Introduction

Weirs are categorized into linear and non-linear types based on their plan shapes, with non-linear weirs distinguishing themselves by a longer crest length within a given width compared to linear weirs. This geometric advantage translates to a higher flow capacity under identical hydraulic heads. Non-linear weirs typically exhibit various geometric shapes, including triangular, trapezoidal, rectangular, or arc forms arranged periodically along the flow width. Among these designs, the Piano Key Weir (PKW) has emerged as a highly effective modification for enhancing discharge capacities (Lempérière and Ouamane 2003). PKWs can be classified into four types—A, B, C, and D—depending on their respective upstream and downstream overhangs, as illustrated in Fig. 1. Type A features overhangs on both sides, type D lacks any overhangs, while type B has an upstream overhang and type C has a downstream overhang.

Fig. 1
figure 1

Types of PKWs (Lempérière et al. 2011)

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The inherent advantages of piano key weirs (PKWs) include their compact space requirements, allowing for smaller foundations and making them suitable for installation on concrete dam crests (Lempérière 2017). Remarkably, PKWs can facilitate up to four times the flow of traditional linear weirs (Machiels 2012). Khassaf and Baghdadi (2015) highlighted these benefits, emphasizing the increased efficiency of the trapezoidal PKW shape compared to its rectangular counterpart (Mehboudi et al. 2014).

Designing hydraulic structures to ensure safety and cost-effectiveness necessitates accurate estimations of maximum scour depths, which can significantly impact structural integrity. Scouring can compromise bed protection and jeopardize the stability of weirs. Research indicates that maximum scour depth is inversely related to tailwater depth, while an increase in discharge can exacerbate scour hole dimensions (Ahmadi Dehrashid and Gohari 2019). Jüstrich et al. (2016) experimentally investigated the scouring of the bed downstream of a rectangular PKW, showing that the maximum scour depth depended on sediment characteristics, discharge, the difference in upstream and downstream flow depths, and tailwater depth. Yazdi et al. (2020) studied the scouring downstream of trapezoidal and rectangular PKWs, revealing that the scour depth in a rectangular PKW was greater than that of a trapezoidal PKW, indicating a relative decrease in scour depth for trapezoidal designs. Ghafouri et al. (2020) demonstrated that an increase in tailwater depth led to a decrease in scour depth downstream of a trapezoidal PKW and presented an equation for the longitudinal profile of the scour hole. Kumar and Ahmad (2020) studied the scour downstream of a rectangular PKW with and without an apron, finding significant effects on scour hole length and depth with higher discharges and lower tailwater depths. They also noted the reducing effect of the apron on scouring. Ghodsian et al. (2021) compared scours downstream of trapezoidal and triangular PKWs, showing that the maximum relative scour depth increased, and the maximum scour depth location occurred farther from the weir, with increased scour hole length and volume as the particle Froude number rose. In general, the average maximum scour depth of a triangular PKW was greater than that of a trapezoidal PKW.

Lantz et al. (2021) investigated the effects of an apron and cutoff wall on the scour downstream of a rectangular PKW, observing significant reductions in scouring with an apron length of 1.5 times the weir height. Bodaghi et al. (2022) examined scouring downstream of a trapezoidal type-A PKW under free and submerged flow conditions, showing that increased submersion ratios led to decreased maximum relative scour depth. Abdi Chooplou et al. (2022a, b) investigated scour downstream of a trapezoidal PKW under varied conditions. Jamal et al. (2022) conducted an experimental study on a type-C rectangular PKW, examining the effects of weir height and different ratios of inlet key width to outlet key width on scour downstream of the weir with and without an apron, finding that scour depth decreased with increased apron length.

Abdi Chooplou et al. (2023) conducted experimental and numerical studies on the influence of lateral wall crest shape on flow field and scour downstream of a rectangular PKW, indicating that lateral crest indentations alleviated scour and improved performance. The PKW with triangular crest indentations caused asymmetric and weaker jets to impact the bed downstream of the outlet keys. Karimi Chahartaghi et al. (2023) investigated local scour downstream of rectangular and trapezoidal PKWs, demonstrating that scour hole dimensions and bed topography patterns are significantly influenced by hydraulic conditions, crest geometry, and bed material gradation. Their analysis revealed that local scour depths downstream of PKWs with a rectangular crest could be larger than those associated with trapezoidal shapes. Rdhaiwi et al. (2023) studied the effectiveness of a stilling basin in reducing scour downstream of C-type trapezoidal PKWs, finding that the presence of a stilling basin significantly mitigated scour, elongating the scour hole and shifting the maximum scour depth further downstream relative to the weir toe. Bodaghi et al. (2024) conducted an experimental investigation on trapezoidal PKWs under free and submerged flow conditions, indicating that maximum scour depth, distance to maximum scour depth from the weir toe, scour hole length, and weir toe scour depth under free flow conditions increased compared to submerged flow conditions, with maximum scour depth decreasing at higher submergence ratios. Abdi Chooplou et al. (2024a) examined scour hole geometry downstream of different PKW designs, finding that while scour parameters followed similar trends, maximum values for scour depth, hole length, area, and volume varied, with trapezoidal PKWs showing reductions in scour depth compared to rectangular and triangular PKWs. Kumar et al. (2024) conducted experimental and computational fluid dynamics studies on flow fields around rectangular PKWs, revealing significant insights into velocity distributions, indicating that mean vertical velocity was higher in front of the outlet key compared to the inlet key, while mean lateral and longitudinal velocities were relatively low. Notably, the maximum velocity was recorded in front of the inlet key, enhancing sediment movement over the PKW. Fathi et al. (2024) compared stepped and non-stepped trapezoidal PKWs, finding that maximum scour depths for stepped weirs were lower than those of non-stepped weirs, with increased steps and decreased bed material size leading to reduced sediment ridge height. Dehghan and Karami (2024) conducted a numerical study comparing scour downstream of trapezoidal and triangular PKWs, finding that trapezoidal PKWs experience significantly less scouring than triangular ones; doubling the tailwater depth reduced scour depth in both types, while a decrease in discharge at constant tailwater depths led to a reduction in maximum scour depth downstream of both trapezoidal and triangular PKWs. Abdi Chooplou et al. (2024b) studied the impact of baffles on energy dissipation and scour reduction downstream of rectangular and trapezoidal PKWs. Their results showed that baffles increased energy dissipation and reduced scour depth, with the development of accurate empirical equations and machine learning models, highlighting the effectiveness of baffles in minimizing scour effects, particularly in trapezoidal PKWs.

Generally, nappe oscillation of the outflow is one of the problems associated with PKWs. Ehsanifar et al. (2022) showed that flow splitters reduced the intensity of nappe oscillation and increased aeration of PKWs. Their results revealed that flow splitters performed best for flow aeration at H/P ≤ 0.6, where H and P are the upstream head and weir height, respectively. In terms of geometrical shape, square and rectangular splitters showed similar performance in water passage and flow aeration, although the separation of flow with rectangular splitters was evaluated as superior to that with square splitters. Ehsanifar et al. (2024) investigate the impact of flow splitters on the hydraulic performance of PKWs, focusing on various designs including circular, square, and rectangular shapes. Their findings highlighted several benefits of using flow separators: they enhance flow separation by connecting trapped air beneath the flow to the free surface, which helps mitigate nappe oscillation—a common issue in PKW flow dynamics. Additionally, while the flow separators did not significantly alter the discharge coefficients under submerged conditions, they proved to be more effective in free-flow scenarios, particularly with rectangular and square designs outperforming circular ones in terms of flow separation and energy dissipation. Souri et al. (2024) conducted a comprehensive numerical study examining the impact of aeration on the performance of PKWs, revealing that aeration plays a significant role in shaping the water jet trajectory and nappe oscillation—both critical factors influencing the overall efficiency of PKW structures.

Flow splitters effectively mitigate nappe oscillation and enhance aeration in Piano Key Weirs (PKWs). They improve flow separation and energy dissipation, without affecting the discharge coefficient or efficiency of trapezoidal PKWs (Ehsanifar et al. 2022, 2024). Additionally, flow splitters play a critical role in PKW performance by enhancing the aeration of flow (Souri et al. 2024). However, there is a notable gap in the literature regarding the impact of flow splitters on scour reduction in trapezoidal PKWs. This study aims to investigate the effects of flow splitters with three geometries (square, rectangular, and circular) on scour hole characteristics downstream of a type-A trapezoidal PKW under various free flow conditions, focusing on bed profiles, weir toe scour depth, maximum scour depth, distance of maximum scour depth from the weir toe, scour hole volume, temporal evolution of scouring, and sediment ridge characteristics. By addressing this gap, the study seeks to provide valuable insights into the potential benefits of flow splitters in reducing scour downstream of trapezoidal PKWs, which is crucial for their effective design and operation.

Materials and methods

Dimensional analysis

The effective parameters on scouring downstream of a type-A trapezoidal PKW (Fig. 2) depend on the hydraulic and geometric parameters, such as the difference between upstream and downstream heads (\(\Delta H\)), tailwater depth (\({\text{H}}_{\text{d}}\)), weir height (P), weir lateral wall length (B), total weir width (\(W\)), weir wall thickness (\({T}_{s}\)), number of weir keys (N), fluid density (\(\rho\)), sediment density (\({\rho }_{s}\)), discharge per unit width (q), mean sediment size (\({d}_{50}\)) and shape of splitters (SP). The scour hole parameters shown in Fig. 2b include the scour depth at the weir toe (\({Z}_{f}\)), the maximum scour depth (\({d}_{sm}\)), and the distance of location of maximum scour depth from the weir toe (\({L}_{s}\)).

Fig. 2
figure 2

Trapezoidal PKW: a plan, and b flow characteristics and geometries of the scour hole and sediment ridge

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Therefore:

$$Z_{f} , d_{sm} , L_{s} = f\left( {\Delta H, H_{d} , P, B, W, T_{s} , N, SP, g, \rho , \rho_{s} , q, d_{50} } \right)$$
(1)

Using the Buckingham Π-theorem with the repeating variables of \(\rho\), ΔH, and q, and after ignoring the constant parameters, the following dimensionless equation is obtained:

$$\frac{{Z_{f} }}{\Delta H}, \, \frac{{d_{sm} }}{\Delta H}, \, \frac{{L_{s} }}{\Delta H} = f\left( { \frac{{H_{d} }}{\Delta H}, \, \frac{{\rho_{s} - \rho }}{\rho }, \, \frac{{d_{50} }}{\Delta H}, \, \frac{q}{{\sqrt {g\Delta H^{3} } }}, SP} \right)$$
(2)

With the combination of \(\frac{q}{\sqrt{g{\Delta H}^{3}}}\), (\(\frac{{\rho }_{s}-\rho }{\rho }\)), and \(\frac{{d}_{50}}{\Delta H}\), the particle Froude number is obtained as follows:

$${\text{Fr}}_{\text{d}}=\frac{q}{\Delta H\sqrt{g{d}_{50}(s-1)}}$$
(3)

here s (= ρs/ρ) is the relative density of sediment particles. Therefore, Eq. 2 reduces to:

$$\frac{{Z_{f} }}{\Delta H}, \, \frac{{d_{sm} }}{\Delta H}, \, \frac{{L_{s} }}{\Delta H} = f\left( {{\text{Fr}}_{d} , \, \frac{{H_{d} }}{\Delta H}, SP} \right)$$
(4)

The experiments were carried out in a rectangular channel of 10 m length, 0.75 m width, and 0.8 m height in the hydraulic laboratory of the water engineering and hydraulic structures group, Tarbiat Modares University, Tehran. The channel bed and side walls were made of metal and glass, respectively. The water was supplied from a sump by using a pump with a maximum discharge of 85 L/s. All experiments were conducted under free flow conditions, which were maintained by keeping the tailwater depth below 0.48H (Dabling and Tullis 2012). A gate installed at the downstream end of the flume was used to adjust the tailwater depths within the range of 0.08–0.18 m.

A type-A trapezoidal PKW with six keys was made of thermoplastic material by a 3D printer and placed at a distance of about 3 m from the channel entrance. The PKW is displayed in Fig. 3 and its characteristics are specified in Table 1.

Fig. 3
figure 3

The used PKW and its parameters

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Table 1 Geometric features of the used type-A PKW
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Flow splitters with three cross-sectional geometries: square, rectangular, and circular were used. The splitters were installed on the downstream corners of the crest of inlet keys (Fig. 4). The specifications of the splitters are shown in Table 2.

Fig. 4
figure 4

Flow splitters with: a circular, b rectangular, and c square cross-section

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Table 2 Geometric features of splitters
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Uniform sediments with an median size of 2.2 mm were utilized as the downstream bed. The tests were performed in two scenarios: with and without splitters, to facilitate comparative analysis. The upstream flow depth in all experiments exceeded 3 cm, rendering the effect of surface tension negligible (Pfister et al. 2013).

After installing and sealing the weir, sediments were placed on the downstream channel bed and leveled with a bed flattener. A metal mesh was placed at the channel’s end to prevent sediment from entering the sump. To mitigate the influence of initial conditions, a thin metal sheet was placed over the bed materials. Discharge and tailwater depth were adjusted using an ultrasonic flow meter and a gate installed at the channel’s end, respectively. Upstream and downstream flow depths were measured with a digital point gauge, accurate to ± 1 mm. Once the flow stabilized, tests commenced by gently removing the metal sheet. Upon completion of each experiment and after draining the channel, bed topography was measured using a laser meter with an accuracy of ± 1 mm. Measurements were taken over a zone of 2 m in length and 0.75 m in width.

The experiments were performed for three tailwater depths (i.e. 8, 13, and 18 cm) and three discharges (i.e. 30, 40, and 50 L/s), both with and without splitters. Table 3 summarizes the test details. In this table, the symbol MQ-\({H}_{d}\) denotes specific test, where M represents the splitter shape (S for square, R for rectangular, and C for circular), Q indicates the discharge in liter per second, and \({H}_{d}\) denotes the tailwater depth in centimeters. For example, S30-8 represents the test with square splitters, a discharge of 30 L/s, and a tailwater depth of 8 cm. The results indicate that the geometric variations in flow splitters did not significantly influence the upstream water head (H).

Table 3 Specifications of the tests
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To determine the appropriate test duration, a 15-h test was conducted with the maximum discharge (i.e. 50 L/s) and minimum tailwater depth (i.e. 8 cm). Scour depth was measured as depicted in Fig. 5. According to this figure, approximately 96% of the scour depth occurred within the first 5 h, with negligible changes thereafter. This duration, which satisfies Chiew’s criteria (1992) for equilibrium state, was adopted for subsequent tests.

Fig. 5
figure 5

Temporal evolution of the scour depth

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Results and discussion

In this research, the features of the scour hole downstream of a trapezoidal PKW were considered for different discharges and tailwater depths with and without the flow splitters. It was observed that the bed materials started to move with the onset of the experiments and erosion took place downstream of the weir. Maximum extension of the scour hole downstream of the weir generally occurred at a length of about 50 cm from the weir toe, which altered depending on the hydraulic conditions of the flow. Figure 6 portrays the downstream bed topography at the end of the tests, for the weir with and without the flow splitters when Q = 30 L/s and Hd = 13 cm. The scouring took place directly downstream of the weir's toe, while the bed at the end of the channel remained largely unchanged. Furthermore, the sediments eroded from the scour hole accumulated to form a downstream sediment ridge. The most significant scouring occurred downstream of the outlet keys. The flow splitters have successfully minimized topographical alterations in the bed downstream of the weir, as demonstrated by the figure provided.

Fig. 6
figure 6

Downstream bed topography in Tests 50-13 and S50-13: a without flow splitters, and b with square flow splitters

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Figure 7 shows the temporal development of the longitudinal bed profile and sediment ridge during the test S30-18. As shown in Fig. 7a, after 0.2 h, the outflow jet from the weir keys initiated the formation of a scour hole and a sedimentary ridge. Over time, the volume of the scour hole increased, and the sediment ridge moved away from the weir toe. Additionally, the slope of the sediment ridge becomes gentler, as evident in Fig. 7b. By the 2nd and 5th hours of the experiments, the volume of the scour hole and the maximum scour depth had increased slightly (Fig. 7c, d), indicating a decreasing rate of scouring over time. This reduction in scouring rate is attributed to the diminishing excess shear stress acting on the bed (Lantz et al. 2022; Bodaghi et al. 2024). It was also observed that the slope of the sediment ridge decreased by the end of the experiment, as the flow washed the upper part of the ridge and transported the sediments to the end of the channel.

Fig. 7
figure 7

Temporal evolution of the longitudinal bed profile for Test S30-18 at the time: a 20 min, b 1 h, c 2 h, and d 5 h from the start of the experiment

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Figure 8 presents the temporal evolution of the longitudinal bed profile for Test S30-18. According to this figure, the greatest amount of scour occurred within the first hour of the test. Also, a high sediment ridge formed downstream of the scour hole. After approximately the 2nd hours, the maximum scour depth increased by only about 6% until the end of the experiment. By the end of the experiment, the location of the maximum sediment ridge height had shifted farther away from the weir toe.

Fig. 8
figure 8

Temporal evolution of the scour profile for Test S30-18

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Experimental observations following the implementation of the square flow splitter, as depicted in Fig. 9, illustrate significant alterations in the flow dynamics. The presence of the splitter facilitated flow separation, creating a crucial connection between the entrapped air beneath the flow and the free surface of the water. This interaction effectively diminishes the intensity of nappe oscillation, which is vital for maintaining the stability and efficiency of the weir. Consequently, the scouring was reduced in scenarios with flow splitters. Furthermore, the strategic incorporation of flow splitters in water management systems has been shown to enhance the overall efficiency of weirs (Ehsanifar et al. 2024).

Fig. 9
figure 9

Separation after the square splitter

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Figure 10 shows the effect of square splitters on the longitudinal bed profiles downstream of the weir for different Frd and Hd/ΔH. Generally, the maximum scour depth in experiments with splitters installed was, on average about 11% less than in those without splitters. The largest difference, approximately 19%, was observed in tests 30-8 and S30-8. The maximum scour depth was also formed downstream of the outlet keys in the presence of square splitters. The distance from the location of the maximum scour depth to the weir was, on average, 22% less in weirs with splitters compared to those without. This reduction is attributed to the aeration of the flow through the gap created downstream of the splitters (Fig. 10). The highest difference in the maximum height of the sediment ridge occurred in tests 50-18 and S50-18 (Fig. 10c), while the lowest difference was observed in tests 30-8 and S30-8 (Fig. 10a). On average, the maximum height of the sediment ridge in weirs with square splitters was 8% less than in those without splitters. The scour depth at the toe of the weir was, on average, 6% less in weirs with square splitters compared to those without.

Fig. 10
figure 10

Comparison of the longitudinal bed profiles downstream of the weir with and without square splitters for tests: a 30-8 and S30-8, b 40-13 and S40-13, and c 50-18 and S50-18

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Figure 11 shows the reducing effect of rectangular splitters on the maximum scour depth. The maximum scour depth in weirs with rectangular splitters was, on average, about 13% less than in those without splitters. The highest difference, approximately 23%, was observed in tests 40-8 and R40-8. The maximum scour depth was again formed downstream of the outlet keys in tests with rectangular splitters. The distance from the location of the maximum scour depth to the weir toe was, on average, 26% less in weirs with splitters compared to those without.

Fig. 11
figure 11

Comparison of the longitudinal bed profiles downstream of the weir with and without rectangular splitters for tests: a 30-8 and R30-8, b 40-18 and R40-18, and c 50-18 and R50-18

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The highest difference in the maximum height of the sediment ridge occurred in tests 50-18 and R50-18 (Fig. 11c). On average, the maximum height of the sediment ridge in the weirs with the rectangular splitters was 6% less than in those without splitters. The maximum difference in the scour depth at the toe of the weir due to rectangular splitters was, on average, 8%, observed in tests 40-18 and R40-18. The length of the scour hole decreased by 16, 2, and 12%, for tailwater depths of 8, 13, and 18 cm, respectively, compared to the case without splitters.

Figure 12 shows the effect of circular splitters on the longitudinal profiles of the scour hole downstream of the outlet keys. It was observed that the maximum scour depth in weirs with circular splitters was, on average, about 11% less than in those without splitters. The highest difference, approximately 26%, was observed in tests 30-18 and C30-18. The maximum scour depth was also formed downstream of the outlet keys. The distance from the location of the maximum scour depth to the weir toe was, on average, 27% less in the weirs with circular splitters compared to those without splitters. The highest difference in the maximum height of the sediment ridge occurred in tests 50-18 and C50-18. On average, the maximum height of the sediment ridge in weirs with circular splitters was 3% lower than in those without splitters. The maximum scour depth at the toe of the weir was, on average, 5% less in weirs with circular splitters compared to those without. This difference was most notable in tests 40-18 and C40-18. In tests with circular splitters, the length of the scour hole was reduced by 4%, 17%, and 12% for tailwater depths of 8 cm, 13 cm, and 18 cm, respectively, compared to tests without splitters.

Fig. 12
figure 12

Comparison of the longitudinal bed profiles downstream of the weir with and without circular splitters for tests: a 30-8 and C30-8, b 40-18 and C40-18, and c 50-18 and C50-18

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According to Ehsanifar et al. (2024) and Table 3, splitters in trapezoidal piano key weirs do not affect the upstream water head and the discharge coefficient. Therefore, the reduction in scour hole parameters in the tests with the splitters is due to the presence of splitters. The reduction of the maximum scour depth can be explained by the fact that the flow after the splitters is separated and divided into several parts (see Fig. 9). This flow separation which is associated with air entrainment, reduces the scouring potential of falling jet. Therefore, the scour depth is reduced. Figures 10, 11 and 12 show that the splitters reduce the distance of the location of maximum scour depth from the weir, especially at lower values of Frd and Hd/ΔH. This is because the splitters reduce the length of the jet trajectory, as shown in Fig. 13. This figure compares the length of the jet trajectory (X) for tests R30-13 and 30-13. It is clear from this figure that the splitters reduced the distance of the location of the jet impact point from the weir. According to Fig. 13, the value of X = 24 cm for the weir without the splitters, while X = 21 cm for the weir with the splitters. Accordingly, the location of maximum scour depth approaches the weir with the splitters. In general, the distance to the location of maximum scour depth from the weir toe was, on average, 12% shorter in the weir equipped with flow splitters. This phenomenon occurs because the splitters alter the flow dynamics, leading to a concentration of erosive forces in the vicinity of the weir. While the overall maximum scour depth is diminished, the proximity of this depth to the weir indicates that the flow splitters, while beneficial in mitigating scour, may also necessitate careful consideration of weir design and maintenance to address potential localized erosion issues that could arise as a result.

Fig. 13
figure 13

Comparison of the location of the jet collision with the downstream flow in tests: a R30-13, and b 30-13

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The volume of the scour hole (Vs) was calculated by using Tecplot software, as given in Table 4 for all the tests. The volume of scour holes in tests with flow splitters is significantly less than in tests without them, highlighting the impact of splitter shape on flow dynamics. Rectangular-shaped flow splitters are more effective than square ones, as they create greater flow separation and turbulence, leading to enhanced energy dissipation. Their larger surface area increases interaction with the water, promoting mixing and chaotic flow patterns that dissipate kinetic energy more effectively. In contrast, circular splitters produce a more streamlined flow, resulting in less disruption and lower energy dissipation. The volume of scour holes in tests with rectangular, square, and circular splitters showed reductions on average of 18.53%, 17.77%, and 14.92%, respectively, compared to tests without splitters. However, as discharge increases, the effectiveness of these flow splitters in reducing scour depth diminishes due to greater hydraulic forces that can overwhelm their mitigating effects. Elevated flow velocities lead to increased energy and momentum, resulting in more intense turbulence and erosive forces that counteract the benefits of the splitters. Consequently, while flow splitters are beneficial for managing scour, their ability to significantly reduce maximum scour depth is less pronounced under high flow conditions, indicating the need for careful consideration of flow rates in the design and implementation of such mitigation measures. The highest changes in the volume of scour holes in the tests with square, rectangular, and circular splitters compared to the test without the splitters are observed in test 30-8, which have led to a reduction of 33.33, 37.62 and 33.65% of the scour hole volume, respectively. Table 4 also shows that, by increasing the discharge, the volume of the scour hole enhanced for the different shapes of splitters. Moreover, the volume of the scour hole decreased by increasing the tailwater depth, indicating a complex interaction between flow dynamics and splitter geometry.

Table 4 Comparison of scour volume without and with the flow splitters
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In Fig. 14, the relationship between relative tailwater depth and relative scour depth is illustrated, showing that at a constant discharge, an increase in relative tailwater depth leads to a decrease in relative scour depth. This phenomenon can be attributed to several key factors: (1) as the relative tailwater depth increases, the effective drop height of the water flowing over the weir decreases, resulting in reduced kinetic energy upon reaching the bed and less erosive force exerted on the sediment; (2) higher tailwater levels contribute to greater energy dissipation in the flow, leading to a diminished capacity for the flow to scour the sediment; (3) the increase in tailwater depth alters the hydraulic conditions downstream of the weir, creating a more stable flow regime that is less favorable for sediment entrainment and erosion; (4) with higher tailwater levels, the flow becomes less capable of entraining sediment particles and more likely to experience deposition; and (5) the increased hydrostatic pressure on the riverbed helps to keep sediment particles in place, making them less susceptible to being scoured away by the flow. The combination of these factors collectively contributes to a more stable riverbed environment, resulting in less scouring activity as the relative tailwater depth increases. Additionally, in all the graphs of Fig. 14, the reducing effect of the splitters on the relative scour depth is clear.

Fig. 14
figure 14

Effect of relative tail-water depth on relative scour depth: a Circular splitter, b Rectangular splitter, and c Square splitter

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Figure 15 compares the effect of the shape of the splitters on the maximum relative scour depth for different particle Froude numbers and tailwater depths. It is again clear that the shape of the splitters influences the relative scour depth. In general, the reduction of the relative scour depth due to the rectangular, square, and circular splitters was 13, 11, and 10%, respectively. The rectangular and square splitters outperformed the circular splitters by 4% and 1.5%, respectively, in reducing relative scour depth due to their geometric design. The sharper edges and corners of rectangular and square splitters allow for better flow diversion and turbulence management, leading to more effective sediment stabilization and reduced scouring compared to the more uniform flow pattern created by circular splitters.

Fig. 15
figure 15

Effect of shape of splitters on relative scour depth in tailwater depths of: a 8 cm, b 18 cm

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In Fig. 16, the variations of relative scour depth against the particle Froude number are illustrated for different relative tailwater depths. In all cases, as the particle Froude number increases, the maximum relative scour depth also rises. This trend can be attributed to the enhanced scouring potential associated with increased discharge, which elevates the relative scour depth. Furthermore, as the particle Froude number increases, indicating higher flow velocities, there is a corresponding increase in the kinetic energy of the flow. This increase in velocity enhances the flow's ability to mobilize sediment, resulting in more significant erosion and deeper scour holes. Consequently, the relationship between the particle Froude number and relative scour depth underscores the importance of flow velocity in influencing sediment transport dynamics and bed stability. Additionally, at a constant particle Froude number, the relative scour depth in the case with the splitters is less than in the case without the splitters.

Fig. 16
figure 16

Effect of particle Froude number on relative scour depth for different relative tailwater depths: a Square splitters, b Rectangular splitters, and c Circular splitters

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To ensure the safety of the weirs, it is necessary to predict the parameters of the scour hole downstream. According to the previous discussions and the dimensional analysis, the following general equation is used for the geometric characteristics of the scour hole downstream of the trapezoidal PKW.

$$\frac{\emptyset }{\Delta H} = a{\text{Fr}}_{{\text{d}}}^{b} { }\left( {\frac{{H_{d} }}{\Delta H}} \right)^{c}$$
(5)

here \(\emptyset\) represents the scour hole parameters (i.e. the weir toe scour depth Zf, the maximum depth of scour dsm, and the distance of location of maximum scour depth from the weir Ls). By using the least squares method, the constants b, and c were determined using experimental data, leading to Eqs. 6, 7 and 8. The constant a considers the influence of splitters. The values of a for estimating the scour characteristics are listed in Table 5. In this table, R2 is the correlation coefficient and RMSE is the root mean square error. A lower RMSE value and a higher R2 indicate that the selected regression equation has a more accurate description of the data.

Table 5 Values of a with corresponding R2 and RMSE
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$$\frac{{d}_{sm}}{\Delta H}=a {{\text{Fr}}_{\text{d}}}^{1.01} {\left(\frac{{H}_{d}}{\Delta H}\right)}^{-0.24}$$
(6)
$$\frac{{Z}_{f}}{\Delta H}=a {{\text{Fr}}_{\text{d}}}^{0.91} {\left(\frac{{H}_{d}}{\Delta H}\right)}^{-0.02}$$
(7)
$$\frac{{L}_{s}}{\Delta H}=a {{\text{Fr}}_{\text{d}}}^{0.43} {\left(\frac{{H}_{d}}{\Delta H}\right)}^{0.13}$$
(8)

The comparison of the calculated values using the developed equations and the observed values of the scour parameters for the PKW with and without the splitters are shown in Fig. 17. These figures, along with the statistical indices presented in Table 5, demonstrate the good accuracy of the presented equations.

Fig. 17
figure 17

Comparison of the observed and calculated a values of relative scour depth (Eq. 6), b values of relative scour depth at the weir toe (Eq. 7), and c values of distance of location of the maximum scour depth from the weir toe (Eq. 8) with and without the splitters

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The values of the scour parameters were calculated for the present data using Eqs. 6, 7 and 8, and compared with the data of Ghafouri et al. (2020) and Yazdi et al. (2020), as shown in in Fig. 18. The results revealed that the values obtained for the maximum scour depth and the distance of its location from the weir toe differed by 5% and 3% with the data of Ghafouri et al. (2020) and by 9% and 12% with data of Yazdi et al. (2020), respectively, which are within acceptable limits. However, the generality of the obtained equations is to be further validated.

Fig. 18
figure 18

Comparison of the present equation with the data of the previous studies: a maximum scour depth, and b distance of the maximum scour depth location from the weir toe

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Conclusion

This experimental study on scouring downstream of a trapezoidal PKW with and without flow splitters has yielded several significant findings. Firstly, the scour depth was observed to increase with a higher particle Froude number and a lower tailwater depth. Notably, while the geometric variations of flow splitters did not substantially affect the upstream water head, rectangular-shaped flow splitters demonstrated greater effectiveness compared to square and circular variants. Specifically, the maximum scour depth for the weir with rectangular, square, and circular splitters was reduced by approximately 13%, 11%, and 10%, respectively, compared to the weir without splitters. Additionally, the volume of scour holes in tests with rectangular, square, and circular splitters showed reductions of 18.53%, 17.77%, and 14.92%, respectively. As discharge decreases, the effectiveness of these flow splitters in minimizing scour depth becomes increasingly evident. It was observed that the distance to the maximum scour depth from the weir toe was, on average, 12% shorter for the weir equipped with flow splitters. Moreover, the maximum height of the sediment ridge with square splitters was 8% less than that of the weir without splitters, and the scour depth at the weir toe was, on average, 6% less with square splitters. When the tailwater depth was kept constant and the particle Froude number was enhanced, both the extension and volume of the scour hole increased, with the maximum scour depth occurring further from the weir toe. The flow splitters resulted in a notable reduction of about 23% in the scour hole volume, with the most significant and least effects observed in tests R30-8 and S50-8, showing differences of 37.62% and 3.04%, respectively. Finally, new equations were developed for predicting scour hole parameters with and without flow splitters, taking into account various splitter geometries. These equations were formulated using non-linear regression and achieved high accuracy with a correction factor. Overall, the findings underscore the importance of flow splitter geometry in mitigating scour effects and provide valuable insights for future engineering applications.