Jet trajectory of flow-separating slot-type flip bucket

h_o

depth of approach flow

v_o

average velocity of approach flow

Fr_o

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

Q_M

maximum available discharge

x_j

jet trajectory

α_j

take-off angle

v_j

take-off velocity

α_U

the upper take-off angle

α_L

the lower take-off angle

v_end

average velocity at the end of bucket

h_end

depth of the flow at the end of bucket

ξ_U

head loss coefficient of the upper jet

ξ_L

head loss coefficient of the lower jet

error between the calculated and measured jet trajectories

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.

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 2 is the definition sketch of the flow through the FSSFB, in which h_o is the depth of an approach (subscript ‘o’) flow, v_o is the average velocity resulting in the approach flow of Froude number Fr_o = v_o/(gh_o)^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 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.

Table 1 lists the cases and the parameters of the FSSFB models. The maximum available discharge is Q_M = 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°.

The model discharges were measured with weir instruments of ±0.1 mm reading accuracy. The depth h_o and the jet trajectory x_j 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.

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.

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.

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

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 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.

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.

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