Bounds on internally heated convection with fixed boundary heat flux
Author(s)
Arslan, Ali
Fantuzzi, giovanni
Craske, john
Wynn, andrew
Type
Journal Article
Abstract
We prove a new rigorous bound for the mean convective heat transport ⟨wT⟩, 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 ⟨wT⟩ is equal to half the ratio of convective to conductive vertical heat transport, and also to 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 ⟨wT⟩≤12(12+13√)−1.6552R−(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 R) bound (Goluskin, Internally Heated Convection and Rayleigh–Bénard Convection, Springer, 2016, 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 12 at high R, which corresponds to the top boundary being hotter than the bottom one on average.
Date Issued
2021-09-10
Date Acceptance
2021-06-09
Citation
Journal of Fluid Mechanics, 2021, 992, pp.R1-R1
ISSN
0022-1120
Publisher
Cambridge University Press
Start Page
R1
End Page
R1
Journal / Book Title
Journal of Fluid Mechanics
Volume
992
Replaces
10044/1/89026
Copyright Statement
© The Author(s), 2021. Published by Cambridge University Press. This paper has been accepted for publication and will appear in a revised form, subsequent to peer-review and/or editorial input by Cambridge University Press.
Sponsor
Engineering and Physical Sciences Research Council
Identifier
https://arxiv.org/abs/2103.16498
Grant Number
EP/L016230/1
Subjects
physics.flu-dyn
Publication Status
Published
Article Number
ARTN R1
Date Publish Online
2021-07-05