A mathematical model for type II profile of concentration distribution in turbulent flows

This paper presents a mathematical model to investigate type II profile of suspension concentration distribution (i.e., the concentration profile where the maximum concentration appears at some distance above the bed surface) in a steady, uniform turbulent flow through open-channels. Starting from the mass and momentum conservation equations of two-phase flow, a theoretical model has been derived. The distribution equation is derived considering the effects of fluid lift force, drag force, particle inertia, particle–particle interactions, particle velocity fluctuations and drift diffusion. The equation is solved numerically and is compared with available experimental data as well as with other models existing in the literature. Good agreement between the observed value and computed result, and minimum error in comparison to other models indicate that the present model can be applied in predicting particle concentration distribution for type II profile for a wide range of flow conditions. The proposed model is also able to show the transition from type I profile to type II profile.     

Effects of secondary current and stratification on suspension concentration in an open channel flow

In this work, a mathematical model on concentration distribution is developed for a steady, uniform open channel turbulent flow laden with sediments by incorporating the effect of secondary current through velocity distribution together with the stratification effect due to presence of sediments. The effect of particle-particle interaction at reference level and the effect of incipient motion probability, non-ceasing probability and pick-up probability of the sediment particles at reference concentration are taken into account. The proposed model is compared with the Rouse equation as well as verified with existing experimental data. Good agreement between computed value and experimental data indicates that secondary current influences the suspension of particles significantly. The direction and magnitude (strength) of secondary current lead to different patterns of concentration distribution and theoretical analysis shows that type II profile (where maximum concentration appears at significant height above channel bed surface) always corresponds to upward direction and greater magnitude of secondary current.     

Modeling and experiments of polydisperse particle clouds

A model for polydisperse particle clouds has been developed in this study. We extended the monodisperse particle cloud model of Lai et al. (Environ Fluid Mech 13(5):435–463, 2013) to the case of polydisperse particles. The particle cloud is first considered to be a thermal or buoyant vortex ring, with the thermal induced velocity field modeled by an expanding spherical Hill’s vortex. The buoyancy of the composite thermal is assumed to be the sum of buoyancy contributed by the all particles inside the thermal. Individual particles (of different particle properties) in the cloud are then tracked by the particle tracking equation using the computed induced velocity field. The turbulent dispersion effect is also accounted for by using a random walk model. Experiments of polydisperse particle clouds were carried out to validate the model. The agreement between model predictions and experiments was reasonable. We further validate our model by comparing it with the LES study of Wang et al. (J Hydraul Eng ASCE 141(7):06015006, 2014). The limitations of our model are then discussed with reference to the comparison. Overall, although some flow details are not captured by our model, the simplicity and generality of the model makes it useful in engineering applications.     

Suspended particulate matter dynamics in a particle framework

Suspended particulate matter (SPM) dynamics in ocean models are usually treated with an advection–diffusion equation for one or more sediment size classes coupled to the hydrodynamical part of the model. Numerical solution of these additional partial differential equations unavoidably introduces numerical diffusion, i.e. in the case of sharp gradients the possible occurrence of artificial oscillations and non-positivity. A Lagrangian particle-tracking model has been developed to simulate short-term SPM dynamics. Modelling individual sediment particles allows a straightforward physical interpretation of the processes. The tracking of large numbers of individual and independent particles (up to 25 million in total in a single sediment class) can be achieved on high performance computer clusters, due to efficient parallelisation of particle tracking. The movement of the particles is described by a stochastic differential equation, which is consistent with the advection–diffusion equation. Here, the concentration profile is represented by a set of independent moving particles, which are advected according to the 3D velocity field, while the diffusive displacements of the particles are sampled from a random distribution, which is related to the eddy diffusivity field. To account for erosion a new parameterisation is proposed. Three numerical particle tracking schemes (EULER, MILSTEIN and HEUN) are presented and validated in idealised test cases. Finally, the particle tracking algorithms are applied to a realistic scenario, a severe winter storm in the East Frisian Wadden Sea (southern North Sea). The comparison with observations and an Eulerian SPM transport model seems to indicate a somewhat better fidelity of the Lagrangian approach.     

Indoor particle dynamics in a school office: determination of particle concentrations,deposition rates and penetration factors under naturally ventilated conditions

Recently, the problem of indoor particulate matter pollution has received much attention. An increasing number of epidemiological studies show that the concentration of atmospheric particulate matter has a significant effect on human health, even at very low concentrations. Most of these investigations have relied upon outdoor particle concentrations as surrogates of human exposures. However, considering that the concentration distribution of the indoor particulate matter is largely dependent on the extent to which these particles penetrate the building and on the degree of suspension in the indoor air, human exposures to particles of outdoor origin may not be equal to outdoor particle concentration levels. Therefore, it is critical to understand the relationship between the particle concentrations found outdoors and those found in indoor micro-environments. In this study, experiments were conducted using a naturally ventilated office located in Qingdao, China. The indoor and outdoor particle concentrations were measured at the same time using an optical counter with four size ranges. The particle size distribution ranged from 0.3 to 2.5 μm, and the experimental period was from April to September, 2016. Based on the experimental data, the dynamic and mass balance model based on time was used to estimate the penetration rate and deposition rate at air exchange rates of 0.03–0.25 h^?1. The values of the penetration rate and deposition velocity of indoor particles were determined to range from 0.45 to 0.82 h^?1 and 1.71 to 2.82 m/h, respectively. In addition, the particulate pollution exposure in the indoor environment was analyzed to estimate the exposure hazard from indoor particulate matter pollution, which is important for human exposure to particles and associated health effects. The conclusions from this study can serve to provide a better understanding the dynamics and behaviors of airborne particle entering into buildings. And they will also highlight effective methods to reduce exposure to particles in office buildings.     

Comparison between remote-sensed data and in situ measurements in coastal waters: The taranto sea case

Monitoring and managing small coastal ecosystems requires a considerable understanding of the temporal dynamics of biophysical factors describing the coastal water systems. For this reason, daily observation from space could be a very efficient tool. The objective of the work described in this paper is to evaluate the contribution of remote sensing to the continuous monitoring of coastal areas. It is well known that in coastal areas, the presence of inorganic suspended sediments and coloured dissolved organic matter can make chlorophyll-concentration measurements from remote sensing difficult. To overcome these difficulties, an alternative approach to the SeaWiFS standard chlorophyll algorithm is presented, based on a semi-analytic model for sea water and on the use of MODIS data as input in a model for atmospheric effects removal. Moreover, land contamination (mixed sea–land pixels) can introduce ambiguities in sea-surface temperature measurements from remote sensing. This paper proposes the use of a hydrodynamic model as a time–space interpolator of in situ campaign data, to extensively validate the temperature values extracted from AVHRR sensor. We validated the proposed approach, using experimental field data collected over a two-year campaign in the Taranto Gulf. The results seem to indicate a good agreement between remote-sensed and in situ data.     

Turbulence suppression by suspended sediment within a geophysical flow

Experiments are performed in a mixing box to evaluate the effect of suspended sediment on turbulence generated by an oscillating grid. Quartz-density sand of varying sizes and concentrations is used, and particle image velocimetry is employed to quantify only the fluid phase. Results show that (1) while a relatively large secondary flow field is present in the box, turbulence is a maximum near the grid and it decreases systematically toward the water surface; (2) relatively high concentrations of fine sediment can markedly alter this secondary flow field and significantly decrease both the time-mean and turbulent kinetic energy within the flow, yet these same sediment concentrations have little effect on the integral time and length scales derived for each velocity component; and (3) the overall turbulence suppression observed can be related to the transfer of energy from the fluid to the sediment and the maintenance of a suspended sediment load rather than commonly employed turbulence modulation criteria. These experimental data demonstrate unequivocally that the presence of a suspended sediment load can significantly reduce overall turbulent kinetic energy, and these results should be applicable to a range of sediment-laden geophysical flows.     

Modelling the distribution and effect of heavy metals in an aquatic ecosystem

Models are reviewed describing the distribution and effect of heavy metals in an aquatic ecosystem. Since a model used for an impact statement should give the maximum concentration level rather than the seasonal variation, a model focussing on this situation is suggested. The basic differential equations describe (1) the variation in concentration of the toxicant per biomass dry matter in a given trophic level, and (2) the exchange of toxicant between sediment and water. Furthermore, since a substantial part of the heavy metal in an aquatic ecosystem is bound to suspended matter, an equation describing the equilibrium between dissolved and suspended matter must be included.A literature review has been carried out on the parameters used in the above mentioned equations and a demonstration, showing how it is possible to find approximate values for such parameters as excretion coefficient and uptake coefficient on the basis of a relationship between these two parameters and the size of an organism, is given.     

Hierarchical modeling of the dilute transport of suspended sediment in open channels

We propose, discuss and validate a theoretical and numerical framework for sediment-laden, open-channel flows which is based on the two-fluid-model (TFM) equations of motion. The framework models involve mass and momentum equations for both phases (sediment and water) including the interactive forces of drag, lift, virtual mass and turbulent dispersion. The developed framework is composed by the complete two-fluid model (CTFM), a partial two-fluid model (PTFM), and a standard sediment-transport model (SSTM). Within the umbrella of the Reynolds-Averaged Navier-Stokes (RANS) equations, we apply K–ε type closures (standard and extended) to account for the turbulence in the carrier phase (water). We present the results of numerical computations undertaken by integrating the differential equations over control volumes. We address several issues of the theoretical models, especially those related to coupling between the two phases, interaction forces, turbulence closure and turbulent diffusivities. We compare simulation results with various recent experimental datasets for mean flow variables of the carrier as well as, for the first time, mean flow of the disperse phase and turbulence statistics. We show that most models analyzed in this paper predict the velocity of the carrier phase and that of the disperse phase within 10% of error. We also show that the PTFM provides better predictions of the distribution of sediment in the wall-normal direction as opposed to the standard Rousean profile, and that the CTFM is by no means superior to the PTFM for dilute mixtures. We additionally report and discuss the values of the Schmidt number found to improve the agreement between predictions of the distribution of suspended sediment and the experimental data.     

Two-phase flow insights into open-channel flows with suspended particles of different densities

The effect of particle density on the turbulent open-channel flow carrying dilute particle suspensions is investigated using two specific gravities and three concentrations of solid particles. The particles, identical in size and similar in shape, were natural sand and a neutrally buoyant plastic. The particles were fully suspended, and formed no particle streaks on the channel’s bed. Accordingly, the changes in the flow are attributed to the interactions between suspended particles and flow turbulence structures. Measurements were obtained by means of image velocimetry enabling simultaneous, but distinct, measurement of liquid and particle velocities. The experimental results show that, irrespective of particle specific gravity, particle suspension influences bulk velocity of flow and the Kármán coefficient, while friction velocity essentially remains constant. The results also show that particles in suspension modify local water turbulence over the flow depth, but in ways not accurately predicted using the customary parameters for characterizing turbulence modification.     

Consideration of translocation into a growth model of a plant organ

Taking account of the Bertalanffy's differential equation on animal growth, plant growth is also considered as the net result of anabolism and catabolism. When we, however, consider the growth of a plant organ, it is necessary to add a term of translocation because it plays an important role in the growth of plant organs, such as leaf and fruit. Considering translocation, therefore, a growth model of a plant organ was proposed on the basis of the compartment model for estimating the carbon balance in the organ by using the experimental data on translocation, photosynthesis and respiration of a tropical fruit of durian (Durio zibethinus Murray). The present growth model of a plant organ belongs to an extended Bertalanffy's growth equation, and was possible to be transformed into the simple Bertalanffy's growth equation on the basis of the proportionality between the growth and translocation rates. The Bertalanffy's growth equation of a plant organ was also possible to apply to that of the whole plant on the assumption of the allomeric relationship between a plant organ and the whole plant.     

Simulation-based optimization of in-stream structures design: rock vanes

We employ a three-dimensional coupled hydro-morphodynamic model, the Virtual Flow Simulator (VFS-Geophysics) in its Unsteady Reynolds Averaged Navier–Stokes mode closed with \(k-\omega\) model, to simulate the turbulent flow and sediment transport in large-scale sand and gravel bed waterways under prototype and live-bed conditions. The simulation results are used to carry out systematic numerical experiments to develop design guidelines for rock vane structures. The numerical model is based on the Curvilinear Immersed Boundary approach to simulate flow and sediment transport processes in arbitrarily complex rivers with embedded rock structures. Three validation test cases are conducted to examine the capability of the model in capturing turbulent flow and sediment transport in channels with mobile-bed. Transport of sediment materials is handled using the Exner equation coupled with a transport equation for suspended load. Two representative meandering rivers, with gravel and sand beds, respectively, are selected to serve as the virtual test-bed for developing design guidelines for rock vane structures. The characteristics of these rivers are selected based on available field data. Initially guided by existing design guidelines, we consider numerous arrangements of rock vane structures computationally to identify optimal structure design and placement characteristics for a given river system.     

乌梁素海叶绿素a与理化因子的统计分析

利用2008年6月—10月乌梁素海水体叶绿素a浓度与环境理化因子的测定结果,分析了叶绿素a的时空分布特征.运用分层聚类分析法将乌梁素海现有的20个测点分成四类,找出各类测点及总测点中与叶绿素a显著相关的理化因子,建立了多元逐步回归方程.2008年乌梁素海各测点叶绿素a的平均值为6.31 mg.m-3,变幅1.54—26.87 mg.m-3;夏季最高,秋季居中,春季最低,整个区域呈现较明显的东北高西南低的分布趋势.应用SPSS统计分析软件进行相关分析的结果表明叶绿素a与BOD5、悬浮物、浊度、总磷都呈极显著相关,与透明度呈极显著负相关,与pH值和溶氧呈显著负相关.综合逐步回归方程表明,影响乌梁素海叶绿素a的理化因子因不同类型的测点各有所不同,但主要的影响因子有浊度、悬浮物、总磷、总氮、硝酸盐氮和亚硝酸盐氮.     

A general random walk model for the leptokurtic distribution of organism movement: Theory and application

The movement of organisms is usually leptokurtic in which some individuals move long distances while the majority remains at or near the area they are released. There has been extensive research into the origin of such leptokurtic movement, but one important aspect that has been overlooked is that the foraging behaviour of most organisms is not Brownian as assumed in most existing models. In this paper we show that such non-Brownian foraging indeed gives rise to leptokurtic distribution. We first present a general random walk model to describe the organism movement by breaking the foraging of each individual into events of active movement and inactive stationary period; its foraging behaviour is therefore fully characterized by a joint probability of how far the individual can move in each active movement and the duration it remains stationary between two consecutive movements. The spatio-temporal distribution of the organism can be described by a generalized partial differential equation, and the leptokurtic distribution is a special case when the stationary period is not exponentially distributed. Empirical observations of some organisms living in different habitats indicated that their rest time shows a power-law distribution, and we speculate that this is general for other organisms. This leads to a fractional diffusion equation with three parameters to characterize the distributions of stationary period and movement distance. A method to estimate the parameters from empirical data is given, and we apply the model to simulate the movement of two organisms living in different habitats: a stream fish (Cyprinidae: Nocomis leptocephalus) in water, and a root-feeding weevil, Sitona lepidus in the soil. Comparison of the simulations with the measured data shows close agreement. This has an important implication in ecology that the leptokurtic distribution observed at population level does not necessarily mean population heterogeneity as most existing models suggested, in which the population consists of different phenotypes; instead, a homogeneous population moving in homogeneous habitat can also lead to leptokurtic distribution.     

A model of vertical distribution of suspended matter in an open channel flow

A quasi-stationary model of vertical distribution of concentration of suspended particular matter in the bottom layer of 1D open channel with a sloped bottom and varying free surface slope is discussed. The model proceeds from the balance between the turbulent diffusion and settling with the buoyancy flux effects on the medium turbulence neglected. The model outcome is formulated in the form of an analytic formula for the vertical distribution of concentration. It is shown that the derived formula embraces two basic types of vertical distribution of concentration, one with a monotonic decrease of concentration gradient and the other with a gradient maximum (lutocline) located at some distance from the bottom. The first distribution type realizes for a relatively large settling velocity or low intensity of turbulence and the second type for a small settling velocity or high intensity of turbulence. The skill of the model to mimic realistic situations is demonstrated on data measured in the Jiaojiang Estuary (China).     

Prediction of the upper tail of concentration distributions of a continuous point source release in urban environments

The peak values observed in a measured concentration time series of a dispersing gaseous pollutant released continuously from a point source in urban environments, and the hazard level associated with them, demonstrate the necessity of predicting the upper tail of concentration distributions. For the prediction of concentration distributions statistical models are preferably employed which provide information about the probability of occurrence. In this paper a concentration database pertaining to a field experiment is used for the selection of the statistical distribution. The inverses of the gamma cumulative distribution function (cdf) for 75th–99th percentiles of concentration are found to be more consistent with the experimental data than those of the log-normal distribution. The experimental values have been derived from measured high frequency time series by sorting first the concentrations and then finding the concentration which corresponds to each probability. Then the concentration mean and variance that are predicted with Computational Fluid Dynamics-Reynolds Averaged Navier–Stokes (RANS) methodology are used to construct the gamma distribution. The proposed model (“RANS-gamma”) is included in the framework of a computational code (ADREA-HF) suitable for simulating the dispersion of airborne pollutants over complex geometries. The methodology is validated by comparing the inverses of the model cdfs with the observed ones from two wind tunnel experiments. The evaluation is performed in the form of validation metrics such as the fractional bias, the normalized mean square error and the factor-of-two percentage. From the above comparisons it is concluded that the overall model performance for the present cases is satisfactory.     

Impact of aquacultures on the marine ecosystem: Modelling benthic carbon loading over variable depth

In order to be used within Environmental Impact Assessment study, we have developed a three-dimensional particle tracking model for prediction of benthic carbon loading (BCL) caused by fish farms. The model is based on stochastic differential equations for particle transport consistent with the well-known semi-empirical advection/diffusion equation. It requires only easily obtainable input data in the form of measured current record, the source location and a specification of local bathymetry. The model accounts for advection by long-term residual and tidal currents, turbulent diffusion, realistic bathymetry and variations in daily (monthly or yearly) emissions from fish farm.Here, we concentrate on the changes in sedimentation pattern caused by various bathymetric shapes. Examination of idealized cases reveals where and why we can expect the worst impact on benthic communities. For future reference, these results will be included into guidelines for fish farming.     

Influences of suspended particles on the runoff of pesticides from an agricultural field at Askim, SE-Norway

Field and laboratory experiments were conducted to study the loss of particles from agricultural fields, and the role of suspended particles in carrying pesticides in surface runoff and drainage water. Propiconazole, a widely used fungicide was applied to experimental fields located at Askim, SE-Norway. Samples from surface runoff and drainage water were collected and analyzed for sediment mass, pesticides, particulate and dissolved organic carbon through a whole year. The surface soil and the runoff material were characterized by its particle size distribution, organic carbon content in size fractions and its ability to bind propiconazole. The results show that (1) particle runoff mostly occurred during the rainfall event shortly after harrowing in autumn. The highest particle concentration observed in the surface runoff water was 4600 mg l(-1), and in the drainage water 1130 mg l(-1); (2) the erosion of surface soil is size selective. The runoff sediment contained finer particle/aggregates rich in organic matter compared to its original surface soil; (3) the distribution coefficient (Kd) of propiconazole was significantly higher in the runoff sediment than in the parent soil. According to our calculation, particle-bound propiconazole can represent up to 23% of the total amount of propiconazole in a water sample with a sediment concentration of 7600 mg l(-1), which will significantly influence the transport behavior of the pesticide.     

A combined suspended particle and phosphorus water quality model: Application to Lake Vänern

A dynamic model describing the flow of phosphorus and suspended particulate matter for large fresh water lakes is proposed and validated against data from Lake Vänern in Sweden. The model was modified to handle two separate sub-basins, but increasing the horizontal resolution further by splitting the basins into coastal area and pelagial failed, as the model fit to experimental data deteriorated. Besides, the scant reference data available for the coastal areas makes this a dubious exercise. Parameters for the nutrient dynamics in the water column required less tuning (up to 60%) than the sedimentation and sediments (up to a factor 70). The fit to experimental data is good for the periods between 1900 and 1940 and that after 1980, but is less satisfactory for the more polluted conditions in the middle of the century.The model is applied to two scenarios: increased emissions from a pulp and paper mill by the lake, and decreased phosphorus emissions achieved by a combination of effects on farmland, woodland, and rural households. These two scenarios demonstrate the usefulness of a dynamic quantitative lake water quality model.     

Some effects of langmuir circulation on the quality of water resource systems

The quantitative extent to which the large-scale organized water motion in the surface waters of lakes and reservoirs, known as Langmuir circulation, affects the distribution and settling of algae and other suspended particles is not known and has thus been ignored in conventionally used water-quality models. Since the distribution and settling of these particles is important in determining water quality, this study set out to investigate these effects. Current literature which discusses this problem is reviewed and a mathematical model is developed based on the two-dimensional advection-diffusion mass transport describing the temporal and spatial distribution of suspended particles in a typical Langmuir cell; the Langmuir circulation flow field and turbulent diffusion coefficients are empirically modelled by relating them to environmental parameters.The results show that Langmuir circulation does affect particle distribution and settling. For particles with small sinking speeds, such as the lighter algae, the circulation causes intense mixing, resulting in essentially uniform distribution of particles over the cell (as assumed in the ‘well-mixed compartment model’). For particles with high sinking velocities, however, aggregation can occur, giving rise to significant reduction in sinking loss when compared with that predicted by conventional models. For diatoms, reductions of 6% and higher can occur depending on which conventionally used model is being considered, while for silt and sand particles in a cell of large width-to-depth ratio a reduction of more than 60% is possible.