Bounds on heat transport for two-dimensional buoyancy driven flows between rough boundaries

Abstract

We consider the two-dimensional Rayleigh-Bénard convection in a layer of fluid between rough Navier-slip boundaries. The top and bottom boundaries are described by the same height function h. We prove rigorous upper bounds on the Nusselt number which capture the dependence on the curvature of the boundary κ and the (non-constant) friction coefficient α explicitly. For h ∈ W 2,∞ and κ satisfying a smallness condition with respect to α, we find Nu Rawhich agrees with the predicted Spiegel-Kraichnan scaling when κ = 0. This bound is obtained via local regularity estimates in a small strip at the boundary. When h ∈ W 3,∞ and the functions κ and α are sufficiently small in L ∞ , we prove upper bounds using the background field method, which interpolate between Ra 1 2 and Ra 5 12 with non-trivial dependence on α and κ. These bounds agree with the result in [DNN22] for flat boundaries and constant friction coefficient. Furthermore, in the regime Pr ≥ Ra 5 7 , we improve the Ra 1 2 -upper bound, showing Nu α,κ Ra 3 7 , where α,κ hides an additional dependency of the implicit constant on α and κ.

Optimal wall shapes and flows for steady planar convection

“…On the theoretical side, Goluskin & Doering (2016) used the 'background method' to obtain upper bounds on the rate of heat transfer in fluid layers between upper and lower rough walls whose profiles correspond to single-valued functions of the horizontal coordinate. Bleitner & Nobili (2022) used a similar approach to derive upper bounds on heat transfer for Navier-slip rough boundaries.…”