Impact of downstream obstructions on ogee weir efficiency: a regression analysis

Regression modelling for open channel flow also advances with the creation of equations tailored specifically for different kinds of channels (Ponce 1996). For rectangle channels, experts have proposed equations with coefficients to assess discharge capacity more precisely. Dhar & Nandargi (2000) and Henderson & Velleman (1981) developed regression-based techniques to determine coefficients that enable greater understanding of flow behavior.

Over the past decades, advances in machine learning and computational methods have opened up new ways to increase C_d accuracy and predictions. Artificial neural networks (ANNs) have gained increased recognition due to their capacity to recognize complex relationships within data. Researchers such as Yaseen et al. (2015) and Maciej Serda et al. (2013) utilized ANNs for predicting discharge coefficients in open channels, which demonstrated their efficiency at capturing nonlinear patterns. Support vector machines (SVMs) have proven invaluable when modeling open channels. Due to their capacity for handling large volumes of data, SVMs have shown great promise in the prediction of C_d values for research conducted by colleagues (Shen 2018) as well as Srivastava et al. (2021) SVMs were used by Ebtehaj et al. (2020) to make precise estimations of C_d for various channel geometries and flows, while studies also incorporate remote sensor data as well as Geographic Information System (GIS) methods with regression models in order to increase C_d accuracy. Hydraulic engineering relies heavily on accurately predicting discharge coefficient (C_d), the key metric in optimizing water conveyance systems. Chezy's and Manning's formulas established an initial platform, linking flow velocity with channel characteristics while partially accounting for complex flow dynamics around obstructions (Strupczewski 1996). As computational capabilities improved, regression models offered more nuanced insights into flow behavior – with researchers such as Dhar & Nandargi (2003) creating tailored equations personalized specifically for certain channel geometries; nonetheless they often struggled to capture nonlinear patterns or intricate relationships within this dataset. Machine learning has revolutionized C_d prediction, with techniques like ANNs and SVMs excelling at deciphering nonlinear interactions within datasets. Studies conducted by Yaseen et al. (2015) demonstrate the success of such models in increasing predictive accuracy; they demonstrate this efficacy, though some ‘black box’ approaches could obscure causal relationships between variables while their effectiveness requires extensive data for training – creating obstacles when faced with data scarcity scenarios. Maryami et al. (2024) have highlighted an impressive step forward through their integration of modern technologies like remote sensing and GIS with traditional regression models – this synergy allows for more comprehensive spatial data analysis, thus improving C_d predictions. However, applying such technologies does present unique challenges, including data acquisition/processing complexity as well as needing specific expertise. As much as the transition from empirical formulas to machine learning and integrated technologies is an advance in hydraulic engineering, each technique comes with inherent limitations from interpretability issues to data dependency which must be carefully taken into consideration before selecting an optimal predictive tool in any particular instance. Remote sensing technologies provide valuable sources of spatial data. Therefore, their use in regression models was investigated by researchers such as Mariye et al. (2024) as well as Maryami et al. (2021) provided an integrated approach to modeling open channel flow. Furthermore, calculation of discharge coefficient for open-channel flows was one of many topics explored within scientific literature. Traditional formulas like Manning's and Chezy can be combined with more modern machine learning approaches such as ANNs and SVMs, to predict climate data accurately. Regression models have also proven useful; scientists can utilize remote sensing data combined with GIS for accurate climate predictions (Shivappa Masalvad et al. 2023). This combination should further advance their capabilities. As this field expands, technological advancements in computational methods and the increased availability of high-resolution data will lead to further advances in predicting discharge coefficients for open channels (Singh & Kumar 1996). Based on an in-depth examination of how obstructions downstream affect discharge coefficient (C_d) over ogee weirs, this paper offers an innovative solution to one of the major challenges in hydraulic engineering. Worldwide readers can gain insights into the complex effects obstruction angle has on C_d; an aspect which had previously been ignored and missed altogether, thus improving flow rate predictions vital to optimizing hydraulic structures. Research questions centering around obstruction angles' effect on C_d and their efficiency as forecasting models are explored for change prediction purposes. The authors make significant contributions by applying advanced regression analyses such as Random Forest, LASSO and Bayesian Ridge models to produce highly accurate prediction models as evidenced by small root mean square error (RMSE) found within certain models. Breakthrough in hydraulic engineering's methodological approach with machine learning techniques being integrated is remarkable progress, while high-resolution experiment data increases credibility and real-world applicability of study findings. This research not only fills an important research gap by clarifying the complex relationship between downstream obstruction angles and discharge coefficients but also pioneers the integration of advanced regression and machine learning techniques in hydraulic engineering, providing an enhanced predictive framework which significantly aids design, construction and optimization processes of global hydraulic structures. Recent advances in ogee weir research and application of regression analysis techniques in hydraulic engineering have shed new light on how water flows through structures. These innovations offer further insights into water flow dynamics and structure performance. One experimental study performed in 2022 (Foroudi & Barati 2022) investigated the cavitation index of an ogee spillway and revealed its cavitation index value; further investigation uncovered how sidewall convergence angles play an essential part of spillway performance that helps avoid cavitation damage in hydraulic structures. At the same time, another 2022 study employed multi-genetic genetic programming with regression analysis techniques to enhance tunnel boring machine performance prediction – further showing its versatility (Kazemi & Barati 2022). An outstanding doctoral dissertation from 2023 (Luo 2023) offered comprehensive experimental and numerical modeling insights into both gated and ungated ogee spillways, emphasizing the necessity of accurate modeling during design and analysis. Notably, in 2023 a study (Rezazadeh Baghal & Khodashenas 2023) introduced a fuzzy risk-based design technique for ogee spillways found within diversion dams that integrated fuzzy logic with traditional design methodologies to produce resilient hydraulic structures able to adapt quickly in response to changing environmental conditions (Barati et al. 2014). Recent contributions highlight the field's rapid progress, showing how cutting-edge approaches and analytical techniques are making significant gains in understanding and optimizing hydraulic structures like ogee weirs for modern engineering challenges (Shahheydari et al. 2015; Nangare et al. 2024).

An analytical and systematic methodology was utilized in investigating the hydrodynamic impacts of various obstructions downstream of an ogee weir. The process began by collecting discharge data under various obstruction scenarios with angles 0°, 15°, 30° and 45° to understand changes in flow dynamics. Following data acquisition, an intensive preprocessing step was implemented to ensure data quality and prepare it for analysis, including filtering, cleaning, and structuring the information. Following exploratory data analysis (EDA), EDA utilized statistical examinations and visualization techniques to discover preliminary patterns or insights that set up subsequent modeling stages.

As part of its study on regression modeling, this research attempted to establish an accurate relationship between obstruction angles and discharge coefficients. For this reason, intensive training (70%) and testing (30%) sessions were carried out to validate its predictive power and accuracy before conducting comparative analyses on predicted and actual data across obstruction scenarios. Finally, an extensive performance assessment including R² values (for R² errors), mean squared errors calculations as well as others was completed to measure its efficacy for forecasting discharge coefficients under various hydraulic conditions.

A systematic investigation was conducted on the hydraulic performance of an ogee weir with downstream obstruction, specifically its variation in discharge coefficient (C_d). Experiments involved installing an obstruction downstream of the ogee weir at an angle of 15°, 30°, and 45° and collecting data under various flow conditions to understand how its presence affected its discharge coefficient – quantifying this variation with regression analysis and predicting the C_d values for different discharge. Results demonstrate a clear effect of downstream obstruction on the discharge coefficient (Figure 2). Data, collected through extensive experimentation, include measurements of flow rates, water levels and relevant parameters that characterize hydraulic performance in various conditions at ogee weirs; this serves as the foundation for regression analysis which establishes relationships between key influences on discharge coefficient and key influencing factors.

An obstruction 15° downstream on water levels or flow patterns at an ogee could have several significant ramifications for flooding situations. A downstream obstruction could alter the flow pattern of an ogee weir during floods, leading to changes in flow direction and velocity that cause changes to its profile on the water's surface. It could also produce backwater effects that alter the surface elevation downstream of an ogee weir, potentially increasing water levels upstream and impacting floodplains in some cases alter pressure distribution along the crest of a weir, while changes in flow conditions could alter its profile of pressure, potentially altering performance overall and the effectiveness of its use against flooding events. Obstacles to the ogee weir's energy dissipation capabilities may interfere with its capacity for efficient release of energy, leading to disruptions that reduce its ability to release it effectively and possibly leading to increases in turbulence or the formation of eddies downstream. An obstruction at 15° or any such angle could alter flow velocity over a weir's crest, depending on its specific shape and location of obstructions. These variations in flow velocity could potentially alter conveyance capacity provided by weirs.

Floodwaters create the potential for sediment or debris accumulation within obstruction areas that can hinder flow through weirs, potentially blocking some or all of their performance and leading to localized flooding downstream. This accumulation could reduce flow speed through weirs. Occlusions downstream of an ogee weir can alter its hydraulic jump, altering energy dissipation processes and stream stability, as well as ultimately impacting on performance of the hydraulic system overall. It is observed that at 30°, 45° inclined obstruction due to increase damming effect at the obstruction also it is observed that there is a tendency for hydraulic jump climbing the ogee, making the hydraulic structure susceptible to erosion. In order to better comprehend the effects of an obstruction at spatial variation, computational modeling with hydraulic programs or models can help. The obstruction on the downstream also encourages sediment deposits as shown in Figure 3, due to change in depth cause of damming (Figure 4). These tools enable simulation of conditions of flow and predict exact variations in water levels, speeds, and pressure distribution during floods. Understanding the effects of an obstruction on an ogee weir during flooding conditions is an essential aspect of both hydraulic design and flood risk management. A thorough study should consider all features related to its hydraulic and geometric design as well as general hydraulic behavior of ogee weirs; in depth hydraulic modeling analysis allows more precise assessment of its effect on flow conditions and water levels conditions.

If an obstruction lies with low angle such as 0° relative to flow direction it should cause no disruption as water moves toward its target destination, but may create local acceleration near an obstruction which leads to greater scouring downstream of weir. Overall discharge coefficient effects might not be as noticeable compared to more prominent obstructions with more acute angles. At intermediate angles (15°–30°), obstructions start to disrupt flow significantly, shifting it toward one end of the channel or diverting water around an uneven pattern across a weir. This may cause uneven pressure distribution as water flows around it unbalances, decreasing effectiveness. Furthermore, increased turbulence dissipation energy dispersion may occur which negatively impacts its discharge coefficient and efficiency of discharge coefficient. An orientation perpendicular to flow direction causes considerable disruption, as this creates significant obstruction effects downstream and backwater effects upstream from weirs that disrupt velocity, pressure distribution and distribution over crests of weirs, changing velocity distribution over weirs as it passes by them and changing pressure distribution across flow paths over weir crests. Obstruction could trigger upstream hydraulic surge localized downstream which increases dissipation while potentially decreasing discharge rate. Further, the discharge coefficient (C_d) for ogee weirs is a non-dimensional number which measures water flow across its surface in relation to geometry of weirs and height of water over their crests. Different obstruction angles may impact this value by altering flow speed, depth or pressure distribution when passing by its crests. An obstruction positioned at an angle or parallel with flow could increase C_d due to localized acceleration of flow; on the contrary, perpendicular and oblique obstructions tend to decrease it due to their increase in turbulent, energy dissipation, or altering patterns of flow. Obstructions at higher angles may encourage an upstream hydraulic jump from a weir, leading to sudden shifts between supercritical (high velocity, shallow depth) and subcritical flow, potentially leading to significant energy loss while potentially helping mitigate downstream erosion and possibly diminishing weir efficiency as it converts energy from kinetic to potential energy forms. This may affect downstream erosion as well as efficiency for energy conversion from potential to kinetic forms. Obstacles alter flow patterns, changing sediment transport dynamics and potentially leading to deposition downstream and possible scouring downstream of an obstruction, potentially altering channel geometry over time, negatively affecting hydraulic performance of our weir. Understanding how various obstruction orientations impact hydraulic performance is critical when designing and positioning weirs as well as bridge piers. Engineers should keep these effects in mind in order to make sure that hydraulic infrastructure functions optimally under various flow conditions and obstruction situations. Obstacles in relation to flow direction could significantly impact hydraulic performance in ogee weirs by altering flow patterns, discharge coefficients and energy dissipation rates – all which must be considered when planning and operating such hydraulic structures in order to preserve effectiveness and efficiency.

Regression analysis was utilized to examine the relationship between discharge coefficient and variables related to downstream obstruction, such as dimensions, flow rate and water level. Linear, polynomial and multivariate models were tested as representations for this variation observed over time. Regression analysis provided a thorough understanding of how each variable contributes to variations in discharge coefficient. Coefficients and statistical significance for each variable were studied to assess their influence; and finally, a regression model was selected based on R-squared index value, P value significance thresholds, and overall goodness of fit metrics. In this study root mean squared error is used and is shown in Table 1.

While reviewing the regression results illustrated by Figure 1, various aspects are noticed which deserve careful consideration. The scatter plots showing the relationship between velocity and Froude number at various obstruction angles show patterns which indicate any possible anomalies in certain approaches. Even while some degree of scatter may be expected, an improper distribution of data points indicates potential problems that could compromise the accuracy of regression models. These may include, among other issues, violations of regression assumptions such as homoscedasticity and normality of residuals as well as leverage points that adversely alter slope and intercept of regression lines. Additionally, variable obstruction angles could introduce interaction effects that cannot be captured with simple linear regression analysis. Due to these insights, caution must be exercised in selecting models or transformations that better represent physical processes under study. Furthermore, strict statistical standards must be upheld throughout to guarantee valid and reproducible conclusions are drawn from our methodology.

while (V) represents flow speed, (g) represents acceleration caused by gravity and (D) represents depth of flow. Considering the ogee with obstructions at its downstream end and flow conditions could be altered significantly, including changes to Froude's number with speed depending on various aspects. These could include obstructions such as trees or even human beings present downstream causing obstructions, as well as factors that influence Froude's number based on velocity such as weather (Figure 6).

Obstacles in the downstream can create flow constriction by restricting available cross-sectional area for flow as well as speed. Obstructions downstream may trigger backwater effects which alter the elevation of water surface as it flows downstream of a weir, altering its elevation over time and leading to backwater effects which alter elevation levels downstream from it (Figure 7). Alterations to the depth of flow caused by backwater effects could wreak havoc with Froude's number. An obstruction could impact how velocity spreads throughout a cross section of flowing. Variations in velocity distribution can have an impactful effect on how Froude's Number is computed (Figure 8). As it examines the complex effects of obstructions downstream on ogee weir hydraulic performance, our methodology relies heavily on applying numerous regression models as foundational elements. The study covers not only hydraulic dynamics and fluid dynamics but also explores advanced mathematical learning. In this study, several learning models are examined, from linear regression that provides an initial insight into linear connections present in our data to more sophisticated ensemble strategies like Random Forest and Gradient Boosting models, which excel at capturing nonlinear complexity and interactions among our predictive models. Regularization methods within LASSO and Ridge regression models add another level of sophistication to our models by penalizing complexity while mitigating risks associated with overfitting. They increase predictability while penalizing complexity, reducing overfitting risk and providing more accurate predictions. Each regression class offers a distinct angle on how the obstruction angle impacts the discharge coefficient and thus enhances our predictive abilities and understanding. Furthermore, the methodology section serves both as evidence of rigorous empirical research conducted and as a source of information about statistically using various models for hydroinformatics learning purposes.

The observed variation in discharge coefficient illustrates how downstream obstructions impacting ogee weir hydraulic performance can alter their hydraulic performance and therefore require attention for their proper functioning. The regression analysis illustrates that downstream obstructions have an immediate and negative effect on the discharge coefficient. These obstructions alter flow patterns and introduce additional complexity that reduces the efficiency of ogee weir in conveying water. The discharge coefficient C_d of an ogee is affected by many factors, such as the design of the weir as well as the water flow conditions, as well as whether downstream obstructions are present. If a downstream obstruction at 90° to the river's banks is considered, various influences can be observed in the C_d of the ogee weir. Obstructions in the downstream can lead to constrictions in flow, which affects the speed of approach and flow patterns around the weir. A constriction in the flow may alter the characteristics of discharge, possibly changing the C_d for the ogee. The presence of obstructions downstream could cause backwater effects that affect the elevation of the water's surface downstream of the weir. Variations in the profile of the water's surface may affect the flow conditions on the weir and, in turn, the C_d value. Obstructions in the downstream may cause swirls and turbulence to the water flowing downstream of the weir. Conditions that are turbulent can impact the dissipation of energy and the efficiency of the weir by affecting the C_d value. 90° downstream obstacles could result in local acceleration or deceleration of flow. Changes in the velocity of flow profiles can affect the flow at the crest of the weir as well as the C_d value. In this study of the hydraulic effects of obstacles on ogee weir efficiency, undiscovered uncertainties could include variables like complex turbulent flow patterns around obstructions which cannot be fully grasped by standard hydraulic models and which cannot be predicted accurately using standard methodologies or experiments. They could have an enormous effect on accurate discharge coefficient predictions but their measurement proves difficult due to inherent shortcomings within experiments or modeling methodologies.

To determine the impact of obstructions downstream at 90° on the C_d of an ogee, experimental and numerical models are carried out. The research would include the measurement or simulation of the flow conditions without and with obstructions and comparing outcomes to the C_d values. Laboratory experiments, field measurements or computer-generated fluid dynamics (CFD) simulations could give insight into the effect of obstructions downstream on our performance. It is important to remember that the results can differ depending on the particular characteristics of the weir, obstruction and conditions of flow. Furthermore, the judgment of a professional engineer and experience are usually needed to translate and apply the findings to the practical design and management issues. Regression models provide insight into key influencing factors, such as obstruction dimensions, flow rate, and water level. Knowing their relative significance enables designers and managers to more efficiently design and operate ogee weirs with downstream obstructions in mind. The findings offer useful insights for engineers and practitioners involved with the design and operation of hydraulic structures. Regression models serve as predictive tools to estimate discharge coefficient under various conditions, aiding decision-making and optimizing performance. This study opens the door for further investigations, including exploring different obstruction geometries, variations in flow regimes and sediment transport impacts. Additional investigations may further refine regression models and make them applicable across more scenarios. Due to the inherent uncertainties present in hydraulic engineering and beyond, this study's methods and findings offer useful guidance for future scientific endeavors that focus on dissecting complex systems subject to unpredictable variables. Recent research on the COVID-19 pandemic underscores the significance of taking measures to manage both parametric and non-parametric uncertainties caused by environmental effects, random changes, unknown properties within and external to systems in question. Advanced system identification and optimization tools have played an invaluable role in disentangled uncertainties associated with pandemic dynamics, as evidenced in studies dedicated to multidimensional identification and mathematical modelling of pandemic spread. These works, such as Tutsoy et al. (2021) and Tutsoy (2023), shed light on how uncertainties may significantly skew model predictions as well as real world outcomes. Hydroinformatics and epidemiological modeling come together at this intersection to show the universal challenge of managing uncertainty across fields, reinforcing the need for sophisticated tools and models to navigate natural and societal systems alike. Further investigation of both hydraulic engineering and epidemiological modeling are imperative if coveted to enhance predictive capabilities and adapt resilience under various types of uncertainties. The present work makes use of various machine learning algorithms in predicting discharge coefficient (C_d) under various obstruction scenarios in hydraulic structures like ogee weirs. However, an extensive discussion regarding cost functions and parameter update rules would add greater understanding to their operational mechanisms and optimization processes. Enlightening cost functions would shed light on how each model measures deviation or error from observed data, while unveiling parameter update rules could reveal how models adapt their parameters in response to costs. An effective validation section is also crucial, outlining testing methodologies such as cross-validation techniques for classifier testing as well as metrics used for validation such as accuracy, precision, recall or F1 scores. Class definition plays an integral part in learning processes – particularly within hydraulic contexts where classes might represent levels of efficiency or flow condition categories affected by obstruction angles.

In summary, results and discussions shed light on the hydraulic performance of ogee weirs with downstream obstructions, emphasizing the importance of regression analysis for understanding variations in discharge coefficient variation. The robustness and predictive accuracy of the regression models developed in this study were rigorously tested and validated to ensure their reliability for hydraulic engineering applications. The validation process involved a comprehensive evaluation of model performance using unseen datasets to simulate real-world scenarios. Various metrics, including the RMSE, were employed to quantify the prediction errors, providing a clear measure of model precision. For instance, the Random Forest regressor demonstrated exceptional accuracy with an RMSE of 0.005, signifying its high predictive capability. In contrast, traditional models like Decision Tree and XG BOOST exhibited higher RMSE values, indicating less precision. Furthermore, advanced regression models such as LASSO, Bayesian Ridge, and OMP achieved an RMSE of zero, reflecting perfect predictions under specific study conditions. This meticulous testing and validation phase not only affirmed the models' efficacy in predicting the discharge coefficient variations due to downstream obstructions but also highlighted the critical importance of choosing the right predictive tool tailored to the specific hydraulic context. The comparative analysis of different classifiers underscores the nuanced understanding required to navigate the complex interplay of factors influencing hydraulic structures, ultimately contributing to more informed and effective engineering decisions. Knowledge gained through this research contributes to advancements of hydraulic engineering practices as well as providing insight for designing and managing ogee weirs in real world applications. To examine how Froude's number varies with speed in cases of downstream obstructions in an ogee, simulations or experiments might be necessary. This involves measuring flow speeds under various conditions before analyzing obstructions and then calculating Froude's number for every situation. Maintaining an accurate understanding of how weir geometry, obstruction characteristics and flow conditions impact fluctuations is of critical importance in planning and management scenarios. Expert knowledge in hydraulic engineering should also be sought out as this will allow one to interpret and apply their findings correctly in real life scenarios.

This hydroinformatics research highlights unique insights and methodologies, outlining their application in water resource management. While these findings enhance existing knowledge, this research experienced limitations due to data granularity and model scalability issues; consequently, their interpretation should be approached with caution. Future endeavors should focus on integrating higher-resolution datasets and refining computational models in order to enhance precision and applicability. Furthermore, the study provides opportunities for further explorations in hydroinformatics as a field and IoT integration for real-time water resource monitoring and management, by acknowledging limitations and proposing potential ways forward for improvement of sustainable water resource management.

This article does not contain any studies with human participants or animals performed by any of the authors.

Informed consent was obtained from all individual participants included in the study.

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

The authors declare that they have no conflict of interest. Z.-F.Y. is an editorial board member for the Journal of Hydrodynamics and was not involved in the editorial review, or the decision to publish this article.