The effect of runner cone design on pressure oscillation characteristics in a Francis hydraulic turbine

Transcription

1 137 The effect of runner cone design on pressure oscillation characteristics in a Francis hydraulic turbine Z D Qian 1 *,WLi 1, W X Huai 1, and YLWu 2 1 State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan, People s Republic of China 2 Thermal Engineering Department, Tsinghua University, Beijing, People s Republic of China The manuscript was received on 4 May 2011 and was accepted after revision for publication on 18 August DOI: / Abstract: Three-dimensional unsteady turbulent flow through the entire flow passage of a model Francis hydraulic turbine is simulated using the transition shear-stress transport turbulence model and verified with experimental data. Pressure oscillations with the original runner cone, an extended runner cone, an extended runner cone with grooves, and a round-top runner cone are analysed under five operating conditions. The computational results show that (1) runner cone design can change the distribution pattern of the vortex rope in the draft tube and narrow the zone of special pressure oscillation; (2) the dominant frequency of pressure oscillation in the draft tube at guide vane opening angle, ¼ 17, is significantly decreased by runner cone design while the peak-to-peak amplitude increased; (3) varying guide vane opening alters the influence of runner cone design on pressure oscillation; (4) the modified runner cones are effective for low frequency pressure oscillations induced by the vortex rope in the draft tube but have little effect on blade frequency pressure oscillation induced by rotor stator interaction; (5) the cavitation performance at ¼ 17 is decreased while that at ¼ 18,19, and 20 is increased by runner cone design; and (6) the round-top runner cone is most effective in decreasing pressure oscillations and increasing turbine cavitation performance. Keywords: Francis hydraulic turbine, pressure oscillation, runner cones, numerical simulation 1 INTRODUCTION Large Francis-type hydraulic turbines are widely used to cope up with the variable consumer power demand from hydroelectric sources. The Francis turbine is chosen for such applications because the design can operate over an extended range of flow rates and is not limited to a narrow optimal operating point [1]. To improve the stable range of operation, much research has been undertaken on the complex nature of turbine pressure oscillation. The Karman vortex near the guide vanes and the runner trailing edges, inter-blade vortices in the turbine runner, and *Corresponding author: State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan , People s Republic of China. zdqian@whu.edu.cn the vortex rope in the draft tube are recognized as major contributors to pressure oscillations [2 4]. The presence of these vortices can lead to variation in power output, shaft vibration, and ultimately to runner blade damage. Some vibrations are hydraulic in origin; low frequency pressure oscillation can be induced by the formation of a helical vortex rope a little downstream of the turbine runner and by extending it to the draft tube [5 7]. Several practical solutions have been proposed to eliminate or mitigate these low frequency pressure oscillations. Papillon et al. [8] and Qian et al. [9] adopted air injection to reduce the pressure difference in the vortex rope. Susan-Resiga et al. [10] suggested a water jet projected from the crown tip to mitigate the vortex rope. Wei et al. [11] conducted model tests with four different runner cone geometries using the original runner cone as a control, and examining the effect

2 138 Z D Qian, W Li, W X Huai, and Y L Wu of a short runner cone, an extended runner cone, and a runner cone with deflectors, analysing the frequency and amplitude of pressure oscillations in the draft tube. Liu et al. [12] simulated unsteady flow with an extended runner cone by computational fluid dynamics. Vevke [13] proposed a semitapered cone to suppress the formation and development of the vortex rope and measured the velocity in the draft tube by laser Doppler velocimetry. Qian [14] discussed the variation of the zone of special pressure oscillation using runner cones with different diameters, lengths, and air injection positions. However, the shape of the runner cone has not still been considered. In this article, three-dimensional (3D) unsteady flow through the entire flow passage of a model Francis turbine is simulated using the transition shear-stress transport (SST) turbulence model. Four different runner cone designs are analysed, by using the original runner cone as a control, and comparing the performance with an extended runner cone, an extended runner cone with grooves, and a round-top runner cone, under five different operating conditions. Pressure oscillations in the draft tube, near the guide vanes, and in the spiral case are calculated and analysed using fast Fourier transform (FFT) methods. 2 NUMERICAL SIMULATION METHOD 2.1 Mathematical model The Reynolds-averaged Navier Stokes (RANS) equations for incompressible flow are ð iþ ¼ 0 ðu i Þ ðu iu j Þ þ i i where the Reynolds stress is ij ¼ u 0 i u0 j. The transition SST turbulence model is used. In this turbulence model, two more transport equations, one for intermittency and another for the transition momentum thickness Reynolds number Ret g, are added to the governing equations of the SST k! model. Menter et al. [15, 16] and Langtry et al. [17] provided detailed descriptions of the mathematical formulations of the transition SST model and the application of the technique to turbo machinery; this model is widely used to investigate the working of a hydraulic machinery. The four governing equations of the transition SST turbulence model are defined as follows. The equation for the intermittency ð j ¼ P 1 E 1 þ P 2 E þ ð3þ t The equation for transition momentum thickness Reynolds number Re g t Re g U jret g " þ ¼ P t t ð þ t g # Re The equations for the turbulence kinetic energy k and the specific dissipation u j k ¼ Pk f f k ð þ k t j ð! P k u j! ¼ D! þ Cd! v t ð þ k t ð6þ The flow passage simulated in this article includes the spiral case, stay and guide vanes, runner, and draft tube. There are two rotor stator interactions, between the guide vanes and the runner, and the runner and the draft tube. The sliding mesh model is used to simulate these rotor stator interactions [18, 19]. The finite volume method is used to discretize the governing equations and a second-order full implicit scheme is adopted for time discretization; a second-order central-differencing scheme is used for the diffusion term and a second-order upwind scheme for the other terms. Pressure Implicit with Splitting of Operators, based on the higher degree of the approximate relation between the corrections for pressure and velocity, is applied to achieve pressure velocity coupling [19 21]. 2.2 Physical model The complete flow passage of the model turbine for this simulation is shown in Fig. 1. The hydraulic head of the model turbine is H ¼ 30 m. The turbine comprises a spiral case, a runner with 15 blades, 24 stay vanes, 24 guide vanes, and an elbow draft tube. The operating conditions for the experiments and simulations are guide vane opening values () of 16, 17, 18, 19, and 20. The runner rotation speed n ¼ 1125 rev/min and thus the rotation frequency, f n ¼ Hz. In this simulation, four different runner cones (Fig. 2) are used to analyse the influence of the runner cone design on pressure oscillation. The runner cones used

3 The effect of runner cone design on pressure oscillation characteristics in a Francis hydraulic turbine 139 Fig. 1 Three-dimensional perspective view of the model turbine are (a) the original runner cone, (b) an extended runner cone the bottom of the original runner cone is extended to the outlet of runner by the addition of a cylinder, (c) a grooved extended runner cone grooves and changes in section are added to the cylinder of the extended runner cone, and (d) a round-top runner cone with a streamlined pyramid added at the bottom of the original runner cone. All three modified runner cones are of the same length. Unstructured meshes were used for the spiral case, turbine runner, and draft tube, and structured ones for the stay and guide vanes. About cells formed the spiral case, grid cells were used to model the stay and guide vane, grid cells formed the runner, and grid cells represented the draft tube. The total number of mesh cells was thus about The y þ value in the walls was approximately 1 to capture the laminar and transitional boundary layers correctly. (a) (b) (c) (d) Fig. 2 Runner cone designs

4 140 Z D Qian, W Li, W X Huai, and Y L Wu of the runner rotational period. The total computation time was s or 20 rotational periods. 2.3 Recording points and sections Fig. 3 Positions of pressure observations Thirty points located in the spiral case, near the guide vanes, and in the draft tube (Fig. 3), were chosen to record the time-varying pressures. Five points were placed on the surfaces S1 (S11 S15) and S2 (S21 S25), two on S3 (S31 and S32), four on S4 (S41 S44), and six on S5 (S51 S56). Three sections S0, S1, and S2 (Fig. 3) were chosen to analyse the distribution of pressure. Section S0 lies below the runner cone and in the runner. The distances between S0 and the bottom of the runner cones were the same for all runner cones. Sections S1 and S2 were in the draft tube and their positions were the same for all cases. Experimental measurements in the model turbine with the original runner cone installed in the test rig were conducted to verify the mathematical model. The operating conditions were guide vane opening ¼ 16,17,18,19, and 20. Three pressure transducers ROSEMOUNT AP6 E22 were located in the spiral case (P101), near the guide vanes (P201), and in the draft tube (S31) (Fig. 3), and an HP 3566A PC Spectrum/Network Analyser system was used to record pressure oscillations for different guide vane openings. 3 COMPARISON OF EXPERIMENTAL AND COMPUTATIONAL RESULTS Fig. 4 Time-domain graph of pressure oscillation at S12 at ¼ 18 The mesh sensitivity was also analysed with three different mesh systems. The mesh system with about grid cells was chosen in order to reduce computer cost. The time step for the unsteady simulation was s, which is related to the turbine runner rotational speed and corresponded to 1/80th The unsteady flows in the model turbines with the original runner cone and three modified runner cones were calculated under five operating conditions. Time-domain graphs of pressure oscillations at all points were obtained and the dominant frequencies and the amplitudes of the pressure oscillations calculated using FFT methods. A graph giving the variation of static pressure with time of point S31 at ¼ 18 with the original runner cone is shown in Fig. 4. After about 0.21 s or four rotational periods, the variation of static pressure tended to be periodic. The data analysed in this article were sampled after regular pressure oscillation began. A comparison between experimental and computational results of the dominant frequencies ( f ) and the peak-to-peak amplitudes (H/H ) at P101, P201, and S31 at ¼ 18 with the original runner cone is listed in Table 1. As shown in Table 1, low frequency pressure oscillations appeared in the draft tube and the spiral case and the blade frequency oscillations near the guide vanes. The low frequency pressure oscillation is

5 The effect of runner cone design on pressure oscillation characteristics in a Francis hydraulic turbine 141 Table 1 Comparison of experimental and computational dominant frequencies and amplitudes ¼ 18 Computational Experimental Recording points f (Hz) H/H (%) f (Hz) H/H (%) P P S PRESSURE OSCILLATION AND UNSTEADY FLOW CHARACTERISTICS The variation of the dominant frequencies and the peak-to-peak amplitudes of pressure oscillations at S31 at ¼ 16,17,18,19, and 20 with the original runner cone are shown in Fig. 5. The calculated results show that the operating point at ¼ 16 is located in the zone of special pressure oscillation while the operating points at ¼ 18,19, and 20 lie outside. The operating point at ¼ 17 is in the transient zone. The biggest peak-to-peak amplitude is at ¼ 18, decreasing with ¼ 19,17,20, and 16. The distributions of vortex rope behaviour at ¼ 17 and 18 with original runner cone at t ¼ s are shown in Fig. 6. The vortex ropes are strong helical swirls which form just downstream of the turbine runner and extend into the draft tube. Two interlacing vortex ropes are found at ¼ 17, and so the dominant frequency is higher than that at ¼ 18,19, and 20, as shown in Fig Pressure oscillation in the draft tube Fig. 5 Dominant frequency and peak-to-peak amplitude values at S31 induced by the vortex rope and transferred upstream [19, 22, 23], while the blade frequency pressure oscillation is induced by the rotor stator interaction between runner blades and guide vanes [24 26]. The computational dominant frequencies of the three points agree well with the experimental ones and the computational peak-to-peak amplitudes at P201 and S31 are also acceptable. Because the upstream pressure pipe was not included in this simulation (to reduce computer costs), the computational peak-to-peak amplitude at P101 is less than the experimental one. The helical vortex rope is recognized as the main contributor of low frequency pressure oscillations in the draft tube. The dominant frequency depends on the shape of the turbine runner, the runner frequency, and the degree of guide vane opening [18, 27, 28]. In Tables 2 to 4, the dominant frequencies and the peak-to-peak amplitudes of the pressure oscillations at S11, S21, and S31 with four runner cone designs at ¼ 16,17,18,19, and 20 are listed. Cases 1 to 4 are used to represent the four turbines with the original runner cone, extended runner cone, extended runner cone with grooves, and round-top runner cone, respectively. The dominant frequencies of pressure oscillations at S11 in the draft tube at ¼ 16,17,18, and 19 at case 1 range from 0.279f n to 0.513f n (f n ¼ Hz), while the dominant frequency at ¼ 20 is the fundamental frequency as shown in Table 2. In all three modified runner cones, the dominant frequencies at ¼ 16,18,19, and 20 remain the same as those in case 1. However, the dominant frequencies at ¼ 17 are decreased in cases 2 to 4. The peak-to-peak amplitudes at ¼ 18 and 19 are decreased in cases 2 to 4, with a minimum in case 4. At ¼ 16 and 20, the peak-to-peak amplitudes are decreased in case 4 and increased in cases 2 and 3. The peak-to-peak amplitudes at ¼ 17 are increased in cases 2 to 4, and that in case 4 is the minimum. The dominant frequencies of pressure oscillations at S21 at ¼ 16,17,18,19, and 20 in case 1 shown in Table 3 are the same as those at S11. The variations

6 142 Z D Qian, W Li, W X Huai, and Y L Wu Table 2 Dominant frequency and peak-to-peak amplitude values at S11 Guide vane opening Types of runner cone Case 1 Case 2 Case 3 Case 4 ¼ 16 f (Hz) H/H (%) ¼ 17 f (Hz) H/H (%) ¼ 18 f (Hz) H/H (%) ¼ 19 f (Hz) H/H (%) ¼ 20 f (Hz) H/H (%) Table 3 Dominant frequency and peak-to-peak amplitude values at S21 Guide vane opening Types of runner cone Case 1 Case 2 Case 3 Case 4 ¼ 16 f (Hz) H/H (%) ¼ 17 f (Hz) H/H (%) ¼ 18 f (Hz) H/H (%) ¼ 19 f (Hz) H/H (%) ¼ 20 f (Hz) H/H (%) Table 4 Dominant frequency and peak-to-peak amplitude values at S31 Guide vane opening Types of runner cone Case 1 Case 2 Case 3 Case 4 ¼ 16 f (Hz) H/H (%) ¼ 17 f (Hz) H/H (%) ¼ 18 f (Hz) H/H (%) ¼ 19 f (Hz) H/H (%) ¼ 20 f (Hz) H/H (%) Fig. 6 Distributions of vortex rope behaviour at ¼ 17,18 and t ¼ s of the dominant frequencies and the peak-to-peak amplitudes at S21 are similar to those at S11 at ¼ 16 and 17 in cases 2 to 4. At ¼ 18 and 19, the variations of the dominant frequencies at S21 are the same as those at S11, while the dominant frequencies at S21 at ¼ 20 are increased in cases 2 to 4. The peak-to-peak amplitude at ¼ 18 is increased in case 3, and decreased in cases 2 and 4 (the minimum). The peak-to-peak amplitude at ¼ 19 is decreased in case 3 and increased in cases 2 and 4. The peak-topeak amplitude at ¼ 20 is decreased in cases 2 to 4, with a minimum in case 2. In Table 4, the variations of the dominant frequencies and the peak-to-peak amplitudes of pressure oscillations at S31 at ¼ 16 and 17 are similar to those at S11. The dominant frequencies at S31 with the four runner cones at ¼ 18 and 19 are the same as those at S11. The dominant frequency at ¼ 20 is increased from 1.7 to 5.11 Hz in case 1 and they

7 The effect of runner cone design on pressure oscillation characteristics in a Francis hydraulic turbine 143 Table 5 Dominant frequency and peak-to-peak amplitude values at P201 Guide vane opening Types of runner cone Case 1 Case 2 Case 3 Case 4 ¼ 16 f (Hz) H/H (%) ¼ 17 f (Hz) H/H (%) ¼ 18 f (Hz) H/H (%) ¼ 19 f (Hz) H/H (%) ¼ 20 f (Hz) H/H (%) Table 6 Dominant frequency and peak-to-peak amplitude values at P101 Guide vane opening Types of runner cone Case 1 Case 2 Case 3 Case 4 ¼ 16 f (Hz) H/H (%) ¼ 17 f (Hz) H/H (%) ¼ 18 f (Hz) H/H (%) ¼ 19 f (Hz) H/H (%) ¼ 20 f (Hz) H/H (%) remain unchanged in cases 3 and 4, while increasing to Hz in case 2. The peak-to-peak amplitude at ¼ 18 is decreased in case 2 and increased in cases 3 and 4. The peak-to-peak amplitude at ¼ 19 is increased in cases 2 to 4. At ¼ 20, the peak-topeak amplitude is decreased in cases 2 and 3 and slightly increased in case 4. From the above analysis of the computational results in Tables 2 to 4, it can be concluded that with varying guide vane openings, the influence of runner cones on low frequency pressure oscillation also varies. With the three modified runner cones, the dominant frequencies are changed in the same way. The dominant frequency can be decreased from 0.404f n to 0.243f n at ¼ 17 and the zone of special pressure oscillation becomes narrow with three modified runner cones. The round-top runner cone is the most effective among the three modified runner cones in decreasing the peak-to-peak amplitude. 4.2 Pressure oscillations in the guide vanes and spiral case The dominant frequencies and amplitudes of the pressure oscillations at P201 near the guide vanes and those at P101 in the spiral case at ¼ 16,17, 18,19, and 20 are listed in Tables 5 and 6. The dominant frequencies of pressure oscillations near the guide vanes are blade frequencies induced by the rotor stator interaction between runner blades and guide vanes. The variation of the peak-to-peak amplitudes is not significant with any of the three modified runner cone designs. The results indicate that the modified runner cones are effective for low frequency pressure oscillations induced by the vortex rope in the draft tube, but not effective for the blade frequency pressure oscillation. The upstream pressure pipe is not included in this simulation, and so the simulated peak-to-peak amplitudes at P101 are very small as shown in Table 6. The variations of the dominant frequencies and the peak-to-peak amplitudes in the spiral case at ¼ 16,17,18,19, and 20 are similar to those in the draft tube. These results provide further evidence that the low frequency pressure oscillations are induced by the vortex rope in the draft tube transfers upstream. 4.3 Analysis of pressure oscillation The three modified runner cones are very effective in changing the dominant frequency and peak-to-peak amplitude of pressure oscillation in the draft tube at ¼ 17 and also effective for the peak-to-peak amplitude in the draft tube at ¼ 18,19, and 20. The vortex rope behaviour and pressure in sections S0, S1, and S2 are analysed as follows. The patterns of vortex ropes at t ¼ s, t ¼ s, and t ¼ s and the time-domain graphs of pressure oscillation at S21 with four runner cones at ¼ 17 are shown in Fig. 7. The distributions of pressure at t ¼ s in sections S0, S1, and S2 at ¼ 17 are sampled, as shown in Fig. 8. In Fig. 7, there are two interlacing vortex ropes in the draft tube with the original runner cone and the dominant frequency at ¼ 17 is 7.58 Hz or 0.404f n, which is higher than that at ¼ 18, 19, and 20. When using any of the three modified runner cones, only one helical vortex rope appears in the draft tube and the distribution pattern of the vortex rope is changed. Accordingly, the dominant frequency is decreased to 4.55 Hz or 0.243f n. The modified runner cones narrow the zone of special pressure oscillation. However, the modified runner cones do increase the length of the vortex rope and the pressure differences in sections S1 and S2 are also increased as shown in Fig. 8. This contributes to an increase in the peak-to-peak amplitude of pressure oscillation at ¼ 17. Comparing the pressure differences in sections S1 and S2 and minimum pressures in S0, the

8 144 Z D Qian, W Li, W X Huai, and Y L Wu Fig. 7 Pattern of vortex rope and time-domain graph of pressure oscillation at S21

9 The effect of runner cone design on pressure oscillation characteristics in a Francis hydraulic turbine 145 Fig. 7 Continued

10 146 Z D Qian, W Li, W X Huai, and Y L Wu (a) (b) (c) (d) Fig. 8 Distributions of pressure at ¼ 17 with four runner cones (a) (b) (c) (d) Fig. 9 Distributions of pressure at ¼ 18 with four runner cones

11 The effect of runner cone design on pressure oscillation characteristics in a Francis hydraulic turbine 147 (a) (b) (c) (d) Fig. 10 Distributions of pressure at ¼ 19 with four runner cone designs (a) (b) (c) (d) Fig. 11 Distributions of pressure at ¼ 20 with four runner cone designs

12 148 Z D Qian, W Li, W X Huai, and Y L Wu pressure difference with the round-top runner cone is the lowest and the minimum pressure the highest. Therefore, in the three modified runner cones, the round-top design is the most effective, in terms of the peak-to-peak amplitude of pressure oscillation in the draft tube and the cavitation performance of the turbine. The distributions of pressure at t ¼ s in sections S0, S1, and S2 with four runner cones at ¼ 18, 19, and 20 are shown in Figs 9 to 11, respectively. The minimum pressures in section S0 increased in all three modified runner cones at ¼ 18, as shown in Fig. 9. In sections S1 and S2, the vortex cores with three modified runner cones are nearer to the centre of the tube. The pressure differences in S1 and S2 slightly increase in cases 2 and 3 and remain unchanged in case 4. In Fig. 10, the minimum pressures in section S0 in the three modified runner cone designs are increased at ¼ 19, reaching a peak in case 3. The vortex cores are similar to those in Fig. 11(a), being nearer to the centre of the tube with the three modified runner cone designs. The pressure differences in sections S1 and S2 are slightly increased in case 2 and decreased in case 4. The pressure difference is decreased in section S1 and remains unchanged in section S2 in case 3. The minimum pressures in section S0 are also increased with the three modified runner cone designs at ¼ 20, as shown in Fig. 11, case 4 being optimal. The pressure difference is slightly increased in section S1 and decreased in S2 in cases 2 and 4. The pressure differences are slightly increased in sections S1 and S2 in case 3. From the analysis of the computational results of pressure in sections S0, S1, and S2 at ¼ 18,19, and 20, it can be concluded that three modified runner cone designs do change the pattern of the vortex rope and consequently influence pressure oscillation. They improve the stability of the Francis turbine by decreasing the amplitude of pressure oscillation. The low-pressure zone below the turbine runner cone is one of the zones where cavitation usually appears [7, 29]. The three runner cone designs improved the cavitation performance of the turbine by increasing the minimum pressure in section S0. The round-top runner cone has advantages over the others in increasing turbine stability and cavitation performance. 5 CONCLUSIONS The comparison of the computational and experimental results has shown that the RANS equations with the transition SST turbulence model can predict pressure oscillations and unsteady flow behaviour inside the turbine reasonably well. Pressure oscillation, unsteady flow behaviour inside the turbine, and the distribution of pressure in sections with four runner cones are analysed. The results are summarized as follows. 1. All three modified runner cone designs can change the distribution patterns of the vortex rope in the draft tube and narrow the zone of special pressure oscillation. The dominant frequency of pressure oscillation in the draft tube at ¼ 17 is significantly decreased, while the peak-to-peak amplitude increased with the three modified runner cones. 2. The modified runner cone designs are effective for low frequency pressure oscillations induced by the vortex rope in the draft tube, but are not effective for the blade frequency pressure oscillations induced by the rotor stator interaction. If the degree of guide vane opening is varied, the influence of the runner cones on pressure oscillation is also changed. 3. The three modified runner cone designs decreased the minimum pressure in section S0 at ¼ 17 but worsened the cavitation performance, while improving the cavitation performance at ¼ 18, 19, and The round-top runner cone design is the most effective one in decreasing the pressure oscillation and increasing the cavitation performance of the Francis turbine. ACKNOWLEDGEMENTS This investigation is funded by the National Nature Science Foundation of China (grant no ) and the Fundamental Research Funds for the Central Universities (grant no ). ß Authors 2011 REFERENCES 1 Baya, A., Muntean, S., Câmpian, V. C., Cuzmoş, A., Diaconescu, M., and Bălan, G. Experimental investigations of the unsteady flow in a Francis turbine draft tube cone. In Proceedings of the 25th IAHR Symposium on Hydraulic Machinery and Systems, Timisoara, Romania, September 2010, pp Skotak, A. Of the helical vortex in the turbine draft tube modeling. In Proceedings of the 20th IAHR Symposium on Hydraulic Machinery and Systems, Charlotte, North Carolina, USA, 7 9 August 2000, pp Scherer, T., Faigle, P., and Aschenbrenner, T. Experimental analysis and numerical calculation of

13 The effect of runner cone design on pressure oscillation characteristics in a Francis hydraulic turbine 149 the rotating vortex rope in a draft tube operating at part load. In Proceedings of the 21st IAHR Symposium on Hydraulic Machinery and Systems, Lausanne, Switzerland, 9 12 September 2002, pp Susan-Resiga, R., Muntean, S., Hasmatuchi, V., Ruprecht, A., and Sandor, B. Development of a swirling flow control technique for Francis turbines operated at partial discharge. In Proceedings of the 3rd German-Romanian Workshop on Turbomachinery Hydrodynamics, Timisoara, Romania, May 2007, pp Susan-Resiga, R. and Ciocan, G. D. Analysis of the swirling flow downstream a Francis turbine runner. J. Fluids Eng., 2006, 128(1), Skotak, A. and Obrovsky, J. Low swirl flow separation in a Kaplan turbine draft tube. In Proceedings of the 2nd IAHR International Meeting of the Workgroup on Cavitation and Dynamic Problems in Hydraulic Machinery and Systems. Timisoara, Romania, October 2007, pp Iliescu, M. S., Ciocan, G. D., and Avellan, F. Analysis of the cavitating draft tube vortex in a Francis turbine using particle image velocimetry measurements in two-phase flow. J. Fluids Eng., 2008, 130(2), Papillon, B., Kirejczyk J., and Sabourin, M. Atmospheric air admission in hydro turbines. In Proceedings of the HydroVision, Charlotte, North Carolina, USA, 8 11 August 2000, paper no. 3C. 9 Qian, Z. D., Yang, J. D., and Huai, W. X. Numerical simulation and analysis of pressure pulsation in Francis hydraulic turbine with gas admission. J. Hydrodyn. Ser B, 2007, 19(4), Susan-Resiga, R., Vu, T. C., Muntean, S., Ciocan, G. D., and Nennemann, B. Jet control of the draft tube vortex rope in Francis turbines at partial discharge. In Proceedings of the 23rd IAHR Symposium on Hydraulic Machinery and Systems, Yokohama, Japan, October 2006, pp Wei, C. X., Han, F. Q., Chen, Z. S., and Xiong, Z. The characteristic test of runner hubs for hydraulic turbine. Large Electr. Mach. Hydraul. Turb., 2001, 1, (in Chinese). 12 Liu, S. H., Wu, X. J., and Wu, Y. L. Analysis of the influence of runner hub shape on the internal flow in Francis turbine. J. Hydroelectr. Eng., 2006, 25(1), (in Chinese). 13 Vevke, T. An experimental investigation of draft tube flow. PhD Thesis, NTNU 2004:36, Norwegian University of Science and Technology, Qian, M. Methods to reduce pressure pulsation in high load zone of Francis turbine models. Dongfang Electr. Rev., 2010, 95(24), (in Chinese). 15 Menter, F. R., Langtry, R. B., Likki, S. R., Suzen, Y. B., Huang, P. G., and Völker, S. A correlationbased transition model using local variables, part 1-model formulation. In Proceedings of the ASME turbo expo, Vienna, Austria, June 2004, paper no. GT (The American Society of Mechanical Engineers, New York). 16 Menter, F. R., Langtry, R. B., and Völker, S. Transition modelling for general purpose CFD codes. Flow Turbul. Combust., 2006, 77(1 4), Langtry, R. B., Menter, F. R., Likki, S. R., Suzen, Y. B., Huang, P. G., and Völker, S. A correlation-based transition model using local variables, part 2-test cases and industrial applications. In Proceedings of the ASME turbo expo, Vienna, Austria, June 2004, paper no. GT (The American Society of Mechanical Engineers, New York). 18 Hassan, O., Probert, E. J., Morgan, K., and Weatherill, N. P. Unsteady flow simulation using unstructured meshes. Comput. Meth. Appl. Mech. Eng., 2000, 189(4), Qian, Z. D., Zheng, B., Huai, W. X., and Lee, Y. H. Analysis of pressure oscillations in a Francis hydraulic turbine with misaligned guide vanes. Proc. IMechE, Part A: J. Power and Energy, 2010, 224(1), Farrant, T., Tan, M., and Price, W. G. Cell boundary element method applied to laminar vortex shedding from circular cylinders. Comput. Fluids, 2001, 30(2), Wang, T., Gu, C. G., Yang, B., and Huang, J. D. PISO algorithm for unsteady flow field. Chin. J. Hydrodyn., 2003, 18(2), (in Chinese). 22 Wang, X. M., Nishi, M., and Tsukamoto, H. A simple method for predicting the draft tube surge. In Proceedings of the 17th IAHR Symposium, Beijing, China, September 1994, pp Wang, X. M. and Nishi, M. Swirling flow with helical vortex core in a draft tube predicted by vortex method. In Proceedings of the 18th IAHR Symposium, Dordrecht, Netherlands, August 1996, pp Brekke, H. A review on oscillatory problems in Francis turbines and simulation of unsteady flow in conduit systems. In Proceedings of the 17th IAHR Symposium, Beijing, China, September 1994, pp Liu, S. H., Shao, Q., Yang, J. M., Wu, Y., and Dai, J. Unsteady turbulent simulation of three gorges hydraulic turbine and analysis of pressure in the whole passage. J. Hydroelectr. Eng., 2004, 23(5), (in Chinese). 26 Pulpitel, L., Vesely, J., and Mikulasek, J. Comments to vibrations and pressure oscillations induced by the rotor stator interaction in a hydraulic turbine. In Proceedings of the 3rd IAHR International Meeting of the Workgroup on Cavitation and Dynamic Problems in Hydraulic Machinery and Systems, Brno, Czech Republic, October 2009, pp Escaler, X., Egusquza, E., Farhat, M., Avellan, F., and Coussirat, M. Detection of cavitation in hydraulic turbines. Mech. Syst. Signal Process., 2006, 20(4), Tsurusaki, H. Unsteady swirling flows arising in straight tubes. In Proceedings of the 23rd IAHR Symposium, Yokohama, Japan, October 2006, pp. 1 8.

14 150 Z D Qian, W Li, W X Huai, and Y L Wu 29 Liu, S. H., Zhang, L., Nishi, M., and Wu, Y. Cavitating turbulent flow simulation in a Francis turbine based on mixture model. J. Fluids Eng., 2009, 131(5), 1 8. H n t operating head (m) runner rotational speed (r/min) flow time (s) APPENDIX Notation f f n frequency (Hz) runner rotational frequency (Hz) guide vane opening ( ) H amplitude (Pa) H/H peak-to-peak amplitude (%)