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.     

Numerical Model for Local Scour under Offshore Pipelines

A numerical model, based on potential-flow theory is proposed for simulating the equilibrium scour hole formed by unidirectional flow underneath offshore pipelines. The model employs a finite-difference method to solve the Laplace equation in terms of velocity potential in a curvilinear coordinate system. A boundary adjustment technique based on the Newton-Raphson method is used to calculate the free boundary formed by the eroded seabed by means of the equilibrium of all forces acting on a sediment particle on a sloping bed. Because the solution of flow field and adjustment of the seabed topography are carried out in an iterative manner, the model takes into account the interactions between the flow, pipe, and the seabed. The comparison of the present model with empirical formulas on the prediction of the maximum scour depth indicates that the present model is useful for approximate estimation of scour depth at a pipeline on the seabed for the case of clear-water scour.     

Coherent Structures in the Flow Field around a Circular Cylinder with Scour Hole

Large-eddy simulation (LES) and laboratory-flume visualizations were used to investigate coherent structures present in the flow field around a circular cylinder located in a scour hole. The bathymetry corresponds to equilibrium scour conditions and is fixed in LES. The flow parameters in the simulation correspond to the experimental conditions in which the approach flow is fully turbulent. Detailed consideration is given to the interaction of the horseshoe vortex (HV) system within the scour hole with the detached shear layers formed from the cylinder, and the near bed turbulence. It is found that the overall structure of the HV system varies considerably in space and time, though a large, relatively stable, primary necklace vortex is present at practically all times inside the scour hole. The simulation captures the presence of bimodal chaotic oscillations inside the HV system, as well as the sharp increase in the resolved turbulent kinetic energy levels and pressure fluctuations reported in prior experimental investigations. High levels of the mean bed shear stress are observed beneath the primary necklace vortex, especially over the region where the bimodal oscillations are strong, as well as beneath the small junction vortex at the base of the cylinder. It is also found that the detachment and advection of patches of vorticity from the downstream part of the legs of the necklace vortices can induce large instantaneous bed shear stress values. When the critical bed shear stress value for sediment entrainment on a flat surface is adjusted for bed slope effects, the LES simulation correctly predicts that the distribution of the mean bed shear stress is consistent with equilibrium scour conditions.     

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.     

Measured and Simulated Flow near a Submerged Spur Dike

To improve knowledge of the flow and scour processes associated with spur dikes, three-dimensional flow velocities were measured at 2,592 points using an acoustic Doppler velocimeter over a fixed flat bed with a trapezoidal shaped submerged spur dike in a laboratory flume. General velocity distribution and detailed near field flow structures were revealed by the measurements and numerical simulations performed using a free surface turbulent flow model with a k–ε closure scheme. The three-dimensional flow separation characterized in this study was found to yield forces on the bed that were significantly different from nonsubmerged vertical obstructions that have been measured in other studies. Values of bed shear stress derived from both measured and simulated values were similar but indicated that local scour would be initiated in one rather than in the two locations of initial local scour measured in previous experiments with a similar flow.     

Phenomenological Characterization of Vortex Induced Scour

A series of experiments, investigating vortex induced scour formation, show that the scour hole geometry induced by a rotating fluid flow, can be explained within the concept that the granular bed is eroded as the imposed shear stress overcomes a certain critical value inherent in the liquid/bed pair. The situation is seemingly kindred to that observed in the pseudoplastic Bingham fluids and the first experiment, investigating cylindrical scour, is conducted using a high viscous Bingham fluid with a low yield stress. Two other experiments investigating the two and three dimensional scour formation for the stationary flow regime are also conducted using a fluid and grain substrate pair. A simple model for vortex induced scour is derived for an infinitely long axisymmetric vortex with circular streamlines surrounded by a deformable boundary. The Navier–Stokes equations are solved by using a polynomial approximation of the tangential velocity. The model is then extended to two and three dimensional stationary flow situations by considering the torque balance between the fluid motion and the critical bed shear stress. The quantified results agree well with experimental results obtained in both the one dimensional Bingham fluid experiment and the two and three dimensional fluid/grain experiments.     

Numerical Simulation of Relatively Wide, Shallow Channels with Erodible Banks

A two-dimensional numerical model was developed to simulate relatively wide, shallow rivers with an erodible bed and banks composed of well-sorted, sandy materials. A moving boundary-fitted coordinate system was used to calculate water flow, bed change, and bank erosion. The cubic interpolated pseudoparticle method was used to calculate flow, which introduced little numerical diffusion. The sediment-transport equation for the streamline and transverse transport was used to estimate bed and bank evolution over time, while considering the secondary flow. Bank erosion was simulated when the gradient in the cross-sectional direction of the banks was steeper than the submerged angle of repose because of bed erosion near the banks. The numerical model reproduced the features of central bars well, such as bar growth, channel widening due to divergence of the flow around the bars, scour holes at the lee of the bars, and the increase of bar size with time. These features were in accordance with the observations for laboratory experiments. It also reproduced the features of braided rivers, such as the generation of new channels and abandonment of old channels, the bifurcation and confluence of channels, and the lateral migration of the channels. The model showed that the sediment discharge rate fluctuated with time, one of the dynamic features observed in braided channels.     

Numerical Modeling of Three-Dimensional Flow Field Around Circular Piers

A three-dimensional numerical model FLUENT is used to simulate the separated turbulent flow around vertical circular piers in clear water. Computations are performed using different turbulence models and results are compared with several sets of experimental data available in the literature. Despite commonly perceived weakness of the k-ε model in resolving three-dimensional (3D) open channel and geophysical flows, several variants of this turbulence model are found to have performed satisfactorily in reproducing the measured velocity profiles. However, model results obtained using the k-ε models show some discrepancy with the measured bed shear stress. The Reynolds stress model performed quite well in simulating velocity distribution on flat bed and scour hole as well as shear stress distribution on flat bed around circular piers. The study demonstrates that a robust 3D hydrodynamic model can effectively supplement experimental studies in understanding the complex flow field and the scour initiation process around piers of various size, shape, and dimension.     

Sediment Threshold under Stream Flow on Horizontal and Sloping Beds

A model is presented to determine the threshold shear stress for noncohesive sediment (uniform and nonuniform) motion on horizontal and stream-wise sloping sedimentary beds, under a unidirectional steady-uniform streamflow. Hydrodynamic and particle-mechanic forces on a solitary sediment particle, resting over a sedimentary bed under the slip-spinning condition, are analyzed including the effect of turbulent fluctuations. Hydrodynamic forces such as drag, shear lift, and Magnus lift are taken into consideration. The drag coefficient is determined using an empirical formula. The inclusion of Magnus lift is significant because spherical particles spin just before dislodging downstream from their original position due to the differential hydrodynamic force along the vertical. The experimental data of sediment threshold are used to calibrate the model making the lift coefficient as a free parameter. The dependency of normalized threshold shear stress on particle parameter for various angles of repose and stream-wise bed slopes is presented graphically. The results obtained using the present model are compared with the curves proposed by different investigators and the experimental data of sediment (uniform and nonuniform) threshold for horizontal and stream-wise sloping beds.     

Flow Velocity and Pier Scour Prediction in a Compound Channel: Big Sioux River Bridge at Flandreau, South Dakota

The two-dimensional (2D) depth-averaged river model Finite-Element Surface-Water Modeling System (FESWMS) was used to predict flow distribution at the bend of a compound channel. The site studied was the Highway 13 bridge over the Big Sioux River in Flandreau, South Dakota. The Flandreau site has complex channel and floodplain geometry that produces unique flow conditions at the bridge crossing. The 2D model was calibrated using flow measurements obtained during two floods in 1993. The calibrated model was used to examine the hydraulic and geomorphic factors that affect the main channel and floodplain flows and the flow interactions between the two portions. A one-dimensional (1D) flow model of the bridge site was also created in Hydrologic Engineering Centers River Analysis System (HEC-RAS) for comparison. Soil samples were collected from the bridge site and tested in an erosion function apparatus (EFA) to determine the critical shear stress and erosion rate constant. The results of EFA testing and 2D flow modeling were used as inputs to the Scour Rate in Cohesive Soils (SRICOS) method to predict local scour at the northern and southernmost piers. The sensitivity of predicted scour depth to the hydraulic and soil parameters was examined. The predicted scour depth was very sensitive to the approach-flow velocity and critical shear stress. Overall, this study has provided a better understanding of 2D flow effects in compound channels and an overall assessment of the SRICOS method for prediction of bridge pier scour.     

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.     

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.     

Local Scour and Riprap Stability at an Abutment in a Degrading Bed

The paper reports on an experimental investigation concerning two important issues: (1) local scour and (2) riprap stability at a 45° wing-wall abutment in a degrading river bed of noncohesive sediment. The abutment considered was short (that is abutment length/flow depth <1). From the experimental observations, no influence of abutment inclusion on bed degradation was evident, as bed profiles with and without abutment were quite identical apart from the immediate vicinity of the abutment. Total scour depth at an abutment is found to be the maximum abutment scour depth in addition to the reduction of bed elevation due to bed degradation. The maximum abutment scour depth can be estimated from the equation given by Kandasamy and Melville in their 1998 paper. For scouring time beyond 24?h, the local abutment scour depth remains independent of time. In a degrading bed, the bed forms cause edge failure of the riprap at an abutment when the dunes propagate over the riprap layer. Initially, the dune height is significant causing the maximum damage of riprap layer. As the flow velocity reduces, the resulting bed-shear stress diminishes with the degrading bed and gradually the formation of dunes ceases. An additional experiment reveals that the damaged riprap layer is significantly vulnerable against a subsequent flood accompanied by large dunes.     

The modeling of transverse solids motion in rotary kilns

Mathematical models have been developed to predict the conditions giving rise to the different forms of transverse bed motion in a rotary cylinder: slumping, rolling, slipping, cascading, cataracting, and centrifuging. Model predictions of the boundaries between these modes of bed motion compare well with previously reported measurements, and can be represented conveniently on a Bed Behavior Diagram which is a plot of pct fill against Froude number (or bed depthvs rotational speed). The location of the boundaries is shown to depend on material variables which characterize frictional conditions in the bed. For the slumping/rolling boundary these are primarily the shear angle and the limiting wedge angle which defines the solids involved in a slump. For the slipping/slumping and slipping/rolling boundaries the governing material variables are the bed/wall friction angle and the upper angle of repose and dynamic angle of repose, respectively. Similarly, the location of the other boundaries related to cascading and cataracting is determined by the dynamic angle of repose. Complete Bed Behavior Diagrams have been calculated for solids having different particle size and particle shape rotated in cylinders having different diameters. H. Henein, formerly Graduate Student, Department of Metallurgical Engineering, University of British Columbia     

Simulation of Scour Process in Plunging Pool of Loose Bed-Material

The scouring process in a plunge pool of loose bed with uniform bed-materials due to a two-dimensional plane impinging jet was simulated computationally. The finite-element-based unsteady three-dimensional model, CCHE3D, with k-ε turbulence closure was employed to solve the flow field. It has long been recognized that the unsteady behavior of the turbulent jet fluctuation plays an important role in scouring and transporting sediment in the plunge pool. In order to model this phenomenon realistically, one has to consider the effects of both shear stress and the life force on sediment particles due to pressure fluctuation. The latter has been taken into account by using empirical relationships of flume data. Both of these effects have been incorporated in the nonequilibrium sediment transport model consisting of sediment pickup rate and step length adopted for the jet scour problem. The model constant relating to the fluctuating lift force was calibrated using an empirical equation to predict the quasi-equilibrium scour depth. The results simulated by the model proposed here agree reasonably well with experimental data.     

3D Unsteady RANS Modeling of Complex Hydraulic Engineering Flows. II: Model Validation and Flow Physics

A chimera overset grid flow solver is developed for solving the unsteady Reynolds-averaged Navier-Stokes (RANS) equations in arbitrarily complex, multiconnected domains. The details of the numerical method were presented in Part I of this paper. In this work, the method is validated and applied to investigate the physics of flow past a real-life bridge foundation mounted on a fixed flat bed. It is shown that the numerical model can reproduce large-scale unsteady vortices that contain a significant portion of the total turbulence kinetic energy. These coherent motions cannot be captured in previous steady three-dimensional (3D) models. To validate the importance of the unsteady motions, experiments are conducted in the Georgia Institute of Technology scour flume facility. The measured mean velocity and turbulence kinetic energy profiles are compared with the numerical simulation results and are shown to be in good agreement with the numerical simulations. A series of numerical tests is carried out to examine the sensitivity of the solutions to grid refinement and investigate the effect of inflow and far-field boundary conditions. As further validation of the numerical results, the sensitivity of the turbulence kinetic energy profiles on either side of the complex pier bent to a slight asymmetry of the approach flow observed in the experiments is reproduced by the numerical model. In addition, the computed flat-bed flow characteristics are analyzed in comparison with the scour patterns observed in the laboratory to identify key flow features responsible for the initiation of scour. Regions of maximum shear velocity are shown to correspond to maximum scour depths in the shear zone to either side of the upstream pier, but numerical values of vertical velocity are found to be very important in explaining scour and deposition patterns immediately upstream and downstream of the pier bent.     

Hydraulics of Submerged Jet Subject to Change in Cohesive Bed Geometry

The results of an experimental investigation of the time variation of scour hole and the flow characteristics of the quasi-equilibrium state of scour of a cohesive bed downstream of an apron due to a submerged horizontal jet issuing from a sluice opening are presented. Experiments were carried out with natural cohesive sediment for various sluice openings, jet velocities, and lengths of apron. Attempts are made to explain the similarity existing either in the process of scour or in the scour profiles that the scour holes follow downstream of an apron. The scour profiles at different times follow a particular geometrical similarity and can be expressed by a polynomial using relevant parameters. The characteristic parameters affecting the time variation of scour depth are identified based on the physical reasoning and dimensional analysis. An equation for time variation of maximum scour depth is obtained empirically. The diffusion characteristics of the submerged jet, growth of boundary layer thickness, velocity distribution within the boundary layer, and shear stress at the quasi-equilibrium state of scour are also investigated. The expression of shear stress is obtained from the solution of the von Kármán momentum integral equation.     

Reynolds Stress and Bed Shear in Nonuniform Unsteady Open-Channel Flow

Expressions for the Reynolds stress and bed shear stress are developed for nonuniform unsteady flow in open channels with streamwise sloping beds, assuming universal (logarithmic) velocity distribution law and using the Reynolds and continuity equations of two-dimensional open-channel flow. The computed Reynolds stress distributions are in agreement with experimental data.     

Erosion Function Apparatus for Scour Rate Predictions

Scour is the number one cause of bridge failures. Scour in coarse grained soils (sand, gravel) is relatively well known, but scour in fine grained soils (silt, clay) and weak rock is not. In coarse grained soils, scour takes place very rapidly and the scour rate is rarely an issue because one flood is likely to create the maximum scour depth. In fine grained soils, the scour process is much slower; as a result, even after a hundred years, a bridge may not experience the maximum depth of scour. Therefore, in fine grained soils it becomes necessary to predict the rate at which scour takes place. A new apparatus called the EFA (Erosion Function Apparatus; 〈http:∕/tti.tamu.edu∕geotech∕scour〉) has been built and tested to measure the erosion rate of fine grained soils; the EFA can also be used to measure the erosion rate of coarse grained soils if necessary. The end of a Shelby tube sample from the bridge site is fitted through a tight opening at the bottom of a pipe with a rectangular cross section. Water flows through the pipe and erodes the soil sample, which protrudes 1 mm above the bottom of the pipe. The rate at which the sample erodes is measured, and the shear stress imposed by the water on the soil is calculated. The plot of erosion rate versus shear stress is the result of an EFA test. It indicates the critical shear stress at which erosion starts and the rate of erosion beyond that shear stress. This function can be used to predict the rate of scour at a bridge.     

Flow Turbulence over Fixed and Weakly Mobile Gravel Beds

Characteristics of turbulence structure in quasi-2D flows with static and weakly mobile gravel beds are presented. Three sets of measurements with acoustic Doppler velocimeters in an irrigation canal were used: two with subcritical bed shear stress (static beds) and one with the bed shear stress τo close to critical τoc (weakly mobile bed). The analyses included vertical distributions of local mean velocities, turbulence intensities, turbulent shear stresses, velocity auto- and cross-spectra, the quadrant method, and high-order velocity moments. A number of properties of turbulence intensities, high-order moments, streamwise bursting parameters, and velocity spectra appeared to be similar for all three flows, but some properties were different. The most important one was an observed reduction in the von Kármán constant for the flow with weakly mobile bed. Comparison of these results with other studies and analogies with drag-reducing flows suggest that at τo∕τoc ≈ 1 the drag on the bed for a given granular material should be minimized.