Flows through energy dissipaters with sudden reduction and sudden enlargement forms

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2010,22(3):360-365 DOI: 10.1016/S1001-6058(09)60065-2

FLOWS THROUGH ENERGY DISSIPATERS WITH SUDDEN REDUCTION AND SUDDEN ENLARGEMENT FORMS* WU Jian-hua, AI Wan-zheng College of Water Conservancy and Hydropower Engineering, Hohai University, Nanjing 210098, China, E-mail: [email protected]

(Received March 23, 2010, Revised June 7, 2010)

Abstract: The energy dissipation of flood discharges has been one of important problems that affect directly the safety of hydropower projects. The energy dissipater with sudden reduction and sudden enlargement forms, used widely in large-scale projects, has been a kind of effective structure for energy dissipation. The concept of critical thickness was defined, which is related to both the geometric parameters and the hydraulic parameters of the energy dissipater, and the factors affecting the critical thickness, were analzsed by means of dimensional analysis. The empirical expression about the critical thickness was obtained and could be used as the criterion to distinguish the flows through the energy dissipater, i.e., the plug flow and the orifice plate flow. The error analysis showed that the critical thickness calculated by the expression has the errors of smaller than 10% in the estimation of the flows for the energy dissipater mentioned above. Key words: critical thickness, energy dissipater, orifice plate, plug, reduction ratio, sudden reduction, sudden enlargement

1. Introduction  Both the plug and the orifice plate, as the two types of energy dissipaters with sudden reduction and sudden enlargement forms, have been successfully used in large-scale hydropower projects. For the Mica dam in Canada the flow velocity of the flood discharge tunnel was decreased from 52 m to 35 m at the head of 175 m, due to the use of the plug energy dissipaters, which were the two plugs with the lengths of 49 m and 37 m[1, 2]. In the Xiaolangdi hydropower project in China the three orifice plates in the flood discharge tunnel got the energy dissipation ratio of 44% and controlled effectively the flow velocity through the gate to lower than 35 m/s under the condition of the head of 145 m[3,4]. Many researches investigated the energy 

* Project supported by the Ministry of Science and Technology of China (Grant No. 2008BAB19B04). Biography: WU Jian-hua (1958-), Male, Ph. D., Professor

dissipaters with sudden reduction and sudden enlargement forms. The interest has been focused on the effects of the geometric parameters on hydraulic characteristics, such as energy dissipation ratio, and cavitation performance and so on. The contraction ratio ( E ), defined as the ratio of the orifice diameter ( d ) of the energy dissipater and the diameter ( D ) of flood discharge tunnel, is an important index affecting all the hydraulic characteristics. Bullen et al.[5], Wang and Yue[6], Chai et al.[7], Fossa and Guglielmini[8], He and Zhao[9] deemed that the energy dissipation ratio increases with the decrease of the contraction ratio. The energy dissipation ratio increases with decreasing contraction ratio. Meanwhile, the contraction ratio influences directly the cavitation performance, and the incipient cavitation number decreases with the increase of contraction ratio according to the results from Ball et al.[10], Huang and Liu[11], Wu et al.[3], Zhang and Cai[12], and Tian et al.[13]. The other geometric parameters, such as the thickness ( T ), the shape, and the edge radius, have

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the effects on the energy dissipation ratio, or/and cavitation performance. The sharp-edged form has larger energy dissipation ratio [14] compared with the square-edged and sloping-approach forms and the orifice plate with ring, but causes the increase of the incipient cavitation number[3] at the same contraction ratio. Cai and Zhang[15] showed that the energy dissipation ratio decreases with the increase of the thickness of the energy dissipater. The thickness ( T ) of the energy dissipater brings about the great changes of the flow regime through it. When this thickness is large enough the flow becomes the tube flow in it, while it is the orifice flow for the thinner energy dissipater. They are usually called as plug energy dissipater and orifice plate one respectively in order to investigate their hydraulic characteristics[16-18]. As a matter of fact, those names, only based on the geometry of the energy dissipaters, are not very reasonable for certain range of thickness, and the flows through the energy dissipaters with sudden reductioin and sudden enlargement forms are closely related not only to the geometric parameters but also the hydraulic parameters. It is possible that the tube flow occurs when the flow velocity is low and then changes into the orifice flow with the increase of the velocity for a given thickness. The use of the names plug flow and orifice plate flow, and the confirmation of the flow regime through them, or the presentation of the criterion of the critical status for the flow regime development, are of significance. The objectives of the present work, therefore, are to define the critical thickness distinguishing the plug and orifice plate flows through the energy dissipaters, to analyze the factors affecting this thickness, and to establish an empirical expression as a criterion to determine the flows through the energy dissipater mentioned above. 2. Definition of critical thickness For the given geometry of energy dissipater with sudden reduction and sudden enlargement forms, there are two kinds of the flows on the basis of their flow regimes when they pass through the energy dissipaters as shown in Fig.1. They could be called as orifice plate flow and plug flow respectively. For the former, the flow reduces when it enters into the orifice and there is a vortex area of ring form to separate the flow and the whole orifice surface, as shown in Fig.1(a). And for the latter the flow also reduces, but it passes along the orifice surface before leaving this orifice, see Fig.1(b). So the flow regimes depend on not only the geometric parameters, such as the contracion ratio ( E ), given by E = d / D , and the ratio of the energy dissipater thickness to the tunnel diameter ( D ), by D = T / D , but also the hydraulic parameters, such as

the velocity of the flow ( u ), the Reynolds number ( Re ) and so on. Thus, there is a critical status between the orifice plate flow and the plug flow, or in other words, a critical thickness ( D c ) for a given contraction ratio and certain conditions of the flow, in which the flow meets just the outlet edge of the orifice when it passes through the energy dissipater. It is the plug flow if the energy dissipater thickness is larger than the critical thickness, whereas it is the orifice plate flow when the thickness is smaller than the critical thickness.

Fig.1 Flows through energy dissipater with sudden reduction and sudden enlargement forms

3. Analysis of factors affecting critical thickness As was stated above, the critical thickness ( Tc ) is an index, or a criterion, to distinguish the flows through the energy dissipaters with sudden reduction and sudden enlargement forms. This thickness, of course, is related closely to geometric and hydraulic parameters, including the contraction ratio ( E = d / D ), the viscosity of fluid ( P ), the density of fluid ( U ), and the flow velocity ( u ). It is a function of the parameters mentioned above, and could be expressed as[19]

Tc = f ( d , D, P , U , u )

(1)

Based on the independent parameters of D , u and U , its dimensionless form is

D c = f ( E , Re)

(2)

which implies that the critical thickness ( D c ) of the

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energy dissipater is only the function of E and Re .

4. Numerical simulation 4.1 Governing equations The RNG k  H model was used to calculate the hydraulic parameters of the flows through the energy dissipaters. For the steady and incompressible flows, the governing equations of this model can be expressed as[20] Continuity equation:

wui =0 wxi

(3)

Momentum equation:

(4)

k equation:

ª wk º 1 «D k Q + Q t » + Gk  H wx j »¼ U «¬

wk w = wxi wx j

(5)

H equation:

ui

ª wH º 1 * H «D H Q +Q t » + C1 Gk  k wx j »¼ U «¬

wH w = wxi wx j C2

H2

(6)

k

where ui is the velocity components in the xi directions, ȡ is the density of water, p is the pressure, Q is the kinematic viscosity of water, Q t is the eddy viscosity and can be given by Q t = CP ( k 2 / H ) , in which k is the turbulence kinetic energy, H is the dissipation rate of CP = 0.085 . The other parameters are

K· K0 ¸¹ Sk * © , K= , C1 = C1  3 1 + OK H §

K ¨1 

k

K0 = 4.377 , O = 0.012 , § wu wu · wu Gk = UQ t ¨ i + j ¸ i , C2 = 1.68 , ¨ wx ¸ © j wxi ¹ wx j

D k = DH = 1.39 4.2 Boundary conditions In simulation, the boundary conditions are treated as follows: in the inflow boundary the turbulence kinetic energy kin and the turbulent dissipation rate

H in can be defined as respectively:

§ wu wu j · º wu 1 wp w ª + uj i =  « Q + Q t ¨¨ i + ¸¸ » wx j U wxi wx j ¬« © wx j wxi ¹ ¼»

ui

1 § wu wu · S = ¨ i + j ¸ , C1 = 1.42 , 2 ¨© wx j wxi ¸¹

and

kin = 0.0144uin2 , H in =

kin1.5 0.25D

(7)

where uin is the average velocity in the inflow boundary. In the outflow boundary the flow is considered as fully developed. The wall boundary is controlled by the wall functions[17]. And the symmetric boundary condition is adopted, that is, the radial velocity on the symmetrical axis is zero. 4.3 Calculation phases The calculation phases include: (1) E = 0.40, 0.50, 0.60, 0.70 and 0.80, (2) D = 0.40 -1.30 , and (3) D = 5.00m . The purposes of the present work are: (1) for the given thickness ( D ) and the contraction ratio ( E ) of the energy dissipater, to determine Reynolds number ( Re ) at the occurrence of the critical status of the flow change from the plug flow to the orifice plate flow, (2) to establish the relationship between the critical thickness ( D c ) and the Reynolds number ( Re ) of the flow for the different values of the contraction ratio ( E ), and (3) to present an empirical expression for determining the flows through the energy dissipater. 5. Results and discussions 5.1 Flow regime control Figure 2 is the case of the flow regime development through the energy dissipater with sudden reduction and sudden enlargement, in which the contraction ratio E = 0.50 and the dimensionless thickness D = 0.60 . It could be seen that the flow

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orifice directly not to meet the orifice surface (see Figs.2(d) and 2(e)). In the process of the flow regime development, for certain cases, there is the critical flow distinguishing those two flows mentioned above, i.e., the plug and orifice plate flows. The flow shown in Fig.2(c), obviously, is just this kind of the critical flow, and there exists a vortex area which has almost the same length as the orifice surface, or we can say, the length of the vortex area equals the thickness of the energy dissipater ( T ) approximately. For this energy dissipater with E of 0.50 and D of 0.60, the critical thickness ( D c ) occurs when Re = 2.63 u 107 . We can get the critical thickness ( D c ) for different Reynolds numbers ( Re ) and reduction ratios ( E ) by means of same procedure.

Fig.3 Variations of critical thickness ( D c ) with the Reynolds number ( Re ) at the different reduction ratios ( E )

Fig.2 Flow regime developments from plug flow to orifice plate flow

regimes depend on not only its geometric parameters but also the hydraulic parameters through the energy dissipater. For the given energy dissipater, the flow regimes develop from the plug flow to orifice plate flow when the Reynolds number ( Re ) increases gradually. At small Re the plug flow appears and the flow entering into the orifice meets the surface of the orifice before it leaves this orifice (see Figs.2(a) and 2(b)). The flow becomes the orifice plate flow when Re is large enough and there is a vortex area of the ring form between the orifice surface and the flow entering into the orifice, so that the flow leaves the

5.2 Characteristics of critical thicknesses Figure 3 is the relationship of the critical thickness ( D c ) of the energy dissipater with sudden reduction and sudden enlargement forms and the Reynolds number ( Re ) at the different reduction ratios ( E ). The lines of the critical thickness divide the aera into two parts, i.e., the left or upper part of the lines belongs to the plug flow for each contraction ratios ( E ), while the right or lower part is the orifice plate flow. It could be seen that the critical thickness ( D c ) approximately linearly varies with the increase of the Reynolds number ( Re ) for each contraction ratio ( E ) of the energy dissipaters. Furthermore, the slopes of the lines decrease with the increase of reduction ratio, that is to say, the critical thickness decreases with the increase of the contraction ratio at the same Reynolds number ( Re ). Meanwhile, it could be seen that the differences of the slopes are relatively small at small contraction ratios, such as E of 0.40, 0.50 and 0.60, while the big changes of the slopes take place at E of 0.70, and 0.80 (see Fig.3). An

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important phenomenon should be noted that only the orifice plate flow occurs when the thickness ( D ) of the energy dissipater is smaller than 0.37 for any contraction ratio ( E ). This figure could be used as the criterion of distinguishing the flows through the energy dissipaters with sudden reduction and sudden enlargement forms. Each line expresses a kind of the critical status of the flow regime development from the plug flow into the orifice plate flow for each contraction ratio. The empirical expression for all the contraction ratio ( E ) could be obtained on the basis of Fig.3:

D c =  3.33E 4  8.01E 3 + 7.02E 2  2.68E + 0.39 u 106 Re  46.42 E 4 +110.52 E 3 

97.20 E 2 + 37.41E  4.94

(8)

It belongs to the plug flow when D ! D c , while the flow is the orifice plate one when D  D c . Naturally, it is the critical status of the flow at the critical thickness, i.e., D = D c . This expression is valid for E = 0.40 - 0.80 and D = 0.37 -1.30 . Let the relative error Er between the calculated critical thickness ( D cal ) by Eq.(8) and the results ( D nu ) of numerical simulations from Fig.3 as:

Er =

D cal  D nu u 100% D cal

(9)

The results of the error analysis are shown in Fig.4. From this figure the maximum error of Eq.(8) is obviously smaller than 10% for each E . Therefore, it is effective to distinguish flows through the energy dissipater with sudden reduction and sudden enlargement forms by means of Eq.(8). 6. Conclusions The concept of critical thickness is useful in the investigation of the flows through the energy dissipater with sudden reduction and sudden enlargement forms. The critical thickness is related to not only the geometric parameters but also the hydraulic parameters for the energy dissipaters mentioned above. The empirical expression has been obtained herein about the critical thickness, reduction ratio and Reynolds number. This expression could be used as the criterion to distinguish the flows through the energy dissipater, i.e., the plug flow and the orifice plate flow. The critical thickness calculated by Eq.(8) has its error smaller than of 10%. References [1] [2]

[3]

[4]

[5]

[6]

[7]

Fig.4 Comparisons of results from Eq.(8) with data from Fig.3

[8]

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