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 $\mathbf{u}$. Given an enstrophy budget $⟨|\nabla \mathbf{u}{|}^2⟩\le P{e}^2$ we construct steady two-dimensional flows that transport at rates $Nu(\mathbf{u})\gtrsim P{e}^{2/3}/(\log Pe{)}^{4/3}$ in the large enstrophy limit. Combined with the known upper bound $Nu(\mathbf{u})\lesssim P{e}^{2/3}$ for any such enstrophy-constrained flow, we conclude that maximally transporting flows satisfy $Nu\sim P{e}^{2/3}$ 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 $Nu\sim R{a}^{1/2}$ 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.

Edited to take into account referee comments. To appear in Phys. Rev. Lett

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