Wall-to-wall optimal transport in two dimensions

Gradient ascent methods are developed to compute incompressible flows that maximize heat transport between two isothermal no-slip parallel walls. Parameterizing the magnitude of the velocity fields by a Peclet number proportional to their root-mean-square rate of strain, the schemes are applied to compute two-dimensional flows optimizing convective enhancement of diffusive heat transfer, i.e. the Nusselt number up to . The resulting transport exhibits a change of scaling from for in the linear regime to for . Optimal fields are observed to be approximately separable, i.e. products of functions of the wall-parallel and wall-normal coordinates. Analysis employing a separable ansatz yields a conditional upper bound as similar to the computationally achieved scaling. Implications for heat transfer in buoyancy-driven Rayleigh-Benard convection are discussed.