On the dissipation rate of temperature fluctuations in stably stratified flows

Environmental Fluid Mechanics - In this study, we explore several integral and outer length scales of turbulence which can be formulated by using the dissipation of temperature fluctuations (...     

Turbulent Prandtl number and characteristic length scales in stably stratified flows: steady-state analytical solutions

Environmental Fluid Mechanics - In this study, the stability dependence of turbulent Prandtl number ( $$Pr_t$$ ) is quantified via a novel and simple analytical approach. Based on the variance and...     

Numerical and experimental studies of turbulence in vegetated open-channel flows

Environmental Fluid Mechanics - A numerical model based on the double-averaged (spatial and ensemble averaged) method has been developed to simulate vegetated free surface flows. The classical $$k...     

Gravity currents and intrusions of stratified fluids into a stratified ambient

We consider high-Reynolds-number Boussinesq gravity currents and intrusions systems in which both the ambient and the propagating “current” are linearly stratified. The main focus is on a current of fixed volume released from a rectangular lock; the height ratio of the fluids H, and the stratification parameter of the ambient S, are quite general. We develop a one-layer shallow-water (SW) model which is an extension of previously used and tested formulations for currents and intrusions of constant density. The internal stratification enters as a new dimensionless parameter, s ? [0,1]{\sigma \in [0,1]} . Analytical results are obtained for the initial “slumping” stage during which the speed of propagation is constant, and finite-difference solutions are presented for the more general time-dependent motion. Overall, this is a versatile and robust self-contained prediction tool, which reduces smoothly to the classical case when σ = 0. We show that, in general, the speed of propagation decreases when the internal stratification becomes more pronounced (σ increases). An interesting non-expected behavior was detected: when the stratification of the ambient is weak and moderate then the height of the current decreases with σ, but the opposite occurs when the stratification of the ambient is strong (S ≈ 1, including the case of an intrusion). Moreover, when the stratification of the ambient is strong a current with internal stratification may “run out” of driving power. We also consider the Benjamin-type steady state current with internal linear stratification in a non-stratified ambient, and show that an analytical solution exists, and that the maximal thickness decreases to below half-channel depth when σ increases.     

Development of similarity relationships for energy dissipation rate and temperature structure parameter in stably stratified flows: a direct numerical simulation approach

In this study, a newly developed direct numerical simulation (DNS) solver is utilized for the simulations of numerous stably stratified open-channel flows with bulk Reynolds number (Re _b ) spanning 3400–16,900. Overall, the simulated bulk Richardson number (\(Ri_b\)) ranges from 0.08 (weakly stable) to 0.49 (very stable). Thus, both continuously turbulent and (globally) intermittently turbulent cases are represented in the DNS database. Using this comprehensive database, various flux-based and gradient-based similarity relationships for energy dissipation rate (ε) and temperature structure parameter (\(C_T^2\)) are developed. Interestingly, these relationships exhibit only minor dependency on Re _b . In order to further probe into this Re _b -effect, similarity relationships are also estimated from a large-eddy simulation (LES) run of an idealized atmospheric boundary layer (very high Re _b ) case study. Despite the fundamental differences in the estimation of ε and \(C_T^2\) from the DNS- and the LES-generated data, the resulting similarity relationships, especially the gradient-based ones, from these numerical approaches are found to be remarkably similar. More importantly, these simulated relationships are also comparable, at least qualitatively, to the traditional observational data-based ones. Since these simulated similarity relationships do not require Taylor’s hypothesis and do not suffer from mesoscale disturbances and/or measurement noise, they have the potential to complement the existing similarity relationships.     

Gravity currents in two-layer stratified media

An analytical, experimental and numerical study of boundary gravity currents propagating through a two-layer stratified ambient of finite vertical extent is presented. Gravity currents are supposed to originate from a lock-release apparatus; the (heavy) gravity current fluid is assumed to span the entire channel depth, H, at the initial instant. Our theoretical discussion considers slumping, supercritical gravity currents, i.e. those that generate an interfacial disturbance whose speed of propagation matches the front speed, and follows from the classical analysis of Benjamin (J Fluid Mech 31:209?C248, 1968). In contrast to previous investigations, we argue that the interfacial disturbance must be parameterized so that its amplitude can be straightforwardly determined from the ambient layer depths. Our parameterization is based on sensible physical arguments; its accuracy is confirmed by comparison against experimental and numerical data. More generally, measured front speeds show positive agreement with analogue model predictions, which remain strictly single-valued. From experimental and numerical observations of supercritical gravity currents, it is noted that this front speed is essentially independent of the interfacial thickness, ??, even in the limiting case where ?? = H so that the environment is comprised of a uniformly stratified ambient with no readily discernible upper or lower ambient layer. Conversely, when the gravity current is subcritical, there is a mild increase of front speed with ??. Our experiments also consider the horizontal distance, X, at which the front begins to decelerate. The variation of X with the interface thickness and the depths and densities of the ambient layers is discussed. For subcritical gravity currents, X may be as small as three lock lengths whereas with supercritical gravity currents, the gravity current may travel long distances at constant speed, particularly as the lower layer depth diminishes.     

The flow of high-Reynolds axisymmetric gravity currents of a stratified fluid into a stratified ambient: shallow-water and box model solutions

We consider the axisymmetric flow (in a full cylinder or a wedge) of high-Reynolds-number Boussinesq gravity currents and intrusions systems in which both the ambient and the propagating “current” are linearly stratified. The main focus is on a current of fixed volume released from a cylinder lock; the height ratio of the fluids H, and the stratification parameter of the ambient S, are quite general. We develop a one-layer shallow-water model. The internal stratification enters as a new dimensionless parameter, ${\sigma \in [0, 1]}$ . In general, the time-dependent motion is obtained by standard finite-difference solutions; a self-similar analytical solution exists for S?= 0. We show that, in general, the speed of propagation decreases when the internal stratification becomes more pronounced (σ increases). We also developed a box-model approximation, and show that the resulting radius of propagation is in good agreement with the more rigorous shallow-water prediction.     

A mesoscale vortex in a small stratified lake

A mesoscale vortex structure in the small stratified Lake Stechlin has been revealed by field experiments with satellite-tracked quasi-lagrangian drifters. The vortex with a radius of about 200 m drifted at 300 m/day along the western bight of the lake with nearly constant rotation speed of 3 cpd. Analysis of kinematical properties of the vortex motion demonstrates solid body character of rotation. Extrapolation of the vortex drift trajectory over the period preceding the observations combined with data on local winds and seiche dynamics has allowed tracing the vortex fate from its generation point. The normal modes analysis of the internal seiching in the lake reveals the vortex generation mechanism to be the interaction of certain seiche modes with local bottom topography and suggests generation of the mesoscale vortices to be the a regular feature of the lake circulation. Analysis of vorticity suggests additional energy supply to rotational flow, possibly from inverse cascading of small-scale turbulent motions—a feature typical for quasi-2D turbulence. The vortices can play an important role in the energy transport from basin-scale motions to small-scale boundary mixing. They can also contribute significantly to the horizontal heterogeneity of phyto- and zooplankton distribution as well as to the transport of dissolved matter such as nutrients between littoral and profundal areas. The topographically generated traveling vortices represent an analog of the synoptic eddies in the Ocean and in the Atmosphere, whereas their role in the lake hydrodynamics is practically unknown.     

The similarity solution for turbulent mixing of two-layer stratified fluid

Experiments are reviewed in which a two-layer salt-stratified tank of water was mixed by turbulence. The density profile began as a single step and evolved to a smooth mixed profile. The turbulence was generated by many excursions of a horizontally moving vertical rod with Richardson number Ri >  0.9 and Reynolds Number Re  >  600. There was almost perfect collapse of all the profiles to one universal profile as a function of a similarity variable. We develop a theoretical model for a simple mixing law with a buoyancy flux that is a function of internal Richardson number Rii. A similarity equation is found. A flux law that increases with small Rii and decreases with large Rii is considered next. Since no analytical solution is known, the similarity concept is tested by numerically integrating the equations in space and time. With buoyancy flux monotonically increasing with internal Richardson number, the similarity approach is valid for a profile starting from a slightly smoothed step. However, a shock forms for a mixing law with higher initial Rii (so that buoyancy flux decreases with Richardson number) and the similarity approach is invalid for those initial conditions.     

The three-dimensionality of obstructed shear flows

Obstructed shear flows (i.e. those over permeable media) are common in the environment. An archetypal example, flow over a submerged vegetation canopy, is investigated here. Like any flow through complex geometry, canopy flows are characterised by strong spatial gradients. The focus of this experimental study is the three-dimensionality of aquatic canopy flow, in particular that of the coherent interfacial vortices that govern mixing into and out of the canopy. It is shown here that the vortices have a finite lateral scale that is comparable to their vertical scale; both are of the order of the drag length scale of the canopy, (C _D a)⁻¹, where a is the frontal area density and C _D is a bulk drag coefficient. The finite lateral extent of the vortices generates strong lateral hydrodynamic gradients, both instantaneously and in the long-term. The instantaneous gradients, which can contribute greatly to the dispersion of dissolved and particulate species, are far more pronounced. Finally, the potential for canopies to generate differential roughness secondary circulation is examined. In the consideration of vertical scalar transport, this circulation can be of the same order as turbulent diffusion.     

Spatially stratified sampling using auxiliary information for geostatistical mapping

This paper presents a method of spatial sampling based on stratification by Local Moran’s I _i calculated using auxiliary information. The sampling technique is compared to other design-based approaches including simple random sampling, systematic sampling on a regular grid, conditional Latin Hypercube sampling and stratified sampling based on auxiliary information, and is illustrated using two different spatial data sets. Each of the samples for the two data sets is interpolated using regression kriging to form a geostatistical map for their respective areas. The proposed technique is shown to be competitive in reproducing specific areas of interest with high accuracy.     

Trace element analysis of soil samples from a stratified archaeological site

Thirty-eight soil samples from a sondage excavated through a 6- metre stratified tell (Gomoiava, in central Yugoslavia) have been analysed for nine biophile elements: Mg, Sr, Zn, Cu, Ni, Mn, Cr, Pb and B. The site was occupied from 7000 BC to 500 AD (calendar years). The results have been collated with earlier estimates of P and Ca, made on a larger series of samples from the same site. The results are discussed in terms of useful information, to supplement that given by phosphate analysis, about stratification and the relationship of the settlement to its environment.It is concluded that leaching probably distorts the results for Ca and Mg. The results for Mn may help to distinguish phosphate accumulation due to man from that due to farm livestock. There is significant accumulation of Sr and B, both of which have potential value, although some of the analytical results for the latter are suspect. The results for Zn, Cu and Ni are highly correlated, with sharp peaks which couid provide precise information about individual strata when more experience of interpretation has been gained. Only Cr and Pb appear to provide little useful information at this site.     

Statistical inference using stratified judgment post-stratified samples from finite populations

This paper develops statistical inference for population mean and total using stratified judgment post-stratified (SJPS) samples. The SJPS design selects a judgment post-stratified sample from each stratum. Hence, in addition to stratum structure, it induces additional ranking structure within stratum samples. SJPS is constructed from a finite population using either a with or without replacement sampling design. Inference is constructed under both randomization theory and a super population model. In both approaches, the paper shows that the estimators of population mean and total are unbiased. The paper also constructs unbiased estimators for the variance (mean square prediction error) of the sample mean (predictor of population mean), and develops confidence and prediction intervals for the population mean. The empirical evidence shows that the proposed estimators perform better than their competitors in the literature.     

Mixing layer in open-channel junction flows

When two open-channel flows merge in a three-branch subcritical junction, a mixing layer appears at the interface between the two inflows. If the width of the downstream channel is equal to the width of each inlet channel, this mixing layer is accelerated and is curved due to the junction geometry. The present work is dedicated to simplified geometries, considering a flat bed and a \(90^{\circ }\) angle where two configurations with different momentum ratios are tested. Due to the complex flow pattern in the junction, the so-called Serret–Frenet frame-axis based on the local direction of the velocity must be employed to characterize the flow pattern and the mixing layer as Cartesian and cylindrical frame-axes are not adapted. The analysis reveals that the centerline of the mixing layer, defined as the location of maximum Reynolds stress and velocity gradient, fairly fits the streamline separating at the upstream corner, even though a slight shift of the mixing layer towards the center of curvature is observed. The shape of the mixing layer appears to be strongly affected by the streamwise acceleration and the complex lateral confinement due to the side walls and the corners of the junction, leading to a streamwise increase of the mean velocity along the centerline and a decrease of the velocity difference. This results in a specific streamwise evolution of the mixing layer width, which reaches a plateau in the downstream region of the junction. Finally, the evaluation of the terms in the Reynolds-Averaged-Navier–Stokes equations reveals that the streamwise and normal acceleration and the pressure gradient remain dominant, which is typical of accelerated and rotational flows.     

Estimation of rare and clustered population mean using stratified adaptive cluster sampling

For many clustered populations, the prior information on an initial stratification exists but the exact pattern of the population concentration may not be predicted. Under this situation, the stratified adaptive cluster sampling (SACS) may provide more efficient estimates than the other conventional sampling designs for the estimation of rare and clustered population parameters. For practical interest, we propose a generalized ratio estimator with the single auxiliary variable under the SACS design. The expressions of approximate bias and mean squared error (MSE) for the proposed estimator are derived. Numerical studies are carried out to compare the performances of the proposed generalized estimator over the usual mean and combined ratio estimators under the conventional stratified random sampling (StRS) using a real population of redwood trees in California and generating an artificial population by the Poisson cluster process. Simulation results show that the proposed class of estimators may provide more efficient results than the other estimators considered in this article for the estimation of highly clumped population.

     

Spatially mixed crops to control the stratified dispersal of airborne fungal diseases

Intraspecific crop diversification is thought to be a possible solution to the disease susceptibility of monocultured crops. We modelled the stratified dispersal of an airborne pathogen population in order to identify the spatial patterns of cultivar mixtures that could slow epidemic spread driven by dual dispersal mechanisms acting over both short and long distances. We developed a model to simulate the propagation of a fungal disease in a 2D field, including a reaction-diffusion model for short-distance disease dispersal, and a stochastic model for long-distance dispersal. The model was fitted to data for the spatio-temporal spread of faba bean rust (caused by Uromyces viciae-fabae) through a discontinuous field. The model was used to compare the effectiveness of eight different planting patterns of cultivar mixtures against a disease spread by short-distance and stratified dispersal. Our combined modelling approach provides a reasonably good fit with the observed data for the spread of faba bean rust. Similar predictive power could be expected for the management of resource-mediated invasions by other airborne fungi. If a disease spreads by short-distance dispersal, random mixtures can be used to slow the epidemic spread, since their spatial irregularity creates a natural barrier to the progression of a smooth epidemic wave. In the context of stratified dispersal, heterogeneous patterns should be used that include a minimum distance between susceptible units, which decreases the probability of infection by long-distance spore dispersal. We provide a simple framework for modelling the stratified dispersal of disease in a diversified crop. The model suggests that the spatial arrangement of components in cultivar mixtures has to accord with the dispersal characteristics of the pathogen in order to increase the efficiency of diversification strategies in agro-ecosystems and forestry. It can be applied in low input agriculture to manage pathogen invasion by intercropping and cultivar mixtures, and to design sustainable systems of land use.     

Front propagation of gravity currents on inclined bottoms in linearly stratified fluids

Environmental Fluid Mechanics - Gravity currents propagating on an inclined bottom into stratified environment can be frequently encountered in nature or engineering fields. However, theoretical...