Statistical modelling and direct numerical simulations of decaying stably stratified turbulence. Part 1. Flow energetics

Abstract

The dynamics of a homogeneous turbulent flow subjected to a stable stratification are studied by means of direct numerical simulations (DNS) and by a two-point closure statistical EDQNM model, adapted for anisotropic flows by Cambon (1989). The purpose of this work is to investigate the validity of the anisotropic statistical model, which we refer to as the EDQNM₂ model. The numerical simulations are of high resolution, 256³, which permits Reynolds numbers comparable to those of recent laboratory experiments. Thus, detailed comparisons with the wind-tunnel experiments of Lienhardt & Van Atta (1990) and Yoon & Warhaft (1990) are also presented.

The initial condition is chosen so as to test the anisotropic closure assumption of the EDQNM₂ model. This choice yields a ratio of kinetic to potential energy of 2[ratio ]1. This important amount of initial potential energy drives the flow dynamics during the first Brunt–Väisälä period. Because stronger transfer rates of potential energy than of kinetic energy occur toward small scales, the heat flux is (persistently) counter gradient at those small scales. The loss of potential energy at large scales is partly made up for by conversion of vertical kinetic energy, and this sets up a down-gradient heat flux at those scales, as if no or little potential energy were present at the initial time. Thus, common features with wind-tunnel experiments (in which there is relatively little potential energy just behind the grid) are found. Interestingly, only one quantity displays a similarity law in the DNS, in the EDQNM₂ model and in the experiments of Lienhardt & Van Atta (1990) and Yoon & Warhaft (1990) as well: this is the ratio of the vertical heat flux to the dissipation rate of kinetic energy, which can also be interpreted as an instantaneous mixing efficiency. Thus, this parameter seems to be independent of initial flow conditions.

Our calculations simulate a longer evolution of the flow dynamics than laboratory experiments (in which the flow develops for at most one Brunt–Väisälä period). We find that the flow dynamics change from about 1.5 Brunt–Väisälä periods. At that time, the heat flux collapses while the dissipation rate of kinetic energy displays a self-similarity law attesting that this quantity becomes driven by buoyancy forces. This permits us to link the collapse of the largest scales of the flow with the smallest scales being influenced by the buoyancy force. We finally discuss the influence of a geometrical confinement effect upon the above results.

The EDQNM₂ model compares remarkably well with the DNS, with respect to previous statistical models of stably stratified turbulent flows. Insufficient decorrelation between the vertical velocity and the temperature fluctuations is however observed, but with no dynamical significance. The vortex part of the flow is also overestimated by the EDQNM₂ model, but the relative difference between the model prediction and the DNS does not exceed 15% after 6 Brunt–Väisälä periods. The EDQNM₂ model offers interesting perpectives because of its ability to predict the dynamics of stratified flows at high Reynolds numbers. Knowledge about small-scale behaviour will be especially useful, to build up parameterization of the subgrid scales for instance.