Transitions and Multi-Scaling in Rayliegh-Benard Convection. Small-Scale Universality

Abstract

Asymptotically large Reynolds number hydrodynamic turbulence is characterized by multi-scaling of moments of velocity increments and spatial derivatives. With decreasing Reynolds number toward $R_{\lambda}=R^{tr}_{\lambda}\approx 9.0$, the anomalous scaling disappears in favor of the "normal" one and close-to-Gaussian probability densities [Yakhot \& Donzis, {\bf 119}, 044501 (2017)]. The nature of this transition and its universality are subjects of this work. Here we consider Benard convection ( Prandtl number $Pr=1$) between infinite horizontal plates. It is shown that in this system the "competition" between Bolgiano and Kolmogorov processes, results in small-scale velocity fluctuations driven by effective "large-scale" Gaussian random temperature field. Therefore, the intermittent dynamics of velocity derivatives are similar or even identical to that in homogeneous and isotropic turbulence generated by the large-scale random forcing. It is shown that low-Rayleigh number instabilities make the problem much more involved and may lead to transition from Gaussian to exponential PDF of the temperature field. The developed {\it mean-field theory} yielded dimensionless heat flux $Nu\propto Ra^{\beta}$ with $\beta\approx 15/56\approx 0.27$, close to the outcome of Chicago experiment. These results point to an unusual small-scale universality of turbulent flows. It is also shown that at $R_{\lambda}\leq 9.0$, a flow "remembers" its laminar background and, therefore, cannot be universal.