Energy spectrum of buoyancy-driven turbulence

Abstract

Using high-resolution direct numerical simulation and arguments based on the kinetic energy flux Πu, we demonstrate that for stably stratified flows, the kinetic energy spectrum Eu(k)∼k−11/5, the entropy spectrum Eθ(k)∼k−7/5, and Πu(k)∼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 Eu(k) follows Kolmogorov's spectrum with a constant energy flux. For Rayleigh Bénard convection, we show that the energy supply rate by buoyancy is positive, which leads to an increasing Πu(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 Πu(k) and Eu(k)∼k−5/3 for a narrow band of wavenumbers.