# 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

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**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