Optimal Wall-to-Wall Transport by Incompressible Flows
Ian Tobasco, Charles R. Doering · Physical Review Letters · 2017
We consider wall-to-wall transport of a passive tracer by divergence-free velocity vector fields $\mathbf{u}$. Given an enstrophy budget $⟨|\ensuremath{ abla}\mathbf{u}{|}^{2}⟩\ensuremath{\le}{\mathrm{Pe}}^{2}$ we construct steady two-dimensional flows that transport at rates $\mathrm{Nu}(\mathbf{u})\ensuremath{\gtrsim}{\mathrm{Pe}}^{2/3}/(\mathrm{log}\mathrm{Pe}{)}^{4/3}$ in the large enstrophy limit. Combined with the known upper bound $\mathrm{Nu}(\mathbf{u})\ensuremath{\lesssim}{\mathrm{Pe}}^{2/3}$ for any such enstrophy-constrained flow, we conclude that maximally transporting flows satisfy $\mathrm{Nu}\ensuremath{\sim}{\mathrm{Pe}}^{2/3}$ 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 $\mathrm{Nu}\ensuremath{\sim}{\mathrm{Ra}}^{1/2}$ 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.