Optimal cooling of an internally heated disc

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.

Minor revisions from review, figure added. To appear in Philos. Trans. R. Soc. A

References in corpus (6)

• Radiative heating achieves the ultimate regime of thermal convection
• Internally heated convection beneath a poor conductor
• The background method: Theory and computations
• Bounds on heat transport for convection driven by internal heating
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• Bounds on heat flux for Rayleigh-Bénard convection between Navier-slip fixed-temperature boundaries

Cited by in corpus (4)

• Bounds on heat transfer by incompressible flows between balanced sources and sinks
• Rigorous scaling laws for internally heated convection at infinite Prandtl number
• Scaling laws for Rayleigh-Bénard convection between Navier-slip boundaries
• Bounds on buoyancy driven flows with Navier-slip conditions on rough boundaries