Weak formulation and scaling properties of energy fluxes in 3D numerical turbulent Rayleigh-B\'enard convection

Abstract

We apply the weak formalism on the Boussinesq equations, to characterize scaling properties of the mean and the standard deviation of the potential, kinetic and viscous energy flux in very high resolution numerical simulations. The local Bolgiano-Oboukhov length $L_{BO}$ is investigated and it is found that its value may change of an order of magnitude through the domain, in agreement with previous results. We investigate the scale-by-scale averaged terms of the weak equations, which are a generalization of the Karman-Howarth-Monin and Yaglom equations. We have not found the classical Bolgiano-Oboukhov picture, but evidence of a mixture of Bolgiano-Oboukhov and Kolmogorov scalings. In particular, all the terms are compatible with a Bolgiano-Oboukhov local H\"older exponent for the temperature and a Kolmogorov 41 for the velocity. This behavior may be related to anisotropy and to the strong heterogeneity of the convective flow, reflected in the wide distribution of Bolgiano-Oboukhov local scales. The scale-by-scale analysis allows us also to compare the theoretical Bolgiano-Oboukhov length $L_{BO}$ computed from its definition with that empirically extracted through scalings obtained from weak analysis. The key result of the work is to show that the analysis of local weak formulation of the problem is powerful to characterize the fluctuation properties.