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

Abstract

We investigate the influence of the thermal properties of the boundaries in turbulent Rayleigh-Benard 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 ηwhich continuously interpolates between the previously studied fixed temperature ($\eta =0$) and fixed flux ($\eta =\infty$) 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 $\eta >0$, as R increases the bound on the Nusselt number Nu approaches that for the fixed flux problem. Specifically, for $0<\eta \le \infty$ and for sufficiently large R ($R>R_s=O({\eta }^{-2})$ for small η) the Nusselt number is bounded as $Nu\le c(\eta )R^{1/3}\le CR{a}^{1/2}$, where C is an η-independent constant. In the $R\to \infty$ limit, the usual fixed temperature assumption is thus a singular limit of this general bounding problem.

Submitted

References in corpus (1)

• A Comparison of Turbulent Thermal Convection Between Conditions of Constant Temperature and Constant Flux