Ultimate State of Two-Dimensional Rayleigh-Bénard Convection between Free-Slip Fixed-Temperature Boundaries

Abstract

Rigorous upper limits on the vertical heat transport in two-dimensional Rayleigh-B\'enard 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 $\mathrm{Nu}\ensuremath{\le}0.2891{\mathrm{Ra}}^{5/12}$ uniformly in the Prandtl number Pr. This scaling challenges some theoretical arguments regarding asymptotic high Rayleigh number heat transport by turbulent convection.