# Investigation on the Impact of Air Admission in a Prototype Francis Turbine at Low-Load Operation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Prototype Site Measurements

#### 2.1. Prototype Unit and Experimental Setup

#### 2.2. Measurement Procedure and Plant Behavior

## 3. Numerical Model

#### 3.1. Geometry and Mesh

#### 3.2. CFD-Setup

#### 3.3. FEM-Setup

#### 3.3.1. Modal Analysis

#### 3.3.2. Transient FEM Analysis

#### 3.4. Life Expectancy

## 4. Results and Discussion

#### 4.1. Cavitation Behavior and Pressure Oscillations

#### 4.2. Trailing Edge Vortex Shedding

#### 4.3. Modal Analysis of the Draft Tube

#### 4.4. Fatigue Damage and Dynamical Stress

## 5. Conclusions

- A huge draft tube vortex, with a frequency corresponding to approximately $0.2\xb7{f}_{0}$, is the most concerning fact in terms of high dynamical stress and runner fatigue damage.
- Both turbulence models showed quite similar behavior and a good agreement with the measurement. However, in contrast to single flow analysis the pressure amplitudes of the simulation tend to be higher than the ones obtained by the sensor.
- An improved runner model, targeting separation effects, showed the appearance of trailing edge vortex shedding.
- A simplified modal analysis of the draft tube confirmed that the vibrations of the machine set are related to vortex shedding.
- The air injection not only significantly reduced the vibrations of the machine set and might have a positive effect on cavitation, but also improved runner fatigue life.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Acronyms | |

CDS | Central deference scheme |

CFD | Computational fluid dynamics |

CFL | Courant number |

D | Pressure side |

DES | Detached Eddy Simulation |

DT | Draft tube |

GGI | General grid interface |

FEM | Finite element method |

FFT | Fast-Fourier-Transformation |

FSI | Fluid-structure-interaction |

GV | Guide vanes |

GVA | Guide vanes with inlets for air injection |

LES | Large Eddy Simulation |

MP | Monitor point |

RANS | Reynolds averaged Navier-Stokes equations |

R1 | T-rosette |

RMS | root mean square |

Acronyms | |

RN | Runner |

RNVS | Refined runner domain |

RP | Rated point |

S | Suction side |

SAS | Scale Adaptive Simulation |

SBES | Stress Blended Eddy Simulation |

SC | Spiral casing |

SRS | Scale-Resolving Simulation |

SST | Shear stress transport |

SV | Stay vanes |

URANS | Unsteady RANS |

VIV | Vortex-induced-vibrations |

Greek Symbols | |

$\alpha $ | Rayleigh-Parameter, ($\frac{1}{s}$) |

$\beta $ | Rayleigh-Parameter, ($\frac{1}{s}$) |

${\rho}_{f}$ | Water density, ($\frac{kg}{{m}^{3}}$) |

${\rho}_{g}$ | Vapor density, ($\frac{kg}{{m}^{3}}$) |

${\sigma}_{a}$ | Stress amplitude, ($\frac{N}{{m}^{2}}$) |

${\sigma}_{y}$ | Yield strength, ($\frac{N}{{m}^{2}}$) |

$\zeta $ | Damping ratio, (-) |

${\omega}_{1}$ | Frequency limit, ($\frac{1}{s}$) |

${\omega}_{2}$ | Frequency limit, ($\frac{1}{s}$) |

Latin Symbols | |

${a}_{con}$ | Acceleration measured at the draft tube cone, ($\frac{m}{{s}^{2}}$) |

${a}_{HUB}$ | Acceleration measured at the hollow hub, ($\frac{m}{{s}^{2}}$) |

${a}_{THB.xy}$ | Acceleration measured at the turbine bearing, ($\frac{m}{{s}^{2}}$) |

$\left(\right)$ | Damping marix, ($\frac{Ns}{m}$) |

C | Damage factor, (-) |

D | Outer diameter of the runner, (m) |

${F}_{condensation}$ | Empirical factor for condensation, (-) |

${F}_{vapor}$ | Empirical factor for vapor, (-) |

f | Frequency, ($Hz$) |

${f}_{0}$ | Rotational frequency, ($Hz$) |

${f}_{SV}$ | Vortex shedding frequency, ($Hz$) |

g | Gravity constant, ($m/{s}^{2}$) |

H | Turbine head, (m) |

$\left[K\right]$ | Stiffness marix, ($\frac{N}{m}$) |

k | Turbulent kinetic energy, (${m}^{2}/{s}^{2}$) |

L | Characteristic lateral dimension, (m) |

$\left[M\right]$ | Mass marix, ($kg$) |

${\dot{m}}_{fg.v}$ | Interphase mass transfer rate, ($\frac{kg}{s}$) |

${N}_{i}$ | Number of load cycles until the S-N curve is reached, (-) |

${n}_{ED}$ | Speed factor, (-) |

${n}_{i}$ | Number of load cycles, (-) |

P | Power, ($MW$) |

${P}_{RP}$ | Rated power, ($MW$) |

p | Pressure, ($Pa$) |

${p}_{a}$ | Pressure amplitude, ($Pa$) |

${p}_{a.con}$ | Draft tube cone pressure amplitude, ($Pa$) |

${p}_{v}$ | Pressure in the bubble, ($Pa$) |

${p}_{con}$ | Draft tube cone pressure, ($Pa$) |

${p}_{DT}$ | Draft tube outlet pressure, ($Pa$) |

${p}_{E}$ | Dynamic pressure (RN outlet), ($Pa$) |

Latin Symbols | |

${R}_{B}$ | Bubble radius, (m) |

${R}_{nuc}$ | Nucleation site radius, (m) |

${r}_{nuc}$ | Nucleation volume fraction, (-) |

${r}_{g}$ | Bubble volume fraction, (-) |

$St$ | Strouhal number, (-) |

${s}_{THB.xy}$ | Relative displacement measured at the turbine bearing, (m) |

${u}_{2}$ | Circumferential velocity (RN outlet), ($\frac{m}{s}$) |

v | Velocity, ($\frac{m}{s}$) |

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**Figure 2.**(

**a**) Runner vibrations obtained by the acceleration sensor ${a}_{HUB}$. (

**b**) Dynamical stresses obtained by the strain gauges [24].

**Figure 3.**Vibrations with (green) and without (black) air admission obtained by (

**a**) ${a}_{TBH.x}$ (

**b**) ${a}_{TBH.y}$ (

**c**) ${s}_{TBH.x}$ (

**d**) ${s}_{TBH.x}$ (

**e**) ${a}_{con}$ (

**f**) ${a}_{HUB}$.

**Figure 4.**(

**a**) Computational domains and boundary conditions as well as (

**b**) mesh and interfaces. The turbine model consists of spiral casing (SC), stay vanes (SV), guide vanes (GV), runner (RN) and the draft tube (DT).

**Figure 6.**Transient finite element method (FEM) setup of the prototype Francis turbine runner. Computational fluid dynamics (CFD).

**Figure 7.**Comparison between experimental and numerical captured pressure pulsations without air injection in (

**a**) time and (

**b**) frequency domain. Scale adaptive simulation (SAS) and the newer stress blended eddy simulation (SBES).

**Figure 8.**Comparison between experimental and numerical captured pressure pulsations with air injection in (

**a**) time and (

**b**) frequency domain.

**Figure 9.**(

**a**) Draft tube Vortex formation shown by Q-criterion and (

**b**) vapor volume fraction as well as air volume in the runner domain caused by aeration.

**Figure 10.**(

**a**) Vortex shedding discovered by the SBES turbulence model highlighted by the Q-criterion ${a}_{HUB}$. (

**b**) Blade-to-blade plot near the shroud at $95\%$ trailing edge span.

**Figure 11.**(

**a**) Pressure pulsations induced by vortex shedding captured at three different MP for the case without air injection and (

**b**) Structural response of the draft tube with air revealed by the prototype site measurement.

**Figure 12.**(

**a**) First 15 natural frequencies of the draft tube model and (

**b**) total deformation of the critical eigenmode.

**Figure 13.**Dynamical stresses at $44\%\xb7{P}_{RP}$ with and without air injection obtained by the strain gauges.

**Figure 14.**(

**a**) Stress hotspot according to the FEM simulation and (

**b**) runner stress amplitudes (S2).

**Figure 15.**Load spectra of the measured and computed stresses (

**a**) with and (

**b**) without air injection.

**Table 1.**Mesh size and quality. In addition to the standard RN model, a second finer mesh (RN VS) with refinements especially around the trailing edge; a slightly refined GV domain (GVA).

Domain | SC | SV | GV | RN | DT | GVA | RN VS |
---|---|---|---|---|---|---|---|

Number of cells (million) | $1.7$ | $1.3$ | $3.2$ | $8.8$ | $10.5$ | $3.7$ | 20 |

Minimum determinant (-) | $0.2$ | $0.28$ | $0.7$ | $0.26$ | $0.7$ | $0.412$ | $0.27$ |

Minimum angle (${}^{\circ}$) | $10.3$ | $24.9$ | $18.2$ | $26.1$ | 27 | $33.75$ | $20.3$ |

${y}_{mean}^{+}$ (-) | $27.3$ | $16.4$ | 32 | $12.4$ | $11.1$ | $25.4$ | $11.6$ |

Description Parameters | Parameters |
---|---|

Damping ratio $\zeta $ (-) | $0.01$ |

Low limit frequency ${f}_{1}$ (Hz) | 1 |

Upper limit frequency ${f}_{2}$ (Hz) | $344.2$ |

Rayleigh coefficient $\alpha $ (-) | $0.1253$ |

Rayleigh coefficient $\beta $ (-) | $9.2216\times {10}^{-6}$ |

Description Parameters | Parameters |
---|---|

Material (-) | $X5$$Cr$$Ni$ 13 4 |

Survival probability (%) | $99.99$ |

Mean stress (N/mm${}^{2}$) | 150 |

Logarithmic standard deviation of ${\sigma}_{a}$ (-) | $0.07$ |

Stress ratio (-) | 1 |

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## Share and Cite

**MDPI and ACS Style**

Unterluggauer, J.; Maly, A.; Doujak, E.
Investigation on the Impact of Air Admission in a Prototype Francis Turbine at Low-Load Operation. *Energies* **2019**, *12*, 2893.
https://doi.org/10.3390/en12152893

**AMA Style**

Unterluggauer J, Maly A, Doujak E.
Investigation on the Impact of Air Admission in a Prototype Francis Turbine at Low-Load Operation. *Energies*. 2019; 12(15):2893.
https://doi.org/10.3390/en12152893

**Chicago/Turabian Style**

Unterluggauer, Julian, Anton Maly, and Eduard Doujak.
2019. "Investigation on the Impact of Air Admission in a Prototype Francis Turbine at Low-Load Operation" *Energies* 12, no. 15: 2893.
https://doi.org/10.3390/en12152893