- Email: [email protected]
Lifting force acting on a gate with high head
379
2011,23(3):379-383 DOI: 10.1016/S1001-6058(10)60126-6
LIFTING FORCE ACTING ON A GATE WITH HIGH HEAD* LIU Xiao-qing, ZHAO Lan-hao, CAO Hui-ying, SUN Xiao-peng College of Water Conservancy and Hydropower Engineering, Hohai University, Nanjing, 210098, China, E-mail: [email protected] (Received December 26, 2010, Revised March 20, 2011) Abstract: The hydrodynamic lifting force acting on a gate with high head is one of the key factors concerning the safety and reliability of gates. The lifting force is closely related to hydrodynamic pressure, and generally, is obtained through the model test. This work presents a method of numerical simulation based on the VOF method for the flow and FEM for the structure of a gate to investigate this kind of the lifting force. The physical model experiments were conducted about the hydrodynamic pressure and the lifting force to verify the numerical results. The comparisons of those two methods show that the maximum relative error is smaller than 11.40 % and the method presented in this paper is feasible and could be used in the designs of hydropower projects. Key words: gate, pressure, high head, lifting force, numerical simulation, physical model experiment
Introduction For a hydropower project, the gates of the intakes of release works are subjected to high hydrodynamic pressure in order to control discharge. In the recent years, several hydropower plants with dams of the height of over 100 m have been constructed[1]. The gates with high head are usually operated under the different conditions, such as partially opening or completely closed, and the high pressure deduced by the head on the gates leads to difficulties of the lifting of them. The hydraulic characteristics of the lifting force are key factors and have direct impacts on the safety and reliability of the gates[2], and the estimation of the lifting force of the gates with high-head is of great importance. There are many factors affecting the gate lifting force, including the dead weight of gate, friction force, and pull force, and hydrodynamic pressure, in which the hydrodynamic pressure is of significance[3]. Since this kind of pressure is impacted by hydraulic and geometric parameters, such as tunnel outlet geometries, top shape of the gate, water sealing structure, * Project supported by the National Natural Science Foundation of China (Grant No. 51079044). Biography: LIU Xiao-qing (1965-), Female, Ph. D., Associate Professor
water head, opening degree of gates, the estimation of the hydrodynamic pressure is very complex through numerical simulation under a practical flow condition with high head[4]. At present, physical model experiments are commonly used to estimate the hydrodynamic pressure, for example, in the investigation of lifting forces of emergency gates in the Three Gorges Hydropower Station and the Xiluodu Hydropower Station[5,6]. This article presents a method of the estimation for the lifting force of a gate by numerical simulation. The physical model experiments are conducted to examine the results of the numerical simulation.
Fig.1 Physical model of gate and discharge tunnel
380
Table 1 Results of hydrodynamic pressure with head 46 m (kPa) No.
e 0.11
0.22
0.33
0.44
0.56
0.67
0.78
0.89
1
297.36
246.24
225.09
220.37
213.90
224.10
234.89
270.04
2
377.45
335.81
306.87
297.30
287.30
293.11
251.72
/
3
385.31
356.40
335.71
327.62
296.93
/
/
/
4
319.62
297.28
278.37
/
/
/
/
/
5
315.70
/
/
/
/
/
/
/
7
0.39
0.42
2.63
3.04
1.70
0.62
0.88
4.08
1. Experiments 1.1 Experimental set-up and methodology The physical model was designed at scale 1/30 according to the discharge tunnel of the Xiaowan Hydropower Station based upon the Froude similitude and the experiments were conducted in the Hydraulic Structure Laboratory at Hohai University. Figure 1 is the photo of the experimental facilities used in this study. It consists of a pump, an approach conduit, a large feeding basin, a pressure circular tunnel, a gate, a free-surface open tunnel and a flow return system. The maximum pump capacity was 800 l/s. The model gate, with the size of 0.4333 m× 0.45 m (width×height), was made of steel SS4 to meet the demands of the Froude similarity and structural dynamic similarity. For approach heads upstream, the gate could be moved from the fully closed to the fully open situation ( d = 0.45 m ) during all phases of experiments of this study. The flow downstream of the gate through the free-surface open tunnel led into the return flow system. The pressure tunnel with circular cross-section with the diameter of 0.50 m was set up and fixed with the longitudinal slope of 1.5%. The total length of the physical model is about 17.30 m. Water discharges were measured with a discharge measurement weir downstream of the free-surface tunnel.
Fig.2 Hydrodynamic pressure points (unit: m)
There are 7 measuring points of hydrodynamic pressure located in the face of the gate as shown in Fig.2, in which No. 7 is set on the bottom of the gate. The set of No. 1, 3 and 5, and set of No. 2, 4 and 6, were located in the right and left sides of the gate central line with 0.025 m to the line respectively, the distance of No.1 to the bottom is 0.015 m, and the vertical distances are all 0.085 m for the rest points. The transducers CYG1505 and CRAS were used to measure hydrodynamic pressures for all phases of experiments. The experimental phases included 8 gatages of the gate at the heads of 1.00 m, 1.33 m and 1.53 m corresponding to prototype heads 30 m, 40 m and 46 m respectively, in which those gatages ( e ) were 0.11, 0.22, 0.33, 0.44, 0.56, 0.67, 0.78 and 0.89, given by e = h1 / d , where h1 is the opening size of different experimental phases, and d = 0.45 m, the height of the gate. 1.2 Experimental results and discussions Table 1 is the result of the hydrodynamic pressure of the prototype gate at different gatages, obtained by physical model experiments at the model head 1.53 m, corresponding to the prototype head of 46 m. The hydrodynamic pressure of the gate decreases below the gatage of 0.56 and then increases with the increase of the gate gatage at the measuring points No. 1 and 2. At the points No. 3 and 4, those pressures show the reduced tendency as the gatage increases because it is smaller than the gatage of 0.56. The pressure at the point No. 7 turns to be near the atmospheric pressure since the position of this point is located just at the interface of pressure and free surface flows. At the same gatage, such as e = 0.11 or 0.22, it could be seen that the values of the biggest pressure are 385.31 kPa and 356.40 kPa, respectively, and they appear at the measuring point No. 3. This position depends only on the distance from this point to the bottom of the gate and is not related to the gatage of the gate. By the way, no data were measured by the point No. 6 due to too high position at the gate for all the gatages.
381
Table 2 Results of lifting force Fl with head 46 m (kN)
e
Fl (kN)
0.11
4 355.02
0.22
4 398.00
0.33
4 462.51
0.44
4 286.23
0.56
3 873.54
0.67
3 697.26
0.78
3 757.45
0.89
3 675.77
Table 2 is the results of the lifting force ( Fl ) of the prototype gate with the head of 46 m, which is converted from the model experiments. It should be noted that the lifting force increases as the gatage is smaller than 0.33, then decreases with the increase of the gate gatage, and the biggest value of lifting force reaches 4 462.51 kN at gatage 0.33. 2. Numerical simulation of hydrodynamic pressure 2.1 Computational models For a given gatage of the gate, the hydrodynamic pressures on gate were computed by standard k − ε model with the VOF method[7,8] which was used to describe the free-surface of turbulent flows[9-11], and the gate was simulated by a FEM model. The length of the computational area is 65 m, and the upstream altitude of design head is 46 m and the downstream altitude 13.50 m, including 7 750 elements and 9 416 nodes in flow mesh, 4 200 elements and 5 278 nodes in solid structure mesh (as shown in Fig.3). The hydrodynamic pressures were studied at 8 gatages of the gate with the same phases as listed in Table 1.
k=
3 k 3/ 2 , Ti = 0.16 Re −1/8 (Ti u0 ) , ε = 0.3L 2
(1)
where u0 is the flow velocity, Ti the turbulence intensity, L characteristic length, Re the Reynolds number. At the outflow boundary the flow was considered fully-developed. The wall boundary is controlled by the wall functions. The free surface is determined by means of the VOF method. For solid boundary, the fixed constrained condition is used at the interfaces between fluid and structure, we have: Displacement compatibility condition (2) u f = δs
Force equilibrium condition
f (t ) = ³ τ f
is the flow velocity, δ s
(3) the solid
displace- ment, τ f the stress tensor of flow, f (t ) the force acting on solid.
Fig.4 Numerical simulation of flow ( e = 0.11 , H = 46 m ) Fig.3 Computational FEM mesh
2.2 Boundary conditions The flow boundary conditions were treated as follows. At the inflow boundary the turbulent kinetic energy k and the turbulent dissipation rate ε were defined, respectively, as[12]
2.3 Results and analysis Flow velocity vectors and free-surface profile in gatage 0.11 and water head 46 m are shown in Fig.4. In Fig.4(b), the fluid is expressed with black colour, air with grey colour, air-fluid mixture with another color.
382
Table 3 Numerical results of hydrodynamic pressure (kPa) No.
e 0.11
0.22
0.33
0.44
0.56
0.67
0.78
0.89
1
291.37
221.37
206.00
193.05
187.10
211.72
217.65
247.61
2
380.15
337.46
298.55
277.76
261.32
257.82
228.91
/
3
380.85
347.30
308.24
289.53
260.14
/
/
/
4
362.15
321.32
290.30
/
/
/
/
/
5
336.64
/
/
/
/
/
/
/
It could be noted from the figure that the flow velocity is great because of reduction of the area when the flow passes through the gate, and then decreases gradually after passing through the gate. The air downstream of gate leads to the anti-clockwise eddy current. Flow velocity vectors and free-surface profile in the another gatages have also the similar characteristics. After obtaining the free surface of the flow, the hydrodynamic pressure on the gate can be solved. The hydrodynamic pressure results at different gatages from numerical simulation are shown in Table 3. The comparisons of the numerical simulation (Table 3) with the physical model experiments (Table 1) show that the hydrodynamic pressure profiles determined by the numerical simulation agree with the experimental results. 3. Numerical simulation of hydrodynamic lifting force 3.1 FEM model A half of the gate was taken to construct the FEM model due to the symmetry of the gate. Eightnode isoparametric element, plate element, beam element and bar element[13,14] were used in the model. There were 16 158 nodes and 18 607 elements, in which 1444 eight-node isoparametric elements, 17 161 plate elements, and 2 bar elements were included. The FEM mesh is shown in Fig.5.
Symmetric constraint is exerted on the plane of the gate symmetry. The top of sling and the bottom of fixed hinge are subjected to fixed contractions. Meanwhile, displacement compatibility condition is considered in the connection between real element and plate element, bar element and plate element. An interface element among fixed hinge, active hinge and connect of hinge axis was set up to simulate the rotational effect of the hinge, and it is solved as a nonlinear interface problem with no friction. 3.2 Algorithm of lifting force The hydrodynamic lifting force[15,16] is closely related to geometric and hydraulic parameters, such as the dead weight of the gate, the friction forces of the hinge pivot and the water seal, the downpull force and the hydrodynamic pressure. The friction force of the water seal and the downpull force were solved through specification[17]. The force acting on hinge pivot was calculated by the method of finite element internal force. As a result, the hydrodynamic lifting force can be given after applying loads on the gate are obtained. For the coupled system, the stress criterion and displacement criterion were employed for the iteration. They are defined as:
τ kf − τ kf −1
{
max τ kf , ε 0
δ sk − δ sk −1
{
max δ sk , ε 0
}
≤ ετ
(4)
}
≤ εd
(5)
where ετ and ε d are tolerances for stress and displacement convergence, respectively, and ε 0 is a pre-determined small constant (10-8). 3.3 Comparisons of lifting force between numerical simulation and physical experiments Numerical results ( Flnum ) and experimental Fig.5 Computational FEM mesh
results ( Fl exp ) of lifting force at the head of 46 m are
383
Table 4 Comparisons of numerical simulation of Fl with experiments
e
Flnum (kN)
Flexp (kN)
Err (%)
0.11
4 416.02
4 355.02
1.40
0.22
4 425.64
4 398.00
0.60
0.33
4 440.33
4 462.51
0.50
0.44
4 343.04
4 286.23
1.30
0.56
4 172.45
3 873.54
7.10
0.67
4 118.9
3 697.26
11.40
0.78
4 046.59
3 757.45
7.10
0.89
3 968.11
3 675.77
7.40
[4] [5]
[6]
shown in Table 4, respectively. The relative error “ Err ” is calculated Err =
Flexp − Flnum Flexp
[3]
[7]
[8]
(6) [9]
From Table 4, we can see that the lifting force ( Fl ) increases at the beginning of the gate opening process and then decreases, the maximum value is 4 440.33 kN, occurring at e = 0.33 , and the lifting force varies in the range of 12.8% for the whole process. For all the gatages of the gate, the maximum relative error of Fl is 11.40% at e = 0.67 . 4. Conclusion The hydrodynamic lifting force of the gate with high-head has been paid much attention to for the design and operation of a hydropower project. This problem has been investigated by means of physical model experiments. The present work presents the method of a numerical simulation to solve the lifting force of the gate, and the verification of this method is conducted by the physical experiments and the results shows that the present method of numerical simulation is feasible and the maximum relative error is less than 11.4 % in comparison of the two kinds of results.
[10]
[11]
[12]
[13] [14]
[15]
[16]
References [1]
[2]
CHINA SOCIETY FOR HYDROPOWER ENGINEERING. The 30th anniversary of reform and opening-up for China water power[M]. Beijing: China Electric Power Press, 2009(in Chinese). LIU Chao-ying, YE Ben-de. Detection and analysis of lifting force in hydraulic gate[J]. Zhejiang Hydrotechnics, 2001, (5): 7-8(in Chinese).
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ZHANG Wen-yuan. Physical model study on the dynamic loads and flow-Induced vibration of emergency sluice gate in Xiluodu tunnel spillway[D]. Master Thesis, Beijing: China Institute of Water Resources and Hydropower Research, 2006(in Chinese). LIU Xi-long, CHEN Fu-rong. Gate and headstock gear[M]. Beijing: China Water Power Press, 2003(in Chinese). WANG Cai-huan, ZHANG Hui and HOU Dong-mei. Research on hydraulic characteristics for intake emergency gate of Three Gorges Hydropower Station[J]. Water Resources and Hydropower Engineering, 2005, 36(10): 17-19(in Chinese). CHEN Duan, WANG Zhong-liang and CHEN Xiu-ling. Model research on lifting force with eccentrically hinged radial gate[J]. Journal of Yangtze River Scientific Research Institute, 2010, 27(5): 44-57(in Chinese). ZHAO Xi-zeng, HU Chang-hong and SUN Zhao-chen. Numerical simulation of extreme wave generation using VOF method[J]. Journal of Hydrodynamics, 2010, 22(4): 466-477. DONG Zhi, ZHAN Jie-min. Comparison of existing methods for wave generating and absorbing in VOFbased numerical tank[J]. Journal of Hydrodynamics, Ser. A, 2009, 24(1): 15-21(in Chinese). WANG Kun, JIN Sheng and LIU Gang. Numerical modelling of free-surface flows with bottom and surface-layer pressure treatment[J]. Journal of Hydrodynamics, 2009, 21(3): 352-359. WANG Li-ying, ZHAO Wei-guo and WEI Da-hua. Application of Lyapunov exponent spectrum in pressure fluctuation of draft tube[J]. Journal of Hydrodynamics, 2009, 21(6): 856-860. PANI P. K., BHATTACHARYYA S. K. Hydrodynamic pressure on a vertical gate considering fluid–structure interaction[J]. Finite Elements in Analysis and Design, 2008, 44(12-13): 759-766. CHEN Yong-ming. Numerical simulation of water release structure turbulent flow with complicated free surface[D]. Master Thesis, Yangzhou: Yangzhou University, 2006(in Chinese). ZHENG Ke-hong. Three dimensional finite element analysis of high-pressure radial steel gate[D]. Ph. D. Thesis, Nanjing: Hohai University, 2005(in Chinese). NI Han-gen, LIU Yong-lin and LIU Bin et al. Prediction of tainter gate vibration using combined physical and numerical models[J]. Journal of Hydrodynamics, Ser. A, 2005, 20(1): 24-32(in Chinese). LIU Hong-bo, HAN ping. Summarization of hydraulic characteristic of the sluice gate[J]. South-to-North Water Tromsfersomd and Water Science and Technology, 2005, 3(2): 56-58(in Chinese). ZHUO Mei-yan, WANG Su-hua. Discussion and analysis of lifting force calculation of small and medium-sized plane steel gate[J]. Journal of Shandong Agricultural University, 2009, 40(4): 615-617(in Chinese). MINISTRY OF WATER CONSERVANCY AND ELECTRIC POWER. Specification for design of steel gate in hydraulic and hydroelectric engineering (DL/ T5039-95)[M]. Beijing: China Water Power Press, 1995(in Chinese).
2011,23(3):379-383 DOI: 10.1016/S1001-6058(10)60126-6
LIFTING FORCE ACTING ON A GATE WITH HIGH HEAD* LIU Xiao-qing, ZHAO Lan-hao, CAO Hui-ying, SUN Xiao-peng College of Water Conservancy and Hydropower Engineering, Hohai University, Nanjing, 210098, China, E-mail: [email protected] (Received December 26, 2010, Revised March 20, 2011) Abstract: The hydrodynamic lifting force acting on a gate with high head is one of the key factors concerning the safety and reliability of gates. The lifting force is closely related to hydrodynamic pressure, and generally, is obtained through the model test. This work presents a method of numerical simulation based on the VOF method for the flow and FEM for the structure of a gate to investigate this kind of the lifting force. The physical model experiments were conducted about the hydrodynamic pressure and the lifting force to verify the numerical results. The comparisons of those two methods show that the maximum relative error is smaller than 11.40 % and the method presented in this paper is feasible and could be used in the designs of hydropower projects. Key words: gate, pressure, high head, lifting force, numerical simulation, physical model experiment
Introduction For a hydropower project, the gates of the intakes of release works are subjected to high hydrodynamic pressure in order to control discharge. In the recent years, several hydropower plants with dams of the height of over 100 m have been constructed[1]. The gates with high head are usually operated under the different conditions, such as partially opening or completely closed, and the high pressure deduced by the head on the gates leads to difficulties of the lifting of them. The hydraulic characteristics of the lifting force are key factors and have direct impacts on the safety and reliability of the gates[2], and the estimation of the lifting force of the gates with high-head is of great importance. There are many factors affecting the gate lifting force, including the dead weight of gate, friction force, and pull force, and hydrodynamic pressure, in which the hydrodynamic pressure is of significance[3]. Since this kind of pressure is impacted by hydraulic and geometric parameters, such as tunnel outlet geometries, top shape of the gate, water sealing structure, * Project supported by the National Natural Science Foundation of China (Grant No. 51079044). Biography: LIU Xiao-qing (1965-), Female, Ph. D., Associate Professor
water head, opening degree of gates, the estimation of the hydrodynamic pressure is very complex through numerical simulation under a practical flow condition with high head[4]. At present, physical model experiments are commonly used to estimate the hydrodynamic pressure, for example, in the investigation of lifting forces of emergency gates in the Three Gorges Hydropower Station and the Xiluodu Hydropower Station[5,6]. This article presents a method of the estimation for the lifting force of a gate by numerical simulation. The physical model experiments are conducted to examine the results of the numerical simulation.
Fig.1 Physical model of gate and discharge tunnel
380
Table 1 Results of hydrodynamic pressure with head 46 m (kPa) No.
e 0.11
0.22
0.33
0.44
0.56
0.67
0.78
0.89
1
297.36
246.24
225.09
220.37
213.90
224.10
234.89
270.04
2
377.45
335.81
306.87
297.30
287.30
293.11
251.72
/
3
385.31
356.40
335.71
327.62
296.93
/
/
/
4
319.62
297.28
278.37
/
/
/
/
/
5
315.70
/
/
/
/
/
/
/
7
0.39
0.42
2.63
3.04
1.70
0.62
0.88
4.08
1. Experiments 1.1 Experimental set-up and methodology The physical model was designed at scale 1/30 according to the discharge tunnel of the Xiaowan Hydropower Station based upon the Froude similitude and the experiments were conducted in the Hydraulic Structure Laboratory at Hohai University. Figure 1 is the photo of the experimental facilities used in this study. It consists of a pump, an approach conduit, a large feeding basin, a pressure circular tunnel, a gate, a free-surface open tunnel and a flow return system. The maximum pump capacity was 800 l/s. The model gate, with the size of 0.4333 m× 0.45 m (width×height), was made of steel SS4 to meet the demands of the Froude similarity and structural dynamic similarity. For approach heads upstream, the gate could be moved from the fully closed to the fully open situation ( d = 0.45 m ) during all phases of experiments of this study. The flow downstream of the gate through the free-surface open tunnel led into the return flow system. The pressure tunnel with circular cross-section with the diameter of 0.50 m was set up and fixed with the longitudinal slope of 1.5%. The total length of the physical model is about 17.30 m. Water discharges were measured with a discharge measurement weir downstream of the free-surface tunnel.
Fig.2 Hydrodynamic pressure points (unit: m)
There are 7 measuring points of hydrodynamic pressure located in the face of the gate as shown in Fig.2, in which No. 7 is set on the bottom of the gate. The set of No. 1, 3 and 5, and set of No. 2, 4 and 6, were located in the right and left sides of the gate central line with 0.025 m to the line respectively, the distance of No.1 to the bottom is 0.015 m, and the vertical distances are all 0.085 m for the rest points. The transducers CYG1505 and CRAS were used to measure hydrodynamic pressures for all phases of experiments. The experimental phases included 8 gatages of the gate at the heads of 1.00 m, 1.33 m and 1.53 m corresponding to prototype heads 30 m, 40 m and 46 m respectively, in which those gatages ( e ) were 0.11, 0.22, 0.33, 0.44, 0.56, 0.67, 0.78 and 0.89, given by e = h1 / d , where h1 is the opening size of different experimental phases, and d = 0.45 m, the height of the gate. 1.2 Experimental results and discussions Table 1 is the result of the hydrodynamic pressure of the prototype gate at different gatages, obtained by physical model experiments at the model head 1.53 m, corresponding to the prototype head of 46 m. The hydrodynamic pressure of the gate decreases below the gatage of 0.56 and then increases with the increase of the gate gatage at the measuring points No. 1 and 2. At the points No. 3 and 4, those pressures show the reduced tendency as the gatage increases because it is smaller than the gatage of 0.56. The pressure at the point No. 7 turns to be near the atmospheric pressure since the position of this point is located just at the interface of pressure and free surface flows. At the same gatage, such as e = 0.11 or 0.22, it could be seen that the values of the biggest pressure are 385.31 kPa and 356.40 kPa, respectively, and they appear at the measuring point No. 3. This position depends only on the distance from this point to the bottom of the gate and is not related to the gatage of the gate. By the way, no data were measured by the point No. 6 due to too high position at the gate for all the gatages.
381
Table 2 Results of lifting force Fl with head 46 m (kN)
e
Fl (kN)
0.11
4 355.02
0.22
4 398.00
0.33
4 462.51
0.44
4 286.23
0.56
3 873.54
0.67
3 697.26
0.78
3 757.45
0.89
3 675.77
Table 2 is the results of the lifting force ( Fl ) of the prototype gate with the head of 46 m, which is converted from the model experiments. It should be noted that the lifting force increases as the gatage is smaller than 0.33, then decreases with the increase of the gate gatage, and the biggest value of lifting force reaches 4 462.51 kN at gatage 0.33. 2. Numerical simulation of hydrodynamic pressure 2.1 Computational models For a given gatage of the gate, the hydrodynamic pressures on gate were computed by standard k − ε model with the VOF method[7,8] which was used to describe the free-surface of turbulent flows[9-11], and the gate was simulated by a FEM model. The length of the computational area is 65 m, and the upstream altitude of design head is 46 m and the downstream altitude 13.50 m, including 7 750 elements and 9 416 nodes in flow mesh, 4 200 elements and 5 278 nodes in solid structure mesh (as shown in Fig.3). The hydrodynamic pressures were studied at 8 gatages of the gate with the same phases as listed in Table 1.
k=
3 k 3/ 2 , Ti = 0.16 Re −1/8 (Ti u0 ) , ε = 0.3L 2
(1)
where u0 is the flow velocity, Ti the turbulence intensity, L characteristic length, Re the Reynolds number. At the outflow boundary the flow was considered fully-developed. The wall boundary is controlled by the wall functions. The free surface is determined by means of the VOF method. For solid boundary, the fixed constrained condition is used at the interfaces between fluid and structure, we have: Displacement compatibility condition (2) u f = δs
Force equilibrium condition
f (t ) = ³ τ f
is the flow velocity, δ s
(3) the solid
displace- ment, τ f the stress tensor of flow, f (t ) the force acting on solid.
Fig.4 Numerical simulation of flow ( e = 0.11 , H = 46 m ) Fig.3 Computational FEM mesh
2.2 Boundary conditions The flow boundary conditions were treated as follows. At the inflow boundary the turbulent kinetic energy k and the turbulent dissipation rate ε were defined, respectively, as[12]
2.3 Results and analysis Flow velocity vectors and free-surface profile in gatage 0.11 and water head 46 m are shown in Fig.4. In Fig.4(b), the fluid is expressed with black colour, air with grey colour, air-fluid mixture with another color.
382
Table 3 Numerical results of hydrodynamic pressure (kPa) No.
e 0.11
0.22
0.33
0.44
0.56
0.67
0.78
0.89
1
291.37
221.37
206.00
193.05
187.10
211.72
217.65
247.61
2
380.15
337.46
298.55
277.76
261.32
257.82
228.91
/
3
380.85
347.30
308.24
289.53
260.14
/
/
/
4
362.15
321.32
290.30
/
/
/
/
/
5
336.64
/
/
/
/
/
/
/
It could be noted from the figure that the flow velocity is great because of reduction of the area when the flow passes through the gate, and then decreases gradually after passing through the gate. The air downstream of gate leads to the anti-clockwise eddy current. Flow velocity vectors and free-surface profile in the another gatages have also the similar characteristics. After obtaining the free surface of the flow, the hydrodynamic pressure on the gate can be solved. The hydrodynamic pressure results at different gatages from numerical simulation are shown in Table 3. The comparisons of the numerical simulation (Table 3) with the physical model experiments (Table 1) show that the hydrodynamic pressure profiles determined by the numerical simulation agree with the experimental results. 3. Numerical simulation of hydrodynamic lifting force 3.1 FEM model A half of the gate was taken to construct the FEM model due to the symmetry of the gate. Eightnode isoparametric element, plate element, beam element and bar element[13,14] were used in the model. There were 16 158 nodes and 18 607 elements, in which 1444 eight-node isoparametric elements, 17 161 plate elements, and 2 bar elements were included. The FEM mesh is shown in Fig.5.
Symmetric constraint is exerted on the plane of the gate symmetry. The top of sling and the bottom of fixed hinge are subjected to fixed contractions. Meanwhile, displacement compatibility condition is considered in the connection between real element and plate element, bar element and plate element. An interface element among fixed hinge, active hinge and connect of hinge axis was set up to simulate the rotational effect of the hinge, and it is solved as a nonlinear interface problem with no friction. 3.2 Algorithm of lifting force The hydrodynamic lifting force[15,16] is closely related to geometric and hydraulic parameters, such as the dead weight of the gate, the friction forces of the hinge pivot and the water seal, the downpull force and the hydrodynamic pressure. The friction force of the water seal and the downpull force were solved through specification[17]. The force acting on hinge pivot was calculated by the method of finite element internal force. As a result, the hydrodynamic lifting force can be given after applying loads on the gate are obtained. For the coupled system, the stress criterion and displacement criterion were employed for the iteration. They are defined as:
τ kf − τ kf −1
{
max τ kf , ε 0
δ sk − δ sk −1
{
max δ sk , ε 0
}
≤ ετ
(4)
}
≤ εd
(5)
where ετ and ε d are tolerances for stress and displacement convergence, respectively, and ε 0 is a pre-determined small constant (10-8). 3.3 Comparisons of lifting force between numerical simulation and physical experiments Numerical results ( Flnum ) and experimental Fig.5 Computational FEM mesh
results ( Fl exp ) of lifting force at the head of 46 m are
383
Table 4 Comparisons of numerical simulation of Fl with experiments
e
Flnum (kN)
Flexp (kN)
Err (%)
0.11
4 416.02
4 355.02
1.40
0.22
4 425.64
4 398.00
0.60
0.33
4 440.33
4 462.51
0.50
0.44
4 343.04
4 286.23
1.30
0.56
4 172.45
3 873.54
7.10
0.67
4 118.9
3 697.26
11.40
0.78
4 046.59
3 757.45
7.10
0.89
3 968.11
3 675.77
7.40
[4] [5]
[6]
shown in Table 4, respectively. The relative error “ Err ” is calculated Err =
Flexp − Flnum Flexp
[3]
[7]
[8]
(6) [9]
From Table 4, we can see that the lifting force ( Fl ) increases at the beginning of the gate opening process and then decreases, the maximum value is 4 440.33 kN, occurring at e = 0.33 , and the lifting force varies in the range of 12.8% for the whole process. For all the gatages of the gate, the maximum relative error of Fl is 11.40% at e = 0.67 . 4. Conclusion The hydrodynamic lifting force of the gate with high-head has been paid much attention to for the design and operation of a hydropower project. This problem has been investigated by means of physical model experiments. The present work presents the method of a numerical simulation to solve the lifting force of the gate, and the verification of this method is conducted by the physical experiments and the results shows that the present method of numerical simulation is feasible and the maximum relative error is less than 11.4 % in comparison of the two kinds of results.
[10]
[11]
[12]
[13] [14]
[15]
[16]
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