Exhausting the background approach for bounding the heat transport in Rayleigh–Bénard convection

We revisit the optimal heat transport problem for Rayleigh–Benard convection in which a rigorous upper bound on the Nusselt number, $Nu$ , is sought as a function of the Rayleigh number, $Ra$ . Concentrating on the two-dimensional problem with stress-free boundary conditions, we impose the time-averaged heat equation as a constraint for the bound using a novel two-dimensional background approach thereby complementing the ‘wall-to-wall’ approach of Hassanzadeh et al. ( J. Fluid Mech. , vol. 751, 2014, pp. 627–662). Imposing the same symmetry on the problem, we find correspondence with their maximal result for $Ra\leqslant Ra_{c}:=4468.8$ but, beyond that, the results from the two approaches diverge. The bound produced by the two-dimensional background field approaches that produced by the one-dimensional background field from below as the length of computational domain $L\rightarrow \infty$ . On lifting the imposed symmetry, the optimal two-dimensional temperature background field reverts to being one-dimensional, giving the best bound $Nu\leqslant 0.055Ra^{1/2}$ compared to $Nu\leqslant 0.026Ra^{1/2}$ in the non-slip case. We then show via an inductive bifurcation analysis that introducing two-dimensional temperature and velocity background fields (in an attempt to impose the time-averaged Boussinesq equations) is also unable to lower the bound. This then exhausts the background approach for the two-dimensional (and by extension three-dimensional) Rayleigh–Benard problem with the bound remaining stubbornly $Ra^{1/2}$ while data seem more to scale like $Ra^{1/3}$ for large $Ra$ . Finally, we show that adding a velocity background field to the formulation of Wen et al.  ( Phys. Rev. E., vol. 92, 2015, 043012), which is able to use an extra vorticity constraint due to the stress-free condition to lower the bound to $Nu\leqslant O(Ra^{5/12})$ , also fails to further improve the bound.