Hydraulic characteristics of a siphon-shaped overflow tower in a long water conveyance system: CFD simulation and analysis*

Kang YU （余康）， Yong-guang CHENG （程永光）， Xiao-xi ZHANG （張曉曦）

1. State Key Laboratory of Water Resources and Hydropower Engineering Science， Wuhan University， Wuhan 430072， China

2. State Key Laboratory of Hydroscience and Engineering， Tsinghua University， Beijing 100084， China，

E-mail: yuk13@mails.tsinghua.edu.cn

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Hydraulic characteristics of a siphon-shaped overflow tower in a long water conveyance system: CFD simulation and analysis*

Kang YU （余康）1，2， Yong-guang CHENG （程永光）1， Xiao-xi ZHANG （張曉曦）1

1. State Key Laboratory of Water Resources and Hydropower Engineering Science， Wuhan University， Wuhan 430072， China

2. State Key Laboratory of Hydroscience and Engineering， Tsinghua University， Beijing 100084， China，

E-mail: yuk13@mails.tsinghua.edu.cn

The siphon-shaped overflow tower is a new type of pressure-suppressing structure used in long water conveyance systems，and it plays a crucial role in guaranteeing the system's stability and safety during hydraulic transient processes. The flow in the tower is characteristic of weir flow in a closed duct， and is thus a complex air-water two-phase flow. Intensive studies of the flow patterns，the pressure pulsations， and the discharge capacity are necessary for better understanding of the flow processes and for the purpose of design. In this paper， we simulate the flow in a siphon-shaped overflow tower under both steady and unsteady flow conditions. Through a steady-flow field simulation， the relationship between the overflow discharge and the pressure in the connected pipeline is analyzed and an empirical formula for evaluating the discharge capacity is provided. Through a transient-flow field simulation， the negative-pressure distributions on the weir crest， the pressure pulsations on the crest and in the falling pond， and the transformation of the air-water two-phase flow in the downstream outlet pipe are analyzed. Moreover， the major influencing factors of the flow patterns， especially， the sectional area of the air vents， are clarified. It is indicated that the siphon-shaped overflow tower can regulate the pressure surge during hydraulic transients and guarantee the safety and stability of the pipeline system， if the shape and the vents are properly designed.

siphon-shaped overflow tower， flow pattern， pressure pulsation， air vent， CFD simulation

Introduction

The pressure-regulating equipment for suppressing the water hammer and preventing the pipe burst is essential for long-distance and large-discharge pressurized-water delivery systems［1-3］， in which the hydraulic transients are normally an serious issue owing to the large water inertia and the complex pipeline layout. Common types of equipment include the simple surge tank， the one-way pressure regulating tower， the air valve， the air tank， and the water release depressurization device［4］. The transient protection devices were widely studied［5-7］， but their selection and installation strongly depend on the specific nature of the particular pipe system and each device has its own problems. For instance， the surge tank is restricted by its topography， the one-way tower cannot depressurize，and the air valve shows a poor exhaust performance and is a potential cause of significant “secondary”high pressure transients［8，9］. In China， the most popular and widely used pressure-suppressing structure is the cylindrical overflow tower. The overflow tower usually consists of a tall well with a circular weir on the top， a water chamber below the weir， an energy dissipation pool， a stilling basin， and a drainage canal，which are all built above the ground. The drawbacks of this structure have come to be known gradually in practice， e.g.， a complex upper structure， complicated techniques， the difficulty in construction， and the large investment. Even worse， the poor capacity of the timely drainage constitutes a great threat to the safety and the stability of the water delivery system， especially under the condition of the valve turnoff in pipeline accidents. To avoid the problems， Pan et al.［10］put forward a new type of overflo w to wer， namely， the siphon-shaped overflow tower.Thepreliminary studyshows that the tower has good features and may be a promising alternative in engineering applications.

The siphon-shaped overflow tower （see Fig.1） is a pressure-suppressing structure with working principles similar to those of normal overflow towers. When the water head in the pipelines exceeds a threshold，the weir overflows so as to restrict the pressure rise. The structure plays a protective role in accidents or under other hydraulic transient conditions. At present，this new type of overflow tower has been first applied to the Dahuofang Reservoir Water Diversion Project in China， and the performance has been satisfactory. The siphon-shaped overflow tower in this project is composed of an inlet pipe， a 90oelbow， a vertical water-rising pipe， a180oelbow， a vertical waterfalling pipe， and a horizontal outlet pipe， as shown in Fig.1. Because the weir flow is in a closed tube， the flow patterns in transient processes are fundamentally different to those of traditional overflow towers. There is a complex and strong air-water two-phase flow，associated with the rapid rising/falling of the water level， the pulsative overflowing， and the violent plunging. An in-depth understanding of the transient-flow characteristics is important for the design of hydraulic structures. It is necessary to carry out studies to guarantee the overflow capacity， enhance the energy dissipation， avoid the cavitation， and keep the flow regime steady in the tower. Moreover， the proper setting of ventilation at the top of the bend is a special issue for this siphon-shaped pipe structure， because it is crucial to avoid a large vacuum， the siphonage phenomenon，and the vibration of the pipe wall.

Fig.1 Schematic diagram of a siphon-shaped overflow tower

At present， the siphon-shaped overflow tower has not been adequetely studied. The only available studies are due to Pan et al.［10］， who experimentally studied seven different schemes by setting several vents at the bend top， adding energy-dissipating plates in the water-falling pipe， and adjusting the outflow conditions of the horizontal outlet pipe， and finally found an optimal scheme for a practical project. The relation between the overflow discharge and the pressure in the connected pipeline was also given. However， only a steady working condition was considered， without discussions on transient-flow characteristics or the optimization of air ventilation. Methodologically， the three-dimensional computational fluid dynamics （3-DCFD） technique is increasingly widely applied in simulations of transient-flows in hydraulic structures with significant 3-D flow features to reveal their comprehensive performance［11，12］. In this paper， we analyze both the steady and transient-flow patterns in a siphon-shaped overflow tower using a 3-D CFD method.

1. Numerical models

1.1 Governing equations

The mass conservation equation and the momentum conservation equation governing the mean flow of incompressible fluid are as follows［13］:

whereμis the dynamic viscosity，F(xiàn)ithe source item，andthe Reynolds stress. Here all variables are mean flow quantities with the time symbols customarily left out. The modeling of the Reynolds stress，based on the Boussinesq's turbulent viscosity hypothesis［14］and for the turbulence closure［15］， are succinctly given below.

1.2 Turbulence model

The renormalization group （RNG）k-εtwoequation model of the Reynolds-averaged Navier-Stokes （RANS） method can be used to simulate flows of high strain rates and large streamline curvatures with good accuracy. Because the rotation of the mean flow and the effect of swirl on turbulence are considered， the RNG model can preferably simulate complex anisotropic flows， as the case in our problem. The transport equations for the RNG k-εmodel［16］are:

where

with the model constants being C1ε=1.42，C2ε= 1.68，η0=4.377，β=0.012and αk=αε=1.39.

1.3 Volume of fluid （VOF） model for air-water two phase flow

The well-known VOF model［17］based on the multiphase flow theory is an effective numerical simulation method with which to track the interface between two or more immiscible fluids. The idea of the VOF method is to use an additional scalar information in each cell to track the ratio of fluids within the whole domain. For air-water two-phase flow，αaand αware defined as the volume fractions of the air phase and the water phase， respectively， in each control cell within the domain. If the cell contains only water， then αw=1， if the cell contains no water， then αw=0. For the cells on the air-water interface，0＜αw＜1.

The air-water interface is tracked by solving a continuity equation for the volume fraction of the water phase

The two fluids share common velocity and pressure fields. In each cell， the average properties are computed according to the volume fraction of each phase. For example， the density and the molecular viscosity in each cell of the two phases are

where ρwand μware the density and the molecular viscosity of the water， while ρaand μaare the density and the molecular viscosity of the air， respectively.

1.4 Numerical schemes and accuracy guarantee

In this work， the commercial software Fluent 6.3 is adopted. The finite volume method is used to discretize the governing equations. The first-order upwind momentum and turbulent kinetic energy discretization scheme is applied. The pressure-implicit with splitting of operators （PISO） pressure-velocity coupling algorithm is used purely because it is specifically designed for transient simulations. It is well known that the hexahedron mesh， adopted here with 1 423 872 cells， is suitable for the multiphase flow in 3-D CFD calculations owing to the higher calculation accuracy. In view of the small mesh size， the transient simulations are performed with variable time steps of about 10-5s-10-4s. A standard wall function［18］is selected to model the turbulence boundary layers. To guarantee the reliability and the accuracy， the numerical model is verified and validated with the published experimental data， a small convergence residual is set， and sufficient calculation time is allowed to stabilize the flow patterns.

Fig.2 Schematic diagram of the 3-D computational domain and boundary conditions

Table 1 Comparison of the relations between the overflow discharge and the connected pipeline pressure

2. Steady-flow patterns and overflow capacity

2.1 Computational conditions

Fig.3 Flow patterns in the tower top for different overflow discharges

The siphon-shaped overflow tower investigated here is equipped with a bend pipe and only one air ven t at the bend top. The inner diameter o2f the pipe is 3.2m，thearea ofthe air vent is 0.283 m at the bend top and the height of the overflow weir is 49.80 m（with the elevation of the centerline of the horizontal inlet pipe being taken as 0 m）. Figure 2 illustrates the 3-D computational domain and the boundary conditions. The boundary conditions are defined as follows. （1）At the inlet of the inlet pipe， the inflow velocity is prescribed. Here， five steady working conditions are chosen according to the different inlet discharges， namely 5 m3/s， 10 m3/s， 15 m3/s， 20 m3/s and 25 m3/s. （2） The atmospheric pressure boundary condition is imposed at the outlet of the air vent pipe. （3） The outlet of the outlet pipe is submerged at a constant depth of 2.00 m.（4） The pipe walls are treated as non-slip wall boundaries using the standard wall function.

The initial conditions are: （1） the water level in the vertical water-rising pipe reaches exactly the overflow level and （2） the inlet velocity corresponds to the aforementioned discharges. Here， we set 20 monitoring points along the overflow weir crest to investigate the negative-pressure distribution and thus judge the possibility of cavitation under the five working conditions.

2.2 Validation

Here we adopt Pan's experimental data［10］to test the performance of the above models in simulating the air-water two-phase flow in the siphon-shaped overflow tower. Note that， the height of the overflow weir of the validation case is higher than that of the case described in Section 2.1. The simulated head-discharge relations are compared with the experimental data in Table 1. A good agreement between the simulated and experimental data is observed， the maximal difference in the head is 0.64 m， which is 1.21% relative to the experimental data. Therefore， the above models are reliable and can be used to investigate the flow patterns in the siphon-shaped overflow tower.

2.3 Results and analyses

Figure 3 shows the flow patterns at the tower top for different inlet discharges. Apparently， the water level upstream the overflow weir rises， the velocity over the weir increases， the overflow water tongue thickens， and the falling water occupies more space of the falling pipe， when the overflow discharge increases.

Fig.4 Relation between the piezometric head at the tower inlet and the overflow discharge

The piezometric head （Hp）is monotonously related to the discharge（Q）， as shown in Fig.4. Here，we use a quadratic function to fit the six points obtained by the CFD， as

This formula may be applied to the hydraulic transient analysis of the pipeline system for defining the boundary condition of the overflow tower.

Fig.5 Pressure distributions on the weir crest for different overflow discharges

Figure 5 shows the distributions of the pressure head（H）on the weir crest under various working conditions. The pressure distributes non-uniformly on the weir crest， decreasing along the first 3/4 of the monitoring points（P）and then increasing along the last 1/4 of the points. With the discharge（Q）increasing， the overall pressure on the weir crest drops and the negative-pressure region expands. The minimum pressure changes from -1.50 m to -6.50 m， the minimum-pressure point retreats from the vicinity of the point 18 to the vicinity of the point 15， and the negative-pressure region enlarges from a half of the crest arc length to nearly the total of the arc length， when the discharge increases from 5 m3/s to 25 m3/s. Moreover， the maximum vacuum degree， approximately 6.50 m under the 25 m3/s discharge condition， is close to the critical value for cavitation， which is around 7.00 m to 8.00 m water head if the local atmospheric pressure is 10.00 m water head. Therefore， in practical applications， the allowable overflow discharge of this siphon-shaped overflow tower should be less than 25 m3/s.

3. Transient-flow patterns and pressure pulsations

3.1 Computational conditions

During the transient process， the overflow discharge， the water head at the tower inlet， the airflow rate through the air vent， and other parameters of the flow field change with time. The transient-flow patterns in the tower see great differences from those under the steady conditions， and are the main issue of interest in this paper.

Fig.6 Computational conditions for the transient flow simulation

Taking the inlet discharge Qinas the boundary condition， the transient flow in the tower is simulated. The given inlet discharge history is shown as a translated sine function in Fig.6. Two periods of simulation are employed to eliminate the influence of initial conditions. According to the above steady-flow results，we select 15 m3/s as the extreme amplitude of the discharge pulsation， to avoid the cavitation to a great extent. At the initial time， the water level in the tower reaches exactly the end of the vertical water-rising pipe， at 49.15 m， which is slightly below the weir crest. Moreover， at the water cushion basin of the vertical water-falling pipe， an additional monitoring point is set to investigate the impact pressure pulsation caused by the plunging jet. The computational domain and other conditions are the same as those in the previous section.

3.2 Results and analyses

The simulation results of the second period of the inlet discharge pulsation are analyzed in this section. For a clear narration， the starting time of the second period is reset to 0 s， although it is actually 200 s.

Fig.7 Histories of the inlet head， overflow discharge and air discharge in the transient process

3.2.1 Water head in the pipeline and air vent discharge

Figure 7 shows the histories of the piezometric head at the inlet （Hp）， the overflow discharge（Q）through the weir crest， and the air vent discharge （Qa）. As prescribed above， here the time of 0 s corresponds to the beginning of the second period， when the piezometric head at the inlet is approximately 37.00 m and the overflow discharge is zero. As shown in Fig.7，there are three phases of the transient process. In the first phase， the water rises rapidly fromt=0 s to 30.0 s. While， the inlet piezometric head rises persistently to fully fill the rising pipe and there is no overflow because the water level is lower than the weir crest. The water level is equal to the crest level at approximatelyt=30.0sand the piezometric head attains almost its maximum level several seconds later. Furthermore， the influent water expels the air out of the air vent， leading to the pulsation of the air discharge.

Fig.8 Pressure distributions on the weir crest at different moments

In the second phase， the water overflows from t=30.0sto approximately 150.0 s， and can be separated into two sub-phases: the surging overflow and the withdrawing overflow. After the onset of the overflow， the overflow discharge through the weir crest section gradually increases after a steep rise， and a negative pressure emerges at the weir surface soon afterwards. For the air vent， the exhausting and suction exchanges increase as a result of the formation of vacuum and the imbalance of the intake discharge and the overflow discharge. This increasing of the discharge sub-phase lasts from t=30.0sto approximately 75.0 s， when the overflow attains its maximum. Afterward， the overflow discharge switches to continually decrease to zero aroundt=150.0sowing to the decrease of the pressure head at the inlet. During this decreasing discharge sub-phase， the airflow rate through the air vent fluctuates acutely owing to strong variations of the flow patterns and pressure in the tower. This process is taken as the most crucial stage for the air vent to regulate and balance the pressure differences. It is worth noting that during the overflowing phase， the piezometric head does not change significantly， which means that the siphon-shaped overflow tower plays an effective role in the pressure stabilization via the overflowing.

In the third phase， the water falls aftert=150.0s. With overflowing being ceased， the water level declines from the weir crest level， and the piezometric head thus declines quickly. For the air vent， the airflow rate fluctuates relatively slightly owing to the gradual balancing of the inside and outside pressures.

Fig.9 Comparison of pressure distributions on the weir crest under the transient and steady flow conditions

3.2.2 Pressure on the weir crest

Figure 8 shows the pressure head（H）distributions on the weir crest at several moments in the transient process. Obviously， the increase and decrease tendencies of the vacuum degree in the weir crest and the overflow discharge（Q）are similar. The negative-pressure values and the negative-pressure regions change with the overflow discharge， and the most intense moments are from t=65.0 s to 75.0 s，during which the lowest pressure is around -4.00 m and the negative-pressure region is between the points 6 and 20. Moreover， the minimum pressures appear atthe points 16-18 among the several moments shown in Fig.8.

Comparing the pressure distribution at t=75.0s（Q=15 m3/s）in the transient process with that of the constant dischargeQ=15 m3/sunder the steady-flow condition of the previous section， it is shown that the deviation of the maximum negative pressures is approximately 1.00 m， and the pressure in the transients is generally higher than that under the steady-flow condition with a similar distribution， as shown in Fig.9. This means that the air vent works well in the current situation and plays an effective role in boosting the pressure and preventing the vacuum by suction in the transient process.

Fig.10 Pressure pulsations on the weir crest and in the water cushion basin for the transient condition

Figure 10 shows the pressure histories on the weir crest at several monitoring points in the transient process. Apparently， the pressure heads and the overflow discharge have opposite varying tendencies， in particular， the pressure pulsation emerges around the time of the highest overflow discharge， reaches a violent state around the fast-decrease time and becomes calm when the discharge vanishes. The characteristic values of the pulsating pressure are summarized in Table 2， from which we see that the pressure pulsations at different points share the same dominant frequency while the amplitude（A）is larger if the point moves towards the right. Actually， for the points in Table 2， the points are numbered in the order of increasing the vacuum degree. Although the pulsation amplitudes are not very large and the maximum vacuum degrees are not as high as to cause cavitation，the data sufficiently indicate that there are three factors that should be taken into account when investigating the maximum vacuum degree on the weir crest during transient processes. They are （1） the locations of the monitoring points， （2） the instant corresponding to the largest discharge， and （3） the instantaneous pulsation influence on pressure values.

Table 2 Characteristic values of the pulsating pressure at different monitoring points

3.2.3 Pressure pulsation in the water cushion basin

In the transient process， with the overflow discharge variations， the pressure pulsations are induced by the plunging jet at the bottom of the vertical waterfalling pipe， which resembles a water cushion basin. Figure 10 shows the pressure pulsation at the aforementioned demersal monitoring point （1） （see Fig.6）. At approximately t=30.0s， with a rapid rise of the overflow discharge， the pressure head at the point increases sharply. Then， the pulsation changes with the overflow discharge. From approximatelyt= 60.0 s to t=70.0s， the pressure decreases considerably owing to the strengthened aerial energy dissipation effect by the breaking and diffusing of the aerated jet，caused by the intensified impacting and mixing of the high-speed airflow and plunging jet. Thereafter， the pressure remains at a lower level until approximately t=130.0s， after which it decreases further with the decreasing overflow discharge. Aftert=150.0s， the pulsating pressure dampens and vanishes finally， and the following wavy oscillation indicates a transientflow surge in the horizontal downstream outlet pipe.

Fig.11 Transitional flow patterns in the tower top and velocity vectors near the vent

Fig.12 Transitional flow patterns in the horizontal outlet pipes

The statistical parameters of the pressure data include the mean pressure head of 12.58 m and the maximum pressure head of 24.52 m at t=51.2s. The power spectral density of the pressure pulsation shows that the pressure pulsation is a narrow-band， low-frequency stochastic process and the main frequency is 0.011 Hz， corresponding to the dominant frequency（fd）of the pressure pulsation on the weir crest.

3.2.4 Flow patterns at the tower top

The transitional flow patterns at the tower top at several instants are displayed in Fig.11. The rising and falling of the water level over the weir and the thickness variation of the water tongue can be seen. Soon after the onset of the overflow， the influent water occupies a certain space， expelling the air out of the pipe through the vent （Fig.11（a））. With an increase of the overflow discharge， the air flow in the vent reverses its direction， owing to the tendency of the siphonage in the tower （Fig.11（b））. When the overflow discharge switches from increasing to decreasing and the flow pattern in the horizontal outlet pipe changes between full and partly full regimes， the airflow rate through the air vent fluctuates frequently to balance the inside and outside pressures （Figs.11（c）-11（e））.

3.2.5 Flow patterns in the horizontal outlet pipe

Figure 12 visualizes the transitional flow patterns in the horizontal outlet pipe. Figure 12（a） shows the incipient full-flow state in the pipe， as a result of a small overflow discharge and the air intake flow rate. With the increase of the overflow discharge， the highspeed plunging jet facilitates the air entrainment and a certain amount of air is entrained into the out-let pipe，resulting in a stratified air-water two-phase flow（Fig.12（b））. Both the air and the air pressure increase owing to the increasing discharge. As the overflow discharge begins to decrease， the pressurized air in the pipe is gradually pumped out， resulting in the drawdown of the air pressure （Fig.12（c））. Because both the overflow discharge and the air pressure in the pipe decrease further， the free-surface flow turns up（Fig.12（d））. When the overflowing ceases， the fullflow state re-establishes in the pipe （Fig.12（e））.

4. Influence of the area of the air vent

As mentioned above， the ventilation is a key factor for the flow patterns in the siphon-shaped overflow tower. Therefore， the effects of the sectional area of the air vents are investigated specifically.

4.1 Computational conditions

Four more air vents with different sizes are simulated. Case 3 is the original vent， the vents of Case 2 and Case 1 decrease in size while those of Case 4 andCase 5 increase in size. The range of the diameter ratio of the vents and the pipe is 0.132-0.265. The boundary and initial conditions are set according to Fig.6.

Fig.13 Comparison of the pipeline head histories in different cases

4.2 Results and analyses

4.2.1 Pipeline head and overflow discharge

Fig.14 Comparison of the air discharge variations for different vents

Despite various vent sizes， different cases share the same overflow process. The histories of the piezometric head at the tower inlet for the five different orifice areas with the same overflow discharge pulsation are compared in Fig.13. Initially， all water levels in the rising pipes are 49.15 m， which is slightly lower than the weir crest. Then， at approximately t=8.0s，the water levels reach the weir crest height and thewater starts to overflow. Different cases see small differences in the inlet piezometric head regardless of the same overflow discharge. From approximately t =75.0sto t=130.0s， the smaller vents lead to more intense pressure pulsations at the tower inlet，especially in Case 1. Thus for a better pressure regulation， the diameter ratio of the vent and the pipe is recommended not smaller than 0.162 （Case 2）.

Fig.15 Comparison of pressure histories at two typical points on the weir crest in different cases

4.2.2 Air vent discharge

The histories of the air discharge in different cases are compared in Fig.14. Before the overflow discharge decreases， all curves have the same tendency， as shown in Fig.7. Significant differences are seen when the overflow discharge reaches its maxi-mum. In the period that the overflow discharge decreases， the air flows of the larger vents fluctuate less than those of the smaller vents. However， in the waterfalling phase， the air flows of the larger vents are dwindled from larger amplitudes while those of the smaller vents are dwindled from smaller amplitudes. Table 3 compares the maximum air speeds in the vents，showing a monotonic decrease with the increase of the vent size. In summary， the smaller vents （Cases 1 and 2） induce more intense water and air fluctuations during overflowing and the larger vents （Cases 4 and 5） induce large amplitudes after overflowing ceases. Therefore， an eclectic design of the vent size （the diameter ratio of the vent and the pipe， 0.162-0.230）， may be indispensable to effectively balance the inside and outside pressures of the tower in the whole transient process.

Table 3 Comparison of the maximum air speeds for different vents

Table 4 Comparison of the maximal negative-pressures on the weir crest

4.2.3 Pressure on the weir crest

During the period of decreasing the overflow discharge， smaller vents （e.g.， Cases 1 and 2） lead to more intense pressure pulsations on the weir crest and thus expand the negative-pressure region， as shown in Fig.15. This again confirms the poor pressure regulation capacity of smaller vents. The maximal negative pressures in different cases are compared in Table 4. Apparently， smaller vents induce higher maximal negative pressures， especially when the vents are smaller than the original. In addition， in Cases 1 and 2， the locations of the maximal negative pressure are in higher points than in other cases and the occurrence times fall into the decreasing overflow discharge subphase， during which the air vents play the most crucial role in regulating and balancing the pressure differences. In summary， for an effective cavitation prevention， the diameter ratio of the vent and the pipe is recommended not smaller than 0.187 （Case 3）.

4.2.4 Pressure pulsation in the water cushion basin

Table 5 summarizes the characteristic parameters of the pressure pulsations in the water cushion basin in different cases. For Case 1， the mean value is lower while the maximum value and the standard deviation are larger， compared with other cases， which indicates more intense pressure pulsations in this case， corresponding to the analogous situation on the weir crest.

Table 5 Comparison of the characteristic parameters of the pulsating pressure in the water cushion basin

Table 5 shows that the maximal pressure at the bottom of the water-falling pipe declines with the increase of the vent size. This is because large vents lead to intensive impacting and mixing between the airflow and the plunging jet， which are more beneficial to the energy dissipation.

Moreover， spectral analysis shows that different cases share the same range of dominant frequency，except that a small amount of energy is concentrated around 1.5 Hz for the intense pressure pulsation in Case 1.

Through the above comparisons， we conclude that comprehensive considerations are required when optimizing the vent size for good performance in terms of pressure balance， water and air fluctuations， cavitation prevention， and energy dissipation， and the diameter ratio of the vent and the pipe， as an important design parameter， is recommended in the range of 0.187-0.230.

5. Conclusions

In this paper， the flow patterns in the siphonshaped overflow tower， particularly the weir crest pressure distributions， the air-water two-phase flow pulsations， and the influences of the air vent on the flow patterns are analyzed in detail. Conclusions are as follows:

（1） During the transient processes， the water in the siphon-shaped overflow tower experiences a rapidrise， a surging overflow， and a withdrawing overflow. The negative-pressure region and the lowest negative pressure on the weir crest are determined mainly by the overflow discharge， and their extreme values correspond to the maximal discharge.

（2） Pressure pulsations are intense after the occurrence of the maximum overflow discharge， and the pulsation frequencies correspond to the interactions of the plunging jet， the pipe walls， and the high-speed airflow.

（3） Air ventilation at the tower top can reduce both the region and the value of negative-pressure，and plays a positive role in preventing cavitation. With due consideration of the pressure difference balance and the air-water pulsations， the diameter ratio of the vent and the pipe is recommended in a range of 0.187-0.230.

（4） If well ventilated， the negative-pressure distributions and the values under the transient-flow condition agree well with those in the steady-flow condition with the same discharge. This means that the discharge capacity can be reasonably ascertained under steady-flow conditions. However， the overflow pulsating characteristics of the two conditions are totally different owing to the unsteady flow patterns and the intensive air pulsations in the transient-flow condition.

（5） In the design process， the allowable maximum overflow discharge can be decided by the steady-flow calculation or the experiment based on the cavitation prevention criteria for the weir crest. Meanwhile， the number and the area of the air vents can be optimized by transient CFD simulations.

（6） The relationship between the overflow discharge and the inlet pressure， obtained by the experiment or by the steady-flow simulation， can be used as the boundary condition defining the siphon-shaped overflow tower in 1-D transient simulations of the pipeline system.

The direct specification of the discharge history at the inlet pipe is just an assumption and may not agree with the actual situation. Therefore， a coupled simulation of the 1-D hydraulic model for the pipeline system and the 3-D two-phase CFD model for the tower is necessary to investigate the overflow behaviors and their influences on the pipeline system.

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［14］ SCHMITT F. G. About Boussinesq's turbulent viscosity hypothesis: Historical remarks and a direct evaluation of its validity［J］. Comptes Rendus Mecanique， 2007，335（9-10）: 617-627.

［15］ POPE S. B. The determination of turbulence-model statistics from the velocity-acceleration correlation［J］. Journal of Fluid Mechanics， 2014， 757: 1-9.

［16］ KIM W. J.， KIM D. H. and VAN S. H. Computational study on turbulent flows around modern tanker hull forms［J］. International Journal for Numerical Methods in Fluids， 2002， 38（4）: 377-406.

［17］ GAO D.， MORLEY N. B. and DHIR V. Numerical simulation of wavy falling film flow using VOF method［J］. Journal of Computational Physics， 2003， 192（2）: 624-642.

［18］ NAZIF H. R.， TABRIZI H. B. Comparison of standard turbulent wall function with a non-equilibrium wall model［J］. International Journal of Fluid Mechanics Research， 2011， 38（6）: 499-508.

10.1016/S1001-6058（16）60660-1

August 23， 2014， Revised January 27， 2015）

* Project supported by the National Natural Science Foundation of China （Grant Nos. 51039005， 50909076 and 51579187）.

Biography: Kang YU （1992-）， Male， Ph. D.

Yong-guang CHENG，

E-mail: ygcheng@whu.edu.cn

2016，28（4）:564-575