Optimal heat transport solutions for Rayleigh–Bénard convection

• Published in: Journal of fluid mechanics 2015-12, Vol.784, p.565-595
• Main Authors: Sondak, David, Smith, Leslie M., Waleffe, Fabian
• Format: Article
• Language: English
• Subjects: Convection; Fluid mechanics; Heat transfer; Heat transport; Steady flow; Temperature
• ISSN: 0022-1120; 1469-7645
• DOI: 10.1017/jfm.2015.615

Steady flows that optimize heat transport are obtained for two-dimensional Rayleigh–Bénard convection with no-slip horizontal walls for a variety of Prandtl numbers $\mathit{Pr}$ and Rayleigh number up to $\mathit{Ra}\sim 10^{9}$ . Power-law scalings of $\mathit{Nu}\sim \mathit{Ra}^{{\it\gamma}}$ are observed with ${\it\gamma}\approx 0.31$ , where the Nusselt number $\mathit{Nu}$ is a non-dimensional measure of the vertical heat transport. Any dependence of the scaling exponent on $\mathit{Pr}$ is found to be extremely weak. On the other hand, the presence of two local maxima of $\mathit{Nu}$ with different horizontal wavenumbers at the same $\mathit{Ra}$ leads to the emergence of two different flow structures as candidates for optimizing the heat transport. For $\mathit{Pr}\lesssim 7$ , 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 downdraughts adjacent to the central updraught. For $\mathit{Pr}>7$ at high enough $\mathit{Ra}$ , the optimal structure is a single updraught lacking significant horizontal structure, and characterized by the larger maximal wavenumber.