Improved convection cooling in steady channel flows

We find steady channel flows that are locally optimal for transferring heat from fixed-temperature walls, under the constraint of a fixed rate of viscous dissipation (enstrophy = $Pe^2$), also the power needed to pump the fluid through the channel. We generate the optima with net flux as a continuation parameter, starting from parabolic (Poiseuille) flow, the unique optimum at maximum net flux. Decreasing the flux, we eventually reach optimal flows that concentrate the enstrophy in boundary layers of thickness $\sim Pe^{-2/5}$ at the channel walls, and have a uniform flow with speed $\sim Pe^{4/5}$ outside the boundary layers. We explain the scalings using physical arguments with a unidirectional flow approximation, and mathematical arguments using a decoupled approximation. We also show that with channels of aspect ratio (length/height) $L$, the boundary layer thickness scales as $L^{3/5}$ and the outer flow speed scales as $L^{-1/5}$ in the unidirectional approximation. At the Reynolds numbers near the turbulent transition for 2D Poiseuille flow in air, we find a 60\% increase in heat transferred over that of Poiseuille flow.