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Wall-to-wall optimal transport in two dimensions
Published online by Cambridge University Press: 28 February 2020
Abstract
Gradient ascent methods are developed to compute incompressible flows that maximize heat transport between two isothermal no-slip parallel walls. Parameterizing the magnitude of the velocity fields by a Péclet number $Pe$ proportional to their root-mean-square rate of strain, the schemes are applied to compute two-dimensional flows optimizing convective enhancement of diffusive heat transfer, i.e. the Nusselt number $Nu$ up to $Pe\approx 10^{5}$. The resulting transport exhibits a change of scaling from $Nu-1\sim Pe^{2}$ for $Pe<10$ in the linear regime to $Nu\sim Pe^{0.54}$ for $Pe>10^{3}$. Optimal fields are observed to be approximately separable, i.e. products of functions of the wall-parallel and wall-normal coordinates. Analysis employing a separable ansatz yields a conditional upper bound ${\lesssim}Pe^{6/11}=Pe^{0.\overline{54}}$ as $Pe\rightarrow \infty$ similar to the computationally achieved scaling. Implications for heat transfer in buoyancy-driven Rayleigh–Bénard convection are discussed.
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- © The Author(s), 2020. Published by Cambridge University Press
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