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.     

Estimation of the Longitudinal Dispersion Coefficient Using the Velocity Profile in Natural Streams

In this study, a theoretical method for predicting the longitudinal dispersion coefficient is developed based on the transverse velocity distribution in natural streams. Equations of the transverse velocity profile for irregular cross sections of the natural streams are analyzed. Among the velocity profile equations tested in this study, the beta distribution equation, which is a probability density function, is considered to be the most appropriate model for explaining the complex behavior of the transverse velocity structure of irregular natural streams. The new equation for the longitudinal dispersion coefficient that is based on the beta function for the transverse velocity profile is developed. A comparison of the proposed equation with existing equations and the observed longitudinal dispersion coefficient reveals that the proposed equation shows better agreement with the observed data compared to other existing equations.     

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.     

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.     

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.     

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.     

Analysis and Prediction of Transverse Mixing Coefficients in Natural Channels

An experimental study has been undertaken using a naturally formed meandering channel to obtain unique tracer and hydrodynamic data. The velocity data presented are from laser Doppler anemometer measurements; tracer data were collected using an array of fluorometers in continuous flow through mode. Techniques for the prediction of the primary and secondary velocity flow fields are explored, and shown to be accurate. Analysis of the tracer data by the generalized method of moments using the cumulative transverse discharge approach is undertaken. The coefficient of transverse mixing is shown to exhibit considerable longitudinal variation over the meander cycle. A new integrated approach for predicting transverse mixing coefficients is developed and explored and has been validated against the data set. This approach requires only three parameters as input, namely, longitudinal planform curvature, cross-sectional shape, and total discharge, and has been shown capable of accurately predicting the longitudinal variation of the transverse mixing coefficient over the meander cycle.     

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.     

Dispersion Coefficient of Streams from Tracer Experiment Data

The paper deals with a method for optimal identification of the dispersion coefficient for streams from observed concentration profiles at downstream sections (at least two sections are required) following injection of an environmentally safe tracer at an upstream section. The method makes use of the exact solution of the one-dimensional advection-dispersion equation. A reliable and accurate procedure is proposed for routing an arbitrary concentration variation further downstream using the convolution equation and pulse kernels. The new method does not require the frozen cloud approximation and avoids the error due to numerical integration of the convolution integral used for routing the concentration. It employs the objective criterion of minimum integral squared errors between observed and computed concentrations. Application of the method to field data sets shows that reliably accurate estimates of the dispersion coefficient are obtained.     

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.     

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.     

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.     

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.     

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.     

Investigation on the Suitability of Two-Dimensional Depth-Averaged Models for Bend-Flow Simulation

A numerical experiment is carried out to study the suitability of two-dimensional (2D) depth-averaged modeling for bend-flow simulation, in which the geometry of the studied channel is rectangular. Two commonly used 2D depth-averaged models for bend-flow simulation are considered in this study of which the bend-flow model includes the dispersion stress terms by incorporating the assumption of secondary-current velocity profile, and the conventional model neglects the dispersion stress terms. The maximum relative discrepancy of the longitudinal velocity, obtained from the comparison of these two models, is used as a criterion to judge their applicability for bend-flow simulation. The analysis of simulation results indicated that the maximum relative difference in longitudinal velocity is mainly related to the relative strength of the secondary current and the relative length of the channel. Empirical relations between the maximum relative difference in the longitudinal velocity, the relative strength of the secondary current, and the relative length of the channel for both the channel-bend region and the straight region following the bend have been established. The proposed relations provide a guideline for model users to determine the proper approach to simulate the bend-flow problem by either using the conventional model or the bend-flow model. Experimental data have been adopted herein to demonstrate the applicability and to verify the accuracy of the proposed relations.     

Diffusive Behavior of Bedform-Induced Hyporheic Exchange in Rivers

Solute transport in natural streams is a complex phenomenon that involves both in-stream dispersion and mass exchange with the porous zones surrounding the water body. Due to the complex nature of the riverine systems several models may be used to simulate and analyze the transport of solutes with different degrees of complexity. The bedform-induced hyporheic transport is a stream-subsurface exchange mechanism that can be reproduced in controlled systems, such as laboratory flumes. Application of a simple Fickian diffusion model to laboratory data obtained with passive solutes and stationary bedforms proves successful within a range of durations of the contamination process. A dimensionless form of the diffusion coefficient, scaled with dynamic, physical, and geometric properties of the system is derived by comparison with another physically based model. A prediction of the dimensionless diffusion coefficient is obtained as a function of the timescale of the exchange process and is validated with a few sets of results from laboratory tests.     

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.     

Master Diagnostic Curve for Dispersion Coefficient of Soils

A master diagnostic curve (MDC) method is proposed for identifying the dispersion coefficient from observed breakthrough curve in soil–column experiments. The method uses matching of diagnostically plotted points to the MDC. Accurate identification of the dispersion coefficient is possible with a parallel shift of only one axis. Another advantage of the method is that the MDC is an invariant curve and once prepared, can be used for different data sets characterized by high Péclet number. The new method is simple, does not require large computations as in conventional least-squares approach, and yields a quick and accurate estimate of the dispersion coefficient. The new method does not require an initial guess for the dispersion coefficient. It can also yield a value of the dispersion coefficient from a single observed concentration. Although subjectivity is involved in matching, there is a transparent visual realization of the reliability of the estimated dispersion coefficient, and the points with substantial errors. The new method has several advantages over the least-squares approach and is suited for an advanced study on the subject. An estimate of the dispersion coefficient obtained using the new method is as good as obtained using a good optimization method.     

The conductance of the muscle nicotinic receptor channel changes rapidly upon gating

We have recorded single-channel currents through fetal-type muscle nicotinic receptor channels at recording bandwidths of approximately 50 and 75 kHz. The time course of the rising phase of aligned and averaged openings can be entirely accounted for if it is assumed that the conductance of the single channel changes instantaneously, and that alignment and averaging introduce a dispersion of 2-3 microseconds. We conclude that we find no evidence for a gradual change in conductance as a channel opens or closes. The shapes of averaged power spectra are consistent with this conclusion, insofar as they exclude an exponential relaxation in the transition with a time constant of 10 microseconds or more.     

New Methods for Aquifer Parameters from Slug Test Data

An objective method and a diagnostic curve method are developed for estimating the aquifer parameters (storage coefficient and transmissivity) from slug test data on a fully penetrating well. In the objective method, explicit equations are developed for estimating the aquifer parameters. In the diagnostic curve method, a set of unimodal diagnostic curves is developed along with a guiding straight line. The rise or fall in water level of the well is plotted diagnostically on a double logarithmic graph and matched to one of the diagnostic curves plotted on the same scale with a parallel shift of axes, to estimate the aquifer parameters from the dual coordinates of a selected point on the matched portion of the graphs. The unimodal shape of the diagnostic curves and the guiding straight line facilitate the matching and limit the subjectivity. The proposed methods can easily identify a nonideal condition. The estimates of the aquifer parameters obtained using the proposed methods are more accurate than those obtained using the prior curve matching methods. The proposed methods are also able to identify nonideal conditions. It is hoped that the new methods will be of help to field and practicing engineers.