Theoretical and numerical study of hydraulic characteristics of orifice energy dissipator

Ning HE*, Zhen-xing ZHAO

1. Institute of High Energy Physics, CAS, Beijing 100049, P. R. China

2. College of Mechanics and Materials, Hohai University, Nanjing 210098, P. R. China

Theoretical and numerical study of hydraulic characteristics of orifice energy dissipator

Ning HE*1, Zhen-xing ZHAO2

1. Institute of High Energy Physics, CAS, Beijing 100049, P. R. China

2. College of Mechanics and Materials, Hohai University, Nanjing 210098, P. R. China

Different factors affecting the efficiency of the orifice energy dissipator were investigated based on a series of theoretical analyses and numerical simulations. The main factors investigated by dimension analysis were identified, including the Reynolds number (Re), the ratio of the orifice diameter to the inner diameter of the pipe (), and the ratio of distances between orifices to the inner diameter of the pipe (). Then, numerical simulations were conducted with ak-εtwo-equation turbulence model. The calculation results show the following: Hydraulic characteristics change dramatically as flow passes through the orifice, with abruptly increasing velocity and turbulent energy, and decreasing pressure. The turbulent energy appears to be low in the middle and high near the pipe wall. For the energy dissipation setup with only one orifice, whenReis smaller than 105, the orifice energy dissipation coefficientKincreases rapidly with the increase ofRe. WhenReis larger than 105,Kgradually stabilizes. Asincreases,Kand the length of the recirculation regionL1show similar variation patterns, which inversely vary with. The function curves can be approximated as straight lines. For the energy dissipation model with two orifices, because of different incoming flows at different orifices, the energy dissipation coefficient of the second orifice (K2) is smaller than that of the first. Ifis less than 5, theKvalue of themodel, depending on the variation ofK2, increases with the spacing between two orificesL, and an orifice cannot fulfill its energy dissipation function. Ifis greater than 5,K2tends to be steady; thus, theKvalue of themodel gradually stabilizes. Then, the flow fully develops, andLhas almost no impact on the value ofK.

orifice energy dissipator; theoretical analysis; numerical simulation; k-ε two-equation turbulent model; hydraulic characteristics

1 Introduction

The orifice energy dissipator is a new energy dissipation method that dissipates the energy within outlet works. The general mechanism of the orifice energy dissipator can be summarized as follows: by using a sharp-edged orifice to generate sudden enlargements in tunnel flow, a large amount of energy can be dissipated over a small distance.

The advantages of the orifice energy dissipator (Li 1999) are as follows: optimization of the project layout; reduction of difficulty in energy dissipation in mountainous areas, which always have a limited spatial area; and resolution of the complex technical problems related tohigh heads and velocities that cannot be solved by conventional energy dissipation methods. Although it is known that many factors may influence the efficiency of the orifice energy dissipator, the mechanisms of orifice energy dissipators are not yet fully understood.

Until now, most research on orifice energy dissipators has been conducted through experiments (Cai et al 1999; Chen et al. 2006; Wei et al. 2006; Yang et al. 2004), which can only provide parameters for total flow. Practical data are also far from sufficient. However, numerical simulation can provide information on the total distribution of each parameter. Moreover, the numerical results are not affected by the scale effect of physical model tests. With advances in computer science and technology, the computational models have become highly efficient and can be used with greater convenience and higher precision. In this study, dimension analysis was conducted to identify the main factors that affect the orifice energy dissipator. Then, numerical simulations were conducted with ak-εtwo-equation turbulence model for energy dissipation to investigate the main factors.

2 Theoretical analysis

Fig. 1 Theoretical analysis model

The theoretical analysis model is shown in Fig. 1, wheredandDare, respectively, the diameter and inner diameter of the orifice and pipe. Considering the incompressible steady flow in an orifice energy dissipator, cross-section 1 is determined to be 0.5Din front of the orifice, and cross-section 2 is 2.5Dbehind it. The flow through cross-sections 1 and 2 is fully developed flow, and the velocity distribution profiles are the same, so the kinetic energy correction coefficients of the two cross-sections,α1andα2, are constants equal to one. The continuity and momentum equations between cross-sections 1 and 2 can be described as follows:

whereQis the flow rate;A1andA2are the areas of cross-sections 1 and 2, respectively;v1andv2are the fluid velocities at two cross-sections;p1andp2are the average water pressures at two cross-sections; andρis the fluid density. The energy loss can be obtained:

wheregis the gravitational acceleration. With the assumption that the pressure distribution at the cross-sections obeys the principle of hydrostatic pressure, Eqs. (1) through (3) are combined and rearranged. ΔEcan be further described as

In order to identify the factors affecting the efficiency of the orifice energy dissipator, dimension analysis is performed to simplify the problem. The following function (Bushell et al. 2002) is assumed:

whereμis the fluid viscosity, Δpis the pressure drop,Lis the spacing between two orifices, andvis the fluid velocity.

According to the ∏ theorem and the principle of dimensional homogeneity, the parametersD,v, andρa(bǔ)re defined as basic dimensions. This problem can be described in the ∏ equations as follows:

The functional relationship can be written as

The complete dimensionless relation for this problem is

The energy loss between cross-sections 1 and 2 is described as

From Eq. (9), we can obtain the following relationship:

According to the pressure distribution in experiments, the head loss can be expressed by the average pressure difference between cross-sections 1 and 2. With Eqs. (4) and (10), the coefficient of the orifice energy dissipator can be expressed as

3 Numerical study

3.1 Mathematical model

In order to simulate the flow in an orifice energy dissipator, it is vital to select a proper turbulence model (Schiestel 1987). In this study, thek-εtwo-equation turbulence model (Liu et al. 1993; Qu et al. 2000; Zhang et al. 2004) was applied to simulate the flow in an orifice energy dissipator. The control volume method (Tao 2004) was employed to obtain the discretization equations by integrating the governing equations over each control volume.

The flow in the pipe is considered incompressible, three-dimensional, and viscous. The governing equations include the continuity equation, the momentum equation, the turbulent kinetic energy equation (kequation), and the turbulent kinetic energy dissipation rate equation (εequation). These equations can be written as follows:

Continuity equation:

Momentum equation:

kequation:

εequation:

whereuiandujare the components of velocity;C1εandC2εare coefficients;σkandσεare the turbulent Prandtl numbers forkandε; and the values of these parameters are as follows:C1ε=1.44,C2ε=1.92,σk=1.0, andσε=1.3. The turbulent viscosityμtis computed by combiningkandεas follows:

whereCμis a coefficient, andCμ=0.09.Gkis defined as

The boundary conditions (Launder and Spalding 1974; Tian et al. 2005; Xia and Ni. 2003) are as follows: (1) The inflow velocityU0and uniform velocity profile form the inlet boundary. (2) The uniform flow condition is the outlet boundary. (3) The wall function andno-slip boundary condition form a wall boundary.

3.2 Model setup

On the basis of previous analytical studies, the main factors that influenceKwere identified:Re,, and. In order to investigate the influence of different factors onK, different numerical models were constructed for each factor: theRemodel (including an idealRemodel and an actualRemodel), themodel, and themodel. In the models, only one factor was changed for a series of numerical experiments.

3.2.1Remodel

TheRemodel for the investigation of the influence of the Reynolds number is described below. The other models are defined in the same way.

(1) IdealRemodel

In order to calibrate the numerical model, an idealRemodel was set up. In this model, onlyRewas altered while the other factors remained the same. The general setting of theRemodel was a 14.5 cm-diameter pipe with a 2 cm-thick orifice. In front of and behind the orifice were 20-cm and 50-cm straight pipes. Flow in front of the orifice was fully developed. The kinematic viscosity wasγ=1.003×10-6m2/s at a temperature 20°C. For different models, slight changes ofγcan be made.Rewas altered within a range of 104to 106in the experiments and the ratio ofwas set at 0.60, 0.65, 0.70, and 0.75.

(2) ActualRemodel

To obtain results with practical significance, an actualRemodel, in which the length unit in the idealRemodel was changed from centimeters to meters, was constructed. The correspondingRevaried from 107to 108.

3.2.2dDmodel

WhenRe, the orifice shape, andLare constant,Kdepends primarily on changes in. The model in whichwas varied but other factors were fixed was called themodel. The setting of themodel was as follows: The pressure pipe and the orifice location were the same as those in the actualRemodel. The value ofvaried from 0.5 to 0.8. The mean velocity inside the pipe was 6.92 m/s and the correspondingReof the flow was 108.

3.2.3model

WhenRe, the orifice shape, andare constant,Kdepends primarily on changes in. The model in which the orifice shape andwere fixed butwas varied was called themodel, and it is shown in Fig. 2:

Fig. 2model

The inner pipe diameter of the pressure pipe with double orifices was 14.5 m. Thediameter of each orifice was 10 m and each had a thickness of 2 m. In front of the first orifice, a straight round pipe with a length of 20 m was connected, so that the flow from the front orifice could be fully adjusted. Behind the second orifice was a 50-m straight round pipe. The mean velocity inside the pipe was 6.92 m/s and the correspondingReof the flow was 108.varied from 0.5 to 10.

3.3 Calculation results and analysis

3.3.1 Hydraulic characteristics analysis

We used the simulation results for=3, as shown in Fig. 3, to analyze the energy dissipation characteristics near orifices. In Fig. 3,lis the distance from the left endpoint of the pressure pipe. The cross-sectional average pressure in the horizontal direction can be seen to have step distribution. Because of the contraction effect of the orifices, part of the flow potential energy transforms into kinetic energy, and the pressure suddenly decreases as flow passes through the first orifice. Later, the flow potential energy gradually recovers, and the pressure curve continues increasing behind the first orifice, which means that the pressure does not completely recover when the flow passes through the second orifice. The same situation occurs as flow passes through the second orifice.

Fig. 3 Pressure distribution along tunnel

The energy dissipation coefficientKof the two orifices was calculated with Eq. (10), and ΔEwas calculated based on the difference in water heads at the locations 0.5Dahead of an orifice and 2.5Dbehind it. Thus, the energy dissipation coefficients of the first and second orifices were obtained; they were 1.08 and 0.786, respectively. The difference in flow conditions leads to the difference in the coefficients. The inflow at the first orifice is more uniform. Therefore, the contraction effect has full play and affects significant energy dissipation as flow passes through the first orifice. However, because of the contraction effect of the first orifice, the second orifice inflow is concentrated in the middle, which means that the second orifice cannot effectively dissipate energy.

In Fig. 4 it can be seen that hydraulic characteristics are distributed symmetrically along the axis of the pipe. Due to the contraction effect of the orifice, the cross-section of the main flow suddenly decreases and all the hydraulic characteristics change dramatically, with an abrupt increase in velocity and turbulent energy and a decrease in pressure. Taking the center line of the pipe as the axis, the velocity and pressure recover gradually behind the orifice, while the turbulent energy decreases gradually and appears to be low in the middle and high near the pipe wall. Behind the orifice there is a whirlpool region between the main flow and the wall, andstrong shear stress exists between the whirlpool and main flow regions, which leads to the conversion of kinetic energy to thermal energy. Then, flow energy dissipates as the thermal energy disappears.

Fig. 4 Hydraulic factor distribution

3.3.2 Total model results

(1)Remodel results

(a) IdealRemodel results:

Fig. 5 shows the relationship betweenReandKfor different values of. WhenReis less than 105, the turbulent flow is not fully developed, and the hydraulic factors vary intensely. Under such conditions,Kincreases significantly with increase ofRe. WhenReis larger than 105, the turbulent flow can be considered fully developed, and the change in hydraulic factors is very small, which means thatRedoes not impact the energy dissipation coefficientK. In Fig. 5 we can see thatKtends to be stable whenReis greater than 105.

(b) ActualRemodel results:

Fig. 6 shows thatKhardly increases with increases ofRefor different values of, which supports the results of the idealRemodel, whereRehas no influence onKin fully developed flow. As real flow can usually be considered fully turbulent flow, the impact ofReis always ignored in hydraulic engineering design.

As shown in Fig. 7, the lengthL1of the recirculation region is defined as a distance from the orifice surface to the zero-velocity point of backflow. Fig. 8 shows the relation betweenReandL1. For different ratios of,L1increases withRe. As the recirculation region is the key area for the exchange of energy and momentum between the main stream region and the recirculation region, the longerL1is, the greater the energy loss and the value ofKare.

Fig. 5 Relation betweenReandKof idealRemodel

Fig. 6 Relation betweenReandKof actualRemodel

Fig. 7 Sketch ofL1

Fig. 8 Relation betweenReandL1

(2)dDmodel results:

Figs. 9 and 10 show that the length of the recirculation regionL1and the orifice energy dissipation coefficientKhave similar variation patterns. The functions can be approximated as straight lines. AsdDincreases,L1andKdecrease rapidly. The reasons are as follows: With the increase ofdD, the energy conversion between kinetic energy and potential energy becomes smaller. Accordingly,K, which represents the efficiency of energy loss during the process of energy conversion, becomes smaller as well.

Fig. 9 Relation betweenandL1

Fig. 10 Relation betweenandK

(3)model results:

Fig. 11 shows the relation betweenandKof the wholemodel. If the distance between the orifices is too short, the recirculation region within that space cannot fully develop. The distribution of hydraulic factors changes intensely, and the total head loss is small. Therefore, the effect on energy dissipation is not very significant. When the distance between the orifices increases, the total head loss increases at the same time. However, whenis larger than 5, the recirculation region within the space between two orifices is fully developed. The hydraulic factors vary uniformly. The value ofKgradually stabilizes. In contrast to the condition=5, the efficiency of energy dissipation in the condition=3 can reach 95.7% .

Fig. 11 Relation betweenandK

Fig. 12 Relation betweenandK2

Simulation results show that the energy dissipation coefficient of the first orifice (K1) increases rapidly whenis less than 1.7, and thatK1is stable around 1.23, whenis larger than 3. Generally, the first and second orifices have different incoming flows. The incoming flow of the first orifice is more uniform and that of the second one is more centralized. Because of this, the coefficient of energy dissipation of the second orificeK2is small. Fig. 12 shows the relation betweenandK2, demonstrating that it is more difficult forK2to reach a stable value. WhenK1reaches a relatively stable value, the variation ofKprimarily depends onK2. In Fig. 11 we can see that theKvalue increases slowly whenchanges from 1.7 to 5. However, whenis larger than 5, the flow velocity and pressure vary slightly within the space between the two orifices (Li 1999) and the recirculation region between the two orifices develops fully; thus,K2tends to be steady. Accordingly, theKof the wholemodel gradually stabilizes andhas almost no impact on its value.

4 Conclusions

The main factors that influence the efficiency of the orifice energy dissipator were examined through numerical simulation. The relation between head loss and the factors was investigated.

(1) WhenReis less than 105,Kincreases rapidly along withRe. WhenReis greater than 105,Kgradually stabilizes. The recirculation region is the key energy dissipation region. TheshorterL1is, the less the energy loss and theKvalue are.

(2)Kis sensitive to the increase ofdD. The relation betweenKanddDcan be approximated as an inverse relation.

(3) The change in theKvalue withLDin theLDmodel can be divided into two phases. IfLDis less than 5,Kincreases rapidly withLD, but ifLDis larger than 5,Kgradually stabilizes, and changes inLDhave almost no influence on bothKandK2.

(4) The numerical results indicate that it is feasible to use ak-εtwo-equation turbulence model to simulate the flow in an orifice energy dissipator, and that the orifice energy dissipator is an effective means for energy dissipation.

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*Corresponding author (e-mail:hening@ihep.ac.cn)

Received Dec. 15, 2008; accepted Dec. 10, 2009