Sustainable Hydraulic Design of Water Structures Through Optimal Technical Pairing of Upstream Wing-Wall Geometry and Canal Inside Slopes: HEC-RAS Numerical Investigation

Abstract

Hydraulic structures disturb natural flow patterns, reducing water conveyance efficiency and increasing hydraulic energy losses, thereby affecting the sustainable management of water structures. Entrance-zone geometry, particularly upstream wing-wall configuration and canal inside slope, plays a critical role in controlling flow behavior, energy dissipation, upstream afflux, and hydraulic performance. However, the coupled effects of these geometric parameters have not been systematically investigated. Therefore, this study employed a validated HEC-RAS model to evaluate the combined influence of canal inside slope and upstream wing-wall configuration on the hydraulic performance of irrigation water structures and to support sustainable hydraulic design. Four wing-wall configurations (box, broken, curved, and splayed) and three canal inside slopes (1:1, 3:2, and 2:1) were analyzed under a fixed contraction ratio of 0.6 and upstream Froude numbers ranging from 0.12 to 0.18 under steady subcritical flow conditions. The model was validated against measurements from a 1:10 laboratory flume, demonstrating excellent agreement, with an average variation of 5.75% and coefficients of determination (R²) ranging from 0.97 to 0.99. Gradual entrance transitions significantly improved hydraulic performance by reducing flow disturbances and enhancing flow uniformity. For a canal inside slope of 1:1, the curved wing-wall configuration reduced relative heading-up and energy loss by 18.02% and 46.83%, respectively, whereas the splayed configuration achieved the best overall performance, with corresponding reductions of 27.63% and 73.11% compared with the conventional box configuration. Furthermore, dimensionless predictive equations were developed for the principal hydraulic performance indicators, achieving R² values of 0.96–0.99 and RMSE values of 0.001–0.01. The proposed framework improves water conveyance efficiency, minimizes hydraulic losses, and provides a validated, cost-effective numerical tool for evaluating alternative design scenarios, reducing reliance on extensive physical experimentation while supporting sustainable irrigation structures and long-term water resources management.

1. Introduction

The hydraulic behavior of water structures is governed by complex flow interactions, such as flow separation, turbulence generation, energy dissipation, and upstream afflux. So, accurate assessment of these hydraulic phenomena is of great importance for achieving optimal performance, minimizing any type of water losses. At the same time, flow transitions and turbulence effects further modulate the hydraulic efficiency and the needed flow stability through the structure. Since field measurements and laboratory experiments are usually time-consuming and costly, in addition to the difficulty of implementing them under the same natural working conditions on site, numerical modeling has become an effective and widely adopted approach for investigating complex hydraulic phenomena, enabling engineers to simulate flow conditions, evaluate alternative design scenarios, and optimize hydraulic performance under various geometric and operational conditions. Among the available hydraulic modeling platforms, the Hydrologic Engineering Center–River Analysis System (HEC-RAS) provides a reliable environment for simulating water-surface profiles, flow hydraulics, sediment transport, and water-quality processes [1,2,3,4,5,6,7]. Although the one-dimensional (1D) version of HEC-RAS computes cross-sectionally averaged hydraulic variables and cannot explicitly resolve three-dimensional flow structures, it remains a practical and computationally efficient tool for predicting bulk hydraulic performance indicators, including water-surface profiles, energy losses, and heading-up. Therefore, the present study employs HEC-RAS to investigate the combined effects of canal inside slopes and upstream wing-wall configurations on the hydraulic performance of water structures. The numerical model was validated against the experimental results of [8], with particular emphasis on relative energy loss and relative heading-up.

Recent experimental studies have highlighted the significant influence of structural geometry on localized hydraulic behavior around water structures. Fakhimjoo et al. [9] reported that flow velocities near piers and abutments may increase by up to 80%, depending on structural geometry. Similarly, Afzalimehr et al. [10] found that compact abutments generate more intense vortices, whereas Setyandito et al. [11] demonstrated that spill-through abutments provide more stable hydraulic conditions than vertical-wall configurations. Although these findings emphasize the importance of entrance geometry in controlling flow stability and hydraulic efficiency, they were obtained mainly for bridge abutments and cannot be directly applied to irrigation canal structures because of differences in geometry, hydraulic conditions, and operational objectives.

The influence of wing-wall geometry on hydraulic performance has also been widely investigated. Ashour et al. [12] showed that a transition angle of approximately 30° for broken-type wing walls provides an effective balance between hydraulic performance and construction cost. More recently, Ashour et al. [13] reported that combining a broken-type upstream wing wall with a 1H:1V canal inside slope reduced the relative velocity and relative heading-up by approximately 12% and 20%, respectively, compared with the conventional box-type configuration. However, that study was limited to a single wing-wall geometry and did not examine other entrance configurations or canal inside slope values.

Advanced numerical approaches for hydraulic analysis generally include one-dimensional (1D) models, such as HEC-RAS, and two- or three-dimensional computational fluid dynamics (CFD) models. CFD techniques, particularly large eddy simulation (LES), can accurately reproduce complex three-dimensional flow structures, turbulence characteristics, vortices, and local shear stresses around hydraulic structures [14,15]. However, these models require detailed geometric representation, extensive calibration, long computational time, and considerable computational resources, especially for large parametric investigations. In contrast, the 1D HEC-RAS model provides computational efficiency, requires relatively limited input data, and has been extensively validated for predicting bulk hydraulic characteristics, including water-surface profiles, energy losses, flow velocities, and heading-up [1,2,3,4,5,6,7]. Therefore, considering that the objective of the present study is to evaluate the overall hydraulic performance of different entrance configurations rather than resolve detailed turbulence structures, HEC-RAS was considered an appropriate and practical numerical tool.

Water structures also exert a significant influence on upstream and downstream flow conditions. Atabay et al. [16] reported that bridge geometry and upstream Froude number strongly affect backwater levels, whereas Abdel-Aal et al. [17] showed that increasing the Froude number increases scour potential downstream of hydraulic transitions. Similarly, [18,19,20] emphasized the influence of seismic loading and hydraulic transients on canal slope stability. Although these loading conditions are beyond the scope of the present study, they highlight the importance of minimizing hydraulic losses and heading-up to improve hydraulic performance and structural stability.

Detailed laboratory investigations have further revealed the complexity of three-dimensional flow around entrance structures. Barbhuiya et al. [21,22,23] identified upstream and downstream vortices, irregular wake regions, and substantial variations in bed shear stress around wing-wall abutments. Although one-dimensional numerical models cannot explicitly reproduce these localized flow structures, their integrated hydraulic effects, including increased energy loss and upstream heading-up, are reflected in cross-sectionally averaged hydraulic parameters that can be reliably predicted by HEC-RAS.

Hydraulic analyses have also examined the influence of geometric parameters on afflux, flow characteristics, and energy dissipation. Hadi et al. [24] demonstrated that increasing the opening ratio improves HEC-RAS predictions and shifts the maximum afflux farther upstream. Similarly, Ghodsian et al. [25] reported that wing-wall angles significantly influence scour development downstream of box culverts. Ahmed et al. [26] highlighted the effects of non-uniform velocity distributions and increased bed shear stress near abutment noses, whereas Xiao et al. [27] demonstrated that steeper canal inside slopes promote more uniform shear-stress distributions by reducing momentum exchange. Collectively, these studies confirm that both entrance geometry and canal inside slopes play important roles in controlling hydraulic losses and sediment-related processes.

From a geotechnical perspective, considerable attention has been devoted to canal inside slopes, particularly regarding slope stability, erosion resistance, and failure mechanisms. Previous investigations examined the influence of soil properties, slope geometry, and hydraulic loading on embankment stability [19,28,29,30,31,32,33]. However, these investigations focused primarily on geotechnical performance, with limited consideration of the hydraulic interaction between canal inside slopes and upstream entrance structures.

Although previous studies have provided valuable insights into the hydraulic performance of water structures, canal inside slopes and upstream wing-wall configurations have generally been investigated independently, with insufficient attention given to their combined interaction within a unified hydraulic system. While Ashour et al. [13] partially addressed this interaction for a broken-type wing wall combined with a canal inside slope of 1:1, the hydraulic performance of other wing-wall configurations over different canal inside slopes has not been systematically investigated. Consequently, a comprehensive understanding of the combined effects of these parameters on energy loss, heading-up, velocity distribution, afflux, and overall hydraulic efficiency remains unavailable. This knowledge gap limits the development of integrated and evidence-based design recommendations for improving the hydraulic performance of irrigation water structures.

To address this knowledge gap, the present study investigates the combined effects of canal inside slopes and upstream wing-wall configurations on the hydraulic performance of water structures using the HEC-RAS numerical model under steady subcritical flow conditions. The study was conducted in two stages. First, the numerical model was validated against physical model measurements obtained from a 1:10 scale recirculating flume to evaluate its capability in predicting relative energy loss and relative heading-up. Second, a comprehensive numerical investigation was performed for four upstream wing-wall configurations (box, broken, curved, and splayed) combined with three canals inside slopes (1:1, 3:2, and 2:1) under a fixed contraction ratio of 0.6 and upstream Froude numbers ranging from 0.12 to 0.18. Finally, empirical predictive equations were developed as functions of the upstream Froude number and canal inside slope to facilitate preliminary hydraulic design. The findings are expected to provide practical guidance for selecting optimized entrance configurations that improve flow conditions, reduce hydraulic losses, and enhance the overall hydraulic performance of irrigation water structures.

The present study also supports sustainable water resources management by providing an integrated hydraulic design framework that improves water conveyance efficiency while reducing hydraulic losses and unnecessary upstream afflux. Furthermore, employing a validated numerical model enables extensive evaluation of alternative design scenarios with substantially lower material consumption, time, and cost than physical modeling alone, thereby contributing to sustainable engineering practice.

2. Materials and Methods

2.1. Methodological Framework

The adopted methodology for investigating the influence of entrance-zone configurations on the hydraulic performance of water structures using the HEC-RAS (Version 6.5) steady-flow model follows the framework illustrated in Figure 1. The study begins with data collection and processing, including the geometric characteristics of the investigated configurations and the associated hydraulic data. Subsequently, the research problem and knowledge gap are identified, and the key governing parameters, namely upstream wing-wall configuration and canal inside slopes, are selected for further investigation. The study is conducted under steady subcritical flow conditions, with the parametric scope limited to four wing-wall configurations (box, broken, curved, and splayed), three canal inside slopes (Z = 1:1, 3:2, and 2:1), and a fixed contraction ratio of 0.6.

A theoretical framework is then established to provide the basis for the hydraulic analysis through dimensional analysis, from which the governing dimensionless parameters are identified. The methodology proceeds through two complementary components. The first component involves numerical modeling using HEC-RAS, where the geometric data and steady-flow conditions are incorporated into the model to perform hydraulic simulations and analyze the numerical results. The second component consists of a physical model program, conducted at 1:10 scale, including the experimental setup, experimental runs, measurement data collection, and experimental data analysis as described in detail in [8].

The results obtained from both approaches are integrated for model validation and performance assessment using appropriate statistical indicators and error metrics, including the average variation (ε%), coefficient of determination (R²), and root mean square error (RMSE). Following the achievement of acceptable agreement between the numerical and physical model results, the validated HEC-RAS model is used to analyze the hydraulic behavior of the investigated configurations across the full parametric matrix. Finally, regression analysis is conducted to develop empirical predictive equations as functions of the upstream Froude number (Fr₁) and canal inside slope (Z), and the study concludes with evidence-based conclusions and practical engineering recommendations applicable within the investigated parametric range.

2.2. Dimensional Analysis

The dimensional analysis was conducted to identify the governing parameters affecting the hydraulic behavior of water structures and to establish dimensionless relationships between the influencing variables. This approach reduces the complexity of the hydraulic problem by expressing the flow characteristics in terms of non-dimensional parameters, allowing a generalized interpretation of the physical model and numerical results that is independent of model scale. In addition, this framework facilitates the development of predictive equations for key hydraulic performance indicators specifically, relative energy loss (ΔE/y₁) and relative heading-up (h_u/y₁) that are applicable across a range of operating conditions within the investigated parametric bounds.

Figure 2 presents a schematic representation of the physical model employed in this study, illustrating the governing parameters affecting canal contraction. These parameters are summarized in

Table 1. Governing hydraulic variables and their fundamental dimensions.

Symbol	Definition	Unit	Fundamental Dimensions (M L T)
B	Canal bed width	m	L
b	Bed width through structure	m	L
Fr	Froude number	–	–
g	Gravitational acceleration	m/s2	L T−2
hu	Afflux (heading-up)	m	L
L	Transition length	m	L
Q	Discharge	m3/s	L3 T−1
r = b/B	Contraction ratio	m/m	–
S	Canal bed slope	m/m	–
v1	Upstream velocity	m/s	L T−1
v2	Velocity through structure	m/s	L T−1
v3	Downstream velocity	m/s	L T−1
y1	Upstream water depth	m	L
y2	Water depth through structure	m	L
y3	Downstream water depth	m	L
Z (H:V)	Canal inside slope	m/m	–
ΔE	Total energy loss	m	L
θ	Wing-wall transition angle	degree	–
ρ	Water density	kg/m3	M L−3
μ	Dynamic viscosity	kg/m·s	M L−1 T−1
σ	Surface tension	kg/s2	M T−2
ΔE/y1	Relative energy loss	m/m	–
hu/y1	Relative heading-up	m/m	–
y2/y1	Relative water depth	m/m	–
v2/v1	Relative water velocity	m/s/m/s	–

; note that all variables marked with an asterisk in Table 1 were held constant throughout the investigation, as discussed below.

The following generic form can be used to describe the hydraulic parameters as functions of the governing variables:

$$f\left(\rho , \mu , \sigma , y_1{, y}_2, {hu, v}_1, v_2, Q, g, r, {\Delta E}_1, B, b, \theta , L, Z, S\right)=0$$

(1)

By choosing $\rho$, $v_1$ and $y_1$ as the repeating variables, Equation (1) is converted into a dimensionless form (Equation (2)) using Buckingham’s π theorem. These variables cover the fundamental dimensions (M, L, and T), that describe the fluid characteristics, flow inertia, and characteristic length scale at the structure entry, and allow the construction of dimensionless groups that control the equilibrium between gravitational and inertial forces.

$$f\left({\displaystyle \frac{h_u}{y_1}}, {\displaystyle \frac{y_2}{y_1}}, {\displaystyle \frac{b}{y_1}}, {\displaystyle \frac{B}{y_1}}, {\displaystyle \frac{L}{y_1}}, {\displaystyle \frac{\Delta E1}{y_1}}, {\displaystyle \frac{v_2}{v_1}}, {\displaystyle \frac{{\rho v_1}^2{y}_1}{\sigma }}, {\displaystyle \frac{\mu }{\rho v_1{y}_1}}, {\displaystyle \frac{v_1}{\surd g{y}_1}}, {\displaystyle \frac{Q}{\sqrt{{g{y}_1}^5}}}, \theta , Z, S\right)=0$$

(2)

where ${\displaystyle \frac{\mathsf{\rho }{\mathrm{v}}_1{\mathrm{y}}_1}{\mathrm{\mu }}}=Re$ (Reynolds number), ${\displaystyle \frac{v_1}{\surd g{y}_1}}=F_{r1}$ (Froude number at the upstream section), ${\displaystyle \frac{Q}{\sqrt{{g{y}_1}^5}}}=Q*$ (discharge factor), ${\displaystyle \frac{{\mathsf{\rho }{v}_1}^2{y}_1}{\mathsf{\sigma }}}=We$ (Weber number), and ${\displaystyle \frac{b}{y_1}}/{\displaystyle \frac{B}{y_1}}=$r (contraction ratio). Then Equation (2) can be written in the following form:

$$f\left({\displaystyle \frac{h_u}{y_1}}, {\displaystyle \frac{y_2}{y_1}}, {\displaystyle \frac{L}{y_1}}, {\displaystyle \frac{\Delta E1}{y_1}}, {\displaystyle \frac{v_2}{v_1}}, r, Re, We, F_{r1}, Q*, \theta , Z, S\right)=0$$

(3)

Since gravity dominates the flow and all physical model runs were carried out under fully turbulent conditions (Re > 10⁴), viscous effects are considered insignificant for the purpose of the current analysis. Surface tension effects are also neglected, as all runs were conducted under controlled laboratory temperature conditions with negligible variation, and the Weber number remained sufficiently large to render surface tension forces insignificant relative to inertial forces.

To isolate the combined effects of upstream wing-wall configuration and canal inside slope, several hydraulic and geometric parameters were intentionally kept constant throughout the dimensional analysis and numerical simulations. These include the contraction ratio (r = 0.6), entrance transition angle (θ = 30°), bed slope (S = 0), and channel roughness. The adopted contraction ratio represents one of the most hydraulically critical conditions reported in previous studies [12], where flow contraction, energy loss, and upstream afflux become more pronounced, while also being widely recognized as an economical design value for irrigation water structures. Similarly, the selected entrance angle corresponds to a commonly adopted engineering configuration, whereas a horizontal bed was considered to eliminate the influence of longitudinal slope and facilitate a clearer interpretation of the investigated variables. Consequently, the proposed dimensionless relationships are applicable within the investigated parameter ranges, and extrapolation beyond these conditions should be undertaken only after appropriate validation.

Under these conditions, the upstream Froude number (Fr₁) emerges as the primary governing dimensionless parameter controlling water surface response, flow stability, heading-up, and energy loss within the investigated subcritical flow range (Fr₁ = 0.12–0.18). The resulting reduced functional relationship is expressed as Equation (4), which forms the basis for the subsequent regression analysis and empirical equation development.

$$f\left({\displaystyle \frac{h_u}{y_1}}, {\displaystyle \frac{y_2}{y_1}}, {\displaystyle \frac{\Delta E1}{y_1}}, {\displaystyle \frac{v_2}{v_1}}, F_{r1}, Q*, Z\right)=0$$

(4)

2.3. HEC-RAS Modeling Procedure

The public-domain software HEC-RAS (Hydrologic Engineering Center–River Analysis System) is widely used for performing hydraulic simulations under both steady and unsteady flow conditions [34,35]. The software is capable of computing water surface profiles for steady flow, in addition to simulating sediment transport and water quality processes. In the present study, the steady-flow module of HEC-RAS was employed, to simulate steady subcritical flow and assess bulk hydraulic behavior under different canal inside slopes (1:1, 3:2, and 2:1) in combination with four upstream wing-wall configurations (box, broken, curved, and splayed), as shown in Figure 3, Figure 4 and Figure 5. The modeling approach was adopted on the basis that the primary performance indicators of interest—relative energy loss (ΔE/y₁) and relative heading-up (h_u/y₁)—are bulk, cross-sectionally averaged parameters that can be adequately estimated using simulations.

The model development process was carried out through four main stages, as described below. First, the geometric data representing the physical model configurations were prepared and incorporated into the model framework, as illustrated in Figure 3. The canal cross-sections were defined as trapezoidal, with bed width B = 0.30 m and the canal inside slope varying across the three investigated values (Z = 1:1, 3:2, and 2:1). The structural opening had a bed width of b = 0.18 m, yielding a contraction ratio of r = b/B = 0.6. Manning’s roughness coefficients (n) of 0.020, 0.040, and 0.014 were adopted for the concrete channel bed, sand layer in the bed, and smooth side slope surface, respectively, in accordance with the guidelines provided in the HEC-RAS Hydraulic Reference Manual [34,35].

Next, the required steady-flow data were defined and entered into the model as boundary and hydraulic conditions, as shown in Figure 4. The six flow discharge values simulated were 5.46, 8.01, 13.01, 17.76, 21.78, and 24.90 L/s, corresponding to the discharge range used in the physical model experiments and covering upstream Froude numbers from approximately 0.12 to 0.18. A normal depth boundary condition was applied at the downstream end of the model. At the upstream boundary, the known flow discharge was specified for each simulation profile. All simulations were performed under subcritical flow conditions. Finally, the simulation outputs, including water surface profiles, cross-sectional velocity distributions, and energy grade line elevations, were extracted, tabulated, and analyzed to assess the flow characteristics of each configuration and provide a systematic interpretation of the hydraulic behavior of the system. A total of 360 simulation runs were conducted, representing four wing-wall configurations, three canal inside slopes, six discharge values, and five water depths.

2.4. Experimental Works

The HEC-RAS model was validated against physical model measurements obtained from a 1:10 scale laboratory model.

A geometric scale of 1:10 was adopted to achieve a suitable balance between laboratory limitations and hydraulic similitude requirements. The selected scale allowed the entire physical model to be accommodated within the available laboratory flume while maintaining adequate flow depths and velocities for accurate measurements and minimizing potential scale effects. Since the present study investigates free-surface flow, Froude similitude was adopted as the governing similarity criterion to ensure dynamic similarity between the model and the prototype. The laboratory flume width of 0.30 m corresponds to a prototype canal width of 3.0 m, which is representative of medium-sized irrigation canals commonly encountered in practice. Accordingly, all geometric dimensions were scaled by the same ratio. The tested model flow depths ranged from 0.125 to 0.255 m, corresponding to prototype depths of 1.25 to 2.55 m, while the tested discharges represent prototype flows ranging from 1.73 to 7.31 m³/s.

The experiments were conducted using a recirculating flume system installed in the Irrigation and Hydraulics Laboratory, Civil Engineering Department, Assiut University, Egypt. A general view of the physical model setup is presented in Figure 6. The physical model focused on key hydraulic parameters under six discharge values of 5.46, 8.01, 13.01, 17.76, 21.78, and 24.90 L/s, corresponding to upstream Froude numbers in the range of approximately 0.12 to 0.18. Water surface elevations were measured at specified cross-sections upstream, within, and downstream of the structure using an electrical point gauge with an accuracy of ±0.1 mm, from which the key dimensionless hydraulic parameters, including ΔE/y₁, h_u/y₁, y₂/y₁, and v₂/v₁, were derived. These physical model measurements served as the reference dataset for evaluating the accuracy and reliability of the numerical simulations. The key validation parameters and their computed agreement statistics are reported in Section 3.1. Detailed descriptions of the physical model configuration, including the flume dimensions, model geometry, instrumentation, and experimental procedures, can be found in [8].

2.5. Uncertainty Assessment and Statistical Analysis

The reliability of the physical model measurements was assessed through an uncertainty analysis of both measured and derived hydraulic parameters. Water depths were measured using an electrical point gauge with an accuracy of ±0.1 mm; flow discharge was regulated and measured using a calibrated V-notch weir. Based on the measured water-depth range (12.5–25.5 cm), the corresponding relative uncertainty in depth measurement varied between ±0.04% and ±0.08%.

The estimated uncertainty ranges were ±(0.01–0.76)% for the relative energy loss (ΔE₁/y₁), ±(0.03–0.69)% for the relative heading-up (h_u/y₁), ±(0.31–0.68)% for the relative water depth ratio (y₂/y₁), and ±(0.88–1.22)% for the relative velocity ratio (v₂/v₁). The relatively higher uncertainty in v₂/v₁ (up to ±1.22%) is noted and is attributable to the combined propagation of velocity and depth measurement errors through the continuity equation. This level of uncertainty is considered acceptable for the present investigation but should be borne in mind when interpreting comparisons between configurations where velocity differences are small. These values indicate an acceptable level of measurement accuracy and support the reliability of the obtained results.

To evaluate the individual and interactive effects of upstream wing-wall configuration and canal inside slope on hydraulic performance, a two-way analysis of variance (ANOVA) was conducted for the four principal dimensionless hydraulic parameters: relative energy loss (ΔE/y₁), relative heading-up (h_u/y₁), relative downstream water depth (y₂/y₁), and relative velocity ratio (v₂/v₁). The analysis employed a 4 × 3 factorial designs comprising four wing-wall configurations and three canal inside slopes, with six discharge levels serving as replicates for each treatment combination (n = 6). The ANOVA model included both main effects and their interaction term (wing-wall configuration × canal inside slope) to assess whether the hydraulic response to entrance geometry depends on canal slope conditions. When statistically significant differences were identified, Tukey’s honestly significant difference (HSD) post-hoc test was applied to determine the specific groups responsible for the observed variations. Statistical significance was evaluated at the 5% level (α = 0.05). Detailed ANOVA results, including degrees of freedom, sums of squares, mean squares, F-values, p-values, and effect sizes, are presented in Section 5.

3. Results

This section presents the results of the HEC-RAS numerical simulations and their comparison with the physical model measurements of Ashour et al. [8]. The analysis was carried out to investigate the bulk hydraulic behavior of water structures under four different upstream wing-wall configurations combined with three different canal inside slope values. Particular attention was given to the influence of the investigated parameters on the following important hydraulic performance indicators: relative energy loss (ΔE/y₁), relative heading-up (h_u/y₁), water surface profile, and cross-sectional flow velocity distribution. It is important to recall that the HEC-RAS model computes cross-sectionally averaged hydraulic variables; therefore, all results presented herein reflect flow behavior. The results are organized into two subsections: Section 3.1 presents the model validation against physical model data, and Section 3.2 presents the parametric HEC-RAS analysis of the combined effects of wing-wall configuration and canal inside slope on hydraulic performance under steady subcritical flow conditions.

3.1. Model Validation and Comparison with Experimental Measurements

To assess the accuracy of HEC-RAS model, the computed relationships between the upstream Froude number (Fr₁) and the key dimensionless hydraulic parameters, namely relative energy loss (ΔE/y₁), relative heading-up (h_u/y₁), relative downstream water depth (y₂/y₁), and relative velocity (v₂/v₁), were compared against the physical model measurements as shown in

Table 2. Experimental results for the box-type wing-wall configuration with a 1:1 canal inside slope.

Run No.	Q (lit/s)	y1 (cm)	y2 (cm)	y3 (cm)	hu (cm)	hu/y1	y2/y1	V1 (m/s)	V2 (m/s)	V2/V1	Fr1 (U.S)	E1 (m)	E2 (m)	∆E/y1
1	5.46	14.50	14.20	14.30	0.20	0.0138	0.9793	0.0846	0.2136	2.5245	0.0709	0.1454	0.1443	0.0072
2	8.01	15.60	15.00	15.20	0.40	0.0256	0.9615	0.1126	0.2967	2.6347	0.0910	0.1566	0.1545	0.0138
3	13.01	18.20	17.05	17.30	0.90	0.0495	0.9368	0.1483	0.4239	2.8584	0.1110	0.1831	0.1797	0.0190
4	17.76	19.90	18.00	18.10	1.80	0.0905	0.9045	0.1789	0.5481	3.0648	0.1280	0.2006	0.1953	0.0267
5	21.78	21.05	18.30	18.65	2.40	0.1140	0.8694	0.2027	0.6612	3.2623	0.1410	0.2126	0.2053	0.0347
6	24.90	21.95	18.35	18.80	3.15	0.1435	0.8360	0.2184	0.7539	3.4523	0.1488	0.2219	0.2125	0.0431

and

Table 3. HEC-RAS results for the box-type wing-wall configuration with a 1:1 canal inside slope.

Run No.	Q (lit/s)	y1 (cm)	y2 (cm)	y3 (cm)	hu (cm)	hu/y1	y2/y1	V1 (m/s)	V2 (m/s)	V2/V1	Fr1 (U.S)	E1 (m)	E2 (m)	∆E/y1
1	5.46	14.54	14.25	14.30	0.24	0.0165	0.9801	0.0930	0.2127	2.2871	0.0778	0.1458	0.1448	0.0069
2	8.01	15.68	15.12	15.21	0.47	0.0300	0.9643	0.1240	0.2941	2.3718	0.1000	0.1574	0.1556	0.0115
3	13.01	18.23	17.07	17.31	0.92	0.0505	0.9364	0.1658	0.4230	2.5513	0.1240	0.1835	0.1799	0.0197
4	17.76	19.71	17.67	18.11	1.60	0.0812	0.8965	0.2044	0.5578	2.7290	0.1471	0.1989	0.1926	0.0320
5	21.78	20.95	17.97	18.65	2.30	0.1098	0.8578	0.2312	0.6728	2.9100	0.1613	0.2118	0.2028	0.0430
6	24.90	21.84	17.89	18.81	3.03	0.1387	0.8191	0.2501	0.7727	3.0896	0.1709	0.2210	0.2093	0.0536

. Figure 7, Figure 8, Figure 9 and Figure 10 present these comparisons for the box-type wing-wall configuration with Z = 1:1, selected as a representative case.

For this configuration, increasing Fr₁ produced consistent increases in both (ΔE/y₁) and (h_u/y₁), a modest increase in (v₂/v₁), and a progressive decrease in (y₂/y₁). This behavior is physically consistent with the expected response of a contracting channel section to increasing flow intensity: higher discharges generate greater flow acceleration and turbulence, leading to increased energy losses and upstream water level rise. Satisfactory agreement was observed between the physical model measurements and the HEC-RAS numerical results, with average variations of 3.56% for h_u/y₁, 0.65% for y₂/y₁, 8.38% for ΔE/y₁, and 10.40% for v₂/v₁, as calculated using Equation (5). The relatively higher variation for v₂/v₁ (10.40%) is consistent with the known limitation of models in resolving cross-sectional velocity distributions, where the model computes a single average velocity rather than the actual non-uniform velocity profile. This value remains within the range considered acceptable for hydraulic modeling in similar studies. Slight deviations are observed at higher Froude numbers, where the model systematically tends to underestimate both energy loss and heading-up. This systematic underestimation at higher Fr₁ values is likely attributable to the increasing importance of three-dimensional flow effects, including stronger vortex formation and flow separation, at higher discharges, which cannot be fully captured by cross-sectional averaging.

The percentage variation (ε) between measured and computed values was calculated using:

$$\mathit{Variation} (\epsilon )\% ={\displaystyle \frac{ Measured-Computed }{Measured}}\times 100$$

(5)

The statistical evaluation demonstrated a good agreement between the experimental measurements and the HEC-RAS predictions. Among the investigated hydraulic parameters, the sequent depth ratio (y₂/y₁) showed the highest level of accuracy, yielding a mean percentage bias (MPB) of 0.65% and a mean absolute percentage error (MAPE) of only 0.77%, which indicates an almost unbiased prediction. The relative heading-up ratio (hu/y₁) and the relative energy loss ratio (ΔE/y₁) also exhibited satisfactory agreement, with MPB values of −3.56% and −8.38% and corresponding MAPE values of 9.32% and 9.72%, respectively. The velocity ratio (v₂/v₁) produced the largest deviation, with both MPB and MAPE equal to 10.40%; however, this level of error remains within acceptable limits for hydraulic modeling applications. Overall, the low MPB values and relatively small MAPE values confirm that the HEC-RAS model is capable of reproducing the hydraulic characteristics of the studied flow conditions with satisfactory accuracy and limited systematic bias.

Figure 11 and Figure 12 present the correlation between physical model measurements and HEC-RAS computed values for ΔE/y₁ and h_u/y₁, respectively, pooled across all investigated configurations and canal inside slope values. The agreement between the measured and computed values of ΔE/y₁ produced a coefficient of determination (R²) of 0.9748 with a RMSE of 0.007, while h_u/y₁ produced an R² of 0.9953 and RMSE of 0.009. These statistics indicate that the HEC-RAS model reproduces the bulk hydraulic behavior of the investigated configurations with acceptable accuracy for the purposes of parametric design comparison. In both cases, the data points were closely distributed around the 1:1 agreement line, further indicating the capability of the HEC-RAS model to adequately reproduce the investigated bulk hydraulic parameters.

3.2. HEC-RAS Analysis of Hydraulic Behavior Under the Combined Effects of Wing-Wall Configuration with Different Canal Inside Slopes

The HEC-RAS results presented in Figure 13, Figure 14, Figure 15 and Figure 16 demonstrate that both the canal inside slope and upstream wing-wall configuration have a significantly influence on the hydraulic performance of the structure, particularly in terms of relative energy loss (ΔE/y₁) and relative heading-up (h_u/y₁). The quantitative comparisons presented in this section are based on average values computed across the six simulated discharge levels (Fr₁ = 0.12–0.18).

For all investigated configurations, ΔE/y₁ and h_u/y₁ increased progressively with increasing Fr₁. This behavior is attributed to the increases in bulk flow velocity and momentum exchange near the transition zone at higher discharges, which intensify flow disturbances. Consequently, the model reflects these effects through greater computed hydraulic resistance and energy losses and larger upstream water surface rise. It should be noted that claims regarding flow separation, vortex formation, and turbulence dissipation as direct mechanisms cannot be verified from HEC-RAS output alone. These are inferred physical explanations consistent with the bulk hydraulic trends observed; confirmation of these localized mechanisms would require higher resolution 2D or 3D numerical modeling.

The results also indicate that a transition from Z = 1:1 to Z = 2:1 causes a consistent increase in both ΔE/y₁ and h_u/y₁ for all wing-wall configurations, as shown in Figure 15 and Figure 16. Physically, wider canal side slopes expand the wetted flow cross-section near the structure entrance, increasing the interaction between the approaching flow and the wing walls. This interaction is associated with greater hydraulic resistance and energy dissipation. Moreover, the wider transition geometry reduces the flow contraction efficiency, causing greater afflux upstream of the structure.

Regarding wing-wall configurations, the box-type configuration produced consistently the highest values of both relative energy loss and heading-up across all investigated canal inside slope values and discharge levels. This is attributed to the abrupt geometric transition associated with the box-type configuration, which produces a sharp change in flow direction at the structure entrance. Compared with the box-type configuration, the broken-type wing wall reduced h_u/y₁ and ΔE/y₁ by approximately 10.89% and 28.62%, respectively; these values represent averages computed across the six discharge levels at Z = 1:1 and are consistent with the trend reported by Ashour et al. [13] for the same configuration. The curved-type wing wall provided greater improvement by reducing h_u/y₁ by approximately 18.02% and ΔE/y₁ by approximately 46.83%. The splayed-type wing wall achieved the best hydraulic performance, with reductions of approximately 27.63% in h_u/y₁ and approximately 73.11% in ΔE/y₁ compared with the box-type configuration. These results apply specifically to the Z = 1:1 canal inside slope condition; the corresponding improvement percentages for Z = 3:2 and Z = 2:1 are presented and discussed in the end of this section.

The performance hierarchy across configurations, splayed > curved > broken > box, was consistent across all three canal inside slope values, indicating that the relative hydraulic advantage of gradual entrance transitions is robust to changes in canal geometry within the investigated range. The superior performance of the splayed-type wing wall in terms of bulk energy loss and heading-up reduction is consistent with the gradual expansion geometry of this configuration, which is expected to produce smoother flow guidance and reduced hydraulic resistance at the structure entrance. In contrast, the broken- and curved-type wing walls exhibited intermediate hydraulic behavior, with the curved configuration consistently outperforming the broken type. This performance difference is reflected in the lower ΔE/y₁ and h_u/y₁ values computed for the curved configuration relative to the broken type across all Fr₁ values and canal inside slopes investigated.

Figure 17 and Figure 18 illustrate the influence of wing-wall configurations on the relative water depth (y₂/y₁) and relative velocity (v₂/v₁) at canal inside slope Z = 1:1. The results indicate that y₂/y₁ decreases with increasing upstream Froude number for all investigated wing-wall configurations. A decreasing y₂/y₁ with increasing Fr₁ indicates that the downstream depth becomes progressively smaller relative to the upstream depth, reflecting greater energy dissipation and flow acceleration through the contracted section. The box-type configuration exhibited the highest y₂/y₁, reflecting greater hydraulic losses associated with abrupt flow contraction. In contrast, the splayed-type wing wall produced the lowest y₂/y₁ values due to its gradual transition geometry, which is associated with reduced hydraulic resistance and improved hydraulic efficiency.

The relative velocity ratio (v₂/v₁) also increases progressively with increasing Fr₁. This trend is physically expected: as Fr₁ increases, the flow accelerates more strongly through the contracted section, producing higher velocity ratios regardless of wing-wall configuration. The splayed-type wing wall consistently produces the lowest velocity ratios compared with the box-type configuration, reflecting the lower hydraulic resistance associated with its gradual entrance geometry. Meanwhile, the box-type configuration generates the highest velocity ratios due to its abrupt entrance geometry and the associated greater contraction effects.

Figure 19 presents the water surface profiles computed for the box-type and splayed-type configurations at Z = 1:1 under the maximum simulated discharge (PF 6). The splayed-type configuration produces a comparatively smoother surface profile with reduced afflux compared with the box-type configuration. The difference in upstream water surface elevation between the two configurations reflects the heading-up reduction quantified in Section 3.2. Moreover, the cross-sectional velocity distributions presented in Figure 20 and Figure 21 show that the maximum velocities are concentrated near the channel centerline and increase within the contracted section due to flow acceleration, while the computed velocity distribution remain relatively uniform across the section. This behavior is consistent with the lower ΔE/y₁ and h_u/y₁ values computed for the splayed-type configuration and supports the conclusion that gradual entrance geometries reduce hydraulic resistance within the investigated parametric range.

Figure 22, Figure 23 and Figure 24 present the percentage improvement in relative energy loss (ΔE/y₁) and relative heading-up (h_u/y₁) for each upstream wing-wall configuration, compared with the box-type configuration, under the three canal inside slopes (Z = 1:1, 3:2, and 2:1). The results confirm that the splayed-type wing wall consistently achieves the greatest reduction in both ΔE/y₁ and h_u/y₁ among all investigated configurations, followed in order by the curved and broken configurations. The maximum percentage reduction in relative energy loss reached approximately 73.11% for the splayed-type configuration under the canal inside slope of Z = 1:1, with a corresponding reduction in relative heading-up of approximately 27.63%; these represent the most favorable hydraulic conditions observed in the study. The percentage improvements in both ΔE/y₁ and h_u/y₁ decrease progressively as the canal inside slope transitions from Z = 1:1 to Z = 2:1, for all non-box configurations. This trend indicates that the hydraulic advantage of gradual entrance transitions relative to the box type is most pronounced under steeper canal inside slope conditions (Z = 1:1) and diminishes as the slope becomes flatter. The physical implication is that, under wider canal geometries (Z = 2:1), the entrance transition geometry plays a comparatively smaller role in determining bulk hydraulic losses, potentially because the enlarged wetted cross-section dominates the hydraulic resistance regardless of wing-wall type. The reported hydraulic improvements substantially exceed both the estimated measurement uncertainty (±0.01–±1.22%) and the average model validation error (approximately 5.75%). Consequently, the observed differences among the investigated entrance configurations are considered to be practically significant and representative of genuine hydraulic performance improvements within the investigated conditions, rather than artifacts of measurement or numerical uncertainty.

4. Development and Statistical Validation of Empirical Equations

This section presents the development and statistical validation of empirical predictive equations for the two primary hydraulic performance indicators: relative energy loss (ΔE/y₁) and relative heading-up (h_u/y₁). The equations were derived through nonlinear regression analysis applied to the HEC-RAS simulation results, following the dimensional analysis framework established in Section 2.2. The statistical evaluation of the developed empirical equations is summarized in

Table 4. Statistical parameters of the developed empirical relationship.

Wing Wall Type	Regression Equation	Adjusted R2	p-Value	Significance F	RMSE
Box type	$\Delta E/y_1=13.18 Z^{2.39} {{Fr}_1}^{3.13}$	0.96	<0.001	$5.17\times {10}^{-20}$	0.006
	$h_{\mathrm{u}}/y_1=33.61 Z^{2.07} {{Fr}_1}^{3.12}$	0.96	<0.001	$5.70\times {10}^{-19}$	0.01
Broken type	$\Delta E/y_1=7.29 Z^{2.22} {{Fr}_1}^{3.04}$	0.98	<0.001	$1.38\times {10}^{-8}$	0.002
	$h_{\mathrm{u}}/y_1=30.50 Z^{2.03} {{Fr}_1}^{3.20}$	0.99	<0.001	$4.26\times {10}^{-8}$	0.005
Curved type	$\Delta E/y_1=4.52 Z^{2.24} {{Fr}_1}^{2.97}$	0.97	<0.001	$1.93\times {10}^{-7}$	0.002
	$h_{\mathrm{u}}/y_1=27.79 Z^{2.10} {{Fr}_1}^{3.21}$	0.99	<0.001	$1.1\times {10}^{-8}$	0.004
Splayed type	$\Delta E/y_1=1.59 Z^{2.83} {{Fr}_1}^{2.80}$	0.97	<0.001	$3.83\times {10}^{-8}$	0.001
	$h_{\mathrm{u}}/y_1=17.60 Z^{2.12} {{Fr}_1}^{3.06}$	0.99	<0.001	$1.7\times {10}^{-8}$	0.004

Note: The equations presented in Table 4 are valid for upstream Froude numbers in the range Fr₁ = 0.12–0.18, canal inside slopes Z = 1:1 to 2:1 (H:V), a contraction ratio r = 0.6, an entrance angle θ = 30°, and steady subcritical flow conditions. Application outside this range is not recommended without additional verification.

, while the graphical comparison between predicted and HEC-RAS computed values for the box-type wing wall, presented as a representative example, is shown in Figure 25 and Figure 26. The proposed relationships show satisfactory agreement between the HEC-RAS values and the empirical predictions for estimating the relative energy loss (ΔE/y₁) and relative heading-up (h_u/y₁) under the four upstream wing-wall configurations combined with the three canal inside slopes investigated. The equations were formulated as power-law functions of the canal inside slope (Z) and upstream Froude number (Fr₁), taking the general form:

$$\Delta E/y_1 or h_{\mathrm{u}}/y_1=A Z^B {{Fr}_1}^C$$

(6)

where A, B, and C are empirically determined coefficients specific to each wing-wall configuration and response variable, as reported in Table 4. These two variables were identified as the dominant governing parameters through the dimensional analysis presented in Section 2.2, where the upstream Froude number (Fr₁) and canal inside slope (Z) were retained as the primary dimensionless variables after the Reynolds number, Weber number, contraction ratio, entrance angle, transition length, and bed slope were either shown to be negligible or held constant throughout the investigation. The physical basis for selecting a power-law form is that both ΔE/y₁ and h_u/y₁ are expected to scale nonlinearly with Fr₁ under subcritical flow contraction conditions, consistent with classical energy loss relationships in open-channel hydraulics.

The reported statistical indicators demonstrate satisfactory predictive capability for all investigated configurations, with the adjusted coefficient of determination (R²) ranging from 0.96 to 0.99 and RMSE values between 0.001 and 0.01. Table 4 shows that equations for h_u/y₁ consistently exhibit slightly higher prediction accuracy than those developed for ΔE/y₁, with adjusted R² values reaching 0.99 for the broken, curved, and splayed-type configurations. This difference in predictive accuracy between h_u/y₁ and ΔE/y₁ equations may reflect the fact that heading-up is a more directly measured and spatially stable parameter, representing a single upstream water level rise, whereas energy loss integrates velocity and depth measurements at multiple cross-sections, introducing greater cumulative uncertainty into the fitted values. The box-type configuration equations produce the lowest adjusted R² values for ΔE/y₁ (R² = 0.96), which is attributed to the greater hydraulic variability associated with the abrupt entrance geometry of this configuration, which produces a wider scatter in ΔE/y₁ values across the investigated Fr₁ and Z range and is less well approximated by a simple two-variable power-law model. This interpretation is consistent with the known behavior of abrupt contractions, where energy losses are more sensitive to local flow conditions and exhibit greater variability than in gradual transitions. In contrast, the broken-, curved-, and splayed-type configurations show consistently improved predictive performance reflected in higher adjusted R² values and lower RMSE, due to the smoother entrance geometry, which produces more regular hydraulic behavior and a tighter relationship between the governing variables (Fr₁, Z) and the computed performance indicators.

Specifically, the broken-type configuration equations yield adjusted R² values of 0.98 and 0.99 for ΔE/y₁ and h_u/y₁, respectively, while the curved-type equations produce adjusted R² values of 0.97 and 0.99 for the same parameters. Similarly, the splayed-type configuration’s equations demonstrate the strongest overall agreement between empirical predictions and HEC-RAS computed values and produce the lowest RMSE values across all configurations (RMSE = 0.001 for ΔE/y₁ and 0.004 for h_u/y₁), indicating the highest prediction accuracy for this configuration and the most regular hydraulic behavior associated with its gradual entrance geometry. The progressive improvement in equation fit from box to broken to curved to splayed configurations, reflected in both increasing R² and decreasing RMSE, is consistent with the performance hierarchy established in Section 3.2 and suggests that the power-law functional form is increasingly appropriate as the entrance geometry becomes more gradually transitional.

Furthermore, all developed equations are statistically significant, as indicated by p-values well below the significance threshold (p < 0.001 for all equations) and correspondingly small Significance F values (ranging from 1.7 × 10⁻⁸ to 5.17 × 10⁻²⁰, as reported in Table 4). The high statistical significance of all equations is expected given the strong monotonic relationships between Fr₁, Z, and the hydraulic performance indicators within the narrow parametric range investigated (Fr₁ = 0.12–0.18; Z = 1:1 to 2:1). It should be noted that statistical significance does not by itself confirm physical validity or generalizability beyond the investigated range; the equations should be treated as empirical approximations calibrated to the specific geometric and flow conditions of this study. These findings confirm the statistical reliability of the proposed empirical relationships within the investigated hydraulic and geometric conditions.

The developed equations provide a practical preliminary design tool for estimating bulk hydraulic performance indicators, specifically ΔE/y₁ and h_u/y₁. The equations are applicable to trapezoidal canal with the following geometric and hydraulic characteristics: upstream Froude number Fr₁ in the range 0.12–0.18, canal inside slope Z between 1:1 and 2:1 (H:V), contraction ratio r = 0.6, entrance angle θ = 30°, and steady subcritical flow conditions. These constraints reflect the parametric scope of HEC-RAS simulations from which the equations were derived. Therefore, within these bounds, these relationships can assist in preliminary hydraulic design, performance evaluation, and systematic comparison between the four upstream wing-wall configurations studied under conditions similar to those considered in the present study.

Therefore, careful engineering judgment and site-specific assessment are recommended before applying the proposed equations under prototype-scale field conditions, particularly given that field factors such as sediment deposition, irregular channel geometry, vegetation, and non-uniform approach flow, which were not considered in the present study, may substantially alter hydraulic performance relative to the idealized laboratory conditions on which these equations are based. In particular, the contraction ratio (r = 0.6) and entrance angle (θ = 30°) were held constant throughout this investigation; sites with substantially different values of these parameters will require either additional calibration or an independent hydraulic analysis. The use of higher resolution 2D or 3D numerical modeling is recommended for detailed design applications where localized flow features, such as scour potential or velocity distribution near the wing walls, are of specific concern.

5. Discussion and Interpretation of Results

The hydraulic performance of water structures is strongly influenced by the interaction between the entrance geometry and upstream approach conditions. Accordingly, understanding the combined effect of canal inside slope and wing-wall configuration is essential for improving hydraulic efficiency, reducing energy loss, minimizing upstream afflux, and enhancing flow stability at water structures. As established in the Introduction, prior studies have predominantly examined wing-wall geometry and canal inside slope in isolation, with only one previous investigation, Ashour et al. [13], partially addressing their combined interaction for a limited wing-wall type. The present study extends this prior work to a broader parametric basis, evaluating four wing-wall configurations across three canals inside slope values. Accordingly, the present study provides a systematic evaluation of this interaction using HEC-RAS simulations validated against physical model measurements. The discussion presented herein focuses on three interconnected themes: interpreting the bulk hydraulic behavior, examining the inferred physical mechanisms that are consistent with the observed flow patterns, and comparing the present findings with previously published investigations.

The present HEC-RAS results generally agree with the findings reported in the previous experimental and numerical investigations concerning the hydraulic behavior of canal contraction structures and the influence of entrance geometry on flow characteristics. The observed increases in ΔE/y₁ and h_u/y₁ with increasing upstream Froude number are consistent with the expected hydraulic response of a contracting channel section: as Fr₁ increases, greater flow momentum transfer occurs near the structure entrance, leading to higher hydraulic resistance and greater upstream water level rise. Similar trends were reported by Ahmed et al. [26], who demonstrated that increasing flow intensity near hydraulic structures is associated with greater turbulence generation and elevated upstream water levels.

The HEC-RAS results also confirm that gradually transitioning wing-wall geometries produce lower bulk ΔE/y₁ and h_u/y₁ values compared with abrupt entrance configurations. The curved- and splayed-type configurations produce lower ΔE/y₁ and h_u/y₁ values than the conventional box-type configuration across all investigated canal inside slope values and Fr₁ levels, with the splayed type consistently achieving the greatest reductions (up to 73.11% in ΔE/y₁ and 27.63% in h_u/y₁ at Z = 1:1). The lower bulk energy losses computed for gradual configurations are consistent with the expected effect of smoother entrance geometry on flow guidance at the structure inlet, which is expected to reduce hydraulic resistance and minimize energy dissipation. Similar findings were reported by Ashour et al. [12], and Ashour et al. [13], who demonstrated that gradual entrance transitions are associated with reduced energy dissipation and improved hydraulic efficiency compared with abrupt wing-wall geometries; these findings were also obtained using HEC-RAS simulations under comparable flow conditions, providing methodologically consistent corroboration of the present results. Additional support comes from Barbhuiya et al. [21,22,23], whose physical measurements around wing-wall abutments demonstrated that abrupt geometries generate stronger vortex structures and greater bed shear stress variability, effects that would be expected to manifest as higher bulk energy losses in simulations.

In contrast, the box-type configuration produces the highest computed bulk energy losses and velocity ratios among all investigated configurations, due to the abrupt entrance geometry and abrupt flow redirection at the structure inlet. The model reflects the greater hydraulic resistance of this configuration through consistently higher computed ΔE/y₁ and h_u/y₁ values across all investigated Fr₁ and Z combinations. The physical interpretation, that the sharp entrance geometry promotes flow separation, secondary circulation, and turbulence dissipation, is inferred from the computed bulk trends and is not directly observable from simulation output. These computed trends are broadly consistent with the experimental findings of Barbhuiya et al. [21,22,23], who identified strong vortex structures and irregular wake regions in physical measurements around abrupt wing-wall abutment geometries, and whose results provide direct physical evidence for the three-dimensional flow mechanisms inferred in the present study.

The influence of canal inside slope on ΔE/y₁ and h_u/y₁ observed in the present study is consistent with the hypothesis that wider canal cross-sections near the entrance zone produce greater hydraulic resistance. Specifically, a transition from Z = 1:1 to Z = 2:1 consistently increased both ΔE/y₁ and h_u/y₁ across all four wing-wall configurations, as quantified in Section 3.2 and Figure 22, Figure 23 and Figure 24. A wider canal cross-section at the structure entrance enlarges the wetted perimeter and the flow interaction area with the side boundaries, which is expected to increase boundary friction effects and hydraulic resistance. As with the mechanisms discussed in the preceding paragraphs, the specific attribution to boundary shear enhancement and turbulence generation represents a physically plausible inference from the computed bulk trends rather than a directly measured or computed result. Consequently, higher ΔE/y₁ and h_u/y₁ values are consistently observed for wider canal inside slope geometries (Z = 2:1). Furthermore, the percentage improvement in ΔE/y₁ and h_u/y₁ afforded by the non-box configurations relative to the box type was found to decrease progressively with increasing canal inside slope (Figure 22, Figure 23 and Figure 24), suggesting that the hydraulic advantage of gradual entrance transitions is most pronounced under steeper canal geometries and diminishes under wider cross-sectional conditions. This trend has practical implications for entrance-zone design: where site or geotechnical constraints necessitate wider canal side slopes, the selection of entrance configuration type becomes comparatively less critical in terms of bulk hydraulic performance, although gradual transitions remain preferable. This interpretation is consistent with the findings reported by Xiao et al. [27], who demonstrated that canal inside slope geometry significantly affects momentum transfer and boundary shear stress distribution within open channels; however, it is noted that Xiao et al. [27] investigated compound channel flow rather than canal contraction structures, and their findings are therefore qualitatively rather than quantitatively applicable to the present context.

The comparison between the HEC-RAS simulations and the physical model measurements, presented in detail in Section 3.1, demonstrated satisfactory agreement for the key bulk hydraulic parameters, with R² values of 0.97–0.99 and overall average variations within 5.75%. This level of agreement supports the use of the model as a practical tool for parametric comparison of entrance-zone configurations under the investigated flow conditions, subject to the limitations noted in Section 2.3 and Section 3.1. The observed progressive increases in ΔE/y₁ and h_u/y₁ with increasing upstream Froude number are consistent across the numerical and physical model results. Similarly, the decrease in y₂/y₁ and the increases in v₂/v₁ agree reasonably well with the expected bulk hydraulic behavior under flow contraction conditions. It is acknowledged, however, that the agreement for v₂/v₁ was comparatively weaker (average variation of 10.40%), consistent with the known limitation of 1D cross-sectional averaging in capturing the non-uniform velocity distribution through contracted channel sections. This limitation should be borne in mind when the present velocity results are interpreted in the context of scour potential or sediment transport, which are sensitive to local rather than averaged velocity values.

The satisfactory agreement between the HEC-RAS simulations and the physical model measurements confirms that the adopted modeling approach is capable of reproducing the bulk hydraulic response of the investigated configurations with acceptable accuracy for the purposes of parametric design comparison. Specifically, the model adequately captures the influence of entrance geometry and canal inside slope on cross-sectionally averaged energy loss and heading-up, which are the primary design-relevant performance indicators assessed in this study. In addition, the numerical model adequately reproduces the differential influence of the four upstream wing-wall configurations and the three canal inside slopes on the bulk hydraulic performance indicators, confirming its suitability as a practical parametric design tool within the investigated range of conditions.

In addition to the hydraulic analysis, power-law empirical equations were developed to estimate ΔE/y₁ and h_u/y₁ as functions of Fr₁ and Z, taking the general form presented in Section 4 (Equation (6)). The developed relationships show satisfactory predictive capability, with adjusted R² values of 0.96–0.99 and RMSE values of 0.001–0.01, indicating satisfactory agreement between the empirical predictions and the HEC-RAS simulation results. These equations provide a practical preliminary design tool for systematically evaluating the combined influence of Fr₁ and Z on bulk hydraulic performance within the investigated parametric range and may assist engineers in selecting optimized entrance configurations that minimize hydraulic losses and reduce upstream afflux. Several important limitations of the developed equations should be reiterated in this context. First, the equations were calibrated against HEC-RAS simulation outputs rather than directly against physical model measurements, meaning their accuracy is contingent on the validity of the modeling approach. Second, the equations are strictly valid within the ranges Fr₁ = 0.12–0.18, Z = 1:1 to 2:1, r = 0.6, and θ = 30°; extrapolation beyond these bounds is not supported by the present data. Third, field factors including sediment deposition, vegetation, and non-uniform approach flow were not accounted for and may alter the performance of the recommended configurations under prototype conditions.

The two-way ANOVA was performed on the results of the four primary dimensionless performance indicators (ΔE/y₁, h_u/y₁, y₂/y₁, and v₂/v₁) using a 4 × 3 factorial design (four wing-wall configurations × three canal inside slopes) with six replicate discharge values per cell, as described in Section 2.5. The analysis confirmed that both canal inside slope and wing-wall configuration significantly influenced the overall hydraulic response of the investigated intake configurations. For ΔE/y₁, the analysis yielded F = 27.92 for the wing-wall configuration factor and F = 24.40 for the canal inside slope factor, with both effects significant at p < 0.001.

The partial eta-squared effect size for wing-wall configuration was η²p = 0.233, indicating a large practical effect, whereas the effect size for canal inside slope was η²p = 0.150, indicating a moderate practical effect. The interaction term (configuration × slope) produced F = 0.912 (p = 0.487), indicating that the performance ranking of the wing-wall configurations remained consistent across the three canal inside slope values. The corresponding interaction effect size was very small (η²p = 0.019), confirming that the combined influence of configuration and canal inside slope was negligible for this hydraulic parameter.

Similarly, h_u/y₁ exhibited a statistically significant response to wing-wall configuration (F = 3.30, p = 0.038), confirming its sensitivity to entrance-zone geometry. However, the relatively lower F-value compared with ΔE/y₁ indicates that heading-up is less strongly affected by configuration type than energy loss. This observation is consistent with the experimental results presented in Section 3.2, where the maximum reduction in h_u/y₁ reached 27.63%, compared with a substantially larger reduction of 73.11% for ΔE/y₁.

The Tukey HSD post-hoc test identified the specific configurations and slope levels responsible for these differences. The canal inside slope Z = 1:1 produced significantly lower ΔE/y₁ than Z = 2:1 (mean difference = 0.0132, p < 0.001, Cohen’s d = 0.89), representing a large effect size according to conventional benchmarks [36]. Likewise, the splayed-type wing-wall configuration resulted in significantly lower ΔE/y₁ than the conventional box-type configuration (mean difference = 0.0131, p < 0.001, Cohen’s d = 1.31), also representing a large effect. The similar meaning differences and effect sizes for these two comparisons (slope effect: Δ = 0.0132, d = 0.89; splayed vs. box: Δ = 0.0131, d = 1.31) indicate that the choice of wing-wall configuration has a practically comparable influence on bulk energy loss to the choice of canal inside slope within the investigated parametric range. This is a practically significant finding that supports the recommendation of gradual entrance configurations as an effective design strategy for reducing hydraulic losses, independent of canal inside slope.

When interpreting the statistical results, both statistical significance and practical significance should be considered. Although some comparisons exhibited relatively small numerical differences, the major hydraulic improvements reported in this study substantially exceeded the estimated measurement uncertainty (±0.01–±1.22%) and the average model validation error (approximately 5.75%). Therefore, the observed reductions in relative energy loss and heading-up represent meaningful engineering improvements rather than variations arising from measurement uncertainty or numerical prediction error. Nevertheless, comparisons involving relatively small differences should be interpreted with appropriate consideration of the reported uncertainty levels.

These findings demonstrate that both canal inside slope and wing-wall configuration are statistically significant and practically important factors governing computed bulk hydraulic performance and support the adoption of gradual entrance configurations, particularly the splayed and curved types, to improve flow conditions and reduce hydraulic losses within the parametric range investigated in this study.

6. Study Limitations and Engineering Applicability

The following limitations should be considered when interpreting the results of the present study and extending their application to engineering practice:

a.

One-dimensional hydraulic modeling:

The study employed the one-dimensional steady-flow module of HEC-RAS, which computes cross-sectionally averaged hydraulic parameters and does not directly resolve three-dimensional flow features such as secondary currents, flow separation, vortex formation, recirculation zones, or lateral velocity distributions.

b.

Steady subcritical flow conditions:

All simulations were performed under steady subcritical flow conditions. Although this represents a limitation of the study, it also reflects the hydraulic regime typically maintained in irrigation canals, where low-velocity subcritical flow is preferred to reduce bed and bank erosion and to ensure stable flow conditions. Consequently, the developed empirical relationships are directly relevant to most practical irrigation canal applications. Nevertheless, their applicability to unsteady, transient, or flood-flow conditions requires further investigation.

c.

Restricted parametric range:

The findings are valid only within the investigated ranges of upstream Froude number (Fr1 = 0.12–0.18), canal inside slope (Z = 1:1–2:1), contraction ratio (r = 0.6), and entrance transition angle (θ = 30°). Since the contraction ratio and entrance angle were held constant throughout the investigation, their individual and combined effects on hydraulic performance were not evaluated. Additional studies are required to assess the applicability of the developed relationships under different geometric configurations.

d.

Idealized laboratory and numerical conditions:

The study assumed smooth rigid boundaries, uniform channel cross-sections, and constant Manning’s roughness coefficients. Field conditions may differ significantly due to sediment deposition, debris accumulation, vegetation growth, non-uniform approach flow conditions, irregular channel boundaries, and seasonal environmental variations. These factors may alter hydraulic performance by introducing additional flow disturbances and energy losses that were not represented in the present modeling framework.

It should be noted that the contraction ratio, entrance angle, channel roughness, and bed slope were intentionally maintained as constant throughout the experimental program and numerical investigation to isolate the interactive effects of the upstream wing-wall configuration and canal inside slope. Although variations in these parameters are expected to influence the magnitude of hydraulic characteristics such as afflux, energy loss, and velocity distribution, previous studies indicate that their primary effect is on the intensity rather than the fundamental trends of the flow behavior. The adopted contraction ratio (r = 0.6) represents one of the most critical hydraulic conditions reported in the literature and is also widely considered an economical design value for irrigation structures, balancing hydraulic efficiency with practical construction requirements. Similarly, the selected entrance angles correspond to the standard configurations most frequently adopted in previous investigations. Therefore, the selected conditions provide a representative and conservative basis for evaluating the investigated variables, while future studies should examine the sensitivity of the proposed relationships to broader ranges of these hydraulic and geometric parameters.

7. Conclusions

Based on validated HEC-RAS steady-flow simulations supported by laboratory measurements, the following conclusions are drawn:

(1)

HEC-RAS was demonstrated to be a reliable and computationally efficient tool for evaluating the hydraulic performance of water structures, producing satisfactory agreement with physical model measurements for the key hydraulic performance indicators (R² = 0.97–0.99, overall average variation ≤ 5.75%). The validated model is suitable for preliminary hydraulic design and evaluation of alternative entrance-zone configurations.

(2)

The combined interaction between canal inside slope and upstream wing-wall configuration significantly affected the hydraulic performance of water structures, particularly in terms of ΔE/y₁, h_u/y₁, y₂/y₁, and v₂/v₁. Two-way ANOVA confirmed statistically significant main effects for both geometric parameters.

(3)

The curved-type wing-wall configuration reduced ΔE/y₁ and h_u/y₁ by approximately 46.83% and 18.02%, respectively, compared with the conventional box-type configuration at Z = 1:1.

(4)

The splayed-type wing-wall configuration achieved the best hydraulic performance, reducing ΔE/y₁ and h_u/y₁ by approximately 73.11% and 27.63%, respectively, relative to the conventional box-type configuration at Z = 1:1. The hydraulic performance ranking remained consistent for all investigated canal inside slopes as splayed > curved > broken > box.

(5)

Among the investigated canal inside slopes, Z = 1:1 consistently provided the highest hydraulic efficiency for all wing-wall configurations. Where flatter canal side slopes are required (e.g., Z = 2:1), gradual entrance geometries or local side-slope transitions are recommended to minimize hydraulic losses.

(6)

Power-law empirical equations were developed for each wing-wall configuration to predict the principal hydraulic performance indicators. The equations showed satisfactory predictive accuracy, with adjusted R² values ranging from 0.96 to 0.99 and RMSE values between 0.001 and 0.01.

(7)

From a sustainability perspective, the proposed combination of optimized wing-wall geometry and canal inside slope can improve hydraulic efficiency, reduce energy losses, upstream afflux and support sustainable hydraulic design and long-term management of irrigation water structures.

(8)

Unlike previous studies that investigated wing-wall geometry or canal inside slope separately, this study evaluated their combined interaction and provides practical guidance for optimizing entrance-zone configurations of irrigation water structures.

8. Practical Recommendations

Based on the results of the present parametric investigation, the following practical recommendations are proposed for the design and evaluation of canal contraction structures. These recommendations are applicable within the investigated range of hydraulic and geometric conditions under steady subcritical flow.

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The power-law empirical equations may serve as a practical preliminary design tool for estimating the key performance indicators, allowing engineers to rapidly compare alternative entrance configurations and quantitatively assess their relative bulk hydraulic performance.

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Splayed and curved upstream wing-wall configurations are recommended when minimizing energy loss and upstream heading-up is a primary design objective, as they consistently produced lower ΔE/y₁ and h_u/y₁ values than the box-type configuration across all investigated canal inside slopes. The splayed configuration is the preferred option where construction feasibility permits, while the curved configuration provides an effective alternative. The broken configuration also performs better than the box type and may be adopted where curved or splayed geometries are impractical. The superior performance of gradual entrance configurations is attributed to smoother flow guidance and reduced hydraulic resistance at the contraction entrance.

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A canal inside slope of Z = 1:1 (1H:1V) is recommended as the most hydraulically efficient option within the investigated range. The greatest hydraulic improvements were achieved when Z = 1:1 was combined with a splayed or curved wing-wall configuration, making this arrangement the preferred design alternative. Nevertheless, the final selection of canal inside slope should also satisfy project-specific geotechnical and embankment stability requirements.

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When the adoption of flatter canal inside slopes (Z = 3:2 or Z = 2:1) is required due to site or geotechnical constraints, splayed or curved wing-wall configurations should be adopted in preference to the conventional box-type configuration. Where feasible, side-slope transitions or suitable slope-stabilization measures, including canal lining, riprap, gabions, geosynthetic reinforcement, and retaining systems, may be employed to permit steeper canal inside slopes and improve the hydraulic efficiency of the contraction structure.