Energy Spectrum of Buoyancy-Driven Turbulence

Using high-resolution direct numerical simulation and arguments based on the kinetic energy flux $\Pi_u$, we demonstrate that for stably stratified flows, the kinetic energy spectrum $E_u(k) \sim k^{-11/5}$, the entropy spectrum $E_\theta(k) \sim k^{-7/5}$, and $\Pi_u(k) \sim k^{-4/5}$, consistent with the Bolgiano-Obukhov scaling. This scaling arises due to the conversion of kinetic energy to the potential energy by buoyancy. For weaker buoyancy, this conversion is weak, hence $E_u(k)$ follows Kolmogorov's spectrum with a constant energy flux. For Rayleigh B\'{e}nard convection, we show that the energy supply rate by buoyancy is positive, which leads to an increasing $\Pi_u(k)$ with $k$, thus ruling out Bolgiano-Obukhov scaling for the convective turbulence. Our numerical results show that convective turbulence for unit Prandt number exhibits a constant $\Pi_u(k)$ and $E_u(k) \sim k^{-5/3}$ for a narrow band of wavenumbers.