Roughness as a Route to the Ultimate Regime of Thermal Convection

Abstract

We use highly resolved numerical simulations to study turbulent Rayleigh-Bénard convection in a cell with sinusoidally rough upper and lower surfaces in two dimensions for $Pr=1$ and $Ra=\left[4\times 10^6,3\times 10^9\right]$. By varying the wavelength $\lambda$ at a fixed amplitude, we find an optimal wavelength ${\lambda }_{\text{opt}}$ for which the Nusselt-Rayleigh scaling relation is $\left(Nu-1\propto R{a}^{0.483}\right)$ maximizing the heat flux. This is consistent with the upper bound of Goluskin and Doering \cite{Goluskin:2016} who prove that $Nu$ can grow no faster than $\mathcal{O}(R{a}^{1/2})$ as $Ra\to \infty$, and thus the concept that roughness facilitates the attainment of the so-called ultimate regime. Our data nearly achieve the largest growth rate permitted by the bound. When $\lambda \ll {\lambda }_{\text{opt}}$ and $\lambda \gg {\lambda }_{\text{opt}}$, the planar case is recovered, demonstrating how controlling the wall geometry manipulates the interaction between the boundary layers and the core flow. Finally, for each $Ra$ we choose the maximum $Nu$ among all $\lambda$, and thus optimizing over all $\lambda$, to find $N{u}_{\text{opt}}-1=0.01\times R{a}^{0.444}$.

arXiv comment8 pages, 6 figures. Accepted for publication in Physical Review Letters

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