# Improved convection cooling in steady channel flows

## TL;DR

This paper identifies steady channel flow configurations that optimize heat transfer under energy constraints, revealing boundary layer and flow speed scalings, and demonstrating significant heat transfer improvements over classical Poiseuille flow.

## Contribution

It introduces a new class of optimal steady flows for heat transfer constrained by viscous dissipation, with detailed scaling laws and practical implications for turbulent transition conditions.

## Key findings

- Optimal flows concentrate enstrophy in boundary layers of thickness ~ Pe^{-2/5}.
- Outer flow speeds scale as ~ Pe^{4/5}.
- Heat transfer increases by 60% compared to Poiseuille flow near turbulent transition.

## Abstract

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.