Optimal heat transport solutions for Rayleigh-Benard convection

Steady flows that optimize heat transport are obtained for two-dimensional Rayleigh-Benard convection with no-slip horizontal walls for a variety of Prandtl numbers Pr and Rayleigh number up to Ra similar to 10(9). Power-law scalings of Nu similar to Ra-gamma are observed with gamma approximate to 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. 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 Pr > 7 at high enough Ra, the optimal structure is a single updraught lacking significant horizontal structure, and characterized by the larger maximal wavenumber.