Modeling the Length of Hydraulic Jump on Sloping Rough Bed Using Hybrid Model based on Adaptive Neuro-Fuzzy Inference Systems and Particle Swarm Optimization

Rapid transition from a supercritical to subcritical flow is characterized by large-scale turbulence and energy dissipation. This transition is called hydraulic jump. The hydraulic jump is a type of rapid varied flows that used for water chlorination in treatment plants and energy dissipation of the flow and other hydraulic purposes. Due to the importance and complex structure of this phenomenon, many experimental, analytical and numerical studies have been carried out in this field. In general, hydraulic jumps occur after ogee spillways. Also, stilling basins are usually situated at downstream of ogee spillways. Therefore, for proper design of the length of stilling basins, accurate determination of the hydraulic jump length has a significant importance. In this study, a hybrid model (ANFIS-PSO) is introduced which uses Adaptive Neuro Fuzzy Inference Systems (ANFIS) and Particle Swarm Optimization (PSO) for predicting the hydraulic jump length on sloping rough beds. The Adaptive neuro fuzzy inference system is a kind of artificial neural network which is based on the Takagi-Sugeno fuzzy inference system. The inference system is a set of fuzzy IF–THEN rules that have learning capability to approximate nonlinear functions. In this model, PSO is applied to enhance the performance of ANFIS by adjusting the membership functions and subsequently minimizing the error. In fact, PSO is considered as an evolutionary computational method, optimizing continues and discontinues making decision functions. Additionally, PSO is considered as a population-based search method in which each potential solution, known as a swarm, represents a particle of a population. In this approach, the particle position changes continuously in a multi-dimensional search space, until reaching the optimal response and/or computational limitations. Also, in the current study, to evaluate the performance of ANFIS-PSO models, the Monte Carlo simulation (MCs) is applied. Monte Carlo simulation is a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. Their main idea is using randomness to solve problems that might be deterministic in principle. They are often used in physical and mathematical problems and are most useful when it is difficult or impossible to use other approaches. The Monte Carlo simulation is mainly used in different problems such as optimization and numerical integration from a probability distribution. Also, in this paper, the k-fold Cross Validation (k=4) is used for examination of the models ability. In k-fold Cross Validation, the original sample is randomly separated into k equal size sub-samples. In k sub-samples, a single sub-sample is retained as the validation data for testing the model, and the remaining k-1 sub-samples are used as training data. The cross-validation process is then repeated k times (the folds), with each of the k sub-samples used exactly once as the validation data. The k results from the folds can then be averaged to produce a single estimation. The advantage of this method over repeated random sub-sampling is that all observations are used for both training and validation, and each observation is used for validation exactly once. At first, five ANFIS-PSO models are defined using effective parameters on length of hydraulic jump. To validate the ANFIS-PSO models, the Kumar and Lodhi’s (2016) experimental measurements were used. The experimental model was conducted in a rectangular channel with a length of 8.0 m, 0.60 m width and 0.60 m depth. Three slopes of a flume, viz. 0.005, 0.010 and 0.016, were observed. Next, by analyzing the ANFIS-PSO models results, the superior model is presented. The superior model predicts the length of hydraulic jump in terms of Froude Number (F1), ratio of roughness bed (Ks/h1), ratio of sequent depth (h2/h1) and slope bed (S0). This model simulates the experimental measurement with suitable accuracy. For superior model, the Mean Absolute Percentage Error (MAPE), Root Mean Square Error (RMSE) and the correlation coefficient (R) were respectively computed equal to 3.750, 0.688 and 0.984. In addition, the scatter index (SI) for superior model was estimated equal to 0.055. Also, in order to more examine the ANFIS-PSO models results, the ratio of the predicted jump length to the observed jump length (λ=(Lr/h1)(Predicted)-(Lr/h1)(Observed)) was introduced. For the superior model, the average of this ratio was calculated equal to 1.003. According to numerical models results, the Froude number is identified as the most effective parameter for modeling the length of the hydraulic jump.