Hydraulic Characteristics of Rotational Flow Shaft Spillway for High Dams

Hydraulic Characteristics of Rotational Flow Shaft Spillway for High Dams

Zhang Xianqi

School of Water Resources, North China University of Water Resources and Electric Power, Zhengzhou, 450045, China

Page: 
167-174
|
DOI: 
https://doi.org/10.18280/ijht.330123
Received: 
N/A
| |
Accepted: 
N/A
| | Citation

OPEN ACCESS

Abstract: 

Flood discharge and energy dissipation of large-scale hydraulic and hydroelectric projects are two key technological issues in project planning, design and construction phases. Hydraulic characteristics research of new type high dam flood discharge and energy dissipation facility is one of the major research directions of high-speed hydraulic. To deal with problems of flood discharge and energy dissipation faced in large-scale hydraulic and hydroelectric projects development such as high dam, large discharge and narrow river valley and so on, this paper proposes a new internal rotational flow shaft spillway which suits high arch dam and analyzes the design criteria and size of each part. Discharge volume and ratio of energy dissipation verifications are conducted to the designed internal rotational flow shaft spillway through physical modeling test. Results show that this new discharge and dissipation facility has simple structure; stable discharge volume with good discharge capacity, high ratio of energy dissipation and its project investment is low. Besides, this paper also analyzes the discharge performance of this new discharge and dissipation facility and proposed its discharge function relational expression based on modeling test results. This project research can offer theoretical basis to further body optimization and engineering application of discharge and dissipation facility.

Keywords: 

rotational flow shaft spillway; discharge volume; ratio of energy dissipation; model.

1. Introduction

To date, the major areas where China develops hydroelectric energy mainly concentrate in southwest part. Heights of constructed large-scale hydropower projects are mostly higher than 150m. From projects constructed and being constructing, high arch dam has become one of the primary dam types of large-scale hydroelectric projects, for example, dams in Ertan, Xiaowan, Xiluodu, Jinping, Goupitan, Baihetan and Wudongde etc. all adopted high arch dam arrangement form. These projects have features of “narrow river valley, high water head and large discharge volume”. The features bring severe issues of flood discharge and energy dissipation. Therefore, in terms of design of release structure, general layout framework combined by dam body trepanning discharge and shore-side spillway or spillway water diversion is mostly adopted. Due to the narrow canyon riverbed and high discharge volume, the construction of flood releasing structure is very difficult. If shore-side spillway cannot be constructed due to landform limitations of alpine and gorge region, except in dam body surface outlet and middle outlet, shore-side spillway is additionally needed to meet discharge needs. That will increase the cost a lot. Moreover, high-speed water flow will bring special phenomena like mixed gas, pulsation, vibration, cavitations, cavitations erosion and pulverization, harming the security of water releasing structure and downstream river channel environment. Therefore, searching for new type internal discharge and dissipation facility for high dams has been always a research key point at home and abroad. Long before in 1960s, C. Drioli from Italy and D. Jeanpierre from France et al. had begun research of rotational flow shaft spillway and obtained some research results related to energy dissipation which were applied to some degree in former Soviet Union, France, Italy and China. Nowadays, these applied rotational spillways are mostly built in medium-low internal with height less than 100m. Discharge capacity of single outlet is generally smaller than 600m3/s, and ratio of energy dissipation less than 60%. Besides, the sizes of producing rotating flow facilities of shaft are relatively larger, so the construction cost is high. And they are not suitable for high arch dam. This research proposed an internal rotational flow shaft spillway which suits high dam. It has simple structure, small size, low building cost, convenient construction and extensive application range. In addition, its discharge volume is large and ratio of energy dissipation is also high. This water discharge and energy dissipation facility is constituted of water inlet section, tangential rotating flow segment, tapered segment, shaft segment, cushion pool segment and water outflow segment. Water flow is formed through water inlet section, and then rotated by rotating parts and tapered segment. The water rotated clings to wall surface in shaft. A stable cavity is formed in the middle. After the energy of water flow being dissipated when water crashes cushion pool in the lower part of shaft, water discharges towards to downstream through horizontal outlet. Hydraulic modeling test verification is conducted to the proposed design scheme, discharge capacity and ratio of energy dissipation of internal rotational shaft spillway is analyzed. Hydraulic characteristics of this scheme are further studied. These all together can offer theoretical basis and technological support to further optimization of this new type water discharge and energy dissipation facility and engineering application.

2. Scheme Design

The proposed internal rotational flow shaft should combine tightly the actual conditions of China high dam construction. On the basis of features of internal rotational flow shaft, the other following conditions should also be satisfied:

(1) Dam height of high arch dam should be 300m;

(2) Discharge capacity of single spillway reaches 2000~3000m3/s;

(3) Intake water head is 20~40m;

(4) Ratio of energy dissipation of this discharge and dissipation facility should be bigger than 70%;

(5) Outlet water velocity should be controlled to stay in the water flow speed range that will not cause harmful secondary phenomenon.

2.1 Water intake and inlet

The cross-section of water intake becomes small gradually from big rectangle. Entrance top is oval curve. Its long axis equals with the height of arch door, short axis is 0.35. It is convenient for connection of water intake and non-pressure conduit and transition section. It also lowers Fr (Froude number) and increases amount of cavitations of flow discharge.

The elevation of conduit entrance baseboard and area of radial gate outlet $A_{0}$ ($A_{0}=a \times b$; $a$ is the width of outlet, $b$ is the height of outlet) are determined by discharge volume $Q_{0}$ required. Their relation is as follow:

$Q_{0}=\varphi_{0} \xi A_{0} \sqrt{2 g\left(H_{0}-\xi b\right)}$   (1)

Where, $\varphi_{0}$ is velocity coefficient of intake; $\xi$ is shrinkage coefficient behind the floodgate when water is discharged through the radial gate, according to existing test results, shrinkage coefficient $\xi$ can be calculated by the equation below:

$\xi=\frac{h_{c}}{a}=\frac{l a}{l+\sqrt{0.5\left(\sin a \cdot \operatorname{tg}^{-1} i_{0}\right) \cdot\left(\frac{a}{H_{0}}\right)^{0.1}}}$  (2)

Where, $h_{c}$ is shrinkage depth of water behind floodgate; $a$ is height of arch gate (m); $i_{0}$ is baseboard gradient of conduit; $H_{0}$ is functional water head of water intake (m); $l$ is length of conduit.

In line with general design criterion, $b / a \geq 1.2$ should be made. Because $A_{0}$ is too big, flow velocity behind the gate related to $H_{0}$ above should be 23~27m/s to satisfy the structural requirement.

2.2Rotating flow generator

Body type of rotating parts of rotational flow shaft spillway constructed abroad is mostly similar with volute of water turbine; its major size is written in table 1.

Table 1. The size of rotating parts of rotational flow shaft spillway abroad

Project name

B

(m)

H

(m)

D

(m)

Rt

(m)

Q

(m3/s)

Fr

Santa Maria

3.00

3.50

2.60

3.45

21.6

0.35

di Monte Argento

5.50

7.86

5.50

6.05

165.0

0.41

di Grotto Companre

5.25

7.23

4.50

5.63

100.0

0.31

di Montemaggiore

3.90

4.68

3.26

4.28

40.0

0.32

di Narni

7.20

7.60

6.00

7.65

180.0

0.38

di Curbans

7.30

6.02

7.30

9.26

140.0

0.41

From table 1, it can be seen that in all existing projects, the maximum discharge amount of rotational flow shaft spillway is 180.0m3/s. Water flows before rotating parts water intake are all sub-critical flows (Froude number Fr<1.0). Because the connection between conduit and rotating parts is not tangency, the cross-section of rotating parts is relatively bigger. The structure is complicated; the distance between conduit and shaft axle wires  is also bigger. When discharge volume is smaller, the conduit water flow is smooth and can acquire bigger ring-directional flow speed and increase ratio of energy dissipation. However, when the discharge volume of water diversion is bigger, supercritical flow will emerge. At this time, water flow will rotate strongly, causing high internal water level of rotating parts. Moving hydraulic jump is generated in water diversion canal and no stable cavity can be created in shaft.

Consequently discharge capacity and ratio of energy dissipation are lowered greatly. For arch dam with thinner body, sub-thread tangential waterlogged rotating method is suggested which is demonstrated in figure 1. The facility structure is simple, construction is convenient, and the occupied internal space is smaller which is beneficial to the structural security of dam body.

(a) Traditional vortex water

(b) Sub-thread tangential waterlogged rotating

Figure 1. Structure schematic diagram of rotational flow generator

2.3 Eccentricity distance $r_{t}$ and radius of shaft $R_{0}$

Confirming eccentricity distance $r_{t}$ and radius of shaft $R_{0}$ in general occasions needs to consider two principles: (1) with given functional water head, discharge volume determined by $Q_{0}=\varphi_{0} \xi A_{0} \sqrt{2 g\left(H_{0}-\xi b\right)}$ is needed; (2) there must be relatively high ratio of energy dissipation.

In order to avoid harmful phenomena caused by high-speed water flow in shaft, it is more appropriate to keep the maximum flow speed in shaft within 30m/s. When the distance between water surface in shaft and baseboard of water intake is less than 50m, the target can be reached without very high ratio of energy dissipation; once the distance reaches 100m or above, very high ratio of energy dissipation is needed. So at this time the eccentricity distance $r_{t}$ must be as big as possible within the structural design limitation range. To realize this, a transition section can be set up between rotating parts and shaft. The upper ends of transition section and rotating parts have equal diameters. The lower end is connected to shaft. Then the radius of rotating parts $R_{1}$ is bigger than radius of shaft $R_{0}$ and bigger eccentricity distance $r_{t}$ can be acquired.

For the purpose to acquire the maximum discharge volume $Q_{\max }$ that can get through internal shaft with given $r_{t}$, $R_{0}$ and functional water head, suppose under conditions of no energy lost from rotating parts, the following relational expressions are obtained according to angular momentum theory and Bernoulli’s equation:

$v_{m}=\mu \sqrt{2 g H_{0}^{\prime}}$    (3)

$H_{0}^{\prime}=\varphi_{0}^{2} \xi^{2}\left(H_{0}-\xi b\right)+\xi b$    (4)

$Q_{\max }=v_{m} \pi R_{0}^{2}$   (5)

$v_{z}=v_{m} / \omega, v_{\theta, R}=S v_{m}$    (6)

$\mu=1 / \sqrt{S_{0}^{2} /\left(1-\omega_{c}\right)+1 / \omega_{c}^{2}}$   (7)

$\omega=1-\left(r_{G} / R_{0}\right)^{2}$    (8)

$S_{0}=\pi R_{0} r_{t} \sin \left(b / \xi A_{0}\right)$     (9)

Where, $v_{m}$ is virtual flow direction speed in shaft; $\mu$ is velocity coefficient corresponded to $v_{m}$ when critical flow emerges in shaft entrance; $v_{z}$, $\mathcal{V}_{c, R}$ are cross-sectional average flow speed of actual flow direction and ring-directional flow speed on shaft internal wall surface; $\omega$ is area of relative cross section of passage; $r_{G}$ is gas cavity radius of spiral flow center; $c$ is subscript representing critical state; $S_{0}$ is initial rotating parameter. $\omega_{c}$ is determined by $1-S_{0}^{2} \omega_{c}^{3} / 2\left(1-\omega_{c}\right)^{2}=0$ in which $S_{0}^{2} \omega_{c}^{3} / 2\left(1-\omega_{c}\right)^{2}$ is the square of propagation velocity and water flow directional speed specific value of surface turbulence at the critical fracture surface. The flow is race if the specific value is smaller than 1, critical flow if it equals to 1 and slow flow if it is bigger than 1.

$r_{t}$ and $R_{0}$ are calculated jointly by equations (3) to (9).

Under supercritical flow conditions, in order to ensure that shaft is cavity rotational flow, diameter of shaft can also be calculated by empirical equation (10):

$D=k\left(Q^{2} / g\right)^{0.2}$    (10)

2.4 Cushion pool and horizontal outlet

Shaft should be connected to the fracture surface of horizontal outlet entirely. There is cushion pool in the lower part. Make diameter of horizontal outflow outlet $D^{\prime}$ equal with diameter of shaft $D$ and cushion pool diameter equal with shaft diameter, then the effects of water flow in shaft have on horizontal outflow outlet will be minimized to ensure that a more stable water flow regime in outflow outlet is maintained. Cushion pool is mainly used to keep a certain buffer layer to prevent baseboard from being eroded by initial falling water in addition to energy dissipation. Besides, pier for rotation resistance can be set in cushion pool to dissipate energy, prevent vibration and improve flow regime within the outlet. And it is also useful for reinforcing baseboard.

When downstream water level is lower than horizontal outflow outlet top, the outlet becomes un-submerged outflow outlet; conversely, when downstream water level is higher than outlet top, discharge outlet becomes submerged outflow outlet. Exit cross-sectional area is controlled by gradually changing method at the discharge outlet for the purpose of controlling outlet velocity and keeping flow speed away from flow speed range within which no harmful secondary phenomena occur.

3. Modeling Test

3.1 Model design

Suppose the height of an arch dam is 300m and that of internal rotational flow shaft spillway intake head is 40m; conduit section bottom slope is plain; the difference of water levels of upstream and downstream is 210m. According to design criterion and requirements mentioned above, scheme design of rotational flow shaft spillway is conducted. In order to verify the rationality of the design scheme, hydraulic modeling test is conducted to the scheme. The testing model of rotational flow shaft spillway is an undistorted model designed in line with gravity similarity criterion with linear ratio being $\lambda$=1/150 and model length being 13m, width being 2m, including upstream measuring weir, flat water segment, reservoir, water inlet section, rotating parts, tapered segment, shaft segment (plug included), water-pillow segment, outflow segment and water return conduit etc. Sizes of each part of internal rotational flow shaft spillway are listed in table 2.

Table 2. The sizes of rotational flow shaft spillway

Water inlet

Rotational flow generator

transition section

shaft

cushion pool

Water outlet

a×b(m)

eccentricity distance $r_{t}$

diameter $R_{1}$

upper ends diameter

diameter

diameter

diameter

10×12m

8.0m

27.0m

27.0m

16.0m

16.0m

16.0m

3.2 Testing scheme

In order to test the rationality and the scientific nature of designed scheme and analyze features of water discharge and

energy dissipation of rotational flow shaft, the model conducted tests of 5 schemes in terms of water in flow conditions, plug layout situation, water outflow conditions and so on. The testing scheme is as demonstrated in table 3.

Table 3. The schemes of model test

Scheme

Mode of water inlet

1#Plug (cm)

2#Plug (cm)

3#Plug (cm)

Mode of water outlet

SchemeⅠ

Pressure flow and tangential inlet

9.33

9.33

9.33

Non pressure and un-submerged outflow

SchemeⅡ

Non pressure flow and tangential inlet

9.33

9.33

9.33

Non pressure and un-submerged outflow

SchemeⅢ

Non pressure flow and tangential inlet

9.33

6.67

9.33

Non pressure and un-submerged outflow

SchemeⅣ

Non pressure flow and tangential inlet

9.33

8.00

9.33

Pressure flow and un-submerged outflow

SchemeⅤ

Non pressure flow and tangential inlet

9.33

8.00

9.33

Pressure flow and submerged outflow

3.3 Discharge capacity

The relational curve of upstream functional water head and discharge volume is acquired through modeling tests of 5 schemes. It is demonstrated in figure 2.

Figure 2. The relational curve of upstream functional water head and discharge volume

From the discharge capacity curve of each scheme above, it can be seen that: 1) when water intake functional water head is 20m, discharge capacity of all rotational flow shaft spillway exceeds 1000m3/s; when it is 30m, discharge capacity reaches 2000m3/s; rising trends of discharge capacity of each scheme along with water head is better; 2) from the fitted curve, discharge capacity of each scheme is basically in direct ratio with 1.5th power of intake head;

slight differences exist between each scheme while Scheme Ⅳ has the biggest and Scheme Ⅰ has the smallest; 3) the reason why discharge capacity of Scheme Ⅲ is smaller than that of Scheme Ⅱ and Scheme Ⅳ is the diameter of its 2#plug is smaller, thus the water-pillow formed by water flow at this part is higher and gets through the above parts of 2#plug very quickly, affecting its discharge volume; 4) from the perspective of discharge capacity, Scheme Ⅱ, Scheme Ⅳ and Scheme Ⅳ are better; when water intake functional water head is 35m, discharge capacity reaches 2800m3/s, three schemes all have big flow discharging potential; 5) Scheme Ⅴ is the submerged outflow situation of Scheme Ⅳ at downstream; the difference between discharge capacity of two schemes is not big which means that Scheme Ⅳ still has bigger discharge capacity under conditions of downstream being submerged. Thus with consideration of discharge capacity, Scheme Ⅳ is the primary choice and Scheme Ⅱ is the secondary.

3.4 Ratio of energy dissipation

The calculation of ratio of energy dissipation is mainly based on the measurement of lower outlet velocity and pressure and then construction of energy equation with inflow and outflow water flows. Through modeling tests of 5 schemes, the relations between different upstream functional water heads, discharge volume and ratio of energy dissipation are acquired as demonstrated in table 4.

Table 4. Ratio of energy dissipation of each scheme

SchemeⅠ

SchemeⅡ

SchemeⅢ

SchemeⅣ

SchemeⅤ

water heads

/m

discharge volume /m3/s

ratio of energy dissipation /%

water heads

/m

discharge

volume /m3/s

ratio of energy dissipation /%

water heads

/m

discharge volume /m3/s

ratio of energy dissipation /%

water heads

/m

discharge volume /m3/s

ratio of energy dissipation /%

water heads

/m

discharge volume /m3/s

ratio of energy dissipation /%

15.6

703

98.2%

21.8

1314

93.7%

20.9

1193

94.8%

21.2

1221

94.6%

21.2

1218

94.6%

21.0

1190

94.9%

24.5

1604

90.8%

24.5

1582

91.0%

24.3

1590

91.0%

24.3

1587

91.0%

24.0

1405

92.9%

27.5

1976

86.2%

27.2

1855

87.9%

28.4

1998

86.0%

28.4

2023

85.6%

27.0

1596

91.0%

30.6

2320

81.2%

30.3

2122

84.3%

29.9

2238

82.5%

29.9

2235

82.5%

30.6

1874

87.8%

32.7

2568

77.2%

33.5

2439

79.5%

32.7

2521

78.0%

32.7

2497

78.5%

36.8

2149

84.3%

35.0

2794

73.3%

36.0

2768

73.2%

36.3

2960

70.2%

36.3

2888

71.6%

4. Hydraulic Characteristics

4.1 Rotating flow generator without a tapered segment

Internal rotational flow shaft in the application as spillway must have certain discharge capacity. This is precondition of it being popularized and applied. When tapered segment is not set under rotating flow generator, height of tapered segment equals with zero. It is relatively simpler, and there is already some experimental data abroad. At this time expression of discharge volume $Q$ can be written as:

$Q=\mu_{0} \pi R^{2} \sqrt{2 g(h+\delta)}$  (11)

$Q=\mu \pi R^{2} \sqrt{2 g(h+\delta)+v_{0}^{2}}=\mu \pi R^{2} \sqrt{2 g h_{v}}$   (12)

Where, discharge volume coefficient $\mu_{0}$ already considered the effects of near velocity $v_{0}$ while discharge

volume coefficient $\mu$ is not included in the effects of near water velocity.

To calculate discharge volume in different situations using limited experimental data, a reference system is brought in. It is a rotational flow shaft that has geometric similarity with radius of shaft being 1 dm. Its discharge volume is the referred discharge volume $Q_{r}$. If radius of the actual shaft is $R$, taking reference system as model, then the ratio is $\lambda_{l}=R(d m) / 1$, relation between actual discharge volume $Q_{P}$ and referred discharge volume $Q_{r}$ can be expressed as $\left(Q_{P} / Q_{r}\right)^{2 / 5}=\lambda_{l}$. Convert the experimental results into referred discharge volume. The referred discharge volume and discharge volume coefficient acquired are listed in table 5.

Table 5. The referred discharge volume and discharge volume coefficient

number

$Q_{P}$(m3/s)

$Q_{r}$(m3/s)

$\mu_{0}$

1

703

4.24

0.370

2

1190

7.18

0.482

3

1405

8.47

0.530

4

1596

9.62

0.570

5

1874

11.30

0.652

6

2149

12.96

0.748

The fitted curve of referred discharge volume and referred discharge volume coefficient is shown in figure 3.

Figure 3. The curve of referred discharge volume and referred discharge volume coefficient

From experimental results, empirical correlation of $\mu_{0}$ and $Q_{r}$ is:

$\mu_{0}=0.148 Q_{r}^{0.6109}$   (13)

The front depth of water of rotating parts $h$ can be derived from discharge volume equation (13) as:

$h=R\left[\frac{1}{2 g R}\left(\frac{Q}{\mu_{0} \pi R^{2}}\right)^{2}-\frac{\delta}{R}\right]$   (14)

Solve jointly equation (14), expression of water diversion segment width $b$ is:

$b=\pi R^{2} / h \sqrt{\frac{1}{\mu^{2}}-\frac{1}{\mu_{0}^{2}}} \approx \frac{\mu \pi R^{2}}{0.224 h}$  (15)

In order to find out relations between distancesof shaft axis and water diversion segment center line and other parameters, theorem of momentum can be similarly applied.

At the front intake outlet cross-section of rotating parts, angular momentum of fluid to shaft axis $K_{1}$ is

$K_{1} \approx \rho Q v_{0} \Delta=\rho Q^{2} \Delta / b h$     (16)

where, $\rho$ is density of water.

Suppose that the ring-directional flow speed $v_{t}$ and vertical velocity $\mathcal{V}_{z}$ at the shaft entrance are constants, and then angular momentum at the shaft entrance $K_{2}$ is:

$K_{2}=\rho \int_{r_{g}}^{R} 2 \pi r^{2} v_{z} v_{\theta} d r=\frac{2 \rho Q v_{\theta} R\left(1-r_{g_{0}}^{* 3}\right)}{3\left(1-r_{g_{0}}^{* 2}\right)}, r_{g_{0}}^{*}=\frac{r_{g_{0}}}{R}$  (17)

It is derived from momentum balance $K_{1}=K_{2}$ that:

$\rho Q^{2} \Delta / b h=\frac{2 \rho Q v_{\theta} R\left(1-r_{g_{0}}^{* 3}\right)}{3\left(1-r_{g_{0}}^{* 2}\right)}$    (18)

Simplified as:$\Delta=\frac{2 b h v_{\theta} R\left(1-r_{g_{0}}^{* 3}\right)}{3 Q\left(1-r_{g_{0}}^{* 2}\right)}$   (19)

4.2 Rotating flow generator with a tapered segment

The calculation of rotational flow shaft spillway discharge capacity of rotating parts with a tapered segment in generally in the following form:

$Q=\mu_{c} \pi R_{w}^{2} \sqrt{2 g\left(h+h_{c}\right)+V_{0}^{2}}$     (20)

Where, $\mu_{c}$ is discharge volume coefficient of rotational flow shaft spillway with a tapered segment; $h$ is depth of water at the water intake of rotating parts; $V_{0}$ is relative flow speed; $h_{c}$ is height of tapered segment; $R_{w}$ is radius of shaft. In this test, plug is added into the shaft, so plug factor should be considered. Suppose an average radius of shaft is used here, so $R_{w}$ is shaft average radius which is

$R_{w}=\frac{16 \times 80+14 \times 30}{110} \times \frac{1}{3}=5.15 \mathrm{cm}$.

This test conducted tests for several schemes but here only scheme 1 is verified, that when$h_{c}$=20cm, $R_{w}$=5.15cm, discharge volume and discharge volume coefficient are listed in table 6.

Table 6. The discharge volume and discharge volume coefficient of the scheme Ⅰ

Water head(cm)

10.41(cm)

14.01(cm)

16.01(cm)

18.01(cm)

20.71(cm)

24.31(cm)

$Q(l / s)$

2.55

4.32

5.10

5.79

6.80

7.80

$\mu_{c}$

0.127

0.207

0.240

0.268

0.306

0.339

In order to build up expression for discharge volume coefficient, situation of rotating parts without a tapered segment needs to be considered first. In this situation, discharge volume calculation can adopt the following formula.

$Q=\mu \pi R_{w}^{2} \sqrt{2 g(h+\delta)+V_{0}^{2}}$   (21)

Apply angular momentum theory, at the cross-section of rotating part water intake, get a discharge volume infinitesimal near the free water surface $\Delta Q$, its angular momentum to shaft axis $K_{1}$ is:

$K_{1}=\int \Delta m \cdot V \cdot x=\int \rho \Delta h \cdot V^{2} x d x=\rho \Delta h \cdot V^{2} b \cdot \Delta=\rho \Delta Q \cdot V \cdot \Delta$    (22)

Angular momentum of this discharge volume infinitesimal at shaft entrance $K_{2}$ is:

$K_{2}=\int \Delta m \cdot V_{T} \cdot T=\int \rho 2 \pi r \Delta r \cdot V_{Z} \cdot V_{T} \cdot T=2 \pi \rho V_{Z} \cdot V_{T} \int r^{2} d r =\frac{\Delta Q}{\pi\left(R_{w}^{2}-r_{g}^{2}\right)} \cdot V_{T} \rho \frac{2 \pi}{3}\left(R_{w}^{3}-r_{g}^{3}\right)$  (23)

Among the equation, $V_{T}$ means tangential velocity.

Since $K_{1}$=$K_{2}$, then:

$V \cdot \Delta=\frac{2}{3} V_{T} \frac{1-R_{*}^{3}}{1-R_{*}^{2}} R_{w}$    (24)

Where, $R_{*}=r_{g} / R_{w}$.

At the cross-section of shaft entrance, relation between overall velocity $V$, tangential velocity $V_{T}$ and vertical velocity $V_{z}$ is:

$V_{T}=\left(V^{2}-V_{z}^{2}\right)^{\frac{1}{2}}=\left[\frac{Q}{\mu^{2} \pi^{2} R_{w}^{4}}-\frac{Q^{2}}{\pi^{2}\left(R_{w}^{2}-r_{g}^{2}\right)^{2}}\right]^{\frac{1}{2}} =\frac{Q}{\pi R_{w}^{2} \mu} \cdot \sqrt{1-\frac{\mu^{2}}{\left(1-R_{*}^{2}\right)^{2}}}$       (25)

From these equations above it can be derived that:

$\frac{b h}{\pi R_{w} \cdot \Delta}=\frac{3}{2} \frac{1-R_{*}^{2}}{1-R_{*}^{3}} \frac{\mu}{\sqrt{1-\frac{\mu^{2}}{\left(1-\frac{\mu^{2}}{\left(1-R_{*}^{2}\right)^{2}}\right.}}}$  (26)

Make $\Phi=\frac{b h}{\pi R_{w} \Delta}$ , which reflects geometric parameter of shaft body type. Then according to experimental results it can be derived that:

$R_{*}=0.8801-1.6866 \mu^{2}$    (27)

Then it is confirmed that relation of $\mu$ and $\Phi$ without tapered segment is:

$\Phi=1.5 \frac{1-\left(0.8801-1.6866 \mu^{2}\right)}{1-\left(0.8801-1.6866 \mu^{2}\right)^{3}} \frac{\mu}{\sqrt{1-\frac{\mu^{2}}{\left[1-\left(0.8801-1.6866 \mu^{2}\right)^{2}\right]^{2}}}}$   (28)

This equation is bit complicated, for convenience, the relation between $\mu$ and $\Phi$ is demonstrated in figure 4:

Figure 4.The relationship curve of φ~μ

For the purpose to translate discharge volume coefficient without tapered segment into discharge volume coefficient with conical gradual shrinkage tapered segment, the following equation is used to express discharge volume of the latter one:

$Q=\mu_{c} \pi R_{c}^{2} \cdot \sqrt{2 g(h+\delta)+V_{0}^{2}}=\mu_{c} \pi R_{c}^{2} \sqrt{2 g H_{v}}$  (29)

Same expression equation is also used in expressing situation without tapered segment but replace $\mu$ and $R_{w}$ with $\mu_{c}$ and $R_{c}$ . Obviously, with given dimensionless parameter $\Phi$, $\mu$ can be calculated directly.

5. Conclusions

Rotational flow shaft spillway can meet hydraulic conditions of different intake and outtake water outlets. From the perspectives of discharge volume and ratio of energy dissipation, rotational flow shaft spillway is able to meet large discharge volume and high energy dissipation effect requirements at the same time. It is suited to be constructed in high arch dam with high water head, large discharge amount and deep canyon for the purpose of flood discharge and energy dissipation. Discharge capacity of rotational flow shaft spillway increases along with increase of water intake functional water head. When water intake functional water head is 35m, discharge capacity can reach 2800m3/s. Ratio of energy dissipation of rotational flow shaft spillway decreases along with increase of discharge volume. When discharge volume reaches approximately 3000m3/s, ratio of energy dissipation is still higher than 70%. Discharge capacity of rotational flow shaft spillway with transition section is related to geometric features of tapered segment and hydraulic conditions. When height of tapered segment reaches certain degree, difference value ofandis smaller and it is closer to situation without tapered tube. If values ofandare given, then asincreases, it is closer to situation without tapered tube; under condition of,,being fixed, as depth of water of rotating parts decreases, water layer of lower discharge flow is thinner, impact of tapered segment is smaller and it is closer to situation without tapered segment.

Acknowledgment

This work is financially supported by NSFC-Henan Provincial People's Government Joint Fund of Personnel Training (No.U1304511), Collaborative Innovation Center of Water Resources Efficient Utilization and Protection Engineering, Henan Province, Program for Science & Technology Innovation Talents in Universities of Henan Province (No.15HASTIT049), and Program for Innovative Research Team (in Science and Technology) in University of Henan Province (No.14IRTSTHN028). Our gratitude is also extended to reviewers for their efforts in reviewing the manuscript and their very encouraging, insightful and constructive comments.

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