"Ultimate state" of two-dimensional Rayleigh-Benard convection between   free-slip fixed temperature boundaries
  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. 
Jared P. Whitehead and Charles R. Doering
physics.flu-dyn math-ph math.MP