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.     

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.     

Exact Solutions for Sequent Depths Problem

Design of a stilling basin downstream of a barrage requires knowledge of various elements of hydraulic jump with known values such as the discharge and the energy lost in the jump. The problem reduces to solution of the implicit equations for sequent depths requiring the tedious method of trial and error. Presented herein are the exact equations for sequent depths.     

Numerical Solution of Boussinesq Equations to Simulate Dam-Break Flows

To investigate the effect of nonhydrostatic pressure distribution, dam-break flows are simulated by numerically solving the one-dimensional Boussinesq equations by using a fourth-order explicit finite-difference scheme. The computed water surface profiles for different depth ratios have undulations near the bore front for depth ratios greater than 0.4. The results obtained by using the Saint Venant equations and the Boussinesq equations are compared to determine the contribution of individual Boussinesq terms in the simulation of dam-break flow. It is found that, for typical engineering applications, the Saint Venant equations give sufficiently accurate results for the maximum flow depth and the time to reach this value at a location downstream of the dam.     

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.     

Hydraulic Modeling of a Mixed Water Level Control Hydromechanical Gate

This article describes the hydraulic behavior of a mixed water level control hydromechanical gate present in several irrigation canals. The automatic gate is termed “mixed” because it can hold either the upstream water level or the downstream water level constant according to the flow conditions. Such a complex behavior is obtained through a series of side tanks linked by orifices and weirs. No energy supply is needed in this regulation process. The mixed flow gate is analyzed and a mathematical model for its function is proposed, assuming the system is at equilibrium. The goal of the modeling was to better understand the mixed gate function and to help adjust their characteristics in the field or in a design process. The proposed model is analyzed and evaluated using real data collected on a canal in the south of France. The results show the ability of the model to reproduce the function of this complex hydromechanical system. The mathematical model is also implemented in software dedicated to hydraulic modeling of irrigation canals, which can be used to design and evaluate management strategies.     

Closed-Form Expression of the Response Time of an Open Channel

Computing accurately the response time of an open channel is of extreme importance for management operations on canal networks, such as feed-forward control problems. The methods proposed in the literature to approximate the response time do not always account for the influence of a cross structure at the downstream end of a canal pool, which may have a significant impact on the response time. This paper proposes a new approach to compute the response time, accounting explicitly for the backwater and the feedback effects due to the downstream cross structure. The method provides a distributed analytical expression of the response time as a function of the characteristics of the canal (geometry, roughness) and of the downstream cross structure. A test canal with a weir or a gate at the downstream end is used to compare the new method with some of the others. Results show that the proposed expression accurately reproduces the response time for various backwater and downstream boundary conditions.     

Preserving Steady-State in One-Dimensional Finite-Volume Computations of River Flow

When using finite-volume methods and the conservative form of the Saint Venant equations in one-dimensional flow computations, it is important to establish the correct balance between the discretized flux vector and the geometric source terms. Over the last few years various improvements to numerical schemes have been presented to achieve this correct balance, focusing on the capability to simulate water at rest on irregular geometries (C-property). In this paper it is shown that common schemes can lead to energy-violating solutions in the case of steady flow. We present developments based on the Roe TVD finite-volume scheme for one-dimensional Saint Venant equations, which results in a method that not only satisfies the C-property, but also preserves the correct steady flow when stationary boundary conditions are used. We also present a totally irregular channel test case for the verification of the method.     

Optimal Design of Parabolic Canal Section

Optimal design equations for a parabolic canal section are presented in this paper. The design equations for a minimum earthwork cost section and a minimum cost lined section are in explicit form and result in optimal dimensions of a canal in single step computations. These have been obtained after applying the Fibbonaci search method on a nonlinear unconstrained optimization problem. The study also addresses the bounds on the canal dimensions and the velocity of flow. A nondimensional parameter approach has been used to simplify the analysis, and a set of graphs for nondimensional parameters are presented as an alternative for design. Design procedures for different cases have been presented to demonstrate the simplicity of the method.     

Cartesian Cut Cell Two-Fluid Solver for Hydraulic Flow Problems

A two-fluid solver which can be applied to a variety of hydraulic flow problems has been developed. The scheme is based on the solution of the incompressible Euler equations for a variable density fluid system using the artificial compressibility method. The computational domain encompasses both water and air regions and the interface between the two fluids is treated as a contact discontinuity in the density field which is captured automatically as part of the solution using a high resolution Godunov-type scheme. A time-accurate solution has been achieved by using an implicit dual-time iteration technique. The complex geometry of the solid boundary arising in the real flow problems is represented using a novel Cartesian cut cell technique, which provides a boundary fitted mesh without the need for traditional mesh generation techniques. A number of test cases including the classical low amplitude sloshing tank and dam-break problems, as well as a collapsing water column hitting a downstream obstacle have been calculated using the present approach and the results compare very well with other theoretical and experimental results. Finally, a test case involving regular waves interacting with a sloping beach is also calculated to demonstrate the applicability of the method to real hydraulic problems.     

Discharge Formulas of Crump-De Gruyter Gate-Weir for Computer Simulation

One of the problems of interest to professionals in the field of irrigation and drainage is the computer simulation of discharge or level control structures. Particularly troublesome are structures that display a marked change of behavior when the downstream water level exceeds a certain limit. The Crump-de Gruyter gate displays several such changes of behavior. Not only does it exhibit a transition from free to drowned flow when the downstream water level rises, it can also go from weir to gate flow. A series of experiments in a laboratory flume provided the basic data to test a simple mathematical model of this structure. The model assumes the structure is located between two reaches with sub-critical flow in the upstream and downstream reach.     

Simple Optimal Downstream Feedback Canal Controllers: ASCE Test Case Results

In a companion paper, a class of downstream-water-level feedback canal controllers was described. Within this class, a particular controller is chosen by selecting which controller coefficients to optimize (tune), the remaining coefficients being set to zero. These controllers range from a series of simple proportional-integral (PI) controllers to a single centralized controller that considers lag times. In this paper, several controllers within this class were tuned with the same quadratic performance criteria (i.e., identical penalty functions for optimization). The resulting controllers were then tested through unsteady-flow simulation with the ASCE canal automation test cases for canal 1. Differences between canal and gate properties, as simulated and as assumed for tuning, reduced controller performance in terms of both water-level errors and gate movements. The test case restrictions placed on minimum gate movement caused water levels to oscillate around their set points. This resulted in steady-state errors and much more gate movement (hunting). More centralized controllers handle unscheduled flow changes better than a series of local PI controllers. Controllers that explicitly account for pool wave travel times did not improve control as much as expected. Sending control actions within a given pool to upstream pools improved performance, but caused oscillations in some cases, unless control signals were also sent downstream. A good compromise between controller performance and complexity is provided by controllers that pass feedback from a given water level to the check structure at the upstream end of its pool (i.e., that used for downstream control of an individual pool) and to all upstream and one downstream check structures.     

Nonlinear Control of Open-Channel Water Flow Based on Collocation Control Model

This paper is devoted to the nonlinear control of open-channel water flow dynamics via a one-dimensional collocation control model for irrigation canals or dam-river systems. Open channel dynamics are based on the well-known Saint-Venant nonlinear partial differential equations. In order to obtain a finite-dimensional model an orthogonal collocation method is used, together with functional approximation of the solutions of Saint-Venant equations based on Lagrange polynomials. This method can give a more tractable model than those obtained from classical finite-difference or finite-element methods (from the viewpoint of both state dimension and structure), and is well suited for control purposes. In particular it is shown how such a model can be used to design a nonlinear controller by techniques of dynamic input–output linearization with the goal of controlling water levels along an open-channel reach. Controller performance and robustness are illustrated in simulations, with a simulated model for the canal chosen as more accurate than the one used for control design.     

Structure of Flow Upstream of Vertical Angled Screens in Open Channels

This paper presents the results of a laboratory study of the structure of flow in a diversion structure with a vertical angled wedge-wire fish screen. This screen had a 10×25?mm mesh and was tested at three angles of 10.4, 17.5, and 26.8°, to the direction of the approaching flow, for two mean velocities of 0.5 and 0.8?m/s, with a depth of flow of about 0.75?m. In this water and fish diversion (channel or) structure, it was found that the depth of flow at any section is approximately constant with a drop at the screen on the side of the canal and decreased towards the bypass located at the downstream end. The distribution of the velocity component u in the direction of the approaching flow as well as the perpendicular component w and the resultant velocity V was uniform in the vertical direction. The depth averaged mean velocity for different verticals at any section in the diversion structure increased with the longitudinal distance x and was correlated with the relative width, bs/b (in the diversion structure) for all five experiments. Correlations have been found for the depth averaged transport velocity and the impinging velocity on the screen in terms of the approach velocity U. A general relation has also been developed for the attack angle of the flow on the screen. The downstream part of the screen carried more flow into the canal compared to the upstream part as a result of the uniform mesh size used in this study. The results of this hydraulic study should be useful, particularly for freshwater adult fish, in designing screens in irrigation canals and for micro-hydro sites that use diversion canals.     

Relationship between Hazen–William and Colebrook–White Roughness Values

Despite advances in computing technology and derivation of explicit approximation formulas, the experimentally verified and widely applicable Colebrook–White friction factor formula is often rejected in favor of the limited and less accurate Hazen–Williams equation. The general reluctance of practicing engineers to embrace the Colebrook–White formula may be due to the relatively large available database for Hazen–Williams C coefficient values versus a relatively small database of the equivalent sand roughness ks values required by the Colebrook–White equation. Until now, converting C to ks required knowledge of both the Reynolds number and pipe diameter originally used to determine C. The current effort derives implicit equations relating C to ks that do not require additional information and compare well with published data. The exact solution is approximated with a single explicit equation, accurate to within 4% error.     

Wavelet-Galerkin Solution to the Water Hammer Equations

In this paper, a wavelet-Galerkin method is utilized to solve the hyperbolic partial differential equations describing transient flow in a simple pipeline. Two wavelets (Haar and Daubechies) are utilized as bases for the Galerkin scheme. The governing equations are solved for the expansion coefficients, which are then used to reconstruct the signal at the downstream end of the pipeline; the computed results are in an excellent agreement with those calculated by using the method of characteristics including laminar or linearized turbulent friction terms. Most importantly, the wavelet-Galerkin approach allows the transient flow equations to be solved directly for the expansion coefficients at a certain level of resolution. This can be used to form the wavelet multiresolution framework that can be utilized for further analysis, such as feature extraction and signal identification.     

Salt River Project Canal Automation Pilot Project: Simulation Tests

The feasibility of automatically controlling water levels and deliveries on the Salt River Project (SRP) canal system through computer-based algorithms is being investigated. The proposed control system automates and enhances functions already performed by SRP operators, namely feedforward routing of scheduled demand changes, feedback control of downstream water levels, and flow control at check structures. Performance of the control system was tested with unsteady flow simulation. Test scenarios were defined by the operators for a 30 km, four-pool canal reach. The tests considered the effect of imperfect knowledge of check gate head-discharge relationships. The combined feedback-feedforward controller easily kept water level deviations close to the target when dealing with routine, scheduled flow changes. Those same routine changes, when unscheduled, were handled effectively by the feedback controller alone. The combined system had greater difficulty in dealing with large demand changes, especially if unscheduled. Because feedback flow changes are computed independently of feedforward changes, the feedback controller tends to counteract feedforward control actions. The effect is unimportant when dealing with routine flow changes but is more significant when dealing with large changes, especially in cases where the demand change cannot be fully anticipated.     

General Characteristics of Solutions to the Open-Channel Flow, Feedforward Control Problem

A dimensionless formulation of the open-channel flow equations was used to study the feedforward control problem for single-pool canals. Feedforward inflow schedules were computed for specified downstream demands using a gate-stroking model. The analysis was conducted for various design and operational conditions. Differences in the shape of the computed inflow hydrographs are largely related to the volume change resulting from the transient, the time needed to supply this volume, and the time needed by the inflow perturbation to travel down the canal. The gate-stroking method will fail to produce a solution or the solution will demand extreme and unrealistic inflow variations if the time needed to supply the canal volume change is much greater than the travel time of the upstream flow change. As an alternative, a simple feedforward-control flow schedule can be developed based on this volume change and a reasonable delay estimate. This volume compensating schedule can deliver the requested flow change and keep water levels reasonably close to the target under the range of conditions tested.     

Pivoting Strategies in the Solution of the Saint-Venant Equations

Pivoting was incorporated in the process of solving the linear system of equations that results after discretizing the Saint- Venant equations using the four-point implicit scheme, and applying the Newton–Raphson algorithm to the resulting set of nonlinear equations. Both exchange of rows only (partial pivoting) and exchange of rows and columns (full pivoting) were investigated using the CanalMan hydraulic model. Partial pivoting was used with the LU (lower and upper) decomposition linear equation solver, whereas full pivoting was used with the Gauss–Jordan elimination algorithm. It was demonstrated that the application of partial and full pivoting to the solution of the linear set of equations during Newton–Raphson iterations can make the difference between convergence and divergence of the solution, and should be applied as needed. However, full pivoting should be used only when needed because it slows the simulation considerably.     

Field Calibration of Submerged Sluice Gates in Irrigation Canals

Four rectangular sluice gates were calibrated for submerged-flow conditions using nearly 16,000 field-measured data points on Canal B of the B-XII irrigation scheme in Lebrija, Spain. Water depth and gate opening values were measured using acoustic sensors at each of the gate structures, and the data were recorded on electronic data loggers. Several gate calibration equations were tested and it was found that the rectangular sluice gates can be used for accurate flow measurement. The Energy-Momentum (E-M) equations proved to be sound. The calibration of the contraction coefficient, to be used in the energy equation, allowed good estimations of the discharge for three of the four gates studied. The gate for which the E-M method did not perform satisfactorily was located at the head of the canal with a unique nonsymmetric approach flow condition. Alternatively, we investigated the performance of the conventional discharge equation. The variation of the discharge coefficient, Cd, with the head differential, Δh, and the vertical gate opening, w, suggests that Cd be expressed as a function of these two variables. For the sluice gates considered in this study, the best empirical fit was obtained by expressing Cd as a parabolic function of w, although an exponential expression tested previously by other writers also produced satisfactory results. The greatest uncertainty in the variables considered in this study was in the calculated coefficient of discharge, and based on the uncertainty analysis, it is possible to quantify the uncertainty in the estimated discharge through a calibrated sluice gate. The discharge uncertainty in each of the four gates in this study decreases with increasing gate opening, and it decreases slightly with increasing head differentials.