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Stepped spillways are structures that dissipate the energy on steps, especially in topography, where building the energy dissipate pool is impossible as long as required. For this reason, stepped spillways are one of the critical topics that have attracted the attention of researchers for many years. It has been the subject of many studies, especially because the step geometry is an effective parameter in the energy dissipation rate. In this study, the effect of the labyrinth and harmonic geometry on the plan on energy dissipation performance is examined using the open-source OpenFOAM software. The 11 different geometries were analyzed in detail and discussed with the literature results. For analysis, the k − ω SST turbulence model and the multi-phase solver interFoam were utilized. The results showed that the number of cycles is an effective parameter for energy dissipation performance; the energy dissipation rate increases when effective crest length increases, and labyrinth and harmonic models could dissipate about 20% more energy than the flat model.

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  • Energy dissipation rate increases with increasing effective crest length in single-cycle labyrinth models.

  • The models with same crest length and higher number of cycles have higher energy dissipation rates in harmonic models.

  • The labyrinth and harmonic models are more effective in energy dissipation compared to literature.

  • The energy dissipation rate is higher in models with more cycles.

Computational fluid dynamics (CFD), energy dissipation rate, harmonic stepped spillway, OpenFOAM, triangular labyrinth stepped spillway
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Spillways are important hydraulic structures that safely transfer the extra water accumulated in the reservoir downstream. Stepped spillways have a history of about 3,500 years due to their simple design and construction (Chanson 2000). They are important spillways that dissipate the flow energy by mixing air into the flow (Chanson 1998). Compared to traditional chute channels, these structures are far more efficient at dissipating energy and lower the potential of cavitation on the channel (Chanson 1993; Boes et al. 2000) (Figure 1). In addition, these spillways are also beneficial for the river ecosystem by providing natural aeration of the flow (Berkün 2007). Due to these features, they are also utilized as cascade structures in water treatment plants (Chanson 1998). Stepped spillways are a widely utilized and cost-effective solution for optimizing energy dissipation performance and extending the operational life of RCC (Roller-Compacted Concrete) dams (Frizell & Mefford 1991; Berkün 2007).
Figure 1
Flow on channels: (a) classical chute channel and (b) stepped spillway channel.
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Flow on channels: (a) classical chute channel and (b) stepped spillway channel.

Figure 1
Flow on channels: (a) classical chute channel and (b) stepped spillway channel.
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Flow on channels: (a) classical chute channel and (b) stepped spillway channel.

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The earliest stepped spillway construction is the Akarnian stepped spillway, constructed in Greece in 1,300 BC. The oldest examples are the Kasserine Dam and the Ajilah Dam, built in 694 BC by the Assyrian King Sennacherib. Stepped spillways are well-exemplified by the 150-m-long crest of the Kasserine Dam (Chanson 2018). The first professionally planned stepped spillway was the New Croton Dam in the United States (Chanson 1998). It was constructed between 1885 and 1890 to provide New York City's growing need for drinking water. The spillway is the biggest erected during this period, with a maximum discharge capacity of 1,550 m3/s (Wegmann 1911; Hager et al. 2020; ).

Since the 1970s, step spillways have become increasingly common because of innovations in design and construction materials, including gabions and reinforced concrete (RCC) (Chanson 2018). These days, the most common materials utilized to build stepped spillways are gabions and RCC (Aras & Berkun 2006). Because traditional concrete is used, building stepped spillways downstream of RCC dams is especially feasible and cost-effective (Frizell & Mefford 1991). Although unit discharges up to q = 10–15 m2/s are usually handled, research shows that the flow rate may be raised to q = 30 m2/s (Boes 2012).

Due to their ease of construction, stepped spillways have become a subject of great interest among researchers. While early studies focused on design criteria (Essery & Horner 1971; Sorensen 1985; Chanson 2001; Boes & Hager 2003), recent years have seen the proposal of new step geometries (Felder et al. 2012a; Felder & Chanson 2014; Mero & Mitchell 2017) and even studies to increase dissolved oxygen in the water (Emiroglu & Baylar 2003, 2006; Baylar et al. 2010, 2015). The development of CFD software has also led to increased numerical studies (Ghaderi et al. 2020, 2021; Ikinciogullari 2021, 2023a, 2023b).

Rice & Kadavy (1996) studied the flow over a stepped spillway by creating a 1:20 scale 2D Salado Creek Site Dam model. They examined the downstream pool performance and energy dissipation at unit flow rates ranging from 5.81 to 14.50 m3/s.m. The researchers found that the models with a threshold at the downstream pool experienced less turbulence than those without one. Peyras et al. (1992) investigated the flow characteristics of gabion-stepped spillways. They concluded that gabion- stepped weirs could operate up to 3 m3/ms unit flow without damage, dissipate 10–30% more energy than classical spillways, and reduce the cost of energy-dissipating structures by 5–10%. Zare & Doering (2012) built stepped spillway models with different configurations. They found that rounding the step tips can dissipate about 3% more energy than the classical step and that 20% less sidewall height is required. However, they also discovered that rounding only the first few steps in the inlet region did not significantly increase energy dissipation. Felder et al. (2012a) examined three different stepped spillway models using flat, pooled, and combined.

Felder et al. (2012b) examined the energy dissipation and aeration rates for three models (Figure 4) compared to the classic stepped spillway. The researchers performed the experiments by varying the discharge rates between 0.02 and 0.155 m3/s in a channel with a width of 52 cm, a step width of 20 cm, and a step height of 10 cm. Thresholds were placed at the ends of the steps, with a height of 0.031 m and a width of 0.015 m for the pooled models. The results showed that the energy dissipation performance of the pooled stepped spillway is lower than the flat stepped model. The researchers pointed out that the new designs were not advantageous regarding aeration. Furthermore, the researchers observed significant differences in flow across the spillway cross section in the pooled and stepped models.

Ghaderi et al. (2020) examined the flow characteristics of three-cycle trapezoidal labyrinth-stepped spillways using Flow3D. The researchers used the flat model results of Felder et al. (2012b) to validate the numerical model. The results showed that the trapezoidal labyrinth model dissipates more energy than the flat model.

Ghaderi et al. (2021) examined the energy-dissipating performance of a notched and classical threshold with Flow3D. The researchers emphasized that the flat model is better than the notched models in energy dissipation and dissipates about 5.80% more energy than the notched models.

Ma et al. (2022) conducted a series of numerical analyses to investigate the flow characteristics of intermittently pooled stepped spillways. The researchers used the RNG k-e turbulence method in the numerical analyses, which they verified with the experimental results of Felder et al. (2012b). They compared the conventional pooled and flat model with the intermittent pooled model and emphasized that the intermittent pooled model has the best energy dissipation performance.

Ikinciogullari (2021, 2023a) investigated the energy dissipation rates of stepped spillways by designing step geometries as trapezoidal and circular. As a result of the analysis carried out using Flow3D software, it was emphasized that trapezoidal and circular shaped steps were more successful than flat model in energy dissipation.

Abdel Aal et al. (2018) aimed to increase the energy dissipation performance with the energy breakers placed downstream of the stepped spillway. The results stated that the energy breakers were more effective in energy dissipation than the classical stepped spillway and that the breakers arranged differently showed maximum performance.

Ibrahim et al. (2022) experimentally examined the effect of different-arranged bed jets placed on the downstream apron of stepped spillways on energy dissipation performance. The results emphasized that bed jets increased energy dissipation performance and that the arrangement did not affect it.

Yalçın et al. (2023) used Flow3D software to examine the turbulence model performance on the outcomes using an actual stepped spillway model. According to the researchers, the k–ω turbulence model was most consistent with discharge-water level, the k–ε turbulence model with pressure, velocity, and energy dissipation performance, and the LES turbulence model with water surface profiles.

Ikinciogullari (2023b) proposed two-cycle in-line and staggered trapezoidal steps to increase the energy dissipation performance of the stepped spillway. Numerical simulations were performed using OpenFOAM and the k-ω SST. The numerical results were validated using literature results (Felder et al. 2012b). According to the results, models using trapezoidal labyrinth steps outperform classical ones regarding energy dissipation by almost 20%.

Daneshfaraz et al. (2024) examined the impact of rough steps on stepped spillways regarding energy dissipation and hydraulic jumps. They used smooth steps for different step arrangements. They found that stepped spillways with rough steps perform better in energy dissipation, with an improvement of approximately 20.97% compared to smooth steps in the nappe flow regime. When all steps have a rough surface, it leads to a decrease in both the hydraulic jump length and roller length compared to smooth steps.

Jahad et al. (2024) examined the flow behavior and total energy dissipation performance of step geometries, namely traditional, sill, and curve stepped geometries. They supplemented their laboratory investigations by numerically modeling the flow and energy behavior in the step using a 2-D CFD model that incorporated the k-ε turbulence model. They pointed out that the energy dissipation performance was highest for the curved steps, at about 10.5%. It was also noted that the skimming flow regime shifted to a higher discharge range with curved steps. The predicted energy dissipation performance demonstrated that the energy dissipation increased with the number of steps. The results also indicated that the energy dissipation increased as the step height was increased within the range of tested heights.

When the literature studies are examined, the step geometry is a significant parameter for energy dissipation. This study aims to dissipate more energy thanks to the different geometry on the plan. For this purpose, novel stepped spillway geometries have been designed as labyrinth and harmonic. The numerical model of the study, in which 66 analyses were performed for 11 different geometries and six different discharges, was validated with the classical step spillway model in Felder et al. (2012b). The numerical model has been run using OpenFOAM software and the k − ω SST turbulence method.

Hydraulic of stepped spillway

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The flow situation in stepped spillways can be summarized as a highly turbulent flow where air enters the flow at the step and kinetic energy is reduced (Özbek 2009). As the velocity of the flow decreases, its momentum decreases. For this reason, the energy dissipation performance of stepped spillways is much higher than that of classical chutes (Rice & Kadavy 1996).

In stepped spillways, the flow has two regimes, the nappe and the skimming flow regimes, according to experimental research (Sorensen 1985; Peyras et al. 1992). The nap flow regime occurs at low flows and is defined as successively decreasing free naps. Steady water flow at high discharges causes the skimming flow regime (Chanson 1996). In the nap flow regime, which occurs at low discharges, the flow energy is dissipated at the toe of the hydraulic jump that occurs in the steps when the steps are long. Larger discharges cause the flow to flow over the step-formed pseudo-bottom as a coherent stream, and strong recirculating vortices can be seen between the step edges below the pseudo-bottom (Gonzalez et al. 2005). Therefore, the criterion for nappe flow is the step height; in the case of skimming flow, the main criterion is the head height (Berkün 2007; Felder et al. 2012a). Ohtsu and Yasuda (Ohtsu I. & Yasuda 1997) first pointed out a transitional regime between the nappe and skimming flow regimes (Figure 2). Temporary hydrodynamic changes in the transition phase may lead to unstable flow behavior and unnecessary vibrations. Therefore, researchers do not recommend the transition regime for safety reasons (Chanson 1996).
Figure 2
Illustration of flow regimes in stepped spillways: (a) nappe regime, (b) transition regime, and (c) skimming regime (Takahashi & Yasuda 2001).
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Illustration of flow regimes in stepped spillways: (a) nappe regime, (b) transition regime, and (c) skimming regime (Takahashi & Yasuda 2001).

Figure 2
Illustration of flow regimes in stepped spillways: (a) nappe regime, (b) transition regime, and (c) skimming regime (Takahashi & Yasuda 2001).
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Illustration of flow regimes in stepped spillways: (a) nappe regime, (b) transition regime, and (c) skimming regime (Takahashi & Yasuda 2001).

Close modal
Chanson (Chanson 2001) proposed an Equation (1) to determine the minimum limit of the skimming regime and another Equation (2) to define the border between the nappe flow regime and the transition flow regime. However, Boes and Hager (Boes & Hager 2003) used a different Equation (3) to establish the line that separates the transition regime from the skimming regime. Boes (Boes 2012) claimed that this equation is applicable for 26 < θ < 55, where θ indicates the angle of the chute channel, h refers to the step height, l refers to the step length, and yc refers to the critical flow depth.
(1)
(2)
(3)
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Geometric model

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This study investigated the energy dissipation rate of labyrinth and harmonic stepped spillways for different configurations. Five labyrinths and three harmonic models were constructed and tested numerically for six discharges (0.049–0.113 m3/s). First, the numerical result of the flat model was validated with the experimental results (Felder et al. 2012b). The laboratory model (Felder et al. 2012b) consisted of 10 steps. The width and length of the steps are 20 and 10 cm, respectively. The dam height and crest length are 100 cm, and the channel width is 52 cm, as illustrated in Figure 3.
Figure 3
Flat stepped model (all measurements are in cm) (Felder et al. 2012b).
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Flat stepped model (all measurements are in cm) (Felder et al. 2012b).

Figure 3
Flat stepped model (all measurements are in cm) (Felder et al. 2012b).
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Flat stepped model (all measurements are in cm) (Felder et al. 2012b).

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Figure 4
Plan view of the models used in this study.
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Plan view of the models used in this study.

Figure 4
Plan view of the models used in this study.
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Plan view of the models used in this study.

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The plan view of 11 different models examined in the study is shown in Figure 4. The models used in this study were inspired by the weir models conducted by Yıldız et al. (2024). The study mentioned investigated the flow characteristics of the labyrinths and harmonic-shaped weirs on the plan. In the current study, the steps are designed as labyrinths and harmonic on the plan. In Models 1–6, while the b and d values are equal, the b value is gradually reduced in Models 2–7 and 3–8. In Model 4, the f and e values are equal, while in Model 5, the f value is designed to be half the e value. Thus, the effect of crest length on the energy dissipation performance in labyrinth-shaped models was observed. The effect of the number of cycles on the energy dissipation performance was investigated in Models 9–10–11. These models are original, as they are not used in the literature. The details of the designed models are shown in Table 1.

Table 1

Geometric properties of models (all measurements are in cm)

ModelsabcDefr1r2r3l1l2l3l4l5l6Effective crest length (lef)
Model 1 8.65 8.65 8.70 8.70 – – – – – 12.23 12.30 – – – – 73.54 
Model 2 7.58 7.58 10.85 10.85 – – – – – 10.72 15.34 – – – – 73.57 
Model 3 6.50 6.50 13.00 13.00 – – – – – 9.19 18.38 – – – – 73.54 
Model 4 – – – – 13.00 13.00 – – – – – 18.38 – – – 73.54 
Model 5 – – – – 13.00 6.50 – – – – – 14.53 – – – 58.14 
Model 6 8.65 8.65 8.70 8.70 – – – – – 12.23 12.30 – – – – 73.54 
Model 7 7.58 7.58 10.85 10.85 – – – – – 10.72 15.34 – – – – 73.57 
Model 8 6.50 6.50 13.00 13.00 – – – – – 9.19 18.38 – – – – 73.54 
Model 9 – – – – – – 13.00 - – – – – 40.84 – – 81.68 
Model 10 – – – – – – - 8.70 – – – – – 27.33 – 82.00 
Model 11 – – – – – – – – 6.50 – – – – – 20.42 81.68 
ModelsabcDefr1r2r3l1l2l3l4l5l6Effective crest length (lef)
Model 1 8.65 8.65 8.70 8.70 – – – – – 12.23 12.30 – – – – 73.54 
Model 2 7.58 7.58 10.85 10.85 – – – – – 10.72 15.34 – – – – 73.57 
Model 3 6.50 6.50 13.00 13.00 – – – – – 9.19 18.38 – – – – 73.54 
Model 4 – – – – 13.00 13.00 – – – – – 18.38 – – – 73.54 
Model 5 – – – – 13.00 6.50 – – – – – 14.53 – – – 58.14 
Model 6 8.65 8.65 8.70 8.70 – – – – – 12.23 12.30 – – – – 73.54 
Model 7 7.58 7.58 10.85 10.85 – – – – – 10.72 15.34 – – – – 73.57 
Model 8 6.50 6.50 13.00 13.00 – – – – – 9.19 18.38 – – – – 73.54 
Model 9 – – – – – – 13.00 - – – – – 40.84 – – 81.68 
Model 10 – – – – – – - 8.70 – – – – – 27.33 – 82.00 
Model 11 – – – – – – – – 6.50 – – – – – 20.42 81.68 
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According to the evaluations conducted by the equations mentioned above, the analyses in this study were analyzed for the skimming flow condition (Table 2) using discharges between 0.049 and 0.113 m3/s, as conducted by Felder et al. (2012b). The step heights and widths of the novel models are fixed at 10 and 20 cm, respectively.

Table 2

The parameters of the flow

Q (m3 / s)q (m3/s.m)yc (m)h (m)l (m)h/l (-)θ (-)yc/h (-)Flow regime
0.049 0.094 0.097 0.20 0.10 0.50 26.6 0.967 Skimming 
0.063 0.121 0.114 0.20 0.10 0.50 26.6 1.144 Skimming 
0.075 0.144 0.128 0.20 0.10 0.50 26.6 1.285 Skimming 
0.090 0.173 0.145 0.20 0.10 0.50 26.6 1.451 Skimming 
0.097 0.187 0.153 0.20 0.10 0.50 26.6 1.525 Skimming 
0.113 0.217 0.169 0.20 0.10 0.50 26.6 1.688 Skimming 
Q (m3 / s)q (m3/s.m)yc (m)h (m)l (m)h/l (-)θ (-)yc/h (-)Flow regime
0.049 0.094 0.097 0.20 0.10 0.50 26.6 0.967 Skimming 
0.063 0.121 0.114 0.20 0.10 0.50 26.6 1.144 Skimming 
0.075 0.144 0.128 0.20 0.10 0.50 26.6 1.285 Skimming 
0.090 0.173 0.145 0.20 0.10 0.50 26.6 1.451 Skimming 
0.097 0.187 0.153 0.20 0.10 0.50 26.6 1.525 Skimming 
0.113 0.217 0.169 0.20 0.10 0.50 26.6 1.688 Skimming 
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Numerical model

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The numerical analyses were run using OpenFOAM software. This software finds wide applications in various engineering problems (www.openfoam.com). Since the flow in stepped spillways involves a two-phase system consisting of water and air, the interFoam solver was employed in the analysis. Based on previous studies (Ikinciogullari 2022), it was found that the k-ω SST turbulence method is more compatible with the experimental results for the closure problem in the numerical analysis.

The flat model used in the experiments (Felder et al. 2012b) has been numerically validated in previous studies (Ikinciogullari 2023c, 2023b). The solid body, designed using the snappyHexMesh method, was placed in a channel that was 4.30 m long, 0.52 m wide, 1.40 m high, and 1.00 m from the inlet of the channel (Figure 3). A three-dimensional uniform grid of 0.04 m dimensions was designed for the background mesh. The channel bottom and walls were designed as noSlip to form the wall boundary conditions. The top of the channel was defined as atmosphere condition, the inlet as constant discharge condition, and the outlet as noGradient. The dimensionless velocity (V/V90) and dimensionless flow depth (y/y90) results with coarse (648,470), medium (1,063,231), and fine (1,719,576) mesh domains are plotted and compared with the laboratory results (Felder et al. 2012b) in Figure 5 (Q = 0.113 m3/s). In these graphs, V refers to the interfacial velocity, V90 refers to the characteristic interfacial velocity, y refers to the flow depth, and y90 refers to the characteristic flow depth (Ikinciogullari 2023b). The V/V90 value was compared with the laboratory study (Felder et al. 2012b) for y/y90 = 0.65 at the last three step edges (eighth, ninth, and tenth). The findings indicate that the maximum error between laboratory (Felder et al. 2012b) and CFD results was 5.5% for coarse mesh, 1.6% for medium mesh, and 0.7% for fine mesh domains (Ikinciogullari 2023b). Consequently, it was concluded that medium mesh was sufficient for this study.
Figure 5
Numerical validation of the dimensionless velocity: (a) for the eighth step, (b) for the ninth step, and (c) for the 10th step (Ikinciogullari 2023b).
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Numerical validation of the dimensionless velocity: (a) for the eighth step, (b) for the ninth step, and (c) for the 10th step (Ikinciogullari 2023b).

Figure 5
Numerical validation of the dimensionless velocity: (a) for the eighth step, (b) for the ninth step, and (c) for the 10th step (Ikinciogullari 2023b).
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Numerical validation of the dimensionless velocity: (a) for the eighth step, (b) for the ninth step, and (c) for the 10th step (Ikinciogullari 2023b).

Close modal
Using the Richardson extrapolation method suggested by Celik et al. (Celik et al. 2008), a grid convergence index (GCI) was utilized to assess computational meshes to determine the best. This method is a commonly recommended method for evaluating discretization errors. According to Celik et al.'s proposal (Celik et al. 2008), 1.30 was chosen as the minimal refinement ratio. In order to increase the reliability of the results, a sensitivity study was carried out for the model with the most curves in the plan (Model 11). As a result of the superficial roughness determined as a feature of the snappyHexMesh method, the total number of new meshes was calculated as 1,056,611 for coarse, 1,339,570 for medium, and 1,971,680 for fine, depending on the mesh domain in the validation study. Figure 6(a) shows the interfacial velocity distribution with flow depth for coarse, medium, and fine mesh grids at the last step. The plotting of the error bar follows (Figure 6(b)). The averaged apparent order (pave) was determined to be 5.51 based on the data, and the greatest discretization uncertainty was determined to be %2.91, which corresponds to ±0.021 m/s. Thus, it was concluded that the medium mesh domain with 0.04 m background mesh, refinementSurface value 5 by 5, and nsmoothPatch value 5 is reliable for the results, which were important parameters in the snappyHexMesh method (Figure 7).
Figure 6
Mesh sensitivity for interfacial velocity distribution of the harmonic labyrinth model (Model 11 and Q = 0.113 m3/s).
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Mesh sensitivity for interfacial velocity distribution of the harmonic labyrinth model (Model 11 and Q = 0.113 m3/s).

Figure 6
Mesh sensitivity for interfacial velocity distribution of the harmonic labyrinth model (Model 11 and Q = 0.113 m3/s).
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Mesh sensitivity for interfacial velocity distribution of the harmonic labyrinth model (Model 11 and Q = 0.113 m3/s).

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Figure 7
Mesh view of Model 11.
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Mesh view of Model 11.

Figure 7
Mesh view of Model 11.
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Mesh view of Model 11.

Close modal
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In the current study, the flat-stepped spillway model was validated using experimental results (Felder et al. 2012b). Then, the energy dissipation performances of the novel steps with labyrinth and harmonic on the plan were examined numerically. The six discharges in the skimming flow regime were selected from the experimental study (Felder et al. 2012b). The flow energy downstream (Hdam) and the residual energy (Hres) in the last step are considered to investigate the energy dissipation performances. The following equations are used respectively:
(4)
(5)
Equation (6) was used to calculate the residual energy in the experimental study (Felder et al. 2012b). The researchers suggested a ratio in this equation by considering 25% of the results at the center point and 37.5% of the edge points measured with three points in the transverse direction (25–50–75 cm). However, since the flow depth was unstable along the transverse direction and quite variable used in this study, averaged measurements were taken from every 5 cm, as shown in Figure 8. Although intervals smaller than 5 cm were also tried, it was observed that it did not have much effect on the results. Thus, the reliability of the results was increased using averaged results. A comparison of the CFD flat model using Eq. 6 (Ikinciogullari 2023b), the experimental results (Felder et al. 2012b), and the current study is also shown in Figure 9. According to the results, the new evaluation results are very compatible with the literature results. Here, dlocal is the local flow depth, θ is the chute angle, Ulocal is the local flow velocity, and g is the gravitational acceleration.
(6)
Figure 8
Measuring points of transverse direction: (a) flat, (b) labyrinth, and (c) harmonic.
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Measuring points of transverse direction: (a) flat, (b) labyrinth, and (c) harmonic.

Figure 8
Measuring points of transverse direction: (a) flat, (b) labyrinth, and (c) harmonic.
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Measuring points of transverse direction: (a) flat, (b) labyrinth, and (c) harmonic.

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Figure 9
Comparison of the results using different methods (Δz represents the height between the crest and the calculated step edge).
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Comparison of the results using different methods (Δz represents the height between the crest and the calculated step edge).

Figure 9
Comparison of the results using different methods (Δz represents the height between the crest and the calculated step edge).
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Comparison of the results using different methods (Δz represents the height between the crest and the calculated step edge).

Close modal
The energy dissipation performances of 11 new models and one classical stepped spillway are shown in Figure 10. According to the results, the energy dissipation performances of all new models except Model 5 are higher than the flat model. Since the crest length of Model 5 is close to the crest length of the flat model, the results are close to the flat model. Since the crest length of Model 4 is about 50% more than Model 5, the energy dissipation rate is also higher than flat and Model 5 (Table 1). In labyrinth models, it was observed that the arrangement of the model according to the direction of flow did not affect the results much in the skimming flow regime. In harmonic models, Model 11 has the highest number of cycles and dissipation rate.
Figure 10
Comparison of the energy dissipation rates.
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Comparison of the energy dissipation rates.

Figure 10
Comparison of the energy dissipation rates.
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Comparison of the energy dissipation rates.

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In Figure 11, the energy dissipation percentages of all models analyzed in the study are plotted according to the dimensionless discharge. According to the results obtained, the energy dissipation percentages of the labyrinth and harmonic models are higher than the flat model. The energy dissipation percentages decrease as the discharge increases for all models. The highest dissipation percentages are observed in Model 11, a multi-cycle harmonic model, while the lowest dissipation percentage is observed in Model 5. In the labyrinth models (Models 1–2–3–6–7–8), where the effective crest lengths are very close to each other, it is concluded that the flow direction is effective on the dissipation percentage. In Model 6–7–8, the energy dissipation percentages are higher than in Model 1–2–3 since the flow jet collides in the middle section. Especially in Model 1–2–3, when a and b values increase, c and d values decrease, and energy dissipation percentages decrease. In Model 6–7–8, which was obtained by rotating the direction of the mentioned models by 180 degrees, no noticeable difference was obtained in the results with the change of the mentioned values.
Figure 11
Energy dissipation percentages vs. dimensionless discharge.
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Energy dissipation percentages vs. dimensionless discharge.

Figure 11
Energy dissipation percentages vs. dimensionless discharge.
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Energy dissipation percentages vs. dimensionless discharge.

Close modal
In the current study, Model 8 showed the highest performance among the labyrinth models. Model 4 has the same effective crest length as Model 8, but the number of cycles is less than that of Model 8. Model 4 and Model- 8 results were compared with the trapezoidal labyrinth models' results in the literature (Ghaderi et al. 2020; Ikinciogullari 2023b) (Figure 12). According to the results, there is not much difference between the results at high discharges. It is observed that the energy dissipation performance of the trapezoidal labyrinth models is higher than that of the triangular labyrinth models as the discharge decreases. It is also observed that the energy dissipation performance increases as the number of cycles increases in triangular and trapezoidal labyrinth models.
Figure 12
Comparison of the energy dissipation percentages of the labyrinth models.
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Comparison of the energy dissipation percentages of the labyrinth models.

Figure 12
Comparison of the energy dissipation percentages of the labyrinth models.
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Comparison of the energy dissipation percentages of the labyrinth models.

Close modal
The residual energy values at the end of the last step of all the models analyzed in this study were compared with the results of many models in the literature (Felder et al. 2012b; Ghaderi et al. 2020; Ikinciogullari 2023b) (Figure 13). In the results, the residual energy of pooled, in-lined, and staggered models proposed by Felder et al. (2012b) are quite high. This situation is also emphasized in the mentioned paper (Felder et al. 2012b). Apart from the models proposed by Ghaderi et al. (2020) and Ikinciogullari (2023b), it is observed that the model with the lowest residual energy is the multi-cycle Model 11. Especially the two-cycle triangular labyrinth models and the multi-cycle harmonic model are better than the flat model in dissipating energy at high discharges.
Figure 13
Residual energy comparison of used models and literature.
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Residual energy comparison of used models and literature.

Figure 13
Residual energy comparison of used models and literature.
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Residual energy comparison of used models and literature.

Close modal
Figure 14 shows the variation of the water surface profiles along the spillway for the models. According to the results, the highest velocity distribution along the spillway was observed in the flat model. It is seen that the flow reaches the maximum velocity value in a section of Model 5, whose crest length is closest to the flat model. In the labyrinth models, the fluctuation on the water surface profile increased as the a and b values increased and the c and d values decreased (Model 1–2–3). In Model 6–7–8, which was obtained by rotating the direction of the mentioned models by 180 degrees, not much change was observed on the water surface profile. However, it was observed that the flow velocity was high in the middle section of these models. In the harmonic models, while an increase in velocity is observed in the regions with internal bends, it is observed that the fluctuation on the water surface profile increases, and the flow velocity decreases with the increase in the number of cycles. Therefore, the images from the water surface profiles are consistent with the energy dissipation rates emphasized above.
Figure 14
Change of the water surface profiles along spillway for Q = 0.113 m3/s: (a) flat, (b) Model 1, (c) Model 2, (d) Model 3, (e) Model 4, (f) Model 5, (g) Model 6, (h) Model 7, (i) Model 8, (j) Model 9, (k) Model 10, and (l) Model 11.
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Change of the water surface profiles along spillway for Q = 0.113 m3/s: (a) flat, (b) Model 1, (c) Model 2, (d) Model 3, (e) Model 4, (f) Model 5, (g) Model 6, (h) Model 7, (i) Model 8, (j) Model 9, (k) Model 10, and (l) Model 11.

Figure 14
Change of the water surface profiles along spillway for Q = 0.113 m3/s: (a) flat, (b) Model 1, (c) Model 2, (d) Model 3, (e) Model 4, (f) Model 5, (g) Model 6, (h) Model 7, (i) Model 8, (j) Model 9, (k) Model 10, and (l) Model 11.
View largeDownload slide

Change of the water surface profiles along spillway for Q = 0.113 m3/s: (a) flat, (b) Model 1, (c) Model 2, (d) Model 3, (e) Model 4, (f) Model 5, (g) Model 6, (h) Model 7, (i) Model 8, (j) Model 9, (k) Model 10, and (l) Model 11.

Close modal
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In this study, labyrinth and harmonic stepped models were proposed to increase the energy dissipation rate of the classical stepped spillways. A total of 66 analyses were run using 11 different models and six different flow rates, and OpenFOAM software was used in the study. The results obtained are summarized below.

  • 1. The energy dissipation rate decreases with increasing flow rate in all models.

  • 2. Two-cycle triangular labyrinth models are more effective in energy dissipation than single-cycle models, up to 8%.

  • 3. The energy dissipation rate increases with increasing effective crest length in single-cycle labyrinth models, up to 5%.

  • 4. In two-cycle labyrinth models, the placement of the model according to the flow direction does not have much effect on the energy dissipation rate.,

  • 5. In harmonic models, the models with the same crest length and higher number of cycles have higher energy dissipation rates, up to 4%.

  • 6. The labyrinth and harmonic models used in the study are more effective in energy dissipation compared to many models in the literature, but the energy dissipation rate of the labyrinth trapezoidal models is better at low discharges.

  • 7. The fluctuations in the water surface profile are higher in more cycle models, and the residual energy is lower in more cycle models. Therefore, the energy dissipation rate is higher in models with more cycles.

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All numerical analyses in this study were conducted at the TUBITAK ULAKBIM High Performance and Grid Computing Center (TRUBA resources).

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The author did not receive support from any organization for the submitted work.

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The author declares that there are no conflicts of interest.

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

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

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

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The authors declare there is no conflict.

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