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Bounds for internally heated convection with fixed boundary heat flux

Published online by Cambridge University Press:  05 July 2021

Ali Arslan Open the ORCID record for Ali Arslan [Opens in a new window] ,
Giovanni Fantuzzi Open the ORCID record for Giovanni Fantuzzi [Opens in a new window] ,
John Craske Open the ORCID record for John Craske [Opens in a new window]  and
Andrew Wynn Open the ORCID record for Andrew Wynn [Opens in a new window]
Ali Arslan*
Affiliation:
Department of Aeronautics, Imperial College London, LondonSW7 2AZ, UK
Giovanni Fantuzzi
Affiliation:
Department of Aeronautics, Imperial College London, LondonSW7 2AZ, UK
John Craske
Affiliation:
Department of Civil and Environmental Engineering, Imperial College London, LondonSW7 2AZ, UK
Andrew Wynn
Affiliation:
Department of Aeronautics, Imperial College London, LondonSW7 2AZ, UK
*
†Email address for correspondence: [email protected]
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Abstract

We prove a new rigorous bound for the mean convective heat transport $\langle w T \rangle$, where $w$ and $T$ are the non-dimensional vertical velocity and temperature, in internally heated convection between an insulating lower boundary and an upper boundary with a fixed heat flux. The quantity $\langle wT \rangle$ is equal to half the ratio of convective to conductive vertical heat transport, and also to $\frac 12$ plus the mean temperature difference between the top and bottom boundaries. An analytical application of the background method based on the construction of a quadratic auxiliary function yields $\langle w T \rangle \leq \frac {1}{2}(\frac {1}{2}+ \frac {1}{\sqrt {3}} ) - 1.6552\, {\textit {R}}^{-(1/3)}$ uniformly in the Prandtl number, where R is the non-dimensional control parameter measuring the strength of the internal heating. Numerical optimisation of the auxiliary function suggests that the asymptotic value of this bound and the $-1/3$ exponent are optimal within our bounding framework. This new result halves the best existing (uniform in $ {\textit {R}}$) bound (Goluskin, Internally Heated Convection and Rayleigh–Bénard Convection, Springer, 2016, table 1.2), and its dependence on $ {\textit {R}}$ is consistent with previous conjectures and heuristic scaling arguments. Contrary to physical intuition, however, it does not rule out a mean heat transport larger than $\frac 12$ at high $ {\textit {R}}$, which corresponds to the top boundary being hotter than the bottom one on average.

JFM classification

Mathematical Foundations: Variational methods
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 922 , 10 September 2021 , R1
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

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