# Characterization of the Effects of Ingested Bodies on the Rotor–Stator Interaction of Hydraulic Turbines

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Case Study

#### 2.2. Geometry and Mesh

^{3}/s flow rate, obtained through similarity analysis of the prototype’s BEP) operating conditions. The tested mesh element sizes ranged from 3 to 10 mm. Mesh independency was achieved at around 4900 kEl for the rotor, and 4300 kEl for the distributor, corresponding to average element sizes of 6 mm and 5 mm, respectively. A boundary layer mesh, with $60<{y}^{+}<300$, was attached to both distributor vanes and rotor blades, and an elbow-type draft tube with varying cross-sections with a 340 kEl hexahedral mesh was included in the computational model. The addition of a draft tube improves the runner pressure results in turbine CFD models by allowing pressure recovery downstream [18].

#### 2.3. Numerical Procedure

^{®}CFX v16.2 CFD software, which simultaneously solves continuity and Navier–Stokes equations for a fluid in motion.

^{®}CFX uses the finite volume method for spatial discretization. A high-resolution advection scheme (which consists of a numerical advection scheme with a calculated blending factor) was selected for the stabilization of the convective term. Time discretization was achieved by a 2nd-order backward Euler scheme. Tri-linear finite element-based functions were used as an interpolation scheme. Ansys

^{®}CFX uses a coupled solver which solves the Navier–Stokes equations (for velocity components and pressure) as a single system. First, non-linear equations are linearized (coefficient iteration), and then these linear equations are solved by an algebraic multigrid (AMG) solver. For convergence criteria, residual types were set to the root mean square (RMS) value of the normalized residuals with a target value of 1 × 10

^{−3}. Simulations were carried out on a 64-bit, Intel core i7 processor (6 core|12 thread) @ 3.3 GHz and 32 GB RAM, with simulation times of about 60 to 110 h (depending on the studied case) to reach quasi-steady flow conditions in all regions of the model.

#### 2.4. Post-Processing and Data Analysis

## 3. Results and Discussion

#### 3.1. Validation of the Numerical Simulations

^{3}/s. The rotating speed of the pump turbine unit is 600 rpm, the runner has a diameter of 2.87 m and its nominal power is about 100 MW.

#### 3.2. Pressure/Velocity Plots for Blockage Cases

#### 3.3. Obstructed Rotor Analysis

#### 3.3.1. Effect of the Blockage Size

#### 3.3.2. Effect of Blockage Position

#### 3.4. Obstructed Distributor Analysis

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

${C}_{P}$ | Coefficient of pressure | ${C}_{P}=\Delta p/\left(\rho {\left(\omega D\right)}^{2}\right)$ |

$D$ | Runner diameter | m |

${f}_{b}$ | Runner blade passing frequency | Hz |

${f}_{f}$ | Runner rotation frequency | Hz |

${f}_{v}$ | Distributor vane passing frequency | Hz |

$g$ | Standard gravity | $9.81\mathrm{m}/{\mathrm{s}}^{2}$ |

$H$ | Pump turbine net head | m |

$N$ | Runner rotation speed | rpm |

${N}_{s}$ | Dimensionless runner specific speed | ${N}_{s}=\omega {V}^{1/2}/{\left(gH\right)}^{3/4}$ |

$p$ | Pressure | Pa |

${t}^{*}$ | Dimensionless time | ${t}^{*}=t/{t}_{cycle}$ |

$t$ | Time | s |

${t}_{cycle}$ | Time of 1 runner revolution | s |

$V$ | Volumetric flow rate | ${\mathrm{m}}^{3}/\mathrm{s}$ |

${y}^{+}$ | Dimensionless wall distance | |

${z}_{b}$ | Number of runner blades | - |

${z}_{v}$ | Number of distributor vanes | - |

Greek letters | ||

$\rho $ | Density | $\mathrm{kg}/{\mathrm{m}}^{3}$ |

$\omega $ | Runner rotation speed | $\mathrm{rad}/\mathrm{s}$ |

Acronyms | ||

AMG | Algebraic multigrid | |

BEP | Best efficiency point | |

CFD | Computational fluid dynamics | |

FFT | Fast Fourier transform | |

RMS | Root mean square | |

RSI | Rotor–stator interaction |

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**Figure 3.**Detail of the geometrical model and computational mesh for the studied cases: (

**A**) distributor channel blockage and (

**B**) rotor channel blockage.

**Figure 6.**Location of the pressure transducers installed for the experimental study [24].

**Figure 8.**Coefficient of pressure (

**C**) vs. normalized time (t

_{p}*****) for pump turbine model (CFD) and prototype (experimental).

**Figure 9.**Example of pressure contours obtained for studied cases: (

**A**) obstructed rotor; (

**B**) obstructed distributor.

**Figure 12.**Pressure signal in the inertial reference frame (monitor point E3) for different rotor blockage sizes.

**Figure 14.**Frequency spectra in the inertial reference frame (monitor point E2) for different rotor blockage sizes.

**Figure 15.**Frequency spectra in the rotating reference frame (monitor point R1) for different rotor blockage sizes.

**Figure 16.**Frequency spectra in the inertial reference frame (monitor point E2) for different rotor blockage positions.

**Figure 17.**Frequency spectra in the rotating reference frame (monitor point R3) for different rotor blockage positions.

**Figure 18.**Pressure signal in the inertial reference frame (monitor point E3) for different distributor blockage sizes.

**Figure 19.**Pressure signal in the rotating reference frame for different distributor blockage sizes.

**Figure 20.**Frequency spectra in the inertial reference frame (monitor point E1) for different distributor blockage sizes.

**Figure 21.**Frequency spectra in the rotating reference frame (monitor point R1) for different distributor blockage sizes.

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**MDPI and ACS Style**

Guardo, A.; Fontanals, A.; Egusquiza, M.; Valero, C.; Egusquiza, E.
Characterization of the Effects of Ingested Bodies on the Rotor–Stator Interaction of Hydraulic Turbines. *Energies* **2021**, *14*, 6669.
https://doi.org/10.3390/en14206669

**AMA Style**

Guardo A, Fontanals A, Egusquiza M, Valero C, Egusquiza E.
Characterization of the Effects of Ingested Bodies on the Rotor–Stator Interaction of Hydraulic Turbines. *Energies*. 2021; 14(20):6669.
https://doi.org/10.3390/en14206669

**Chicago/Turabian Style**

Guardo, Alfredo, Alfred Fontanals, Mònica Egusquiza, Carme Valero, and Eduard Egusquiza.
2021. "Characterization of the Effects of Ingested Bodies on the Rotor–Stator Interaction of Hydraulic Turbines" *Energies* 14, no. 20: 6669.
https://doi.org/10.3390/en14206669