Discharge formula for rectangular sharp-crested weirs

Sharp-crested rectangular weirs used for discharge measurement in channels and laboratories are experimentally investigated. Height and width of weir plate are the two parameters characterizing the head-discharge relationship. Laboratory experiments are conducted by measuring the discharge and the head over the weir for variable weir heights and widths. Applicability of various formulations for the discharge coefficient are investigated. Experiments indicate that discharge is independent of weir height, when the weir is operated within an appropriate discharge range. Average velocity over the weir plotted against the weir head displays universal characteristics such that it can be used in the expression of discharge over the weir, eliminating the need for a discharge coefficient. The head-discharge relationship for a rectangular weir has distinct features for the partially contracted weirs and for the fully contracted slit weirs.     

Discharge prediction for rectangular sharp-crested weirs by machine learning techniques

The stage-discharge relationship of a weir is essential for posteriori calculations of flow discharges. Conventionally, it is determined by regression methods, which is time-consuming and may subject to limited prediction accuracy. To provide a better estimate, the machine learning models, artificial neural network (ANN), support vector machine (SVM) and extreme learning machine (ELM), are assessed for the prediction of discharges of rectangular sharp-crested weirs. A large number of experimental data sets are adopted to develop and calibrate these models. Different input scenarios and data management strategies are employed to optimize the models, for which performance is evaluated in the light of statistical criteria. The results show that all three models are capable of predicting the discharge coefficient with high accuracy, but the SVM exhibits somewhat better performance. Its maximum and mean relative error are respectively 5.44 and 0.99%, and 99% of the predicted data show an error below 5%. The coefficient of determination and root mean square error are 0.95 and 0.01, respectively. The model sensitivity is examined, indicative of the dominant roles of weir Reynolds number and contraction ratio in discharge estimation. The existing empirical formulas are assessed and compared against the machine learning models. It is found that the relationship proposed by Vatankhah exhibits the highest accuracy. However, it is still less accurate than the machine learning approaches. The study is intended to provide reference for discharge determination of overflow structures including spillways.     

Experimental study of discharge formulas for rectangular sharp-crested weirs under free flow condition

Laboratory experiments were carried out to investigate the discharge characteristics of rectangular sharp-crested weirs under free flow condition. The performances of available discharge formulas have been evaluated by using the experimental data sets of present and previous studies. Error statistics of our experimental data indicate that the recent stage-discharge relationships show satisfactory performances. Discharge formula in terms of weir Reynolds number proposed by Vatankhah gives the highest accuracy among the existing slit weir equations, with $E_{\pm 4}=100.00\text{\%}$ (i.e. percent error less than or equal to $\pm 4$) and a mean absolute error ${\left|E\right|}_m=0.88\text{\%}$. The full-range discharge equation presented by Bijankhan and Mahdavi Mazdeh shows the highest accuracy among the relationships in terms of weir contraction ratio, with $E_{\pm 4}=100.00\text{\%}$, ${\left|E\right|}_m=0.91\text{\%}$ for slit weirs and, $E_{\pm 4}=94.64\text{\%}$, ${\left|E\right|}_m=1.60\text{\%}$ for partially contracted weirs, respectively. The weir velocity formulae suggested by Gharahjeh et al. exhibit the relatively better performance, with $E_{\pm 4}=98.41\text{\%}$, ${\left|E\right|}_m=1.34\text{\%}$ for slit weirs and, $E_{\pm 4}=91.07\text{\%}$, ${\left|E\right|}_m=1.91\text{\%}$ for contracted weirs, respectively. Statistical results of this study confirm the weir velocity approach presented by Aydin et al. and show that, the weir velocity is a predominant quantity for rectangular sharp-crested weirs, unique characteristics of the weir velocity curves make it more suitable for expressing the discharges. Moreover, it is important to point out that the performance of weir velocity formulae can be further improved.     

Mean velocity and turbulent characteristics of flow over half-cycle cosine sharp-crested weirs

The classical sharp-crested weirs are not suitable to implement in irrigation canals with high suspended sediment flow. Over the time, sediment deposition occurs in the upstream of sharp-crested weirs and causes a time variant water level raise in the upstream and consequently variable discharge coefficients. A series of laboratory experiments was carried out to test the hypothesis of strong turbulent formation and three-dimensional flow in the upstream of half-cosine sharp-crested weirs to prevent sediment deposition. To verify this hypothesis, a series of laboratory experiment was carried out for different weir heights and top widths in free and submerged flow conditions. Mean flow structure and turbulent characteristics of half-cosine and rectangular sharp-crested weirs were measured using a micro-Acoustic Doppler Velocimeter (ADV) probe. It was found that the streamwise velocities of half-cosine weirs were higher than the corresponding rectangular weirs, however, the velocity fluctuations in this direction were similar for both types of weirs. Velocity fluctuations in vertical and transverse directions were found to be significant in half-cosine weirs. The results showed higher turbulent kinetic energy below the crest level for half-cosine weirs which made them capable of sediment resuspension and sediment removal. In was found that the introduced weir model are suitable structures for passage of small floating debris and sediments. A head-discharge formulation was also developed based on the geometry of half-cosine weir in free flow and a constant discharge coefficient was obtained by solving the Fresnel integrals. For practical purposes, semi-empirical formulations were also developed to estimate flow discharge in both free and submerged flow conditions.     

Robust Extreme Learning Machine for estimation of triangular,rectangular, and parabolic weirs

In this paper, for the first time, the DC of triangular, rectangular, and parabolic weirs is simulated by a new learning machine called “Robust Extreme Learning Machine” (RELM). The used laboratory data are divided into two categories: training (70% data) and testing (30% data). In the next step, the number of neurons inside the hidden layer is examined. For the structure of the proposed RELM algorithm, 10 hidden layer neurons are embedded. For the learning machine used in the present study, six different activation functions are evaluated that the Sigmoid activation function has better performance and is used for the RELM structure. Next, the calibration parameter of the RELM algorithm is discussed. The optimal regularization parameter is selected for the present study equal to 0.0001. Then, using parameters affecting the DC, four RELM models are developed. By performing various analyzes, the superior RELM model and the most effective input parameters are identified. Also, comparing the performance of the RELM model with ELM shows the superiority of the RELM algorithm. For the superior model, a relative derivative sensitivity analysis is performed to investigate the behaviour of the input parameters on the DC. Finally, an equation for estimating the DCs of triangular, rectangular, and parabolic weirs is proposed.     

Discharging capacity of rectangular side weirs in straight open channels

A side weir is a hydraulic control structure used in irrigation and drainage systems and combined sewer systems. A comprehensive laboratory study, including 843 tests for the discharge coefficient of a sharp-crested rectangular side weir in a straight channel, was conducted in a large physical model under subcritical flow conditions. The discharge coefficient is a function of the upstream Froude number, the ratios of weir length to channel width, weir length to flow depth, and weir height to flow depth. An equation was developed considering all dimensional parameters for discharge coefficient of the sharp-crested rectangular side weir. The average error of the proposed equation is 4.54%. The present study data were compared with ten different discharge coefficient equations developed by several researchers. The study also presents water surface profile and surface velocity streamlines.     

Discharge coefficient of rectangular sharp-crested side weirs,Part I: Traditional weir equation

A comprehensive study was performed to examine the flow characteristics over rectangular sharp-crested side weirs based on the traditional weir equation. To obtain a generally convenient discharge coefficient relationship, series of experiments were conducted according to manipulation of different prevailing parameters. The flow regime was consistently subcritical for upstream Froude numbers ranging from 0.08 to 0.91. Furthermore, experimental data sets of the former investigators were also applied. In order to identify the most important parameters affecting the discharge coefficient of rectangular sharp-crested side weirs, a sensitivity analysis was carried out based upon an artificial neural network modeling. Results of the sensitivity analysis indicated the Froude number to be the most influential parameter on discharge coefficient. Accordingly, a power equation is derived for estimating the discharge coefficient, which is applicable for both sub- and supercritical flow conditions simultaneously. Moreover, considering all the influential parameters, a nonlinear correlation was obtained with the highest precision to determine the discharge coefficient of sharp-crested rectangular side weirs.     

Deducing the stage-discharge relationship of rectangular broad and sharp-crested contraction devices

Flumes with a local contraction of the channel width are a common way for accurately measuring the flow discharge. In this paper, the outflow process of rectangular broad and sharp-crested contraction devices is modeled using the dimensional analysis and the incomplete self-similarity condition. The proposed theoretical stage-discharge relationship is tested using measurements available in the literature. The proposed power equation is characterized by a value of the exponent close to 1.5 and a coefficient depending on the contraction ratio. The proposed flow-measuring flumes are characterized by a good accuracy of the measured discharge (ranging from −2.71 to 3.28% for the broad-crested contraction device and in the range from −0.24 to 0.31% for the sharp-crested one). Even if both flume types have good accuracy of the measured discharge (less than the limit of ±5% suggested in the literature), the developed analysis demonstrated that the sharp-crested device is characterized by the lowest values of the errors in the estimate of discharge, and this result increases the interest of this device in practical applications. The advantage of the proposed approach is combining an accurate estimate of the flow discharge and working regardless of the discharge coefficient estimate.     

Comments on “Mean velocity and turbulent characteristics of flow over half-cycle cosine sharp-crested weirs” by Salehi S., Esmaili K., Azimi A.H.

In this paper the stage-discharge equation of a half-cycle cosine weir is theoretically deduced applying the Π-Theorem of dimensional analysis and the self-similarity theory. The coefficients of the new stage-discharge relationships are estimated using the results of the experimental runs by Salehi et al..     

Experimental investigation of discharge coefficient over novel kind of sharp-crested V-notch weir

Thin-plate weirs are the simplest and least expensive devices, which frequently have been employed in many water projects. In this research, a unique type of Sharp-Crested V-notch weir, entitled SCVW, has been introduced. The hydraulic characteristics of the present weir were investigated theoretically and experimentally under free aerated and non-aerated flow conditions in an open channel for large physical models. To investigate the variations of the discharge coefficient of SCVW versus weir height and vertex angle, a comprehensive laboratory experiments were conducted by measuring the discharge and the water head over the crest of weir. Possibility of different formulations for the head- discharge relationship of SCVW was examined and suitable analytical equation was proposed. The computed discharge using the suggested equation was within 0–10% of the observed ones. According to the experimental observations, the SCVW showed better performance in comparison with normal wire.     

Flow velocity pattern around trapezoidal piano key side weirs

In general, the side weirs are the structures installed along a channel or river. When the flow depth rises above the weir crest, the overflow passes through these weirs and enters the lateral canal. Nowadays, piano key weirs are considered as an important alternative to labyrinth weirs to modify the weirs encountering with difficulty to pass the maximum flow discharges. The present study investigates the hydrodynamic performance and the effect of the uniformity of velocity field on the resultant kinetic energy in the trapezoidal piano key side weirs with 90° installed laterally in the main channel wall. These weirs are classified as A-Type piano key weirs and two approaches (main: Mode 1 and adverse: Mode 2) were used to investigate the effect of the weirs' placement on their performance. The results showed that for velocity vectors in both modes, on average, the maximum flow discharge through the side weir occurred in the x and y directions (V_x and V_y) at Z*<0.2 and 0.2<Y*<0.7. The results also showed that at the control surface of X* = 1, the maximum values of α occur due to existing the inverse flow and increasing the deflection angle of the velocity vectors. The performance of the weir in Mode 2 was more appropriate Mode 1 due to the lack of weir base at the flow inlet, which is an obstacle for the deflection angle of the velocity vectors.     

Flow characteristics over asymmetric triangular labyrinth side weirs

Side weir is placed at the channel bank as a head regulator or a diversion device. Flow over a side weir has been the subject of many research studies considering its three dimensional and complicated characteristics. However, the labyrinth side weirs warrant further research due to their higher efficiency compared to linear side weirs. In this paper, subcritical flow characteristics and discharge coefficient for both symmetric and asymmetric triangular labyrinth side weirs were studied experimentally. The results show that asymmetric labyrinth side weirs have higher discharge coefficient compared to symmetric labyrinth side weirs; since a larger portion of the crest is orthogonal to the flow. Using the present laboratory data, general equations were proposed for the estimation of discharge coefficient of both symmetric and asymmetric triangular labyrinth side weirs. The results of this study can be useful to design side weirs with high hydraulic performance.     

A new approach to improve the discharging capacity of sharp-crested triangular plan form weirs

Conventional weirs are inherited with afflux and submergence of area upstream of the weir. Various weirs of modified plan form have been suggested in the past to enhance their discharging capacity and to restrict the afflux. Presented in this paper are results of the experimental study carried out to investigate the discharging capacity of a sharp-crested triangular plan form weir under free flow conditions in a rectangular channel. The efficiency of the triangular plan form weirs is better than the normal weir and also high for low vertex angle and decreases with the increase of ratio of head over the crest of the weir and crest height due to interference of the water jets downstream. The computed discharge using the proposed equation is within ±5% of the observed ones. Sensitivity of the weir, i.e., change of discharge due to unit change in head is also carried out which indicates that the weir is more sensitive at the low head and low vertex angle.     

Discharge equation of semi-circular side weirs: An experimental study

Side weirs are diverting structures and usually used for diverting and controlling the water flow into the side open channel. The present study deals with an experimental study regarding the hydraulic performance of side weirs with semi-circular vertical sections along the main channel. As flow depth of the main channel increases the top flow width of the semi-circular side weir (SCSW) increases which is an advantage when high discharge enters the main channel and should be immediately diverted for safety reasons. In this study, the flow discharge of semi-circular sharp-crested side weirs and their affecting parameters are investigated. To investigate the hydraulic behavior and geometric characteristics of the SCSWs, a comprehensive laboratory study including 155 tests (for three weir diameters 0.25, 0.30 and 0.40 m) was conducted in a physical model under subcritical flow conditions. Flow discharge of the SCSW was investigated in relation to height, diameter and flow head of side weir, also approach Froude number (Froude number at upstream end of the side weir) and main channel width. Three different discharge models were developed based on; purely dimensional analysis technique, classical weir equation with linear water surface and classical weir equation with horizontal water surface profile (conventional weir theory along with dimensional analysis technique). The presented mathematical discharge models enable estimation of discharge along the SCSW with acceptable accuracy (best model has an average error of 1.87% with a maximum error of 6.31%) compared with the measured data under subcritical flow conditions. Additionally, a relationship was proposed for computing the limiting flow depth at the downstream end of the SCSW. Experimental results confirm that the proposed relationship well explains the behavior of flow over the SCSW regarding the downstream flow conditions.     

Discharge characteristics and structure of flow in labyrinth weirs with a downstream pool

A new design of a labyrinth weir is introduced in this study by adding a square pool to the vertex of a one-cycle triangular labyrinth weir with a sidewall angle of 45°. The addition of the square pool increased weir length without causing an excessive nappe interaction, and as a result, reduced the head water over the weir with the same discharge. Laboratory experiments were carried out to investigate the hydraulic performance of the new design with a potential application in pool-weir fishways. Mean and turbulence characteristics of flow for different weir geometries and in both free and submerged flow regimes were measured to be used for prediction of fish behaviour in the upstream of the proposed weir models. Discharge coefficients based on channel width and weir length were calculated. It was found that the new design can significantly increase the capacity of triangular labyrinth weirs and provide financial advantages in construction over triangular labyrinth weirs without pools in low discharges. In submerged flow conditions, the proposed model performed better than sharp-crested linear weirs in low discharges. Contour plots of the three-dimensional velocity components showed a region of strong mean flow around the neck of the new weir model. Turbulent characteristics such as turbulent kinetic energy, power spectra, exuberance ratio, and joint probability distribution functions of velocity fluctuations were extracted from instantaneous three dimensional velocities for different weir depths and flow regimes. Two vertical planes were identified based on the highest turbulent mixing in free and submerged flow regimes. The depths contributing the most to turbulent mixing were identified; active depths decreased as the flow regime changed from free to submerge flow regime.     

A generalized stage-discharge relationship for sharp-crested power-law weirs by dimensional analysis and self-similarity

Weirs are characterized by a stage-discharge relationship which mainly depends on the shape and dimensions of the hydraulic structure. A weir with symmetrical power-law sides is a versatile weir that can produce some known weir types (rectangular, triangular, parabolic) as special cases. In this paper, the outflow process of sharp-crested power-law weirs is investigated using the dimensional analysis and the incomplete self-similarity theory. A new generalized theoretical stage-discharge relationship is proposed, and its applicability is tested using measurements available in the literature for different weir types.     

Evaluation of flow characteristics in labyrinth weirs using response surface methodology

Labyrinth weirs can increase the flow discharge capacity for a specific head and width by increasing the weir length. This paper studies the flow behavior of labyrinth weirs using Flow-3D® software. The main goal is to evaluate the capability of the response surface methodology (RSM), especially central composite design (CCD), to describe the performance of labyrinth weirs. Traditional and RSM-CCD methodologies are applied using experimental data of previous researchers and numerical data of the present study, respectively. Results show that RSM-CCD can produce an acceptably accurate model for the discharge coefficient of labyrinth weirs with only a small fraction of the data required for the traditional model. In addition, the discharge coefficient of a labyrinth weir decreases by increasing head and decreasing sidewall angle due to the inflow contraction and nappe interference in inlet and outlet cycles, respectively. The discharge efficiency can be up to 4.5 times higher for a labyrinth weir compared to a linear weir. Finally, a new method is introduced for designing labyrinth weirs.     

Unified discharge coefficient formula for free and submerged triangular labyrinth weirs

The flow through a triangular plan labyrinth weir is studied for both free and submerged flow conditions experimentally and theoretically. The free flow condition is studied using a new experimental data set collected in this study. For the submerged flow condition, the threshold between free and submerged flow regimes is studied experimentally. Then Buckingham analysis is employed to determine the submerged head-discharge formula of the triangular plan labyrinth weir. Finally, a step by step calibration method is proposed to find the unified discharge coefficient. The proposed discharge coefficient can be used for both free and submerged flow conditions continuously and within the transition zone.     

Study of the hydraulic properties of the cylindrical crested weirs

Since the cylindrical and circular crested weirs are economical and easily made compared to some other weirs, they can be used to measure the flow velocity, the water discharge and flow surface control structures in the canals and reservoirs. In this paper, the hydraulic properties of the cylindrical and circular crested weirs, such as the discharge coefficient, the depth on the crest of the weir and the energy loss in 18 laboratory models categorized in 5 categories have been investigated. The results of the experiments indicated that in both the cylindrical and circular crested weirs by increasing the total partial head, the discharge coefficient increases and any changes in the upstream wall slope has no effect on the discharge coefficient. Also, the partial energy loss more occurs in the cylindrical weirs than the half shaped cylindrical weirs and by increasing the downstream inclination angle, the partial energy loss increases subsequently. The partial flow depth on the crest of both the cylindrical and half shaped cylindrical weirs is equal to 0.7 and by increasing of the upstream inclination angle, the partial flow depth on the crest in the values greater than 0.6, increases slightly and the downstream inclination angle changes of the partial flow depth has no effect on the crest.     

Overflow characteristics of streamlined weirs based on model experimentation

Weirs are a type of hydraulic structures that are used in water supply systems and irrigation networks for different purposes. These structures are categorized as different types and must be selected in respect to the hydraulic conditions to satisfy all the objectives of a project. The present study subjects to investigate the hydraulic characteristics of flow over the streamlined weirs, based on model experimentation. The streamlined weirs are a special type of weirs, designed based on the airfoil theory. These weirs have some merits compared to the other types of weirs. They have high discharge coefficient, more stability and less fluctuations of water free surface. According to the objectives of the present study, an experimental investigation was performed, using different experimental models of the streamlined weirs, designed according to the Joukowsky transform function. Experimental results inferred that for a constant flow discharge, the upstream flow heads over the streamlined weirs with different relative eccentricities are almost constant, showing an approximately constant discharge coefficient by the variation of the relative eccentricities. Furthermore, adding a downstream sloped face does not change the flow discharge coefficient. However, increasing the weir height, results in the increase of the flow head and the decrease of the flow discharge coefficient, especially, for the higher heads and the lower relative eccentricities.