Effects of Pr on Optimal Heat Transport in Rayleigh-B\'{e}nard Convection

Steady flows that optimize heat transport are obtained for two-dimensional Rayleigh-B\'{e}nard convection with no-slip horizontal walls for a variety of Prandtl numbers $Pr$ and Rayleigh number up to $Ra\sim 10^{9}$. 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 where the Nusselt number $Nu$ is a non-dimensional measure of the vertical heat transport. For $Pr \leq 7$, optimal transport is achieved at the smaller maximal wavenumber whereas for $Pr > 7$ at high-enough $Ra$ the optimal structure occurs at the larger maximal wavenumber. Three regions are observed in the optimal mean temperature profiles, $\overline{T}\left(y\right)$: 1.) $d\overline{T}/dy < 0$ in the boundary layers, 2.) $d\overline{T}/dy > 0$ ($Pr\leq 7$) or $d\overline{T}/dy < 0$ ($Pr>7$) in the central region, and 3.) $d\overline{T}/dy > 0$ between the boundary layers and central region. We also search for a signature of these optimal structures in a fully-developed turbulent flow by employing modal decompositions such as the proper orthogonal decomposition and the Koopman mode decomposition.