Bounds for internally heated convection with fixed boundary heat flux - Arslan, Ali; Fantuzzi, Giovanni; Craske, John; Wynn, Andrew (Journal of Fluid Mechanics, 2021)
We prove a new rigorous bound for the mean convective heat transport, where and 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 is equal to half the ratio of convective to conductive vertical heat transport, and also to 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 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 exponent are optimal within our bounding framework. This new result halves the best existing (uniform in) bound (Goluskin, Internally Heated Convection and Rayleigh-Bénard Convection, Springer, 2016, table 1.2), and its dependence on 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 at high, which corresponds to the top boundary being hotter than the bottom one on average.
CRIS-Team Scopus Importer:2022-09-26
Keywords: variational methods