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Wall-to-wall optimal transport in two dimensions

Published online by Cambridge University Press:  28 February 2020

Andre N. Souza Open the ORCID record for Andre N. Souza [Opens in a new window] ,
Ian Tobasco  and
Charles R. Doering
Andre N. Souza*
Affiliation:
Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Ian Tobasco
Affiliation:
Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S. Morgan Street, Chicago, IL 60607-7045, USA
Charles R. Doering
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, USA Center for the Study of Complex Systems, University of Michigan, Ann Arbor, MI 48109-1042, USA Department of Physics, University of Michigan, Ann Arbor, MI 48109-1140, USA
*
†Email address for correspondence: [email protected]
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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.

JFM classification

Mathematical Foundations: Variational methods Flow Control: Mixing enhancement
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 889 , 25 April 2020 , A34
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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