#
Multiscale Simulation of the Hydroabrasive Erosion of a Pelton Bucket: Bridging Scales to Improve the Accuracy^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Modeling Methodology

#### 2.1. Multiscale Model of Erosion

#### 2.2. Microscale Model: Sediment Impacts

#### 2.3. Macroscale Model: Turbulent Sediment Transport

#### 2.4. Sequential Multiscale Coupling Strategy

#### 2.5. The Finite Volume Particle Method

## 3. Test Case Description

## 4. Results

#### 4.1. Distributions of Impact Conditions on the Bucket Surface

#### 4.2. Erosion Distribution of the Bucket Surface

#### 4.3. Erosion Depth Validation

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Variables | |

${A}_{i}$ | Surface area |

${B}_{2}$ | Maximum internal bucket width |

${C}_{\circ}$ | Jet velocity |

${D}_{1}$ | Runner pitch diameter |

${D}_{2}$ | Jet diameter |

${L}_{j}$ | Jet length |

${L}_{s}$ | Splitter length |

c | Sediment concentration by mass |

${d}_{x}$ | Sediment diameter below which x% of the cumulative mass distribution lies |

${e}_{d}$ | Eroded depth |

${f}_{\mathrm{extrap}.}$ | Extrapolation factor |

${m}_{e}$ | Eroded mass |

v | Sediment impact velocity |

${z}_{r}$ | Number of buckets |

$\alpha $ | Jet impinging angle relative to the bucket plane |

$\rho $ | Base material density |

Abbreviations | |

ALE | Arbitrary Lagrangian-Eulerian |

FVPM | Finite Volume Particle Method |

GPU | Graphics Processing Unit |

## References

- Grein, H.; Schachenmann, A. Solving problems of abrasion in hydroelectric machinery. Water Power Dam Construct.
**1992**, 44, 19–24. [Google Scholar] - Chitrakar, S.; Neopane, H.; Gunnar, O. Study of the simultaneous effects of secondary flow and sediment erosion in Francis turbines. Renew. Energy
**2016**, 97, 881–889. [Google Scholar] [CrossRef] - Felix, D.; Abgottspon, A.; Albayrak, I.; Boes, R. Hydro-abrasive erosion on coated Pelton runners: Partial calibration of the IEC model based on measurements in HPP Fieschertal. In Proceedings of the 28th IAHR Symposium on Hydraulic Machinery and Systems, Grenoble, France, 4–8 July 2016; Volume 49, p. 122009. [Google Scholar] [CrossRef]
- Finnie, I. Erosion of surfaces by solid particles. Wear
**1960**, 3, 87–103. [Google Scholar] [CrossRef] - Finnie, I. Some observations on the erosion of ductile materials. Wear
**1972**, 19, 81–90. [Google Scholar] [CrossRef] - Shewmon, P.; Sundararajan, G. The erosion of metals. Annu. Rev. Mater. Sci.
**1983**, 13, 301–318. [Google Scholar] [CrossRef] - Rai, A.; Kumar, A.; Staubli, T. Hydro-abrasive erosion in Pelton buckets: Classification and field study. Wear
**2017**, 392–393, 8–20. [Google Scholar] [CrossRef] - Mann, B.; Arya, V. Abrasive and erosive wear characteristics of plasma nitriding and HVOF coatings: Their application in hydro turbines. Wear
**2001**, 249, 354–360. [Google Scholar] [CrossRef] - Thapa, B.S.; Thapa, B.; Dahlhaug, O.G. Empirical modelling of sediment erosion in Francis turbines. Energy
**2012**, 41, 386–391. [Google Scholar] [CrossRef] - Wang, Y.F.; Yang, Z.G. Finite element model of erosive wear on ductile and brittle materials. Wear
**2008**, 265, 871–878. [Google Scholar] [CrossRef] - Balu, P.; Kong, F.; Hamid, S.; Kovacevic, R. Finite element modeling of solid particle erosion in AISI 4140 steel and nickel-tungsten carbide composite material produced by the laser-based power deposition process. Tribol. Int.
**2013**, 62, 18–28. [Google Scholar] [CrossRef] - Wang, Y.F.; Yang, Z.G. A coupled finite element and meshfree analysis of erosive wear. Tribol. Int.
**2009**, 42, 373–377. [Google Scholar] [CrossRef] - Takaffoli, M.; Papini, M. Material deformation and removal due to single particle impacts on ductile materials using smoothed particle hydrodynamics. Wear
**2012**, 274, 50–59. [Google Scholar] [CrossRef] - Wang, M.H.; Huang, C.; Nandakumar, K.; Minev, P.; Luo, J.; Chiovelli, S. Computational fluid dynamics modelling and experimental study of erosion in slurry jet flows. Int. J. Comput. Fluid Dyn.
**2009**, 23, 155–172. [Google Scholar] [CrossRef] - Grewal, H.S.; Singh, H.; Yoon, E.S. Interplay between erodent concentration and impingement angle for erosion in dilute water-sand flows. Wear
**2015**, 332, 1111–1119. [Google Scholar] [CrossRef] - Messa, G.V.; Malavasi, S. The effect of sub-models and parametrizations in the simulation of abrasive jet impingement tests. Wear
**2017**, 370–371, 59–72. [Google Scholar] [CrossRef] - Leguizamón, S.; Jahanbakhsh, E.; Maertens, A.; Alimirzazadeh, S.; Avellan, F. A multiscale model for sediment impact erosion simulation using the finite volume particle method. Wear
**2017**, 392–393, 202–212. [Google Scholar] [CrossRef] - Abdulle, A.; Weinan, E.; Engquist, B.; Vanden-Eijnden, E. The heterogeoeous multiscale method. Acta Numer.
**2012**, 21, 1–87. [Google Scholar] [CrossRef] - Walther, J.H.; Praprotnik, M.; Kotsalis, E.M.; Koumoutsakos, P. Multiscale simulation of water flow past a C-540 fullerene. J. Comput. Phys.
**2012**, 231, 2677–2681. [Google Scholar] [CrossRef] - Saye, R.I.; Sethian, J.A. Multiscale modeling of membrane rearrangement, drainage, and rupture in evolving foams. Science
**2013**, 340, 720–724. [Google Scholar] [CrossRef] [PubMed] - Yildirim, B.; Muftu, S.; Gouldstone, A. Modeling of high velocity impact of spherical particles. Wear
**2011**, 270, 703–713. [Google Scholar] [CrossRef] - Leguizamón, S.; Jahanbakhsh, E.; Alimirzazadeh, S.; Maertens, A.; Avellan, F. FVPM numerical simulation of the effect of particle shape and elasticity on impact erosion. Wear
**2019**, 430–431, 108–119. [Google Scholar] [CrossRef] - Johnson, G.R.; Cook, W.H. Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures. Eng. Fract. Mech.
**1985**, 21, 31–48. [Google Scholar] [CrossRef] - Leguizamón, S.; Jahanbakhsh, E.; Maertens, A.; Vessaz, C.; Alimirzazadeh, S.; Avellan, F. Impact erosion prediction using the finite volume particle method with improved constitutive models. In Proceedings of the 28th IAHR Symposium on Hydraulic Machinery and Systems, Grenoble, France, 4–8 July 2016; Volume 49, p. 122010. [Google Scholar] [CrossRef]
- Dehbi, A. Turbulent particle dispersion in arbitrary wall-bounded geometries: A coupled CFD-Langevin-equation based approach. Int. J. Multiphase Flow
**2008**, 34, 819–828. [Google Scholar] [CrossRef] - Jahanbakhsh, E.; Vessaz, C.; Maertens, A.; Avellan, F. Development of a finite volume particle method for 3-D fluid flow simulations. Comput. Methods Appl. Mech. Eng.
**2016**, 298, 80–107. [Google Scholar] [CrossRef] - Vessaz, C.; Jahanbakhsh, E.; Avellan, F. Flow simulation of jet deviation by rotating Pelton buckets using finite volume particle method. ASME J. Fluids Eng.
**2015**, 137, 074501. [Google Scholar] [CrossRef] - Jahanbakhsh, E.; Maertens, A.; Quinlan, N.; Vessaz, C.; Avellan, F. Exact finite volume particle method with spherical-support kernels. Comput. Methods Appl. Mech. Eng.
**2017**, 317, 101–127. [Google Scholar] [CrossRef] - Alimirzazadeh, S.; Jahanbakhsh, E.; Maertens, A.; Leguizamón, S.; Avellan, F. GPU-accelerated 3-D finite volume particle method. Comput. Fluids
**2018**, 171, 79–93. [Google Scholar] [CrossRef] - Alimirzazadeh, S.; Kumashiro, T.; Leguizamón, S.; Maertens, A.; Jahanbakhsh, E.; Tani, K.; Avellan, F. GPU-accelerated Pelton turbine simulation using finite volume particle method coupled with linear eddy viscosity models. In Proceedings of the 29th IAHR Symposium on Hydraulic Machinery and Systems, Kyoto, Japan, 17–21 September 2018; Volume 240, p. 072018. [Google Scholar] [CrossRef]
- Perrig, A. Hydrodynamics of the Free Surface Flow in Pelton Turbine Buckets. Ph.D. thesis, École Polytechnique Fédérale de Lausanne (EPFL), Lausanne, Switzerland, 2007. [Google Scholar] [CrossRef]
- Leguizamón, S.; Jahanbakhsh, E.; Maertens, A.; Alimirzazadeh, S.; Avellan, F. Simulation of the hydroabrasive erosion of a bucket: A multiscale model with projective integration to circumvent the spatio-temporal scale separation. In Proceedings of the 29th IAHR Symposium on Hydraulic Machinery and Systems, Kyoto, Japan, 17–21 September 2018; Volume 240, p. 072014. [Google Scholar] [CrossRef]
- Leguizamón, S.; Alimirzazadeh, S.; Jahanbakhsh, E.; Avellan, F. Multiscale simulation of erosive wear in a prototype-scale Pelton runner. Renew. Energy
**2019**, submit. [Google Scholar]

**Figure 1.**Time and length scales involved in the erosion process of a Pelton turbine [17]. The time bounds represent the simulation time step required and its overall duration, whereas the length bounds represent the discretization size necessary for convergence and the overall domain size.

**Figure 2.**Finite volume particle method (FVPM) discretizations used in the multiscale model. (

**a**) Microscale domain, where a sharp sediment is about to impact the solid. (

**b**) Macroscale domain, where the water (with sediments too small to be seen) is being evacuated from the Pelton bucket.

**Figure 3.**Average sediment impact velocity and angle distributions on the bucket surface (

**a**) and on the pitch diameter position (

**b**). The values presented in (

**b**) are taken from the rectangular stripe illustrated in (

**a**); the shaded region represents ± one standard deviation.

**Figure 4.**Erodent mass (i.e., the amount of sediment that impacted) and eroded mass (i.e., the amount of base material removed), on the bucket surface (

**a**) and on the pitch diameter position (

**b**). The values presented in (

**b**) are taken from the rectangular stripe illustrated in (

**a**).

**Figure 5.**Eroded depth distribution on the pitch diameter position after 3180 h of operation; the multiscale simulation results compare well with the corresponding experimental data of Rai et al. [7].

**Figure 6.**Eroded depth distribution on the bucket splitter after 3180 h of operation, including the experimental data of Rai et al. [7].

${\mathit{B}}_{2}$ | ${\mathit{D}}_{1}$ | ${\mathit{D}}_{2}$ | ${\mathit{z}}_{\mathit{r}}$ | ${\mathit{C}}_{\circ}$ | $\mathit{\alpha}$ | ${\mathit{L}}_{\mathit{j}}$ | c | ${\mathit{d}}_{10}$ | ${\mathit{d}}_{50}$ | ${\mathit{d}}_{90}$ |
---|---|---|---|---|---|---|---|---|---|---|

375 | 1089 | 140 | 17 | 28.5 | 80 | 201 | 1174 | 6 | 27 | 134 |

$\left[\mathrm{mm}\right]$ | $\left[\mathrm{mm}\right]$ | $\left[\mathrm{mm}\right]$ | $[-]$ | $\left[\mathrm{m}\phantom{\rule{4.pt}{0ex}}{\mathrm{s}}^{-1}\right]$ | ${[}^{\circ}]$ | $\left[\mathrm{mm}\right]$ | $\left[\mathrm{mg}\phantom{\rule{4.pt}{0ex}}{\mathrm{L}}^{-1}\right]$ | $\left[\mathsf{\mu}\mathrm{m}\right]$ | $\left[\mathsf{\mu}\mathrm{m}\right]$ | $\left[\mathsf{\mu}\mathrm{m}\right]$ |

© 2019 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 NonCommercial NoDerivatives (CC BY-NC-ND) license (https://creativecommons.org/licenses/by-nc-nd/4.0/).

## Share and Cite

**MDPI and ACS Style**

Leguizamón, S.; Jahanbakhsh, E.; Alimirzazadeh, S.; Maertens, A.; Avellan, F. Multiscale Simulation of the Hydroabrasive Erosion of a Pelton Bucket: Bridging Scales to Improve the Accuracy. *Int. J. Turbomach. Propuls. Power* **2019**, *4*, 9.
https://doi.org/10.3390/ijtpp4020009

**AMA Style**

Leguizamón S, Jahanbakhsh E, Alimirzazadeh S, Maertens A, Avellan F. Multiscale Simulation of the Hydroabrasive Erosion of a Pelton Bucket: Bridging Scales to Improve the Accuracy. *International Journal of Turbomachinery, Propulsion and Power*. 2019; 4(2):9.
https://doi.org/10.3390/ijtpp4020009

**Chicago/Turabian Style**

Leguizamón, Sebastián, Ebrahim Jahanbakhsh, Siamak Alimirzazadeh, Audrey Maertens, and François Avellan. 2019. "Multiscale Simulation of the Hydroabrasive Erosion of a Pelton Bucket: Bridging Scales to Improve the Accuracy" *International Journal of Turbomachinery, Propulsion and Power* 4, no. 2: 9.
https://doi.org/10.3390/ijtpp4020009