The effects of splitters on the downstream scour hole of overflow spillways: application of support vector regression Open Access

The following symbols are used in this paper:

Support vector regression

SVM

Support vector machine

MLP

Multi-layer perceptron

Maximum depth of scour hole (m)

Maximum length of scour hole (m)

Maximum width of scour hole (m)

Position of occurrence of the maximum depth in scouring hole (m)

Total value of h,

Froude number for bed material

Froude number of current

Flow specific-discharge (m³/m.s)

Falling height (m) (water surface elevation upstream of the spillway corresponding to spillway crest)

Height of water on spillway crest (m)

Tail-water depth (m)

V_i

Velocity of hitting of jet with tail-water (m/s)

C, ε

Parameters of SVR

δ

Kernel bandwidth is a function of the radial basis

d

Polynomial degree of the kernel

Length of splitter (m)

Width of splitter (m)

Difference between the level of the top of the overflow crest and the upper side of the splitter (m)

Distance among splitters (m)

Coefficient of determination

Root mean squared error

Mean absolute percentage error

Mean particle size (m)

g

Gravitational acceleration

d_m

Materials characteristic diameter (m)

ρ_s

Density of bed materials (kg/m³)

ρ

Fluid density (kg/m³)

The downstream scour phenomenon in hydraulic structures is one of the major problems in hydraulic engineering. The quantification of scouring is a complicated engineering issue, as both the flow properties and the soil physical characteristics affect this phenomenon. Some human-made hydraulic structures in rivers (e.g., bridges, spillways, breakwater, etc.) cause scouring, which may, in turn, cause significant structural damage. The quantification of this phenomenon is therefore of high importance for the preservation of the rivers and hydraulic systems (Sharafati et al. 2021). This issue becomes especially important in high arc dams with free overflow spillways. The downstream scour hole of these dams may threaten the safety of the dam body and foundation (Bollaert & Schleiss 2007). Prediction of scour hole characteristics has always been a topic in the hydraulic engineering research works (Bollaert & Schleiss 2007).

Schoklitsch (1932) was the first one who proposed an empirical formula for determining the maximum scour depth downstream of a vertical drop (Dey & Raikar 2007). To date, numerous experiments have been carried out to investigate scouring downstream of the free fall jet (Bollaert & Schleiss 2007). Some of the common formulas applied for determining the downstream scour rate in spillway jets are listed in Table 1.

In these formulas, denotes the maximum scour hole depth (m), q represents specific discharge (m³/m.s), H is depth of tail-water (m), H is water surface elevation upstream of the spillway crest (m), g is gravitational acceleration, dm is materials characteristic diameter (m), is density of bed materials (kg/m³); and is fluid density (kg/m³), (Figure 2).

These researchers have shown that the formula in which flow discharge (q), spillway height (H), tail-water depth (h) and particle size (d_m) are used to predict the potential depth of scouring is reasonably reliable although other parameters such as density do not substantially improve the predictions (Mason & Arumugam 1985). Canepa & Hager (2003) studied the impact of jet aeration during the scouring phase and concluded that improved airflow within the jet could significantly reduce scouring rates (Canepa & Hager 2003). Rajaratnam & Mazurek (2003) studied the scouring of non-cohesive materials with low tail-water values caused by the circular jet and found that the scouring depth in this state is the result of densimetric Froude number of bed particles (⁠⁠) (Rajaratnam & Mazurek 2003).

George & Annandale (2008) studied the effect of crest modification on decreasing the scouring potential downstream of overtopping dams. Their analysis showed that the crest improvement could decrease the average coefficient of dynamic pressure, but that the fluctuating coefficient of dynamic pressure remains almost unchanged. To increase the strength of dam base against scouring, use is made of various forms of crest modification to reduce the overall pressure change (George & Annandale 2008, 2017).

Pagliara et al. (2010) analyzed 3D plunge pool scour hydraulics in the presence of basic protective steps. Specific formulae for the calculation of scour hole dimensions are based on the most efficient scour parameters like the Densimetric Froude Particulate Number (Pagliara et al. 2010).

Other researchers such as Bollaert & Schleiss (2007), Dey & Raikar (2007), Pagliara et al. (2009), Ghodsian et al. (2012), Asadi et al. (2015), Karbasi & Azmathulla (2016), Pagliara & Palermo (2017), Samma et al. (2019), Daneshfaraz et al. (2019), Eghlidi et al. (2019), Latifi (2020) also demonstrated this issue (Bollaert & Schleiss 2007; Dey & Raikar 2007; Pagliara et al. 2010; Pagliara et al. 2012; Asadi et al. 2015; Eghlidi et al. 2019).

One of the techniques used for scour management is reduction of erosive force in the jet. One of the strategies used in this respect is to use splitters on the overflow spillway (Figure 1) (Mason & Arumugam 1985).

Presence of these splitters causes the moving flow through the overflow spillway at the middle portion to encounter a serrated barrier.

Roberts (1943) first suggested the idea of utilizing such splitters on the free overflow spillway in the construction of Loskop and Vaalbank dams in South Africa, and since then this idea has been utilized by many designers around the world. Also in Iran the use of this device has been highly recommended by many designers. This method was first used in the construction of the Tajan Dam. It has been used in several important projects since then, particularly in the last decade, including Karun 3, Karun 4, Ali Delvari Head, Seymareh, Koohrang 3, and Khersan 3 dams (Calitz 2016).

Asadi et al. (2015), in their research, have examined the effects of splitters on the location and slope of the downstream scour in the free-fall overflow (Asadi et al. 2015).

Presence of a splitter causes the jet fragmentation before reaching the bed material, thus reducing its erosive force and consequently decreasing the scour at the jet's edge (Palmieri et al. 2003).

So far, not enough research has been done on the design of these splitters, and in this field, we can look at the research works of Roberts (1943), Mason (1983), Khatsuria (2005), Annandale (2006), Annandale (2006), Safavi et al. (2011), Asadi et al. (2015), Castillo & Carrillo (2016), Movahedi et al. (2018), Zulfan et al. (2019) and a few other studies (Mason & Arumugam 1985; George & Annandale 2008; Asadi et al. 2015; Castillo & Carrillo 2016; George & Annandale 2017; Movahedi et al. 2018; Zulfan et al. 2019). These researchers have focused on providing criteria for the optimal design of the dimensions and layout of splitters.

Investigating the scouring phenomenon downstream of the free-fall jet, one would confront a completely turbulent and three-phase flow with severe mixing of water, air and bed particles. These issues complicate the scouring phenomenon, and do not allow it to be assessed theoretically.

Application of hydraulic models is among the most valuable approaches to study this phenomenon; these models are used for the quantitative estimation and determination of the efficiency of the solutions to protect the substrate against erosion.

The use of splitters on the free overflow spillways often affects the scour pattern, thus raising downstream scour depth with respect to the case of not using splitters. So far, no significant research work has been performed on the scour holes and the interaction between the patterns and the geometry of the splitters. Also, no study has been performed on the application of a downstream scour model, and the knowledge available on this topic is negligible. In the current situation, a comprehensive study is required to analyze the scour hole pattern downstream of the overflow spillway, as well as its effects on the free fall jets, and to provide an optimum design for splitter arrangement and dimensions (Zulfan et al. 2019).

Various modeling methods have been developed and implemented in water resources structures, such as the operation of reservoirs, hydraulic structures and algorithmic developments (Aboutalebi et al. 2015). Estimating the scour downstream of a free-falling jet has been a research topic among hydraulic engineers. Several empirical models are incorporated for scour calculation downstream of dams. In recent years, focus has been on design of models that are capable of accurate scour simulation. Use of the Artificial Neural Network (ANN) approach to model scour hole depth, width and length has exhibited that ANN model efficiency is much greater than the available empirical models. In various disciplines, the use of SVMs and experimental data is currently being discussed to further enhance the efficiency of ANN models as an alternative method. With this in mind, the present study deals with the creation of regression models using SVR to compute different scour hole parameters. (Azamathulla & Ahmad 2011; Kumar & Ojha 2011).

This study aims to provide an optimal design for the arrangement of splitters to reduce the amount of scour and energy and thus reduce the risk of dam instability. For this reason, experimental studies using a large-scale physical model have been performed at the Iranian Water Research Institute. These include all the required components for modeling the downstream free-falling jet process, overflows in high dams and simulation using the SVR method.

As a result, increase of the airflow inside the jet increases the jet cross-section and for this reason, the erosive force of jet is reduced and the scour depth is decreased (Pagliara et al. 2004) using splitters on the free overflow spillway was proposed for the first time by Roberts (1943) in the design of two dams (Loskop and Vaal bank) in South Africa and thereafter this idea was followed by many designers in various parts of the world. Few studies have been conducted concerning design of splitters until now. The existing studies are done by Roberts (1943), and Mason (1983), as well as a study by Safavi et al. (2011) and a few other studies.

Roberts presented a three-step procedure for design of splitters: the first step is determining the P dimension to obtain the required throw distance for the jet; the second step is to select the width of the splitter teeth (W) from a graph of W/P plotted against 1.2H_d/P (derived by Roberts from his model tests); and the third step is to base all other dimensions on the W value by assuming S = T = L = 1.33 × W and 1.25 L < M < 1.5 L.

In these formulae, H_d denotes the maximum depth of the water passed over the overflow spillway crest, S is the distance between the splitters, W is the width of the splitter, and L is the length of splitter (Figure 1).

According to the hydraulic model studies conducted on various dams, Mason (1983) proposed an optimal geometric design for calculation of the dimensions and arrangement of the splitters. During this investigation, he investigated the effects of splitters on the variance of the exerted dynamic pressure on the downstream apron by the free-fall jet. He presented an optimal design based on the minimization of dynamic pressure on the downstream apron (Mason 1983). Based on the relations between different variables affecting the splitter, the range for each of the variables are defined as follows.

These researchers have reported that these standards contribute to reducing the scour depth by about 30 percent in comparison to the standards proposed by Mason (1983).

It is interesting that the majority of researchers including Roberts (1943), Yen (1987) and Safavi et al. (2011) obtained the same results (1.33) for the length-to-width ratio of the splitter (L/W) (Mason 1983; Mason & Arumugam 1985; Asadi et al. 2015).

If splitters are used on the overflow spillways, the scour hole pattern will change, in addition to reducing the scour depth, compared to the case where no splitter is used. Till now, no study has been conducted concerning the downstream scour pattern in the presence of splitters. Most or all of the previous works had concentrated on the depth of the scour hole and not its dimensions, even without using the splitter, but we have studied all these factors. Also, no study has proposed any equation for calculation of the scour hole dimensions as well as the scour hole depth with splitters.

Therefore, in the present research, using the splitters on the overflow spillway, it was planned to undertake a detailed review of the 3D scour hole pattern downstream of the free overflow spillways. Also a number of equations are suggested for calculation of the maximum depth, length, and width of the scour hole as well as determining where the maximum scour hole depth occurs (with and without presence of splitters). Finally, the effects of a splitter on the longitudinal and cross-section profiles in the scour hole were also investigated.

The experimental dataset used to validate the numerical model is presented in the first part of this section, and the SVR algorithm is defined and explained in the second part.

The tests were conducted on a large scale model of an arch dam with a height of 6 meters at the Water Research Institute of Iran. The layout featured the concrete dam body, the spillway system has a 6 m height, and 5 m wide and a 3 m high rectangular masonry channel, which was filled with a layer of sediment with 1.5 m thickness (Figure 2). The length of the crest is 15 m and the length of the stilling basin is 20 m. In the experiments, two specific non-cohesive granular sediments were used with the characteristics given in Table 2. The coefficient of uniformity for the particle size distribution curve corresponding to a sediment sample is described as and given in Table 2.

Alternative splitter arrangements were used with various dimensions and configurations (Table 3). Three alternatives were formulated according to the requirements of previous researchers such as Roberts (1943), Mason (1983) and Safavi et al. (2011). The other alternatives have been planned to accommodate a broad variety of splitter geometric dimensional parameters. Dimensionless geometric parameters of splitters are described in Figure 2, and their corresponding ranges are shown in Table 3 of this article. The experiments were performed with varying ratios, and (⁠⁠, ⁠). Also a series of tests were performed without splitters as references.

Following previous studies (Mason & Arumugam 1985; Pagliara et al. 2012; Asadi et al. 2015; Calitz 2016), in all the present tests, the summation of the spacing around the spillway opening was set equal to the summation of the splitter width of that opening and the distance between the splitters ‘S’ was set equal to their width (W) (see Figure 1).

The experimental method was such that the channel was filled with water to the height of h above the bed level after the bed was prepared. The depth of tail-water was regulated using the adjustable tailgate positioned downstream of the channel. To calculate the tail-water depth, a point gauge with a precision of 0.5 was used. The water discharge was measured using a straight rectangular crested weir that was placed downstream of the model. The discharge from the water ranged from 0.035 to 0.3 ⁠. The height to tail water ratio, which in fact describes the falling height to tail-water height ratio (⁠⁠), ranged from 0.05 to 0.25. The bed topography was measured with a laser meter with a precision of 1 after each experiment.

Two long tests with and without using splitters were carried out and it was found that the average depth of the scour hole and its expansion did not alter after 180 min (Figure 3). Again all the tests were performed for 180 min. The duration of processing the scour hole equilibrium profiles for two conditions; with or without splitters on the spillway, are given in Figure 3(a) and 3(b), respectively.

It is clear that splitters didn't induce major shifts in the scour hole temporal variations; nevertheless, the equilibrium period was shortened.

The efficient scour parameters are divided into four specific groups in the general mode, for the case of using splitters as follows (see Figure 2):

• 1

  The parameters which introduce hydraulic flow, including: flow specific-discharge (q), falling height (H), tail-water depth (h), height of water on the spillway crest (H_d), density of fluid (ρ), gravity acceleration (g).

• 2

  The parameters which express bed materials, including: mean particle size (d₅₀) and density of bed materials (ρ_s).

• 3

  The parameters which denote splitters comprising the length of splitter in parallel flow (L), the width of splitter placement (W) and the height of splitter placement (P).

• 4

  Time (T)

Parameter ψ denotes one of the characteristics of the scour hole including the maximum depth (D_s), length (L_s) of the scour hole, as well as the location of maximum scour hole depth (X_s).

The first term on the right side of the formula is the Densimetric Froude Number for bed material (⁠⁠) and it expresses the erosive power of jet. Thus it is one of the major effective factors in the scouring process. In this research, 130 tests were performed in total, of which 40 were with splitters and 90 were without splitters. Table 4 shows the range of dimensional parameters in the present analysis compared to the previous investigations.

Due to the nature of the flow patterns that form around the hydraulic structures, it is difficult to accurately estimate the scouring depth. In most research works, the features of scouring are tested using the experimental data and tests (Sharafati et al. 2021).

Support Vector Machines (SVMs) are supervised learning models with associated learning algorithms (Figure 4). These models have been widely used in various fields, including water management and forecasting of sediment loads. The SVM is an efficient learning system based on a restricted theory of optimization that uses the induction principle of minimizing structural error and leads to a general optimal response. The SVM generally consists of two categories: the first category is the SVM, which classifies the data, and the second category is the back vector regression, which performs the regression and fit function (Burges 1997).

In classical models, such as artificial neural networks, the network structure is specific before training and is not practically optimized, but in SVM models, the network structure is also optimized along with the weights. In addition, SVMs can also have potential outputs. Support vector machine is a relatively new and powerful method for clustering (classification) and pattern recognition. This approach is one of the relatively modern methods which have shown good performance in recent years, compared to the older regression methods, including the neural networks of Multi-Layer Perceptron (MLP) (Vapnik et al. 1995).

In this regard, the SVR algorithm for calculating the scouring depth was developed by regressing the experimental data using appropriate relevant parameters. Yen (1987) developed the SVM method, which incorporated two divisions of classification and regression. Support vector regression (SVR) is an important application branch of SVM. Over the years, numerous studies have been conducted on application of the SVR algorithm in solving civil engineering problems, especially in the field of water and hydraulic structures, including (Sihag et al. 2018). The results of this research showed that use of the SVR approach has many advantages including greater accuracy and speed.

The main purpose of this study is to develop a simulation model to predict the scour hole dimensions in the presence and absence of splitters, using the SVR method. In this analysis, the inputs are taken from the spillway non-dimensional parameters, splitter geometry and hydraulic conditions. The purpose is to present the relationship between the scour hole dimensions as the dependent variables (⁠⁠) and those affecting the problem as the independent variables (⁠⁠). The study is done for the two conditions of using splitters and not using them. The following is a summary of the steps for building and implementing the SVR algorithm.

In this study, the research was carried out according to the method mentioned above in the SVR model. After completing the experiments on the physical model, as well as utilizing the previous experiences in this respect, then we evaluated the results by examining the different kernel functions of the outputs.

Several different approaches have been used to determine the certain parameters used in the present research. Furthermore, in the first method, the used algorithm also demonstrated outstanding predictive performance due to implementing the concept of systemic risk minimization. This algorithm considers both the training error and model generalization (Hoang et al. 2018). The experiences of previous studies were also incorporated to estimate the values of constant parameters in the SVR algorithm. In the second method, optimization algorithms were used to estimate the optimum values of the SVR constants.

The SVR model was also used to solve the kernel function in the above problem. For this purpose, various relationships have been described, and three kernel relationships have been used in the present analysis, as follows.

Kernel functions come in a number of forms, and depending on the nature of the problem, each can have its own application. The most common functions for the kernel are given in Table 5. Coefficients of Equation (14) are given in Table 6.

A special case of the polynomial kernel function is the linear kernel function, which is a common and widely used feature in problems. Multi-sentence kernel function can be considerably more efficient in complex issues. RBF functions in the SVM are among the most popular and commonly used kernel functions.

They are used in situations where no information is available on the type of data and its existence. The characteristics of the SVM method, i.e. the values of C and ε, are optimized and the base characteristic γ is optimized for the RBF kernel function.

To solve the SVR model, it is important to properly select the decision variables of the problem. The input parameters of the SVR model, in accordance with the desired target are as follows:

q = special flow discharge, h= shallow dpth, P = height of the splitter placement

H = fall height, L= splitter length, H_d= water level on the overflow

As is seen in the input values, all the input parameters introduced to the model are considered dimensionless. The values used for the inputs of the SVR model also include 116 samples obtained from the experimental model. In addition, the data set was divided into two parts for accurate comparison and assessment of the results: a set of 93 values as the training set, and 23 random data as the testing set.

Thus, after deciding the inputs for SVR model, the output parameters values would include the scour depth, length and width and scour distance from the dam body.

As can be seen in the preceding section on the general structure of the SVR model, in order to implement this model, it is necessary to determine the values of the constant parameters C, σ and ε.

In this section, the configuration of splitters are set using the results of experimental and data-driven models.

Figure 5 Displays a scatter diagram of the dimensionless scour hole (⁠⁠) versus variable Fr_d50. It is noticeable that the variation of the scour hole dimensions variable is expressed exponentially with Fr_d50. This problem is also applied to variable (Figure 6).

The assessment results concerning Equation (14) show the sufficient precision of this equation in approximating the scour hole dimensions (Table 7). This problem can be seen in the scatter diagram between the observed values (O) and the predicted values (P) (see Figure 7).

The splitter independent variables (L), (W), and (P) affect the scour hole depth and dimensions. The scatter diagram of dimensionless variables of the scour hole versus variable Fr_d50 and are shown in Figures 8 and 9, respectively. It can be seen from these figures that these relationships are also exponential, similar to the results for the case of using no splitters (Figures 5 and 6).

The scatter diagram between scour hole dimensions and length of splitters is shown in Figure 10. According to the diagram, there is a critical limit for the length of the splitter in all cases, whereby increase in the L value, the dimensions of the scour hole would increase.

Table 8 depicts the parameters used in Equations (15) and (16). Also the details of their assessment are given in Table 9. The test results obtained using this equation show the sufficient precision of this equation in estimating the scour hole dimensions. This also is seen in Figure 11, which shows the scatter diagram of observed values (O) and expected values (P).

Figure 12 illustrates the scatter diagram of versus variable assuming that all the other variables are constant. According to this diagram, there is a critical limit for variable ⁠, where will have the minimum value and this result agrees well with the experimental observations (Figure 10). Mason (1983) also stated that increase or decrease in the value of from a certain limit causes increase in the exerted dynamic pressures on the downstream floor.

In Equation (16), the variation of is expressed as a plane derived from two variables and ⁠. In Figures 13–15, the diagram of this plane is drawn for three constant values of Frd50 and ⁠. From these diagrams, it can be inferred that the variance pattern of for all values of is such that the scour dimensions within a certain small range would be at a minimum level.

The improvements in the erosive power of the jet, and the proportions of its cross-section with the construction of the splitters change the layout of the scour hole, including the longitudinal and cross-sectional profiles. Also it could affect the ratio of maximum length of hole to maximum depth of hole and ratio of the maximum width of hole to maximum depth of hole ⁠, as well as the location of maximum scour depth (X_s).

Under different flow hydraulic conditions, application of the splitters causes reduction in the maximum depth of the scour hole. The reduced value depends on the flow hydraulic conditions, including Fr_d50 and as well as the geometry of the splitter comprising the and values. In the best condition, the presence of a splitter reduced the maximum scour depth by 31%. For instance in Figure 16, the length and width profiles of the scour hole are plotted for the constant values of Fr_d50 = 0.933 and = 13.2 and various ⁠, values.

The splitter performance is such that it improves jet dimensions directly in parallel to the flow direction. As a result, the presence of the splitter causes elongated flow path of the scour opening. In Figure 17, the scatter diagram has been shown for the maximum scour hole length versus maximum scour hole depth for both the non-splitter (17-a) and splitter (17-b) situations. Within the study limits of this research, the ratio of =K ranged from 3.2 to 5.5 for the case of no splitters and from 3.5 to 6.2 for the case of using splitters. On average, ratio increased by 12% where splitters were installed.

It should also be remembered that increase in ratio in the presence of splitters is associated with reduced scour hole depth. The increase in ⁠, however, does not mean increase in the scour hole length in the presence of splitters compared with the case of not using them. Therefore it is observed that the scour depth is decreased in the presence of splitter by increase in the ratio.

The involved parameters are listed in Table 10, and their evaluation results are given in Table 11. Figure 18 shows a scatter diagram between the predicted and evaluated values using this equation.

Table 12 lists the parameters of this equation as follows:

To validate this equation, use was made of the scatter diagram between the observed (O) and expected (P) values which depicts the accuracy of this equation in determining the location of maximum scour hole depth (as shown in Figure 19). The test results from this equation are presented in Table 13.

When splitters are used on the spillway, the jet trajectory differs considerably from that obtained from the theoretical values. Data analysis and comparison of Equations (17) and (18) reveals that, based on the specific flow and the splitter conditions, the average scour depth location would be closer to the dam body by 16–23%.

In this section, a number of relationships presented by previous researchers are compared with the results of our study. To compare the validity of each model, all available data that were not used in the regression analysis are used in this section. The results of examining various relationships are listed in Table 14. As can be seen in this table, the proposed model has a lower error comparing to the other relationships.

Use was made of the SVR algorithm in the MATLAB software environment to solve the simulation problem described in this report. One of the most critical modeling steps is selection of the correct combination of input variables. As in the models the performance could be improved by choosing the correct and efficient initial inputs to train the consistent nature of the phenomenon.

This model is defined by totally 11 parameters, and is developed using four separate objective functions. In each of these functions, the length, width and depth values of the scour hole as well as the scour hole distance from the downstream according to 7 different dimensionless parameters (⁠⁠), are calculated.

In this section, the results obtained from solution of the simulation model determine the splitters parameters based on the proposed design model and used hydraulic parameters.

Also in this section, the results predicted using the algorithm mentioned in the previous section are presented.

The SVR algorithm was used to estimate the scour hole dimensions and its location and also to estimate the equations. The optimal values are calculated for the model characteristics, which include ε and C. In general, the linear, polynomial and RBF functions were investigated in the SVR. In this method, the optimal values of the parameters needed for the vector regression model were investigated through the trial and error method, and the kernel was obtained.

The parameters coefficients of the SVR model, which were obtained from the performed experiments, are shown in Table 16. The obtained C and γ values for the model per each input combination are shown in Table 15. It is very important to take the right and accurate values for parameters C and ε, as a small value of parameter C contributes to over-fitting, and a high value of parameter ε leads to an inaccurate prediction and a higher error. Several models were used for a better comparison of the results. For this purpose, the network search optimization algorithm was used by incorporating the ε and C values and γ was determined by the trial and error method.

Accuracy and control of predicted results is one of the most important aspects of this process. There are different criteria for measuring the efficiency of a model. The model was tested and analyzed with certain statistical indicators. The performance assessment of the SVR model was determined by the coefficient of determination (⁠⁠), Mean Square Error (MSE), root of the mean error squares (⁠⁠) and Mean Absolute Percentage Error (⁠⁠).

Table 16 presents the effects of estimating the scour hole dimensions and its location and the data validation in the SVR model for the experimental data. The SVR method for the validation phase has an appropriate coefficient of correlation. The results of this analysis at the training and testing stages are given in Table 17.

Finally, the best values were chosen for the radial base kernel function (RBF). The radial base kernel is more accurate than any other function in estimating and predicting. As SVM is based on the minimization of structural error induction principle, using the learning method of monitoring radial base functions, leads to higher speed predictions with lower error parameters with respect to other types of kernels. This is one of the outstanding features of the radial base functions.

The SVR method provided satisfactory results for this analysis. The more accurate the results correlate with the observed results, the more effective the model might be. The distribution of observed and estimated scour hole dimensions and its location values such as (H), (I), (K) in the SVR model are shown in Supplementary Figures 1–3 for the training and testing sets.

Chart of the observational and computational values obtained from the SVR method in (a, c, e, g) for the TESTING stage and (b, d, f, h) for the TRAINING stage with Linear Kernel, Polynomials Kernel and RBF Kernel are explained in the appendix.

The result show that the SVR method has a high efficiency in determining the scour hole dimensions and its location. The Input and output variables are the same as those used in the SVR model.

The criteria used for evaluating the model in the test phase and checking the parameters were the coefficient of determination (⁠⁠),Mean Square Error (MSE), root of the mean error squares (⁠⁠) and Mean Absolute Percentage Error (⁠⁠) Also the sensitivity analysis result shows that the parameters ⁠, and have the highest effect on the downstream scour of free fall overflows, respectively.

Scouring is one of the major external hazards that threatens the downstream of dams. Estimation of scouring characteristics is one of the main concerns in the field of hydraulic and hydrological engineering.

Physical hydraulic models should be used as important tools to improve the precision of theoretical scour predictions and to present appropriate crest improvements, which could efficiently minimize erosion at downstream of dams.

In this research, the effect of splitters on downstream of the dam has been investigated. Both experimental and SVR simulation methods are used for determining the location of splitters and their performance.

High precision and expertise are needed when using the numerical methods and solutions, which have been under focus of attention by researchers. Even minor variations in some of the parameters that impact the numerical model could cause a bad architecture and lead to many forecasting problems. But in this analysis, the intelligent SVM model used as SVR had a remarkable capacity in modelling the dynamic processes such as the erosion phenomenon.

In matters related to water engineering, river engineering and hydraulic structures, estimating the downstream erosion of dams and its location requires experimental and numerical studies. The latest changes in the parameters of this study are taken into account due to their influential and important role in this model.

In the current study use is made of SVR to estimate the scour hole length, width, and depth, as well as its distance from the downstream, according to seven different parameters (specific discharge, fall height, average bed particle diameter, splitter length, splitter width, splitter position height, overflow water level).

The model is defined using 11 parameters in total and is developed using four separate objective functions. The results indicate that the parameter optimization is done based on a highly accurate SVR model (RMSE = 7.1859, MRD = 0.1197, and R² = 0.9540 for the testing set and RMSE = 7.1859, MRD = 0.1197, and R² = 0.9540 for the training set), which has been used to reduce the maximum depth of the scour hole and also maximize the distance of the scour hole from the dam body.

According to the results, the main findings are as follows:

• 1.

  The parameters ⁠, are the most important variables that affect scouring.

• 2.

  The effect of variable on the scour rate is greater when splitters are used.

• 3.

  The use of a splitter on the overflow section causes aeration of the jet and increases its cross-sectional area. Therefore, the erosive power of flow is reduced.

• 4.

  In the best condition, the presence of splitters can reduce the maximum scour depth by 31%.

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