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.     

Experimental and Numerical Investigation of Bed-Load Transport under Unsteady Flows

The dynamic behavior of bed-load sediment transport under unsteady flow conditions is experimentally and numerically investigated. A series of experiments are conducted in a rectangular flume (18?m in length, 0.80?m in width) with various triangular and trapezoidal shaped hydrographs. The flume bed of 8?cm in height consists of scraped uniform small gravel of D50 = 4.8??mm. Analysis of the experimental results showed that bed-load transport rates followed the temporal variation of the triangular and trapezoidal hydrographs with a time lag on the average of 11 and 30?s, respectively. The experimental data were also qualitatively investigated employing the unsteady-flow parameter and total flow work index. The analysis results revealed that total yield increased exponentially with the total flow work. An original expression which is based on the net acceleration concept was proposed for the unsteadiness parameter. Analysis of the results then revealed that the total yield increased exponentially with the increase in the value of the proposed unsteadiness parameter. Further analysis of the experimental results revealed that total flow work has an inverse exponential variation relation with the lag time. A one-dimensional numerical model that employs the governing equations for the conservation of mass for water and sediment and the momentum was also developed to simulate the experimental results. The momentum equation was approximated by the diffusion wave approach, and the kinematic wave theory approach was employed to relate the bed sediment flux to the sediment concentration. The model successfully simulated measured sedimentographs. It predicted sediment yield, on the average, with errors of 7% and 15% of peak loads for the triangular and trapezoidal hydrograph experiments, respectively.     

Three-Dimensional Modeling of Nonuniform Sediment Transport in an S-Shaped Channel

A three-dimensional numerical model was applied to compute uniform and nonuniform sediment transport and bed deformation in an S-shaped laboratory channel located at the University of Innsbruck, where detailed measurements of the velocity field and bed elevation changes were made. The channel had two bends, a trapezoidal cross section, and a slope of S = 0.005. Gravel with a mean diameter of 4.2?mm was used as movable bed material and for sediment feeding. Wu’s formula for multiple grain sizes was compared with van Rijn’s formula using one grain size. Fairly good agreement was found between the computed and measured bed elevations for both approaches, whereas Wu’s formula could further improve the numerical results. Looking at the physics of the erosion pattern, the computed scour areas were located slightly more downstream than what was observed in the physical model. The current study also includes several parameter tests: grid distribution in vertical, lateral, and longitudinal direction; time step; number of inner iterations/time step; active sediment layer thickness; and the Shields coefficient. The variation of those parameters gave some differences in the results, but the overall pattern of bed elevation changes remained the same.     

One-Dimensional Model for Transient Flows Involving Bed-Load Sediment Transport and Changes in Flow Regimes

This study presents a novel, simple, but rather accurate approximation of the eigenvalues of the system formed by the Saint-Venant–Exner equations, based on the comparison between eigenvalues for the complete system and eigenvalues for the water phase only. Moreover, a strategy is proposed to compute efficiently the intercell fluxes by properly adapting a Harten, Lax, and van Leer scheme for each equation. Two examples of transient transcritical flows are developed: the erosive migration of a knickpoint induced by an increase in the bed slope, and the evolution of a hydraulic jump over a mobile bed.     

Modeling of One-Dimensional, Unsteady, Free-Surface, and Pressurized Flows

The utilization of mathematical models in hydraulic engineering for the analysis of one-dimensional, unsteady free-surface, and pressurized flows is discussed, with an emphasis when the models performed well as well as when they did not. For illustration purposes, the applications to a number of real-life projects are presented, outlining limitations, successes, and failures.     

Three-Dimensional Numerical Study of Flows in Open-Channel Junctions

An open-channel junction flow is encountered in many hydraulic structures ranging from wastewater treatment facilities to fish passage conveyance structures. An extensive number of experimental studies have been conducted but a comprehensive three-dimensional numerical study of junction flow characteristics has not been performed and reported. In this paper, a three-dimensional numerical model is developed to investigate the open-channel junction flow. The main objective is to present the validation of a three-dimensional numerical model with high-quality experimental data and compare additional simulations with classical one-dimensional water surface calculations. The three-dimensional model is first validated using the experimental data of a 90° junction flow under two flow conditions. Good agreement is obtained between the model simulation and the experimental measurements. The model is then applied to investigate the effect of the junction angle on the flow characteristics and a discussion of the results is presented.     

Analysis of Dynamic Wave Model for Unsteady Flow in an Open Channel

The models for flood propagation in an open channel are governed by Saint-Venant’s equations or by their simplified forms. Assuming the full form of hyperbolic type nonlinear expressions, the complete or dynamic wave model is obtained. Hence, after first-order linearization procedure, the dispersion relation is obtained by using the classical Fourier analysis. From this expression, the phase and group speed and the variations of the amplitude of the waves are defined and investigated. Adopting Manning’s resistance formula, the effects of the variations of the Froude number, Courant number, and friction parameter are examined in the wave number domain for progressive and regressive waves. For small and high wave numbers, the simplified kinematic and gravity wave models are recovered, respectively. Moreover, the analysis confirms, according to the Vedernikov criterion, the Froude number value corresponding to the stability condition to contrast the development of roll waves. In addition, for stable flow on the group speed versus wave number curves, the results show critical points, maximum and minimum for progressive and regressive waves, respectively.     

Simulating Sediment Transport in a Flume with Forced Pool-Riffle Morphology: Examinations of Two One-Dimensional Numerical Models

One-dimensional numerical sediment transport models (DREAM-1 and DREAM-2) are used to simulate seven experimental runs designed to examine sediment pulse dynamics in a physical model of forced pool-riffle morphology. Comparisons with measured data indicate that DREAM-1 and -2 closely reproduce the sediment transport flux and channel bed adjustments following the introduction of fine and coarse sediment pulses, respectively. The cumulative sediment transport at the flume exit in a DREAM-1 simulation is within 10% of the measured values, and cumulative sediment transport at flume exit in a DREAM-2 simulation is within a factor of 2 of the measured values. Comparison of simulated and measured reach-averaged aggradation and degradation indicates that 84% of DREAM-1 simulation results have errors less than 3.3?mm, which is approximately 77% of the bed material geometric mean grain size or 3.7% of the average water depth. A similar reach-averaged comparison indicates that 84% of DREAM-2 simulation results have errors less than 7.0?mm, which is approximately 1.7 times the bed material geometric mean grain size or 11% of the average water depth. Simulations using measured thalweg profiles as the input for the initial model profile produced results with larger errors and unrealistic aggradation and degradation patterns, demonstrating that one-dimensional numerical sediment transport models need to be applied on a reach-averaged basis.     

Numerical Model for Sediment Transport and Bed Degradation in the Yangtze River Channel Downstream of Three Gorges Reservoir

This paper presents a two-dimensional morphological model for unsteady flow and both suspended-load and bed-load transport of multiple grain size to simulate transport of graded sediments downstream from the Three Gorges Reservoir. The model system includes a hydrodynamic module and a sediment module. The hydrodynamic module is based on the depth-averaged shallow water equations in orthogonal curvilinear coordinates. The sediment module describing nonuniform sediment transport is developed to include nonequilibrium transport processes, bed deformation, and bed material sorting. The model was calibrated using field observations through application to a 63-km-long alluvial river channel on the middle Yangtze River in China. A total of 16 size groups and a loose layer method of three sublayers were considered for the transport of the nonuniform bed materials in a long-term simulation. Predictions are compared with preliminary results of field observations and factors affecting the reliability of the simulated results are discussed. The results may be helpful to the development of more accurate simulation models in the future.     

Analysis of Suspended Sediment Transport in Open-Channel Flows: Kinetic-Model-Based Simulation

This paper presents a comprehensive analysis of suspended sediment transport in open channels under various flow conditions through a kinetic-model-based simulation. The kinetic model, accounting for both sediment-turbulence and sediment-sediment interactions, successfully represents experimentally observed diffusion and transport characteristics of suspended sediments with different densities and sizes. Without tuning any model coefficients, the nonmonotonic concentration distribution and the noticeable lag velocity with a negative value close to the wall are reasonably reproduced. Examination of flow conditions typical of suspension dominated rivers shows that the conventional method may overestimate or underestimate unit suspended-sediment discharge, depending on the Rouse number, sediment size, as well as shear velocity. The error may be less than 20% for dp<0.5?mm and might exceed 60% for dp>1.0?mm under typical flow conditions where shear velocity ranges from 1.0?to?12.5?cm/s and flow depth ranges from 1.0?to?5.0?m.     

Finite Volume Model for Two-Dimensional Shallow Water Flows on Unstructured Grids

A numerical model based upon a second-order upwind finite volume method on unstructured triangular grids is developed for solving shallow water equations. The HLL approximate Riemann solver is used for the computation of inviscid flux functions, which makes it possible to handle discontinuous solutions. A multidimensional slope-limiting technique is employed to achieve second-order spatial accuracy and to prevent spurious oscillations. To alleviate the problems associated with numerical instabilities due to small water depths near a wet/dry boundary, the friction source terms are treated in a fully implicit way. A third-order total variation diminishing Runge–Kutta method is used for the time integration of semidiscrete equations. The developed numerical model has been applied to several test cases as well as to real flows. Numerical tests prove the robustness and accuracy of the model.     

Local Time Stepping for Modeling Open Channel Flows

This paper presents two time accurate local time stepping (LTS) algorithms developed within aeronautics and develops the techniques for application to the Saint-Venant equations of open channel flow. The LTS strategies are implemented within an explicit finite volume framework based on using the Roe Riemann solver together with an upwind treatment for the source terms. The benefits of using an LTS approach over more traditional global time stepping methods are illustrated through a series of test cases, and a comparison is made between the two LTS algorithms. The results demonstrate how local time stepping can reduce computer run times, increase the reliability of the error control, and also increase the accuracy of the solution in certain regions.     

Numerical Modeling of Bed Evolution in Channel Bends

A two-dimensional numerical model is developed to predict the time variation of bed deformation in alluvial channel bends. In this model, the depth-averaged unsteady water flow equations along with the sediment continuity equation are solved by using the Beam and Warming alternating-direction implicit scheme. Unlike the present models based on Cartesian or cylindrical coordinate systems and steady flow equations, a body-fitted coordinate system and unsteady flow equations are used so that unsteady effects and natural channels may be modeled accurately. The effective stresses associated with the flow equations are modeled by using a constant eddy-viscosity approach. This study is restricted to beds of uniform particles, i.e., armoring and grain-sorting effects are neglected. To verify the model, the computed results are compared with the data measured in 140° and 180° curved laboratory flumes with straight reaches up- and downstream of the bend. The model predictions agree better with the measured data than those obtained by previous numerical models. The model is used to investigate the process of evolution and stability of bed deformation in circular bends.     

3D Unsteady RANS Modeling of Complex Hydraulic Engineering Flows. I: Numerical Model

A general-purpose numerical method is developed for solving the full three-dimensional (3D), incompressible, unsteady Reynolds-averaged Navier-Stokes (URANS) equations in natural river reaches containing complex hydraulic structures at full-scale Reynolds numbers. The method adopts body-fitted, chimera overset grids in conjunction with a grid-embedding strategy to accurately and efficiently discretize arbitrarily complex, multiconnected flow domains. The URANS and turbulence closure equations are discretized using a second-order accurate finite-volume approach. The discrete equations are integrated in time via a dual-time-stepping, artificial compressibility method in conjunction with an efficient coupled, block-implicit, approximate factorization iterative solver. The computer code is parallelized to take full advantage of multiprocessor computer systems so that unsteady solutions on grids with 106 nodes can be obtained within reasonable computational time. The power of the method is demonstrated by applying it to simulate turbulent flow at R ? 107 in a stretch of the Chattahoochee River containing a portion of the actual bridge foundation located near Cornelia, Georgia. It is shown that the method can capture the onset of coherent vortex shedding in the vicinity of the foundation while accounting for the large-scale topographical features of the surrounding river reach.     

Three-Dimensional Modeling of Sediment Transport in a Narrow 90° Channel Bend

A three-dimensional numerical model was used for calculating the velocity and bed level changes over time in a 90° bended channel. The numerical model solved the Reynolds-averaged Navier-Stokes equations in three dimensions to compute the water flow and used the finite-volume method as the discretization scheme. The k-ε model predicted the turbulence, and the SIMPLE method computed the pressure. The suspended sediment transport was calculated by solving the convection diffusion equation and the bed load transport quantity was determined with an empirical formula. The model was enhanced with relations for the movement of sediment particles on steep side slopes in river bends. Located on a transversally sloping bed, a sediment particle has a lower critical shear stress than on a flat bed. Also, the direction of its movement deviates from the direction of the shear stress near the bed. These phenomenona are considered to play an important role in the morphodynamic process in sharp channel bends. The calculated velocities as well as the bed changes over time were compared with data from a physical model study and good agreement was found.     

Influence of Vertical Resolution and Nonequilibrium Model on Three-Dimensional Calculations of Flow and Sediment Transport

This note, using a three-dimensional model of river flow and sediment transport, examines the effect of the vertical resolution and the choice a nonequilibrium adaptation length Ls in predicting flow and sediment transport around groins in China’s Yongding River. The results show that a fine vertical grid and nonequilibrium sediment transport model provide good predictions, especially on the river bed profile with an obvious main channel and flood plain.     

Applicability of Sediment Transport Capacity Formulas to Dam-Break Flows over Movable Beds

In this paper, we investigate the extent to which well-known sediment transport capacity formulas can be used in one-dimensional (1D) numerical modeling of dam-break waves over movable beds. The 1D model considered here is a one-layer model based on the shallow-water equations, a bed update (Exner) equation, a space-lag equation for the nonequilibrium sediment transport and an empirical formula calculating the sediment transport capacity of the flow. The model incorporates a variety of sediment transport capacity formulas proposed by Meyer-Peter and Müller, Bagnold, Engelund and Hansen, Ackers and White, Smart and Jaeggi, van Rijn, Rickenmann, Cheng, Abrahams and Camenen, and Larson. We examine the performance of each formula by simulating four idealized laboratory cases on dam-break waves over sandy beds. Comparisons between numerical results and measurements show that for each case better predictions are obtained using a particular formula, but overall, formulas proposed by Meyer-Peter and Müller (with the factor 8 being replaced by 12), Smart and J?ggi, Cheng, Abrahams and Camenen, and Larson rank as the best predictors for the entire range of conditions studied here. Moreover, results show that in the cases where a bed step exists, implementing a mass failure mechanism in the numerical modeling plays an important role in reproducing the bed and water profiles.     

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.     

Modeling Unsteady Open-Channel Flow for Controller Design

An approximated linear model of unsteady open-channel flow is necessary to design the water-level controller for irrigation open channels. Toward this end, this paper presents the matrix approach to derive the linear model of open-channel system in analytical form mainly according to the Saint Venant equations and the backwater profile at the steady state of open channel. The hydraulic model of the check structure at the downstream end of open channel is also incorporated into the linear model. A practical example indicates that the frequency response of the open-channel system can be accurately analyzed with the linear model. The simulation results in the time domain show that the dynamic behavior of the linear model approximates to that of the nonlinear model of the open-channel system. Finally, the limitations of the linear model are discussed.