Optimal wall-to-wall transport by incompressible flows

TL;DR

This paper constructs steady 2D flows that nearly achieve the maximum wall-to-wall passive tracer transport rate under enstrophy constraints, revealing the optimal scaling behavior and its implications for related physical systems.

Contribution

It establishes the asymptotic optimal transport rate for divergence-free flows with bounded enstrophy and links the problem to variational models in pattern formation.

Findings

Constructed flows achieve transport rate $Nu \,\sim\, Pe^{2/3}$ up to logarithmic factors.

Proved the upper bound $Nu \lesssim Pe^{2/3}$ for enstrophy-constrained flows.

Connected the transport problem to singularly perturbed variational problems in materials science.

Abstract

We consider wall-to-wall transport of a passive tracer by divergence-free velocity vector fields $\mathbf{u}$. Given an enstrophy budget $\langle |\nabla \mathbf{u}|^{2} \rangle \le Pe^{2}$ we construct steady two-dimensional flows that transport at rates $Nu(\mathbf{u}) \gtrsim Pe^{2/3}/(\log Pe)^{4/3}$ in the large enstrophy limit. Combined with the known upper bound $Nu(\mathbf{u})\lesssim Pe^{2/3}$ for any such enstrophy-constrained flow, we conclude that maximally transporting flows satisfy $Nu\sim Pe^{2/3}$ up to possible logarithmic corrections. Combined with known transport bounds in the context of Rayleigh-B\'enard convection this establishes that while suitable flows approaching the "ultimate" heat transport scaling $Nu\sim Ra^{1/2}$ 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.