Scale-by-scale energy budget in turbulent convection

Turbulent free convection is driven by buoyancy. A footprint of buoyancy is thus expected in the energy cascade. The existence of this so-called Bolgiano--Obukhov (BO) scaling is a long-standing open question. We use DNS of Rayleigh--B\'enard convection in a horizontally periodic domain to address this question. Moderate Rayleigh numbers $2.6\times 10^6$ and $2.5\times 10^7$ are applied, at three different Prandtl numbers $1$, $3$ and $10$. We show that the length scale bounding the convective scaling regime from below, the Bolgiano scale $L_B$, is typically large relative to the domain size. Scale-by-scale energy budgets are calculated based on Yakhot's equivalent of Kolmogorov's isotropic four-fifths law for convection. They reveal that buoyancy is active on many scales, obscuring the classical Kolmogorov scaling for scales smaller than $L_B$. Only at very large separations a buoyancy-dominated scaling range could exist. Close to the plates, where $L_B$ is smaller, anisotropy complicates the detection of scaling.