Conservative bounds on Rayleigh-Bénard convection with mixed thermal boundary conditions

Abstract

Using the background field variational method developed by Doering and Constantin, we obtain upper bounds on heat transport in Rayleigh-Bénard convection assuming mixed (Robin) thermal conditions of arbitrary Biot number η at the fluid boundaries, ranging from the fixed temperature (perfectly conducting, η = 0) to the fixed flux (perfectly insulating, η = ∞) extremes. Solving the associated Euler-Lagrange equations, we numerically find optimal bounds on the averaged convective heat transport, measured by the Nusselt number $\mathit{Nu}$, over a restricted one-parameter class of piecewise linear background temperature profiles, and compare these to conservative analytical bounds derived using elementary functional estimates. We find that analytical estimates fully capture the scaling behaviour of the semi-optimal numerical bounds, including a clear transition from fixed temperature to fixed flux behaviour observed for any small nonzero η as the usual Rayleigh number $\mathit{Ra}$ increases, suggesting that in the strong driving limit, all imperfectly conducting boundaries effectively act as insulators. The overall bounds, optimized over piecewise linear backgrounds, are $\mathit{Nu}$ ≤ 0.045 $\mathit{Ra}^{1/2}$ in the fixed temperature case η = 0, and $\mathit{Nu}$ ≤ 0.078 $\mathit{Ra}^{1/2}$ in the large-$\mathit{Ra}$ limit in all other cases, 0 < η ≤ ∞.