## Abstract

Stepped spillways are suitable and economical options for high-storage dams that require significant energy dissipation, structural adaptation to roller-compacted concrete dams, and spontaneous flow aeration. In this study, a numerical analysis of the hydraulic characteristics for the skimming flow regime of the stepped spillway of Zirdan Dam is carried out. A comparison between the flow characteristics for stepped and smooth spillways is provided. By preparing numerical models of the spillway using the k-*ε* RNG turbulence model and the multiphase mixture method, a hydraulic analysis of the flow was completed. To verify the performance of the numerical model, field data were collected and used for validation. The results show that the presence of steps along the spillway cause a significant reduction in the length of the boundary layer and faster aeration occurs. In a stepped spillway, the cavitation index is higher than the critical limit along the entire length of the spillway. Thus, the risk of cavitation and destruction is negligible. On the other hand, with a smooth spillway, the possibility of cavitation may occur. This negative effect (negative pressures) occurs 56 m from the crest of the spillway on the downstream side.

## HIGHLIGHTS

The presence of steps along the spillway causes a significant reduction in the length of the boundary layer.

In a stepped spillway, the cavitation index is higher than the critical limit along the entire length of the spillway.

The negative pressures occur 56 m from the crest of the spillway on the downstream side.

The difference in energy dissipation on the stepped and smooth spillway is 47%.

## INTRODUCTION

In recent years, one of the most important developments in the field of storage dam construction technology is the use of roller compacted concrete (RCC), a method that made dams much cheaper and easier to implement. RCC dams require more efficient spillways and energy dissipation systems due to their greater height compared to earth dams. The use of stepped spillways with RCC dams leads to greater energy dissipation among other advantages (Salmasi & Abraham 2022a, 2022b). Due to the severe turbulence of the flow, the entry of air into the flow and the formation of a two-phase flow of water and air in different regimes, the hydraulics governing the flow of these overflow spillways is very complex. Despite extensive research in this field, there is still a need to investigate and understand the hydraulic characteristics of the stepped spillways.

*et al.*2022). In a stepped spillway, the roughness is greater than for smooth spillways. The place where the boundary layer meets the free surface of water marks the onset of aeration, and the presence of air bubbles in the flow reduces or eliminates the risk of the destruction of cavitation. Therefore, knowing how the boundary layer develops and the onset of aeration is of particular importance. Figure 1 shows the schematic representation of a boundary layer on a stepped spillway.

Chanson (1995) presented two relationships to calculate the onset of aeration in stepped spillways. Knowing the location of the onset of aeration is important because from this point, the flow swells and thus higher concrete side walls will be needed in design and construction. The swelling/bulking of water happens because of air entrainment in the flow. In order to understand flow behavior on stepped spillways, many researchers have used physical models in their studies. Among these, Christodoulou (1993) and Chanson (1994) can be mentioned. Other studies used mathematical and numerical models. In the field of numerical studies, the research of Chen *et al.* (2002), Tabbara *et al.* (2005), Salmasi & Samadi (2018) and Kaouachi *et al.* (2021) can be mentioned.

Among studies that focused on geometric optimization are Essery & Horner (1971), Sorensen (1985) and Salmasi & Abraham (2022a, 2022b). They investigated the flow behavior over stepped spillways using physical models and a wide range of step geometries and flow conditions. The effects of discharge, spillway slope, step size, and step slope were analyzed. It was found that the flow on stepped spillways can be divided into two types: (i) nappe and (ii) skimming flow regimes. The result of the Sorensen (1985) study showed that the skimming flow regime starts when the ratio of the critical depth of the flow to the height of the step exceeds 0.8 (*y _{c}*/

*h*> 0.8).

Rajaratnam (1990) presented a criterion to describe the onset of the skimming flow regime based on data available from the experiments of Essery & Horner (1971) in the range of spillway slopes *h*/*l* = 0.4–0.9, where *l* is the horizontal length of the step and *h* is the height of the step. Rajaratnam (1990) provided relations to predict the shear stress between the main flow and the rotating flow below it by using the frictional energy loss of the skimming flow regime.

Felder *et al.* (2012) conducted a physical study of three different forms of stepped spillways, including a simple stepped type, a pool and stepped spillway with alternating pool, and a simple step with a longitudinal slope of 8.9° and a step height of 0.05 m. Comparing the hydraulic characteristics of water–air flow in these three geometries showed some differences in flow behavior. 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 stronger aeration for the same discharge. The greatest energy loss is related 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. Reisi *et al.* (2015) investigated the effect of the convergence angle of the overflow chute walls on the flow characteristics using Flow-3D numerical software. The results showed that with a greater convergence angle of the side walls, the depth and velocity increase along the chute. Also, by examining the cavitation index, it was found that with the increase of the convergence angle, the cavitation index increased; the greatest risk of cavitation is for the model with minimum convergence. Roshangar *et al.* (2018) estimated the discharge coefficient of stepped spillways under skimming and nappe flow regimes by using data-based techniques. They used Gene Expression Programming (GEP) and Support Vector Machine (SVM) methods. The obtained results showed that the methods are able to model the discharge coefficient in stepped spillways and despite the better performance of the SVM compared to the GEP method, the second method is recommended because of the simplified derivation. In that study, it was found that the discharge coefficient for the skimming flow regime can be determined by using the Froude number, the height of the stepped spillway, the channel width and the flow depth on the stepped spillway, and in the nappe flow regime by using the Froude number, the height of the stepped spillway, the Reynolds number, the channel width, and the flow depth on the stepped spillway.

Zhang & Chanson (2019) investigated the effects of a cavity and step edge shapes on energy dissipation and aeration in a skimming flow regime over stepped spillways using laboratory studies. A significant reduction of the friction factor was achieved by the use of rounded/chamfered edges on the steps. It was found that the sharp edges have superior performance with flow aeration and energy loss. In steps with rounded edges, the starting point of ventilation is extended downwards. Wan *et al.* (2019) investigated the effect of the height and geometry of stepped spillways on the position of the air inlet point using volume of fluid (VOF) and the k-*ε* realizable method. They concluded that the flow aeration from the free surface is closer to the spillway crest with round-shaped steps than in a pool, and closer to the flat edge in a pool. Also, they showed that with an increase in discharge, decrease in the height of the steps, and increase in the number of steps, the position of the air entry point moved downstream and therefore the length of the airless area on the spillway increases.

In the study of Cea *et al.* (2007), the use of turbulence models is reviewed and a depth-averaged algebraic stress model was presented. A finite volume model for solving the depth-averaged shallow water equations (SWEs) coupled with several turbulence models is described with special attention to the modeling of wet-dry fronts. Pu *et al.* (2012) investigated the k-*ε* model to improve the ability of the SWEs model to capture the flow turbulence. The combined SWEs k-*ε* model was improved by a surface gradient upwind method (SGUM) to facilitate the extra turbulent kinetic energy (TKE) source terms in the simulation. In another study, Pu (2015) conducted a turbulence modeling of shallow water flows using the Kolmogorov approach. The comparison outcomes showed that the proposed Kolmogorov k-*ε* SWEs model can capture the flow turbulence characteristics reasonably well in all the investigated flows.

In a river, when a flood occurs, the excess water upstream of a dam overflows the dam spillway. High flow velocity and energy of flow downstream of a dam can erode river bed materials. Consequently, some numerical investigations are related to the study of the mechanism of generation of local scour in the river bed. For example, Bihs & Olsen (2011) studied the abutment scour with a focus on the incipient motion on sloping beds. Pu *et al.* (2014) studied the shallow sediment transport flow using time-varying sediment adaptation length.

An alternative of a stepped spillway can be considered as a suitable choice for the design of storage dams. The decrease in water velocity downstream of the spillway, high flow aeration along spillway, and the high energy dissipation of flow over a stepped spillway are the positive effects of this type of spillway. These benefits motivated the present investigation of the hydraulics of the Zirdan Dam spillway for stepped and smooth spillways.

In this study, a numerical model of a stepped and smooth spillway of Zirdan Dam is investigated. The hydraulic characteristics of the flow are discussed, including the depth of the flow, the longitudinal profile of the free surface, the flow velocity, the cavitation index, energy dissipation, and the development of the boundary layer. In addition, a comparison between the stepped and smooth spillways is carried out.

## MATERIALS AND METHODS

### Dam location

^{3}m

^{3}of RCC materials, 200 × 10

^{3}m

^{3}of normal concrete and nearly 1.2 × 10

^{2}m

^{3}of excavation in soil and rock. The volume of the dam reservoir is over 207 × 10

^{6}m

^{3}. Zirdan Dam was built with the purpose of irrigating downstream agricultural fields and supplying drinking water to Chabahar villages. Figure 3 provides a downstream view of Zirdan Dam and its spillway.

The chute spillway of Zirdan Dam has a length of 55 m with a slope of 1H:1.2 V (horizontal:vertical). This slope makes an angle of 50.19° with the horizon. The hydraulic characteristics of the flow passing over the stepped spillway, including the average flow depth, velocity, and average pressure at two different discharges of 540 and 760 m^{3}/s are determined. In addition, the results obtained from the numerical model are compared with those obtained from the physical model.

It should be mentioned that for the flow depth, measurements on the physical model at 10 cross-sections and perpendicular to the steps invert were made. The average velocity in the laboratory was measured by pitot tube and the pressure by piezometers which were installed on the steps. Table 1 shows the geometric and hydraulic characteristics of the Zirdan Dam spillway.

Design spillway discharge | 760 |

Width of the spillway (m) | 55 |

Number of steps | 64 |

Spillway slope (degrees) | 50.19 |

Volume of the dam reservoir (m^{3}) | 207 × 10^{6} |

Dam height (m) | 65 |

Design spillway discharge | 760 |

Width of the spillway (m) | 55 |

Number of steps | 64 |

Spillway slope (degrees) | 50.19 |

Volume of the dam reservoir (m^{3}) | 207 × 10^{6} |

Dam height (m) | 65 |

### Flow equations

*et al.*2022).

In the above equations *U _{i}* and

*U*are the components of the velocity vector in the spatial directions

_{j}*i*and

*j*(

*i*and

*j*equal 1, 2, and 3, respectively, for

*x*,

*y*, and

*z*directions),

*P*is the pressure,

*ρ*is the fluid density, and

*δ*is the Kronecker delta (if

_{i, j}*i*=

*j*, its value is one and otherwise its value is zero).

*k*-

*ε*Renormalization Group (

*RNG*) turbulence model was used. Readers are directed to Gorman

*et al.*(2021) and Abraham

*et al.*(2021) for a review of computational fluid dynamic models, turbulence approaches, and the development of computational fluid dynamics. For solving the free surface flow equation, the VOF method was used. To discretize the pressure expression, the pressure-implicit with splitting of operator (PISO) method and a second-order upward (SOU) method were used to discretize the momentum expression. Flow passing from a gate in an open channel is a two-phase and turbulent flow. The transfer ratio of the fluid fraction is expressed by Equation (5) (Hirt & Nichols 1981).

This method is predicated on the fact that the two fluids will not mix. The value of *F* in each cell represents the concentration of the species; *F* takes on values that range from 0 to 1. For cells with *F* = 0, it is water filled. Cells that have values of *F* = 1 are completely air filled. Intermediate values of *F* indicate a mixture (Ansys Fluent 2015).

### Numerical simulation

*et al.*1998). In addition, in all simulated models, the quality of the elements is considered with appropriate conditions (0.7 < Orthogonal Quality <1.0). The geometry of the spillway and the boundary conditions (BCs) used in the numerical model are presented in Figures 4 and 5.

Upstream BCs were chosen as the velocity inlet, the free surface of the reservoir, and the chute that are connected to the atmosphere are pressure inlets and the BC downstream of the spillway was the pressure outlet. For the other parts like the reservoir invert, steps, and chute bottom, a wall boundary condition was used.

For the spillway without steps, the BCs and input data are similar to the stepped spillway solution and were selected to compare the results.

*ε*turbulence models including Standard, RNG, and Realizable were tested. Based on Figure 6, the k-

*ε*RNG turbulence model provided more accurate results in the prediction of the water surface profile. The Relative Error (RE) for the k-

*ε*RNG turbulence model was less than 3%. In addition, for the chute without steps, the k-

*ε*RNG turbulence model was used, which according to the study by Salmasi & Samadi (2018) can successfully calculate the associated flows. That is, the k-

*ε*RNG turbulence model gives more accurate results in flows with strong shear stress zones (on the chute wall).

The k-*ε* turbulence model is based on the first numerical approaches that comprehensively dealt with turbulence. It relies upon two new transport equations for the TKE (*k*) and for turbulent dissipation (*ε*). These transport equations are coupled to the regular Navier–Stokes equations. The solutions of these two extra transport equations are fed into the Navier–Stokes equations through the eddy viscosity. The development of the k-*ε* model incorporated constants that are based on experimental measurements – the constants were chosen to best match experimental results (Nourani *et al.* 2021).

Since the development of the k-*ε* model, other approaches have been set forth. A number of the alternative approaches are similar to the k-*ε* model in that they require extra transport equations for turbulent fluctuations that ultimately affect the viscosity. These so-called two-equation RANS models (Reynolds-Averaged Navier–Stokes equations) are relatively straightforward to be implemented within modern CFD software. Other approaches, such as Reynolds Stress Models, Large Eddy Simulation (LES) models, Unsteady Reynolds-Averaged Navier–Stokes equations (URANS), approaches, and Direct Numerical Simulation (DNS) differ slightly in their approach and are sometimes used instead of the more traditional RANS methods. On the other hand, the k-*ε* model in particular and RANS models in general are calibrated and validated against a large database of solutions. The specific approach taken here was k-*ε* Renormalization Group theory (RNG) which is a modification of the traditional k-*ε* model that was developed to handle small flow phenomena. The turbulence production term is modified to account for these motions. The RNG approach has advantages over the traditional k-*ε* model for flows that have rotation and curved motion – mathematically the difference between the approaches appears in the definition of the turbulent viscosity. The history of these models and their excellent performance motivate their use here. Interested readers are directed to Gorman *et al.* (2021) and Abraham *et al.* (2021) for more detail about the historical development of these approaches.

A time step of 0.001 s was used to solve the unsteady equations. After assigning the BCs, selecting the numerical model, and performing the discretization, the program is executed. The time required to run the program depends on various factors such as the size of the problem, the number and type of elements, the type of equations, the processor system used and its memory.

*σ*) is used. It is defined as:where

*σ*is the cavitation index,

*p*is the fluid pressure at the desired point,

*p*

_{v}is the water vapor pressure,

*V*is the average water velocity, and

*ρ*is the water density.

### Validation

^{3}/s in a real spillway, these discharges change to 0.30 and 0.42 m

^{3}/s in the physical model. Longitudinal profiles of depth along with error for the experimental and numerical values at a flow rate of 540 m

^{3}/s are shown in Figures 8 and 9, respectively. In Figure 8, the percent of error denotes the difference between experimental and numerical flow depth divided by the experimental flow depth.

After the flow passes over the spillway crest, supercritical flow is formed and at a constant discharge, the velocity gradually increases, in contrast to the depth of the flow until the starting point of aeration (point *x* = 9.8 m in Figure 8). The aeration point is indicated by the dashed line in Figure 8. After the aeration point, due to the entry of air from the free surface into the water flow, and the increase in fluid volume, the depth of the flow increases, and finally it becomes a uniform flow (developed aeration) which is transformed into a continuous line in Figure 8. Establishment of a constant depth of *y* = 1.5 m happens at the discharge of 540 m^{3}/s.

Considering the small difference between the numerical and experimental data shown in Figure 9, it can be concluded that there is an acceptable match between the data. The maximum RE is 11.1% and can be attributed to the error of the second measurement point, which is slightly higher compared to other points.

There is a good agreement between the velocity values in the numerical model and the experimental values, and the difference obtained is less than 10%. Examination of the relationship between the depth and velocity changes is shown in Figures 8 and 10, respectively. The figures show that from *x* = 0 to the dashed line, the flow is a single-phase flow, and in this area, the flow velocity increases as the depth decreases. In the area between the dotted line and the continuous line, the two-phase flow of air and water is developing. The area after the continuous line is called the developed two-phase flow of air and water (uniform flow). In this region, the depth and velocity of the flow are stabilized.

## RESULTS AND DISCUSSION

### Development of the boundary layer and the starting point of aeration

^{3}/s.

Due to the significant roughness caused by the presence of steps in the stepped spillway, the growth of the boundary layer occurs faster than in the smooth chute and it reaches the free surface of the flow sooner. As a result, the starting position of aeration in this type of spillway is closer to the crest of the spillway with respect to a chute spillway (without steps).

### Average flow velocity and depth

*L*= 31.3 m), a slight difference between the flow depth in the two spillways is observed, which is caused by a slight decrease in the flow velocity in the stepped spillway compared to the chute spillway. After the air enters, the depth of the flow in the stepped spillway increases with the increase of the fluid volume, and it is necessary to increase the height of the side walls of the stepped spillway accordingly. The starting point of aeration in the smooth spillway is at the end of the chute (

*L*= 79.7 m) and therefore the two-phase (uniform) developed flow is not completely formed in this spillway.

### Cavitation

^{3}/s for both spillways are shown in Figure 16. In the stepped spillway, due to the formation of the establishment flow region from

*L*= 43.6 m to the downstream side of the chute, the flow velocity remained constant (Figure 15) and the flow depth did not change much (Figure 14). As a result, the values of the cavitation index in this area are almost constant around 0.5 and always higher than the critical value

*σ*= 0.2 (Figure 16). Therefore, cavitation does not happen on this spillway. For a smooth spillway, due to the non-establishment of the flow on the chute, the velocity increases and is very different from that of a stepped spillway at the end of the chute. So that at a distance of 57.3 m from the crest of the spillway, the cavitation index has reached its critical value (

*σ*= 0.2), and from this distance to the end of the chute, the invert of the spillway is exposed to cavitation and possible damage.

### Energy dissipation

^{3}/s.

*ΔE*/

*E*

_{0}) is obtained based on the following equations:where

*E*is the energy at upstream of the spillway and

_{0}*E*

_{1}is the energy at downstream of the spillway,

*y*is the depth of water over spillway crest,

_{0}*H*is the spillway height,

_{w}*V*is the approach velocity, and

*g*is the gravitational acceleration.

The relative energy dissipation at the spillway toe for the stepped and smooth spillways is 78.5 and 31.5%, respectively. This means that the stepped spillway has 47% more energy dissipation than the smooth spillway. The high energy dissipation in the stepped spillway can reduce the need for a stilling basin downstream of the spillway. Stilling basins are usually used downstream of a dam spillway and are a place for flow energy dissipation following a hydraulic jump. Otherwise, the high velocity of the water flow can cause severe erosion of the downstream river bed and creates serious risks for the dam structure. The present study shows that the positive effect of a stepped spillway in reducing water velocity and high energy dissipation can reduce the cost of constructing a stilling basin.

## CONCLUSIONS

A 2D numerical model for the Zirdan Dam spillway was developed for both smooth and stepped spillway types. The multiphase flow method and k-*ε* RNG turbulence model were used. The numerical model was calibrated applying experimental data extracted from the physical model of the Zirdan spillway. The following outcomes are obtained for this numerical simulation:

The depth and average velocity of the flow increase with the increase in the discharge of the stepped spillway. The starting point of aeration in the smooth spillway is located at a further distance from the crest in comparison with the stepped spillway. The ratio of the distance of inception aeration from the crest in the smooth spillway to the stepped type is 2.4. Sufficient aeration along stepped spillways reduces the risk of cavitation near the invert of a spillway. In a stepped spillway, from the middle of the spillway to the downstream side, a uniformly developed flow is formed. In the smooth spillway, the flow is not established, and flow velocity shows an incremental trend till the spillway toe. In a stepped spillway, the depth of the flow is greater than that of a smooth spillway, and the flow velocity is lower compared to it. In smooth spillways for their design discharge, from a distance of 56 m from the spillway crest to the downstream side, it is exposed to the occurrence of cavitation. But in a stepped spillway, cavitation does not occur. The difference in energy loss on the stepped spillway and the smooth spillway is 47%, which indicates the effectiveness of the presence of steps in the energy loss.

Recommendations for the future studies are as follows:

The selection of step sizes along spillway invert to generate high energy dissipation. Finding a suitable range for discharge per unit width of a stepped spillway to maximize energy dissipation. Investigation of spillway material resistance against impinging of jet flows in stepped spillways is another issue, especially in nappe flow regime. In this study, the k-*ε* RNG turbulence model was used for numerical simulations. The combination and comparison of different numerical models/methods are beyond this paper and can be carried out in future studies.

## 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.