Longitudinal Dispersion Coefficient in Straight Rivers

An analytical method is developed to determine the longitudinal dispersion coefficient in Fischer's triple integral expression for natural rivers. The method is based on the hydraulic geometry relationship for stable rivers and on the assumption that the uniform-flow formula is valid for local depth-averaged variables. For straight alluvial rivers, a new transverse profile equation for channel shape and local flow depth is derived and then the lateral distribution of the deviation of the local velocity from the cross-sectionally averaged value is determined. The suggested expression for the transverse mixing coefficient equation and the direct integration of Fischer's triple integral are employed to determine a new theoretical equation for the longitudinal dispersion coefficient. By comparing with 73 sets of field data and the equations proposed by other investigators, it is shown that the derived equation containing the improved transverse mixing coefficient predicts the longitudinal dispersion coefficient of natural rivers more accurately.     

Transverse Dispersion Caused by Secondary Flow in Curved Channels

A new theoretical equation is proposed to describe the streamwise variations of the transverse velocity along a curved channel with a constant curvature. Furthermore, based on this theoretical equation for the transverse velocity, a new equation for the transverse dispersion coefficient is developed to incorporate the effect of the secondary flow on the transverse dispersion in curved channels. The new equations for the transverse velocity and dispersion coefficient are verified with experimental data sets that were obtained from laboratory experiments conducted in two different channels. The results show that the proposed velocity equation properly describes the streamwise variations of the secondary flow developed in the curved channels. The reach-averaged values of the transverse dispersion coefficient calculated by the new equation are in relatively good agreement with the observed values from the laboratory channels. Sensitivity analysis reveals that both the secondary flow and the transverse dispersion coefficient are proportional to the roughness factor, and in inverse proportion to the aspect ratio of the channel.     

Longitudinal Dispersion Coefficient in Single-Channel Streams

Using a new channel shape equation for straight channels and a more versatile channel shape or local flow depth equation for natural streams a method is developed for prediction of the longitudinal dispersion coefficient in single-channel natural streams, including straight and meandering ones. The method involves derivation of a new triple integral expression for the longitudinal dispersion coefficient and development of an analytical method for prediction of this coefficient in natural streams. The proposed method is verified using 70 sets of field data collected from 30 streams in the United States ranging from straight manmade canals to sinuous natural rivers. The new method predicts the longitudinal dispersion coefficient, where more than 90% calculated values range from 0.5 to 2 times the observed values. The advantage of the new method is that it is capable of accurately predicting the longitudinal dispersion coefficient in single-channel natural streams without using detailed dye concentration test data. A comparison between the new method and the existing methods shows that the new method significantly improves the prediction of the longitudinal dispersion coefficient.     

Evaluation of Dispersion Coefficients in Meandering Channels from Transient Tracer Tests

Mixing characteristics of conservative pollutants were examined two-dimensionally in a laboratory meandering channel, and a method to compute the dispersion coefficients was developed based on the measured concentration data. To investigate how the hydrodynamics influences pollutant mixing in meandering channels, both flow and tracer experiments were conducted in an S-curved laboratory channel. A two-dimensional routing procedure was presented to evaluate the longitudinal dispersion coefficient as well as the transverse dispersion coefficient under the unsteady concentration condition. The results of the tracer experiments showed that the tracer cloud behaved quite differently depending on whether or not the tracer cloud was transported following the filament of maximum velocity. Also, separation and reemerging of the tracer cloud were promoted by secondary currents. The observed transverse dispersion coefficients obtained by the routing procedure were close to those obtained by existing moment methods. The transverse dispersion coefficient tended to increase with an increasing aspect ratio, whereas it is not sensitive to the injection location. However, the longitudinal dispersion coefficient was sensitive to the injection location as well as the aspect ratio.     

2D Modeling of Heterogeneous Dispersion in Meandering Channels

Two types of dispersion coefficient tensor for meandering channels were examined. The first type was estimated using measured vertical velocity profile in an S-curved channel, and the second type was based on the depth-averaged velocity field. A Petrov-Galerkin type finite element scheme was used in the numerical modeling, and the simulation results were compared with the experimental results from tracer tests in an S-curved channel. Comparison of the results show that the dispersion coefficient tensor obtained directly from velocity profiles provided a more realistic solution that can describe the abrupt expansion of tracer clouds in the transverse direction. Heterogeneous longitudinal and transverse dispersion coefficients were inversely estimated from the calculated dispersion coefficient tensor based on the velocity profiles. Extremely large transverse dispersion coefficients were formed at the apex of the channel bend, where there was a well-developed secondary current. The dimensionless transverse dispersion coefficient (DT/hu*) in the apex of the bend ranges from 0.495 to 2.60, which is about four times larger than that of the straight region.     

Predicting Longitudinal Dispersion Coefficient in Natural Streams by Artificial Neural Network

An artificial neural network (ANN) model was developed to predict the longitudinal dispersion coefficient in natural streams and rivers. The hydraulic variables [flow discharge (Q), flow depth (H), flow velocity (U), shear velocity (u*), and relative shear velocity (U/u*)] and geometric characteristics [channel width (B), channel sinuosity (σ), and channel shape parameter (β)] constituted inputs to the ANN model, whereas the dispersion coefficient (Kx) was the target model output. The model was trained and tested using 71 data sets of hydraulic and geometric parameters and dispersion coefficients measured on 29 streams and rivers in the United States. The training of the ANN model was accomplished with an explained variance of 90% of the dispersion coefficient. The dispersion coefficient values predicted by the ANN model satisfactorily compared with the measured values corresponding to different hydraulic and geometric characteristics. The predicted values were also compared with those predicted using several equations that have been suggested in the literature and it was found that the ANN model was superior in predicting the dispersion coefficient. The results of sensitivity analysis indicated that the Q data alone would be sufficient for predicting more frequently occurring low values of the dispersion coefficient (Kx<100?m2/s). For narrower channels (B/H<50) using only U/u* data would be sufficient to predict the coefficient. If β and σ were used along with the flow variables, the prediction capability of the ANN model would be significantly improved.     

Elastoplastic 3D Deformation and Stress Analysis of Strip Rolling

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Effect of Applying Different Distribution Shapes for Velocities and Pressure on Simulation of Curved Open Channels

Most of the computational models of curved open channel flows use the conventional depth averaged De St. Venant equations. De St. Venant equations assume uniform velocity and hydrostatic pressure distributions. They are thus applicable only to cases of meandering rivers and curved open channels where vertical details are not of importance. The two-dimensional vertically averaged and moment equations model, developed by the writers, is used to study the effect of applying different distribution shapes for velocities and pressure on the simulation of curved open channels. Linear and quadratic distribution shapes are proposed for the horizontal velocity components, while a quadratic distribution shape is considered for the vertical velocity. Linear hydrostatic and quadratic nonhydrostatic distribution shapes are proposed for the pressure. The proposed model is applied to problems involved in curved open channels with different degrees of curvature. The implicit Petrov–Galerkin finite element scheme is applied in this study. Computed values for depth averaged longitudinal and transverse velocities across the channel width and vertical profiles of longitudinal and transverse velocities are compared to the observed experimental data. A fairly good agreement is attained. Predictions of overall flow characteristics suggest that the results are not very sensitive to different approximations of the preassumed applied velocity and pressure distribution shapes.     

Hybrid-Cells-in-Series Model for Solute Transport in Streams and Relation of Its Parameters with Bulk Flow Characteristics

The conceptualized hybrid-cells-in-series model, consists of a plug flow zone and two thoroughly mixed unequal reservoirs, all connected in series, has three time parameters, namely: (1) residence time of solute in the plug flow zone; and (2) residence times of solute in the two thoroughly mixed reservoirs. The model simulates closely advection-dispersion solute transport in natural streams. The resident time parameters are related to the velocity of flow, width of water surface, and depth of flow in the stream. Through the Péclet number, defined as Pe = (Δxu)/DL (in which Δx=process unit size; u=mean flow velocity; and DL=longitudinal dispersion coefficient), the relations of the model parameters with the longitudinal dispersion coefficient and with the bulk stream flow characteristics have been established. For a given reach of a stream, the parameters are inversely proportional to the flow velocity. By decoupling of pure advection by the plug flow component and dispersion of tracer by the two thoroughly mixed reservoir components, a robust fitting to the observed concentration-time data in natural streams was achieved.     

Solute Transport Modeling in Overland Flow Applied to Fertigation

A model of solute transport in overland flow is developed and applied to the simulation of surface fertigation. Water flow is simulated using the depth-averaged, 1D shallow water equations. Solute flow is represented by an advection-diffusion model. The resulting set of three partial differential equations is sequentially solved at each time step. First, water flow is computed using the explicit two-step McCormack method. Based on the obtained velocity field, solute transport is explicitly determined from the advection-diffusion equation using the operator split technique. Four field experiments involving fertigation events on an impervious free-draining border were performed to validate the proposed model and to obtain estimates of Kx, the longitudinal dispersion coefficient. A value of Kx = 0.075 m2 s?1 satisfactorily reproduces the field experiments. The model is also applied to the simulation of a fertigation event on a pervious border. A sensitivity analysis is performed to assess the dependence of fertilizer distribution uniformity on the value of Kx. Finally, the proposed model is compared with a previous model based on pure advection.     

Moment-Based Calculation of Parameters for the Storage Zone Model for River Dispersion

Theoretical methods have been developed to calculate values of parameters of the storage zone model for river mixing. Analytical solutions of the Laplace-transformed equations of the storage zone model are related to the observed concentration distribution in order to determine model parameters in both the moment matching method and the maximum likelihood method, which were developed in this study. The results obtained by comparison with experimental data show that the parameters calculated by the moment matching method are in good agreement with the observed values of storage zone model parameters, whereas results from the maximum likelihood method and several existing methods are not in good agreement with the experimentally observed values. Dispersion data from natural streams show that the calculated concentration curves from the numerical solutions of the storage zone model with the parameters calculated by the moment matching method fit the observed concentration curves very well. It can be concluded that parameters of the storage zone model calculated using the moment matching method can properly explain the natural dispersion processes in real streams.     

Dispersion in Varying-Geometry Rivers with Application to Methanol Releases

Most analyses of turbulent mixing in rivers assume constant hydraulic geometry (width, depth, and velocity), despite the fact that in natural rivers these variables typically increase downstream. A comprehensive set of data for the rivers and streams in the United States is used to derive generalized equations for variations in hydraulic geometry. As a preliminary investigation of the importance of these variations, an approximate analytical solution to the one-dimensional advective-dispersion equation is derived for rivers with variable velocity, cross-sectional area, and dispersion coefficient. The solution compares well with previous analyses, and is used to assess the potential environmental impacts of methanol releases into a hypothetical river. The resulting downstream concentrations of methanol are considerably lower than those calculated assuming constant hydraulic geometry.     

Treatment of Stagnant Zones in Riverine Advection-Dispersion

Advection-dispersion in streams encounters pockets of stagnant or dead zones in the flow, which trap the injected tracer. Treatment of stagnant or dead zones for dispersion is presented using one-dimensional advection-dispersion equation. A method is suggested for simultaneous estimation of dispersion coefficient, apparent (or effective) velocity, and effective injected mass of tracer, from a temporal concentration profile observed at a downstream section. The method is robust and uses a nonlinear optimization. Using the method procedure for estimation of adsorption coefficient for riverine advection-dispersion has also been suggested. The effective velocity is related to the stagnant zone fraction (average fraction of cross-sectional area attributed to stagnant zones) and adsorption. The application of the method on published data sets show that the parameter-estimates are reliable and the observed concentration profiles are closely reproduced. The analytical procedure described for the treatment of stagnant zones may have a wide application in civil engineering as well as other fields. The amount of chemicals released from the industrial units or by an accident can be estimated.     

Effects of Velocity Gradients and Secondary Flow on the Dispersion of Solutes in a Meandering Channel

Two contrasting mechanisms, created by channel curvature which strongly affect longitudinal dispersion of solutes in rivers are examined. In natural channels the large cross-sectional variability of the primary velocity component tends to increase longitudinal dispersion by providing a large difference between adjacent fast and slow moving zones of fluid. By contrast secondary circulation tends to decrease longitudinal dispersion by enhancing transverse mixing. A series of tests have been carried out in a very large flume containing a meandering water-formed sand bed channel to measure the longitudinal dispersion coefficient at various locations around a meander. These experimental observations are compared with experimental data obtained from meandering channels with smooth, fixed sides and regular cross-sectional shapes. All the data has been compared against predictions from two current modeling approaches. Finally, the significance of the two competing mechanisms in curved channels is discussed with regard to their relative influence on longitudinal mixing.     

Eulerian–Lagrangian Method for Constituent Transport in Water Distribution Networks

The transport and mixing of contaminants in conduits is governed by advection, dispersion, and decay. Several models are available to trace the transport of such constituents and most assume that the principal mechanisms for transport are advection and reaction only. However in pipes where low velocities prevail, longitudinal dispersion is significant and models that neglect the dispersion effects fail to properly simulate the observed concentrations in low velocity pipes. This work presents a method for simulating the advection-dispersion-reaction process of constituent transport in water networks. A Eulerian–Lagrangian method is employed whereby the dispersion term in the governing equation is approximated using finite differences and the resulting first-order partial differential equation is then integrated using the method of characteristics. Analytical solutions of the transport equation are also derived to quantify the effect of neglecting dispersion at pipe junctions and to assess the accuracy of the proposed method. The Eulerian-Lagrangian method is tested on benchmark networks and on the field study at the Cherry Hill/Brushy Plains network. Results show that the model developed is capable of simulating transport with equal accuracy for low and high velocity flows with and without significant dispersion effects. It also performs better than other models because of the nonuniform grid distribution and the interpolation schemes used.     

Dispersion in Sediment-Laden Stream Flow

A theory is developed to demonstrate the effects of sorptive exchange on the transport of a chemical in a sediment-laden open-channel flow. Based on the multiple-scale method of homogenization, a depth-averaged transport equation is deduced up to a long time-scale. The dispersion coefficient is the sum of a modified Taylor dispersion coefficient and a dispersion coefficient due to a finite rate of mass exchange between dissolved phase in the water column and sorbed phase on suspended particles. These coefficients are functions of the suspension number and the bulk solid-water distribution ratio. It is shown that, for sufficiently large particles and solid fractions, enhancement of the longitudinal dispersion coefficient due to the sorptive exchange can be significant and should be included in a comprehensive model.     

Flow and Velocity Distribution in Meandering Compound Channels

An investigation of flow and velocity distribution in meandering compound channels with over bank flow is described. Equations concerning the three-dimensional variation of longitudinal, transverse, and vertical velocity in the main channel and floodplain of compound section in terms of channel parameters are presented. The flow and velocity distributions in meandering compound channels are strongly governed by interaction between flow in the main channel and that in the floodplain. The proposed equations take adequate care of the interaction affect. Results from the formulations, simulating the three-dimensional velocity field in the main channel and in the floodplain of meandering compound channels are compared with their respective experimental channel data obtained from a series of symmetrical and unsymmetrical test channels with smooth and rough sections. The aspect ratio of the test channels varies from two to five. The equations are found to be in good agreement with the experimental data. The formulations are verified against the natural river and other meandering compound channel data. The power laws used for simulating the three-dimensional velocity structure are found to be quite adequate.     

Empirical Relations for Longitudinal Dispersion in Streams

Although several methods are available for dispersion in natural streams, no method is accurate enough to satisfactorily predict the time variation of stream pollution concentration. Further, limited studies exist for dispersion of nonconservative pollutants. In this paper a six-parameter concentration equation for dispersion of conservative and nonconservative pollutants has been proposed. The parameters of the equation have been related to hydraulic variables and stream geometry. Using these predictors, the equation is fairly accurate for concentration predictions. It is hoped that the equation is useful in water quality management studies.     

Finite Difference Method for Computation of 1D Pollutant Migration through Saturated Homogeneous Soil Media

The physical processes such as advection, dispersion, and diffusion and interaction between the solution and the soil solids such as sorption, biodegradation, and retention processes have been considered in the governing equation used in the present study. Finite difference method has been adopted herein to solve the one-dimensional contaminant transport model to predict the pollutant migration through soil in waste landfill. In the finite difference technique, the velocity field is first determined within a hydrologic system, and these velocities are then used to calculate the rate of contaminant migration by solving the governing equation. A total of seven contaminants have been chosen for analysis to represent a wide variety of wastes both organic and inorganic. A computer software CONTAMINATE has been developed for solution of the contaminant transport model. Results of this study have been compared with existing analytical solution for validation of the proposed solution technique. Design charts for liners have also been developed to facilitate the designers. The liner thickness has been optimized by considering the effect of velocity of advection, dispersion coefficient, and geochemical reactions for all the contaminants of this study.     

Discrete Particle Distribution Model for Advection-Diffusion Transport

A new methodology, named DisPar, based on a discrete probability distribution for a particle displacement, was developed to solve 1D advection-diffusion transport problems in water bodies. The discrete probability distribution for the particle displacement was developed as an average and variance function. These probabilities were used to predict the deterministic mass transfer between cells in one time step, and therefore the particle concentration in each cell was considered the state variable. The state equation was found to be similar to an explicit finite-difference formulation with a Eulerian grid. The model stability, positivity, and mass conservation are guaranteed by the probability distribution concept. DisPar produces solutions without numerical dispersion for constant velocity, diffusion coefficient, and cross-sectional area. In these conditions, DisPar was also developed as a function of space and time for an instantaneous mass spill. When the stability and positivity restrictions were respected, the model produced excellent results when compared to analytical solutions and other methods. The discrete particle displacement distribution concept differs from other numerical formulations, and therefore it represents a new modeling technique.