Onset of "hard" turbulence in Benard Convection and small-scale universality

Anomalous scaling of small-scale fluctuations of velocity and its derivatives, a feature of "strong" ("hard") turbulence, is directly related to extreme rare events in turbulent flows and other stochastic processes. The {\bf direct} transition from the low Reynolds number "weak" Gaussian turbulence to fully developed "strong" turbulence at a critical Reynolds number $R_{\lambda, tr}\approx 8.91$ was recently theoretically predicted and tested in high resolution numerical simulations of V. Yakhot \& D. A. Donzis, Phys. Rev. Lett. {\bf 119}, 044501 (2017) \& PhysicaD, {\bf 384-385}, 12 (2018) on an example of a flow excited by a Gaussian random force. In this paper we study the onset of "hard" turbulence in Benard (RB) convection, where, depending on the Rayleigh number, turbulence is produced by both instabilities of the bulk flow and of the wall boundary layers. The developed theory predicts nonmonotonic behavior of the low-Reynolds number moments of velocity derivatives $M_{2n}(Re)\propto Re^{\rho_{2n}}$, observed in the direct numerical simulations of Schumacher et.al (Phys.Rev.E, {\bf 98},033120 (2018)), The calculated magnitudes of anomalous exponents $\rho_{2n}$ in flows stirred by forces obeying Gaussian or exponential statistics are slightly different, which may indicate existence of universality classes defined by production mechanisms.