Exhausting the background approach for bounding the heat transport in Rayleigh-Benard convection

We revisit the optimal heat transport problem for Rayleigh-Benard convection in which a rigorous upper bound on the Nusselt number, , is sought as a function of the Rayleigh number, . 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 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 . On lifting the imposed symmetry, the optimal two-dimensional temperature background field reverts to being one-dimensional, giving the best bound compared to 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 while data seem more to scale like for large . 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 , also fails to further improve the bound.