Abstract:
Background and Objective: One of the new methods to receive and distribute flow in floodwater spreading systems is utilizing side weirs. These weirs are employed extensively in environmental, irrigation and hydraulics engineering to control the flow direction. This work evaluates the hydraulic behaviour of the rectangular broad-crested side weir in a trapezoidal shape channel numerically and experimentally to calculate the coefficient of discharge. Materials and Methods: In this investigation, gene expression programming is employed to predict the discharge coefficient under various conditions. Results: The study results revealed that six key factors affect the side weir discharge coefficient, which is the froude number (Fr), main channel’s bed slope, side weir step’s height to side weir’s length
INTRODUCTION
In general, side weirs are structures employed to divert the fluid flow and are located at the side of an open channel that steers the flow stream to the side channel's direction. Side weirs are primarily deployed for irrigation, land drainage and hydraulic engineering to adjust heads in distributaries and escapes. They are generally designed to operate as a control structure in the open channels and divert a certain amount of discharge. Hence, in the area of measurement of water flows, an approximate evaluation of discharge over the side weirs is still a problem of importance and a novel issue. They are easy to use, precise and traditional devices employed in the laboratory and for measuring flow in open channels1. The hydropower aspect and the coefficient of discharge of these side weirs for the various kinds of main channels, flow conditions and weirs have been investigated extensively.
Side weirs come in different types for different applications. Depending on their specific type and usage, side weirs can have various shapes (i.e., triangular, rectangular, trapezoidal, etc.). Additionally, the crest of side weirs can be made broad or sharp. Taking an experimental approach Abbasi et al.2 investigated the effect of geometric variables of the antivortex on a triangular side weir. In addition, the coefficient of discharge of side weir as related to the angle at the climax of the triangular weir and Froude number was described in their effort. The hydraulic performance of sharp-crested rectangular side weirs has been broadly studied in the literature for various scenarios combining different flow conditions and a variety of main channels. Previously3-7 evaluated the flow behaviour over sharp-crested rectangular side weirs in rectangular main channels. They concluded that the experimental data for subcritical flow agrees well with the derived analytical discharge coefficient. Also, they developed a relationship for the rectangular sharp-crested side weir’s coefficient of discharge. In addition, Granata and Marinis8 studied a dynamic relationship for flow in a rectangular channel along a side weir. They used De Marchi’s equation extensively for calculating the rectangular side weir’s coefficient of discharge.
Maranzoni et al.9 studied the lateral outflow over rectangular side weir placed in a straight channel based on the specific energy principle. The flow behaviour of a developed oblique side weir in subcritical flow was studied by Borghei and Parvaneh10 using the principle of specific energy. Their experimental results indicated that the developed oblique side weir is more efficient in comparison to other conventional side weirs. Azimi et al.11 investigated the discharge coefficient of rectangular-shaped side weir placed in a circular channel. To evaluate the coefficient of discharge, they studied the ratio of the side weir’s height to the weir’s upstream flow depth and the Froude number of upstream. Finally, Afshar and Hosein12 experimentally investigated a broad-crested rounded corner/rectangular shape side weir. In their investigation, a CFD simulation alongside the laboratory model was performed to determine the free surface profile in broad-crested rectangular shape side weir.
It is clear that nearly all researchers have investigated the fluid flow specifications of triangular/rectangular side weirs; however, little attention has been given to specifications of the hydraulic behavior of trapezoidal shape side weirs. In practical terms, trapezoidal shape channels along a side weir can be employed to manage drainage water and to deliver water to farm fields. Azimi et al.13 investigated the discharge coefficient of the rectangular side weir placed in a trapezoidal channel using support vector machines. They proposed an optimal model based on effective parameters on the discharge coefficient. Nezami et al.14 studied the trapezoidal lateral opening in a rectangular main channel intended for subcritical flow and discovered that the coefficient of discharge is related to the Froude number (Fr1). This correlation was also confirmed with experimental tests. Using the principles of specific energy. Vatankhah15 developed an analytical method to calculate the free surface flow profile along a side weir installed in a trapezoidal shape channel.
Previous studies performed on the typical side weirs suggest that three primary factors influencing their discharge coefficient (Cm). These parameters are the froude number (Fr1), the ratios of weir’s height to upstream depth and weir’s length to channel’s width. Earlier, it was assumed that the Froude number was the main parameter affecting the value of Cm. Namaee et al.16 experimentally investigated the fluid flow behaviour of a broad-crested side weir in a rough bed trapezoidal channel under the subcritical flow condition. Their results demonstrated that four primary factors affect the side weir coefficient of discharge and a new equation was proposed. Their equation included the parameters noted above and also the influence of the main channel’s bed slope.
The advances in computational fluid dynamics have made numerical simulation a suitable method for studying the free surface flow of devices used for water measurement in free surface channels17. Bilhan et al.18 and Emiroglu et al.19,20 investigated the hydraulic characteristics of rectangular, triangular and different trapezoidal shape labyrinth side weirs in subcritical flows. Their results showed that the labyrinth side weir has a noticeably higher coefficient of discharge compared to standard side weirs. A relationship for the calculation of the coefficient of discharge in labyrinth side weirs was also provided. Aydin and Emiroglu21 studied the hydrodynamic performance of a two-cycle side weir numerically. They showed that an increase in the upstream Froude number caused a decline in the discharge coefficient. Wang et al.22 studied the hydraulic parameters of side weirs of trapezoidal sharp-crested shape, including the froude number, velocity distribution, water profiles and discharge coefficient. Their results indicated that the simulated water depths were similar to the observation, with a lower than 5.63% relative error.
In the last few years, artificial intelligence (AI) techniques have been applied to predict the behaviour of non-linear systems in different scientific fields. In addition, many scientists have employed soft computing to recognize patterns of various hydraulic phenomena. Taking an experimental approach, Gopakumar and Mujumdar 23 studied the discharge coefficient of rectangular sharp-crested side weirs in a straight channel. A neural network model was developed to calculate the discharge coefficient and the results showed that these techniques could be effectively used in predicting the coefficient of discharge. Ebtehaj et al.24 derived a model for the discharge coefficient of side weirs placed in rectangular shape channels employing the gene expression programming (GEP) model. The value of the correlation coefficient in their presented model was 0.947. Optimization was carried out by Khoshbin et al.25 by employing the singular value analysis and the genetic algorithm (GA) for a neuro-fuzzy inference system for estimating the discharge coefficient of side weirs. Jahanpanah et al.26 predicted the coefficient of discharge of channels with rectangular shapes by applying four soft computing models. The parameters used included the roughness coefficient, the depth of the brink, the channel’s bed slope and the channel’s width. Experimental data were compared with the computed discharge coefficients for the accuracy test. The results also indicated that the artificial neural network model is more efficient than the other soft computing methods. Azimi et al.27 analyzed the sensitivity of variables that affect the coefficient of discharge of weirs installed in channels with trapezoidal shape by employing the extreme learning machines (ELM) method. It was also shown that the efficacy of the Froude number in estimating the coefficient of discharge is more than other variables. The side weirs installed on channels with trapezoidal shape and their conducting capacity were investigated by Azimi et al.28 using the GEP. It was indicated in their effort that the AI model of 81% of results had an error of less than 5%.
In summary, sharp-crested side weirs are instruments employed to determine the discharge of the water from the small rivers and channels, while in the case of the broad-crested side weirs, large discharge coefficients are measured. Compared with the other main channel shapes, the trapezoidal main channel is quite common due to the advantages such as the capability of large discharge of water, performing very effectively, suitable structure and ease of arrangement. However, so far, the numerical simulations on broad-crested rectangular side weir in trapezoidal channels for studying their hydraulic performance are scarce.
In this study, the flow behaviour of rectangular broad-crested shape side weir in the trapezoidal channel was investigated. The experimental data were used to develop an equation for predicting the discharge coefficient of rectangular broad-crested side weir. Also, computational hydraulic models and gene expression programming were developed to simulate the hydraulic behaviour of rectangular broad-crested side weir under various hydraulic conditions. The developed three-dimensional numerical model effectively captured the principal hydraulic features of broad-crested side weirs. In addition, applying the gene expression programming (GEP) method, the range of the geometrical parameters was extended and the effects of additional discharge coefficients were evaluated. To consider the effect of geometrical parameters (step height, side weir length and side weir angle) on the discharge coefficient, six gene expression programming models were created and their influences on the coefficient of discharge estimation were evaluated. Eventually, the optimal model is achieved and an equation of this model is calculated for use in real applications.
MATERIALS AND METHODS
Basic theory: In Granata and de Marinis8 investigation, De Marchi’s equations theoretically demonstrated that, while the flow profile is curved, the energy head alongside the crest’s spillway stays the same, increasing in subcritical flow and decreasing in supercritical flow. An example of a varied flow with decreased discharge in space is the flow over a side weir. The energy equation is generally used for deriving the governing equations of flow over a side weir. French and French29 derived the main differential equations of a varied flow in space alongside a side weir with a reduction in discharge as:
| (1) |
In this equation, the is the distance along weir from the end of upstream, represents the flow depth, S0 is main channel’s bed slope, Sf is the slope of friction, is the correction factor of kinetic energy, the discharge of the channel is represented by Q, dQ/dx is the variation of discharge along the side weir length, g is the gravitational acceleration, A is the flows cross-sectional area and T is the channel section top width.
In trapezoidal channels, the cross-section of flow is evaluated using:
| (2) |
Also, for a rectangular shape broad-crested side weir assuming that S0-Sf = 0 and α = 1 the discharge Q per unit length of a rectangular side weir of heights, Eq. 1 is simplified as29,30 :
| (3) |
In which, x is weir’s distance from upstream (horizontal distance), y is the height of the side weir, is the depth of flow at the main channel, (y-s) is the pressure head on the weir and Cm is the coefficient of discharge.
In general, introducing the dimensional groups, Cm can be expressed as:
| (4) |
In this equation, S0 is main channel’s bed slope, p is the weir height, B is the main channel width, y1 is the upstream flow depth at the section of the side weir, Fr1 is the upstream froude number. The functional form of Eq. 4 was evaluated in the experimental phase of the study.
Additional new dimensional parameters are used to assess the influence of all variables affecting the discharge coefficient of side weir in the numerical (CFD/GEP) study of the proposed side weir of broad-crested rectangular shape. These are side weir step’s height to side weir’s length ratio, side weir angle (θ), ratio of side weir’s flow depth upstream to the height of side weir step and the ratio of the side weir’s flow depth upstream to side weir length are considered. Therefore, the dimensionless variables influencing the prediction of the coefficient of discharge using dimensional analysis are expressed as:
| (5) |
| Table 1: | Efficient dimensionless variables on the coefficient of discharge for 1-6 models |
| Number of the models | Variables (dimensionless) |
| First | |
| Second | |
| Third | |
| Forth | |
| Fifth | |
| Sixth |
where, Fr is the Froude number, H is the side weir step’s height, L is the side weir’s length, θ is the side weir’s angle and y1 is the side weir’s flow depth upstream.
In this work, the functional form of Eq. 4 is evaluated in the experimental investigation. In the numerical study, six models are presented for estimating the coefficient of discharge based on Eq. 5. The data in Table 1 lists the dimensionless variables influencing the coefficient of discharge for all six models.
Experimental setup and procedure: The experimental setup included a main channel and a side-channel parallel to the main channel, as shown in Fig. 1. The trapezoidal cross-section main channel is 30 m long with a depth of 0.6 m. At the bottom, the main channel width is 1 m, with different bed slopes of 1-1000 and 1-3000. The rectangular side weir had a length of 4 m, a width of 0.85 m and was placed at the main channel's mid-section as indicated in Fig. 2. The distance between the side weir and both ends of the channel was 13 m.
The main channel’s water supply came from a large 1.5 m deep feeding basin. The inlet discharge rates were measured by a sharp-crested rectangular weir. To measure the outlet discharge rates of the side weir, a calibrated rectangular weir at the end of the side channel was used. The measurements were taken with an accuracy of Lit/s. Three-point gauges with ±0.1 mm sensitivity were employed to measure the water depth. These gauges were placed at the side weir section and the centerline of the main channel. To be able to measure the depth of flow, a specific type of rail machine was used that was able to navigate in the x and y dimensions, as shown in Fig. 2. Pressures were measured using eight piezometers located on the centerline of the side weir. A sluice gate was installed at the end of the main channel to regulate the depth of flow as the downstream boundary condition. Nearly 300 experimental tests were performed for subcritical and stable flow conditions.
| Fig. 1: | A picture of the experimental setup |
| Fig. 2: | Point gauge is mounted on the side weir |
| Table 2: | Experimental parameters | |||
| Bed slope of main channel (S0) | Froude number (Fr1) |
Height of weir (p) (cm) |
Inlet discharge rate (Qin) (lit/s) |
Number of runs |
| 0.736111111 | 0.53-0.95 |
02-Oct |
32-96 |
198 |
| 2.125 | 0.53-0.95 |
02-Oct |
32-96 |
99 |
The measurement results were compiled and used to develop a correlation for estimating the rectangular side weir’s discharge coefficient. Table 2 shows the range of experimental parameters.
Numerical simulation
Computational fluid dynamic: The numerical simulations were carried out with the CFD software to estimate the side weir of broad crested shape’s coefficient of discharge. The flow conditions and dimensions of the numerical model are the same as the aforementioned experimental model. The schematic of the simulated broad-crested side weir is illustrated in Fig. 3.
The equations employed in the simulations are the conservation of mass and momentum. The finite volume method was employed to model the multiphase flows. For a multiphase flow with components of velocities (u, v, w) along the cartesian coordinate axes (x, y, z), the governing Eq.:
| (6) |
| (7) |
Here, VF is the open volume parameter for conducting the flow in the fast volume-reconstruction (favor) algorithm, which is specified in the next section. Ax is the open surface parameter conducting the flow along the x direction Ay and Az and represent the corresponding surface parameters along the y and z directions, respectively.
| Fig. 3: | Schematic of the simulated broad-crested side weir and the computational mesh |
| Table 3: | Range of various parameters used in the simulation | ||
| Side weir’s length (L) (m) | Side weir step’s height (H) (cm) |
Side weir’s angle (θ) (°) |
Number of simulations |
| 3, 4 and 5 | 12.62, 25.24 and 37.86 |
10, 20, 30, 45 and 60 |
45 |
In the conservation of momentum equations, ρ is the density of the fluid and (Gx, Gy, Gz) are the components of body acceleration. In these equations, (fx, fy, fz) are the effects of both molecular viscosity and viscosity because of turbulence. Generally, two-equation models are employed in hydraulics for turbulence modelling. The RNG k-ε three-dimensional turbulence model was used in the present study to solve the Reynolds-averaged equations31.
To determine the appropriate geometry in the finite volume method, the surface/volume elements expressing obstacles were employed in the FAVOR algorithm. In this algorithm, values between 0 (air) and 1 (water) are assigned to phases, i.e., water and air in the computational cell domains. If water fills the entire cell, then the surface/volume element is given the value of 1. The Volume of Fluid (VOF) algorithm can determine the free surface of the flow. The numerical simulations were conducted using the VOF method with open channel options, which allows for defining free surface levels and velocities (and so froude numbers) at the inlet boundary. Modified high-resolution interface capturing (HRIC) method with an implicit scheme was employed to achieve a sharp interface and greater accuracy between phases. To simulate the face fluxes for all elements, the implicit scheme employed a standard finite-difference interpolation scheme as follows:
| (8) |
In which n+1 is the superscript of the current time step, is the superscript of the previous time step, aq.f is the face value of the th volume fraction estimated from the modified HRIC method, V is cell volume, Uf is volume flux from the face. The time step size was 0.01s in the solutions. To obtain the iteration convergence for the unsteady cases, the solutions continue until the residuals drop at least three order magnitudes for each equation.
In the numerical simulations, three specific geometry parameters, side weir length (L), side weir step height (H) and side weir angle (θ) are changed and various parameter ranges are presented in Table 3. These changes in geometry parameters and boundary conditions are shown in Fig. 4. The inlet boundary condition for the main channel was a specified volume flow rate. The outflow condition was considered at the end of the side and main channels. Furthermore, a pressure outlet boundary condition was specified at the top of the domain. Other boundaries were considered as a wall boundary condition.
As shown in Fig. 5, several structured meshes were used to achieve a reliable accuracy and convergence of the numerical solution. Roache32 recommended that the grid refinement factor should be set to a value of approximately 10%. For grid study, simulations are performed for the flow rate of 52 lit/s in the main channel with a slop of 0.001 using 106, 4×106, 75×105 and 9×106 cells. The flow depth at a point 13 meters from the beginning of the main channel is presented in Fig. 5. This figure shows that the simulation results with more than 75×105 grids are independent of the grid size.
| Fig. 4: | Three-dimensional sketch, main geometric variables of a broad-crested side weir and boundary conditions |
| Fig. 5: | Grid independence tests (water depth in the centerline of the main channel for different grid numbers, Qin = 52 Lit/s, bed slope of 0.001) |
The number of elements in the x, y and z directions is 1000, 150 and 50, respectively, with the element sizes of 0.03 m in the x-direction, 0.024 m in the y-direction and 0.016 m for z-direction.
Gene expression programming: Gene expression programming (GEP), as a novel method based on Genetic Programming (GP), was first developed by Ferreira33. GEP, alongside the GP and the genetic algorithm (GA), are transformative methods capable of establishing programs of complicated nature. The GEP technique is similar to the evolution theory of Darwin, in which natural selection research methods and optimization are applied to attain excellence. Various computer programs, which are models of quasi-mathematical nature and established upon the specified function of fitness to resolve the issue of evolution, are applied to develop the process of GEP.
Commonly, GEP comprises five basic elements, the terminal set, fitness function, control parameters, function set and termination conditions34. To show the technique used for problem-solving, GEP utilizes string characters of fixed-length parse trees shapes of several sizes and forms. Expression trees (ETs) is the name given to these trees. One significant benefit of the GEP is its ordinary variety in genetics, whereby employing genetic operators of various kinds, GEP takes action at various levels of the chromosome. Also, the mutagenic character, a remarkable aspect of GEP, guides to a program with a convened nature consisting of subprograms to attain an optimal solution. Several symbols are assigned to all of the available genes in GEP. The genes can be employed as part of the terminal set while their length is permanent, for example (Y. X. Z. 5) and the determined set of function, for instance (+. ×. -. /. √). All functions must be suitable to get any data, reinstate a function, or be taken into account by a terminal. A typical gene of GEP with a terminal function presumed as mentioned is represented as the following:
+. ×. √. X. -. +. ×. Y. X. Z. 5. Y. Z. |
(9) |
Here, 5 is a constant and Z, Y and X are parameters. K-notation is the given name to the expression of Eq. 928. The expression tree diagram can be one of the representatives of the K-notation.
In this effort, based on GEP, a model is introduced employing optimization programming software. Next, the coefficient of discharge modeling for side weirs in channels with trapezoidal shape is presented.
| Table 4: | Values of various variables in the practical gene expression programming (GEP) |
| Parameters | Setting |
| General | |
| Chromosomes’ quantity | 45 |
| Gene | 3 |
| Size of the head | 10 |
| Rate of mutation | 0.01 |
| Rate of inversion | 0.1 |
| Genetic operators | |
| Rate of IS transposition | 0.1 |
| Rate of RIS transposition | 0.1 |
| Rate of one-point recombination | 0.3 |
| Two-point recombination | 0.3 |
| Recombination of gene | 0.1 |
| Transposition of gene | 0.1 |
| Generations’ quantity | 300 |
| Tires’ quantity | 3 |
| Increase in complexity | |
| Maximum complexity | 5 |
| Table 5: | Functions employed for GEP model | ||
| Name of the functions | Label in expression tree |
Name of the function |
Label in expression tree |
| Multiplication | * |
Inverse |
Inv |
| Addition | + |
x to the power of 2 |
X2 |
| Division | / |
Arctangent |
tan–1 |
| Subtraction | - |
hyperbolic tangent |
tanh |
| Exponential | Exp |
Complement |
Not |
| Natural logarithm | Ln |
First, the function of fitness for this particular problem must be specified. In the present study, RMSE (root mean squared error) is utilized as proposed by Shiri and Kisi35. The function of fitness of the RMSE (Ei) for each computer program is expressed as:
| (10) |
where, Tj is the fitness case target value j, Pij is the individual program’s anticipated value employing GEP for the fitness case j and n represents the samples’ total number. The most effective answer is achieved when Tj the Pij and values are equal to each other consequently, Ei= 0. To evaluate the fi function or the fitness of the individual computer program i, a formula is utilized. That is:
| (11) |
As illustrated in 0-1000 range, the equation conversions and the most effective result is obtained when Ei = 0 and thus fi = 100036. Subsequently, when the fitness function is defined, the function set (F) and the terminal set (T) are determined. Concerning the present investigation, the T comprises
RESULTS AND DISCUSSION
Experimental results: The flow over a rectangular broad-crested side weir in the trapezoidal channel is studied and a new correlation for the corresponding discharge coefficient as a function of non-dimensional variables defined in Eq. 4 was developed. As noted before, the discharge coefficient, Cm is a function of several dimensionless groups, including froude number, Fr1 channel slope and the geometry of the weir and the channel, p/y1 ,y1/b . To measure the effect of each of these variables, several experiments are performed and the influence of each parameter is determined. Figure 6 shows the variation of measured values of Cm against Fr1 for different values of S0. The values of p/y1 and y1/B are fixed as 1m, 1.25 and 0.1, respectively. The Cm of side weir with S0 = 1-1000 was 1.3-1.7 times higher than that of side weir’s with S0 = 1-3000. This figure shows that by increasing flow velocity with changing the bed slope, the discharge coefficient increases.
| Fig. 6: | Variation of measured Cm versus for various Fr1 values of bed slope S0 |
| Fig. 7: | Variation of measured Cm versus for various values of bed slope S0 |
| Table 6: | Different hydraulic/geometric variables of broad-crested side weir |
| Hydraulic and geometric variables | Values |
| Fr1 | 0.53-0.95 |
| S0 | 1-1000 and 1-3000 |
| p/y1 | 0.13-2.22 |
| y1/B | 0.045-0.15 |
Figure 6 also shows that with the increase of the upstream Froude number, the values decrease. This result means that higher Cm values are achieved at lower upstream Froude numbers due to the decrease of flow velocity and increase of depth of flow.
In Fig. 7, the values of Cm are plotted against the ratio of weir height to water depth,
As expressed in previous studies, the upstream Froude number is the key influential factor affecting the discharge coefficient. Therefore, all the obtained experimental data was used and regression in different bed slopes for obtaining the best fit which is given as:
| (12) |
| (13) |
Figure 9 compares the measured values of Cm against the computed values Cm across the entire range of the included parameters. As can be seen from Fig. 9, Eq. 12 provides a reasonably accurate estimate of the experimental data.
The statistical indices Root Mean Square Error (RMSE), Mean Absolute Relative Error (MARE), Scatter Index (SI) and BIAS methods are employed to investigate the accuracy of the used models. They are defined as:
| (14) |
| (15) |
| (16) |
| (17) |
| Fig. 8: | Variation of measured Cm versus for various values of bed slope S0 |
| Fig. 9: | Comparison of the computed and measured values of Cm |
Here, Cm (predicted)i-Cm (predicted)i-Cm (observd)i and are, respectively, the experimental discharge coefficient, the predicted coefficient of discharge, the average coefficient of discharge and the number of measurements. The statistical indices RMSE, MARE, SI and BIAS for two different bed slopes are shown in Table 7.
As noted before, the bed slope and the Froude number have major effects. As already mentioned, due to the experimental results, the purpose of this work was to analyze the hydraulic performance of the rectangular broad crested side weir based on limited parameters such as Fr1. S0.
Numerical results
Validation: The results of the numerical simulations were validated using the experimental data. The comparison of the simulation and experimental surface profiles of water through the trapezoidal main channel centerline for Qin = 110 Lit/s, bed slope of 1-1000, side weir length of 4 m and height step of 12.62 cm as shown in Fig. 10. It is seen that the flow depth in the main channel decreased significantly due to the side weir. In addition, decreasing the flow depth in the main channel caused an ascending trend in the flow depth along the side weir length. Figure 10 indicates that the numerical results are in good agreement with the experimental data. Furthermore, the percent error of the numerical results compared with the experimental data was within a ±6% error limit, which is acceptable.
Water surface profile: The Water surface profile is an essential variable describing flow patterns in an open channel. Water levels along the side weir and main channel lengths are simulated and the water surface profiles are obtained. These simulations were performed for the domain that is 4 m before the upstream edge of the side weir to 2 m after the downstream of the side weir for an inlet discharge rate of 52 Lit/s and various bed slopes. Along the centerline, Water surface profiles for flow over the main channel and the side weir for different bed slopes and an inlet discharge rate of 52 Lit/s are presented in Fig. 11a-b, respectively. The data in Fig. 11a shows the water profile along the 10 m length in the middle-point of the main channel. The data in Fig. 11a shows a sharp decrease in the water level at the start of the side weir in the main channel. Namaee et al.16 have investigated the flow behaviour along the side weir in a trapezoidal channel. Namaee et al.16 have reported the presence of secondary flow as the reason for this behaviour. As seen in Fig. 11b, free surface levels gradually rose towards the downstream end of the side weir due to the secondary flow. The secondary flow disperses main channel flow energy and causes a rise in water level along the side weir length. Also, with an increase in the bed slope, the mainstream velocity in the entire flow regime increases accordingly. Therefore, the water surface level of both the main and side channels is increased.
The velocity field: In common side weirs that are located perpendicular to the flow direction, the passing flow is considered approximately two-dimensional.
| Fig. 10: | Experimental and numerical water surface profile through the centerline of the main channel for Qin = 110 Lit/s, bed slope of 1-1000, side weir of 4 m length and height step of 12.62 cm |
| Fig. 11(a-b): | Flow profile for different bed slopes and inlet discharge rate of 52 Lit/s, (a) Main channel and (b) Side weir |
| Table 7: | Statistical indices results | |||
| Bed slope (S0) | RMSE |
MARE |
SI |
BIAS |
| 1-1000 | 0.0158 |
0.0341 |
0.0398 |
0.01 |
| 1-3000 | 0.0395 |
0.0687 |
0.074 |
0.022 |
Azimi et al.39 have investigated the longitudinal velocity distribution passing over typical side weirs in circular channels. They found that the maximum velocity occurs near the side weir crest. Bagheri and Heidarpour5, in their studies, indicated that the maximum longitudinal velocity happens at the upstream end of the side weir. Figure 12 shows the longitudinal velocity contours in the main channel and on the side weir for an inlet discharge rate of 52 Lit/s and bed slope of 1-1000. This figure shows that the maximum longitudinal velocity occurs near the downstream end of the side weir and the free surface. Then immediately after the peak, the velocity drops and reaches its lowest value. In Fig. 12, the stagnation zone near the right side wall of the side channel downstream of the side weir is seen. The stagnation region was also seen in the experimental study. In addition, Fig. 12 shows the secondary flow at the cross-section of the main channel due to the lateral flow presented by Y-component velocity vectors. The secondary flow related to the mean kinetic energy of lateral motion in the main channel significantly influences the discharge efficiency. Streamlines are also diverted to the side weir outlet due to the secondary flow.
| Fig. 12: | Simulated longitudinal and lateral velocity contours in the main channel and side weir for Qin = 52 Lit/s, S0 = 1-1000 |
Emiroglu et al.20 suggested that the intensity of the secondary flow is related to the froude number, the location of the side weir and the height of the weir crest.
Discharge rate of the side weir: The key problem in practice is determining the discharge behaviour of a side weir, which is primarily described by the discharge coefficient. The average specific energy difference between the downstream and upstream ends of the side weir must be approximately less than 5%8 to use the De Marchi equation, which is based on a constant energy theory. Thus, the following analysis is performed using the assumption of constant specific energy. The effect of bed slope on the outlet discharge rate of broad crested side weir is calculated numerically in Fig. 13. It is seen that increasing the bed slope of the channel increases the side weir discharge rate in the side weir.
Statistical indices and superior geometry: To assess the correctness of 1-6 models in the Table 1, RMSE, MARE, R2 (coefficient of correlation), SI and BIAS are evaluated:
| (18) |
| (19) |
| (20) |
| (21) |
| (22) |
Here, Cm(GEP)i, Cm(GEP)i, m(GEP)i and are the coefficient of discharge estimated by gene expression programming, the coefficient of discharge estimated in the experiments (EXP), the average experimental discharge coefficient and the total number of experimental tests, respectively. The statistical indices MARE, RMSE, R2, SI and BIAS indicate the errors between the experimental tests and GEP coefficients of discharge. The RMSE index only shows a root mean square error between the model prediction and experiment8. Thus, the models should be assessed employing threshold statistics and indices of other kinds40,41. The TSx index indicates the distribution of error in the amounts estimated by each of the models for X% of anticipations. This variable is specified for different mean absolute relative error amounts. TSx index’s amount for X% of the anticipations is provided as the following:
| (23) |
In the equation above, Yx represents the predicted quantities for all of the MAREs lower than X% of the total amount of data. The presented statistical indices are described in references42,43.
The MARE, RMSE, BIAS and SI values obtained for the first model are, respectively, 0.025, 0.01, 0.013 and 0.036.
| Fig. 13: | Calculated side weir discharge rate for broad-crested rectangular side weir for different bed slopes |
| Fig. 14: | Error distributions of different models |
The first model is a function of the side weirs upstream Froude number (Fr), the height of side weir step to side weir length ratio
The second model is a function of Fr, θ, and and the MARE and RMSE values calculated for the second model are 0.029 and 0.012, respectively. The BIAS and SI indices estimated for the second model are 0.010 and 0.034, respectively. In the mentioned model, 10.5% of data have an error of 5-10%, while 85.3% of the error is below 5%. In addition, 4.2% of the second model data have an error of more than 10%.
For the third model, the side weir discharge coefficient is a function of
For the fourth model, MARE and RMSE are 0.035 and 0.013, respectively. The computed BIAS and SI for the mentioned model are 0.010 and 0.029, respectively. The fourth model’s coefficient of discharge is a function of
The fifth model’s MARE and RMSE values are 0.033 and 0.015, respectively. The BIAS and SI of this model are 0.012 and 0.04, respectively, while 70% of data have an error of lower than 5%. Finally, the prediction error of all of the discharge coefficients is less than 10%.
Investigating the modelling results for the coefficient of discharge of a side weir in a channel of trapezoidal shape using GEP, the first model is the most suitable. The correlation given by Eq. 20 is developed using the results of the coefficient of discharge of the side weir for the first model. In addition, according to the experimental results, the GEP results demonstrated that the discharge coefficient of broad-crested rectangular side weir is improved. The optimized geometry model is a 5 m in length side weir step with a height of 37.86 cm and a side weir angle of 60°:
| (24) |
| Fig. 15(a-d): | Comparison of the discharge coefficient vs- attained along the side weir for various side weir angles |
| Table 8: | Statistical indices and error distribution for different models | |||||
| Statistical indices models | MARE |
RMSE |
BIAS |
SI |
R2 |
Error<5% |
| 1 | 0.025 |
0.01 |
0.013 |
0.036 |
0.964 |
88.5 |
| 2 | 0.029 |
0.012 |
0.01 |
0.034 |
0.919 |
85.3 |
| 3 | 0.032 |
0.013 |
0.012 |
0.035 |
0.892 |
84.7 |
| 4 | 0.035 |
0.013 |
0.01 |
0.029 |
0.864 |
86.28 |
| 5 | 0.033 |
0.015 |
0.012 |
0.04 |
0.837 |
70 |
| 6 | 0.039 |
0.02 |
0.015 |
0.045 |
0.812 |
62.35 |
As noted before, Cm is a function of parameters
CONCLUSION
In the present work, the experimental study, CFD and GEP simulations were used to analyze the hydraulic characteristics of the rectangular broad-crested side weir in the trapezoidal channel. The analysis of the experimental data shows that the discharge coefficient of a broad-crested rectangular side weir is a function of and and a new correlation for the discharge coefficient was developed. In the CFD and GEP simulations, the dimensionless variables and and six models were employed to predict the discharge coefficient. Therefore, the most appropriate correlation was also identified. Finally, based on CFD and GEP simulations, a practical discharge coefficient for rectangular broad-crested side weir is introduced for hydraulic applications.
SIGNIFICANCE STATEMENT
The presented comprehensive investigation evaluated the major hydraulic parameters of flow around the rectangular broad-crested side weir in the trapezoidal-type main channel that can be beneficial for developing novel discharge coefficients. The results of this study help the researcher uncover the critical area of designing an optimum rectangular side weir that is of significant interest in many hydraulic and irrigation management projects.