https://orcid.org/0000-0001-7891-3502
Abstract
To improve the performance of stepped spillways, their combination with labyrinth spillways is an interesting topic. In this study, several labyrinth configurations of stepped spillways were presented. Validation of the numerical model was done using the results of the previous physical models. After that, three configurations including: conventional stepped, trapezoidal-labyrinth, and rectangular-labyrinth were modeled using the OpenFOAM model for the skimming flow regime. For simulation, the InterFOAM solver and RNG k–ε turbulence model were used. The results showed an increase of 34.7 and 21.1% in energy dissipation in the trapezoidal and rectangular stepped–labyrinth spillways compared to the conventional stepped type, for dc/h = 1.45 (the range of dc is between 8 and 14.5 cm). The flow velocity in the end step of the trapezoidal- and rectangular-labyrinth configuration is reduced by 50.5 and 31.1%, respectively. Furthermore, in the trapezoidal configuration, a 14.7% reduction in flow velocity has been achieved compared to the rectangular stepped–labyrinth configuration. The results showed that the minimum pressure on the vertical faces of the steps occurred in their upper half and the rectangular configuration has resulted in the highest amount of negative pressure. The turbulence kinetic energy, especially in trapezoidal configuration, has increased toward the downstream.
HIGHLIGHTS
The combination of stepped and labyrinth spillways is effective in improving the performance of spillways in terms of energy dissipation.
The trapezoidal stepped–labyrinth combined spillway has the best performance among the combined spillways in terms of energy dissipation.
The minimum pressure on the horizontal face of the steps occurred in the central axis and the maximum occurred near the outer edge of the steps.
INTRODUCTION
In the design of spillways and chutes, if the flow energy is well dissipated along the structure, the dimensions of the stilling basin will be reduced. One of the cases that can fulfill this condition is the use of stepped spillways (Chanson 2002). The stepped spillway has increasingly become an effective energy dissipator. When the hydraulic performance of the overflow is clearly known, the energy dissipation could be increased (Chen et al. 2002). The flow resistance of stepped spillways is significantly larger than for smooth chutes due to the macro roughness of the steps (Boes & Hager 2003). Due to their shape, stepped spillways are used to dissipate water energy, reduce its erosive power, and also reduce the cost of energy dissipation structures downstream of the spillway of large dams (Khatsuria 2005).
Tabbara et al. (2005) presented numerical simulations of water flow over stepped spillways with different step configurations. The finite element computational fluid dynamics module of the ADINA software was used to predict the main characteristics of the flow, including the determination of the water surface, the development of skimming flow over corner vortices, and the determination of energy dissipation. Two-phase flow properties were measured by Felder & Chanson (2009) in high-velocity open channel flows above a stepped chute. Their results implied that small-size models did underestimate the rate of energy dissipation and the aeration efficiency of prototype stepped spillways for similar flow conditions.
Felder et al. (2012a) conducted a physical study of three different forms of stepped spillways, including a flat stepped chute, a pooled step chute, and a chute with an alternation of flat and pooled steps. The density distribution of air bubbles on the pool and the simple stepped spillways were almost the same, while the stepped spillway with the combination of pool and simple step provided better aeration for the same discharge. The greatest energy loss belongs to a stepped spillway with an alternating pool and simple step, but this geometry is not recommended as an optimal design due to the severe flow instabilities. In addition, Felder et al. (2012b) experimentally investigated four stepped chute configurations including a stepped spillway with flat horizontal steps, a pooled stepped spillway, and two stepped spillways configurations with in-line and staggered arrangements of flat and pooled steps. The results showed that the rate of energy dissipation was smaller on the pooled stepped spillway compared to that on the flat stepped chute. Conversely, the residual energy was larger at the downstream end of the pooled stepped chute.
In a numerical study Abbasi & Kamanbedast (2012) presented the relationships between relative energy loss and relative critical flow depth on stepped spillways. It was found that increase in stairs length and heights cause an increase in energy dissipation. Hamedi et al. (2012) conducted an experimental procedure and laboratory work to scrutinize whether the rate of energy loss on stepped spillways using inclined steps together with end sill increases. Results showed that using inclined steps together with end sill have considerable effects on both nappe and skimming flows, however, energy dissipation in nappe flows is greater than that in skimming flows. Morovati et al. (2016) studied the effect of different number of steps and discharge on flow pattern, especially energy dissipation. Results showed that decreasing the number of the steps of pooled stepped spillways reduced flow velocity and increased the relative energy dissipation at the end of the spillway. Using the finite volume method and Flow-3D, Rezapour Tabari & Tavakoli (2016) investigated the effect of number of steps, step height, step length and discharge in width unit on energy dissipation in the simple stepped spillway and the relationship between energy dissipation and flow critical depth was achieved.
In an experimental and numerical study, Salmasi & Samadi (2018) investigated fluctuation of velocity vectors, shear stress, and pressure during the flow on each step. Experiments were performed with 10 different flow rates. The numerical simulations were also performed using FLUENT software and RNG k–ε turbulence model. Li & Zhang (2018) employed the RNG k–ε turbulence model with the VOF method to simulate the hydraulic characteristics of the stepped spillways with four types of pool weirs included: fully pooled steps, fully pooled and two-sided pooled steps, fully pooled and central pooled steps, and two-sided pooled and central pooled steps. It was found that the maximum velocities for all cases were approximately the same. Li et al. (2019) devised distorted step faces and pool weirs to improve the hydraulic behaviors for traditional stepped spillways. By numerical modeling, comparative studies were conducted to look into the flow features. They observed that symmetrical secondary flows were formed in V- and inverted V-shaped cases with different patterns.
In an experimental study, the effects of the transverse slope of steps on flow properties over the overtopping protection systems were explored by Ali & Yousif (2019). The results showed that the models with steps having transverse slope had higher energy dissipation capability compared to the model with steps having zero transverse slopes (flat steps); up to 14% higher. In the interest of maximum energy dissipation, Ashoor & Riazi (2019) numerically studied non-uniform stepped spillways. Then, within the range of skimming flow, four different types of non-uniform step lengths, including convex, concave, random, and semi-uniform configurations, were tested in InterFOAM. The results showed that in semi-uniform stepped spillways, when the ratio between the lengths of the successive steps is 1:3, a vortex interference region occurs within the two adjacent cavities of the entire stepped chute, and as a result, the energy dissipation increases by up to 20%. In this study, the effect of the presence of bed-block roughness in an ogee spillway on energy dissipation and jet length is investigated. Daneshfaraz et al. (2020) conducted a series of experimental and numerical tests to investigate the effect of block roughness in ogee spillways with flip buckets. The relationships for the estimation of relative energy dissipation and jet length were extracted.
Ghaderi et al. (2020b) experimentally investigated the effective scouring parameters downstream from stepped spillways. The results indicated that the flow regime plays an important role in scour-hole dimensions. The combination of stepped and labyrinth spillways, as a very strong energy dissipater, was examined experimentally by Banishoaib et al. (2020). Analysis of the results showed a greater energy loss reduction in triangular rather than rectangular-labyrinth stepped spillway (RLSS) or trapezoidal-labyrinth stepped spillway (TLSS). In addition, energy loss was greater in labyrinth spillways with two cycles than those with one cycle. Nouri et al. (2020) developed some data-based models to explain the relationships between hydraulic parameters. Multiple linear and nonlinear regression-based equations were developed based on dimensional analysis theory to compute energy dissipation over cascade spillways. Salmasi & Abraham (2021) used the genetic algorithms for optimizing stepped spillways to maximize energy dissipation. In addition, a stepped spillway with optimal dimensions was proposed as a replacement of the smooth spillway of Sarogh Dam located in West Azerbaijan province, Iran.
In an experimental study, Salmasi et al. (2021a) studied the effect of stepped spillways on increasing dissolved oxygen in water. It was found that at high unit discharges, aeration efficiency increases with increasing spillway slope. However, for low discharges, the effect of slope on aeration efficiency is negligible. By conducting a series of laboratory experiments, Salmasi et al. (2021b) developed a genetic programming (GP) approach for estimating energy dissipation of flow over cascade spillways. The results showed that formulation based on the GP approach in solving energy dissipation problems over cascade spillways is more successful than the method based on the regression equation.
Samadi et al. (2022) experimentally studied the trapezoidal, triangular, and rectangular-labyrinth weirs. It was found that increasing the number of cycles leads to a reduction in flow rate and discharge coefficient. These hydraulic performance reductions were most significant for trapezoidal followed by rectangular and triangular configurations. Using the k–ε RNG turbulence model and the multiphase mixture method, Salmasi & Abraham (2023) provided numerical models of the stepped spillway. It was found that the presence of steps along the spillway causes a significant reduction in the length of the boundary layer and faster aeration occurs.
A review of previous research indicates that the combination of other spillways with the stepped spillway can be effective in improving their performance. However, very limited studies have been carried out by researchers in this field. Therefore, in this research using the OpenFOAM open-source model, a three-dimensional evaluation of the performance of TLSS and RLSS, including the parameters of velocity, pressure, turbulence kinetic energy (TKE), and energy dissipation, and comparing it with conventional stepped spillways is done.
MATERIALS AND METHODS
Experimental model
The geometric parameters of the stepped spillway in the experiment done by Felder et al. (2012a).
The geometric parameters of the stepped spillway in the experiment done by Felder et al. (2012a).
Numerical model
is one of the components of the Cartesian coordinate system (i = 1, 2, 3); t, u, and 
are time, mean fluid velocity, and deviatoric stress tensor, respectively; p and μ represent flow pressure and molecular viscosity; ρ and g are density and gravitational acceleration. To solve the discretized Navier–Stokes equations, the PIMPLE algorithm was used. This algorithm is an implicit method for pressure-dependent equations and is based on a combination of SIMPLE and PISO solution algorithms (Hu et al. 2016). The Gambit model was used to draw the geometry of the spillways and generate the mesh. Figure 2 shows three-dimensional configurations of the conventional, the TLSS and the RLSS that were investigated in this research. Configurations include flat stepped spillway (FSS), RLSS, and TLSS. The dimensional details of spillways are also presented in Figure 3. 
Different configurations of stepped spillways: flat, rectangular-labyrinth, and the trapezoidal-labyrinth.
Different configurations of stepped spillways: flat, rectangular-labyrinth, and the trapezoidal-labyrinth.
Dimensional characteristics of different configurations of investigated spillways.
Dimensional characteristics of different configurations of investigated spillways.
A turbulence model should be added to model the nonlinear Reynolds stress term in the Navier–Stokes equation. For this purpose, in this research, the RGN k– turbulence model is used due to its high ability to simulate flow separation areas and the suggestion of researchers such as Rezapour Tabari & Tavakoli (2016), Daneshfaraz & Ghaderi (2017) and Ghaderi et al. (2019).
Meshing and boundary conditions applied in the numerical model.
In order to analyze the sensitivity of the model meshing, the flow velocity parameter in the numerical model was compared at the same points with the experimental model for different sizes of mesh cells from 0.008 to 0.04 m (Table 1).
Characteristics of cells in four mesh scenarios of the numerical model
| Scenario no. | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Mesh size (m) | 0.008 | 0.01 | 0.03 | 0.04 |
| Mesh number | 510,000 | 430,000 | 360,000 | 220,000 |
| Numerical and experimental result error (%) | 3.7 | 4 | 9 | 13 |
| Scenario no. | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Mesh size (m) | 0.008 | 0.01 | 0.03 | 0.04 |
| Mesh number | 510,000 | 430,000 | 360,000 | 220,000 |
| Numerical and experimental result error (%) | 3.7 | 4 | 9 | 13 |
Accordingly, a finer mesh than scenario (2) with a mesh size of 0.01 m and a mesh number of 430,000 does not change the accuracy of the results, which indicates the adequacy of the mesh size.
) for each step of the stepped spillway and compare different configurations in this regard, the following equation has been used:
In which, 
is the head of the flow in the upstream, 
is the head of the water in the downstream of the spillway before the jump, 
and 
.
RESULTS AND DISCUSSION
Numerical results verification
and 
, where y is the flow depth on the steps, dc is the critical depth and Vc is the critical flow velocity. Figure 5 shows a comparison of the velocity distribution in FSSs between the numerical model and the experimental model for the flow rate of 0.09 m3/s. The results show that the maximum relative error between the results of numerical and experimental models for steps 7, 8, and 9 is 4, 7.47, and 6.3%, respectively. The root mean square error (RMSE) values between the experimental and numerical results for steps 7, 8, and 9 are 0.23, 0.32, and 0.27, respectively. This indicates the appropriate matching of the results of these two models. 
Velocity distribution and error percentage of numerical and experimental data in steps 7, 8, and 9 for Q = 0.09 m3/s.
Velocity distribution and error percentage of numerical and experimental data in steps 7, 8, and 9 for Q = 0.09 m3/s.
Flow pattern
Water volume fraction contours and flow pattern at the axial plane in different configurations of stepped spillways: flat, rectangular-labyrinth, and trapezoidal-labyrinth.
Water volume fraction contours and flow pattern at the axial plane in different configurations of stepped spillways: flat, rectangular-labyrinth, and trapezoidal-labyrinth.
Water surface profiles
Water surface profiles and velocity changes on stepped spillways: flat, rectangular-labyrinth, and trapezoidal-labyrinth.
Water surface profiles and velocity changes on stepped spillways: flat, rectangular-labyrinth, and trapezoidal-labyrinth.
Flow velocity
The depth profile of the flow velocity on different configurations of the stepped spillway in different axes of the spillway in steps 8 and 9.
The depth profile of the flow velocity on different configurations of the stepped spillway in different axes of the spillway in steps 8 and 9.
Flow pressure
) and the maximum pressure occurred near the outer edge of the step (
). Comparison of the results of the pressure on the horizontal axis of the steps in the mentioned configurations indicates that the labyrinth configurations have led to a higher maximum pressure than the conventional flat configuration in both the central and lateral axes. However, in the central axis, the higher maximum pressure and the displacement of the position of the maximum pressure toward the edge of the steps, as well as the development of the mentioned pressure toward the lower steps, caused higher negative pressures in the vertical face of the labyrinth spillway steps and especially in the rectangular type of it. 
Pressure changes on the horizontal face of the eighth and ninth steps in different configurations of the stepped spillway in different axes of the spillway.
Pressure changes on the horizontal face of the eighth and ninth steps in different configurations of the stepped spillway in different axes of the spillway.
), the pressure values tended to zero or even negative values, and in the areas close to the horizontal face (
), the pressure values increased. In addition, both in the central axis and in the edge axis of the trapezoidal- and rectangular-labyrinth configuration of stepped spillways, more negative pressure values have been created in the vertical face than in the conventional stepped spillway. Among the stepped spillways of the trapezoidal- and rectangular-labyrinth types, the rectangular configuration has more negative pressure values in the vertical face (Figure 10). From this point of view, trapezoidal stepped spillways will be more favorable. 
The pressure profile on the vertical face of steps 8 and 9 from different configurations of stepped spillways in the central and side axes.
The pressure profile on the vertical face of steps 8 and 9 from different configurations of stepped spillways in the central and side axes.
Usually, the flow pressure is positive on the horizontal side of the steps due to the flow hitting the bottom. However, in the vertical face and especially in the upper half of the vertical faces of the steps and in the cavity area, pressure decreases and this issue is associated with negative pressures in some cases. Naturally, the configurations that have the lowest negative pressure on the vertical faces of the steps among the stepped–labyrinth configurations are more favorable in terms of the possibility of cavitation. Based on the graphs in Figure 10, the highest value of negative pressure in all configurations and in all axes occurred in the RLSS configuration. This makes the RLSS configuration less favorable than the TLSS configuration. It should be noted that one of the solutions to deal with negative pressures and cavitation in spillways is flow aeration. The flow on conventional and labyrinth type of stepped spillways is disturbed and highly aerated due to the roughness in the bed. Therefore, it can be resistant to negative pressures and cavitation to a great extent.
TKE of flow
), whose mean and variance are as given in the following:
Graphical changes of TKE in different configurations of stepped spillways.
The average value of the TKE steps 7, 8, and 9 for different configurations.
Flow energy dissipation
was compared with conventional, trapezoid-labyrinth, and rectangular-labyrinth forms of stepped spillway. Figure 13 shows the values of relative energy dissipation for different forms of stepped spillways in percentage terms. 
The amount of relative energy dissipation in the conventional, trapezoidal-labyrinth, and rectangular-labyrinth stepped spillways.
The amount of relative energy dissipation in the conventional, trapezoidal-labyrinth, and rectangular-labyrinth stepped spillways.
Energy dissipation rate in the current research model and comparison with models studied by other researchers.
Energy dissipation rate in the current research model and comparison with models studied by other researchers.
In Figure 14, Hmax = 1.5dc + Hdam and Δz is the vertical distance of the spillway crest to the last step.
CONCLUSION
To improve the performance of stepped–labyrinth combined spillways, in the present research, an attempt was made to evaluate their hydraulic performance in three dimensions with trapezoidal and rectangular configurations using the OpenFOAM model. In this regard, the following results were obtained:
The labyrinth and, especially, the trapezoidal configuration has significantly reduced the flow velocity at the end of the spillway compared to the conventional form of the stepped spillway, so that in the last step of the trapezoidal- and rectangular-labyrinth configurations, the flow velocity was decreased by 50.5 and 31.1%, respectively, compared to the conventional stepped spillway configuration.
The minimum pressure on the vertical face of the steps has been created in the upper half of this face
and the rectangular configuration has resulted in the highest value of negative pressure on the vertical face.The TKE over the spillway in all configurations is increased downstream. The trapezoidal configuration of stepped–labyrinth spillway also has the highest amount of TKE.
In stepped–labyrinth spillways, more energy dissipation has been achieved than the conventional stepped spillway, so that for the flow conditions of 0.09 m3/s and dc/h = 1.45, trapezoidal and rectangular stepped–labyrinth spillways had higher energy dissipation than the conventional stepped spillway, 34.7 and 21.1%, respectively.
DATA AVAILABILITY STATEMENT
Data cannot be made publicly available; readers should contact the corresponding author for details.
CONFLICT OF INTEREST
The authors declare there is no conflict.
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