Large Reynolds number self-similar states of unstably stratified homogeneous turbulence

Abstract

We study the influence of the large scale energy distribution on the long term dynamics of unstably stratified homogeneous turbulence at high Reynolds number Re = 106, using a statistical two-point spectral model based on the eddy-damped quasi-normal closure. We consider several initial spectral scalings ks in the infrared range with s ∈ [1; 5] and we establish that the resulting kinetic energy growth rates are controlled by s, with the appearance of backscatter effects for s ≳ 3.5. We then assess that only for s ≤ 4 do we observe self-similarity in the infrared and in the inertial ranges, but not in the dissipative range. Compensated energy and buoyancy spectra exhibit the expected Kolmogorov-Obukhov k−5/3 scaling at long time, and a trend to the theoretically predicted k−7/3 scaling for velocity-buoyancy cross-correlation spectrum thanks to the very large Reynolds number. We also show a direct link between the late-time anisotropy of the flows and the infrared spectrum, thus demonstrating long-lasting effect of initial conditions on unstably stratified turbulence. We show that, in addition to the Kolmogorov k−5/3 scaling, the kinetic energy spectrum inertial range includes a k−3 zone due to polarization anisotropy, and we confirm the clear sin2θ dependence of the velocity-buoyancy spectrum in the inertial range, where θ is the orientation of the wave vector to the axis of gravity. However, an unexpected quick return to isotropy of the scalar spectra has been identified, which cannot be explained by a standard dimensional analysis.