Introduction

Structures that allow water to be used for various purposes of control are called hydraulic structures. Engineering applications such as hydroelectric and wave energy generation, environmental regulation and recreation areas, irrigation water supply, drinking and utility water supply, coastal and harbor structures, urban drainage, wastewater management and flood control are among the main areas of use of water engineering structures (Novák and Cabelka 1981). Scale effect in hydraulic structures is vital for analyzing these structures (El Baradei et al. 2022; Yamini et al. 2021). Excellent and reliable results are obtained in the laboratory models with the appropriate scale (Al-Bedyry et al. 2023; Qasim et al. 2022).

Side weirs are vital hydraulic applications specifically designed to regulate the flow or water level of the stream. Side weirs are frequently utilized to remove excess water from any channel or to provide the required flow rate. Side weirs are applied as lateral intake spillways in irrigation systems, land drainage, surface water removal, urban combined sewerage systems, removal of water considered clean above a certain depth, hydroelectric dams and flood protection structures. Placing the spillway on the dam body is impossible in dams constructed in narrow valleys. The spillway crest at normal water level can be designed as a lateral intake spillway parallel to the dam reservoir to solve this problem. In the late 90 s, Ahmed (1985) pioneered the study of combined weirs and gates in open channels. Since then, many studies have been conducted on combined weir–gate models for frontal flow (Alhamid et al. 1997; Alhamid 1999; Altan-Sakarya and Kökpinar 2013; Askeroğlu 2006; Negm et al. 2002; Parsaie et al. 2017, 2018, 2018; Samani and Mazaheri 2009; Salehi and Azimi 2019; Nouri and Hemmati 2020; Altan-Sakarya et al. 2020; Azamathulla et al. 2019).

Combined weir–gate structures can be built to regulate the water level, control the flow, take water and measure the flow rate in rivers and open channels. Combined weir–gate structures can be constructed in different geometric shapes. Combined weir–gate structures are also used to obtain hydrological data, the control section of water supply channels and diversion structures. Many studies in the literature on weirs and gates show that they operate separately from each other. Weirs and gates are applied in many hydraulic engineering applications in open channel flows, especially for frontal flows. They have advantages and disadvantages. Namely, the accumulation of sediment from the stream or reservoir from which the water is taken over time behind the weir and the accumulation of floating materials behind the gates cause measurement errors and operation and maintenance difficulties and reduce the operating life of these structures. It aims to reduce the problems that have occurred or will occur by ensuring the passage of the material that will accumulate behind the weir and gate. In this way, both the safety and lifetime of the structure will increase, and at the same time, eliminating the problems caused by accumulation will increase the preferability of these structures (Altan-Sakarya and Kökpinar 2013; Askeroğlu 2006; Azamathulla et al. 2019). For this purpose, many weir–gate structures with different geometries have been studied in the literature. However, in practice, combined rectangular weir–gate structures are more common. There are many studies on combined weir–gate in the literature. However, these studies are related to structures with frontal flows. It is not easy to make precise statements about the performance of the sediment passage through the gate and the passage of floating materials through the weir in lateral flows. Researchers have already noted these advantages of combined weir–gate structures for frontal flows. Detailed experimental studies are needed for sediment passage in lateral flows.

For this reason, it is also clear that there will be differences in sediment passage in lateral flows. Weirs and gates have been used independently of each other in open channel flows for a long time in hydraulic engineering. Combined weir–gate structures in frontal flow are frequently mentioned in the literature for the passage of accumulated materials downstream under gates and the passage of floating materials downstream over the weir (Altan-Sakarya and Kökpinar 2013; Askeroğlu 2006; Negm et al. 2002; Parsaie et al. 2017, 2018).

Alhamid (1999) studied the combined flow using triangular weir and rectangular gate structures. The author stated that the combined flow is complex because of the mixing of over-weir and under-gate flows and the number of parameters affecting the flow characteristics. Askeroğlu (2006) investigated the combined flow over trapezoidal weirs and under rectangular gates. The combined flow was evaluated by examining the dimensionless unit discharge value obtained from the dimensional analysis. It was determined that the variation of the combined flow with the dimensionless parameters (b/a, Reynolds and Weber numbers) in this equation is negligible. As a result, the combined flow was analyzed by considering the hydraulic parameters (y1/a and h/b) and the geometric parameters (d/a and b/B) (Fig. 1). The researcher proposed equations to estimate the dimensionless unit discharge using multiple regression. The author proposed the following equation to estimate the discharge of combined flow.

Fig. 1
figure 1

Flow characteristics of H-side weir: a plan, b cross-sectional view and c side view

Full size image
$$\frac{{Q}_{{\text{c}}}/b}{\sqrt{2g}{a}^{1.5}}=-1.9808+1.5240\left(\frac{{y}_{1}}{a}\right)+0.0949\left(\frac{h}{b}\right)-1.3918\left(\frac{d}{a}\right)+0.0840\left(\frac{B}{b}\right)$$
(1)

Negm et al. (2002) investigated experimentally combined flow characteristics using sharp-crested rectangular weirs and gates for free-flow conditions. The researchers stated that the viscosity effect could be neglected for Reynolds number Re > 200,000, and surface tension can be ignored for Weber number We > 40. Negm et al. (2002) discussed hydraulic and geometric effects on the simultaneous discharge, i.e., the combined flow rate. Negm et al. (2002) proposed the following equation to estimate combined discharge.

$$\frac{{O}_{{\text{c}}}/b}{\sqrt{2g}{a}^{1.5}}=-0.3863+0.8764\left(\frac{{y}_{1}}{a}\right)-0.1494\left(\frac{h}{b}\right)-0.7341\left(\frac{d}{a}\right)$$
(2)

Altan-Sakarya and Kökpinar (2013) experimentally investigated the simultaneous flow over the weir and under the gate in frontal flows under free-flow conditions. They named this structure, a combination of a rectangular weir with a sharp crest and a rectangular gate, an H-weir and considered it a flow measurement structure. They defined H-weirs by the weir length and gate opening, gate opening and the vertical distance between the bottom of the weir, i.e., the crest and the top of the gate opening. This experimental study measured water depth upstream and flow rates using different H-weir models.

Parsaiea and Haghiabi (2021) predicted the discharge coefficient of the combined weir–gate via soft computing techniques. The authors analyzed the combined weir–gate by considering the contraction ratio (b/B), the ratio of the vertical distance from the top of the gate opening to the crest (d/a), the ratio of the weir length to the opening of the weir–gate (b/a) and the ratio of the upstream flow depth to the gate opening (y1/a).

Altan-Sakarya et al. (2020) numerically investigated combined weir–gate models. The researchers emphasized that accumulating weir and gate formulations cannot calculate the combined discharge. Furthermore, they stated that the discharge coefficient values of the weirs and gates are different and larger than the experimental values. Soft computing methods have been widely used in estimating discharge capacity due to the complex relationships of discharge capacity between various geometric and hydraulic conditions. Many researchers attempted to use soft computing techniques to solve the drawbacks of extensive calculations and inappropriate use of empirical (Ayaz et al. 2023; Bonakdari and Zaji 2018; Elshaarawy and Hamed 2024; Ismael et al. 2021; Li et al. 2023; Norouzi et al. 2019; Parsaie 2016; Parsaie et al. 2018, 2023; Parsaie and Haghiabi 2016; Razmi et al. 2022; Roushangar et al. 2018; Salmasi et al. 2021; Seyedian et al. 2023; Simsek et al. 2023; Zaji et al. 2016). Tao et al. (2022) used the learning model to predict the discharge coefficient patterns of the gate flow under free-flow and submerged flow conditions. Ismael et al. (2021) used neural network technology to predict the discharge coefficient of inclined cylindrical weirs with different diameters. Chen et al. (2022) proposed to predict the discharge coefficient of aerodynamic weirs, and the results showed that hybrid deep data-driven algorithms provide more accurate results than classical ones. Simsek et al. (2023) used an artificial neural network (ANN) to estimate the discharge coefficient of a trapezoidal broad-crested weir. They found that the ANN method was more successful than other methods in determining the flow coefficients. Balouchi and Rakhshandehroo (2018) applied soft computing models to assess the discharge coefficient for combined weir–gate and found that the multilayer perceptron outperformed.

When the literature is examined, we have found that limited studies are related to using combined weir and gate structure as a water intake structure in lateral flows in open channels (Ghodsian et al. 2020; Kartal 2022; Kartal and Emiroglu 2022). Ghodsian et al. (2020) studied the effect of the upstream Froude number (F1), the ratio of the vertical distance between the gate top and the crest to the gate opening (d/a), the ratio of the upstream flow depth to the gate opening (y1/a) and the angle of curvature (α) on the discharge capacity for subcritical flow regime. The researchers found that the discharge coefficient of the combined side weir–gate is a decreasing function of F1 and d/a. It was also found that as y1/a increases, the discharge coefficient increases, i.e., y1/a is also an increasing function. The researchers emphasized that the results obtained from this study are limited to 0.12 < F1 < 0.63, 53 ≤ α ≤ 115°, 0.75 ≤ d/a ≤ 2.0, 1.9 ≤ y1/a ≤ 6.9 and b/B = 0.616. While analyzing the combined flow, the researchers took the combined side weir–gate discharge as \({Q}_{{\text{c}}}={Q}_{{\text{g}}}+{Q}_{{\text{w}}}\).

$${Q}_{{\text{c}}}={C}_{{\text{dg}}}a{b}_{{\text{g}}}\sqrt{2g{y}_{1}}+\frac{2}{3}{C}_{{\text{dw}}}{b}_{{\text{w}}}\sqrt{2g}{\left({y}_{1}-p\right)}^\frac{3}{2}$$
(3)
$${\text{Discharge}}\;{\text{coefficient}}\;{\text{of}}\;{\text{combined}}\;{\text{side}}\;{\text{weir - gate:}}\;C_{{\text{d}}} = \frac{{Q_{{\text{c}}} }}{{Q_{{\text{w}}} + Q_{{\text{g}}} }}$$
(4)

It is aimed to increase the discharge capacity by using such structures in lateral flows. However, to our knowledge, there are limited studies on using combined side weir–gate in lateral flows in straight channels (Ghodsian et al. 2020; Kartal 2022; Kartal and Emiroglu 2022). Therefore, this study analyzes H-side weir flow and determines its advantages over classical side weir and gate structures based on dimensional analysis.

The present study experimentally investigated H-side weir flow for different Froude numbers and different flow depths in lateral flows under free-flow conditions and subcritical regimes in a straight channel. Moreover, the study aims to determine the discharge capacity, reveal the flow characteristics, determine the hydraulic performance and increase the discharge capacity for the same length and estimate the H-side weir discharge capacity by combining various artificial intelligence, bio-inspired optimization and data decomposition techniques using geometrical parameters (d/a) and flow parameters. Furthermore, least-squares support-vector machines (LSSVM), particle swarm optimization (PSO) and variable mode decomposition (VMD) algorithms are combined. Moreover, it aims to contribute to hydraulic engineering by using a new hydraulic structure in addition to classical weirs and gates in lateral flows. The effect of flow and geometric properties on the discharge capacity was analyzed with a wide range of experiments on the combined models of a rectangular weir and a rectangular gate (H-weir). The results will contribute to the limited literature on the subject, pioneer future studies and fill the gaps in the related field.

Materials and methods

Weirs and gates have been used in hydraulic engineering for a long time independently of each other in open channel flows. H-weir is a combination of a gate and a weir. Figure 1 shows the flow characteristics of the H-side weir.

Experiments

The experiments were conducted in the Hydraulic Laboratory at Firat University. The experimental setup used in this study is a straight channel with a width of 0.40 m and a height of 0.50 m (Fig. 2). A collection channel with a width of 0.40 m and a height of 0.40 m is also mounted on the main channel (Fig. 2). The bottom slope of the main channel and the collection channel is 0.10%. The experiments were conducted for H-side weirs in a rectangular cross-sectional straight channel under steady flow conditions for the subcritical regime. According to Novák and Cabelka (1981), surface tension effects are practical if the piezometric head over the weir is less than 3 cm. Therefore, all heads were taken larger than 3 cm, and thus, surface tension effects were neglected. Sieves were used as breakers and placed upstream of the main channel and toward the end of the collection channel to ensure stable flow conditions (i.e., unsteady flow). While a digital flowmeter was used to measure Q total, a triangular V-notch weir was used for Qc. A cross-sectional view of the experimental setup and flowchart of the present study is shown in Fig. 2.

Fig. 2
figure 2

Experimental setup and the flowchart of the present study

Full size image

As mentioned above, since the conducted the study of Ahmed (1985), several studies have been carried out. Since then, many researchers have investigated combined weirs and gates for frontal flows and introduced different combined weir–gate structures (Alhamid 1999; Alhamid et al. 1996; Altan-Sakarya and Kökpinar 2013; Negm et al. 2002; Negm 2002; Nouri and Hemmati 2020; Salehi and Azimi 2019; Samani and Mazaheri 2009; Severi et al. 2014).

Discharge of combined weir–gate, Qc (Alhamid 1999; Alhamid et al. 1996; Altan-Sakarya and Kökpinar 2013; Ferro 2000; Negm et al. 2002; Negm 2002; Nouri and Hemmati 2020; Salehi and Azimi 2019; Samani and Mazaheri 2009; Severi et al. 2014) is:

$${Q}_{{\text{c}}}={Q}_{{\text{g}}}+{Q}_{{\text{w}}}$$
(5)

in which Qg and Qw are the discharge of the gate and weir (m3/s), respectively.

Equation (6) uses lateral flows as the discharge equation for the under-gate flow (Ghodsian 2003; Swamee et al. 1993). However, Eq. (7) is the weir discharge equation for the Domínguez approach (Bagheri et al. 2014; Domínguez and Francisco 1959).

$${Q}_{{\text{g}}}={C}_{{\text{dg}}}a{b}_{{\text{g}}}\sqrt{2g{y}_{1}}$$
(6)
$${Q}_{{\text{w}}}=\frac{4}{15}{C}_{{\text{dw}}}{b}_{{\text{w}}}\sqrt{2g}\left(\frac{{h}_{2}^{2.5}-{h}_{1}^{2.5}}{{h}_{2}-{h}_{1}}\right)$$
(7)

If Eqs. (6) and (7) are substituted in Eq. (5). We will get Eq. (8)

$${Q}_{{\text{c}}}={C}_{{\text{dg}}}a{b}_{{\text{g}}}\sqrt{2g{y}_{1}}+\frac{4}{15}{C}_{{\text{dw}}}{b}_{{\text{w}}}\sqrt{2g}\left(\frac{{h}_{2}^{2.5}-{h}_{1}^{2.5}}{{h}_{2}-{h}_{1}}\right)$$
(8)

Negm et al. (2002) and Altan-Sakarya and Kökpinar (2013) reported that applying the classical weir and gate discharge coefficient leads to intolerable mistakes in predicting the combined flow. Salehi and Azimi (2019) also said that interactions were between weir and gate flow. So, it results in excessive energy losses in the combined system. Consequently, using classical equations, the measured combined discharge differs from the calculated discharge separately for the weir and gates. Therefore, a different approach is necessary to estimate the discharge of the H-side weir rather than conventional equations. As stated above, an H-side weir is a system in which a side weir and a side gate are combined, as given in Fig. 1.

Dimensional analysis

The parameters affecting the H-side weir flow are presented in Eq. (9).

$${\Phi }\left( {Q_{{\text{c}}} ,a, b,B,h, d,S_{{\text{o}}} ,{\text{y}}_{1} ,\varphi ,\mu ,\sigma ,g,\rho { }} \right) = 0$$
(9)

where \({Q}_{{\text{c}}}\)=discharge of H-side weir (m3/s), a = gate opening (m), b (bw = bg = b) = length of the H-side weir (m), B = main channel width (m), h = head over the weir (m), d = vertical distance between the crest height of the weir and the top of the gate (m), \({S}_{{\text{o}}}\)= main channel slope (–), \({y}_{1}\) =water depth at the upstream section of H-side weir (m), φ = deflection angle of the streamlines, σ = surface tension (kg/s2), μ = dynamic viscosity (kg/m s), g = gravitational acceleration (m/s2) and ρ = mass density of the water (kg/m3).

As a result, Eq. (10) is obtained (see “Appendix”).

$$\frac{\frac{{Q}_{{\text{c}}}}{b}}{\sqrt{g} {a}^{1.5}}={f}_{1}\left(\frac{{y}_{1}}{a},\frac{b}{B},\frac{b}{a},\frac{h}{b},\frac{d}{a}, {{\text{F}}}_{1}\right)$$
(10)

y1/a and h/b are hydraulic parameters affecting the flow, and b/B, b/a and d/a are geometric parameters affecting the flow. Among the geometric parameters, d/a is called the obstruction ratio. Values and ranges of the physical and hydraulic conditions of the study are given in Table 1.

Table 1 Physical and hydraulic conditions of the present study
Full size table

Least-squares support-vector machine (LSSVM)

The LSSVM algorithm is an enhanced version of SVM. For example, consider a given set of n data points {xi, yi} where xi is the ith input vector, and yi is the corresponding output. Examples of a nonlinear mapping ϕ express the LSSVM algorithm as in Eq. (11).

$$f\left({x}_{i}\right)={w}^{T}\phi \left({x}_{i}\right)+b$$
(11)

where w and b are the variables defined by minimizing.

The objective function:

$$R=\frac{1}{2}{w}^{2}+\frac{1}{2}C\sum_{i=1}^{N}{e}_{i}^{2}$$
(12)

where C is the regularization constant, and ei is the training data error. Equation (3) is used to solve the optimization problem (Mellit et al. 2013; Suykens and Vandewalle 1999).

$$\left\{ {\begin{array}{*{20}l} {{\text{Minimize}}:J\left( {w, e_{i} } \right) = \frac{1}{2}w^{2} + \frac{1}{2}C\mathop \sum \limits_{i = 1}^{N} e_{i}^{2} } \\ {{\text{Subject}}\;{\text{to}}:y_{i} = w^{{T\left( {x_{i} } \right)}} + b + e_{i} i = 1, \ldots N} \\ \end{array} } \right.$$
(13)

Particle swarm optimization (PSO)

PSO is an approach that tries to solve by mimicking the behavior of a flock of birds or fish. To find the most suitable solution, it tries to reach the best solution by updating the positions and velocities of the particles according to their neighbors. The PSO bio-inspired approach is a stochastic population-based technique. In this technique, the motion of each i in the search space is carried out by vectors as calculated in Eqs. (17) and (18).

$$v_{i} = \omega v_{i} + c_{1} \eta_{1} \circ \left( {p_{i} - x_{i} } \right) + c_{2} \eta_{2} \circ \left( {g_{{{\text{best}}}} - x_{i} } \right)$$
(14)
$${x}_{i}={x}_{i}+{v}_{i}$$
(15)

in which ◦: entry-wise products, xi, vi, pi, gbest values indicate position, velocity, best individual solution position and best neighbor solution position, respectively. In addition, c1 and c2 are acceleration constants, ω: inertia weight, η1 and η2: random vectors (Katipoğlu and Sarıgöl 2023; Novoa-Hernández et al. 2011).

Variational mode decomposition (VMD)

VMD is a novel signal decomposition technique for separating a time series into various intrinsic mode functions and residuals. VMD is superior to wavelet transform and Hilbert–Huang transform because it has no modal aliasing effect and is sensitive to noise (Chaitanya et al. 2021; Sarıgöl and Katipoğlu 2023). This study aims to strengthen the VMD signal processing algorithm by integrating it into LSSVM.

Results and discussion

Experimental results

The variation of upstream and downstream flow depth is plotted in Fig. 3. Variations in specific energy at upstream and downstream sections in a straight channel in the H-side weir for the subcritical regime (\({E}_{1}={y}_{1}+\frac{{V}_{1}^{2}}{2g}, {E}_{2}={y}_{2}+\frac{{V}_{2}^{2}}{2g})\) are almost equal to each other and compatible with each other (\({E}_{1}\cong {E}_{2}\)). Similarly, upstream and downstream flow depths are almost equal to each other. Therefore, it is observed that the assumption of De Marchi (1934) and Domínguez and Francisco (1959) used to obtain the discharge coefficients for the gate and weir is correct, namely that the specific energy along the main channel is constant. As a result, specific energy at the upstream and downstream almost being equal to each other shows that De Marchi and Domínguez approaches can be safely used to study the H-side weir flow characteristics (\({E}_{1}\cong {E}_{2}\)).

Fig. 3
figure 3

Variation of upstream and downstream flow depth

Full size image

Figure 4 shows the variation of d/y1 with the parameter dimensionless unit discharge (\(\frac{\frac{{Q}_{{\text{c}}}}{b}}{{g}^{0.5}{a}^{1.5}}\)). When the figure is analyzed, it is seen that the geometry of the H-side weir is practical on the dimensionless unit discharge (qt) value. The slope of the fitted line in Fig. 4 varies with increasing d/a value. The variation has a linear trend (y = ax + b) and a good agreement (R2 = 0.95–0.99).

Fig. 4
figure 4

Variation of dimensionless unit discharge (\(\frac{\frac{{Q}_{{\text{c}}}}{b}}{{g}^{0.5}{a}^{1.5}}\)) with d/y1

Full size image

Figure 5 shows the variation of \(\frac{\frac{{Q}_{{\text{c}}}}{b}}{{g}^{0.5}{a}^{1.5}}\) with the a/y1 parameter. As demonstrated in Fig. 5, geometry effectively affects the dimensionless unit discharge value. The geometrical parameters (d/a and b/B) are shown in Fig. 5. Figure 5 demonstrates that the data can be scattered as different categories based on d/a. The variation of each category is clarified by the effects of geometrical parameters (d/a and b/B). The effect of the parameter d/a is thus significant because it allows discharge through the H-side weir. At a particular value of b/B, the discharge increases sharply as d/a increases. It can be related to increasing the head over the weir. For a particular a/y1, dimensionless unit discharge increases as d/a decreases. More fluctuations form at large lengths (b/B > 0.2) than at small lengths (b/B < 0.2). The deviation between the weir and gate jet trajectory is minimal at small Froude numbers but increases with F1. The weir flow and the gate flow are in contact at large lengths (b/B > 0.2), but this does not appear at small lengths (b/B < 0.2) and high F1 values. The weir flow is in contact with the gate flow, leading to a breaking of the energy of the gate flow, and this is particularly noticeable for small obstruction ratios (d/a). Therefore, hydraulic designers need to consider such situations when using the H-weir in lateral flows for purposes such as water intake and flow regulation.

Fig. 5
figure 5

Variation of qt with a/y1

Full size image

Consequently, the total discharge through the H-side weir increases because the side gate flow is higher than that of the side weir flow for larger a. The slope of the fitted line increases with the decrease of d/a value. The smallest values for the variation of the dimensionless unit discharge with a/y1 are obtained for d/a = 0.50. Their variation is a polynomial function (y = ax2 + bx + c) with a good agreement (R2 = 0.99).

Figure 6 demonstrates the variation of \(\frac{\frac{{Q}_{{\text{c}}}}{b}}{{g}^{0.5}{a}^{1.5}}\) with the b/y1. As demonstrated in Fig. 6, the H-side weir geometry is practical on the dimensionless unit discharge value. The slope of the fitted line varies with the decrease of d/a value. Their relationship is a polynomial function (y = ax2 + bx + c), and its coefficient of determination is 0.99.

Fig. 6
figure 6

Variation of qt with b/y1

Full size image

Figure 7 shows the variation of \(\frac{\frac{{Q}_{{\text{c}}}}{b}}{{g}^{0.5}{a}^{1.5}}\) with b/h1. Figure 7 shows that qt increases with increasing b/a. As shown in Fig. 7, the value of b/a is quite effective on qt. However, after a certain point (b/h1 = 2), it has the same effect on the change between qt and b/h1. It is observed that an exponential relationship can be fitted for this variation. The variation of \({q}_{{\text{t}}}\) with b/h1 is an exponential function (y = a + becx)). It is seen that the relationship with b/h1 has good agreement (R2 = 0.99).

Fig. 7
figure 7

Variation of qt with b/y1

Full size image

A new equation was proposed for hydraulic designers to estimate the discharge capacity of an H-side weir (\(\frac{\frac{{Q}_{{\text{c}}}}{b}}{{g}^{0.5}{a}^{1.5}}\)) using both hydraulic and geometrical parameters. The contribution of a/y1, b/h1, b/B, F1 and d/a to qt is thus significant. As shown in Fig. 8, the measured and calculated values of qt are compatible. Therefore, it can be reported that the proposed equation [Eq. (16)] is in good agreement. This equation was obtained using the least-squares error method (R2 = 0.995) for H-side weir flow under 0 < F1 < 1 and 0.10 \(\le b/B\le\) 0.40 and 0.50 \(\le d/a\le\) 5 conditions.

Fig. 8
figure 8

Variation of measured and calculated qt values

Full size image
$${q}_{{\text{t}}}=0.18{{\text{F}}}_{1}^{-0.04}{\left(\frac{a}{{y}_{1}}\right)}^{-2.56}{\left(\frac{b}{{y}_{1}}\right)}^{0.50}{\left(\frac{b}{a}\right)}^{-0.56}{\left(\frac{d}{a}\right)}^{-0.60}{\left(\frac{b}{B}\right)}^{-0.021094}$$
(16)

Figure 8 shows the comparison of the predicted and measured values of \(\frac{\frac{{Q}_{{\text{c}}}}{b}}{{g}^{0.5}{a}^{1.5}}\). As demonstrated in Fig. 8, it is clear that the scatter data are compatible, i.e., there is a perfect correlation between the measured and predicted values, and therefore, the equation is very reliable (R2 = 0.995).

Figure 9 shows the interpretation of the residuals with measured values of \(\frac{\frac{{Q}_{{\text{c}}}}{b}}{{g}^{0.5}{a}^{1.5}}\) at H-side weir. The figure shows tiny differences between the measured and predicted values. So, it can be stated that the study is reliable.

Fig. 9
figure 9

Variation of residuals with qt

Full size image

Figure 10 shows the variation of \(\frac{\frac{{Q}_{{\text{c}}}}{b}}{{g}^{0.5}{a}^{1.5}}\) with F1. As shown in Fig. 10, qt is large at small upstream Froude numbers (i.e., from F1 = 0.05 to F1 = 0.20). It is observed that after a certain value of F1 (from F1 = 0.40), qt nearly remains the same.

Fig. 10
figure 10

Variation of qt parameter with F1

Full size image

Figure 11 shows the variation of qt parameter with b/B. As shown in Fig. 11, b/B ratio affects the dimensionless unit discharge.

Fig. 11
figure 11

Variation of qt with b/B

Full size image

The interpretation of the discharge coefficient of the H-side weir with \(\frac{\frac{{Q}_{{\text{c}}}}{b}}{{g}^{0.5}{a}^{1.5}}\) is plotted in Fig. 12. Generally, Cd increases logarithmic with increasing value of qt. The discharge coefficient values range from around 0.50–0.68. The data in Fig. 12 are scattered due to the variation of the H-side weir models' geometrical parameters (d/a and b/a). It was observed that large discharge coefficient values are related to high b/a and d/a values and vice versa.

Fig. 12
figure 12

Variation of discharge coefficient with qt

Full size image

The comparison of the present study with the literature is plotted in Fig. 13. The present study is compared with the studies of Negm et al. (2002) and Askeroğlu (2006). They conducted their studies for frontal flows. However, the present study is conducted for lateral flow. Negm et al. (2002) stated that the study is valid for 0.47 ≤ d/a ≤ 4.00, 0.65 ≤ b/a ≤ 5.00, 1.80 ≤ y1/a ≤ 7.50 and 0 ≤ S0 ≤ 0.0161 for frontal flow. Moreover, Askeroğlu (2006) stated that the study is valid for 1.34 ≤ y1/a ≤ 8.30, 0.25 ≤ d/a ≤ 4.40, 0.06 ≤ h/b ≤ 2.08, 2.50 ≤ B/b ≤ 6.00, 1.00 ≤ b/a ≤ 2.50 conditions. Although the flow type differs, the present study is compatible with Negm et al. (2002) and Askeroğlu (2006). The present study was conducted for 0.10 \(\le b/B\le\) 0.40, 0 < F1 < 1, 2 ≤ b/a ≤ 4 and 0.50 \(\le d/a\le\) 5. Trend of variation of qt with a/y1 is similar to the literature. The differences between qt values can be due to different geometric (d/a, b/a and b/B) and hydraulic conditions (a/y1, b/y1, b/h1 and d/y1) and flow type.

Fig. 13
figure 13

Comparison of the present study with the literature

Full size image

Statistical values for the current study are given in Table 2. As shown in Table 2, the data have a good agreement. Bias refers to a systematic deviation from the actual value. The root mean square errors (RMSE), average percent error (APE), mean absolute percentage errors (MAPE), scatter index (SI), BIAS and coefficient of determination (R2) were calculated below.

$${\text{RMSE}}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}{\left({{\text{Predicted}}}_{i}-{{\text{Observed}}}_{i}\right)}^{2}}$$
(17)
$${\text{SI}}=\frac{{\text{RMSE}}}{\overline{x} }$$
(18)
$${\text{BIAS}}=\frac{1}{N}\sum_{i=1}^{N}\left({{\text{Observed}}}_{i}-{{\text{Predicted}}}_{i}\right)$$
(19)
$${\text{MAE}}=\frac{1}{N}\sum_{i=1}^{N}\left|{{\text{Observed}}}_{i}-{{\text{Predicted}}}_{i}\right|$$
(20)
$${\text{MAPE}}=\frac{100}{N}\sum_{i=1}^{N}\left|\frac{{{\text{Observed}}}_{i}-{{\text{Predicted}}}_{i}}{{{\text{Observed}}}_{i}}\right|$$
(21)

in which N is the total number of runs, i is its run and \(\overline{x }\) is the mean value observed.

Table 2 Statistical values of proposed Eq. (19)
Full size table

Moreover, Qc increases with increasing b/a at constant d/a = 2. For narrow range of d/a (i.e., d/a = 0.5 and 1), little variations are observed; for a wide range of y/d (i.e., d/a = 2 and 5), high variations of Qc with a/y1 are observed. The decreasing trend of the change between a/y1 and Qc decreases with the increase of b/a. For instance, the slope line is 0.1041 for b/a = 4 but 0.0612 for b/a = 2. The slope line decreases by 41.2% with the decrease of b/a. The variation has a linear trend (y = ax + b), and the coefficient of determination is from 0.95 to 0.99. The trend of the change between d/y1 and Qc increases with the increase of b/a. For instance, the slope of the line is 0.0297 for b/a = 4 and 0.031 for b/a = 2. The slope line increases by 4.4% with the increase of b/a. The variation is a linear trend (y = ax + b), and the coefficient of determination is from 0.95 to 0.99. Moreover, the discharge capacity of the H-side weir is 1.3–4 times higher than conventional side weirs and 1.6–9.0 times higher than side gates under the same conditions.

LSSVM, PSO and VMD results

Moreover, the study aims to estimate the H-side weir discharge capacity by combining various artificial intelligence, bio-inspired optimization and data decomposition techniques using geometrical parameters (d/a) and flow parameters. Furthermore, least-squares support-vector machines (LSSVM), particle swarm optimization (PSO) and variable mode decomposition (VMD) algorithms are combined. This study selected 70% of the data for training and 30% for testing in setting up the prediction model. PSO updates particle velocities and positions using global best result (gbest) values and pbest, and the search process ends when the maximum number of iterations is reached. PSO was applied with 300 iterations. Different experiments were conducted as the number of particles (N) and c constants varied depending on the problem to be solved. Therefore, N = 20 and c1 = c2 = 2 values were selected as they yielded very prosperous results. The SVM model was set up using a radial basis kernel function.

Experimental data used to estimate the H-side weir discharge capacity were sub-banded by VMD. VMD-LSSSVM and PSO-VMD-LSSVM hybrid models were established by presenting them as inputs to such LSSVM and PSO-LSSVM models in Fig. 14.

Fig. 14
figure 14

Separation of experimental data used in the study into sub-bands with VMD

Full size image

Taylor diagrams are used to evaluate the performance of the artificial intelligence models established in the H-side weir capacity estimation in Fig. 15. Taylor diagrams are widely used to visualize the similarities and relationships between time series. Accordingly, during the training phase, the VMD-LSSVM hybrid approach with the highest correlation coefficient (R) and the lowest root mean square error (RMSE) value showed the most accurate estimation results. In addition, the fact that this model is closest to the actual data and has the same standard deviation value supports the predictive success of the model. In the testing phase, it was concluded that the PSO-LSSVM model was the best model with the lowest R and highest RMSE values. In addition, it is noteworthy that all models' predictions are promising and satisfactory.

Fig. 15
figure 15

Comparison of H-side weir discharge capacity estimation models with Taylor diagram

Full size image

Boxplot graphics were used during the training phase to evaluate the H-side weir discharge capacity estimates in Fig. 16. These graphs visually select the most suitable model by comparing the actual data's spread, distribution, percentiles and outliers with the artificial intelligence prediction results. According to this, it is deduced that the properties are the most suitable for the actual data and the LSSVM and VMD-LSSVM models because their structures are similar.

Fig. 16
figure 16

Comparing the accuracy of the prediction models in the training phase with Boxplot plots

Full size image

Boxplot plots were used in the testing phase to compare the estimation results of the H-side weir discharge capacity in Fig. 17. Accordingly, the PSO-LSSVM model shows the most compatible distribution with the actual data. It is also noteworthy that the structure of the PSO-VMD-LSSVM models is quite similar to the actual data.

Fig. 17
figure 17

Comparing the accuracy of the prediction models in the test phase with Boxplot plots

Full size image

The success of the hybrid estimation models of H-side weir discharge capacity was compared based on various statistical indicators in Table 3. The optimum model was indicated to have the lowest error, Bias factor closest to 1 and highest determination coefficient value (R2). Therefore, statistical indicators were ranked to determine the optimum model, with the highest rank value indicating the best model. Accordingly, the success ranking of the models was presented as VMD-LSSVM > PSO-LSSVM > PSO-VMD-LSSVM > LSSVM.

Table 3 Testing the accuracy of the established hybrid prediction models according to various statistical indicators
Full size table

Conclusion

In this study, the flow characteristics of H-side weirs were investigated in detail for free-flow conditions and subcritical flow regimes in a straight channel. The discharge capacity of the H-side weir is 1.3–4.0 times higher than conventional side weirs and 1.6–9.0 times higher than side gates under the same conditions. The side flow breaks the gate flow in low d/a values, while the side flow does not contact the gate flow and thus does not break the gate flow. The geometrical parameter (d/a) and the flow parameters (a/y1, h/b and F1) significantly affect discharge capacity, while the others are negligible. Therefore, a discharge equation was proposed using a/y1, h/b, b/B, F1 and d/a with a mean absolute percentage error (MAPE) of about 5.84%. The success ranking of the models was presented as VMD-LSSVM > PSO-LSSVM > PSO-VMD-LSSVM > LSSVM. It was concluded that the PSO-LSSVM model was the best model with the lowest R and highest RMSE values in the testing phase. The developed equation [Eq. (16)] is compatible with observations within the limitations of the present experimental work. Moreover, the most successful model for estimating the H-side weir discharge capacity values was VMD-LSSVM based on statistical analyses. In addition, it was concluded that the established LSSVM, PSO-LSSVM and PSO-VMD-LSSVM models produced very satisfactory predictions.