## Abstract

In this research, the models of Inflatable weirs with internal pressures of 0.5 and 1.4 at three heights of 0.15, 0.2 and 0.25 m were constructed. The deflectors were used in two geometric sections, including rectangular and triangular shapes, and they were installed at angles of 30, 45, and 60 degrees downstream of the weir body. The results showed that the value of the *C _{d}* changes between 0.28 and 0.35 while the range of the ratio of the upstream flow head on the crest to the radius of the downstream side of the inflatable weir varies between 0.2 and 1.5. The flow by crossing over the inflatable weir and colliding with the surface of the downstream stilling basin loses its energy by about 10–85%. The relative horizontal distance of the projectile-jet from the crest varies between 1.6 and 2.8. The effect of the deflector on the

*C*is negligible due to being located in the supercritical part of the weir. Examination of the obtained results declared that the deflector has a considerable effect on energy dissipation.

_{d}## HIGHLIGHTS

The hydraulic properties of the Inflatable weir, including discharge coefficient (Cd), the flow energy dissipation ratio (EDR), and the profile of the outlet jet of the deflector were investigated.

The stage-discharge relation (rating curve) of inflatable weirs was provided.

The effects of defector and its installation angle on the properties of outlet jet and energy dissipation have been investigated.

## NOTATION

*The following symbols are used in this paper:*

*C*_{d}discharge coefficient;

*d*_{crest}flow depth above the weir crest;

*d*_{0}jet thickness;

*d*_{C1}critical depth at the upstream of the weir;

*d*_{C2}critical depth over the weir crest;

- Fr
Froude number;

*g*acceleration due to the gravity;

*H*_{dwn}total head at the downstream of the weir;

*H*_{up}total head at the upstream of the weir;

*h*deflector height;

*h*_{S}flow depth over the weir crest;

*h*_{V}total head over the weir crest;

*L*_{impact}length of the projectile-jet;

*P*weir height;

*P*_{i}the internal pressure of the weir;

*q*discharge per unit width;

*R*radius of curvature of inflatable weir;

- Re
Reynolds numbers;

*u*streamwise velocity component;

*V*flow velocity at the upstream of the weir;

*V*_{0}projected jet velocity;

*V*_{1}flow velocity at the downstream of the weir;

- We
Weber numbers;

*x*longitudinal distance from the origin of the coordinates;

*y*transverse distance from the origin of the coordinates;

*Y*_{0}flow depth at the upstream of the weir;

*y*_{1}downstream ﬂow depth;

*y*_{2}sequence depth;

*α*internal pressure coefficient;

*θ*angular of the deflector's shape;

*θ*_{0}deflected jet angle;

*μ*fluid viscosity;

*ρ*fluid density;

*σ*surface tension; and

*ϕ*_{def}angular position of the deflector on the body of the weir.

## INTRODUCTION

*et al.*2021). Inflatable weirs, also known as rubber dams, are flexible elliptical structures made of rubberized material attached to a concrete foundation and inflated by air, water, or a combination of them (Kumar & ul Islam 2019). In recent years, using rubber dams in developing countries has substantially grown due to the more durable rubber membrane (Saleh & Mondal 2001). Inflatable weirs can be deflated if not needed and inflated again in flood situations with water or air. These are relatively easy to install, do not corrode, and require less maintenance than other weirs. Therefore, it can be said that the construction of this type of weir is more cost-effective (or economical) than small dams (Anwar 1967). Also, these kinds of weirs have a higher discharge than broad and sharp-crested weirs for the given upstream head (Chanson 1997). Due to the ease of construction, implementation, and operation, using inflatable weirs is increasing (Cheraghi-Shirazi

*et al.*2014). Implementation of this type of weir requires the study of their hydraulic behavior based on fluid-structure interaction. In fact, more accurate understanding about the hydraulic behavior of this structure located in the flow path is important to increase the efficiency of this type of structure. According to the relationship between the flow head and the discharge overflow the weir, the discharge relationship in the inflatable weir is defined as (Anwar 1967).where

*q*is the discharge per unit width;

*C*is the discharge coefficient;

_{d}*h*is the total head on the crest as shown in Figure 1; and

_{v}*g*is acceleration due to gravity.

Anwar (1967) performed extensive studies about inflatable weirs and presented a discharge coefficient equation for them. He found that generally the discharge coefficient (*C _{d}*) depends on the ratio and for different kinds of inflatable weirs, it varies as this ratio changed. Al-Shami (1983) experimentally investigated the

*C*of inflatable weirs. He presented two separate equations to calculate the

_{d}*C*of inflatable weirs inflated by air and water. Chanson (1997) mentioned that the nappe adherence at the downstream of the rubber dam may lead to instability of the flow and failure of the dam. Chanson (1998) studied the hydraulics of inflatable weirs in two ways, with and without a deflector. He found that to have an effective deflector with an excellent operation, it is necessary to aerate the lower layer of flow by aerator systems installed at the deflector position. The length of the water projectile (

_{d}*L*

_{impact}) is also calculated by studying the hydraulics of the flow over the inflatable weir, with and without a deflector. The nappe trajectory and wall pressure distribution of circular inflatable dams with deflectors was analyzed and it was found that the adherence pressure is reduced due to the operation of deflectors and they can avoid the dams’ instability. Alhamati

*et al.*(2005a) determined the

*C*of inflatable weirs designed at different internal pressures. Their experimental data were combined with the results of Anwar (1967), finally, they provided equations mentioned in the next sections to calculate the discharge coefficient. They declared that

_{d}*C*increases by increasing . Cheraghi-Shirazi

_{d}*et al.*(2014) showed that the hydraulic flow characteristics over the equilibrium shape of the flexible rubber dam are similar to those of the rigid circular-crested weirs.

In this study, the discharge coefficient, three potential of the weir in energy dissipation, suitable shape and location of deflector and its effects on projectile-jet length are investigated. To this aim, inflatable weirs with minimum and maximum internal pressure, in a fully upright position, are studied. Two different shapes of deflectors, rectangular and triangular, are installed at 3 different angular positions of the weir's body. The study of inflatable weir in fully standing conditions by considering the shape of the real upstream curve of the weir and using the triangular and rectangular shape for deflector are the innovations of the present study. Also, in this research, the results obtained for the discharge coefficient and also the length of the water projectile are compared with the results presented by previous researchers for a circular-crested weir and inflatable weir (Table 1).

Reference . | C – Circular crested weir
. _{d} | Comments . |
---|---|---|

Fawer (1937) | ||

Chanson & Montes (1998) | Partially-developed inflow | |

Fully-developed inflow | ||

Bagheri & Kabiri-Samani (2020) | ||

C – Inflatable weir _{d} | ||

Anwar (1967) | ||

Al-Shami (1983) | Inflated with air | |

Inflated with water | ||

Alhamati et al. (2005a) | ||

L_{impact}/R | ||

Chanson (1998) | Analytical solution |

Reference . | C – Circular crested weir
. _{d} | Comments . |
---|---|---|

Fawer (1937) | ||

Chanson & Montes (1998) | Partially-developed inflow | |

Fully-developed inflow | ||

Bagheri & Kabiri-Samani (2020) | ||

C – Inflatable weir _{d} | ||

Anwar (1967) | ||

Al-Shami (1983) | Inflated with air | |

Inflated with water | ||

Alhamati et al. (2005a) | ||

L_{impact}/R | ||

Chanson (1998) | Analytical solution |

## MATERIALS and METHODS

### Dimensional analysis

*C*and relative

_{d}*L*

_{impact}of these weirs are shown in Figure 1. According to this figure, the main parameters affecting the hydraulic properties of the inflatable weir (

*C*and

_{d}*L*

_{impact}) are given in Equation (2):where

*R*is the radius of the downstream part of the body of the weir ;

*V*is the flow velocity at upstream;

*ρ*is density of the fluid;

*σ*is the water surface tension;

*μ*is the water viscosity;

*ϕ*is the angular position of the deflector;

_{def}*α*is the internal pressure coefficient;

*θ*is the deflector angle; and

*L*

_{impact}is the length of the projectile-jet. Using the Buckingham

*Π*-theorem, the involved dimensionless parameters in the

*C*and the

_{d}*L*

_{impact}of the flow are derived as described in Equation (3). Three parameters:

*V*,

*ρ*, and

*R*are considered as repetitive parameters to perform the dimensional analysis:where is the ratio of upstream head over the crest to its radius, Fr, We, and Re are the Froude, Weber, and Reynolds numbers, respectively. The depth of flow was kept at more than 3 cm during the laboratory test to remove the effect of surface tension. There is a fully turbulent and sub-critical flow upstream of the weir, so the effects of Reynolds and Froude number could be ignored. According to the points mentioned above, Equation (3) can be written as Equation (4):

*Π*-theorem of dimensional analysis:where

*dc*is the critical depth . According to the description in the previous section, finally, Equation (6) is rewritten as follows:

*H*is the upstream total energy and

_{up}*H*is the downstream total energy (Parsaie & Haghiabi 2019). According to the explanations mentioned above and using the

_{dwn}*Π*-theorem, the dimensionless parameters affected on energy dissipation in inflatable weirs are as Equation (10):where

*d*is the critical depth on the crest of the weir.

_{C2}### Experimental model

Experiments were conducted in a flume with glass walls and a metal bed with a length of 11.9 m, depth of 0.5 m, and width of 0.25 m. The inflow is supplied with a pump. A special valve is used to adjust the discharge of inflow. The discharge was measured by a 53° triangular weir with an accuracy measurement of 1%, located at the end of the flume (Figure 2). A point gauge with 1 mm accuracy was used to record the water surface level, and a Micro Moline with the propeller's diameter of 20 mm giving an accurate measurement of 0.01 m/s was used to collect longitudinal velocity data. It should be noted that the data collected for this study were not complicated and the devices used to collect the required data are in the good quality category. Figure 3 shows a model of the inflatable weir tested in the laboratory.

*α*, which is defined as:where

*P*is the internal pressure of the weir and

_{i}*P*is the weir's height. In this study, we considered the inflatable weir by considering the minimum (0.5) and maximum (1.4) amount for

*α*when both are at standing position. Anwar's equation to determine the upstream curve of the inflatable weir's body is as follows (Anwar 1967):where

*x*is the longitudinal distance from the origin of the coordinates and

*y*is the transverse distance from the origin of the coordinates. As mentioned, by using the values of 0.5 and 1.4 for

*α*and considering the three heights of 25, 20, and 15 cm (

*P*= 15, 20, and 25 cm) for the weir, the equation of the upstream curve of the body was determined. Poplar wood was used to make the desired models. The deflectors were also made in two shapes, triangular (

*θ*= 45°) and square (

*θ*= 90°) with a ratio of using Tooka wood and studied at three angles of

*ϕ*= 30, 45, and 60° on the body of the weir. All models were insulated against water penetration.

_{def}One of the effective ways to make flow stable is aeration near the downstream body of the inflatable weir. The air under the water jet will be gradually evacuated and enter the downstream flow. Due to the air evacuation, the water jet is inclined towards the weir body of the weir and the flow sticks to the body over time, consequently, the projectile jet will disappear. The vibration of the weir and flow instability are the most important consequences of eliminating the projectile jet. Therefore, in order to prevent flow instability, some rubber tubes were also placed in the groove on the model and on the flume walls simultaneously for aeration. Figure 4 shows the aeration equipment and the position of the aeration tube on the weir's body.

The geometric characteristics of the models and the planning of the experiments are given in Table 2. The discharge of flow changed between 1 and 20 L/s in all experiments. The flow head on the weir's crest ranges from 3 to 12 cm. The walls of the flume cause some surface waves. Due to the low rate of velocity in the selected test range, the range of effect of the waves is very small and the effect of sidewall friction on the flow is negligible in a narrow flume.

Num . | P
. | α
. | ϕ
. _{def} | θ
. | Num
. | P
. | α
. | ϕ
. _{def} | θ
. |
---|---|---|---|---|---|---|---|---|---|

. | . | . | . | . | . | . | . | . | . |

1 | 0.15 | 0.5 | 30 | 90 | 16 | 0.15 | 0.5 | 30 | 45 |

2 | 0.15 | 0.5 | 45 | 90 | 17 | 0.15 | 0.5 | 45 | 45 |

3 | 0.15 | 0.5 | 60 | 90 | 18 | 0.15 | 0.5 | 60 | 45 |

4 | 0.2 | 0.5 | 30 | 90 | 19 | 0.2 | 0.5 | 30 | 45 |

5 | 0.2 | 0.5 | 45 | 90 | 20 | 0.2 | 0.5 | 45 | 45 |

6 | 0.2 | 0.5 | 60 | 90 | 21 | 0.2 | 0.5 | 60 | 45 |

7 | 0.15 | 1.4 | 30 | 90 | 22 | 0.15 | 1.4 | 30 | 45 |

8 | 0.15 | 1.4 | 45 | 90 | 23 | 0.15 | 1.4 | 45 | 45 |

9 | 0.15 | 1.4 | 60 | 90 | 24 | 0.15 | 1.4 | 60 | 45 |

10 | 0.2 | 1.4 | 30 | 90 | 25 | 0.2 | 1.4 | 30 | 45 |

11 | 0.2 | 1.4 | 45 | 90 | 26 | 0.2 | 1.4 | 45 | 45 |

12 | 0.2 | 1.4 | 60 | 90 | 27 | 0.2 | 1.4 | 60 | 45 |

13 | 0.25 | 1.4 | 30 | 90 | 28 | 0.25 | 1.4 | 30 | 45 |

14 | 0.25 | 1.4 | 45 | 90 | 29 | 0.25 | 1.4 | 45 | 45 |

15 | 0.25 | 1.4 | 60 | 90 | 30 | 0.25 | 1.4 | 60 | 45 |

Num . | P
. | α
. | ϕ
. _{def} | θ
. | Num
. | P
. | α
. | ϕ
. _{def} | θ
. |
---|---|---|---|---|---|---|---|---|---|

. | . | . | . | . | . | . | . | . | . |

1 | 0.15 | 0.5 | 30 | 90 | 16 | 0.15 | 0.5 | 30 | 45 |

2 | 0.15 | 0.5 | 45 | 90 | 17 | 0.15 | 0.5 | 45 | 45 |

3 | 0.15 | 0.5 | 60 | 90 | 18 | 0.15 | 0.5 | 60 | 45 |

4 | 0.2 | 0.5 | 30 | 90 | 19 | 0.2 | 0.5 | 30 | 45 |

5 | 0.2 | 0.5 | 45 | 90 | 20 | 0.2 | 0.5 | 45 | 45 |

6 | 0.2 | 0.5 | 60 | 90 | 21 | 0.2 | 0.5 | 60 | 45 |

7 | 0.15 | 1.4 | 30 | 90 | 22 | 0.15 | 1.4 | 30 | 45 |

8 | 0.15 | 1.4 | 45 | 90 | 23 | 0.15 | 1.4 | 45 | 45 |

9 | 0.15 | 1.4 | 60 | 90 | 24 | 0.15 | 1.4 | 60 | 45 |

10 | 0.2 | 1.4 | 30 | 90 | 25 | 0.2 | 1.4 | 30 | 45 |

11 | 0.2 | 1.4 | 45 | 90 | 26 | 0.2 | 1.4 | 45 | 45 |

12 | 0.2 | 1.4 | 60 | 90 | 27 | 0.2 | 1.4 | 60 | 45 |

13 | 0.25 | 1.4 | 30 | 90 | 28 | 0.25 | 1.4 | 30 | 45 |

14 | 0.25 | 1.4 | 45 | 90 | 29 | 0.25 | 1.4 | 45 | 45 |

15 | 0.25 | 1.4 | 60 | 90 | 30 | 0.25 | 1.4 | 60 | 45 |

## RESULTS AND DISCUSSION

In this section, the results obtained from the experiments are presented and analyzed. Initially, the water surface profile (WSP) over the inflatable weir at the different discharges is shown in Figure 5. The smooth passage of the flow without any disturbance over the body is an essential point to consider. The disturbance and fluctuations of the flow increase the energy loss, which in turn reduces the hydraulic efficiency (discharge capacity) and causes erroneous measurement of flow discharge. According to the experimental results and WSP in Figure 5, in all discharges, the water surface over the inflatable weir is smooth and regular and there is no disturbance or fluctuation. Hence, the flow measurement by this structure can be performed easily.

As it is clear from Equation (1), the flow depth is directly dependent on the flow rate so as it increases, *Q* will be increased. The upstream water surface is uniform and by passing over the weir and leaving the deflector, it is thrown away from the weir body in the form of a jet. For proper operation of inflatable weirs, aeration (under outlet-jet leaving the deflector) is essential. In the experiments, it was observed that if the aeration is not performed well, the outlet jet downstream of the weir disappeared and stuck to the weir body. Figure 6 shows the aeration details.

As the flow passes over the downstream circular side of the weir, momentum is transferred between the fluid and the air trapped between the fluid and the weir body. The momentum transfer accelerates the air, and due to preventing the weir body from aeration, the pressure under the outlet-jet decreases, hence the outlet-jet adheres to the weir body. Adhesion of the outlet jet to the downstream part of the weir negatively affects the hydraulic performance of the inflatable weir.

### Stage-discharge relation

The stage-discharge relation (rating curve) of any flow measuring structure, especially in weirs (or small dams), is an important factor for the selection and design of such structures in water engineering projects.

Having the rating curve and measuring the water level in the dam's reservoir (upstream of the dam), the outlet flow is easily estimated. For this purpose, the rating curve for inflatable weirs as shown in Figure 7 was prepared based on the values measured in the laboratory. The values of versus the dimensionless flow depth upstream of the weir crest are plotted in Figure 7.

*a*and

*m*are numerical constants that should be determined experimentally (Bijankhan & Ferro 2017). Considering this, a power function was fitted on the dataset presented in Figure 7. The mathematical formula as described in Equation (15) is obtained for the stage-discharge relation. Another proposed relationship is presented in Equation (16) with R-squared correlation (R

^{2}) and mean relative error percentage (Equation (17)):where

*f*is the observed value,

*O*is the calculated value by equation, and

*N*is the total number of data. As it is clear from the relations and their accuracy, the stage-discharge relationship in the form of power relationship (Equation (15)) and linear relationship (Equation (16)) have good accuracy.

If aeration is not performed and consequently nappe is eliminated, the flow head will increase at a constant flow rate at the upstream. Also, the slope of the stage-discharge curve will decrease.

Figure 8 shows a comparison of the stage-discharge relation in the inflatable, circular-crested weir (Ramamurthy & Vo 1993), and broad-crested weir (Hager & Schwalt 1994). According to this diagram, for a given value of , the amount of in a circular-crested weir is more than the other two cases and the lowest amount of occurred in a broad-crested weir.

### Discharge coefficient

One of the important goals of this research is the development of a mathematical equation for the discharge coefficient *C _{d}* of the inflatable weir. The most important and effective parameter in determining the

*C*is the relative upstream head . In Figure 9 the values of the

_{d}*C*versus are plotted for all experiments in this study.

_{d}*C*changes between 0.28 and 0.35 for the inflatable weir and 0.37 and 0.48 for the circular crested weir while the range of varies between 0.2 and 1.5. The overlap of the points in Figure 9 shows that the three parameters of the take-off of outlet-jet of the deflector (

_{d}*θ*), the angular position of the deflector (

*ϕ*) and the internal pressure coefficient (

_{def}*α*) do not have significant effects on the

*C*. When the flow passes over the crest and is crossing the critical point, it enters the supercritical zone where the deflector is located. In the supercritical flow, the control point is at the upstream. In this flow condition, the flow characteristics are not transmitted upstream and all the effects received are due to the information sent from the upstream to the downstream. The location of the deflector in the supercritical flow zone is the reason that the two geometric parameters of the deflector (i.e. and

_{d}*θ*) can not affect the

*C*. Internal pressure

_{d}*α*of the inflatable weir also does not affect the

*C*. This point is also presented in the results of Alhamati

_{d}*et al.*(2005b). The greatest effect of the internal pressure ratio (

*α*) is on the formation of the longitudinal cross-section of the inflatable weir. The upstream flow of the inflatable weir is always subcritical and as a result, the values of the Fr number are very small and its effect can be ignored. In this study, to provide a suitable mathematical formula for the

*C*, different functions were fitted to the points of Figure 9, and finally, Equation (18) was founded with the lowest error:

_{d}As is shown in Figure 10, the relation of the flow head and the depth is linear. So, to estimate the *C _{d}* (Equation (18)), if the flow depth is used instead of the flow head, the value of error can be dissembled.

Figure 11 is the longitudinal velocity profiles taken at crests of the inflatable weirs at maximum discharge (Q = 20 L/s). As can be seen, the velocity profiles on the inflatable weir crests from the highest value at minimum depth to the lowest value at the water surface are in the form of a nonlinear curve. The study of longitudinal velocity is also showing that the *θ* and *ϕ _{def}* do not have any significant effect on the velocity profile. The velocity is highest near the crest of the weir and by decreasing this distance from the crest and moving towards the water surface, the velocity decreases. The intensity of deceasing is decreased because of the presence of air friction.

### Energy dissipation

The structural design of the downstream stilling basin requires the determination of the amount of pressure and hydrodynamic force that the falling jet exerts on its concrete slab. Therefore, determining the amount of energy dissipation is one of the most important issues in the study of dams and weirs. When the flow-jet leaves the deflector, a small amount of its energy is lost due to air resistance (flow-air interaction) and a significant amount of its energy is lost due to collision with the surface of the downstream stilling basin. Examination of laboratory data was performed to determine energy dissipation, the results of which are shown in Figure 12. In this figure, the range of changes in energy dissipation, and a 20 confidence interval is specified.

*Q*, a layer of water is created on the floor of the stilling basin (water blanket) which reduces the intensity of the collision of falling jet to the surface of the concrete slab, thus EDR reduces. As

*Q*increases, the thickness of this layer increases, thus the turbulence caused by the collision falling jet with the floor reduces more. Therefore, less amount of energy is dissipated. On the other hand, because the increasing of

*Q*and the increasing of kinetic energy of flow-jet are not in the same ratio, hence, the velocity and kinetic energy of the outlet jet decreases, as a result, the spreading of the falling-jet is decreased, therefore, the thickness of the falling-jet and water blanket increases. Figure 12 shows that this structure can dissipate the flow energy between 10 and 85% in the relative upstream head ranges from 0.1 to 0.7. According to the flow pattern around the deflectors, a vortex is formed in front of the rectangular deflector that causes energy dissipation. Due to the lack of formation of this vortex in the triangular deflector, less EDR occurs and as a result, the horizontal length of the projectile-jet increases. As the length of the projectile-jet increases, the spreading of the flow jet increases, which in turn reduces the thickness of the layer of water blanket and its effect on reducing the turbulence of the collision is reduced, therefore, the EDR increases. The results show that increasing the (increasing the flow discharge at a constant weir's height) reduces the EDR (%). The weir's height, the geometric shape of the deflector, and its location affect the amount of EDR. By increasing the weir's height at different angles of deflectors, the amount of EDR increases. Also, in each model, the amount of EDR is directly related to the deflector angle. The maximum amount of EDR was obtained at 60

^{o}and the lowest value of EDR was obtained at 30

^{o}. Results show that in the presence of a rectangular deflector, the flow energy has a greater energy reduction than a triangular deflector. Finally, a mathematical formula for EDR is developed as Equation (19):where

*θ*and

*ϕ*is in degrees.

_{def}The flow condition at section 1 (in which the depth is shown with *y _{1}*) is sub-critical. Then the flow arrived at section 2 (which the depth is shown with

*y*) by creating a hydraulic jump. The amount of energy in section 2 is significantly reduced compared to section 1 due to the hydraulic jump and hydraulic jump conditions. Figure 13 shows the energy dissipation data of an inflatable weir and circular weir (Chanson & Montes 1998) and drop structure (Rand 1955).

_{2}According to Figure 13, the amount of energy dissipation in an inflatable weir is more than in a circular-crested weir and the drop structure. The presence of a deflector in the stream flow path causes the deflector to act as an obstacle and the flow energy will be decreased due to their collision. On the other hand, the presence of a deflector prevents sticking of the flow to the downstream part of the body of the weir, so the downstream part of the weir acts as a drop. Consequently, energy dissipation in inflatable weirs with deflector is significant compared to the circular-crested wires and drop structures.

### Horizontal length of the projectile-jet

One of the most important issues in inflatable weirs is the horizontal distance of the falling jet to the crest of the weir. The upstream head and geometry properties of an inflatable weir such as type of deflector and its angle of installation, angle of flow-jet take-off are involved in the horizontal length of the projectile-jet. The observed data of relative distance of projectile-jet versus the relative upstream head in the presence of geometric properties of inflatable weir models, including *α*, *ϕ _{def}*, and

*θ*are shown in Figure 14.

*α*did not have much effect on the horizontal length of the projectile-jet and by fitting an equation to the data, the following equation was obtained:where

*θ*and

*ϕ*are in degrees.

_{def}As can be seen from Figure 14, in the range of between 0.2 and 0.6, by increasing the flow head, the length of the projectile-jet is greatly increased. For values of more than 0.6, by an increasing head, the ascent rate of the length of the projectile-jet decreases sharply. Due to the supply of flow energy from upstream, with increasing the flow head, the amount of potential energy increases gradually and dominates over the kinetic energy. As a result, the rate of increasing the projectile-jet length will no longer be proportional to the increase of the flow head.

Figure 15 shows the trajectory jet curve obtained from the experimental data of this study (Equation (20)) and the theoretical relationship proposed by Chanson (1998). The increasing trend of projectile length in the inflatable weir follows the theoretical solution and the two graphs are matched by using the correction factor of 0.78.

### Comparison and analysis of former equations by obtained results

*f*is the observed value,

*O*is the calculated value by equations,

*e*is the difference between the two observed and calculated values, and

*N*is the total number of data.

Equation . | ME . | RMSE . | MSE . | MAE . | |
---|---|---|---|---|---|

Q | Anwar (1967) | −0.0023 | 0.0029 | 8.59 × 10^{−6} | 0.0023 |

Al-Shami (1983) – Air filled | 0.0022 | 0.0028 | 7.91 × 10^{−6} | 0.0022 | |

Al-Shami (1983) – Water filled | 0.0016 | 0.0019 | 3.73 × 10^{−6} | 0.0016 | |

Alhamati et al. (2005b) | 0.0016 | 0.0022 | 4.99 × 10^{−6} | 0.0016 | |

L_{impact}/R | Chanson (1998) | 1.414 | 1.441 | 2.077 | 1.414 |

Equation . | ME . | RMSE . | MSE . | MAE . | |
---|---|---|---|---|---|

Q | Anwar (1967) | −0.0023 | 0.0029 | 8.59 × 10^{−6} | 0.0023 |

Al-Shami (1983) – Air filled | 0.0022 | 0.0028 | 7.91 × 10^{−6} | 0.0022 | |

Al-Shami (1983) – Water filled | 0.0016 | 0.0019 | 3.73 × 10^{−6} | 0.0016 | |

Alhamati et al. (2005b) | 0.0016 | 0.0022 | 4.99 × 10^{−6} | 0.0016 | |

L_{impact}/R | Chanson (1998) | 1.414 | 1.441 | 2.077 | 1.414 |

*et al.*(2019) for a cylindrical weir and by Hager & Schwalt (1994) for broad crested weir to compare with amounts for inflatable weirs. It should be noted that Equation (1) is used for broad crested weir and inflatable weir and Equation (25) is used for the circular crested weir to calculate the flow rate. The value is multiplied by the discharge coefficients of the circular crested weir for assimilation (Bos 1989):

*R* is half of the total height of the inflatable weir, half of the total height of the broad-crested weir is defined as *R*.

According to the experimental results and comparison with previous researches in Figure 16, it can be said that the discharge coefficient in the short-crested weirs and the circular-crested weir is higher than the broad-crested weir. The discharge coefficient in the inflatable weir is closed to the simple broad-crested weir at the low amount of . The efficiency of inflatable weirs is increased more by increasing the amount of in comparison to the simple broad-crested weir. The reason for the higher amount of discharge coefficient in circular-crested weir in comparison to the inflatable weir for the given range of is the circular curve at the upstream of their body.

## CONCLUSION

In this study the hydraulic properties of inflatable weirs included the profile of the water surface and jet profile leaving the deflector, discharge coefficient, and flow energy dissipation were investigated. The results are summarized as follow:

The surface profile is smooth from upstream to downstream and no oscillating waves were seen.

The discharge coefficient is independent of the angular position of the deflector on the body of the weir (

*ϕdef*), the internal pressure coefficient (*α*), and the angle of the deflector's shape (*θ*), it means that these parameters do not have significant effects on the*C*; the most effective parameter on_{d}*C*is the ratio of ._{d}The shape of the deflector impacts on the energy dissipation. In fact, rectangular deflector models are more effective than triangular models; the inflatable weir can dissipate the energy of flow between 15 and 85% for the range of between 0.7 and 0.1.

There is a nappe re-attachment when the installation angle of the deflector is 30°. No nappe re-attachment was observed at an installation angle of 45 and 60°. The results also show that the length of the projectile-jet in the installation angle of 45° is longer than the length of the projectile-jet in the other two installation angles. The calculation of the horizontal length of the projectile-jet showed that at low flow discharge, increasing the flow head greatly increases the length of the projectile-jet, and with further increase of the flow head, the projectile length is less affected by the head.

## ACKNOWLEDGEMENTS

The authors thank Dr. AmirHamzeh Haghiabi (Lorestan University) for sharing his experiments about the laboratory modeling and studies of dams.

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.

28