Bounds on Rayleigh - Bénard convection with general thermal boundary conditions . Part 1 . Fixed Biot number boundaries By Ralf

We investigate the influence of the thermal properties of the boundaries in turbulent Rayleigh-Bénard convection on analytical bounds on convective heat transport. Using the Doering-Constantin background flow method, we systematically formulate a bounding principle on the Nusselt-Rayleigh number relationship for general mixed thermal boundary conditions of constant Biot number η which continuously interpolates between the previously studied fixed temperature (η = 0) and fixed flux (η = ∞) cases, and derive explicit asymptotic and rigorous bounds. Introducing a control parameter R as a measure of the driving which is in general different from the usual Rayleigh number Ra, we find that for each η > 0, as R increases the bound on the Nusselt number Nu approaches that for the fixed flux problem. Specifically, for 0 < η ≤ ∞ and for sufficiently large R (R > Rs = O(η−2) for small η) the Nusselt number is bounded as Nu ≤ c(η)R ≤ CRa, where C is an η-independent constant. In the R → ∞ limit, the usual fixed temperature assumption is thus a singular limit of this general bounding problem.