Optimizing scalar transport with branching pipe flows

Abstract

We consider the problem of "wall-to-wall optimal transport" in which we attempt to maximize the transport of a passive temperature field between hot and cold plates. Specifically, we optimize the choice of the divergence-free velocity field in the advection-diffusion equation amongst all velocities satisfying an enstrophy constraint (which can be understood as a constraint on the power required to generate the flow). Previous work established an a priori upper bound on the transport, scaling as the 1/3-power of the flow's enstrophy. Recently, Tobasco & Doering (Phys. Rev. Lett. vol.118, 2017, p.264502) and Doering & Tobasco (Comm. Pure Appl. Math. vol.72, 2019, p.2385--2448) constructed self-similar two-dimensional steady branching flows saturating this bound up to a logarithmic correction to scaling. This logarithmic correction appears to arises due to a topological obstruction inherent to two-dimensional steady branching flows. We present a construction of three-dimensional ``branching pipe flows" that eliminates the possibility of this logarithmic correction and therefore identifies the optimal scaling as a clean 1/3-power law. These branching pipe flows are not merely mathematical construct but in principle can be engineered. Our flows resemble previous numerical studies of the three-dimensional wall-to-wall problem by Motoki, Kawahara & Shimizu (J. Fluid Mech. vol.851, 2018, p.R4). However, using an unsteady branching flow construction, we also show that the 1/3 scaling is, in fact, optimal in two dimensions as well. This unsteady flow construction challenges the general belief that in the class of all incompressible flows, steady flows are optimal in transporting heat. We discuss the underlying physical mechanism that makes the branching flows "efficient" in transporting heat. Finally, we discuss the implications of our result to the heat transfer problem in the Rayleigh--B\'enard convection and the problem of anomalous dissipation in a passive scalar, and we propose a few conjectures in these directions.