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}⟩≤Pe^{2} we construct steady two-dimensional flows that transport at rates Nu(u)≳Pe^{2/3}/(logPe)^{4/3} in the large enstrophy limit. Combined with the known upper bound Nu(u)≲Pe^{2/3} for any such enstrophy-constrained flow, we conclude that maximally transporting flows satisfy Nu∼Pe^{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∼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.