Optimal Wall-to-Wall Transport by Incompressible Flows

Optimal Wall-to-Wall Transport by Incompressible Flows We consider wall-to-wall transport of a passive tracer by divergence-free velocity vector fields u. Given an enstrophy budget ⟨|∇u|2⟩≤Pe2 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 Nu∼Pe2/3 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 Nu∼Ra1/2 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Physical Review Letters American Physical Society (APS)

Optimal Wall-to-Wall Transport by Incompressible Flows

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Optimal Wall-to-Wall Transport by Incompressible Flows

Abstract

We consider wall-to-wall transport of a passive tracer by divergence-free velocity vector fields u. Given an enstrophy budget ⟨|∇u|2⟩≤Pe2 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 Nu∼Pe2/3 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 Nu∼Ra1/2 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.
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Publisher
The American Physical Society
Copyright
Copyright © © 2017 American Physical Society
ISSN
0031-9007
eISSN
1079-7114
D.O.I.
10.1103/PhysRevLett.118.264502
Publisher site
See Article on Publisher Site

Abstract

We consider wall-to-wall transport of a passive tracer by divergence-free velocity vector fields u. Given an enstrophy budget ⟨|∇u|2⟩≤Pe2 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 Nu∼Pe2/3 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 Nu∼Ra1/2 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.

Journal

Physical Review LettersAmerican Physical Society (APS)

Published: Jun 30, 2017

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