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MiS Preprint

Rayleigh-Bénard convection: Improved bounds on the Nusselt number

Felix Otto and Christian Seis


We consider Rayleigh--Bénard convection as modelled by the Boussinesq equations in the infinite-Prandtl-number limit. We are interested in the scaling of the average upward heat transport, the Nusselt number $Nu$, in terms of the non-dimensionalized temperature forcing, the Rayleigh number $Ra$. Experiments, asymptotics and heuristics suggest that $Nu\sim Ra^{1/3}$.

This work is mostly inspired by two earlier rigorous work on upper bounds of $Nu$ in terms of $Ra$:
1.) The work of Constantin and Doering establishing $Nu\lesssim Ra^{1/3}\ln^{2/3}Ra$ with help of a (logarithmically failing) maximal regularity estimate in $L^\infty$ on the level of the Stokes equation.
2.) The work of Doering, Reznikoff and the first author establishing $Nu\lesssim Ra^{1/3}\ln^{1/3}Ra$ with help of the background temperature method.

The paper contains two results:
1.) The background temperature method can be slightly modified to yield $Nu\lesssim Ra^{1/3}\ln^{1/15}Ra$ .
2.) The estimates behind the temperature background method can be combined with the maximal regularity in $L^\infty$ to yield $Nu\lesssim Ra^{1/3}\ln^{1/3}\ln Ra$ --- an estimate that is only a double logarithm away from the supposedly optimal scaling.

MSC Codes:
76R10, 76E06, 76F99
Rayleigh-Benard convection, heat transport, turbulence

Related publications

2011 Repository Open Access
Felix Otto and Christian Seis

Rayleigh-Bénard convection : improved bounds on the Nusselt number

In: Journal of mathematical physics, 52 (2011) 8, p. 083702