Pressure pulsation prediction by 3D turbulent ...

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Pressure pulsation prediction by 3D turbulent unsteady flow simulation through whole flow passage of Kaplan turbine Shuhong Liu

Received 4 April 2008 Revised 1 September 2008 Accepted 9 September 2008

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

Jianqiang Mai WSP Buildings, Salford, Manchester, UK, and

Jie Shao and Yulin Wu State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing, China Abstract Purpose – The purpose of this paper is to predict pressure pulsation in Kaplan hydraulic turbines. Design/methodology/approach – State of the art numerical simulation techniques are employed to simulate three-dimensional flows in the whole flow passage of a Kaplan turbine so that pressure pulsations can be computed in both time domain and frequency domain. Numerical results are verified by experiments carried out on the most advanced experimental platform in China. Findings – It is found that the proposed numerical model is a viable tool for prediction of pressure pulsations. The simulation shows that the model turbine and prototype turbine have the same pressure pulsation frequencies and rotating frequencies and the same transmission patterns under similar operation conditions. However, there is no similarity for the amplitude of the pressure pulsation between the model turbine and the prototype turbine. Therefore pressure pulsations in a prototype turbine cannot be obtained by scaling the experimental results of the model turbine using a similarity relationship. Practical implications – The findings will be very valuable for the design of hydraulic turbines and large-scale hydraulic power stations. Originality/value – The proposed numerical method provides a viable tool for hydraulic turbine and power station designers to predict the pressure pulsations in prototype turbines. It is a useful tool to help improve the performance of hydraulic turbines. The findings made in the numerical simulation have been verified by experiments, which is also a valuable reference for hydraulic turbine designers. Keywords Pressure, Turbulent flow, Simulation, Turbulence, Turbines, Hydraulic engineering Paper type Research paper

Engineering Computations: International Journal for ComputerAided Engineering and Software Vol. 26 No. 8, 2009 pp. 1006-1025 # Emerald Group Publishing Limited 0264-4401 DOI 10.1108/02644400910996871

1. Introduction The Kaplan turbine is a great development of early 20th century, which is of the propeller type, similar to an airplane propeller. The Kaplan’s blades are adjustable for pitch and will handle a great variation of flow very efficiently. The Kaplan turbine is 90 per cent or better in efficiency. It is very expensive and used principally in large installations. Unlike all other propeller turbines, the Kaplan turbine runner’s blades are movable. The guide vanes can also be turned and are automatically adjusted to any angle suitable to that of the blades by The research work was funded by Chinese National Foundation of Natural Science (Key project No. 90410019).

a combiner, so the turbine is efficient at different work loads. A circular stay collar absorbed the compressive forces acting on the flume casing. All low-head, high discharge propeller turbines have to be given amply dimensioned draft tubes since the efficiency of the turbine depends on a strong pressure recovery in the tubes. If the height of the draft tube was too great, the water pressure around the runner became so low that cavitation posed a serious problem. It is necessary to mount the runner below tail-race level. Accompanying the steady increment in unit output and runner diameters of hydraulic turbine units in recent years, stability problems such as hydraulic vibration and runner blade cracks in large hydraulic turbines have become prominent. Vibrations, swings, and pressure fluctuations are three major parameters to characterize the stability of hydraulic turbine units. Among them, pressure fluctuation is produced by the unsteady flow field exists in the process of running the unit. This is the major hydraulic source that leads to vibration and non-steady operation of the hydraulic turbine unit. It can excite turbine unit vibration, produce blade cracks, and even lead to resonance of the powerhouse, which directly threaten the safe operation of the whole power station. It is of great importance in engineering to study the pressure fluctuations in hydraulic turbine units. Before a hydraulic power station is constructed, it is not possible to test the performance of a prototype hydraulic turbine. The only viable way to test the prototype performance is to do model tests and scale the results to predict the performance of the prototype. However, due to lack of similarity laws in scaling vibration and pressure fluctuations, the prediction of turbine stability can only be done by computational fluid dynamics (CFD) simulations. Therefore the correctness and accuracy of a numerical simulation become extremely important. From the end of the 1990s, pioneer studies on the calculation of fluid flow field in the whole flow passage of hydraulic turbines were started on super computers. Nowadays the CFD method has been widely applied to design of hydraulic turbines (Vu, 2006; Gehrer, 2006). Models such as the Reynolds Stress Model and the Large Eddy Simulation (LES) model have been employed to simulate unsteady flows in draft tubes and study the formation of vortex rope and corresponding pressure fluctuations. Jaeger and Seidel (1999) performed unsteady flow simulations in the whole flow passage including the runner and draft tube. They obtained the pulsation information and draft tube vortex train produced by dynamic and static interference between the runner and the draft tube. Yang (1998) established a LES – double equation model based on LES idea but the equation structure is similar to the time average k-" model. Ruprecht (2000) carried out static and dynamic interference calculations for the whole flow passage and obtained more comprehensive pulsation information. Chen (2000) employed a LES in simulating 3-D turbulent flow through a pump-turbine system, which consists of spiral case, stay vanes, wicket gates, and runner. Their simulation results in the runner and the draft tube were very close to experimental results. Recently Wang (2006) solved the three-dimensional unsteady Reynolds-averaged Navier-Stokes equation in a Francis turbine flow passage, which consists of the guide vanes, runner, and draft tube. They obtained the characteristics of draft tube pressure pulsation and vortex ropes under partial load and different guide vane openings in the Francis turbine. For Kaplan turbines, most researches have conducted 3-D steady flow simulations to improve the turbines’ design or to study the special flow in the turbines. Nilsson (2000, 2002) studied the internal flow inside the runner and flows through the gap between the runner and its chamber under four different operation conditions using a numerical flow simulation method. Muntean (2004) completed the flow analysis in the spiral case and distributor of a Kaplan turbine and obtained the information of channel vortex at different

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operation conditions. Tomas (2004) used the flow simulation to improve the hydraulic design of a Kaplan turbine. Lindsjo¨ (2004) calculated the movement of bubbles in a Kaplan runner. Gehrer (2006) conducted Kaplan turbine runner optimizations by numerical flow simulations (CFD). Prediction of pressure pulsation in the whole flow passage of a Kaplan turbine by a numerical method still cannot satisfy engineering requirement. The present paper reports our computational work on the three-dimensional unsteady turbulent flow in the whole flow passage of a Kaplan turbine using the RNG k-" turbulence model. The calculated results are compared with experimental results to test the robustness of the numerical model. Then the numerical model is used to predict the pressure pulsation of a prototype hydraulic turbine. The computational results of both the model turbine and the prototype turbine were compared. Possible similarity relationship between the pressure pulsations of the model and prototype were analysed. The transmission characteristics of the pressure pulsation in the Kaplan turbine were also studied. It has been found that the proposed numerical method can successfully predict pressure pulsations in the whole flow passage of a Kaplan turbine. It is noticed that in the present work, the pressure pulsation prediction is treated under non-cavitating conditions. So that, the single phase unsteady turbulent flow has been computed in this paper. In order to predict the pressure pulsation under cavitating conditions, the complex cavitating two-phase unsteady flow simulation should be carried out. Under such non-cavitating condition, the pressure pulsation may be caused by the whirl of vortex rope and the interaction of guide vane-runner blades interaction. Although it is discussed at the end of the paper, it will be better to mention also in the Introduction so that the reader can be sure what is treated in the paper. 2. Turbulence model and numerical computational methods The incompressible continuity equation and Reynolds-averaged Navier-Stokes equation are adopted to simulate the flow through the turbine, and the RNG k-" double equations turbulence model is selected to make the equations closed (Speziale, 1992). The continuity and momentum equations are as follows: @uj ¼0 @xj

ð1Þ

where u¯j is the Reynolds averaged velocity component along the Cartesian coordinate axis xj. @ui @ui @p @ 2 ui @ 0 0 þ uj ¼ Fi þ ðu u Þ @xi @xj i j @t @xj @xj @xj

ð2Þ @ui @uj where the Reynolds stress of turbulent flow, u0i u0j ¼ t þ @xj @xi 2 @ui k þ t ij ; p is the averaged pressure, is the fluid density, and F is the body 3 @xi force acting on the unit volume fluid. The turbulence kinetic energy, k, equation and the turbulence kinetic energy dissipation rate, ", equation in the RNG k-" turbulence model are as follows: Dk @ @k @ui ¼ ak eff " ð3Þ þ 2t Sij Dt @xj @xj @xj

D" @ @" " @ui "2 ¼ ð4Þ a" eff C2" R þ 2C1" t S ij Dt @xj @xj k @xj k @ui @uj where the strain tensor components: S ij ¼ þ , the effective viscosity k2 @xj @xi eff ¼ t þ and the eddy viscosity t ¼ C , and is the molecular viscosity of fluid. " And the additional term R: R¼

C h3 ð1 h=ho Þ "2 k 1 þ bh3


ð5Þ

k where h ¼ S ; ho ¼ 4:38; C ¼ 0:0845; b ¼ 0:012; C1" ¼ 0:42 (original in the model), " C2" ¼ 1.68, k ¼ 1.0, " ¼ 0.769. The constants in the turbulence model have a great influence on improving the numerical simulation, especially C1" variation is important to improve the prediction of pressure pulsation. In the present simulation, C1" ¼ 1.45 is selected after many times pre-computations. The governing equations are discretized to algebra equations by the finite volume method in spatial domain at each time step. They are also discretized by second-order implicit formula in temporal domain and integrated within one time step. The discretized equations reflect flow field parameters at each time step. The algebra equations obtained by discretization in spatial domain are solved by sub-relaxation method. Frequency analyses of time-dependent results of unsteady flow were carried out by Fast Fourier Transform (FFT) method. Figure 1 shows the calculating algorithm in the present work, which is based on the Fluent software. Figures 2(a) and 2(b) show the flow passage and the runner in the Kaplan turbine. Before formal calculation is started, aimed at the effects of grid numbers and

Figure 1. Calculation algorithm

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calculation time on computational results, grid independence and calculation time independence were studied. Optimum number of grids and optimum calculation time were obtained after a variety of calculation strategy test. In order to validate the effect of total grid number on the computation, three different grid systems, such as case I, II, and III, were performed to the steady and unsteady flow simulation of the Kaplan turbine. The total grid element numbers in case I, II, and III are 1,933,265, 2,287,573, and 2,643,148, respectively. In the final computation, a mesh of 2,578,000 elements and 543,000 crunodes all over the flow passage was employed. It is considered that the elements were dense enough to obtain the detailed pressure data and velocity data needed. The grid system of the turbine is shown in Figure 2(c). In the calculating algorithm, the time step was 0.001s. Rotation speed of the runner was 1,267.9 rpm, therefore the converged turbulent flow solution were obtained by rotating the mesh in the runner region by 7.61 per time step. The converged solutions at all times form the unsteady solutions that are the pressure and velocity variations with time in the whole flow passage. 3. Prediction of pressure pulsation of a Kaplan model turbine and verification in experiments 3.1 Measurement of pressure pulsation in a model turbine In order to study the pressure pulsation, experiments were carried out on the No. II experimental rig of the Harbin Institute of Large Turbine and Generator Research. The flow parameters were measured in the following measuring range and the accuracy, as shown in Table I. In the Kaplan turbine model pressure pulsation experiments, pressure transducers were located at the inner walls of places 0.3D1 under the runner, the draft tube inlet at the point (-X), which is located on the left side of the draft tube, and the outside of the draft tube elbow as shown in Figure 3(b).

Figure 2. Schematic diagram of the Kaplan turbine

Table I. Measuring accuracy in experiments

Number

Performance

Unit

1 2 3 4 5 6

Flow rate (Q) Head (H) Force on arm (K) Angular speed (!) Pressure (p) Pressure fluctuation

m3/s MPa E Pound r/s kPa kPa

Measuring range

Uncertainty

0-1 0-2.07 0-1,000 0-3,000 0-200 0.5-200 kHz

±0.2% 0.075% ±0.02% ±0.05% 0.075% 2.0 106 so that the drag force on the vanes does not depend on the Reynolds number. (3) The gravity similarity requires the water head of model experiment or model simulation equals the water head of the prototype or 350/8000 of the water head of the prototype. (4) Elastic similarity requires the water head of the model turbine equals the water head of the prototype. (5) Surface tension similarity requires the water head of the model turbine equals the water head of the prototype turbine or 8000/250 times of the water head of the prototype. However the requirements (2), (3), (4), and (5) mentioned above are not satisfied in the present study. Similarly study of the pressure pulsations of hydraulic turbines by experiments or numerical simulations generally cannot satisfy the five similarity requirements mentioned above. Consequently there will be no similarity law for pressure pulsation between a model turbine and its prototype. Due to some systems affect each other or the flow characteristics of the model turbine and its prototype being different, there is a huge error by directly scaling the measured pressure pulsation and frequency of a model turbine to its prototype. IEC


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60193 especially explains the scaling of pressure pulsation from a model turbine experimental results to its prototype: ‘‘Conversion from model to prototype value can involve error caused by the following factors: deviation in Froude similitude; interaction with water conduit; interaction with the electric machine. In case that significant influence caused by the above factors is involved, it is difficult to estimate the prototype values accurately.’’ The significance of pressure pulsation study by model experiments or numerical simulations is not to predict the pressure pulsation and its frequencies in prototype turbines but to compare pressure pulsations between different models. The attempt to predict whether the prototype turbine will produce hydraulic resonance through model experiments or numerical simulation of a model turbine is still not well developed yet at the present time. 4.3 Relationship between the predicted pressure pulsation and output of the prototype turbine In order to study the relationship between pressure pulsation and output of the prototype Kaplan turbine in the operational region, the design parameters of the prototype turbine and its operation in the power station were referenced. The present paper chooses eight different coordinated operation conditions for unsteady turbulent flow calculations, see Table VI for the parameters. Table VII gives the time domain analysed pressure pulsation results at all measurement points, where P is the Operational conditions

Table VI. Parameters of the coordinated operation conditions for unsteady calculations

Table VII. Time domain characteristic pressure pulsation results at each measurement point

1 2 3 4 5 6 7 8

Water head H (m)

Blade angle ( )

Guide vane opening A0 (%)

Output P (MW)

42 47 52.2 57.8 47 47 57.8 52.2

5 5 5 5 10 15 15 20

41 39 38 37 57 71 65 78

58 68 77 88 115 160 202 226

Operational conditions

After stay vane P H/H (Pa) (%)

After guide vane P H/H (Pa) (%)

1 2 3 4 5 6 7 8

1,300 1,200 1,000 1,280 750 1,100 1,500 2,600

2,500 2,250 1,750 2,000 920 1,520 2,200 2,850

0.32 0.26 0.20 0.23 0.16 0.24 0.26 0.51

0.61 0.49 0.34 0.35 0.20 0.33 0.39 0.56

0.3D1 under the runner P H/H (Pa) (%) 12,000 11,000 11,500 12,500 10,600 11,800 15,000 16,000

2.91 2.39 2.25 2.20 2.30 2.56 2.65 3.12

Inlet of the draft tube P H/H (Pa) (%) 32,500 33,300 33,500 34,400 32,200 35,100 47,000 46,500

7.89 7.22 6.54 6.07 6.99 7.62 8.30 9.08

0.3D1 under the inlet of the draft tube P H/H (Pa) (%) 24,400 24,600 24,800 25,100 23,200 25,600 33,700 34,000

5.93 5.34 4.85 4.43 5.04 5.56 5.95 6.65

absolute peak value of pressure pulsation at time domain, H/H is the relative pressure pulsation amplitude at time domain, which is defined above. The following laws can be obtained from the time domain pressure pulsation results listed in Table VII: .

The time domain pressure pulsation amplitude is the maximum at the inlet of the draft tube and second strong at 0.3D1 under the inlet of the draft tube, which is near the draft tube inlet. It transmits to upstream direction in the sequence of 0.3D1 under the runner, behind the guide vane, in front of the guide vane, and in front of the stay vane, in which process the time domain amplitude of the pressure pulsation decreases faster and faster. It is considered that the source of the pressure pulsation is at the inlet area of the draft tube for the whole flow passage. The major factors affect the hydraulic stability of the Kaplan turbine are pressure pulsation induced by the swirling flow out of the runner and the draft tube, and. flow separations induced by the interference between the out flow of the guide vane and the inlet of the runner. If the wake flow direction at the guide vane exit does not coincide with the inlet angle of the runner blade, the separation and vortex will occur in the baled channel, and they will develop to the exit of runner, especially the separation flow near blade suction side, which should affect the pressure pulsation at the inlet of draft tube.

.

It is seen from Figure 10 that when the turbine is operating in the region of 50 to 90 MW the time domain pressure pulsation at the inlet of the draft tube decreases with the increasing of the output. The pressure pulsation affects the hydraulic stability of the turbine very much when the turbine is operating at very low output. When the turbine is operating in the region between 90 and 230 MW, the time domain pressure pulsation amplitude at the inlet of the draft tube gradually increases with the increasing of the output.

.

Under the same water head, accompany the increasing of the output, the angle of the runner blades and the openings of guide vanes are increasing, the time domain amplitude of the pressure pulsation at the inlet of the draft tube tends to increase. In all the calculated operation conditions, the operation condition with 52.2 m water head, 20 blade angle, has the highest amplitude in time domain pressure pulsation at the inlet of the draft tube. The pressure pulsation amplitudes in time domain on the section with 0.3D1 distance under the runner and at the inlet of the draft tube are 3.12 and 9.08 per cent, respectively.


Figure 10. Relationship between the time domain pressure pulsation at the inlet of the draft tube and the output of the turbine

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4.4 Principles of production the pressure pulsation of the major frequencies In all the above calculations for the coordinated operation condition, frequencies of 0.348 Hz or 0.195 times of the rotating frequency, 1.94 Hz or the rotating frequency, 3.73 Hz or double rotating frequency, 7.41 Hz or four times of the rotating frequency, 10.89 Hz or six times of the rotating frequency, and other pressure pulsation frequencies were obtained. The principles of the production of the major frequency pressure pulsations are analysed as below: (1) Low frequency pressure pulsation in the draft tube The strength of the pressure pulsation of approximately 0.195 times the rotating frequency f ¼ 0.348 Hz is the strongest at 0.3D1 under the inlet of the draft tube. It decreases in both upstream and downstream directions. Zhang et al. (2005) pointed out that the physical origin of the spontaneous unsteady flow under off-design conditions was identified as due to the absolute instability feature of the swirling flow in the draft-tube conical inlet, which either occupied a large portion of the inlet and caused a strong helical vortex rope at small flow rate condition, or started midway and caused a breakdownlike vortex bubble followed by weak helical waves at large flow rate condition. It was worth emphasizing that the axial-flow velocity profile in the inlet played a key role in the absolute instability/convective instability characters of the swirling flow. Rheingans W I of the American company Allis-Chalmers suggested an empirical equation to estimate the frequency of the vortex train in the draft tube before 1969, that is: f ¼ n=c

ð6Þ

where n is the rotating speed of the turbine, r/s, and c is a constant between 3.2 and 4.0. In recent years, it has been realized further through more experimental studies that the low frequency pressure pulsation in the draft tube has a frequency of 0.15 to 0.33 times the rotating frequency. Analysis showed that pressure pulsation of this frequency is mainly induced by the flow at the area near the inlet of the draft tube. When the natural frequency of water vibration f0 closes or equals to the pressure pulsation frequency of the draft tube, resonance would be induced in the draft tube which is the major reason why the pressure pulsation is relatively strong in a prototype turbine. The natural frequency of the water body can be calculated by (IEC 60193): vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 1 u 1 ð u ð7Þ f0 ¼ 2p t@VVAP dL @NPSE A where VVAP is the air volume in the vortex rope, NPSE is the suction head of the power station, dL is arc length of the centreline of the draft tube, and A is the area of the draft tube cross section. (2) Pressure pulsation of one times rotating frequency The pressure pulsation frequency f ¼ 1.94 Hz is approximately one times of the rotating frequency. Its energy spectrum is the highest at the inlet of the draft

tube and decreases in both downstream and upstream directions. This pressure pulsation is mainly induced by the shaft system of the turbine. (3) Pressure pulsation of two times rotating frequency The pressure pulsation frequency f ¼ 3.73 Hz is approximately two times of the rotating frequency. Its energy spectrum is the highest at the inlet of the draft tube and decreases in both downstream and upstream directions. It is seen from the pressure distribution on the inlet of the draft tube in Figure 11 that this pressure pulsation is mainly induced by the rotations of the two low pressure areas accompany the rotation of the runner.


(4) Pressure pulsation of four times rotating frequency The pressure pulsation frequency f ¼ 7.41 Hz is approximately four times of the rotating frequency. Its energy spectrum is also the highest at the inlet of the draft tube and decreases in both downstream and upstream directions. It is seen from the pressure distribution on the inlet of the draft tube in Figure 11 that this pressure pulsation is mainly induced by the rotations of the two low pressure areas and the two sub-low pressure areas accompany the rotation of the runner. (5) Pressure pulsations of multi-times rotating frequency in runner and draft tube region The pressure pulsation frequency f ¼ 10.89 Hz is approximately six times of the rotating frequency. Its energy spectrum is the highest at 0.3D1 under the runner and decreases in both downstream and upstream directions. When water goes into the runner, pressure pulsations of the frequencies of blade number times rotating frequency can be induced due to the pressure applied on the water by the blades. Empirical equation gives the pressure pulsation frequencies behind the runner fb ¼ Zrn/60 ¼ Zr fr, where Zr is the number of runner blades, fr is the rotating frequency. It is seen from the pressure distribution on the cross section 0.3D1 under the runner, Figure 12, that the six times rotating frequency pressure pulsation is induced mainly by the rotation of the six blades in the runner which affect the flow field. (6) Pressure pulsation of multi-times of rotating frequency in guide vane area Under the optimum operation condition, water goes into the guide vane cascade without colliding with the guide vanes. However, when the turbine unit is

Figure 11. Pressure distributions on the inlet cross section of the draft tube at different time

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Figure 12. Pressure distributions on the cross section 0.3D1 under the runner at different time

operating under low load conditions, due to vortices induced by flow separations at the outlets of the guide vanes, pressure distributions between the guide vanes at the outlet become non-uniform. The non-uniform water pressures are not worn out before water goes into the runner, which leads to the inlet of the runner blades being affected by pressure pulsations of frequency fg ¼ Zgn/60 ¼ Zgfr, where is Zg the guide vane number. For the Kaplan turbine under study in the present paper fg ¼ 24 1.783 ¼ 42.79 Hz. 5. Conclusions The robustness of the numerical method used in the unsteady turbulent flow pressure pulsation calculations has been verified by model experiments. Based on the verification, the pressure pulsations in a prototype Kaplan turbine have been predicted by the numerical method. It is found that the model turbine and prototype turbine have pressure pulsation frequencies of the same times rotating frequencies and the same transmission patterns under similar operation conditions. However, there is no similarity for the amplitude of the pressure pulsation between the model turbine and the prototype turbine. Therefore pressure pulsations in a prototype turbine cannot be obtained by scaling the experimental results of the model turbine using a similarity relationship. The time domain amplitude of pressure pulsation in a Kaplan turbine is maximum at the inlet cross section of the draft tube. Under the same water head, accompany the increasing of the output, the angle of the runner blades and the openings of the guide vanes are increasing, the time domain amplitude of the pressure pulsation at the inlet of the draft tube tends to increase. References Chen, X., Song, C.C.S., et al. (2000), ‘‘Simulation of pressure fluctuations in pump-turbines induced by runner-guide vane interactions’’, Proceedings of the 20th IAHR Symposium on Hydraulic Machinery and Systems, Charlotte, NC, 6-9 August, No. CFD-S04. Gehrer, A., Schmidl, R. and Sadnik, D. (2006), ‘‘Kaplan turbine runner optimization by numerical flow simulation (CFD) and an evolutionary algorithm’’, Proceedings of the 23rdIAHR Symposium on Hydraulic Machinery and Systems, Yokohama, l7-21 October, No. 125.

Jaeger, E.U. and Seidel U. (1999), ‘‘Pressure fluctuations in Francis turbines’’, Voith Hydro Internal Report, ID. Lindsjo¨, H., Lo¨rstad, D. and Fuchs, L. (2004), ‘‘Modelling of the unsteady transport of bubbles past a Kaplan turbine model’’, Proceedings of the 22nd IAHR Symposium on Hydraulic Machinery and Systems, Stockholm, 29 June-2 July, No. B3(1). Muntean, S., Balint, D. and Susan-Resiga, R. (2004), ‘‘3D flow analysis in the spiral case and distributor of a Kaplan turbine’’, Proceedings of the 22nd IAHR Symposium on Hydraulic Machinery and Systems, Stockholm, June 29-2 July, No. A10 (2). Nilsson, H. and Davidson, L. (2000), ‘‘A numerical comparison of operation in a Kaplan water turbine, focusing on tip clearance flow’’, Proceedings of the 20th IAHR Symposium on Hydraulic Machinery and Systems, Charlotte, 6-9 August, No. CFD-F12. Nilsson, H. and Davidson, L. (2002), ‘‘Validations and investigations of the computed flow in the GAMM Francis runner and the Ho¨lleforsen Kaplan runner’’, Proceedings of the XXIst IAHR Symposium on Hydraulic Machinery and Systems, Lausanne, 9-12 September, No. A-35. Ruprecht, A., Heitele, M. and Helmrich, T. (2000), ‘‘Numerical Simulation of a complete francis turbine including unsteady rotor/stator interactions’’, Proceedings of the 20th IAHR Symposium on Hydraulic Machinery and Systems, Charlotte, 6-9 August, No. CFD-S03. Speziale, C.G. and Thangam, S. (1992), ‘‘Analysis of a RNG based turbulence model for separated flows’’, International Journal of Engineering Science, Vol. 30 No. 11, pp. 1379-88. Tomas, L., Traversaz, M. and Sabourin, M. (2004), ‘‘An approach for Kaplan turbine design, hydraulic machinery and systems’’, Proceedings of the 22nd IAHR Symposium on Hydraulic Machinery and Systems, Stockholm, 29 June-2 July, No. A2-(1). Vu, T.C. and Nennemann, B. (2006), ‘‘Modern trend of CFD application for hydraulic design procedure’’, Proceedings of the 23rd IAHR Symposium on Hydraulic Machinery and Systems, Yokohama, l7-21 October, No. 169. Wang, Z.W. and Zhou, L.J. (2006), ‘‘Simulations and measurements of pressure oscillations caused by vortex ropes’’, Journal of Fluids Engineering, Transactions of the ASME, Vol. 128 No. 4, pp. 649-55. Yang, J.M., Wu, Y.L. and Cao, S.L. (1998), ‘‘Large eddy simulation of high Reynolds number flow in fluid machinery’’, Journal of Engineering Thermophysics, Vol. 19 No. 2, pp. 184-8 (in Chinese). Zhang, R.K. et al. (2005), ‘‘The physical origin of severe low-frequency pressure fluctuation in giant Francis turbines’’, Modern Physics Letters B, Vol. 19 Nos 28/29, pp. 1527-30. Further reading CENELEC. (1999), ‘‘Hydraulic turbines, storage pumps and pump-turbines-model acceptance tests’’, IEC 60193:1999, EN 60193-1999. Qian, Z.D., Yang, J.D. and Wen-xin Huai, W.X. (2007), ‘‘Numerical simulation and analysis of pressure pulsation in Francis hydraulic turbine with air admission’’, Journal of Hydrodynamics, Series B, Vol. 19 No. 4, pp. 467-72. Corresponding author Jianqiang Mai can be contacted at: [email protected]

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