Bounds for Rayleigh–Bénard convection between free-slip boundaries with an imposed heat flux

Fantuzzi G (2018)

Publication Type: Journal article

Publication year: 2018

Journal

Journal of Fluid Mechanics Cambridge University Press (CUP)

Book Volume: 837

Article Number: 837

DOI: 10.1017/jfm.2017.907

Abstract

We prove the first rigorous bound on the heat transfer for three-dimensional Rayleigh–Bénard convection of finite-Prandtl-number fluids between free-slip boundaries with an imposed heat flux. Using the auxiliary functional method with a quadratic functional, which is equivalent to the background method, we prove that the Nusselt number Nu is bounded by Nu 6 0.5999R1/3 uniformly in the Prandtl number, where R is the Rayleigh number based on the imposed heat flux. In terms of the Rayleigh number based on the mean vertical temperature drop, Ra, we obtain Nu 6 0.4646Ra1/2. The scaling with Rayleigh number is the same as that of bounds obtained with no-slip isothermal, free-slip isothermal and no-slip fixed-flux boundaries, and numerical optimisation of the bound suggests that it cannot be improved within our bounding framework. Contrary to the two-dimensional case, therefore, the Ra-dependence of rigorous upper bounds on the heat transfer obtained with the background method for three-dimensional Rayleigh–Bénard convection is insensitive to both the thermal and the velocity boundary conditions.

Authors with CRIS profile

Giovanni Fantuzzi

Involved external institutions

Imperial College London / The Imperial College of Science, Technology and Medicine GB United Kingdom (GB)

How to cite

APA:

Fantuzzi, G. (2018). Bounds for Rayleigh–Bénard convection between free-slip boundaries with an imposed heat flux. Journal of Fluid Mechanics, 837. https://dx.doi.org/10.1017/jfm.2017.907

MLA:

Fantuzzi, Giovanni. "Bounds for Rayleigh–Bénard convection between free-slip boundaries with an imposed heat flux." Journal of Fluid Mechanics 837 (2018).

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