Modeling Bed-Load Rates from Fine Grain-Size Patches during Small Floods in a Gravel-Bed River

This study investigates the applicability of five bed-load-transport formulas (the Meyer-Peter and Müller, Schoklitsch, Bagnold, Smart and Jaeggi, and Rickenmann equations) to predict bed-load transport rates of frequent, low-magnitude flood events (maximal bankfull discharge) for a mountainous, poorly sorted gravel-bed river characterized by a bimodal sediment-size distribution and spatially distributed patches. For model parametrization, special emphasis was placed on the spatial composition of the grain-size distribution (GSD) to evaluate the impact of preferential removal of sediments from patches with finer sediments on bed-load transport. Three parametrization approaches to the choice of an appropriate sediment size that considered the apparent bimodality of the GSD to varying degrees were tested. The modeling study demonstrated that the incorporation of spatial structure of GSD and its bimodal character has an important impact on model performance—a unimodal parametrization failed to reproduce measured bed-load rates for all tested bed-load formulas; a threshold parametrization approach that considered only finer sediments from the small patches as bed-load source material in combination with the Schoklitsch, Smart and Jaeggi, and Rickenmann equations yielded the best results, whereas the Meyer-Peter and Müller and the Bagnold equations failed to predict bed-load rates for all parametrization approaches. The modeling study thus showed that bed-load formulas are sensitive to the spatial structure of the GSD, which should not be treated as a continuum of sediment size fractions but rather as composition of finer sediment patches to enable an adequate reproduction of measured bed-load data from low-magnitude floods in gravel-bed rivers.     

Reanalysis and Correction of Bed-Load Relation of Meyer-Peter and Müller Using Their Own Database

The pioneering predictor of fluvial bed-load transport rate proposed by Meyer-Peter and Müller in 1948 is still extensively used in basic research and engineering applications. A review of the basis for its formulation reveals, however, that an unnecessary bed roughness correction was applied to cases of plane-bed morphodynamic equilibrium. Its inclusion followed a flow resistance parameterization in terms of the Nikuradse roughness height, which has been shown (well after the publication of their work) to be inappropriate for the characterization of mobile bed rough conditions in rivers. Removing the unnecessary correction and incorporating an improved correction of the boundary shear stress due to sidewall effects allow elucidation of the most parsimonious form of the bed-load relation of Meyer-Peter and Müller that is dictated by their own data set. The new predictor is presented in terms of two alternative power law forms. These amended forms show that, in the case of lower-regime plane-bed equilibrium transport of uniform bed sediment, the new estimates of volume bed-load transport rates are less than or equal to half the values that would be obtained with the original relation of Meyer-Peter and Müller in the absence of the unnecessary bed roughness correction. The meticulous database and clear analysis of the original work of Meyer-Peter and Müller greatly aided the present writers in their reanalysis, which liberally uses the hindsight offered by 58 years of subsequent research.     

Simple Model for Downstream Variation of Median Sediment Size in Chilean Rivers

A simple theory is developed to account for the observed downstream variation of the median sediment size in Chilean rivers based on a reach-wise equilibrium sediment transport concept. The theory makes use of a bedload transport equation linked with a resistance equation to estimate the median sediment diameter as a function of channel slope, with the flow Reynolds number and bedload concentration as parameters. Both, Meyer-Peter and Müller’s and Ackers and White’s formulas are used alternatively as bedload equations. A Manning–Strickler type of formulation is used as a resistance relationship. The resulting model is validated against field data corresponding to 150 rivers in Central Chile, covering slopes in the range of 0.04–8.61%, with median sediment size in the range of 0.3–250 mm. Despite the simplicity of the present theory and the somewhat bold assumptions made in its derivation, the estimated variation of the median sediment size with channel slope follows the same trend as the field data. Most of the scatter of these data falls within the theoretical limits given by the estimated range of values of the parameters of the model.     

Numerical Modeling of Abutment Scour with the Focus on the Incipient Motion on Sloping Beds

A three-dimensional computational fluid dynamics model is applied to predict local scour around an abutment in a rectangular laboratory flume. When modeling local scour, steep bed slopes up to the angle of repose occur. To predict the depth and the shape of the local scour correctly, the reduction of the critical shear stress due to the sloping bed must be taken into account. The focus of this study is to investigate different formulas for the threshold of noncohesive sediment motion on sloping beds. Some formulas only take the transversal angle (perpendicular to the flow direction) into account, but others also consider the longitudinal angle (streamwise direction). The numerical model solves the transient Reynolds-averaged Navier-Stokes equations in all three dimensions to compute the water flow. Sediment continuity in combination with an empirical formula is used to capture the bed load transport and the resulting bed changes. When the sloping bed exceeds the angle of repose, the bed slope is corrected with a sand-slide algorithm. The results from the numerical simulations are compared with data from physical experiments. The reduction of the bed shear stress on the sloping bed improves the results of the numerical simulation distinctly. The best results are obtained with the formulas that use both the transversal and the longitudinal angle for the reduction of the critical bed shear stress.     

Concentration Profiles and Solids Transport above Stationary Deposit in Enclosed Conduit

Concentration profiles and overall flow parameters were measured in a 0.37-mm-sand-mixture flow above a plane stationary deposit in a pressurized circular pipe of the inner diameter of 150 mm. The new data and additional data from the literature are used to propose a tentative transport formula of the Meyer-Peter and Müller type, and a slip-ratio formula, for flows at high shear stress in enclosed conduits.     

One-Dimensional Modeling of Dam-Break Flow over Movable Beds

A one-dimensional model has been established to simulate the fluvial processes under dam-break flow over movable beds. The hydrodynamic model adopts the generalized shallow water equations, which consider the effects of sediment transport and bed change on the flow. The sediment model computes the nonequilibrium transport of bed load and suspended load. The effects of sediment concentration on sediment settling and entrainment are considered in determining the sediment settling velocity and transport capacity. In particular, a correction factor is proposed to modify the Van Rijn formulas of equilibrium bed-load transport rate and near-bed suspended-load concentration for the simulation of sediment transport under high-shear flow conditions. The governing equations are solved by an explicit finite-volume method with the first-order upwind scheme for intercell fluxes. The model has been tested in two experimental cases, with fairly good agreement between simulations and measurements. The sensitivities of the model results to parameters such as the sediment nonequilibrium adaptation length, Manning’s roughness coefficient and the proposed correction factor have been verified. The proposed model has also been compared to an existing model and the results indicate the new model is more reliable.     

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.     

Prediction of Concerted Sediment Flushing

A proprietary one-dimensional numerical model was developed for predicting the amounts of sediment flushed and deposited in the reservoirs in series, the bed evolutions, and variations of the suspended solids concentrations along a river during the concerted sediment flushing events. The model consists of a flow movement module and sediment transport module in which the bed material load is taken as sediment mixture. The nonuniform property of the bed material load is modeled by the introduction of a mixing layer, transition layer, and deposition strata. The model was calibrated on the basis of the field data at Dashidaira and Unazuki reservoirs on the Kurobe River in Japan. The calculated results are in good agreement with the measurements. For the reservoirs out of Japan, the Ashida and Michiue bed load formula used in the model should be verified or replaced by other formulas.     

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.     

Dam-Break Wave-Front Celerity

This work is concerned with the role that friction and inertia effects can play on the magnitude of dam-break wave-front celerity. Classic analytical solutions are presented. A large collection of experimental data is used, covering a wide range of different initial conditions: fixed bed or mobile bed (five types of bed material), dry or wet bed downstream, and with or without bed step. To overcome the limitations of analytical solutions, a numerical model is used. The model is based on the shallow-water approach with contact-load dominated sediment transport, and it makes use of developments recently made in the study of sheet flows. The analytical and numerical results are compared with experimental data. It was found that the celerity is mainly dictated by the friction coefficient, by the sediment inertia, by the initial downstream water depth, and by the initial bed step height. For good data fitting, the friction coefficient must be influenced by the type of bed, fixed or mobile. In the latter, the coefficient must vary with the bed material characteristics. The dissimilarities between the experimental, analytical, and numerical results are interpreted.     

One-Dimensional Numerical Model for Nonuniform Sediment Transport under Unsteady Flows in Channel Networks

In this study, the proposed one-dimensional model simulates the nonequilibrium transport of nonuniform total load under unsteady flow conditions in dendritic channel networks with hydraulic structures. The equations of sediment transport, bed changes, and bed-material sorting are solved in a coupling procedure with a direct solution technique, while still decoupled from the flow model. This coupled model for sediment calculation is more stable and less likely to produce negative values for bed-material gradation than the traditional fully decoupled model. The sediment transport capacity is calculated by one of four formulas, which have taken into consideration the hiding and exposure mechanism of nonuniform sediment transport. The fluvial erosion at bank toes and the mass failure of banks are simulated to complement the modeling of bed morphological changes in channels. The tests in several cases show that the present model is capable of predicting sediment transport, bed changes, and bed-material sorting in various situations, with reasonable accuracy and reliability.     

Sand Transport in Nile River, Egypt

Measurements of bed-load and suspended-load transport rates were carried out successfully at four cross sections of the Nile River, in Egypt, along the entire length from Aswan to Cairo using a mechanical sampler called the Delft Nile Sampler. The measured transport rates were compared to similar data sets from two other large scale rivers: the Rhine-Waal River in the Netherlands and the Mississippi River in the USA. The bed-load transport rates in the Nile River and in the Rhine-Waal River are in very good agreement. Comparison of suspended transport rates in the Nile River and in the Mississippi River shows that both data sets are complementary, revealing a very consistent trend of suspended transport against current velocity; suspended transport is roughly proportional to (Vav)3?to?4. Three formulas for the prediction of bed-load transport were tested using the Nile data: Meyer-Peter–Muller, Bagnold, and Van Rijn. The prediction formula of Van Rijn produced significantly better results than the other two formulas; the average relative error was about 60%. The formula of Van Rijn was modified to extend it to conditions with slightly nonuniform sediment mixtures by introducing a correction factor for the bed shear parameter. Based on a limited number of flume experiments, the correction factor was found to be dependent on the characteristics of the sediment mixture (d10, d50, d90, and σg). Comparison of bed-load transport measured in the Nile River with computed transport rates of the modified formula showed improved results; the average relative error decreased to about 30%. The formulas of Bagnold and Van Rijn were also used to compute the suspended transport rates in the Nile River. The computed transport rates were found to be within a factor of 2 of measured values; the formula of Bagnold performed slightly better. The total load transport formula of Engelund–Hansen was also successfully used (computed values within a factor of about 2 of measured values).     

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.     

Three-Dimensional Numerical Model for Flow and Bed Deformation around River Hydraulic Structures

This paper describes a numerical model developed to simulate flow and bed deformation around river hydraulic structures. The model solved the fully three-dimensional, Reynolds-averaged Navier–Stokes equation expressed in a moving boundary-fitted coordinate system to calculate the flow field with water and bed surfaces varying in time. A nonlinear k-ε turbulence model was employed in order to predict flow near the structure where three-dimensional flow is dominant. The temporal change in bed topography was calculated by coupling a stochastic model for sediment pickup and deposition using a momentum equation of sediment particles in order to account for the effect of nonequilibrium sediment transport. In validating the numerical model, a spur dike and a bridge pier, which are considered to be typical river-engineering structures, were selected. By comparing the numerical results with observed laboratory experimental data, the model was found to reproduce flow and scour geometry around these structures with sufficient accuracy.     

Assessment of Methods Used in 1D Models for Computing Bed-Load Transport in a Large River: The Danube River in Slovakia

Comprehensive measurements of bed-load sediment transport through a section of the Danube River, located approximately 70?km downstream from Bratislava, Slovakia, are used to assess the accuracy of bed-load formulas implemented in 1D modeling. Depending on water discharge and water level, significant variations in the distribution of bed load across the section were observed. It appeared that, whatever the water discharge, the bed shear stress τ is always close to the estimated critical bed shear stress for the initiation of sediment transport τcr. The discussion focuses on the methods used in 1D models for estimating bed-load transport. Though usually done, the evaluation of bed-load transport using the mean cross-sectional bed shear stress yields unsatisfactory results. It is necessary to use an additional model to distribute the bed shear stress across the section and calculate bed load locally. Bed-load predictors also need to be accurate for τ close to τcr. From that point of view, bed-load formulas based on an exponential decrease of bed-load transport close to τcr appear to be more appropriate than models based on excess bed shear stress. A discussion on the bed-load formula capability to reproduce grain sorting is also provided.     

Two-Dimensional Total Sediment Load Model Equations

An unsteady total load equation is derived for use in depth-averaged sediment transport models. The equation does not require the load to be segregated a priori into bed and suspended but rather automatically switches to suspended load, bed load, or mixed load depending on a transport mode parameter consisting of local flow hydraulics. Further, the sediment transport velocity, developed from available data, is explicitly tracked, and makes the equation suitable for unsteady events of sediment movement. The equation can be applied to multiple size fractions and ensures smooth transition of sediment variables between bed load and suspended load for each size fraction. The new contributions of the current work are the consistent treatment of sediment concentration in the model equation and the empirical definition of parameters that ensure smooth transitions of sediment variables between suspended load and bed load.     

Maximum Level and Time to Peak of Dam-Break Waves on Mobile Horizontal Bed

This experimental study focuses the influence of bed material mobility and initial downstream water level on maximum water level and time to peak of dam-break waves. It covers horizontal bed conditions on fixed bed, sand bed, and pumice bed. Results include water surface level time evolution, maxima wave levels and time to peak. The influence of bed material mobility and downstream water level was identified and characterized, stressing the importance of using mathematical models with appropriate sediment transport formulations instead of purely hydrodynamic models to simulate dam-break waves on mobile bed channels.     

Modeling Sediment Transport Using Depth-Averaged and Moment Equations

A 1D mathematical model to calculate bed variations in alluvial channels is presented. The model is based on the depth-averaged and moment equations for unsteady flow and sediment transport in open channels. Particularly, the moment equation for suspended sediment transport is originally derived by the assumption of a simple vertical distribution for suspended sediment concentration. By introducing sediment-carrying capacity, suspended sediment concentration can be solved directly from sediment transport and its moment equations. Differential equations are then solved by using the control-volume formulation, which has been proven to have good convergence. Numerical experiments are performed to test the sensitivity of the calibrated coefficients α and k in the modeling of the bed deposition and erosion. Finally, the computed results are compared with available experimental data obtained in laboratory flumes. Comparisons of this model with HEC-6 and other numerical models are also presented. Good agreement is found in the comparisons.     

Effect of Bed Armoring on Bed Topography of Channel Bends

The two-dimensional numerical model previously developed by the writers for modeling the bed variations in a channel bend with uniform sediment is upgraded to incorporate the nonuniformity of sediment particles as well as bed armoring. In this model, the two-dimensional, depth-averaged, unsteady flow equations along with the bed-load mass conservation equation are solved in a body-fitted coordinate system by using the Beam and Warming alternating-direction implicit (ADI) scheme. A one-dimensional bed surface armoring approach is extended herein for application to a two-dimensional domain. The model is applied to a 180° bend with a constant radius under unsteady flow conditions. Numerical simulations are carried out to study the effect of bed armoring on the bed deformations in channel bends. Results show that bed armoring reduces scour in channel bends.