Comparison of Gradually Varied Flow Computation Algorithms for Open-Channel Network

This paper presents a comparison of two algorithms—the forward-elimination and branch-segment transformation equations—for separating out end-node variables for each branch to model both steady and unsteady flows in branched and looped canal networks. In addition, the performance of the recursive forward-elimination method is compared with the standard forward-elimination method. The Saint–Venant equations are discretized using the four-point implicit Preissmann scheme, and the resulting nonlinear system of equations is solved using the Newton–Raphson method. The algorithm using branch-segment transformation equations is found to be at least five times faster than the algorithm using the forward-elimination method. Further, the algorithm using branch-segment transformation equations requires less computer storage than the algorithm using the forward-elimination method, particularly when only nonzero elements of the global matrix are stored. Comparison between the Gauss-elimination method and the sparse matrix solution technique for the solution of the global matrix revealed that the sparse matrix solution technique takes less computational time than the Gauss-elimination method.     

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.     

Efficient Algorithm for Gradually Varied Flows in Channel Networks

This paper presents an efficient solution technique for one-dimensional unsteady flow routing through a general channel network system—dendritic, looped, divergent, or any combination of such networks. The finite difference method is used to solve the de St. Venant equations in all the branches of the network simultaneously. The number of equations to be solved at a time during any iteration is reduced to only four times the number of branches of the network. This results in a significant reduction in storage requirements and solution time. Importantly, the algorithm does not require any special node numbering schemes and the nodes can be numbered independently for each branch. The algorithm is also suitable for programming on a parallel-processing computer.     

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.     

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.     

Hydraulic Jet Control for River Junction Design of Yuen Long Bypass Floodway, Hong Kong

The Yuen Long Bypass Floodway (YLBF) was designed to collect flows from the Sham Chung River (SCR) and the San Hui Nullah (SHN) and to serve as a diversion channel of the Yuen Long Main Nullah (YLMN). Under a 200-year return period design condition, the floodway was designed (1) to divert a flow of approximately 38?m3/s from the supercritical YLMN flow and (2) to convey a total combined flow of 278?m3/s to downstream within acceptable flood levels. The success of the design depends critically on complicated junction flow interactions that cannot be resolved by 1D unsteady flow models. These features include the supercritical-subcritical flow transition at the San Hui-Floodway (SHN-YLBF) junction and the diversion of part of the supercritical flow from the Main Nullah (YLMN). A laboratory Froude scale physical model was constructed to study water stages and flow characteristics in the floodway and to investigate optimal design arrangements at channel junctions and transitions. This paper summarizes the main features of the unique river junction network, in particular the use of the hydraulic jet principle at the SHN-YLBF junction to lower flood levels. In addition, a numerical flow model is employed to study flow details at the river junctions. The model is based on the general 2D shallow water equations in strong conservation form. The equations are discretized using the total variation diminishing finite-volume method which captures the discontinuity in hydraulic jumps. The numerical model predictions are well supported by the laboratory data, and the theoretical and experimental results offer useful insights for the design of urban flood control schemes under tight space constraints.     

Kinematic and Diffusion Waves: Analytical and Numerical Solutions to Overland and Channel Flow

Flood wave propagation is the unifying concept in representing open channel and overland flow. Therefore, understanding flood wave routing theory and solving the governing equations accurately is an important issue in hydrology and hydraulics. In an attempt to contribute to the understanding of this subject, in this study: (1) an analytical solution is derived for diffusion waves with constant wave celerity and hydraulic diffusivity applied to overland flow problems; and (2) an algorithm is developed using the MacCormack explicit finite difference method to solve the kinematic and diffusion wave governing equations for both overland and open channel flow. The MacCormack method is particularly well suited to approximate nonlinear differential equations. The analytical solutions provide the practicing engineer with computational speed in obtaining results for overland flow problems, and a means to check the validity of the numerical models. On the other hand, for larger scale catchment-stream problems, the verified numerical methods provide efficient and accurate algorithms to obtain solutions. Both the analytical approaches and the MacCormack algorithm are used to solve the same synthetic examples. Comparison of results shows that the numerical and analytical solutions are in close agreement. Furthermore, the MacCormack algorithm is applied to a real catchment: a segment of the Duke University West Campus storm water drainage system. In order to check the accuracy of the results obtained by the MacCormack method, the results are compared to predictions of the Environmental Protection Agency storm water management model (SWMM) as calibrated with measured rainfall and surface runoff flow data. The results obtained from SWMM are in good agreement with the results obtained from applying the MacCormack algorithm.     

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.     

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.     

Depth-Averaged Simulation of Supercritical Flow in Channel with Wavy Sidewall

Supercritical flow in a channel with a wavy sidewall is numerically simulated by solving the two-dimensional (2D) depth-averaged equations using two different second-order accurate finite-difference schemes: ADI and MAC. ADI is an implicit model that uses an alternating-direction-implicit (ADI) scheme to solve the governing equations. MAC is an explicit model employing the MacCormack two-step predictor-corrector scheme. To accurately simulate the wavy sidewall, both models solve the governing equations in transformed computational coordinates. Bottom friction is computed using the Manning formula and the effective stresses are modeled with a constant eddy-viscosity turbulence model. As is customary, the stresses due to depth-averaging are neglected. The computed water depth in the channel is compared with experimental data obtained by Mizumura. The effect of bottom friction, effective stresses, artificial viscosity, grid geometry, boundary conditions, and the Courant-Friedrichs-Lewy (CFL) number are investigated. Similarities and differences in the behavior of the models are observed and discussed.     

Large‐Eddy Simulations and Particle‐Image Velocimetry Measurements of Tundish Flow

Large‐eddy simulations (LES) and digital particle‐image velocimetry (DPIV) of a tundish flow are performed to investigate the turbulent flow structures and vortex dynamics. The LES is carried out using an implicit approach. In implicitly filtered LES, the computational grid and the discretization operators are considered as the filtering tools of the governing equations. The numerical computations are performed by solving the viscous conservation equations for compressible fluids. An implicit dual‐time stepping scheme combined with low Mach number pre‐conditioning and a multigrid accelerating technique is implemented for LES computations. The impact of jet spreading, jet impingement on the wall, and wall jets on the flow field and steel quality is investigated. The characteristics of the flow field in a one‐strand tundish such as the time‐dependent turbulent flow structure and vortex dynamics are analysed and compared with experimental results. To validate the numerical results, DPIV measurements are performed in a reduced 1:1.7 scaled water model. The investigations focus on steady‐state casting conditions for the flow in a tundish. The results evidence a good agreement between the LES and experimental data. The LES solutions provide an extremely detailed insight into the highly intricate turbulent flow structure. Even phenomena like funnel‐shaped vortices downward the shroud jet are well captured.     

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.     

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.     

Gradually Varied Flow Computation in Cyclic Looped Channel Networks

A novel and computationally efficient algorithm is presented to compute the water surface profiles in steady, gradually varied flows of open channel networks. This algorithm allows calculation of flow depths and discharges at all sections of a cyclic looped open channel network. The algorithm is based on the principles of (1) classifying the computations in an individual channel as an initial value problem or a boundary value problem; (2) determining the path for linking the solutions from individual channels; and (3) an iterative Newton–Raphson technique for obtaining the network solution, starting from initial assumptions for discharges in as few channels as possible. The proposed algorithm is computationally more efficient than the presently available direct method by orders of magnitude because it does not involve costly inversions of large matrices in its formulation. The application of this algorithm is illustrated through an example network.     

Computer and Experimental Models of Transient Flow in a Pipe Involving Backflow Preventers

Transient flow in a pipe was studied using both experimental and computer models. In the present study, three different numerical models: The method of characteristics model, the axisymmetrical model, and the implicit scheme model are utilized and compared. Experiments for transient flow in a simple pipeline have been conducted to verify the results from the computer models. It was found that head loss coefficient for the 1D models, such as the method of characteristics model and the implicit scheme model, should be much bigger than the Darcy-Weisbach frictional coefficient. Experiments for transient flow with the backflow preventer in a pipe were conducted. Results show that backflow preventer serves as a strong damper to the water hammer generated by the hydraulic transients. Numerical investigation simulating a backflow preventer in transient flow has been performed in this study. It was found that different values of head loss coefficient should be applied for the upstream and downstream of backflow preventer. All of the numerical models were compared with the experiments. The results of different computer models developed in the present study agree well with the experimental data.     

Multilayer Depth-Averaged Flow Model with Implicit Interfaces

Using a depth-averaged model to obtain the velocity and pressure distributions in the vertical direction is difficult. A multilayer model is an option that can be used to improve on the depth-averaged model. However, the unknown flow depth needs to be predicted first and then divided into layers as an input for the multilayer model. An improved multilayer model is proposed here by introducing an implicit layer dividing interfaces that are associated with the flow velocity and pressure distribution. The formulation of interfaces also applies to boundary faces: Free surface and channel beds. Therefore, each flow layer behaves like that in the classical depth-averaged model. Subsequently the governing equations are also simplified due to the vanishing terms related to interfacial flow exchanges. This improved model has been satisfactorily applied to steady flow simulations in three cases: Flow over a slope transition from mild to steep, from steep to mild, and over a trapezoidal weir. The results demonstrate the efficiency and validity of the proposed models to simulate open channel flows with bed slope changes.     

Numerical Study of Flow Affected by Bendway Weirs in Victoria Bendway, the Mississippi River

It has been observed that submerged weirs in bendways realign the flow and in general improve navigation conditions. This qualitative observation has been the basis for field design. This paper presents a study of hydrodynamics in the Victoria Bendway in the Mississippi River using three-dimensional numerical simulations. A numerical model, CCHE3D, was applied and computational results were compared to three-dimensional velocity data provided by the U.S. Army Corps of Engineers with reasonable agreement. The numerical simulation results were then used to analyze helical currents due to the channel curvature and the presence of submerged weirs. The simulated flow realignment near the free surface indicates that the flow conditions in the bendway were improved by the submerged weirs, however, the effectiveness of each weir depends on its alignment, local channel morphology, and flow conditions.     

连铸结晶器内电磁制动过程的数值模拟

电磁制动的磁场分布与电磁制动的冶金效果密切相关，建立了描述连铸结晶器内电磁制动过程的三维数学模型，在交错网络下利用半隐式压力校正算法求解了动量方程和感生电流方程，研制了带形电磁制动结晶器内的磁场和流场，并将计算机结果与实测值进行了比较，结果表明，合理的电磁制动磁场强度为0.04～0.24T。     

Numerical Model of Sedimentation∕Thickening with Inertial Effects

A numerical model of gravity sedimentation and thickening was developed from the governing two-phase flow equations for the liquid and solid phases. The inertial and gravity terms in the solid and liquid momentum equations were retained in the gravity sedimentation and thickening model. An implicit, space-staggered finite-difference algorithm was developed for the resulting coupled partial differential equations. Constitutive relationships describing the physical properties of the slurry were required to solve the numerical model. These constitutive properties describing the relationship between effective stress and porosity and between permeability and porosity were determined experimentally and by model calibration. The model was calibrated and verified using the data of dynamic porosity profiles of gravity sedimentation and thickening of kaolin suspensions in distilled water.     

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.