Modification of the Einstein Bed-Load Formula

The original derivation of the Einstein bed-load formula involves several oversimplified assumptions concerning step length of a particle, exchange time, and probability of being lifted into motion. A modified version of the Einstein formula that relaxes these assumptions and incorporates a nonuniform sediment model is proposed and tested with both field and laboratory data. The modified formula shows promise as being more reliable than the original version in estimating bed-load transport rates.     

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.     

Exponential Formula for Bedload Transport

An exponential formula that does not involve the concept of the critical shear stress is derived in this study for computing bedload transport rates. The formula represents well various experimental data sets ranging from the weak transport to high shear conditions. Comparisons of the present study are also made with many previous bedload formulas commonly cited in the literature.     

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).     

Bed-Load Transport Flume Experiments on Steep Slopes

Experiments were conducted over uniform gravel bed materials to obtain 143 friction factor values under bed-load equilibrium flow conditions in an attempt to add to the scarce data available on slopes between 1 and 9% for Shields numbers between 0.08 and 0.29. Analyses showed that when only flows over flat beds are considered, a distinction must be made between flows with and without bed load. More particularly, fitting flow resistance equations indicated that the roughness parameter increases by a factor of 2.5 from clear water flow to intense bed-load transport. Between these two states, the flow resistance can be approximated by a constant for a given slope.     

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.     

Wash Load and Bed-Material Load Transport in the Yellow River

It has been the conventional assumption that wash load is supply limited and is only indirectly related to the hydraulics of a river. Hydraulic engineers also assumed that bed-material load concentration is independent of wash load concentration. This paper provides a detailed analysis of the Yellow River sediment transport data to determine whether the above assumptions are true and whether wash load concentration can be computed from the original unit stream power formula and the modified unit stream power formula for sediment-laden flows. A systematic and thorough analysis of 1,160 sets of data collected from 9 gauging stations along the Middle and Lower Yellow River confirmed that the method suggested by the conjunctive use of the two formulas can be used to compute wash load, bed-material load, and total load in the Yellow River with accuracy.     

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.     

Closed-Conduit Bed-Form Initiation and Development

Nonintrusive measurement of closed-conduit erodible-bed development was undertaken for 12 experiments of ranges of flow strengths and sediment (solids) sizes. Analogous to open-channel flows, wavelets on the sediment bed of a closed-conduit are instigated by discontinuities in the bed, with wavelet lengths λ for laminar and turbulent open-channel and closed-conduit flows given by λ = 175d0.75, where λ and sediment size d are in millimeters. For closed-conduit flows, ripples, and dunes grow from these wavelets (at rates increasing with increasing flow strength, and utilizing the mechanisms of bed-form coalescence and throughpassing) to limiting lengths, heights, steepnesses, and bed friction factors that are approximately maintained or possibly decrease thereafter. Limitation of free-surface deformation results in increased rates of bed-wave development for closed-conduit flows in comparison to open-channel flows. Measured results indicate that equilibrium closed-conduit ripple and dune magnitudes can be predicted using relations derived for equivalent open-channel flows. The present findings are of particular relevance for understanding and modeling engineering activities ranging from dredging to transport of solids in stormwater and sewer systems, bed-form transport of solids in closed conduits influencing (potentially markedly) conduit conveyance, rate of solids transport, and system head losses for such flows.     

Evaluation and Improvement of Bed Load Discharge Formulas based on Helley–Smith Sampling in an Alpine Gravel Bed River

Bed load discharge formulas have been evaluated by analyzing them in relation to measured Helley–Smith data for the gravel-bedded armored Drau River, Austria. Comparison of calculations with measurements leads to ranking of the formulas that depends on the evaluation parameters. The choice of formula is made with respect to our specific aims: the investigation of individual floods requires a different approach from that of long-term budgets. Formula performance is consistently improved when conditions for the threshold of motion are modified according to data measured up on the initiation of motion. Formulas such as those reported by Parker in 1990, Zanke in 1999, and Sun and Donahue in 2000 are capable of coping with partial transport, which is commonly found in Alpine rivers. These formulas therefore provide encouraging results, particularly after the introduction of modifications. The augmentation of field measurements, even if limited in scope, considerably improves the performance of bed load discharge formulas.     

Validation of Existing Bed Load Transport Formulas Using In-Sewer Sediment

Granular sediment in pipe inverts has been reported in a number of sewer systems in Europe. Given the range of flow conditions and particle characteristics of inorganic sewer sediments the mode of transport may normally be considered as bed load. Current commercial software for modeling the erosion and transport of sediments in sewer pipes still utilizes well-known, or modified versions of transport equations that were derived for transport of noncohesive sediment in alluvial streams. In this paper the performances of the equations of Ackers and White (originally developed for the transport of river sediments) and of May (derived from laboratory pipe experiments) are examined against two separate data sets. One set is from laboratory erosion experiments on sewer sediment obtained in Paris. A second data set has bed load transport rate measurements recorded in a sewer inlet pipe. The formulas were selected because of their widespread use in the prediction of in-sewer sediment transport both in commercial software and in the latest United Kingdom design guidance for new sewers. The results indicated that both the relationships performed poorly, even in such well-controlled conditions. These formulas have significant difficulties in predicting the erosion thresholds and fractional transport rates for non-uniformly sized in-sewer sediments. An empirical formula to adjust the threshold of motion for individual grain size fractions was developed which significantly improved predictions. Although such techniques have been used in gravel bed rivers, the threshold adjustment function for in-sewer deposits was significantly different from these previously published for fluvial gravels, indicating that a direct transfer of fluvial relationships to sewers may be inappropriate without further research.     

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.     

Numerical Simulation of Flow over a Rough Bed

This paper presents results of a direct numerical simulation (DNS) of turbulent flow over the rough bed of an open channel. We consider a hexagonal arrangement of spheres on the channel bed. The depth of flow has been taken as four times the diameter of the spheres and the Reynolds number has been chosen so that the roughness Reynolds number is greater than 70, thus ensuring a fully rough flow. A parallel code based on finite difference, domain decomposition, and multigrid methods has been used for the DNS. Computed results are compared with available experimental data. We report the first- and second-order statistics, variation of lift/drag and exchange coefficients. Good agreement with experimental results is seen for the mean velocity, turbulence intensities, and Reynolds stress. Further, the DNS results provide accurate quantitative statistics for rough bed flow. Detailed analysis of the DNS data confirms the streaky nature of the flow near the effective bed and the existence of a hierarchy of vortices aligned with the streamwise direction, and supports the wall similarity hypothesis. The computed exchange coefficients indicate a large degree of mixing between the fluid trapped below the midplane of the roughness elements and that above it.     

Response of Velocity and Turbulence to Sudden Change of Bed Roughness in Open-Channel Flow

The experimental study shows how an open-channel flow would respond to a sudden change (from smooth to rough) in bed roughness. Using a two-dimensional acoustic Doppler velocimeter and a laser Doppler velocimeter, the velocity, turbulent intensities, and Reynolds stress profiles at different locations along a laboratory flume were measured. Additionally, the water surface profile was also measured using a capacitance-type wave height meter. The experimental data show the formation of an internal boundary layer as a result of the step change in bed roughness. The data show that this boundary layer grows much more rapidly than that formed in close-conduit flows. The results also show that the equivalent bed roughness, bed-shear stress, turbulent intensities, and Reynolds stress change gradually over a transitional region, although the bed roughness changes abruptly. The behavior is different from that observed in close-conduit flows, where an overshooting property—which describes the ability of the bed-shear stress to attain a high-peak value over the section with the larger roughness, was reported. A possible reason for the difference is the variation of the water surface profile when an open-channel flow is subjected to a sudden change in bed roughness.     

Case Study: Bed Stability around the East Caisson of the Tacoma Narrows Bridge

A study on the hydraulic and sediment conditions at the Tacoma Narrows Bridge, in Washington State, was carried out to examine the stability of the bed material around the bridge caissons. Specifically, this was conducted around the east caisson, where the maximum velocities around either of the two caissons are experienced. This was performed for the peak tidal exchange event of May 27th to 28th, 2002. During this max flow event, multibeam surveyed bathymetry and three-dimensional acoustic doppler current profiler velocity data were collected around the east caisson in the course of both the flood and ebb. The surface of the bed material surrounding the east caisson was videotaped during the slack conditions following the yearly maximum flow event, and used to determine the particle size distribution and spatial arrangement of those distributions around the caisson. This was done by lowering a submersible video camera and appropriate lighting to the bottom of the Narrows, a depth of approximately 45?m. Flow and sediment observations were coupled to determine the commencement of sediment motion for different size classes of sediment. Two methods were utilized to calculate friction velocity in order to assess the stability of different bed particle size fractions during these high flow conditions. Friction velocity was first calculated from measured velocity profiles at various locations around the east caisson. The second method was based on the concept of a free stream power-law expression for depth-averaged velocity. Stability was then examined using the critical shear stress concept and captured video data of the bed. General results showed the particles ? 30?mm in diameter were in motion during the flood and ebb. The work is here presented as a case study because of the unique large-scale flow conditions that are present around the east caisson of the Tacoma Narrows Bridge.     

Entrainment Probabilities of Mixed-Size Sediment Incorporating Near-Bed Coherent Flow Structures

In this work we incorporate the effect of near-bed coherent flow structures into the entrainment of randomly configured mixed-size sediments. The fourth-order Gram–Charlier type probability density function (GC pdf) of near-bed streamwise velocity is employed to account for the higher-order correlations associated with turbulent bursting. A compilation of the published data over a wide range of bed roughness is used to analyze the near-bed coherent flow structures, including the second-, third-, and fourth-order moments of velocity fluctuation (i.e., turbulence intensity, skewness, and flatness factors) required in the fourth-order GC pdf. An important result of this study is a set of quantitative relations used to predict these higher-order moments as a function of the roughness Reynolds number. The random grain protrusion is parameterized with the exposure and friction heights, and an existing probabilistic approach is used to correct the hiding effect of mixed-size sediment. The above factors are all incorporated into the formulation of entrainment (rolling and lifting) probabilities. As compared to the previous normal and lognormal models, the present results demonstrate significantly improved agreement with the observed data for the unisize and mixed-size sediments under partial- and full-transport conditions. The results also reveal that the third-order GC pdf can be used to approximate the fourth-order one for the fully rough beds, however, for smooth beds the fourth-order GC pdf should be used to adequately incorporate the effects of higher-order correlations. This paper offers some new insights into the processes of sediment entrainment.     

Formation Processes and Configuration of Channel-Flow Dominated Alluvial Deltas by Numerical Simulation

A horizontal two-dimensional mobile bed model for simulating the formation of river-dominated deltas in the river mouth or reservoir is presented, which is composed of shallow water equations, sediment transport formula, and a sediment continuity equation. Geometry similarity of river deltas during the processes of formation is discussed. Stability analysis and sensitivity analysis of parameters in the model are analyzed, which indicates that bed configuration is sensitive to the incipient-motion criteria of bed–load particles. The effect of gravity component on the initiation of sediment movement, therefore, is recommended to be considered in the modeling. The bed configuration including the reverse slope in the longitudinal profile and concave in the transverse profiles are correctly simulated with help from the correction of incipient-motion criteria. Simulation results are verified with a series of experiments and are consistent with series geometric functions and dimensionless profiles inducted from experimental data. This reflects the great reliability of the model. Historical topographical records of two typical in-land deltas depicting their earlier developmental stages are discussed to show the usefulness of this study.     

Dynamic Routing of Flow Resistance and Alluvial Bed-Form Changes from the Lower to the Upper Regime

On the basis of a Strouhal number and the definition of the control factor, m, a new routing to calculate the energy slope in the lower and upper alluvial regimes is proposed. The control factor, m, representing the interactions in alluvial rivers, is reckoned as a bed-form index: while the flow evolves through transition, the control factor, m, decreases from m = 2, associated with two-dimensional fully developed dunes, to m = 1, associated primarily with in-phase waves. The way to predict the value of the control factor, m, is drawn from a previously published criterion for delineating the upper regime and is calibrated with experimental data. On several data from flumes and rivers, the routing is tested and compared with other methods from the literature. It appears that the new routing is the most robust because it allows researchers to obtain low averages of the discrepancy ratio for a wide range of ratios between the water depth and the median sediment diameter. On a selection of contrasted freshet events, the new routing allows for the capture of the primary dynamic of the flow resistance decrease.