Aspect-Ratio-Dependent Upper Bounds for Two-Dimensional Rayleigh--B\'{e}nard Convection between Stress-Free Isothermal Boundaries

One of the central challenges in studies of Rayleigh--B\'{e}nard convection is the determination of the heat transport enhancement factor, i.e. the Nusselt number $Nu$, as a function of the Rayleigh number $Ra$, Prandtl number $Pr$, and domain aspect ratio $L$. Although the functional relation between $Nu$, $Pr$ and $Ra$ is usually presumed to be $Nu \sim Pr^\alpha Ra^\beta$ in the ``ultimate" high-$Ra$ regime, experiments and simulations have yielded different scaling exponents. Here, we investigate this scaling relationship for two-dimensional Rayleigh--B\'{e}nard convection between stress-free isothermal boundaries by computing rigorous upper bounds on the heat transport in domains of varying aspect ratio. Using a novel two-step algorithm (Wen et al. PLA 2013), we numerically solve the full ``background field" variational problem arising from the upper bound analysis of Whitehead \& Doering (PRL 2011) to obtain the optimal bound for $Ra \le 10^{10}$ as a function of $L$. Our results show that $Nu \le 0.106Ra^{5/12}$ at fixed $L = 2\sqrt{2}$ uniformly in $Pr$, confirming that molecular transport \emph{cannot} be neglected even at extreme values of $Ra$. Moreover, for large $Ra$, the aspect ratio has little impact on the bounds until the domain becomes sufficiently small.