Analysis of Piano Key Weir Drainage Characteristics

Abstract

The piano key weir is a new type of labyrinth weir structure form, its advantages such as stable structure, upstream and downstream inverted overhang and small base area greatly improve the scope of application, and it has high discharge capacity and overflow efficiency, which is considered as a very effective method to solve the shortage of reservoir and dam discharge flow, and has more applications abroad. In this paper, the experimental observation of the hydraulic characteristics of the discharge from the hydraulic physics model of the piano key weir confirms its good overflow capacity, but also limits the study of its complex flow. Combined with the numerical simulation software, the three-dimensional flow field of the piano key weir is numerically simulated based on the RNG k-ɛ turbulence model and the free-surface VOF technique to analyze the changes of hydraulic characteristics such as flow patterns, flow lines and flow velocities, and to clarify the hydraulic characteristics of the spillway and the mechanism of the improved spillway capacity by combining the experimental results of the physical model. In addition, a comparative analysis of the overflow capacity and flood discharge efficiency between the piano key weir and the traditional thin-walled weir is carried out to better reflect the advantages of the piano key weir in flood discharge by using the over-discharge ratio, and to discuss the application of the piano key weir in actual diseased reservoir and dam projects.

1 Introduction

In recent years, global climate change has intensified and extreme weather has emerged frequently, resulting in severe flooding, frequent occurrence of mega-floods and super-standard floods, and the emergence of extreme hydrological events has raised higher requirements for flood safety of reservoirs and dams. Piano key weir is a new type of labyrinth weir, which is structurally wide at the top and narrow at the bottom, with upside down weir walls hanging upside down, invented by French Hydroccop Association in 2003, the first piano key weir was constructed in the summer of 2006 in Goulours, Southwest France (Guo et al. 2014). As the weir axis is elongated along the longitudinal direction, several connected side weirs are formed and the overflow front is increased accordingly. When the head on the weir is the same, the overflow capacity of the piano key weir is increased several times compared with the ordinary linear weir, which can significantly improve the discharge capacity and flood discharge efficiency.

The piano key weir is wide at the top and narrow at the bottom, resembling the piano keys in appearance. The piano key weir is a high super-stationary structure from the structural mechanics point of view, with good overall balance and it has a wide range of applications (Lempérière and Ouaman 2003) due to its small footprint and upside down design structure. The symbols and meanings of the geometric parameters of the piano key weir are as follows: L is the total length of the overflow leading edge of the piano key weir; W is the weir width of the piano key weir, Wi is the weir inlet width, W_o is the weir outlet width; B is the total upstream and downstream length of the piano key weir, the side weir length, B_i is the downstream inverted overhang length of the inlet; B_o is the upstream inverted overhang length of the outlet; B_b is the length of the base; P is the weir height, P_d is the base height; S_i is the inlet inverted overhang angle, S_o is the inverted overhang angle of the outlet (Neelakantan et al. 2019). The geometric schematic diagram of the piano key weir is shown in Fig. 1.

Fig. 1.

Geometric diagram of the piano key weir

In this paper, we observe the over-weir flow pattern and drainage capacity of the piano key weir through physical model tests, and analyze the advantages of the piano key weir in terms of drainage compared to traditional thin-walled weirs (Jiang et al. 2019). Due to the complex flow pattern of the weir, the model test alone cannot fully analyze its superior drainage efficiency mechanism. numerical simulation is used as a supplement to study the drainage characteristics in terms of flow pattern and flow velocity, and research the mechanism for improving the drainage of the weir, so as to provide a reference for the optimization of the weir body shape and parameter sensitivity analysis.

2 Physics Model Experiment

2.1 Experimental Setup

The experimental device was built in the hall of Tiexinqiao test base of Nanjing Hydraulic Research Institute, and the experimental device consisted of water measuring weir and water tank, as shown in Fig. 2 below. During the experiment, the water from the groundwater reservoir is pumped to the upstream water measuring weir, and the water flow is stabilized by the water measuring weir free overflow sink, and finally the water flow is stabilized again by the elimination wall to the downstream sink, and the water flows through the sink to reach the model of the piano key weir, and then the overflow flows through the model and finally flows into the groundwater reservoir through the sink, forming a circulatory system throughout.

Fig. 2.

Experiment layout

The model of the piano key weir is made of 3 mm organic plastic board, and the specific parameters are shown in Table 1. The model is 6 inlet chambers and 5 outlet chambers with inverted A-type inverted structure upstream and downstream, and the model is installed in the middle of the 16 m-long downstream flume, and the model centerline coincides with the flume centerline in the incoming flow direction, and the model is spaced 0.81 m from both ends of the flume to the left and right. The experiments were conducted to study the hydraulic characteristics of the weir and to compare the discharge capacity with that of the traditional thin-walled weir, and reasonable body parameters were selected according to the conclusions of previous studies by experts and scholars (Lemperiere and Guo 2005): the spreading ratio L/W = 5.34, the tangent of the upstream and downstream inversion angle Tan Si = 0.53, and the tangent of the upstream and downstream inversion angle Tan S_o = 0.53. Tan S_o = 0.53, upstream/downstream inversion ratio B_i/B_o = 1.06, inlet/outlet width ratio W_i/W_o = 1.23, and weir height to weir width ratio P/W = 0.16.

Table 1. Dimensions of the piano key weir model (Unit: m)

2.2 Experimental Results

In order to observe the flow pattern and discharge capacity of the piano weir through physical model tests and to compare it with the traditional thin-walled weir, 12 sets of water crossing experiments were conducted throughout the experiment, and data such as flow rate, syringe reading, and travel velocity were measured and recorded, and the head on the weir was obtained from the water level syringe reading, and the total head on the weir was calculated from the measured travel velocity. The ratio of the total head on the weir to the weir height can be obtained by combining the total head on the weir with the weir height, which is known from the previous measurements of piano key weir. The specific data are shown in Table 2 below.

Table 2. Experimental data table

In order to compare the overflow capacity of the piano key weir and the traditional linear weir, it is assumed that the position of the piano key weir in the flume is replaced by a thin-walled weir of equal width and height (Qi et al. 2018), and the overflow flow of the thin wall is projected with the same total head on the weir, and the ratio of the flow of the piano key weir to the flow of the thin-walled weir is expressed as the over-discharge ratio: r, $r=\frac{{Q}_{PKW}}{{Q}_{S}}$ and the over-discharge ratio of the two is shown in Table 3 below.

The discharge expression of the piano key weir is similar to the thin-walled weir flow equation through the dimensional analysis, and the flow equation of both is as follows.

$${Q}_{PKW}={C}_{d}W\sqrt{2\mathrm{g}}{H}^{3/2}$$

(1)

$${Q}_{S}={m}_{0}W\sqrt{2g}{H}^{3/2}$$

(2)

where: ${C}_{d}$ is the piano key weir flow coefficient, W is the piano key weir width, H is the total head on the piano key weir, $H={h}_{0}+\frac{{V}^{2}}{2g}$, ${h}_{0}$ is the head on the piano key weir, V is the traveling flow rate, ${m}_{0}$ is the thin walled weir flow coefficient, ${m}_{0}=0.4034+0.0534\frac{H}{{P}_{1}}+\frac{0.0007}{H}$, ${P}_{1}$ is the sum of the weir height and the base height, ${P}_{1}=P+{P}_{d}$.

Table 3. Over-discharge ratio of piano key weir and thin-walled weir

2.3 Analysis of Drainage Capacity

In order to compare the discharge capacity of the piano key weir and the traditional thin-walled weir, the flow rate of the piano key weir and the thin-walled weir at the same head on the weir, and the flow coefficient of the piano key weir and the thin-walled weir at the same head on the weir are made as follows, and the relationship between the over-discharge ratio and the head on the weir and the weir height ratio is made as follows (Barcouda et al. 2006) which can visually show the comparative relationship between the discharge of the piano weir and the thin-walled weir.

Fig. 3.

The relationship between the piano key weir and thin-walled weir flow coefficient, flow rate and total water head on the weir

Fig. 4.

The relationship between over-discharge ratio and H/P

As can be seen from Fig. 3, the flow coefficient and discharge volume of the piano key weir are significantly higher than those of the thin-walled weir at the same total head on the weir, and the piano key weir has a better discharge capacity than the thin-walled weir. As can be seen from Fig. 4, the piano key weir has a high discharge efficiency than the thin-walled weir, and the overflow ratio is basically maintained at 3–4 times, with the increase of upstream flow, the overflow ratio of the piano key weir shows a decreasing trend, the decrease of the piano key weir discharge capacity is because the overflow effect of the outlet chamber and the side weir of the inlet chamber receives a weakening when the upstream flow increases, but overall the piano key weir has a better overflow capacity compared with the thin-walled weir.

3 Mathematical Model Setup and Result Analysis

3.1 Model Building and Numerical Processing Methods

3.1.1 Model Setup

In order to study the hydraulic characteristics of the discharge of the piano key weir in more detail, the above physical model was used as a prototype and modeled 1:1 using software, and finally imported into numerical simulation software for simulation calculations. The simulated calculation model of the piano key weir is as follows, which consists of upstream channel, weir body and downstream channel (Li et al. 2015). The length of the piano key weir body in the direction of water flow is 630 mm, the inlet court is 135 mm, the outlet court is 110 mm, and the width of the weir body is 785 mm. In order to ensure the smooth flow over the weir and better simulate the water tongue as well as the actual situation, the length of the upstream channel is 800 mm and the length of the downstream channel is 800 mm in the calculation model 2000 mm.

The grid quality, quantity and division method have an important influence on the accuracy of the simulation results. In the numerical simulation of pian key weir, the total area grid number is 387.7946 million, the calculation file size asks 12 GB, the finish time is 50 s, and the calculation time is about 60 h, among which the grid number of PKW area is 107.2537 million and the grid size is 0.005 m.

The simulated area is divided into the upstream channel area, the pian key weir area and the downstream channel area according to the overall area and the piano key parts. The upstream channel area and the downstream channel area have relatively regular flow patterns, so the grid is relatively sparse. The weir area and the area behind the weir are the piano key parts of the study, so a denser grid is generated to ensure the accuracy of the simulation (Li et al. 2016).

Fig. 5.

The mathematical model setting of the piano key weir

3.1.2 Turbulent Flow Equation and VOF

Combined with the flow characteristics of the experimental piano key weir, the turbulent flow mathematical model is selected as the RNG k-ε equation model (Jiang 2020). The RNG k-ε equation is a mathematical model derived using the statistical method of reformed group theory for the instantaneous N--S equation, compared with the standard k-ε model, the RNG k-ε model adds a term to the ε equation that reflects the mainstream time-averaged strain rate and improves the accuracy of high-speed motion (Zhao et al. 2020). In addition, the RNG k-ε model takes into account the effect of vortices on turbulent flow and eliminates the singularity of the standard k-ε model, which combined makes the RNG k-ε equation more accurate and reliable in simulating jet diffusion, etc. The k and ε equations are as follows.

$$\frac{\partial (\rho k)}{\partial t}+\frac{\partial (\rho k{u}_{i})}{\partial {x}_{i}}=\frac{\partial }{\partial {x}_{i}}\left[\frac{{\mu }_{t}}{{\sigma }_{k}}\frac{\partial k}{\partial {x}_{i}}+\mu \right]+{G}_{k}-\rho \varepsilon $$

(3)

$$\frac{\partial (\rho \varepsilon )}{\partial t}+\frac{\partial (\rho \varepsilon {u}_{i})}{\partial {x}_{i}}=\frac{\partial }{\partial {x}_{j}}\left[\frac{{\mu }_{t}}{{\sigma }_{\varepsilon }}\frac{\partial \varepsilon }{\partial {x}_{j}}+\mu \right]+\rho {C}_{1}S\varepsilon -\rho {C}_{2}\frac{{\varepsilon }^{2}}{k+\sqrt{\nu \varepsilon }}$$

(4)

where: ρ is the fluid density, t is time, k is the turbulent kinetic energy, ε is the turbulent kinetic energy dissipation rate, ${u}_{i}$ is the velocity component, $x_{i}$, $x_{j}$ are the coordinate components, $\nu$ is the viscosity, $\nu =\frac{\mu }{\rho }$, $\mu$ is the coefficient of viscosity, ${\sigma }_{k}=1.0$, ${\sigma }_{\varepsilon }=1.2$, ${G}_{k}$ is the turbulent kinetic energy generation term due to the average velocity gradient, ${G}_{k}={\mu }_{t}{S}^{2}$, $S=\sqrt{\left(2{E}_{ij}{E}_{ij}\right)}$, ${E}_{ij}=\frac{1}{2}\left(\frac{\partial {u}_{i}}{\partial {x}_{j}}+\frac{\partial {u}_{j}}{\partial {x}_{i}}\right)$, the coefficient ${C}_{1}=max\left[0.43,\frac{\eta }{\eta +5}\right]$, $\eta =\frac{Sk}{\varepsilon }$, ${C}_{2}=1.9$, ${\mu }_{t}$ is the turbulent viscosity, ${\mu }_{t}=\rho {C}_{\mu }\frac{{k}^{2}}{\varepsilon }$, ${C}_{\mu }=\frac{1}{{A}_{0}+\frac{{A}_{s}{U}^{*}k}{\varepsilon }}$, ${A}_{0}$ = 4.0, ${A}_{s}=\sqrt{6}\mathrm{cos}\phi$, $\phi ={\mathrm{cos}}^{-1}\left(\sqrt{6}\omega \right)$, $\phi =\frac{{E}_{ij}{E}_{jk}{E}_{ik}}{\sqrt{\left({E}_{ij}{E}_{ij}\right)}}$, ${U}^{*}=\sqrt{{E}_{ij}{E}_{ij}+{\stackrel{\sim }{\Omega }}_{ij}{\stackrel{\sim }{\Omega }}_{ij}}$, ${\stackrel{\sim }{\Omega }}_{ij}={\Omega }_{ij}-2{\varepsilon }_{ijk}{\omega }_{k}$, ${\Omega }_{ij}={\stackrel{\sim }{\Omega }}_{ij}-{\varepsilon }_{ijk}{\omega }_{k}$.

The flow pattern is complex during the release of the piano key weir, and the method VOF (Fan et al. 2022) is used in the capture of the free liquid surface, which is constructed using the volume ratio function $F\left( {x_{i} ,\,t} \right)$, and the $F\left( {x_{i} ,\,t} \right)$ function is as follows.

$$\frac{\partial F}{\partial t}+{u}_{i}\frac{\partial F}{\partial {x}_{i}}=0$$

(5)

When: $F\left( {x_{i} ,\,t} \right) = 0$, it means the place is water phase; when $0 < F\left( {x_{i} ,\,t} \right) < 1$, it means the place is water-gas phase intersection; when $F\left( {x_{i} ,\,t} \right) = 1$, it means the place is gas phase.

3.1.3 Model Boundary Settings

Set the boundary conditions of the model before simulation, set the specified pressure at the upstream inlet, and set the fluid elevation; set the outflow at the outlet: the two walls and the bottom are set as non-sliding wall; the top is set as atmospheric pressure inlet boundary condition, see Fig. 5 above for the specific settings (Li 2020).

3.2 Numerical Simulation Results and Analysis

3.2.1 Drainage Capacity Model Validation

In order to verify the numerical simulation of the discharge capacity of the piano key weir, by giving different working conditions of the head on the weir, the respective discharge volume was calculated by simulation, and the error between the simulated and experimental values was calculated, the experimental data and the data derived from the simulation are shown in Table 4 below, and Fig. 6 can more visually reflect the error between the experimental and simulated values.

Table 4. Data table of experimental and simulated values

Fig. 6.

Relationship between the experimental and simulated values of the piano key weir

By comparing the flow value and simulation value of this model experiment, it can be found that the flow value calculated by numerical simulation of head on the same weir is basically linearly distributed with the experimental value and the values of the two curves are very close, and the error values of both are distributed around the straight line with slope k = 1, and the error values are between the two straight lines of −5% and +5%, and the results of numerical simulation are more reliable and meet the requirements of numerical simulation of hydraulics.

3.2.2 Flow Analysis

The following figure shows the profiles of the inlet chamber and outlet chamber of the weir and four profiles in the direction of the incoming flow of the weir. The flow pattern has the characteristics of ternary flow. When the water flows through the piano key weir, the water flows partly from the inlet, and the water flows downstream along the bottom plate in the inlet chamber and freely overflows to the downstream sink at the outlet, and the water flows in the inlet chamber also overflows from the side weir to the outlet chamber. Some of the water flows in from the exit overflow, collide and mix with the water flowing down the side weir of the incoming palace chamber, flow downstream along the bottom plate of the outgoing palace chamber, and are ejected at the exit. The overall flow over the weir is a mixture of forward and lateral flow, the flow pattern is complex, and the water flows collide and mix with each other, the flow pattern of the whole process can be seen in Fig. 7 below.

Fig. 7.

Schematic diagram of the flow pattern of water over the weir of the piano key weir

During the simulation, it was found that when the flow rate of water over the weir was small, the water surface line upstream of the piano key weir was basically the same as the water surface line of the flume, and the water flowed smoothly through the piano key weir, producing only a small drop in water level at the entrance and exit of the palace, and it could be found from the profile feature map in the direction of incoming flow that the water tongue overflowing from the side weir to the exit palace chamber was detached from the side wall, and the lateral water flow was free to flow out, and the lateral water tongue was thin, and the adjacent side walls were free to flow out Tongue of water does not interfere with each other into the exit palace room, in the middle of the front of the exit palace room to stimulate the final convergence of water outflow. The flow pattern at the exit of the exit chamber is more complex, the water jet at the exit while the water will spread down and left and right, and again collide and mix with the water flowing down from the overflow of the incoming palace, the water diffusion at the exit of the exit chamber is intense, and the flow pattern is chaotic.

3.2.3 Flow Rate Distribution

The flow pattern of the water over the weir is complex, and the velocity of the water will change when passing the weir. The velocity contour clouds of the inlet and outlet chambers are derived from the numerical simulation, which can reflect the magnitude of the fluid velocity visually. From the simulation results, it can be seen that the upstream of the weir is smooth and the velocity is small. The water flows into the inlet chamber first and the velocity starts to increase, and then picks up at the end of the inlet chamber in the form of a water tongue (Fig. 8).

Fig. 8.

Schematic diagram of the inlet and outlet speed of the water head on different weirs

From the above figure, it can be seen that the overall flow velocity of the piano key weir is small at low head, and the water can flow smoothly and evenly along the palace chamber and side walls, and finally overflows out from the weir at the outlet, with good overall flow. With the increase of the head on the weir, the flow velocity of the water over the weir also increases, indicating that the increase of the head on the weir can increase the flow velocity of the water over the weir. However, while the head on the weir increases, some areas with lower flow velocity appear at the entrance and exit of the palace, and these areas obstruct the water flow under the action of the inertia of the water flow, which reduces the drainage capacity of the piano key weir, which is the reason why the head on the piano key weir increases and the flow velocity increases but the drainage capacity decreases.

4 Conclusion

As a new type of labyrinth weir, the piano key weir has a good application prospect in flood control and drainage. In this paper, the flow pattern and discharge capacity of the piano key weir are studied through physical model tests, and the discharge capacity of the piano key weir is compared with that of the thin-walled weir and the over-discharge ratio of the piano key weir. The numerical simulations of the discharge capacity of the piano key weir were compared with the physical simulations, and the flow patterns matched with the experimental flow patterns, which verified the reliability of the simulations. The simulation analysis of the flow velocity of the piano key weir shows that the increase of head on the weir can increase the flow velocity of the water over the weir, but the increase of head on the weir also decreases the discharge efficiency, which provides a reference for the optimization of the piano key weir shape to improve the discharge efficiency in the future.