# Numerical Study of a Francis Turbine over Wide Operating Range: Some Practical Aspects of Verification

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## Abstract

**:**

## 1. Introduction

## 2. Test Case

## 3. Verification and Validation

^{±0.1 m}. However, the reduced numerical error and the minimized fluctuations in runner torque are very important as they reflect the global performance of the runner. Figure 5 shows the standard deviation of torque for the HydroFlex runner. The torque is normalized by the corresponding value at the maximum efficiency point. The standard deviation $\sigma $ corresponds to samples acquired during one complete rotation ($\phi ={360}^{\circ}$) of the runner. The deviation is high for some operating conditions, especially the 40%, 50%, 130% and 140% guide vane positions and ${n}_{ED}=0.19-0.22$. Unsteady torque fluctuations at some of these operating points are presented in Figure 6. The scale on the y-axis is different so as to preserve clarity in the high frequency fluctuations. The pattern of torque fluctuations across different operating conditions is quite interesting. Two operating points at 40% guide vane position, two operating points at 70% guide vane position and one operating point (${n}_{ED}=0.18$) at each 90%, 100%, 120% and 140% guide vane positions are shown in the figure. Two operating points at 40%, which guide vane opening, show quite different behaviors. At ${n}_{ED}=0.18$, the amplitudes ($f\approx 3{f}_{n}$) of torque fluctuations are small; however, at ${n}_{ED}=0.20$, the amplitudes ($f=0.5{f}_{n}$) are predominantly high. While comparing the results at the same speed factor (${n}_{ED}=0.18$), but different load/guide vane positions, the results are even more surprising, especially the frequency variation. Stochastic fluctuations are predominant at high load conditions and the 120% and 140% guide vane positions. At the location of maximum efficiency (Figure 6f), the amplitudes of torque fluctuations are moderate, and the frequency corresponds to the runner blades and the rotor–stator interactions. While comparing the signature of torque fluctuations, low frequency oscillations ($f=0.5-0.7$ Hz) were recorded at almost all operating points. Presently, it is unclear what causes such low frequency oscillations. This could be valuable to investigate in the future with longer simulation time, 20 – 25 revolutions of the runner, which would allow a large enough number of samples for spectral analysis and for examining the signature of low frequency fluctuations.

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

LES | Large eddy simulation |

RANS | Reynolds-averaged Navier–Stokes equation |

SAS | Scale adaptive simulation |

SBES | Stress blended eddy simulation |

SST | Shear stress transport |

D | Runner reference diameter (m), $D=0.349$ m |

${\widehat{e}}_{r-\eta}$ | Relative error in efficiency with respect to maximum efficiency |

f | Frequency (Hz) |

g | Gravity (m s^{−2}), $g=9.821465$ m s^{−2} |

H | Head (m) |

N | Number of data points |

${N}_{QE}$ | Specific speed, ${N}_{QE}={n}_{ED}{Q}_{ED}$ |

${N}_{s}$ | Specific speed, ${N}_{s}=N\sqrt{P}/{H}^{5/4}$ (rpm, kW, m) |

${n}_{ED}$ | Speed factor |

P | Power (W) |

p | Pressure (Pa) |

Q | Flow rate (m^{3} s^{−1}) |

${Q}_{ED}$ | Discharge factor |

s | Runner pitch |

T | Torque (Nm) |

t | Time (s) |

v | Velocity (m s^{−1}) |

$\alpha $ | Guide vane opening position (%) |

$\eta $ | Efficiency |

$\sigma $ | Standard deviation |

## Appendix A

${n}_{ED}$ | ${Q}_{ED}$ | n (rpm) | Q (m^{3} s^{−1}) | ${p}_{1}$ (Pa) | ${p}_{2}$ (Pa) | H (m) | T (Nm) | $\eta $ (%) | $\alpha $ (°) |
---|---|---|---|---|---|---|---|---|---|

0.180 | 0.063 | 531.50 | 0.133 | 355,025 | 60,633 | 29.99 | 603.40 | 86.120 | 3.95 |

0.180 | 0.080 | 531.40 | 0.168 | 355,825 | 61,526 | 30.04 | 792.5 | 89.280 | 5.01 |

0.180 | 0.095 | 531.40 | 0.199 | 356,325 | 62,523 | 30.03 | 955.5 | 90.970 | 6.02 |

0.180 | 0.110 | 531.40 | 0.230 | 353,725 | 60,322 | 30.06 | 1126.10 | 92.290 | 6.99 |

0.180 | 0.125 | 531.50 | 0.261 | 353,625 | 61,310 | 30.02 | 1288.90 | 93.230 | 8.00 |

0.180 | 0.139 | 531.40 | 0.291 | 353,725 | 62,217 | 30.01 | 1436.90 | 93.470 | 9.01 |

0.180 | 0.153 | 531.50 | 0.319 | 350,725 | 60,143 | 30.00 | 1578.40 | 93.520 | 10.02 |

0.180 | 0.166 | 531.60 | 0.347 | 352,025 | 62,756 | 30.02 | 1710.60 | 93.160 | 10.99 |

0.180 | 0.179 | 531.60 | 0.374 | 349,825 | 60,740 | 30.09 | 1843.80 | 92.970 | 12.00 |

0.180 | 0.191 | 531.70 | 0.399 | 348,925 | 61,145 | 30.04 | 1952.30 | 92.400 | 13.05 |

0.180 | 0.202 | 531.70 | 0.423 | 348,325 | 61,549 | 30.03 | 2054.20 | 91.760 | 14.02 |

${n}_{ED}$ | ${Q}_{ED}$ | n (rpm) | Q (m^{3} s^{−1}) | ${p}_{1}$ (Pa) | ${p}_{2}$ (Pa) | H (m) | T (Nm) | $\eta $ (%) | $\alpha $ (°) |
---|---|---|---|---|---|---|---|---|---|

0.180 | 0.057 | 531.50 | 0.118 | 353,840 | 60,579 | 29.98 | 549.87 | 88.120 | 3.95 |

0.180 | 0.073 | 531.40 | 0.153 | 354,390 | 61,472 | 29.99 | 722.04 | 90.638 | 5.01 |

0.180 | 0.087 | 531.40 | 0.181 | 354,400 | 62,508 | 29.93 | 879.27 | 92.137 | 6.02 |

0.180 | 0.100 | 531.40 | 0.209 | 351,120 | 60,390 | 29.86 | 1020.40 | 92.662 | 6.99 |

0.180 | 0.113 | 531.50 | 0.237 | 349,950 | 61,536 | 29.98 | 1247.70 | 92.625 | 8.00 |

0.180 | 0.130 | 531.40 | 0.272 | 348,800 | 62,658 | 29.91 | 1353.40 | 92.095 | 9.01 |

0.180 | 0.148 | 531.50 | 0.310 | 371,850 | 59,703 | 30.01 | 1530.00 | 95.309 | 10.02 |

0.180 | 0.156 | 531.60 | 0.326 | 350,450 | 62,288 | 29.89 | 1641.90 | 95.295 | 10.99 |

0.180 | 0.169 | 531.60 | 0.352 | 348,000 | 60,145 | 29.94 | 1771.20 | 95.148 | 12.00 |

0.181 | 0.180 | 531.70 | 0.374 | 346,090 | 60,756 | 29.75 | 1838.10 | 93.906 | 13.05 |

0.181 | 0.191 | 531.70 | 0.397 | 345,390 | 61,150 | 29.72 | 1930.70 | 93.008 | 14.02 |

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**Figure 3.**Iso-efficiency hill diagrams of newly designed HydroFlex runner (the present numerical simulations).

**Figure 4.**Relative deviation in hydraulic efficiency for all operating points of a performance hill diagram: (

**a**) Francis-99 runner—model acceptance tests; (

**b**) HydroFlex runner—numerical simulations; and (

**c**) difference between the Francis-99 and HydroFlex runners.

**Figure 6.**Signature of torque fluctuations at selected operating points across the hill diagram of the HydroFlex runner. The scale of the y-axis is kept different to maintain clarity in the high frequency fluctuations and the amplitude level.

**Table 1.**Boundary conditions and the numerical parameters enabled for the simulations. SAS-SST: scale adaptive simulation-shear stress transport.

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

Mesh | spiral casing—3.56 million, guide vanes—4.75 million, |

runner—5.63 million, draft tube—2.9 million. $0.1<{y}^{+}<30$ | |

Boundary types | Total pressure inlet and static pressure outlet |

Turbulence model | SAS-SST |

Advection scheme | High Resolution |

Time marching scheme | Second order backward Euler |

Time | Time step: 1° of runner rotation. Total time: three rotations |

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**MDPI and ACS Style**

Trivedi, C.; Iliev, I.; Dahlhaug, O.G. Numerical Study of a Francis Turbine over Wide Operating Range: Some Practical Aspects of Verification. *Sustainability* **2020**, *12*, 4301.
https://doi.org/10.3390/su12104301

**AMA Style**

Trivedi C, Iliev I, Dahlhaug OG. Numerical Study of a Francis Turbine over Wide Operating Range: Some Practical Aspects of Verification. *Sustainability*. 2020; 12(10):4301.
https://doi.org/10.3390/su12104301

**Chicago/Turabian Style**

Trivedi, Chirag, Igor Iliev, and Ole Gunnar Dahlhaug. 2020. "Numerical Study of a Francis Turbine over Wide Operating Range: Some Practical Aspects of Verification" *Sustainability* 12, no. 10: 4301.
https://doi.org/10.3390/su12104301