Keywords

1 Introduction

As global climate changes intensify, extreme floods occurred frequently. Once the discharge capacity of diversion tunnel is insufficient, flow overflows the top of cofferdam, which has a certain impact on the construction safety of the permanent building. Therefore, horizontal and vertical cross inlet is proposed to improve the discharge capacity under a certain water head. As a new type, the inlet can not only make full use of the shaft in discharge structure during the operation period and the horizontal hole during the construction period as the diversion structure during the construction period, to meet the needs of the low water level and the large flow discharge, but also can be quickly converted into the flood discharge tunnel at the operation period through the blocking of the horizontal hole. When the horizontal and vertical cross inlet discharge flow, there exists the two-layer discharge characteristics distinguished by horizontal hole discharge and vertical shaft discharge, so hydraulic characteristics are more complex. Since the inlet is a new type, there is a relative lack of systematic researches in this field, and previous studies have been carried out mainly on separate horizontal inlets or shaft inlets. As for the horizontal inlets, Deng [1], for example, summarized the effects which various water flow boundary conditions and shapes of inlets had on the water surface vortex in front of the horizontal inlets, and proposed a method to improve the state of the inlet flow to overcome the vortex. Zhao et al. [2] studied the flow rate, flow regime, pressure, full flow boundary and body shape through model tests of the diversion tunnel, and proposed that V-type eddy-eliminating beam arranged in front of the inlet can effectively eliminate inlet vortex. Shi et al. [3] conducted hydraulic modeling tests aimed for steep-slope diversion tunnel and found that the head plate of the horizontal inlet was changed from the typical elliptical type to a sharp-edge form, which could eliminate the phenomenon of mixed free-surface-pressure flow in the delivery tunnel at the downstream of the inlet. Based on the Baihetan inflow model, Fu et al. [4] investigated the effect of different heights of residual cofferdam on the discharge capacity of horizontal inflow and outflow holes. Zheng [5] conducted numerical simulation of hydraulic characteristics of the spillway at low and atmospheric pressures and found that atmospheric pressure mainly affected cavitation number of the spillway. For the vertical shaft inlet, Guo et al. [6] proposed that the reasonable size of the energy dissipation well should be that the depth of the well was equal to 1.69 times the diameter of the shaft by analyzing the effect of the change of the shape on each hydraulic parameter in the energy dissipation well. Zhang [7] proposed a new internal rotational flow shaft spillway for high arch dams. The simple structured design provided not only a stable higher capacity water discharge, but also a high-energy dissipation rate and low construction costs. Guo et al. [8] proposed a new vortex drop shaft spillway, which not only improved energy dissipation, but also protected ecology and reduced investment. Liu et al. [9] proposed new shaft spillways types to generate swirling flow by using piers and circular piano-keys on the annular crest. Their findings demonstrated that the swirling flow strength in the modified spillway was several times lower when compared to that of the conventional spillway. Based on modeling tests, Kabiri-Samani et al. [10] investigated the hydraulic characteristics including head-discharge relationship, discharge coefficient, and critical submergence depths of a vertical shaft spillway with an innovative inlet (namely marguerite-shaped inlet) and derived empirical equations related to critical submergence depths and discharge coefficients of the marguerite-shaped inlet for different flow regimes. Aydin et al. [11] employed the novel labyrinth-shaft spillway which was better in terms of discharge capacity for the same weir head compared to the conventional shaft spillway. Aydin et al. [12] explored siphon-shaft spillway which could make an effective discharge through siphonic feature in dam reservoirs with narrow valleys.

To summarize, the current scholars carry out the study of the diversion tunnel basically based on the specific horizontal inlet or vertical inlet project cases, lacking systematic and regular analysis of the horizontal and vertical cross inlet. This paper utilizes Fluent, to establish the two-dimensional hydrodynamic model of the diversion tunnel, and to study the discharge characteristics of the vertical cross inlet by varying head ratio and area ratio of different horizontal inlet as well as shaft inlet.

2 Analytical Modeling and Validation

2.1 Structural Model of Inlet

The structural arrangement of the novel inlet is shown in Fig. 1. The scope of the analysis includes the reservoir area, the inlet, and the delivery tunnel. The inlet consists mainly of horizontal and shaft inlets, as well as flared and horizontal sections on the downstream of the shaft. The dimension of the inlet of the shaft are 10 × 17 m, and the elevation of the bottom of the shaft is lower than the floor of the horizontal hole, in order to form an energy-dissipating water cushion for the flow water from the shaft. The shaft downstream is connected with a flared section of 6 m in length, followed by a horizontal section of 5 m in length. The length of the cave of the diversion tunnel is 85 m and the slope of the bottom plate is 3.10%. In the figure, h stands for the height of the horizontal hole inlet and H means the depth of water. In addition, l represents the length of the shaft. HV expresses the depth of water above the leading-edge of the overflow at the inlet of the shaft. CS1 and CS2 sections are defined as the horizontal hole and shaft inlet section respectively. CS3 section is termed as the end of flare section (maximum average flow velocity section at the inlet).

Fig. 1
A diagram of a reservoir represents an inlet and a delivery tunnel. The horizontal hole inlet and shaft are labeled. The height of the horizontal hole inlet h, depth of water H, length of shaft l, slope of bottom plate 3.1%, C S 1, C S 2, C S 3, and other dimensions are labeled.

Structural arrangement of the inlet and the analysed scope (unit: m)

Full size image

2.2 Numerical Analysis Model

The numerical simulation is based on the SST k-omega, and the gas–liquid two-phase flow is calculated by Volume of Fluid. The control equations are discretized by the finite volume method. The control equations are as follows.

Turbulent kinetic energy:

$$\begin{array}{c}\frac{\partial }{\partial t}\left(\rho k\right)+\frac{\partial }{\partial {x}_{i}}\left(\rho k{u}_{i}\right)=\frac{\partial }{\partial {x}_{j}}\left({\Gamma }_{k}\frac{\partial k}{\partial {x}_{j}}\right)+{G}_{k}-{Y}_{k}+{S}_{k}\end{array}$$
(1)

Turbulent dissipation rate

$$\begin{array}{c}\frac{\partial }{\partial t}\left(\rho \omega \right)+\frac{\partial }{\partial {x}_{i}}\left(\rho \omega {u}_{i}\right)=\frac{\partial }{\partial {x}_{j}}\left({\Gamma }_{\omega }\frac{\partial \omega }{\partial {x}_{j}}\right)+{G}_{\omega }-{Y}_{\omega }+{D}_{\omega }+{S}_{\omega }\end{array}$$
(2)

In the formula: i, j are parameters of coordinate directions, i, j = 1, 2 correspond to x, y coordinate directions, respectively; \({x}_{i}\), \({x}_{j}\) are the Cartesian coordinates in the i, j directions; \({u}_{i}\) is the time-mean flow velocity in the i direction; \(\rho \) is the density; \({G}_{k}\) is the turbulent kinetic energy which is generated by the velocity gradient of laminar flow; \({\Gamma }_{k}\) and \({\Gamma }_{\omega }\) are the active diffusion terms of \(k\) and \(\omega \); \({Y}_{k}\) and \({Y}_{\omega }\) are the divergent terms of \(k\) and \(\omega \); \({D}_{\omega }\) is the Orthogonal divergence term; \({S}_{k}\) and \({S}_{\omega }\) are defined by users.

2.3 Grid Division and Solution Method

The pre-processing software Gambit is utilized to perform grid generation of the research area, and a combination of structural and non-structural grids are used for grid division. In this paper, four grid scales are used: 10, 15, 20 and 25 cm. The boundary condition of inlet and outlet are the same for the different grid scales. Grid independence analysis is performed by comparing values of flow rate and velocity at the CS3 section for different grid scales. The grid independence results are shown in Table 1, and the results indicate that the calculation results tended to be stable. Therefore, grid scale of 15 cm is selected for numerical simulation in this paper.

Table 1 The results of grid independence analysis
Full size table

The coupled algorithm is applied to deal with pressure–velocity coupling. Momentum, Turbulent kinetic energy and turbulent dissipation rate are based on second order upwind. Least squares cell based and PRESTO! are utilized in gradient and pressure interpolation, respectively. Standard initialization is used to set the volume fraction of water in the calculation region below the free surface to be assigned a value of 1, and the flow velocity and pressure are treated as stationary flow field. The inlet uses pressure-inlet with additional water level boundary condition. The outlet is set as pressure-outlet and average pressure of cross section was 0. The near-wall is treated with wall function and non-slip condition is adopted. The convergence criterions are set as 1 × 10–4.

2.4 Model Verification

A physical model with a length scale of 1:50 is chosen to validate the mathematical simulation. The test results under H = 30 m and h = 4 m are used to verify the reasonableness of the established mathematical model by comparing the flow rate and velocity at the inlet and the three measuring points of the bottom, middle and surface on the perpendicular bisector in the cross sections of CS1. Table 2 shows that the relative errors of the flow rate and flow velocity results of numerical simulation and physical model are all within 3%. Considering the influence of sidewall resistance in the physical model, it is reasonable that experimental values are smaller than the calculated values, which indicate that the accuracy of the mathematical model meet the requirements of the calculations.

Table 2 Comparison of hydraulic parameters between numerical and physical models
Full size table

3 Results Analysis

3.1 Research Program

The study program takes water depths H of 30, 35, 40 and 50 m. The height of the horizontal hole h adopts 4, 5, 6, 7 and 8 m. Therefore, there exists 20 sets of calculation conditions.

3.2 Analysis and Discussion of Results

Analysis of flow regime

For \({\lambda }_{H}\) = 0.75 and \({\lambda }_{A}\) = 2.5, the streamlines and flow velocity distribution profile at the inlet is shown in Fig. 2. The chart indicates that due to the deflection of flow direction in the shaft, horizontal vortex is be formed on the upstream side of the shaft, and the upstream velocity is less than the downstream velocity. In addition, the clockwise horizontal vortex is formed in the water cushion pool at the bottom of the shaft.

Fig. 2
A contour map represents the velocity of streamlines at an inlet. Horizontal vortexes are formed on the upstream side and at the bottom of the shaft. The downstream velocity is higher than the upstream velocity.

Streamlines and flow velocity distribution profile at the inlet

Full size image

Analysis of discharge capacity

In terms of horizontal and vertical cross inlet, flow rates of horizontal hole and vertical shaft as well as the former two total sum are set as QH, QV and QT respectively. So as to explore discharge capacity of this inlet, the definitions of relevant terms in this analysis are as follows.

The head ratio of the shaft to the horizontal hole is defined as:

$$\begin{array}{c}{\lambda }_{H}=\frac{{H}_{V}}{H}\end{array}$$
(3)

The ratio of the overflow area of the shaft to the horizontal hole is normed as:

$$\begin{array}{c}{\lambda }_{A}=\frac{l}{h}\end{array}$$
(4)

The confluence ratio of the shaft to the horizontal hole is as:

$$\begin{array}{c}{\lambda }_{Q}=\frac{{Q}_{V}}{{Q}_{H}}\end{array}$$
(5)

Effect of head ratio on confluence ratio

In order to analyze the confluence ratios of the inlet under different head ratios, λA = 1.25, 1.43, 1.67, 2.00 and 2.50 are selected, and the relationship curves of λH and λQ with different area ratios are plotted, as shown in Fig. 3.

Fig. 3
A multi-line graph of lambda H versus lambda Q. It plots 5 differently shaded vertically upward lines for lambda A = 2.50, 2.00, 1.67, 1.43, and 1.25 starting at (3.3, 0.57), (2.7, 0.57), (2.25, 0.57), (2.0, 0.57), and (1.7, 0.57), respectively. Values are estimated.

Varying principles between λQ and λH under different area ratios

Full size image

Seen from the graph, λQ is proportional to λH, which means the former increase as the latter grew; however, the increase is inversely proportional to λH. The bigger λH becomes, the stronger blocking effect of the shaft inflow on the horizontal hole discharge is, which in turn lead more water to flow into flow into the downstream delivery tunnel through the shaft. In the meantime, as λH increases, λQ is less and less affected by λH since the discharge capacity is mainly controlled by the shape of the delivery tunnel section, and λQ tends to a constant.

Effect of area ration on confluence ratio

With respect to analyze the confluence ratios of the inlet under different area ratios, this paper selects λH = 0.583, 0.643, 0.688 and 0.750, and plots the relationship curves between λA and λQ under different head ratios, as shown in Fig. 4.

Fig. 4
A multi-line graph of lambda Q versus lambda A. It plots 4 differently shaded increasing lines for lambda H = 0.583, 0.643, 0.688, and 0.750 ending at (2.5, 3.25), (2.5, 3.5), (2.5, 3.6), and (2.5, 3.7), respectively. Values are estimated.

Varying principles between λQ and λA under different head ratios

Full size image

According to the chart, λQ increases as λA inclines. In view of areas, essential factors affecting results, controlled by the CS3 section, the total flow rates are essentially the same when head ratios of the inlet are equal, significantly varying only the confluence ratios. In addition, due to the interaction of the horizontal and vertical flow, as λA increases, more water enters the shaft and less water flows out of the horizontal hole, and λA and λQ show a roughly good linear relationship.

Analysis of the Froud number

In order to quantitatively investigate the extent, to which the horizontal hole and the shaft interact with each other when the shaft is in the regime of orifice flow for combined inlet, FrH and FrV are introduced.

The Froude Number of the horizontal hole is defined as:

$$\begin{array}{c}{{\text{Fr}}}_{{\text{H}}}=\frac{{v}_{H}}{\sqrt{gH}}\end{array}$$
(6)

The Froude Number of the inlet of the shaft is normed as:

$$\begin{array}{c}{{\text{Fr}}}_{{\text{V}}}=\frac{{v}_{V}}{\sqrt{g{H}_{V}}}\end{array}$$
(7)

In the formula: \({v}_{H}\) and \({v}_{V}\) are as the cross-sectional average flow rates of the horizontal hole and shaft inlet in sequence.

Effect of head ratio on the Froud number

With respect to analyze the Froud Number \({\text{Fr}}_{\text{H}}\) and \({\text{Fr}}_{\text{V}}\) of the inlet under head ratios, λA = 1.25, 1.43, 1.67, 2.00 and 2.50 are chosen, and the relationship curves between λH and \({\text{Fr}}_{\text{H}}\), λH and \({\text{Fr}}_{\text{V}}\) under different area ratios are plotted, as shown in Figs. 5 and 6.

Fig. 5
A multi-line graph of lambda H versus F r H. It plots 5 differently shaded downward lines for lambda A = 2.50, 2.00, 1.67, 1.43, and 1.25 ending at (0.295, 0.57), (0.27, 0.57), (0.257, 0.57), (0.24, 0.57), and (0.228, 0.57), respectively. Values are estimated.

Curves about the relationship between FrH and λH

Full size image
Fig. 6
A multi-line graph of lambda H versus F r V. It plots 5 differently shaded downward lines for lambda A = 2.50, 2.00, 1.67, 1.43, and 1.25 ending at (0.51, 0.57), (0.49, 0.57), (0.465, 0.57), (0.444, 0.57), and (0.43, 0.57), respectively. Values are estimated.

Curves about the relationship between FrV and λH

Full size image

The figures reveal that \({\text{Fr}}_{\text{H}}\) and \({\text{Fr}}_{\text{V}}\) are inversely proportional to λH; however, the decreases are inversely proportional to λH. The relationship shows that with the increase of λH, the water flow in the shaft has more and more obvious blocking effect on the discharge of the horizontal hole, the role of the horizontal hole in the discharge of the flow gets smaller and smaller and with the increase of λH, \({\text{Fr}}_{\text{H}}\) and \({\text{Fr}}_{\text{V}}\) tend to a constant.

Effect of area ratio on the Froud number

With respect to analyse the Froud Number \({\text{Fr}}_{\text{H}}\) and \({\text{Fr}}_{\text{V}}\) of horizontal and vertical cross inlet under different λA, in this paper, λH = 0.583, 0.643, 0.688 and 0.750 are selected, the relationship curves are plotted, as shown in Figs. 7 and 8.

Fig. 7
A multi-line graph of F r H versus lambda A. It plots 5 differently shaded increasing curves for lambda H = 0.583, 0.643, 0.688, and 0.750 ending at (2.5, 0.29), (2.5, 0.27), (2.5, 0.265), and (2.5, 0.254). Values are estimated.

Curves about the relationship between FrH and λA

Full size image
Fig. 8
A multi-line graph of F r V versus lambda A. It plots 5 differently shaded increasing curves for lambda H = 0.583, 0.643, 0.688, and 0.750 ending at (2.5, 0.50), (2.5, 0.484), (2.5, 0.47), and (2.5, 0.45), respectively. Values are estimated.

Curves about the relationship between FrV and λA

Full size image

Seen from the chat, \({\text{Fr}}_{\text{H}}\) and \({\text{Fr}}_{\text{V}}\) are proportional to λA; however, the increases are inversely proportional to λA.

4 Conclusion

Horizontal and vertical cross inlet can greatly improve the discharge capacity, and effectively solve the insufficient discharge capacity caused by extreme floods. In this paper, a two-dimensional hydrodynamic mathematical model is used to analyze the discharge characteristics such as confluence ratio and the Froud number, the conclusions are followed as.

Under different head ratios, confluence ratios are proportional to head ratios, which mean the former increases as the latter grows; however, the Froud Number of the horizontal hole and shaft are inversely proportional to head ratios. The results reveal that the increase of head ratios, the water flow in the shaft has more and more obvious blocking effect on the discharge of the horizontal hole. In addition, the head ratios reach a certain level, the discharge capacity of the inlet is mainly controlled by the shape of the delivery tunnel.

Under different area ratios, confluence ratios and the Froud Number are proportional to area ratios. According to the results, controlled by the CS3 section, the total flow rates are essentially the same when the head ratios of the inlet are equal, varying confluence ratios and the Froud Number.