Asymptotic behaviour of heat transfer in two-dimensional turbulent convection with high-porosity fluid-saturated media

We study the asymptotic behaviour of convective heat transfer for turbulent flows in high-porosity fluid-saturated media by two-dimensional high-resolution numerical simulation. The generalized Navier-Stokes equations for incompressible fluid flow and the heat transport equation in porous media at the representative element volume scale are solved by the lattice Boltzmann method, wherein the non-Darcian effects are taken into consideration. The asymptotic behaviour of the Nusselt number N has been revealed for Rayleigh numbers 10(4) <= R <= 10(11) and Darcy numbers 10(-6) <= xi <= 10(6): all the data for various Darcy numbers gradually collapse onto a unique line with increasing Rayleigh number. The asymptote can be well represented by N = 0.146 x R-0.286 for R > 2 x 10(7), which approaches the relationship for the Rayleigh-Benard turbulent convection of free fluid flows. The transition can be characterized by a scaling analysis with R xi(3/2) similar to 1, below which, however, the data collapse onto the Darcy limit for porous media. The Reynolds number and the Nusselt number both increase with Darcy number above the onset of convection, whereas a premature saturation of the Nusselt number is observed in comparison with that of the Reynolds number. The counter-gradient heat transport by the large-scale flows is quantified, which compensates for the increase of the gradient heat transport with Darcy number. The heat transfer in high-porosity fluid-saturated media with a very small Darcy number xi >= 10(-6) can be comparable to that of free fluid flows for a sufficiently high Rayleigh number R >= 10(11).