## Abstract

Screens are one of the recent energy dissipator structures that can be used downstream of small hydraulic structures. In this study, screens were used horizontally at the brinks of the vertical drop with downstream smooth and rough bed to investigate the energy loss of the drop. Experiments were performed on two porosities of screens, a relative critical depth of 0.13–0.39 and a median size of 1.9 cm aggregates. The results showed that for a relative critical depth of more than 0.3 in a vertical drop equipped with a screen with a rough bed, the drop length with respect to smooth bed increases. Compared to applying a Type I stilling basin, a vertical drop equipped with a screen with downstream smooth and rough bed reduces the drop length by approximately 50%. Although a rough bed increases air entrainment, it has no effect on the energy loss and pool depth of a vertical drop equipped with a horizontal screen with smooth bed. The use of horizontal screens at the brinks of the vertical drop causes maximum energy loss in the downstream of drop. Equations were provided to estimate the flow parameters with a *R*^{2} value of more than 0.925 and a normalized root mean square error of less than 0.04.

## HIGHLIGHTS

Introducing a relatively new structure to create energy loss is presented.

The importance of energy loss in open channels flow is stated.

Using accurate laboratory results, the amount of energy loss in the channels has been investigated.

To understand the problem, proper hydraulic analysis is done.

The results of this research can be generalized in the design of hydraulic structures.

## NOTATION

The following symbols are used in this paper:

*B*flume width (m)

*Q*flow discharge (m

^{3}/s)*g*acceleration due to gravity (m/s

^{2})*ρ*density of water (kg/m

^{3})*μ*dynamic viscosity (kg/m·s)

*E*_{u}total energy upstream of the drop (m)

*E*_{d}downstream specific energy (m)

*L*_{ds}total drop length (m)

*L*_{d}drop length (m)

*L*_{b}basin length (m)

*P*screen porosity (-)

*y*_{c}critical depth (m)

*q*unit discharge of flow (m

^{3}/s·m)*h*drop height (m)

*y*_{u}upstream depth of drop (m)

*y*_{d}downstream depth of drop (m)

*y*_{p}pool depth (m)

*L*_{wet}wetting length of screen (m)

- Δ
*E* energy loss (m)

*Fr*_{u}upstream Froude number (–)

*Fr*_{d}downstream Froude number (–)

*Re*_{u}Reynolds number (–)

*k*_{s}median size of rough bed sands (m)

- σ
surface tension (ML

^{−1})

## INTRODUCTION

Vertical drops are among the most commonly used water management structures because of their ease of construction compared to options. Vertical drops are commonly used in mountainous areas to reduce steep slopes. Flow downstream of the drop often has a destructive kinetic energy. If this destructive energy is not controlled and reduced, downstream structures will be exposed to erosion and potential damage. A hydraulic jump in the stilling basin is commonly used to reduce this energy, as discussed, for example, in Daneshfaraz *et al.* (2020a) and Abbaspour *et al.* (2019).

The first studies of vertical drops with the upstream subcritical flow were made by Bakhmeteff (1932). He presented an equation to calculate the downstream depth of the drop by assuming a hydrostatic pressure distribution and a uniform velocity distribution in the energy equation. Subsequently, extensive experimental studies have been performed to characterize the flow over a simple vertical drop. Many of these researchers investigated the total energy loss and the hydraulic parameters for plain vertical drops (Gill 1979; Rajaratnam & Chamani 1995; Chamani *et al.* 2008).

Square steps with different heights were investigated by Esen *et al.* (2004). The results of that study showed that as the step height increased, energy loss and the vertical downstream depth of the drop also increased. Hong *et al.* (2010) utilized a positive slope in the downstream bed of a drop and showed that by increasing the bed slope, the drop length and force on the downstream bed is increased.

Chiu *et al.* (2017) carried out a numerical investigation on the various plunge pool lengths with a vertical drop and found that the three types of flow regimes (skimming, nappe and periodic oscillatory flows) occurred. The flow regime depends on the discharge and geometry of the pool. Kabiri-Samani *et al.* (2017) performed experiments to study energy loss for a vertical drop equipped with grid dissipaters. They showed that using a dissipater increased the energy loss and the authors proposed equations to predict energy loss.

Abrari *et al.* (2017) developed a theoretical method to investigate the brink depth for a vertical drop in an inverted semi-circular channel with a steep slope. The results of that study provided an equation to estimate the discharge using brink depth. The influence of tailwater depth on the vertical hydraulic performance of a vertical drop equipped with grid dissipaters was also studied experimentally by Sharif & Kabiri-Samani (2018). The results showed that as the tailwater depth increases, air entrainment decreased. Recently, Shubing & Sheng (2019) developed a theoretical model of a vertical drop with a steep slope and bed friction. Their predictions of the water surface profile, discharge, and brink depth were in good agreement with the laboratory results.

Air entrainment is a phenomenon commonly employed to cause energy loss. In fact, the use of vertical screens promotes air–water mixing and two-phase flow has been one of the methods in recent years (Daneshfaraz *et al.* 2017). The first studies on vertical screens were carried out by Rajaratnam & Hurtig (2000). They showed that the Froude number downstream of the screens was approximately 1.65 for all laboratory conditions. Results also showed that the thickness of the screen had no effect on the energy loss, but modifications to the number of screens and the shape of the apertures did have an impact (Mahmoud *et al.* 2013; Daneshfaraz *et al.* 2017; Sadeghfam *et al*. 2019).

Recently, the application of a horizontal screen at the brink of a vertical drop has been considered as an energy dissipater. Horizontal screens increase energy loss by creating several falling jets and increasing downstream turbulence. Daneshfaraz *et al.* (2021a) investigated the effect of a horizontal screen on the energy loss of a vertical drop with the upstream subcritical flow. The experiments were performed with a 15 cm drop height, screens whose porosity was 40 and 50%, and a relative critical depth that ranged from 0.128 to 0.4. The results show that the utilization of a horizontal screen at the brink of a vertical drop increases the flow energy loss.

The stability of rivers and channels is often related to the stability of its bed. Creating a natural rough bed is a method that increases bed roughness and decreases the flow rate, thereby protecting against erosion (Keller 2004). Rough beds were first studied in 1968 (Rajaratnam 1968). Subsequently, various researchers have studied the effect of a rough bed on a hydraulic jump and have showed that height, density, and the shape of roughness elements can affect the hydraulic jump (Ead & Rajaratnam 2002; Tokyay 2005; Carollo *et al.* 2007; AboulAtta *et al.* 2011; Parsamehr *et al.* 2017).

By simultaneously applying a vertical drop and a horizontal screen, we hope to show a significant increase in energy loss downstream of these structures. On the other hand, no studies have been carried out on vertical drop equipped with a horizontal screen and a downstream rough bed. This combination is expected to increase energy loss and reduce bed erosion. Consequently, this combination will be the focus of the current study.

## METHODOLOGY

### Dimension analysis

*σ*is the surface tension (),

*ρ*is the density of water (), is the dynamic viscosity (),

*g*is gravitational acceleration (),

*q*is the unit discharge of flow (),

*B*is the flume width (L),

*h*is the drop height (L),

*P*is the screen porosity ratio (–),

*y*is the critical depth (L),

_{c}*y*is the upstream depth of drop (L),

_{u}*k*is the median size of rough bed sands (L) and

_{s}*E*is the total energy upstream of the drop (L) (). Using the π- Buckingham's theorem, the total relative energy loss was obtained on the basis of the independent dimensionless parameters in Equation (2):

_{u}*E*/

*E*is the total energy loss,

_{u}*y*/

_{c}*h*is the relative critical depth,

*P*is the screen porosity,

*y*/

_{p}*h*is the relative pool depth

*, L*/

_{wet}*y*is the relative wetting length of screen,

_{c}*L*/

_{d}*h*is the relative drop length,

*k*/

_{s}*h*is the relative grain height of rough bed,

*Fr*is the Froude number at the upstream of drop,

_{u}*Re*is the upstream Reynolds number and

_{u}*We*is the Weber number. Since the Reynolds number range is between 8,335 and 38,900, the flow is quite turbulent, viscosity and surface tension effects can be neglected (Hager & Bremen 1989; Bagherzadeh

*et al.*2022). The flow upstream of the drop is subcritical, and the Froude numbers are low (0.69 <

*Fr*< 0.86), so the effect of this parameter on Equation (4) is neglected (Daneshfaraz

_{u}*et al.*2021b, 2021c). Finally, Equations (4) and (5) are modified as follows:

The relative critical depth in the present study ranged from 0.13 to 0.39.

### Laboratory characteristics

*et al.*2020a, 2020b, 2021a). The distance between screen holes and its edge is half the diameter of its hole (5 mm). Figure 3 shows a schematic of the experimental model in the present study.

After constructing the laboratory model and installing the horizontal screen, the pump was turned on. Then, when stable flow conditions were achieved, the parameters of discharge (*Q*), upstream drop depth (*y _{u}*), wetted length of the horizontal screen (

*L*), the pool depth (

_{wet}*y*), drop length (

_{p}*L*), and downstream depth (

_{d}*y*) were measured. A rotameter installed on the pump was used to measure the discharge. For measuring depth, a point gauge with an accuracy of 1 mm was used and the desired depth was measured at five locations along a traverse and the average value was determined. A ruler with 1 mm accuracy was used to measure the wetted length and drop length. In all experiments, the upstream flow of the drop was considered subcritical. In total, 108 different experiments were carried on rough and smooth bed; the ranges of variables are presented in Table 1.

_{d}P
. | Q (L/s)
. | y (cm)
. _{u} | L (cm)
. _{wet} | y (cm)
. _{p} | L (cm)
. _{d} | y (cm)
. _{d} |
---|---|---|---|---|---|---|

40% | 2.5–13.3 | 2.45–6.5 | 9–45 | 3.6–10.2 | 18–56 | 2.6–7.1 |

50% | 2.5–13.3 | 2.45–6.5 | 8–37.5 | 3.7–10.3 | 16–55.5 | 2.6–7.1 |

P
. | Q (L/s)
. | y (cm)
. _{u} | L (cm)
. _{wet} | y (cm)
. _{p} | L (cm)
. _{d} | y (cm)
. _{d} |
---|---|---|---|---|---|---|

40% | 2.5–13.3 | 2.45–6.5 | 9–45 | 3.6–10.2 | 18–56 | 2.6–7.1 |

50% | 2.5–13.3 | 2.45–6.5 | 8–37.5 | 3.7–10.3 | 16–55.5 | 2.6–7.1 |

### Drop length theory

*et al.*(2004) have presented predictive calculations for the relative downstream depth as in Equation (9).

## RESULTS AND DISCUSSION

### Wetted length of a horizontally oriented screen

*R*

^{2}) and normalized root mean square error (NRMSE) that are equal to 0.925 and 0.04, respectively.

### Pool depth

As seen in Figure 8, for the vertical drop with a horizontal screen with and without rough beds and for both screen porosities, the pool depth is an adaptation with results of Daneshfaraz *et al.* (2021a) and is increased compared to the simple vertical drop. This increase is due to the increase in the angle of the falling jet for the vertical drop equipped with a horizontal screen, compared to the absence of screens. Horizontal screens positioned at the brink of a vertical drop create a large number of submerged jumps in the pool and these submerged jumps also increase the relative pool depth. The average increase in the relative pool depth in the present study for porosities of 40 and 50% was 13.5 and 9.5%, respectively, compared to the plain vertical drop situation. Also, the relative pool depth of the rough bed, compared to smooth bed in both porosities, is not different.

*R*

^{2}and NRMSE that are equal to 8%, 0.99 and 0.021, respectively. It is worth noting that due to non effect of bed roughness on the pool depth of drop,

*k*has not been considered in the presented equation.

_{s}/h### Drop length

Passing strip jets through the horizontal screen and colliding them with the flume bed cause numerous jumps in the pool and create a uniform depth downstream of the falling jets. The longitudinal distance from the brink of the drop to the downstream depth of the drop is called the ‘drop length’ or the ‘mixing length’. This length is always greater than the wetted length of the screens (Figure 2).

*et al.*(2021a). Also, for values of relative critical depth greater than 0.3, the drop length of rough bed is slightly increased compared to smooth bed and results of Daneshfaraz

*et al.*(2021a). The reason for this behavior is the increase in air entrainment due to the collision of falling jets with the rough bed. The air bubbles created by the air entrainment tend to move along with the flow, the movement of these bubbles to downstream causes and the increase in the drop length (Figure 5).

In Figure 10, it is observed that the utilization of screens significantly reduces the drop length compared to a Type 1 stilling basin. Also, the vertical drop length gradient of the present study is also lower than the vertical drop with a Type 1 stilling basin. This means that the greatest decrease in the length of the vertical drop occurs in the higher critical depths compared to using a stilling basin. Comparison of the values of the horizontal screen drop length for the two porosities in Figure 10(a) and 10(b) also shows that a 40% horizontal screen causes a larger drop length than a more porous screen. As noted earlier, the drop length created inside the pool is influenced by the wetted length of the horizontal screen. The results also show that vertical drops equipped with horizontal screens decrease the drop length by about 47.5 and 49.5%, respectively, for the 40 and 50% porosities (compared to a Type I stilling basin).

*R*

^{2}and NRMSE are equal to 0.997 and 0.02, respectively.

### Energy loss

*E*), if the flow energy downstream, (

_{u}*E*) is a minimum value (

_{d}*E*=

_{min}*E*), and Equation (18) will have its maximum value. So, for the rectangular channel:

_{d}*et al.*(2017) and Daneshfaraz

*et al.*(2021) is shown in Figure 12. A comparison of the energy loss in the figures shows that the porosity of the screens does not have as large of an effect on the energy loss.

In Figure 12(a), it is seen that the energy loss for a vertical drop equipped with a 40% horizontal screen and with a downstream rough bed compared to the smooth bed and results of Daneshfaraz *et al.* (2021a) overlap each other. The bed roughness has little effect on the energy loss of a vertical drop equipped with a horizontal screen in comparison to the smooth bed. This is due to the maximum energy loss that occurs in the vertical drop (Equation (19)). Also, it can be seen that energy loss from Kabiri-Samani *et al.* (2017) is greater than the maximum amount that can occur in vertical drops and is not in agreement with predictions from Equation (19). It seems that the approach presented by Kabiri-Samani *et al.* (2017) is not capable of accurately estimating the flow energy loss and this failure needs further investigation.

In Figure 12(b), for 50% porous screens, the energy loss of a vertical drop equipped with the horizontal screens with rough and smooth bed, and Daneshfaraz *et al.* (2021a) have values close to the maximum energy loss. When the energy loss for screens of 50% porosity is compared to the results of Kabiri-Samani *et al.* (2017), the comparison shows that a vertical drop equipped with a horizontal screen with a rough and smooth bed has greater energy loss. The differences in the shape and size of the aperture and the zig-zagging arrangement of the horizontal screen apertures in the present study are some of the parameters that might explain the increase in energy loss of the present study compared to Kabiri-Samani *et al.* (2017).

### Sensitivity analyzed

The results of sensitivity analysis are presented in the Table 2. In this table, for independent parameters, the amount of increase and decrease of 10–40% is considered, and based on the presented relationships, the sensitivity of the model to decrease and increase of the parameter is calculated. As can be seen, for all the studied parameters, the changes in the porosity of the screen have the least effect on the model compared to the other parameter and the maximum effect for the wetted length of the screen (Equation (14)) is related to the upstream Froude number of the drop. In addition, in other case (Equations (15) to (20)), the relative critical depth has the greatest impact on the model.

Equation number . | Percentages applied (%) . | Increase . | Decrease . | ||||
---|---|---|---|---|---|---|---|

Parameters . | Parameters . | ||||||

y/_{c}h
. | Fr
. _{u} | P
. | y/_{c}h
. | Fr
. _{u} | P
. | ||

14 | 10 | – | 1.220 | 0.940 | – | 0.802 | 1.070 |

20 | – | 1.464 | 0.889 | – | 0.627 | 1.155 | |

30 | – | 1.730 | 0.844 | – | 0.475 | 1.259 | |

40 | – | 2.020 | 0.805 | – | 0.344 | 1.391 | |

15 | 10 | 1.092 | – | 0.984 | 0.907 | – | 1.018 |

20 | 1.183 | – | 0.970 | 0.814 | – | 1.038 | |

30 | 1.274 | – | 0.958 | 0.719 | – | 1.061 | |

40 | 1.365 | – | 0.946 | 0.624 | – | 1.088 | |

16 or 17 | 10 | 1.188 | – | 0.982 | 0.826 | – | 1.020 |

20 | 1.391 | – | 0.966 | 0.668 | – | 1.043 | |

30 | 1.608 | – | 0.951 | 0.524 | – | 1.070 | |

40 | 1.839 | – | 0.938 | 0.397 | – | 1.102 | |

20 | 10 | 0.976 | – | 0.997 | 1.028 | – | 1.003 |

20 | 0.954 | – | 0.994 | 1.060 | – | 1.007 | |

30 | 0.934 | – | 0.991 | 1.097 | – | 1.012 | |

40 | 0.916 | – | 0.989 | 1.142 | – | 1.017 |

Equation number . | Percentages applied (%) . | Increase . | Decrease . | ||||
---|---|---|---|---|---|---|---|

Parameters . | Parameters . | ||||||

y/_{c}h
. | Fr
. _{u} | P
. | y/_{c}h
. | Fr
. _{u} | P
. | ||

14 | 10 | – | 1.220 | 0.940 | – | 0.802 | 1.070 |

20 | – | 1.464 | 0.889 | – | 0.627 | 1.155 | |

30 | – | 1.730 | 0.844 | – | 0.475 | 1.259 | |

40 | – | 2.020 | 0.805 | – | 0.344 | 1.391 | |

15 | 10 | 1.092 | – | 0.984 | 0.907 | – | 1.018 |

20 | 1.183 | – | 0.970 | 0.814 | – | 1.038 | |

30 | 1.274 | – | 0.958 | 0.719 | – | 1.061 | |

40 | 1.365 | – | 0.946 | 0.624 | – | 1.088 | |

16 or 17 | 10 | 1.188 | – | 0.982 | 0.826 | – | 1.020 |

20 | 1.391 | – | 0.966 | 0.668 | – | 1.043 | |

30 | 1.608 | – | 0.951 | 0.524 | – | 1.070 | |

40 | 1.839 | – | 0.938 | 0.397 | – | 1.102 | |

20 | 10 | 0.976 | – | 0.997 | 1.028 | – | 1.003 |

20 | 0.954 | – | 0.994 | 1.060 | – | 1.007 | |

30 | 0.934 | – | 0.991 | 1.097 | – | 1.012 | |

40 | 0.916 | – | 0.989 | 1.142 | – | 1.017 |

## CONCLUSION

In the present study, the energy loss flow passing through vertical drops equipped with a horizontal screen with downstream rough and smooth bed was investigated. Experiments were performed using a constant drop height, two different screen porosities and a relative critical depth ranging from 0.13 to 0.39. The results show that by increasing the upstream Froude number and decreasing the screen porosity, the relative wetted length of screens increases. The results also show that a vertical drop equipped with a horizontal screen with rough bed had a negligible effect on the energy loss and pool depth compared to the smooth bed. However, for a relative critical depth of more than 0.3, the vertical drop length equipped with a horizontal screen with a rough bed increases compared to situations with smooth beds. Also, the use of horizontal screens at the brink of a vertical drop with a rough and smooth bed reduces the total drop length by approximately 50% compared to a Type 1 stilling basin and creates maximum energy loss. Another advantage of applying horizontal screens at the brink of a vertical drop is to avoid the need for a tailwater depth to form a hydraulic jump. A decrease in the screen performance due to blockage of their apertures by debris flow is a disadvantage that can occur with horizontal screens. If the pores can be prevented from being blocked, these screens can be a good alternative for a stilling basin downstream of the vertical drop. It should be noted that turbulence and air entrainment are increased through the use of horizontal screens. Such structures can be used for aeration in fish farming pools, mixing chemicals in water for water treatment and agricultural as well as in wastewater chlorination.

## DATA AVAILABILITY STATEMENT

If needed, all data will be made available to the readers.

## CONFLICT OF INTEREST

The authors declare there is no conflict.

## REFERENCES

**12**(4), 3451–3459. https://doi.org/10.1016/j.asej.2021.03.013.