Statistical modelling and direct numerical simulations of decaying
stably
stratified turbulence. Part 1. Flow energetics
Authors: STAQUET C. , GODEFERD F. S. 
Published: July 2002

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.