Improved convection cooling in steady channel flows

A fundamental problem in heat transfer is the convective cooling of the heated walls of a channel. We find steady two-dimensional (2D) flows that maximize the heat removed from fixed-temperature walls for a given rate of energy used to drive the flow, Pe (Pe is the Peclet number, a dimensionless flow speed). For parabolic (Poiseuille) flow, the heat transfer scales as Pe. Starting from Poiseuille flow, we compute a sequence of optima using Newton’s method with continuation. Computed optimal flows are found to be approximately unidirectional and nearly uniform outside of sharp boundary layers at the channel walls. A linear approximation near Poiseuille flow shows how the thermal boundary layer generates a boundary layer in the optimal flow. We are thus led to compute optimal unidirectional flows with boundary layers, for which the heat transfer scales as Pe, an improvement over the Pe Poiseuille flow scaling. The optimal flows have viscous dissipation concentrated in boundary layers of thickness ∼Pe−2/5 at the channel walls, and have a uniform velocity ∼ Pe outside the boundary layers. We explain the scalings using physical and mathematical arguments. We also show that with channels of aspect ratio (length/height) L, the outer flow speed scales as L−1/5 and the boundary layer thickness scales as L3/5 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.