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Introduction
The effect on the conservational and efficient use of water resources now has an impact on improving water management. One of the components of integrated water resources management (IWRM) is the ability to accurately measure and control water flow at specific points in a basin or in a distribution system, such as: Channel bifurcations, control structures, and water delivery points. An efficient water measuring system allows accurate accounting of water use, and the supply of water at optimal rates to areas of demand (Clemmens, Wahl, Bos, & Replogle, 2001). Several types of structures can be used for flow gauging: weirs, Parshall gauges, gates, orifices, among others (Skertchly, 1988).
Weirs are elevated barriers located perpendicular to the direction of water movement so that water rises above the obstruction through an opening on a regular basis (Bautista, Robles, Júnez, & Playán, 2013). The geometric parameters defining the hydraulic behavior of weirs are the length of the ridge and the shape of the flow control section (Emiroglu, Kaya, & Agaccioglu, 2010; USBR, 1997). Thin-walled weirs are studied using the principles of classical physics with experimental results to understand flow characteristics and determine the coefficient of discharge (Cd) under different configurations; this coefficient integrates the effects not considered in the derivation of equations used to estimate discharge, such as viscosity, capillarity, surface tension, velocity distribution and aerodynamic curvature attributable to weir contraction (Aydin, Altan-Sakarya, & Sisman, 2011; Rady, 2011).
Design Specifications and proper installation of weirs have been documented by the British Standards Institute (BSI, 1965), the International Standardization Organization (ISO, 1980), the American Society for Testing Materials (ASTM, 1993), and the United States Bureau of Reclamation (USBR, 1997). Normally, the measurement accuracy of the flow rate depends of the geometry of the weir; in areas with unevenness, triangular weirs are recommended for flow rates lower than 30 l s-1 for its greater precision for small discharges, and for higher flow rates rectangular weirs are used (Sotelo, 1997).
Shen (1981) described the procedure for designing sharp-crest triangular weirs supported by experimental results to determine the coefficient of discharge reported in the literature. El-Alfy (2005) experimentally evaluated the effect of the vertical flow curvature on the coefficient of discharge (Cd ), in triangular weirs, indicating that it is inversely proportional to the angle of the weir notch (θ) and directly proportional to the relative load (h/P). Bagheri and Heidarpour (2009) obtained a discharge coefficient equation for thin-walled rectangular weirs at the top and bottom of the sheet profiles using the free vortex theory. Bautista et al. (2013) used a low-speed photographic technique to characterize the upper and lower profiles of the flow draft on a triangular weir. Chen, Fu, Chen, and Cui (2018) analyzed the effects on the discharge coefficient by the variation of the slope of the ramps implemented on the upstream and downstream faces in a rectangular short-crest weir. Fu, Cui, Dai, and Chen (2018) proposed an equation to determine the discharge coefficient in structures of the orifice-weir type. Tian, Wang, Bai, and Li (2018) investigated the flow pattern and discharge coefficient at finite crest length combined with parallel walls with the flared entry (FGPs) on the crest of the weir.
It has been reported that the use of compound weirs combining triangular and rectangular shapes for flow measurement gives good results (Zahiri, Tang, & Azamathulla, 2014). Negm, Al-Brahim, and Alhamid (2002) proposed a structure composed of an upper rectangular weir with a lower gate to avoid the accumulation of sediments upstream of the structure. Another type of compound weir consists of a rectangular notch with a triangular notch in the center of the rectangular crest; the triangular weir could accurately measure the normal range of flows less than 30 l s-1, but larger flows can also be measured by operating the rectangular weir (USBR, 1997). In irrigation channels with the variability of discharge, the use of compound weirs is recommended to avoid the use of separate structures, which can represent the variability of flow with greater precision; to meet this need, weir structures described in this document play a key role in irrigation zones.
Numerical simulation techniques have been used to solve Fluid Mechanics problems, particularly with the use of general solution tools generically called Computational Fluid Dynamics (CFD). Advances in computer storage and processing capacity allowed the process of creating and fitting a CFD model to evolve and facilitate the analysis of results, both in time and cost. In particular, CFD techniques numerically solve the governing equations of the flow, such as the Navier-Stokes equations, and offer the possibility of hydraulically evaluating a weir under different geometries and hydraulic conditions. In fact, once validated and evaluated, CFD allows the extraction of additional information, such as: Forces, velocity fields, and pressures. Several studies have shown the advantages of CFD techniques, for example, Ho, Boyes, and Donohoo (2001), and Ho, Boyes, Donohoo, and Cooper (2003) compared numerical results from a Creager crest weir and reported results very similar to experimental studies using physical models. Rady (2011) reported a maximum deviation of ±3 % when comparing the coefficient of discharge of a thin-walled rectangular weir obtained numerically with CFD versus, those calculated with the general weir equation. Zuhair and Zubaidy (2013) compared the CFD results, using the VOF (Volume of Fluid) model, with the experimental data of a prototype of the Mandali weir, the results indicate that the flow profile obtained with CFD adjusts to the experimental results with a Pearson correlation coefficient close to 1. Therefore, the CFD methodology constitutes an effective tool to solve fluid mechanics problems related to hydraulic weirs.
Representation of physics-free surface phenomena, in the experimental or analytical models, represent one of the most difficulties in this area. CFD simulation can be an excellent option in order to investigate modeling on weirs and simulate numerical flow and tracking in detail of the free surface. In analyzing this problem, the use of CFD has been reported, in particular, the computational platform ANSYS-FLUENT, known for its ability to accurately track the free surface using the Volume Of Fluid (VOF) method (Duró, Dios, López, Liscia, & Angulo, 2012).
The aim of this work was (1) to build and evaluate a numerical model of a compound weir using Computational Fluid Dynamics (CFD) and (2) to propose a global discharge coefficient equation for the compound weir (triangular-rectangular). The numerical simulation of compound weirs will facilitate the understanding of its hydraulic behavior, which depends on the geometric characteristics of the weir, its hydraulics conditions, and the estimated discharge coefficient.
Materials and methods
Reference Prototype model
Experimental data and prototype models used are reported by Sotelo (1997), to using rectangular and triangular weir thin-walled, and Jan, Chang, and Lee (2006) in compound weirs. Figure 1 shows the geometric dimensions of the rectangular and triangular weirs, both weirs are independent in a channel with a hydraulic load of 0.27 m for the rectangular weir, and 0.50 m for the triangular weir. Both weirs are sharp-crest for the operation, and for the simulation standard wall value were used (Skertchly, 1988).
Figure 1 Dimensions of the rectangular and triangular prototypes weirs (Sotelo, 1997; Skertchly, 1988).
Analytic weir equations
Experimental and analytical studies were performance on the flow behavior upstream and downstream of the weir, for instance, upstream near the weir there is a speed varied movement and present a backwater of depression originating the transformation of potential energy into kinetic energy, there is a zone of stagnation of the water called dead water. Upstream of the weir, there is gradually varied flows; this area is up to a distance of 4h, h being the static charge on the crest of the weir (Sotelo, 1997).
Equation (1) shows the general form that allows obtaining the fluid flow through a weir (Bos, 1989).
Where Q = flow (m3 s-1); k = coefficient depending on the dimensions and shape of the weir (m1.5 s-1) for a rectangular weir (m0.5 s-1) for a triangular weir), and n = dimensionless number depending on the shape of the weir; for a rectangular and triangular weir n is equal to 1.5 and 2.5, respectively (Bos, 1989).
For Henderson (1966) the discharge equation for a rectangular weir can be simplified as (Equation (2)):
Where Cd is the coefficient of discharge (dimensionless); b, the length of the weir crest (m); g, the acceleration of gravity (m s-2), and h is the static load on the weir crest (m). The coefficient of the discharge depends on the flow characteristics and geometry of the channel and weir (Kumar, Ahmad, & Mansoor, 2011).
The triangular section weirs are recommended for the gauging of flow rates less than 30 l s-1 and loads from 6 cm to 60 cm; their precision is better than that of a rectangular weir for small flows. For larger discharges, a rectangular weir is recommended, because the triangular weir is more sensitive to any change in the roughness of the plate and because it requires greater accuracy in the measurement of loads. The flow equation for the triangular weir is Equation (3) (Sotelo, 1997):
Where θ is the angle of the notch of the triangular section (°).
The discharge coefficients are calculated with Equation (4) for the rectangular weir and Equation (5) for the case of a triangular weir whose notch angle, θ, is 90° (Sotelo, 1997).
Where B is channel width (m); b, the length of the rectangular weir (m); P, the height of the weir (m); h, the static load on the weir (m), and Cd is the coefficient of discharge (dimensionless).
Figure 2 shows the compound weirs with rectangular and triangular sections. The discharge equation can be derived using the linear combination method, according to Jan et al. (2006), and is expressed with Equation (6):
The compound weir equation (Equation (6)) contains two basic terms: The first one is the discharge of the triangular section
The value of Cdt can be estimated using the empirical relationship (Jan et al., 2006) (Equation (7)):
The value of Cdr depends on load h1 (m), crest height P (m), length of rectangular section b (m), and width of channel B (m), it is described as Equation (8) (Jan et al., 2006):
It should be clarified that the Cd values of Equation (4) and Equation (5) are different from Cdr and Cdt of Equation (7) and Equation (8), the latter corresponds to a compound weir of rectangular and triangular sections. Sotelo (1997) analyzed just two cases of weirs; the first one using a triangular weir and the second one was a rectangular weir, both with different hydraulics loads, and subsequently evaluated their discharge coefficients based on the level of hydraulics loads obtained in each weir. The results and experimental data are summarized in Table 1.
Table 1 Results and experimental data of the rectangular and triangular weirs were reported by Sotelo (1997).
| Weir | Hydraulic load h (m) | Coefficient of discharge C d | Flow rates Q (l s-1) |
|---|---|---|---|
| Rectangular | 0.27 | 0.602 | 249 |
| Triangular | 0.50 | 0.590 | 246 |
On the other hand, for the weir with combined triangular-rectangular section, Jan et al. (2006) reported experimental results of discharge for three cases of weirs of different width of the rectangular section (b1) and different flow rates (Q); Table 2 show the experimental conditions and the results obtained.
Table 2 Experimental conditions and results of the compound weir reported by Jan et al. (2006).
| θ (°) | Channel width (m) | b1 (m) | b2 (m) | h1 (m) | h2 (m) | Cdr | Cdt | Analytical flow rates Q (l s-1) | Experimental flow rates Q (l s-1) |
|---|---|---|---|---|---|---|---|---|---|
| 90 | 1.49 | 0.25 | 0.20 | 0.013 | 0.113 | 0.590 | 0.579 | 7.13 | 8.06 |
| 90 | 1.49 | 0.25 | 0.20 | 0.031 | 0.131 | 0.591 | 0.579 | 13.02 | 14.36 |
| 90 | 1.49 | 0.25 | 0.20 | 0.036 | 0.136 | 0.591 | 0.579 | 14.95 | 16.33 |
| 90 | 1.49 | 0.40 | 0.20 | 0.022 | 0.122 | 0.594 | 0.579 | 11.59 | 13.35 |
| 90 | 1.49 | 0.40 | 0.20 | 0.041 | 0.141 | 0.595 | 0.579 | 21.41 | 23.35 |
| 90 | 1.49 | 0.40 | 0.20 | 0.069 | 0.169 | 0.597 | 0.579 | 39.90 | 45.50 |
| 90 | 1.49 | 0.50 | 0.20 | 0.01 | 0.110 | 0.595 | 0.579 | 7.23 | 8.06 |
| 90 | 1.49 | 0.50 | 0.20 | 0.023 | 0.123 | 0.597 | 0.579 | 13.30 | 14.36 |
| 90 | 1.49 | 0.50 | 0.20 | 0.027 | 0.127 | 0.598 | 0.579 | 15.53 | 16.33 |
Computational Fluid Dynamics (CFD) modeling
ANSYS WORKBENCH V14.5 was used to perform the simulations. CFD is a numerical tool that predicts the behavior of a fluid by solving the general transport equations, also called Navier-Stokes (NS) equations. Because there is no analytical solution, CFD uses the Finite Volume Method (FVM), which divides the domain into a finite number of cells over which discrete conservation of the variable is imposed.
General Equations and sub-models
Equation (9) is the differential form of the mass equation balance for incompressible flow (constant density), where
Regarding the incompressible fluid, the momentum differential equation is shown as (Equation (10)):
The Volume of Fluid (VOF) model is a surface tracking technique by defining calculation cells that can be empty, partially filled, or full of fluid. It was reported by Hirt and Nichols (1981) and is based on a concept that the volume occupied by one material cannot be occupied by the other. For each additional phase added to the model, a new variable is introduced: the volume fraction of the phase in the computational cell. In each control volume, the volume fractions of all phases conform the unit. The volume fraction of the fluid qth in the cell is denoted as αq , then the following three conditions are possible: if 𝛼𝑞 = 0, the cell is empty of fluid 𝑞𝑡ℎ; if 𝛼𝑞 = 1, the cell is full of fluid 𝑞𝑡ℎ, and if 0 <𝛼𝑞< 1, the cell contains the interface between the fluid 𝑞𝑡ℎ and the rest of the fluids.
The follow-up of the interface between phases is carried out by the solution of the continuity equation (Equation (11), for the volume fraction of one (or more) of the phases. For the qth phase this equation has the following form (Hirt & Nichols, 1981):
Where ρq is the density of the phase q (kg m-3); vq is the velocity of the phase q (m s-1);
The turbulence model K-ε is a semi-empirical model based on the transport equations for turbulent energy (K) and for the dispersion of turbulent kinetic energy (ε) (Launder & Spalding, 1974; Spalart & Almaraz, 1992). In obtaining this model, it is assumed that the turbulent flow is fully developed and that the effects of molecular viscosity are neglected (Jones & Launder, 1972), parameters K (m2 s-2) and ε (m2 s-3) are obtained from the Equation (12) and Equation (13):
Turbulent kinetic energy (K) (Equation (12)):
Dispersion of turbulent kinetic energy (ε) (Equation (13)):
μt is the turbulent viscosity (kg m-1 s-1) and it is shown as (Equation (14)):
Pkb and Pεb represent the influence of floatability force. PK is the turbulent generation due to the viscous force (kg m-1 s-3), as shown below (Equation (15)):
The values of the coefficients Cμ, C1ε, C2ε, σk, and σε were adjusted by Launder and Spalding (1974), which are shown as follows: Cμ = 0.09, C1ε = 1.44, C2ε = 1.92, σε = 1.3, σk = 1.0.
Numerical CFD simulation
Numerical simulation and solution using CFD consider three basic steps: a) Preprocess, b) Process, and c) Post-process. This a sequential activities, quality, and results depend of each one its activities.
The first step consists of digital generation as a solid of the construction of the physical model under study (Figure 3), to hydraulically analyze weirs in CFD the computational model must be defined as a three-dimensional geometry. The geometry module was used to perform the numerical model. The characteristics of the numerical model are based in a prototype triangular, rectangular, and compound model. Jan et al. (2006) and Sotelo (1997) do not mention the slope of the channel on which the tests were carried out; it was assumed that the slope is totally horizontal and the flow gradually varied so that the discharge coefficient is not affected by the slope of the channel.
After the generation of geometry, the domain is divided into a sufficient number of cells or elements that do not overlap and cover all the geometry, where general conservation equations were applied. Meshing module was used to generate mesh in the model, for spatial discretization, it was decided to employ a predominantly hexahedral mesh; among the advantages of hexahedral meshing is the reduction of the number of domain elements and the improvement of the convergence of the solution (Sánchez & Elsitidié, 2011). The most important parameter to define de viability of the computational model is the relationship between experimental and simulate data.
The boundary conditions of the computational model are established (Table 3). The flow inlet condition is located upstream of the weir at a minimum distance of two times the width of the channel (Boss, Replogle, & Clemmens, 1984) to avoid that the hydraulic load is not affected by the flow approach condition, the most common inlet condition in open channels is the mean flow velocity (inlet velocity) obtained from the flow rate and the continuity equation; the vertical plane located downstream of the spillway is defined as the free outlet of the flow to the atmosphere, the outlet condition assigned for open channels is atmospheric pressure (outlet pressure); as the computational model is symmetric in the axis of the spillway, the condition of symmetry is assigned in the lateral planes perpendicular to the axis of the spillway; the weir, the platform and the lateral walls of the channel are defined as solid stationary and non-slip with the roughness of 1.6*10-9m (roughness of high-density polyethylene).
Table 3 Boundary conditions (Figure 3).
| Boundary conditions | |
|---|---|
| Input | Velocity inlet (upstream) |
| Atmospheric pressure (upper area of the channel) | |
| Output | Atmospheric pressure (downstream) |
| Solid | Weir |
| Side walls and channel platform | |
| Symmetry | Side boundary and perpendicular plane to the axis of the weir |
The basis of an adjusted and adequate development to the physical conditions of a computational model is the numerical scheme that approximately solves the equations that describe the flow. Because it is a two-phase (water-air) and free surface flow, it is assigned to the multiphase VOF model and the turbulence K-ε (Launder & Spalding, 1974) model because it is the most complete simple method with the lowest computational cost to simulate turbulence demonstrating its advantage in confined and internal flows (Fernández, 2012) and in free-surface flows (Chanel & Doering, 2008; Olsen, Nils, & Kjellesvig, 1998; Jiang, Diao, Sun, & Ren, 2018).
The pressure-velocity coupling of the SIMPLE algorithm was used (Semi-Implicit Method for Pressure Linked Equations), it approaches convergence through a series of intermediate fields of pressure and velocity which satisfies continuity (Fernández, 2012). The use of the "Upwind" spatial discretization system ensures stable schemes, but its first-order characteristic makes it sensitive to numerical diffusion errors. Such errors can be minimized using higher-order discretization schemes.
Several simulations were performed to evaluate the precision of the model according to the previous results reported in the bibliography. Statistical methods were used to compare simulated results with experimental and analytical results. As an indicator of the quality of measurement of numerical results, the relative percentage error was used, which indicates how far from the experimental value is from the numerical estimation with CFD and it is defined with Equation (16):
Table 4 shows simulation scenarios according to experimental conditions.
Table 4 Simulation scenarios.
| Weir | Experimental conditions | |||||
|---|---|---|---|---|---|---|
| Channel width (m) | Width of rectangular section b 1(m) | Width of triangular section b 2(m) | Notch angle θ (°) | Load h (m) | Flow rates Q (l s-1) | |
| Rectangular | 2.00 | 1.00 | - | - | 0.27 | 249.00 |
| Triangular | 2.00 | - | 2.00 | 90 | 0.50 | 246.00 |
| Compound | 1.49 | 0.25 | 0.20 | 90 | 0.113 | 8.06 |
| 0.25 | 0.20 | 90 | 0.131 | 14.36 | ||
| 0.25 | 0.20 | 90 | 0.136 | 16.33 | ||
| 0.40 | 0.20 | 90 | 0.122 | 13.35 | ||
| 0.40 | 0.20 | 90 | 0.141 | 23.35 | ||
| 0.40 | 0.20 | 90 | 0.169 | 45.50 | ||
| 0.50 | 0.20 | 90 | 0.110 | 8.06 | ||
| 0.50 | 0.20 | 90 | 0.123 | 14.36 | ||
| 0.50 | 0.20 | 90 | 0.127 | 16.33 | ||
Results are analyzed and they are exposed in a qualitatively (maps, distributions, vectors) and quantitatively (graphs, integrals, values, averages) way.
Results analysis and discussion
Evaluation of the numerical model
Table 5 reports the simulation parameters and shows the statistical comparison of the experimental results reported by Sotelo (1997) with those simulated with CFD. The discharge coefficients of the rectangular and triangular weirs were obtained from Equation (4) and Equation (5), respectively.
Table 5 Hydraulic variables and relative percentage errors between experimental and numerical values of the rectangular and triangular weirs.
| Weir | Method | Load h (m) | Notch angle θ (°) | Discharge coefficient C d | Flow rates Q (l s-1) |
|---|---|---|---|---|---|
| Rectangular | Experimental | 0.270 | - | 0.6017 | 249 |
| CFD | 0.271 | - | 0.6016 | 251 | |
| Relative error (%) | 0.37 | - | -0.01 | 0.66 | |
| Triangular | Experimental | 0.500 | 90 | 0.5902 | 246 |
| CFD | 0.499 | 90 | 0.5902 | 245 | |
| Relative error (%) | -0.2 | - | 0.00 | -0.30 |
According to the results, the CFD model simulates both weirs very well. Although the CFD model predicted in the best way the discharge flow in a triangular weir than in the case of the rectangular weir, the relative error of the triangular weir is slightly less than the case of the rectangular weir. The use of the triangular weir is recommended for flow rates lower than 30 l s-1, however, Sotelo (1997) also documents that its accuracy is greater than a rectangular weir even for flow rates of 30 to 300 l s-1.
Once the CFD model was evaluated, nine scenarios were simulated for the combined weir, which are statistically compared with the experimental results reported by Jan et al. (2006). The discharge coefficients, Cdt and Cdr, of the triangular and rectangular sections of the compound weir are calculated with Equation (7) and Equation (8). Table 6 reports the flow rates obtained numerically for the three sections of compound weirs.
Table 6 Coefficients of discharge and numerical flow rates in compound weirs obtained by means of the CFD tool.
| Notch angle θ (°) | Rectangular section width b1 (m) | Triangular section width b2 (m) | Load on rectangular section CFD h1 (m) | Load on triangular section CFD h2 (m) | Discharge coefficient in rectangular section C dr | Discharge coefficient in triangular section C dt | CFD Flow rates Q (l s-1) |
|---|---|---|---|---|---|---|---|
| 90 | 0.25 | 0.20 | 0.014 | 0.114 | 0.590 | 0.579 | 8.08 |
| 90 | 0.25 | 0.20 | 0.033 | 0.133 | 0.591 | 0.579 | 14.31 |
| 90 | 0.25 | 0.20 | 0.040 | 0.140 | 0.591 | 0.579 | 16.31 |
| 90 | 0.40 | 0.20 | 0.028 | 0.128 | 0.594 | 0.579 | 13.27 |
| 90 | 0.40 | 0.20 | 0.042 | 0.142 | 0.594 | 0.579 | 23.24 |
| 90 | 0.40 | 0.20 | 0.079 | 0.179 | 0.596 | 0.579 | 45.30 |
| 90 | 0.50 | 0.20 | 0.010 | 0.110 | 0.595 | 0.579 | 8.11 |
| 90 | 0.50 | 0.20 | 0.025 | 0.125 | 0.596 | 0.579 | 14.47 |
| 90 | 0.50 | 0.20 | 0.029 | 0.129 | 0.597 | 0.579 | 16.33 |
The relative errors, calculated with Equation (16), are presented in Table 7; the analytical flow rates (obtained with Equation (6)) and the numerical flow rates (obtained in CFD) are compared against the experimental flow rates.
Table 7 Relative errors of analytical and numerical flow rates versus experimental flow rates.
| Rectangular section width b1(m) | Experimental flow rates Q (ls-1) | Analytical flow rates Q (l s-1) | CFD flow rates Q (l s-1) | Relative error analytical vs. experimental (%) | Relative error CFD vs. experimental(%) |
|---|---|---|---|---|---|
| 0.25 | 8.06 | 7.13 | 8.08 | -11.49 | 0.21 |
| 0.25 | 14.36 | 13.02 | 14.31 | -9.33 | -0.35 |
| 0.25 | 16.33 | 14.95 | 16.31 | -8.47 | -0.14 |
| 0.40 | 13.35 | 11.59 | 13.27 | -13.21 | -0.59 |
| 0.40 | 23.35 | 21.41 | 23.24 | -8.30 | -0.46 |
| 0.40 | 45.50 | 39.90 | 45.30 | -12.30 | -0.44 |
| 0.50 | 8.06 | 7.23 | 8.11 | -10.28 | 0.65 |
| 0.50 | 14.36 | 13.30 | 14.47 | -7.41 | 0.74 |
| 0.50 | 16.33 | 15.53 | 16.33 | -4.93 | 0.00 |
Estimate of discharge coefficient
The discharge equation of the compound weir (Equation (6)) can be rewritten as a function of a single discharge coefficient (Equation (17)):
Equating Equation (7) and Equation (17) it is possible to obtain a single discharge coefficient (Equation (18)).
The discharge coefficient of the rectangular section (Cdr ) is calculated with Equation (4) and that of the triangular section (Cdt ) is obtained with Equation (7). Equation (4) considers the geometry of the weir and the load of the rectangular section, while Equation (7) is independent of the load and its value is constant for each angle of the notch, and the effects of surface tension and viscosity are especially negligible.
Equation (18) is only valid for loads greater than h1, that is, h2 > h1. The discharge coefficients in Table 8 are calculated with Equation (18) for values of h2 / P ≤ 1.
Table 8 Discharge coefficient calculation.
| θ (°) | Channel width (m) | P (m) | b1 (m) | b2 (m) | h1 (m) | h2 (m) | Cd | h2/P |
|---|---|---|---|---|---|---|---|---|
| 90 | 1.49 | 0.2 | 0.25 | 0.20 | 0.013 | 0.113 | 0.634 | 0.565 |
| 90 | 1.49 | 0.2 | 0.25 | 0.20 | 0.031 | 0.131 | 0.626 | 0.655 |
| 90 | 1.49 | 0.2 | 0.25 | 0.20 | 0.036 | 0.136 | 0.623 | 0.68 |
| 90 | 1.49 | 0.2 | 0.25 | 0.20 | 0.02 | 0.12 | 0.632 | 0.60 |
| 90 | 1.49 | 0.2 | 0.25 | 0.20 | 0.06 | 0.16 | 0.613 | 0.80 |
| 90 | 1.49 | 0.2 | 0.25 | 0.20 | 0.10 | 0.20 | 0.603 | 1.00 |
| 90 | 1.49 | 0.2 | 0.40 | 0.20 | 0.022 | 0.122 | 0.655 | 0.61 |
| 90 | 1.49 | 0.2 | 0.40 | 0.20 | 0.041 | 0.141 | 0.641 | 0.705 |
| 90 | 1.49 | 0.2 | 0.40 | 0.20 | 0.069 | 0.169 | 0.629 | 0.845 |
| 90 | 1.49 | 0.2 | 0.40 | 0.20 | 0.01 | 0.11 | 0.662 | 0.55 |
| 90 | 1.49 | 0.2 | 0.40 | 0.20 | 0.08 | 0.18 | 0.626 | 0.90 |
| 90 | 1.49 | 0.2 | 0.40 | 0.20 | 0.10 | 0.20 | 0.622 | 1.00 |
| 90 | 1.49 | 0.2 | 0.50 | 0.20 | 0.01 | 0.110 | 0.680 | 0.55 |
| 90 | 1.49 | 0.2 | 0.50 | 0.20 | 0.023 | 0.123 | 0.669 | 0.615 |
| 90 | 1.49 | 0.2 | 0.50 | 0.20 | 0.027 | 0.127 | 0.665 | 0.635 |
| 90 | 1.49 | 0.2 | 0.50 | 0.20 | 0.04 | 0.14 | 0.656 | 0.70 |
| 90 | 1.49 | 0.2 | 0.50 | 0.20 | 0.06 | 0.16 | 0.647 | 0.80 |
| 90 | 1.49 | 0.2 | 0.50 | 0.20 | 0.10 | 0.20 | 0.639 | 1.00 |
The discharge coefficients in Table 8 are presented as a function of the ratio h2/P (Figure 4) and are adjusted to Equation (19) to facilitate the design of the weir.
The coefficients of Equation (19) are calculated with equations (20)-(25).
It was used as a criterion to evaluate the quality of the equations (19)-(25) the coefficient of determination (R2), the coefficient R2 allows to measure the agreement between two linearly related values. The values of the coefficient R2 of equations (19)-(25) is 0.999, which allows us to conclude that there is a high degree of concordance between the measured and calculated values. Finally, the flow rates that pass over the weir and the new discharge coefficients obtained with the equations (19)-(25) are shown in Table 9.
Table 9 Calculation of the flow rates and the discharge coefficient.
| θ (°) | Channel width (m) | b1 (m) | b2 (m) | h1 (m) | h2 (m) | Cd | Analytical flow rates Q (l s-1) | Experimental flow rates Q (l s-1) |
|---|---|---|---|---|---|---|---|---|
| 90 | 1.49 | 0.25 | 0.20 | 0.013 | 0.113 | 0.634 | 7.79 | 8.06 |
| 90 | 1.49 | 0.25 | 0.20 | 0.031 | 0.131 | 0.626 | 14.00 | 14.36 |
| 90 | 1.49 | 0.25 | 0.20 | 0.036 | 0.136 | 0.623 | 15.99 | 16.33 |
| 90 | 1.49 | 0.40 | 0.20 | 0.022 | 0.122 | 0.655 | 13.01 | 13.35 |
| 90 | 1.49 | 0.40 | 0.20 | 0.041 | 0.141 | 0.641 | 23.40 | 23.35 |
| 90 | 1.49 | 0.40 | 0.20 | 0.069 | 0.169 | 0.629 | 42.56 | 45.50 |
| 90 | 1.49 | 0.50 | 0.20 | 0.01 | 0.110 | 0.680 | 8.45 | 8.06 |
| 90 | 1.49 | 0.50 | 0.20 | 0.023 | 0.123 | 0.669 | 15.18 | 14.36 |
| 90 | 1.49 | 0.50 | 0.20 | 0.027 | 0.127 | 0.665 | 17.61 | 16.33 |
Discussion
Results revealed that the flow rates obtained through the CFD tool better predict the discharge than the analytical flow rates. Analytically there is inaccuracy of the discharge measurement during the transition between geometric sections, that is, in the region where the discharge exceeds the boundary between the triangular and rectangular sections. Lee, Chan, Huang, and Leu (2012) mentioned that the discharge discontinuity during the transition is associated with changes in mass and momentum; these are effects of variables, of conservation equations, which are not considered in the theoretical equations for the measurement of flow rates, as are the effects of viscosity and velocity distribution in proximity to the weir.
To improve the fit of the analytical model, Jan, Chang, and Kuo (2009) proposed an adjustment factor α, which multiplies the analytically calculated flow rates to estimate the discharge flow. α is defined as the average of ratio of experimental flow rates to the analytical flow rates (α was not considered in the results of Table 8). The adjustment value was α = 1.11, which indicates that the analytical method is undervalued concerning the experimental results.
The difficulty of describing the flow in the transition zone between the triangular and rectangular sections is well known. CFD must assume hypotheses to simulate and test sub-models to describe the flow, such as turbulence and phase change. Flow rates are very similar in several scenarios. Under these conditions, CFD simulations will allowed to know in detail the behavior of the flow in critical sections of the weir.
Hydraulic analysis of the coefficient of discharge
The proposed equations for obtaining the discharge coefficient (equations (19)-(25)) made it possible to analytically predict the flow rates of the compound weir, these equations consider the geometry of the weir (width, height, angle, and height of the triangular notch, width of the rectangular section) and the level of static charge. Table 10 compares the analytical flow rates with the experimental flow rates, the relative errors between both flow rates vary from 0.2 % to 7.8 %, the results of the flow rates obtained analytically from the discharge coefficient are acceptable.
Table 10 Comparison between analytical and experimental flow rates.
| Cd | Analytical flow rates Q (l s-1) | Experimental flow rates Q (l s-1) | Relative error (%) |
|---|---|---|---|
| 0.634 | 7.79 | 8.06 | -3.36 |
| 0.626 | 14.00 | 14.36 | -2.50 |
| 0.623 | 15.99 | 16.33 | -2.06 |
| 0.655 | 13.01 | 13.35 | -2.59 |
| 0.641 | 23.40 | 23.35 | 0.22 |
| 0.629 | 42.56 | 45.50 | -6.47 |
| 0.680 | 8.45 | 8.06 | 4.82 |
| 0.669 | 15.18 | 14.36 | 5.72 |
| 0.665 | 17.61 | 16.33 | 7.80 |
The graph in Figure 4 shows that the weirs have a higher discharge coefficient for small hydraulic loads h2 / P < 0.6 and increasing the width of the rectangular section. The reduction of the discharge coefficient values in the weir b1 = 0.25 m is smooth compared to the other weirs, from h2 ≈ 0.7 values the slope of the coefficient discharge curve of weirs b1 = 0.40 m and b1 = 0.50 m is softer and tends to stabilize from h2 / P ≥ 1. It is possible that at the h2 / P ≈ 0.7 inflection point where the change in the slope occurs, it is due to the presence of the pressure difference between the triangular and rectangular sections as well as the variation in the velocity of the discharge flow. Although the proposed discharge coefficient equation allows a better prediction of the flow rates, the results are not accurate to the experimental values, the charge-discharge relationship is directly affected by the discontinuity of the flow for loads above the horizontal crest of the weir (Lee et al., 2012).
Nappe characteristics
In the compound weir there are two nappes (Figure 5) as a consequence of the pressure difference that exists during the discharge in the triangular and rectangular sections (Figure 6). On the other hand, the rectangular section presents subatmospheric pressures and lower discharge speed, while in the triangular section there are pressures greater than the atmosphere and higher discharge velocity together with a nappe thicker than that of the rectangular section (Figure 5). The CFD tool allows to identify the presence of depression in the surface free of the water as a consequence of the high discharge speed generated in the triangular zone of the compound weir (Figure 7).
Figure 5 Particle tracking monitoring during discharge: 
X = 0.3 m, 
X = 0.6 m; b1 = 0.40 m; Q = 23.35 l s-1
Figure 6 Relative pressures: b1 = 0.40 m; Q = 23.35 l s-1; X in meters; 
x = 0.3; 
x = 0.6; 
free surface.
Figure 7 Profile of the free surface during discharge. Free surface
, weir edge
; b1 = 0.40 m; Q = 45.30 l s-1.
Speed measurements were made, its behavior is affected by the hydraulic load and the geometry of the compound weir. An important consequence of the distribution of the velocity field (Figure 8) is the impact on the discharge in certain sections of the weir. In general, the speed contours of the rectangular section are presented almost parallel to the crest of the weir. The velocities near the sloping regions of the crest triangular are lower than the rest of the triangular section, near the bottom of that section the velocity is relatively high, the length of the discharge flow in that area is greater relative to the other areas of the weir. The velocity profiles in Figure 8 present velocity gradients at the transition from the triangular to the rectangular section called the mixing zone (Lee et al., 2012), the velocity gradients decrease, and the velocities in the mixing zone increase as increases the hydraulic load. The velocity profile along the width of the channel (Figure 8) shows a reduction in discharge speed before approaching the triangular section, this phenomenon (red circle) is more visible for small loads on the rectangular section (Y = 0.31 m) and it is possible that it is due to the discontinuity of the flow near the mixing zone where according to Lee et al. (2012) presents mass and momentum exchange, for higher loads the speed reduction is less abrupt and with a tendency to stabilize (Y = 0.34 m).
Conclusions
The experimental and analytical studies reported in the bibliography allowed to reproduce numerically, by CFD simulations, the hydraulic behavior of compound (rectangular-triangular) sharp-crest weirs. Based on the objective of this work, as well as the results achieved, it is worth noting that the flow rates obtained numerically are statistically in agreement with their experimental values. Furthermore, it has been possible to improve the estimation of the analytically obtained flows. Therefore, the main conclusions of this study can be summarized as:
There is an error of up to 7.8 % in the discharge flow estimate with an analytical calculation compared to the laboratory flow measurements, and the error is reduced to 0.74 % when the estimate is made with CFD.
Regarding weir design, an expression of the global discharge coefficient was proposed as a function of the h2/P ratio and the weir geometry. Furthermore, the simulation results reveal that the value and the variation of the discharge coefficient increase with the increasing width of the rectangular section (b1). In fact, the discharge coefficient values are higher for ratios h2 / P < 0.6 and from h2/P≥1 it tends to stabilize. From h2 / P ≈ 0.7 the decrease in the discharge coefficient values is less drastic as a consequence of the reduction of the speed gradients that occur in the mixing zone and the pressure difference.
It was also possible to characterize the water nappe. CFD allowed identifying sub-atmospheric pressure in the rectangular region, while in the triangular zone the pressure is higher than atmospheric. In addition, during discharge, a thick nappe was observed in the area dominated by the triangular section and depression in the surface free. The speed distribution was characterized, the speed contours of the rectangular section are almost parallel to the crest of the weir, in the lower area of the triangular section there are high discharge speeds than in the rest of the weir where the nappe is more thickness and depression in the free surface is originated, the speed gradients that present during the transition from the triangular to the rectangular section decrease with the increase of the hydraulic load and the speeds in the mixing zone increases. In the rectangular section, the speed reduction was observed and later the sudden increase near the area dominated by the triangular section this caused by the discontinuity of the flow and the proximity to the mixing area, the speed increases and tends to stabilize as the hydraulic load increases.
According to the results obtained, it is possible to assume that the analytical error in the estimation of the discharge flow is due to the variation of the discharge velocity in the transition from the triangular section to the rectangular one and discontinuity of the flow.
Finally, the results allow us to conclude, on the one hand, that a triangular-rectangular compound weir largely satisfies the precision required to efficiently deliver water to irrigation users considering the important variation of delivery of water volumes. On the other hand, the experimentation and modeling with CFD are complementary strategies to understand and analyze details of the hydraulic operation of compound weirs.
Nomenclature
B |
channel width |
b |
weir width |
b1 |
width of the rectangular section of the compound weir |
b2 |
width of the triangular section of the compound weir |
Cd |
coefficient of discharge |
Cdr |
coefficient of discharge of the section rectangular the compound weir |
Cdt |
coefficient of discharge of the section triangular the compound weir |
Cμ, C1ε, C2ε |
constants of turbulent model K-Ꜫ |
f |
forces of the body per unit mass |
g |
acceleration of gravity |
h |
static load on the weir |
h1 |
static load on the section rectangular the compound weir |
h2 |
static load on the section triangular the compound weir |
K |
turbulent kinetic energy |
k |
coefficient depending on the dimensions and shape of the weir |
n |
dimensionless number depending on the shape of the weir |
mass transfer from phase p to phase q |
|
the mass transfer from phase q to phase p |
|
P |
height of the weir |
PK |
turbulent generation due to the viscous force |
Pkb, Pεb |
influence of floatability force |
p |
dynamic pressure |
Q |
dischage over weir |
qth |
phase q |
term source |
|
U |
velocity field |
αq |
volume fraction of the phase q |
Ꜫ |
dispersion of turbulent kinetic energy |
θ |
notch angle of the triangular weir |
μ |
dynamic viscosity |
μt |
turbulent viscosity |
ρ |
density of the fluid |
ρq |
density of the phase q |
σk, σε |
constants of turbulent model K-ε |
