Transition to the ultimate regime in a radiatively driven convection experiment

We report on the transition between two regimes of heat transport in a radiatively driven convection experiment, where a fluid gets heated up within a tunable heating length ` in the vicinity of the bottom of the tank. The first regime is similar to that observed in standard Rayleigh-Benard experiments, the Nusselt number Nu being related to the Rayleigh number Ra through the power law Nu similar to Ra-1/3. The second regime corresponds to the 'ultimate' or mixing-length scaling regime of thermal convection, where Nu varies as the square root of Ra. Evidence for these two scaling regimes has been reported in Lepot et al. (Proc. Natl Acad. Sci. USA, vol. 115, 2018, pp. 8937-8941), and we now study in detail how the system transitions from one to the other. We propose a simple model describing radiatively driven convection in the mixing-length regime. It leads to the scaling relation Nu similar to (l/H)(PrRa1/2)-Ra-1/2, where H is the height of the cell and Pr is the Prandtl number, thereby allowing us to deduce the values of Ra and Nu at which the system transitions from one regime to the other. These predictions are confirmed by the experimental data gathered at various Ra and l. We conclude by showing that boundary layer corrections can persistently modify the Prandtl number dependence of Nu at large Ra, for Pr greater than or similar to 1.