# Understanding the Influence of Wake Cavitation on the Dynamic Response of Hydraulic Profiles under Lock-In Conditions

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Modal Work Approach

#### 2.2. Numerical Setup

_{L}and C

_{D}. The expressions used to calculate the C

_{L}and C

_{D}are presented as follows:

_{L}and C

_{D}for a cavitating condition.

_{21}and r

_{43}, being greater than 1.3, met the GCI requirements. Table 2 shows that the extrapolated value (ɸ

^{31}

_{ext}) was 882 Hz, the approximate relative error (ɸ

^{31}

_{a_E}) and the extrapolated relative error (ɸ

^{31}

_{ext_E}) were 2.2% and 0.68% respectively, and the GCI was 0.84%.

## 3. Results

#### 3.1. Validation of the CFD Model

#### 3.2. Validation of the Modal Work Approach

#### 3.3. Discussion

## 4. Conclusions

- The modal work approach can be employed to compute the vibration amplitude of a hydrofoil under a torsional lock-in condition for both cavitation-free and incipient cavitation conditions.
- The drop in the vibration amplitude when wake cavitation occurs is around 45%.
- The deviation in the numerically simulated vibration amplitude relative to the experimental results is approximately 24% for a cavitation-free condition.

- The von Karman vortex cavitation seems to create an additional extension of the trailing edge solid boundary, which results in a significant alteration of the near-wake flow dynamics.
- When wake cavitation occurs, the vortex formation location is displaced downstream, which results in an additional space between the vortex and the trailing edge. This additional space allows an early occurrence of an opposite vortex and hence a drop in pressure at this location.
- The increase in the modal work, which corresponds to a decrease in the vibration amplitude, seems not to be as dependent on viscous forces as it is on pressure ones. Specifically, the early occurrence of the vortex is believed to be responsible for a positive pressure gradient in the trailing edge wall displacement direction and thus for the increment of the modal work.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Strouhal, V. Uber eine besondere Art der Tonerregung. Ann. Phys. Chem.
**1878**, 10, 216–251. [Google Scholar] [CrossRef] [Green Version] - Von Kármán, T. Ueber den Mechanismus des Widerstandes, den ein bewegter Körper in einer Flüssigkeit erfährt. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen Mathematisch-Physikalische Klasse
**1911**, 1911, 509–517. [Google Scholar] - Von Kármán, T. Ueber den Mechanismus des Widerstandes, den ein bewegter Körper in einer Flüssigkeit erfährt. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen Mathematisch-Physikalische Klasse
**1912**, 1912, 547–556. [Google Scholar] - Gerrard, J.H. The mechanics of the formation region of vortices behind bluff bodies. J. Fluid Mech.
**1966**, 25, 401–413. [Google Scholar] [CrossRef] - Perry, A.E.; Chong, M.S.; Lim, T.T. The vortex-shedding process behind two-dimensional bluff-bodies. J. Fluid Mech.
**1982**, 116, 77–90. [Google Scholar] [CrossRef] - Ausoni, P. Turbulent Vortex Shedding from a Blunt Trailing Edge Hydrofoil. Ph.D. Thesis, EPFL, Lausanne, Switzerland, 2009. [Google Scholar]
- Naudascher, E.; Rockwell, D. Flow Induced Vibrations: An Engineering Guide; Dover Publications Inc.: Mineola, NY, USA, 2005. [Google Scholar]
- Blevins, R. Flow-Induced Vibrations; Krieger Publishing Company: Malabar, FL, USA, 2001. [Google Scholar]
- Arioli, G.; Gazzola, F. Torsional instability in suspension bridges: The tacoma narrows bridge case. Commun. Nonlinear Sci. Numer. Simul.
**2017**, 42, 342–357. [Google Scholar] [CrossRef] [Green Version] - Hoskoti, L.; Misra, A.; Sucheendran, M.M. Frequency lock-in during vortex induced vibration of a rotating blade. J. Fluids Struct.
**2018**, 80, 145–164. [Google Scholar] [CrossRef] - Gummer, J.H.; Hensman, P.C. A review of stay vane cracking in hydraulic turbines. Int. Water Power Dam Constr.
**1992**, 44, 32–42. [Google Scholar] - Lockey, K. Flow induced vibrations at stay vanes: Experience at site and CFD simulation of von Kármán vortex shedding. In Proceedings of the Hydro 2006, Porto Carras, Greece, 25 September 2006; pp. 25–28. [Google Scholar]
- Dörfler, P.; Sick, M.; Coutu, A. Flow Induced Pulsation and Vibration in Hydroelectric Machinery; Springer Science & Business Media: London, UK, 2013. [Google Scholar] [CrossRef]
- Pang, L.J.; Lu, G.P.; Zhong, S.; Liu, J.S. Vortex shedding simulation and vibration analysis of stay vanes of hydraulic turbine. J. Mech. Eng.
**2011**, 47, 159–166. [Google Scholar] [CrossRef] - Vu, T.C.; Nennemann, B.; Ausoni, P.; Farhat, M.; Avellan, F. Unsteady CFD prediction of von Karman vortex shedding in hydraulic turbine stay vanes. In Proceedings of the Hydro 2007, Granada, Spain, 15–17 October 2007. [Google Scholar]
- Alexandre, A.N.; Gissoni, H.; Gonçalves, M.; Cardoso, R.; Jung, A.; Meneghini, J. Engineering diagnostics for vortex-induced stay vanes cracks in a Francis turbine. IOP Conf. Ser. Earth Environ. Sci.
**2016**, 49, 072017. [Google Scholar] - Nennemann, B.; Monette, C. Prediction of vibration amplitudes on hydraulic profiles under von Karman vortex excitation. Conf. Ser. Earth Environ. Sci.
**2019**, 240, 062004. [Google Scholar] [CrossRef] - Arndt, R.E. Cavitation in fluid machinery and hydraulic structures. Annu. Rev. Fluid Mech.
**1981**, 13, 273–326. [Google Scholar] [CrossRef] - Franc, J.P.; Michel, J.M. Fundamentals of Cavitation; Springer Science & Business Media: Berlin, Germany, 2006; pp. 1–13, 247–262. [Google Scholar]
- Young, J.O.; Holl, J.W. Effects of cavitation on periodic wakes behind symmetric wedges. J. Fluids Eng.
**1966**, 88, 163–176. [Google Scholar] [CrossRef] - Belahadji, B.; Franc, J.P.; Michel, J.M. Cavitation in the rotational structures of a turbulent wake. J. Fluid Mech.
**1995**, 287, 383–403. [Google Scholar] [CrossRef] - Chen, J.; Geng, L.; Excaler, X. Numerical investigation of the cavitation effects on the vortex shedding from a Hydrofoil with blunt trailing edge. Fluids
**2020**, 5, 218. [Google Scholar] [CrossRef] - Ramamurthy, A.S.; Balachandar, R. Characteristics of constrained cavitating bluff body wakes. J. Eng. Mech.
**1991**, 117, 513–531. [Google Scholar] [CrossRef] - Zeng, Y.; Yao, Z.; Gao, J.; Hong, Y.; Wang, F.; Zhang, F. Numerical investigation of added mass and hydrodynamic damping on a blunt trailing edge hydrofoil. J. Fluids Eng.
**2019**, 141, 081108. [Google Scholar] [CrossRef] - Liaghat, T.; Guibault, F.; Allenbach, L.; Nennemann, B. Two-Way Fluid-Structure Coupling in Vibration and Damping Analysis of an Oscillating Hydrofoil. ASME
**2014**, 46476, V04AT04A073, Paper No. IMECE2014-38441. [Google Scholar] - Zeng, Y.S.; Yao, Z.F.; Yang, Z.J.; Wang, F.J.; Hong, Y.P. The Prediction of Hydrodynamic Damping Characteristics of a Hydrofoil With Blunt Trailing Edge. IOP Conf. Ser. Earth Environ. Sci.
**2017**, 163, 012041. [Google Scholar] [CrossRef] - Alexandre, A.N.; Saltara, F. Study of Stay Vanes Vortex-Induced Vibrations with different Trailing-Edge Profiles Using CFD. J. Fluid Mach. Syst.
**2009**, 2, 4. [Google Scholar] - Miyagawa, K.; Fukao, S.; Kawata, Y. Study on stay vane instability due to Vortex shedding. In Proceedings of the 22th IAHR Symposium on Hydraulic Machinery and Systems, Stockholsm, Sweden, 29 June–2 July 2004. [Google Scholar]
- Williamson, C.H. Vortex dynamics in the cylinder wake. Annu. Rev. Fluid Mech.
**1996**, 28, 477–539. [Google Scholar] [CrossRef] - Kim, J.; Choi, H. Distributed forcing of flow over a circular cylinder. Phys. Fluids
**2005**, 17, 103–116. [Google Scholar] [CrossRef] - Roshko, A. On the Drag and Shedding Frequency of Two-Dimensional Bluff Bodies; National Aeronautics and Space Administration: Washington, DC, USA, 1954. [Google Scholar]
- Farhadi, M.; Sedighi, K.; Korayem, A.M. Effect of wall proximity on forced convection in a plane channel with a built-in triangular cylinder. Int. J. Therm. Sci.
**2010**, 49, 1010–1018. [Google Scholar] [CrossRef] - Liang, Q.W.; Rodriguez, 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] - Escaler, X.; De La Torre, O. Axisymmetric vibrations of a circular Chladni plate in air and fully submerged in water. J. Fluids Struct.
**2018**, 82, 432–445. [Google Scholar] [CrossRef] - Zhu, W.R.; Gao, Z.X.; Lu, L.; Wang, F.J. Analysis and Optimization on Natural Frequencies Depreciation Coefficient of Centrifugal Pump Impeller in Water. J. Hydraul. Eng.
**2013**, 44, 1455–1461. [Google Scholar] - Celik, I.B.; Ghia, U.; Roache, P.J.; Freitas, C.J.; Coleman, H.; Raad, P.E. Procedure for Estimation and Reporting of Uncertainty Due to Discretization in CFD Applications. J. Fluids Eng.
**2008**, 130, 078001. [Google Scholar] [CrossRef] [Green Version] - Zeng, Y.S.; Yao, Z.F.; Zhou, P.J.; Wang, F.J.; Hong, Y.P. Numerical investigation into the effect of the trailing edge shape on the added mass and hydrodynamic damping for a hydrofoil. J. Fluids Struct.
**2019**, 88, 167–184. [Google Scholar] [CrossRef] - Neidhardt, T.; Jung, A.; Heyneck, S.; Gummer, J. An alternative approach to the von Karman vortex problem in hydraulic turbines. In Proceedings of the Hydro 2017, Seville, Spain, 2 November 2017. [Google Scholar]

**Figure 1.**(

**a**) Normalized mode shape with the selected section highlighted in black and (

**b**) the CFD model with the imported mode shape section.

**Figure 5.**(

**a**) Selected mesh for the entire fluid domain (N

_{1}) and (

**b**) detailed zooms showing the mesh topology around the hydrofoil surface and near the wake region.

**Figure 6.**Comparison of the vortex shedding frequencies estimates between the numerical simulations and the experimental measurements (Ausoni et al. [6]) as a function of the relative sigma value.

**Figure 7.**Predicted maximum vibration amplitudes at the trailing edge corresponding to the points with zero work located near the dashed horizontal line (

**a**); and displacement time histories of the first torsion mode in resonance with the von Karman vortex street for no cavitation and incipient cavitation regimes (

**b**).

**Figure 8.**Computed modal work (

**a**) and acceleration vibration amplitude (

**b**) measured by Ausoni et al. [6] as a function of cavitation level.

**Figure 9.**Hydrofoil initial position at t/T = 0 (

**left**) and at t/T = 0.25 when it reaches its maximum deformation (

**right**). Red arrows indicate the displacement direction.

**Figure 10.**Streamwise velocity for $\frac{\sigma}{{\sigma}_{i}}=2$ (

**a**,

**b**), $\frac{\sigma}{{\sigma}_{i}}=0.98$ (

**c**,

**d**), and $\frac{\sigma}{{\sigma}_{i}}=0.69$ (

**e**,

**f**) regimes at t/T = 0 (

**a**,

**c**,

**e**) and t/T = 0.25 (

**b**,

**d**,

**f**).

**Figure 11.**Streamwise velocity superimposed on the vapor fraction (contour plot) for $\frac{\sigma}{{\sigma}_{i}}=2$ (

**a**), $\frac{\sigma}{{\sigma}_{i}}=0.98$ (

**b**), and $\frac{\sigma}{{\sigma}_{i}}=0.69$ (

**c**) regimes at t/T = 0 (

**a**,

**b**,

**c**).

**Figure 12.**Absolute pressure for $\frac{\sigma}{{\sigma}_{i}}=2$ (a,b), $\frac{\sigma}{{\sigma}_{i}}=0.98$ (

**c**,

**d**), and $\frac{\sigma}{{\sigma}_{i}}=0.69$ (

**e**,

**f**) regimes at t/T = 0 (

**a**,

**c**,

**e**) and t/T = 0.25 (

**b**,

**d**,

**f**).

**Figure 13.**Cp coefficient over the NACA 0009 profile for $\frac{\sigma}{{\sigma}_{i}}=2$ (

**a**,

**b**), $\frac{\sigma}{{\sigma}_{i}}=0.98$ (

**c**,

**d**), and $\frac{\sigma}{{\sigma}_{i}}=0.69$ (

**e**,

**f**) regimes at t/T = 0 (

**a**,

**c**,

**e**) and t/T = 0.25 (

**b**,

**d**,

**f**).

**Figure 14.**Contour plots of the Eulerian Q criterion in the range from 1,000,000 to 3,000,000 s

^{−2}for no cavitation, $\frac{\sigma}{{\sigma}_{i}}=2$ (

**a**), and cavitation, $\frac{\sigma}{{\sigma}_{i}}=0.69$ (

**b**), regimes at t/T = 0.25.

**Figure 15.**Velocity curl for $\frac{\sigma}{{\sigma}_{i}}=2$ (

**a**,

**b**), $\frac{\sigma}{{\sigma}_{i}}=0.98$ (

**c**,

**d**), and $\frac{\sigma}{{\sigma}_{i}}=0.69$ (

**e**,

**f**) regimes at t/T = 0 (

**a**,

**c**,

**e**) and t/T = 0.25 (

**b**,

**d**,

**f**).

Case | Number of Elements | Max y+ [-] | Mesh Size Resolution [mm] | Frequency (Hz) |
---|---|---|---|---|

N1 | 461,600 | 0.24 | 0.2 | 888 |

N2 | 427,800 | 2.4 | 0.2 | 888 |

N3 | 175,616 | 2.4 | 0.5 | 908 |

N4 | 80,860 | 2.4 | 1 | 839 |

Case | r_{21} | r_{43} | ɸ_{1} (Hz) | ɸ_{3} (Hz) | ɸ_{4} (Hz) | p | ɸ^{31}_{ext} (Hz) | ɸ^{31}_{a_E} (%) | ɸ^{31}_{ext_E} (%) | GCI (%) |
---|---|---|---|---|---|---|---|---|---|---|

N1-N3-N4 | 1.62 | 1.47 | 888 | 908 | 839 | 3.04 | 882 | 2.2 | 0.68 | 0.84 |

Group 1 | Group 2 | Group 3 | Group 4 | |
---|---|---|---|---|

$\frac{\sigma}{{\sigma}_{i}}$ | [2, 1.2, 0.98, 0.9, 0.84, 0.77, 0.69] | 2 | 0.98 | [2, 1.2, 0.98, 0.9, 0.84, 0.77, 0.69] |

${q}_{0}$ [mm] | - | [0.08, 0.076, 0.07] | [0.045, 0.041, 0.038] | 0.076 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Roig, R.; Chen, J.; de la Torre, O.; Escaler, X.
Understanding the Influence of Wake Cavitation on the Dynamic Response of Hydraulic Profiles under Lock-In Conditions. *Energies* **2021**, *14*, 6033.
https://doi.org/10.3390/en14196033

**AMA Style**

Roig R, Chen J, de la Torre O, Escaler X.
Understanding the Influence of Wake Cavitation on the Dynamic Response of Hydraulic Profiles under Lock-In Conditions. *Energies*. 2021; 14(19):6033.
https://doi.org/10.3390/en14196033

**Chicago/Turabian Style**

Roig, Rafel, Jian Chen, Oscar de la Torre, and Xavier Escaler.
2021. "Understanding the Influence of Wake Cavitation on the Dynamic Response of Hydraulic Profiles under Lock-In Conditions" *Energies* 14, no. 19: 6033.
https://doi.org/10.3390/en14196033