Numerical simulation study on spread angle in oblique cut flip bucket

Abstract

With the advantages of simple structure, good energy dissipation effect and so on, the sky jump energy dissipation is widely used in energy dissipation of water conservancy project. But at present, the prediction about the range of the free nappe is not complete. The studies show that the ejection angle is different at different locations of the outlet section, but the specific distribution of the outlet section velocity is still unclear. The measurement of free nappe range is mostly the direct measurement, how to accurately control is still difficult, and no systematic study has been made. Especially, the studies on the diffusion width of free nappe are few, the understanding of free nappe diffusion is not enough. In this article, the deflection degree of flow velocity at the outlet section in the lateral direction is quantified by the spread angle. Studied the distribution characteristic of spread angle through numerical simulation. The results show that the distribution characteristics and the maximum value of spread angle are affected by the change of flip bucket size and approach flow condition, and presented a set of equations for calculating the maximum angle.

KEYWORDS:

• Spread angle
• numerical simulation
• oblique cut flip bucket
• angle distribution
• outlet section

1. Introduction

In water conservancy projects, suitable design of energy dissipation and erosion control are often built to protect downstream. Ski jump energy dissipation is a common energy dissipator, which is widely used in high dam discharge structures. The origins of ski jump energy dissipation can be dated back as far as the mid-1930s, the scheme designed a circular arc spillway to discharge the high velocity flow in the form of a trajectory jet and keep the flow away from the building to mitigate potential structural damage (Fraser, 2016). The flow which jets into the air produces aeration, fragmentation, and atomisation under the influence of air friction, in addition, when the flow hitting the pond, the energy will be consumed and the pond can buffer the impact on the riverbed, thus protecting the riverbed downstream. At present, there are still many studies about the energy dissipation efficiency of sky jump (Fraser, 2016; Yavuz, 2021, 2022), among them, the study (Qian et al., 2016) has raised the energy dissipation efficiency to 75%. As an economic energy dissipation method, ski jump energy dissipation is widely used in hydropower projects with dam height greater than 100 m, according to the statistics, the proportion of energy dissipator by ski jump energy dissipation is more than 80% in the high dam with height of more than 70 m (Mason & Arumugam, 1985; Wu et al., 2010), it has great value to be researched. When the flood discharges, it is often necessary to consider whether the lateral spread of the free nappe will cause scour to the mountains on both banks. About the spread of the free nappe, from the following studies, it can be known the factors affecting the diffusion of nappe. The research (X. L. Liu & J. Liu, 1989) studied that deformation, diffusion and aeration of the free nappe mainly depend on the surface tension, inertial force and turbulent force, while the effect of the surface tension can be basically eliminated for the sky jump flow due to the thick nappe. The research (Y. Wang et al., 2022) studied that the diffusion width of nappe is mainly related to Reynolds number, Weber number and Froude number, the sensitivity of diffusion width to Froude number is greater than other parameters. The research (Jiang & Zhang, 1984) also believes that the diffusion of nappe is related to Froude number. From above articles, the diffusion width of nappe mainly depends on the effect of surface tension, turbulent force, and inertial force. They are also studied that Reynolds number, Weber number, and Froude number are the main parameters affecting the width of the nappe, and the Froude number is more sensitive to the influence of width, and Reynolds number, Weber number determine the diffusion and aeration of nappe behind the bucket. The research (Mo et al., 2013) studied the inflow width of the oblique cut flip bucket under different oblique angles through model experiment, obtained the distance from the lower edge short jump to the long jump and the lateral inflow width affected by the oblique angle. The research (Lian et al., 2019) studied nappe wind of continuous bucket and tongue-shape bucket in different angles and mentioned that the trajectory width of water tongue is affected by the side wall. So, the shape of flip bucket can also affect the diffusion of nappe.

The inflow range is an important design parameter in the design of sky jump, which determines the safety of dam and the economy of downstream protection measures. As mentioned in the article (Borthakur et al., 2019; Wu et al., 2015b), the pressure in the tunnel has a pressure drop trend near the outlet, and the pressure above and below nappe is atmospheric pressure, the free nappe follows the trajectory of projectile motion. It shows that the trajectory of the free nappe and the inflow range are mainly determined by the acceleration of gravity and the initial conditions of flow velocity at the bucket outlet, including the magnitude and direction. The articles (Dey, 2002; Guha et al., 2010) show that the actual ejection angle of free nappe is not equal to the angle of bucket, the researches (Heller et al., 2005; Steiner et al., 2008) found through model experiment that the ejection angles of the upper and lower edges of free nappe are both smaller than the flip angle of the bucket, and the ejection angle of the lower edge is smaller than the ejection angle of the upper edge. So, it can be known that the distribution of ejection angle at different positions is different. However, the above articles have not studied the lateral spreading of the free nappe and cannot form a three-dimensional overall understanding of the trajectory. These article (Juon & Hager, 2000; Khalifehei et al., 2022; Lucas et al., 2014; Yamini et al., 2015), including the ones mentioned above, are mostly similar studies on the outlet with different shapes, have not analysed the lateral distribution of ejection angle. The above articles all did not recognise the difference in lateral distribution of ejection angle, this is why there are few studies on the diffusion of nappe, it also cannot form a systematic study of ejection angle.

The research on outflow spreading is mostly focused on the pressure outflow (Cleaver et al., 1995; Guha et al., 2010; Naib, 1974; Puentes et al., 2012), and the research on outflow spreading in open channels is less. In addition to the research on diffusion of nappe mentioned above. The research (Heidari et al., 2019) studied the flow spreading in the channel with slit outlet through the model experiment, obtained that the shape of inflow range is approximately circular or elliptical, and the variation of the inflow range under different slope, outlet contraction angle, and inflow time. In order to control the spreading and inflow range of nappe, various shape of bucket have been grown to adapt to different river conditions (Deng et al., 2016; Mo et al., 2013; Wang & Luo, 2011; Wu et al., 2015a; Yamini et al., 2015). The change of spread angle caused is mainly to control the angle of side wall and the shape of bottom plate. At present, the diffusion of the nappe is a direct measurement of the inflow range, the distribution of the outlet velocity direction and pressure has not been known, so it is impossible to analyse the cause of spreading, the understanding of the spread angle is still insufficient, and further research is needed (Figure 1).

Figure 1. A sample ski jump flow of oblique cut flip bucket.

To sum up, this study wants to obtain the diffusion of nappe after losing the binding of side wall and bottom plate, and understand the influence conditions of spreading, finally analyse the cause of the formation of diffusion angle through the pressure of approach flow. The purposes of this paper are to understand the influence of flip bucket size and approach flow conditions on the spread angle of oblique cut flip bucket, and obtain the relevant equation of spread angle, which providing a reference basis for engineering design and the subsequent cause analysis.

2. Method

The design of the oblique cut flip bucket is shown in Figure 2. The position coordinate adopts the right-handed coordinate system, the origin of the position coordinate is the bottom of the long side wall at the entrance of the bucket, the flow direction is the X-axis, the gravity direction is the Z-axis, and the lateral direction is Y-axis. The values of x, y and z represent the distance from the origin in each direction. The design parameters of flip bucket size refer to the relevant specifications for flip bucket energy dissipation and spillway design. The spillway width is fixed at 0.5 m, the flip bucket width is consistent with the spillway width, so the width both spillway and bucket are denoted by b. The shape of the bucket is arc, the flip angle θ is fixed at 20°, that is the angle from inlet to outlet of the bucket, and the radius r of flip bucket is 2.0–3.5 m. The oblique angle is the angle between the outlet section and the y-axis, it is denoted by β, its range is 0–45°. The approach flow velocity is denoted by v₀, it is used to represent the initial velocity condition of the flow in front of the bucket, the value range is 4–12 m/s, and v_x, v_y and v_z represent the velocity components in each direction. The vertical distance from the water point to the floor is denoted by h. The approach flow depth is used to describe the distance between the free surface and the floor, it is denoted by h₀, the value range is 0.05–0.2 m. For the sky jump flow, the approach flow velocity v₀ affects the inertial force, and the approach flow depth h₀ reflects the quality characteristics of flow. For the bucket shape and size, the bucket size determines the constraint of the boundary, oblique angle β affects the non-uniform sky jump flow along the process, and the radius r affects the flow characteristics of flow under centrifugal action. According to the calculation by the above values, the ratio of width to approach flow depth b/h₀ ranges from 2.5–10, it represents the dimension characteristics of the spillway, determines the characteristic length of the approach flow, and the ratio of width to radius b/r ranges 5–7. The ratio of radius to approach flow depth r/h₀ ranges 12.5–70. The parameters of bucket size satisfy the values commonly used in the Chinese design specification for spillway (SL 253-2018). The approach flow was characterised by the approach flow Froude number (Fr), it represents the relative relation between inertial force and gravity of the flow, the range of Fr is from 3 to 17, this range includes the range of Fr commonly used in prototypes, such as the prototypes mentioned in the references (Kawakami, 1973; Yavuz, 2022). Through the calculation of the equation from the article (Heller et al., 2005), it can be proved that the above parameters satisfy the requirements of outflow in the continuous bucket, and the research (M. Liu et al., 2017) obtained the method to improve the chocking in discharge tunnel through the model experiment, it can prove that the oblique cut flip bucket is easier to outflow than the continuous bucket. In view of the influence of scale effect, the study (Heller et al., 2005) considered that scale effects due to viscosity and surface tension are small for flow depths greater than 40 mm. The study (Kobus, 1984) proposed Reynolds numbers (Re) should be at least 10⁵ to minimise the viscous effect. The studies (Rutschmann, 1988; Speerli, 1999) studied the spillway aerators and bottom outlets respectively, concluded that the Weber number (We) should be at least 110 for surface tension effects to be negligible. The research (Pfister & Chanson, 2012) summarised the limitations to avoid significant scale effects in two-phase air–water flows under Froude similitude and conclude that the limitations relevant for high-speed air–water two-phase flows using the Froude similitude are either We ^0.5 > 140 or Re > 2 to 3×10⁵. Reynolds number (Re) represents the ratio of inertial force to viscous force level and the Weber number (We) represents the ratio of the inertial force to the surface tension effect, it can be used to determine whether viscous force and surface tension have an effect on the flow. The formula for calculating Froude number (Fr), Reynolds number (Re), Weber number (We) is as follows:

Froude number: (1) $\begin{array}{r}Fr=\frac{v}{\sqrt{gh}}\end{array}$(1)

Reynolds number: (2) $\begin{array}{r}Re=\frac{\rho vl}{\mu }\end{array}$(2)

Weber number: (3) $\begin{array}{r}We=\frac{\rho v^{2}l}{\sigma }\end{array}$(3) where v = approach flow velocity; g = acceleration of gravity; h = approach flow depth; ρ =average flow density; l = characteristic length of spillway; μ =viscosity coefficient; σ = surface tension coefficient.

According to the calculation of the above parameters, the ranges of Reynold number (Re) and Weber number (We) are respectively 6.6×10⁵–5.28×10⁶ and 3.7×10⁴–8.8×10⁵, these values all satisfy the requirements of the above reference. Therefore, the above values can make the water fully outflow in all working conditions and the scale effect of the model can be ignored. The specific values of variable parameters are shown in Table 1. Since the main research object is the lateral deflection degree of flow velocity at the outlet section, the spread angle is chosen to quantify it. The spread angle is the angle between v_x and v_y. The size of spread angle is denoted by α, according to the length of the side wall, the spread angle at the long side is denoted as α_l (subscript l), and the spread angle at the short side is denoted as α_s (subscript s).

Figure 2. Definition sketch of the oblique cut flip bucket and the nappe.

Table 1. Test conditions with basic parameter variation.

Parameter	Value
Oblique angle β (°)	0–45
Bucket radius r (m)	2.0–3.5
Approach flow velocity v0 (m/s)	4–12
Approach flow depth h0 (m)	0.05–0.20
Froude number Fr (-)	2.86–17.13
Reynolds number Re (-)	6.6×10^5–5.28×10^6
Weber number We (-)	3.7×10^4–8.8×10^5

This article uses the method of numerical simulation to research. On the one hand, the main object of the research is the interior of the approach flow, and it is difficult to accurately measure the data inside the water flow in model experiment. On the other hand, the model experiment has limitations on the selection of parameters due to the limitations of the site and flow, while the numerical simulation can select the desired parameters according to the needs. It can be known from these articles that the numerical simulation results are consistent with the model experiment results and have good accuracy (Deng et al., 2020; Hamedi et al., 2016; Zhang et al., 2020). The research (Li et al., 2019) has simulated the flow of the oblique cut flip bucket and compared it with the model test, the calculation results are consistent with the model test. This research uses commercial computational fluid dynamics (CFD) package FLUENT computer code (Fluent, Pittsburgh) to simulate the flow across the flip bucket, the version of FLUENT is 2020R2, released on 15 July 2020. The aeration of water flow on the flip bucket is relatively weak, and the aeration effect on the transverse pressure difference is assumed to be small. Thus, the RNG k–ε turbulence model with the VOF method is introduced to improve the simulation rationality and efficiency (Yakhot & Orszag, 1986; Yakhot et al., 1992), considering the strong anisotropism and the 3D transverse-converging movement in high turbulent flows. The previous investigations using the RNG k–ε turbulence model show good agreements with respect to the free surface and interior flow structures (Abbaspour et al., 2009; Viti et al., 2019). To sum up, this calculation model of this research uses renormalisation group k-ε (RNG k-ε).

2.1. Equations

The governing equations (Yakhot & Orszag, 1986; Yakhot et al., 1992) can be written in a Cartesian coordinate system. The continuity equation, momentum equation, turbulent kinetic energy k-equation, and dissipation rate ε are as follows:

continuity equation (4) $\begin{array}{r}\frac{\mathrm{\partial }\rho }{\mathrm{\partial }t}+\frac{\mathrm{\partial }\rho u_i}{\mathrm{\partial }{x}_i}=0\end{array}$(4)

momentum equation (5) $\begin{array}{rl}& \frac{\mathrm{\partial }\rho u_i}{\mathrm{\partial }t}+\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}(\rho u_i{u}_j)\\ & =-\frac{\mathrm{\partial }p}{\mathrm{\partial }{x}_i}+\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_j}\\ & \times \left[(\mu +{\mu }_t)\left(\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}+\frac{\mathrm{\partial }{u}_j}{\mathrm{\partial }{x}_i}\right)\right]\end{array}$(5)

k-equation (6) $\begin{array}{rl}\frac{\mathrm{\partial }\rho k}{\mathrm{\partial }t}+\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}(\rho u_{i}k)& =\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}\left[\left(\mu +\frac{{\mu }_t}{{\sigma }_k}\right)\frac{\mathrm{\partial }k}{\mathrm{\partial }{x}_i}\right]\\ & +G_k-\rho \epsilon \end{array}$(6)

ε-equation (7) $\begin{array}{rl}\frac{\mathrm{\partial }\rho \epsilon }{\mathrm{\partial }t}+\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}(\rho u_i\epsilon )& =\frac{\mathrm{\partial }}{\mathrm{\partial }{x}_i}\left[\left(\mu +\frac{{\mu }_t}{{\sigma }_{\epsilon }}\right)\frac{\mathrm{\partial }\epsilon }{\mathrm{\partial }{x}_i}\right]\\ & +C_{1\epsilon }\rho \frac{\epsilon }{k}{G}_k-C_{2\epsilon }\rho \frac{{\epsilon }^2}{k}\end{array}$(7) where ρ = average flow density and μ =dynamic viscous co-efficient. They are deduced by $\rho =F_w{\rho }_w+(1-F_w){\rho }_a$ and $\mu =F_w{\mu }_w+(1-F_w){\mu }_a$, respectively, where F_w = water volume fraction; ρ_w and ρ_a = density of water and air, respectively; μ_w and μ_a = viscosity of water and air, respectively; p = pressure; t = time; k = turbulent kinetic energy; ε = dissipation rate of turbulent kinetic energy; and μ_t = turbulent viscous coefficient, which can be derived from (8) $\begin{array}{r}{\mu }_t=\rho C_{\mu }\frac{k^2}{\epsilon }\end{array}$(8) G_k is the turbulent kinetic energy produced by the mean velocity gradients. (9) $\begin{array}{rl}{G}_k& ={\mu }_t\left(\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}+\frac{\mathrm{\partial }{u}_j}{\mathrm{\partial }{x}_i}\right)\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}\end{array}$(9) (10) $\begin{array}{rl}{C}_{1\epsilon }& =1.42-\frac{\eta \left(1-\frac{\eta }{{\eta }_0}\right)}{1+\beta {\eta }^3}\end{array}$(10) (11) $\begin{array}{rl}\eta & =\frac{Sk}{\epsilon }\end{array}$(11) (12) $\begin{array}{rl}S& =\sqrt{2\overline{S_{ij}}\overline{S_{ij}}}\end{array}$(12) (13) $\begin{array}{rl}\overline{S_{ij}}& =\frac{1}{2}\left(\frac{\mathrm{\partial }{u}_i}{\mathrm{\partial }{x}_j}+\frac{\mathrm{\partial }{u}_j}{\mathrm{\partial }{x}_i}\right)\end{array}$(13)

The values of the parameters are shown in Table 2. The semi-implicit method for pressure linked equations (SIMPLE) algorithm is utilised for the pressure-velocity coupling of incompressible turbulent flows.

Table 2. Constants of governing equations.

Parameter	Value
η0	4.38
β	0.012
Cμ	0.0845
C1ε	1.42
C2ε	1.68
σk	0.7179
σε	0.7179

The VOF method is introduced to track the free surface (Brackbill et al., 1992; Hirt & Nichols, 1981), in which water and air phases share the same velocity and pressure field. When the water fraction ratio F_w = 0, it implies that there is no water phase in the controlling body; when F_w = 1, the controlling body is full of water; when 0 < F_w < 1, the controlling body contains some portion of water in the free surface section. The governing equation of VOF method is as follows:

continuity equation: (14) $\begin{array}{r}\mathrm{\nabla }\cdot \overrightarrow{u}=0\end{array}$(14)

momentum equation: (15) $\begin{array}{rl}\rho \frac{\mathrm{\partial }\overrightarrow{u}}{\mathrm{\partial }t}+\rho \overrightarrow{u}\cdot \mathrm{\nabla }\overrightarrow{u}& =-\mathrm{\nabla }\overrightarrow{p}+\rho \overrightarrow{g}\\ & +\mathrm{\nabla }\cdot \mu (\mathrm{\nabla }\overrightarrow{u}+{\mathrm{\nabla }}^T\overrightarrow{u})+\overrightarrow{f}\end{array}$(15)

momentum equation: (16) $\begin{array}{r}\frac{\mathrm{\partial }F}{\mathrm{\partial }t}+\overrightarrow{u}\cdot \mathrm{\nabla }F=0\end{array}$(16) where $\overrightarrow{u}$ = velocity; ρ = average flow density; $\overrightarrow{p}$ =pressure; $\overrightarrow{g}$ = average flow density; μ =viscosity; F =volume fraction; $\overrightarrow{f}$ = interfacial force, which is the increase in pressure caused by interfacial force at the interface of two phases, it can be derived from (17) $\begin{array}{r}\overrightarrow{f}=-\sigma \left(\mathrm{\nabla }\cdot \left(\frac{\mathrm{\nabla }F}{|\mathrm{\nabla }F|}\right)\right)\mathrm{\nabla }F\end{array}$(17) σ is the surface tension coefficient.

2.2. Modelling mesh

The model and mesh of oblique cut flip bucket is shown in Figure 3. The mesh is modelled by the ICEM CFD version 2020 R2 software for the numerical simulation of an oblique cut flip bucket. The model consists of spillway, flip bucket, and approximate shape of nappe. The total length of spillway is 5 m, and the width of spillway is 0.5 m, the length is 25–100 times of the approach flow depth. There are 1 m long pressure inlet and 4 m long channel for stabilising approach flow. In order to ensure the velocity to remain unchanged when the flow reach the entrance of the flip bucket, the spillway slope is 0°. The shape of bucket is determined by the corresponding parameters of each working condition. The shape of nappe obtained is roughly fitted by gravity acceleration and velocity from the pre-calculation, and extend 0.01–0.15 m to the outside, ensure that the shape is similar to actually nappe, and make the grid completely cover the nappe. The purpose of this is to make the flow direction perpendicular to the mesh direction and reduce the calculation error. The mesh adopts structured mesh, the flip bucket and the position near the outlet is the key research position, and the position with water here is encrypted. The mesh quantity is 1.7 million – 3 million. In the X direction, the mesh size of the first 4 m of the spillway is 0.04 m, and the mesh size of the last 1 m is 0.02 m, as a transition position to the bucket. The mesh size of the bucket is 0.01 m. The mesh size of nappe position is variable, from the outlet of the bucket, the mesh size gradually increased from 0.01 m to 0.04 m, it ensures the density of mesh at the outlet, while making the mesh quantity not too large. In the Y direction, the mesh size is all 0.01 m. In the Z direction, the position with water is encrypted, the encryption range is determined by the depth of incoming water, the size encryption from 0.01 m to 0.002 m.

Figure 3. Typical computational mesh.

2.3. Boundary conditions

Inlet velocity is used as the inlet boundary condition, the velocity magnitude depends on the working conditions, it is set to v₀ = 4–12 m/s. The approach flow depth h₀ is controlled by the height of pressure section baffle, the magnitude depends on the working conditions. The nonslip boundary condition is used on the wall, and the near-wall condition is given by the standard wall functions, with the normal velocity set to zero on the wall. The pressure outlet is the outlet boundary, with ∂u / ∂x = 0, ∂k / ∂x = 0, and ∂ε / ∂x = 0. The atmospheric pressure is applied at the free surface of the water flow.

2.4. Sensitivity analysis

Figure 4 shows the spread angle α distribution at the outlet under different encryption conditions, Figure 4(a) shows the lateral distribution when the mesh size is encrypted laterally, Figure 4(b) shows the vertical distribution when the mesh size is encrypted vertically. The selected working condition is r = 2.0 m, β = 45°, v₀ = 5 m/s, h₀ = 0.2 m. The mesh size of the flip bucket is 0.01 m when it is not encrypted, in the case, the mesh quantity is approximately 8.2 million, and the range of spread angle α is 0.03°–10.63°, the trend in Figure 4(a) is to increase first and then decrease from y = 0 m to y = 0.5 m, and the trend in in Figure 4(b) is decreasing from h = 0 m to h = 0.2 m. When the mesh size is encrypted to 0.005 m in lateral direction, the mesh quantity is approximately 15.4 million, and the range of spread angle α is 0°–10.67°. After lateral encryption, the maximum angle has changed by about 0.4%, the minimum angle has changed by about 0.03%, the range is basically remained unchanged, and the two curves almost coincide, this shows that lateral encryption will not affect the distribution of spread angle α, there is no need to encrypt laterally. After vertical encryption, the trend in different mesh size is unchanged. When the mesh size is encrypted to 0.005 m in lateral direction, the mesh quantity is approximately 12.1 million, and the range of spread angle α is 0.01°–12.38°, the minimum angle has changed by about 0.02%, the maximum angle has changed by about 14%. The minimum angle has basically not changed, while the maximum angle has a large change, which indicates that there is a large gap between the mesh size of 0.01 m in the vertical direction. So further encryption is still needed. When the mesh size is encrypted to 0.002 m in lateral direction, the mesh quantity is approximately 24 million, and the range of spread angle α is 0°–13.96°, the minimum angle has basically not changed, the maximum angle has changed by about 11%, the maximum angle has a large change, which indicates that there is a large gap between the mesh size of 0.005 m in the vertical direction. So further encryption is still needed. When the mesh size is encrypted to 0.001 m in lateral direction, the mesh quantity is approximately 43.6 million, and the range of spread angle α is 0°–14.75°, the minimum angle has basically not changed, the maximum angle has changed by about 5%, the change less than 10%, so we can think that when the mesh size is 0.002 m, the magnitude range of diffusion angle can be fully reflected. The computing CPU uses AMD Ryzen 9 5900X, there are 24 codes, each with 2 processes, and the main CPU frequency is 3.7 GHz. The number of processes used in numerical simulation is 16–20, and the average calculation time of one working condition is 1.5 days.

Figure 4. Check of mesh sensitivity (r = 2.0 m, β = 45°, v₀ = 5 m/s, h₀ = 0.2 m): (a) spread angle distribution on the lateral direction at h = 0.01 m; (b) spread angle distribution on the vertical direction at y = 0.46 m.

2.5. Model validation

Under the condition that approach flow depth h₀ is 0.1 m and oblique angle β is 0°, the simulated data of floor pressure in the central section along the flow are compared with the measured data, as shown in Figure 5. The predicted results of CFD model were in good agreement with the measurements, and the average error between the results is 1.35%. In the figure, the approach flow velocity v₀ of the two working conditions is respectively 4 and 5 m/s, the coordinate of the bucket inlet is x = 0 m. When the approach flow velocity v₀ is 4 m/s, the pressure along the flow in front of the inlet does not change, then, when the flow approaching the inlet, the pressure begins to increase, and the increase rate slows down when reaching the inlet, finally, the pressure within the bucket becomes stable for a certain distance, and then decrease to about 0 kPa when approaching the outlet. When the approach flow velocity v₀ is 5 m/s, the pressure distribution characteristics does not change, but the pressure inside the bucket is greater. Whether the approach flow velocity v₀ is 4 m/s or 5 m/s, the simulation data of the two working conditions are consistent with the measured data. The average error is 1.06% when the approach flow velocity v₀ is 5 m/s and 1.64% when the approach flow velocity v₀ is 4 m/s. Therefore, it can be proved that the simulation results accord with the realistic conditions, and it is feasible to use this model for numerical simulation.

Figure 5. Comparison between numerical and measured pressure data at the central of bucket bottom.

3. Results and discussion

The numerical simulation uses the model size. The main research content of this study is to understand the influence of the bucket size and initial flow conditions at the bucket inlet on the distribution of the velocity at the outlet section. In order to ensure the initial flow conditions at the bucket inlet can be controlled, the spillway slope is 0, and there is no specific prototype engineering as a reference object, so there is no fixed scale, but the ratio of the sizes commonly used in prototype engineering is considered. There are the reasons for choosing the model size: first, the main object of this study is the flow velocity distribution of the outlet section, the jet over the bucket and interaction between water and air will not be considered too much; second, when dividing the grid, if the quantity of grids is the same, the size of a grid in prototype is much larger than that in the model, and the simulation accuracy of prototype is weaker than that in the model, if achieved the same accuracy, the quantity of grids will be increased and the calculation time will be greatly increased; third, in the subsequent research plan, relevant model experiments will be carried out, so that better verification and analysis can be carried out under the same size conditions, without considering the influence of scale effect. The definition of the model size satisfies the Froude number similarity criterion, and how to minimise the scale effect is considered, which can satisfy the research requirements.

3.1. Comparison of flow regime between continuous flip bucket and oblique cut flip bucket

The flow regime of continuous flip bucket and oblique cut flip bucket is shown in Figure 6. The figure shows the iso-surface of volume fraction VOF = 0.9. The selected working condition is r = 2.5 m, v₀ = 4 m/s, h₀ = 0.2 m, oblique angle β is 0° (a) and 30° (b) respectively. From Figure 6(a), the nappe of continuous flip bucket is symmetrical, and the nappe has a spreading to the outside, the spread angle on both sides α_s and α_l are about 3°–4°. It is consistent with the results measured by the model experiment in the article (Pfister et al., 2014). The level of spread angle is not too large, but there is spread phenomenon. On the free surface, it can be clearly seen that the flow on each side has phenomenon of being higher, about 0.15 m away from the side wall (y = 0.15 m & y = 0.35 m). The two flow gradually approach the centre, finally converges during the flow process at the position of x = 2 m. From Figure 6(b), the nappe of oblique cut flip bucket falls ahead of time at the short side. The main change occurs at the short side, the range of jet trajectory distance is lengthened compared with the other side, and the spread angle α_s is greater than that of the continuous bucket, about 10°–11°, more than 2 times, and is about 1/3 of oblique angle β. The spread angle at the long side α_l is close to 0°, which is reduced. It indicates that the nappe deflected to the short side. On the free surface, there is still an increase in water depth, also at y = 0.15 m, but it is not symmetrical. Due to the drop of the approach flow at the short side, the position where the water depth increased has changed to y = 0.43 m, the two streams do not converge.

Figure 6. The flow regime (r = 2.5 m, v₀ = 4 m/s, h₀ = 0.2 m): (a) continuous flip bucket (β = 0°); (b) oblique cut flip bucket (β = 30°).

The velocity vector distribution of approach flow is shown in Figure 7. The two figures show the velocity vector distribution of continuous flip bucket and oblique cut flip bucket respectively. The selected working condition is also r = 2.5 m, v₀ = 4 m/s, h₀ = 0.2 m, and with oblique angle β is 0° (a) and 30° (b). Taking the outlet section as the benchmark, the slice distance is 0.3 m, one section is sliced into the bucket, and sliced 5 sections outward on the nappe. The vectors in the large image are the composite vector of v_z and v_y, and the vectors in the small image are the composite vector of v_x and v_y. From Figure 7(a), the velocity vector on each section is symmetrical along the centre, in the bucket, the velocity at centre is about 5–5.5 m/s, the velocity near the boundary is about 3–5 m/s, the velocity at centre of approach flow is greater than the velocity near the boundary. The approach flow in the bucket moves towards the free surface under the action of the side wall, and the velocity direction deflects to the middle after moving to the free surface, resulting in the phenomenon of local water depth increase. With the outflow of water, the velocity on both sides deflects outward, and the water falls downward after moved upward to the highest. The velocity magnitude gradually increases under the action of gravity. From Figure 7(b), the velocity vector in the bucket also points to the free surface from the bottom under the action of the side wall. At the section of end of the short side wall (y = 0.5 m), the direction of velocity on the left bank is obviously deflected, and the velocity on the right bank does not deflect. At the end of the long side wall (y = 0 m), the flow at the short side (y = 0.5 m) falls earlier than that on the other side and has obvious deflection. The velocity magnitude on the oblique cut flip bucket is also greater than that on the boundary, in the bucket, the velocity at centre is 3.5–4.5 m/s, the velocity near the boundary is about 2.5–3.5 m/s, but the velocity magnitude is about 1 m/s less than that of continuous bucket, and the velocity magnitude on the left bank is greater than that on the right bank. The velocity also increases gradually under the action of gravity along the way.

Figure 7. Velocity vector distribution of approach flow (r = 2.5 m, v₀ = 4 m/s, h₀ = 0.2 m): (a) continuous flip bucket (β = 0°); (b) oblique cut flip bucket (β = 30°).

Figure 8 shows the distribution of velocity vector on outlet section, the vectors are the composite vector of v_z and v_y. The two figures show the velocity vector distribution of continuous flip bucket and oblique cut flip bucket respectively. The selected working condition is also r = 2.5 m, v₀ = 4 m/s, h₀ = 0.2 m, and with oblique angle β is 0° (a) and 30° (b). From Figure 8(a), velocity vector distribution of continuous flip bucket is symmetrical along the centre and move to the free surface under the influence of the side wall. From the middle of the flip bucket to the side walls on both sides, the velocity component v_yz increases from 1.17 m/s to 1.22 m/s first and then decreases to 1.09 m/s. The velocity vector gradually deflects outward from the vertical direction, and finally restore to the vertical direction. Take the spread angle α at the bottom as an example, α increases from 0° to 3.8°, and decreases sharply to 0° near the side wall. From bottom to free surface, the velocity component v_yz decreases from 1.22 m/s to 0.84 m/s, the vectors gradually change from outward deflection to inward deflection, α_s and α_l changes from 3.8° to 0° at the position where h is about 0.16 m, then start to deflect inwards. From Figure 8(b), the degree of velocity vector deflection of the oblique cut flip bucket is greater than that of the continuous flip bucket. From the long side (y = 0 m) to the short side (y = 0.5 m), the velocity component v_yz be kept at 0.88 m/s and began to decrease to 0.63 m/s at y = 0.25 m. The spread angle at the short side α_s increases from 0° to 11°, then the deflection decreases at y = 0.35 m. From bottom to free surface, the velocity component v_yz decreases from 0.9 m/s to 0.54 m/s. The vectors at the short side gradually changes from outward deflection to inward deflection, α_s changes from 11° to 0° at the position where h is about 0.18 m, and the vectors at the long side α_l keep at 0°–1°, then it increases to 4° near h = 0.1 m.

Figure 8. Velocity vector distribution of outlet section (r = 2.5 m, v₀ = 4 m/s, h₀ = 0.2 m): (a) continuous flip bucket (β = 0°); (b) oblique cut flip bucket (β = 30°).

The purpose of this section is to have an overall understanding of the velocity distribution of the water flow in the oblique cut flip bucket, especially the deflection of velocity in the lateral direction. Because the velocity condition of the bucket outlet is the main research content, the velocity distribution of the bucket outlet is focused on. The purpose of the analysis of continuous bucket is to understand whether lost the bondage of side wall and bottom plate will have a substantial influence on the spread angle and the way of influence.

3.2. Distribution characteristics of spread angle at outlet section of oblique cut flip bucket

This section focuses on the spread angle distribution at the bucket outlet, which has the same effect as Figure 8. The difference is that this section quantifies the lateral deflection degree of velocity direction through spread angle, so that its distribution can be directly reflected from specific data. As mentioned in Figure 2 above, the spread angle at the outlet can be calculated as: (18) $\begin{array}{r}\alpha =|atan(v_y/v_x)|\end{array}$(18) The lateral distribution of spread angle α in different water depth is shown in Figure 9, the selected working condition is r = 2.5 m, β = 30°, v₀ = 5 m/s, h₀ = 0.2 m. Take the curve when h = 0.01 m as an example, from the long side (y = 0 m) to the short side (y = 0.5 m), the spread angle α decrease from 1° to 0° at 0.05 m, the peak value with a trend of steep first and then slow, the peak value is 7.1°. Finally, there is a steep drop about 0.05 m away from the side wall (y = 0.45 m). When the water depth h changes, the trend of distribution has not changed much, from the long side (y = 0 m) to the short side (y = 0.5 m), the value at y = 0 m decreases from 1° to 0°, each curve decreases to 0° and then increases. The position of α = 0° approach to the side wall (y = 0 m). When h = 0.1 m, the position of α = 0° is y = 0 m, after h = 0.1 m, there is no decrease trend in the curve. The peak value is gradually decreases as it approaches the free surface. From h = 0.01 m to h = 0.12 m, the spread angle decreases from 8.36° to 2.1°. At the position of y = 0.45 m, the spread angle at each water depth has a steep decline trend. The position of steep decline does not change.

Figure 9. The lateral distribution of spread angle in different water depth (r = 2.5 m, β = 30°, v₀ = 5 m/s, h₀ = 0.2 m).

The vertical distribution of spread angle in different lateral section is shown in Figure 10, the selected working condition is also r = 2.5 m, β = 30°, v₀ = 5 m/s, h₀ = 0.2 m. Combined with the previous flow pattern analysis, from the curve of y = 0.45 m, the spread angle α at the bottom is the maximum, the value is 11.3°. The closer to the free surface, the smaller the spread angle, the angle decreases to 0° at h = 0.177 m, and then deflects 2° inward. As the section approaches to the long side (y = 0 m), the distribution of spread angle along the water depth does not change, is the trend of decreasing along the increase of water depth. The difference is that the spread angle α in the bottom and free surface, from y = 0.45 m to y = 0.05 m, the spread angle α in the bottom is getting smaller and smaller, and has decreased to 1° at y = 0.05 m. The spread angle α in the free surface is getting larger and larger. At y = 0.05 m, the spread angle α decrease from 1° to 0°, then increase to 4°, it is consistent with the velocity vector distribution in Figure 8.

Figure 10. The vertical distribution of spread angel in different lateral section (r = 2.5 m, β = 30°, v₀ = 5 m/s, h₀ = 0.2 m).

In practical engineering, the actual spread angle may be caused by the maximum spread angle α_max at the outlet, so selected α_max as the characteristic value for research. From the above result, the maximum spread angle α_max is at the position of the bottom near the side wall at the short side, the specific position is the peak value in the position of one mesh higher than bottom (h = 0.002 m).

3.3. Influence of flip bucket size on spread angle distribution

Since the change of water depth will not change the lateral distribution of spread angle, and the maximum spread angle α_max is located at the bottom of bucket, so the research object in this part selects the distribution at the bottom of bucket. The influence of oblique angle β on the distribution of spread angle α at the outlet is shown in Figure 11, the selected working condition is r = 2.5 m, v₀ = 5 m/s, h₀ = 0.1 m, and different β = 0°–45°. The curve of β = 0° represent the lateral distribution of the spread angle at the outlet of the continuous flip bucket, the distribution of spread angle α is symmetrical along the flip bucket centre. From y = 0 m to y = 0.5 m, there is a steep increase of spread angle α near the side wall, and achieved peak in y = 0.04 m, the peak value is 4.2°, then α decreases to 0° at y = 0.25 m, finally increased to the other peak value at y = 0.46 m and have steep decrease after achieved peak, the peak value is 4.2°. Use α_smax (subscript smax) and α_lmax (subscript lmax) to represent the peak values of the left and right banks respectively, α_smax is equal to α_lmax when β = 0°. When the bucket is changed into oblique cut flip bucket, from the curve at 15°, the spread angle starts to be biased. From y = 0 m to y = 0.5 m, the distribution characteristics is resembled, there is a steep increase of α near the side wall, and achieved peak in y = 0.04 m, α_lmax = 2°, then α decreases to 0° at y = 0.07 m, finally increased to the other peak at y = 0.46 m and have steep decrease after achieved peak, α_smax = 6.3°, the maximum spread angle α_max = α_smax. But the increase trend is to increase first, then remain unchanged and then increase, different from that in continuous flip bucket. It can be seen that α_lmax is less than that with β = 0°, and α_smax is more than that with β = 0°, the changed value Δα = 2°, about 50% change. The position of α_smax and α_lmax is not changed, but the position of α = 0° start approaching y = 0 m. When the oblique angle β increases to 30° and 45°, the distribution is also not changed, the trend is also decreased first and then increased, with a steep increase and decrease neat the side wall. But as the angle β increases, the unchanged part in the process of increasing is getting closer to the peak, and extremely close to peak position when β = 45°. The position of α_max is not changed, but the position of α = 0° has changed to 0.05 m when β = 30° and disappeared when β = 45°. So when β = 45°, there is no longer a peak and a process of decrease on the right bank, increase from α = 1° at y = 0 m. The peak value of spread angle on the right bank α_lmax decreases from 4.2° to 1.6°, disappear when β = 45°. And the peak value of spread angle on the left bank α_smax increases from 4.2° to 10°, about 58% reduction, have a large distinguish.

Figure 11. Lateral distribution of spread angle at the outlet under different oblique angles (r = 2.5 m, v₀ = 5 m/s, h₀ = 0.1 m).

The influence of radius r on the distribution of spread angle α at the outlet is shown in Figure 12, the selected working condition is β = 30°, v₀ = 5 m/s, h₀ = 0.1 m, and different r = 2.0–3.5 m. When the radius r = 2.0 m, from y = 0 m to y = 0.5 m, there is a steep increase of α near the side wall, and achieved peak in y = 0.02 m, α_lmax = 2°, then α decreases to 0° at y = 0.05 m, and then increased to the other peak at y = 0.46 m, the increase trend is to increase first, then remain unchanged and then increase, finally have steep decrease after achieved peak, α_smax = 7.4°, the maximum spread angle α_max = α_smax. As the radius r increases, the four curves almost overlap, the position of the peak value in left and right bank α_smax and α_lmax is not changed, located at y = 0.02 m and y = 0.46 m respectively. And the position of α = 0° is also not changed, located at y = 0.05 m. The peak value in left bank α_smax decreased from 7.4° to 6.8°, the changed valued of α_smax between two curves is 0.2°, the deviation is less than 10%, so the change of radius will not change α_max. The peak value in right bank α_lmax decreased from 1.74° to 1.02°, the change value compared to α_max ­is less than 10%. Therefore, the change of radius will not affect the distribution of spread angle on the outlet.

Figure 12. Lateral distribution of spread angle at the outlet under different radius (β = 30°, v₀ = 5 m/s, h₀ = 0.1 m).

3.4. Influence of inflow condition on spread angle distribution

The influence of approach flow velocity v₀ on the distribution of spread angle α at the outlet is shown in Figure 13, the selected working condition is r = 2.5 m, β = 30°, h₀ = 0.1 m, and different v₀ = 4–12 m/s. When the approach flow velocity v₀ = 4 m/s, from y = 0 m to y = 0.5 m, the spread angle α increases from 0.5° to 8.81° with the gradient decreases gradually, the spread angle α remain unchanged after y = 0.3 m, then increases at y = 0.4 m, and finally have steep decrease after y = 0.46 m. As the velocity v₀ increases, the distribution characteristic has some changes. The reduction of the intermediate spread angle is greater than that near the side wall, so the peak value on the left bank α_smax is more and more obvious, and the position of α_smax is not changed, at y = 0.46 m. The position of α = 0° is y = 0 m when v₀ = 4 m/s, as the velocity v₀ increases, the position gradually far away from the side wall (y = 0 m), and the peak value on the right bank α_lmax began to appear, the position of α_lmax is all y = 0.02 m. The peak value in right bank α_lmax increased from 1.6° to 2.5°, there is no peak value when v₀ = 4 m/s. The peak value in left bank α_smax decreased from 11.7° to 5.3°, about 53% reduction, have a large distinguish. And the maximum spread angle α_max = α_smax. The influence of the velocity v₀ change on the spread angle becomes smaller and smaller as the velocity v₀ increases, when v₀ = 10 m/s and v₀ = 12 m/s, the two curves close to overlap. The maximum spread angle α_max is 5.8° and 5.3° when v₀ = 10 m/s and v₀ = 12 m/s, only 0.5° difference, the deviation is just 9%, only a small difference.

Figure 13. Lateral distribution of spread angle at the outlet under different approach flow velocity (r = 2.5 m, β = 30°, h₀ = 0.1 m).

The influence of approach flow depth h₀ on the distribution of spread angle α at the outlet is shown in Figure 14, the selected working condition is r = 2.5 m, β = 30°, v₀ = 5 m/s, and different h₀ = 0.05–0.2 m. When the approach flow depth h₀ = 0.2 m, from y = 0 m to y = 0.5 m, there is a steep increase of α near the side wall, and achieved peak in y = 0.02 m, α_lmax = 1.1°, then α decreases to 0° at y = 0.05 m, and then increased to the other peak at y = 0.45 m with the gradient decreases gradually, finally have steep decrease after achieved the peak, α_smax = 9.1°, the maximum spread angle α_max = α_smax. As the depth h₀ decreases, the distribution characteristic has some changes. The reduction of the intermediate spread angle is greater than that near the side wall, so the peak value on the left bank α_smax is more and more obvious, and the position of α_smax is not changed, at y = 0.46 m. The position of α = 0° is not changed, all are located at y = 0.05 m, and the peak value in right bank α_{lmax ­}is also not changed, at y = 0.02 m. The peak value in right bank α_lmax increased from 1.1° to 1.6°, the change value compared to α_max ­is less than 10%. The peak value in left bank α_smax decreased from 9.1° to 5.3°, about 45% reduction, have a large distinguish.

Figure 14. Lateral distribution of spread angle at the outlet under different approach flow depth (r = 2.5 m, β = 30°, v₀ = 5 m/s).

These two sections are to study the change of the bucket size and approach flow condition how to influent the distribution of lateral spread angle α at the bucket outlet. The influence on the distribution characteristics, the position of the maximum spread angle α_max and 0° spread angle is the focus. On the one hand, it is to determine the characteristic position of the spread angle, so that these effects can be quantitatively analysed later. On the other hand, in the subsequent research, the pressure distribution in the oblique cut flip bucket will be studied, and the reasons for the formation of the spread angle will be combined with the distribution characteristics for analysis.

3.5. Influence of flip bucket size on maximum spread angle

From the above analysis of spread angle α distribution, it can be known that the maximum diffusion angle is located in the bottom near the side wall of the short side, only a few working conditions have a large range of maximum values, its specific location cannot be determined. So the maximum spread angle select the maximum value keep a mesh from the bottom, is h = 0.02 m. The relationship between the maximum spread angle α_max and the radius r is shown in Figure 15, The vertical axis compares the maximum spread angle with oblique angle to obtain the specific spreading degree, the expression is α_max / (90° – β), the reason why the denominator selects (90° – β) is to indicate the working condition at β = 0°. The selected working condition is the approach flow velocity v₀ = 5 m/s, oblique angle β = 15°–45° and the approach flow depth h₀ = 0.05–0.2 m. When the approach flow depth h₀ = 0.2 m, and oblique angle β = 45°. As the radius r is from 2.0 m to 3.5 m, the maximum spread angles α_max are respectively 0.30, 0.31, 0.30 and 0.31 times of (90° – β), only changed 0.01 times of (90° – β), about 3%, can almost be ignored. The other working conditions is also like this, the change of the maximum spread angle α_max between two points is 0.01° – 0.93°, just changed 0.2% – 10%, no more than 10%. The working conditions in this article all adopted that can be fully discharge, so it can be considered that the change of radius r will not lead to the change of maximum spread angle α_max when the flow can be fully discharge.

Figure 15. The relationship between the ratio of the maximum spread angle to oblique angle and the radius.

The relationship between the maximum spread angle α_max and the oblique angle β is shown in Figure 16. The vertical axis also is α_max / (90° – β), made α_max dimensionless, and the horizontal axis is sin β, made β dimensionless. The selected working condition is the radius r = 2.5 m, the approach flow velocity v₀ = 4–6 m/s and the approach flow depth h₀ = 0.05–0.2 m. In the figure, different lines represent different approach flow depths, and different point patterns represent different approach flow velocities. When the approach flow velocity v₀ = 5 m/s, and the approach flow depth h₀ = 0.2 m, as the β from 0° to 45°, the angle is larger and larger. The maximum spread angle α_max is 0.05 times of (90°- β) when β = 0°, α_max ­ 4.5°. When β increases to 45°, the maximum spread angle α_max increases to 0.31 (90°- β), α_max ­ 14.05°, the changed value is 10°. From other curves, when the oblique angle β is small, the influence of approach flow condition is also not too much. There is almost no change when β = 0°. The all maximum spread angle α_max when β = 0° is 0.05 (90°- β) or 0.04 (90°- β). As the β increases, the gap of α_max in different approach flow depth or approach flow velocity with same oblique angle β is larger and larger. This shows that, when β is large, the influence of the change of approach flow condition will be larger. Influence of approach flow condition on maximum spread angle.

Figure 16. The relationship between the ratio of the maximum spread angle to oblique angle and the oblique angle.

3.6. Influence of inflow condition on maximum spread angle

In this article, the approach flow condition selected approach flow velocity v₀ and approach flow depth h₀. From the previous analysis, the increase of v₀ and h₀ has opposite effect on the maximum spread angle α_max, in combination with the formula of the Froude number Fr = v₀ / (gh₀) ^1/2, so it can be using the Froude number Fr to represent the approach flow condition. Froude number Fr represent the ratio of inertial force to gravity, to a certain extent, the Fr can explain the effect of approach flow velocity and approach flow water depth on spread angle. Therefore, Fr is selected as the dimensionless parameter of h₀ and v₀. The relationship between the maximum spread angle α_max and the Froude number Fr is shown in Figure 17. The vertical axis also is α_max / (90° – β), and the horizontal axis is Fr. The selected working condition is the radius r = 2.5 m, the oblique angle β = 0°–45°. When β = 45°, as the Fr decrease from 3.85 to 17.13, the maximum spread angle α_max gradually increases from 0.35 (90° – β) to 0.1 (90° – β), and the gradient gradually decreases, α_max does not change after Fr = 12. The change degree of α_max will decrease after the velocity is too large or the water depth is too small, as mentioned in the previous artis. When the oblique angle β increases, the trend also decreases as the Fr decreases with the gradient gradually decreases. The initial gradient of the curve gradually decreases with the increase of β, there is almost no change of the maximum spread angle α_max after β = 0°, currently, α_max is 0.04 (90° – β) or 0.05 (90° – β), about 3.0° – 4.5°. It can be known from Figure 17 that the maximum spread angle α_max decreases as the Froude number Fr increase with the trend of slope decreasing. The curve under different oblique angles β have the same decreasing trend. The maximum diffusion angle distribution under different oblique angles β is fitted through Eq. 19. The coefficients K₁ and K₂ in Eq. 19 are calculated by Eq. 20 and Eq. 21. (19) $\begin{array}{rl}{\alpha }_{max}/({90}^{\circ }-\beta )& =K_{1}F{r}^{K_2}\end{array}$(19) (20) $\begin{array}{rl}{K}_1& =0.065e^{3.5sin\beta }\end{array}$(20) (21) $\begin{array}{rl}{K}_2& =0.82sin\beta +0.21\end{array}$(21) where α_max is the maximum spread angle at the outlet section of flip bucket. The β is oblique angles. The K₁ and K₂ are coefficients related to the oblique angle. Fr is the Froude number.

Figure 17. The relationship between the ratio of the maximum spread angle to oblique angle and the Froude number.

Figure 18. Comparison between original value and calculated value.

Set Fr and β into the above equation and compare the calculated data with the original data obtained from numerical simulation. The result is shown in Figure 18. In Figure 18, the solid line is the reference line, and its equation is y = x, which is used to verify the difference between the original simulation results and the calculated results. If the point set is closer to the reference line, then the fitting result of the formulas is good; if the point set is far away from the reference line, then the fitting result of the calculated formula is poor, and maximum spread angle α_max cannot be accurately calculated. The horizontal axis is the calculated value, and the vertical axis is the original simulation value. If the point is below the line, the calculated result is too large; if it is above the line, the calculated result is too small. Different point styles represent the maximum spread angles α_max under different oblique angles. As shown in the Figure 18, both the original value and the calculated value of the maximum spread angle α_max at the bucket outlet can be well distributed on the line y = x. The error between the calculated value and the original value is 0.02%–18%, none greater than 20%, the average error is 6%, the average angle difference is 0.11°, only a few cases have an error of more than 10%, and the large error mainly located at the position of small maximum spread angles α_max. The error calculation method is: |(calculated value – original value) / original value ×100%|. The result shows that Eq. 19 to Eq. 21 fit well and the maximum spread angles α_max of the bucket outlet can be calculated accurately.

4. Conclusion

In this study, the discharge of oblique cut flip bucket is calculated by numerical simulation. Compare and analyse the calculation results under deferent working conditions. The main objects are flow pattern, flow field and the distribution characteristics of spread angle. The findings are summarised as follows:

From the analysis of the flow pattern and flow field of the oblique cut flip bucket, it can be known that the oblique cut flip bucket can effectively increase the diffusion of the nappe, the diffusion at the short side increase, the diffusion at the long side decrease. After losing the bound of the side wall and bottom plate, the flow falls ahead of time, and the velocity direction begins to gradually deflect inside the bucket. The deflection degree of the velocity direction along the flow gradually increases, and the water near the side wall will be forcibly squeezed to the free surface from both sides. The overall velocity v is smaller than that in the continuous bucket.

Obtained the distribution characteristics of the spread angle at the outlet of bucket, had an overall understanding of the distribution of the velocity direction at the outlet. From the long side wall to the short side wall, the spread angle α is basically 0° near the long side wall, and the flow moves from the bottom plate to the free surface. The closer getting to the short side wall, the greater spread angle α is, until it approaches the short side wall, spread angle α decreases. From the bottom plate to the free surface, spread angle α decreases gradually. The distribution of velocity magnitude v is similar to the spread angle. From distribution characteristics of spread angle, it can be known the position of maximum spread angle α_max is at the bottom near the short side wall.

The change of oblique angle β, approach flow depth h₀ and approach flow velocity v₀ will affect the distribution of spread angle. Bucket size and approach flow condition both can be not affect the position of maximum spread angle α_max, but they can affect the magnitude of maximum spread angle α_max. Besides the radius r, the oblique angle β, approach flow velocity v₀ and approach flow depth h₀ can affect the distribution characteristic, among them, the oblique angle β and approach flow velocity v₀ can affect the position of spread angle α = 0°.

For the influence of bucket size, the influence of the bucket radius r on the maximum spread angle α_max is small, and its change rate is less than 10%, which can be considered as the error. Therefore, it is believed that the bucket radius r does not affect the maximum diffusion angle α_max when the approach flow can be fully discharge. The oblique angle β is positively correlated with the maximum spread angle α_max. For the influence of flow condition, the approach flow velocity v₀ and approach flow depth h₀ are positively and negatively correlated with the maximum spread angle respectively. If the flow condition is represented by Froude number Fr, the larger the Froude number Fr, the smaller the maximum spread angle α_max. The calculation equations of maximum spread angle α_max related to Froude number Fr and oblique angle β can be fitted, they are Eq.19 to Eq.21. The mean fitting error is 6%. The equations provide some basis for the calculation method of spread angle.

This research studies the deflection of the velocity direction at the outlet of the bucket on the horizontal plane, obtained the distribution characteristics of the inner spread angle, and provided a calculation method of maximum spread angle α_max related to the oblique angle β and Froude number Fr, which provides a part of basis for the calculation of nappe width. However, this study only used numerical simulation for analysis, there is no complete model experiment to prove the relationship between the actual nappe width and the velocity distribution at the bucket outlet, it is still in the hypothesis stage. In addition, due to the scale effect and the air–water interaction, the specific diffusion range of nappe cannot be accurately predicted in this study.

Subsequent studies will start with the pressure distribution characteristics in the bucket, analyse the causes of the velocity distribution characteristics at the bucket outlet, so as to analyse the velocity distribution characteristics under other bucket types from the perspective of force. At the same time, relevant model experiment will be carried out to determine the relationship between the actual nappe width and the velocity distribution at the bucket outlet. After completing the relevant research on the model scale, the difference between the prototype and the model will be studied at last, obtained a complete prediction scheme of the nappe diffusion width.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the National Natural Science Foundation of China under grant [number 52079084; number 51939007].