Developing improved machine learning methods to predict the flow characteristics through vertical and horizontal transitions in open channels

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

h₁

The number of neurons in hidden layer one

h₂

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

ΔZ

Transition height

Δb

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

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

R²

Coefficient of determination

STD

Standard deviation

SVM

Support vector machine

TSO

Tuna swarm optimization

WOA

Whale optimization algorithm

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.

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.

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.

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.

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.

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.

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 ΔZ/L were 0.025, 0.05, and 0.075, and Δb/B were 0.17, 0.33, and 0.5.

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 ΔZ/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 ΔZ/L as listed in Table 2.

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

Figure 8(c) shows a variation of F_u with the F_s for different contraction ratios Δb/B. The F_s value increases with the increasing values of F_u and the contraction ratio Δb/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

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*h₁+h₁)+(n*h₂+h₁*h₂+h₂)+(n+h₁+h₂+1)* m) where n is the number of inputs, h₁ and h₂ 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 h₁ = 10 and h₂ = 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).

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.

In the vertical transition case, R² = 0.7715, 0.6334, and 0.7193 which agrees with the R² = 0.9785, 0.9822, and 0.9803 calculated in the present study for ΔZ/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.

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.

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.

In the horizontal transition case, R² = 1 (Ashour et al. 2018) which agrees with the R² = 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.

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.

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.

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.

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 ΔZ/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 R² = 0.769 and RMSE = 0.177) and 0.91705 for the calculation of the transition water depth (with R² = 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 (R² = 0.8478, RMSE = 0.028) and 0.9198 for the Froude number calculation in the transition zone (R² = 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 (R²) 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 ΔZ/L were 0.025, 0.05, and 0.075, and Δb/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.

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’.

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

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

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.

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

The authors declare there is no conflict.