Application of meta-heuristic methods in the optimization of geometrical sections in trapezoidal channels in jump energy loss

According to the findings of the previous studies, it can be inferred that the majority of this research focused on the hydraulic jump phenomenon within the rectangular channels for understanding the jump characteristics. The hydraulic jump is generally defined as the flow mode transformation from critical state to subcritical state, during which significant energy dissipation is expected and the flow depth mitigates considerably within a relatively short distance. Fatehi-Nobarian et al. (2022) conducted a study on semicircular channels entitled ‘Investigation of the Effect of Velocity on Secondary Currents in Semicircular Channels on Hydraulic Jump Parameters’. The modeling results showed that the velocity of the cells of the secondary currents in the horizontal direction (perpendicular to the axis of flow) (X) was greater in the semicircular sections with a radius of 0.3 m. Figure 1 depicts the general structure of a hydraulic jump within a horizontal channel. Omid et al. (2007) have carried out research on energy loss induced by the hydraulic jump, and they numerically studied the various trapezoidal sections in terms of jump and energy loss. Among the studies on the hydraulic jump phenomenon, we can refer to studies conducted by Chanson (2009) on the classic hydraulic jump and Wanoschek & Hager (1989) and Muhsun (2012) on the hydraulic jump in the trapezoidal sections. The results of the aforementioned experimental studies on a trapezoidal flume (channel) with a wall slope of 45° and numbers range from 5.5 to 15. It was evident that the hydraulic jump in the channel with trapezoidal sections exhibited a different behavior compared to that of the rectangular channel and focused on the characteristics of the trapezoidal channel and they employed the Newton Raphson method for cross-sectional momentum requirements of these channels. They found that the non-rectangular sections exhibited high efficiency in energy dissipation. In a numerical study on the behavior of trapezoidal channels during the hydraulic jump, Fatehi-Nobarian et al. (2018) determined the proper geometrical section based on the Renormalization Group (RNG) turbulence model for the secondary currents within the channels. Kim et al. (2015) carried out a study on the hydraulic jump and investigated the associated properties regarding the energy depreciation at the downstream of a sliding gate. Javan & Eghbalzadeh (2013) reported their findings for the submerged hydraulic jump using a turbulence model ⁠. Gualtieri & Chanson (2010) studied the impact of Froude number on the air bubbles developed in the hydraulic jump and they concluded that the presence of the air bubbles during the hydraulic jump in the jump rolling part may play an important role in energy depreciation. During the past few years, a new method has been added to those optimization techniques inspired by natural phenomena like genetic algorithms. The harmony search algorithm (HSA)-based optimization algorithm is a novel, powerful optimization technique. Hassanvand et al. (2019) have used multi-criteria decision-making for selecting the spillway type and optimizing dimensions by applying the harmony search algorithm. Geem (2006) used the optimal cost design of water distribution networks based on a harmony search. Geem (2009) has studied the music-inspired harmony search algorithm, theory and applications. Houichi et al. (2013) have studied the length of a hydraulic jump in the U-shaped channel by the Artificial Neural Network (ANN) method. Abbaspour et al. (2013) have estimated the hydraulic jump on a corrugated bed using ANNs and genetic programming, and finally they concluded that the proposed ANN models are much more accurate than the GP models. Azimi et al. (2018) have used the neuro-fuzzy inference system–firefly algorithm for the estimation of hydraulic jump length on rough channels. Geem et al. (2001) investigated HSA. In this paper, two test systems of the Economic Load Dispatch (ELD) problems are solved by adopting the Cuckoo Search (CS) algorithm. A comparison of obtained simulation results by using the CS algorithm is carried out against six other swarm intelligence algorithms: Particle Swarm Optimization, Shuffled Frog Leaping Algorithm, Bacterial Foraging Optimization, Artificial Bee Colony, Harmony Search, and Firefly Algorithm. Fatehi-Nobarian et al. (2019) have experimentally researched trapezoidal channels and the results obtained from the comparison of experimental measurements and numerical models of the rate of secondary currents in different Froude numbers demonstrated that there exists an opposite relationship between the secondary velocity in the direction perpendicular to the axis of flows (V_x) and the velocity in a direction perpendicular to the flow level (V_z) in trapezoidal channels. Moreover, at a 45° angle, there has been a remarkable energy loss during the hydraulic jump. The ratio of the increase in the secondary current velocity in the X direction in Froude number 10 of 45° angle is higher than that of two other sections, which is equal to 71%, compared to the 75° angle in numerical models, and in Froude number 9, it was 91% during experimental tests. Then, the secondary current velocity in the Z direction for Froude number 2 of 75° angle is higher than that of two other sections, which is 88%, compared to the 45° angle in numerical models, and in Froude number 1.5, it was equal to 74.5% for experimental tests. The results of a metaheuristic investigation by Mahdavi-Meymand & Sulisz (2022) indicate that the meta-heuristic algorithms substantially improve the performance of Support Vector Regression (SVR). The results show that the integrative methods, Support Vector Regression-Multi Tracker Optimization Algorithm (SVR-MTOA), Support Vector Regression-Particle Swarm Optimization (SVR-PSO), and Support Vector Regression-Differential Evolution (SVR-DE), are more accurate than the Multilayer Perceptron Neural Network (MLPNN) and the Multiple Linear Regression (MLR). On average, the integrative methods provide 39.63% more accurate results than the MLPNN and 79.34% more accurate results than the MLR. The average Root Mean Square Error (RMSE) and R² for the integrative methods are 0.0054 m and 0.977, respectively. Among all integrative methods, the SVR-MTOA yields the best results, with RMSE = 0.0044 m and R² = 0.986. Wang et al. (2022) researched the Artificial Intelligence algorithm. In this study, we proposed a novel approach to improve centrifugal pump performance with regard to the pump head, pump efficiency, and power. Firstly, to establish constraints, an optimal numerical model accounted for factors such as pump efficiency and the head was considered. The pump was designed, and an Artificial Intelligence algorithmic approach was applied to the pump before performing experiments. Singh & Roy (2023) have investigated energy dissipation in hydraulic jumps. The experiments showed that the energy of the supercritical flows can be dissipated effectively by using perforated screens. The difference in energy dissipation between the upstream and downstream of the screen was more significant than the energy dissipation caused by classical hydraulic jumps. Comparing the results of the present study with the previous research, it was found that the energy loss in the case of the present study was more than the previous research. The relative energy loss in the present study was found to vary from 74 to 94%. The value of the Froude number downstream of the screen, F₂, varied from 1.1 to 1.81, with an average value of 1.35. The tailwater deficit parameter, D, varied from 0.66 to 0.90. Laishram et al. (2022) have investigated the hydraulic jump. The experimentation was performed in a channel flume with a rectangular cross-section (16 m long, 0.6 m wide, and 0.8 m deep) in the Hydraulics Laboratory, Department of Civil Engineering, NIT, Manipur. A sluice gate and a radial gate are used in the channel to generate a hydraulic jump at the downstream of the flow. The characteristics of hydraulic jumps, such as length of jump, upstream and downstream head, velocity, sequent depth ratio, and amount of energy dissipated were measured with different heights of gate opening and flow rate. The results show that the radial gate is much safer for the structure and dissipates more energy, and the control of the flow is much easier as compared to the sluice gate and can be considered as an appropriate model for dissipating energy. Benabdesselam et al. (2022) have researched hydraulic jumps in a straight rectangular compound channel. In this paper, the general regression neural network (GRNN) was applied to predict the basic characteristics of hydraulic jumps in a straight rectangular compound channel: (i) the sequent depth ratios, (ii) the relative energy losses, and (iii) the relative lengths. Experiments were carried out with three different values of the ratio between the main channel width and the flood plain one (W_y). The W_y values were: (1/4, 1/3, and 1/2). For each W_y ratio, several values of the inflow Froude number were considered according to the five inflow ratio depths' (W_z) values (0.167, 0.200, 0.253, 0.287, and 0.333) between the first sequent depth and the main channel one. The predicted values in the testing stages using the GRNN followed the experimental ones with a correlation coefficient (R) of 0.990 for the sequent depth ratios, 0.982 for relative energy losses, and 0.873 for the relative lengths of hydraulic jumps. The best models have been selected among several input configurations for each of three considered characteristics of the jump. In this paper, the energy depreciation in the hydraulic jump of the trapezoidal channels with continuous lateral wall angles ranging from 45° to 75° is investigated using one of the new meta-heuristic approaches, so-called HSA, which was able to outline the relative energy dissipation in the hydraulic jump within the trapezoidal channel for the continuous angles ranging from 45° to 75°. The novelty of this research can be justified by the research gap on the application of this algorithm in solving hydraulic problems. One of the innovations of the meta-heuristic optimization method with the continuous method in channels, which is used in this research, is the investigation of the full range of side wall angles in a trapezoidal channel. In this sense, all the side angles in a trapezoidal channel have been found to have much more accurate results than the previous methods, and in the old optimization methods, only a few specified angles were examined, which definitely did not have more accurate results. In general, the method used in this research can be used in spillways in dams and also can be used on the outer edges of coastal piers.

All algorithms involve a search mechanism, and their emergence philosophy is related to the likelihood of several solutions for the optimization problem, dimension magnitude, and discrete of the search space since there is a probability of the fact that the optimization problem may possess several locally or globally optimal solutions. Figure 2 presents the locally and globally optimal points (Fatehi-Nobarian et al. 2022).

Using harmony memory, the initial acceptable solutions are generated. Indeed, the harmony memory is a matrix encompassing our harmony which is outlined by Equation (1). Each harmony is regarded as a vector including the decision variables of the optimization problem. Until the desired number, the harmonies are created and stored in the harmony memory during the initial step of the algorithm.where nvar represents the variable count and IMS denotes the number of primary harmonies. The harmony memory consideration rate (HMCR) controls the solutions of the new harmonies against the initial solutions. This coefficient determines the citation likelihood of the harmony memory.The closer the value of this coefficient is to 1, the better the answer will be. The concentration power on the former solutions increases and the reduction in this coefficient leads to an increase in the search power within the search space. This algorithm consists of five phases and all phases of the harmony search, from start to the end, are presented by a flowchart in Figure 3.

To determine the relative energy dissipation of the hydraulic jump within trapezoidal channels, it is feasible to use energy relations. This analysis is done by considering the uniform flow and the prismatic channel between the two sections where the beginning section is the start point of the hydraulic jump and the end section is known as the shallow depth of the hydraulic jump. Generally, it is worth noting that the use of an energy relationship is suggested when the amount of energy loss between the two sections is equal to or greater than zero. However, due to the limitations of the execution phase and equipment in the laboratory, it is not possible to use many lateral angles for the channels. Equation (3) yields the energy dissipation of jump along the hydraulic jump.where V₁ and V₂ refer to the flow velocity within the channel before and after the hydraulic jump, respectively. While g denotes the gravitational acceleration, and y₁ and y₂ are conjunctive depths linked to the section #1 and section #2, respectively, which is written as Equation (4):where A₁ and A₂ are the cross-sections of the flow in section #1 and section #2, respectively. Fr₁ is the Froude number of the flow for section #1; starting section of the hydraulic jump. D₁ denotes the hydraulic depth of flow, which is obtained from Equation (5). Finally, E_L represents the energy difference of the flow between two sections. In Equation (5), T₁ is the flow level width in the trapezoidal channel for section #1.

The relative energy dissipation of flow is obtained from Equation (6).

With regard to the preceding studies on the trapezoidal channel properties in dealing with phenomena, like hydraulic jump (1), it can be stated that the variation in the lateral angle of the trapezoidal channel wall plays an important role in push energy dissipation, and these shifts are highly limited in the experimental studies. Due to the limitations of the execution phase and the equipment in the laboratory, it is not possible to use many lateral angles for the channels. This means that the researchers cannot test or experiment with many different angles for the channels due to the limitations of the laboratory equipment. It should be noted that the m parameter represents the lateral angle of the trapezoidal channel (Figure 4).

Therefore, this is one of the most important problems related to the trapezoidal channels deserving in-depth study using the meta-heuristic approach like the harmony search optimization algorithm. So far, such an algorithm is not applied to the hydraulic and fluid mechanic field for optimization purposes.where m is side slope of the channel wall.

The lateral angle range for the trapezoidal channel wall in this research follows the following inequality:

The pseudocode for the implemented algorithm is presented in Figure 5 for the proper perception of the coding phases. It is worth noting that the entire coding using MATLAB programming language consisted of 260 lines.

The entire optimized space for the desired range is stored as an output with the title of ‘Hyper Memory,’ and the best results are stored with the title of ‘Output,’ which are applied to curves of relative energy dissipation of hydraulic jump relative to the variation in the channel geometrical section for various discharge rates. The algorithm results are presented in Figures 6–10.

Of course, for discharge rates of 70, 50, 30, and 10 L/s, the amount of energy loss during hydraulic jump is equal to 71.3, 53, 20, and 9%, respectively, which implies that the energy loss for trapezoidal sections with more open lateral walls is much more than that of trapezoidal sections with more vertical walls. Before the complete comparison of the results of this research with the previous research, which is shown in Figure 11, first, an explanatory and thematic comparison between the research conducted in this field with the results of the current research has been made, which is given in Table 3. To study the lateral angles of the wall of trapezoidal channels continuously (angle infinity) in the algorithm used in this research, Figure 11 illustrates a proper comparison of this research with previous research.