## Abstract

The hydraulic behavior of a hydraulic jump can be challenging to estimate in order to design gradually expanding stilling basins with roughness elements on the bed. In this study, five dependent variables were identified that comprise: (i) the sequent depth ratio; (ii) the relative length of the jump; (iii) the relative roller length of the jump; (iv) the relative energy dissipation; and (v) the water surface profile. This study undertook a set of formulations based on the regression analysis and Sugeno Fuzzy Logic (SFL) to predict these variables based upon experimental data. Results demonstrate that the trained SFLs predicted the behavior of the dependent variables with the Nash–Sutcliffe Coefficient (NSC) greater than 0.96 in the testing phase. In contrast, the NSC values for the regression models are greater than 0.79. The higher accuracy of SFL is attributed to its capability in managing uncertainty and imprecise data owing to water surface profile oscillations of the hydraulic jump. Also, the results indicate that the prediction residuals for SFL are homoscedastic for all hydraulic parameters investigated except for the water surface profile, the prediction residuals of which for the regression equations are heteroscedastic.

## HIGHLIGHTS

Hydraulic jump characteristics are experimentally investigated in a gradually expanding basin.

Five empirical equations are fitted to predict variables related to hydraulic jump characteristics.

Five SFLs are trained and tested to predict the same variables and manage data uncertainty.

SFLs provide homoscedastic results in most cases, but there is room for further improvements.

### Graphical Abstract

## NOTATIONS

Width of the stilling basin in upstream

Width of the stilling basin in downstream

Specific energy upstream the hydraulic jump

Specific energy downstream the hydraulic jump

Energy loss in the hydraulic jump

Relative energy loss

Inflow Froude number, where

Gravitational acceleration

Length of the hydraulic jump

Relative length of the hydraulic jump

Roller length of the hydraulic jump

Relative roller length of the hydraulic jump

*m*,_{k}*m*,_{j}*m*,_{p}*m*_{q}Coefficients for membership functions

Weighted average of the SFL output

Discharge per unit width

Height of roughness elements

Inflow Reynolds number

*V*_{1}Inflow velocity

*V*_{2}Sequent velocity of the hydraulic jump

*w*_{kj}Firing strength

Inflow depth of the hydraulic jump

Sequent depth of the hydraulic jump

Sequent depth ratio

Mass density of water

Kinematic viscosity of water

- AI
Artificial intelligence

- ANN
Artificial neural network

- GEP
Gene expression programming

- GP
Genetic programming

- MR
Multiple regression

- NF
Neuro-fuzzy

- RMSE
Root mean square error

- SVM
Support vector machine

- SVR
Support vector regression

- SFL
Sugeno Fuzzy Logic

## INTRODUCTION

Energy dissipation is one of the challenges downstream of hydraulic structures and generally is accomplished, such as in the case of many free-surface spillways, with flow regime transformation within the structure from supercritical to subcritical through a hydraulic jump (Rajaratnam 1967; Montes 1998). Classic equations describe hydraulic jump characteristics (Rouse *et al.* 1958; Schröder 1963; Peterka 1984), including energy dissipation, jump length, and sequent depth, in which the sequent depth refers to the paired depths upstream and downstream of a hydraulic jump. However, these equations prevail for simplified conditions such as rectangular cross-sections, prismatic, and smooth-bed channels. Rectangular channels are the most common water conveying structure due to simplicity in construction and have been merited and constructed at specific project sites.

Hydraulic jumps are classified based on the Froude number (*Fr*_{1}) of the approaching flow. Low values for the Froude number (1.7 < *Fr*_{1} < 2.5) correspond to undular hydraulic jumps, which are characterized by a series of surface waves (Chanson & Montes 1995; Ohtsu *et al.* 2001; Ohtsu *et al.* 2003). The transitional hydraulic jump (2.5 < *Fr*_{1} < 4.5) is associated with pulsating action, in which the entering jet oscillates heavily from the basin floor to the surface without a regular period (Hager 2013). Higher values of the Froude number (4.5 < *Fr*_{1} < 9) result in a strongly defined roller and stable hydraulic jumps; this range is often considered in basin design due to the significant energy dissipated by the jump. Finally, high values of the Froude number (*Fr*_{1} > 9.0) tend to instabilities with a highly chaotic, rough water surface and intense bubble and spray formation (Hager 2013).

Previous experimental studies on hydraulic jumps are categorized into classic hydraulic jumps with simplified conditions and hydraulic jumps through a channel with complexities in geometry such as non-smooth bed, non-rectangular cross-section, and expansion in sidewalls. In the first category, a selection of noteworthy studies on the classic hydraulic jump includes Rouse *et al.* (1958), Wu & Rajaratnam (1995), Carollo *et al.* (2009), Hager (2013), and Retsinis & Papanicolaou (2020).

Studies in the second category considered some variations in channel geometry such as roughness elements, expanding sidewalls, and non-rectangular cross-sections. For example, roughness elements via a corrugated bed have been studied by Elsebaie & Shabayek (2010), Samadi-Boroujeni *et al.* (2013), Pagliara *et al.* (2008), and Kumar & Lodhi (2016). Their results show that the corrugated bed significantly reduces the sequent depth and hydraulic jump length. Peterka (1984) and Blaisdell (1959) are two classic studies from the USA that focused on basin elements in specific geometric schemes to reduce the length of the classic hydraulic jump, *L _{r}*. Literature also includes studies on non-rectangular cross-sections and gradually or abruptly expanding channels by appurtenances in the stilling basins (e.g., Negm 2000; Roushangar

*et al.*2017). The results show that these appurtenances dissipate the energy of the flow and reduce the length of stilling basins, and deflect the high-velocity jets away from the stilling basins’ bottom (Hassanpour

*et al.*2017; Aal

*et al.*2018). For example, Varaki

*et al.*(2014) and Hassanpour

*et al.*(2017) investigated the characteristics of the hydraulic jumps in stilling basins with expanded sidewalls. They show that the tailwater depth is necessary to impose a hydraulic jump, and also, the length of the hydraulic jump was smaller compared with a rectangular basin.

The previous studies indicate that any variation in channel geometry causes variations in the characteristics of the hydraulic jumps and introduces uncertainties in the prediction of hydraulic properties via equations of classical hydraulic jumps. This reveals the necessity and complimentary nature of data-driven models such as regression equations and artificial intelligence (AI)-based models to estimate the characteristics of hydraulic jumps. Table 1 summarizes a set of studies on the data-driven modeling of hydraulic jumps, demonstrating the application of statistical and AI models. Table 1 also compares studies in terms of bed, cross-section, sidewall configurations, models, and predicted parameters of hydraulic jump. Despite the variety of AI models summarized in Table 1, including artificial neural network (ANN), gene expression programming (GEP), and support vector machine (SVM), the table provides evidence that AI models have been used to estimate characteristics of hydraulic jumps in complex geometries of channels.

Researcher (s) . | Bed . | Cross-section . | Wall . | Model . | Predicted parameter . | ||||
---|---|---|---|---|---|---|---|---|---|

Smooth . | Rough . | Slope . | Normal . | Expanding . | Statistical . | AI . | |||

Omid et al. (2005) | ✓ | – | • Horizontal | • Rectangular | – | Gradually | • ANN^{a} | • Sequent depth | |

• Trapezoidal | • MR^{b} | • Jump length | |||||||

Naseri & Othman (2012) | ✓ | – | • Horizontal | • Rectangular | ✓ | – | – | • ANN | • Jump length |

Houichi et al. (2013) | ✓ | – | • Horizontal | • U-shape | ✓ | – | ✓ | • ANN | • Jump length |

• MR | |||||||||

Abbaspour et al. (2013) | – | ✓ | • Horizontal | • Rectangular | ✓ | – | – | • ANN | • Sequent depth |

• GP^{c} | • Jump length | ||||||||

• Energy loss | |||||||||

Abdeen et al. (2015) | ✓ | – | • Horizontal | • Rectangular | ✓ | – | – | • ANN | • Jump length |

• Sloping | • with drop | ||||||||

Karbasi & Azamathulla (2016) | – | ✓ | • Horizontal | • Rectangular | ✓ | – | – | • GEP^{d} | • Sequent depth |

• ANN | • Jump length | ||||||||

• SVR^{e} | |||||||||

Roushangar & Ghasempour (2018) | – | ✓ | • Horizontal | • Rectangular | – | Sudden | – | • GEP | • Jump length |

• without appurtenances | • Energy loss | ||||||||

• with a central sill | |||||||||

• with a negative step | |||||||||

Azimi et al. (2019) | – | ✓ | • Horizontal | • Rectangular | ✓ | – | – | • GEP | • Roller length |

Kumar et al. (2019) | – | ✓ | • Sloping | • Rectangular | ✓ | – | – | • ANN | • Sequent depth |

Sadeghfam et al. (2019) | – | – | • Horizontal | • Rectangular | ✓ | – | – | • SVM^{f} | • Depth and length of a pit induced by hydraulic jump |

• NF^{g} | |||||||||

• SFL^{h} | |||||||||

Roushangar & Ghasempour (2019) | – | ✓ | • Horizontal | • Rectangular | – | Sudden | – | • SVM | • Energy loss |

• Trapezoidal | |||||||||

• Expanding |

Researcher (s) . | Bed . | Cross-section . | Wall . | Model . | Predicted parameter . | ||||
---|---|---|---|---|---|---|---|---|---|

Smooth . | Rough . | Slope . | Normal . | Expanding . | Statistical . | AI . | |||

Omid et al. (2005) | ✓ | – | • Horizontal | • Rectangular | – | Gradually | • ANN^{a} | • Sequent depth | |

• Trapezoidal | • MR^{b} | • Jump length | |||||||

Naseri & Othman (2012) | ✓ | – | • Horizontal | • Rectangular | ✓ | – | – | • ANN | • Jump length |

Houichi et al. (2013) | ✓ | – | • Horizontal | • U-shape | ✓ | – | ✓ | • ANN | • Jump length |

• MR | |||||||||

Abbaspour et al. (2013) | – | ✓ | • Horizontal | • Rectangular | ✓ | – | – | • ANN | • Sequent depth |

• GP^{c} | • Jump length | ||||||||

• Energy loss | |||||||||

Abdeen et al. (2015) | ✓ | – | • Horizontal | • Rectangular | ✓ | – | – | • ANN | • Jump length |

• Sloping | • with drop | ||||||||

Karbasi & Azamathulla (2016) | – | ✓ | • Horizontal | • Rectangular | ✓ | – | – | • GEP^{d} | • Sequent depth |

• ANN | • Jump length | ||||||||

• SVR^{e} | |||||||||

Roushangar & Ghasempour (2018) | – | ✓ | • Horizontal | • Rectangular | – | Sudden | – | • GEP | • Jump length |

• without appurtenances | • Energy loss | ||||||||

• with a central sill | |||||||||

• with a negative step | |||||||||

Azimi et al. (2019) | – | ✓ | • Horizontal | • Rectangular | ✓ | – | – | • GEP | • Roller length |

Kumar et al. (2019) | – | ✓ | • Sloping | • Rectangular | ✓ | – | – | • ANN | • Sequent depth |

Sadeghfam et al. (2019) | – | – | • Horizontal | • Rectangular | ✓ | – | – | • SVM^{f} | • Depth and length of a pit induced by hydraulic jump |

• NF^{g} | |||||||||

• SFL^{h} | |||||||||

Roushangar & Ghasempour (2019) | – | ✓ | • Horizontal | • Rectangular | – | Sudden | – | • SVM | • Energy loss |

• Trapezoidal | |||||||||

• Expanding |

^{a}Artificial neural network.

^{b}Multiple regression.

^{c}Genetic programming.

^{d}Gene expression programming.

^{e}Support vector regression.

^{f}Support vector machine.

^{g}Neuro-fuzzy.

^{h}Sugeno Fuzzy Logic.

The novel features of the present study are outlined as follows: (i) the experimental investigation of hydraulic jump characteristics through a channel with a gradual expansion of sidewalls with quantification of the sequent depths (*y*_{1}, *y*_{2}), the length of the jump (*L _{j}*), the roller length of the jump (

*L*), and subsequent energy dissipation and water surface profile and (ii) the estimation of these characteristics through a multiple regression (MR) model and Sugeno Fuzzy Logic (SFL). To the best of the authors’ knowledge, the aforementioned features have not been investigated in gradually expanding stilling basins with rough elements. Also, the capability of SFL in managing imprecise and uncertain data due to water surface oscillations makes it a rational choice for modeling, which thus far has not been employed to estimate the characteristics of hydraulic jump.

_{r}## METHODOLOGY

### Experimental setup

The physical experiments were carried out in a flume (see Figure 1) with a cross-section of 0.5 × 0.5 m^{2} and a length of 10 m in the hydraulic laboratory of the University of Tabriz. The sidewalls for expanding section were constructed using two vertical Plexiglas plates and continued up to 2.1 m toward the channel downstream, and finally reached the main channel. The flume was equipped with an ultrasonic flow meter (model Trodeks with TM-1 transducer, ±0.5 mm) and two sluice gates upstream and downstream of the flume to control upstream and downstream conditions (i.e., discharge, flow depth). Water surface profiles were measured using ultrasonic sensors (Data Logic US30, ±0.1 mm) mounted on a carriage. The Froude number of input flows studied for 6 ≤ *Fr*_{1}*≤*12.5. The tailwater gate installed at the end of the channel was used to control and adjust the location of the hydraulic jump at the beginning of the gradual expansion section. Four expansion ratios (*B*) (defined as *b*_{1}/*b*_{2}, where *b*_{1} and *b*_{2} are the widths of the flume at the beginning and end of expanded sections, respectively) equal to 0.4, 0.6, 0.8, and 1 were considered (see Figure 1). Roughness elements evenly distributed throughout the expansion section were created by discontinuous lozenge shape elements of three heights (*r* = 14 and 28 cm) fastened to the flume bed. Note that the elements were oriented with a corner facing the approaching flow field. The experimental observations indicate that this orientation of elements increases the flow turbulence and bed shear stresses; otherwise, the flow skimmed the elements, and the energy dissipation decreased. Also, the roughness elements are located in the depression form, where the channel bed within the expanding section was lowered as much as the height of the elements (e.g., Ead & Rajaratnam 2002). Another alternative for the elements is the block form without depression in the channel bed (e.g., Daneshfaraz *et al.* 2017). Although the depression form insignificantly changes the cross-section, it produces higher shear stress and energy dissipation than the block form (see Ead & Rajaratnam 2002).

### Dimensional analysis

The dimensional analysis through the Buckingham-π theorem (Buckingham 1915) was employed to identify the effective parameters, referred to as independent parameters. These parameters were used to predict the characteristics of the hydraulic jump (dependent variables) on an expanding stilling basin and rough bed. In this study, dependent variables were considered as the sequent depth (), the length of hydraulic jump (), the roller length of hydraulic jump (), and energy dissipation (), in which and are specific energy at upstream and downstream of the hydraulic jump.

*g*is the gravitational acceleration; is the kinematic viscosity of water; is the density of water;

*r*is the height of roughness elements; and and are the widths of the channel upstream and downstream of the hydraulic jump, respectively.

### Sugeno Fuzzy Logic (SFL)

Fuzzy logic (Zadeh 1988) is an appropriate data-driven model when there are uncertainties with noisy and imprecise datasets. The literature highlights its robustness to uncertainties (see Gonzalez *et al*. 1999; Sadeghfam *et al.* 2018), and in the case of this study, noisy datasets are common to hydraulic jumps due to the highly fluctuating free surface, intense turbulence and aeration, and corresponding measurement techniques (Hager 1993; Mossa 1999; Montano *et al.* 2018; Bung *et al.* 2021). To consider the turbulent hydraulic jump and corresponding data, membership functions (MFs) with a gradual transition between defined sets are used to represent the fuzzy sets according to the fuzzy theory developed by Zadeh (1996). Fuzzy models typically are implemented by three steps, which comprise (i) fuzzification; (ii) inference system based on the fuzzy rule; and (iii) defuzzification. These steps are implemented through the ‘genfis’ function in the MATLAB platform (for further information, see https://la.mathworks.com/help/fuzzy/genfis.html).

The present study employs an SFL model, in which the output MFs are constant or linear (Takagi & Sugeno 1985). Also, the subtractive clustering (SC) technique is utilized to identify rules and parameters of MFs as it is adapted in SFL (Chiu 1994). The cluster radius is an essential parameter in SC to control fuzzy rules; it varies between 0 and 1. Notably, a lower cluster radius renders a higher number of clusters and overly complicates system behavior. In contrast, a large cluster radius generates a lower number of clusters, which may result in decreased performance. Its optimum value is identified using a trial-and-error procedure until meeting the minimum root mean squared error (RMSE) between predicted and observed values.

*i*th cluster of the input ; is the MF of the

*i*th cluster of the input ; is the MF of the

*i*th cluster of the input ; is the MF of the

*i*th cluster of the input . Also, are coefficients for the

*i*th cluster. The final output

*Out*is the weighted average of all outputs as follows:where

_{j}*w*is firing strength for the rule

_{ij}*i*th and output

*j*, which is obtained using the ‘AND’ (minimize) operator.

### Performance metrics

#### Goodness-of-fit test

The performance of regression equation and SFLs is evaluated using the Nash–Sutcliffe Efficiency (NSE) and RMSE at calibration/training and validation/testing phases. The perfect performance of a model has the Nash–Sutcliffe Coefficient (NSC) values close to 1, and RMSE is close to 0, whereas poor performances are reflected in negative NSC values and positive RMSE values.

#### Residual analysis

In addition to the goodness-of-fit test, the residual analysis provides a deeper insight into the modeling performance. The homoscedasticity test analyzes residual errors for the model outputs based on the null hypothesis, referred to as the homoscedasticity of residuals at a given level of significance. Notably, residual errors refer to differences between the observed and predicted values. A perfect model has homoscedastic residuals, which render constant variance for residual errors, and it means that models obtain all the information from an input dataset. Whereas heteroscedastic residual for the rejected null hypothesis denotes that further investigations are required for understanding possible trends, seasonality, and noise in data.

## RESULTS

The total number of 72 experiments were carried out by the combination of three independent parameters, which comprise 6 ≤ *Fr*_{1} ≤ 12.5, the relative height of the roughness elements (*r*/*y*_{1} = 0.67 and 1.33), and the expansion ratio of walls (*B* = *b*_{1}/*b*_{2} = 0.4, 0.6, 0.8, 1). The experimental data were used for predicting hydraulic jump characteristics, specifically (i) the sequent depth ratio *y*_{2}/*y*_{1}; (ii) the relative length of hydraulic jump *L _{j}*/

*y*

_{1}; (iii) the relative roller length of the hydraulic jump

*L*/

_{r}*y*

_{1}; and (iv) the relative energy dissipation

*E*/

_{L}*E*

_{1}. The data relating to these characteristics were predicted using the regression-based models and SFLs. In addition to these characteristics, the water surface profile is also predicted. The experimental data were divided randomly into the calibration (or training) dataset and validation (or testing) dataset with a ratio of 70–30. Notably, the regression equation is fitted by the SPSS (Statistical Package for the Social Sciences) software.

### Sequent depth ratio

Figure 2(a)–2(c) illustrates the behavior of the sequent depth ratio *y*_{2}/*y*_{1} versus *Fr*_{1}, *r*/*y*_{1}, and *B*, which can be summarized as follows:

- (i)
*y*_{2}/*y*_{1}increases by increasing*Fr*_{1}, in which this behavior is observed in all expansion ratios with and without roughness elements (see Figure 2(a)); - (ii)
*y*_{2}/*y*_{1}decreases as the height of the roughness elements increases (see Figure 2(b)). Visual observations indicate that the flow separation and recirculation vortex are formed between the roughness elements and by increasing the height of roughness elements, the recirculation vortexes are developed (further details are available in Hassanpour*et al.*(2017)); - (iii)
*y*_{2}/*y*_{1}increases by increasing the expansion ratio (see Figure 2(c)). The experimental results show that*y*_{2}/*y*_{1}for the hydraulic jump on gradually expanding stilling basins was smaller than that of the corresponding classic hydraulic jump on a rectangular stilling basin (*B*= 1). In addition, by increasing the expansion ratio, flow separation and consequently flow turbulence increase, therefore*y*_{2}/*y*_{1}increases (further details are available in Hassanpour*et al.*(2021)).

A first-order polynomial equation was fitted to the experimental data and is presented in Table 2 (see Row 1). Notably, the type of fitted equation was determined using trial-and-error and literature (e.g., Torkamanzad *et al.* 2019). An SFL model was trained by the same data as per Equation (2a) with a cluster radius of 0.9 and the Gaussian MF. Figure 2(d) and 2(e) compare the experimental and predicted values of the subsequent depth ratio by scattering diagram and residual errors for the regression model and SFL. SFL has lower dispersity across the agreement lines compared with the regression model. Notably, the agreement line in Figure 2(d) is the solid line with the slope of 1:1, and in Figure 2(e) it is the horizontal solid line. A closer glimpse at Figure 2(d) indicates that the predicted values by the regression model located between the lines have ±15% error, whereas the corresponding error for SFL is ±6%. Also, the high residual values in the regression model are observed in both positive and negative ranges of residual, whereas residual values are reduced significantly by SFL (see Figure 2(e)). The residual values for SFL are generally lower than ±0.3, while the residual error above ±0.5 is evident for the regression model. Table 2 (Row 1) represents the performance metrics for the regression model and SFL in terms of NSE and RMSE for both training and testing phases. These results provide evidence that SFL significantly improves prediction accuracy compared with the regression model in both training and testing phases.

Hydraulic jump characteristics . | Model . | NSE . | RMSE . | ||
---|---|---|---|---|---|

Training . | Testing . | Training . | Testing . | ||

Subsequent depth ratio (Row 1) | Regression eq. | 0.950 | 0.925 | 0.406 | 0.462 |

SFL | 0.995 | 0.983 | 0.134 | 0.220 | |

The fitted regression equation: | |||||

Relative length of the hydraulic jump (Row 2) | Regression eq. | 0.883 | 0.832 | 6.627 | 6.551 |

SFL | 0.982 | 0.949 | 2.566 | 3.610 | |

The fitted regression equation: | |||||

Relative roller length (Row 3) | Regression eq. | 0.952 | 0.870 | 2.975 | 4.333 |

SFL | 0.996 | 0.981 | 0.882 | 1.651 | |

The fitted regression equation: | |||||

Relative energy loss (Row 4) | Regression eq. | 0.969 | 0.973 | 0.008 | 0.009 |

SFL | 0.995 | 0.951 | 0.003 | 0.013 | |

The fitted regression equation: | |||||

Water surface profile (Row 5) | Regression eq. | 0.779 | 0.793 | 0.093 | 0.093 |

SFL | 0.987 | 0.966 | 0.023 | 0.038 | |

The fitted regression equation: |

Hydraulic jump characteristics . | Model . | NSE . | RMSE . | ||
---|---|---|---|---|---|

Training . | Testing . | Training . | Testing . | ||

Subsequent depth ratio (Row 1) | Regression eq. | 0.950 | 0.925 | 0.406 | 0.462 |

SFL | 0.995 | 0.983 | 0.134 | 0.220 | |

The fitted regression equation: | |||||

Relative length of the hydraulic jump (Row 2) | Regression eq. | 0.883 | 0.832 | 6.627 | 6.551 |

SFL | 0.982 | 0.949 | 2.566 | 3.610 | |

The fitted regression equation: | |||||

Relative roller length (Row 3) | Regression eq. | 0.952 | 0.870 | 2.975 | 4.333 |

SFL | 0.996 | 0.981 | 0.882 | 1.651 | |

The fitted regression equation: | |||||

Relative energy loss (Row 4) | Regression eq. | 0.969 | 0.973 | 0.008 | 0.009 |

SFL | 0.995 | 0.951 | 0.003 | 0.013 | |

The fitted regression equation: | |||||

Water surface profile (Row 5) | Regression eq. | 0.779 | 0.793 | 0.093 | 0.093 |

SFL | 0.987 | 0.966 | 0.023 | 0.038 | |

The fitted regression equation: |

### Relative length of the hydraulic jump

Figure 3(a)–3(c) shows the behavior of the relative length of the hydraulic jump *L _{j}*/

*y*

_{1}versus

*Fr*

_{1},

*r*/

*y*

_{1}, and

*B*, which can be outlined as follows:

- (i)
*L*/_{j}*y*_{1}increases as the inflow Froude number increases. This trend is observed in all expansion ratios with and without roughness elements (see Figure 3(a)). - (ii)
*L*/_{j}*y*_{1}decreases by increasing the height of roughness elements (see Figure 3(b)). The primary role of the roughness elements in an expanding stilling basin can be regarded as a way to stabilize the hydraulic jump to prevent it from moving downstream. Thus,*L*decreases by increasing the height of roughness elements._{j} - (iii)
According to Figure 3(c) and experimental observations in the expansion ratio of

*B*= 0.8 and 0.6, the*L*increases compared with_{j}*B*= 1 and 0.4, because the hydraulic jump moves downstream of the stilling basin due to the flow turbulence in the expanding stilling basin. Therefore, the*L*is greater than the expansion ratio of_{j}*B*= 1 and 0.4.

A first-order polynomial equation fitted to the data related to the relative length of the hydraulic jump *L _{j}*/

*y*

_{1}is presented in Table 2 (see Row 2). Also, an SFL model was trained by the same datasets as per Equation (2b) with a cluster radius of 0.85 and the Gaussian MF. Figure 3(d) and 3(e) compare the experimental and predicted values of the relative length of the hydraulic jump by scattering diagram and residual errors for the regression model and SFL. According to the figure, the predicted data by the SFL model are close to the line of agreement compared with the regression model. Furthermore, Figure 3(d) illustrates that the predicted values by the regression model are within a ±22% margin of the measured values. In contrast, the data predicted by the SFL model are close to the line of agreement with a maximum error of ±14% from the line of agreement. Also, the high residual values in the regression model are observed in both positive and negative ranges of residual, whereas residual values are reduced significantly by SFL (see Figure 3(e)). The regression model has minimum and maximum residual values of −12.8 to 11.4, respectively, while these are in the range of −6.83 to 9.4 for the SFL. Table 2 (Row 2) presents the performance metrics for the regression model and SFL in terms of NSE and RMSE. The results prove that SFL significantly improves prediction accuracy compared with the regression model in both training and testing phases.

### Relative roller length of the hydraulic jump

Figure 4(a)–4(c) illustrate the behavior of the relative roller length of the hydraulic jump *L _{r}*/

*y*

_{1}versus

*Fr*

_{1},

*r*/

*y*

_{1}, and

*B*, which can be summarized as follows:

- (i)
A clear behavior can easily be observed among all the data, in which the relative roller length of the hydraulic jump increases by increasing the inflow Froude number. This behavior is observed in all expansion ratios with and without roughness elements (see Figure 4(a)).

- (ii)
Figure 4(b) shows that the relative roller length of the hydraulic jump on the rough bed in all expansion ratios decreases as the height of the roughness elements increases. According to the visual observations, installing roughness elements stabilizes the hydraulic jump on the stilling basin and causes the relative roller length of the hydraulic jump to be decreased.

- (iii)
According to Figure 4(c) and experimental observations in expansion ratios of

*B*= 0.8 and 0.6, the roller length increases compared with*B*= 1 and 0.4. By increasing the expansion ratio, flow separation and flow turbulence increase in expansion ratios of*B*= 0.8 and 0.6. These turbulences cause the hydraulic jump transmitted to the downstream of the stilling basin. Therefore, the roller length of the hydraulic jump is greater than that on the expansion ratio of*B*= 1 and 0.4 (further details are available in Hassanpour*et al.*2017).

Table 2 (see Row 3) presents the fitted quadratic equation to the data related to *L _{r}*/

*y*

_{1}. Also, an SFL model was trained using the same datasets as per Equation (2c) with a cluster radius of 0.8 and the Gaussian MF. According to the NSC and RMSE values in the table, SFL performs better than the regression models. A considerable improvement in the performance of SFL is presented in the testing phases. Figure 4(d) and 4(e) compare the experimental and predicted values of

*L*/

_{r}*y*

_{1}by scattering diagram and residual errors for the regression model and SFL. According to Figure 4(d) and 4(e), the predicted data by the SFL model are close to the line of agreement compared with the regression model. According to Figure 4(d), the predicted values by the regression equation lie between two lines with ±30% error, while this error for the SFL model is ±11%. Also, Figure 4(e) shows that the minimum and maximum residual values in the regression model range from −8.9 to 7.9, respectively, which are in the range of −4.12 to 2.4 for the SFL model. These results indicate that the SFL model has less dispersity and is more accurate in the estimation of

*L*than the regression model.

_{r}### Relative energy dissipation

Figure 5(a)–5(c) illustrates the behavior of the relative energy dissipation *E _{L}*/

*E*

_{1}versus

*Fr*

_{1},

*r*/

*y*

_{1}, and

*B*, which can be summarized as follows:

- (i)
*E*/_{L}*E*_{1}increases by increasing the inflow Froude number for all expansion ratios with and without roughness elements (see Figure 5(a)). - (ii)
*E*/_{L}*E*_{1}for the same inflow Froude number decreases as the height of the roughness elements increases (see Figure 5(b)). Further details are available in Hassanpour*et al.*(2017). - (iii)
Figure 5(c) indicates that

*E*/_{L}*E*_{1}increased by decreasing*B*. The experimental results show that the stilling basin with an expansion ratio of*B*=0.4 is more effective in terms of energy dissipation. This refers to the lateral force, high turbulence, and rolling flow along the longitudinal section (further details are available in Hassanpour*et al.*2017).

An equation is fitted in the form of a combination of the logarithmic and quadratic equation to the experimental data, and the derived equation is presented in Table 2 (see Row 4). Also, an SFL model was trained by the same data as per Equation (2d) with a cluster radius of 0.9 and the Gaussian MF. Figure 5(d) and 5(e) compares the experimental and predicted values of the relative energy dissipation by scattering diagram and residual errors for the regression model and SFL. The figure shows that SFL has lower dispersity across the agreement lines compared with the regression model. From the comparison between the results in Figure 5(d), the predicted values by the regression model located between the lines with ±3% error. In contrast, the corresponding error for SFL is ±4%. Figure 5(e) illustrates that the minimum and maximum residual values in the regression model are in the range of −0.026 to 0.014. These values are in the range of −0.028 to 0.012 for the SFL model and indicate an insignificant difference between the results of the two models. However, a close look suggests that SFL has less dispersity in the residual error diagram. Table 2 (Row 4) represents the performance metrics for the regression model and SFL in terms of NSE and RMSE for both training and testing phases. These results provide evidence that SFL improves prediction accuracy, despite the fit-for-purpose prediction by the regression model in the training and testing phases.

### Water surface profile

Figure 6(a)–6(c) indicate the behavior of the water surface and corresponding profile in the expanding stilling basin versus *Fr*_{1}, *r*/*y*_{1}, and *B*, which can be summarized as follows:

- (i)
A behavior can easily be observed in which the water surface increases by increasing

*Fr*_{1}. This behavior is observed in all*B*ratios regardless of the effect of roughness elements (see Figure 6(a)). - (ii)
Figure 6(b) shows that the water surface on the rough bed in all

*B*ratios decreases as*r*of the roughness elements increases. According to the experimental data, installing roughness elements stabilizes the hydraulic jump on the stilling basin and decreases the water surface fluctuations. - (iii)
The water surface increases by increasing

*B*(see Figure 6(c)). The results show that the water surface for the hydraulic jump on gradually expanding stilling basins was smaller than the classical hydraulic jump in a rectangular stilling basin because the gradual expansion walls provide a greater width for the flow and decrease the flow turbulence (further details are available in Hassanpour*et al.*2021).

An equation selected in the form of a combination of the logarithmic and quadratic equation is fitted to predict the water surface profile. The derived equation is presented in Table 2 (see Row 5). The same data trained the SFL model as per Equation (2e) with a cluster radius of 0.88 and the Gaussian MF. Figure 6(d) and 6(e) compare the experimental and predicted values of the water surface profile by scattering diagram and residual errors for the regression model and SFL. As shown, the SFL has lower dispersity across the agreement lines compared with the regression model. The results in Figure 6(d) show that the predicted values by the regression model are within a ±32% margin of the measured values. In contrast, the predicted data by the SFL model are close to the line of agreement with a maximum error of ±28% from the line of agreement. Figure 6(e) shows that the minimum and maximum residual error values in the regression model are in the range of −0.336 to 0.326, respectively, while these values are in the range of −0.137 to 0.186 for the SFL model. A close look at the result indicates that SFL has less dispersity in the residual error diagram. Table 2 (Row 5) represents the performance metrics for the regression model and SFL in terms of NSE and RMSE for both training and testing phases. The figure and the table provide evidence that SFL improves the accuracy of prediction, despite the fit-for-purpose prediction by the regression model.

### Model comparison in terms of the homoscedasticity test

In the study, the homoscedasticity test is applied based on the MATLAB platform (R2015b, MathWorks) at a significance level of 5%. Table 3 presents some statistical features of residuals for the regression models and SFLs and shows the results of the homoscedasticity test by the Breusch–Pagan and White tests. The table indicates that SFL significantly reduces the standard deviation of residuals compared with the regression models for the first three models (subsequent depth, jump length, and roller length). This means that the dispersity of SFL residuals has become narrower across the agreement line with zero residual error (see also residual diagrams in Figures (3)–(5)). A similar improvement is observed in other statistical features such as min, max, and mean. According to the Breusch–Pagan and White tests, SFLs provide homoscedastic residual, while regression equations present heteroscedastic results. Therefore, some information has been retained in the residual errors of regression equations, unlike the SFLs. Notably, the regression model for predicting the roller length is homoscedastic in terms of the Breusch–Pagan Test, but it is heteroscedastic in terms of the White Test, which is signified by mixed performance in the table.

Models . | Min . | Max . | Mean . | Std. . | Breusch–Pagan Test . | White Test . | Comments . | |||
---|---|---|---|---|---|---|---|---|---|---|

p-value
. | Risk% . | p-value
. | Risk% . | |||||||

Subsequent depth | Reg. | −0.802 | 1.355 | 0.000 | 0.427 | 0.009 | 0.89 | < 0.0001 | 0.001 | Heteroscedastic |

SFL | −0.502 | 0.339 | 0.001 | 0.166 | 0.762 | 76.18 | 0.954 | 95.38 | Homoscedastic | |

Jump length | Reg. | −13.527 | 20.757 | 0.088 | 6.650 | 0.002 | 0.17 | < 0.0001 | 0.001 | Heteroscedastic |

SFL | −6.833 | 9.447 | 0.137 | 2.942 | 0.134 | 13.4 | 0.122 | 12.16 | Homoscedastic | |

Roller length | Reg. | −8.906 | 7.934 | −0.422 | 3.446 | 0.621 | 62.11 | 0.008 | 0.76 | ^{a}Mixed performance |

SFL | −4.125 | 2.391 | −0.103 | 1.175 | 0.963 | 96.35 | 0.781 | 78.05 | Homoscedastic | |

Energy loss | Reg. | −0.026 | 0.014 | −0.002 | 0.008 | < 0.0001 | 0.001 | < 0.0001 | 0.001 | Heteroscedastic |

SFL | −0.028 | 0.012 | −0.001 | 0.008 | 0.248 | 24.76 | 0.217 | 21.74 | Homoscedastic | |

Water surface profile | Reg. | −0.490 | 0.326 | 0.001 | 0.093 | < 0.0001 | 0.001 | < 0.0001 | 0.001 | Heteroscedastic |

SFL | −0.325 | 0.186 | 0.001 | 0.028 | < 0.0001 | 0.001 | < 0.0001 | 0.001 | Heteroscedastic |

Models . | Min . | Max . | Mean . | Std. . | Breusch–Pagan Test . | White Test . | Comments . | |||
---|---|---|---|---|---|---|---|---|---|---|

p-value
. | Risk% . | p-value
. | Risk% . | |||||||

Subsequent depth | Reg. | −0.802 | 1.355 | 0.000 | 0.427 | 0.009 | 0.89 | < 0.0001 | 0.001 | Heteroscedastic |

SFL | −0.502 | 0.339 | 0.001 | 0.166 | 0.762 | 76.18 | 0.954 | 95.38 | Homoscedastic | |

Jump length | Reg. | −13.527 | 20.757 | 0.088 | 6.650 | 0.002 | 0.17 | < 0.0001 | 0.001 | Heteroscedastic |

SFL | −6.833 | 9.447 | 0.137 | 2.942 | 0.134 | 13.4 | 0.122 | 12.16 | Homoscedastic | |

Roller length | Reg. | −8.906 | 7.934 | −0.422 | 3.446 | 0.621 | 62.11 | 0.008 | 0.76 | ^{a}Mixed performance |

SFL | −4.125 | 2.391 | −0.103 | 1.175 | 0.963 | 96.35 | 0.781 | 78.05 | Homoscedastic | |

Energy loss | Reg. | −0.026 | 0.014 | −0.002 | 0.008 | < 0.0001 | 0.001 | < 0.0001 | 0.001 | Heteroscedastic |

SFL | −0.028 | 0.012 | −0.001 | 0.008 | 0.248 | 24.76 | 0.217 | 21.74 | Homoscedastic | |

Water surface profile | Reg. | −0.490 | 0.326 | 0.001 | 0.093 | < 0.0001 | 0.001 | < 0.0001 | 0.001 | Heteroscedastic |

SFL | −0.325 | 0.186 | 0.001 | 0.028 | < 0.0001 | 0.001 | < 0.0001 | 0.001 | Heteroscedastic |

^{a}The model is homoscedastic in terms of the Breusch–Pagan Test, but it is heteroscedastic in terms of the White Test.

Table 3 indicates that the statistical features of models for predicting energy loss are approximately identical for the regression model and SFL. The goodness-of-fit test in Table 2 (Row 4) also confirms this identical performance. However, the homoscedasticity test results show heteroscedastic residual for the regression model but homoscedastic residual for SFL. In this case, the regression equation cannot extract all the information in input data, and SFL has a better performance.

The comparison between the regression equation and SFL for predicting water surface profile indicates that the statistical features of residuals for SFL are considerably better than the regression model, so that the standard deviation for SFL residuals is about one-third of the regression equation. Also, the result of the goodness-of-fit test in Table 2 (Row 5) confirms that SFL performs better than the regression model. However, the result of the homoscedasticity test provides evidence that both models are heteroscedastic. A closer look at the SFL residual indicates no trend in the residual, but the heteroscedasticity stems from the noise in data due to water surface oscillation, which is not solved despite the performance improvement.

## DISCUSSION

Predicting hydraulic jump characteristics in complicated geometries is an ongoing research activity. Any complexity in the flow geometry makes the classic equation of hydraulic jumps invalid because the assumptions used to derive these equations are no longer valid. One of the most crucial assumptions is disruption of the one-dimensional nature of the flow, and consequently, the water surface oscillation, turbulence, and the air–water interface. In this case, AI models can predict hydraulic jump characteristics with acceptable performance. The type of complexity is broad and can be categorized as cross-sectional complexities in a compound channel (Khozani *et al.* 2019) and circular cross-section (Roushangar *et al.* 2021), some appurtenant as complexity on ordinary channels such as flow passing through screens (Sadeghfam *et al.* 2019), and complexity in the channel bed such as rough (Zounemat-Kermani & Mahdavi-Meymand 2021) or movable bed (Najafzadeh *et al.* 2017). However, the trained AI models in these complexities are case-specific, and training and testing phases should be carried out if the case of study is changed.

There is no theoretical basis for selecting data-driven models, and the study aims to compare the regression models as one of the classic approaches, specifically the SFL, which is an AI model with a capability of managing uncertainty and imprecise data. The uncertainty associated with the oscillation of water surface profile stems from roughness elements, hydraulic jump, and gradual expansion. Our laboratory measurements could not capture the contribution of each of these factors. However, the effect of these factors was reflected in water fluctuations, affecting the quality of the prediction of the studied parameters.

There are limited studies in the literature related to uncertainty analysis with hydraulic jumps. However, the existing literature emphasizes the role of uncertainty with data or models in predicting hydraulic jump characteristics (Macián-Pérez *et al.* 2020; Roushangar *et al.* 2022). Roushangar *et al.* (2022) carried out an uncertainty analysis based on Monte Carlo simulation to capture the uncertainty in the parameters of AI models and select the more appropriate model. Although the present papers do not quantify uncertainty, it employs the SFL model capable of managing uncertain data. Given the importance of uncertainty with both data and models, further investigations can be carried out to gain a deeper insight into the uncertainty by using different statistical techniques or AI models.

Although the regression equations presented herein are not novel techniques and this approach is limited by the supporting data ranges and values, the explicitly derived equations can be employed in future studies or practical problems. Also, some regression equations’ performances are close to SFL (see the prediction of energy dissipation). Meanwhile, a significant difference in the performance of models was also observed in predicting the relative length of the hydraulic jump or the water surface profile. However, future studies can consider further models regarding the complexities in flow induced by appurtenance in gradually expanding stilling basins.

To the best of the authors’ knowledge, this study is a first attempt to predict water surface profiles by AI models through stilling basins with gradually expanding walls. However, water surface profiles are widely investigated by mathematical modeling (e.g., Bayon *et al.* 2016). After testing different forms of equations, the parabolic form was selected for the regression equations. Due to severe oscillation in the water surface and the complicated flow geometry, the regression equations had not achieved higher accuracy. Still, SFL managed the uncertainty and significantly increased the prediction accuracy.

Unlike the regression equations, the homoscedastic test indicated that SFLs provided homoscedastic results in predicting the subsequent depth, jump length, roller length, and energy dissipation. However, the results of the water surface profile prediction are still heteroscedastic for both the regression equation and SFL. To resolve the heteroscedasticity in the result, authors undertook some actions as follows: (i) conducting different outlier tests; (ii) training different AI models such as ANN and SVM; and (iii) implementing a modeling strategy based on the inclusive multiple modeling (see Sadeghfam *et al.* 2019). However, the results were still heteroscedastic. The authors believe that this is attributable to the water surface oscillations, despite using an ultrasonic gauge for water surface measurement. There is room to eliminate the noisy signals by using more advanced statistical techniques on data or utilizing other devices to measure water surface profiles with more accuracy.

The range of dependent variables and independent parameters identify the limitations of the study. For example, the independent parameters vary in different ranges as follows: (i) 6 ≤ *Fr*_{1} ≤ 12; (ii) 0 ≤ *r* ≤ 2.8; and (iii) the 0.4 ≤ *B* ≤ 1. Although the regression equations and SFLs predict the hydraulic jump characteristics, they are data-driven and results highly depend on the ranges of independent parameters. Therefore, there is room to extend the range of experimental data and consequently increase the extent of results for practical purposes.

## CONCLUSION

The paper experimentally investigates the hydraulic jump behavior through stilling basins with gradual expanding sidewalls and roughness elements. Five dependent variables were derived through dimensional analysis to describe the hydraulic jump behavior, which comprises (i) the sequent depth ratio; (ii) the relative length of the hydraulic jump; (iii) the relative roller length of the hydraulic jump; (iv) the relative energy dissipation; and (v) the water surface profile. The dimensional analysis also demonstrated that the dependent variables are as a function of three independent parameters as follows: (i) Froude number of supercritical flow (*Fr*_{1}); (ii) the height of the roughness elements; and (iii) the expansion ratio. The experimental results describe the behavior of the dependent variables as a function of the independent parameters. The results showed that the dependent variables increase by increasing the supercritical *Fr*. Also, the dependent variables decrease by increasing the height of the roughness elements except for the relative energy dissipation. The dependent variables versus the expansion ratio showed a combined behavior of increasing or decreasing.

The paper predicted the behaviors of dependent variables using the regression equations and SFL. A set of equations were derived through the regression analysis by combinations of the logarithmic and quadratic equations. Also, different SFL models were trained and tested for the same data used for the regression analysis. The results indicated that the trained SFLs predicted the behavior of the dependent variables with the NSC greater than 0.96 in the testing phase, but the NSC values for the regression models were greater than 0.79. The higher performance of SFL refers to its capability in managing uncertainty and imprecise data owing to water profile oscillations. Also, the residual analysis by the homoscedasticity test indicated that the prediction residuals for SFL were homoscedastic except for the water surface profile, whereas the results by the regression equations were heteroscedastic.

## CONFLICT OF INTEREST STATEMENT

The authors have no conflicts of interest.

## DATA AVAILABILITY STATEMENT

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

## REFERENCES

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*Die turbulente Strömung im freien Wechselsprung*