Open AccessArticle

Energy Dissipation Assessment in Flow Downstream of Rectangular Sharp-Crested Weirs

Water Resources Research Center, Disaster Prevention Research Institute, Kyoto University, Uji 6110011, Japan

Faculty of Civil Engineering, Babol Noshirvani University of Technology, Shariati Ave., Babol 4714871167, Iran

Institute for Geophysics and Meteorology, University of Cologne, Pohligstr. 3, D-50969 Köln, Germany

Author to whom correspondence should be addressed.

Water 2024, 16(23), 3371; https://doi.org/10.3390/w16233371

Submission received: 26 October 2024 / Revised: 18 November 2024 / Accepted: 21 November 2024 / Published: 23 November 2024

(This article belongs to the Section Hydraulics and Hydrodynamics)

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Graphical abstract
"> Figure 1
Different types of sharp-crested weir: (a) rectangular; (b) triangular; (c) trapezoidal; (d) circular. "> Figure 2
Effective parameters of energy dissipation downstream of RSCWs. "> Figure 3
Geometry of rectangular sharp-crested weir. (a) 3D view; (b) downstream view; (c) section A–A (side View). "> Figure 4
Experimental flume components. (a) Side view; (b) Layout. "> Figure 5
Experimental model RSCW. "> Figure 6
Flow nappe in rectangular sharp-crested weir. (a) Schematic sketch; (b) laboratory observation; (c) nappe flow thickness. "> Figure 7
Upstream (V0) and downstream (V1) velocities vs. discharge (Q) for different opening ratios (b/B) of a sharp-crested weir: (a) b/B = 6/24; (b) b/B = 7/24; (c) b/B = 8/24; (d) b/B = 9/24; (e) b/B = 10/24. "> Figure 8
Comparative analysis of relative energy dissipation (∆Er) with (a) headwater ratio (h0/P); (b) discharge per unit width (q). "> Figure 9
Comparative analysis of relative residual energy (E1/E0) with (a) headwater ratio (h0/P); (b) discharge per unit width (q). "> Figure 10
Comparative analysis of (a) relative residual energy (E1/E0) with Cd; (b) relative energy dissipation (∆Er) with Cd. "> Figure 11
Comparison of the measured and calculated relative energy dissipation values. "> Figure 12
Comparison of the frequency occurrence of errors for the equations proposed in the present study and Amin et al. [<a href="#B39-water-16-03371" class="html-bibr">39</a>]. ">

Versions Notes

Abstract

Sharp-crested weirs are commonly used in hydraulic engineering for flow measurement and control. Despite extensive research on sharp-crested weirs, particularly regarding their discharge coefficients, more information is needed via research on their energy dissipation downstream. This study conducted experimental tests to assess the influence of contraction ratio (b/B) of rectangular sharp-crested weirs (RSCWs) on energy dissipation downstream under free flow conditions. Five RSCWs with different b/B equals 6/24, 7/24, 8/24, 9/24, and 10/24 were used. The results showed a consistent decrease in relative energy dissipation (ΔE_r) with an increase in the head over the weir. Furthermore, as the discharge per unit width (q) increased, the relative energy dissipation (ΔE_r) decreased, indicating more efficient discharge over the weir. A higher b/B further reduces ΔE_r, suggesting that wider weirs are more effective in minimizing energy losses. The maximum relative residual energy (E₁/E₀) and relative energy dissipation (ΔE_r) occurred at b/B = 10/24 and 6/24, with values of 0.825 and 0.613, respectively. Additionally, the maximum discharge coefficient (C_d) of RSCWs is found at b/B = 6/24, with an average value of 0.623. The results support the accuracy of the proposed equation with R² = 0.988, RMSE = 0.0083, and MAPE = 1.43%.

Keywords:

rectangular sharp-crested weir; laboratory flume; contraction ratio; energy dissipation; discharge coefficient

Graphical Abstract

1. Introduction

Weirs are ancient hydraulic structures that engineers use for various hydraulic purposes [1]. Among the different types of weirs, the sharp-crested weir is used to measure flow precisely in open channels [2,3,4,5], downstream of the wall of the dam [6], in irrigation systems [7], and in full-scale monitoring of Green Infrastructure (GI) [8]. In these weirs, the crest thickness in the direction of flow is minimal, ensuring that, even for low-flow discharges, water does not make contact with the crest edge [9]. These weirs typically appear in cross-sections as rectangular, triangular, trapezoidal, and circular shapes [10,11,12,13,14,15] (Figure 1). More intricate notch designs, such as those employing polynomial [16,17] and logarithmic [18] features, have also garnered significant interest. When the weir crest length is shorter than the channel width, it is an unsuppressed rectangular weir, known as a contracted rectangular sharp-crested weir [9].

The flow velocity of water and the energy downstream of hydraulic structures are high, and it is important to reduce them significantly in high-velocity flows [19,20]. The dissipation of energy was studied downstream of various types of weirs, including labyrinth weirs [21,22,23], piano key weirs [24,25,26], stepped spillways [27,28,29,30], ogee spillways [31,32], and drops [33].

The relative energy dissipation (ΔE_r) varies at different flow rates or discharges for different types of weirs. For instance, in piano key weirs, energy dissipation is greater at lower discharge flows and decreases logarithmically as the discharge increases [24]. In ogee spillways, a gentler slope provides greater energy dissipation than a steep slope (0.6:1) and, as the flow discharge increases, energy dissipation decreases [31]. For trapezoidal-triangular labyrinth weirs (TTLW), increasing the sidewall angle and weir height reduces energy dissipation. TTLW has the highest energy dissipation compared to vertical drop, and values for the relative residual energy (E₁/E₀) are at their lowest when discharge increases [21]. In stepped spillways, there are minor differences in the energy dissipation rate for uniform and nonuniform stepped configurations [28]. Increasing the effective length of the weir crest increases energy dissipation, and labyrinth or harmonic geometry increases the efficiency of energy dissipation up to 20% more than the flat one [30]. Additionally, stepped spillways with rough steps under the nappe flow regime have about 21% more energy dissipation than smooth ones [34].

Sharp-crested weirs are almost low-head weirs; hence it is required to know about the flow behavior of low-head weirs. At a constant low-head weir, flow over the weir generates a nappe determined by horizontal waves and audible low-frequency acoustic energy, which may affect the downstream environment. The stable projection angle was −17° for rectangular and quarter-round upstream, and −35° for half-round and quarter-round downstream weirs [35]. The Mechanism of energy loss in rectangular free jets incorporates stages including spreading of the jet during falling (aeration and atomization), air entrainment and diffusion of the jet in the water cushion, impact on the basin bottom, and recirculation and diffusion in the basin [36,37]. Rectangular sharp-crested weirs, due to their opening, can be influenced by the energy loss variations. Although energy dissipation downstream of sharp-crested weirs was reported in a few studies, the rectangular sharp-crested weir is a novel topic to evaluate, due to the opening ratio impacts on it. Amin et al. [38] studied the energy loss downstream of sharp-crested weirs and the corresponding energy dissipation occurring immediately below the weir under free-flow conditions. They reported that, at low head, increasing the tailwater depth leads to the formation of a hydraulic jump on the falling nappe, with most of the energy dissipation attributed to mixing within the jump. Amin et al. [39] observed that primary energy dissipates in supercritical flow at the top phase of impinging jet flow. In these conditions, the high velocity of the water results in significant turbulence and energy dissipation as the flow shifts to subcritical conditions. Conversely, the energy loss in the submerged flow regime is minimal, as the water flow remains relatively stable and less turbulent, allowing for more efficient energy transfer downstream with reduced dissipation.

Several parameters influence energy dissipation in rectangular sharp-crested weirs and can be categorized into (I) weir geometry, including weir height (P), the opening ratio or opening crest width to weir total width (b/B), and the thickness of weir crest (T_s); (II) hydraulic parameters, including head over the weir (h₀), the downstream flow depth (h₁), flow discharge (Q), and Froude number (Fr); (III) water characteristics, including temperature, viscosity, and density. Figure 2 illustrates the effective parameters mentioned for energy dissipation downstream of rectangular sharp-crested weirs (RSCWs).

It could be concluded from the literature review that studies calculating energy dissipation parameters downstream of sharp-crested weirs, especially with a rectangular opening, are unique, and it is necessary to focus on this topic. The present study evaluates the influence of key parameter variations, including the contraction ratio (b/B), the ratio of head on the weir to the weir height (h₀/P), and the weir discharge coefficient (C_d), on energy dissipation downstream of RSCWs. In particular, it investigates how varying the contraction ratio (b/B = 6/24, 7/24, 8/24, 9/24, 10/24) affects energy dissipation parameters including relative residual energy (E₁/E₀) and relative energy dissipation (ΔE_r) downstream of RSCWs under free-flow conditions. A new equation was developed based on observed data to predict this procedure for ΔE_r. Error analyses were applied via frequencies of occurrence and by calculating suitable statistical indexes for the proposed equation. Furthermore, the comparison between the new equation and the available equation in the literature for free flow over thin weirs has been illustrated.

2. Materials and Methods

2.1. Dimensional Analysis

The various aspects that contribute to weir behavior are classified into three categories: the physical properties of the fluid, the hydrodynamic features of the flow, and the geometric characteristics of the weir (Figure 3). The effective parameters that influence these weirs are as in Equation (1):

F (B, b, P, V₀, V₁, Q, Ts, g, S₀, ρ, σ, μ, h₀, h₁) = 0

(1)

where B = channel width; b = opening width; P = height of RSCW; h₀ = head on RSCW; T_s = thickness of RSCW; V₀ = flow velocity upstream of the weir, V₁ = flow velocity downstream of the weir, Q = flow discharge, g = gravitational acceleration; S₀ = slope of the channel floor; ρ = specific mass of water; σ = surface tension; μ = water dynamic viscosity, h₀ = upstream flow head, h₁ = downstream flow depth.

Based on the dimensional analysis, Equation (1) can be written as Equation (2).

F (b/B, B/P, h₀/P, T_s/B, T_s/P, g, S₀, We, Re, ∆E_r, C_d) = 0

(2)

where ∆E_r is relative energy dissipation (∆E_r = (E₀ − E₁)/E₀) [22,24,40], E₀ is upstream flow energy (E₀ = P + h₀ + (V₀)²/2g), and E₁ is downstream flow energy (E₁ = h₁ + (V₁)²/2g). Additionally, C_d is discharge coefficient (C_d = Q/(2/3. (2g)^0.5. b. h₀^1.5)) [41,42,43,44,45]. C_d can consider the factors not considered in the calculation of a flow discharge formula, including the surface tension, viscous effect, flow structures near the sharp-crested weirs, and flow profile curvature due to weir contraction [9].

B/P, T_s/P, T_s/B, S₀, and g are constant in all tests and can be deleted. In Equation (2), We= (ρV²b)/σ, and Re = ρVb/μ. Since viscosity effects are negligible (turbulent flow with a Reynolds number greater than 25,000), the Reynolds number is omitted from Equation (2) [46,47]. Viscous forces, characterized by the Reynolds number, and surface tension forces, represented by the Weber number, are of minor significance. This is due to the operational conditions of the weir, where the relatively higher head (2 cm in this instance) prevents surface tension effects (clinging), and the smooth design of the weir plate effectively mitigates viscous effects [48].

Similar to the study of Amin et al. [39], Equation (3) presents the effective parameters achieved in studying energy dissipation of flow downstream of RSCWs.

F (b/B, h₀/P, ∆E_r, C_d) = 0

(3)

2.2. Experimental Setup

Experimental tests were conducted in the hydraulic laboratory of Babol Noshirvani University of Technology (BNUT) using a rectangular flume with 4 m in length, 0.6 m in width, and 0.2 m in height. This system was used to investigate physical hydraulic models in some studies, such as Dadamahalleh et al. [49], Hamidi et al. [50], and Mahdian Khalili et al. [51]. The flume consists of upstream and downstream tanks, a pump, and a channel (Figure 4). The flume pump can provide a flow discharge of up to 0.0083 m³/s, measured and calibrated using a flowmeter. A flow straightener (mesh plate) at the channel’s beginning provides the steady flow inlet with the lowest disturbance. The length of the developing flow (L) was measured at 1.86 m to achieve a fully developed flow in the channel. Thus, the weir was located 2 m from the beginning of the flume. This study performed forty tests on four different weir widths (b = 15, 0.175, 0.200, 0.225, 0.250 m) under the flow discharges ranging from 0.0007 to 0.0049 m³/s. The weir height (P) and thickness (T_s) were kept constant throughout the tests at 0.1 m and 3 mm, respectively. Upon activating the pump, water from the upstream tank flowed into the channel, calmed with the flow straightener, and passed over the weir width under free flow conditions. The hydraulic conditions and test parameters’ values are provided in Table 1. The illustration of the laboratory models performed in this study is presented in Figure 5, which shows the downstream view.

2.3. Scale Effect

The Froude similitude is commonly used to scale hydraulic data from models to prototypes in open-channel flows, like weirs. In hydraulic models, the scale effect could deviate in results from the prototype [20]. This happens when the prototype variables are not scaled properly leading to force ratios not similar in the model and prototype [52]. Differences in performance between models and prototypes are often due to size–scale effects caused by fluid viscosity and surface tension, especially when the flow nappe coheres to the weir body [53]. Therefore, according to Erpicum et al. [54], surface tension is negligible in most prototypes.

In the present study, hydraulic models were performed based on the laboratory model with a constant width and height. Therefore, the scale effect has not been examined, and the results and proposed equation are only usable under hydraulic and geometric conditions with ranges of dimensionless parameters according to the experimental setup [55,56,57,58].

3. Flow Behavior

3.1. Flow Nappe

Figure 6a shows the flow nappe over a sharp-crested weir, as reported in the studies by Rao [59] and Chanson [60] in low discharge. Laboratory observations of sharp-crested weirs reveal distinct flow behaviors at varying discharge rates. At low discharge, the nappe initially remains close to the weir body (Figure 6b). As the discharge and hydraulic head increase, the water blade moves further away from the weir body, permitting the flow to pass freely over the downstream crest and along the lateral length of the spillway (Figure 6c). Further rise in discharge increases the blade thickness while, at constant discharge, increasing weir crest width (b) decreases blade thickness.

3.2. Hydraulic Jump

Immediately after the weir, a hydraulic jump often occurs when the flow encounters the downstream bed, especially at high discharge rates. This phenomenon is a sudden transition from high-velocity, low-depth (supercritical) flow to low-velocity, high-depth (subcritical) flow. The high velocity of the water exiting the weir generates this abrupt change.

4. Results

4.1. Flow Velocity

Figure 7 presents the relationship between the discharge Q (lit/s) and the upstream (V₀) and downstream (V₁) velocities of a sharp-crested weir for different values of the opening ratio b/B. As the discharge Q increases from 0.7 to 4.9 lit/s, the upstream velocity V₀ generally increases for all b/B ratios. The increase is consistent across all the velocity profiles, showing that, as the flow rate increases, the upstream velocity also rises to accommodate the increased flow over the weir. For example, at Q = 4.9 lit/s, the highest V₀ is 3.29 cm/s for b/B = 10/24, while the lowest V₀ is 3.09 cm/s for b/B = 6/24.

Downstream velocities (V₁) follow a similar increasing trend with discharge, though at a higher magnitude compared to V₀. This is expected due to the acceleration of water flow as it passes over the weir. For instance, at Q = 4.9 lit/s, V₁ for b/B = 6/24 is around 6.86 cm/s while V₁ for b/B = 10/24, this is around 6.64 cm/s. The differences between upstream and downstream velocities highlight the energy dissipation across the weir.

The difference between V₀ and V₁ indicates the energy dissipation occurring as water flows over the weir. This is critical in hydraulic structures for controlling flow energy to prevent erosion or damage downstream. The larger the difference between V₀ and V₁, the greater the energy dissipation. For example, at Q = 0.7 lit/s, V₀ for b/B = 10/24 is 0.547 cm/s and V₁ for b/B = 10/24 is 2.33 cm/s, showing significant dissipation. For lower b/B ratios (e.g., b/B = 6/24), the velocities are generally higher than for larger b/B ratios, indicating that narrower openings experience higher velocities due to flow constriction.

4.2. Relative Energy Dissipation

Figure 8a illustrates the ∆E_r curve for the RSCW to h₀/P. The ratio h₀/P (head to weir height) consistently increases as the discharge increases for each opening ratio b/B. This is a typical observation in weir flow, as higher discharges lead to a higher water head above the weir crest. For example, at b/B = 6/24, h₀/P increases from 0.189 at the lowest discharge to 0.672 at the highest. Similar trends are observed across all other b/B values, although the magnitude of the increase differs depending on the opening ratio. The opening ratio has a noticeable effect on the h₀/P values. As b/B increases, the h₀/P values tend to decrease slightly for the same discharge. For instance, at the highest discharge, h₀/P = 0.672 for b/B = 6/24, but only h₀/P = 0.483 for b/B = 10/24. This indicates that wider openings (larger b/B) lead to a lower head for the same flow rate, which is expected because larger openings allow water to pass through more easily, reducing the build-up of water upstream.

The relative energy dissipation ∆E_r decreases as the flow rate increases for all opening ratios. For example, at b/B = 6/24, ∆E_r decreases from 0.613 at the lowest flow rate to 0.288 at the highest. This trend indicates that, as the discharge increases, the proportion of energy dissipated relative to the total energy available decreases. This is typical in hydraulic systems as the velocity of the flow increases and more kinetic energy is carried downstream. The data show that, for higher opening ratios b/B, the energy dissipation ∆E_r is lower at all flow rates. This suggests that larger openings result in less energy dissipation, likely because the wider opening reduces turbulence and flow constriction, allowing more energy to pass through the weir without being dissipated. At lower discharges, the differences in energy dissipation between different opening ratios are less pronounced while, at higher discharges, these differences become more significant. This indicates that, at higher flows, the design of the weir (specifically the opening ratio) plays a more critical role in how much energy is dissipated.

Figure 8b illustrates the ∆E_r curve for the rectangular sharp-crested weir to q. Across all opening ratios, the relative energy dissipation

{∆ E}_{r}

decreases as the discharge per unit width q increases. This indicates that, for larger discharges, less energy is dissipated relative to the flow’s total energy, suggesting that higher flow rates pass more efficiently through the weir. The rate of energy dissipation decreases more rapidly at smaller discharges and stabilizes as q increases. As discharge increases, the differences in energy dissipation between different b/B ratios become smaller. At q = 0.00817 m²/m, the ∆E_r values across the different opening ratios converge, indicating that at higher discharges, the impact of the opening ratio on energy dissipation diminishes.

4.3. Relative Residual Energy

Figure 9a shows the relative residual energy (E₁/E₀) to the head above the crest weir. For each opening ratio (b/B), the relative residual energy (E₁/E₀) tends to increase with the head over the weir (h₀). This indicates that, as the headwater increases, the energy retained in the flow also rises, reflecting a more efficient energy transfer through the weir. Each opening ratio displays distinct trends. The overall pattern shows that higher opening ratios (e.g., b/B = 10/24) consistently yield higher E₁/E₀ values compared to lower opening ratios (e.g., b/B = 6/24). This trend suggests that wider openings lead to a reduction in energy losses, enhancing the relative residual energy for given heads over the weir. The E₁/E₀ values range from 0.403 to 0.726 as the head over the weir (h₀) increases from 0.14 to 0.64. The increase is moderate, indicating that, while there is some residual energy, it is less pronounced compared to wider openings.

The findings emphasize the importance of selecting an appropriate opening ratio (b/B) in weir design to optimize the relative residual energy in flow applications. Wider openings may facilitate improved residual energy, which can enhance the performance of hydraulic structures. The consistent increase in E₁/E₀ with h₀ across all b/B ratios suggests that greater heads over the weir allow for better energy recovery. This can positively impact the overall efficiency of hydraulic systems, particularly in situations involving varying flow conditions.

Figure 9b indicates the relative residual energy (E₁/E₀) to the discharge per unit width (q). The dataset consists of measurements of relative residual energy (E₁/E₀) corresponding to varying discharge rates (q) across different opening ratios (b/B) ranging from 6/24 to 10/24. The values of E₁/E₀ tend to increase with both the increase in discharge and the opening ratio, suggesting a relationship between these parameters. As q increases from 0.00117 m²/s to 0.00817 m²/s, the relative residual energy (E₁/E₀) generally shows an upward trend. This indicates that higher discharges are associated with greater residual energy, which can be attributed to increased turbulence and energy dissipation in the flow. The increase in E₁/E₀ across all opening ratios signifies that higher discharges enhance the capacity for residual energy in the flow, which could influence the performance of hydraulic structures. The data indicate that, for a given discharge rate, the relative residual energy values are consistently higher at larger opening ratios. This trend suggests that wider openings facilitate better energy recovery within the flow, as they allow for reduced hydraulic losses and improved flow patterns.

The differences in E₁/E₀ values across the different opening ratios are notable, especially at higher discharge rates. For example, the relative residual energy at b/B = 6/24 is significantly lower than at b/B = 10/24 for equivalent discharge conditions, indicating a substantial effect of the opening ratio on the flow dynamics. The convergence of the E₁/E₀ values at higher discharge rates suggests that, while the opening ratio has a significant influence, at elevated discharges, the effects may become less pronounced, possibly reaching a limit of residual energy capacity.

Table 2 presents the characteristics of each test. Results indicate that Model F1 has the highest discharge coefficient at 0.82. Following Model F1, the average discharge coefficients for Models A, B, C, D, and F are 0.623, 0.609, 0.595, 0.612, and 0.622, respectively. The relative residual energy (E₁/E₀) for Models A, B, C, D, and F are 0.555, 0.58, 0.598, 0.62, and 0.642, respectively. Furthermore, the relative energy dissipation (∆E_r) for Models A, B, C, D, and F are 0.445, 0.42, 0.402, 0.382, and 0.358, respectively. These findings highlight the variations in discharge efficiency and energy characteristics across the different models. Table 2 defines the discharge coefficient.

Figure 10 shows the illustration of energy dissipation values with C_d changes for all models. It can be observed from Figure 10a that, for all b/B, the relative residual energy (E₁/E₀) increases with a steep slope when discharge coefficients increase. Furthermore, Figure 10b indicates that relative energy dissipation (∆E_r) reduces by increasing discharge coefficients for all models. Generally, it can be concluded that relative energy dissipation has a direct relationship with a discharge coefficient, while the results present an adverse relationship between the relative residual energy and discharge coefficient.

4.4. Prediction Equation of Energy Dissipation

Chamani and Rajaratnam [33] proposed Equation (4) based on C_d and h₀/P to calculate ∆E_r downstream of drops.

∆E/E₀ = [(0.09. (h₀/P)^−0.77/(C_d)^0.77]

(4)

Amin et al. [39], based on these findings, added P/h₀ to Equation (4) and developed it as Equation (5):

∆E_r/h₀ = [(0.09.(1 + P/h₀))/(h₀/P) ^0.77.(C_d)^0.77]

(5)

To leverage the current research findings, Equation (6) is proposed for estimation of the relative energy dissipation of RSCWs. This equation was generated using 40 items of experimental data obtained in the present study via nonlinear regression analysis. This is developed using the dimensionless parameters introduced in Equation (6), which are similar in form to the proposed equation by Amin et al. [39], except for b/B, which was added to the new equation:

∆E_r = ∆E/E₀ = i (^h₀/^P) ^j + k (b/B) ^l + m (C_d) ⁿ

(6)

where parameters i, j, k, l, m, and n are constant parameters determined by regression analysis, having values of −0.96, 0.443, −1.394, 1.634, 1.121, and −0.194, respectively. Several statistical parameters were applied to validate the developed equation, including correlation coefficient (R²), root mean square error (RMSE), mean absolute percentage error (MAPE), Nash–Sutcliffe efficiency (NSE), relative error (RE%), and percent bias (PBIAS). The relationship in Equation (6) correlates with the relative energy dissipation (∆E_r) with effective parameters obtained through dimensional analysis. Figure 11 indicates the calculated and measured relative energy dissipation achieved from tests and the proposed Equation. The R² value of 0.988 indicates a strong correlation.

Table 3 presents the errors computed from Equation (6) proposed in the present study compared with Equation (5), suggested by Amin et al. [39].

Error index values R² = 0.988, RE = 4.14%, RMSE = 0.0083, MAPE = 1.43%, PBIAS = 0.51%, and NSE = 0.97 indicate good compatibility of experimental data and the proposed equation (Equation (6)). R² = 0.854, RE = 28.13%, RMSE = 0.1224, MAPE = 21.37%, PBIAS = 28.15%, and NSE = 0.29 also present acceptable results of the present study when compared with the data from Amin et al. [39]. It can be observed that the results of relative energy dissipation computed in these two equations show some differences, because the new equation is assigned for RSCWs and the effect of opening ratio (b/B) was considered in it. Figure 12 shows the frequency distribution of the errors for calculated relative energy dissipation from Equation (5) and the modified equation with values measured in the experiments. It can be concluded from the values of the frequency distributions of errors in Figure 12 that the modified equation is more consistent with the experimental data and reduces the errors compared to the previous equation.

Notably, the proposed equation improved the accuracy of relative energy dissipation prediction, although it has some limitations and uncertainties. First, it could be applied to RSCWs, and sharp-crested weirs with other plan shapes, such as triangular, circular, or irregular cross-section openings require further investigation. Second, it is usable within the ranges mentioned for dimensionless parameters according to Table 4.

4.5. Uncertainties and Limitations of the Model

This study, similar to most experimental studies, has some limitations and uncertainties that have been attempted to be partially addressed by considering criteria to improve the generalization of the methodology and results. First is the measurement of flow depth. A longitudinal distance of 3 to 4 times the weir head should be considered from the weir crest for headwater measurement to avoid the effect of flow curvature and water level oscillation [61]. Therefore, the headwater was measured 0.3 m upstream of the weir crest, via a moving point gauge with a measurement accuracy of 0.1 mm, the same as in Li et al. [62]; and Wang et al. [63]. Second is the limited range of parameters according to the channel dimensions. Accordingly, the experiments were conducted for a flow rate of 0.7 to 4.9 L/s, a channel width of 0.6 m, a contraction ratio (b/B) of 6.24 to 10.24, and b/P of 1.5 to 2.5.

5. Discussion and Conclusions

The current study provides important insights into the dynamics of energy dissipation and the residual energy dissipation relating to effective hydraulic parameters over rectangular sharp-crested weirs. In the experimental behavior indicated at low discharges, the nappe clings closely to the weir but, as discharge and head increase, it moves away, allowing freer flow over the crest and along the weir. As discharge increases, the thickness of the water blade also increases. However, with a constant discharge, expanding the weir crest width (b) results in a thinner water blade. Downstream of the weir, a high discharge often causes a hydraulic jump, shifting the flow from supercritical to subcritical.

The velocity of upstream of the weir (V₀) and downstream (V₁) indicate a clear trend of increasing flow velocities with higher discharge rates (Q). The notable difference between V₀ and V₁ underscores the energy dissipation occurring at the weir, which is essential for effective flow management and minimizing potential erosion downstream.

The results demonstrate that increasing the weir head and discharge per unit width (q) decreases relative energy dissipation. Furthermore, a larger contraction ratio (b/B) correlates to reduced energy dissipation, emphasizing the significance of the weir opening ratio in enhancing performance. According to the results, the highest and lowest relative energy dissipation belongs to b/B = 6/24 and b/B = 10/24 with values of 0.613 and 0.175, respectively. Regarding relative residual energy (E₁/E₀), the results show that increasing the head over the weir (h₀) leads to higher relative residual energy (E₁/E₀). Additionally, larger opening ratios (b/B) are associated with improved relative residual energy. Furthermore, a rise in discharge per unit width (q) also correlates with greater relative residual energy, suggesting that higher discharge rates enhance energy dissipation capabilities, particularly at larger opening ratios. Results indicate that the highest and lowest relative residual energy belongs to b/B = 10/24 and b/B = 6/24 with values of 0.825 and 0.387, respectively.

Regarding the discharge coefficient, results indicate that the highest discharge coefficient (C_d) belongs to b/B = 6/24, with a value of 0.635. To predict energy dissipation downstream of the weir, Equation (6) was proposed. Results present good compatibility between experimental data and the suggested Equation with R² = 0.988, RE = 1.87%, RMSE = 0.0083, MAPE = 1.43%, PBIAS = 0.51%, and NSE = 0.97. Additionally, R² = 0.854, RE = −27.67%, RMSE = 0.1224, and MAPE = 21.37% confirm the results of the present study compared with Amin et al. [39].

Author Contributions

Conceptualization, H.S.A. and A.M.K.; validation, H.S.A. and S.A.K.; formal analysis, H.S.A. and S.A.K.; investigation, H.S.A., S.A.K., A.M.K. and M.H.; resources, H.S.A.; data curation, H.S.A. and A.M.K.; writing—original draft preparation. H.S.A.; writing—review and editing, H.S.A., S.A.K., A.M.K. and M.H.; visualization; H.S.A. and A.M.K.; supervision; S.A.K. and M.H.; project administration, H.S.A. and A.M.K.; funding acquisition, M.H. All authors have read and agreed to the published version of the manuscript.

Funding

This research received no external funding.

Data Availability Statement

The data presented in this study are available on request from the corresponding author.

Acknowledgments

M.H. gratefully acknowledges the Alexander von Humboldt Foundation for awarding the George Forster Experienced Researcher Fellowship and its support. M.H. also acknowledges the Babol Noshirvani University of Technology. The data used in the research are adopted from the experiments performed in the research project Experimental Investigation of Discharge Coefficient of Rectangular Sharp-crested Weirs with Fully Contracted (Slit) and Partially Contracted in Small Scale Channels at Babol Noshirvani University of Technology (BNUT).

Conflicts of Interest

The authors declare no conflicts of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

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Figure 1. Different types of sharp-crested weir: (a) rectangular; (b) triangular; (c) trapezoidal; (d) circular.

Figure 2. Effective parameters of energy dissipation downstream of RSCWs.

Figure 3. Geometry of rectangular sharp-crested weir. (a) 3D view; (b) downstream view; (c) section A–A (side View).

Figure 4. Experimental flume components. (a) Side view; (b) Layout.

Figure 5. Experimental model RSCW.

Figure 6. Flow nappe in rectangular sharp-crested weir. (a) Schematic sketch; (b) laboratory observation; (c) nappe flow thickness.

Figure 7. Upstream (V₀) and downstream (V₁) velocities vs. discharge (Q) for different opening ratios (b/B) of a sharp-crested weir: (a) b/B = 6/24; (b) b/B = 7/24; (c) b/B = 8/24; (d) b/B = 9/24; (e) b/B = 10/24.

Figure 8. Comparative analysis of relative energy dissipation (∆E_r) with (a) headwater ratio (h₀/P); (b) discharge per unit width (q).

Figure 9. Comparative analysis of relative residual energy (E₁/E₀) with (a) headwater ratio (h₀/P); (b) discharge per unit width (q).

Figure 10. Comparative analysis of (a) relative residual energy (E₁/E₀) with C_d; (b) relative energy dissipation (∆E_r) with C_d.

Figure 11. Comparison of the measured and calculated relative energy dissipation values.

Figure 12. Comparison of the frequency occurrence of errors for the equations proposed in the present study and Amin et al. [39].

Table 1. Test parameter values.

Test	Q (L/s)	b (cm)	P (cm)	B (cm)	b/B	b/P
A1	0.7	15	10	60	6/24	1.5
A2	1	15	10	60	6/24	1.5
A3	1.4	15	10	60	6/24	1.5
A4	2.1	15	10	60	6/24	1.5
A5	2.8	15	10	60	6/24	1.5
A6	3.5	15	10	60	6/24	1.5
A7	4.2	15	10	60	6/24	1.5
A8	4.9	15	10	60	6/24	1.5
B1	0.7	17.5	10	60	7/24	1.75
B2	1	17.5	10	60	7/24	1.75
B3	1.4	17.5	10	60	7/24	1.75
B4	2.1	17.5	10	60	7/24	1.75
B5	2.8	17.5	10	60	7/24	1.75
B6	3.5	17.5	10	60	7/24	1.75
B7	4.2	17.5	10	60	7/24	1.75
B8	4.9	17.5	10	60	7/24	1.75
C1	0.7	20	10	60	8/24	2
C2	1	20	10	60	8/24	2
C3	1.4	20	10	60	8/24	2
C4	2.1	20	10	60	8/24	2
C5	2.8	20	10	60	8/24	2
C6	3.5	20	10	60	8/24	2
C7	4.2	20	10	60	8/24	2
C8	4.9	20	10	60	8/24	2
D1	0.7	22.5	10	60	9/24	2.25
D2	1	22.5	10	60	9/24	2.25
D3	1.4	22.5	10	60	9/24	2.25
D4	2.1	22.5	10	60	9/24	2.25
D5	2.8	22.5	10	60	9/24	2.25
D6	3.5	22.5	10	60	9/24	2.25
D7	4.2	22.5	10	60	9/24	2.25
D8	4.9	22.5	10	60	9/24	2.25
F1	0.7	25	10	60	10/24	2.5
F2	1	25	10	60	10/24	2.5
F3	1.4	25	10	60	10/24	2.5
F4	2.1	25	10	60	10/24	2.5
F5	2.8	25	10	60	10/24	2.5
F6	3.5	25	10	60	10/24	2.5
F7	4.2	25	10	60	10/24	2.5
F8	4.9	25	10	60	10/24	2.5

Table 2. Experimental results, including discharge coefficients, and energy dissipation.

Test	Q (m³/s)	q (m²/s)	b/B	b/P	h₀/b	h₀/P	h₀ (cm)	h₁ (cm)	E₁/E₀	∆Er	C_d
A1	0.0007	0.001167	6/24	1.5	0.126	0.189	1.89	4.61	0.387	0.613	0.608
A2	0.0010	0.001667	6/24	1.5	0.157	0.236	2.36	5.18	0.419	0.581	0.623
A3	0.0014	0.002333	6/24	1.5	0.198	0.297	2.97	6.20	0.477	0.523	0.618
A4	0.0021	0.0035	6/24	1.5	0.256	0.384	3.84	7.47	0.539	0.461	0.630
A5	0.0028	0.004667	6/24	1.5	0.316	0.474	4.74	8.81	0.597	0.403	0.612
A6	0.0035	0.005833	6/24	1.5	0.359	0.538	5.38	9.83	0.638	0.362	0.633
A7	0.0042	0.007	6/24	1.5	0.408	0.612	6.12	10.89	0.674	0.326	0.626
A8	0.0049	0.008167	6/24	1.5	0.448	0.672	6.72	11.93	0.712	0.288	0.635
B1	0.0007	0.001167	7/24	1.75	0.098	0.171	1.71	4.81	0.410	0.590	0.606
B2	0.0010	0.001667	7/24	1.75	0.123	0.216	2.16	5.39	0.443	0.557	0.610
B3	0.0014	0.002333	7/24	1.75	0.155	0.271	2.71	6.34	0.498	0.502	0.607
B4	0.0021	0.0035	7/24	1.75	0.202	0.354	3.54	7.72	0.569	0.431	0.610
B5	0.0028	0.004667	7/24	1.75	0.246	0.431	4.31	8.97	0.625	0.375	0.606
B6	0.0035	0.005833	7/24	1.75	0.284	0.497	4.97	9.79	0.652	0.348	0.611
B7	0.0042	0.007	7/24	1.75	0.321	0.561	5.61	11.03	0.704	0.296	0.612
B8	0.0049	0.008167	7/24	1.75	0.355	0/621	6.21	12.04	0.740	0.260	0.613
C1	0.0007	0.001167	8/24	2	0.080	0.159	1.59	4.91	0.423	0.577	0.591
C2	0.0010	0.001667	8/24	2	0.100	0.201	2.01	5.52	0.459	0.541	0.594
C3	0.0014	0.002333	8/24	2	0.126	0.252	2.52	6.37	0.508	0.492	0.593
C4	0.0021	0.0035	8/24	2	0.164	0.329	3.29	7.71	0.578	0.422	0.596
C5	0.0028	0.004667	8/24	2	0.200	0.399	3.99	9.11	0.649	0.351	0.595
C6	0.0035	0.005833	8/24	2	0.231	0.462	4.62	10.08	0.687	0.313	0.597
C7	0.0042	0.007	8/24	2	0.261	0.522	5.22	11.14	0.729	0.271	0.596
C8	0.0049	0.008167	8/24	2	0.288	0.577	5.77	11.87	0.749	0.251	0.599
D1	0.0007	0.001167	9/24	2.25	0.064	0.144	1.44	4.96	0.433	0.567	0.610
D2	0.0010	0.001667	9/24	2.25	0.081	0.182	1.82	5.62	0.474	0.526	0.613
D3	0.0014	0.002333	9/24	2.25	0.102	0.229	2.29	6.51	0.528	0.472	0.608
D4	0.0021	0.0035	9/24	2.25	0.133	0.299	2.99	7.60	0.583	0.417	0.611
D5	0.0028	0.004667	9/24	2.25	0.161	0.363	3.63	9.08	0.663	0.337	0.609
D6	0.0035	0.005833	9/24	2.25	0.186	0.419	4.19	10.22	0.716	0.284	0.614
D7	0.0042	0.007	9/24	2.25	0.210	0.473	4.73	11.27	0.761	0.239	0.614
D8	0.0049	0.008167	9/24	2.25	0.232	0.523	5.23	12.24	0.799	0.214	0.617
F1	0.0007	0.001167	10/24	2.5	0.053	0.133	1.33	5.03	0/443	0.557	0.618
F2	0.0010	0.001667	10/24	2.5	0.067	0.168	1.68	5.58	0/476	0.524	0.622
F3	0.0014	0.002333	10/24	2.5	0.084	0.211	2.11	6.60	0/543	0.457	0.619
F4	0.0021	0.0035	10/24	2.5	0.110	0.276	2.76	7.93	0/618	0.382	0.620
F5	0.0028	0.004667	10/24	2.5	0.134	0.335	3.35	9.37	0/698	0.302	0.619
F6	0.0035	0.005833	10/24	2.5	0.155	0.388	3.88	10.44	0/747	0.253	0.620
F7	0.0042	0.007	10/24	2.5	0.175	0.437	4.37	11.39	0/787	0.213	0.629
F8	0.0049	0.008167	10/24	2.5	0.193	0.483	4.83	12.33	0/825	0.175	0.625

Table 3. Error indexes of the present study compared with Amin et al. [39].

Model	Prediction Equation	Error Indexes
Model	Prediction Equation	R²	RE%	RMSE	MAPE%	PBIAS%	NSE
Present Study	Equation (6)	0.988	4.14	0.0083	1.43	0.51	0.97
Amin et al. [39]	Equation (5)	0.854	28.13	0.1224	21.37	28.15	0.29

Table 4. Applicable ranges of the proposed equation for ∆E_r of RSCWs.

Parameter	Minimum	Maximum
h₀/P	0.133	0.672
b/B	6/24	10/24
C_d	0.591	0.635
b/P	1.5	2.5

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Sohrabzadeh Anzani, H.; Kantoush, S.A.; Mahdian Khalili, A.; Hamidi, M. Energy Dissipation Assessment in Flow Downstream of Rectangular Sharp-Crested Weirs. Water 2024, 16, 3371. https://doi.org/10.3390/w16233371

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Sohrabzadeh Anzani H, Kantoush SA, Mahdian Khalili A, Hamidi M. Energy Dissipation Assessment in Flow Downstream of Rectangular Sharp-Crested Weirs. Water. 2024; 16(23):3371. https://doi.org/10.3390/w16233371

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Sohrabzadeh Anzani, Hossein, Sameh Ahmed Kantoush, Ali Mahdian Khalili, and Mehdi Hamidi. 2024. "Energy Dissipation Assessment in Flow Downstream of Rectangular Sharp-Crested Weirs" Water 16, no. 23: 3371. https://doi.org/10.3390/w16233371

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Sohrabzadeh Anzani, H., Kantoush, S. A., Mahdian Khalili, A., & Hamidi, M. (2024). Energy Dissipation Assessment in Flow Downstream of Rectangular Sharp-Crested Weirs. Water, 16(23), 3371. https://doi.org/10.3390/w16233371

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