## Abstract

A multi-segment sharp-crested V-notch weir (SCVW) was used both theoretically and experimentally in this study to evaluate the length of the hydraulic jump at the downstream of the weir. For this aim, a SCVW with three triangular segments at different tail-water depths (tailgate angles), and ten different discharges at a steady flow condition were investigated. Then, the most effective parameters on the length of the hydraulic jump are defined and several parametric and nonparametric regression models, namely multi-linear regression (MLR), additive non-linear regression (ANLR), multiplicative non-linear regression (MNLR), and generalized regression neural network (GRNN) models are compared with two semi-empirical regression models from the literature. The results indicate that the GRNN model is the best model among the selected models. These results are also linked to the nature of the hydraulic jump and the turbulent behavior of the phenomenon, which masks the experimental results with outliers.

## NOMENCLATURE

*B*Channel width (m);

*Fr*_{1}Approach Froude number [–];

*Fr*_{2}Froude number after hydraulic jump [–];

*L*Crest width (m);

*L*_{j}Length of the hydraulic jump (m);

*L*_{j}/BRelative length of the hydraulic jumps [–];

*P*_{1}Crest height (m);

*P*_{2}Height of triangular section (m);

*P*_{3}Weir height (m);

*Q*Discharge (lps);

*R*Correlation coefficient [–];

*R*^{2}Determination coefficient [–];

*Re*_{1}Incoming Reynolds number [–];

*Re*_{2}Reynolds number after hydraulic jumps [–];

*S*_{0}Slope of main channel bed [–];

*β*Intercept in MLR and ANLR;

*C*and_{i}*ξ*_{i}Coefficients assigned for ANLR and MNLR;

*n*Number of independent variables;

*N*Number of data set;

*y*_{i}Estimated

*i*th value for the hydraulic jump;*μ*_{y}Average of the estimated values for the hydraulic jump;

*σ*_{y}Standard deviation of the estimated values;

*g*Acceleration due to gravity (m/s

^{2});*h*Water head over the crest of weirs (mm);

*n*Roughness of the main channel (s/m

^{1/3});*y*_{1}Depth of flow before the hydraulic jumps (mm);

*y*_{2}Depth of flow after the hydraulic jumps (mm);

*y*_{c}Critical depth in open channel (mm);

*ɛ*Roughness of the surface (s/m

^{1/3});*η*Efficiency of hydraulic jumps [–];

*ɸ*Tailgate angles (degree);

*μ*Dynamic viscosity of the fluid (kg/s · m);

*v*_{1}Mean velocity of flow before hydraulic jumps (m/s);

*v*_{2}Mean velocity of flow after hydraulic jumps (m/s);

*ρ*Density of the water (kg/m

^{3});*σ*Surface tension (N/m);

*θ*Vertex angle (degree);

*α*_{i}Constants in MLR and ANLR;

*x*_{i}*i*th dependent variable in MLR, ANLR and MNLR;*b*Constant in MNLR;

*x*_{i}Observed

*i*th value for the hydraulic jump;*μ*_{x}Average of the observed values;

*σ*_{x}Standard deviation of the observed values.

## INTRODUCTION

The hydraulic jump is a natural phenomenon in which the entrained bubbles are advected into regions, collisions, and coalescence leading to larger air entities that are driven towards the free surface together with the combination of buoyancy and turbulent advection (Chanson 2007).

*et al.*2014). For instance, Gupta

*et al.*(2013) made an effort similar to the previous studies in order to obtain a semi-empirical model with optimized coefficients for defining the length of a hydraulic jump in horizontal channels regarding a rectangular cross-section as:where the

*L*

_{j}is calculated using the approach Reynolds number (

*Re*

_{1}), approach depth (

*y*

_{1}), and Froude number (

*Fr*

_{1}).

*L*

_{j}/

*y*

_{1}equation as:while due to the adequacy and applicability of the diagrams suggested by Bradley & Peterka (1957), Equations (1) and (2) were not widely used in practice. The United States Bureau of Reclamation (USBR 2016) has also introduced several equations for calculating the length of the hydraulic jump in stilling basins based on the length and the existence of any limiting wall of the stilling basin. Similarly, Hager (1992a, 1992b) and Wang & Chanson (2015) also proposed empirical laws related to the dimensionless jump length and the Froude number. In addition to the physical experiments, analytical development of jump roller length was reported by Valiani (1997), which became a point of interest for other researchers as well.

After the development of computers and fast calculating techniques, numerical models had a breakthrough and several techniques were developed in computational fluid dynamics (e.g. Gharehbaghi 2016, 2017; Gharehbaghi *et al.* 2017). The artificial neural network (ANN) is one of the evolutionary optimization techniques, which is used in many branches of science. Many researchers have used this technique in practice (e.g. Safari *et al.* 2016; Vaheddoost *et al.* 2016; Mehdizadeh *et al.* 2018; Valero & Bung 2018). For instance, Omid *et al.* (2005) developed an ANN approach for modeling the hydraulic jumps in rectangular and trapezoidal sections. Naseri & Othman (2012) announced the Levenberg–Marquardt (LM) as a more precise ANN technique to determine the *L*_{j} in a rectangular section with a horizontal apron. Houichi *et al.* (2013) used ANN techniques in the estimation of the length of the hydraulic jump in U-shaped channels. More recently, Safari *et al.* (2016), Khosravinia *et al.* (2018), and Azimi *et al.* (2018) used ANN techniques in open channel hydraulics to evaluate several phenomena like sediment motion and deposition.

In this respect, the aim and scope of this study is to (i) conduct experimental studies to investigate the behavior of the SCVW in open channels and (ii) develop several parametric and nonparametric regression models in comparison with several equations from the literature to define the credibility of physical and statistical approaches in practice.

## EXPERIMENTAL SET-UP AND MEASURING TECHNIQUES

The experimental parts of the present study were carried out on a multi-segment SCVW in the applied hydraulic laboratory of the Water Engineering Department in Urmia University (Ghaffari *et al.* 2011). The experimental studies were conducted over a model with three segments across a 1 m flume of width *L* and vertex angle *θ**=* 128*°,* in steady and free overflow conditions. A compound weir criterion of *P*_{2}/*h* < 1 was also applied over the model to satisfy the modeling conditions. Thus, *h* as the water head over the crest of the weir, *P*_{1} = 0.254 m as the crest height, *P*_{2}*=* 0.08 m as the height of the triangular cross-section, and *P*_{3}*=* 0.334 m as the weir height are used (data are given in Appendix A).

The characteristics of the model with *θ**=* 128° are given in Figure 1(a). Figure 1(a)–1(c) show the experiment that was conducted with a 3 mm steel plate with sharpened edges and supported by another 5 cm plate to avoid it from bending through overtopping of the flow and was located 3 m from the inlet of the channel at the downstream. Wave suppressors, grid walls, and a tailgate at the outlet were also used respectively to break the large eddies, dissipate the water surface disturbances, and control the water depths at the downstream. The jumps were initiated after the weir, whilst the end of the jump was observed at the point which was determined by the beginning of the jump plus the length of the jump, *L*_{j} (Bradley & Peterka 1957; Hager 1992a, 1992b). Figure 2 illustrates the location and length of the skewed hydraulic jump, while Table A1 in Appendix A shows the exact measurements. This length was measured using a ruler with a precision of ±5 mm. At the end of the jump, macro-scale vortices developed into the jump roller and interacted with the free surface, leading to air entrainment, splashes, and droplet formation in the two-phase flow region (Chanson & Brattberg 2000; Chanson 2007; Murzyn & Chanson 2009). The air entrainment occurred in the form of air bubbles and air packets entrapped at the impingement of the upstream jet flow with the roller (Chanson 2007). Based on experimental observations and Froude number, two types of hydraulic jump were formed. First was an undular jump (N_{Fr}: 1–3), which was not well formed, together with turbulence. The second one was a weak jump (N_{Fr}: 3–6), which occurred when the water velocity was low. It is important to mention that in this study, according to experimental observation, none of the formed hydraulic jumps were submerged.

### Dimensional analysis

Although Borghei *et al.* (1999) stated that the effects of *S*_{0}, *n*, *σ*, *ɛ*, and *μ* on *L*_{j} can be negligible since surface tension is very small in nappe heights (i.e. the minimum nappe height over the weir was taken as 20 mm), in the experimental study, the effect of *μ* was employed to investigate the effect of *Re*_{1} and *Re*_{2} on the results.

By using the Buckingham *π-*theorem and applying the properties of dimensional analysis, non-dimensional equations can be obtained as *Π*_{1} = *L*_{j}/*y*_{1}, *Π*_{2} = *v*_{1}^{2}/*gy*_{1}, *Π*_{3} = *v*_{2}^{2}/*gy*_{2}, *Π*_{4} = *ρ**v*_{1}*y*_{1}/*μ*, *Π*_{5} = *ρ**v*_{2}*y*_{2}/*μ*, *Π*_{6} = *y*_{2}/*B*, *Π*_{7} = *θ*.

### Variable selection and data analysis

*et al.*2013) and the dimensional analysis,

*Fr*

_{2},

*Re*

_{2},

*θ*,

*ɸ*, and

*y*

_{2}/

*B*are negligible and Equation (5) can be rewritten as

Since the aim of the study is to develop models to compare with equations obtained by Gupta *et al.* (2013) and Silvester (1965), in the rest of the study Equation (6) and its variables are used to study the length of the hydraulic jump, *L*_{j}.

A data set of 42 observations, recorded for *L*_{j}, *Fr*_{1}, *Re*_{1}, and *y*_{1}, were used (see Table A1 in Appendix A). To avoid misinterpretation, providing iso-dimensionality on both sides of the equation, and minimizing the precision, all data sets were normalized (Table 1). Figure 3(a) shows the nonlinear nature of the relationship between *Fr*_{1} and *y*_{1}. Similarly, in Figure 3(b) and 3(c) the relationship between *Re*_{1} is given compared with *y*_{1} and *Fr*_{1} respectively. In addition, Figure 3(d)–3(f) show the relationship between *L*_{j} compared with *y*_{1}, *Fr*_{1}, and *Re*_{1} respectively. In Figure 3(d), results are very similar to those obtained for Figure 3(a), and the relationship of *L*_{j} with *Fr*_{1} and *Re*_{1} in Figure 3(e)–3(f) seems quite complicated. Based on the results of Figure 3, it is obvious that a high precision in modeling would be inevitable.

Variable . | Mean (m) . | Standard deviation (m) . | Coefficient of variation (−) . | Skewness (−) . | Kurtosis (−) . |
---|---|---|---|---|---|

L_{j} | 0.15 | 0.28 | 1.89 | 1.99 | 3.19 |

Fr_{1} | 0.26 | 0.27 | 1.04 | 1.00 | 0.07 |

Re_{1} | 0.40 | 0.31 | 0.76 | 0.76 | −0.73 |

Y_{1} | 0.36 | 0.24 | 0.67 | 0.84 | 0.04 |

Variable . | Mean (m) . | Standard deviation (m) . | Coefficient of variation (−) . | Skewness (−) . | Kurtosis (−) . |
---|---|---|---|---|---|

L_{j} | 0.15 | 0.28 | 1.89 | 1.99 | 3.19 |

Fr_{1} | 0.26 | 0.27 | 1.04 | 1.00 | 0.07 |

Re_{1} | 0.40 | 0.31 | 0.76 | 0.76 | −0.73 |

Y_{1} | 0.36 | 0.24 | 0.67 | 0.84 | 0.04 |

It is noteworthy that the zero values associated with the *L*_{j} could be explained when the hydraulic jump is not formed. Therefore, the selected models also have to consider whether the jump is formed or not. As seen in Table 1, deviations from means caused a lot of variations. It is obvious that the *L*_{j} has the most unstable behavior by means of the highest coefficients of variation and skewness. Particularly, in Figure 3(d)–3(f) a lot of deviations occurred when *L*_{j} is expressed by *y*_{1}, *Fr*_{1}, or *Re*_{1}. In Figure 3, the dashed line (in black) as the linear relation of the data versus the continuous line (in red) as the second-degree polynomial fit of the data show several similarities, whilst the dissimilarities between linear and non-linear lines are too coarse to be detailed easily. Since non-excludable outliers were included (i.e. data related to hydraulic jump in Figure 3(d)–3(f); Table A1), outlier-resistant nonparametric approaches are also used in the analysis to deal with the rank of the data set.

## MODELING

*et al.*(2013). The nonparametric generalized regression neural network (GRNN) was also used along with the selected parametric models due to its non-linear outlier-resistance nature in function approximation. Models were developed to estimate the length of hydraulic jump based on

*Re*

_{1},

*Fr*

_{1}, and

*y*

_{1}as:

First, a set of 33 randomly selected observations (80% of the data) is used in model calibration and model approximation. Then, the remaining sets of the data with nine sets of observations (20%) for each variable, namely test data, is used in evaluation of the models. In this respect, several criteria are used to evaluate each model.

### Multi-linear regression (MLR)

Coefficients and the intercept for linear regression are obtained by minimizing the difference between the observed values and model outputs using the ordinary least squares method.

### Non-linear regression (NLR)

*et al.*(2016) introduced them as useful key models in defining non-linearity in water resource studies. These models respectively for additive (ANLR) and multiplicative (MNLR) regression are:

In this study, a Levenberg–Marquardt (LM) algorithm (Moré 1978) is used in coefficient estimation.

### Generalized regression neural network (GRNN)

*S*-summation and

*D*-summation. The third layer covers the output vector obtained for each input vector, which can be expressed as:

### Performance evaluation criteria

### Model grading

*et al.*(2016) is used. Hence, success grade (

*SG*) and failure grade (

*FG*) of the performance criteria are defined as:

The total grade of each model is obtained by adding up the *SG*s and *FG*s of each model individually, which can differ between +20 and −20.

## RESULTS AND DISCUSSION

### Results of the experimental studies

Based on the laboratory experiments, when using SCVWs, by increasing *Fr*_{1} and *Re*_{1}, the values of *L*_{j} would increase. Moreover, the current weir causes formation of additional blades in water jets at the downstream. Hence, by collision of the falling water jets at the downstream, *L*_{j} decreases, noticeably. As a result, by decreasing *L*_{j}, the length of the stilling basin, effects of the scour, and erosion phenomena in downstream channels will reduce. This advantage induces a decrease in the construction costs of the stilling basin. The interested reader may also refer to Saadatnejadgharahassanlou *et al.* (2017) for more details about the experimental study.

### Evaluation of models in train data (model fitting)

#### MLR model

#### NLR model

#### GRNN model

The GRNN model is also developed by a set of 12 different spread parameters. Figure 4 shows the results of different spread parameters on model results. The *PI* criterion gradually increased through different spread parameters from 1 to 0.2 and decreased afterward. Based on the results of the *PI* and *MAE*, a model with spread parameter 0.2 could be selected in GRNN calibration.

### Evaluation of models in test data

Results of all models including MLR, MNLR, GRNN, Silvester (1965) and Gupta *et al.* (2013) were compared using a scatter-plot (Figure 5), *R*^{2}, *CC*, *PI*, *RMSPE*, and the total grades obtained by Equations (17)–(20) (Table 2).

Model . | R^{2}
. | SG (R^{2})
. | CC
. | SG (CC)
. | PI
. | FG (PI)
. | RMSPE (%)
. | FG (RMSPE)
. | Total grade . |
---|---|---|---|---|---|---|---|---|---|

MLR | 0.25 | +2.55 | 0.29 | +3.24 | 0.75 | −9.76 | 131.59 | −7.75 | −11.71 |

MNLR | 0.31 | +3.16 | 0.27 | +3.00 | 0.63 | −9.83 | 132.80 | −7.77 | −11.44 |

GRNN | 0.98 | +10.00 | 0.88 | +10.00 | 1.01 | −0.00 | 29.62 | −0.00 | +20.00 |

Silvester | 0.73 | +7.48 | 0.83 | +9.34 | 1.81 | −9.92 | 101.94 | −7.09 | −0.20 |

Gupta | 0.28 | +2.84 | −0.53 | +5.93 | 0.35 | −9.91 | 250.35 | −8.82 | −9.94 |

Model . | R^{2}
. | SG (R^{2})
. | CC
. | SG (CC)
. | PI
. | FG (PI)
. | RMSPE (%)
. | FG (RMSPE)
. | Total grade . |
---|---|---|---|---|---|---|---|---|---|

MLR | 0.25 | +2.55 | 0.29 | +3.24 | 0.75 | −9.76 | 131.59 | −7.75 | −11.71 |

MNLR | 0.31 | +3.16 | 0.27 | +3.00 | 0.63 | −9.83 | 132.80 | −7.77 | −11.44 |

GRNN | 0.98 | +10.00 | 0.88 | +10.00 | 1.01 | −0.00 | 29.62 | −0.00 | +20.00 |

Silvester | 0.73 | +7.48 | 0.83 | +9.34 | 1.81 | −9.92 | 101.94 | −7.09 | −0.20 |

Gupta | 0.28 | +2.84 | −0.53 | +5.93 | 0.35 | −9.91 | 250.35 | −8.82 | −9.94 |

According to Figure 5 and Table 2, GRNN is the best model, whilst the results obtained by the Silvester (1965) equation are the second best. Figure 5 also shows that the GRNN and the Silvester (1965) models are eligible, while the results obtained by the other models have significant bias in estimation. On the other hand, the equation suggested by Gupta *et al.* (2013) has the worst results. Since the experiments are conducted on a SCVW, it is not obvious if the type of weir could cause deviation from the results obtained by Silvester (1965) or Gupta *et al.* (2013). Thereby, along with the data analysis given in Table 1 and Figures 3–5, the obtained results indicate strong deviations that mislead the parametric modeling in practice. The moments of the distribution and the parametric approaches seem to be less desirable in this approach. Due to the high accuracy and existence of outliers, nonparametric GRNN shows the best performance. Its superior performance can also be related to the higher number of individually taken input variables in a one-way pass classification that is lumped in parametric regression models. Thereby, along with the results of Specht (1991), it is obvious that the justification of linearity or non-linearity in the parametric approach is vital. In addition, GRNN is bounded to the minimum and maximum values of the observation data, which could not be achieved by the ANLR model when estimating negative values.

It is concluded that the GRNN as the representative of the nonparametric models is superior in comparison to those parametric approaches used in this study. Obviously, outliers associated with the hydraulic jump can only be carried out based on the associated rank of each observation in the data set (Table 1 and Table A1). The idea of the parametric reduction in this sense may need an update due to the high uncertainty associated with phenomena like hydraulic jump.

## CONCLUSIONS

In this study, a multi-segment sharp-crested V-notch weir, termed SCVW, has been used. Extensive laboratory experiments were conducted to measure the length of hydraulic jump at the downstream of the SCVW. It was observed that by increasing the approach *Fr* and *Re* the length of the hydraulic jump would increase and vice versa. As a result, by decreasing *L*_{j}, the length of the stilling basin, effects of the scour, and the erosion phenomena at the downstream of the channels will decrease, which also decreases the construction costs of the stilling basin.

In addition, three types of parametric regression models, (i.e. MLR, ANLR, and MNLR), two models from the literature (i.e. Silvester 1965; Gupta *et al.* 2013), and a non-parametric regression model (i.e. GRNN) are used in modeling. Evaluation of the models was conducted in two stages, training and testing. For this aim several performance criteria, namely *R*^{2}, *CC*, *PI*, *RMSPE*, *MAE*, and total grade are used. The results indicate that the GRNN model with a nonparametric regression approach is the superior model whilst the Silvester (1965) model is the second best. It is concluded that the high uncertainty and skewness associated with the length of *L*_{j}, together with the presence of outliers, is better expressed in nonparametric regression, i.e. GRNN. It is obvious that the rank of the data in each allocated set is more dependable than the moments of the distribution.

## ACKNOWLEDGEMENT

The authors appreciate the cooperation of Urmia University in conducting the experiments.

## CONFLICT OF INTEREST

None.

## SUPPLEMENTARY MATERIAL

The Supplementary Material for this paper is available online at https://dx.doi.org/10.2166/ws.2019.198.