In this paper, the discharge coefficient prediction model for this structure in a subcritical flow regime is first established by extreme learning machine (ELM) and Bayesian network, and the model's performance is analyzed and verified in detail. In addition, the global sensitivity analysis method is introduced to the optimal prediction model to analyze the sensitivity for the dimensionless parameters affecting the discharge coefficient. The results show that the Bayesian extreme learning machine (BELM) can effectively predict the discharge coefficients of the symmetric stepped labyrinth side weir. The range of 95% confidence interval [−0.055,0.040] is also significantly smaller than that of the ELM ([−0.089,0.076]) and the Kernel extreme learning machine (KELM) ([−0.091,0.081]) at the testing stage. The dimensionless parameter ratio of upstream water depth of stepped labyrinth side weir p/y1 has the greatest effect on the discharge coefficient Cd, accounting for 55.57 and 54.17% under single action and other parameter interactions, respectively. Dimensionless step number bs/L has little effect on Cd, which can be ignored. Meanwhile, when the number of steps is less (N = 4) and the internal head angle is smaller (θ = 45°), a larger discharge coefficient value can be obtained.

  • The discharge coefficient model of a symmetrical stepped labyrinth side weir is developed.

  • The quantitative model of the discharge coefficient is established.

  • The discharge characteristics of a symmetrical stepped labyrinth side weir are analyzed comprehensively.

  • The effects of different dimensionless parameters on the discharge coefficient are compared.

As a new type of weir, the labyrinth weir can greatly increase the discharge for hydraulic structures with a limited overflow front (Karimi et al. 2019). Its excellent discharge capacity and low project cost have made it popular for reservoir reinforcement, landscape engineering, flood control engineering, and spillway reconstruction in recent years. Also, considering factors such as power supply guarantee and management cost in practical engineering, compared with the broad-crested weir controlled by gate control, the labyrinth weir can greatly reduce the operation and management cost. However, in practical engineering, the design parameters of labyrinth weir are generally estimated based on experience, and then verified by model experiment. The whole design process lasts for a long time. If the design parameters are not properly formulated, it will lead to repeated optimization and adjustment of engineering design and modeling experiments, resulting in a waste of time and cost (Yan 2023).

At present, there are mainly rectangular, triangular, circular, and other shapes for the crest shape of labyrinth weirs (Mahmoud et al. 2021). Alfatlawi et al. (2023) presented a symmetrical stepped labyrinth side weir. The discharge capacity of the weir can be increased by 15–35% using the larger length of the weir, small head angles, and steps for the triangular labyrinth side weir. The discharge coefficient (Cd) is an important hydraulic parameter of the structure. With the change in the number of steps, the water flows through the crest of the weir, there are both positive water flow and lateral water flow, which makes the overflow nappe collide with each other to produce secondary flow and vortex. Its discharge characteristics are more complex than the traditional weir type. Its discharge characteristics are more complex than the traditional labyrinth side weir. Thus, it is important to systematically study the discharge characteristics of the structure and reveal the interaction mechanism of hydraulic variables for the application of the structure in practical engineering.

Many scholars have studied the Cd for labyrinth side weirs through modeling experiments and numerical simulations (Ben Said & Ouamane 2022; Alfatlawi & Alkafaji 2023; Saffar et al. 2023; Zare et al. 2023). So far, some researchers have adopted soft computing methods to predict Cd for efficient calculation. Bijanvand et al. (2023) assessed the discharge in compound open using the soft computing method, and the most important parameters affecting the discharge were the relative roughness, the ratio in terms of flow dimensions, the relative depth, the convergence angle, and the relative radius. Simsek et al. (2023) developed a Cd prediction model for trapezoidal broad-crested weirs, and the Artificial Neural Network model outperformed M5Tree and the Support Vector Machines model. Roushangar et al. (2023) established a Cd prediction model using different flow conditions with radial gates. The values of R = 0.940, RMSE = 0.022, and R = 0.927, RMSE = 0.018 were obtained for KELM-GWO and Support Vector Machines models for flow under submerged and free conditions, respectively. Azma et al. (2023) developed the Gabion weir flow model, and while estimating the ratio of flow discharge, the CatBoost model achieves the best performance; when calculating the Cd, the performance of the XGBoost and CatBoost models is very close to each other. Emami et al. (2022) evaluated the Labyrinth Weir's Cd using the Walnut algorithm and support vector regression (Walnut-SVR), and the Walnut-SVR model has higher accuracy than other counterparts. Seyedian et al. (2023) state that Gaussian process regression (GPR) provides improved performance and robustness (RMSE = 0.009 and R2 = 0.986) for a triangular labyrinth weir. Zaji et al. (2020) developed the neural network and genetic algorithm to evaluate the labyrinth side weir's Cd, and the Genetic Algorithm Radial Basis neural network model could still predict the Cd for this structure with the limited dataset. Zounemat-Kermani et al. (2019) predicted the Cd of the triangular arc labyrinth weir, and the Multi-Layer Perceptron Neural Network has the highest precision and generalization capability (R2 = 0.999, RMSE = 0.00385, and bias coefficient bias < 0.0001). Akbari et al. (2019) developed a Cd calculation model for the Piano Key weir from 156 experimental datasets, and they showed that the GPR model significantly outperforms other intelligent models. Karami et al. (2018) demonstrated that the ELM model outperforms ANN and genetic programming (GP) in predicting the Cd for a triangular labyrinth weir.

However, according to the current literature research, no scholars have proposed an intelligent calculation model for the Cd of the symmetrical stepped labyrinth side weir. Also, the Cd for the structure is affected by the slope of the steps, the height of the steps, the width of the steps, and the number of steps. In addition, many scholars have explored the impact of hydraulic parameters on the Cd through the sensitivity method. Samadi et al. (2022) researched the influence of crest height, cycle number, and plane weir configuration on the Cd for a labyrinth weir through the model experiments, and results show that under the same geometric parameters, the rectangular has a higher discharge for each head as compared to trapezoidal and triangular. Dogan & Kaya (2023) experimentally explored the influence of effective crest length on the Cd for the labyrinth side weir, and by reducing the upstream effective crest length by 1/3, 2/3, and 3/3, the discharge reduced about 10, 23, and 48%, respectively. It can be seen that the influence on Cd of hydraulic parameters has also been the focus of attention for many researchers. For the stepped labyrinth side weir, hydraulic parameters are obviously more than other labyrinth weirs. Furthermore, the stability of the model can be improved by reducing parameter uncertainties and analyzing the influence on Cd from parameter interactions. However, for the impact of hydraulic parameters on Cd, most scholars use local sensitivity studies to obtain the influence of individual parameters on Cd.

This paper is designed to address the discharge characteristics for the stepped labyrinth side weir in a subcritical flow regime. Firstly, the Cd calculation model is developed based on the experimental dataset and artificial intelligence algorithm. The dimensional analysis of the hydraulic parameters affecting the Cd is carried out to define the model input parameters and output parameters. The optimal intelligent model is synthesized by different evaluation indexes, and the model uncertainty is further analyzed. On this basis, the global sensitivity method is introduced to quantify the sensitivity coefficients between the dimensionless parameters and the discharge coefficients, and the insensitive parameters are removed. Finally, the variation of important parameters about the discharge coefficients is analyzed.

Experimental model

The experimental data were from Alfatlawi et al. (2023). The experimental model is composed of a rectangular flume of 10 m in length, 0.3 m in width, and 0.45 m in height, and the structure consists of a metal bed and glass side walls. The main and evacuation channels have a calibrated rectangular gauging weir positioned at each channel's end, and a point gauge with ±0.1 mm accuracy is fixed at the upstream end of the weir, to provide for precise discharge measurement, 0.40 m away from the gauging weir. The stepped labyrinth side weir is located 5.50 m above the main channel. This experiment was performed on different stepped labyrinth side weirs with different lengths, heights, number of steps, and internal head angles for a total of 275 experiments. The experiment was in a subcritical flow regime, and the data are characterized as shown in Figure 1.
Figure 1

Characteristics of experimental data.

Figure 1

Characteristics of experimental data.

Close modal

Dimensionless parameters

For a symmetric stepped labyrinth side weir, the discharge is calculated as:
formula
(1)
where Lf is the weir crest length, m; Cd is the discharge coefficient; y1 is the upstream water depth of the channel center, m; g is the acceleration of gravity, m/s2.
As shown in Figure 2, the parameters that affect the value of Cd are as follows:
formula
(2)
Figure 2

Symmetrical stepped labyrinth side weir: (a) layout plan, (b) three-dimensional structure diagram, and (c) flow regime diagram (Alfatlawi et al. 2023).

Figure 2

Symmetrical stepped labyrinth side weir: (a) layout plan, (b) three-dimensional structure diagram, and (c) flow regime diagram (Alfatlawi et al. 2023).

Close modal
According to Buckingham π theorem, the dimensional analysis shows that:
formula
(3)
where L is the side weir width, m; B is the channel width, m; p is the side weir height, m; bs is the step width, m; as is the step height, m; bs/as is the dimensionless step slope (θ); bs/L is the dimensionless steps number (N); Fr is the upstream Froude number; We is the Weber number; Re is the Reynolds number. The experiment was carried out under subcritical flow and steady flow, and the water head above the weir was higher than 30 mm. Thus, the influence of the Re and We on the Cd can be neglected (Mohammed & Golijanek-Jedrzejczyk 2020). Refer to Subramanya & Awasthy (1972) and Borghei et al. (1999) for the dimensional analysis of discharge coefficients. Equation (3) can be expressed as the following equation.
formula
(4)

Models

Considering that the dataset used is the experimental dataset of the physical model. The dimensional analysis shows that the model has six input parameters, while the extreme learning machine (ELM) has only one hidden layer, so its learning speed is faster. Additionally, in the process of using the ELM, the output weight of a Moore–Penrose generalized inverse matrix is a least squares problem, it is easy to overfit. Therefore, the kernel function is introduced to avoid randomly generating weights and thresholds. Secondly, ELM accuracy is very sensitive to the number of hidden neurons, which leads to a very time-consuming and labor-intensive process by manually tuning the parameters in the training process. So far, the Bayesian network optimization ELM has shown good prospects in many engineering fields (Olyaie et al. 2019; Quilty et al. 2023), and the ability of this method to solve hydraulic engineering problems has not been explored. Therefore, in this study, a Bayesian network is chosen to find the optimal hyperparameters for ELM automatically.

Extreme learning machine

ELM as an improved neural network model using the least squares method for network development. ELM has a very fast learning speed because it is a simple untuned algorithm, which avoids the slow iterative learning process and easily falls into the local minimum (Luo & Zhang 2014). Compared with traditional intelligent algorithms, ELM has better scalability, generalization performance, lower computational complexity, and less manual intervention (Huang et al. 2011).

Kernel extreme learning machine (KELM)

The kernel function mainly relies on Mercer's theorem, and kernel-based learning methods are widely used for their powerful generalization performance. The KELM introduces the concept of kernel matrix and adopts the kernel function matrix instead of the random matrix HHT for the ELM model, which transforms the linearly indivisible data in the low-dimensional space into linearly divisible data, which not only preserves the speed of the ELM operation but also abandons the principle of randomly generating weights and biases of the ELM (Jalil-Masir et al. 2022). After determining the kernel function, KELM only needs to adjust the penalty factor, and the training results are stable and unchanged. The same accuracy can be obtained when the same samples are trained (Parida et al. 2021).

Bayesian extreme learning machine

The Bayesian extreme learning machine (BELM) uses Bayesian networks for optimizing the output layer weights for ELM models, which helps to overcome the model falling into overfitting. The joint probability distribution is represented by encoding the independent relationships of random variables using a graph. Generally, Bayesian modeling is divided into two steps (Soria-Olivas et al. 2011; Olyaie et al. 2019)

  • (1)
    The model parameter inference from the posterior distribution. In order to scale it, multiply the likelihood function by the prior distribution:
    formula
    (5)
    where P represents the probability distribution, w represents the parameter set, and D represents the dataset.
  • (2)
    Calculate a new distribution for the input xnew and the output ynew. It is defined as the integration over the posterior distribution for parameter w.
    formula
    (6)

The linear modeling follows

where h = [h (x, o1, r2), …, (x, oM, rM) is the hidden layer's output vector for the input x; β is an output weight vector; ε obeys a Gaussian distribution with zero mean and σ2 variance; and then, the output conditional distribution is as follows:
formula
(7)
For most applications, the parameters are distributed as shown in the following equation:
formula
(8)
where I represent the unit matrix; α represents the hyperparameter. Assuming that the distribution of the prior and the likelihood function follow a Gaussian distribution and that the posterior distribution is also Gaussian, and the mean m and variance S are defined as follows:
formula
(9)
where y = [y1, y2, …, yn], x = [x1, x2, …, xN] are the model input vector and output vector matrices, respectively.

Global sensitivity analysis method

The Sobol method is used as a global sensitivity analysis method, which can quantitatively give the parameter sensitivity of the model (Sobol 1990). Sobol's method is centered on decomposing the overall variance in the objective function into the variance for individual parameters and the variance between combinations of parameters, and the sensitivity coefficient is obtained. More importantly, this method can simultaneously reflect the direct and interactive effects of parameters. Compared with other methods, the Sobol method has higher stability. The specific calculation principle can be found in our previous research (Shanshan et al. 2023).

Uncertainty analysis

Model uncertainty analysis can effectively evaluate the prediction error of the model and validate the data relationship for the model. Through the mean error () and standard error (Se) of the model, a 95% confidence interval was created around the error value (Pradeep et al. 2022). Positive and negative mean error values represent overestimation and underestimation of the model, respectively.
formula
(10)
formula
(11)
formula
(12)
where ei represents the prediction error of individual data; N is the total number of samples in the dataset.

Model evaluation indices

Seven statistical methods were used in this study to evaluate model quality. The indices are root mean square error (RMSE), variance account factor (VAF), Nash–Sutcliffe efficiency (NSE), Willmott's agreement index (WIA), mean absolute percentage error (MAPE), coefficient of determination (R2), bias coefficient (Bias), respectively. The specific calculation process for each indicator is described in the following equations (Pradeep & Samui 2022; Shen et al. 2022).
formula
(13)
formula
(14)
formula
(15)
formula
(16)
formula
(17)
formula
(18)
formula
(19)
where Oi represents the experimental value of Cd, Pi represents the predicted value of Cd, Oa represents the mean of the experimental value of Cd, and Pa represents the mean of the predicted value of Cd.
According to the dimensional analysis, the dimensionless parameters affecting the symmetrical stepped labyrinth side weir are Fr, L/B, L/y1, p/y1, bs/as, and bs/L, respectively. The dimensionless parameters above are used as parameters for the inputs of ELM, KELM, and BELM, respectively, and Cd is used as the output parameter for the model. Randomly selected 70% (192) and 30% (83) of the model experimental datasets were used as the training and testing sets for all models, respectively. The performance of the ELM is mainly affected by neuron quantity, and when neurons are 20 in number, the model performs best. For the KELM model, the best performance is achieved when the kernel parameter C = 1, the regular coefficient λ = 10, and the weights and thresholds do not change randomly. For the BELM model, as shown in Figure 3, firstly, some hyperparameters are randomly tried to obtain their outputs, and then the next observation point is obtained in the search space, and finally, the minimum feasible point of the model is obtained. At this time, the hyperparameters k = 1.23, λ = 830.07.
Figure 3

Bayesian network performance diagram.

Figure 3

Bayesian network performance diagram.

Close modal

Model performance comparison

Tables 1 and 2 show the evaluation indicators for all the models in the training and testing phases. The smaller the values of MAPE, RMSE, and bias, the higher the values of R2, WIA, NSE, and VAF, indicating that the model has less error and better computing efficiency and accuracy. The ELM model has the worst performance in all evaluation indicators. The KELM has slightly better performance indicators than the ELM. Among them, in the training phase, all evaluation indicators are better than the ELM. In the testing phase, the values of WIA and VAF are very close to those of the ELM model, which indicates that the prediction abilities of the ELM and KELM models are close to each other. For the BELM model, compared with KELM, the MAPE is reduced by about 8.95 and 50.77% in the training and testing phases, respectively, which indicates that the hyperparameters obtained through Bayesian network optimization are suitable for the model, and improved modeling accuracy. The value of VAF is significantly better than that of KELM and ELM, indicating that the prediction ability and generalization ability of BELM are higher than those of KELM and ELM.

Table 1

Performance indicators for all models in the training phase

ModelMAPERMSER2WIANSEVAFBias
ELM 3.164 0.036 0.957 0.989 0.956 95.733 0.0013 
KELM 2.815 0.035 0.963 0.990 0.961 96.140 0.0012 
BELM 2.563 0.024 0.990 0.998 0.990 99.027 0.0003 
ModelMAPERMSER2WIANSEVAFBias
ELM 3.164 0.036 0.957 0.989 0.956 95.733 0.0013 
KELM 2.815 0.035 0.963 0.990 0.961 96.140 0.0012 
BELM 2.563 0.024 0.990 0.998 0.990 99.027 0.0003 
Table 2

Performance indicators for all models in the testing phase

ModelMAPERMSER2WIANSEVAFBias
ELM 3.701 0.043 0.942 0.985 0.941 96.258 0.0019 
KELM 3.455 0.042 0.943 0.984 0.941 94.272 0.0018 
BELM 1.701 0.023 0.981 0.995 0.979 98.035 0.0006 
ModelMAPERMSER2WIANSEVAFBias
ELM 3.701 0.043 0.942 0.985 0.941 96.258 0.0019 
KELM 3.455 0.042 0.943 0.984 0.941 94.272 0.0018 
BELM 1.701 0.023 0.981 0.995 0.979 98.035 0.0006 

Figure 4 shows the scatter fitting plots for ELM, KELM, and BELM in the training and testing phases. As can be seen from Figure 4, the experimental and predicted values for BELM are closer to the 1:1 line. The R2 values of BELM are greater than those of ELM and KELM in the training and testing phases, indicating that the predicted values of the discharge coefficient obtained by BELM have less error with the experimental values, and the fitting ability of the function is significantly better than that of KELM and ELM in the running process.
Figure 4

Scatter plot of all models (a) Training phase (b) Testing phase.

Figure 4

Scatter plot of all models (a) Training phase (b) Testing phase.

Close modal
To further visualize the comprehensive performance for different models. Taylor diagram can statistically quantify the similarity between different models. In the Taylor diagram, the three different evaluation indexes of standard deviation, RMSE, and correlation coefficient are represented by one point in the two-dimensional space, and a model that is closest to the target point is the best. It can be seen from Figure 5 that the black line represents the standard deviation, the green line indicates the RMSE and the blue line indicates the correlation coefficient. In the training and testing phase, the points representing the BELM are closer to the target points. Therefore, from the perspective of visualization, BELM is the best model.
Figure 5

Taylor diagram for ELM, KELM, and BELM: (a) training phase and (b) testing phase.

Figure 5

Taylor diagram for ELM, KELM, and BELM: (a) training phase and (b) testing phase.

Close modal

Model uncertainty analysis

Tables 3 and 4 show uncertainty analysis results for all models. The mean error values for all the models are negative, indicating that all the models are underestimated in the calculation process of Cd of the symmetric stepped labyrinth side weir, but in the training and testing phases, the average error values are very smaller, and it can be considered that the models are more stable. In addition, the BELM has the smallest average error and the narrowest confidence bandwidth at the same confidence level. Meanwhile, it can be seen that the BELM has the smallest range of uncertainty widths in different phases, and the range of the 95% confidence intervals is also significantly smaller than that of the ELM and the KELM.

Table 3

Uncertainty analysis results for all models in the training phase

ModelMean errorStandard errorBandwidth95% confidence interval
ELM −0.0020 0.036 0.070 −0.070 0.070 
KELM −0.0012 0.035 0.068 −0.069 0.067 
BELM −0.0002 0.017 0.034 −0.034 0.034 
ModelMean errorStandard errorBandwidth95% confidence interval
ELM −0.0020 0.036 0.070 −0.070 0.070 
KELM −0.0012 0.035 0.068 −0.069 0.067 
BELM −0.0002 0.017 0.034 −0.034 0.034 
Table 4

Uncertainty analysis results for all models in the testing phase

ModelMean errorStandard errorBandwidth95% confidence interval
ELM −0.0051 0.044 0.086 −0.091 0.081 
KELM −0.0063 0.042 0.082 −0.089 0.076 
BELM −0.0045 0.024 0.048 −0.055 0.040 
ModelMean errorStandard errorBandwidth95% confidence interval
ELM −0.0051 0.044 0.086 −0.091 0.081 
KELM −0.0063 0.042 0.082 −0.089 0.076 
BELM −0.0045 0.024 0.048 −0.055 0.040 

Sensitivity analysis for the parameters

In summary, it can be seen that the BELM has the best performance, and it can be used as the Cd prediction model for a symmetric stepped labyrinth side weir. However, six dimensionless parameters affect the discharge coefficient. Sensitivity analysis for the input parameters of the BELM model is performed by the Sobol method. For comparing the influence on sensitivity analysis results from different sampling methods, firstly, the sampling method is verified. In this study, Halton sequence (Ermakov 2017), and Sobol sequence (Wang et al. 2022) are used. Halton sequence is a quasi-random low difference sequence, which uses different bases on each dimension, and the bases on each dimension must be coprime, resulting in Halton sequence. Sobol sequence is a numerical sequence method for generating efficient uniformly distributed values, which uses a set of predefined dummy points, which are referred to as the generating elements. Each value in the sequence is computed by bitwise operations in combination with the original polynomial. Figure 6 shows the sampling results of the two sampling methods for sample size N = 1,000 with two-dimensional sampling and value space [0,1]. It can be seen that the Sobol sequence obtains a more uniform sampling distribution.
Figure 6

Data point distribution from different sampling methods: (a) Halton sequence and (b) Sobol sequence.

Figure 6

Data point distribution from different sampling methods: (a) Halton sequence and (b) Sobol sequence.

Close modal
Figure 7 shows the sensitivity coefficient of dimensionless parameters calculated by the Sobol method for the BELM model. It is observed that the coefficient of first-order sensitivity S1 of Fr, L/B, L/y1, p/y1, bs/as, and bs/L are 0.35503, 0.0276, 0.00242, 0.54203, 0.05110, and 0.00100, respectively. It means that p/y1 > Fr >bs/as > L/B > L/y1 > bs/L when only considering the influence of individual parameters on the model output Cd. The global sensitivity coefficients of Fr, L/B, L/y1, p/y1, bs/as, and bs/L are 0.36082, 0.03467, 0.01622, 0.55653, 0.05822, and 0.00100, respectively. The results show that when considering the influence of individual parameters on the model output parameter Cd after interacting with other parameters, p/y1> Fr > bs/as > L/B > L/y1 > bs/L. Tang et al. (2007) used 0.01 as the sensitivity threshold of the Sobol method and considered that when the sensitivity of the parameter is greater than 0.01, it is a sensitive parameter, and when it is less than 0.01, it is an insensitive parameter. Therefore, p/y1 had the greatest impact on Cd, accounting for 55.57 and 54.17% under single and interaction, respectively. The effect of bs/L on Cd is very small and can be ignored.
Figure 7

Sensitivity coefficients for input parameters.

Figure 7

Sensitivity coefficients for input parameters.

Close modal
Figure 8 depicts the relationship between Cd and p/y1 for different geometric parameters. It can be seen that Cd increases with the increase of p/y1. In particular, when the number of steps is low, the discharge coefficient is larger. Meanwhile, the smaller θ is, the larger the corresponding Cd value is. As the θ increases, the trend of Cd increase also decreases for the same p/y1, when N = 10, the corresponding Cd value is greater than or close to N = 8. This may be because as the N increases, the vortex above the step and the overflow nappe collide with each other, increasing the secondary flow along the lateral flow, and making the discharge coefficient larger.
Figure 8

The relationship between Cd and p/y1.

Figure 8

The relationship between Cd and p/y1.

Close modal

The flow characterization of this structure is complicated due to the interaction of multiple factors affecting the symmetric stepped labyrinth side weir. In this study, the discharge coefficient estimation model was developed by ELM and Bayesian Network to comprehensively evaluate the stability and generalization ability from multiple perspectives. Also, uncertainty and sensitivity analyses of the model are analyzed, and the relationship between the variations of the geometric parameters. The following conclusions are obtained:

  • (1)

    The model performance of BELM in evaluating the Cd for the symmetric stepped labyrinth side weir is better than that of ELM and KELM, which can be used as the discharge coefficient prediction model for this structure. In the testing phase, Bias = 0.0006, R2 = 0.981, with the smallest average error and 95% confidence interval range [−0.055, 0.040], indicating that the model has robust performance, data consistency, and high confidence.

  • (2)

    The global sensitivity indicates that the first-order sensitivity coefficient S1 for Fr, L/B, L/y1, p/y1, bs/as, and bs/L are 0.35503, 0.0276, 0.00242, 0.54203, 0.05110, and 0.00100, respectively. Global sensitivity coefficient ST is 0.36082, 0.03467, 0.01622, 0.55653, 0.05822, and 0.00100, respectively. Therefore, the influence of each dimensionless parameter on the Cd is p/y1 > Fr > bs/as > L/B > L/y1 > bs/L.

  • (3)

    Cd increases with increasing p/y1. In particular, the discharge coefficient is larger when the number of steps N = 4. Also, the smaller internal head angle θ = 45° corresponds to a larger value of Cd, and the tendency of increasing Cd decreases as the internal angle increases.

In summary, the dataset for this study is small, and only flow characteristics under subcritical flow are analyzed. Secondly, this paper is carried out under the condition of clear water and does not involve the change of flow characteristics of the structure when the flow contains sediment. In future research, it can be further studied.

This study was supported by the National Natural Science Foundation of China (Grant Nos. 52079122, 52379080).

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.

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