Experimental and numerical investigation of the trajectories of free and pressurized jets through storage dams

The following symbols are used in this text

x and y

Coordinates of the lower edge of the free jet

v₀

initial jet velocity

θ₀

the initial angle of the jet from the horizon

g

gravitational acceleration

t₀

thickness of watery jet

Q

discharge

C

discharge coefficient

L

weir crest length

q

discharge per unit crest length

D

outlet width of free jet

d

diameter of the gate

P

the height of the fall from the ground

H_overtop

head passing through the structure (Overtopping head)

y_c

critical depth on the spillway

y_b

depth at the end of the spillway (spill site)

V_0p

initial velocity of the pressurized jet

V_0f

initial velocity of the free jet

The outlet water flow from a dam has a lot of energy and if this energy is not reduced, the flowing water can cause irreparable damage to the dam and downstream structures. One way to waste extra energy is by creating jets (pressurized jets from flood discharge systems and free-falling jets from dam weirs) that descend into the plunge pool.

Falling jets are important in hydraulic structures; water jets often cause erosion and scouring downstream of the structure, affecting the abutments and the downstream canal (Tu & Wood 1996).

In Figure 1, x and y are the coordinates of the lower edge of the jet, V₀ is the initial velocity of the jet, H is the height of the dam reservoir water from the center of the gate, H_overtop is the water head over the spillway (overtopping head), D is the width of the outlet of the free jet, d is the diameter of the gate, and X_max is the maximum range of the pressurized jet.

In the above equations, θ is the initial angle of the jet from the horizon, V₀ is the initial velocity of the jet, V₀ (x) is the initial velocity along the x-axis, and V₀ (y) is the initial velocity along the y-axis.

In Equation (6), h_v is the velocity head.

This equation is proposed to calculate the trajectory of pressurized jets that outlets from flood discharge systems (USBR 1960, 1976, 1977).

In Equation (10), h₀ is the initial velocity head of the projectile.

In Equation (11), x and y are the coordinates of the lower edge of the jet, θ is the initial angle of the jet from the horizon, d_c is the depth of flow, h_v is the velocity head, and K is the coefficient in the jet trajectory equation.

Ahadiyan & Musavi-Jahromi (2009) investigated the effect of jet hydraulic properties on the shape of the jet trajectory at the top of its arc. The findings of these researchers show that the length of the downward curve depends on the diameter of the nozzle. Zhang & Zhu (2014) injected air into the jet through a nozzle to study the downward curve of the air–water bubbly jets in cross-flow. They then analyzed the center line of the downward curve for both phases of the air and water in the bubbly jet. After the fluid separates from the discharger, the jet flow creates a downward curve after falling through a horizontal and straight trajectory. Heller et al. (2005), performed a laboratory study of ski jump projectiles and they checked the maximum pressure and its location. The results showed that the Froude number, the relative curvature of the ski, and the angle of the ski had a significant effect on the maximum pressure and its location. Lauria & Alfonsi (2020a) numerically investigated the flow pressure behavior downstream of ski jumps. The result shows that particular attention should be paid to the maximum pressure head at the tail water. The results of this study provide useful suggestions for the design of ski jump spillways in dam construction. Lauria & Alfonsi (2020b) conducted a numerical investigation of ski jump hydraulics to form a detailed and systematic understanding of the phenomena. The results of the simulations showed that the pressure head characteristics on the bucket centerline are adequately reproduced and that the upper and lower jet trajectory geometries follow the standard parabolic profile. Also, their other results can be useful in the design of dams.

Karami Golbaghi et al. (2017) designed a series of different experiments to investigate the erosion of plunge pools by changing the conditions of vertical water jets. They found that the change in scouring rate, the critical shear stress of the bed, and the erodibility coefficient depended on the water jet conditions. He et al. (2017) investigated the scouring of sandy beds under the influence of vertical water jets. They found that with an increasing jet flow rate, the depth and volume of the scour hole gradually increased. When the flow reached a critical value, the scour hole behavior was affected by a change in the outflow flow characteristics of the water jet. Despite these prior studies, there have been no studies using physical models on the trajectory of outlet jets that pass through the gates. However, the discharge coefficients for different openings of the bottom outlet gates of Narmasheer (Hosseini & Saneie 2010) and Molasadra (Karimi Chahartighi & Nazari 2018) Dams have been investigated.

Fluent software is one of the most powerful applied computational fluid dynamics (CFD) programs; it has been used to model fluid flow and heat transfer in complex geometries. This software is based on balances of momentum and mass over control volumes. These control volumes are often referred to as elements or cells (Ansys Fluent 2015).

Bahrebar et al. (2021) numerically and experimentally studied the combination of a labyrinth weir/orifice and its effect on the discharge coefficient. Salmasi et al. (2021a) investigated the stage-discharge relationship and discharge coefficient in sharp-crested weirs with triangular shapes. Ghaffari & Eghbalzadeh (2017) investigated the effects of rectangular side orifice crest height on the flow around an orifice. Despite this past research, it appears that no studies have been performed on the trajectory of outlet jets from gates and weirs (pressurized and free-falling jets).

In the present study, the characteristics of pressurized and free-falling jets are investigated using physical models and CFD software. The effect of air resistance on the pressurized and free-falling jet trajectory is investigated with the goal of improving the accuracy of jet trajectory predictions. To calculate the jet more accurately, air resistance is incorporated into the projectile equations. In addition, the effect of the outlet gate diameter and the height of the dam reservoir water from the center of the gate on the pressurized jet was investigated. Also the effect of the width of the outlet cross-section of the free jet and the head passing through the dam to the jet was examined.

In Figure 3, x and y are the horizontal and vertical coordinates from the lower edge of the pressurized jet, H is the height of the dam reservoir water from the center of the gate, H_overtop is the water head over the spillway (overtopping head), d is the diameter of the gate, and X_max is the maximum range of the pressurized jet.

Table 1 shows the range of parameters that affect the launch jet trajectory and Table 2 shows the structure specifications and performed tests in the present study.

In Table 1, H is the height of the dam reservoir water above the center of the gate, V_0p is the initial velocity of the pressurized jet, V_0f is the initial velocity of the free jet, and D is the width of the outlet of the free jet. H_overtop is the water head over the spillway (overtopping head) and d is the diameter of the gate.

A camera was used to capture the images of the pressurized and free-falling jet's route. After establishing the flow through the dam gate and weir (circular orifice and a rectangular broad-crested weir installed on the tank) photographs were taken of the trajectories of the pressurized and free-falling jets separately. The x and y coordinates of the pressurized and free-falling jets in these photos were extracted using Plot-Digitizer software (Rohatgi 2015). Four different locations were identified and quantified and entered into the Plot-Digitizer software and the coordinates were extracted. Thus, the coordinates of the trajectories of the pressurized jets and free-falling jets are obtained. It should be noted that the discharge of the jet was measured from the volumetric flow method.

It is necessary to explain that in the present study, the initial angle of the jet from the horizon was equal to zero. The broad-crested weir and gate are installed horizontally in the laboratory. In truth, only the bottom streamline of the jet actually parallels the crest; the mid-depth streamline is deflected about 7° downward at the brink (Henderson 1966), further shortening the real trajectory, but only by a small amount. Also, according to Wahl et al.'s (2008) results, the initial deflection angle of the jet up to 7° has very little effect on the trajectory of the falling jet. Therefore, our assumption that the initial angle from the horizon is zero has very little effect on the results and can be ignored.

In the above equations, U_i and U_j are the components of the velocity vector in the spatial directions i and j, (i and j equal to 1, 2, and 3 , respectively, for the x, y, and z directions), P is the pressure, ρ is the fluid density, and δ_{i, j} is the Kronecker delta (if i = j, its value is one and otherwise its value is zero).

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. 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).

In a numerical simulation, the computational mesh can affect the model results. In this section, the effect of the computational mesh on the modeling results is investigated. Gambit was used to create the mesh.

It is necessary to explain that Figure 4 is for a pressurized jet; the results of free-falling jets are also similar to Figure 4. The number of elements for a free-falling jet is 15,860. It should be noted that in this condition, the average size of the elements is approximately 0.0021 m².

Another important issue in a numerical simulation is the proper definition of boundary conditions (BCs). The BCs at the inlet are an applied pressure value (pressure inlet). At the outlet flow location, zero pressure is applied (pressure outlet). For the downstream side of the structure, an applied pressure was set (pressure inlet). In these BCs, the hydrostatics pressure was taken into account on the walls.

In Equation (21), x and y are the coordinates of the lower edge of the jet, D is the width of the outlet of the free jet, P is the height of the drop from the ground, a, b, c, d, e, f and h are fitted coefficients and H_overtop is the head passing through the structure.

In Equations (22) and (23), P_i and O_i are the measured and calculated values, respectively, and are the average measured and calculated values, and n is the number of data.

After establishing the flow in the model made in the hydraulic laboratory, about 350 jet coordinates were extracted from the trajectory of pressurized jets (jets exiting from the circular gate). In the present study, with 75% of the data, the correction coefficient (adjustment coefficient) was extracted by a trial-and-error method and the obtained coefficient was tested using the remaining 25% of the data. The results showed that a correction coefficient of 0.65 resulted in the best trajectory projection.

According to Figure 7, it can be seen that most of the data are located near the bisecting line, which indicates the excellent ability of the models to calculate the trajectory of the pressurized jet. These conclusions are reinforced by the values of the correlation coefficient.

It was found that the trajectory of the pressurized jet extracted from the projectile equations (Equation (16b)), compared to its measured value, has an average error of about 26%. By importing a correction coefficient of 0.65 into the projectile equation, this error is reduced to approximately 8%. Also, Equation (25) calculates the trajectory of the jet with an error of 9%, a reduction in the error by approximately 66%.

According to Figure 8(a), it can be seen that the pressurized jet hits the ground 0.9 m from the structure after reaching a steady state. A comparison between Figure 8(a) and 8(b) shows that the highest pressure is formed at the impact location of the jets to the ground, at which point the velocity is also maximum. Inattention to this issue, incorrect prediction of the jet impact location which will affect the pressure and velocity of the jet, and the subsequent impacts downstream can damage structures and cause erosion.

According to Figure 9, it can be seen that Equation (10) (the equation presented by Salmanzadeh & Ahadiyan (2016)) initially has a large error. This large error at the beginning of the pressurized jet is due to the existence of a fixed constant in that model. Also, because the height of the fall in their research work is low, the air resistance has not been able to influence the trajectory of the pressurized jet. Also, according to Figure 9, if the laboratory data is considered as a basis, the highest error occurs with Equation (9) and the lowest error is related to Equation (24), with a correction coefficient of 0.65. In addition, the simulated trajectory is superior to the projectile equation to estimate the trajectory of the pressurized jet. In general, it can be seen that air resistance has a great effect on the trajectory of the pressurized jet issuing from a gate. These results show that it is necessary to correct predictive equations to incorporate the effects of air resistance.

Figure 10(a) shows that by increasing the diameter of the gate at a constant discharge (Q = 1 L/s), the maximum range of the jet decreases. Also, Figure 10(b) shows that for a fixed diameter (gate diameter equal to 5 cm) with increasing upstream discharge (increasing the height of the dam reservoir water from the center of the gate), the maximum range of the pressurized jet increases. In addition, Figure 10(c) shows that for a fixed diameter (gate diameter equal to 5 cm) with increasing the Froude number, the maximum range of the pressurized jet increases.

It can be seen that the trajectory length calculated using the projectile equation (Equation (6)) is greater than its actual value because air resistance is ignored. In other words, the actual value of the projectile range is shorter than the value obtained from Equation (6) due to air resistance. By analyzing the laboratory results and comparing them with the results of the projectile equation, it was found that the trajectory of the falling jet extracted from the equations compared to its true value (laboratory data) has an average error of about 21%.

Equation (28) is a relation for predicting the trajectory of the jet plane (range of the falling jet from any altitude). In fact, Equation (28) is the transformation of Equation (19) with a modification term (ξ) for air resistance.

According to Figure 12, it can be seen that most of the data are located near the semicircle of the first region, which indicates the accuracy of the relationships for calculating the trajectory of the free jet. Also, the high value of the correlation coefficient (R²) of these graphs shows the high accuracy of these graphs in predicting the trajectory of the free jet.

In general, it is necessary to explain that the equations presented in the present study for predicting the trajectory of free and pressurized jets have been fitted using the laboratory data of the present study (Equations (24), (25), (28), and (29)). The use of the equations presented in this study is completely valid for the number range of the present study experiments. Some care should be used when expanding these values beyond their original range of applicability.

Figure 13 shows the water surface profile in which the two phases (air and water) are separable. According to Figure 13, it can be seen that after about 0.25 s, the jet reaches the end of the broad-crest weir and after about 1 s, the falling jet hits the spillway toe. After 10 s the jet reaches a steady state condition. Also, according to Figure 13, some of the free-falling jets return toward the spillway face (rectangular black color in Figure 13) and some move downstream of the canal. It is interesting that the concentration of the jet when it hits the bottom is quite low–it is a water–air mixture at that point.

As shown in Figure 14, the velocity is increased over the spillway and the flow velocity at the end of the spillway crest is 2.51 m/s. After the flow separates from the spillway and forms a free-fall jet and the jet collides free-fall to the bottom of the canal, the velocity at the end of the canal and in a position above the point of impact to the ground reaches its maximum value of 6.2 m/s. It can be noted that upstream of the broad-crested weir has been simulated as a dam reservoir and thus, the approach velocity behind the weir is near zero, as indicated in the legend of Figure 14.

According to Figure 15(a), it is observed that downstream of the dam, the static pressure at the point of impact of the jet is higher than other points downstream of the dam. Figure 15(b) shows that the dynamic pressure increases due to the increase in velocity downstream of the channel. The total pressure, which is the sum of dynamic and static pressure, is presented in Figure 15(c). It can be seen that the total pressure upstream is mainly due to static pressure and downstream it is caused by both static and dynamic pressures.

In addition, according to Figure 16, the trajectory of the free-falling jet extracted from Equation (6) and the equation presented by the USACE (Equation (12)) corresponds very well with the curve of the downstream crest of the Ogee spillway, and in the simulated trajectory in Ansys Fluent software, these three trajectories are close. The trajectory calculated using Equation (11) with k = 0.9 and also assuming d + h_v = H_overtop has a large error compared to laboratory data.

Equation (11) may seem simple at first because it is obtained by a change in Equation (6). In the case of flow passing through a dam, if the losses are ignored, d + h_v is approximately equal to the head passing through the dam. Comparing Equations (6) and (11), which are obtained directly from the projectile motion equation, it can be seen that these equations are not equal even when K = 1. When K = 1, Equations (6) and (11) are dimensionally similar but are not numerically equal. Equation (11) will be true if the entire head passing through the dam (H_overtop) is converted to the velocity head (h_v). As a first approximation, for flow through the dam, the depth and velocity must be critical. Further confusion arises because most citations that use Equation (11) do not clearly define the terms d and h_v in the text (most citations of Equation (11): USBR 1960, 1977; Annandale 2006). If Equation (11) is used, the falling jet profile becomes less steep and gentler (the distance between the jet colliding point in the plunging pool and the dam body increases).

None of the publications presenting this equation has a specific reason for including the term depth. In general, the accuracy of the equations presented in the present study is superior to other equations over the range covered by the present experiments. Also, the trajectories extracted from Equations (6) and (12), as well as the Ogee spillway curve, provide more acceptable results compared to numerical simulations and other equations.

Following the exit of the falling jet from the overflow or gate, a significant amount of air enters the jet. This air reduces the energy of the pressurized and free jets. In the present study, due to the relatively high elevation of the free jets, aeration was observed throughout the experiments. In all the tests of this study, the breakup length did not exceed 20 cm and the breakup length of the falling jets in the prototypes is very short. Therefore, the data extracted from the laboratory includes a jet aeration effect. It is clear that because of the existence of limitations in the physical model height, the model-prototype scale effects on aeration rate will happen. In this study, the height of the weir from the ground is 2.3 m. In future research, more height for the weir is recommended. In addition, three-dimensional numerical simulations of falling jets are suggested for future studies.

According to Figure 17(a), it can be seen that for all three cases studied (laboratory data, data obtained from numerical simulation, and projectile equations), by increasing the cross-sectional width of the rectangular broad crest weir, the lower the maximum range of the jet. With increasing discharge through the spillway, the maximum range of the jet increases (Figure 17(b)). In other words, for a constant discharge with increasing width, the place where the jet falls to the ground surface is closer to the toe of the dam; and for a constant width with increasing discharge, the place where the jet falls to the ground surface is further from the toe of the dam. In addition, for a constant width with increasing discharge, the Froude number increases, and the place where the jet impacts the ground surface is further from the toe of the dam (Figure 17(c)).

With the development of dam construction technology, we have seen an increase in the height of dams in recent years. Increasing the height of the dam can cause problems related to the discharge of the water as a jet from the structure. By increasing the velocity at the outlet of pressurized/free jets, the water jet can cause irreparable damage to the downstream structures of dams. Here, the trajectories of pressurized and free jets were investigated experimentally and then numerically simulated. The results showed that:

• The trajectory of these jets is affected by air resistance.

• The range of the pressurized jets observed in the experiments is less than that calculated using the projectile equations.

• Projectile equations calculating the trajectory of falling jets have an average error of 21 and 26% for free-falling and pressurized jets, respectively.

• The numerical simulation predicts the trajectories of the free and pressurized jets on average with a RE of 25.5 and 23.5%, respectively, compared to the laboratory data.

• The effect of the height of the dam reservoir water from the center of the gate and the diameter of the flow outlet section on the maximum range of the pressurized jet was investigated. Results showed that with increasing the height of the dam reservoir water from the center of the gate and decreasing the bottom outlet diameter in constant discharge, the range of the pressurized jet increases. In addition, it was found that by increasing the Froude number, the location of the falling jet impact moves further from the dam toe.

• The numerical simulations show that the dynamic pressure and velocity at the point of falling jet impact have the maximum value that should be considered in designs.

• The effect of the passing head on the dam and the width of the flow passage section on the range of the free-falling jet was investigated and the results showed that decreasing the width at constant discharge and increasing the passing head over the dam at constant width increased the falling jet range.

• To use the results of the present study, readers are encouraged to use Equations (24) and (25) for pressurized jets and Equations (28) and (29) for free jets. These are modified projectile equations. Also, the researchers can compare the results of their study with the trajectories of falling jets in Figures 9 and 16.

The following suggestions are set forth for future research:

• In pressurized jets, what is the impact of the gate shape?

• How do variations in overflow affect free-fall jets?

• A real dam should be modeled and compared with the results of the present study.

• More attention should be paid to the high velocities that are present in both jet types.

• In future research, a three-dimensional falling jets should be simulated.

This paper is the outcome of a research project supported by the University of Tabriz research affairs office.

This article does not contain any studies with human participants or animals performed by any of the authors.

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

The authors declare there is no conflict.