Temperature fluctuations in the Ultimate Regime of Convection

A new regime of turbulent convection has been reported nearly one decade ago, based on global heat transfer measurements at very high Rayleigh numbers. We examine the signature of this “Ultimate Regime” from within the flow itself. A systematic study of probe-size corrections shows that the earlier temperature measurements within the flow were altered by an excessive size of thermometer, but not according to a theoretical model proposed in the literature. Using a probe one order of magnitude smaller than the one used previously, we find evidence that the transition to the Ultimate Regime is indeed accompanied with a clear change in the statistics of temperature fluctuations in the flow. In 1997, a new turbulent regime of thermal convection was observed by Chavanne et al. for Prandtl number of order unity (Pr ∼ 1) and above a threshold Rayleigh number of order Ra ≃ 10 (the definitions of Pr and Ra are recalled later) [1]. Such conditions are found in environmental flows, including atmospheric and oceanic, giving a major practical importance to this result, beyond the theoretical challenge it raises. This new regime is characterised by an improved heat transfer, which is usually assessed by the dimensionless conductivity Nu of the flow. This Nusselt number Nu is defined as the total heat flux across the cell normalised by the diffusive heat flux that would settle in a quiescent fluid. Right above the reported transition, Nu scales like Nu ∼ Ra while the Ra exponent reaches at most 1/3 right below. A second signature of this regime was recently reported : enhanced fluctuations of the temperature drop across the boundary layer covering the plate used to force heat through the flow. This observation is consistent with an hydrodynamic boundary layer instability [2]. Both observations, as well as specific tests (in particular the observation of a Nu ∼ Ra heat transfer law [3]) are fully consistent with a 1962 prediction by R. Kraichnan [4]. This prediction states that an asymptotic convection regime will settle at high enough Ra once the boundary layer have undergone a laminar to turbulent transition. To the best of our knowledge, no alternative interpretation of all these observations is proposed any longer. Despite the good agreement between observations and the theoretical prediction, two important issues still remain open. The first concerns the precise nature of the regime observed at very high Ra. In particular, what is the degree of overlapping between this observed “Ultimate Regime” (following the naming introduced in 1997) and Kraichnan’s prediction ? Beside the experimental difficulty of reaching very high Ra in laboratory experiments, this comparison is delicate due to the ill-defined concept of laminar-to-turbulent transition in unsteady boundary layers, such as the ones present in turbulence convection (for example, see [5,6]) and due to Kraichnan’s renouncement to treat the “join” between his asymptotic regime and the so-called “hard turbulence” regime present at lower Ra. The second important open issue is the experimental conditions for the triggering of this Ultimate Regime, which is observed in some experiments but not in all. Indeed, if we consider heat transfer measurements reaching at least Ra = 10 and fulfilling the Boussinesq approximations, the litterature reports two sets of results in apparent contradiction : a clear transition is found in some [1, 3, 7–10] and not in others [11–14]. Adding to the complexity of the present situation, two “in-between” results evidenced some features of transitions at very high Ra but without increase in heat transfer. The first is a simulation showing that the friction coefficient on the thermal plates departs from the typical scaling of laminar boundary layers above Ra ≃ 10, “presumably marking the transition to turbulence” [14] . The second is an experiment which identifies two transitions for Ra ≃ 10