Optimal Heat Transport in Rayleigh-B\'enard Convection

TL;DR

This study identifies optimal steady flow configurations in two-dimensional Rayleigh-Bénard convection that maximize heat transport, revealing weak Prandtl number dependence and distinct flow structures for different Prandtl regimes.

Contribution

It provides the first detailed analysis of optimal heat transport flows in 2D Rayleigh-Bénard convection across a range of Prandtl and Rayleigh numbers, highlighting different flow structures.

Findings

Nu scales as Ra^{0.31} with weak Prandtl dependence.

Two local maxima of Nu lead to different flow structures.

Optimal flow structures vary with Prandtl number.

Abstract

Steady flows that optimize heat transport are obtained for two-dimensional Rayleigh-B\'enard convection with no-slip horizontal walls for a variety of Prandtl numbers $Pr$ and Rayleigh number up to $Ra\sim 10^9$. Power law scalings of $Nu\sim Ra^{\gamma}$ are observed with $\gamma\approx 0.31$, where the Nusselt number $Nu$ is a non-dimensional measure of the vertical heat transport. Any dependence of the scaling exponent on $Pr$ is found to be extremely weak. On the other hand, the presence of two local maxima of $Nu$ with different horizontal wavenumbers at the same $Ra$ leads to the emergence of two different flow structures as candidates for optimizing the heat transport. For $Pr \lesssim 7$, optimal transport is achieved at the smaller maximal wavenumber. In these fluids, the optimal structure is a plume of warm rising fluid which spawns left/right horizontal arms near the top of the channel, leading to downdrafts adjacent to the central updraft. For $Pr > 7$ at high-enough Ra, the optimal structure is a single updraft absent significant horizontal structure, and characterized by the larger maximal wavenumber.