ABSTRACT
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.
HIGHLIGHTS
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.
INTRODUCTION
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
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).
MATERIALS AND METHODS
Geometric model
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.
Models . | a . | b . | c . | D . | e . | f . | r1 . | r2 . | r3 . | l1 . | l2 . | l3 . | l4 . | l5 . | l6 . | Effective 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 |
Models . | a . | b . | c . | D . | e . | f . | r1 . | r2 . | r3 . | l1 . | l2 . | l3 . | l4 . | l5 . | l6 . | Effective 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 |
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.
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 |
Numerical model
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.
RESULTS AND DISCUSSION
CONCLUSION
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.
ACKNOWLEDGEMENTS
All numerical analyses in this study were conducted at the TUBITAK ULAKBIM High Performance and Grid Computing Center (TRUBA resources).
FUNDING
The author did not receive support from any organization for the submitted work.
CONFLICTS OF INTEREST
The author declares that there are no conflicts of interest.
ETHICAL APPROVAL
This article does not contain any studies with human participants or animals performed by any of the authors.
INFORMED CONSENT
Informed consent was obtained from all individual participants included in the study.
DATA AVAILABILITY STATEMENT
All relevant data are included in the paper or its Supplementary Information.
CONFLICT OF INTEREST
The authors declare there is no conflict.