Uncertainty analysis regarding evaluating effective parameters on the hydraulic jump characteristics of different shape channels

The hydraulic jump as an inherent phenomenon happens when the flow changes from supercritical to subcritical type (Corry et al.1975). The water surface promptly increases during this transition, forming surface rollers and leading to intense mixing. Then, the air is entrained while dissipating a huge deal of energy. Hydraulic jump, as an effective phenomenon in rivers, open channels, and energy dissipater systems, is typically employed for kinetic energy dissipation downstream of several hydraulic structures such as gates, drops, spillways, and chutes. Thus, estimating the jump properties such as the sequent depth ratio and hydraulic jump length plays an essential role in designing hydraulic structures. A hydraulic jump can happen in submerged or free circumstances on sloped and horizontal beds. Accordingly, further examination of some basic jump features such as jump length and sequent depth ratio is necessary. For instance, the hydraulic jump length is considered one of the most vital variables in the design of stilling basins.

Nevertheless, it cannot be determined only by mathematical calculations. Thus, laboratory and experimental results require assessment. Numerous experimental works have focused on hydraulic jump formation regarding the estimating of sequent depth ratio and length of the jump. In their study, Abbaspour et al. (2009) investigated hydraulic jumps on corrugated beds. Results demonstrated that the hydraulic jumps on corrugated beds have a smaller length and tail-water depth compared with those on smooth beds. Likewise, Gupta et al. (2013) studied free hydraulic jump length and energy losses in a horizontal prismatic channel. Based on their findings, the relative length and energy losses of the free jump increased with increasing the Froude number, although it reduced with increasing the Reynolds number. Similarly, Ead & Rajaratnam (2002) assessed the hydraulic jump on corrugated beds and indicated that a jump on corrugated beds had a length of half the jump over smooth beds. They further found that the hydraulic jumps had a smaller tail-water depth over corrugated beds compared with smooth beds. In a study, Ahmed et al. (2014) evaluated the submerged hydraulic jump over a triangle strip rough bed. It was concluded that the corrugated beds have permanently superior behavior compared with smooth beds. Also the length and tail-water depth of a jump over a corrugated bed were smaller in comparison with smooth beds. Carollo et al. (2007) assessed hydraulic jump features on a bed roughened by strictly packed bottom-cemented crushed gravel particles and revealed that boundary roughness decreases the sequent and the length of the hydraulic jump depth. Recently, soft computing methods such as Adaptive Neuro-Fuzzy Inference System (ANFIS), Artificial Neural Networks (ANN), Gene Expression Programming (GEP), Genetic Programming (GP), and Gaussian Process Regression (GPR) have been utilized in open channel hydraulics and water resources studies. In this regard, machine learning approaches can be evaluated for different purposes such as capacity prediction for the reservoir (Foddis et al. 2015), groundwater level fluctuation prediction (Gholami et al. 2015; Rajaee et al. 2019; Di Nunno & Granata 2020; Pandey et al. 2020), and roughness coefficient and sediment transport in sewer pipes (Roushangar & Ghasempour 2017; Roushangar et al. 2019), and tide level predictions (Granata & Di Nunno 2021). Other areas of focus are modeling side weir discharge (Azamathulla et al. 2016; Azimi et al. 2017; Granata et al. 2019), predicting daily flows and stage-discharge (Wu et al. 2009; Kumar et al. 2020; Malik et al. 2020), lake water level (Yaseen et al. 2020), water quality (Wang 2016), and energy dissipation in channels with rough beds (Roushangar & Ghasempour 2018), evaluating soft computing methods for predicting river bed load (Azamathulla et al. 2010; Chiang & Tsai 2011), and forecasting downstream river flows (Chen et al. 2015).

Based on literature reviews, no research has comprehensively focused on predicting hydraulic jump features in various shape channels by kernel-based methods. Due to the nonlinearity and uncertainties of the hydraulic jump phenomenon, the existing regression methods often do not show the desired accuracy. The existing semi-empirical equations rely on a limited database, untested model assumptions, and a general lack of field data, and do not show the same results under variable flow conditions. Consequently, the applications of many of these methods are limited to special cases of their development, and therefore do not show uniform results under different conditions. On the other hand, the correct estimation of hydraulic jump characteristics is of great importance in hydraulic engineering, as it directly affects the planning, design and management of hydraulic structures. However kernel-based approaches based on quadratic optimization of convex functions can easily switch from linear to nonlinear separation. This is realized by nonlinear mapping using so-called kernel functions. Therefore, the present study employed kernel-based approaches to predict the variable of interest. Among artificial models, kernel-based approaches such as SVM and GPR are relatively new and important methods based on the different kernel types which are based on statistical learning theory. Such models are capable of adapting themselves to predict any variable of interest via sufficient inputs. The training of these methods is fast, they have high accuracy, and the probability of occurrence of data overtraining in these methods is less. Also, the most important principle of SVM in comparison with other techniques (such as ANN, ANFIS, GP, and GEP) is the application of minimizing an upper bound to the generalization error instead of minimizing the training error. The current study aimed to expand former studies by representing the capability of kernel-based methods (i.e., SVM and GPR) in order to predict submerged and free hydraulic jump features in channels with various bed conditions and shapes (i.e., trapezoidal, rectangular, expanding, sloped, and with negative step channels with rough and smooth beds). Several models were prepared under various input combinations (based on hydraulic characteristics and geometry of the channels and appurtenances) in order to find the most appropriate input combination for modeling hydraulic jump characteristics in suddenly expanding channels. Then the accuracy of the best applied models was compared with the accuracy of several semi-empirical equations. In addition, the most essential parameters in the prediction procedure were determined by utilizing sensitivity analysis, followed by employing Monte Carlo uncertainty analysis (UA) to evaluate the dependability of the applied models.

In this research, two hydraulic jump types were considered, including the free jump in trapezoidal, horizontal rectangular, expanding, rectangular with slope, and with negative step channels with rough and smooth beds and the submerged jump in horizontal and sloped rectangular channels with smooth beds. Bhutto (1987) performed hydraulic jump experiments on a rectangular smooth bed in the hydraulic laboratory of Mehran University of Engineering and Technology, Jamshoro. Accordingly, Bhutto's measurements yielded smooth-bed datasets. The experiments were conducted in a channel 16 m long. The cross-section of the channel was rectangular, 0.3 m wide and 0.4 m high. The channel included steel and glass-sided floors (70, 82, and 65 data for the submerged jump–horizontal, free jump–horizontal, and submerged jump–sloped smooth beds, respectively). The free-type hydraulic-jump datasets on the rectangular smooth bed channel with slope were achieved from the experiments conducted by Li (1995), including an adjustable channel (with length of 2.43 m, width of 0.15 m, and depth of 0.305 m) on different channel bed slopes in the range of – (88 data). Hughes & Flack (1984) performed hydraulic jump experiments over a rectangular rough bed by artificially roughened test beds in a horizontal rectangular flume with a width of 0.305 m, and smooth Plexiglas side walls and length of 2.13 m. They utilized two kinds of roughness elements, including some parallel square bars perpendicular to the flow direction, along with strictly packed gravel particles cemented to the base (168 data). Additionally, free jump experimental data provided by Bremen (1990) and Evcimen (2012) were applied to predict the objectives. Bremen's experiments (1990) were anticipated for smooth (178 data) and with negative step (129 data) sudden diverging basins. Accordingly, 17 m³ upstream basins were supplied by two 0.30-m-diameter conduits (with a maximum total discharge of 0.375 m³/s) during the experiments. Further, a horizontal and prismatic rectangular channel (with a width of 0.5 m and length of 10.8 m) was linked to the basin. Similarly, Evcimen (2012) performed some experiments at the hydraulic laboratory of the Middle East Technical University. The trapezoidal shape channels were used during the experiments and the impacts of prismatic roughness on the characteristics of hydraulic jump were evaluated (107 data). In Figure 1 a hydraulic jump in various shape channels is illustrated.

Kernel-based methods such as Support Vector Machine (SVM) and Gaussian Process Regression (GPR) are considered as relatively innovative and significant techniques in terms of various kernel types and statistical learning theory. These models can adapt themselves for predicting any parameter of interest by adequate inputs. Furthermore, they can model non-linear decision boundaries, and numerous kernels exist in this regard. These methods are also objectively strong against overfitting, particularly in high-dimensional spaces. Nevertheless, proper selection of the kernel kind is the most essential step in GPR and SVM methods because of its direct effect on classification precision and training. In this study the SVM and GRP methods were implemented using the MATLAB computer programming language.

In GPR models, it is assumed that the nearby observations require the conveying of data regarding each other. Gaussian processes are a way of specifying a prior directly over function space. This is a natural generalization of the Gaussian distribution, whose mean and covariance are a vector and matrix, respectively. Although the Gaussian distribution is over vectors, its procedure is considered over functions. Accordingly, generalization requires no validation process considering the former knowledge on functional dependencies and data. Moreover, GP regression models can recognize the predictive distribution equivalent to test inputs (Rasmussen & Williams 2006). A GP is referred to as a combination of random variables for which any finite number possesses a mutual multivariate Gaussian distribution. For instance, denote input and output domains, respectively, and n pairs are independently drawn and identically distributed from these domains. As regards regression, it is assumed that ⁠. Then, a GP on is determined by the covariance function of and the mean function of ⁠.

To determine hyperparameters, the partial derivative of Equation (3) can be obtained based on and k, and the gradient descent should be considered to get minimization. Due to higher effects on regression model accuracy, calculating the optimal value of the capacity constant (C), the size of the error-intensive zone (ɛ) in SVM and Gaussian noise in GPR are necessary. The optimal values for setting parameters of each kernel-based model should be obtained following the trial-and-error procedure.

SVM as an intelligence method is utilized in information and dataset categorization. Such a method, which was first introduced by Vapnik (1995), is recognized as structural risk minimization, minimizing an upper bound on the expected risk contrary to the traditional empirical risk that reduces the error on the training data. The SVM technique relies on the optimal hyperplane concept separating the samples of two classes while taking into account the widest gap between the two groups. Support Vector Regression (SVR) is an expanded version of SVM regression that aims to identify a function with the most deviation from the actual target vectors for all considered training data and the flatness as possible (Smola 1996). Vapnik (1995) represented the kernel function concept for non-linear SVR. It should be noted that proper selection of the kernel kind is the most significant step in the SVM owing to its direct effect on classification precision and training. Generally, various kinds of kernel functions exist, including linear, polynomial, and sigmoid types, along with radial basis function (RBF).

In this study, the GPR and SVM models' capabilities were examined for measuring hydraulic jump characteristics such as jump length and sequent depth ratio. Several datasets from previous studies were utilized to estimate hydraulic jump features (Figure 2). The intended datasets were classified into testing and training data types comprising 30% and 70% of samples. It should be noted that the order of the datasets was selected in a way such that the training dataset contains a representative sample of all the behavior in the data in order to obtain a model with higher accuracy. One method for finding a good training set which can give good accuracy both in training and testing sets is an instance exchange (Bolat & Yildirim 2004). The process starts with a randomly selected training set. After the initial training process, the test data is applied to the network. A randomly selected true classified instance in the training set (I1) is thrown into the test set, a wrongly classified instance in the test set (I2) is put into the training set and the network re-trained. The process is repeated until reaching the maximum training and test accuracy. On the other hand, selecting appropriate variables as inputs for the models is essential through the modeling procedure. Several hydraulic jump parameters were considered in the present study, including flow features such as upstream flow depth, tail-water depth, immersion factor, and bed specifications (e.g., bed slope and roughness). Moreover, the Froude number is considered as a major parameter extensively utilized in a wide variety of hydraulic studies. Different arrangements of such variables were studied for the considered kinds of the jump, followed by evaluating the effects of these variables on features of hydraulic jumps. Tables 1–3 present the recommended input arrangements. The applied parameters for modeling include Y₁, Y₂, tanθ, Fr_1,Y₄, w, and k_s, indicating the initial depth of hydraulic jump, the second depth of hydraulic jump, the slope of the channel's bed, upstream Froude number, tail-water depth, rough element distance, and rough element height, along with S and S’, which denote the immersion factor in free and submerged states, respectively.

In this research, in order to determine the best performance of the SVM and GPR models and select the best kernel functions, several developed models were tested via the SVM and GPR considering various kernels. Also, the optimal value of capacity constant (C) and the size of the error-intensive zone (ε) in these methods are required due to their high impact on the accuracy of the mentioned regression approaches. The coefficient C is a positive constant that influences a trade-off between the approximation error and the regression and must be selected by the user, and ε has an effect on the smoothness of the SVM and GPR responses, so both the complexity and the generaliz-ation capability of the network depend on its value. If epsilon is larger than the range of the target values, we cannot expect a good result. If epsilon is 0, we can expect overfitting. Also, the variable parameter used with a kernel function considerably affects the flexibility of the function. These parameters should be selected by the user. In the current study, optimization of these parameters has been done by a systematic grid search of the parameters using cross-validation on the training set of each considered state. For selecting the optimum parameters and assessing average developed model performance, the RMSE was used based on some well-known studies. Also, designing SVM- and GP-based regression methods involves using the kernel function concept. In this study, to determine the best structure of GPR and SVM models and choose the best kernel function, the depth ratio variable in the sloped smooth bed channel with the free jump was estimated by utilizing different kernels and using the model with the input parameters of tanθ and Fr₁. Figure 3 displays the findings of the statistical parameters of various kernels for this model. Based on the findings, both the SVM and GPR models resulted in better prediction accuracy using the kernel function of RBF compared with other kernels. Thus, the RBF kernel was utilized as a core instrument of GPR and SVM applied for the remaining models.

Various models were established for evaluating hydraulic jump features in channels with various shapes based on the geometry and flow condition of rough elements and the applied channels. The expanded models were examined with the GPR and SVM models in order to perform depth ratio and jump length estimation in such channels. The best model was identified for each dataset. The model with the closest RMSE to 0 and the correlation coefficient (R) and determination correlation (DC) with closest value to 1 was selected as the best one. Table 1 and Figure 4 represent the findings of the SVM and GPR models. According to the results, the established models for the case of the sloped smooth bed channel performed more positively among all kinds of channels in the state of a free hydraulic jump compared with the other case. In most practical uses, it is desired and occasionally essential to create the hydraulic jump on sloped beds since this can present effective design information. According to Table 1, it can be seen that the combination of tanθ and Fr₁ parameters was the best model. This model predicted the depth ratio more accurately in the case of the channel with a sloped smooth bed (DC = 0.952, R = 0.974, and RMSE = 0.625). Nonetheless, analytical studies reveal that including the gravitational forces over the flow is essential for predicting the free jump characteristics over the sloped bed channels. As expected, adding the bed slope variable to input combinations enhanced the sequent depth ratio and the jump length prediction accuracy. Furthermore, according to the findings, utilizing Y₁/K_s as an input parameter increased the modeling performance for channels with the rough bed. Likewise, the expanding channel with no appurtenances resulted in better prediction compared with the expanding channel with steps. Given the results of the established model regarding the trapezoidal channel with the rough bed, higher accuracy was presented by the model with parameters Fr₁ and w/k_s. As regards channels with a step or rough bed, it was found that Y₁/k_s and w/k_s increased the efficiency of the models and the impacts of w/k_s in further enhancing model accuracy compared with Y₁/k_s. Furthermore, favorable accuracy was found for the model, including only the input parameter of Fr₁. Therefore, the applied methods are capable of successfully predicting the sequent depth ratio by utilizing only upstream flow features as input data. The results of GPR models are negligibly accurate compared with those of the SVM model. Figure 4 depicts the verification between estimated and measured values of the test series for the best models of each state. In the form of a submerged hydraulic jump, the model with parameters Fr₁ and S in the case of the smooth horizontal channel functioned more successfully in comparison with the other models. Based on the comparison of the obtained results for free and submerged states, models for the submerged state led to better outcomes.

The statistical criteria obtained for the developed models for jump length in channels with a varied shape are listed in Table 2. Regarding channels with appurtenances, the horizontal rough bed channel resulted in better estimation accuracy compared with the expanding channel with step and trapezoidal channel and a rough bed, the maximum R and DC, and the lowest RMSE values. Based on the obtained data, the model with parameters Y₂/k_s and Y₁/k_s demonstrated a higher rate of accuracy. In addition, adding (Y₂ − Y₁)/Y₁, Y₁/k_s, w/k_s, and Y₂/Y₁ as input parameters increased the model's capability, confirming the impact of the geometry (i.e., roughness and step elements) of the applied appurtenances on the jump length in channels with various appurtenances. As regards the channels without appurtenances, the model including Fr₁, Y₁, and tan θ as input parameters and using the sloped smooth bed channel gave optimal results. It was observed that in the jump length prediction in the expanding channel with a negative step the model with input parameters of Fr₁, S/Y₁ led to more accurate predictions, while for the case of the asymmetric channel the model L(II) with parameters of Fr₁, (Y₂ − Y₁)/Y₁ represented higher efficiency. Based on the results it could be inferred that for modeling the length of hydraulic jump in basins with a negative step, using parameters of Y₂/Y₁ and Y₁/k_s increased the accuracy of the models. Parameter Y₁/k_s confirms the importance of the relative height of the applied step in the hydraulic jump characteristic predicting process in channels with a negative step.

Table 3 provides validation results for predicting the submerged jump length over the horizontal channels with smooth beds. Investigation of the results confirms that the input combination with parameters S, Y₄/Y₁, and Fr₁ for length prediction led to superior performance regarding other combinations. Therefore, the immersion factor and the upstream Froude number were the most critical variables in modeling the submerged jump features over a horizontal channel with a smooth bed. Moreover, the addition of Y₂/Y₁ to the input parameters increased the model accuracy. Contrary to former cases, the objective function was L_sj/Y₂ regarding modeling the submerged jump length in the sloped channels with smooth beds. Generally, the literature review indicated that submerged jump on the sloped channel is nearly untouched, and restricted empirical equations exist for estimating the length of such jumps. Statistical parameters confirmed that the application of the upstream Froude number as an input parameter led to poor predictions in jump modeling. Accordingly, the slope of the bed and the immersion factor (S’ and tan θ) played a crucial role in modeling the submerged jump length over the channels with sloped beds. Moreover, investigating input parameters to estimate the submerged jump sequent depth ratio in these channels confirmed that adding the slope of the bed (tan θ) to the preliminary Froude number enhances the findings (Table 3). Figure 5 shows the experimental values against the length values of the simulated jump with regard to the GPR best models in the channel with a varied shape for the test series.

The effects of various parameters on hydraulic jump features were assessed by a sensitivity analysis using the RMSE parameter. In this regard, the superior model was rerun via the GPR model, and the importance of each variable was evaluated by omitting the variable. Based on the RMSE values presented in Figure 6, it can be seen that the omitting of Fr₁ led to a significant reduction in modeling accuracy. Therefore, this parameter is the most effective variable in predicting the free hydraulic jump length over horizontal smooth bed channels, smooth and with negative step expanding channels, and trapezoidal channels with rough and sloped smooth beds. The secondary depth ratio normalized by the roughness diameter strongly affects the models' accuracy for predicting the length of a free hydraulic jump over horizontal rough beds. The immersion factor statistically is significant for predicting the length of the submerged hydraulic jumps over the sloped and horizontal smooth bed channels. As shown in Figure 6, Fr₁ was the essential factor for predicting the sequent depth ratio in all types of jumps, although it may not be the first leading parameter regarding modeling the hydraulic jump lengths.

Various semi-empirical and analytical equations have been presented for forecasting hydraulic jump features having simple and complex structures. In this regard, empirical equations mentioning free hydraulic jumps are the most famous and valuable equations. The basis of the equations is obtained using the momentum principle regarding the sequent depth ratio. Nonetheless, using experimental equations, some references calculated several parts of the equation in some types of hydraulic jumps. In addition, some of the studies focused on submerged hydraulic jumps. The stated equations for such a jump are valid for estimating jump features. In the present study, the accuracy of the best-introduced models and several available semi-theoretical formulae in the literature were compared to investigate the performance of the employed approaches. Table 4 summarizes the obtained data from the comparison. Based on three evaluated criteria (i.e., R, DC, & RMSE), the estimated values by the GPR and SVM models gave more precise results compared with equations. The results further revealed that the formulae used for computing the sequent depth ratio presented a rational fit to the experimental data. The length of the equations of the hydraulic jump represented no desired consistency between the approximated values and the observed sets in comparison with the sequent depth ratio formulae. It is worth noting that the present equations are based on untested model assumptions and a restricted database. Moreover, they lack field data and do not yield the same findings under variable flow conditions. Based on the results, it can be seen that the use of the Fr₁ parameter in the semi-empirical equations led to an increment in the equations' efficiency; however, the obtained results from the equations differ from each other and their application is limited to special cases of their development. Nonetheless, regardless of the uncertainty and complexity of the hydraulic jump phenomenon, the obtained data show the workability of GPR and SVM as kernel-based machine learning approaches for modeling the characteristics of free and submerged hydraulic jumps.

The most important characteristics of hydraulic jump, length, and sequent depth ratio play a key role in the design of hydraulic structures. However, the results of the semi-empirical equations which have been developed to estimate the length and sequent depth ratio of hydraulic jumps are not general and acceptable due to the uncertainty of the phenomenon. Therefore, artificial intelligence models, especially kernel-based approaches, can be used to improve the modeling efficiency. In the present investigation, the SVM and GPR modeling approaches were applied because of their high learning ability and information processing potentiality, which make them suitable for complex nonlinear modeling without prior knowledge about the input–output relationships, which is difficult to handle with the statistical approach. The kernel-based approaches have better generalization ability, less susceptibility to noise and outliers than the regression models, and can handle incomplete data. As can be seen from the obtained results, these methods led to more accurate outcomes compared with the semi-empirical equations.

In the present study, novel data-mining approaches (i.e. GPR and SVM) were used in order to predict the sequent depth ratio and the length in various kinds of channels and hydraulic jumps in trapezoidal, rectangular, expanding, sloped, and with negative step channels with smooth and rough beds. The applied meteorological data-sets for establishing the models were found from former valid studies. The results from the presented GPR and SVM methods and the obtained values from other semi-empirical equations were compared, and it was demonstrated that the SVM- and GPR-based models overwhelm the employed semi-empirical equations regarding predicting the sequent depth ratio and the length of the jump in submerged and free jumps. Thus, some notable conclusions were drawn as follows.

In the channels with smooth horizontal beds, it was observed that the upstream Froude number increased the modeling accuracy and led to better prediction of free hydraulic jump length. Regarding free hydraulic jumps over horizontal rough bed channels, using both the ratio of primary and secondary depths to roughness diameter (Y₁/k_s and Y₂/k_s) plays a key role in predicting the length of this jump. Regarding the free hydraulic jump over sloped smooth bed channels, the addition of the bed slope to the upstream Froude number and the initial depth results in an enhancement in the findings. Moreover, for channels with a step or rough bed, parameters w/k_s and Y₁/k_s incremented model efficiency, revealing the effect of the geometry of the applied appurtenances (i.e., roughness elements and step) over hydraulic jump features in channels with numerous appurtenances. Based on the obtained results, the developed models for the sloped smooth bed channel were more positive between all types of channels in the state of a free hydraulic jump than other cases.

Concerning the submerged jumps on channels with sloped and horizontal smooth beds, the immersion factor was considered as the most influential variable. It had a bold contribution to the submerged jump length predictions.

In general, it was confirmed that the upstream Froude number is the most contributing factor regarding predicting the sequent depth ratio in all kinds of hydraulic jumps. It was further found that the results of the GPR models are slightly more precise compared with the SVM models. On the other hand, it should be noted that although GPR and SVM can develop an extremely precise model for predicting the objective function, they cannot establish explicit rules. Moreover, the dependability of the superior applied models was assessed through UA, and it was revealed that the GPR model possessed an acceptable uncertainty degree in the modeling of hydraulic jump characteristics in channels with shapes. It should, however, be noted that the AI models are data-driven models, and thus, they are data sensitive. The prediction of data using SVM and GPR depends on the availability of laboratory data. Therefore, different types of channels to be used (as the ones in the present study) may affect the fitting and prediction accuracy of applied models, which should be further discussed in the future. Also, it is suggested to use other methods of artificial intelligence and compare the obtained results with the present research.

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