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

Abstract

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 $Ra \,{=}\, 4\pi^2$ up to $Ra\,{=}\,10^4$. Over an intermediate range of increasing Rayleigh numbers, the simulations display the ‘classical’ heat transport $\hbox{\it Nu} \,{\sim}\, Ra$ scaling. As the Rayleigh number is increased beyond $Ra \,{=}\, 1255$, we observe a sharp crossover to a form fitted by $\hbox{\it Nu} \,{\approx}\, 0.0174 \times Ra^{0.9}$ 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: $\hbox{\it Nu} \,{\le}\, 0.0297 \times Ra$. 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.