study convection and heat transport in a basic model of porous-medium convection over a broad range of Rayleigh number $Ra$ . High-resolution direct numerical simulations are performed to explore the modes of convection and measure the heat transport, i.e. the Nusselt number Nu , from onset at Ra \,=\, 4π² up to Ra\,=\,10⁴ . Over an intermediate range of increasing Rayleigh numbers, the simulations display the ‘classical’ heat transport Nu \,~\, Ra scaling. As the Rayleigh number is increased beyond Ra \,=\, 1255 , we observe a sharp crossover to a form fitted by Nu \,≈\, 0.0174 × Ra0.9 over nearly a decade up to the highest $Ra$ . New rigorous upper bounds on the high-Rayleigh-number heat transport are derived, quantitatively improving the most recent available results. The upper bounds are of the classical scaling form with an explicit prefactor: Nu \,≤\, 0.0297 × Ra . The bounds are compared directly to the results of the simulations. We also report various dynamical transitions for intermediate values of $Ra$ , including hysteretic effects observed in the simulations as the Rayleigh number is decreased from $1255$ back down to onset.