# Wall-to-wall optimal transport in two dimensions

## TL;DR

This paper develops gradient ascent methods to find two-dimensional incompressible flows that maximize heat transfer between walls, revealing a transition in scaling laws and near-separable optimal flow structures up to very high Péclet numbers.

## Contribution

It introduces a novel gradient ascent approach to compute optimal flows for heat transfer, demonstrating a change in scaling behavior and near-separable flow structures at high Péclet numbers.

## Key findings

- Transport scaling transitions from Pe^2 to Pe^0.54
- Optimal flows are approximately separable functions
- Derived an upper bound consistent with computational results

## 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 P\'eclet 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 $\text{Pe} \rightarrow \infty$ similar to the computationally achieved scaling. Implications for heat transfer in buoyancy-driven Rayleigh-B\'enard convection are discussed.