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Stratified turbulence and mixing processes

Periodic Reporting for period 3 - STAMP (Stratified turbulence and mixing processes)

Reporting period: 2020-10-01 to 2022-03-31

The project investigates turbulence and mixing in stably stratified fluid. Mixing is central to a wide range of questions from the heat uptake in the global ocean, the transport and dilution of pollutants in the atmosphere, the efficient cooling of buildings, to the homogenising of products in the food industry. However, the mechanisms that are responsible and their physical and dynamical aspects are largely unknown, and it is not possible to predict mixing rates from a knowledge of the overall flow and density fields.

All fluid bodies in nature (the atmosphere, oceans, lakes, the air within a room) have significant regions that are stably stratified, making the study of fluxes and mixing in stratified turbulence essential for applications to a wide range of environmental problems from climate predictions to the design of low-energy buildings. Unstratified turbulence, itself an unsolved problem, is specified by a single parameter, the Reynolds number, which describes the ratio of inertia to viscous forces. Stable stratification adds the complication that vertical motions are inhibited by buoyancy forces, but it also provides an opportunity. Stratification introduces two additional parameters, the Richardson number, which provides a measure of the stability of the flow, and the Prandtl number, characterising the molecular properties of the fluid. Consequently, the physics of stratified turbulence is much richer with anisotropic motions at large scales, interactions with internal waves, and turbulent events localised in space and time. The central question we propose to answer is how these features determine mixing in stratified flows.

The overall objectives are
I. Identify the structures responsible for mixing turbulent mixing events in a maintained stratified shear at high Reynolds numbers, in particular their connection to classical flow instabilities.
II. Using highly resolved 3D measurements over a volume, study their dynamics and life-cycles.
III. Determine their contributions to mixing in transitional, intermittent and fully turbulent flow, investigating the differences and similarities between these contributions and the equivalent contributions for nonlinear structures in classical unstable initial value problem studies.
IV. Determine the entrainment and relate it to the prevailing dynamical processes.
The work performed has been to conduct experiments in a stratified inclined duct (SID) and to measure the structures involved in the mixing processes. This has been done and the results are published. We have explained the scaling of the transitions between flow regimes in the two-dimensional plane of input parameters. We have improved upon previous studies of this problem by providing a firm physical basis and non-dimensional scaling laws that are mutually consistent and in good agreement with the empirical transition curves we inferred from 360 experiments. To do so, we employed state-of-the-art simultaneous volumetric measurements of the density field and the three-component velocity field, and analysed these experimental data using time- and volume-averaged potential and kinetic energy budgets. We showed that regime transitions are caused by an increase in the non-dimensional time- and volume-averaged kinetic energy dissipation within the duct, which scales with Reynolds number at high enough angles. As the power input is increased above zero, the two-dimensional, parallel-flow dissipation (power output) increases to close the budget through an increase in the magnitude of the exchange flow, incidentally triggering Holmboe waves above a certain threshold in interfacial shear. However, once the hydraulic limit of two-layer exchange flows is reached, two-dimensional dissipation plateaus and three-dimensional dissipation at small scales (turbulence) takes over, at first intermittently, and then steadily, in order to close the budget. This general understanding of regime transitions and energetics in the SID experiment may serve as a basis for the study
of more complex sustained stratified shear flows found in the natural environment.

In a second paper we present the first extensive, unified set of experimental data where we varied systematically all five input parameters and measured all three output variables with the same methodology. Our results reveal a variety of scaling laws, and a non-trivial dependence of all three variables on all five parameters, in addition to a sixth elusive parameter. We further develop three classes of candidate models to explain the observed scaling laws: (i) the recent volume-averaged energetics of Lefauve et al. (J. Fluid Mech., vol. 848, 2019, pp. 508–544); (ii) two-layer frictional hydraulics; (iii) turbulent mixing models. While these models provide significant qualitative and quantitative descriptions of the experimental results, they also highlight the need for further progress on shear-driven turbulent flows and their interfacial waves, layering, intermittency and mixing properties.

We have also made simultaneous two-dimensional velocity and buoyancy field measurements on a central vertical plane in two-dimensional line plumes: a free plume distant from all vertical boundaries and a wall plume, adjacent to a vertical wall. Data are presented in both an Eulerian frame of reference and a plume coordinate system that follows the instantaneous turbulent/non-turbulent interface (TNTI) of the plume. We present velocity and buoyancy measurements in both coordinate systems and compare the entrainment in the two flows and find that the value of the entrainment constant in the free plume is slightly less than double that of the wall plume. We show that this difference results from the wall plume physically entraining less than the free plume, for a given buoyancy flux. This reduction in entrainment resulting from the presence of the wall is investigated by considering the wall shear stress, the statistics of the TNTI and the conditional vertical transport of the ambient and engulfed fluid in both flows. We show that the wall shear
stress in the wall plume is non-negligible and that the wall also inhibits the lateral meandering of the plume, compared to that observed for the free plume. The effect of the plume meandering on the entrainment process is quantified in terms of the stretching of the TNTI where it is shown that the total length of the TNTI in the free plume is greater than the wall plume by a factor of 2.5 and the relative vertical transport of the engulfed ambient fluid in the free plume is observed to be greater than in the wall plume by 15%. Finally, the turbulent velocity and buoyancy fluctuations and the turbulent velocity and buoyancy fluxes are presented for both flows in both an Eulerian and plume coordinate system. The conditional statistics show that the Reynolds stresses near the TNTI are larger in the wall plume consistent with a larger momentum jump at the edge of the wall plume. We also find that, at the viscous scale of the TNTI, the local entrainment velocity is on average greater in the wall plume than the free plume.
The main additional progress has been the construction of a new stratified inclined duct with increased spatial and temporal resolution for the flow measurements. This will allow us to achieve higher turbulence levels than has been possible to date and to make more accurate measurements of the flow. This makes the new apparatus beyond the current state of the art and we expect to obtain new insights into the mixing processes as a result.