Optimal cooling of an internally heated disc

TL;DR

This paper investigates optimal flow configurations for cooling an internally heated disc, establishing bounds and constructing near-optimal flow patterns under certain constraints, with implications for heat exchanger design.

Contribution

It provides new bounds on energy-constrained cooling and introduces a family of near-optimal branching flows for enstrophy constraints, advancing understanding of heat transfer optimization.

Findings

Constructed a family of self-similar branching flows that are near-optimal.

Proved bounds on cooling efficiency under enstrophy constraints.

Developed a variational principle for bounding solutions of the advection-diffusion equation.

Abstract

Motivated by the search for sharp bounds on turbulent heat transfer as well as the design of optimal heat exchangers, we consider incompressible flows that most efficiently cool an internally heated disc. Heat enters via a distributed source, is passively advected and diffused, and exits through the boundary at a fixed temperature. We seek an advecting flow to optimize this exchange. Previous work on energy-constrained cooling with a constant source has conjectured that global optimizers should resemble convection rolls; we prove one-sided bounds on energy-constrained cooling corresponding to, but not resolving, this conjecture. In the case of an enstrophy constraint, our results are more complete: we construct a family of self-similar, tree-like ``branching flows'' whose cooling we prove is within a logarithm of globally optimal. These results hold for general space- and time-dependent source-sink distributions that add more heat than they remove. Our main technical tool is a non-local Dirichlet-like variational principle for bounding solutions of the inhomogeneous advection-diffusion equation with a divergence-free velocity.