# Investigation of the Time Resolution Set Up Used to Compute the Full Load Vortex Rope in a Francis Turbine

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Test Case

^{3}/s, E the total energy in J/kg equal to 262.83, D the outlet runner diameter in m equal to 0.35 m, n the runner frequency in s

^{−1}equal to 13.33, and $NPSE$ the Net Positive Suction Energy.

## 3. Numerical Settings

## 4. Results

#### 4.1. Integrated Variables

#### 4.2. Local Pressure

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Computational domain (left top), view of the surface mesh of blade (left bottom) and side view of the structured mesh in a mid-plane (right).

**Figure 3.**Instantaneous iso-surface of the liquid volume fraction ${\alpha}_{L}=0.9$. Downstream view (top) and side view (bottom).

**Figure 4.**Time-history of the efficiency, the dimensionless volume of vapour, the torque factor ${T}_{ED}$ (measured value: 0.127) and the speed factor ${n}_{ED}$ (measured value: 0.288).

**Figure 5.**Time-history of the dimensionless volume of vapour in the runner (left) and in the draft tube (right).

**Figure 6.**Magnitude of the Fourier transform of the efficiency (left) and volume of vapour (right) signals.

**Figure 7.**Positions of pressure probe in the computational domain (left). Zoom on the runner probe locations (right).

**Figure 8.**Time history of the pressure fluctuations in the runner and the draft tube cone. The zero of experimental signal has been set arbitrary.

**Figure 9.**Magnitude of the Fourier transform of the pressure signal in the runner and the draft tube.

**Figure 11.**Time history of the dimensionless mass flow at the runner inlet section (left) and at the outlet of the computational domain (right).

**Figure 12.**Time history of the left- and right- hand terms of Equation (2) for each case.

**Table 1.**Parameters set for the time integration. Bold terms refer to the imposed values by the user.

Simulation | Time Step | CFL | Internal Loop | Number of Revolution | CPU Effort |
---|---|---|---|---|---|

(deg/$\mathbf{\Delta}\mathit{t}$) | (-) | (-) | (-) | (Hours) | |

Case 1 | 0.96 | 80 | 5 | 2.3 | 3${}^{\prime}$897 |

Case 2 | 0.96 | 80 | 15 | 13.3 | 76${}^{\prime}$861 |

Case 3 | adaptive | ≤2 | 3 | 3.15 | 137${}^{\prime}$444 |

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**MDPI and ACS Style**

Decaix, J.; Müller, A.; Favrel, A.; Avellan, F.; Münch-Alligné, C.
Investigation of the Time Resolution Set Up Used to Compute the Full Load Vortex Rope in a Francis Turbine. *Appl. Sci.* **2021**, *11*, 1168.
https://doi.org/10.3390/app11031168

**AMA Style**

Decaix J, Müller A, Favrel A, Avellan F, Münch-Alligné C.
Investigation of the Time Resolution Set Up Used to Compute the Full Load Vortex Rope in a Francis Turbine. *Applied Sciences*. 2021; 11(3):1168.
https://doi.org/10.3390/app11031168

**Chicago/Turabian Style**

Decaix, Jean, Andres Müller, Arthur Favrel, François Avellan, and Cécile Münch-Alligné.
2021. "Investigation of the Time Resolution Set Up Used to Compute the Full Load Vortex Rope in a Francis Turbine" *Applied Sciences* 11, no. 3: 1168.
https://doi.org/10.3390/app11031168