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An algorithm for calculating air demand in gated tunnels using a 3D numerical model
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Journal of Hydro-environment Research 5 (2011) 3e13 www.elsevier.com/locate/jher
Research paper
An algorithm for calculating air demand in gated tunnels using a 3D numerical model Jafar Yazdi, Amir Reza Zarrati* Department of Civil and Environmental Engineering, Amirkabir University of Technology, Tehran, Iran Received 10 December 2008; revised 20 January 2010; accepted 5 July 2010
Abstract Aeration of high-speed flows issued from gates in tunnels is necessary to prevent high negative pressures and their consequences such as cavitation and vibration. In this research work 3D numerical modeling of complicated airewater flow in a gated tunnel was performed employing FLUENT computer code. Simulation of the high-speed free surface flow was conducted using VOF method with Young’s scheme. Owing to high Reynolds numbers associated with the flow through the gates, the standard two equation kee turbulence model was employed. The airewater flow was simulated for both circular and rectangular tunnel cross-sections. Using the results of the numerical model, the rate of air discharge through the vent was determined and validated by available experimental data. The comparison of results with experimental data showed very good agreement. From calculation of the air flow along the tunnel at different conditions, an equation was developed for the velocity distribution of the dragged air over the free surface and the air supplied from the tunnel outlet. The amount of dragged air and air supplied from the tunnel outlet was then calculated knowing the velocity distribution and the area of the air flow. Subsequently an algorithm was presented to design the air supply system in gated tunnels both for circular and rectangular cross-sections. The algorithm was then employed to calculate air demand in number of bottom outlets and comparison with experimental data showed good agreement. Ó 2010 International Association of Hydro-environment Engineering and Research, Asia Pacific Division. Published by Elsevier B.V. All rights reserved. Keywords: Airewater flow; Gated tunnel; Numerical modeling; Simulation; FLUENT; Air vent; Air demand
1. Introduction Gated tunnels are used in dams for various purposes such as regulating the reservoir water surface in impounding, drawdown of the reservoir, sediment flushing and flood release (Vischer and Hager, 1997). Downstream of the gate, highspeed water flow, and as a result, motion of air cause negative pressure. Negative pressure can increase possibility of cavitation and vibration. To prevent these phenomena, usually an air vent is installed immediately downstream of the gate for aeration and prevention of cavitation. Estimating the vent air
discharge is a fundamental aspect in design of the air vent. So far, numerous researches have been conducted to determine the vent air discharge. The first study related to air entrainment in pipes was presented by Kalinske and Robertson (1943). They studied the formation of hydraulic jump and its impact on the air entrainment in pipes. Campbell and Guyton (1953) presented the following relationship for air discharge ratio, b ¼ Qair/Qwater, where Qair is vent air discharge and Qwater is the flow discharge, using the Froude number at the contracted section Frc downstream of the gate: b ¼ 0:04ðFrc 1Þ
0:85
* Corresponding author. Tel.: þ98 21 64543002; fax: þ98 21 66414213. E-mail addresses: [email protected] (J. Yazdi), [email protected] (A.R. Zarrati).
ð1Þ
U. S. Army Corps of Engineers (USACE, 1964) based on prototype observations presented a similar relationship for air discharge ratio:
1570-6443/$ - see front matter Ó 2010 International Association of Hydro-environment Engineering and Research, Asia Pacific Division. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jher.2010.07.002
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b ¼ 0:03ðFrc 1Þ1:06
ð2Þ
Sharma (1976) classified three different patterns of flow in conduits: spray flow, free surface flow and hydraulic jump. By experiment in a rectangular tunnel, Sharma showed that air demand in spray flow is more than free surface flow with a similar Froude number as is indicated below: spray flow : b ¼ 0:2Frc
ð3Þ
free surface flow : b ¼ 0:09Frc
ð4Þ
Speerli and Volkart (1997) studied air demand in a rectangular tunnel and related air vent discharge to various parameters and presented the following relationship by analysis of experimental data: qav ¼
Qav
0:5 gb2 HE3
¼ 0:022
b HE
0:5
L Ht
1=6
S0:5 x00:43
ð5Þ
where b ¼ tunnel width, HE ¼ upstream energy head, L ¼ length of the tunnel, Ht ¼ height of tunnel, S ¼ relative gate opening, x0 ¼ total loss coefficient of air vent, Qav ¼ discharge of air vent and qav ¼ dimensionless air discharge. Speerli and Hager (2000) also measured air-concentration profiles in the free surface flow downstream of the gate. Downstream of the gate air is dragged along the water surface and is then transported towards downstream. Air is also entrained into the high-speed water flow owing to turbulence and rough free surface. Therefore the total air demand of the flow Qademand ¼ Qaen þ QaD where Qaen is air discharge entrained to the flow and QaD is the discharge of the dragged air. Part of the air demand is supplied from the air vent (Qav). Part of air demand may also be supplied from the tunnel outlet (Qat). Therefore one can write: Qaen þ QaD ¼ Qav þ Qat ¼ Qademand
ð6Þ
Safavi et al. (2008) experimentally studied the effect of parameters such as gate opening, vent diameter and tunnel dimensions on the vent discharge and air supplied from the tunnel outlet in a circular tunnel. Results of this work showed that in addition to Frc, relative flow depth (defines as the ratio of flow depth to tunnel diameter) and vent opening have important effect on air supplied from the tunnel outlet. They also presented a relationship to determine negative pressure downstream of the gate. Most of the previous research works were based on physical model studies. Therefore the validity of empirical equations is limited considering the experimental limitations, scale effects and measurement accuracy. Conducting more experiments with larger facilities and wider range of flow parameters is also very expensive and time consuming. In the present paper, employing the 3D computer code “FLUENT”, air flow downstream of the gate was simulated numerically, and air demand was calculated. Numerical model results were then verified using experimental data. In the next step, air flow in gated tunnels with circular and rectangular cross-sections was simulated in a wide range of flow parameters and air discharges from the vent and the tunnel outlet were
calculated. Based on numerical results, an algorithm was presented to design air supply system in gated tunnels. 2. Numerical modeling In the present work, FLUENT (FLUENT Inc, 2005) computer code was used to simulate airewater flow after the gate in the tunnel. FLUENT solves full three-dimensional Reynolds Averaged NaviereStokes equations. This computer code is also able to calculate the location of the interface between different phases using a VOF (Volume of Fluid) technique. For airewater flow the governing equations can be written as given below (Streeter, 1961). 2.1. The continuity equation v ðrÞ þ V$ðr! vÞ¼0 ð7Þ vt ! ! ! In this equation v is the mixture velocity: v ¼ ðaw rw v w þ v a Þ=r where ! v w and ! v a are velocity of water and aa ra ! air, respectively. r is the mixture density defined as: r ¼ aw rw þ aa ra ¼ aw rw þ ð1 aw Þra where rw and ra are density and aw and aa are the volume fraction of water and air, respectively. 2.2. The momentum equation In the numerical model, a single momentum equation is solved throughout the entire air and water domain, and the resulting velocity field is shared among the phases. The momentum equation, shown below, is dependent on the volume fractions of air and water through the properties r and m0. v ! T ðr v Þ þ V$ðr! v! v Þ ¼ Vp V$ m V! v þ V! v vt ! þ r! g þF
ð8Þ
where p ¼ pressure, m ¼ m0 þ mt. m0 is the mixture viscosity of airewater flow and is calculated similar to density, mt is ! turbulence viscosity and F is the body force. In this research, to determine mt, kee, standard model was employed (Launder and Spalding, 1974). To treat the free surface, the VOF method and Young’s (1982) scheme was used. In this method, the capturing of the interface between the phases is accomplished by the solution of a continuity equation for the volume fraction of one of the phases. In this research, water is assumed as the first and air is assumed as the second phase. For air, the transport equation has the following form: v v aa Þ ¼ 0 ð9Þ ðaa Þ þ V$ð! vt This equation is solved in the entire domain and volume fraction is computed for all cells. Then the interface between air and water can be calculated by assuming a specific volume fraction as the free surface. In the present study, it was noticed that flow velocity increased with distance from the tunnel invert till volume fraction of 0.5. From this point away, flow
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velocity decreased which showed entering the air phase (Fig. 1). It was therefore assumed that volume fraction of 0.5 represents the free surface. After solution of the governing equations, the vent air discharge was also calculated. The first step in the numerical modeling is mesh generation. In the present research, both structured and unstructured meshes were used. Unstructured mesh was used at the gate area because of the complex geometry (Fig. 2). Structured mesh was used at other regions. In all computations, mesh size was refined till the results were independent of the mesh size. For each of the models, to ensure the independency of results from the mesh sizes, the selected computational cells were first refined by two times and then by four times and it was observed that the calculated air discharge did not change more than 1.3%. Fig. 1 shows the geometry of a typical gated tunnel. At the upstream section, the operating head of the reservoir was introduced as total pressure with reference to atmospheric pressure. At the entrance to the air vent which is in contact with atmosphere, only air phase is present and total pressure was assumed equal to zero. For all solid boundaries, no-slip condition was applied (Rodi, 1980). Owing to symmetry of the tunnel and flow, only half of the tunnel was modeled and symmetric boundary condition was applied at the tunnel centerline. The zero piezometric pressure was imposed at the outlet assuming discharge of flow to the atmosphere. Since the flow was simulated in unsteady state, the PISO algorithm was used for coupling pressure and velocity fields (Versteeg and Malalasekera, 1995). Computations were conducted by an AMD 3800þ processor. Each run of the numerical model took 5e30 h depending on tunnel dimensions, mesh numbers and flow characteristics. 3. Model verification Before employing the numerical model to study airewater flow in gated tunnels, it was necessary to ensure the model accuracy by comparing its results with experimental data. For this purpose, Speerli and Hager (2000) and Safavi et al. (2008) experiments were used. Speerli and Hager (2000) studied air demand in a rectangular tunnel in various gate openings. The experimental set-up included an approach conduit, a sluice gate and a tailrace tunnel (see Fig. 1). The approach conduit had a width and height of 0.3 m. The tailrace tunnel height increased to 0.45 m. The tailrace tunnel had a width of 0.3 m and a maximum length of 21 m. The approach energy head was 10 m and the
5
relative gate openings varied from 0 to 45%. Fig. 3 shows the comparison of numerical model results with experimental data of Speerli and Hager (2000). In this figure, y/Ht is relative depth of the flow where y is the flow depth close to the outlet and Ht is the tunnel height. The maximum discrepancy was 28% in the gate opening of 10%. At small gate openings, flow after the gate is in the form of spray (Sharma, 1976). Therefore, results of the VOF model, which is not suitable for modeling spray, show more deviation from the physical model results. At other gate openings numerical results under-estimated vent air discharge by maximum of about 8%. It should be noticed that the amount of air entrained into the flow Qaen is not simulated by the numerical model. Experimental data of air-concentration profiles was available in 13.33%, 26.7% and 40% gate openings from Speerli and Hager (2000). Air discharge in water flow from the tunnel invert to a distance where volume fraction of air was 50% (and was considered as the free surface) was therefore calculated. From these calculations, maximum Qaen was found to be about 10% of the vent discharge. This amount was approximately equal to the difference between numerical and physical model results for air vent discharge. Therefore, it can be concluded that under-estimation of vent air discharge is because Qaen is not simulated by the numerical model. Similar results were reported by Safavi et al. (2009). In these experiments air dragged above the water flow was prevented by a plate at the tunnel outlet. It was concluded that the vent air discharge which in this test was supplying only the air entrained to the water flow reduced about 80%. These experiments showed that the dragged air includes a considerable amount of the air demand. Safavi et al. (2008) studied air demand in a circular tunnel in various gate openings. Experiments were conducted in a 17.5 cm diameter Perspex pipe. A slide gate was installed upstream of the tunnel (Fig. 4). Length of the tunnel after the gate was 7 m. Energy head upstream of the gate was constant and equal to 2.2 m. To conform to prototype design a step was considered in the model after the gate so that a jet forms over this step. A 3 cm diameter vent made from Perspex was installed just after the bottom outlet gate. Fig. 5 shows comparison of numerical and experimental results. In this figure y/D is relative depth of flow ( y is depth of flow and D is tunnel diameter). As can be seen in this figure, results have a good agreement with experimental data. In average the numerical model under-estimated the air vent discharge by about 7%. This difference can also be attributed to air entrainment to the flow which is not simulated by the numerical model.
Fig. 1. Free surface and dragged air (output of the numerical model)
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Fig. 2. A sample of computational mesh in the gate area: (a) rectangular tunnel, and (b) circular tunnel.
In the next stage, calculated air discharge ratio (b) was compared with few empirical equations in Fig. 6. In this figure Frc is the Froude number at the contracted section downstream of the gate. This figure shows that the numerical model results are closer to experimental measurements than empirical equations. In addition, numerical model results have a good agreement with Campbell and Guyton (1953) and USACE (1964) equations (with average difference of 10.44% and 19.32%, respectively). These equations are based on prototype data. Sharma (1976) equation overestimates the air discharge ratio. Overestimation of Sharma was also reported by Safavi et al. (2008). These results show the validity of the numerical model in calculating the flow field of air and water. In the next stage by changing various parameters including flow depth and discharge, tunnel length and diameter of the vent, the numerical model was used to calculate velocity field of air and water flow. Results of these computations were then used to study the air flow pattern and the parameters that affect this pattern. In the following section these results are discussed.
increase in the air demand. In a constant reservoir head and gate opening, however, if energy losses in the air vent increase, the ratio of the air supply from the tunnel outlet to that from the air vent increases (Safavi et al., 2008). In addition to hydraulic parameters such as reservoir head, flow depth, flow velocity and the loss coefficient of air vent which are discussed above, diameter and length of the tunnel affect the flow air demand. For example the relative flow depth y/D affects the amount of dragged air by changing the area of air passage above the water flow. Moreover, increasing the length of the tunnel decreases the amount of air supplied from the tunnel outlet by increasing the energy loss in the air passage above the flow (Garcia and Fuentes, 1984; Speerli and Volkart, 1997; Safavi et al., 2008). The numerical model which was validated with available experimental data was employed to study the influence of various parameters on air demand. These studies are presented first for a tunnel with a circular and in the next stage for a tunnel with a rectangular cross-section. 4.1. Circular tunnels
4. Air flow simulation in gated tunnels If the reservoir head increases when the gate opening is constant, flow velocity increases in the tunnel and consequently air demand increases. On the other hand, in a constant reservoir head with opening the gate, flow discharge and consequently flow velocity along the tunnel increases and this also causes an
For simulation of flow in various conditions in circular tunnels, dimensions of the physical model conducted by Safavi et al. (2008) were considered. Range of different parameters considered in this study was as below: 0:2 < y=D < 0:43; 20 < L=D < 60; 0:1 < d=D < 0:3 and 0:0066 < Q2 =gD5 < 0:23
0.14 0.12
Qav (m3/s)
0.10 0.08 0.06 0.04
Experimental, Speerli et al. 1997
0.02
Numerical
0.00 0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
y/Ht
Fig. 3. Comparison between numerical results and experimental data in a rectangular cross-section.
where d is the vent diameter and Q is the flow discharge. Studies showed that there is a circulation of air above the water flow as air is dragged along the water surface towards downstream and on the other hand is flowing from the tunnel outlet towards the gate at distances close to the tunnel ceiling (Fig. 7). Results showed that in the range of parameters considered, the velocity distribution of air which is dragged towards downstream follows the following equation: n ya u ¼ u0 1 h0
ð10Þ
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7
Fig. 4. Details of the gate in the experimental model, Safavi et al. (2008).
In this equation u0 is the water surface velocity, ya is distance from the flow surface and h0 is the distance from water surface to the point where velocity is zero (Fig. 7). In this equation, n is a constant factor that depends on the shape of the cross-section. In the circular cross-section, by regression analysis n was obtained equal to 1.5. Comparison of Eq. (10) at various longitudinal sections is shown in Fig. 8a. Numerical results also indicated that velocity distribution of air flow from the tunnel outlet follows the following equation: n0 0 ya h0 u ¼ u0 Dy
ð11Þ
In this equation u00 is the maximum velocity of air, flowing from the end of the tunnel. Since the location of maximum velocity u00 was very close the tunnel ceiling (Fig. 7), the distance of u00 from the free surface can be approximated by D y. Regression analysis showed that in Eq. (11) n0 ¼ 0.5. Fig. 9a shows the comparison of numerical results and Eq. (11) for a typical cross-section near the tunnel outlet. For other sections similar results are obtained. The numerical results showed that h0/D is independent of the y/D, Q and L but is a function of position in the transversal cross-section z and also x0. The following relationship was fitted for determining
h0 from the numerical data with a correlation factor of 0.998 for the whole range of data: z 2 z h0 ¼ 1:54 þ0:137 þC D D D
ð12Þ
In this equation, z is the distance of the longitudinal crosssection from the tunnel axis and C is a function of x0 and can be calculated from the following equation: 2 2 C ¼2:22 0:1133 þ 19:742ðx0 0:6525Þ 2 þ 0:13 0:1133 þ 19:742ðx0 0:6525Þ þ 0:25
ð13Þ
Knowing the velocity distribution of air above the free surface, dragged air discharge (Qav) and air entrained from the tunnel outlet (Qat) can be calculated as: Za ZD QaD ¼2
udydz ¼ 0
X
y
2u0 nþ1
D y nþ1 Dzi if : h0 > D y h0i 1 1 h0i
ð14Þ
0.010
0.006
3
Qav (m /s)
0.008
0.004 Experimental, Safavi et al. 2008
0.002
Numerical
0.000 0.15
0.2
0.25
0.3
0.35
0.4
0.45
y/D
Fig. 5. Comparison between numerical results and experimental data.
Fig. 6. Comparison of air discharge ratio b in the present research with previous works.
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Fig. 7. Velocity distribution of the dragged air over the water surface.
where a is the lateral distance from the tunnel axis to the tunnel wall (Fig. 7) and is a function of ya which is the distance from the water surface. Air vent discharge can therefore be calculated as:
Za Zh0 QaD ¼ 2
udya dz 0
¼
0
2u0 1:54 3 0:137 2 a þ a þ kDa if : h0 < D y n þ 1 3D 2
ð15Þ
Qav ¼ QaD Qat
If calculated h0 is more than D y there will be no air entrainment from the tunnel outlet and therefore all air demand is supplied from the air vent (Eq. (14)). There is no analytical solution to Eqs. (14) and (16) and therefore they should be solved numerically. To determine Qat from Eq. (16), u00 must be estimated. Numerical data showed that sum of u0 and u00 is a function of u0, L and x0. In the present study the following
Z a ZDy Z Z Qat ¼ 2
udya dz 0
¼
h0
1
2u00 ðD yÞ X nþ1
h0i ðD yÞ
a
nþ1 Dzi
ð17Þ
ð16Þ
0.08
y/D=0.2
y/D=0.25
0.06 0.04 0.02 0
0
3
2
1
y a (m )
0.04
y/D=0.43
y/D=0.31
0.02 0 1
0
b
2
1
0
3
2
4
3
0.24
y/Ht=0.07
y/Ht=0.2
0.16 0.08 0.04
0
1
2
3
0
4
2
6
4
10
8
0.16
y/Ht=0.4
y/Ht=0.27 0.08 0 0
5
10
0
5
10
u (m / s ) Numerical
Eq. (10)
Fig. 8. Air velocity distributions above the free surface, in a typical longitudinal cross-section at various relative depths: (a) circular tunnel, and (b) rectangular tunnel.
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a
9
0.06 0.04
y/D=0.2
y/D=0.25
y/D=0.31
y/D=0.41
0.02 0 0.06
y a (m )
0.04
b
0.02 0 -1.3
-0.8
-0.3
-0.9
0
-0.4
0
0.16 0.12
y/Ht=0.07
y/Ht=0.09
0.08 0.04 0.15 0.12
y/Ht=0.27
y/Ht=0.2
0.09 0.06 0.03 -2.5
-2
-1.5
-1
-0.5
-1
0
-0.5
0
u (m / s ) Numerical
Eq. (11)
Fig. 9. Velocity distributions of air flow from the tunnel outlet in a typical section at different relative depths: (a) circular tunnel, and (b) rectangular tunnel.
equation was found for u00 þ u0 with a correlation factor of 0.95: L 0 u0 þ u0 ¼1:24u0 0:008 D 5:91 0:1133 þ 19:742ðx0 0:6525Þ2 þ 1:41 ð18Þ
of the tunnel is also valid for rectangular cross-section with n ¼ 0.9 and D z Ht. Figs. 8b and 9b show the comparison of numerical results and Eqs. (10) and (11) in a typical section along the tunnel. Similar results were obtained at other sections. The following equations were found for the rectangular cross-section using numerical data:
For a real case, L, D and x0 are known or easily can be determined. u0 can be calculated from logarithmic velocity distribution law and knowing the flow depth. u00 can then be computed from Eq. (18).
2 h0 ¼ 4:13 eð3x0 þ1:1Þ 0:45 eð3x0 þ1:1Þ þ 0:3 Ht
4.2. Rectangular tunnels Numerical analysis was also conducted for tunnels with rectangular cross-section. To simulate the flow in various conditions in rectangular tunnels, geometry of the physical model conducted by Speerli and Hager (2000) as described above was considered. Range of different parameters considered in this study was as follows: 0.07 < y/Ht < 0.53, 20 < L/Ht < 60, 0.01 < Q2/gHt5 < 0.9 and 0 < d/Ht < 0.44. Similar to a circular cross-section numerical results indicated that velocity distribution of dragged air over the free surface can also be expressed by Eq. (10), with n ¼ 1.8. Eq. (11) for velocity distribution of air supplied from the end
ð19Þ
Knowing velocity distribution QaD and Qat can be calculated as: Zb=2 ZHt QaD ¼ 2
udydz 0
y
nþ1
h0 u0 b Ht y if : h0 > D y 1 1 ¼ nþ1 h0
Zb=2 Zh0 QaD ¼ 2
udya dz ¼ 0
0
h0 u0 b if : h0 < D y nþ1
ð20Þ
ð21Þ
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Zb=2 ZHt Qat ¼ 2
udya dz ¼ 0
h0
u00 ðHt h0
nþ1
yÞb
ð22Þ
where b is width of the tunnel. To calculate Qat from Eq. (22), it is necessary to estimate u00 (maximum velocity of air flow from the tunnel outlet). The following equation was suggested to estimate u00 for rectangular sections using numerical results.
u00 þu0 ¼1:186u0 0:02
L 6:09 eð3x0 þ1:1Þ þ2:38 Ht
ð23Þ
Based on the above equations an algorithm was suggested for designing the vent in gated tunnels with circular or rectangular cross-sections. This algorithm is demonstrated in Fig. 10 and is presented below. 1. Assume a diameter (d0) and discharge (Qav) for the air vent. For estimating Qav existing empirical equations can be used.
Fig. 10. Flowchart for calculating air demand in gated tunnels.
J. Yazdi, A.R. Zarrati / Journal of Hydro-environment Research 5 (2011) 3e13 0.024
0.010
0.020
Q av (m /s)
0.008
0.016
3
0.006
3
Qa v (m /s)
11
0.004
0.008 Experiments (Gavoshan dam)
Experimental, Safavi et al. 2008
0.002
0.012
0.004
Present Algorithm
Present Algorithm
0.000 0.15
0.2
0.25
0.3
0.35
0.4
0.000 0.23
0.45
0.28
0.33
0.38
0.43
0.48
y/D
y/D
Fig. 11. Comparison of calculated vent air discharge with Safavi et al. (2008) experimental data.
2. x0 is calculated regarding d0 and Qav knowing the length of the vent la. (x0 ¼ ke þ fla/d where ke is coefficient of local losses, f is DarcyeWeisbach coefficient). 3. The height h0 is calculated from Eqs. (12) or (19) depending on the shape of the tunnel cross-section. 4. Knowing Qw, the surface velocity u0 is computed from logarithmic velocity distribution law and the flow depth. 5. QaD and Qat are calculated from Eqs. (14)e(16) for a circular and Eqs. (20)e(22) for a rectangular crosssection. 6. To find the total air demand, entrained air to the flow that is Qaen should be estimated and added to QaD. Based on Speerli and Hager (2000) measurements, it was shown that when water surface is defined at 50% air concentration, Qaen is about 10% of the vent discharge. More experiments are needed to find a more accurate value for Qaen. 7. Vent air discharge is calculated as: Qav ¼ QaD þ Qaen Qat. 8. The losses and the value of x0 in the air vent is calculated and compared to its value in Step 1. If these values are not equal, calculations must be repeated from Step 1 with a new x0. 9. The negative pressure is calculated immediately after the vent. If negative pressure is high, diameter of vent should be increased and computations should be repeated again from Step 1. Speerli and Hager (2000) suggested that the negative pressure should be less than 1.5 m water column.
Fig. 13. Comparison of calculated vent air discharge with experimental data of Gavoshan model study.
5. Comparison of the present model with measured data The present algorithm was employed to calculate vent air discharge for experiments of Safavi et al. (2008) and Speerli and Hager (2000) in different gate openings. Vent discharge calculated was then compared with experimental data in Figs. 11 and 12. In these calculations Qaen was also considered. The average difference was found to be 8.3% for Safavi et al. (2008) and 6% for Speerli and Hager (2000) which is considered to be acceptable. The present algorithm was also used to calculate the vent air discharge in two more cases of Gavoshan and Gotvand dam bottom outlets. Gavoshan dam is a rock fill dam in the west of Iran. Its bottom outlet is a 4.4 m diameter circular tunnel with 650 m length. The design discharge of this bottom outlet is 133.7 m3/s. The physical model of this tunnel was built in Water Research Institute affiliated to Ministry of Power in Iran with 1/15 scale (WRI, 2003). Gotvand the highest embankment dam in Iran is 175 m high and its bottom outlet is 1640 m long with 9.5 m diameter. The design discharge of this bottom outlet, one of the highest in its type is about 520 m3/s. The physical model of this bottom outlet was also built in Water Research Institute with 1/25 scale (WRI, 2008). Air flow velocity of the vents in these two tunnels was measured by a hot wire anemometer at different gate openings and air discharge was then calculated from the air velocity. The vent air discharges were also calculated by the
0.05
0.14 0.12
0.04
Q av (m 3/s)
3
Qa v (m /s)
0.10 0.08 0.06 0.04 Experimental, Speerli et al. 1997
0.02 0.00 0.00
0.03 0.02 Experiments (Gotvand dam)
0.01
Present Algorithm
Numerical
0.05
0.10
0.15
0.20
0.25
0.30
0.35
y/Ht
Fig. 12. Comparison of calculated vent air discharge with Speerli and Volkart (1997) experimental data.
0.00 0.18
0.26
0.34
0.42
0.5
0.58
0.66
y/D
Fig. 14. Comparison of calculated vent air discharge with experimental data of Gotvand model study.
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J. Yazdi, A.R. Zarrati / Journal of Hydro-environment Research 5 (2011) 3e13
present algorithm and were compared with the measured data in Figs. 13 and 14. It can be seen that results of calculations are close to measurements. In small flow depths (lower gate openings) owing to spray flow conditions, calculations show more deviation from the physical model results. In general results of the present algorithm are satisfactory with an average error of 13.4% for Gavoshan dam and 11.4% for Gotvan dam bottom outlets. These results show the ability of the suggested algorithm to predict air demand in gated tunnels. 6. Conclusion In the present paper airewater flow in gated tunnels was studied with a 3D numerical model. The numerical modeling involved the solution of the NaviereStocks equations using the FLUENT computer code. The kee turbulence model with the VOF method was used to simulate complicated airewater flow both in circular and rectangular cross-sections. Flow of air above high-speed water flow was studied and the effect of different parameters on air demand was investigated. These parameters included reservoir head, discharge, flow depth, flow velocity, the loss coefficient of air vent and diameter and length of the tunnel. From the results of the numerical model, vent air discharge (Qav) was calculated at different gate openings and was validated by the experimental data. Numerical results showed a good agreement with experiments. The numerical model however under-estimated the vent air discharge by about 10%. This is probably owing to air entrainment to the water flow which was not be simulated by the numerical model. Calculating the air velocity field in a wide range of flow conditions, air velocity distribution of the dragged air along the water flow as well as the air flow from the tunnel outlet was estimated. A relationship was then found for the height of the dragged air area above the water surface as a function of transverse position in the cross-section, energy losses in air vent and diameter of the tunnel. Having the air velocity distribution and the area of air flow along the water surface and also air entrained from the tunnel outlet, dragged air discharge (QaD) and supplied air from open end of tunnel (Qat) were computed for both circular and rectangular cross-sections. An algorithm was then presented for designing the aeration systems in gated tunnels. The algorithm was validated with a number of available experimental data, which were conducted for measuring air demand in circular and rectangular tunnels and the agreement was good. References Campbell, F.B., Guyton, B., 1953. Air-demand in gated outlet works. In: Proceedings of the Fifth IAHR congress, Minnesota, U.S.A., pp. 529e533. FLUENT Inc., 2005. Fluent v. 6.2. FLUENT Inc., Lebanon, NH, USA. Garcia, J., Fuentes, R., 1984. Influence of the tunnel length on the hydraulic modeling of the air entrainment in the flow downstream of a high head gate. In: Symposium on Scale Effects in Modeling Hydraulic Structures, Esslingen, Germany, pp. 4-14-1e4-14-2.. Kalinske, A.A., Robertson, J.M., 1943. Closed conduit flow. ASCE Transactions 108, 1435e1447.
Launder, B.E., Spalding, D.B., 1974. The numerical computation of Turbulent flows. Computational Methods in Applied Mechanics and Engineering 3, 269e289. Rodi, W., 1980. Turbulence models and their application in hydraulic engineering. In: IAHR Monograph. Safavi, K.H., Zarrati, A.R., Attari, J., April 2008. Experimental study of airdemand in high head gated tunnels. Journal of Water Management 161 (WM2). Safavi, K.H., Zarrati A.R., Attari, J., 2009. Experimental study of air-demand in bottom outlets. In: 8th Iranian Conference on Hydraulics, Tehran University, Faculty of Engineering, Tehran, Iran. Sharma, H.R., 1976. Air entrainment in high head gated conduits. Journal of Hydraulics Division, ASCE 102 (No. 11). Speerli, J., Volkart, P.U., 1997. Air entrainment in bottom outlet tailrace tunnels. Proceedings of the 27th IAHR Congress, San Francisco, Theme D, pp. 613e618. Speerli, J., Hager, W.H., 2000. Air-water flow in bottom outlets. Canadian Journal of Civil Engineering 27, 454e462. Streeter, V.L., 1961. Handbook of Fluid Dynamics. McGraw-Hill Inc. U.S. Army Corps of Engineers, 1964. Air-demand Regulated Outlet Works. Hydraulic Design Criteria, Chart 050 -1. Versteeg, H.K., Malalasekera, W., 1995. An Introduction to Computational Fluid Dynamics. The Finite Volume Method. Longman Publications. Vischer, D.L., Hager, W.H., 1997. Dam Hydraulics. John Wily & Sons, Chichester, pp. 190e213. Water Research Institute, WRI, 2003. Hydraulic Model Study of Bottom Outlet Gates of Gavoshan Dam. Technical Report. Water Research Institute affiliated to Ministry of Power, Iran. Water Research Institute, WRI, 2008. Hydraulic Model Study of Upper Gotvand Dam Bottom Outlets. Technical Report. Water Research Institute affiliated to Ministry of Power, Iran. Young, D.L., 1982. Time-dependent multi-material flow with large fluid distortion. In: Morton, K.W., Baines, M.J. (Eds.), Numerical Methods for Fluid Dynamics. Academic Press.
Notation a: lateral distance from the tunnel axis to the tunnel wall; b: tunnel width; D: tunnel diameter; d: vent diameter; F: body force; Frc: Froude number at the contracted section; G: percentage of gate opening; HE: reservoir head; Ht: tunnel head; h0: distance from water surface to the point where velocity is zero; h: full size of gate opening; L: tunnel length; p: pressure; QaD: dragged air; Qademand: air demand; Qaen: air discharge entrained to the flow; Qat: tunnel outlet air discharge; Qav: vent air discharge; qav: dimensionless air discharge; Qw: water flow discharge; Re: Reynolds number; Rh: hydraulic radius; S: relative gate opening; u: air velocity upon the flow surface; u0: water surface velocity; u00 : maximum velocity of air flow from the end of the tunnel; v: mixture velocity; va: air velocity; vw: water velocity; We: Weber number; w: gate opening; y: flow depth;
J. Yazdi, A.R. Zarrati / Journal of Hydro-environment Research 5 (2011) 3e13 ya: distance from the flow surface; z: position in the transversal cross-section. Greek symbols aa: volume fraction of air; aw: volume fraction of water; b: relative air vent discharge;
DP: sub-atmospheric pressure after the gate; gw: water specific weight; n: kinematic viscosity of water; r: mixture density; ra: air velocity; rw: water density; m0: mixture viscosity; mt: turbulent viscosity; x0: total loss coefficient of air vent.
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Journal of Hydro-environment Research 5 (2011) 3e13 www.elsevier.com/locate/jher
Research paper
An algorithm for calculating air demand in gated tunnels using a 3D numerical model Jafar Yazdi, Amir Reza Zarrati* Department of Civil and Environmental Engineering, Amirkabir University of Technology, Tehran, Iran Received 10 December 2008; revised 20 January 2010; accepted 5 July 2010
Abstract Aeration of high-speed flows issued from gates in tunnels is necessary to prevent high negative pressures and their consequences such as cavitation and vibration. In this research work 3D numerical modeling of complicated airewater flow in a gated tunnel was performed employing FLUENT computer code. Simulation of the high-speed free surface flow was conducted using VOF method with Young’s scheme. Owing to high Reynolds numbers associated with the flow through the gates, the standard two equation kee turbulence model was employed. The airewater flow was simulated for both circular and rectangular tunnel cross-sections. Using the results of the numerical model, the rate of air discharge through the vent was determined and validated by available experimental data. The comparison of results with experimental data showed very good agreement. From calculation of the air flow along the tunnel at different conditions, an equation was developed for the velocity distribution of the dragged air over the free surface and the air supplied from the tunnel outlet. The amount of dragged air and air supplied from the tunnel outlet was then calculated knowing the velocity distribution and the area of the air flow. Subsequently an algorithm was presented to design the air supply system in gated tunnels both for circular and rectangular cross-sections. The algorithm was then employed to calculate air demand in number of bottom outlets and comparison with experimental data showed good agreement. Ó 2010 International Association of Hydro-environment Engineering and Research, Asia Pacific Division. Published by Elsevier B.V. All rights reserved. Keywords: Airewater flow; Gated tunnel; Numerical modeling; Simulation; FLUENT; Air vent; Air demand
1. Introduction Gated tunnels are used in dams for various purposes such as regulating the reservoir water surface in impounding, drawdown of the reservoir, sediment flushing and flood release (Vischer and Hager, 1997). Downstream of the gate, highspeed water flow, and as a result, motion of air cause negative pressure. Negative pressure can increase possibility of cavitation and vibration. To prevent these phenomena, usually an air vent is installed immediately downstream of the gate for aeration and prevention of cavitation. Estimating the vent air
discharge is a fundamental aspect in design of the air vent. So far, numerous researches have been conducted to determine the vent air discharge. The first study related to air entrainment in pipes was presented by Kalinske and Robertson (1943). They studied the formation of hydraulic jump and its impact on the air entrainment in pipes. Campbell and Guyton (1953) presented the following relationship for air discharge ratio, b ¼ Qair/Qwater, where Qair is vent air discharge and Qwater is the flow discharge, using the Froude number at the contracted section Frc downstream of the gate: b ¼ 0:04ðFrc 1Þ
0:85
* Corresponding author. Tel.: þ98 21 64543002; fax: þ98 21 66414213. E-mail addresses: [email protected] (J. Yazdi), [email protected] (A.R. Zarrati).
ð1Þ
U. S. Army Corps of Engineers (USACE, 1964) based on prototype observations presented a similar relationship for air discharge ratio:
1570-6443/$ - see front matter Ó 2010 International Association of Hydro-environment Engineering and Research, Asia Pacific Division. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.jher.2010.07.002
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J. Yazdi, A.R. Zarrati / Journal of Hydro-environment Research 5 (2011) 3e13
b ¼ 0:03ðFrc 1Þ1:06
ð2Þ
Sharma (1976) classified three different patterns of flow in conduits: spray flow, free surface flow and hydraulic jump. By experiment in a rectangular tunnel, Sharma showed that air demand in spray flow is more than free surface flow with a similar Froude number as is indicated below: spray flow : b ¼ 0:2Frc
ð3Þ
free surface flow : b ¼ 0:09Frc
ð4Þ
Speerli and Volkart (1997) studied air demand in a rectangular tunnel and related air vent discharge to various parameters and presented the following relationship by analysis of experimental data: qav ¼
Qav
0:5 gb2 HE3
¼ 0:022
b HE
0:5
L Ht
1=6
S0:5 x00:43
ð5Þ
where b ¼ tunnel width, HE ¼ upstream energy head, L ¼ length of the tunnel, Ht ¼ height of tunnel, S ¼ relative gate opening, x0 ¼ total loss coefficient of air vent, Qav ¼ discharge of air vent and qav ¼ dimensionless air discharge. Speerli and Hager (2000) also measured air-concentration profiles in the free surface flow downstream of the gate. Downstream of the gate air is dragged along the water surface and is then transported towards downstream. Air is also entrained into the high-speed water flow owing to turbulence and rough free surface. Therefore the total air demand of the flow Qademand ¼ Qaen þ QaD where Qaen is air discharge entrained to the flow and QaD is the discharge of the dragged air. Part of the air demand is supplied from the air vent (Qav). Part of air demand may also be supplied from the tunnel outlet (Qat). Therefore one can write: Qaen þ QaD ¼ Qav þ Qat ¼ Qademand
ð6Þ
Safavi et al. (2008) experimentally studied the effect of parameters such as gate opening, vent diameter and tunnel dimensions on the vent discharge and air supplied from the tunnel outlet in a circular tunnel. Results of this work showed that in addition to Frc, relative flow depth (defines as the ratio of flow depth to tunnel diameter) and vent opening have important effect on air supplied from the tunnel outlet. They also presented a relationship to determine negative pressure downstream of the gate. Most of the previous research works were based on physical model studies. Therefore the validity of empirical equations is limited considering the experimental limitations, scale effects and measurement accuracy. Conducting more experiments with larger facilities and wider range of flow parameters is also very expensive and time consuming. In the present paper, employing the 3D computer code “FLUENT”, air flow downstream of the gate was simulated numerically, and air demand was calculated. Numerical model results were then verified using experimental data. In the next step, air flow in gated tunnels with circular and rectangular cross-sections was simulated in a wide range of flow parameters and air discharges from the vent and the tunnel outlet were
calculated. Based on numerical results, an algorithm was presented to design air supply system in gated tunnels. 2. Numerical modeling In the present work, FLUENT (FLUENT Inc, 2005) computer code was used to simulate airewater flow after the gate in the tunnel. FLUENT solves full three-dimensional Reynolds Averaged NaviereStokes equations. This computer code is also able to calculate the location of the interface between different phases using a VOF (Volume of Fluid) technique. For airewater flow the governing equations can be written as given below (Streeter, 1961). 2.1. The continuity equation v ðrÞ þ V$ðr! vÞ¼0 ð7Þ vt ! ! ! In this equation v is the mixture velocity: v ¼ ðaw rw v w þ v a Þ=r where ! v w and ! v a are velocity of water and aa ra ! air, respectively. r is the mixture density defined as: r ¼ aw rw þ aa ra ¼ aw rw þ ð1 aw Þra where rw and ra are density and aw and aa are the volume fraction of water and air, respectively. 2.2. The momentum equation In the numerical model, a single momentum equation is solved throughout the entire air and water domain, and the resulting velocity field is shared among the phases. The momentum equation, shown below, is dependent on the volume fractions of air and water through the properties r and m0. v ! T ðr v Þ þ V$ðr! v! v Þ ¼ Vp V$ m V! v þ V! v vt ! þ r! g þF
ð8Þ
where p ¼ pressure, m ¼ m0 þ mt. m0 is the mixture viscosity of airewater flow and is calculated similar to density, mt is ! turbulence viscosity and F is the body force. In this research, to determine mt, kee, standard model was employed (Launder and Spalding, 1974). To treat the free surface, the VOF method and Young’s (1982) scheme was used. In this method, the capturing of the interface between the phases is accomplished by the solution of a continuity equation for the volume fraction of one of the phases. In this research, water is assumed as the first and air is assumed as the second phase. For air, the transport equation has the following form: v v aa Þ ¼ 0 ð9Þ ðaa Þ þ V$ð! vt This equation is solved in the entire domain and volume fraction is computed for all cells. Then the interface between air and water can be calculated by assuming a specific volume fraction as the free surface. In the present study, it was noticed that flow velocity increased with distance from the tunnel invert till volume fraction of 0.5. From this point away, flow
J. Yazdi, A.R. Zarrati / Journal of Hydro-environment Research 5 (2011) 3e13
velocity decreased which showed entering the air phase (Fig. 1). It was therefore assumed that volume fraction of 0.5 represents the free surface. After solution of the governing equations, the vent air discharge was also calculated. The first step in the numerical modeling is mesh generation. In the present research, both structured and unstructured meshes were used. Unstructured mesh was used at the gate area because of the complex geometry (Fig. 2). Structured mesh was used at other regions. In all computations, mesh size was refined till the results were independent of the mesh size. For each of the models, to ensure the independency of results from the mesh sizes, the selected computational cells were first refined by two times and then by four times and it was observed that the calculated air discharge did not change more than 1.3%. Fig. 1 shows the geometry of a typical gated tunnel. At the upstream section, the operating head of the reservoir was introduced as total pressure with reference to atmospheric pressure. At the entrance to the air vent which is in contact with atmosphere, only air phase is present and total pressure was assumed equal to zero. For all solid boundaries, no-slip condition was applied (Rodi, 1980). Owing to symmetry of the tunnel and flow, only half of the tunnel was modeled and symmetric boundary condition was applied at the tunnel centerline. The zero piezometric pressure was imposed at the outlet assuming discharge of flow to the atmosphere. Since the flow was simulated in unsteady state, the PISO algorithm was used for coupling pressure and velocity fields (Versteeg and Malalasekera, 1995). Computations were conducted by an AMD 3800þ processor. Each run of the numerical model took 5e30 h depending on tunnel dimensions, mesh numbers and flow characteristics. 3. Model verification Before employing the numerical model to study airewater flow in gated tunnels, it was necessary to ensure the model accuracy by comparing its results with experimental data. For this purpose, Speerli and Hager (2000) and Safavi et al. (2008) experiments were used. Speerli and Hager (2000) studied air demand in a rectangular tunnel in various gate openings. The experimental set-up included an approach conduit, a sluice gate and a tailrace tunnel (see Fig. 1). The approach conduit had a width and height of 0.3 m. The tailrace tunnel height increased to 0.45 m. The tailrace tunnel had a width of 0.3 m and a maximum length of 21 m. The approach energy head was 10 m and the
5
relative gate openings varied from 0 to 45%. Fig. 3 shows the comparison of numerical model results with experimental data of Speerli and Hager (2000). In this figure, y/Ht is relative depth of the flow where y is the flow depth close to the outlet and Ht is the tunnel height. The maximum discrepancy was 28% in the gate opening of 10%. At small gate openings, flow after the gate is in the form of spray (Sharma, 1976). Therefore, results of the VOF model, which is not suitable for modeling spray, show more deviation from the physical model results. At other gate openings numerical results under-estimated vent air discharge by maximum of about 8%. It should be noticed that the amount of air entrained into the flow Qaen is not simulated by the numerical model. Experimental data of air-concentration profiles was available in 13.33%, 26.7% and 40% gate openings from Speerli and Hager (2000). Air discharge in water flow from the tunnel invert to a distance where volume fraction of air was 50% (and was considered as the free surface) was therefore calculated. From these calculations, maximum Qaen was found to be about 10% of the vent discharge. This amount was approximately equal to the difference between numerical and physical model results for air vent discharge. Therefore, it can be concluded that under-estimation of vent air discharge is because Qaen is not simulated by the numerical model. Similar results were reported by Safavi et al. (2009). In these experiments air dragged above the water flow was prevented by a plate at the tunnel outlet. It was concluded that the vent air discharge which in this test was supplying only the air entrained to the water flow reduced about 80%. These experiments showed that the dragged air includes a considerable amount of the air demand. Safavi et al. (2008) studied air demand in a circular tunnel in various gate openings. Experiments were conducted in a 17.5 cm diameter Perspex pipe. A slide gate was installed upstream of the tunnel (Fig. 4). Length of the tunnel after the gate was 7 m. Energy head upstream of the gate was constant and equal to 2.2 m. To conform to prototype design a step was considered in the model after the gate so that a jet forms over this step. A 3 cm diameter vent made from Perspex was installed just after the bottom outlet gate. Fig. 5 shows comparison of numerical and experimental results. In this figure y/D is relative depth of flow ( y is depth of flow and D is tunnel diameter). As can be seen in this figure, results have a good agreement with experimental data. In average the numerical model under-estimated the air vent discharge by about 7%. This difference can also be attributed to air entrainment to the flow which is not simulated by the numerical model.
Fig. 1. Free surface and dragged air (output of the numerical model)
6
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Fig. 2. A sample of computational mesh in the gate area: (a) rectangular tunnel, and (b) circular tunnel.
In the next stage, calculated air discharge ratio (b) was compared with few empirical equations in Fig. 6. In this figure Frc is the Froude number at the contracted section downstream of the gate. This figure shows that the numerical model results are closer to experimental measurements than empirical equations. In addition, numerical model results have a good agreement with Campbell and Guyton (1953) and USACE (1964) equations (with average difference of 10.44% and 19.32%, respectively). These equations are based on prototype data. Sharma (1976) equation overestimates the air discharge ratio. Overestimation of Sharma was also reported by Safavi et al. (2008). These results show the validity of the numerical model in calculating the flow field of air and water. In the next stage by changing various parameters including flow depth and discharge, tunnel length and diameter of the vent, the numerical model was used to calculate velocity field of air and water flow. Results of these computations were then used to study the air flow pattern and the parameters that affect this pattern. In the following section these results are discussed.
increase in the air demand. In a constant reservoir head and gate opening, however, if energy losses in the air vent increase, the ratio of the air supply from the tunnel outlet to that from the air vent increases (Safavi et al., 2008). In addition to hydraulic parameters such as reservoir head, flow depth, flow velocity and the loss coefficient of air vent which are discussed above, diameter and length of the tunnel affect the flow air demand. For example the relative flow depth y/D affects the amount of dragged air by changing the area of air passage above the water flow. Moreover, increasing the length of the tunnel decreases the amount of air supplied from the tunnel outlet by increasing the energy loss in the air passage above the flow (Garcia and Fuentes, 1984; Speerli and Volkart, 1997; Safavi et al., 2008). The numerical model which was validated with available experimental data was employed to study the influence of various parameters on air demand. These studies are presented first for a tunnel with a circular and in the next stage for a tunnel with a rectangular cross-section. 4.1. Circular tunnels
4. Air flow simulation in gated tunnels If the reservoir head increases when the gate opening is constant, flow velocity increases in the tunnel and consequently air demand increases. On the other hand, in a constant reservoir head with opening the gate, flow discharge and consequently flow velocity along the tunnel increases and this also causes an
For simulation of flow in various conditions in circular tunnels, dimensions of the physical model conducted by Safavi et al. (2008) were considered. Range of different parameters considered in this study was as below: 0:2 < y=D < 0:43; 20 < L=D < 60; 0:1 < d=D < 0:3 and 0:0066 < Q2 =gD5 < 0:23
0.14 0.12
Qav (m3/s)
0.10 0.08 0.06 0.04
Experimental, Speerli et al. 1997
0.02
Numerical
0.00 0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
y/Ht
Fig. 3. Comparison between numerical results and experimental data in a rectangular cross-section.
where d is the vent diameter and Q is the flow discharge. Studies showed that there is a circulation of air above the water flow as air is dragged along the water surface towards downstream and on the other hand is flowing from the tunnel outlet towards the gate at distances close to the tunnel ceiling (Fig. 7). Results showed that in the range of parameters considered, the velocity distribution of air which is dragged towards downstream follows the following equation: n ya u ¼ u0 1 h0
ð10Þ
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7
Fig. 4. Details of the gate in the experimental model, Safavi et al. (2008).
In this equation u0 is the water surface velocity, ya is distance from the flow surface and h0 is the distance from water surface to the point where velocity is zero (Fig. 7). In this equation, n is a constant factor that depends on the shape of the cross-section. In the circular cross-section, by regression analysis n was obtained equal to 1.5. Comparison of Eq. (10) at various longitudinal sections is shown in Fig. 8a. Numerical results also indicated that velocity distribution of air flow from the tunnel outlet follows the following equation: n0 0 ya h0 u ¼ u0 Dy
ð11Þ
In this equation u00 is the maximum velocity of air, flowing from the end of the tunnel. Since the location of maximum velocity u00 was very close the tunnel ceiling (Fig. 7), the distance of u00 from the free surface can be approximated by D y. Regression analysis showed that in Eq. (11) n0 ¼ 0.5. Fig. 9a shows the comparison of numerical results and Eq. (11) for a typical cross-section near the tunnel outlet. For other sections similar results are obtained. The numerical results showed that h0/D is independent of the y/D, Q and L but is a function of position in the transversal cross-section z and also x0. The following relationship was fitted for determining
h0 from the numerical data with a correlation factor of 0.998 for the whole range of data: z 2 z h0 ¼ 1:54 þ0:137 þC D D D
ð12Þ
In this equation, z is the distance of the longitudinal crosssection from the tunnel axis and C is a function of x0 and can be calculated from the following equation: 2 2 C ¼2:22 0:1133 þ 19:742ðx0 0:6525Þ 2 þ 0:13 0:1133 þ 19:742ðx0 0:6525Þ þ 0:25
ð13Þ
Knowing the velocity distribution of air above the free surface, dragged air discharge (Qav) and air entrained from the tunnel outlet (Qat) can be calculated as: Za ZD QaD ¼2
udydz ¼ 0
X
y
2u0 nþ1
D y nþ1 Dzi if : h0 > D y h0i 1 1 h0i
ð14Þ
0.010
0.006
3
Qav (m /s)
0.008
0.004 Experimental, Safavi et al. 2008
0.002
Numerical
0.000 0.15
0.2
0.25
0.3
0.35
0.4
0.45
y/D
Fig. 5. Comparison between numerical results and experimental data.
Fig. 6. Comparison of air discharge ratio b in the present research with previous works.
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Fig. 7. Velocity distribution of the dragged air over the water surface.
where a is the lateral distance from the tunnel axis to the tunnel wall (Fig. 7) and is a function of ya which is the distance from the water surface. Air vent discharge can therefore be calculated as:
Za Zh0 QaD ¼ 2
udya dz 0
¼
0
2u0 1:54 3 0:137 2 a þ a þ kDa if : h0 < D y n þ 1 3D 2
ð15Þ
Qav ¼ QaD Qat
If calculated h0 is more than D y there will be no air entrainment from the tunnel outlet and therefore all air demand is supplied from the air vent (Eq. (14)). There is no analytical solution to Eqs. (14) and (16) and therefore they should be solved numerically. To determine Qat from Eq. (16), u00 must be estimated. Numerical data showed that sum of u0 and u00 is a function of u0, L and x0. In the present study the following
Z a ZDy Z Z Qat ¼ 2
udya dz 0
¼
h0
1
2u00 ðD yÞ X nþ1
h0i ðD yÞ
a
nþ1 Dzi
ð17Þ
ð16Þ
0.08
y/D=0.2
y/D=0.25
0.06 0.04 0.02 0
0
3
2
1
y a (m )
0.04
y/D=0.43
y/D=0.31
0.02 0 1
0
b
2
1
0
3
2
4
3
0.24
y/Ht=0.07
y/Ht=0.2
0.16 0.08 0.04
0
1
2
3
0
4
2
6
4
10
8
0.16
y/Ht=0.4
y/Ht=0.27 0.08 0 0
5
10
0
5
10
u (m / s ) Numerical
Eq. (10)
Fig. 8. Air velocity distributions above the free surface, in a typical longitudinal cross-section at various relative depths: (a) circular tunnel, and (b) rectangular tunnel.
J. Yazdi, A.R. Zarrati / Journal of Hydro-environment Research 5 (2011) 3e13
a
9
0.06 0.04
y/D=0.2
y/D=0.25
y/D=0.31
y/D=0.41
0.02 0 0.06
y a (m )
0.04
b
0.02 0 -1.3
-0.8
-0.3
-0.9
0
-0.4
0
0.16 0.12
y/Ht=0.07
y/Ht=0.09
0.08 0.04 0.15 0.12
y/Ht=0.27
y/Ht=0.2
0.09 0.06 0.03 -2.5
-2
-1.5
-1
-0.5
-1
0
-0.5
0
u (m / s ) Numerical
Eq. (11)
Fig. 9. Velocity distributions of air flow from the tunnel outlet in a typical section at different relative depths: (a) circular tunnel, and (b) rectangular tunnel.
equation was found for u00 þ u0 with a correlation factor of 0.95: L 0 u0 þ u0 ¼1:24u0 0:008 D 5:91 0:1133 þ 19:742ðx0 0:6525Þ2 þ 1:41 ð18Þ
of the tunnel is also valid for rectangular cross-section with n ¼ 0.9 and D z Ht. Figs. 8b and 9b show the comparison of numerical results and Eqs. (10) and (11) in a typical section along the tunnel. Similar results were obtained at other sections. The following equations were found for the rectangular cross-section using numerical data:
For a real case, L, D and x0 are known or easily can be determined. u0 can be calculated from logarithmic velocity distribution law and knowing the flow depth. u00 can then be computed from Eq. (18).
2 h0 ¼ 4:13 eð3x0 þ1:1Þ 0:45 eð3x0 þ1:1Þ þ 0:3 Ht
4.2. Rectangular tunnels Numerical analysis was also conducted for tunnels with rectangular cross-section. To simulate the flow in various conditions in rectangular tunnels, geometry of the physical model conducted by Speerli and Hager (2000) as described above was considered. Range of different parameters considered in this study was as follows: 0.07 < y/Ht < 0.53, 20 < L/Ht < 60, 0.01 < Q2/gHt5 < 0.9 and 0 < d/Ht < 0.44. Similar to a circular cross-section numerical results indicated that velocity distribution of dragged air over the free surface can also be expressed by Eq. (10), with n ¼ 1.8. Eq. (11) for velocity distribution of air supplied from the end
ð19Þ
Knowing velocity distribution QaD and Qat can be calculated as: Zb=2 ZHt QaD ¼ 2
udydz 0
y
nþ1
h0 u0 b Ht y if : h0 > D y 1 1 ¼ nþ1 h0
Zb=2 Zh0 QaD ¼ 2
udya dz ¼ 0
0
h0 u0 b if : h0 < D y nþ1
ð20Þ
ð21Þ
10
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Zb=2 ZHt Qat ¼ 2
udya dz ¼ 0
h0
u00 ðHt h0
nþ1
yÞb
ð22Þ
where b is width of the tunnel. To calculate Qat from Eq. (22), it is necessary to estimate u00 (maximum velocity of air flow from the tunnel outlet). The following equation was suggested to estimate u00 for rectangular sections using numerical results.
u00 þu0 ¼1:186u0 0:02
L 6:09 eð3x0 þ1:1Þ þ2:38 Ht
ð23Þ
Based on the above equations an algorithm was suggested for designing the vent in gated tunnels with circular or rectangular cross-sections. This algorithm is demonstrated in Fig. 10 and is presented below. 1. Assume a diameter (d0) and discharge (Qav) for the air vent. For estimating Qav existing empirical equations can be used.
Fig. 10. Flowchart for calculating air demand in gated tunnels.
J. Yazdi, A.R. Zarrati / Journal of Hydro-environment Research 5 (2011) 3e13 0.024
0.010
0.020
Q av (m /s)
0.008
0.016
3
0.006
3
Qa v (m /s)
11
0.004
0.008 Experiments (Gavoshan dam)
Experimental, Safavi et al. 2008
0.002
0.012
0.004
Present Algorithm
Present Algorithm
0.000 0.15
0.2
0.25
0.3
0.35
0.4
0.000 0.23
0.45
0.28
0.33
0.38
0.43
0.48
y/D
y/D
Fig. 11. Comparison of calculated vent air discharge with Safavi et al. (2008) experimental data.
2. x0 is calculated regarding d0 and Qav knowing the length of the vent la. (x0 ¼ ke þ fla/d where ke is coefficient of local losses, f is DarcyeWeisbach coefficient). 3. The height h0 is calculated from Eqs. (12) or (19) depending on the shape of the tunnel cross-section. 4. Knowing Qw, the surface velocity u0 is computed from logarithmic velocity distribution law and the flow depth. 5. QaD and Qat are calculated from Eqs. (14)e(16) for a circular and Eqs. (20)e(22) for a rectangular crosssection. 6. To find the total air demand, entrained air to the flow that is Qaen should be estimated and added to QaD. Based on Speerli and Hager (2000) measurements, it was shown that when water surface is defined at 50% air concentration, Qaen is about 10% of the vent discharge. More experiments are needed to find a more accurate value for Qaen. 7. Vent air discharge is calculated as: Qav ¼ QaD þ Qaen Qat. 8. The losses and the value of x0 in the air vent is calculated and compared to its value in Step 1. If these values are not equal, calculations must be repeated from Step 1 with a new x0. 9. The negative pressure is calculated immediately after the vent. If negative pressure is high, diameter of vent should be increased and computations should be repeated again from Step 1. Speerli and Hager (2000) suggested that the negative pressure should be less than 1.5 m water column.
Fig. 13. Comparison of calculated vent air discharge with experimental data of Gavoshan model study.
5. Comparison of the present model with measured data The present algorithm was employed to calculate vent air discharge for experiments of Safavi et al. (2008) and Speerli and Hager (2000) in different gate openings. Vent discharge calculated was then compared with experimental data in Figs. 11 and 12. In these calculations Qaen was also considered. The average difference was found to be 8.3% for Safavi et al. (2008) and 6% for Speerli and Hager (2000) which is considered to be acceptable. The present algorithm was also used to calculate the vent air discharge in two more cases of Gavoshan and Gotvand dam bottom outlets. Gavoshan dam is a rock fill dam in the west of Iran. Its bottom outlet is a 4.4 m diameter circular tunnel with 650 m length. The design discharge of this bottom outlet is 133.7 m3/s. The physical model of this tunnel was built in Water Research Institute affiliated to Ministry of Power in Iran with 1/15 scale (WRI, 2003). Gotvand the highest embankment dam in Iran is 175 m high and its bottom outlet is 1640 m long with 9.5 m diameter. The design discharge of this bottom outlet, one of the highest in its type is about 520 m3/s. The physical model of this bottom outlet was also built in Water Research Institute with 1/25 scale (WRI, 2008). Air flow velocity of the vents in these two tunnels was measured by a hot wire anemometer at different gate openings and air discharge was then calculated from the air velocity. The vent air discharges were also calculated by the
0.05
0.14 0.12
0.04
Q av (m 3/s)
3
Qa v (m /s)
0.10 0.08 0.06 0.04 Experimental, Speerli et al. 1997
0.02 0.00 0.00
0.03 0.02 Experiments (Gotvand dam)
0.01
Present Algorithm
Numerical
0.05
0.10
0.15
0.20
0.25
0.30
0.35
y/Ht
Fig. 12. Comparison of calculated vent air discharge with Speerli and Volkart (1997) experimental data.
0.00 0.18
0.26
0.34
0.42
0.5
0.58
0.66
y/D
Fig. 14. Comparison of calculated vent air discharge with experimental data of Gotvand model study.
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J. Yazdi, A.R. Zarrati / Journal of Hydro-environment Research 5 (2011) 3e13
present algorithm and were compared with the measured data in Figs. 13 and 14. It can be seen that results of calculations are close to measurements. In small flow depths (lower gate openings) owing to spray flow conditions, calculations show more deviation from the physical model results. In general results of the present algorithm are satisfactory with an average error of 13.4% for Gavoshan dam and 11.4% for Gotvan dam bottom outlets. These results show the ability of the suggested algorithm to predict air demand in gated tunnels. 6. Conclusion In the present paper airewater flow in gated tunnels was studied with a 3D numerical model. The numerical modeling involved the solution of the NaviereStocks equations using the FLUENT computer code. The kee turbulence model with the VOF method was used to simulate complicated airewater flow both in circular and rectangular cross-sections. Flow of air above high-speed water flow was studied and the effect of different parameters on air demand was investigated. These parameters included reservoir head, discharge, flow depth, flow velocity, the loss coefficient of air vent and diameter and length of the tunnel. From the results of the numerical model, vent air discharge (Qav) was calculated at different gate openings and was validated by the experimental data. Numerical results showed a good agreement with experiments. The numerical model however under-estimated the vent air discharge by about 10%. This is probably owing to air entrainment to the water flow which was not be simulated by the numerical model. Calculating the air velocity field in a wide range of flow conditions, air velocity distribution of the dragged air along the water flow as well as the air flow from the tunnel outlet was estimated. A relationship was then found for the height of the dragged air area above the water surface as a function of transverse position in the cross-section, energy losses in air vent and diameter of the tunnel. Having the air velocity distribution and the area of air flow along the water surface and also air entrained from the tunnel outlet, dragged air discharge (QaD) and supplied air from open end of tunnel (Qat) were computed for both circular and rectangular cross-sections. An algorithm was then presented for designing the aeration systems in gated tunnels. The algorithm was validated with a number of available experimental data, which were conducted for measuring air demand in circular and rectangular tunnels and the agreement was good. References Campbell, F.B., Guyton, B., 1953. Air-demand in gated outlet works. In: Proceedings of the Fifth IAHR congress, Minnesota, U.S.A., pp. 529e533. FLUENT Inc., 2005. Fluent v. 6.2. FLUENT Inc., Lebanon, NH, USA. Garcia, J., Fuentes, R., 1984. Influence of the tunnel length on the hydraulic modeling of the air entrainment in the flow downstream of a high head gate. In: Symposium on Scale Effects in Modeling Hydraulic Structures, Esslingen, Germany, pp. 4-14-1e4-14-2.. Kalinske, A.A., Robertson, J.M., 1943. Closed conduit flow. ASCE Transactions 108, 1435e1447.
Launder, B.E., Spalding, D.B., 1974. The numerical computation of Turbulent flows. Computational Methods in Applied Mechanics and Engineering 3, 269e289. Rodi, W., 1980. Turbulence models and their application in hydraulic engineering. In: IAHR Monograph. Safavi, K.H., Zarrati, A.R., Attari, J., April 2008. Experimental study of airdemand in high head gated tunnels. Journal of Water Management 161 (WM2). Safavi, K.H., Zarrati A.R., Attari, J., 2009. Experimental study of air-demand in bottom outlets. In: 8th Iranian Conference on Hydraulics, Tehran University, Faculty of Engineering, Tehran, Iran. Sharma, H.R., 1976. Air entrainment in high head gated conduits. Journal of Hydraulics Division, ASCE 102 (No. 11). Speerli, J., Volkart, P.U., 1997. Air entrainment in bottom outlet tailrace tunnels. Proceedings of the 27th IAHR Congress, San Francisco, Theme D, pp. 613e618. Speerli, J., Hager, W.H., 2000. Air-water flow in bottom outlets. Canadian Journal of Civil Engineering 27, 454e462. Streeter, V.L., 1961. Handbook of Fluid Dynamics. McGraw-Hill Inc. U.S. Army Corps of Engineers, 1964. Air-demand Regulated Outlet Works. Hydraulic Design Criteria, Chart 050 -1. Versteeg, H.K., Malalasekera, W., 1995. An Introduction to Computational Fluid Dynamics. The Finite Volume Method. Longman Publications. Vischer, D.L., Hager, W.H., 1997. Dam Hydraulics. John Wily & Sons, Chichester, pp. 190e213. Water Research Institute, WRI, 2003. Hydraulic Model Study of Bottom Outlet Gates of Gavoshan Dam. Technical Report. Water Research Institute affiliated to Ministry of Power, Iran. Water Research Institute, WRI, 2008. Hydraulic Model Study of Upper Gotvand Dam Bottom Outlets. Technical Report. Water Research Institute affiliated to Ministry of Power, Iran. Young, D.L., 1982. Time-dependent multi-material flow with large fluid distortion. In: Morton, K.W., Baines, M.J. (Eds.), Numerical Methods for Fluid Dynamics. Academic Press.
Notation a: lateral distance from the tunnel axis to the tunnel wall; b: tunnel width; D: tunnel diameter; d: vent diameter; F: body force; Frc: Froude number at the contracted section; G: percentage of gate opening; HE: reservoir head; Ht: tunnel head; h0: distance from water surface to the point where velocity is zero; h: full size of gate opening; L: tunnel length; p: pressure; QaD: dragged air; Qademand: air demand; Qaen: air discharge entrained to the flow; Qat: tunnel outlet air discharge; Qav: vent air discharge; qav: dimensionless air discharge; Qw: water flow discharge; Re: Reynolds number; Rh: hydraulic radius; S: relative gate opening; u: air velocity upon the flow surface; u0: water surface velocity; u00 : maximum velocity of air flow from the end of the tunnel; v: mixture velocity; va: air velocity; vw: water velocity; We: Weber number; w: gate opening; y: flow depth;
J. Yazdi, A.R. Zarrati / Journal of Hydro-environment Research 5 (2011) 3e13 ya: distance from the flow surface; z: position in the transversal cross-section. Greek symbols aa: volume fraction of air; aw: volume fraction of water; b: relative air vent discharge;
DP: sub-atmospheric pressure after the gate; gw: water specific weight; n: kinematic viscosity of water; r: mixture density; ra: air velocity; rw: water density; m0: mixture viscosity; mt: turbulent viscosity; x0: total loss coefficient of air vent.
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