Important water structures known as stepped spillways are used to dissipate energy, aerate rivers, and reduce the cost of the downstream pool. The aim of this study is to increase the energy dissipation rate by proposing an alternative design to the flat stepped spillways, which are frequently used in practice. For that reason, the Circular Stepped Spillway (CSS) has been constructed and numerically investigated in this paper. The energy dissipation rate of the CSS and the flat stepped spillway has been compared using three different models and discharges. According to the simulation findings, the CSS performs better as the step radius gets smaller. The CSS models with step depths of 0.50, 1.50, and 2.50 cm have dissipated up to 21.4, 38, and 50.7% more energy than the flat stepped spillway, respectively.

  • Circular stepped spillway (CSS) is more efficient than the flat stepped spillway for energy dissipation.

  • As the radius of the step on the CSS decreases, the energy dissipation rate increases.

  • As the step depth of the CSS increases, the energy dissipation rate increases.

  • The flow depth of the CSS is higher than the flat stepped spillway due to the amount of air entering the flow.

circular stepped spillway, computational fluid dynamics (CFD), energy dissipation, Flow3D®
α

chute angle

ρ

specific mass of the fluid

A

average flow area

B

channel width

E0

energy over the dam

E1

energy downstream of the dam

EL

energy difference

F

fluid ratio function

g

gravity due to acceleration

Hdam

dam height

h

step height

l

step length

lr

depth of the step for CSS

Q

discharge of the dam

q

unit flow rate of the dam

R

radius of circle

V1

average velocity downstream of the dam

Vc

critical velocity

y1

flow depth downstream of the weir

yc

critical depth

x, y, and z

Cartesian coordinates

X, Y, and Z

mass acceleration on the respective axis

As the human population increases, our need for energy increases daily. To balance this increasing energy need, the design of high dams has also become necessary. The destruction that occurs downstream of the dam attracted the attention of researchers, and ideas for dissipating the energy of the flow on the spillway emerged. However, stepped spillways cause less destruction at the dam downstream due to the steps. The development of this natural ventilation method benefits the economy by reducing the costs of energy-breaking structures in the downstream pool.

The stepped spillways are mainly used in Roller-Compacted Concrete (RCC) dams. Generally, the unit flow rate used in stepped spillways is q = 10 to 15 m3/s m. It has also been argued that the unit flow can be increased to 30 m3/s m (Boes 2012). However, as RCC dams are built with conventional concrete, it is very economical and practical to make cascading spillways downstream of these dams (Frizell & Mefford 1991).

In the literature review, the stepped spillways were generally researched for design criteria and the development of fundamental equations (Chanson 1993, 2001; Essery & Horner 1971; Sorensen 1985; Boes & Hager 2003). For example, some researchers examined the flow characteristics of the stepped spillways using box gabions (Peyras et al. 1992; Wuthrich & Chanson 2015; Zuhaira et al. 2020). Some researchers have also examined the amount of scouring downstream stepped spillways (Tuna & Emiroglu 2013; Aminpour & Farhoudi 2017; Eghlidi et al. 2020).

Felder et al. (2012a, 2012b) conducted many experimental studies about pooled stepped spillways that used different step heights and chute channel angles. According to their findings, pooling stepped spillways dissipate energy more effectively than flat stepped spillways.

Zare & Doering (2012a, 2012b) used various threshold designs to experimentally investigate the air inception points of the stepped spillway. The researchers emphasized that taking rounded steps results in a 3% increase in energy loss compared to standard steps.

Experimental research was done by Mero & Mitchell (2017) to determine the impact of step shapes on energy dissipation rates using five different stepped spillway models. The researchers modified the step geometry and added reflectors to them. They concluded that stepped spillways with reflectors have substantially higher energy dissipation rates than conventional ones. For all models, the energy dissipation rate declines as the discharge rises.

The Computational Fluid Dynamics (CFD) method has attracted many researchers because of saves time and money and applies to different hydraulic models. For example, the numerical studies on stepped spillways have increased daily. The researchers used various step and threshold geometries or chute angles to analyze the stepped spillway flow dynamics and energy dissipation (Tabbara et al. 2005; Shahheydari et al. 2015; Mohammad et al. 2016; Hekmatzadeh et al. 2018; Li et al. 2018, 2020; Ashoor & Riazi 2019; Reeve et al. 2019; Arjenaki & Sanayei 2020; Ghaderi et al. 2020, 2021; Ikinciogullari 2021).

As mentioned above, many researchers have tried to increase the energy dissipation rate by working on the step geometry to dissipate more energy on the spillway. Therefore, circular step geometry, which can be an alternative to the proposed designs and dissipate more energy than the classical step geometry, has been numerically investigated in this study. The Circular Stepped Spillway (CSS) has been built for three-circle radii using Flow3D® software to examine the radius effect on steps.

The nappe and skimming flow regimes on the stepped spillways occur at low and high flow rates, respectively. The transition flow zone between the nappe and skimming regimes was initially described by Ohtsu & Yasuda (1997). The flow characteristics have lost the nappe flow characteristic in the transition flow regime but have not yet entirely adapted to the skimming flow characteristic (Figure 1). However, due to the increased vibrations in the transition phase, it is an unpleasant scenario for designers (Chanson 1996).
Figure 1
The following flow regimes are present on a stepped spillway: (a) nappe flow; (b) transition flow; and (c) skimming flow (Takahashi et al. 2001; Ikinciogullari 2021).
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The following flow regimes are present on a stepped spillway: (a) nappe flow; (b) transition flow; and (c) skimming flow (Takahashi et al. 2001; Ikinciogullari 2021).

Figure 1
The following flow regimes are present on a stepped spillway: (a) nappe flow; (b) transition flow; and (c) skimming flow (Takahashi et al. 2001; Ikinciogullari 2021).
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The following flow regimes are present on a stepped spillway: (a) nappe flow; (b) transition flow; and (c) skimming flow (Takahashi et al. 2001; Ikinciogullari 2021).

Close modal
The maximum and lower limits of the nappe and skimming flow regimes were provided by certain studies (Chanson 2001; Boes & Hager 2003) using Equations (1) and (2).
formula
(1)
formula
(2)
where h is the step height (m), l is the step length, and yc is the critical flow depth (m).
The step geometry (height and length of the steps) and discharge affect the flow regime through stepped spillways (Chanson 2001; Boes & Hager 2003). By employing a dimensionless variable of steps (h/l) and a normalized critical depth (yc/h), the flow regimes were constrained (Boes & Hager 2003) (Figure 2). Discharges ranged from 0.0121 to 0.00684 m3/s in this investigation. Two analyses were done in the skimming regime, and one was done in the transition flow regime, as shown in Figure 2.
Figure 2
Flow regime conditions of this study.
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Flow regime conditions of this study.

Figure 2
Flow regime conditions of this study.
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Flow regime conditions of this study.

Close modal

Numerical model

A general-purpose CFD tool named Flow3D® software is available for modeling fluid flow phenomena (Flow Science Incorporated 2022). Utilizing the Navier–Stokes and general mass equations, this program generates solutions in two or three dimensions. The continuity equation and the Navier–Stokes equations are presented in Equations (3)–(6).
formula
(3)
formula
(4)
formula
(5)
formula
(6)

Navier–Stokes equations are explained according to x, y, and z cartesian coordinates. ρ is the specific mass of the fluid, p is the pressure, and υ is the kinematic viscosity. X, Y, and Z specified in these equations show the mass acceleration on the respective axis.

Flow3D® uses the Volume of Fluid (VOF) and FAVOR (Fractional Area–Volume Obstacle Representation) methods for modeling, as shown in Equation (7). The VOF method indicates the fluid behavior on the free surface.
formula
(7)

Here, A is the average flow area along the x, y and z directions, and F is the fluid ratio function between 0 and 1. When F is equal to zero, a cell is empty; when F is equal to 1, the cell is full (Flow Science Incorporated 2022).

Geometrical model

In this study, the flat stepped spillway simulation solutions were validated using experimental data from Mero and Mitchell (Mero & Mitchell 2017). They utilized a five-step stepped spillway with a 0.250 m height and 0.296 m width. This spillway has steps that are 10 × 5 cm in size (Figure 3). The numerical studies were performed using three distinct discharges (0.0121, 0.00831, 0.00684 m3/s) and three distinct circular step geometries (lr = 0.50, 1.50, and 2.50 cm, and R = 25, 9, and 6.25 cm). According to the literature, the flow regimes were established (Boes & Hager 2003). The mentioned models were produced using Solidworks software and uploaded with .stl extensions to Flow3D® software (Figure 4). The circular step's depth and radius are indicated by the characters lr and R, respectively. Table 1 summarizes the hydraulic features of the investigation.
Table 1

Hydraulic characteristics of the study

Analyze No.Model No.Q (m3·s−1)Step height (h) (m)Step length (l) (m)lr (m)R (m)Flow regime
1 Model-1 0.01210 0.05 0.10 0.005 0.250 Skimming 
2 Model-1 0.00831 0.05 0.10 0.005 0.250 Skimming 
3 Model-1 0.00684 0.05 0.10 0.005 0.250 Transition 
4 Model-2 0.01210 0.05 0.10 0.015 0.090 Skimming 
5 Model-2 0.00831 0.05 0.10 0.015 0.090 Skimming 
6 Model-2 0.00684 0.05 0.10 0.015 0.090 Transition 
7 Model-3 0.01210 0.05 0.10 0.025 0.0625 Skimming 
8 Model-3 0.00831 0.05 0.10 0.025 0.0625 Skimming 
9 Model-3 0.00684 0.05 0.10 0.025 0.0625 Transition 
Analyze No.Model No.Q (m3·s−1)Step height (h) (m)Step length (l) (m)lr (m)R (m)Flow regime
1 Model-1 0.01210 0.05 0.10 0.005 0.250 Skimming 
2 Model-1 0.00831 0.05 0.10 0.005 0.250 Skimming 
3 Model-1 0.00684 0.05 0.10 0.005 0.250 Transition 
4 Model-2 0.01210 0.05 0.10 0.015 0.090 Skimming 
5 Model-2 0.00831 0.05 0.10 0.015 0.090 Skimming 
6 Model-2 0.00684 0.05 0.10 0.015 0.090 Transition 
7 Model-3 0.01210 0.05 0.10 0.025 0.0625 Skimming 
8 Model-3 0.00831 0.05 0.10 0.025 0.0625 Skimming 
9 Model-3 0.00684 0.05 0.10 0.025 0.0625 Transition 
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Figure 3
Schematic design of the model used by Mero & Mitchell (2017).
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Schematic design of the model used by Mero & Mitchell (2017).

Figure 3
Schematic design of the model used by Mero & Mitchell (2017).
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Schematic design of the model used by Mero & Mitchell (2017).

Close modal
Figure 4
The design of CSS: (a) two-dimensional and (b) three-dimensional.
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The design of CSS: (a) two-dimensional and (b) three-dimensional.

Figure 4
The design of CSS: (a) two-dimensional and (b) three-dimensional.
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The design of CSS: (a) two-dimensional and (b) three-dimensional.

Close modal

Grid and boundary conditions for simulation

For this investigation, four different mesh sizes were used. First, 341,055 cells were chosen and gradually increased to 2,622,000 cells. The energy downstream of the spillway (E1), which is reported in Table 2 for Q = 0.00684 m3/s, was compared. Standard k–ε and RNG k–ε (Renormalization Group) turbulence models were contrasted for the correct solution. One of the most reliable turbulence models in CFD is the k–ε model. The RNG k–ε model of turbulence is an enhanced version of the k–ε model. Renormalization group theory, a rigorous statistical method, was used to develop the RNG k–ε model, which considers various turbulent motion scales by varying the production term. Although this model has additional components for the dissipation rate, the k and ε transport equations are still quite similar to the traditional k–ε model (Rodi 1993; Qian et al. 2009; Hekmatzadeh et al. 2018). Up until steady-state flow conditions, analyses were conducted. Table 2 indicates that a 2,622,000 cell computational mesh and the RNG k–ε turbulence model with an absolute error of 5.88% have been chosen for further analysis. The domain's maximum and minimum dimensions were planned to be 0.58 and 0.25 cm, respectively. In addition, six more mesh planes were added for the best render (Figure 5).
Table 2

The mesh domain characteristics

Turbulence modelMax.-min. dimension of the mesh domain (cm)Total mesh cell for the axis (x/y/z)Total mesh cellE1 (Experimental result) mE1 (Flow3D) mAbsolute Error
Standard k–ε 0.78–0.75 165/39/53 341,055 0.119 0.219 84.42 
Standard k–ε 1.70–0.25 290/44/59 752,840 0.119 0.186 56.63 
Standard k–ε 0.98–0.25 330/49/80 1,293,600 0.119 0.140 17.89 
Standard k–ε 0.58–0.25 380/60/115 2,622,000 0.119 0.128 7.79 
RNG k–ε 0.58–0.25 380/60/115 2,622,000 0.119 0.126 5.88 
Turbulence modelMax.-min. dimension of the mesh domain (cm)Total mesh cell for the axis (x/y/z)Total mesh cellE1 (Experimental result) mE1 (Flow3D) mAbsolute Error
Standard k–ε 0.78–0.75 165/39/53 341,055 0.119 0.219 84.42 
Standard k–ε 1.70–0.25 290/44/59 752,840 0.119 0.186 56.63 
Standard k–ε 0.98–0.25 330/49/80 1,293,600 0.119 0.140 17.89 
Standard k–ε 0.58–0.25 380/60/115 2,622,000 0.119 0.128 7.79 
RNG k–ε 0.58–0.25 380/60/115 2,622,000 0.119 0.126 5.88 
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Figure 5
The mesh domain: (a) two-dimensional and (b) three-dimensional.
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The mesh domain: (a) two-dimensional and (b) three-dimensional.

Figure 5
The mesh domain: (a) two-dimensional and (b) three-dimensional.
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The mesh domain: (a) two-dimensional and (b) three-dimensional.

Close modal
According to the requirements of the experimental investigation, the boundary conditions have been chosen. Firstly, the flow region equal to the weir height has been designed upstream of the weir. Then, the specific discharge (Q) was used at the beginning of the spillway channel (xmin). The ‘outflow’ boundary condition (O) was used at the downstream flow (xmax), and the ‘wall’ boundary condition (W) was used at the channel bed and walls (zmin, ymin, and ymax). Then, the ‘symmetry’ condition (S) (does not require any other input due to the relative atmosphere pressure) was used to determine the upper boundary condition (zmax) (Figure 6).
Figure 6
Boundary conditions.
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Boundary conditions.

Figure 6
Boundary conditions.
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Boundary conditions.

Close modal

Energy dissipation

Equation (8) was employed in the experimental investigation (Mero & Mitchell 2017) to determine the critical depth of the rectangular channel.
formula
(8)
where Q refers to the discharge (m3/s), yc refers to the critical flow depth (m), q refers to the unit flow rate (m3/s·m), g refers to the gravity due to acceleration (m/s2), and B refers to the channel width (m). As shown in Figure 7, the energies at (0) and (1) points were calculated by Equations (9) and (10). Then, using Equation (11), the energy difference (EL) between the two points was calculated.
formula
(9)
formula
(10)
formula
(11)
where E0 refers to the energy over the dam (m), y1 refers to the flow depth for downstream of the dam (m), Hdam refers to the dam height (m), V1 refers to the average velocity for downstream of the dam (m/s), Vc refers to the critical flow velocity (m/s) ⁠, and E1 refers to the energy for downstream of the dam (m).
Figure 7
Overview of the longitudinal section of the spillways.
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Overview of the longitudinal section of the spillways.

Figure 7
Overview of the longitudinal section of the spillways.
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Overview of the longitudinal section of the spillways.

Close modal

Validation of numerical model

Experimental data from Mero and Mitchell (Mero & Mitchell 2017) were used to validate the numerical estimates for the total energy head downstream of the spillway for the flat stepped spillway (E1). As shown in Figure 8, the energy downstream of the spillway (E1) was illustrated as a discharge function (Q). The results of the numerical analysis with used standard k–ε turbulence model were compared with four mesh cells. According to Figure 8, the optimum mesh domain for the standard k–ε turbulence model was the excellent mesh (2,622,000 elements). The numerical research was then done using the 2,622,000 element RNG k–ε turbulence model (Figure 9) to compare turbulence models. Standard k–ε and RNG k–ε turbulence models had maximum absolute error ratios of 7.79 and 5.88%, respectively. The results show good agreement between the experimental and numerical results.
Figure 8
Convergence of the numerical results for different mesh domains.
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Convergence of the numerical results for different mesh domains.

Figure 8
Convergence of the numerical results for different mesh domains.
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Convergence of the numerical results for different mesh domains.

Close modal
Figure 9
Comparison of the turbulence models.
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Comparison of the turbulence models.

Figure 9
Comparison of the turbulence models.
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Comparison of the turbulence models.

Close modal

Flow characteristics

In Figure 10, three-dimensional velocity profiles of the models have been illustrated for Q = 0.0121 m3/s. According to the results, the flow velocity on the CSS was less than the classical one. As the lr rises, the flow velocity downstream of the spillway reduces. As a result, the water surface profile becomes flattened in the flat spillway. However, in the CSS, the water surface becomes wavy due to the local jump on each step. The fluctuation on the water surface increase as the R decreases. The stepped shape of the CSS is considered to cause more air to enter the flow. A similar issue arose in the literature investigations for various step geometries (Felder et al. 2012b; Arjenaki & Sanayei 2020; Ghaderi et al. 2020). The two-dimensional velocity vectors of all models were given as shown in Figure 11 (at the center of the channel). Generally, the stepped spillways constrain the flow in the recirculation zones below the pseudo-bottom. However, the CSS's steps experience greater vortex flow than the flat-stepped spillway's steps.
Figure 10
Three-dimensional velocity profiles of the models.
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Three-dimensional velocity profiles of the models.

Figure 10
Three-dimensional velocity profiles of the models.
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Three-dimensional velocity profiles of the models.

Close modal
Figure 11
Two-dimensional velocity vectors of the models.
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Two-dimensional velocity vectors of the models.

Figure 11
Two-dimensional velocity vectors of the models.
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Two-dimensional velocity vectors of the models.

Close modal
The streamlined versions of the examined models are shown in Figure 12. The models’ streamlines are often parallel to the pseudo-bottom. As a result, Model 3 has more streamlined aggregation on the steps.
Figure 12
Streamlines of the models.
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Streamlines of the models.

Figure 12
Streamlines of the models.
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Streamlines of the models.

Close modal
Figure 13 illustrates the pressure distribution on steps for all models. The low-pressure value was observed at the beginning of the steps. However, the highest-pressure value was seen at the end of the steps for the CSSs. At the steps of the models, negative pressure values never existed. Therefore, the phenomenon of cavitation was not observed for CSSs.
Figure 13
Pressure distribution of the models.
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Pressure distribution of the models.

Figure 13
Pressure distribution of the models.
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Pressure distribution of the models.

Close modal

Energy dissipation rate

The total hydraulic head profiles of all models have been compared in Figure 14 for Q = 0.0121 m3/s. It has been discovered that the CSS's energy downstream of the dam is lower than the flat-stepped one. The flat stepped spillway has been shown to have the highest total energy, whereas Model 3 has been found to have the lowest total energy. Besides, in the CSS, the smaller R, the more energy dissipation observed, and the more fluctuations occurred in the water surface profile. The energy dissipation rate, on the other hand, increases as the lr rises.
Figure 14
Total head profiles in three dimensions for all the models in the regime of the skimming flow.
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Total head profiles in three dimensions for all the models in the regime of the skimming flow.

Figure 14
Total head profiles in three dimensions for all the models in the regime of the skimming flow.
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Total head profiles in three dimensions for all the models in the regime of the skimming flow.

Close modal
Figure 15 shows all models’ two-dimensional simulation outputs of the turbulence dissipation. Because of changing the step shape to circular and increasing the step depth, the turbulence dissipation value rises.
Figure 15
Comparison of the turbulent dissipation of all models.
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Comparison of the turbulent dissipation of all models.

Figure 15
Comparison of the turbulent dissipation of all models.
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Comparison of the turbulent dissipation of all models.

Close modal
The variation of the energy dissipation rates versus yc/h (dimensionless critical flow depth) is shown in Figure 16. The energy dissipation rates of the circular and flat stepped spillways have been contrasted in this figure under various flow regimes. The limit of the flow regime has been defined in the literature (Boes & Hager 2003). In the skimming flow regime, the CSS dissipates energy at a noticeably higher rate than the flat stepped spillway. Upon examination, it is observed that this rate reached 50.7% for Model 3. However, this ratio gradually decreased with transit from the skimming regime to the transition regime. The flow jumps in each step and progresses without touching the next step in the transition regime (like a flip bucket), so its energy is not dissipated to the desired extent. Hence, the CSSs are more efficient in the skimming flow regime. Model 3 showed a more effective dissipation performance than other models due to the highest step depth (lr).
Figure 16
Variation of the energy dissipation rate versus the dimensionless critical flow depth.
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Variation of the energy dissipation rate versus the dimensionless critical flow depth.

Figure 16
Variation of the energy dissipation rate versus the dimensionless critical flow depth.
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Variation of the energy dissipation rate versus the dimensionless critical flow depth.

Close modal
Figure 17 displays the dimensionless energy (E1/yc) downstream of the spillway versus yc/h. The findings show that under a similar flow situation (yc/h = 1.11), E1/yc is equivalent to 3.24 for the flat stepped spillway, while E1/yc is 2.58 for Model 3. As the lr increases, E1/yc decreases. Furthermore, E1/yc has been compared with experimental results of different step shape spillways conducted by Mero and Mitchell (Mero & Mitchell 2017). However, except for the model with reflectors, the results show that CSSs are more effective than other types.
Figure 17
The relationship between the variation of the dimensionless energy downstream of the spillways and the dimensionless critical flow depth.
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The relationship between the variation of the dimensionless energy downstream of the spillways and the dimensionless critical flow depth.

Figure 17
The relationship between the variation of the dimensionless energy downstream of the spillways and the dimensionless critical flow depth.
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The relationship between the variation of the dimensionless energy downstream of the spillways and the dimensionless critical flow depth.

Close modal

Stepped spillways are utilized for more energy dissipation of the flow. Designating the stepped spillways benefits the economy by reducing the costs of energy-breaking structures. Using Flow3D® software, the CSS is the subject of this study. Three different radii and flow rates were examined for this study. The simulation results were verified with experimental data for flat stepped spillway by Mero and Mitchell (Mero & Mitchell 2017). The results of this study are as follows:

The numerical findings from the RNG k–ε turbulence model agree with the experimental data (Mero & Mitchell 2017), with the highest error being 5.88%. Generally, as the discharge decreases, energy dissipation rates increase in all models. Under the same flow conditions, the CSS might dissipate more energy than the flat stepped spillways. In the case of CSS with step depths of 0.5, 1.5, and 2.5 cm, respectively, the energy dissipates 21.4, 38, and 50.7% more than in the case of the flat stepped spillway, respectively. The energy dissipation rate increases as the circle's radius on the step geometry decrease. The dimensionless energy downstream of the flat stepped spillway is equal to 3.24, which is equal to 2.58 on Model 3 for the same flow condition. Thus, increasing the step depth decreases the energy downstream of the spillway. Positive effects of using the CSS are the concrete volume to be used less than flat stepped spillways and the reduction in the size of the energy breaker pool that needs to be built in the downstream pool. Due to the high fluctuation in the flow over the CSS, the flow depth is higher compared to the flat stepped spillway, considering that the amount of air entering the flow also increases. So, the weir sidewall height and labor costs also disadvantage the CSS. These structures may be employed in chute channels created in tourist locations for a chic design or in downstream places with limited terrain constraints.

The author thanks Firat University, Department of Civil Engineering (Turkey) for use of the Flow3D® and Solidworks software.

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

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

The author declares there is no conflict.

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