Bound on Vertical Heat Transport at Large Prandtl Number


We prove a new upper bound on the vertical heat transport in Rayleigh-Bénard convection of the form c Rafrac(1, 3) (ln Ra)frac(2, 3) under the assumption that the ratio of Prandtl number over Rayleigh number satisfies frac(Pr, Ra) ≥ c0 where the non-dimensional constant c0 depends on the aspect ratio of the domain only. This new rigorous bound agrees with the (optimal) Rafrac(1, 3) bound (modulo logarithmic correction) on vertical heat transport for the infinite Prandtl number model for convection due to Constantin and Doering [P. Constantin, C.R. Doering, Infinite Prandtl number convection, J. Stat. Phys. 94 (1) (1999) 159-172] and Doering, Otto and Reznikoff [C.R. Doering, F. Otto, M.G. Reznikoff, Bounds on vertical heat transport for infinite Prandtl number Rayleigh-Bénard convection, J. Fluid Mech. 560 (2006) 229-241]. It also improves a uniform (in Prandtl number) Rafrac(1, 2) bound for the Nusselt number [P. Constantin, C.R. Doering, Heat transfer in convective turbulence, Nonlinearity 9 (1996) 1049-1060] in the case of large Prandtl number. © 2007 Elsevier Ltd. All rights reserved.


Mathematics and Statistics


National Science Foundation, Grant None

Keywords and Phrases

Boussinesq Equations; Nusselt Number; Prandtl Number; Rayleigh Number; Rayleigh-Bénard Convection

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Document Type

Article - Journal

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© 2023 Elsevier, All rights reserved.

Publication Date

15 May 2008