## Abstract

The aim of the current research is to investigate the effect of geometry and different angles (180°, 150°, 120° and 90°) of the edge of a vertical drop with a height of 0.15 m with and without a vertical screen on energy dissipation. The finding showed that the triangulation plane forms a vertical drop at different angles of 180°, 150°, 120° and 90° without a screen that increases the energy dissipation of the drop, which is highest at an angle of 90°, by 50.5%. In the vertical drop with a screen, reducing the angle also leads to a decrease in energy dissipation, so the highest energy dissipation occurs at an angle of 180° with a 45.2% increase in energy dissipation compared to the vertical drop without a screen. The relative depth of the edge and the relative depth of the vertical drop pool of the triangular plane are different at the angle vertex and its wings, in such a way that the relative depth of the edge at the vertex is less than the wings and the relative depth of the pool at the vertex is greater than the wings.

## HIGHLIGHTS

This study investigated the hydraulic parameters of a triangular plan forming a vertical drop.

The energy dissipation is different at different angles in the triangular plan forming a vertical drop.

The length of the edge of the triangular plan forms a vertical drop that affects the energy dissipation.

The simultaneous use of the screen affects the hydraulic parameters of the triangular plan that forms a vertical drop.

## INTRODUCTION

It is very important to depreciate the kinetic energy of water when the slope in its bed is high, in order to prevent possible damages to the downstream. The vertical drop is one of the structures that dissipate the destructive energy of water and prevents damage. Also, researchers have used additional structures such as screens to increase the energy dissipation (ED) in the vertical drop. Rajaratnam & Hurtig (2000) were the first to use vertical screens to improve ED, they found that the screen with a porosity of 40% had higher ED. These structures are placed perpendicular to the flow and create a hydraulic jump in the flow regime and cause an increase in ED (Abbaszadeh *et al.* 2024). Rouse (1936) was one of the first to study the drop, using the results obtained from the edge depth measurements. He presented an equation to estimate the discharge, which was later used by Blaisdell (1980) and was corrected by Rajaratnam & Chamani (1995). Rajaratnam & Chamani (1995), Akram Gill (1979) and Chamani *et al.* (2008) by conducting studies on ED in the vertical drop found that the impact of the falling jet with the bottom of the channel and creating turbulence in the pool are the main factors of ED. Hong *et al.* (2010) studied the impingement force of the flow jet and the drop length in the presence of a positive slope at the downstream bed of the drop. The result of their research is an equation to estimate the jet force and the length of the drop. Liu *et al.* (2014) investigated the effect of positive slope upstream of the vertical drop on the hydraulic parameters of the flow, according to the results obtained from their research, the edge depth, pool depth, and falling jet angle decrease due to the presence of the upstream slope. Parsaie & Haghiabi (2019) using the M5 algorithm and Multilayer Perceptron Neural Network (MLPNN), investigated the ability of staircase overflows in terms of ED. Their results showed that, based on the M5 algorithm, Drop and Froude numbers play an important role in ED modeling and approximation. Also, the evaluation of the results of both applied methods showed that the accuracy of MLPNN is slightly lower than the M5 algorithm.

Daneshfaraz *et al.* (2020) studied the effect on the hydraulic parameters of the flow by placing double horizontal screens on the edge of the vertical drop. Their results showed that the distance between the two screens did not affect the relative depth and downstream residual energy and also the increase in the relative length of the drop resulted in the decrease of the relative depth of the downstream. Parsaie & Haghiabi (2021) experimentally investigated the hydraulic properties of the finite crested stepped spillway (FCSS) including discharge coefficient (*C _{d}*) and energy dissipation ratio (EDR). The results showed that

*C*in FCSS varies between 0.9 and 1.2, while the ratio of upstream depth to crown length (

_{d}*h*

_{up}/

*L*

_{c}) varies between 0.25 and 1.8. FCSS performance in relation to EDR varies between 40 and 95%. With the increase in the flow rate and the formation of skimming flow, the performance of FCSS related to ED decreases dramatically.

Daneshfaraz *et al.* (2021) investigated the effect of a vertical drop at three different heights and a horizontal screen with two porosity ratios on the hydraulic parameters of the flow. According to the results of their experiments, the relative depth of the pool, the relative depth of the downstream and the ED increase in the drop equipped with a horizontal screen. They also found that the use of a horizontal screen in a vertical drop reduces the downstream Froude number and increases the percentage of porosity of the horizontal screen, causing a decrease in the relative mixing length and the wetted relative length of the horizontal screen. Norouzi *et al.* (2021) investigated the performance of the ability of artificial intelligence methods including ANN, ANFIS, GRNN, SVM, GP, LR, and MLR to predict the relative ED for vertical drops equipped with a horizontal screen. The results showed that the performance of ANFIS_gbellmf is more acceptable than all the models. Daneshfaraz *et al.* (2022a) experimentally researched the hydraulic parameters of vertical gabion drops with different heights and lengths. The results of their research showed that at a constant relative length, increasing the relative height of the gabion drop causes an increase in the flow through the porous medium and results in an increase in ED. Daneshfaraz *et al.* (2022b) studied the effect of the screen downstream of the vertical drop at different distances and porosity percentages and investigated its effect on ED. According to their findings, the downstream relative depth, the relative depth of the pool, and the relative ED increase in the drop equipped with a vertical screen compared to the drop without a vertical screen. Haghiabi *et al.* (2022) studied triangular and trapezoidal overflows with one and two cycles. Their results showed that labyrinth overflows can reduce the flow energy by between 70 and 85%.

Shamsi *et al.* (2022) have investigated the discharge coefficient (*C*_{d}) of cylindrical weirs, its flowability and ED in it. The results showed that in the range of relative height (ratio of flow depth on the crown to weir diameter: *H*/*D*) between 0.15 and 2.0, *C*_{d} is variable between 1.0 and 1.4. At the same *H*/*D* range, such structures can dissipate flow energy between 15 and 80%. The optimum value (*C*_{d}) for economic design purposes is about 1.3, which occurs in the *H*/*D* range between 0.5 and 0.7.

Parsaie *et al.* (2023) used the group data processing method (GMDH) to estimate the energy loss of flow passing over labyrinth weirs with triangular and trapezoidal planes. Examining the GMDH network structure shows that ho/P, Ncy and Mr play more significant roles in the development network.

Afaridegan *et al.* (2023) evaluated and modified the performance of semi-cylindrical weirs (SCWs) considering a downstream ramp. Modified semi-cylindrical weirs (MSCWs) in 12 variations, with four different slopes (*θ*) and three radii (*R*) were fabricated and tested through laboratory experiments. The findings show that the discharge coefficient (*C*_{d}) of MSCWs is dependent on *R* (negative correlation) and independent of *θ*. When the ratio of the depth of upstream flow (*y*_{up}) to *R* changes from 0.306 to 1.36, the value of *C*_{d} changes in the range of 1.1–1.45. Investigating the bed pressure distribution on MSCW models shows that decreasing *θ* effectively controls the negative pressure on the crown surface.

Yonesi *et al.* (2023) investigated the vertical drop equipped with a horizontal screen embedded in the edge in two states of rough and smooth bed, according to their results, at a critical depth of more than 0.3, the length of the drop in the drop equipped with a horizontal screen with a rough bed is more than the drop with a smooth bed.

Reviewing the literature has shown that the vertical drop consumes water energy and the use of screen significantly increases the ED. So far, there has been no research that can be used to increase the ED by changing the geometry of the vertical drop and also using the screen. This research is focused for the first time with the aim of increasing the ED in the vertical drop by creating the edge of the triangular plan.

## METHODS

### Experimental setup

Table 1 shows the range of parameters measured in the laboratory for four drops, with an angle of 150°, 180°, 120° and 90°, without screen and with screen.

Measured variables . | Vertical plane drop . | θ = 150̊. | θ = 120̊. | θ = 90̊. | ||||
---|---|---|---|---|---|---|---|---|

Without screen . | With screen . | Without screen . | With screen . | Without screen . | With screen . | Without screen . | With screen . | |

Q (L/min) | 150–600 | 150–600 | 150–750 | 150–600 | 150–750 | 150–600 | 150–750 | 150–600 |

y_{0} (m) | 0.0213–0.0545 | 0.0213–0.0545 | 0.022–0.055 | 0.022–0.0515 | 0.0221–0.0532 | 0.0221–0.0526 | 0.0235–0.058 | 0.0235–0.0503 |

y_{1} (m) | 0.0055–0.0202 | 0.0404–0.0906 | 0.0048–0.0295 | 0.0406–0.0886 | 0.0039–0.0385 | 0.0414–0.09 | 0.021–0.0452 | 0.0414–0.0903 |

y_{2} (m) | 0.019–0.0768 | 0.028–0.0603 | 0.0192–0.0822 | 0.0285–0.0555 | 0.0233–0.0798 | 0.0284–0.0582 | 0.0252–0.0615 | 0.0281–0.0588 |

Measured variables . | Vertical plane drop . | θ = 150̊. | θ = 120̊. | θ = 90̊. | ||||
---|---|---|---|---|---|---|---|---|

Without screen . | With screen . | Without screen . | With screen . | Without screen . | With screen . | Without screen . | With screen . | |

Q (L/min) | 150–600 | 150–600 | 150–750 | 150–600 | 150–750 | 150–600 | 150–750 | 150–600 |

y_{0} (m) | 0.0213–0.0545 | 0.0213–0.0545 | 0.022–0.055 | 0.022–0.0515 | 0.0221–0.0532 | 0.0221–0.0526 | 0.0235–0.058 | 0.0235–0.0503 |

y_{1} (m) | 0.0055–0.0202 | 0.0404–0.0906 | 0.0048–0.0295 | 0.0406–0.0886 | 0.0039–0.0385 | 0.0414–0.09 | 0.021–0.0452 | 0.0414–0.0903 |

y_{2} (m) | 0.019–0.0768 | 0.028–0.0603 | 0.0192–0.0822 | 0.0285–0.0555 | 0.0233–0.0798 | 0.0284–0.0582 | 0.0252–0.0615 | 0.0281–0.0588 |

### Calculation of ED

*E*,

_{0}*E*and

_{1}*E*, respectively, represent the upstream total energy, the specific energy downstream of the drop (before the hydraulic jump and the screen) and the specific energy after the hydraulic jump and the screen, and

_{2}*y*,

_{0}*y*and

_{1}*y*is the depth in the upstream, the depth in the downstream (before the hydraulic jump and screen) and the depth after the hydraulic jump and screen, also

_{2}*q*shows the discharge per unit width. The equation related to the total ED, drop ED is also according to Equations (4) and (5).

*ΔE _{Total}* and

*ΔE*show the total ED and ED of the vertical drop.

_{Drop}### Dimensional analysis

*ρ*is the density of the fluid,

*g*is the gravitational acceleration of the earth,

*μ*is the dynamic viscosity of the fluid,

*Q*is the inlet flow rate,

*h*is the height of the drop,

*θ*is the angle of the drop,

*y*is the depth upstream of the breakwater,

_{0}*y*is the critical depth,

_{c}*y*is the depth at the edge of the drop,

_{b}*y*is the depth of the drop pool,

_{p}*y*is the depth downstream and the primary depth of the jump,

_{1}*y*is the depth downstream and the secondary depth of the jump,

_{2}*E*is the total energy upstream of the drop,

_{0}*E*is the specific energy downstream and before the hydraulic jump,

_{1}*E*is the specific energy downstream the drop and after of the hydraulic jump and

_{2}*ΔE*represents the total ED.

_{Total}*ρ*,

*g*and

*h*as repeated variables, the independent dimensionless parameters are obtained according to Equation (7).

In relation (7), *Re*_{0} is the Reynolds number upstream of the drop, *Fr*_{0} is the landing number upstream of the drop, is the relative initial depth, is the relative critical depth, is the relative depth of the edge of the drop, is the relative depth of the pool, is the relative primary depth of the hydraulic jump, is the relative depth of the secondary hydraulic jump, is the energy relative upstream, is the relative energy downstream and before the hydraulic jump, is the relative energy downstream and after the hydraulic jump, is the relative ED and *θ* represents the angle of the drop edge.

_{0}< 115,296), the flow is turbulent, and as a result, the effect of viscosity is neglected (Daneshfaraz

*et al.*2021). The upstream Froude number is also in the range of (0.76 < Fr < 0.95), which can be ignored due to the non-change of the flow regime and being in the range of insignificant changes (Bagherzadeh

*et al.*2022), as a result of the dimensionless parameter dependent on the relative ED, based on independent dimensionless parameters, they are obtained according to Equation (8).

*p*is the percentage of porosity of the screen,

*t*is the thickness of the screen,

*d*is the distance of the screen from the drop,

*y*is the depth downstream and the depth before the screen,

_{1}*y*is the depth downstream and the depth after the screen,

_{2}*E*is the specific energy of the water before the screen,

_{1}*E*shows the specific energy of water after the screen and

_{2}*ΔE*shows the total ED. Due to the constancy of the thickness, percentage of porosity and the distance of the screen from the drop, their influence can be ignored, according to this Equation (10) is obtained from the theory of π–Buckingham.

_{Total}_{0}< 115,296) shows the turbulent flow, the effect of viscosity is ignored. Also, because in this experiment, a screen with a type of porosity percentage was used, the effect of

*p*can be ignored. The thickness of the screen also has no effect in the experiments because it is constant (Çakir 2003; Balkış 2004). After simplifying the dimensionless parameters for the drop equipped with a screen according to Equation (11):

### Evaluation criteria

## RESULTS AND DISCUSSION

### Energy dissipation

*et al.*(2008), which fluctuates around linear regression.

### The relative depth of the drop edge

### Relative depth of the pool

Figure 7(c) is a comparison of the relative depth of the pool between the present research and the research of researchers such as Rajaratnam & Chamani (1995), Chamani *et al.* (2008) and Moore (1943), according to Figure 7(c), the present study has the process of the previous researchers.

### Relative depth of the downstream

## CONCLUSIONS

Increasing the ED in the areas with greater slope has been considered, for this purpose, the change in the geometry of the drop combined with the hydraulic jump, as well as the use of the screen in the downstream, has been to achieve this goal. Changing the geometry of the edge of the drop and its triangular plan at angles of 150°, 120° and 90° has increased the ED of the drop compared to the simple drop, in such a way that ED has increased with the decrease of the angle. The highest ED occurred in the drop with a 90-degree angle with an increase of 50.5% in the relative critical depth of 0.093. By examining the total ED, it can be seen that in both cases of drop with screen and without screen, the total ED decreases with the decrease of the angle. The reason for this is the increase in the downstream depth due to the triangular plan of the drop edge, because the decrease in the angle leads to an increase in the length of the edge, as a result, an increase in the volume of passing water. Finally, the triangular plan forms a vertical drop with a vertex angle of 90°, with the highest ED of the drop, which is the most favorable mode of the vertical drop for ED. According to the present research, the relative depth of the edge of the drop is different at the vertex of the angle and its wings, so that it is less at the vertex of the angle than at the wings and also less than the depth of the edge in the simple drop. Also, the relative depth of the edge of the drop at the vertex of the angle has decreased with the decrease of the angle of the drop, and it has also increased slightly at the wings compared to the simple drop. The relative depth of the pool is also different due to the triangular plan of the drop at the vertex of the angle and its wings, and on the contrary, the relative depth at the edge is more at the vertex than at the wings. The relative depth of the pool at the vertex of the angle increases with the decrease of the angle, and at a 90-degree angle, it is 11.47% more than the simple drop, while at the wings, the decrease of the angle does not have much effect on the relative depth of the pool. The relative depth downstream of the vertical drop without a screen at an angle of 120° has increased by 3.7% compared to the simple drop, and in the vertical drop with a screen, the relative depth downstream has increased with the decrease of the angle, and the maximum relative depth of the downstream at the angle of 90° and happens with an increase of 5.1% compared to the simple vertical drop. In future research, it is possible to study different types of drop edge geometry (trapezoidal, semicircular and congressional), and simultaneously with the change of the drop edge geometry, the effect of the presence of a slope in the bottom of the channel on the hydraulic parameters can be investigated. Investigating the effect of the supercritical flow upstream of the drop on the hydraulic parameters of the flow in the triangular plan form vertical drop is one of the other studies that can be done.

## 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.

## REFERENCES

*Experimental Investigation of Energy Dissipation Through Inclined Screens*

*Experimental Investigation of Energy Dissipation Through Screens*