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

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Ding, Zijing 
Kerswell, Rich R 


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 RaRac:=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. On lifting the imposed symmetry, the optimal 2-dimensional temperature background field reverts back to being 1-dimensional giving the best bound Nu≤0.055Ra1/2 compared to Nu≤0.026Ra1/2 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 Ra1/2 while data seems more to scale like Ra1/3 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.



Benard convection, variational methods

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Journal of Fluid Mechanics

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Cambridge University Press


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Engineering and Physical Sciences Research Council (EP/P001130/2)
EPSRC under grant EP/P001130/1.