Abstract

A flip bucket is a common element used to dissipate energy for release works. For the purpose of avoiding excessive scour and flow choking, the slot-type flip bucket was developed. In this paper, a flow-separating slot-type flip bucket (FSSFB) is proposed on this basis, which can divide the approach flow into three branches by dividing walls, and thus generate two small, completely separated jets resulting in better energy dissipation performance and reduced scour. Based on model tests, the jet trajectory of the FSSFB is investigated. Considering the local head loss from the flow passing the dividing walls, the take-off velocity is amended for calculating the jet trajectory using the projectile method. Based on fitting analysis, the head loss coefficient is a function of the relative width b/B, the relative angle θ/β of the slot and the Froude number Fro of the approach flow. Finally, an empirical relationship for the head loss coefficient is provided, and the error in the calculation of jet trajectory is less than 10% for the FSSFB.

NOTATION

     
  • ho

    depth of approach flow

  •  
  • vo

    average velocity of approach flow

  •  
  • Fro

    Froude number of approach flow

  •  
  • S

    difference of elevation between the bottom of the open channel upstream and the tailwater channel

  •  
  • B

    total width of bucket

  •  
  • b

    width of slot

  •  
  • width of the flow-dividing wall

  •  
  • w

    bucket height

  •  
  • R

    radius of side bucket

  •  
  • β

    deflection angle of side bucket

  •  
  • θ

    deflection angle of slot

  •  
  • QM

    maximum available discharge

  •  
  • xj

    jet trajectory

  •  
  • αj

    take-off angle

  •  
  • vj

    take-off velocity

  •  
  • αU

    the upper take-off angle

  •  
  • αL

    the lower take-off angle

  •  
  • vend

    average velocity at the end of bucket

  •  
  • hend

    depth of the flow at the end of bucket

  •  
  • ξU

    head loss coefficient of the upper jet

  •  
  • ξL

    head loss coefficient of the lower jet

  •  
  • error between the calculated and measured jet trajectories

INTRODUCTION

With the rapid development of large-scale dam projects, the flip bucket has become a crucial ski-jump dissipater (Khatsuria 2005). For upstream flow with high water head and large discharge, especially when the geological condition of the downstream riverbed is not good, various types of flip buckets have been adopted for enhancing energy dissipation. Some hydropower projects have used flip buckets, such as Xiluodu, Xiangjiaba, and Longyangxia Projects in China, the Baells Project in Spain and the Tucurui Project in Brazil.

The jet trajectory of the flow passing through a flip bucket is a significant issue in the investigation with respect to the hydraulic performance of the dissipater. In practical projects, the air will be entrained into the flow when it leaves the bucket, and the trajectory will be greatly affected by the air entrainment and the air resistance (Kramer et al. 2006; Schmocker et al. 2008). A comprehensive resistance coefficient has been proposed to estimate the jet trajectory accurately (Wu et al. 2015). In addition, projectile theory for a rigid body is also used to estimate the jet trajectory when the air resistance is neglected, but undeniably, the deviation around it is considerable. What is more, the method of replacing approach flow velocity with take-off velocity or replacing geometric angle with take-off angle is often used to calculate an accurate jet trajectory (Heller et al. 2005, 2006; Steiner et al. 2008; Pfister et al. 2014; Ma et al. 2016; Wu et al. 2016).

Recently, in order to improve energy dissipation and make the jet flow smoothly back to the downstream river channel, some irregular-shaped flip buckets have been proposed, such as the triangular-shaped (Steiner et al. 2008), the differential (Zhang et al. 2013; Sun et al. 2017), the tongue-type (Zhu et al. 2004), the beveled flip bucket (Xue et al. 2013; Yin et al. 2016), the slit-type or multiple slit-type (Wu et al. 2012, 2014; Huang et al. 2017) and the deflector (Juon & Hager 2000; Pfister & Hager 2009; Lucas et al. 2013, 2014; Omidvarinia & Jahromi 2013). However, the jet trajectories of these buckets mentioned above are difficult to estimate because of the irregular geometry.

The slot-type flip bucket (Deng et al. 2016; Ma 2016; Wu et al. 2018) has no expansion or contraction of the outlet, with the advantages of the longitudinal flow expanding and high energy dissipation. When the water flows through this bucket, one stream is jetted from the side buckets while the other stream flows out from the slot. Nevertheless, the jets are not thoroughly separated by the slot-type flip bucket, making it difficult to control the impact onto the downstream channel. Consequently, in the present work, a kind of slot-type flip bucket with two flow-dividing walls, called a flow-separating slot-type flip bucket (FSSFB), is proposed. The flow-dividing walls have the advantage of separating the jets completely to avoid mixing the two stream jets.

This bucket not only separates the approach flow but also helps the control of the jet impact upon the downstream riverbed. Moreover, the location of the impact point of the jet flow estimated by the jet trajectory is closely related to the safety and stability of the upstream hydraulic structures. Some of failures resulting from jet impact upon the downstream demonstrate the potential of destruction such as the spillway scour erosion failure that occurred in Oroville reservoir in the USA. On this basis, the approach flow is divided into multiple jets by the FSSFB, and then the corresponding impact points are different, which can achieve the purpose of adjusting the downstream flow velocity and flow pattern, and further ensure the efficiency and safety of energy dissipation (Erpicum et al. 2010; Sharif & Ravori 2014).

As for the irregular-shaped flip buckets, there is no research on calculating the jet trajectory by using the projectile theory. For the FSSFB, the take-off angle of the jet flow is obviously smaller than the deflection angle, which needs to be corrected. In addition, due to the significant local head loss, the take-off velocity is lower than the approach flow velocity, which also needs to be corrected. In this investigation, the coefficient of local head loss at the slot is determined by physical model tests. Based on the energy equation with coefficient, the take-off velocity is estimated and the jet trajectory is obtained.

The objectives of this paper are to introduce the new type of flip bucket, verify the rationality and feasibility of this method of introducing the coefficients, and obtain the empirical expression for estimating the jet trajectory of the FSSFB.

EXPERIMENTAL SETUP AND METHODOLOGY

Experimental setup

The experiments were conducted in the High-Speed Flow Laboratory, Hohai University, Nanjing, China. The experimental setup consisted of a pump, an approach conduit, a tailwater channel, a test model, and a flow return system (Figure 1). The approach conduit with orifice flow was 0.15 m wide and 0.135 m high. There was pressure flow in the approach conduit and free surface flow in the model. The conduit, with a pressure slope of 1:5 at the end, was connected to the model by an arc gate. The test model, made of Perspex, included two parts with an open channel upstream and a flip bucket. The open channel upstream was 0.70 m long, 0.15 m wide and 0.38 m high, and the end of it was closely attached on the flip bucket. The bucket was divided into three sections in the transverse direction by two flow-dividing walls (two side buckets and one center bucket), and the upstream of the walls was connected to triangular guide walls. The length of the triangular guide wall with an angle of 25° between the sloping top and the horizontal was 0.70 m. The dividing walls were designed only for smoothly separating the approach flow into three jets that did not affect each other, increasing the expansion effect, while the slot gave the jets different trajectories.

Figure 1

Experimental setup.

Figure 1

Experimental setup.

Figure 2 is the definition sketch of the flow through the FSSFB, in which ho is the depth of an approach (subscript ‘o’) flow, vo is the average velocity resulting in the approach flow of Froude number Fro = vo/(gho)1/2, and S = 0.938 m is the difference of elevation between the bottoms of the open channel upstream and the tailwater channel. The origin of the coordinate system (x, y) is at the bottom of the edge of the flip bucket. The approach flow is divided into the lower jet (impact between x1 and x2) and the upper jet (impact between x3 and x4).

Figure 2

Definition sketch of the flow through the FSSFB.

Figure 2

Definition sketch of the flow through the FSSFB.

Figure 3 shows the geometry of the dissipater, where B = 0.15 m is the total width of bucket; R = 0.50 m and β= 35° are the radius and the deflection angle of the side bucket, respectively; b and θ are the width and deflection angle of the slot, respectively; and w=R(1 − cosβ) is the bucket height. The width of the flow-dividing wall is fixed as = 0.01 m, and the width of the tailwater channel is 0.70 m for each experiment.

Figure 3

Geometry of the FSSFB: (a) section A–A view; (b) plane view.

Figure 3

Geometry of the FSSFB: (a) section A–A view; (b) plane view.

Experimental methodology

Table 1 lists the cases and the parameters of the FSSFB models. The maximum available discharge is QM = 96.5 L/s. Cases M01–M03 are for the effects of b, and Cases M02–M22 are for the effects of θ. The width b was 0.030, 0.045 and 0.060 m respectively, whereas the deflection angle θ varied from 10° to 35°.

Table 1

Cases and parameters of the models

Casesb/mθRemarks
M01 0.060 35 For each case, two depths of approach flow are designed, ho = 0.18 m and 0.10 m. 
M02 0.045 35 
M12 0.045 20 
M22 0.045 10 
M03 0.030 35 
Casesb/mθRemarks
M01 0.060 35 For each case, two depths of approach flow are designed, ho = 0.18 m and 0.10 m. 
M02 0.045 35 
M12 0.045 20 
M22 0.045 10 
M03 0.030 35 

The model discharges were measured with weir instruments of ±0.1 mm reading accuracy. The depth ho and the jet trajectory xj were directly measured with a ruler. Affected by the swing of the jet, the measurement accuracy was ±1 cm. In the tests, each experimental value was obtained by taking the average after five measurements.

RESULTS AND DISCUSSION

Jet trajectory observation

Figure 4 shows a photograph of jet trajectories of flow discharged by the FSSFB. It can be seen that the approach flow is divided into upper and lower jets, the former from the two side buckets and the latter from the slot. In addition, the upper and lower jets are clearly demarcated and separated from each other completely. The trajectory of each jet follows almost a parabolic trajectory. However, the upper and lower trajectories of the upper and lower jets are not suitable to be estimated only by the previous approaches, which may be due to the apparent decrease of take-off velocity affected by the local head loss compared with the approach flow velocity.

Figure 4

Observation of jet trajectories of flow discharged by the FSSFB.

Figure 4

Observation of jet trajectories of flow discharged by the FSSFB.

Determination of local head loss coefficient

In order to estimate the jet trajectories of flow discharged by the FSSFB through using the projectile formula, this paper considers the effect of significant local head loss on the take-off velocities of the jet flow, and obtains the local head loss coefficients by inverse calculation of the measured jet trajectories and energy equation.

On the basis of projectile theory, the jet trajectory can be expressed as: 
formula
(1)
where αj is the take-off angle corresponding to trajectory xj (j = 1, 2, 3, 4), and y is the vertical distance between the starting point of the jet trajectory and the bottom of the tailwater channel. It is worth noting that the vo in this formula is not appropriate here because the actual take-off velocity vj (corresponding to xj) has changed.
Previous studies have shown that the take-off angle of the jet is generally smaller than the deflection angle of the flip bucket. The virtual take-off angle expression has high accuracy and reliability (Ma et al. 2016), and is used to correct αj of Equation (1): 
formula
(2)
 
formula
(3)
 
formula
where αU and αL are the upper and lower take-off angles of the jet, respectively. The geometric angle α here is equal to θ when estimating the corresponding upper take-off angle, while α is equal to β when estimating the lower. The width of the flow-dividing walls is neglected in the calculation.
In addition to Equations (1)–(3), the depth of flow hend at the outlet of the bucket, contained in y, is also needed to calculate the upper and lower take-off velocities, which is established as follows (Figure 5): 
formula
(4)
where vend is the average velocity at section 1-1 at the end of bucket, and hend stands for the depth of the flow at this section. Section 0-0 is a flow section at the front of the triangular guide wall.
Figure 5

Definition sketch of the energy equations of the different jets: (a) the upper jet; (b) the lower jet.

Figure 5

Definition sketch of the energy equations of the different jets: (a) the upper jet; (b) the lower jet.

The take-off velocity vj obtained from the experimental jet trajectory xj is calculated by Equations (1)–(4). Since the difference between v1 and v2 (v3 and v4) is small, the average velocity vendL (vendU) is used to estimate the calculated jet trajectory instead. Then the head loss coefficient ξ is calculated by the average value.

The horizontal plane of the bottom of the open channel section was taken as the reference plane with the energy equation of the water between section 0-0 and section 1-1 by introducing the head loss coefficient ξ: 
formula
(5)
where the average velocity vendL = (v1 + v2)/2 for the lower jet, similarly vendU = (v3 + v4)/2 for the upper jet; hendL and hendU are the flow depths corresponding to vendL and vendU, respectively. The parameter Δz is the vertical difference of the surface of the water above the outlet with the bottom of the open channel.

It is worth mentioning that the head loss coefficient calculated by the method above may not represent the actual value but include the effect of air resistance and so forth. However, the jet trajectory is investigated while other influencing factors are neglected in our work.

Figure 6 shows the relationship between ξU and the comprehensive parameter E1, and the best fit is (R2 = 0.94): 
formula
(6)
Similarly, Figure 7 gives the relationship between ξL and the comprehensive parameter E2, and the best fit is (R2 = 0.90): 
formula
(7)
where the parameters E1 and E2 can be calculated as E1 = (b/B)0.22Fro1.12 when 1.40 ≤ E1 ≤ 2.22, E2 = (b/B)−0.17(θ/β)−0.13Fro0.28 when 1.66 ≤ E2 ≤ 4.30. In addition, Equations (6) and (7) are valid for 0.20 ≤ b/B ≤ 0.40, 0.29 ≤ θ/β ≤ 1 and 1.88 ≤ Fro ≤ 4.61. Accordingly, the head loss coefficient ξU is closely related to parameters b/B and Fro, while ξL is closely related to b/B, θ/β and Fro.
Figure 6

Relationship between ξU and E1.

Figure 6

Relationship between ξU and E1.

Figure 7

Relationship between ξL and E2.

Figure 7

Relationship between ξL and E2.

Comparison of jet trajectories with previous approaches

The errors between the calculated and experimental jet trajectories are investigated to verify the rationality of replacing experimental with calculated values. The calculated jet trajectory is estimated by Equations (1)–(5), where the take-off velocity vj is replaced by the average velocity vend. The upper and lower trajectories of each jet flow are different only in the take-off angles and vertical distance y when calculating. The comparison between the calculated values and experimental results of the jet trajectories is shown in Figure 8. The dotted lines in the figure stand for the 10% error lines, and the errors between the calculated and measured jet trajectories may be expressed as: 
formula
(8)
where xjexp and xjcal are the calculated and measured jet trajectories, respectively.
Figure 8

Comparison between the calculated values obtained by different approaches and the measured results of jet trajectories.

Figure 8

Comparison between the calculated values obtained by different approaches and the measured results of jet trajectories.

Figure 8 shows the jet trajectories estimated by the method presented in this paper compared with those calculated with previous approaches. It can be seen that the trajectories calculated by the method of replacing geometric angle with take-off angle (Heller et al. 2005; Steiner et al. 2008; Pfister et al. 2014) and replacing approach flow velocity with take-off velocity (Wu et al. 2016) are close to the actual trajectories, but there is a certain gap between them. As expected, the jet trajectories estimated by the local head loss coefficients and revised velocities are obviously more accurate, matching most closely the actual values. The error of the method of this paper is within the margins of essentially ± 10%, so it is more suitable for estimating jet trajectory in practical engineering.

CONCLUSION

The FSSFB proposed in this paper can divide the approach flow into two jets, and then disperse the kinetic energy of a high-speed flow and reduce the scour of the riverbed. According to experimental observation, projectile theory can be used to estimate the jet trajectory of the FSSFB. However, not only the take-off angle, but also the take-off velocity need to be revised considering the local head loss. Based on the experimental results, the local head loss coefficients can be expressed by Equations (6) and (7). Compared with previous methods, the calculated results of the method provided in this paper are significantly in good agreement with the experimental data.

ACKNOWLEDGEMENT

The work is financially supported by the National Natural Science Foundation of China (51579076).

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