Modeling Noncohesive Suspended Sediment Transport in Stream Channels Using an Ensemble-Averaged Conservation Equation

The governing conservation equation for the transport of noncohesive suspended sediment in erodible channels is recognized as a stochastic partial differential equation due to the uncertainties in the parameters, and a deterministic ensemble-averaged equation is developed. Variables in this one-dimensional equation are represented as averaged quantities, and their covariances are also taken into account. Lateral inflows and deposition and entrainment of sediment are incorporated in the formulation. A hypothetical test problem is constructed to examine the model behavior. Manning’s coefficient, bed slope and bottom width are taken as the primary random parameters. Results from the solution of the ensemble-averaged equation are compared to results from Monte Carlo simulations. For comparison purposes, predicted values are also obtained by solving the deterministic transport equation without the covariance terms. It is found that predictions obtained from this latter approach deviate significantly from Monte Carlo simulation results. On the other hand, the ensemble-averaged predictions compare favorably to the Monte Carlo simulation results indicating that this promising technique needs further exploration.     

3D Numerical Simulation for Water Flows and Sediment Deposition in Dam Areas of the Three Gorges Project

This paper presents a three-dimensional (3D) mathematical model for suspended load transport in turbulent flows. Based on the stochastic theory of turbulent flow proposed by Dou, numerical schemes of Reynolds stresses for anisotropic turbulent flows are obtained. Instead of a logarithmic law, a specific wall function is used to describe the velocity profile close to wall boundaries. The equations for two-dimensional suspended load motion and sorting of bed material have been improved for a 3D case. Numerical results are in good agreement with the measured data of the Gezhouba Project. The present method has been employed to simulate sediment erosion and deposition in the vicinity of the Three Gorges Dam. The size distribution of the deposits and bed material, and flow and sediment concentration at different times and elevations, are predicted. The results agree well with the observations in physical experiments. Thus, a new method is established for 3D simulation of sediment motion in the vicinity of dams.     

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.     

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.     

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.     

Unified View of Sediment Transport by Currents and Waves. II: Suspended Transport

The problem of suspended sediment transport in river and coastal flows is addressed. High-quality field data of river and coastal flows have been selected and clustered into four particle size classes (60–100, 100–200, 200–400, and 400–600?μm). The suspended sand transport is found to be strongly dependent on particle size and on current velocity. The suspended sand transport in the coastal zone is found to be strongly dependent on the relative wave height (Hs/h), particularly for current velocities in the range 0.2–0.5?m/s. The time-averaged (over the wave period) advection–diffusion equation is applied to compute the time-averaged sand concentration profile for combined current and wave conditions. Flocculation, hindered settling, and stratification effects are included by fairly simple expressions. The bed-shear stress is based on a new bed roughness predictor. The reference concentration function has been recalibrated using laboratory and field data for combined steady and oscillatory flow. The computed transport rates show reasonably good agreement (within a factor of 2) with measured values for velocities in the range of 0.6–1.8?m/s and sediments in the range of 60–600?μm. The proposed method underpredicts in the low-velocity range (<0.6?m/s). A new simplified transport formula is presented, which can be used to obtain a quick estimate of suspended transport. The modeling of wash load transport in river flow based on the energy concept of Bagnold shows that an extremely large amount of very fine sediment (clay and very fine silt) can be transported by the flow.     

Statistically Derived Bedload Formula for Any Fraction of Nonuniform Sediment

Based on a method of combining stochastic processes with mechanics, a new bedload formula for the arbitrary kth size fraction of nonuniform sediment is theoretically developed by using a stochastic model of sediment exchange and the probabilistic distribution of fractional bedload transport rates. The relations, proposed recently by Sun, for the probability of fractional incipient motion and for the average velocity of particle motion are introduced to bedload formula. Plenty of experimental data for the bedload transport rate of uniform sediment are used to determine two constants. The theoretical bedload formula for any fraction of nonuniform sediment possesses several advantages, including a clear physical concept, a strict mathematical derivation, and a self-adaptability to uniform sediment. The formula is verified with natural data expressing the transport of nonuniform sediment under full motion in laboratory flume. The result shows that the experimental observations agree well with the predicted fractional bedload transport rates. Comparison of the theory with field data finds that the proposed formula still applies to partial transport of bedload in gravel-bed streams as long as the immobile percentage of bed material is taken into account.     

On a stochastic theory of grain growth

A theory of isothermal grain growth in polycrystalline solids, which treats grain growth as a statistical or stochastic process, is presented. In this treatment deterministic equation for the rate of grain growth is made stochastic by the addition of a “noise” term. The noise or fluctuations are used to model the effect of complex topologically connected structure of the specimen on grain boundary motion, in addition to such motion directed by surface tension forces. Such considerations lead to a second order partial differential equation (Fokker-Planck equation) for the grain size distribution. Many of the major attributes of grain growth are shown to be a natural consequence of this equation. The solution obtained for this equation is a modified form of Rayleigh distribution which in many respects is similar of log normal distribution. Grain size distribution is also obtained from independent statistical consideration and is shown to be approximately log normal. Extension of the mathematical analysis to the case of Ostwald ripening is indicated.     

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.     

Numerical Model of Sediment Pulses and Sediment-Supply Disturbances in Mountain Rivers

Sediment pulses in rivers can result from many mechanisms including landslides entering from side slopes and debris flows entering from tributaries. Artificial sediment pulses can be caused by the removal of a dam. This paper presents a numerical model for the simulation of gravel bedload transport and sediment pulse evolution in mountain rivers. A combination of the backwater and quasi-normal flow formulations is used to calculate flow parameters. Gravel bedload transport is calculated with the surface-based bedload equation of Parker in 1990. The Exner equation of sediment continuity is used to express the mass balance at different grain size groups and lithologies, as well as the abrasion of gravel. The river is assumed to have no geological controls such as bedrock outcrops and immobile boulder pavements. The results of nine numerical experiments designed to study various key parameters relevant to the evolution of sediment pulses are reported here. Results of the numerical runs indicate that the evolution of sediment pulses in mountain rivers is dominated by dispersion rather than translation. Here dispersion is an expression for the observation that a sediment pulse aggrades both upstream and downstream of its apex whereas its amplitude decreases in time. The results also indicate that grain abrasion is an important and yet often neglected mechanism in removing the excess sediment associated with pulse inputs from some mountain rivers.     

Movement of Finite Amplitude Sediment Accumulations

The movement of finite amplitude sediment accumulations is studied using a simple advection-diffusion relation derived from the sediment continuity equation and using some heuristic reasoning. The movement of a finite amplitude sediment accumulation is found to be strongly diffusive with a small advection component due to the increase in transport rate of the sediment accumulation relative to the transport rate of the original bed material. A semianalytical solution to the advection-diffusion equation is found and the equation is applied to two laboratory experiments. The equation is found to predict the general movement of finite amplitude sediment accumulations with a minimal number of parameters.     

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.     

Influence of Coherent Flow Structures on the Dynamics of Suspended Sediment Transport in Open-Channel Flow

The influence of suspended sediments on coherent flow structures has been studied by simultaneously measuring the longitudinal and vertical components of the instantaneous velocity vector and the instantaneous suspended particle concentration with an acoustic particle flux profiler. The measurements were carried out in clear water and in particle-laden open-channel flows. In both cases, they clearly show the predominance of ejection and sweep phases that are part of a burst cycle. The analysis further demonstrates the importance of the ejection and sweep phases in sediment resuspension and transport. Ejections pick up the sediment at the bed and carry it up through the water column close to the surface. It is shown that ejections and sweeps are in near equality in the near-bottom layer, whereas ejections clearly dominate in the remaining water column. The implications of these results for sediment transport dynamics are discussed.     

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.     

An improved method for evaluating the seasonal variability of total suspended sediment flux field in the Yellow and East China Seas

The suspended sediment flux field in the Yellow and East China Seas (YECS) displays its seasonal variability. A new method is introduced in this paper to obtain the flux field via retrieval of ocean color remote sensing data, statistical analysis of historical suspended sediment concentration data, and numerical simulation of three-dimensional (3D) flow velocity. The components of the sediment flux field include (i) surface suspended sediment concentration inverted from ocean color remote sensing data; (ii) vertical distribution of suspended sediment concentration obtained by statistical analysis ofhistorical observation data; and (iii) 3D flow field modeled by a numerical simulation. With the improved method, the 3D suspended sediment flux field in the YECS has been illustrated. By comparison with the suspended sediment flux field solely based on the numerical simulation of a suspended sediment transport model, the suspended sediment flux field obtained by the improved method is found to be more reliable. The 3D suspended sediment flux field from ocean colour remote sensing and in situ observation are more closer to the reality. Furthermore, by quantitatively analyzing the newly obtained suspended sediment flux field, the quantity of sediment erosion and deposition within the different regions can be evaluated. The sediment exchange between the Yellow Sea and the East China Sea can be evident. The mechanism of suspended sediment transport in the YECS can be better understood. In particular, it is suggested that the long-term transport of suspended sediment is controlled mainly by the circulation pattern, especially the current in winter.     

Two-Dimensional Nonequilibrium Noncohesive and Cohesive Sediment Transport Model

The purpose of this paper is to develop an unsteady 2D depth-averaged model for nonuniform sediment transport in alluvial channels. In this model, the orthogonal curvilinear coordinate system is adopted; the transport mechanisms of cohesive and noncohesive sediment are both embedded; the suspended load and bed load are treated separately. In addition, the processes of hydraulic sorting, armoring, and bed consolidation are also included in the model. The implicit two-step split-operator approach is used to solve the flow governing equations and the coupling approach with iterative method are used to solve the mass-conservation equation of suspended sediment, mass-conservation equation of active-layer sediment, and global mass-conservation equation for bed sediment simultaneously. Three sets of data, including suspension transport, degradation and aggradation cases for noncohesive sediment, and aggradation, degradation, and consolidation cases for cohesive sediment, have been demonstrated to show the rationality and accuracy of the model. Finally, the model is applied to evaluate the desilting efficiency for Ah Gong Diann Reservoir located in Taiwan to show its applicability.     

Sand Transport on Steeply Sloping Plane and Rippled Beds

Experiments on sand transport have been carried out in the Sloping Sediment Duct at HR Wallingford. The aim of the experiments was to investigate sediment transport mechanisms, for sand of varying degree of grading, on sloping beds. The Sloping Sediment Duct is a steady flow, recirculating duct, capable of generating mean flow speeds of up to 1 m/s and tilting to +/?30°. Twenty-two tests with two different sediments were conducted. Both sediments had a median grain size of about 0.23 mm but different standard deviations. Bed slopes up to +/?20° were used in the experiments. The results show that bedforms have a significant effect on the transport rate. Since the bedforms, in turn, are affected significantly by the slope, the relation between transport rate and slope is not a monotonic function. Maximum suspended transport rates were attained for downslope flows at angles of about 10°. The transport rate for widely graded sediment was significantly larger than that for well-sorted sediment for almost all flows and slopes.     

Three-Dimensional Mathematical Model of Suspended-Sediment Transport

This paper presents the basic equations for a mathematical model of sediment-laden flow in a nonorthogonal curvilinear coordinate system. The equations were derived using a tensor analysis of two-phase flow and incorporate a natural variable-density turbulence model with nonequilibrium sediment transport. Correspondingly, a free-surface and the bottom sediment concentration are employed to provide the boundary conditions at the river surface and the riverbed. The finite analytic method is used to solve the equations of mass and momentum conservation and also the transport equation for suspended sediment. To demonstrate the method, the sediment deposition for the Three Gorges Project is considered. The mathematical model specifies the boundary conditions for the inlet and outlet using data from physical model experiments. The results for the mathematical model were tested against laboratory measurements from the physical model experiment. Good agreement and accuracy were obtained.