Maximal heat transfer between two parallel plates

TL;DR

This paper determines the three-dimensional velocity fields that maximize heat transfer between two plates under enstrophy constraints, revealing scaling laws and hierarchical structures relevant to high-Rayleigh-number turbulence.

Contribution

It extends the variational problem to three dimensions, deriving new optimal states and scaling laws for heat transfer in turbulent convection.

Findings

Scaling of Nu with Pe as Nu ~ Pe^{2/3}

Emergence of hierarchical vortical structures at high Pe

Logarithmic temperature profiles near walls at high Pe

Abstract

The divergence-free time-independent velocity vector field has been determined so as to maximise heat transfer between two parallel plates of a constant temperature difference under the constraint of fixed total enstrophy. The present variational problem is the same as that first formulated by Hassanzadeh $\it et{\ }al$. (2014); however, a search range of optimal states has been extended to a three-dimensional velocity field. The scaling of the Nusselt number $Nu$ with the P\'eclet number $Pe$ (i.e., the square root of the non-dimensionalised enstrophy with thermal diffusion timescale), $Nu\sim Pe^{2/3}$, has been found in the three-dimensional optimal states, corresponding to the asymptotic scaling with the Rayleigh number $Ra$, $Nu\sim Ra^{1/2}$, in extremely-high-$Ra$ convective turbulence, and thus to the Taylor energy dissipation law in high-Reynolds-number turbulence. At $Pe\sim10^{0}$, a two-dimensional array of large-scale convection rolls provides maximal heat transfer. A three-dimensional optimal solution emerges from bifurcation on the two-dimensional solution branch at higher $Pe$. At $Pe\gtrsim10^{3}$, the optimised velocity fields consist of convection cells with hierarchical self-similar vortical structures, and the temperature fields exhibit a logarithmic mean profile near the walls.