Preprint 6/2011

Rayleigh-Bénard convection: Improved bounds on the Nusselt number

Felix Otto and Christian Seis

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Submission date: 02. Mar. 2011
Pages: 30
published in: Journal of mathematical physics, 52 (2011) 8, art-no. 083702 
DOI number (of the published article): 10.1063/1.3623417
MSC-Numbers: 76R10, 76E06, 76F99
Keywords and phrases: Rayleigh-Benard convection, heat transport, turbulence
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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 Ra13.

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 Ra13 ln23Ra with help of a (logarithmically failing) maximal regularity estimate in L on the level of the Stokes equation. 2.) The work of Doering, Reznikoff and the first author establishing Nu Ra13 ln13Ra with help of the background temperature method.

The paper contains two results: 1.) The background temperature method can be slightly modified to yield Nu Ra13 ln115Ra . 2.) The estimates behind the temperature background method can be combined with the maximal regularity in L to yield Nu Ra13 ln13 lnRa — an estimate that is only a double logarithm away from the supposedly optimal scaling.

04.01.2023, 02:15