This paper asks which two-dimensional incompressible flows transfer the most heat between two fixed-temperature, no-slip walls for a prescribed flow intensity. Gradient-ascent methods search for velocity fields that maximize the Nusselt number under a constraint on their rate of strain. The computed optima develop organized structures and show a high-intensity transport scaling close to a power of 0.54 in the Péclet number. A separable approximation to those structures yields a conditional upper bound with a closely matching exponent. Readers get optimization methods for designing efficient transport and an example of how numerical discoveries can motivate analytical bounds for convection.