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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 Péclet number $Pe$ 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 $Nu$ up to $$Pe\approx 10^{5}$$ . The resulting transport exhibits a change of scaling from $$Nu-1\sim Pe^{2}$$ for $Pe<10$ in the linear regime to $$Nu\sim Pe^{0.54}$$ for $$Pe>10^{3}$$ . 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 $${\lesssim}Pe^{6/11}=Pe^{0.\overline{54}}$$ as $$Pe\rightarrow \infty$$ similar to the computationally achieved scaling. Implications for heat transfer in buoyancy-driven Rayleigh–Bénard convection are discussed.