# Exhausting the background approach for bounding the heat transport in Rayleigh-B\'enard convection

## TL;DR

This paper explores the limits of bounding heat transport in 2D Rayleigh-Bénard convection using a background field approach, showing that the bounds cannot be improved beyond a certain scaling despite complexifying the models.

## Contribution

It exhaustively analyzes the background approach for bounding heat transport, demonstrating its limitations and the inability to lower the established bounds with more complex models.

## Key findings

- Optimal bounds approach $Ra^{1/2}$ scaling.
- Imposing full heat and velocity constraints does not lower the bound.
- Adding a velocity background field does not improve the bound.

## Abstract

We revisit the optimal heat transport problem for Rayleigh-B\'enard 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 2-dimensional problem with stress-free boundary conditions, we impose the full heat equation as a constraint for the bound using a novel 2-dimensional background approach thereby complementing the `wall-to-wall' approach of Hassanzadeh \etal \,(\emph{J. Fluid Mech.} \textbf{751}, 627-662, 2014). Imposing the same symmetry on the problem, we find correspondence with their result for $Ra \leq Ra_c:=4468.8$ but, beyond that, the optimal fields complexify to produce a higher bound. This bound approaches that by a 1-dimensional background field as the length of computational domain $L\rightarrow\infty$. On lifting the imposed symmetry, the optimal 2-dimensional temperature background field reverts back to being 1-dimensional giving the best bound $Nu\le 0.055Ra^{1/2}$ compared to $Nu \le 0.026Ra^{1/2}$ in the non-slip case. % We then show via an inductive bifurcation analysis that imposing the full time-averaged Boussinesq equations as constraints (by introducing 2-dimensional temperature {\em and} velocity background fields) is also unable to lower this bound. This then exhausts the background approach for the 2-dimensional (and by extension 3-dimensional) Rayleigh-Benard problem with the bound remaining stubbornly $Ra^{1/2}$ while data seems more to scale like $Ra^{1/3}$ for large $Ra$. % Finally, we show that adding a velocity background field to the formulation of Wen \etal\, (\emph{Phys. Rev. E.} \textbf{92}, 043012, 2015), which is able to use an extra vorticity constraint due to the stress-free condition to lower the bound to $ Nu \le O(Ra^{5/12})$, also fails to improve the bound.