How much stuff can be transported by an incompressible flow containing a specified amount of kinetic energy or enstrophy? We study this problem for steady 2D flows focusing on passive tracer transport between two parallel impermeable walls, employing the calculus of variations to find divergence-free velocity field 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. Results are reported in terms of the Nusselt number Nu, the convective enhancement of transport normalized by the flow-free diffusive transport, and the Peclect number Pe, the dimensionless gauge of the strength of the flow. For both energy and enstrophy constraints we find that as Pe increases, the maximum transport is achieved by cells of decreasing aspect ratio. For each of the two flow intensity constraints, we also consider buoyancy-driven flows the same constraint to see how the scalings for transport reported in the literature compare with the absolute upper bounds. This work provides new insight into both steady 2D optimal transport and turbulent transport, an increasingly lively area of research in geophysical, astrophysical, and engineering fluid dynamics. This is joint work with Pedram Hassanzadeh (UC Berkeley/Harvard) and Gregory P. Chini (University of New Hampshire).
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