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