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We perform a numerical study of the heat transfer and flow structure ofRayleigh-B\'enard (RB) convection in (in most cases regular) porous media,which are comprised of circular, solid obstacles located on a square lattice.This study is focused on the role of porosity $\phi$ in the flow propertiesduring the transition process from the traditional RB convection with $\phi=1$(so no obstacles included) to Darcy-type porous-media convection with $\phi$approaching 0. Simulations are carried out in a cell with unity aspect ratio,for the Rayleigh number $Ra$ from $10^5$ to $10^{10}$ and varying porosities$\phi$, at a fixed Prandtl number $Pr=4.3$, and we restrict ourselves to thetwo dimensional case. For fixed $Ra$, the Nusselt number $Nu$ is found to varynon-monotonously as a function of $\phi$; namely, with decreasing $\phi$, itfirst increases, before it decreases for $\phi$ approaching 0. Thenon-monotonous behaviour of $Nu(\phi)$ originates from two competing effects ofthe porous structure on the heat transfer. On the one hand, the flow coherenceis enhanced in the porous media, which is beneficial for the heat transfer. Onthe other hand, the convection is slowed down by the enhanced resistance due tothe porous structure, leading to heat transfer reduction. For fixed $\phi$,depending on $Ra$, two different heat transfer regimes are identified, withdifferent effective power-law behaviours of $Nu$ vs $Ra$, namely, a steep onefor low $Ra$ when viscosity dominates, and the standard classical one for large$Ra$. The scaling crossover occurs when the thermal boundary layer thicknessand the pore scale are comparable. The influences of the porous structure onthe temperature and velocity fluctuations, convective heat flux, and energydissipation rates are analysed, further demonstrating the competing effects ofthe porous structure to enhance or reduce the heat transfer.

From Rayleigh-B\'enard convection to porous-media convection: how porosity affects heat transfer and flow structure