Roughness as a Route to the Kraichnan Regime in Thermal Convection

We use highly resolved numerical simulations to study turbulent Rayleigh-B\'enard convection in a cell with sinusoidally rough upper and lower walls in two dimensions. By varying the wavelength at a fixed amplitude, we find an optimal wavelength for which the Nusselt-Rayleigh scaling relation is $\left(Nu-1 \propto Ra^{0.482}\right)$. This is consistent with (i) the upper bound of Goluskin and Doering (2016) who prove that $Nu$ can grow no faster than ${\cal O} (Ra^{1/2})$ as $Ra \rightarrow \infty$, and thus (ii) the concept that roughness facilitates the attainment of the so-called ultimate regime of Kraichnan (1962). In the limits of very small and very large wavelengths we recover the planar case results, demonstrating how controlling the wall geometry manipulates the interaction between the boundary layers and the core flow.