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

TL;DR

This paper investigates the transition from anomalous to normal scaling in Rayleigh-Bénard convection, revealing a small-scale universality in turbulent flows at high Reynolds numbers and the influence of large-scale temperature fluctuations.

Contribution

It introduces a mean-field theory for Rayleigh-Bénard convection, connecting small-scale turbulence behavior with large-scale temperature fields and identifying a universal regime at high Reynolds numbers.

Findings

Small-scale velocity fluctuations mimic homogeneous turbulence driven by large-scale temperature fields.

Transition from Gaussian to exponential temperature PDFs at low Rayleigh numbers.

Heat flux scaling exponent close to experimental results, indicating 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.