Wall to wall optimal transport

How much stuff can be transported by an incompressible flow flow containing a specified amount of kinetic energy (mean square velocity) or enstrophy (mean square vorticity)? We study this problem for steady 2D flows focusing on passive tracer transport between parallel impermeable walls, employing the calculus of variations to find divergence-free velocity fields with a given intensity budget that maximize transport between the walls. The maximizing velocity fields, i.e., the optimal flows, consist of arrays of (convection-like) cells of a particular form. Results are reported in terms of the Nusselt number Nu, the convective enhancement of transport normalized by the flow-free diffusive transport, and the Péclect number Pe, the dimensionless gauge of the strength of the flow. For each of the two flow intensity constraints, we also consider buoyancy-driven Rayleigh-Bénard flows with the same flow intensity constraints to see how parameter scalings for Nu (the heat transport in this context) reported in the literature compare with the bounds. We also consider the time-dependent optimal transport problems coming from Galerkin trunctions of a Rayleigh's original convection model, namely the Lorenz equations (a three Fourier-mode truncation) and another eight-mode model.

Much of this is joint work with Pedram Hassanzadeh (Berkeley/Harvard) and Gregory P. Chini (New Hampshire) published in /Journal of Fluid Mechanics /*751*, 627-662 (2014), and Andre Souza (Michigan) published in /Physics Letters A /*379*, 518-523 (2015).