Numerical modal analysis of a Kaplan turbine runner

Proceedings of the 1st IAHR Young Professionals Congress 17-18 November 2020 ©YEAR, IAHR. Used with permission / ISSN (Online) - ISSN (Print) - ISSN (USB) NUMERICAL MODAL ANALYSIS OF A KAPLAN TURBINE RUNNER RAFEL ROIG, OSCAR DE LA TORRE, ESTEVE JOU, XAVIER ESCALER Universitat Politècnica de Catalunya, Barcelona, Spain e-mail rafel.ro[email protected] ABSTRACT The aim of the research has been to investigate the dynamic behavior of a Kaplan turbine runner when submerged in still water. More specifically, the effects of the added mass on the modes of vibration have been quantified. For that, the modes of vibration of the runner in vacuum, in air and in water have been simulated with a coupled Acoustic-Structural modal analysis and from their comparison the reduction of the natural frequencies and the possible changes of the mode shapes of the runner have been obtained. The results show that the typically assumed invariance of all the mode shapes of a structure when submerged in a fluid is not fully accomplished in our case. And regards to the frequency reduction ratios, they are similar for most of the first modes with an average value of about 37%. Keywords: Kaplan runner, modal analysis, fluid structure interaction, frequency reduction ratio, mode shape. 1 METHOD The numerical model is based on a coupled Acoustic-Structural Finite Element Model (FEM), where the fluid is modeled using acoustic elements and the runner using structural elements. The acoustic elements are based on the Helmholtz equation so that a potential flow modeling approach is used. Consequently, the added mass due to viscous effect is not taken into account. Other considerations are that despite of the fact that the flow velocity affects the fluid-added stiffness value in a way that it increases for higher flow velocities, it can be considered negligible in the case studied here and that the mode shape can be assumed not to be affected by the additional stiffness introduced by the flow (J.P. Gauthier et al., 2017). So, taking into account the above considerations, it can be concluded that an Acoustic-Structural approach is valid to calculate natural frequencies and mode shapes of the Kaplan turbine runner. The frequency reduction radio (FRR) between the natural frequencies of the structure in vacuum and in water has been computed using the following formula: ( ) Where Wv is the natural frequency of the structure in vacuum and Ww in water. Once the mode shapes have been computed, the Modal Assurance Criterion (MAC) has been used to compare the modal vectors between the runner in vacuum, in air and in water. The MAC is calculated as the normalized scalar product of two sets of vectors {Vr} and {Vq} (M. Pastor et al., 2012). ( ) |{ } { }| ({ } { })({ } { }) Where {Vr} is the mode shape of the structure in vacuum and {Vq} is the mode shape of the structure in air or water. The MAC can take a value between 0 (representing no consistent correspondence) and 1 (representing perfect correspondence). Values larger than 0.9 indicate a consistent correspondence whereas small values indicate a very poor resemblance of the two shapes. 2 RESULTS 2.1 Frequency Reduction Ratio In table 1, the frequencies of the first modes in vacuum, in air and in water are indicated. It can be observed that the natural frequencies decrease when the density of the fluid increases. Because of the fact that the added mass effect of the air is almost negligible, it has not been computed. Proceedings of the 1st IAHR Young Professionals Congress 17-18 November 2020 Table 1. Natural Frequencies of the runner in vacuum, air and water; and the frequency reduction ratio (vacuum-water). MODE VACUUM (HZ) AIR (HZ) WATER (HZ) FREQUENCY REDUCTION RATIO (VACUUM-WATER) (%) M1 1081 1081 736 32 M2 1257 1257 768 39 M3 1262 1261 775 39 M4 1269 1268 779 39 M5 1280 1279 833 35 M6 1285 1284 836 35 2.2 Mode Shapes Despite of the common assumption that the mode shapes are invariant between structures in vacuum and water, it seems that this correlation between mode shapes is lost for complex structures which are submerged in a dense fluid (with lower correspondence for higher mode shapes). As an example, Figure 1 shows M5 of the Kaplan runner in water. Figure 1. Isometric view (left) and top view (right) of the M5 of the Kaplan runner in water. The wireframe shows undeformed geometry. Table 2 indicates the MAC values for the first vibration modes and it can be observed that they are exactly 1 between vacuum and air as expected, but they are significantly lower between vacuum and water especially for M6. Table 2. MAC values for mode shape: vacuum-water and vacuum-air. 3 CONCLUSIONS It can be concluded that a coupled Acoustic-Structural FEM model is a valid method to estimate the natural frequencies and mode shapes of complex submerged structures such as Kaplan turbine runners. It has been found that the FRR are around 37% for the first modes and that the mode shapes when submerged in water present differences with the ones in vacuum. REFERENCES Gauthier, J.P., Giroux, A.M., Etienne, S., Gosselin, F.P., 2017. A numerical method for the determination of flow-induced damping in hydroelectric turbines. J. Fluids Struct, 69 (2017), 341-354. Pastor, M., Binda, M., Harcarik, T., 2012. Modal Assurance Criterion. Procedia Engineering, 48 (2012), 543- 548. water M1 M2 M3 M4 M5 M6 vacuum M1 0,8 0,7 0,7 0,6 0,7 0,6 M2 0,5 0,9 0,4 0,2 0,3 0,2 M3 0,9 0,8 0,9 0,3 0,7 0,3 M4 0,5 0,4 0,6 0,8 0,6 0,3 M5 0,8 0,5 0,9 0,4 0,8 0,3 M6 0,3 0,4 0,3 0,7 0,3 0,4 air M1 M2 M3 M4 M5 M6 vacuum M1 1 0,5 0,8 0,7 0,7 0,6 M2 0,5 1 0,6 0,2 0,3 0,3 M3 0,8 0,6 1 0,6 0,8 0,3 M4 0,7 0,2 0,6 1 0,7 0,5 M5 0,7 0,3 0,7 0,7 1 0,2 M6 0,6 0,4 0,4 0,5 0,2 1