Bounds for internally heated convection with fixed boundary heat flux

Abstract

We prove a new rigorous bound for the mean convective heat transport $⟨wT⟩$, 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 $⟨wT⟩$ is equal to half the ratio of convective to conductive vertical heat transport, and also to $\frac{1}{2}$ 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 $⟨wT⟩\le \frac{1}{2}(\frac{1}{2}+\frac{1}{\sqrt{3}})-1.6552R^{-\frac{1}{3}}$ 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 $\frac{1}{2}$ at high $R$, which corresponds to the top boundary being hotter than the bottom one on average.

arXiv comment11 pages, 3 figures

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Cited by in corpus (8)

• The background method: Theory and computations
• Analytical bounds on the heat transport in internally heated convection
• Bounds on heat transfer by incompressible flows between balanced sources and sinks
• Optimal cooling of an internally heated disc
• Rigorous scaling laws for internally heated convection at infinite Prandtl number
• Internal heating profiles for which downward conduction is impossible
• Internally heated convection with rotation: bounds on heat transport
• Bounds on dissipation in three-dimensional planar shear flows: reduction to two-dimensional problems