## Abstract

Experimental tests were carried out to investigate the effective scouring parameters downstream from stepped spillways with different flow rates and step sizes. The results indicated that the flow regime plays an important role in scour-hole dimensions such that the minimum scouring depth happens in the nappe flow regime. Moreover, step size and tailwater depth are essential parameters for maximum scouring depth. Increasing tailwater depth from 6.31 cm to 8.54 cm and then to 11.82 cm decreases the scouring depth by 18.56% and 11.42%, respectively. These alterations also decrease the scouring length by 31.43% and 16.55%, respectively. By increasing the flow rate, the particle Froude number will increase, and the increased momentum of the flow promotes scouring. In addition, the results show that scouring at the sidewalls is higher than in the middle of the cross-section. Finally, an empirical formula with root mean square error = 0.107 and R^{2} = 0.974 is proposed to predict the maximum scouring depth downstream from the stepped spillways. Comparisons were made between the proposed formula and experimental results. This comparison demonstrated that the formula can predict souring depth to within 3.86% and 9.31% relative and maximum errors, respectively.

## HIGHLIGHTS

The flow regime on stepped spillway plays an important role in scour-hole dimensions.

Step size and tailwater depth are essential parameters for maximum scouring depth.

The particle's Froude number affects the downstream scouring of the spillway in respect to velocity.

An empirical formula with good accuracy to predict the maximum scouring depth downstream from the stepped spillways.

## NOTATIONS

The following symbols were used in this study:

*q(Q/B)*Water discharge per unit width [L

^{2}T^{−1}]*B*Width of the flume [L]

*h*The step's height [L]

*l*The length of step [L]

*L*_{d}The length of the stilling basin [L]

*H*_{s}Height of stepped spillway [L]

*d*_{smax}The maximum scouring depth [L]

*α*Chute angle [–]

*h*_{c}The critical depth [L]

*h*_{t}The tail water depth [L]

*ρ*_{s}Density of sediments [ML

^{−3}]*ρ*_{w}Density of the water [ML

^{−3}]*μ*Dynamic viscosity of water [ML

^{−1}T^{−1}]*Re*The Reynolds number [–]

*g*Gravitational acceleration [LT

^{−2}]*S*The relative density of sediments [–]

*D*_{50}Bed material diameter of which is finer by weight [L]

*D*_{10}Grain size for which 10% of material is finer [L]

*D*_{30}Grain size for which 30% of material is finer [L]

*D*_{60}Grain size for which 60% of material is finer [L]

*C*_{u}Coefficient of uniformity [–]

*C*_{c}Coefficient of curvature [–]

*Fr*_{d}Particle Froude number [–]

*σ*_{g}Geometric standard deviation [–]

## INTRODUCTION

Stepped spillways with a basin are one of the best solutions to transfer high-energy water and dissipate water flow energy (Ghaderi *et al.* 2020). Hence, the use of stepped spillways has become increasingly widespread due to low manufacturing costs and high efficiency (Daneshfaraz *et al.* 2016). Some energy will be dissipated by the hydraulic jump and as the flow transitions from supercritical to subcritical in the basin. The remaining energy can cause a scouring hole in the spillway toe, downstream from the structure. Severe scouring can impact structural stability.

Significant research has been conducted on downstream scouring of structures (Kells *et al.* 2001; Verma & Goel 2005; Roshan *et al.* 2010; Tuna & Emiroglu 2011; Xie & Lim 2015; Zahabi *et al.* 2018; Ben Meftah & Mossa 2019; Daneshfaraz *et al.* 2019a, 2019b; Ghaderi & Abbasi 2019; Ghaderi *et al.* 2019; Sadeghfam *et al.* 2019). Tuna & Emiroglu (2013) investigated the effect of step size on scouring downstream from stepped spillways. These results indicate that the step height influences the depth of scouring. By increasing the step height with the same flow chute angle, scouring is shown to decrease. Moreover, scouring depths grow with increasing flow rate and chute angle. Aminpour & Farhoudi (2017) investigated the scouring profile downstream from different stepped spillways. This study yielded important information about appropriate dimensionless parameters. By using a dimensionless scouring profile, geometrical characteristics and the volume of transported sediments can be estimated. Elnikhely (2018) worked on scouring downstream from spillways. They placed blocks in the spillway chute to investigate critical parameters and to decrease downstream scouring. Awad *et al.* (2018) worked on minimizing scouring in contracting stepped spillways. They used 60% of the flume width contraction ratios with a 10% opening area in the breaker. Their results indicated that four rectangular openings is the ideal number of openings and can reduce scouring by ∼55%. A 10-degree divergent angle can reduce scouring by ∼65%.

Ghaderi *et al.* (2018) added a nanomaterial and silicafume additive to the downstream moveable bed of a stepped spillway. Their results showed that these additives can decrease the depth and length of a scouring hole by ∼41% and ∼38%, respectively. Aminpour *et al.* (2018) investigated the time scale of local scouring downstream of stepped spillways. They showed that in certain circumstances, the dimensions of the scour hole increase with increasing particle Froude number. They also observed that an increase in the slope of a spillway causes a reduction in the size of the scour hole. It can be seen, based on the research discussed here, that the governing parameters of scouring have been investigated.

There is a significant need to better understand the physics of the scouring process so that design engineers can predict accurately with a high degree of certainty the magnitude and location of local scour around hydraulic structures. The main purpose of the present research is to understand effective scouring parameters downstream from stepped spillways, which were investigated experimentally for different unit discharges (flow regime), step geometries (step sizes), and tailwater depth.

## DIMENSIONAL ANALYSIS

*q*is the flow rate per unit width,

*H*is the height of stepped spillway,

_{s}*h*is the step height,

*l*is the length of each step,

*α*is the chute angle of the spillway,

*h*is the tail water depth,

_{t}*d*is the sediment particle's dimension,

_{50}*g*is gravitational acceleration,

*ρ*is the density of sediments,

_{s}*ρ*is the density of water,

_{w}*μ*is dynamic viscosity of water, and

*L*is the length of the stilling basin. By combining

_{d}*ρ*and

_{s}*ρ*, a parameter Δ can be defined that is related to the interaction between sediments and the water flow. It is defined as:

_{w}*S*is the ratio of sediment to water density (Dey & Raikar 2007). This parameter can be incorporated into the analysis so that:

*π*theorem, the dependent dimensionless parameter is based on independent parameters (Equation (4)). Previous researchers, for instance Hamedi

*et al.*(2014), used the same dimensional analysis equation. where

*Fr*is particle Froude number

_{d}*Fr*

_{d}*=*

*q/(gΔd*. The model will not take into account changes in the Reynolds number, chute angle, or still basin length.

_{50})^{0.5}d_{50}## METHODS AND MATERIALS

### Experimental facilities

Tests were carried out at the rectangular flume with a 5 m length, a 0.3 m width, and a 0.45 m height. The flume's bed and walls are plexiglass to improve flow visibility and reduce friction. With plexiglass, the influence of sidewall effects is considered to be negligible based upon the findings of Johnson (1996) and Moradinejad *et al.* (2019). The flow in the flume is generated by two pumps with a maximum flow rate of 7.5 L/s connected to two rotameters with ±2% accuracy (see Appendix-A1). To eliminate turbulence in the entrance region, a planar mesh has been added. The moveable bed is 1 m long and 12 cm deep. Three different stepped spillways, with two, three, and four steps of the same height and a 45-degree chuting angle were used. The stepped spillway was fabricated from dense polyethylene. Table 1 lists the geometric and hydraulic conditions of the experiment.

Q(L/s) . | α°
. | h_{c}(cm)
. | h_{t}(cm)
. | l(cm) . | h(cm) . | L_{d(cm)}
. | H_{s}(cm)
. |
---|---|---|---|---|---|---|---|

3–15 | 45 | 2.1–6.3 | 2.96–12.56 | 3, 4, 6 | 3, 4, 6 | 80 | 12 |

Q(L/s) . | α°
. | h_{c}(cm)
. | h_{t}(cm)
. | l(cm) . | h(cm) . | L_{d(cm)}
. | H_{s}(cm)
. |
---|---|---|---|---|---|---|---|

3–15 | 45 | 2.1–6.3 | 2.96–12.56 | 3, 4, 6 | 3, 4, 6 | 80 | 12 |

*et al.*2007). The equations are based on the ratio between the critical depth and the step height. where

*h*is the step's height and

*h*is the critical depth upstream of the spillway. Figure 2 indicates the flow regimes over stepped spillways.

_{c}By decreasing the number of steps and the flow rate, the flow will be in the transitional region and close to the nappe flow regime. By increasing the steps and decreasing the height of the steps, the flow regime will enter the skimming regime. Considering the time needed for reaching a balanced condition within the scouring pit, the duration of the experiment was decided to be 120 min. After this time, the scouring reached a steady state with no further changes in the dimensions of the pit. For the first 20 min, the depth and length of scouring was measured every 2 min. Thereafter, measurements were made every 4 min. To determine the length and depth of the scoured bed, a 3D-laser scanner was used. The 3D scanner used in this research had both a laser and an ultrasonic sensor. These sensors allowed the device to measure distances with an accuracy of 1 mm. To find the longitudinal scouring profile, five points from the mid-section were taken and averaged to eliminate sidewall errors. This technique was adopted from previous studies, such as Torabi & Shafieefar (2015).

### Bed sedimentation characteristics

Attention was given to the appropriate sediment dimensions (Yalin 1971). According to Raudkivi & Ettema (1983), the minimum dimension of sediment was 0.7 mm to prevent the formation of ripples and to avoid particle adhesion during the scouring process. Uniform particle sizes have a standard deviation (*σ _{g}*

*=*

*(d*) and a coefficient of uniformity of less than 1/3 and 2, respectively. The scouring depth in non-uniform sediment is less than for the uniform sediment case (Lambe & Whitman 1994; Melville 1997). In this study, the average particle size, with the density, a standard deviation and a coefficient of uniformity and the density were 1.8 mm, 2,400 kg/m

_{84}/d_{16})^{0.5}^{3}, 1.26 and 1.56, respectively. With this sediment, scouring takes place without ripple formation. Table 2 lists the sediments characteristics.

Bed particles . | D_{10} (mm)
. | D_{30} (mm)
. | D_{60} (mm)
. | σ_{g}
. | C_{u}
. | C_{c}
. |
---|---|---|---|---|---|---|

1.3 | 1.7 | 2.1 | 1.26 | 1.56 | 0.67 |

Bed particles . | D_{10} (mm)
. | D_{30} (mm)
. | D_{60} (mm)
. | σ_{g}
. | C_{u}
. | C_{c}
. |
---|---|---|---|---|---|---|

1.3 | 1.7 | 2.1 | 1.26 | 1.56 | 0.67 |

Before discussing the results, a comment should be made about the applicability of laboratory-scale experiments to provide accurate information for large-scale flow systems. The use of scaled experiments is helpful in this regard. We are also guided by past success of other researchers in extending laboratory-scale experiments to larger scales (Azmathullah *et al.* 2005; Emiroglu & Tuna 2011; Tuna & Emiroglu. 2011; Tuna & Emiroglu 2013). The use of scaled experiments also motivates the presentation of dimensionless results (as already revealed in the dimensionless governing equations). Of course, in reality it may be that small-scale laboratory experiments are not suitable for scaled investigation in this situation. With this as a possibility, there are some caveats when considering the results. Future work will be performed to ensure validity of small-scale experiments, but such a comprehensive comparison is beyond the scope of the present work.

## RESULTS AND DISCUSSION

### The effect of the number of steps on downstream scouring

According to the Chanson (1994) studies on small experimental canals, the nappe flow regime can dissipate more energy than the skimming regime. This effectively decreases scouring downstream from the structure. Figure 3 shows the scouring hole downstream of the stepped spillway for two, three, and four steps and for different flow rates. In the figure, the flow passes from right to left. The figures correspond to different flow rates and each figure contains various step heights.

For similar flow rates, an increasing number of steps and a decreasing step height increases the scouring. Consequently, scouring in a stepped spillway with fewer large steps is less than scouring in a spillway with more smaller steps. This finding agrees with prior research (Chanson 1994; Ghaderi 2016). A comparison of different flow regimes shows that the nappe flow regime has the least scouring, again in agreement with prior research (Tuna & Emiroglu 2013). Figure 4 shows the scouring profile downstream of the stepped spillway with a constant flow rate of 11 L/s and three different step heights. The photographs were taken in the 6th 20-min segment (the last time period of the 2-h test).

The figure shows that the stepped spillway with two steps decreases the length and depth of scouring downstream of the stepped spillways compared with spillways with three or four steps. The two-step spillway decreases scouring by up to 20.2% compared to the three steps and 48.28% compared to the four steps. The reason for this energy dissipation is the nappe flow regime in two-step spillways. In other research studies, researchers assumed the longitudinal scouring profile existed next to the sidewalls. This assumption is not correct. Scouring near the sidewalls is more extensive than in the middle of the channel. The reason for this phenomenon is the interaction and impact of the transverse flow with sidewalls that transports sediments to the mid-section (see Appendix-A2). By increasing the flow rate, momentum will increase and enhance the length and depth of downstream scouring of the stepped spillway, although the scouring depth in the sidewalls will be more intense.

### Tailwater depth effect in downstream scouring

Tailwater depth is one of the important parameters for downstream scouring and its study is important for understanding spillway performance. Based on the Farhoudi & Smith (1985) study, the shape of the scoured bed is independent of the sediment properties and depends instead on the flow regime and flow rate. The most intensive bed scouring takes place with a stepped spillway with four steps of 3 cm height and with skimming flow regime. Figure 5 indicates the longitudinal scouring profile of the above-mentioned spillway with a 13 L/s flow rate.

According to the longitudinal scouring profile shown in Figure 5, with the same hydraulic characteristics and increasing tailwater depth, the size of the scouring hole decreases. By increasing tailwater depth from 6.31 cm to 8.54 cm and then to 11.82 cm, the maximum depths and lengths of the scouring hole will decrease by 18.56% and 12.62% for 8.54 cm and by 31.43% and 16.55% for 11.82 cm, respectively. The results are summarized in Table 3.

Tailwater depth (cm) . | Maximum scour depth (cm) . | Reduction (%) . | Maximum scour length (cm) . | Reduction (%) . |
---|---|---|---|---|

6.31 | 9.48 | NA | 47.19 | NA |

8.54 | 7.72 | 18.56 | 41.80 | 11.42 |

11.82 | 6.50 | 31.43 | 39.38 | 16.55 |

Tailwater depth (cm) . | Maximum scour depth (cm) . | Reduction (%) . | Maximum scour length (cm) . | Reduction (%) . |
---|---|---|---|---|

6.31 | 9.48 | NA | 47.19 | NA |

8.54 | 7.72 | 18.56 | 41.80 | 11.42 |

11.82 | 6.50 | 31.43 | 39.38 | 16.55 |

The reason for this is the damping effect of the tailwater on the incident flow. This effect reduces energy and decreases scouring (Johnston 1990; Emiroglu & Tuna 2011). The tailwater depth also affects the downstream vortex. Figure 6 shows the impact of two tailwater depths (6.31 cm and 11.82 cm) on the vortex with a 13 L/s flow rate.

By decreasing the flow depth with maintenance of the same flow rate, the velocity and momentum will increase and cause more downstream scouring. With increasing tailwater depth, the intensity and size of the vortex will decrease. Consequently, by increasing the tailwater depth and maintaining the same hydraulic characteristics, the scoured bed (the length and depth of the scouring) will decrease (see Appendix-A3).

### The effect of the particle Froude number on scouring

The Froude number of a particle is an important parameter for downstream scouring (Tuna & Emiroglu 2011). In this study, to find the Froude number effect, the velocity at the end of the still basin upstream of the scour bed was measured. Figure 7 shows the relationship between the Froude number and the downstream scouring depth. It is seen that, in general, smaller but more numerous steps increase the Froude number significantly.

In all tests, the scouring depth increased by increasing the particle's Froude number. The higher the Froude number, the greater the velocity, and consequently larger fluid momentum hitting the bed. The increased momentum increases the vortex intensity and enhances the scouring rate. Scouring rarely took place in models with *Fr _{d}* < 4. The maximum scouring rate occured for 6 <

*Fr*< 9. This finding is in agreement with previous studies (Tuna & Emiroglu 2011) that also showed that a higher Froude number increases scouring.

_{d}^{2}= 0.974.

Equation (8) was developed based on the 70–30% rule for experimental results. In this method, 70% of the data was used to train the coefficients and the remaining 30% was used to test the correlation accuracy. To quantify the accuracy, three statistical indices were used. The RMSE, R^{2}, and the relative error between the experimental data and the proposed formula. The definitions of these metrics are provided in Table 4.

Equation . | Statistical index . |
---|---|

Root mean square error | |

R-squared | |

Relative error |

Equation . | Statistical index . |
---|---|

Root mean square error | |

R-squared | |

Relative error |

In Table 4, *n* is the number of experimental tests, *X _{Exp}* is the experimental value, and

*X*is the calculated value of the hydraulic parameter. To ensure the accuracy of the proposed experimental data, predicted values were compared with measurements from Emiroglu & Tuna (2011). The comparison shows that the relative and maximum errors were 3.86% and 9.31%, respectively. Figure 8 provides a comparison between the maximum relative depth obtained from experimental data and the predicted formula.

_{Cal}## CONCLUSIONS

Scouring is an important issue to consider when designing hydraulic structures. In this research, a scouring phenomenon downstream from the stepped spillway was investigated and the following results were obtained:

In small canals (the flume), the nappe flow regime can dissipate energy better than the skimming flow.

The flow regime on a stepped spillway can affect the scouring hole. In this case, nappe flow minimizes scouring.

The size of the steps is important in the maximum scouring depth. As step height increases, the maximum scouring depth falls.

The 3D-scoured profile indicates that scouring near the walls is greater than in the middle. The reason for this is the interaction of the cross-sectional flow with the sidewalls that transports sediments to the mid-section.

Tailwater depth also affects the scouring depth. By decreasing tailwater depth, the bed experiences a larger vortex and a flow with higher momentum.

By increasing tailwater depth while maintaining the other hydraulic characteristics, the scoured hole will decrease. By increasing the tailwater from 6.31 cm to 8.54 cm, and then to 11.82 cm, the depth and length of the scoured bed decreases. This happens because the water acts like a damper on incident flow, decreasing the scouring rate.

The particle's Froude number affects the downstream scouring of the spillway in respect to velocity. By increasing the flow rate and the Froude number, flow with higher momentum will impact the bed and cause more scouring.

Finally, a formula was proposed to investigate the maximum relative scouring depth downstream from the stepped spillway, considering effective parameters. The proposed formula was also used to test the predictions against previous experimental studies (Emiroglu & Tuna 2011), yielding relative and maximum errors of 3.86% and 9.31%.

## ACKNOWLEDGEMENTS

We would like to thank the editor and reviewers for their valuable and useful comments and suggestions.

## SUPPLEMENTARY MATERIAL

The Supplementary Material for this paper is available online at https://dx.doi.org/10.2166/ws.2020.113.