## Turbulent mixing and dispersion in environmental flows.

##### Abstract

Stably stratified flows are common in the environment such as in the atmospheric·
boundary layer, the oceans, lakes and estuaries. Understanding mixing and dispersion
in these flows is of fundamental importance in applications such as the prediction of
pollution dispersion and for weather and climate prediction/models.
Mixing efficiency in stratified flows is a measure of the proportion of the turbulent kinetic
energy that goes into increasing the potential energy of the fluid by irreversible mixing.
This can be important for parameterizing the effects of mixing in stratified flows. In this
research, fully resolved direct numerical simulations (DNS) of the Navier-Stokes
equations are used to study transient turbulent mixing events. The breaking of internal
waves in the atmosphere could be a source of such episodic events in the
environment. The simulations have been used to investigate the mixing efficiency
(integrated over the duration of the event) as a function of the initial turbulence
Richardson number Ri = N2L2/U2, where N is the buoyancy frequency, L is the
turbulence length scale, and u is the turbulence velocity scale. Molecular effects on the
mixing efficiency have been investigated by varying the Prandtl number Pr = V/K, where
v is the viscosity and K is the scalar diffusivity. Comparison of the DNS results with grid
turbulence experiments has been carried out. There is broad qualitative agreement
between the experimental and DNS results.· However the experiments suggest a
maximum mixing efficiency of 6% while our DNS gives values about five times higher.
Reasons for this discrepancy are investigated. The mixing efficiency has also been
determined using linear theory. It is found that the results obtained for the very stable
cases converge on those obtained from DNS suggesting that strongly stratified flows
exhibit linear behaviour.
Lagrangian analysis of mixing is fundamental in understanding turbulent diffusion and
mixing. Dispersion models such as that of Pearson, Puttock & Hunt (1983) are based
on a Lagrangian approach. A particle-tracking algorithm (using a cubic spline
interpolation scheme following Yeung &Pope, 1988) was developed and incorporated
into the DNS code to enable an investigation into the fundamental aspects of mixing
and diffusion from a Lagrangian perspective following fluid elements. From the
simulations, the ensemble averaged rate of mixing as a function of time indicates
clearly that nearly all the mixing in these flows occurs within times of order 3 Vu. The
mean square vertical displacement statistics show how the stable stratification severely
inhibits the vertical displacement of fluid elements but has no effect on displacements in the transverse direction. This is consistent with the Pearson, Puttock & Hunt model.
The important link that asymptotic value of the mean square vertical displacement is a
measure of the total irreversible mixing that has occurred in the flow is made. However
the results show that the change in density of the fluid elements is only weakly
correlated to the density fluctuations during the time when most of the mixing occurs,
which contradicts a key modeling assumption of the PPH theory. Improvements to the
parameterization of this mixing are investigated.
Flow structures in stably stratified turbulence were examined using flow visualization
software. The turbulence structure for strong stratification resembles randomly
scattered pancakes that are flattened in the horizontal plane. It appears that
overturning motions are the main mechanism by which mixing occurs in these flows.