Role of Artificial Dissipation Scaling and Multigrid Acceleration in Numerical Solutions of the Depth-Averaged Free-Surface Flow Equations

A numerical model is developed for solving the depth-averaged, open-channel flow equations in generalized curvilinear coordinates. The equations are discretized in space in strong conservation form using a space-centered, second-order accurate finite-volume method. A nonlinear blend of first- and third-order accurate artificial dissipation terms is introduced into the discrete equations to accurately model all flow regimes. Scalar- and matrix-valued scaling of the artificial dissipation terms are considered and their effect on the accuracy of the solutions is evaluated. The discrete equations are integrated in time using a four-stage explicit Runge–Kutta method. For the steady-state computations, local time stepping, implicit residual smoothing, and multigrid acceleration are used to enhance the efficiency of the scheme. The numerical model is validated by applying it to calculate steady and unsteady open-channel flows. Extensive grid sensitivity studies are carried out and the potential of multigrid acceleration for steady depth-averaged computations is demonstrated.     

Weighted Averaged Flux-Type Scheme for Shallow-Water Equations with Fractional Step Method

A numerical model describing two-dimensional fluid motions has been developed on an unstructured grid system. By using a fractional step method, a two-dimensional problem governed by the two-dimensional shallow-water equations is treated as two one-dimensional problems. Thus it is possible to simulate two-dimensional numerical problems with a higher computational efficiency. One-dimensional problems are solved by using an upwind total variation diminishing version of the second-order weighted averaged flux method with an approximate Riemann solver. Numerical oscillations commonly observed in second-order numerical schemes are controlled by exploiting a flux limiter. For the general purpose, the model can simulate on an arbitrary topography, treat a moving boundary, and resolve a shock. Five ideal and practical problems are tested. Very accurate results are observed.     

Case Study: Malpasset Dam-Break Simulation using a Two-Dimensional Finite Volume Method

The accuracy, stability, and reliability of a numerical model based on a Godunov-type scheme are verified in this paper, through a comparison between calculated results and observed data for the Malpasset dam-break event, which occurred in southern France in 1959. This event is an unique opportunity for code validation because of the availability of extensive field data on the flooding wave due to the dam failure. In the code the shallow water equations are discretized using the finite volume method, and the numerical model allows second order accuracy, both in space and time. The classical Godunov approach is used. More specifically, the Harten, Lax, and van Leer Riemann solver is applied. The resulting scheme is of high resolution and satisfies the total variation diminishing condition. For the numerical treatment of source terms relative to the friction slope, a semi-implicit technique is used, while for the source terms relative to the bottom slope a new explicit method is developed and tested.     

Finite-Volume Model for Shallow-Water Flooding of Arbitrary Topography

A model based on the finite-volume method is developed for unsteady, two-dimensional, shallow-water flow over arbitrary topography with moving lateral boundaries caused by flooding or recession. The model uses Roe’s approximate Riemann solver to compute fluxes, while the monotone upstream scheme for conservation laws and predictor-corrector time stepping are used to provide a second-order accurate solution that is free from spurious oscillations. A robust, novel procedure is presented to efficiently and accurately simulate the movement of a wet/dry boundary without diffusing it. In addition, a new technique is introduced to prevent numerical truncation errors due to the pressure and bed slope terms from artificially accelerating quiescent water over an arbitrary bed. Model predictions compare favorably with analytical solutions, experimental data, and other numerical solutions for one- and two-dimensional problems.     

Finite Volume Model for Shallow Water Equations with Improved Treatment of Source Terms

A simple yet precise relation between the flux gradient and the bed slope source term is presented, which produces a net force within the cell with an inclined water surface, but ensures still water condition when there is no flow across the boundaries. The proposed method consists of calculating the pressure term based on the water depths at the cell vertices, which may be computed by a higher order scheme and the bed slope source term by a centered discretization technique. The methodology is demonstrated with a Godunov-type upwind finite volume formulation. The inviscid fluxes are calculated using Roe’s approximate Riemann solver and a second-order spatial accuracy is obtained by implementing multidimensional gradient reconstruction and slope limiting techniques. The accuracy and applicability of the numerical model is verified with a couple of test problems and a real flow example of tidal water movement in a stretch of River Hooghly in India.     

Explicit Integration Method for Extended-Period Simulation of Water Distribution Systems

Extended-period simulation of incompressible and inertialess flow in water distribution systems is normally done using numerical integration techniques, although regression methods are also sometimes employed. A new method for extended-period simulation, called the explicit integration (EI) method, is proposed. The method is based on the premise that a complex water distribution system can be represented by a number of simple base systems. The simple base systems are selected in such a way that their dynamic equations can be solved through explicit integration. In this paper a simple base system consisting of a fixed-head reservoir feeding a tank through a single pipeline is analyzed. It is then illustrated how a complex water distribution system can be decoupled into simple base systems and its dynamic behavior simulated using a stepwise procedure. The EI method is compared to the commonly used Euler numerical integration method using two example networks. It is shown that the accuracy of the EI method is considerably better than that of the Euler method for the same computational effort.     

WAF Method and Splitting Procedure for Simulating Hydro- and Thermal-Peaking Waves in Open-Channel Flows

Hydro- and thermal-peaking waves, generated by hydroelectric power generation, have a strong impact on the ecological integrity of aquatic ecosystems. In order to reduce such effects, mitigation procedure must be studied and implemented. To this end a one-dimensional model which solves the coupling of hydrodynamics with heat transport is developed. The solution is obtained advancing simultaneously the hydrodynamic and thermal module with the same accuracy. For the numerical solution of the governing advection-reaction/diffusion problem a splitting procedure is adopted: the advection-reaction part is solved by means of the weight average flux (WAF) finite volume explicit method, while the diffusion part is solved using a nonlinear version of the implicit Crank-Nicolson method. The WAF method is extended to second-order in the presence of reaction terms. Numerical results are presented for different test examples, which demonstrate the accuracy and robustness of the scheme and its applicability in predicting temperature transport by shallow water flows. Application to the Adige River (Northern Italy) of this framework proves that the model is an effective tool for designing hydro- and thermal-peaking waves mitigation procedures.     

Numerical Modeling of Bed Evolution in Channel Bends

A two-dimensional numerical model is developed to predict the time variation of bed deformation in alluvial channel bends. In this model, the depth-averaged unsteady water flow equations along with the sediment continuity equation are solved by using the Beam and Warming alternating-direction implicit scheme. Unlike the present models based on Cartesian or cylindrical coordinate systems and steady flow equations, a body-fitted coordinate system and unsteady flow equations are used so that unsteady effects and natural channels may be modeled accurately. The effective stresses associated with the flow equations are modeled by using a constant eddy-viscosity approach. This study is restricted to beds of uniform particles, i.e., armoring and grain-sorting effects are neglected. To verify the model, the computed results are compared with the data measured in 140° and 180° curved laboratory flumes with straight reaches up- and downstream of the bend. The model predictions agree better with the measured data than those obtained by previous numerical models. The model is used to investigate the process of evolution and stability of bed deformation in circular bends.     

Lagrangian Modeling of Weakly Nonlinear Nonhydrostatic Shallow Water Waves in Open Channels

A Lagrangian, nonhydrostatic, Boussinesq model for weakly nonlinear and weakly dispersive flow is presented. The model is an extension of the hydrostatic model—dynamic river model. The model uses a second-order, staggered grid, predictor-corrector scheme with a fractional step method for the computation of the nonhydrostatic pressure. Numerical results for solitary waves and undular bores are compared with Korteweg-de Vries analytical solutions and published numerical, laboratory, and theoretical results. The model reproduced well known features of solitary waves, such as wave speed, wave height, balance between nonlinear steepening and wave dispersion, nonlinear interactions, and phase shifting when waves interact. It is shown that the Lagrangian moving grid is dynamically adaptive in that it ensures a compression of the grid size under the wave to provide higher resolution in this region. Also the model successfully reproduced a train of undular waves (short waves) from a long wave such that the predicted amplitude of the leading wave in the train agreed well with published numerical and experimental results. For prismatic channels, the method has no numerical diffusion and it is demonstrated that a simple second-order scheme suffices to provide an efficient and economical solution for predicting nonhydrostatic shallow water flows.     

Numerical Modeling of Helical Static Mixers for Water Treatment

To improve the understanding of how static mixers work and how to better utilize them in environmental engineering (or, specifically, drinking water treatment), a numerical model for simulating turbulent flows in helical static mixers is developed. The model solves the three-dimensional, Reynolds-averaged Navier-Stokes equations, closed with the k-ω turbulence model, using a second-order-accurate finite-volume numerical method. Numerical simulations are carried out for a two-element helical static mixer, and the computed results are analyzed to elucidate the complex, three-dimensional features of the flow. The results show that the flow field within the mixer is characterized by the presence of pockets of reversed flow and the growth and interaction of strong longitudinal vortices. As an example of the kind of practical insights that can be gained from such detailed three-dimensional computations, the simulated flow field is used to investigate two quantities that are often used to characterize mixing within a static mixer and to discuss the merits of these quantities for coagulant mixing in drinking water treatment.     

Flux-Difference Splitting Schemes for 2D Flood Flows

Two numerical models for 2D flood flows are presented. One model is first-order accurate and another is second-order accurate. Roe's numerical flux is used to develop the first-order accurate model, while second-order accuracy, in space and time, is obtained by using the Lax-Wendroff numerical flux. A simple operator splitting is found to yield the same results as that obtained by using more complicated, and thus, time consuming, operator splitting. Roe's approximate Jacobian is used for conservative properties and Harten and Hyman's procedure is followed for the entropy inequality condition. Flux limiter is used in the second-order accurate model that removes oscillations while maintaining the order of accuracy. The models are verified against available experimental data of a 2D flood wave due to partial dam-break. Numerical experiments are conducted to verify the models' ability to correctly predict behavior of the free surface, in addition to prediction of depth and velocity.     

3D Unsteady RANS Modeling of Complex Hydraulic Engineering Flows. I: Numerical Model

A general-purpose numerical method is developed for solving the full three-dimensional (3D), incompressible, unsteady Reynolds-averaged Navier-Stokes (URANS) equations in natural river reaches containing complex hydraulic structures at full-scale Reynolds numbers. The method adopts body-fitted, chimera overset grids in conjunction with a grid-embedding strategy to accurately and efficiently discretize arbitrarily complex, multiconnected flow domains. The URANS and turbulence closure equations are discretized using a second-order accurate finite-volume approach. The discrete equations are integrated in time via a dual-time-stepping, artificial compressibility method in conjunction with an efficient coupled, block-implicit, approximate factorization iterative solver. The computer code is parallelized to take full advantage of multiprocessor computer systems so that unsteady solutions on grids with 106 nodes can be obtained within reasonable computational time. The power of the method is demonstrated by applying it to simulate turbulent flow at R ? 107 in a stretch of the Chattahoochee River containing a portion of the actual bridge foundation located near Cornelia, Georgia. It is shown that the method can capture the onset of coherent vortex shedding in the vicinity of the foundation while accounting for the large-scale topographical features of the surrounding river reach.     

Frequency Modeling of Open-Channel Flow

A good model is necessary to design automatic controllers for water distribution in an open-channel system. The frequency response of a canal governed by the Saint-Venant equations can be easily obtained in the uniform case. However, in realistic situations, open-channel systems are usually far from the uniform regime. This paper provides a new computational method to obtain a frequency domain model of the Saint-Venant equations linearized around any stationary regime (including backwater curves). The method computes the frequency response of the Saint-Venant transfer matrix, which can be used to design controllers with classical automatic control techniques. The precision and numerical efficiency of the proposed method compare favorably with classical numerical schemes (e.g., Runge–Kutta). The model is compared in nonuniform situations to the one given by a finite difference scheme applied to Saint-Venant equations (Preissmann scheme), first in the frequency domain, then in the time domain. The proposed scheme can be used, e.g., to validate finite difference schemes in the frequency domain.     

Hydrodynamics of Turbid Underflows.?I: Formulation and Numerical Analysis

A mathematical model is developed for unsteady, two-dimensional, single-layer, depth-averaged turbid underflows driven by nonuniform, noncohesive sediment. The numerical solution is obtained by a high-resolution, total variation diminishing, finite-volume numerical model, which is known to capture sharp fronts accurately. The monotone upstream scheme for conservation laws is used in conjunction with predictor-corrector time-stepping to provide a second-order accurate solution. Flux-limiting is implemented to prevent the development of spurious oscillations near discontinuities. The model also possesses the capability to track the evolution and development of an erodible bed, due to sediment entrainment and deposition. This is accomplished by solving a bed-sediment conservation equation at each time step, independent of the hydrodynamic equations, with a predictor-corrector method. The model is verified by comparison to experimental data for currents driven by uniform and nonuniform sediment.     

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.     

High-Resolution Method for Modeling Hydraulic Regime Changes at Canal Gate Structures

A method for modeling flow regime changes at gate structures in canal reaches is presented. The methodology consists of using an approximate Riemann solver at the internal computational nodes, along with the simultaneous solution of the characteristic equations with a gate structure equation at the upstream and downstream boundaries of each reach. The conservative form of the unsteady shallow-water equations is solved in the one-dimensional form using an explicit second-order weighted-average—flux upwind total variation diminishing (TVD) method and a Preissmann implicit scheme method. Four types of TVD limiters are integrated into the explicit solution of the governing hydraulic equations, and the results of the different schemes were compared. Twelve possible cases of flow regime change in a two-reach canal with a gate downstream of the first reach and a weir downstream of the second reach, were considered. While the implicit method gave smoother results, the high-resolution scheme—characteristic method coupling approach at the gate structure was found to be robust in terms of minimizing oscillations generated during changing flow regimes. The complete method developed in this study was able to successfully resolve numerical instabilities due to intersecting shock waves.     

Two-Dimensional Simulation Model for Contour Basin Layouts in Southeast Australia. I: Rectangular Basins

Contour basin irrigation layouts are used to irrigate rice and other cereal crops on heavy cracking soils in Southeast Australia. In this study, a physically based two-dimensional simulation model that incorporates all the features of contour basin irrigation systems is developed. The model’s governing equations are based on a zero-inertia approximation to the two-dimensional shallow water equations of motion. The equations of motion are transformed into a single nonlinear advection–diffusion equation in which the friction force is described by Manning’s formula. The empirical Kostiakov equation and the quasi-analytical Parlange equation are used to model the infiltration process. The governing equations are solved by using a split-operator approach. The numerical procedure described here is capable of modeling rectangular basins; a procedure for irregular shaped basins is presented in Paper II. The model was validated against field data collected on commercial lasered contour layouts.     

Two-Dimensional Simulation of Subcritical Flow at a Combining Junction: Luxury or Necessity?

Classically, in open-channel networks, the flow is numerically approximated by the one-dimensional Saint Venant equations coupled with a junction model. In this study, a comparison between the one-dimensional (1D) and two-dimensional (2D) numerical simulations of subcritical flow in open-channel networks is presented and completely described allowing for a full comprehension of the modeling of water flow. For the 1D, the mathematical model used is the 1D Saint Venant equations to find the solution in branches. For junction, various models based on momentum or energy conservation have been developed to relate the flow variables at the junction. These models are of empirical nature due to certain parameters given by experimental results and moreover they often present a reduced field of validity. In contrast, for the 2D simulation, the junction is discretized into triangular cells and we simply apply the 2D Saint Venant equations, which are solved by a second-order finite-volume method. In order to give an answer to the question of luxury or necessity of the 2D approach, the 1D and 2D numerical results for steady flow are compared to existing experimental data.     

Advances in Modeling Completely Mixed Flow Reactors for Ion Exchange

This work advances the mathematical modeling of ion exchange treatment in completely mixed flow reactors in which there is recycle and regeneration of the ion exchange resin. The most common application of this process is magnetic ion exchange resin to remove dissolved organic carbon from raw drinking water. The motivation for this work was the complex distribution of resin particle ages and sizes that result from the recycle and regeneration processes. The newly developed model uses a single “age-averaged” diffusion equation to represent resin particle age as compared with the previous Monte Carlo model that uses a large number of diffusion equations to represent various resin particle ages. Advantages of the age-averaged model over the Monte Carlo model include a closed-form analytical solution for the steady-state case of the model, advanced numerical techniques used for the nonsteady-state case of the model, and model simulations require much less computational time and yield more accurate results. The age-averaged model is a robust numerical tool that can be used to evaluate a range of treatment scenarios as a result of these advancements.