# Cavitation Effects on the Structural Resonance of Hydraulic Turbines: Failure Analysis in a Real Francis Turbine Runner

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## Abstract

**:**

## 1. Introduction

## 2. Finite Element Formulation for FSI

## 3. Description of Francis Turbine and Failure

#### 3.1. Unit Description and Damage Found

#### 3.2. Operating Conditions before and after the Damage

#### 3.3. Hydraulic Excitations

#### 3.3.1. Deep Part Load Flow and Inter Blade Vortices

#### 3.3.2. Vortex Rope

#### 3.3.3. Von Karman Vortices

#### 3.3.4. RSI

## 4. Flow Conditions through CFD Simulation

#### 4.1. Simulated Cavities at Deep Part Load

_{vap}= 50, condensation coefficient F

_{cond}= 0.01, bubble radius ${\vartheta}_{B}=2\times {10}^{-6}\mathrm{m}$, nucleation site volume fraction α

_{nuc}= 5 × 10

^{−4}. P is the local pressure, and P

_{v}the vapor pressure. ${\rho}_{l}$ is the density of liquid (water).

_{V}and ρ

_{L}have been taken as 1.74 × 10

^{−2}(saturated vapor at 20 °C) and 1000 kg/m

^{2}respectively. The void ratio ${\alpha}_{v}$ is a local variable that determines the density, sonic velocity, geometry and size of the simulated cavities. Therefore, a realistic selection of this value is critical, as it will have a great influence on the added mass. In previous works [21,47], this parameter adopts a wide range of values and thus, a unique specific value cannot be selected. Furthermore, the experimental conditions in those studies are quite different as in the present case. Instead of selecting one value, in this paper we carry out a sensitivity analysis varying this value in a range from 0.1–0.9 (with a step of 0.1). In this way, it can be seen how the simulated cavities characteristic change. Therefore, its influence on the added mass effect can be also determined as will be discussed in the following section.

_{v}= 0.1 and for 16 MW (deep part load condition in Table 1). In this situation the total volume V of the cavities is maximum. In every runner channel, a single inter blade vortex cavity is observed. This vortex detaches from the band, and extends towards the suction side of the blade close to the trailing edge.

_{v}was, but only small cavitation nuclei (typical travelling bubbles at high loads [53]). Therefore, for the FSI simulation of this particular operating condition (Section 5), water will be modelled as a homogeneous fluid around the turbine runner.

#### 4.2. Hydraulic Excitations from the CFD

## 5. Natural Frequencies and Mode Shapes of the Runner without and with Cavitation Effects

^{3}, Young Modulus of 201 GPa and Poisson Ratio of 0.3 has been adopted.

^{3}and speed of sound of 1450 m/s, outside of the possible cavitation volumes. The fluid mesh was built by extending the structure mesh so that the nodes on the interface are shared by both domains.

#### 5.1. Without Cavitation Effects (Conditions Close to BEP)

_{air}have been also obtained in a parallel simulation eliminating the surrounding flow. Then, the frequency reduction ratios (FRR), which are also shown in this table can be defined as:

#### 5.2. With Interblade Vortices (Deep Part Load Condition)

_{v}= 0.1.

_{v}values. Figure 7 shows the increase of the different natural with respect the natural frequencies in homogenous water $({f}_{cav}-{f}_{wat})/{f}_{wat}$, where ${f}_{cav}$ are the natural frequencies considering cavitation effects (depending on α

_{v}) and ${f}_{wat}$ are the natural frequency in homogenous water (runner working close to BEP). As in other studies [21], when cavitation exist, natural frequencies of the structure are substantially increased and this increase depends on the cavitation volumes and its properties. For the smallest α

_{v}considered, the volumes are the largest one and the increase in the natural frequencies can be 10–20%, depending on the mode shape considered. For the 4ND mode, the increase in frequency with respect the non-cavitating case is much higher as for the rest of the modes.

## 6. Discussion

_{v}considered and particularly for the values calibrated in by De la Torre et al. [21]. Therefore, to evaluate the risk of a structural resonance in a turbine runner, cavitation effects on the added mass, which depend on the operating condition, should be considered.

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Brennen, C. A Review of Added Mass and Fluid Inertial Forces; Brennen (C.E.): Sierra Madre, CA, USA, 1982. [Google Scholar]
- Blevins, R.D. Flow-Induced Vibration; Van Nostrand Reinhold Co.: New York, NY, USA, 1990. [Google Scholar]
- Lamb, H. On the vibrations of an elastic plate in contact with water. Proc. R. Soc. Lond.
**1921**, A98, 205–216. [Google Scholar] [CrossRef] - Amabili, M.; Kwak, M.K. Free vibrations of circular plates coupled with liquids: Revising the Lamb problem. J. Fluids Struct.
**1996**, 10, 743–761. [Google Scholar] [CrossRef] - Amabili, M. Effect of finite fluid depth on the hydroelastic vibration of circular and annular plate. J. Sound Vib.
**1996**, 193, 909–925. [Google Scholar] [CrossRef] - Kwak, M.K. Vibration of circular plates in contact with water. J. Appl. Mech.
**1991**, 58, 480–483. [Google Scholar] [CrossRef] - Rodriguez, C.G.; Flores, P.; Pierart, F.G.; Contzen, L.R.; Egusquiza, E. Capability of structural-acoustical FSI numerical model to predict natural frequencies of submerged structures with nearby rigid surfaces. Comput. Fluids
**2012**, 64, 117–126. [Google Scholar] [CrossRef] - Jeong, K.-H.; Kang, H.-S. Free vibration of multiple rectangular plates coupled with a liquid. Int. J. Mech. Sci.
**2013**, 74, 161–172. [Google Scholar] [CrossRef] - Valentín, D.; Presas, A.; Egusquiza, E.; Valero, C. Experimental study on the added mass and damping of a disk submerged in a partially fluid-filled tank with small radial confinement. J. Fluids Struct.
**2014**, 50, 1–17. [Google Scholar] [CrossRef] - Presas, A.; Valentin, D.; Egusquiza, E.; Valero, C.; Seidel, U. Influence of the rotation on the natural frequencies of a submerged-confined disk in water. J. Sound Vib.
**2015**, 337, 161–180. [Google Scholar] [CrossRef] - Valentín, D.; Presas, A.; Egusquiza, E.; Valero, C.; Egusquiza, M. Experimental Study of a Vibrating Disk Submerged in a Fluid-Filled Tank and Confined With a Nonrigid Cover. J. Vib. Acoust.
**2017**, 139, 021005–021011. [Google Scholar] [CrossRef] - Bossio, M.; Valentín, D.; Presas, A.; Martin, D.R.; Egusquiza, E.; Valero, C.; Egusquiza, M. Numerical study on the influence of acoustic natural frequencies on the dynamic behaviour of submerged and confined disk-like structures. J. Fluids Struct.
**2017**, 73, 53–69. [Google Scholar] [CrossRef] - Egusquiza, E.; Valero, C.; Huang, X.; Jou, E.; Guardo, A.; Rodriguez, C. Failure investigation of a large pump-turbine runner. Eng. Fail. Anal.
**2012**, 23, 27–34. [Google Scholar] [CrossRef] [Green Version] - Lais, S.; Qwenguei, L.; Henggeler, U.; Weiss, T.; Escaler Puigoriol, F.X.; Egusquiza Estévez, E. Dynamic analysis of Francis Runners-Experiment and numerical simulation. In Proceedings of the 24th IAHR Symposium on Hydraulic Machinery and Systems, Foz do Iguassu, Brazil, 27–31 October 2008. [Google Scholar]
- He, L.; Wang, Z.; Kurosawa, S.; Nakahara, Y. Resonance investigation of pump-turbine during startup process. In IOP Conference Series: Earth and Environmental Science; IOP Publishing: Bristol, UK, 2014; p. 032024. [Google Scholar]
- Xiao, R.; Wang, Z.; Luo, Y. Dynamic Stresses in a Francis Turbine Runner Based on Fluid-Structure Interaction Analysis. Tsinghua Sci. Technol.
**2008**, 13, 587–592. [Google Scholar] [CrossRef] - Liang, Q.W.; Rodríguez, C.G.; Egusquiza, E.; Escaler, X.; Farhat, M.; Avellan, F. Numerical simulation of fluid added mass effect on a francis turbine runner. Comput. Fluids
**2007**, 36, 1106–1118. [Google Scholar] [CrossRef] [Green Version] - Dompierre, F.; Sabourin, M. Determination of turbine runner dynamic behaviour under operating condition by a two-way staggered fluid-structureinteraction method. IOP Conf. Ser. Earth Environ. Sci.
**2010**, 12, 012085. [Google Scholar] [CrossRef] [Green Version] - Hübner, B.; Seidel, U.; Roth, S. Application of fluid-structure coupling to predict the dynamic behavior of turbine components. In IOP Conference Series: Earth and Environmental Science; IOP Publishing: Bristol, UK, 2010; p. 012009. [Google Scholar]
- Liu, X.; Luo, Y.; Karney, B.; Wang, Z.; Zhai, L. Virtual Testing for Modal and Damping Ratio Identification of Submerged Structures using the PolyMAX Algorithm with Two-way Fluid-Structure Interactions. J. Fluids Struct.
**2015**, 54, 548–565. [Google Scholar] [CrossRef] - De La Torre, O.; Escaler, X.; Egusquiza, E.; Farhat, M. Experimental investigation of added mass effects on a hydrofoil under cavitation conditions. J. Fluids Struct.
**2013**, 39, 173–187. [Google Scholar] [CrossRef] [Green Version] - Ducoin, A.; André Astolfi, J.; Sigrist, J.-F. An experimental analysis of fluid structure interaction on a flexible hydrofoil in various flow regimes including cavitating flow. Eur. J. Mech. B/Fluids
**2012**, 36, 63–74. [Google Scholar] [CrossRef] [Green Version] - Benaouicha, M.; Astolfi, J.-A. Analysis of added mass in cavitating flow. J. Fluids Struct.
**2012**, 31, 30–48. [Google Scholar] [CrossRef] [Green Version] - Lelong, A.; Guiffant, P.; Astolfi, J.A. An Experimental Analysis of the Structural Response of Flexible Lightweight Hydrofoils in Cavitating Flow. J. Fluids Eng.
**2018**, 140, 021116. [Google Scholar] [CrossRef] - ANSYS Inc. ANSYS Theory Reference; ANSYS Inc.: Canonsburg, PA, USA, 2013. [Google Scholar]
- Junger, M.C.; Feit, D. Sound, Structures, and Their Interaction; MIT Press: Cambridge, MA, USA, 1986; p. 460. [Google Scholar]
- Ewins, D.J. Modal Testing: Theory and Practice; Research Studies Press: Letchworth, UK, 1984; Volume 15. [Google Scholar]
- Wilhelm, W.; Florian von, L.; Philipp, C.; Jiri, K. Dynamic stresses in a Francis model turbine at deep part load. J. Phys. Conf. Ser.
**2017**, 813, 012014. [Google Scholar] [Green Version] - Mende, C.; Weber, W.; Seidel, U. Progress in load prediction for speed-no-load operation in Francis turbines. IOP Conf. Ser. Earth Environ. Sci.
**2016**, 49, 062017. [Google Scholar] [CrossRef] [Green Version] - Morissette, J.F.; Chamberland-Lauzon, J.; Nennemann, B.; Monette, C.; Giroux, A.M.; Coutu, A.; Nicolle, J. Stress predictions in a Francis turbine at no-load operating regime. IOP Conf. Ser. Earth Environ. Sci.
**2016**, 49, 072016. [Google Scholar] [CrossRef] [Green Version] - Sofien Bouajila, J.B. Emmanuel Flores, Claire Segoufin, Thierry Maitre. In Advances in Hydroinformatics, 1st ed.; Gourbesville, P., Cunge, J., Caignaert, G., Eds.; Springer: Signapore, 2018. [Google Scholar]
- Yamamoto, K.; Müller, A.; Favrel, A.; Landry, C.; Avellan, F. Guide vanes embedded visualization technique for investigating Francis runner inter-blade vortices at deep part load operation. In Proceedings of the 6th IAHR International Meeting of the Workgroup on Cavitation and Dynamic Problems in Hydraulic Machinery and Systems, Ljubljana, Slovania, 9–11 September 2015. [Google Scholar]
- Yamamoto, K.; Müller, A.; Favrel, A.; Landry, C.; Avellan, F. Pressure measurements and high speed visualizations of the cavitation phenomena at deep part load condition in a Francis turbine. IOP Conf. Series: Earth Environ. Sci.
**2014**, 22, 022011. [Google Scholar] [CrossRef] - Huang, X.; Oram, C.; Sick, M. Static and dynamic stress analyses of the prototype high head Francis runner based on site measurement. IOP Conf. Ser. Earth Environ. Sci.
**2014**, 22, 032052. [Google Scholar] [CrossRef] [Green Version] - Seidel, U.; Mende, C.; Hübner, B.; Weber, W.; Otto, A. Dynamic loads in Francis runners and their impact on fatigue life. IOP Conf. Ser. Earth Environ. Sci.
**2014**, 22, 032054. [Google Scholar] [CrossRef] [Green Version] - Monette, C.; Marmont, H.; Chamberland-Lauzon, J.; Skagerstrand, A.; Coutu, A.; Carlevi, J. Cost of enlarged operating zone for an existing Francis runner. IOP Conf. Ser. Earth Environ. Sci.
**2016**, 49, 072018. [Google Scholar] [CrossRef] - Favrel, A.; Müller, A.; Landry, C.; Yamamoto, K.; Avellan, F. Study of the vortex-induced pressure excitation source in a Francis turbine draft tube by particle image velocimetry. Exp. Fluids
**2015**, 56, 215. [Google Scholar] [CrossRef] - Duparchy, F.; Brammer, J.; Thibaud, M.; Favrel, A.; Lowys, P.; Avellan, F. Mechanical impact of dynamic phenomena in Francis turbines at off design conditions. In Journal of Physics: Conference Series; IOP Publishing: Bristol, UK, 2017; p. 012035. [Google Scholar]
- Bouajila, S.; De Colombel, T.; Lowys, P.; Maitre, T. Hydraulic Phenomena Frequency Signature of Francis Turbines Operating in Part Load Conditions. In IOP Conference Series: Earth and Environmental Science; IOP Publishing: Bristol, UK, 2016; p. 082001. [Google Scholar]
- Müller, A.; Favrel, A.; Landry, C.; Yamamoto, K.; Avellan, F. On the physical mechanisms governing self-excited pressure surge in Francis turbines. In IOP Conference Series: Earth and Environmental Science; IOP Publishing: Bristol, UK, 2014; p. 032034. [Google Scholar]
- Valentín, D.; Presas, A.; Egusquiza, E.; Valero, C.; Egusquiza, M.; Bossio, M. Power Swing Generated in Francis Turbines by Part Load and Overload Instabilities. Energies
**2017**, 10, 2124. [Google Scholar] - Presas, A.; Valentin, D.; Egusquiza, M.; Valero, C.; Egusquiza, E. Sensor-Based Optimized Control of the Full Load Instability in Large Hydraulic Turbines. Sensors
**2018**, 18, 1038. [Google Scholar] - Fisher, R.K.; Seidel, U.; Grosse, G.; Gfeller, W.; Klinger, R. A case study in resonant hydroelastic vibration: The causes of runner cracks and the solutions implemented for the Xiaolangdi hydroelectric project. In Proceedings of the XXI IAHR Symposium on Hydraulic Machinery and Systems, Lausanne, Switzerland, 9–12 September 2002. [Google Scholar]
- Neidhardt, T.; Jung, A.; Hyneck, S.; Gummer, J. An Alternative Approach to the Von Karman Vortex Problem in Modern Hydraulic Turbines. Available online: https://www.researchgate.net/publication/320811766 (accessed on 15 August 2018).
- Presas, A.; Egusquiza, E.; Valero, C.; Valentin, D.; Seidel, U. Feasibility of Using PZT Actuators to Study the Dynamic Behavior of a Rotating Disk due to Rotor-Stator Interaction. Sensors
**2014**, 14, 11919–11942. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Zwart, P.J.; Gerber, A.G.; Belamri, T. A two-phase flow model for predicting cavitation dynamics. In Proceedings of the Fifth International Conference on Multiphase Flow, Yokohama, Japan, 30 May–4 June 2004. [Google Scholar]
- Coutier-Delgosha, O.; Reboud, J.L.; Delannoy, Y. Numerical simulation of the unsteady behaviour of cavitating flows. Int. J. Numer. Methods Fluids
**2003**, 42, 527–548. [Google Scholar] [CrossRef] - Yamamoto, K.; Müller, A.; Favrel, A.; Avellan, F. Experimental evidence of inter-blade cavitation vortex development in Francis turbines at deep part load condition. Exp. Fluids
**2017**, 58. [Google Scholar] [CrossRef] - Yamamoto, K.; Müller, A.; Favrel, A.; Landry, C.; Avellan, F. Numerical and experimental evidence of the inter-blade cavitation vortex development at deep part load operation of a Francis turbine. In IOP Conference Series: Earth and Environmental Science; IOP Publishing: Bristol, UK, 2016. [Google Scholar]
- Favrel, A.; Müller, A.; Landry, C.; Yamamoto, K.; Avellan, F. LDV survey of cavitation and resonance effect on the precessing vortex rope dynamics in the draft tube of Francis turbines. Exp. Fluids
**2016**, 57, 168. [Google Scholar] [CrossRef] - Müller, A.; Favrel, A.; Landry, C.; Avellan, F. Fluid-structure interaction mechanisms leading to dangerous power swings in Francis turbines at full load. J. Fluids Struct.
**2017**, 69, 56–71. [Google Scholar] [CrossRef] - Presas, A.; Valentin, D.; Egusquiza, E.; Valero, C. Detection and analysis of part load and full load instabilities in a real Francis turbine prototype. In IOP Conference Series: Earth and Environmental Science; IOP Publishing: Bristol, UK, 2017; p. 012038. [Google Scholar]
- Escaler, X.; Egusquiza, E.; Farhat, M.; Avellan, F.; Coussirat, M. Detection of cavitation in hydraulic turbines. Mech. Syst. Signal Process.
**2006**, 20, 983–1007. [Google Scholar] [CrossRef] [Green Version] - De La Torre, O.; Escaler, X.; Egusquiza, E.; Farhat, M. Numerical and experimental study of a nearby solid boundary and partial submergence effects on hydrofoil added mass. Comput. Fluids
**2014**, 91, 1–9. [Google Scholar] [CrossRef] [Green Version] - Presas, A.; Valentin, D.; Egusquiza, E.; Valero, C.; Egusquiza, M.; Bossio, M. Accurate Determination of the Frequency Response Function of Submerged and Confined Structures by Using PZT-Patches. Sensors
**2017**, 17, 660. [Google Scholar] [CrossRef] [PubMed] - Presas, A.; Valentin, D.; Egusquiza, E.; Valero, C.; Seidel, U. Dynamic response of a rotating disk submerged and confined. Influence of the axial gap. J. Fluids Struct.
**2016**, 62, 332–349. [Google Scholar] [CrossRef] - De Siervo, F.; De Leva, F. Modern trends in selecting and designing Francis turbines. Water Power Dam Constr.
**1976**, 28, 28–35. [Google Scholar] - Presas, A.; Valentin, D.; Egusquiza, E.; Valero, C.; Seidel, U. On the detection of natural frequencies and mode shapes of submerged rotating disk-like structures from the casing. Mech. Syst. Signal Process.
**2015**, 60, 547–570. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**(

**a**) Main components of the Francis Turbine and (

**b**) the bottom view of the Francis turbine with broken trailing edges in some blades.

**Figure 2.**Visualization of cavitation vortices in the turbine runner by means of isocontours of void ratio 0.1 at 16 MW.

**Figure 4.**Mesh of the Francis turbine runner with the surrounding water (

**left**) and mesh independence analysis results (

**right**).

**Figure 5.**First mode shapes obtained with the FSI numerical situation. Bottom view. XND, X-nodal diameters.

**Figure 6.**Mapping of cavity shapes into FEM (

**right**) from CDF simulation results (

**left**) at deep part load condition.

**Figure 7.**Relative increase of the natural frequencies of the runner under cavitation effects depending on α

_{v}considered.

Condition | Operating Point with Respect to Rated Power | Head Water (m) | Rotating Speed (rpm) | Discharge (kg/s) | Torque (Nm) | Power (MW) |
---|---|---|---|---|---|---|

Deep part load | 36% | 102 | 250 | 21,779 | 610,371 | 16 |

BEP | 81% | 102 | 250 | 40,595 | 1,366,638 | 35.8 |

**Table 2.**Main excitation frequencies (rotating and stationary point of view) and corresponding excitation shapes.

Harmonics m,n | ${\mathit{f}}_{\mathit{r}\mathit{o}\mathit{t}}$ (Hz) | ${\mathit{f}}_{\mathit{s}\mathit{t}\mathit{a}\mathit{t}}$ (Hz) | Excitation Shape k |
---|---|---|---|

1,2 | 100 | 116.77 | 4 |

2,3 | 200 | 175 | −6 |

2,4 | 200 | 233.33 | 8 |

Phenomenon | Frequency (Range) | Type | Operating Points |
---|---|---|---|

Chaotic flow and inter-blade vortices | ≈0–5 Ω (0–20 Hz) | Stochastic | DPL |

Vortex rope | ≈0–1 Ω (0–4 Hz) | Periodic | Between DPL and BEP (Part Load) |

Von Karman shedding | 240–290 Hz for Deep Part Load 450–550 Hz for BEP | Periodic | Stronger at BEP |

RSI | 100 Hz, 200 Hz | Periodic | Stronger at BEP |

**Table 4.**Natural frequencies in air, water and frequency reduction ratios (FRR). No cavitation condition and machine working close to BEP.

Mode | 0ND | 1ND | 2ND | 3ND | 4ND |
---|---|---|---|---|---|

f_{air} | 83.67 | 117.9 | 149.81 | 228.83 | 270.52 |

f_{water} | 59.87 | 62.99 | 69.03 | 84.85 | 90.51 |

FRR | 28.4% | 46.6% | 53.9% | 62.9% | 66.5% |

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## Share and Cite

**MDPI and ACS Style**

Liu, X.; Luo, Y.; Presas, A.; Wang, Z.; Zhou, L.
Cavitation Effects on the Structural Resonance of Hydraulic Turbines: Failure Analysis in a Real Francis Turbine Runner. *Energies* **2018**, *11*, 2320.
https://doi.org/10.3390/en11092320

**AMA Style**

Liu X, Luo Y, Presas A, Wang Z, Zhou L.
Cavitation Effects on the Structural Resonance of Hydraulic Turbines: Failure Analysis in a Real Francis Turbine Runner. *Energies*. 2018; 11(9):2320.
https://doi.org/10.3390/en11092320

**Chicago/Turabian Style**

Liu, Xin, Yongyao Luo, Alexandre Presas, Zhengwei Wang, and Lingjiu Zhou.
2018. "Cavitation Effects on the Structural Resonance of Hydraulic Turbines: Failure Analysis in a Real Francis Turbine Runner" *Energies* 11, no. 9: 2320.
https://doi.org/10.3390/en11092320