Wall-to-wall optimal transport in two dimensions

Abstract

Gradient ascent methods are developed to compute incompressible flows that maximize heat transport between two isothermal no-slip parallel walls. Parameterizing the magnitude of velocity fields by a Péclet number $\text{Pe}$ 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 $\text{Nu}$ up to $\text{Pe}\approx 10^5$. The resulting transport exhibits a change of scaling from $\text{Nu}-1\sim {\text{Pe}}^2$ for $\text{Pe}<10$ in the linear regime to $\text{Nu}\sim {\text{Pe}}^{0.54}$ for $\text{Pe}>10^3$. 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 $\lesssim {\text{Pe}}^{6/11}={\text{Pe}}^{0.\overline{54}}$ as $\text{Pe}\to \infty$ similar to the computationally achieved scaling. Implications for heat transfer in buoyancy-driven Rayleigh-Bénard convection are discussed.