We compute incompressible two-dimensional fluid flows that maximize the rate of heat transfer from the walls of a straight channel given a specified flow input power ${\text{Pe}}^2$, where $\text{Pe}$ is the Péclet number. We use the Broyden-Fletcher-Goldfarb-Shanno algorithm together with an adjoint method to compute gradients. The optimal flows are approximately unidirectional, consistent with [S. Alben, Phys. Rev. Fluids 2, 104501 (2017)], up to a critical $\text{Pe}\approx 2^{12}$. Our improved numerical scheme allows us to find a new class of wavy optimal flows for $\text{Pe}\ge 2^{12}$, characterized by fingerlike flow structures emanating from the top and bottom walls of the channel. The rate of heat transfer for these wavy flows is 3–30 % greater than that of the previously identified unidirectional optima [S. Alben (2017)] for $2^{13}\le \text{Pe}\le 2^{17}$. The wavy flows have a much smaller flux through the channel than the unidirectional flows, with regions of slow-moving fluid at nearly uniform temperature interspersed with serpentine regions of fast-moving fluid. Consequently, the area of the interface between hot and cold fluid is increased.

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• Boundary layers
• Optimization problems
• Thermal convection
• Transport phenomena