Optimal wall-to-wall transport by incompressible flows
Ian Tobasco, University of Michigan

The wall-to-wall optimal transport problem asks for the design of an incompressible flow between parallel walls that most efficiently transports heat from one wall to the other with a given flow intensity budget. In the energy-constrained case, where kinetic energy is prescribed, optimal designs are known to be convection rolls in the large energy limit. In the enstrophy-constrained case, however, previous numerical studies indicate a much more complicated flow structure is favorable, and observe the emergence of near-wall recirculation zones beyond a certain critical enstrophy level. After a brief introduction to the wall-to-wall optimal transport problem, we describe a useful reformulation inspired by related questions in homogenization. This leads to a perhaps unexpected connection between the wall-to-wall problem and questions arising originally in the study of energy-driven pattern formation in materials science. The result is a new multiple scales construction for the enstrophy-driven wall-to-wall problem which goes beyond the complexity observed in the numerical studies, and achieves the optimal rate of transport in the large enstrophy limit up to logarithmic corrections. We discuss implications for the problem of finding the best absolute upper limits on the rate of heat transport in turbulent Rayleigh-Bénard convection. 
This is joint work with C. Doering.