## ABSTRACT

Transitions in an open channel refer to the change in flow behavior due to changes in the channel geometry. Determining flow characteristics through transitions is an important topic as it is necessary to guarantee the ideal hydraulic performance of water structures with low costs. This research focuses on the flow characteristics through vertical and horizontal transitions through experimental study and then utilizing machine learning to predict the flow characteristics. The proposed framework aims to develop both the cascade-forward artificial neural network (CFANN) model and the regression model to enhance the prediction of flow characteristics. The first model developed modifies the CFANN using dandelion optimizer (DO) to determine the ideal CFANN configuration. The second model used gene expression programming to develop statistical equations. The obtained CFANN–DO model has proven high accuracy in predicting the flow rates at various water loads and speeds achieving a coefficient of determination of approximately 100% for training data and 99.5% for testing data. Finally, predicting the characteristics of vertical and horizontal transitions in open channels is a critical issue. Manipulating these transitions can create a habitat for aquatic organisms, reduce erosion, and improve the overall environment.

## HIGHLIGHTS

Determining the flow characteristics through vertical and horizontal transitions then utilization of machine learning to predict the flow characteristics.

Using machine learning models where the flow characteristics are complex to be predicted using the traditional engineering methods.

Using machine learning models to improve the accuracy of predictions and enable more informed decision-making.

## NOTATION

*A*Cross-sectional area

*B*Main channel bed width

*B*Bed width in transition zone

*E*_{s}Energy loss in the transition zone

*E*_{u}Upstream energy loss

*F*_{s}Froude number in the transition zone

*F*_{u}Upstream Froude number

*g*Acceleration due to gravity

*h*_{1}The number of neurons in hidden layer one

*h*_{2}The number of neurons in hidden layer two

*L*Transition length

*m*The number of outputs.

*n*The number of inputs

*Q*Discharge of flow

- Δ
*Z* Transition height

- Δ
*b* The difference between the original width and contracted width

*R*_{e}Reynold number

*S*Bed slope

*v*_{s}Mean velocity in transition zone

*v*_{u}Upstream mean velocity

*y*_{s}Water depth in transition zone

*y*_{u}Upstream water depth

*σ*Surface tension

*μ*Dynamic viscosity

## ABBREVIATIONS

- AHA
Artificial hummingbird algorithm

- ANN
Artificial neural network

- CFANN
Cascade-forward artificial neural network

- CFD
Computational fluid dynamics

- DO
Dandelion optimizer

- GEP
Gene expression programming

- GP
Genetic programming

- GWO
Gray wolf optimizer

- MLR
Multiple linear regression

- PSO
Particle swarm optimization

- RANS
Reynolds-averaged Navier–Stokes

- RMSE
Root mean square error

- RNG
Renormalization group

- RSM
Reynolds stress model

*R*^{2}Coefficient of determination

- STD
Standard deviation

- SVM
Support vector machine

- TSO
Tuna swarm optimization

- WOA
Whale optimization algorithm

## INTRODUCTION

Water resources nowadays are a crucial issue for many countries, thus maintaining water areas is quite important. Water open channels represent the agriculture and irrigation arteries, so studies are oriented to assure higher efficiency of the water open channels. Flow characteristics through vertical and horizontal transitions in open channels are critical in fluid dynamics and hydraulic engineering. These transitions are points along the channel where cross-section geometry of the channel changes, which can have a substantial impact on the behavior of flowing water. Understanding the flow patterns and qualities at these transitions is critical for a wide range of technical and environmental applications, including developing efficient irrigation systems, regulating flood control, and maintaining aquatic ecosystems. When the orientation of a channel or the gradient of its bed level or cross-sectional area changes, this is known as a channel transition.

Sobeih & Rashwan (2002) presented new equations for the solution of the vertical transition problems for rectangular open channels. In their work, the specific energy adopts a dimensionless form to facilitate problem-solving. The new dimensionless equation was extremely easy to utilize. If the increase in the bed was more than the critical rise, the problem was also solved using their developed equations. Additionally, Zhao *et al.* (2022) presented a novel solution to the issue of horizontal transition for rectangular open channels. Their study covered the issue of the channel section when contracts are made for amounts higher than those that are necessary. Negm *et al.* (2003) carried out an experimental investigation to determine the impact of lateral and vertical contraction on energy loss along the restricted length in sloped open channels. Vatankhah & Bijankhan (2010) addressed the transitions through circular sections of open-channel flow for the three cases of circular cross-sectional channel transitions.

In open-channel subcritical flow, an expansion transition was demonstrated by Alauddin & Basak (2006). Haque (2009) carried out an experimental study in a rectangular open channel using a gradually rising hump on the bed of an expansion. To predict the flow characteristics of an open channel, computational fluid dynamics (CFD) was used. After proper validation, these numerical methods are preferred over experimental methods due to their lower time demand and lower cost. The experimental results validate the CFD solution. The commercial software ANSYS-CFX was used to complete a limited number of CFD simulations. The three-dimensional Reynolds-Averaged Navier–Stokes (RANS) equations, as well as the two equations k-varepsilon models, were used to accomplish this. The model validation using test data was reasonable. Ladopoulos (2010) presented numerical solutions for transitions by simultaneously using singular integral equation methods for subcritical and supercritical flow. Najafi Nejad Nasser (2011) studied channel expansions in open channels. He joined a sizable downstream segment of the river with an upstream piece that was relatively tiny. His research sought to determine the amount of energy lost during lateral expansions and to determine how the hump installed on the expansion's channel bed may help cut down on energy losses.

The effects of divergence angle, hump crest height, and Froude number of subcritical flow on turbulent flow in open-channel extensions were investigated by Najmeddin (2012) using CFD modeling. The model results are validated for a limited number of cases using existing analytical solutions under simplified conditions and available experimental data. The use of a hump was demonstrated to effectively reduce flow separation and eddy motion in the transition. Because the flow is forced to accelerate over the hump, the otherwise adverse pressure gradient, which is known to cause flow separation, diminishes. Alhashimi (2018) predicted the flow characteristics of open-channel expansion using CFD. After being properly validated, these numerical methods were preferred over experimental methods due to their lower time demand and lower cost. Experiment results validated the CFD solution.

A limited number of CFD simulations were completed using the commercial software FLUENT. The two-dimensional Reynolds-averaged Navier–Stokes (RANS) equations and the two equations RNG k-models were used. The model's validation with test data was reasonable. The velocity distributions of flow through the transition models are created, and the efficiencies of the transitions evolved by different discharge values are evaluated. Ashour *et al.* (2018) used experimental data to examine how the contraction and transition angle affected downstream water structures' ability to flow and avoid scour. The study demonstrated that the ideal transition angle (*θ*) for the approaches of water structures is 30° with a relative contracted width ratio (*r* = *b*/*B*) of at least 0.6 for effective hydraulic performance and economical design.

Nikpour *et al.* (2018) studied the supercritical flow in contraction and expansion, and wave formation patterns. For 3D simulation, turbulence models of k-*ε* renormalization group (RNG) and Reynolds stress model (RSM) based on Fluent software were used. For calculating free-surface wave profiles using the k-*ε* RNG and RSM models, the mean relative errors were obtained as 30.3 and 98.2% in contraction: 13 and 2.58% in expansion, respectively. Also, the mean relative errors of the *k*-*ε* RNG and RSM models for wave velocity computation of contraction were calculated as 5.26 and 2.87%, respectively. The results revealed that the RSM turbulence model performs better than *k*-ε RNG in simulating velocity profiles of shock waves.

Pandey *et al.* (2022) formulated a generic transition issue in terms of alternate-depth ratio for exponential channels. An algebraic equation that governs the exponential channel has been constructed to show the beginning stages of choking (rectangular, parabolic, and triangular). An empirical solution has been constructed for the post-choking depth at the upstream portion of the exponential channel. For all channel types, the empirical relationship between the upstream Froude number and shape factor for the incipient state has been determined. Musa & Rusaldy (2020) focused their analysis on the properties of ignition (sudden narrowing, transition, and radius). The analysis proved that the water flow's specific energy was changing as the feeling of narrowing did. The form of constriction that occurs suddenly has the biggest specific energy loss. There is a strong link between the experimental data from this and other research projects and the anticipated outcomes.

Kumari *et al.* (2022) conducted an experimental work using a rectangular hydraulic tilting channel with the primary goal of modeling grain velocity based on the experimental data. They used four input variables; shear velocity, exposed area to base area ratio, relative depth, and sediment particle weight to simulate grain velocity using a variety of soft computing techniques, including support vector machine (SVM), artificial neural network (ANN), and multiple linear regression (MLR). Three separate common statistical indices were used to perform a quantitative performance evaluation of the predicted values. The SVM model has more accurate predictions than the MLR and ANN models, according to the testing phase results. Kisi *et al.* (2023) compared four data-driven methods; Gaussian process regression (GPR), multivariate adaptive regression spline (MARS), M5 model tree (M5Tree), and multilinear regression (MLR), to estimate the mean velocity upstream and downstream of bridges. Utilizing tests in a rectangular laboratory flume, the mean velocity upstream of the bridge model could be most accurately estimated by the MARS, according to the results. Simultaneously, the M5Tree demonstrated the best performance in determining the downstream mean velocity. To estimate mean velocities upstream and downstream of the bridge, the study suggests using the MARS and M5Tree.

In general, the problem statement is the channel transitions are relatively short features, yet they may affect the flow for a great distance upstream and downstream. Expansions of channels are typical in both natural and manufactured open channels. Flow decelerates when cross-sectional dimensions increase in expansion. Physical models alone cannot offer a good understanding of the mechanics underlying the flow field due to the complexity of flow patterns. This phenomenon must be studied in the field, experimentally and statistically (Asnaashari *et al.* 2016). Mir & Patel (2024) evaluated machine learning models for predicting Manning's roughness coefficient in alluvial channels with bedforms, using Pearson's coefficient, sensitivity analysis, Taylor's diagram, box plots, and K-fold method, revealing the energy grade line's impact. Bonakdari *et al.* (2020) presented a self-adaptive extreme learning machine-based model for predicting velocity fields in open channels, demonstrating higher accuracy and sensitivity analysis compared to existing equations. Meddage *et al.* (2022) presented a novel approach to predicting bulk-average velocity in open channels with rigid vegetation using tree-based machine learning models and Shapley's Additive explanation.

Accurately estimating flow characteristics over vertical and horizontal transitions in open channels is a fundamental difficulty in hydraulic engineering. These transitions, which occur in both natural and man-made watercourses, have an impact on flow characteristics, sediment transport, and channel stability. Because these transitions are nonlinear and complicated, conventional hydraulic models and forecasting methods frequently fail to represent their complexities. As a result, there is an urgent need for more accurate and efficient prediction models.

Several studies have investigated machine learning methods for predicting flow characteristics in open-channel (Kitsikoudis *et al.* 2015; Farhadi *et al.* 2019; Mir & Patel 2024). These studies have shown the use of machine learning techniques to improve flow discharge predictions in compound open channels. However, there is a lack of research on applying these methods to predict flow through vertical and horizontal transitions; while ML methods have been successfully applied to predict flow characteristics in open channels, existing research primarily focuses on straight sections. There is a lack of research investigating the application of ML methods specifically for predicting flow characteristics through vertical and horizontal transitions in open channels. These transitions can cause significant changes in flow velocity, depth, and turbulence compared to straight sections.

Accurately predicting flow-through transitions is crucial for the safe and efficient design of open-channel systems like canals, irrigation channels, and spillways. This research addresses this gap by developing improved ML methods specifically tailored for predicting flow characteristics through vertical and horizontal transitions in open channels.

The current study aims to contribute to the advancement of predictive modeling techniques, enabling more accurate and efficient analysis of flow behavior in rectangular channels. The application of enhanced machine learning techniques to forecast flow characteristics in open channels advances comprehension of intricate hydraulic procedures, optimizes hydraulic structure design and operation, and ultimately enhances the resilience and sustainability of water resource management systems.

The objectives of this research are outlined as follows:

1. Utilize the collected experimental data to train machine learning algorithms effectively.

2. Predict flow characteristics during vertical and horizontal transitions within a rectangular channel.

3. Analyze data using statistical techniques such as regression analysis, decision trees, and neural networks.

4. Identify underlying patterns and relationships between input variables (e.g., channel geometry, flow rate, fluid properties) and output variables (e.g., flow velocity, depth).

5. Develop a cascade-forward artificial neural network (CFANN) model and a regression model based on experimental data to enhance the accuracy of flow characteristic predictions.

6. Utilize the design of experiments dandelion optimizer (DO) to determine the optimal configuration for the CFANN model, thereby improving its performance.

7. Create statistical equations using gene expression programming (GEP) as an alternative approach for predicting flow characteristics.

The state of the art of this study is the employment of machine learning techniques to predict flow characteristics among both vertical and horizontal transitions. To increase the precision of flow characteristic predictions, a CFANN model and a regression model are to be developed.

## METHODOLOGY

In this study, once the experimental data have been collected, it can be used to train machine learning algorithms to predict the flow characteristics through vertical and horizontal transitions in rectangular channels. The data can be analyzed using techniques such as regression analysis, decision trees, and neural networks to identify the underlying patterns and relationships between the input variables such as channel geometry, flow rate, and fluid properties, and the output variables such as flow velocity and depth. The experimental results are used to create a CFANN model as well as a regression model to improve the prediction of flow characteristics in the vertical and horizontal transition sections. The goal of using DO is to determine the best CFANN configuration. The second model developed statistical equations through gene expression programming (GEP).

There are some limitations of this approach:

1. The performance and reliability of CFANN and regression models are highly dependent on the quality and representativeness of the experimental data used for training.

2. Differential Evolution's (DE's) ability to discover the optimal global setting is influenced by parameter selection and goal function.

3. Excessive training on experimental data might lead to overfitting in CFANN and regression models.

4. Training and optimizing sophisticated neural network models, such as CFANN, needs significant computer resources, including processing and memory.

## MATERIAL AND METHODS

### Experimental work description

. | . | Q (L/s)
. | F_{r1}
. | y_{u} (Cm)
. | y_{s} (Cm)
. |
---|---|---|---|---|---|

Vertical Transitions | Minimum | 9.0 | 0.0739 | 12.00 | 4.10 |

Maximum | 16.0 | 0.332 | 26.30 | 23.70 | |

Horizontal Transitions | Minimum | 9.0 | 0.086 | 10.56 | 8.30 |

Maximum | 16.0 | 0.435 | 25.67 | 25.60 |

. | . | Q (L/s)
. | F_{r1}
. | y_{u} (Cm)
. | y_{s} (Cm)
. |
---|---|---|---|---|---|

Vertical Transitions | Minimum | 9.0 | 0.0739 | 12.00 | 4.10 |

Maximum | 16.0 | 0.332 | 26.30 | 23.70 | |

Horizontal Transitions | Minimum | 9.0 | 0.086 | 10.56 | 8.30 |

Maximum | 16.0 | 0.435 | 25.67 | 25.60 |

### ANN model

*et al.*2013). CFANN is composed of three different layer types, i.e., input, hidden, and output layers, as shown in Figure 3. Each layer has artificial neurons identified as nodes. All known parameter information is data to input nodes (), which forward these data to all hidden layer nodes. Any hidden node () collects the data received from all input nodes and the input layer bias node. The summarized input is then processed mathematically by a transfer function that may be sigmoid or linear. Like from input nodes, the values collected in the hidden layer are transferred to nodes of the output layer. The data obtained from input nodes, all the hidden nodes, and the hidden layer's bias node at every output node () are also summarized and afterward compared with target values. The mathematical equation of the CFANN in Figure 4 can be written as follows:where and are the hidden and output transfer functions; is the hidden and input layer neuronal weight; is the input and output layer neuronal weight; is the hidden and output layer neuronal weight; is the neuronal weight between the input layer bias node and hidden layer; is the neuronal weight between the hidden layer bias node and output layer;

*i*is the node of the input layer and is the input layer bias node; is the hidden layer bias node;

*n*is the number of input nodes,

*k*is the number of hidden nodes; and is the slope parameter of the sigmoid function.

CFANN needs the training to demonstrate productive results. Training means that the network weights and biases are determined such that the minimal average squared error (MSE) error between targets and outputs occurs. This study used DO as a backpropagation algorithm to train the CFANN.

### Dandelion optimizer

In artificial intelligence systems, there is considerable attention to computational algorithms inspired by nature that can solve complicated engineering issues. The DO was utilized to determine the CFANN's optimum biases and weights. Zhao *et al.* (2022) proposed the DO algorithm, which takes inspiration from nature. The seeds of dandelion plants are dispersed by wind. This is a list of the three phases dandelion seeds experience.

1. During the rising stage, a vortex forms above the dandelion seed and rises while being lifted higher by wind and sunshine. On a rainy day, though, there are no eddies above the seeds. Only local searches are available in this circumstance.

2. During the descending stage, seeds start to descend steadily once they reach a certain height.

3. In the third stage, dandelions finally fall to the ground at random, where the effect of the wind and weather will cause them to grow into new dandelions.

In the following subsection, the initialization procedure of DO is addressed concerning the optimization of DO. The methods for balancing the benefits of exploration and exploitation that are based on search operators and inspired by the rising, falling, and landing stages of dandelion seeds come next.

#### DO population initialization

*i*th individual

*X*

_{i}can be written as:where

*i*is an integer in the range of 1 to pop, and rand is a random number between 0 and 1.

#### The rising stage of DO

Dandelion seeds need to reach a certain height in the buoyancy stage to float away from the mother plant. Dandelion seeds will germinate at different heights depending on humidity and wind speed. These two weather conditions can be expressed mathematically as follows.

**: On a clear day, wind speeds can be considered to have a lognormal distribution, . A dandelion seed's height is influenced by the wind speed. The dandelion flies higher if the wind is stronger, and the seeds scatter farther. For any (**

*Case 1**t*

*+*1)th iteration with a clear day, the new position of each

*i*th dandelion seed can be updated as follows:where is a random perturbation between (0,1); and demonstrate the dandelion's lift component coefficients; is random variation between [−, ]; is the randomly selected position at iteration

*t*; is the dandelion seed's position at iteration

*t*; is a lognormal distribution subject to

*μ*= 0 and

*σ*

^{2}= 1;

*y*is the stander normal distribution (0,1); is the maximum number of iterations.

*X*+ 1 of each

_{it}*i*th dandelion seed during the growing stage of DO are combined as:where the exploration and exploitation intensities of DO throughout the rising stage are controlled by a random number called randn that is produced using a typical normal distribution of the interval

*N*(0, 1). In Equation (16), a cutoff value of 1.5 is established for DO to have better exploration strength in the first stage.

#### Descending stage of DO

*β*

_{t}is the Brownian motion and represents a random number generated from the standard normal distribution of

*N*(0, 1);

*X*

_{mean-t}is the mean position of the population in the

*t*th iteration.

#### Landing stage of DO

*X*

_{elite}is the seed's optimal position; levy(

*λ*) is Levy flight coefficient denoted as is considered to simulate the step size of dandelion seeds during the landing stage of DO; ̃

*β*is a random number, and its value may vary between 0 and 2;

*ω*and ϑ are arbitrary numbers in the range [0,1]; s is fixed constant with the value of 0.01; and

*Γ*(·) is an operation used to represent Gamma distribution.

#### Gene expression programming

An evolutionary algorithm for producing computer programs or models is called gene expression programming (GEP). These computer programs are complex tree structures that, like living things, learn and adapt by changing their sizes, forms, and composition, Gad *et al.* (2022). Moreover, GEP's computer programs are stored in simple linear chromosomes with predetermined lengths, exactly like in living things. GEP is a genotype-phenotype system as a result, utilizing a simple genome to store and transmit genetic information and a sophisticated phenotype to explore and adapt to the environment.

The genome or chromosome in GEP is a linear, symbolic string made up of one or more genes that have a set length. A fixed-length string made up of different primitives makes up each gene on its own. According to genetic programming (GP) terminology, GEP has two types of primitives, referred to as function and terminal. In a given program, a terminal is a primitive that can accept one or more arguments and provide a result after evaluation, as opposed to a constant or variable. A gene is split into two sections in GEP. Functions and terminals combine to produce the first component, known as the head, whereas terminals alone make up the second component, known as the tail.

### DIMENSIONAL ANALYSIS

*ρ*is the water density;

*y*

_{u}is the upstream water depth;

*y*

_{s}is the water depth in the middle of the transition;

*V*

_{u}is the upstream velocity;

*V*

_{s}is the velocity above the transition;

*μ*is the dynamic viscosity;

*σ*is the surface tension;

*Q*is the discharge;

*g*is the acceleration due to gravity;

*E*

_{u}is the upstream energy loss;

*E*

_{s}is the energy loss in the transition zone;

*S*is the channel bed slope;

*B*is the channel width;

*b*is the transition width;

*L*is the transition length equal to 1.0 m in this study, and Δ

*Z*is the transition height.

*ρ*,

*V*

_{u,}and

*y*

_{u}, Equation (23) can be written as follows:where the ratio of

*b*/

*B*is the contraction ratio;

*R*

_{e}is Reynolds's number ;

*W*

_{e}is Weber's number ;

*F*

_{u}is Froude's number in the upstream section ;

*F*

_{s}is Froude's number in the middle of the transition; and

*Q**is the discharge factor .

## RESULTS AND DISCUSSION

The importance of open-channel transitions lies in their ability to impact flood control, erosion control, habitat creation, recreational use, esthetics, water supply, pollution control, landscape connectivity, and climate change adaptation. By understanding the importance of these transitions and designing open channels accordingly, the authors can create functional and resilient ecosystems that provide a range of benefits for human communities and the environment. There are numerous real-world applications in a variety of fields that result from the practical implications of comprehending vertical and horizontal transitions in open channels. They guarantee the effective use of water resources, lower risks, safeguard the environment, and encourage safe and sustainable open-channel management. The preservation of natural ecosystems and community well-being are greatly impacted by these implications. The study's limitation is that the discharge values ranged from 2.5 to 16 L/s, values of Δ*Z*/*L* were 0.025, 0.05, and 0.075, and Δ*b*/*B* were 0.17, 0.33, and 0.5.

### Experimental results of vertical transition

*E*

_{u}

*/y*

_{u}) with the transition relative depth (

*y*

_{s}

*/y*

_{u}) for all used discharges and Δ

*Z*/

*L*= 0.03, 0.06, and 0.09. It can be noticed that the value of (Δ

*E*

_{u}/

*y*

_{u}) increases with the decreasing value of (

*y*

_{s}/

*y*

_{u}) at constant Δ

*Z/L*. The value of (Δ

*E*

_{u}

*/y*

_{u}) increases with the increasing value of Δ

*Z/L*. Figure 5(b) shows a variation of the upstream Froude number (

*F*

_{u}) with the transition relative depth (

*y*

_{s}

*/y*

_{u}) for a relative hump (Δ

*Z/L*) of 0.025, 0.05, and 0.075 under different values of discharge. The relationship between

*F*

_{u}and (

*y*

_{s}

*/y*

_{u}) takes a linear form. The value of

*F*

_{u}increases with the decreasing value of (

*y*

_{s}

*/y*

_{u}) and increases with the increasing value of discharge (

*Q*). Also, the value of (

*y*

_{s}

*/y*

_{u}) decreases with the increasing value of Δ

*Z/L*; for example: at upstream Froude number

*F*

_{u}= 0.1, (

*y*

_{s}

*/y*

_{u}) = 0.88 at Δ

*Z*/

*L*= 0.025; (

*y*

_{s}

*/y*

_{u}) = 0.77 at Δ

*Z/L*= 0.05; and (

*y*

_{s}

*/y*

_{u}) = 0.68 at Δ

*Z*/

*L*= 0.075.

*F*

_{s}) and the relative upstream energy loss (

*E*

_{u}/

*y*

_{u}) is represented in Figure 6(a). The figure shows relative upstream energy loss increases by increasing the transition Froude number. The values of relative upstream energy loss increase with decreasing (Δ

*Z*/

*L*). That means the higher vertical transition led to decreasing energy loss upstream. So, hydraulic engineers must carefully consider these factors when designing and operating hydraulic structures to optimize the efficiency of water resource management and minimize energy loss. Figure 6(b) presents a relationship between (

*F*

_{s}) and transition relative depth (

*y*

_{s}/

*y*

_{u}) under all used discharge values for Δ

*Z*/

*L*= 0.025, 0.05, and 0.075. For any case of Δ

*Z*/

*L*, the change of

*y*

_{s}/

*y*

_{u}with

*F*

_{s}has a declining trend. The variability of the upstream Froude number (

*F*

_{u}) and transition Froude number (

*F*

_{s}) for all discharge values and various Δ

*Z/L*situations is displayed in Figure 6(c). It is obvious that as both the upstream Froude number (

*F*

_{u}) and the relative hump height increase, so does the value of the transition Froude number (

*F*

_{s}). This is because the transition Froude number is influenced by the upstream flow conditions and the geometry of the transition, and changes in these factors can affect the Froude number in the transition zone.

The GEP program is used to generate an empirical equation to determine the transition relative depth for vertical transition based on the experimental data gathered in the current study as a function of both *F*_{u}, *E*_{u}*/y*_{u}, and Δ*Z/L* as tabulated in Table 2.

Also, an empirical equation is obtained to determine the value of the transition Froude number for vertical transition as a function of both *F*_{u} and Δ*Z/L* as listed in Table 2.

*R*

^{2}) = 0.84098, root mean square error (RMSE) = 0.0776 as shown in Figure 7(a) and correlation 0.8716,

*R*

^{2}= 0.769, and RMSE = 0.177 as shown in Figure 7(b).

These results are accurate when the vertical height of the transition ranges from 2.5 to 7.5 cm.

### Experimental results of horizontal transition

*F*

_{u}with the ratio of

*y*

_{s}

*/y*

_{u}for contraction ratio

*b/B*= 0.17, 0.33, and 0.5. As the values of

*y*

_{s}

*/y*

_{u}and

*b*/

*B*decline, the value of

*F*

_{u}increases. When comparing the effects of horizontal and vertical transitions on the water depth of the transition zone in Figure 8(b), it can be observed that the impact of horizontal transition is more significant. The relationship between

*F*

_{s}and

*E*

_{u}

*/y*

_{u}for horizontal transition is demonstrated in Figure 8(b). The figure illustrates how increasing the transition Froude number results in an increase in relative upstream energy loss. The relative upstream energy loss values increase with a decreasing ratio of Δ

*b/B*. That means the increasing transition width led to reduced energy loss upstream.

Figure 8(c) shows a variation of *F*_{u} with the *F*_{s} for different contraction ratios Δ*b*/*B*. The *F*_{s} value increases with the increasing values of *F*_{u} and the contraction ratio Δ*b/B*. Compared with Figure 6 in vertical transition, it is observed that the effect of horizontal transition on the Froude number in the transition zone is higher than vertical transition.

Using the GEP program, the following empirical formulas were introduced for horizontal transition to determine the transition relative depth and the value of the transition Froude number

*R*

^{2}= 0.8478, and RMSE = 0.028 as shown in Figure 9(a) and a correlation of 0.9198,

*R*

^{2}= 0.8846, and RMSE = 0.079 as shown in Figure 9(b). These results are accurate when the contracted width is in the range between 15 and 25 cm.

### Statistical results of different algorithms used in training the CFANN model

The CFANN model is trained to predict the transition relative depth and Froude number for vertical and horizontal transition. The CFANN output is crucially affected by choosing the ANN structure, the number of neurons/hidden layers, transfer functions, and training algorithms. Very few neurons in the hidden layer cannot achieve accurate results, whereas overfitting may result in too many neurons (Shibata *et al.* 2022). As illustrated in the approach diagram shown in Figure 3, CFANN is trained using the DO algorithm by computing the best values of different weights and biases to decrease RMSE between the measured and predicted outputs. Before the training, the input and target data were divided into two sets; 70% of the total data points were used for training and 30% for testing. Then a trial-and-error technique is applied to designing a CFANN architecture since there are no specific rules.

The input layer, two hidden layers, and the output layer are the layers from which the CFANN architecture is chosen. So the number of unknown weights (*w*) in Equation (3) was calculated as *((n*h*_{1}*+**h*_{1}*)**+**(n*h*_{2}*+**h*_{1}**h*_{2}*+**h*_{2}*)**+**(n**+**h*_{1}*+**h*_{2}*+**1)* m)* where *n* is the number of inputs, *h*_{1} and *h*_{2} are the number of neurons in hidden layer one and two, respectively; and *m* is the number of outputs. The smallest unknown weights number (number of neurons) that can still offer good fitting accuracy and extrapolate well were used in the end. Thus, a suitable number of weights was 129 when *h*_{1} = 10 and *h*_{2} = 5. The three transfer functions were also tried, and the *Tansig* function for the hidden and output layers was found to be the best transfer function for this study.

A comparison has been made with five types of stochastic meta-heuristic algorithms to evaluate the quality of the results obtained from DO in the training of the CFANN model. The five types are particle swarm optimization (PSO), gray wolf optimizer (GWO), tuna swarm optimization (TSO), whale optimization algorithm (WOA), and artificial hummingbird algorithm (AHA). For implementation, the population size used is 60, and the maximum iteration count is 200. Weights and biases values are kept in the range [−10, 10]. Each function was constructed independently 10 times to guarantee fair competition between the algorithms. This comparison is based on the best answer, the worst solution, the standard deviation (STD), and the average results (AVG). The most recently optimized algorithms in the comparisons were the TSO, WOA, and AHA algorithms. On the other hand, the PSO and GWO algorithms have been chosen as the most often utilized algorithm for solving multiple optimization issues. MATLAB R2022b implements the proposed DO-CFANN trainer and other trainers. Parameter settings used for all algorithms are taken from Mirjalili (2015), Mirjalili *et al.* (2017), and Zhao *et al.* (2022).

Model . | Input combinations . | Outputs . | . | DO-CFANN RMSE . | PSO-CFANN . | TSO-CFANN . | WOA-CFANN . | GWO-CFANN . | AHA-CFANN . |
---|---|---|---|---|---|---|---|---|---|

Vertical transition | F_{u}, E_{u}/y_{u}, and ΔZ/L | y_{s}/y_{u} | Minimum | 0.0168 | 0.0303 | 0.0534 | 0.0248 | 0.0513 | 0.0329 |

Maximum | 0.0340 | 0.0541 | 0.0620 | 0.0377 | 0.0570 | 0.0469 | |||

Average | 0.0272 | 0.0450 | 0.0591 | 0.0297 | 0.0540 | 0.0405 | |||

Standard deviation | 0.0059 | 0.0095 | 0.0036 | 0.0051 | 0.0022 | 0.0054 | |||

F_{u} and ΔZ/L | F_{s} | Minimum | 0.0316 | 0.0451 | 0.1210 | 0.0395 | 0.0995 | 0.0440 | |

Maximum | 0.0527 | 0.0611 | 0.1388 | 0.0555 | 0.1283 | 0.0905 | |||

Average | 0.0409 | 0.0508 | 0.1270 | 0.0482 | 0.1204 | 0.0648 | |||

Standard deviation | 0.0063 | 0.0061 | 0.0072 | 0.0067 | 0.0118 | 0.0219 | |||

Horizontal transition | F_{u}, E_{u}/y_{u}, and Δb/B | y_{s}/y_{u} | Minimum | 0.0021 | 0.0036 | 0.0178 | 0.0023 | 0.0167 | 0.0033 |

Maximum | 0.0032 | 0.0179 | 0.0425 | 0.0078 | 0.0227 | 0.0172 | |||

Average | 0.0026 | 0.0112 | 0.0251 | 0.0039 | 0.0203 | 0.0123 | |||

Standard deviation | 0.0005 | 0.0062 | 0.0099 | 0.0022 | 0.0025 | 0.0054 | |||

F_{u} and Δb/B | F_{s} | Minimum | 0.0063 | 0.0087 | 0.0219 | 0.0063 | 0.0372 | 0.0084 | |

Maximum | 0.0088 | 0.0101 | 0.0495 | 0.0095 | 0.0539 | 0.0351 | |||

Average | 0.0076 | 0.0095 | 0.0419 | 0.0082 | 0.0476 | 0.0151 | |||

Standard deviation | 0.0008 | 0.0006 | 0.0117 | 0.0013 | 0.0069 | 0.0113 |

Model . | Input combinations . | Outputs . | . | DO-CFANN RMSE . | PSO-CFANN . | TSO-CFANN . | WOA-CFANN . | GWO-CFANN . | AHA-CFANN . |
---|---|---|---|---|---|---|---|---|---|

Vertical transition | F_{u}, E_{u}/y_{u}, and ΔZ/L | y_{s}/y_{u} | Minimum | 0.0168 | 0.0303 | 0.0534 | 0.0248 | 0.0513 | 0.0329 |

Maximum | 0.0340 | 0.0541 | 0.0620 | 0.0377 | 0.0570 | 0.0469 | |||

Average | 0.0272 | 0.0450 | 0.0591 | 0.0297 | 0.0540 | 0.0405 | |||

Standard deviation | 0.0059 | 0.0095 | 0.0036 | 0.0051 | 0.0022 | 0.0054 | |||

F_{u} and ΔZ/L | F_{s} | Minimum | 0.0316 | 0.0451 | 0.1210 | 0.0395 | 0.0995 | 0.0440 | |

Maximum | 0.0527 | 0.0611 | 0.1388 | 0.0555 | 0.1283 | 0.0905 | |||

Average | 0.0409 | 0.0508 | 0.1270 | 0.0482 | 0.1204 | 0.0648 | |||

Standard deviation | 0.0063 | 0.0061 | 0.0072 | 0.0067 | 0.0118 | 0.0219 | |||

Horizontal transition | F_{u}, E_{u}/y_{u}, and Δb/B | y_{s}/y_{u} | Minimum | 0.0021 | 0.0036 | 0.0178 | 0.0023 | 0.0167 | 0.0033 |

Maximum | 0.0032 | 0.0179 | 0.0425 | 0.0078 | 0.0227 | 0.0172 | |||

Average | 0.0026 | 0.0112 | 0.0251 | 0.0039 | 0.0203 | 0.0123 | |||

Standard deviation | 0.0005 | 0.0062 | 0.0099 | 0.0022 | 0.0025 | 0.0054 | |||

F_{u} and Δb/B | F_{s} | Minimum | 0.0063 | 0.0087 | 0.0219 | 0.0063 | 0.0372 | 0.0084 | |

Maximum | 0.0088 | 0.0101 | 0.0495 | 0.0095 | 0.0539 | 0.0351 | |||

Average | 0.0076 | 0.0095 | 0.0419 | 0.0082 | 0.0476 | 0.0151 | |||

Standard deviation | 0.0008 | 0.0006 | 0.0117 | 0.0013 | 0.0069 | 0.0113 |

The best result is shown in bold.

In summary, numerical simulations and mathematical models are developed and improved according to theoretical considerations. Precise models are necessary to forecast flow characteristics, which are necessary for managing water resources and hydraulic structure design.

### Comparison with previous studies

*et al.*(2022) for vertical transition to confirm these findings. In this comparison, Results applied the previous equations and Pandey

*et al.*equation to the present data.

In the vertical transition case, *R*^{2} = 0.7715, 0.6334, and 0.7193 which agrees with the *R*^{2} = 0.9785, 0.9822, and 0.9803 calculated in the present study for Δ*Z*/*L* = 0.025, 0.05, and 0.075, respectively. A sample of the experimental data and calculations for the vertical transition of the present study and Pandey *et al.*’s study is provided in Table 3.

ΔZ (cm)
. | ΔZ/y_{u}
. | F_{u}
. | Observed y_{s}/y_{u}
. | Present studyy_{s}/y_{u}
. | Pandey et al. (2022) y_{s}/y_{u}
. |
---|---|---|---|---|---|

2.5 | 0.17241 | 0.17347 | 0.81379 | 0.81838 | 0.79167 |

2.5 | 0.14205 | 0.12972 | 0.85113 | 0.86438 | 0.83071 |

2.5 | 0.11848 | 0.09882 | 0.8710 | 0.90459 | 0.85842 |

2.5 | 0.10730 | 0.08516 | 0.88540 | 0.92597 | 0.87223 |

2.5 | 0.09766 | 0.07394 | 0.89843 | 0.94614 | 0.88408 |

5.0 | 0.23969 | 0.17872 | 0.74304 | 0.66599 | 0.71244 |

5.0 | 0.30675 | 0.25875 | 0.64417 | 0.58973 | 0.60912 |

5.0 | 0.34722 | 0.31161 | 0.51388 | 0.54710 | 0.48352 |

5.0 | 0.28902 | 0.23664 | 0.67630 | 0.60907 | 0.64116 |

5.0 | 0.26616 | 0.12624 | 0.71102 | 0.61036 | 0.67043 |

7.5 | 0.32258 | 0.16845 | 0.64516 | 0.55152 | 0.59809 |

7.5 | 0.40000 | 0.23259 | 0.50857 | 0.48167 | 0.46328 |

7.5 | 0.41916 | 0.24951 | 0.35928 | 0.46559 | 0.32515 |

7.5 | 0.42424 | 0.25406 | 0.36969 | 0.46138 | 0.33148 |

ΔZ (cm)
. | ΔZ/y_{u}
. | F_{u}
. | Observed y_{s}/y_{u}
. | Present studyy_{s}/y_{u}
. | Pandey et al. (2022) y_{s}/y_{u}
. |
---|---|---|---|---|---|

2.5 | 0.17241 | 0.17347 | 0.81379 | 0.81838 | 0.79167 |

2.5 | 0.14205 | 0.12972 | 0.85113 | 0.86438 | 0.83071 |

2.5 | 0.11848 | 0.09882 | 0.8710 | 0.90459 | 0.85842 |

2.5 | 0.10730 | 0.08516 | 0.88540 | 0.92597 | 0.87223 |

2.5 | 0.09766 | 0.07394 | 0.89843 | 0.94614 | 0.88408 |

5.0 | 0.23969 | 0.17872 | 0.74304 | 0.66599 | 0.71244 |

5.0 | 0.30675 | 0.25875 | 0.64417 | 0.58973 | 0.60912 |

5.0 | 0.34722 | 0.31161 | 0.51388 | 0.54710 | 0.48352 |

5.0 | 0.28902 | 0.23664 | 0.67630 | 0.60907 | 0.64116 |

5.0 | 0.26616 | 0.12624 | 0.71102 | 0.61036 | 0.67043 |

7.5 | 0.32258 | 0.16845 | 0.64516 | 0.55152 | 0.59809 |

7.5 | 0.40000 | 0.23259 | 0.50857 | 0.48167 | 0.46328 |

7.5 | 0.41916 | 0.24951 | 0.35928 | 0.46559 | 0.32515 |

7.5 | 0.42424 | 0.25406 | 0.36969 | 0.46138 | 0.33148 |

Table 4 displays the proposed relative water depth and compared with Pandey *et al.* (2022) for vertical transition based on 43 runs (run the algorithm 43 times). This is to demonstrate the suggested method's consistency when compared to other approaches.

. | Pandey et al. (2022)
. | Present study . | Observed values . |
---|---|---|---|

Number of values | 43 | 43 | 43 |

Minimum | 0.2637 | 0.4614 | 0.2994 |

25% Percentile | 0.5265 | 0.5897 | 0.5659 |

Median | 0.6663 | 0.6805 | 0.7110 |

75% Percentile | 0.8330 | 0.8184 | 0.8511 |

Maximum | 0.8861 | 0.9461 | 0.9011 |

Range | 0.6224 | 0.4848 | 0.6018 |

10% Percentile | 0.3277 | 0.4926 | 0.3634 |

90% Percentile | 0.8738 | 0.8860 | 0.8904 |

Actual confidence level | 96.85% | 96.85% | 96.85% |

Lower confidence limit | 0.5981 | 0.6104 | 0.6442 |

Upper confidence limit | 0.7576 | 0.7440 | 0.7826 |

Mean | 0.6522 | 0.6886 | 0.6835 |

Std. Deviation | 0.1841 | 0.1392 | 0.1765 |

Std. Error of Mean | 0.02807 | 0.02122 | 0.02691 |

Coefficient of variation | 28.23% | 20.21% | 25.82% |

Geometric mean | 0.6213 | 0.6748 | 0.6563 |

Geometric SD factor | 1.399 | 1.227 | 1.358 |

Harmonic mean | 0.5840 | 0.6611 | 0.6234 |

Quadratic mean | 0.6771 | 0.7022 | 0.7054 |

Skewness | −0.5479 | 0.1699 | −0.6901 |

Kurtosis | −0.5347 | −1.071 | −0.3235 |

Sum | 28.04 | 29.61 | 29.39 |

. | Pandey et al. (2022)
. | Present study . | Observed values . |
---|---|---|---|

Number of values | 43 | 43 | 43 |

Minimum | 0.2637 | 0.4614 | 0.2994 |

25% Percentile | 0.5265 | 0.5897 | 0.5659 |

Median | 0.6663 | 0.6805 | 0.7110 |

75% Percentile | 0.8330 | 0.8184 | 0.8511 |

Maximum | 0.8861 | 0.9461 | 0.9011 |

Range | 0.6224 | 0.4848 | 0.6018 |

10% Percentile | 0.3277 | 0.4926 | 0.3634 |

90% Percentile | 0.8738 | 0.8860 | 0.8904 |

Actual confidence level | 96.85% | 96.85% | 96.85% |

Lower confidence limit | 0.5981 | 0.6104 | 0.6442 |

Upper confidence limit | 0.7576 | 0.7440 | 0.7826 |

Mean | 0.6522 | 0.6886 | 0.6835 |

Std. Deviation | 0.1841 | 0.1392 | 0.1765 |

Std. Error of Mean | 0.02807 | 0.02122 | 0.02691 |

Coefficient of variation | 28.23% | 20.21% | 25.82% |

Geometric mean | 0.6213 | 0.6748 | 0.6563 |

Geometric SD factor | 1.399 | 1.227 | 1.358 |

Harmonic mean | 0.5840 | 0.6611 | 0.6234 |

Quadratic mean | 0.6771 | 0.7022 | 0.7054 |

Skewness | −0.5479 | 0.1699 | −0.6901 |

Kurtosis | −0.5347 | −1.071 | −0.3235 |

Sum | 28.04 | 29.61 | 29.39 |

The comparative and recommended technique test results using the Wilcoxon signed-rank test are discussed in Table 5. This statistical test reveals the substantial difference between the indicated results and those of other methods with a *P*-value of less than 0.05.

. | . | Pandey et al. (2022)
. | Present study . | Observed values . |
---|---|---|---|---|

Wilcoxon signed-rank test | Actual median | 0.6663 | 0.6805 | 0.7110 |

Number of values | 43 | 43 | 43 | |

The sum of signed ranks (W) | 946.0 | 946.0 | 946.0 | |

The sum of positive ranks | 946.0 | 946.0 | 946.0 | |

The sum of negative ranks | 0.000 | 0.000 | 0.000 | |

P-value (two-tailed) | <0.0001 | <0.0001 | <0.0001 | |

Exact or an estimate? | Exact | Exact | Exact | |

Significant (alpha = 0.05)? | Yes | Yes | Yes | |

How big is the discrepancy? | Discrepancy | 0.6663 | 0.6805 | 0.7110 |

95% confidence interval | 0.5981–0.7576 | 0.6104–0.7440 | 0.6442–0.7826 | |

Actual confidence level | 96.85 | 96.85 | 96.85 |

. | . | Pandey et al. (2022)
. | Present study . | Observed values . |
---|---|---|---|---|

Wilcoxon signed-rank test | Actual median | 0.6663 | 0.6805 | 0.7110 |

Number of values | 43 | 43 | 43 | |

The sum of signed ranks (W) | 946.0 | 946.0 | 946.0 | |

The sum of positive ranks | 946.0 | 946.0 | 946.0 | |

The sum of negative ranks | 0.000 | 0.000 | 0.000 | |

P-value (two-tailed) | <0.0001 | <0.0001 | <0.0001 | |

Exact or an estimate? | Exact | Exact | Exact | |

Significant (alpha = 0.05)? | Yes | Yes | Yes | |

How big is the discrepancy? | Discrepancy | 0.6663 | 0.6805 | 0.7110 |

95% confidence interval | 0.5981–0.7576 | 0.6104–0.7440 | 0.6442–0.7826 | |

Actual confidence level | 96.85 | 96.85 | 96.85 |

*et al.*(2018) for horizontal transition to confirm these findings. In this comparison, results applied the previous equations and Ashour

*et al.*’s equation to the present data.

In the horizontal transition case, *R*^{2} = 1 (Ashour *et al.* 2018) which agrees with the *R*^{2} = 0.99, calculated in the present study for all values of horizontal transition. An example of experimental data and calculations for the current study's horizontal transition and Ashour *et al.*’s study is provided in Table 6.

Δb/B
. | F_{u}
. | Observed y_{s}/y_{u}
. | Present studyy_{s}/y_{u}
. | Ashour et al. (2018) y_{s}/y_{u}
. |
---|---|---|---|---|

0.17 | 0.15436 | 0.985 | 0.96297 | 0.96973 |

0.19351 | 0.985 | 0.94027 | 0.96153 | |

0.19781 | 0.892 | 0.93777 | 0.96060 | |

0.23495 | 0.944 | 0.91623 | 0.95227 | |

0.25143 | 0.902 | 0.90667 | 0.94837 | |

0.33 | 0.27291 | 0.940 | 0.86071 | 0.91817 |

0.22019 | 0.963 | 0.89129 | 0.94815 | |

0.18667 | 0.976 | 0.91073 | 0.96032 | |

0.16243 | 0.981 | 0.92479 | 0.96724 | |

0.50 | 0.24927 | 0.864 | 0.84092 | 0.82260 |

0.21846 | 0.895 | 0.85880 | 0.90065 | |

0.18935 | 0.936 | 0.87567 | 0.93647 | |

0.16553 | 0.944 | 0.88949 | 0.95395 |

Δb/B
. | F_{u}
. | Observed y_{s}/y_{u}
. | Present studyy_{s}/y_{u}
. | Ashour et al. (2018) y_{s}/y_{u}
. |
---|---|---|---|---|

0.17 | 0.15436 | 0.985 | 0.96297 | 0.96973 |

0.19351 | 0.985 | 0.94027 | 0.96153 | |

0.19781 | 0.892 | 0.93777 | 0.96060 | |

0.23495 | 0.944 | 0.91623 | 0.95227 | |

0.25143 | 0.902 | 0.90667 | 0.94837 | |

0.33 | 0.27291 | 0.940 | 0.86071 | 0.91817 |

0.22019 | 0.963 | 0.89129 | 0.94815 | |

0.18667 | 0.976 | 0.91073 | 0.96032 | |

0.16243 | 0.981 | 0.92479 | 0.96724 | |

0.50 | 0.24927 | 0.864 | 0.84092 | 0.82260 |

0.21846 | 0.895 | 0.85880 | 0.90065 | |

0.18935 | 0.936 | 0.87567 | 0.93647 | |

0.16553 | 0.944 | 0.88949 | 0.95395 |

Table 7 compares the projected relative water depth for the horizontal transition with Ashour *et al.* (2018) based on 45 runs (run the algorithm 45 times). This is done to demonstrate that the suggested method is stable when compared to other ways.

. | Ashour et al. (2018)
. | Present study . | Observed . |
---|---|---|---|

Number of values | 45 | 45 | 45 |

Minimum | 0.8001 | 0.8226 | 0.6148 |

25% Percentile | 0.8700 | 0.9381 | 0.9155 |

Median | 0.9107 | 0.9606 | 0.9676 |

75% Percentile | 0.9435 | 0.9748 | 0.9854 |

Maximum | 0.9968 | 0.9838 | 0.9973 |

Range | 0.1967 | 0.1612 | 0.3825 |

10% Percentile | 0.8404 | 0.8860 | 0.8730 |

90% Percentile | 0.9796 | 0.9804 | 0.9927 |

Actual confidence level | 96.43% | 96.43% | 97.41% |

Lower confidence limit | 0.8895 | 0.9484 | 0.9402 |

Upper confidence limit | 0.9325 | 0.9689 | 0.9806 |

Mean | 0.9091 | 0.9480 | 0.9433 |

Std. deviation | 0.04933 | 0.03743 | 0.06635 |

Std. error of mean | 0.007353 | 0.005580 | 0.009783 |

Coefficient of variation | 5.426% | 3.949% | 7.034% |

Geometric mean | 0.9078 | 0.9472 | 0.9406 |

Geometric SD factor | 1.056 | 1.042 | 1.083 |

Harmonic mean | 0.9065 | 0.9464 | 0.9374 |

Quadratic mean | 0.9104 | 0.9487 | 0.9456 |

Skewness | −0.1463 | −1.674 | −2.998 |

Kurtosis | −0.6699 | 2.571 | 12.67 |

Sum | 40.91 | 42.66 | 43.39 |

. | Ashour et al. (2018)
. | Present study . | Observed . |
---|---|---|---|

Number of values | 45 | 45 | 45 |

Minimum | 0.8001 | 0.8226 | 0.6148 |

25% Percentile | 0.8700 | 0.9381 | 0.9155 |

Median | 0.9107 | 0.9606 | 0.9676 |

75% Percentile | 0.9435 | 0.9748 | 0.9854 |

Maximum | 0.9968 | 0.9838 | 0.9973 |

Range | 0.1967 | 0.1612 | 0.3825 |

10% Percentile | 0.8404 | 0.8860 | 0.8730 |

90% Percentile | 0.9796 | 0.9804 | 0.9927 |

Actual confidence level | 96.43% | 96.43% | 97.41% |

Lower confidence limit | 0.8895 | 0.9484 | 0.9402 |

Upper confidence limit | 0.9325 | 0.9689 | 0.9806 |

Mean | 0.9091 | 0.9480 | 0.9433 |

Std. deviation | 0.04933 | 0.03743 | 0.06635 |

Std. error of mean | 0.007353 | 0.005580 | 0.009783 |

Coefficient of variation | 5.426% | 3.949% | 7.034% |

Geometric mean | 0.9078 | 0.9472 | 0.9406 |

Geometric SD factor | 1.056 | 1.042 | 1.083 |

Harmonic mean | 0.9065 | 0.9464 | 0.9374 |

Quadratic mean | 0.9104 | 0.9487 | 0.9456 |

Skewness | −0.1463 | −1.674 | −2.998 |

Kurtosis | −0.6699 | 2.571 | 12.67 |

Sum | 40.91 | 42.66 | 43.39 |

The comparative and recommended technique test results using the Wilcoxon signed-rank test are discussed in Table 8. This statistical test reveals the substantial difference between the indicated results and those of other methods with a *P*-value of less than 0.05.

Wilcoxon signed-rank test . | Ashour et al. (2018)
. | Present study . | Observed . |
---|---|---|---|

Actual median | 0.9107 | 0.9606 | 0.9676 |

Number of values | 45 | 45 | 46 |

The sum of signed ranks (W) | 1,035 | 1,035 | 1,081 |

The sum of positive ranks | 1,035 | 1,035 | 1,081 |

The sum of negative ranks | 0.000 | 0.000 | 0.000 |

P-value (two-tailed) | <0.0001 | <0.0001 | <0.0001 |

Exact or estimate? | Exact | Exact | Exact |

Significant (alpha = 0.05)? | Yes | Yes | Yes |

Discrepancy | 0.9107 | 0.9606 | 0.9676 |

95% confidence interval | 0.8895–0.9325 | 0.9484–0.9689 | 0.9402–0.9806 |

Actual confidence level | 96.43 | 96.43 | 97.41 |

Wilcoxon signed-rank test . | Ashour et al. (2018)
. | Present study . | Observed . |
---|---|---|---|

Actual median | 0.9107 | 0.9606 | 0.9676 |

Number of values | 45 | 45 | 46 |

The sum of signed ranks (W) | 1,035 | 1,035 | 1,081 |

The sum of positive ranks | 1,035 | 1,035 | 1,081 |

The sum of negative ranks | 0.000 | 0.000 | 0.000 |

P-value (two-tailed) | <0.0001 | <0.0001 | <0.0001 |

Exact or estimate? | Exact | Exact | Exact |

Significant (alpha = 0.05)? | Yes | Yes | Yes |

Discrepancy | 0.9107 | 0.9606 | 0.9676 |

95% confidence interval | 0.8895–0.9325 | 0.9484–0.9689 | 0.9402–0.9806 |

Actual confidence level | 96.43 | 96.43 | 97.41 |

*R*

^{2}= 0.9849 (Mohamed 2005) which agrees with

*R*

^{2}= 1 is calculated in the present study.

Empirical equations are used to estimate various flow parameters and predict the hydraulic behavior through transitions. The accuracy and limitations of these equations are assessed by the results in this section. An alternate technique for creating statistical equations that forecast flow characteristics is GEP. These results clarify the relationship between the observed hydraulic properties and the application of empirical equations, offering perceptions into the situations in which these equations are suitable and those in which more intricate modeling is necessary. Furthermore, the characteristics of flow are predicted through machine learning techniques. The development of models for regression and CFANN is intended to increase the precision of flow characteristic predictions.

## CONCLUSIONS

Open-channel transitions are important in many engineering contexts, especially when designing and managing hydraulic structures such as spillways, sluice gates, and weirs. It is critical to comprehend these transitions to guarantee the effective management of water resources as a whole, as well as the safe and efficient operation of such structures. Studies show that, in comparison to vertical transitions, horizontal transitions have a greater impact on the Froude number and energy dissipation in transition zones. As Δ*Z*/*L* increases, so does the relative upstream energy loss, and this trend also increases as *b*/*B* ratio increases. There is a significant degree of agreement between the results of the current study and earlier research, indicating the validity of the current inquiry. The GEP model exhibits high accuracy in terms of vertical transitions, yielding correlations of 0.8716 for the calculation of the Froude number within the transition zone (with *R*^{2} = 0.769 and RMSE = 0.177) and 0.91705 for the calculation of the transition water depth (with *R*^{2} = 0.84098 and RMSE = 0.0776). Likewise, the model demonstrates remarkable accuracy for horizontal transitions, exhibiting correlations of 0.9208 for the transition water depth calculation (*R*^{2} = 0.8478, RMSE = 0.028) and 0.9198 for the Froude number calculation in the transition zone (*R*^{2} = 0.8846, RMSE = 0.079). Additionally, the utilization of a trained and tested CFANN-DO model allows for the accurate prediction of wear rates for all manufactured composites, with coefficients of determination (*R*^{2}) approximately reaching 100 and 99.5%, demonstrating impressive accuracy. The study's limitation is that the discharge values ranged from 2.5 to 16 L/s, values of Δ*Z*/*L* were 0.025, 0.05, and 0.075, and Δ*b*/*B* were 0.17, 0.33, and 0.5. Future research on this topic could go beyond flow prediction and more thoroughly examine the hydrodynamic behavior at transitions in open channels. This may require further investigation of patterns of sediment transport, turbulence, or flow separation. Hybrid modeling frameworks should also be developed to capitalize on the complementary benefits of both machine learning and physics-based models (such as CFD and hydraulic models). Additionally, real-time monitoring systems and forecasting tools that integrate machine learning models, sensor networks, and data assimilation strategies should be developed to enable accurate predictions of flow characteristics in open channels in real time. Additionally, it examines how climate change affects flow characteristics in open channels and develops forecasting models that account for long-term trends, extreme events, and fluctuating hydrological regimes.

## ACKNOWLEDGEMENTS

This work was supported by the Slovak Research and Development Agency under the Contract no. APVV-20-0281, and a project funded by the Ministry of Education of the Slovak Republic VEGA1/0308/20 ‘Mitigation of hydrological hazards, floods, and droughts by exploring extreme hydroclimatic phenomena in river basins’.

## STATEMENTS AND DECLARATIONS

The authors declare that this research is an original report and that this research has not been submitted to any journal.

## FUNDING

No organization provided funding to the writers for the work they submitted.

## AUTHOR CONTRIBUTIONS

The study's inception and design involved input from all authors. Material preparation was performed by M. G. and H. S. M., data analysis was performed by H. A.-E. and M. G., and the first draft and final versions were performed by M. Z. and W. N. All authors read and approved the final research.

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.

## CONFLICT OF INTEREST

The authors declare there is no conflict.