Simulation of Flow and Mass Dispersion in Meandering Channels

This paper reports the development of an enhanced two-dimensional (2D) numerical model for the simulation of flow hydrodynamics and mass transport in meandering channels. The hydrodynamic model is based on the solution of the depth-averaged flow continuity and momentum equations where the density of flow varies with the concentration of transported mass. The governing equation for mass transport model is the depth-averaged convection and diffusion equation. The dispersion terms arisen from the integration of the product of the discrepancy between the mean and the actual vertical velocity distribution were included in the momentum equations to take into account the effect of secondary current. Two laboratory experimental cases, flow in mildly and sharply curved channels, were selected to test the hydrodynamic model. The comparison of the simulated velocity and water surface elevation with the measurements indicated that the inclusion of the dispersion terms has improved the simulation results. A laboratory experiment study of dye spreading in a sine-generated channel, in which dye was released at the inner bank, centerline, and outer bank, respectively, was chosen to verify the mass transport model. The simulated concentration field indicated that the Schmidt number can be used as a calibration parameter when dispersion is computed using a 2D approach with a simplified turbulence model.     

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.     

Three-Dimensional CFD Modeling of Self-Forming Meandering Channel

A three-dimensional CFD model was used to compute the formation of the meandering pattern in an initially straight alluvial channel. The numerical model was based on the finite volume method using an unstructured grid with dominantly hexahedral cells. The k-ε model was used to predict turbulence and the SIMPLE method was used to compute the pressure. The sediment transport was computed as bed load in addition to solving the convection-diffusion equation for suspended sediment transport. The bed changes were calculated and the grid was altered during the computation as channel erosion and deposition caused wetting and drying. The model was tested by comparing with results from physical model studies carried out at Colorado State Univ., Fort Collins, Colo. The results showed successfully the replication of many of the meander characteristics, including secondary currents, cross-sectional profiles, meander planform, meander wavelength, downstream meander migration, and chute formation.     

Modeling Bed Changes in Meandering Rivers Using Triangular Finite Elements

A two-dimensional depth-averaged model was used for the simulation of scour and deposition in sand-bed meandering channels with fixed banks. The model employs unstructured meshes based on triangular elements and incorporates the effects of curvature-induced helical flow and transverse bed slope in the direction of bed-load sediment transport. The model was tested using experimental data from a well-known laboratory curved channel and a full scale meandering river. The numerical results agreed well with observed data, demonstrating that the model can reproduce the main features of bed profiles along meandering rivers, such as the formation of point bars and pools.     

Mathematical Modeling of Meandering Channels with a Generalized Depth Averaged Model

A two-dimensional depth averaged model is developed in a nonorthogonal curvilinear coordinate system. The ability of the model in handling complex mesh arrangements with high aspect ratios and skewness is investigated. It is shown that model formulation makes it able to handle large skewness in the grid lines. It is also shown that in predicting the water surface profile with a very distorted mesh, most of the errors arise from the large aspect ratio rather than the skewness of the grid lines. The model is then applied to three meandering channels (two simple and one compound), with specifications similar to those found in nature and the results are compared with experimental data. The comparison shows that the model predicted the water surface profile and velocity distribution well in simple channels. Predictions of the model in the main channel of the compound meandering channel were also in general agreement with the experimental results.     

Behavior of Meandering Overbank Channels with Graded Sand Beds

Measurements of velocity distributions, depth variation, and sediment transport have been made under bankfull and overbank flow conditions in meandering channels with a graded sand bed, using the large-scale U.K. Flood Channel Facility. The overbank conditions depend upon the relative strength of opposing secondary circulation cells generated by shear at the channel crossover and centrifugal forces around the meander bend. Generally the shear-generated secondary flow either reversed or weakened the centrifugal circulation around the next downstream bend. This led to considerable modification of the main channel bed morphology, which, in turn, altered flow distributions. Measurements of the lateral distribution of bed load were made using a ?-scale Helley–Smith sampler. This demonstrated that the bed load was generally concentrated within a limited width of the channel and tended to take the shortest route through the meanders. Comparisons of observed and calculated bed material load gives an indication of how secondary circulation around meanders, under both bankfull and overbank conditions, affects the predictive performance of formulas derived for predominantly one-dimensional flow.     

Validation of a Three-Dimensional Numerical Code in the Simulation of Pseudo-Natural Meandering Flows

Validation of a three-dimensional finite volume code solving the Navier–Stokes equations with the standard k-ε turbulence model is conducted using a high quality and high spatial resolution data set. The data set was collected from a large-scale meandering channel with a self-formed fixed bed, and comprises detailed bed profiling and laser Doppler anemometer velocity measurements. Comparisons of the computed primary and secondary velocities are made with those observed and it is found that the lateral momentum transfer is generally under predicted. At the apices this results in the predicted position of the primary velocity maximum having a bias towards the channel center, compared to the position where it has been measured. Using a simplified two zone roughness distribution whereby a separate roughness height was prescribed for the channel center and channel sides relative to a single distributed roughness height, generally led to a slightly improved longitudinal velocity distribution; the higher velocities were located nearer to the outside of the bend. Improving both the free surface calculation and scheme for discretization of the convection terms led to no appreciable difference in the computed velocity distributions. A more detailed study involving turbulence measurements and bed form height distribution should discriminate whether using distributed roughness height is a precursor to using an anisotropic turbulence representation for the accurate prediction of three-dimensional river flows.     

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.     

Numerical Modeling of Local Scour below a Piggyback Pipeline in Currents

Local scour below a piggyback pipeline in steady currents is investigated numerically. A piggyback pipeline comprises two pipelines that are arranged in the so-called piggyback configuration with the small pipeline being located directly above the large pipeline. The Reynolds-averaged Navier–Stokes equations and the transport equation for suspended sediment concentration are solved using a finite element method. The bed scour profile is determined through solving sediment mass conservation equation. The numerical model is validated against experimental data available in literature on scour below a single pipeline. Computations are carried out for the diameter ratio [the small pipe diameter (d) to the larger one (D)] of 0.2 and the gap (G, between the two pipelines) to the large diameter ratio G/D ranging from 0.0 to 0.5. It is found that the flow and the scour profiles are influenced significantly by the gap ratio.     

Numerical Modeling of Bed Deformation in Laboratory Channels

A depth-average model using a finite-volume method with boundary-fitted grids has been developed to calculate bed deformation in alluvial channels. The model system consists of an unsteady hydrodynamic module, a sediment transport module and a bed-deformation module. The hydrodynamic module is based on the two-dimensional shallow water equations. The sediment transport module is comprised of semiempirical models of suspended load and nonequilibrium bedload. The bed-deformation module is based on the mass balance for sediment. The secondary flow transport effects are taken into account by adjusting the dimensionless diffusivity coefficient in the depth-average version of the k–ε turbulence model. A quasi-three-dimensional flow approach is used to simulate the effect of secondary flows due to channel curvature on bed-load transport. The effects of bed slope on the rate and direction of bed-load transport are also taken into account. The developed model has been validated by computing the scour hole and the deposition dune produced by a jet discharged into a shallow pool with movable bed. Two further applications of the model are presented in which the bed deformation is calculated in curved alluvial channels under steady- and unsteady-flow conditions. The predictions are compared with data from laboratory measurements. Generally good agreement is obtained.     

Variation of Flow Pattern with Sinuosity in Sine-Generated Meandering Streams

The effect of channel sinuosity on flow pattern in meandering streams is investigated. The centerlines of the idealized meandering streams under consideration follow sine-generated curves, and the banks are rigid; the flow is turbulent and subcritical. This study focuses on the vertically averaged flow over a flat (horizontal at any cross section) bed formed by a granular material. The “flat bed” is viewed as the initial surface of a moveable bed at the beginning of an experiment (at time t = 0). A series of laboratory flow measurements involving the systematic variation of the deflection angle θ0 from 30 to 110°, is used. It is found that every different sinuosity (every different θ0) has its own convective flow pattern, i.e., its own distribution in plan of (the L/2 long) convergence–divergence zones of flow. As θ0 increases, a gradual change in flow pattern is observed. Two expressions defining the observed θ0 variation of the convective flow pattern are introduced. It is shown, with the aid of the sediment transport continuity equation, that the geometry of the developed bed at the end of an experiment is strongly related to the convective behavior of the vertically averaged (initial) flow over the flat bed at t = 0. In particular, information on the θ0 variation of the convective pattern of the initial flow can be used to estimate the location of erosion–deposition zones and the location(s) of the most intense erosion–deposition corresponding to any θ0.     

Experimental Observations of Flow and Bed Processes in Large-Amplitude Meandering Flume

Meanders of large amplitude often exhibit asymmetric planform shape or subsidiary bends. The present work is aimed at improving on understanding of the morphodynamic phenomena affecting the bed evolution of large amplitude meandering channels. Attention is focused on the development of the steady point bar-pool configuration and of the superimposed large-scale migrating bed forms; of particular interest is the role of the changing channel curvature and bed topography variation on flow pattern. A series of experiments was carried out in a sine-generated large-amplitude meandering flume, for two values of width-to-depth ratio. Maps documenting the bed topography and the flow pattern along the meandering bends are reported. Two point bars per bend were observed and seem to be part of a series of damped oscillations developing in response to the changing channel curvature. In response to the bed deformation, the maximum flow velocity moves at the outer bank at the entrance of the bend.     

Diffusive Modeling of Aggradation and Degradation in Artificial Channels

The unsteady flow and solid transport simulation problem in artificial channels is solved using a three-equation model, coupled with a local erosion law. The three equations are the water mass and momentum balance equations, as well as the total solid load balance equation. It is shown that even during severe hydrological events inertial terms can be neglected in the momentum equation without any substantial change in the solution sought. Empirical equilibrium formulas were used to estimate the solid load as a function of the flow variables. Local erosion, due to the scour generated at the jump between two channels connected at different bottom elevations, was estimated adapting a literature formulation. The double order approximation time and space marching scheme, previously proposed for the solution of the unsteady flow problem in the fixed-bed case, is applied to the solution of the new system. The model was validated with both literature and new laboratory experimental data. No parameter calibration was used to fit the computed results to the experimental ones.     

Modeling Noncohesive Suspended Sediment Transport in Stream Channels Using an Ensemble-Averaged Conservation Equation

The governing conservation equation for the transport of noncohesive suspended sediment in erodible channels is recognized as a stochastic partial differential equation due to the uncertainties in the parameters, and a deterministic ensemble-averaged equation is developed. Variables in this one-dimensional equation are represented as averaged quantities, and their covariances are also taken into account. Lateral inflows and deposition and entrainment of sediment are incorporated in the formulation. A hypothetical test problem is constructed to examine the model behavior. Manning’s coefficient, bed slope and bottom width are taken as the primary random parameters. Results from the solution of the ensemble-averaged equation are compared to results from Monte Carlo simulations. For comparison purposes, predicted values are also obtained by solving the deterministic transport equation without the covariance terms. It is found that predictions obtained from this latter approach deviate significantly from Monte Carlo simulation results. On the other hand, the ensemble-averaged predictions compare favorably to the Monte Carlo simulation results indicating that this promising technique needs further exploration.     

Depth-Averaged Two-Dimensional Numerical Modeling of Unsteady Flow and Nonuniform Sediment Transport in Open Channels

A depth-averaged two-dimensional (2D) numerical model for unsteady flow and nonuniform sediment transport in open channels is established using the finite volume method on a nonstaggered, curvilinear grid. The 2D shallow water equations are solved by the SIMPLE(C) algorithms with the Rhie and Chow’s momentum interpolation technique. The proposed sediment transport model adopts a nonequilibrium approach for nonuniform total-load sediment transport. The bed load and suspended load are calculated separately or jointly according to sediment transport mode. The sediment transport capacity is determined by four formulas which are capable of accounting for the hiding and exposure effects among different size classes. An empirical formula is proposed to consider the effects of the gravity on the sediment transport capacity and the bed-load movement direction in channels with steep slopes. Flow and sediment transport are simulated in a decoupled manner, but the sediment module adopts a coupling procedure for the computations of sediment transport, bed change, and bed material sorting. The model has been tested against several experimental and field cases, showing good agreement between the simulated results and measured data.     

Depth-Dependent Dispersion Coefficient for Modeling of Vertical Solute Exchange in a Lake Bed under Surface Waves

Variable pressure at the sediment/water interface due to surface water waves can drive advective flows into or out of the lake bed, thereby enhancing solute transfer between lake water and pore water in the lake bed. To quantify this advective transfer, the two-dimensional (2D) advection-dispersion equation in a lake bed has been solved with spatially and temporally variable pressure at the bed surface. This problem scales with two dimensionless parameters: a “dimensionless wave speed” (W) and a “relative dispersivity” (λ). Solutions of the 2D problem were used to determine a depth-dependent “vertically enhanced dispersion coefficient” (DE) that can be used in a 1D pore-water quality model which in turn can be easily coupled with a lake water quality model. Results of this study include a relationship between DE and the depth below the bed surface for W>50 and λ ? 0.1. The computational results are compared and validated against a set of laboratory measurements. An application shows that surface waves may increase the sediment oxygen uptake rate in a lake by two orders of magnitude.     

Three-Dimensional Modeling of Negatively Buoyant Flow in Diverging Channels

Diverging channels, also known as diffusers, represent common natural and industrial outlets to lakes, reservoirs, and rivers. If the outflow in a diffuser has a larger density than the ambient water, the inflow may plunge and form a density underflow. In this paper, a three-dimensional numerical study was conducted to gain insight into the mechanism of negatively buoyant flows in diffusers with a sloping bottom. Of particular interest is the formation of separated flows such as wall-jet and free-jet flows. Various cases of plunging and the associated density current in a diffuser with different divergence angles and inflow densimetric Froude numbers are considered. The model successfully simulates the formation of attached flow, wall jets, and free jets in a negatively buoyant environment. The onset, evolution, and stabilization of a stall and the subsequent development of a wall jet in a negatively buoyant flow are investigated in detail. Computed results also show favorable agreement with some published experimental data on density current generated by the plunging of cold water in ambient warm water in a diverging channel.     

On the Quantification of the Bed Development Time of Alluvial Meandering Streams

This paper concerns the quantitative evaluation of the time of bed development of alluvial meandering streams. In agreement with the prevailing approach, it is assumed that the stream centerline follows a sine-generated curve; the banks are rigid. The flow is turbulent and subcritical, and the flow width is much larger than the flow depth. The movable bed is flat at time t = 0; at t = Tb, the bed reaches its equilibrium or developed state. With the aid of dimensional and physical considerations, an expression is found for the duration of bed development Tb. According to this expression, Tb is proportional to the square of the flow width B and inversely proportional to the specific volumetric bed-load rate corresponding to the channel-averaged flow. The proportionality factor is found to be a function of the initial deflection angle θ0 alone. The form of this function is investigated on the basis of a series of experimental runs carried out by the writers in a sine-generated channel having an intermediate value of θ0 (i.e., an intermediate value of sinuosity), as well as data available in the literature.     

Dispersion Model for Varying Vertical Shear in Vegetated Channels

A dispersion model for a wide range of depthwise vertical shear is derived by using perturbation analysis and power (m) law velocity profile. For m = 1, the velocity profile provides linear shear, whereas m>1 provides nonlinear shear, and for m>20, the velocity profile resembles the flow through emergent vegetation. The power law represented parametrically simulates well the complex shear profiles involved in emergent and submerged vegetated flows. The proposed model shows reasonable agreement with past data on vegetated flows for a wide range of nonlinear shear velocities.     

3D Modeling of Ripping Process

Due to environmental constraints and limitations on blasting, ripping as a ground loosening and breaking method has become more popular than drilling and blasting method in both mining and civil engineering applications. The best way of estimating the rippability of rocks is to conduct direct ripping runs in the field. However, it is not possible to conduct direct ripping runs in all sites using different dozer types. Therefore, the utilization of numerical modeling of ripping systems becomes unavoidable. A complex ripping system can better be understood with three-dimensional (3D) models rather than two-dimensional models. In this study, 3D distinct element program called 3DEC was used to investigate the ripping process. First, the ripping mechanisms were investigated and then the individual factors that affect the rippability performance of dozers were reviewed. The rippabilities of rocks depend not only on the rock properties, but also machine or dozer properties. Thus, ripper production and rock rippability with D8 type of dozers were also determined by direct ripping runs on different open pit lignite mines within the scope of this research. Production values obtained from numerical modeling were compared with field production values obtained from the case studies. This comparison shows that the model gives consistent and adequate results. Hence, a link has been established between the field results and the 3D models.