  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.
