Three dimensional branching pipe flows for optimal scalar transport between walls

Abstract We are interested in the design of forcing in the Navier–Stokes equation such that the resultant flow maximises the transport of a passive temperature between two differentially heated walls for a given power supply budget. This problem in the community is also known as ‘wall-to-wall optimal transport’ and can be reduced to optimizing the choice of the divergence-free velocity field in the advection-diffusion equation subject to an enstrophy constraint (which can be understood as a constraint on the power required to generate the flow). Previous work established that the transport cannot scale faster than 1/3-power of the power supply. Recently, Tobasco and Doering (2017 Phys. Rev. Lett. 118 264502) and Doering and Tobasco (2019 Commun. Pure Appl. Math. 72 2385–448) constructed self-similar two-dimensional steady branching flows saturating this bound up to a logarithmic correction. This correction appears to arise due to a topological obstruction inherent to two-dimensional steady flows. In this paper, we present a novel design of three-dimensional ‘branching pipe flows’ that bypasses this obstruction and, consequently, eliminates this logarithmic correction and therefore identifies the optimal scaling as a clean 1/3-power law. Our flows resemble the ones obtained in previous numerical studies of the three-dimensional wall-to-wall problem by Motoki, Kawahara and Shimizu (2018 J. Fluid Mech. 851 R4). We also discuss the implications of this result to the heat transfer problem in Rayleigh–Bénard convection and the problem of anomalous dissipation in a passive scalar.