Transition to turbulence scaling in Rayleigh-Bénard convection

If a fluid flow is driven by a weak Gaussian random force, the nonlinearity in the Navier-Stokes equations is negligibly small and the resulting velocity field obeys Gaussian statistics. Nonlinear effects become important as the driving becomes stronger and a transition occurs to turbulence with anomalous scaling of velocity increments and derivatives. This process has been described by Yakhot and Donzis [Phys. Rev. Lett. 119, 044501 (2017)PRLTAO0031-900710.1103/PhysRevLett.119.044501] for homogeneous and isotropic turbulence. In more realistic flows driven by complex physical phenomena, such as instabilities and nonlocal forces, the initial state itself, and the transition to turbulence from that initial state, is much more complex. In this paper, we discuss the Reynolds-number dependence of moments of the kinetic energy dissipation rate of orders 2 and 3 obtained in the bulk of thermal convection in the Rayleigh-Bénard system. The data are obtained from three-dimensional spectral element direct numerical simulations in a cell with square cross section and aspect ratio 25 by Pandey et al. [Nat. Commun. 9, 2118 (2018)2041-172310.1038/s41467-018-04478-0]. Different Reynolds numbers 1 Re 1000 which are based on the thickness of the bulk region and the corresponding root-mean-square velocity are obtained by varying the Prandtl number Pr from 0.005 to 100 at a fixed Rayleigh number Ra=105. A few specific features of the data agree with the theory. The normalized moments of the kinetic energy dissipation rate En show a nonmonotonic dependence for small Reynolds numbers before obeying the algebraic scaling prediction for the turbulent state. Implications and reasons for this behavior are discussed.