Universal scaling law in turbulent Rayleigh–Bénard convection with and without roughness

Heat-transfer measurements published in the literature seem to be contradictory, some showing a transition for the dependance of the Nusselt number ( ${\textit{Nu}}$ ) with the Rayleigh number ( ${\textit{Ra}}$ ) behaviour at ${\textit{Ra}} \approx 10^{11}$ , some showing a delayed transition at higher ${\textit{Ra}}$ , or no transition at all. The physical origin of this discrepancy remains elusive, but is hypothesised to be a signature of the multiple possible flow configurations for a given set of control parameters, as well as the sub-critical nature of the transition to the ultimate regime (Roche 2020 New J. Phys. vol. 22, 073056; Lohse & Shishkina 2023 Phys. Today vol. 76, no. 11, 26–32). New experimental and numerical heat-flux and velocity measurements, both reaching ${\textit{Ra}}$ up to $10^{12}$ , are reported for a wide range of operating conditions, with either smooth boundaries, or mixed smooth–rough boundaries. Experiments are run in water at $40\,^\circ \textrm {C}$ (Prandtl number, ${\textit{Pr}}$ is 4.4), or fluorocarbon at $40\,^\circ \textrm {C}$ ( ${\textit{Pr}}$ is 12), and aspect ratios 1 or 2. Numerical simulations implement the Boussinesq equations in a closed rectangular cavity with a Prandtl number 4.4, close to the experimental set-up, also with smooth boundaries, or mixed smooth–rough boundaries. In the new measurements in the rough part of the cell, the Nusselt number is compatible with a ${\textit{Ra}}^{1/2}$ scaling (with logarithmic corrections), hinting at a purely inertial regime. In contrast to the ${\textit{Nu}}$ vs ${\textit{Ra}}$ relationship, we evidence that these seemingly different regimes can be reconciled: the heat flux, expressed as the flux Rayleigh number, ${\textit{Ra}}\textit{Nu}$ , recovers a universal scaling with Reynolds number, which collapses all data, both our own and those in the literature, once a universal critical Reynolds number is exceeded. This universal collapse can be related to the universal dissipation anomaly, observed in many turbulent flows (Dubrulle 2019 J. Fluid Mech. vol. 867, no. P1, 1).