Effect of Bed Armoring on Bed Topography of Channel Bends

The two-dimensional numerical model previously developed by the writers for modeling the bed variations in a channel bend with uniform sediment is upgraded to incorporate the nonuniformity of sediment particles as well as bed armoring. In this model, the two-dimensional, depth-averaged, unsteady flow equations along with the bed-load mass conservation equation are solved in a body-fitted coordinate system by using the Beam and Warming alternating-direction implicit (ADI) scheme. A one-dimensional bed surface armoring approach is extended herein for application to a two-dimensional domain. The model is applied to a 180° bend with a constant radius under unsteady flow conditions. Numerical simulations are carried out to study the effect of bed armoring on the bed deformations in channel bends. Results show that bed armoring reduces scour in channel bends.     

3D Numerical Modeling of Flow and Sediment Transport in Laboratory Channel Bends

The development of a fully three-dimensional finite volume morphodynamic model, for simulating fluid and sediment transport in curved open channels with rigid walls, is described. For flow field simulation, the Reynolds-averaged Navier–Stokes equations are solved numerically, without reliance on the assumption of hydrostatic pressure distribution, in a curvilinear nonorthogonal coordinate system. Turbulence closure is provided by either a low-Reynolds number k?ω turbulence model or the standard k?ε turbulence model, both of which apply a Boussinesq eddy viscosity. The sediment concentration distribution is obtained using the convection-diffusion equation and the sediment continuity equation is applied to calculate channel bed evolution, based on consideration of both bed load and suspended sediment load. The governing equations are solved in a collocated grid system. Experimental data obtained from a laboratory study of flow in an S-shaped channel are utilized to check the accuracy of the model’s hydrodynamic computations. Also, data from a different laboratory study, of equilibrium bed morphology associated with flow through 90° and 135° channel bends, are used to validate the model’s simulated bed evolution. The numerically-modeled fluid and sediment transportation show generally good agreement with the measured data. The calculated results with both turbulence models show that the low-Reynolds k?ω model better predicts flow and sediment transport through channel bends than the standard k?ε model.     

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 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.     

Oblique Flow over Dredged Channels.?I: Flow Description

A 3D investigation of flow across long, straight channels aligned obliquely to the flow direction has been conducted. The applied mathematical model solves the Reynolds-averaged Navier-Stokes equations using a k-ε model for turbulence closure in a curvilinear coordinate system. The uniformity along the channel alignment allows the three momentum equations to be solved in a 2D computational domain. With respect to a steady current entering a channel obliquely, two important flow features arise: (1) The flow will be refracted in the direction of the channel alignment, which may be described by depth-averaged models; and (2) a secondary flow will be introduced due to shear in the velocity profile. This can only be described using a 3D approach. The secondary flow will cause a horizontal deflection of streamlines over the vertical. Only by capturing the 3D flow behavior can the direction and magnitude of the bed shear stress be well modeled. When crossing a channel obliquely, the flow is gradually accelerated in the direction of the channel alignment. Results of the numerical flow model are compared with existing experimental data and good agreement is found.     

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.     

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

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

Momentum Transport in Sharp Open-Channel Bends

Flow in open-channel bends is characterized by cross-stream circulation, which redistributes the velocity and the boundary shear stress and thereby shapes the characteristic bed topography. Besides a center-region cell, classical helical motion, a weaker counterrotating outer-bank cell often exists. In spite of its engineering importance, the mechanisms underlying distributions of the velocity and the boundary shear stress in open-channel bends, and especially the role of both circulation cells, are not yet fully understood. In order to investigate these mechanisms, an evaluation is made of the various terms in the momentum equations based on the data measured, which gave the following results. The outer-bank cell forms a buffer layer that protects the outer bank from any influence of the center-region cell and keeps the core of maximum velocity a distance from the bank. Advective momentum transport by the center-region cell is a dominant mechanism; it significantly contributes to the observed outward shift of the downstream velocity and the bed shear stress and to flattening of the vertical profiles of the velocity. This important advective momentum redistribution has to be included in the depth-integrated flow models often used in engineering practice. Commonly used linear models overpredict the effects of the center-region cell. Based on results of the analysis of experimental data, these models are extended by accounting for the feedback between the center-region cell and the downstream velocity. The nonlinear model obtained clearly reveals the mechanisms of the center-region cell and its advective momentum transport. An analysis of nonlinear model results confirms and complements the analysis of experimental data. A true quasithree-dimensional flow model is obtained by coupling this nonlinear model to depth-integrated flow models, thus providing an engineering tool for morphodynamical investigations.     

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.     

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.     

Stream Temperature Dynamics in Upland Agricultural Watersheds

A numerical model to compute the free-surface flow hydrodynamics and stream temperature dynamics by solving the depth-averaged, 1D unsteady flow and heat transport equations is presented. The hydrodynamics model considers the effects of arbitrary stream geometry, variable slopes, variable flow regimes, and unsteady boundary conditions. The thermal transport model accounts for the effects of solar radiation, air temperature, relative humidity, cloud cover, wind speed, heat conduction between water and streambed, subsurface flow, and shading by riparian vegetation. The model is verified with measurements in a stream in an upland agricultural watershed located in Indiana. Diurnal variations in the streamflow and stream temperatures are highly transient. The proposed model predicted well the streamflow and stream temperatures that were measured every 15 min over 25 days. The results of this study demonstrate that the solar (shortwave) radiation and subsurface inflow are the most significant contributors to the stream heat budget.     

Multiple Time Scales of Fluvial Processes with Bed Load Sediment and Implications for Mathematical Modeling

Fluvial bed load transport is often considered to assume a capacity regime exclusively determined by local flow conditions, but its applicability in naturally occurring unsteady flows remains to be theoretically justified. In addition, mathematical river models are often decoupled, being based on simplified conservation equations and ignoring the feedback impacts of bed deformation to a certain extent. So far whether the decoupling could have considerable impacts on the fluvial processes with bed load transport remains poorly understood. This paper presents a theoretical investigation of both issues. The multiple time scales of fluvial processes with bed load sediment are evaluated to examine the applicability of bed load transport capacity and decoupled models. Numerical case studies involving active bed load transport by highly unsteady flows complement the analysis of the time scales. It is found that bed load transport can sufficiently rapidly adapt to capacity in line with local flow because sediment exchange with the bed overwhelms the advection of bed load sediment by the mean flow. The present work provides theoretical justification of the concept of bed load transport capacity in most circumstances, which is underpinned by existing observations of bed load transport by flash floods. For fluvial processes with bed load transport, the feedback impacts of bed deformation are limited; therefore, decoupled modeling is, in this sense, appropriate.     

Efficient Treatment of the Vardy–Brown Unsteady Shear in Pipe Transients

An accurate, simple, and efficient approximation to the Vardy–Brown unsteady friction equation is derived and shown to be easily implemented within a one-dimensional characteristics solution for unsteady pipe flow. For comparison, the exact Vardy–Brown unsteady friction equation is used to model shear stresses in transient turbulent pipe flows and the resulting waterhammer equations are solved by the method of characteristics. The approximate Vardy–Brown model is more computationally efficient (i.e., requires one-sixth the execution time and much less memory storage) than the exact Vardy–Brown model. Both models are compared with measured data from different research groups and with numerical data produced by a two-dimensional turbulence waterhammer model. The results show that the exact Vardy–Brown model and the approximate Vardy–Brown model are in good agreement with both laboratory and numerical experiments over a wide range of Reynolds number and wave frequencies. The proposed approximate model only requires the storage of flow variables from a single time step while the exact Vardy–Brown model requires the storage of flow variables at all previous time steps and the two-dimensional model requires the storage of flow variables at all radial nodes.     

Dam Break in Channels with 90° Bend

In practice, dam-break modeling is generally performed using a one-dimensional (1D) approach for its limited requirements in data and computation. However, for valleys with multiple sharp bends, such a 1D model may fail for predicting as well the maximum water level as the wave arrival time. This paper presents an experimental study of a dam-break flow in an initially dry channel with a 90° bend, with refined measurements of water level and velocity field. The measured data are compared to some numerical results computed with finite-volume schemes associated with Roe-type flux calculation. The 1D approach reveals the expected limits, while a full two-dimensional (2D) approach provides fine level prediction and rather satisfactory information about the arrival time. A hybrid approach is now proposed, mixing the 1D model for the straight reaches and local 2D models for the bends. The compatibility of the Roe fluxes at the interfaces requires a careful formulation, but the resulting scheme seems able to capture reflection and diffraction processes in such a way that the results are really good in what concerns the water level.     

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.     

Bend-Flow Simulation Using 2D Depth-Averaged Model

The purpose of this paper is to present a 2D depth-averaged model for simulating and examining flow patterns in channel bends. In particular, this paper proposes a 2D depth-averaged model that takes into account the influence of the secondary flow phenomenon through the calculation of the dispersion stresses arisen from the integration of the products of the discrepancy between the mean and the true velocity distributions. The proposed model uses an orthogonal curvilinear coordinate system to efficiently and accurately simulate the flow field with irregular boundaries. As for the numerical solution procedure, the two-step split-operator approach consisting of the dispersion step and the propagation step with the staggered grid is used to numerically solve the flow governing equations. Two sets of experimental data from de Vriend and Koch and from Rozovskii were used to demonstrate the model's capabilities. The former data set was from a mildly curved channel, whereas the latter was from a sharply curved channel. The simulations considering the secondary flow effect agree well with the measured data. Furthermore, an examination of the dispersion stress terms shows that the dispersion stresses play a major role in the transverse convection of the momentum shifting from the inner bank to the outer bank for flows in both mild and sharp bends.     

Physically Based Model for Simulating Flow in Furrow Irrigation. I: Model Development

Researchers developed several mathematical models for simulating furrow irrigation using the Saint-Venant equations. Most of these irrigation models use numerical techniques to solve these equations, which in general, require extensive programming and computational skills. Moreover, several of these models consider uniform soil and use empirical equations for modeling infiltration. In this article, a physically based furrow irrigation model was presented for simulating flow in irrigated furrows under both uniform and layered soils. The model consisted of an overland flow and an infiltration module that are modeled using analytical solution of the zero-inertia and the Green and Ampt [one-dimensional and two-dimensional infiltration equations) equations, respectively. Furthermore, the infiltration was also modeled using the Kostiakov-Lewis infiltration equation. The model considered all possible furrow shapes and included graphical user interface. The developed model was evaluated using the field data and the model performance was discussed in the second part of the article.     

Fluid Mechanism of Wing Rock for Configurations with Chine-Shaped Forebodies

The increasing demand to employ sharp-edged geometries in airframe design of advanced fighters to meet stealth requirements has nudged research to explore the new aerodynamic and dynamic characteristics of such configurations. In this paper, the wing rock oscillation is numerically simulated for a generic wing-body model consisting of cropped delta wing of 65°-sweep and chine-shaped forebody. The purpose is to develop a complete understanding of the complex flow interactions that drive the wing rock oscillation. The numerical simulation is based upon coupling the unsteady Euler fluid dynamic equations with the rigid-body dynamic equations in roll. A subiteration algorithm is employed to simultaneously solve the coupled equations. The numerical model exhibits a limit cycle oscillation in roll at an angle of attack of 35° with 16° peak-to-peak amplitude in roll angle. The complex interactions of the forebody-induced flow and the wing leading-edge vortices during wing rock are fully investigated and visualized. Analysis of the observed vortex breakdown dynamics during one entire cycle of the oscillation is also presented.     

Numerical Model for Sediment Transport and Bed Degradation in the Yangtze River Channel Downstream of Three Gorges Reservoir

This paper presents a two-dimensional morphological model for unsteady flow and both suspended-load and bed-load transport of multiple grain size to simulate transport of graded sediments downstream from the Three Gorges Reservoir. The model system includes a hydrodynamic module and a sediment module. The hydrodynamic module is based on the depth-averaged shallow water equations in orthogonal curvilinear coordinates. The sediment module describing nonuniform sediment transport is developed to include nonequilibrium transport processes, bed deformation, and bed material sorting. The model was calibrated using field observations through application to a 63-km-long alluvial river channel on the middle Yangtze River in China. A total of 16 size groups and a loose layer method of three sublayers were considered for the transport of the nonuniform bed materials in a long-term simulation. Predictions are compared with preliminary results of field observations and factors affecting the reliability of the simulated results are discussed. The results may be helpful to the development of more accurate simulation models in the future.     

Numerical Model for Channel Flow and Morphological Change Studies

In this paper a depth-integrated 2D hydrodynamic and sediment transport model, CCHE2D, is presented. It can be used to study steady and unsteady free surface flow, sediment transport, and morphological processes in natural rivers. The efficient element method is applied to discretize the governing equations, and the time marching technique is used for temporal variations. The moving boundaries were treated by locating the wet and dry nodes automatically in the cases of simulating unsteady flows with changing free surface elevation in channels with irregular bed and bank topography. Two eddy viscosity models, a depth-averaged parabolic model and a depth-averaged mixing length model, are used as turbulent closures. Channel morphological changes are computed with considerations of the effects of bed slope and the secondary flow in curved channels. Physical model data have been used to verify this model with satisfactory results. The feasibility studies of simulating morphological formation in meandering channels and flows in natural streams with in-stream structures have been conducted to demonstrate its applicability to hydraulic engineering research∕design studies of stream stabilization and ecological quality among other problems.