Upper bounds on the heat transport in Rayleigh-Benard convection

We study the scaling of bounds on heat transport in Rayleigh-Benard convection of a layer of fluid between two infinite horizontal plates under various thermal boundary conditions. First we demonstrate how to establish an upper bound on the heat transport, measured by the Nusselt number Nu, as a function of the Rayleigh number Ra, using the Doering-Constantin approach of background profiles. Then we numerically compute the bounds using optimal piecewise linear background profiles. For each boundary condition we find that the Nu is bounded above by a constant C times the square root of Ra. In the fixed temperature case, we get C = 0.045; in the fixed flux case, we get C = 0.078; while for general thermal boundary conditions, we find numerically that the prefactor C is similar to that in the fixed flux case, and depends, at best, weakly on the Biot number.