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 are interested in the design of forcing in the forced Navier—Stokes equation that maximizes this transport for a given power supply budget. One can equivalently formulate this problem as the design of a divergence-free flow field that maximizes scalar transport under an enstrophy constraint (which can be understood as a constraint on the power supply). Previous work established that the transport cannot scale faster than 1/3-power of the power supply. 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 upper bound up to a logarithmic correction to scaling. This logarithmic correction appears to arise 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 for which the corresponding passive scalar transport scales as a clean 1/3-power law in power supply. 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, it appears that the 1/3 scaling is also optimal in two dimensions. This unsteady flow design challenges the general belief that steady flows are optimal for transporting heat in the family of all incompressible flows. After carefully examining these designs, we extract the underlying physical mechanism that makes the branching flows “efficient.” We present the relevance of branching in naturally occurring buoyancy-driven flows and discuss if these flows are optimal for transporting a scalar. We also present a design of mechanical apparatus, which in principle, can achieve the best possible case scenario of heat transfer.