Optimal wall shapes and flows for steady planar convection

TL;DR

This paper determines optimal wall shapes and flow patterns in steady planar convection to maximize heat transfer for a given viscous dissipation, revealing scaling laws and regime transitions.

Contribution

It introduces a theoretical and computational framework for optimizing wall shapes and flows in steady convection, identifying key scaling laws and regime behaviors.

Findings

Optimal walls are flat at zero flow and remain so up to a critical flow.

Flow and wall shape converge to invariant forms with Pe^{-1/3} scaling.

Heat transfer rate scales as Pe^{2/3} in the convection regime.

Abstract

We compute steady planar incompressible flows and wall shapes that maximize the rate of heat transfer (Nu) between and hot and cold walls, for a given rate of viscous dissipation by the flow (Pe$^2$). In the case of no flow, we show theoretically that the optimal walls are flat and horizontal, at the minimum separation distance. We use a decoupled approximation to show that flat walls remain optimal up to a critical nonzero flow magnitude. Beyond this value, our computed optimal flows and wall shapes converge to a set of forms that are invariant except for a Pe$^{-1/3}$ scaling of horizontal lengths. The corresponding rate of heat transfer Nu $\sim$ Pe$^{2/3}$. We show that these scalings result from flows at the interface between the diffusion-dominated and convection-dominated regimes. We also show that the separation distance of the walls remains at its minimum value at large Pe.