J. Fluid Mech. (2015), vol. 765, pp. 17–44. c  Cambridge University Press 2015 doi:10.1017/jfm.2014.726 17 Spectral modelling of high Reynolds number unstably stratified homogeneous turbulence A. Burlot 1, 2 , B.-J. Gréa 1, †, F. S. Godeferd 2 , C. Cambon 2 and J. Griffond 1 1 CEA, DAM, DIF, F-91297 Arpajon, France 2 LMFA, Université de Lyon, École centrale de Lyon, CNRS, INSA, UCBL, F-69134 Écully, France (Received 10 July 2014; revised 4 December 2014; accepted 11 December 2014) 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. Key words: stratified turbulence, turbulence modelling, turbulent mixing 1. Introduction The Rayleigh–Taylor instability occurs for variable-density fluids in which a net acceleration applies from the lighter to the heavier fluid, or when heavy fluid is placed above light fluid in a gravitational field (Rayleigh 1882; Taylor 1950; Chandrasekhar 1961; Sharp 1984). The corresponding destabilization of the interface between the two fluids thus produces a mixing zone. These configurations are found in many applications, for instance in inertial confinement fusion (Lindl 1995), in astrophysics for supernovae (Cook & Cabot 2006), in geophysical flows with upwelling sites in the ocean (Cui & Street 2004) or with turbulence in the ionosphere (Molchanov 2004). After the linear regime of initiation of the instability, nonlinearities gain importance, and the mixing zone becomes fully turbulent so that its width L eventually reaches an asymptotic self-similar state (Youngs 1984) L = 2αA gt 2 (1.1) † Email address for correspondence: benoit-joseph.grea@cea.fr