Experimental study and modeling of hydraulic jump for a suddenly expanding stilling basin using different hybrid algorithms

Hydraulic jump is a highly important phenomenon for dissipation of energy. This event, which involves flow regime change, can occur in many different types of stilling basins. In this study, hydraulic jump characteristics such as relative jump length and sequent depth ratio occurring in a suddenly expanding stilling basin were estimated using hybrid extreme learning machine (ELM). To hybridize ELM, imperialist competitive algorithm (ICA), firefly algorithm (FA) and particle swarm optimization (PSO) metaheuristic algorithms were implemented. In addition, six different models were established to determine effective dimensionless (relative) input variables. A new data set was constructed by adding the data obtained from the experimental study in the present research to the data obtained from the literature. The performance of each model was evaluated using k-fold crossvalidation. Results showed that ICA hybridization slightly outperformed FA and PSO methods. Considering relative input parameters, Froude number (Fr), expansion ratio (B) and relative sill height (S), effective input combinations were Fr–B–S and Fr–B for the prediction of the sequent depth ratio (Y ) and relative hydraulic jump length (Lj/h1), respectively.


INTRODUCTION
Hydraulic jump is an important phenomenon that is widely used in hydraulic engineering. It is a rapid transition from a high-velocity flow to a slower stream movement. It usually occurs downstream of dam spillways, in streams and rivers and in industrial channels. There are many different types of stilling basins. According to plan geometry, they are classified as gradually expanding and suddenly expanding.
In the literature, hydraulic jump has been widely studied. Bakhmeteff  Ebtehaj & Bonakdari () used evolutionary algorithms,

Data set
A total of 165 data points were used to model the suddenly expanding stilling basin, ten of which were new. These data In Figure 1, b 1 is the width of the first section before sudden expansion, b 2 is the width of the second section with sudden expansion, b s is the width of the central sill, x 1 is the length of the hydraulic jump before sudden expansion and x j is the length of the hydraulic jump that occurs within a sudden expansion. Adding this value to x 1 gives the jump length L j and Y is the ratio of the water depth after the hydraulic jump (h 2 ) to the water depth before the hydraulic jump (h 1 ); s is the height of the step or sill.
Hydraulic jump and dissipation of energy are influenced by the following parameters: where μ is the dynamic viscosity of water, g is gravity acceleration, s is the sill height and V 1 is the velocity before   (2) can be represented as follows: Equation (3) can be expressed as: respectively, as shown in Table 2. Six different model input combinations were examined to analyze the sensitivity of the variables. The established models are summarized in Table 3.    ELM is as follows: where β is output weights, L is the number of hidden nodes, w is the input weights, b is the bias values and g(x) is the activation function. Expressions of the hidden layer output matrix and compact view are as follows, respectively: where T is the target vector. The hidden layer output weight vector (β) is calculated with the Moore-Penrose generalized inverse of matrix G as in Equation (7): Particle swarm optimization The particle swarm optimization (PSO), meta-heuristic algorithm was inspired by the actions of fish and birds, and was developed by Kennedy & Eberhart (). The PSO, a population-based approach to stochastic optimization, starts with a random solution or a particle population in the search area and updates optima iteratively. The consequence of this simulation of social behavior is a search mechanism by which particles travel to appropriate locations. Particles learn from each other in the community based on information gained; they move towards better neighbors. At any moment, a particle changes its location in the search space to the best position by far and the best position in the neighborhood. Particle (i) is considered to be a vector and position vector in an area of the n-dimensional space. Growing particle update uses two demonstrative particles. First, the best solution to date, called 'pbest', is found by particles. Another is the best ever between all particles in the 'gbest' group. The PSO algorithm structure is shown in Figure 3.
The first step in Figure 3 is an arbitrary distribution of speeds and sites to begin the initial population. The next step is to test this particle using a statistical approach in a regression analysis. One can stop the scheme and export the parameters specified, once the best fitness standard of particulates meets the stop criterion. If the level of operation is insufficient for interruptions, the particle speed and position will be changed in two cases (Kennedy & Eberhart Every particle in PSO is a candidate solution in the ndimensional search space. The position of a particle i at any The new velocity of each particle in the search space is calculated as follows: (8) where j is the dimension of the search space (j ∈ [1, 2, . . . , n]), i is the number of iterations, w is the inertia weight, y i (t) is the pbest position,ŷ(t) is the gbest position, the parameters C 1 and C 2 are acceleration coefficients and the terms r j1 and r j2 are The new position is calculated as follows: Firefly algorithm The firefly algorithm was inspired by the social behavior of fireflies (Yang ). Fireflies are considered to be unisex with a distinctive characteristic that attracts each other.
The attraction rate of a firefly depends on the light intensity emitted by the firefly. In this way, the light intensity produced by the firefly is directly related to the radiation it emits. The mathematical representation of the intensity and attraction of a firefly is given in Equations (10) and (11). Every firefly's ability to attract another firefly depends on its sequence similarity I (Yang ; Tao et al. ): where attraction at a distance is represented by w(r) and the light intensity is represented by I. I 0 and w 0 represent the intensity of the emitted light and the attraction at a distance r ¼ 0 from the firefly and y represents the light absorption coefficient. Equation (12) provides the distance r between any two fireflies j and m (Yang ): where d is the population of the fireflies, and x j and x m are the location of the fireflies in the Cartesian coordinate system. As explained, each firefly is attracted by the others and vice-versa so the movement of fireflies is expressed in Equation (13) for the jth firefly by the mth firefly: where α is a constant between 0 and 1 and ε m is a random number vector obtained from the Gaussian distribution.

Imperial competitive algorithm
Through this newly introduced meta-heuristic methodology where c n is the imperialist nth cost. The normalized imperialist cost (C n ) is determined using the formula: where max (c i ) is the imperialist with maximum cost (weakest imperialist).

k-fold cross validation
Cross-validation tests the performance of a predictive model and is applied to a specific data set in statistical analyses. In the first stage, the data set to be evaluated is divided into subsets equal to k. Up to kÀ1 subsets are selected as training data for the model. The fold-t subset is selected as test data. The calculated accuracy value for the fold-t  subset is added to the cross validation (CV) array. This process is repeated for the number of subsets (k). All accuracies calculated in the final process are averaged. Either this average or the lowest accuracy is used to indicate the performance of the model (see Figure 6).

RESULTS AND DISCUSSION Hybridization
The aforementioned optimization algorithms (PSO, FA and ICA) were implemented for the hybridization of ELM. A number of hidden layer neurons between ten and 20 was selected for each model to provide the optimum performance. With bias and weight values collected in a vector, firstly, the initial population was created for metaheuristic algorithms. Secondly, the population was searched for the best solution, and according to the best weight and bias values, the test data was investigated. This process was performed in each fold. Table 4 shows the initial parameters of the evolutionary algorithms.

Evaluation of model performance
The models were compared using standard statistical

Comparison of models
All the simulations were conducted in the MATLAB 2016 environment running on a PC with 2.67 GHz CPU and 4 Gb memory space. ELM was tuned using PSO, FA, and ICA during the training phase. RMSE was used as the best objective function in the process. The evaluation was continued using 100 iterations. The stability of the machine learning models highly depends on the properties of the data. As can be seen in Figure 7, stability of folds for Y is better than L j /h 1 . As can be seen in the VAF results in Table 5, this may be due to the lower variance of Y. ELM-PSO, ELM-FA and ELM-ICA became almost stable after the 20th iteration. When the behavior of each fold is analyzed, almost all folds during the training lead to very close error rates for the Y prediction, while there are differences in the behavior of the folds for the L j /h 1 prediction ( Figure 7). In Table 6 In ELM-ICA was slightly better than ELM-PSO and ELM-FA.
Scatter plots of all models are presented in Figure 8. As can be seen in Figure 8, Y is in the shaded area (± 10% con- In this study, the percentage of error (RMSE p ) was considered for each of the folds as follows: where CV is the total fold number. Figure 9 displays the RMSE p for each of the models for ELM-PSO, ELM-FA and ELM-ICA. Fold 10 consists of new experimental data.
The results indicate that for prediction of Y, fold 10 adapted well. When considering Model 1, which is the best model,  Figure 9).
As can be seen in Table 5 and Figure 9, Model 1 and Model 6, which are the best models, had worse performance for fold 10 in ELM-PSO (RMSE p ¼ 17% and RMSE p ¼ 23%), ELM-FA (RMSE p ¼ 17% and RMSE p ¼ 23%) and ELM-ICA (RMSE p ¼ 17% and RMSE p ¼ 23%), respectively. When the best models, Model 1 and Model 6, were investigated, fold 7 and fold 10 showed almost the same performance for all machine learning algorithms.

Experimental validation
The design of a suddenly expanding stilling basin depends on different hydraulic jump types such as ( (27)): where Y* is sequent depth ratio for classical hydraulic jump (CHJ). Also, Herbrand () suggested an equation for hydraulic jump length (L j ) as follows:   where δ ¼ S=LS; δ is a dimensionless parameter to control sill height and sill location; LS ¼ L s =h 0 ; L s is the distance from the initial hydraulic jump length to the sill, h 0 is the opening of the gate and S ¼ s=h 0 , respectively.
The performance of various models for the new experimental data obtained from the experiments in the present research for sequent depth ratio and relative jump length is presented in Table 7.  (Table 7).
Measurement of the hydraulic jump length is very difficult because it has a dynamic length. Therefore, the variance of the values of L j /h 1 was higher than the value of Y (Table 7).

CONCLUSIONS
In this study, the hydraulic jump characteristics for a sud-