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 $λ$ at a fixed amplitude, we find an optimal wavelength $λ_{\text{opt}}$ for which the Nusselt-Rayleigh scaling relation is $\left(Nu-1 \propto Ra^{0.483}\right)$ 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 ${\cal O} (Ra^{1/2})$ as $Ra \rightarrow \infty$, 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 $λ\ll λ_{\text{opt}}$ and $λ\gg λ_{\text{opt}}$, 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 $λ$, and thus optimizing over all $λ$, to find $Nu_{\text{opt}} - 1 = 0.01 \times Ra^{0.444}$.

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