## Abstract

Channels with different shapes and bed conditions are used as useful appurtenances to dissipate the extra energy of a hydraulic jump. Accurate prediction of hydraulic jump energy dissipation is important in design of hydraulic structures. In the current study, hydraulic jump energy dissipation was assessed in channels with different shapes and bed conditions (i.e. smooth and rough beds) using the support vector machine (SVM) as an intelligence approach. Five series of experimental datasets were applied to develop the models. The results showed that the SVM model is successful in estimating the relative energy dissipation. For the smooth bed, it was observed that the sloping channel models with steps performed more successfully than rectangular and trapezoidal channels and the step height is an effective variable in the estimation process. For the rough bed, the trapezoidal channel models were more accurate than the rectangular channel. It was found that rough element geometry is effective in estimation of the energy dissipation. The result showed that the models of rough channels led to better predictions. The sensitivity analysis results revealed that Froude number had the more dominant role in the modeling. Comparison among SVM and two other intelligence approaches showed that SVM is more successful in the prediction process.

## INTRODUCTION

For the transition of a supercritical flow into a subcritical flow in an open channel, the hydraulic jump phenomenon is used. Hydraulic jumps can occur downstream of hydraulic structures, such as normal weirs, gates and ogee spillways. It is considered as a rapidly varying flow, and this type of flow regime transformation is associated with severe turbulence and flow energy dissipation (Hager 1992). Based on the energy dissipating action of hydraulic jumps, a stilling basin is one of the possible solutions which may be adopted. In order to design an optimal hydraulic structure, different devices such as sills, baffle blocks, end sills, roughness elements, and roller buckets are used in hydraulic structures. However modeling hydraulic jump characteristics has great importance since it plays an important role in designing hydraulic structures. So far, hydraulic jumps have been extensively studied in order to explain the complex phenomenon of the hydraulic jump and to estimate its characteristics. Wanoschek & Hager (1989) investigated the internal flow features of hydraulic jumps in trapezoidal channels. Finnemore & Franzini (2002) stated that Froude number has significant impact on the characteristics of hydraulic jumps. Ayanlar (2004) investigated hydraulic jump properties in channels with corrugated beds. Bilgil (2005) collected some experiments in a channel with a smooth bed in order to investigate the distribution of shear stress for turbulent flow. However, due to the complexity and uncertainty of the hydraulic jump phenomenon, the classical models often do not show the desired accuracy and the application of many formulas are limited to special cases of their development. In fact, the physical-based approaches 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. These issues cause uncertainty in the prediction of energy dissipation; therefore, it is essential to use other methods which are more accurate in predicting hydraulic jump parameters such as energy dissipation.

The meta-model approaches such as artificial neural networks (ANNs), neuro-fuzzy models (NF), genetic programming (GP), and support vector machine (SVM), have been applied in investigating hydraulic and hydrologic complex phenomena in recent decades. Prediction of total bedload (Chang *et al.* 2012), prediction of daily river flow (Delafrouz *et al.* 2018), modeling flow resistance in open channels with dune bedforms (Roushangar *et al.* 2018), water level prediction (Ghorbani *et al.* 2018), and prediction of sediment transport in circular channels (Roushangar & Ghasempour 2017) are some examples of meta-model approach applications. The SVM method is based on the concept of an optimal hyperplane that separates samples of two classes by considering the widest gap between two classes. SVM was originally developed for binary decision problems. This classification method has also been extended to solve prediction problems. Artificial neural networks are computing systems inspired by the biological neural networks that constitute animal brains. Such systems learn (progressively improve performance) to do tasks by considering examples, generally without task-specific programming. Adaptive neuro-fuzzy inference systems (ANFIS) are considered as a kind of artificial neural network. In this system different functions are applied to express the conclusions. The basis of the fuzzy logic method is condition–result rules. ANFIS merges the principles of neural networks and fuzzy logic; therefore, in a single framework it can take the advantages of both methods.

The SVM, ANN and ANFIS techniques have been used for predicting various hydraulic and hydrologic phenomena. Lan (2014) applied SVM for long-term prediction of lake water levels. Wang (2016) used the SVM method for assessing raw water quality. Azamathulla *et al.* (2016) applied SVM for predicting side weir discharge coefficients. Talei *et al.* (2010) used the ANFIS method for modeling the rainfall–runoff process. Saxena & Yadav (2017) used ANFIS in capacity prediction for Ukai reservoir. Foddis *et al.* (2015) used ANN for the estimation of aquifer pollutant source behavior. In the present study, the relative energy dissipation is assessed in differently shaped channels (i.e. rectangular, trapezoidal, and sloping channel with step) with smooth and rough beds using the SVM as the effective meta-model approach. Different input combinations were considered to determine the most effective combination for predicting the relative energy dissipation. Then, a sensitivity analysis was performed in order to determine the most important variables in the prediction process. Also, the capability of the SVM approach was compared with two other meta-model approaches (i.e. ANN and ANFIS).

## MATERIALS AND METHODS

### Used datasets

The experimental data presented by Wanoschek & Hager (1989), Carollo *et al.* (2007), Sultana (2012), and Evcimen (2012) were used for prediction of relative energy dissipation. The Wanoschek & Hager (1989) experiments were intended for hydraulic jump in smooth trapezoidal channels. The experiments were conducted in a prismatic, symmetrical trapezoidal channel of 0.2 m base width and side slope of 45°. Carollo *et al.* (2007) studied hydraulic jump in smooth and rough rectangular channels. A total of 372 tests were performed under different hydraulic conditions. The Sultana (2012) experiments were performed using three different drop heights of 2, 4.5, and 6 cm. For each drop height, four channel slopes of 0, 0.0042, 0.0083 and 0.0125 and three gate openings of 6.5, 4.5, and 3.5 cm were used. The experiments of Evcimen (2012) were intended for hydraulic jumps in trapezoidal channels and the impact of prismatic roughness on hydraulic jump was assessed. The ranges of experimental data used in the experiments are given in Table 1. In this table *Fr*_{1} is downstream flow Froude number, *S* is space between rough elements, *Y* is sequence depth ratio and *H* is height of drop of rough elements. Also, the schematic view of the experiment channels is show in Figure A in the Appendix (available with the online version of this paper).

Researcher . | Channel type . | Parameters . | No. of data . | |||
---|---|---|---|---|---|---|

Fr_{1}
. | S (cm)
. | Y
. | H (cm)
. | |||

Carollo et al. (2007) | Smooth rectangular | 1.87–8.78 | – | 2.8–10 | – | 72 |

Rough rectangular | 0.1–10 | – | 2.8–10 | 0.46–3.2 | 300 | |

Evcimen (2012) | Rough trapezoidal | 3.92–13.28 | 2–10 | 4.15–14.9 | 1–3 | 107 |

Sultana (2012) | Sloping smooth channel with step | 0.109–0.381 | – | 4.5–6.9 | 2–6 | 108 |

Wanoschek & Hager (1989) | Smooth trapezoidal | 2.35–14.7 | – | 2.34–14 | – | 40 |

Researcher . | Channel type . | Parameters . | No. of data . | |||
---|---|---|---|---|---|---|

Fr_{1}
. | S (cm)
. | Y
. | H (cm)
. | |||

Carollo et al. (2007) | Smooth rectangular | 1.87–8.78 | – | 2.8–10 | – | 72 |

Rough rectangular | 0.1–10 | – | 2.8–10 | 0.46–3.2 | 300 | |

Evcimen (2012) | Rough trapezoidal | 3.92–13.28 | 2–10 | 4.15–14.9 | 1–3 | 107 |

Sultana (2012) | Sloping smooth channel with step | 0.109–0.381 | – | 4.5–6.9 | 2–6 | 108 |

Wanoschek & Hager (1989) | Smooth trapezoidal | 2.35–14.7 | – | 2.34–14 | – | 40 |

### Support vector machine

*ɛ*deviation from the actually obtained objectives for all training data

*y*and at the same time is as flat as possible. The SVR formulation is as follows: where

_{i}*φ*(

*x*) denotes a nonlinear function in feature of input

*x*,

*b*is called the bias and the vector

*w*is known as the weight. The coefficients of Equation (1) are predicted by minimizing the regularized risk function as expressed below: where The constant

*C*is the cost factor and represents the trade-off between the weight factor and approximation error;

*ɛ*is the radius of the tube within which the regression function must lie. represents the loss function in which

*y*is the forecast value and

_{i}*t*is the desired value in the period

_{i}*i*. Since some data may not lie inside the

*ɛ*-tube, the slack variables (

*ξ*,

*ξ**) must be introduced. These variables represent the distance from actual values to the corresponding boundary values of the

*ɛ*–tube. Therefore, it is possible to transform Equation (2) into:

*α*and

_{i}*α** are Lagrange multipliers and

_{i}*l*(

*α*,

_{i}*α**) represents the Lagrange function.

_{i}*K*(

*x*,

_{i}*x*) is a kernel function to yield the inner products in the feature space

_{j}*φ*(

*x*) and

_{i}*φ*(

*x*). In general, there are several types of kernel function, namely linear, polynomial, radial basis function (RBF) and sigmoid functions. It should be noted that the performance of the SVM is highly dependent on the choice of kernel as well as the kernel and cost parameters. In fact SVM prediction accuracy depends on a good setting of meta-parameters

_{j}*C*and

*ɛ*and the kernel parameters. SVM is a powerful tool that can be successfully applied to predict any variable of interest where: the interrelationships among the relevant variables are poorly understood, finding the size and shape of the ultimate solution is difficult and a major part of the problem, and conventional mathematical analysis methods do not (or cannot) provide analytical solutions.

### Adaptive neuro-fuzzy inference system

ANFIS was first introduced by Jang (1993). An ANFIS is a network structure consisting of a number of nodes connected through directional links. Each node is characterized by a node function with fixed or adjustable parameters. The learning or training phase of a neural network is a process to determine parameter values to sufficiently fit the training data. The basic learning rule is the well-known back-propagation method which seeks to minimize some measure of error, usually the sum of the squared differences between the network's outputs and the desired outputs. Depending on the types of inference operations upon ‘if–then rules’, most fuzzy inference systems can be classified into three types: Mamdani's system, Sugeno's system and Tsukamoto's system. Mamdani's system is the most commonly used, meanwhile, Sugeno's system is more compact and computationally efficient; the output is crisp, so, without the time-consuming and mathematically intractable defuzzification operation, it is by far the most popular candidate for sample-data-based fuzzy modeling and it lends itself to the use of adaptive techniques (Takagi & Sugeno 1985).

### Artificial neural networks

The ANN is a learning systems that has solved a large amount of complex problems related to different areas (classification, clustering, regression, etc.) (Cybenko 1989; Haykin 1998), and is a system loosely modeled on the human brain. The field goes by many names, such as connectionism, parallel distributed processing, neuro-computing, neural intelligent systems, machine learning algorithms and artificial neural networks. It is an attempt to simulate within specialized hardware or sophisticated software. This simulation is achieved through multiple layers of simple processing elements called neurons. Each neuron is linked to certain of its neighbors with varying coefficients of connectivity that represent the strengths of these connections. Learning is accomplished by adjusting these strengths to cause the overall network to output appropriate results. The parameters to be found by training are the weight vectors connecting the different nodes of the input, hidden, and output layers of the network by the so-called error-back-propagation method (a specialized version of the gradient-based optimization algorithm). During training the values of the parameters (weights) are varied so that the ANN output becomes similar to the measured output on a known dataset.

### Performance criteria

*R*), determination coefficient (DC), and root mean square error (RMSE), as depicted in Equation (6): where , , , ,

*N*respectively are: the measured values, predicted values, mean measured values, mean predicted values and number of data samples.

*x*

_{n},

*x*,

*x*

_{max},

*x*

_{min}respectively are: the normalized value of variable

*x*, the original value, the maximum and minimum of variable

*x*.

### Simulation and model development

#### Input variables

*et al.*(2007), the important variables which affect energy dissipation can be a function of the parameters: in which

*y*

_{1}and

*y*

_{2}: sequent depth of upstream and downstream,

*V*: upstream flow velocity,

*μ*: water dynamic viscosity,

*g*: gravity acceleration,

*L*: length of jump,

*ρ*: density of water,

*E*

_{j}(=E_{1}−

*E*

_{2}) in which

*E*

_{1}and

*E*

_{2}are energy per unit weight before and after the jump,

*H*: rough element or step height, and S: space between rough elements. Using dimensional analysis and considering

*y*

_{1},

*g*and

*μ*as repeating variables, Equation (9) can be expressed as follows:

According to the experimental studies by Elevatorski (2008) and Ranga Raju *et al.* (1980), hydraulic jump characteristics only depend on Froude number, and Reynolds number has no effective role in the prediction process. Therefore, in this study, the models of Table 2 were considered for modeling the energy dissipation in channels with different shapes and bed conditions. Figure 1 illustrates a flowchart of the simulation process considered in this study. It should be noted that for each state of bed condition (i.e. smooth and rough beds), 75% of the whole dataset was used for training the models and 25% of the data (which was from the same set) was used for testing the models. The orders of the datasets were selected in the way that the training dataset contains a representative sample of all the behavior in the data in order to obtain the model with higher accuracy. The quality of the training data is essential for the evolution of good solutions. To find a good training set which can give good accuracy both in training and testing sets, one method is instance exchange (Bolat & Yildirim 2004). The process starts with a randomly selected training set.

Smooth channels . | Rough channels . | ||
---|---|---|---|

Model . | Input variable(s) . | Model . | Input variable(s) . |

S(I) | Fr_{1} | R(I) | Fr_{1} |

S(II) | Fr_{1}, y_{2}/y_{1} | R(II) | Fr_{1}, y_{2}/y_{1} |

S(III) | Fr_{1}, (y_{2}-y_{1})/y_{1} | R(III) | Fr_{1}, (y_{2}-y_{1})/y_{1} |

S(V) | Fr_{1}, H/y_{1} | R(IV) | Fr_{1}, H/y_{1} |

R(V) | Fr_{1}, S/H |

Smooth channels . | Rough channels . | ||
---|---|---|---|

Model . | Input variable(s) . | Model . | Input variable(s) . |

S(I) | Fr_{1} | R(I) | Fr_{1} |

S(II) | Fr_{1}, y_{2}/y_{1} | R(II) | Fr_{1}, y_{2}/y_{1} |

S(III) | Fr_{1}, (y_{2}-y_{1})/y_{1} | R(III) | Fr_{1}, (y_{2}-y_{1})/y_{1} |

S(V) | Fr_{1}, H/y_{1} | R(IV) | Fr_{1}, H/y_{1} |

R(V) | Fr_{1}, S/H |

## RESULTS AND DISCUSSION

### Developed models for smooth channels

For determining the best performance of SVM and selecting the best kernel function, model S(III) of sloping channel with step was predicted via SVM using various kernels. Table 3 shows the results of the statistical parameters of different kernels. According to the results, using the kernel function of RBF led to better prediction accuracy in comparison with the other kernels. Therefore, the RBF kernel was used as the core tool of SVM, which was applied for the rest of the models.

Kernel function . | SVM model . | Performance criteria . | |||||
---|---|---|---|---|---|---|---|

Train . | Test . | ||||||

R . | DC . | RMSE . | R . | DC . | RMSE . | ||

Linear | S(III) sloping channel with step | 0.835 | 0.781 | 0.099 | 0.812 | 0.701 | 0.110 |

Polynomial | 0.915 | 0.823 | 0.069 | 0.908 | 0.740 | 0.078 | |

RBF | 0.969 | 0.931 | 0.042 | 0.949 | 0.905 | 0.064 | |

Sigmoidal | 0.567 | 0.109 | 0.232 | 0.446 | 0.101 | 0.248 | |

Channel type . | SVM models . | Performance criteria . | |||||

Train . | Test . | ||||||

R . | DC . | RMSE . | R . | DC . | RMSE . | ||

Rectangular | S(I) | 0.855 | 0.681 | 0.078 | 0.834 | 0.652 | 0.083 |

S(II) | 0.872 | 0.751 | 0.067 | 0.871 | 0.719 | 0.079 | |

S(III) | 0.879 | 0.756 | 0.065 | 0.876 | 0.721 | 0.069 | |

Trapezoidal | S(I) | 0.899 | 0.783 | 0.072 | 0.883 | 0.755 | 0.079 |

S(II) | 0.912 | 0.854 | 0.062 | 0.904 | 0.811 | 0.073 | |

S(III) | 0.938 | 0.869 | 0.059 | 0.939 | 0.829 | 0.068 | |

Sloping channel with step | S(I) | 0.905 | 0.812 | 0.058 | 0.898 | 0.805 | 0.076 |

S(II) | 0.925 | 0.885 | 0.053 | 0.923 | 0.816 | 0.070 | |

S(III) | 0.969 | 0.931 | 0.042 | 0.949 | 0.905 | 0.064 | |

S(IV) | 0.948 | 0.915 | 0.049 | 0.945 | 0.898 | 0.066 |

Kernel function . | SVM model . | Performance criteria . | |||||
---|---|---|---|---|---|---|---|

Train . | Test . | ||||||

R . | DC . | RMSE . | R . | DC . | RMSE . | ||

Linear | S(III) sloping channel with step | 0.835 | 0.781 | 0.099 | 0.812 | 0.701 | 0.110 |

Polynomial | 0.915 | 0.823 | 0.069 | 0.908 | 0.740 | 0.078 | |

RBF | 0.969 | 0.931 | 0.042 | 0.949 | 0.905 | 0.064 | |

Sigmoidal | 0.567 | 0.109 | 0.232 | 0.446 | 0.101 | 0.248 | |

Channel type . | SVM models . | Performance criteria . | |||||

Train . | Test . | ||||||

R . | DC . | RMSE . | R . | DC . | RMSE . | ||

Rectangular | S(I) | 0.855 | 0.681 | 0.078 | 0.834 | 0.652 | 0.083 |

S(II) | 0.872 | 0.751 | 0.067 | 0.871 | 0.719 | 0.079 | |

S(III) | 0.879 | 0.756 | 0.065 | 0.876 | 0.721 | 0.069 | |

Trapezoidal | S(I) | 0.899 | 0.783 | 0.072 | 0.883 | 0.755 | 0.079 |

S(II) | 0.912 | 0.854 | 0.062 | 0.904 | 0.811 | 0.073 | |

S(III) | 0.938 | 0.869 | 0.059 | 0.939 | 0.829 | 0.068 | |

Sloping channel with step | S(I) | 0.905 | 0.812 | 0.058 | 0.898 | 0.805 | 0.076 |

S(II) | 0.925 | 0.885 | 0.053 | 0.923 | 0.816 | 0.070 | |

S(III) | 0.969 | 0.931 | 0.042 | 0.949 | 0.905 | 0.064 | |

S(IV) | 0.948 | 0.915 | 0.049 | 0.945 | 0.898 | 0.066 |

For evaluating the relative energy dissipation in smooth channels with different shape, several models were development based on flow and channels geometry and analyzed with SVM. Table 3 and Figure 2 show the results of SVM models. From the obtained results of statistical parameters (*RMSE, R* and *DC*) it can be induced that between three types of channels, developed models for the case of sloping channel with step, in modeling of the relative energy dissipation ratio performed more successful than two other cases. For all cases, the model S(III) with input parameters of *Fr*_{1}, (*y*_{2}*−**y*_{1})/*y*_{1} led to more accurate outcome than the other models. A comparison between the results of the models S(I), S(II) and S(III) showed that using parameter *y*_{2}/*y*_{1} and *(y*_{2}*−**y*_{1})/*y*_{1} as input variables, caused an increment in models efficiency. The impact of parameter (*y*_{2}*−**y*_{1})/*y*_{1} confirms the importance of the jump height in the energy dissipation estimating process. Considering the results of the models S(I) and S(IV) in the case of channel with step, it could be inferred that *H/y*_{1} (relative height of the step) is effective variable in energy dissipation in this type of channels. Also, the model S(I) with only input parameter *Fr*_{1} showed the desired accuracy in the sloping channel with step. Figure 2 indicates the scatter plots of SVM prediction and measured values for the S(III) model in three channels.

### Developed models for rough channels

The obtained results from SVM models for predicting the hydraulic jump relative energy dissipation in rectangular and trapezoidal channels with rough beds are indicated in Table 4 and Figure 3. The superior performance for the rectangular channel was obtained from the model R(III) with input parameters of *Fr*_{1}, (*y*_{2}*−**y*_{1})/*y*_{1}. According to the obtained results, it could be inferred that adding *y*_{2}/*y*_{1} and (*y*_{2}*−**y*_{1})/*y*_{1} and *H/y*_{1} as input parameters caused an increment in the model's efficiency. However, for this state the variable (*y*_{2}*−**y*_{1})/*y*_{1} was more effective than variables *y*_{2}/*y*_{1} and *H/y*_{1} in improving the model's efficiency. For the trapezoidal channel, the superior performance was obtained from the model R(V) with input parameters of *Fr*_{1}, *S/H*. The variable *S/H* shows the impact of the rough elements' geometry on predicting the relative energy dissipation. It could be stated that the applied method in the case of a trapezoidal channel can successfully predict the relative energy dissipation using only the upstream flow characteristic (*Fr*_{1}) as input data. From comparison between the results of Tables 3 and 4, it can be stated that the developed models for rough channels performed more successfully than for smooth channels.

Channel type . | SVM models . | Performance criteria . | |||||
---|---|---|---|---|---|---|---|

Train . | Test . | ||||||

R . | DC . | RMSE . | R . | DC . | RMSE . | ||

Rectangular | R(I) | 0.858 | 0.683 | 0.066 | 0.814 | 0.661 | 0.073 |

R(II) | 0.903 | 0.815 | 0.048 | 0.892 | 0.768 | 0.055 | |

R(III) | 0.913 | 0.824 | 0.042 | 0.896 | 0.804 | 0.049 | |

R(IV) | 0.887 | 0.785 | 0.064 | 0.886 | 0.745 | 0.067 | |

Trapezoidal | R(I) | 0.905 | 0.822 | 0.049 | 0.904 | 0.804 | 0.052 |

R(II) | 0.910 | 0.825 | 0.043 | 0.906 | 0.809 | 0.050 | |

R(III) | 0.912 | 0.831 | 0.041 | 0.909 | 0.811 | 0.045 | |

R(IV) | 0.938 | 0.879 | 0.038 | 0.927 | 0.857 | 0.040 | |

R(V) | 0.942 | 0.885 | 0.036 | 0.935 | 0.858 | 0.039 |

Channel type . | SVM models . | Performance criteria . | |||||
---|---|---|---|---|---|---|---|

Train . | Test . | ||||||

R . | DC . | RMSE . | R . | DC . | RMSE . | ||

Rectangular | R(I) | 0.858 | 0.683 | 0.066 | 0.814 | 0.661 | 0.073 |

R(II) | 0.903 | 0.815 | 0.048 | 0.892 | 0.768 | 0.055 | |

R(III) | 0.913 | 0.824 | 0.042 | 0.896 | 0.804 | 0.049 | |

R(IV) | 0.887 | 0.785 | 0.064 | 0.886 | 0.745 | 0.067 | |

Trapezoidal | R(I) | 0.905 | 0.822 | 0.049 | 0.904 | 0.804 | 0.052 |

R(II) | 0.910 | 0.825 | 0.043 | 0.906 | 0.809 | 0.050 | |

R(III) | 0.912 | 0.831 | 0.041 | 0.909 | 0.811 | 0.045 | |

R(IV) | 0.938 | 0.879 | 0.038 | 0.927 | 0.857 | 0.040 | |

R(V) | 0.942 | 0.885 | 0.036 | 0.935 | 0.858 | 0.039 |

### Validation of proposed best SVM models using ANFIS and ANN

The experimental data for smooth and rough bed channels were used to evaluate the performance of proposed best SVM models in comparison with other data-driven models. In this regard, for each channel with different bed conditions (i.e. smooth and rough beds) the superior model was run using AFIS and ANN models and the results were compared with the SVM. Table 5 shows the results of this comparison. As can be seen from Table 5, both the ANN and ANFIS models led to the desired accuracy and the efficiency of the ANFIS model was more than the ANN. However, the SVM model yielded slightly better results in comparison with the ANFIS and ANN models.

Channel condition . | Model . | Method . | Performance criteria . | |||||
---|---|---|---|---|---|---|---|---|

Train . | Test . | |||||||

R . | DC . | RMSE . | R . | DC . | RMSE . | |||

Smooth | Rectangular | |||||||

S(III) | SVM | 0.879 | 0.756 | 0.065 | 0.876 | 0.721 | 0.069 | |

ANFIS | 0.877 | 0.760 | 0.064 | 0.872 | 0.711 | 0.075 | ||

ANN | 0.833 | 0.748 | 0.066 | 0.843 | 0.708 | 0.078 | ||

Trapezoidal | ||||||||

S(III) | SVM | 0.938 | 0.869 | 0.059 | 0.939 | 0.829 | 0.068 | |

ANFIS | 0.935 | 0.828 | 0.061 | 0.914 | 0.817 | 0.071 | ||

ANN | 0.902 | 0.813 | 0.063 | 0.882 | 0.808 | 0.074 | ||

Sloping channel with step | ||||||||

S(III) | SVM | 0.969 | 0.931 | 0.042 | 0.949 | 0.905 | 0.064 | |

ANFIS | 0.941 | 0.905 | 0.047 | 0.911 | 0.884 | 0.068 | ||

ANN | 0.915 | 0.901 | 0.051 | 0.902 | 0.870 | 0.071 | ||

Rough | Rectangular | |||||||

R(III) | SVM | 0.913 | 0.824 | 0.042 | 0.896 | 0.804 | 0.049 | |

ANFIS | 0.886 | 0.808 | 0.046 | 0.860 | 0.793 | 0.052 | ||

ANN | 0.882 | 0.805 | 0.047 | 0.837 | 0.779 | 0.056 | ||

Trapezoidal | ||||||||

R(V) | SVM | 0.942 | 0.885 | 0.036 | 0.935 | 0.858 | 0.039 | |

ANFIS | 0.904 | 0.858 | 0.039 | 0.893 | 0.828 | 0.042 | ||

ANN | 0.899 | 0.853 | 0.039 | 0.888 | 0.814 | 0.045 |

Channel condition . | Model . | Method . | Performance criteria . | |||||
---|---|---|---|---|---|---|---|---|

Train . | Test . | |||||||

R . | DC . | RMSE . | R . | DC . | RMSE . | |||

Smooth | Rectangular | |||||||

S(III) | SVM | 0.879 | 0.756 | 0.065 | 0.876 | 0.721 | 0.069 | |

ANFIS | 0.877 | 0.760 | 0.064 | 0.872 | 0.711 | 0.075 | ||

ANN | 0.833 | 0.748 | 0.066 | 0.843 | 0.708 | 0.078 | ||

Trapezoidal | ||||||||

S(III) | SVM | 0.938 | 0.869 | 0.059 | 0.939 | 0.829 | 0.068 | |

ANFIS | 0.935 | 0.828 | 0.061 | 0.914 | 0.817 | 0.071 | ||

ANN | 0.902 | 0.813 | 0.063 | 0.882 | 0.808 | 0.074 | ||

Sloping channel with step | ||||||||

S(III) | SVM | 0.969 | 0.931 | 0.042 | 0.949 | 0.905 | 0.064 | |

ANFIS | 0.941 | 0.905 | 0.047 | 0.911 | 0.884 | 0.068 | ||

ANN | 0.915 | 0.901 | 0.051 | 0.902 | 0.870 | 0.071 | ||

Rough | Rectangular | |||||||

R(III) | SVM | 0.913 | 0.824 | 0.042 | 0.896 | 0.804 | 0.049 | |

ANFIS | 0.886 | 0.808 | 0.046 | 0.860 | 0.793 | 0.052 | ||

ANN | 0.882 | 0.805 | 0.047 | 0.837 | 0.779 | 0.056 | ||

Trapezoidal | ||||||||

R(V) | SVM | 0.942 | 0.885 | 0.036 | 0.935 | 0.858 | 0.039 | |

ANFIS | 0.904 | 0.858 | 0.039 | 0.893 | 0.828 | 0.042 | ||

ANN | 0.899 | 0.853 | 0.039 | 0.888 | 0.814 | 0.045 |

### Sensitivity analysis

To investigate the impacts of different employed parameters from the best proposed models on hydraulic jump energy dissipation prediction via SVMs, sensitivity analysis was performed. In order to evaluate the effect of each parameter, the model was run with all input parameters and then, one of the input parameters was eliminated and the SVM model was re-run. Table 6 shows the sensitivity analysis results. From Table 6, it can be deduced that variable *Fr*_{1} is the most important variable in hydraulic jump relative energy dissipation prediction in both the smooth and rough channels.

Channel type . | Eliminated variable . | Performance criteria for test series . | ||
---|---|---|---|---|

R . | DC . | RMSE . | ||

Smooth | ||||

Rectangular | Fr_{1}, (y_{2}-y_{1})/y_{1} | 0.876 | 0.721 | 0.069 |

Fr_{1} | 0.501 | 0.472 | 0.138 | |

(y_{2}-y_{1})y_{1} | 0.834 | 0.652 | 0.083 | |

Trapezoidal | Fr_{1}, (y_{2}-y_{1})/y_{1} | 0.939 | 0.829 | 0.068 |

Fr_{1} | 0.645 | 0.510 | 0.128 | |

(y_{2}-y_{1})y_{1} | 0.883 | 0.755 | 0.079 | |

Sloping channel with step | Fr_{1}, (y_{2}-y_{1})/y_{1} | 0.949 | 0.90 | 0.064 |

Fr_{1} | 0.612 | 0.528 | 0.112 | |

(y_{2}-y_{1})y_{1} | 0.898 | 0.805 | 0.076 | |

Rough | ||||

Rectangular | Fr_{1}, (y_{2}-y_{1})/y_{1} | 0.896 | 0.804 | 0.049 |

Fr_{1} | 0.547 | 0.511 | 0.115 | |

(y_{2}-y_{1})y_{1} | 0.814 | 0.661 | 0.073 | |

Trapezoidal | Fr_{1}, S/H | 0.935 | 0.858 | 0.039 |

Fr_{1} | 0.635 | 0.551 | 0.099 | |

S/H | 0.904 | 0.804 | 0.052 |

Channel type . | Eliminated variable . | Performance criteria for test series . | ||
---|---|---|---|---|

R . | DC . | RMSE . | ||

Smooth | ||||

Rectangular | Fr_{1}, (y_{2}-y_{1})/y_{1} | 0.876 | 0.721 | 0.069 |

Fr_{1} | 0.501 | 0.472 | 0.138 | |

(y_{2}-y_{1})y_{1} | 0.834 | 0.652 | 0.083 | |

Trapezoidal | Fr_{1}, (y_{2}-y_{1})/y_{1} | 0.939 | 0.829 | 0.068 |

Fr_{1} | 0.645 | 0.510 | 0.128 | |

(y_{2}-y_{1})y_{1} | 0.883 | 0.755 | 0.079 | |

Sloping channel with step | Fr_{1}, (y_{2}-y_{1})/y_{1} | 0.949 | 0.90 | 0.064 |

Fr_{1} | 0.612 | 0.528 | 0.112 | |

(y_{2}-y_{1})y_{1} | 0.898 | 0.805 | 0.076 | |

Rough | ||||

Rectangular | Fr_{1}, (y_{2}-y_{1})/y_{1} | 0.896 | 0.804 | 0.049 |

Fr_{1} | 0.547 | 0.511 | 0.115 | |

(y_{2}-y_{1})y_{1} | 0.814 | 0.661 | 0.073 | |

Trapezoidal | Fr_{1}, S/H | 0.935 | 0.858 | 0.039 |

Fr_{1} | 0.635 | 0.551 | 0.099 | |

S/H | 0.904 | 0.804 | 0.052 |

## CONCLUSION

In the current study, the SVM approach was used to predict hydraulic jump energy dissipation in smooth and rough rectangular and trapezoidal channels and a sloping channel with a step. The SVM was applied for different models based on flow characteristics and channel bed conditions. For the state of a smooth bed, the obtained results showed that in predicting the relative energy dissipation the model S(III) with input parameters of *Fr*_{1}, (*y*_{2}*−**y*_{1})/*y*_{1} as input variables performed more successfully than the other models. It was found that using *y*_{2}/*y*_{1} and (*y*_{2}*−**y*_{1})/*y*_{1} as input parameters caused an increment in the model's efficiency. Also, in the case of the channel with a step, it was observed that the relative height of the step is an effective variable in energy dissipation estimation. Among the three types of smooth channels, the developed models for the case of the sloping channel with a step led to better predictions. For rough bed channels, the superior performance in the rectangular channel was the model R(III) with parameters of *Fr*_{1}, (*y*_{2}*−**y*_{1})/*y*_{1}. It was found that *H/y*_{1} as an input parameter improved the model's efficiency. For the case of the trapezoidal channel, the model R(V) with parameters of *Fr*_{1}, *S/H* was superior. It was found that the rough element's geometry has an impact on predicting relative energy dissipation. It showed that the applied method in the case of the trapezoidal channel can successfully predict relative energy dissipation using only the upstream flow characteristic (*Fr*_{1}) as input data. Comparison between the results of smooth and rough bed channels revealed that the developed models in the case of rough channels led to better predictions. The sensitivity analysis showed that *Fr*_{1} had the most effective role in estimation of relative energy dissipation. A comparison was also done between the SVM results and ANN and ANFIS models. The results showed that SVM is more accurate than the two other meta-model approaches. The applied technique was found to be able to predict hydraulic jump energy dissipation in both smooth and rough bed channels.