Transitional natural convection with conjugate heat transfer over smooth and rough walls

We study turbulent natural convection in enclosures with conjugate heat transfer. The simplest way to increase the heat transfer in this flow is through rough surfaces. In numerical simulations often the constant temperature is assigned at the walls in contact with the fluid, which is unrealistic in laboratory experiments. The DNS (Direct Numerical Simulation), to be of help to experimentalists, should consider the heat conduction in the solid walls together with the turbulent flow between the hot and the cold walls. Here the cold wall, $0.5h$ thick (where $h$ is the channel half-height) is smooth, and the hot wall has two- and three-dimensional elements of thickness $0.2h$ above a solid layer $0.3h$ thick. The independence of the results on the box size has been verified. A bi-periodic domain $4h$ wide allows to have a sufficient resolution with a limited number of grid points. It has been found that, among the different kind of surfaces at a Rayleigh number $Ra \approx 2 \cdot 10^6$, the one with staggered wedges has the highest heat transfer. A large number of simulations varying the $Ra$ from $10^3$ to $10^7$ were performed to find the different ranges of the Nusselt number ($Nu$) relationship as a function of $Ra$. Flow visualizations allow to explain the differences in the $Nu(Ra)$ relationship. Two values of the thermal conductivity were chosen, one corresponding to copper and the other ten times higher. It has been found that the Nusselt number behaves as $Nu=\alpha Ra^\gamma$, with $\alpha$ and $\gamma$ independent on the solid conductivity, and dependent on the roughness shape.