Sediment Transport on Arbitrary Slopes: Simplified Model

The paper presents a study on the influence of gravity on the incipient motion and the bed-load transport of sediment. The computation of critical bed-shear stress is revisited considering the balance of forces (hydrodynamic forces and submerged self-weight) acting on a solitary sediment particle lying on an arbitrary sloping bed. Modified effective bed-shear stress and the corresponding critical bed-shear stress, which are defined to assess the incipient motion of sediment in the direction of resultant force, are applied for the estimation of bed-load transport rate in the direction of resultant force. The sediment transport induced by the gravitational force, which is oblique to the direction of the drag force induced by flow, is incorporated into the bed-load transport equation. This modified model provides a reasonable prediction of the critical bed-shear stress and the bed-load transport rate. The model is validated by experimental data. It can be applied to steep slopes and can also avoid the problem of singularity that arises in numerically calculation of sediment transport rate. Additionally, the vectorial transport rate obtained in the model calculation can be implemented in a numerical simulation of channel bed evolution.     

Method for Estimating Sediment Transport in Ice-Covered Channels

A method is proposed for estimating rates of sediment transport in ice-covered alluvial channels. The method extends existing, open-water procedures for estimating rates of sediment transport to conditions of ice-covered flow. A key aspect of the method is the assessment of flow resistance attributable to bed-surface drag. That assessment is used to estimate rates of bed load and suspended load, and thereby total bed-sediment transport rate. Estimation of ice-covered suspended load additionally entails an approximation whereby open-water suspended load is scaled in proportion to the ratio of a reference sediment concentration for ice-covered flow relative to that for open-water flow. The reference concentration is calculated in terms of bed-load rate and shear velocity attributed to bed-surface drag. Flume data are used to develop the method and tentatively verify it. Field verification of the method presently is hampered by the absence of field data on bed sediment transport in ice-covered channels.     

Effects of Vanes and W-Weir on Sediment Transport in Meandering Channels

Sediment transport patterns in a meandering channel with instream restoration structures (vane and W-weir) have been studied. Laboratory experiments were conducted in a large-scale mobile-bed channel with graded materials under bank-full and overbank flow conditions. Bed-load samples were collected with a calibrated minisampler. Vanes, constructed against the outer bank in a meander bend, relocated the scour hole toward midchannel, thereby protecting the bank from erosion. The sediment sizes (d50,d90) in the bend became slightly more coarse and more uniform in the center of the channel. The W-weir installed immediately below a riffle section created two scour holes without affecting the upstream bed or the natural sediment transport of the channel. Predictions of bed-load transport by selected deterministic and stochastic methods showed large deviation from measurements using Helley–Smith sampler in sections downstream of the bend apex. In addition to creating local scour holes, the structures also relocated the locus of sediment transport at downstream sections. This issue should be considered when installing vanes and weirs in meandering rivers with significant bed-load transport.     

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.     

Calculation of Bed Changes in Mountain Streams

Simulation of flow and sediment transport in mountain streams is complicated by the presence of high gradients, abrupt changes in geometry, variations in regime of flow, and large roughness elements. Most of the numerical models to predict aggradation and degradation in alluvial channels have been developed for low-gradient rivers. This paper is devoted to the development of a numerical model to calculate bed elevation and grain size distribution changes in mountain streams where the maximum bed material size is in the range of boulders. An attempt is made to validate the model by using observed field data collected upstream from a small retention dam in a Venezuelan stream. After calibration of the sediment transport equation, reasonable agreement is obtained for the variations in the grain size distribution of the bed-surface material. An additional application is presented in the Cocorotico River, a small mountain stream located in the northwest region of Venezuela, which illustrates the adaptability of the model to handle a case of coarsest-bed-material removal from the active channel and to simulate the armoring process.     

Numerical Simulation of Contraction Scour in an Open Laboratory Channel

A finite-volume computer code developed at the Institute for Hydromechanics, University of Karlsruhe, has been used to calculate the flow and sediment transport in a laboratory channel with constriction and movable bed. The flow is calculated by solving the fully three dimensional Reynolds-averaged Navier-Stokes equations with k?ε turbulence model. The bed deformation is obtained from an overall mass-balance equation for sediment transport and the bed-load transport is simulated with a nonequilibrium model. The calculated results for flow and scour development in the laboratory channel are compared with experimental measurements. The sensitivity of the simulated results to the nonequilibrian adaptation-length parameter in the nonequilibrium bed-load transport model is investigated systematically, which represents the main contribution of this paper.     

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.     

Bed-Load Sediment Transport on Large Slopes: Model Formulation and Implementation within a RANS Solver

Standard bed-load sediment-transport formulas are extended using basic mechanical principles to include gravitational influence on large slopes of arbitrary orientation. The resulting sediment fluxes are then incorporated into a morphodynamics model in a general-purpose, three-dimensional, finite-volume, Reynolds-averaged Navier–Stokes (RANS) code. Major features are: (1) the downslope component of weight is combined with the fluid stress to form an effective bed stress (similar to the work of Wu in 2004); (2) the critical effective stress is reduced in proportion to the component of gravity normal to the slope; (3) a simple flux-based model for avalanching is implemented as a numerical means of preventing the local slope from exceeding the angle of repose; (4) an entirely vectorial formulation of bed-load transport is developed to account for arbitrary surface orientation; and (5) methods for reducing numerical instability in the morphodynamics equation are described. Sample computations are shown for scour and accretion in a channel bend and for the movement of sand mounds on erodible and nonerodible bases.     

Field Assessment of Alternative Bed-Load Transport Estimators

Measurement of near-bed sediment velocities with acoustic Doppler current profilers (ADCPs) is an emerging approach for quantifying bed-load sediment fluxes in rivers. Previous investigations of the technique have relied on conventional physical bed-load sampling to provide reference transport information with which to validate the ADCP measurements. However, physical samples are subject to substantial errors, especially under field conditions in which surrogate methods are most needed. Comparisons between ADCP bed velocity measurements with bed-load transport rates estimated from bed-form migration rates in the lower Missouri River show a strong correlation between the two surrogate measures over a wide range of mild to moderately intense sediment transporting conditions. The correlation between the ADCP measurements and physical bed-load samples is comparatively poor, suggesting that physical bed-load sampling is ineffective for ground-truthing alternative techniques in large sand-bed rivers. Bed velocities measured in this study became more variable with increasing bed-form wavelength at higher shear stresses. Under these conditions, bed-form dimensions greatly exceed the region of the bed ensonified by the ADCP, and the magnitude of the acoustic measurements depends on instrument location with respect to bed-form crests and troughs. Alternative algorithms for estimating bed-load transport from paired longitudinal profiles of bed topography were evaluated. An algorithm based on the routing of local erosion and deposition volumes that eliminates the need to identify individual bed forms was found to give results similar to those of more conventional dune-tracking methods. This method is particularly useful in cases where complex bed-form morphology makes delineation of individual bed forms difficult.     

Unified View of Sediment Transport by Currents and Waves. I: Initiation of Motion, Bed Roughness, and Bed-Load Transport

Attention is given to the properties of sediment beds over the full range of conditions (silts to gravel), in particular the effect of fine silt on the bed composition and on initiation of motion (critical conditions) is discussed. High-quality bed-load transport data sets are identified and analyzed, showing that the bed-load transport in the sand range is related to velocity to power 2.5. The bed-load transport is not much affected by particle size. The prediction of bed roughness is addressed and the prediction of bed-load transport in steady river flow is extended to coastal flow applying an intrawave approach. Simplified bed-load transport formulas are presented, which can be used to obtain a quick estimate of bed-load transport in river and coastal flows. It is shown that the sediment transport of fine silts to coarse sand can be described in a unified model framework using fairly simple expressions. The proposed model is fully predictive in the sense that only the basic hydrodynamic parameters (depth, current velocity, wave height, wave period, etc.) and the basic sediment characteristics (d10, d50, d90, water temperature, and salinity) need to be known. The prediction of the effective bed roughness is an integral part of the model.     

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.     

Hydraulic Interpretation of Cross-Stream Variations in Bed-Load Transport

The hydraulic control of bed-load transport rates in Nahal Yatir and Nahal Eshtemoa, two coarse-grained ephemeral channels in the semiarid northern Negev, Israel, provides a rare opportunity to infer the spanwise variation in bed-shear stress from an analysis of cross-stream variations in bed-load transport rate. Automatic sediment transport monitoring stations were used to obtain synchronous measurements of bed-load discharge at a number of locations across the widths of two straight channel reaches. In both streams, channel-average bed-load fluxes demonstrated a common and well-defined response to changing channel-average shear stress and approximated the transporting capacity of the flow over much of the range of monitored discharges. However, transport rates measured at the channel margins are only half those at the channel centerline, and, at high discharges, a marked asymmetry in the pattern of bed-load transport develops across the central section of the widest channel. This variation in bed-load discharge over the two channel cross sections is thought to reflect lateral variations in shear stress induced by sidewall drag and, more tentatively, the generation and disposition of cellular secondary currents. But no systematic relation is found for the ratios of sediment fluxes at off-center sampling locations and those recorded at the channel center, even though the off-center locations are thought to move into and out of the region affected by sidewall drag as aspect ratio of the flow decreases and increases with changing water-stage. The results suggest that it is difficult to generalize about the changing influence of the sidewall on local shear and bed load as aspect ratio changes during the course of a flood.     

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.     

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.     

Approach to Separate Sand from Gravel for Bed-Load Transport Calculations in Streams with Bimodal Sediment

The bed material found in gravel-bed streams is nonuniform in terms of grain size and can typically be classified as unimodal or bimodal. The latter type of sediment distribution is usually represented by two modes, one of sand size and another of gravel. For this case, the movement of one mode becomes nonlinearly influenced by the other. As a result, the presence of the two modes in a bimodal material complicates the calculation of bed-load transport rates. The present study proposes an approach to separate the calculation of bed-load transport rates for bimodal materials into two independent fractions of sand and gravel, thereby rendering the bed sediment into two unimodal components. This approach is accomplished by decoupling the two fractions through scaling the reference Shields stresses of the sand and gravel modes to match the value of the mode of unimodal materials. Consequently, the contribution of each fraction to bed load can be estimated using a suitable relation derived for unimodal materials. Laboratory and field bed-load data available in the literature are used to examine the validity of the overall approach.     

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.     

Effect of Coarse Surface Layer on Bed-Load Transport

Existing bed-load transport formulas may overestimate the transport rate in mountain rivers by two orders of magnitude or more. Recently published field data sets provide an opportunity to take a fresh look at the bed-load transport relationship and it is hypothesized that the overestimate is due to a failure to account for the effect of a coarse surface layer of bed material inhibiting the release of fine subsurface material. Bed-load transport is determined as gs = aρ(q?qc) where q=water discharge per unit width; qc=critical value for initiation of bed material movement; ρ=water density; and a=coefficient. The gs/q relationship is typically piecewise linear, characterized by two transport phases with, respectively, low and high rates of change. Twenty-one flume and 25 field data sets were used to quantify the relationship for Phase 2. The flume data confirm the dependence of a on S1.5, where S=channel slope, in agreement with earlier studies. The field data additionally show that a varies inversely with the degree of bed armoring, given by the ratio of surface to subsurface bed material size. The finding is consistent with the hypothesis and suggests the need to account for the bed material supply limitation in the bed-load transport formula. However, the available data are not entirely sufficient to rule out an alternative dependency, or codependency, on flow resistance. The critical conditions for initiation of Phase 2 transport are also quantified as a function of bed material size and channel slope. The resulting set of equations allows a more accurate estimation of Phase 2 bed-load transport rates. However, the equations are empirical and should be restricted for use within the range of conditions used in their development, to determine mean rather than instantaneous transport rates and to determine bulk transport rates, not transport by size fraction.     

Probabilistic Modeling of Bed-Load Composition

This paper proposes that the changes which occur in composition of the bed load during the transport of mixed-grain-size sediments are largely controlled by the distributions of critical entrainment shear stress for the various size fractions. This hypothesis is examined for a unimodal sediment mixture by calculating these distributions with a discrete particle model and using them in a probabilistic calculation of bed-load composition. The estimates of bed-load composition compare favorably with observations of fractional transport rates made in a laboratory flume for the same sediment, suggesting that the hypothesis is reasonable. The analysis provides additional insight, in terms of grain mechanics, into the processes that determine bed-load composition. These insights strongly suggest that better prediction methods will result from taking account of the variation of threshold within size fractions, something that most previous studies have neglected.     

Bed-Load Transport Equation for Sheet Flow

When open-channel flows become sufficiently powerful, the mode of bed-load transport changes from saltation to sheet flow. Where there is no suspended sediment, sheet flow consists of a layer of colliding grains whose basal concentration approaches that of the stationary bed. These collisions give rise to a dispersive stress that acts normal to the bed and supports the bed load. An equation for predicting the rate of bed-load transport in sheet flow is developed from an analysis of 55 flume and closed conduit experiments. The equation is ib = ω where ib = immersed bed-load transport rate; and ω = flow power. That ib = ω implies that eb = tan?α = ub/u, where eb = Bagnold’s bed-load transport efficiency; ub = mean grain velocity in the sheet-flow layer; and tan?α = dynamic internal friction coefficient. Given that tan?α ≈ 0.6 for natural sand, ub ≈ 0.6u, and eb ≈ 0.6. This finding is confirmed by an independent analysis of the experimental data. The value of 0.60 for eb is much larger than the value of 0.12 calculated by Bagnold, indicating that sheet flow is a much more efficient mode of bed-load transport than previously thought.     

3D Numerical Modeling of Flow and Sediment Transport in Open Channels

A 3D numerical model for calculating flow and sediment transport in open channels is presented. The flow is calculated by solving the full Reynolds-averaged Navier-Stokes equations with the k ? ε turbulence model. Special free-surface and roughness treatments are introduced for open-channel flow; in particular the water level is determined from a 2D Poisson equation derived from 2D depth-averaged momentum equations. Suspended-load transport is simulated through the general convection-diffusion equation with an empirical settling-velocity term. This equation and the flow equations are solved numerically with a finite-volume method on an adaptive, nonstaggered grid. Bed-load transport is simulated with a nonequilibrium method and the bed deformation is obtained from an overall mass-balance equation. The suspended-load model is tested for channel flow situations with net entrainment from a loose bed and with net deposition, and the full 3D total-load model is validated by calculating the flow and sediment transport in a 180° channel bend with movable bed. In all cases, the agreement with measurements is generally good.