Abstract
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.
HIGHLIGHTS
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.
NOMENCLATURE
- 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
INTRODUCTION
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.
MODEL DEVELOPMENT
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).
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).
Numerical model
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.
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).
METHODOLOGY AND MATERIALS
Geometrical model
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 |
Grid and boundary conditions for simulation
The mesh domain characteristics
Turbulence model . | Max.-min. dimension of the mesh domain (cm) . | Total mesh cell for the axis (x/y/z) . | Total mesh cell . | E1 (Experimental result) m . | E1 (Flow3D) m . | Absolute 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 model . | Max.-min. dimension of the mesh domain (cm) . | Total mesh cell for the axis (x/y/z) . | Total mesh cell . | E1 (Experimental result) m . | E1 (Flow3D) m . | Absolute 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 |
Energy dissipation

FINDINGS AND DISCUSSION
Validation of numerical model
Flow characteristics
Energy dissipation rate
Total head profiles in three dimensions for all the models in the regime of the skimming flow.
Total head profiles in three dimensions for all the models in the regime of the skimming flow.
Variation of the energy dissipation rate versus the dimensionless critical flow depth.
Variation of the energy dissipation rate versus the dimensionless critical flow depth.
The relationship between the variation of the dimensionless energy downstream of the spillways and the dimensionless critical flow depth.
The relationship between the variation of the dimensionless energy downstream of the spillways and the dimensionless critical flow depth.
CONCLUSIONS
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.
ACKNOWLEDGEMENTS
The author thanks Firat University, Department of Civil Engineering (Turkey) for use of the Flow3D® and Solidworks software.
FUNDING
The author did not receive support from any organization for the submitted work.
DATA AVAILABILITY STATEMENT
All relevant data are included in the paper or its Supplementary Information.
CONFLICT OF INTEREST
The author declares there is no conflict.