We study unconfined homogeneous turbulence with a destabilizing background density gradient in the Boussinesq approximation. Starting from initial isotropic turbulence, the buoyancy force induces a transient phase toward a self-similar regime accompanied by a rapid growth of kinetic energy and Reynolds number, along with the development of anisotropic structures in the flow in the direction of gravity. We model this with a two-point statistical approach using an axisymmetric eddy-damped quasi-normal Markovian (EDQNM) closure that includes buoyancy production. The model is able to match direct numerical simulations (DNS) in a parametric study showing the effect of initial Froude number and mixing intensity on the development of the flow. We further improve the model by including the stratification timescale in the characteristic relaxation time for triple correlations in the closure. It permits the computation of the long-term evolution of unstably stratified turbulence at high Reynolds number. This agrees with recent theoretical predictions concerning the self-similar dynamics and brings new insight into the spectral energy distribution and anisotropy of the flow.

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 k^s 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.

We characterise the properties of unstably stratified homogeneous turbulence by means of high-resolution direct numerical simulations and a two-point statistical spectral model based on a quasi-normal closure proposed by Burlot et al. Both approaches agree very well regarding the evolution of one- and two-point turbulent statistics, showing that the model is valid at even higher Reynolds numbers than previously considered. From a parametric study with different initial conditions, we confirm that the energy distribution at large scale influences strongly the late time dynamics of the flow. In particular, we assess the existence of backscatter transfer of energy, and evaluate its role in the growth rate of several turbulent quantities. Moreover, thanks to the statistical model, we analyse the scale-by-scale anisotropy of the flow through the decomposition of turbulent spectra in terms of directional anisotropy and polarisation anisotropy, for a refined characterisation of the structure of the flow which is strongly anisotropic in the large scales. This also allows us to study how isotropy is restored in the inertial scales.

In this paper, we investigate the decay of incompressible homogeneous isotropic turbulence in a variable viscosity fluid. The viscosity coefficient is assumed to depend linearly on a scalar, representing either a temperature or a concentration, and obeying a simple advection-diffusion equation. At high Reynolds numbers, Direct Numerical Simulations (DNS) allow us to confirm the validity of Taylor's postulate that the dissipation is independent from the viscosity and its fluctuations. At low Reynolds numbers, we report the presence of extra energy at small scales due to these variable viscosity effects. This implies that the turbulent kinetic energy decreases less rapidly as a function of time in variable viscosity fluids. In order to explain this phenomenon and quantify its importance on the turbulent flow, we propose a statistical approach based on an eddy-damped quasi-normal Markovian (EDQNM) spectral closure which takes into account the nonlinearity introduced by variable viscosity. It is shown that this latter additional term is of constant sign in the energy spectrum equation and reduces the dissipation of the flow as observed. Also, by assuming the dominance of distant interactions between wave numbers, we can propose a simple formula expressing that variable viscosity effects lead to an effective reduction of the mean viscosity proportional to the variance of viscosity fluctuations.