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arXiv · 2103.16498

Bounds for internally heated convection with fixed boundary heat flux

Ali Arslan·Giovanni Fantuzzi·John Craske·Andrew Wynn
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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 nondimensional 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 $\frac12$ 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 $\langle w T \rangle \leq \tfrac{1}{2}\big(\tfrac{1}{2}+ \tfrac{1}{\sqrt{3}} \big) - 1.6552\, R^{-\frac13}$ uniformly in the Prandtl number, where $R$ is the nondimensional 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 $R$) bound (Goluskin 2016, Springer, Table 1.2) and its dependence on $R$ 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 $\frac12$ at high $R$, which corresponds to the top boundary being hotter than the bottom one on average.

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BibTeXRIS

Ali Arslan, Giovanni Fantuzzi, John Craske, Andrew Wynn. 2021-06-10. Bounds for internally heated convection with fixed boundary heat flux. https://doi.org/10.1017/jfm.2021.527

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