PDF modelling of turbulent mixing in stratified fluids

HAL (Le Centre pour la Communication Scientifique Directe), 2016

Predicting how much mixing occurs when a given amount of energy is injected into a Boussinesq fluid is a longstanding problem in stratified turbulence. Here we address this problem with the point of view of equilibrium statistical mechanics. Assuming random evolution through turbulent stirring, the theory predicts that the unforced, inviscid, adiabatic dynamics is attracted irreversibly towards a state characterised by wild small scale velocity fluctuations carrying kinetic energy, and by a smooth buoyancy profile superimposed with wild small scale buoyancy fluctuations. It is then possible to compute how much of the injected energy has been irreversibly lost into small scale kinetic energy, the remaining part being used to irreversibly raise the potential energy of the system. This yields to quantitative predictions for a global, cumulative mixing efficiency in freely evolving configurations. We argue that this approach may be useful to the understanding of mixing in stratified turbulence in the limit of large Reynolds and Péclet numbers.

Probability distribution of basic instabilities appearing in stratified flows and point density fluctuations have been studied. Various parameters of the mixing process have been changed in the experiments, to investigate mixing. Detailed flow visualization as well as density measurements have been used in zero-mean-flow laboratory experiments involving grid-stirred turbulent mixing across a density interface and bubble-induced mixing. The overall mixing efficiency of the processes depends on the local Richardson number as well as on the local vorticity. Parameter distributions of low and high mixedness corresponding to different instabilities are presented, showing that dipolar vortices penetrating the interface are the most efficient mixing instabilities

Physical Review Fluids, 2020

Understanding how turbulence leads to the enhanced irreversible transport of heat and other scalars (such as salt and pollutants) in density-stratified fluids is a fundamental and central problem in geophysical and environmental fluid dynamics. There is a wide range of highly important applications, an important example being the description and parameterization of diapycnal transport in the world's oceans, a key area of uncertainty in climate modeling. Recently, possibly due to the proliferation of data obtained through direct observation, numerical simulation, and laboratory experimentation, there has been an explosion in research activity directed at improving community understanding, modeling, and parametrization of the subtle interplay among energy conversion pathways, turbulence, and irreversible mixing in density-stratified fluids. However, there are still leading-order open questions and areas of profound uncertainty concerning this interesting and important research challenge.

Journal of Fluid Mechanics, 1998

This paper formulates a model of mixing in a stratified and turbulent fluid. The model uses the horizontally averaged vertical buoyancy gradient and the density of turbulent kinetic energy as variables. Heuristic ‘mixing-length’ arguments lead to a coupled set of parabolic differential equations. A particular form of mechanical forcing is proposed; for certain parameter values the relationship between the buoyancy flux and the buoyancy gradient is non-monotonic and this leads to an instability of equilibria with linear stratification. The instability results in the formation of steps and interfaces in the buoyancy profile. In contrast to previous ones, the model is mathematically well posed and the interfaces have an equilibrium thickness that is much larger than that expected from molecular diffusion.The turbulent mixing process can take one of three forms depending on the strength of the initial stratification. When the stratification is weak, instability is not present and mixing...

Journal of the Brazilian Society of Mechanical Sciences and Engineering, 2006

The user has requested enhancement of the downloaded file. All in-text references underlined in blue are added to the original document and are linked to publications on ResearchGate, letting you access and read them immediately. The transition to turbulence in a stably stratified flow is a problem of considerable interest in fluid dynamics with applications in both geophysical sciences and engineering. This transition is controlled by competition between the vertical shear of the base flow and the buoyancy forces due to the density stratification. The present work investigates numerically the effect of stable stratification on the development of a Kelvin-Helmholtz (KH) instability and the formation of streamwise vortices, which are developed after the saturation of the primary billows of KH. The Direct Numerical Simulation (DNS) technique was used to solve the complete Navier-Stokes equations, in the Boussinesq approximation. Numerical tests were done with different Richardson numbers and forced initial conditions for velocity fluctuations. The results showed that high stratification inhibits the pairing process, reduces the buoyancy flux, weakens the vertical motions, decreases the thickness of the mixing layer and affects the formation of streamwise vortices. The three-dimensional results demonstrated that the streamwise vortices are clearly formed in non-stratified cases. In the stratified cases, on the other hand, the streamwise vortices are weakened, due to the streamwise density gradient, which decrease the levels of vorticity in the billows of KH, while increase in the braid zone.

Journal of Fluid Mechanics, 1995

The results of an experimental study of shear-free decaying grid-generated turbulence on both sides of a sharp interface between two homogeneous layers of different densities are presented. The evolution of turbulence and mixing were examined by simultaneously mapping the velocity (u, w) and density fields @) and the vertical mass flux F(= pW/p'w') together with flow visualization in a low-noise water tunnel. Buoyancy was induced by salinity differences so the value of the Schmidt number S, = 700. Density stratification altered the inertial-buoyancy force balance (most simply expressed by Nt, the product of the buoyancy frequency N and turbulent timescale t ) so as to attenuate turbulent velocity fluctuations, vertical motions and interfacial convolutions, normalized density fluctuations, vertical flux mass, and mean interfacial thickness. Vertical velocity fluctuations w' were found to increase with distance from the interface, whereas the u'-distribution can be non-monotonic. The maximum value of the mass flux, F, was found to be about 0.5 which was less than the typical value of 0.7 for thermally stratified wind tunnel experiments for which S, = 0.7. The vertical mass flux can be a combination of down-gradient and counter-gradient transport with the ratio varying with Nt (e.g. at Nt z 5, the flux is counter-gradient). The flux Richardson number Rf was found to increase monotonically to values of approximately 0.05.

Physical Oceanography, 1994

The mixing efficiency of unsheared homogeneous turbulence in flows stratified by one or two active scalars was calculated with rapid distortion theory (RDT). For the case with one scalar the mixing efficiency η depends on the Schmidt number Sc = ν/D and the Grashof number Gr = NL 2 /ν, where ν is the kinematic viscosity, D is the molecular diffusivity, N is the buoyancy frequency, and L is proportional to the longitudinal integral length scale. For the case with two scalars, the efficiency also depends on the density ratio R ρ , which compares the density difference caused by temperature and the density difference caused by salt. In the one scalar case when Gr is large, η decreases as Sc increases. The mixing efficiency increases with Gr up to a maximum value, as shown in numerical simulations and experiments. The maximum mixing efficiency of approximately 30% for low Sc is consistent with simulations, while the maximum efficiency of 6% for heated water is consistent with laboratory measurements. However, RDT underpredicts the maximum efficiency for saltwater and also the value of Gr at which the efficiency becomes constant. The predicted behavior of the mixing efficiency for two active scalars is similar to that for one scalar, and the efficiency decreases as R ρ decreases, as in experiments and semi-empirical models. These calculations show that results from simulations with low Sc likely overestimate the efficiency of turbulence in strongly stratified flows in lakes and oceans.

The effects of stratification and rotation in mixing are apparent in the environment, the body forcesinvolved produce a strong anysotropy in the behaviour of scalars in the fluid leading to the relaminarization of turbulent patches, to anomalous diffusion and to propagation of internal waves. Basic aspects of stratification in turbulent flows are reviewed, providing some background on the interesting problem of mixing in stratified fluids as well as some of the techniques used to investigate stratified turbulent flows in the laboratory. Stratification reduces vertical transport of scalars by up to several orders of magnitude. In both the ocean and the atmosphere, the existence of many relatively sharp density interfaces produces most of the control of the global vertical mixing. Horizontal transport, on the other hand, is at a maximum near density interfaces. The ways in which stratification affects, and some of the previous attempts to understand flow structure and mixing near density interfaces are reviewed. We show experimental results to relate the different experimental approaches used in the laboratory to model mixing across density interfaces. The effect of density stratification on vertical diffusion and the different behaviour that the mixing efficiency has for different circumstances is discussed in the light of previous experimental observations.

Journal of Fluid Mechanics, 2009

By identifying the stratification which leads to maximal buoyancy flux in a stablystratified plane Couette flow, we make a prediction of what bulk stratification (as a function of the shear) is optimal for turbulent mixing. A previous attempt to do this (Caulfield, Tang & Plasting, J. Fluid Mech., vol. 498, 2004, p. 315) failed due to an unexpected degeneracy in the variational problem. Here, we overcome this issue by parameterizing the variational problem implicitly with the overall mixing efficiency which is then optimized across to return a rigorous upper bound on the buoyancy flux. We find that the bulk Richardson number quickly approaches 1/6 in the asymptotic limit of high shear with the associated mixing efficiency tending to 1/3. The predicted mean profiles associated with the bound appear to have a layered structure, with the gradient Richardson number being low both in the interior, and in boundary layers near the walls, with a global maximum, also equal to 1/6, occurring at the edge of the boundary layers. †