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
Bibtex
MSC-Numbers: 76R10, 76E06, 76F99
Keywords and phrases: Rayleigh-Benard convection, heat transport, turbulence
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Abstract:
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 Ra1∕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 ≲ Ra1∕3 ln2∕3Ra 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 ≲ Ra1∕3 ln1∕3Ra with help of the background temperature method.
The paper contains two results: 1.) The background temperature method can be slightly modified to yield Nu ≲ Ra1∕3 ln1∕15Ra . 2.) The estimates behind the temperature background method can be combined with the maximal regularity in L∞ to yield Nu ≲ Ra1∕3 ln1∕3 lnRa — an estimate that is only a double logarithm away from the supposedly optimal scaling.