"Ultimate state" of two-dimensional Rayleigh-Benard convection between free-slip fixed temperature boundaries

TL;DR

This paper derives rigorous upper bounds on heat transport in 2D Rayleigh-Benard convection with free-slip boundaries, challenging existing theories on turbulent convection at high Rayleigh numbers.

Contribution

It provides the first rigorous upper limit on the Nusselt number scaling in this specific convection setup, with a novel bound of Nu ≤ 0.2295 Ra^{5/12}.

Findings

Nu is bounded by 0.2295 Ra^{5/12}

The bound applies uniformly across Prandtl numbers

Challenges existing theoretical predictions for high Ra convection

Abstract

Rigorous upper limits on the vertical heat transport in two dimensional Rayleigh-Benard convection between stress-free isothermal boundaries are derived from the Boussinesq approximation of the Navier-Stokes equations. The Nusselt number Nu is bounded in terms of the Rayleigh number Ra according to $Nu \leq 0.2295 Ra^{5/12}$ uniformly in the Prandtl number Pr. This Nusselt number scaling challenges some theoretical arguments regarding the asymptotic high Rayleigh number heat transport by turbulent convection.