Inhomogeneous temperature boundary conditions in Rayleigh-B\'enarrd convection

In the ideal Rayleigh-B\'enard problem it is assumed that the fluid layer is heated and cooled, respectively from below and above, by isothermal surfaces that strictly maintain the temperature constant regardless of the flow dynamics. This is however only an approximation since all thermal sources have properties that couple with those of the fluid. In this study we analyze, by three-dimensional direct numerical simulations, the heat and flow dynamics when the plates have non homogeneous temperature boundary conditions in order to understand to what degree the Nusselt number is affected. Several ``temperature patterns'' have been imposed on the plates with the Nusselt number that has shown a surprising robutness to the temperature inhomogeneity. The investigation is further extended to cases in which the plates are made by a sandwich of different materials and the temperature is inhomogeneous also within the plate thickness.