Development of an empirical equation for the transverse dispersion coefficient in natural streams

In this study, a new empirical equation for the transverse dispersion coefficient has been developed based on the hydraulic and geometric parameters in natural streams using a regression technique. First, a total of 32 data sets in 16 streams were collected. Among those sets, 16 sets were used for deriving the new equation, and the other 16 sets were used for verifying the equation. Then, through dimensional analysis, it was found that the normalized transverse dispersion coefficient is associated with several parameters such as sinuosity, aspect ratio, and a friction term. The robust least square method was applied to estimate regression coefficients. The newly proposed equation was proven to be superior in explaining the dispersion characteristics of natural streams more precisely compared to the existing equations.     

Comments on “Development of an empirical equation for the transverse dispersion coefficient in natural streams” by Tae Myoung Jeon,Kyong Oh Baek and Il Won Seo

In a recent paper published in this journal, Jeon et al. ((2007), Environ Fluid Mech 7(4): 317–329) have presented a new empirical equation for the transverse dispersion coefficient in natural streams that was developed based on the hydraulic and geometric parameters using a regression technique. A total of 32 data sets collected in 32 streams was used. Among them, 16 data sets were used for deriving the new equation, and the other 16 were used for verifying the equation. Starting from dimensional analysis the authors found that transverse dispersion coefficient depends on three parameters, such as sinuosity, aspect ratio and a friction term. The robust least square method was applied to estimate regression coefficients resulting in an equation which allows better prediction of transverse dispersion coefficient than previous literature equations. The discussers would like to highlight some points raised in the paper.     

Analytical solution of two-dimensional advection–dispersion equation with spatio-temporal coefficients for point sources in an infinite medium using Green’s function method

In the present study analytical solutions of a two-dimensional advection–dispersion equation (ADE) with spatially and temporally dependent longitudinal and lateral components of the dispersion coefficient and velocity are obtained using Green’s Function Method (GFM). These solutions describe solute transport in infinite horizontal groundwater flow, assimilating the spatio-temporal dependence of transport properties, dependence of dispersion coefficient on velocity, and the particulate heterogeneity of the aquifer. The solution is obtained in the general form of temporal dependence and the source term, from which solutions for instantaneous and continuous point sources are derived. The spatial dependence of groundwater velocity is considered non-homogeneous linear, whereas the dispersion coefficient is considered proportional to the square of spatial dependence of velocity. An asymptotically increasing temporal function is considered to illustrate the proposed solutions. The solutions are validated with the existing solutions derived from the proposed solutions in three special cases. The effect of spatially/temporally dependent heterogeneity on the solute transport is also demonstrated. To use the GFM, the ADE with spatio-temporally dependent coefficients is reduced to a dispersion equation with constant coefficients in terms of new position variables introduced through properly developed coordinate transformation equations. Also, a new time variable is introduced through a known transformation.     

RANS-based simulation of transverse turbulent mixing in a 2D geometry

Although transverse mixing is a significant process in river engineering when dealing with the discharge of pollutants from point sources or the mixing of tributary inflows, no theoretical basis exists for the prediction of its rate, which is indeed based upon the results of experimental works carried on in laboratory channels or in streams and rivers. The paper presents the preliminary results of a numerical study undertaken to simulate the transverse mixing of a steady-state point source of a tracer in a two-dimensional rectangular geometry, which is expected to reproduce a shallow flow. This geometry is that of Lau and Krishnappan (J Hydraul Div 13(HY10):1173–1189, 1977), who collected turbulent mixing data for a shallow flow. In the numerical study an approach based on the Reynolds Averaged Navier–Stokes (RANS) equations was applied, where the closure problem was solved by using turbulent viscosity concept. Particularly, the classical two-equations k–? model was used. Two methods were applied to the model results to evaluate the turbulent transverse mixing coefficient. The effect on transverse mixing of a grid located upstream the tracer source was also studied. Numerical results were generally higher than the experimental data. This overestimation could be explained considering the hypothesis of isotropic turbulence underlying the k–? model, which can lead to large turbulent viscosities and rate of mixing. However, RANS-based results may still be considered acceptable also providing the large uncertainties associated with literature predictive equations.     

Transport of Pollutant in Open Channel with Short Waves

This study examines the effect of short period water waves on the longitudinal mixing of pollutants in open channel flow. These waves create orbital motions and therefore increase the magnitude of the dispersion coefficient. Experiments are conducted for non-wavy and wavy flow. The values of the longitudinal dispersion coefficients are determined by applying the method of least squares to the measured solute concentrations at various time intervals. For non-wavy flow, the measured values of longitudinal dispersion coefficient match closely with those computed from the empirical equation given by Seo [1]. For wavy flow, a new factor called the wave parameter (a/TU _*, a=wave amplitude, T=wave period, U _*=shear velocity) is found important and a nonlinear multiple regression analysis is used to derive a new expression for the wave induced longitudinal dispersion coefficient (WILDC). An uncertainty analysis is conducted as per IS Code 5168 and the confidence interval is determined. Linear water wave theory is applied to modify the existing expression of the longitudinal dispersion coefficient of Seo [1] by including the effect of short waves. A mathematical model for WILDC is then developed. Comparative study between wavy and non-wavy flow cases has been conducted. The results clearly show an increase in the magnitude of longitudinal dispersion coefficient in the presence of waves.     

Improved formulae of velocity distributions along the vertical and transverse directions in natural rivers with the sidewall effect

The distribution of flow velocity is a basis for the research into the transport of flow and sediment in natural rivers. Characteristics of velocity distribution in narrow-deep natural rivers are different from those in wide-shallow open channels, and the effect of sidewalls on the distribution of flow velocity is considerable, which leads to a large transverse gradient of the depth-averaged velocity, with the maximum velocity occurring below the water surface. Based on the Reynolds equation of turbulence flow and the analysis of the features of velocity distribution in natural rivers, improved formulae with two empirical parameters α and β have been proposed for the velocity distributions along the vertical and transverse directions, with the effect of sidewalls being considered, through solving the definite solution problem by the method of variable separation. The proposed formulae were validated fully through comparisons between the calculated and measured velocity profiles and depth-averaged velocities at several sections in the Yangtze and Baitarani Rivers, with close agreement between them being obtained. The formula of velocity distribution along the transverse direction in natural rivers with the sidewall effect was also compared with previous studies, and the calculation accuracy of this formula at a section with a narrow-deep geometry was higher than the accuracy of the previous equations. It is confirmed that the proposed formulae can reproduce well the distribution characteristics of flow velocity along the vertical and transverse directions in narrow-deep natural rivers, with a more wide application in practice.     

Enhancing transverse mixing in streams with submerged vanes

In order to maintain the water quality of moving streams, it is essential to know the process of pollutant mixing. The transverse mixing is very important which is needed to be modeled to understand mixing phenomenon. It was observed that transverse mixing is a strong function of secondary currents, thus, submerged vanes, which are aerofoil skewed at angle of 10°–40° with respect to flow, generate transverse circulations that can be utilized to induce secondary currents in the flow to enhance transverse mixing. Present study is an attempt to utilize submerged vanes as an instrument to enhance the transverse mixing by incorporating various vane configurations. In order to study the effect of vane generated circulations on transverse mixing, experimentations were conducted on three vane sizes and for various row arrangements of vanes attached to bed. An attempt is made to investigate the effect of submerged vane size and rows on transverse velocity, concentration profile and transverse mixing coefficient. It was observed by measurement of concentration profile that transverse mixing was more enhanced for submerged vanes of higher height. It was also observed that as the number of rows is proportional to the transverse mixing. By measuring the transverse velocity profile, it was observed that more and more fluid was advected in transverse direction for higher rows of vanes. By utilizing the observed transverse mixing coefficients, number of vane rows and relative height of vane, a predictor was derived to predict transverse mixing coefficient in the presence of submerged vane rows. It was observed that the derived predictor shows a fair amount of agreement in the result predicted.     

The challenge of providing environmental flow rules to sustain river ecosystems.

Accounting for natural differences in flow variability among rivers, and understanding the importance of this for the protection of freshwater biodiversity and maintenance of goods and services that rivers provide, is a great challenge for water managers and scientists. Nevertheless, despite considerable progress in understanding how flow variability sustains river ecosystems, there is a growing temptation to ignore natural system complexity in favor of simplistic, static, environmental flow "rules" to resolve pressing river management issues. We argue that such approaches are misguided and will ultimately contribute to further degradation of river ecosystems. In the absence of detailed empirical information of environmental flow requirements for rivers, we propose a generic approach that incorporates essential aspects of natural flow variability shared across particular classes of rivers that can be validated with empirical biological data and other information in a calibration process. We argue that this approach can bridge the gap between simple hydrological "rules of thumb" and more comprehensive environmental flow assessments and experimental flow restoration projects.     

Methodological approaches for delimitating mixing zones in rivers: establishing admissibility criteria and flow regime representation

The present paper addresses defining the extent of the mixing zones of effluents discharged into rivers, which is a problem that should be analyzed based on probabilistic terms, as it is governed by several random processes. A river’s flow regime is one of the main variables, and it has a high dependence on hydrological processes. Additionally, after calculating the extent of the mixing zone, it is necessary to determine if the resulting dimensions are admissible or not. Common practice includes the admissibility criteria associated with the geometry of the river. However, this practice does not consider the environmental characteristics of the river that make it capable of absorbing the impact of the effluent (the biological and hydromorphological status of the river and the presence of structures that can change the river flow conditions, ecologically sensitive area or threatened species). This paper presents work on two important topics: on the one hand, the development of a methodology to establish the admissible extent of the mixing zone as a function of the environmental vulnerability of the river to the discharged effluent and, on the other hand, the proposal of a procedure to perform the calculations of the effluent mixing considering the variability of the river’s flow regime. The proposed methodological approaches are illustrated with an application to a real case, including a numerical simulation of the hydrodynamic and effluent evolution of the river during a year, to test the proposed methodology’s suitability and demonstrate the important savings in computational effort that can be achieved.     

Estimating river flow effects on water ages by hydrodynamic modeling in Little Manatee River estuary, Florida, USA

The water age in a tidal river in Florida, Little Manatee River, has been investigated in this study by the application of a three-dimensional hydrodynamic model. In response to a pulse dye release in the upper end of the river boundary, the hydrodynamic model determines the water age for a given location by recording the time for the dye to reach the given river location. The hydrodynamic model uses horizontal curvilinear orthogonal grids to represent the complex river system that includes several bayous and tributaries. The model has was calibrated and verified in previous study by using two continuous data sets for a 6 month period. Satisfactory model verifications indicate that the hydrodynamic model is capable of quantifying the mixing and transport process for calculating the water age in the tidal river. For 17 freshwater inflow scenarios in the Little Manatee River, the hydrodynamic model was applied to simulate water ages along the main channel of the river at 2-km interval. Flow rates in the 17 scenarios varying from 0.26 to 76.56 m³/s cover the range of the observed flows in the Little Manatee River. Water ages from model predictions range from the minimum 1.2 days under the maximum 76.56 m³/s inflow condition to the 50 days under the minimum 0.26 m³/s inflow condition. Empirical regression equations at three selected stations, with the correlation coefficient R² above 0.96, were derived from numerical model simulations to correlate water ages to freshwater inflows. The empirical water-age equation derived from hydrodynamic model simulations can be used to provide quick and low-cost estimations of water ages in response to various inflow scenarios for studying physical–chemical and biological processes in the river.     

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.     

Prediction of longitudinal dispersion coefficients in natural rivers using artificial neural network

An artificial neural network (ANN) model is developed for predicting the longitudinal dispersion coefficient in natural rivers. The model uses few rivers’ hydraulic and geometric characteristics, that are readily available, as the model input, and the target output is the longitudinal dispersion coefficient (K). For performance evaluation of the model, using published field data, predictions by the developed ANN model are compared with those of other reported important models. Based on various performance indices, it is concluded that the new model predicts the longitudinal dispersion coefficient more accurately. Sensitive analysis performed on input parameters indicates stream width, flow depth, stream sinuosity, flow velocity, and shear velocity to be the most influencing parameters for accurate prediction of the longitudinal dispersion coefficient.     

Environmental Impact of Undular Tidal Bores in Tropical Rivers

A tidal bore impacts significantly on the estuarine ecosystem, although little is known on the flow field, mixing and sediment motion beneath tidal bores. In the absence of detailed systematic field measurements, a quasi-steady flow analogy was applied to investigate undular tidal bores with inflow Froude numbers between 1.25 and 1.6. Experimental results indicated that rapid flow redistributions occur beneath the free-surface undulations, with significant variations in bed shear stress between wave crests and troughs. Dynamic similarity was used to predict detailed flow characteristics of undular tidal bores. The effects of periodic loading on river sediments, scour of river bed and flow mixing behind the bore are discussed. A better understanding of these processes will contribute to better management practices in tidal bore affected rivers, including the Styx and Daly rivers in tropical Australia.     

Deterministic particle tracking simulation of pollutant discharges in rivers and estuaries

A two-dimensional deterministic particle tracking model, in which the anisotropic-dispersive process is described by a particle strength exchange scheme, is established for the simulation of pollutant transport in vertically well-mixed rivers and estuaries. By simulating two benchmark problems with analytic solutions, the PSE scheme is shown to be accurate even if the anisotropic ratio of dispersion coefficients is very high. Further simulations of two specific problems concerning the optimal effluent discharge location and procedure are presented. The major conclusion is that in a tidal estuary with a relatively large fresh-water flow, setting the discharge position at the mixing center and making the discharge rate proportional to flow speed may minimize the peaks of concentration.     

Identification of pollution sources in rivers using a hydrodynamic diffusion wave model and improved Bayesian-Markov chain Monte Carlo algorithm

● A hydrodynamic-Bayesian inference model was developed for water pollution tracking. ● Model is not stuck in local optimal solutions for high-dimensional problem. ● Model can estimate source parameters accurately with known river water levels. ● Both sudden spill incident and normal sewage inputs into the river can be tracked. ● Model is superior to the traditional approaches based on the test cases. Water quality restoration in rivers requires identification of the locations and discharges of pollution sources, and a reliable mathematical model to accomplish this identification is essential. In this paper, an innovative framework is presented to inversely estimate pollution sources for both accident preparedness and normal management of the allowable pollutant discharge. The proposed model integrates the concepts of the hydrodynamic diffusion wave equation and an improved Bayesian-Markov chain Monte Carlo method (MCMC). The methodological framework is tested using a designed case of a sudden wastewater spill incident (i.e., source location, flow rate, and starting and ending times of the discharge) and a real case of multiple sewage inputs into a river (i.e., locations and daily flows of sewage sources). The proposed modeling based on the improved Bayesian-MCMC method can effectively solve high-dimensional search and optimization problems according to known river water levels at pre-set monitoring sites. It can adequately provide accurate source estimation parameters using only one simulation through exploration of the full parameter space. In comparison, the inverse models based on the popular random walk Metropolis (RWM) algorithm and microbial genetic algorithm (MGA) do not produce reliable estimates for the two scenarios even after multiple simulation runs, and they fall into locally optimal solutions. Since much more water level data are available than water quality data, the proposed approach also provides a cost-effective solution for identifying pollution sources in rivers with the support of high-frequency water level data, especially for rivers receiving significant sewage discharges.     

A Computational Fluid Dynamics Approach for Urban Area Transport and Dispersion Modeling

A simulation tool has been developed to model the wind fields, turbulence fields, and the dispersion of Chemical, Biological, Radiological and Nuclear (CBRN) substances in urban areas on the building to city blocks scale. A Computational Fluid Dynamics (CFD) approach has been taken that naturally accounts for critical flow and dispersion processes in urban areas, such as channeling, lofting, vertical mixing and turbulence, by solving the steady-state, Reynolds-Averaged Navier–Stokes (RANS) equations. Rapid generation of high quality cityscape volume meshes is attained by a unique voxel-based model generator that directly interfaces with common Geographic Information Systems (GIS) file formats. The flow and turbulence fields are obtained by solving the steady-state RANS equations using a collocated, pressure-based approach formulated for unstructured and polyhedral mesh elements. Turbulence modeling is based upon the Renormalization Group variant of the k–ε model (k–ε RNG). Neutrally buoyant simulations are made by prescribing velocity boundary condition profiles found by a power–law relationship, while turbulence quantities boundary conditions are defined by a prescribed mixing length in conjunction with the assumption of turbulence equilibrium. Dispersion fields are computed by solving an unsteady transport equation of a dilute gas, formulated in a Eulerian framework, using the velocity and turbulence fields found from the steady-state RANS solution. In this paper the model is explained and detailed comparisons of predicted to experimentally obtained velocity, turbulence and dispersion fields are made to neutrally stable wind tunnel and hydraulic flume experiments.     

河流放射性核素水体弥散模型及其应用

随着我国核能及核技术利用产业的迅速发展,放射性物质运输对饮用水水源地的潜在放射性污染健康风险越来越大,阐明事故情况下放射性物质在地表水体中的迁移弥散行为可为居民饮水健康风险防范提供科学依据。针对含钴-60、铯-137、锶-90货包公路运输在事故工况下放射性物质释放造成黄河兰州段河流型饮用水源放射性污染,讨论了河流放射性物质弥散模型和居民饮用水的健康风险。结合国内现有放射源活度水平及国内外放射性物质事故资料,考虑泥沙的影响,设定铯-137不同活度水平、不同流量情形,根据饮用水中铯-137的死亡风险系数和患病风险系数对距离事故点为64 km、114 km的河流下游两个饮用水源地居民饮用受污染自来水存在的健康风险进行评估,同时选择钴-60、锶-90两种核素以30年间河水平均流量偏小值为例进行对比评估。放射性核素在黄河河流的稀释作用下,由饮用受钴-60、铯-137、锶-90污染的水随放射源活度、河水流量及泥沙含量的变化存在不同程度的死亡风险与患病风险。因此,应当对受到放射性污染的水源及时采取适当措施,预防放射性核素造成的风险。     

分子表面积的精确和经验计算及其在QSAR中的应用

本文列出了精确计算分子表面积的公式，在分子几何结构优化的基础上，可用此式得到准确的分子表面积，用该法对２０种氯代酚进行了计算，所得分子表面积能很好地预言其正辛醇／水分配系数，并且用分子表面积（或取代基数）加上ＯＨ基的表面积，能很好预言其对发光菌的产性。此外，还提出了计算公子表面积的经验方法－碎片加和法，此法应用于某些取代烃化合物，也得到正辛醇／水分配系数很好的相关性。     

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.     

Stochastic theory of ecological interactions

It is possible, using the distribution of random numbers of random variables, to derive bimodal vertical distributions of phytoplankton biomass similar to those observed in nature. On the basis of the distribution of pseudorandom events, equations for the probability of individual mortality from one of several reasons, and for the capture of one of several preys by the predators were obtained. From these theoretical formulations the empirical equations of Beverton and Holt, Ricker and Ivlev were derived. An equation for the average feed intake of one individual interacting in a swarm, feeding on aggregated food, is derived from an estimate of the average interval between the random encounters of aggregates, in the nodes of an α-dimensional net.