Optimal Wall-to-Wall Transport by Incompressible Flows

Ian Tobasco and Charles R. Doering
Phys. Rev. Lett. 118, 264502 – Published 27 June 2017

Abstract

We consider wall-to-wall transport of a passive tracer by divergence-free velocity vector fields u. Given an enstrophy budget |u|2Pe2 we construct steady two-dimensional flows that transport at rates Nu(u)Pe2/3/(logPe)4/3 in the large enstrophy limit. Combined with the known upper bound Nu(u)Pe2/3 for any such enstrophy-constrained flow, we conclude that maximally transporting flows satisfy NuPe2/3 up to possible logarithmic corrections. Combined with known transport bounds in the context of Rayleigh-Bénard convection, this establishes that while suitable flows approaching the “ultimate” heat transport scaling NuRa1/2 exist, they are not always realizable as buoyancy-driven flows. The result is obtained by exploiting a connection between the wall-to-wall optimal transport problem and a closely related class of singularly perturbed variational problems arising in the study of energy-driven pattern formation in materials science.

  • Figure
  • Received 21 December 2016

DOI:https://doi.org/10.1103/PhysRevLett.118.264502

© 2017 American Physical Society

Physics Subject Headings (PhySH)

Fluid Dynamics

Authors & Affiliations

Ian Tobasco1 and Charles R. Doering1,2

  • 1Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1043, USA
  • 2Department of Physics, University of Michigan, Ann Arbor, Michigan 48109-1040, USA and Center for the Study of Complex Systems, University of Michigan, Ann Arbor, Michigan 48109-1107, USA

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Vol. 118, Iss. 26 — 30 June 2017

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