Surface-Based Fractional Transport Predictor: Deterministic or Stochastic

Stochastic bed load transport formulas for nonuniform sediment exist, but most of them do not account for the composition of surface material to predict fractional transport rate. This study transformed a surface-based bed-load transport predictor to a stochastic one by approximating the fluctuation of bed-shear stress with a standard log-normal distribution. The deterministic predictor underpredicts fractional transport rate at low values of bed-shear stress and Reynolds number. The modified stochastic predictor predicts fractional transport rate more accurately and converges to the deterministic one at high shear stresses.     

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.     

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.     

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.     

Stream Bank Erosion: In Situ Flume Tests

A new technique for testing the erodibility of cohesive stream banks using an in situ flume is presented. The erosion rate is estimated from direct measurements of bed surface elevations by acoustic sensors. The sediment resuspension rate is obtained using data on sediment concentrations measured by optical backscatter sensors and from water samples. The bed-load contribution to the total erosion rate is evaluated from the conservation equation for sediments. Temporal patterns of erosion and resuspension rates are studied employing stepwise increments of bed-shear stress. The data show that bed load plays a significant role in cohesive bank erosion. The data analysis suggests that erosion and resuspension thresholds observed in experiments were very low or equal to zero. The data support the power type equation for the erosion and resuspension rates with bed-shear stress as the key factor. The data also highlights the potential importance of mud content and water content on erosion.     

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.     

Unified View of Sediment Transport by Currents and Waves. III: Graded Beds

The problem of suspended load and bed load transport in river and coastal flows over graded beds is addressed. Two effects are important: the degree of exposure of the sediment particles of unequal size within a mixture (hiding of smaller particles resting or moving between the larger particles) and the nonlinear dependence of transport on particle diameter. The former effect can be modeled by modifying the critical bed-shear stress through a correction factor and by modifying the effective grain roughness through another correction factor. The modeling of the effective bed-shear stress parameter is studied by using various alternative methods. Based on comparison with suspended load and bed load transport data for graded beds in steady and oscillatory flow, the most promising method is selected. The proposed prediction method is found to work well for the fine sand bed range as well as the coarse sand-gravel bed range.     

Effects of Lifting Force on Bed Topography and Bed-Surface Sediment Size in Channel Bend

In bed-load sediment transport, the lifting force plays an important role in reducing the friction between sediment particles and the bed surface, and it makes particle transportation by the shear force easier. Because the lifting force is related to vorticity, a three-dimensional (3D) numerical model incorporating large eddy simulations was applied to simulate the vorticity field in a channel bend. The results show that the distribution of vorticity is highly nonuniform, and it can lead to significant variations in lifting force and bed-load sediment transport per unit width in a channel bend. Relevant theories are modified on the basis of physical reasoning and then incorporated into numerical models to investigate the lifting-force effects on the bed topography and bed-surface sediment size gradation in a channel bend. With the lifting-force effects considered, it is shown that the errors in simulated bed topography can be reduced by approximately 40% and in bed-surface sediment size by 50%.     

Experimental Study of Bed Load Transport through Emergent Vegetation

Vegetation is an important agent in fluvial geomorphology and sedimentary processes, through its influence on the local hydraulics that determine sediment transport. Within stands of emergent vegetation, bed shear is substantially reduced through the absorption of momentum by drag on the stems. This stimulates deposition of sediment and reduces capacity for bed load transport. The effect of emergent vegetation on hydraulic parameters (including equilibrium bed gradient, flow depth, and velocity) and on bed load transport rate has been investigated experimentally for one sediment size, stem diameter, and stem spacing. Bed load transport rate was found to be closely related to bed-shear stress, which must be estimated by partitioning total flow resistance between stem drag and bed shear.     

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.     

Incipient Motion of Bivalve Shells on Sand Beds under Flowing Water

A model is presented to determine the critical shear stress for the incipient motion of bivalve shells on a horizontal sand bed, under a unidirectional flow of water. Hydrodynamic forces on a solitary bivalve shell, resting over a sand bed, are analyzed for the condition of incipient motion including the effect of turbulent fluctuations. Hydrodynamic forces such as drag and lift are taken into consideration. Three types of bivalve shells, namely Coquina Clam, Cross-barred Chione and Ponderous Ark, were tested experimentally for the condition of incipient motion. The shape parameter of bivalve shells is defined appropriately. Experiments were conducted in a flume with a horizontal bed, and the critical shear stresses were computed using Vanoni’s side-wall correction. The experimental data are used to calibrate the model making lift coefficient a free parameter. The results obtained using the present model agree satisfactorily with the experimental data.     

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.     

Critical Bed Shear Stress for Unisize Sediment

The overall, spatially averaged, mean magnitude of local, spatially averaged (over a small area enclosing the particles’ projected area), instantaneous, critical Shields shear-stress parameters required for incipient motion of uniform-sized sand grains, independent of the bed shear-velocity particle Reynolds number, equal to 0.16, is obtained from calibration of a theory for bed load sediment transport, by minimizing the sum of the squares of the deviations between theoretical and experimental bed load rates. Additionally, optimized expressions for a proposed probability density distribution of the bed shear stresses, for its standard deviation, for finite, maximum, and minimum bed shear stresses, and a bed load rate are obtained. In terms of the mean fluid shear stress, a dimensionless, critical, shear-stress parameter equal to 0.0513 is obtained. Investigation of the probability density distribution of the spatially varying, critical shear stresses would allow a more accurate formulation for the case of low transport rates.     

Stochastic Incipient Motion Criterion for Spheres under Various Bed Packing Conditions

Incipient motion criteria based solely on time-averaged bed shear stress may underpredict sediment transport. The focus of this study is on the stochastic aspect of the problem of incipient motion. Specifically, the role of near-bed turbulent structures and bed packing density on the commencement of sediment motion is investigated. The cornerstone of the proposed model is based on the concept that the particle motion is governed by the intermittent nature of near-bed turbulence. Based on this mechanism, in this article we provide a quantitative model for predicting the commencement of sediment entrainment for the first time under three representative bed packing densities corresponding to the isolated, wake interference, and skimming flow regimes. The performance of the proposed model is compared to published experimental data and the “conventional” approach that is based on the consideration that flow parameters are statistically well represented by a normal distribution.     

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.     

Design Method of Time-Dependent Local Scour at Circular Bridge Pier

A reliable prediction of local scour depth related to hydrological characteristics such as peak discharge, and time corresponding to the equilibrium scour depth is essential for the efficient design of bridge pier foundation. In this paper, a design method to predict the local scour depth with time is proposed. An experimental program was carried out using a cylindrical pier placed in uniform beds under clear-water flows. The pier scour depth was calculated on the basis of a sediment transport equation. Equilibrium local scour depth is reached when the bed-shear stress tends to critical bed-shear stress in the scour hole. Hence, changes to bed-shear stress at the circular bridge pier should be incorporated in the sediment transport theory. The proposed method follows experimental data of various sources.     

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.     

Stochastic Prediction of Sediment Transport in Sand-Gravel Bed Rivers

Classical deterministic bedload transport predictors are applied to sand-gravel bed rivers. The turbulent bed shear stress is modeled according to a probability distribution to obtain realistic bedload transport rates at incipient motion. In extending the predictors to stochastic predictors for nonuniform sediment, many parameters that represent near-bed turbulence and the particle size distribution must be chosen. The parameters that give realistic results are chosen by analyzing the results of a new experimental flume dataset with relatively large water depths. Choosing other combinations of parameters may give equal total bedload transport rates, but at the cost of large errors in fractional transport rates. Attention is given to the hiding-exposure phenomenon and a hindrance effect related to nonuniform sediment. Validation based on two independent field datasets shows that successful predictions of particle sizes near the threshold for motion are feasible using the stochastic approach, while the deterministic approach gives successful predictions well above incipient motion.     

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.