# Comparison of Pressure Pulsation Characteristics of Francis Turbine with Different Draft Tube Arrangement Direction

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## Abstract

**:**

## 1. Introduction

## 2. Francis Turbine Unit

#### 2.1. Basic Parameters

_{rn}is 5.80 m. The rated output power P

_{r}is 183.7 MW. The rated flow rate Q

_{r}is 301 m

^{3}/s. The rated rotation speed n

_{r}is 100 rpm. The rated water head H

_{r}is 68.0 m. The maximum output power P

_{max}is 204.1 MW. The turbine installation height H

_{ins}is 895 m.

#### 2.2. Asymmetric Arrangement of Draft Tube

## 3. Numerical Setup

#### 3.1. Governing Equations

_{ij}is the Kroneker delta. The term named Reynold stress, represented as ρu

_{i}’u

_{j}’, is not closed, so Boussinesq introduced the turbulence isotropy assumption, and it is developed to build the relationship between Reynolds stress and eddy viscosity μ

_{t}[31]:

_{ij}is the mean rate of strain tensor:

_{t}is connected to turbulence kinetic energy k and turbulence eddy frequency ω based on the experiment, and the shear stress transport (SST) model [32] is established by:

_{k}and P

_{ω}are the production terms, F

_{1}is the coefficient of the production term, σ

_{k}is the blending function, and σ

_{ω}and β

_{k}are model constants. l

_{k−ω}is the turbulence scale, formulated as follows:

#### 3.2. CFD Setup with Monitoring Points

^{−5}for the residuals of both continuity and momentum equations. Figure 3 shows the monitoring points set in the draft tube. To investigate the pressure pulsation caused by vortex rope, P1~P4 are set as indicated. P1 and P2 are on the same higher plane and located on the left side and right side near the wall. P3 and P4 are on the same lower plane and located on the left side and right side also near the wall. In this CFD simulation work, we used a high-performance parallel computing workstation. The AMD EPYV 7813 64-core processor is used as the CPU with 128 threads. The frequency is 2.25 GHz. The memory of the workstation is 128 GB. For each transient simulation, the CPU time is approximately 82,800~90,000 s.

#### 3.3. Determination of Mesh

## 4. Results of Performance Analysis

#### 4.1. Efficiency Comparison

_{alw}, C

_{rst}, and C

_{phb}with the unit rotation speed n

_{11}of 77.85, 78.55, and 91.21 and the unit flow rate Q

_{11}of 0.7864, 0.3678, and 0.3495. The definitions for n

_{11}and Q

_{11}are as follows:

_{alw}, the CFD efficiency is 91.05% for LT and 91.08% for RT, and the experimental efficiency is 90.51%. At C

_{rst}, the CFD efficiency is 71.72% for LT and 71.69% for RT, and the experimental efficiency is 70.32%. At C

_{phb}, the CFD efficiency is 55.25% for LT and 55.23% for RT, and the experimental efficiency is 52.88%. RT and LT show a good match on the performance. The CFD value shows a good prediction of unit performance and can be used for subsequent analysis.

#### 4.2. The Internal Flow Pattern

_{alw}point in the allowed region, the flow is relatively stable and the streamline is smooth. Velocity in runner is uniform and up to about 20 m/s. At the C

_{rst}point in the restricted region, the velocity in the runner increases to a maximum of about 38 m/s and the distribution of velocity becomes non-uniform. The flow in the draft tube develops many vortexes. The vortical flow dominates the component downstream to the runner and seems out of control. At the C

_{phb}point in the prohibited region, the velocity in the runner is still relatively high (maximum value is about 35 m/s). The vortical flow in the draft tube further intensifies. In comparison, the flow status is relatively worse in prohibited and restricted operating regions.

## 5. Results of Pressure Pulsation Analysis

#### 5.1. Analysis of Time Domain and Frequency Domain

_{r}(1.67 Hz), the peak value of the frequency domain diagram of IMF4 is 1032 Pa, and the peak frequency is 2.78 Hz. When the draft tube tilts to the right, the peak value of the frequency domain diagram of IMF5 is 5126 Pa, the peak frequency is 1.67 Hz, the peak value of the frequency domain diagram of IMF4 is 1041 Pa, and the peak frequency is 2.78 Hz. It can be seen that the pressure pulsation of the draft tube is mainly affected by the rotating frequency of runner f

_{r}, while the left and right tilts of the draft tube have little influence on the IMF1 and IMF2 components. The frequency domain diagram of IMF1–IMF3 is composed of multiple high-frequency signals with low amplitude. When the draft tube tilts to the left, the center frequencies of IMF1–IMF3 components are 137.1 Hz, 81.36 Hz, and 6.24 Hz, respectively. When the draft tube tilts to the right, the center frequencies of IMF1–IMF3 components are 87.7 Hz, 42.3 Hz, and 5.51 Hz, respectively. Different draft tube tilt conditions have a greater impact on the center frequency of high-frequency components. When the draft tube tilts to the left, the center frequency of the IMF1–IMF3 component of pressure pulsation at the P1 monitoring point is significantly greater than that of the draft tube tilts to the right.

_{r}. Compared with the draft tube tilting to the right, the pressure pulsation amplitude of the draft tube tilting to the left can be reduced by 11.9%. When the draft tube is tilted to the left, the center frequencies of IMF1–IMF3 components are 141.41 Hz, 99.89 Hz, and 46.85 Hz, respectively. When the draft tube tilts to the right, the center frequencies of IMF1–IMF3 components are 78.28 Hz, 42.06 Hz, and 5.78 Hz, respectively. At this point, there is a significant difference in the center frequency of the pressure pulsation component between the P4 monitoring point and the P3 monitoring point under both operating conditions. This shows that as the monitoring point moves downstream of the draft tube, the larger the draft tube tilt, the greater the impact of the tilt on the draft tube pressure pulsation, and the more significant the impact of the left tilt of the draft tube on reducing the amplitude of pressure pulsation.

#### 5.2. The Law of Pressure Pulsation

_{r}(1.67 Hz), and <1 Hz. The frequency value of the same mode of left tilt is higher than that of the right tilt. In addition, the intensity of the runner frequency f

_{r}is very important, as analyzed in Figure 13. On P1, the amplitudes of LT and RT are similar. On P2, RT has a higher amplitude of f

_{r}than that of LT. On P3, RT has a slightly higher amplitude of f

_{r}than that of LT. On P4, RT has a higher amplitude of f

_{r}than that of LT and the difference becomes bigger. Generally, the right tilt draft tube may trigger higher pulsation of the pressure field.

## 6. Conclusions

- (1)
- Left-tilt arrangement and right-tilt arrangement of the draft tube have little impact on the unit performance of the Francis turbine. By comparing the CFD calculation and analysis results with the prototype measurement results, it can be found that the direction of the asymmetric arrangement of the draft tube has little effect on the efficiency. As long as the cross-sectional area remains unchanged, the variation in total pressure remains unchanged, and the impact of left and right tilts on different working conditions is relatively small, typically less than 1% on efficiency. Therefore, from the perspective of hydraulic performance, the tilt direction of the draft tube can be left or right.
- (2)
- From the internal flow, it can be seen that there are significant differences in the flow inside the unit under different operating conditions. In the allowed region with higher efficiency, the flow smoothness is relatively high, and there are almost no obvious vortices or other forms of adverse flow patterns visible. In restricted and prohibited regions, the velocity distribution will be uneven, and the local velocity may be very high. Flow in the draft tube has also become chaotic, mainly with large-scale strong rotating flow. Judging from the characteristics of the flow direction, both the left-tilt and right-tilt draft tubes are acceptable. The impact of the left-tilt arrangement and right-tilt arrangement is still not significant, and the difference is almost invisible.
- (3)
- Based on the mode decomposition of the pressure fluctuation signal by VMD, it can be seen that different arrangement directions of the draft tube will have some effects on the flow. Overall, the frequency characteristics of the two are relatively similar with no huge differences. However, the dominant mode of the left tilt corresponds to a higher frequency, while the right tilt corresponds to a lower frequency. In addition, in terms of the amplitude of the important runner frequency f
_{r}, the value corresponding to the right tilt is slightly higher than that of the left tilt. From a hydrodynamic perspective, the geometric features of flow-passing components that are more adaptable to flow are more acceptable. If the left tilt is used, the fluctuation in the flow field may be more stable due to the same arrangement direction of the draft tube and volute.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Kose, F.; Kaya, M.N. Analysis on meeting the electric energy demand of an active plant with a wind-hydro hybrid power station in Konya, Turkey: Konya water treatment plant. Renew. Energy
**2013**, 55, 196–201. [Google Scholar] [CrossRef] - Wu, Y.; Zhang, J.; Yuan, J.; Geng, S.; Zhang, H. Study of decision framework of offshore wind power station site selection based on ELECTRE-III under intuitionistic fuzzy environment: A case of China. Energy Convers. Manag.
**2016**, 113, 66–81. [Google Scholar] [CrossRef] - Tao, R.; Lu, J.; Jin, F.; Hu, Z.; Zhu, D.; Luo, Y. Evaluation of the rotor eccentricity added radial force of oscillating water column. Ocean Eng.
**2023**, 276, 114222. [Google Scholar] [CrossRef] - Dong, J.; Feng, T.-T.; Yang, Y.-S.; Ma, Y. Macro-site selection of wind/solar hybrid power station based on ELECTRE-II. Renew. Sustain. Energy Rev.
**2014**, 35, 194–204. [Google Scholar] - Jafari, B.; Seddiq, M.; Mirsalim, S.M. Impacts of diesel injection timing and syngas fuel composition in a heavy-duty RCCI engine. Energy Convers. Manag.
**2021**, 247, 114759. [Google Scholar] [CrossRef] - Ping, X.; Yang, F.; Zhang, H.; Zhang, J.; Xing, C.; Yan, Y.; Yang, A.; Wang, Y. Information theory-based dynamic feature capture and global multi-objective optimization approach for organic Rankine cycle (ORC) considering road environment. Appl. Energy
**2023**, 348, 121569. [Google Scholar] [CrossRef] - Jafari, B.; Seddiq, M.; Mirsalim, S.M. Assessment of the impacts of combustion chamber bowl geometry and injection timing on a reactivity controlled compression ignition engine at low and high load conditions. Int. J. Engine Res.
**2021**, 22, 2852–2868. [Google Scholar] [CrossRef] - Zhou, M.; Ye, Q.; Liu, Z. Climate impacts on hydro-power development in China. Bot. Mar.
**2005**, 19, 1–4. [Google Scholar] - Pereira, J.G.; Vagnoni, E.; Favrel, A.; Landry, C.; Alligne, S.; Nicolet, C.; Avellan, F. Prediction of unstable full load conditions in a Francis turbine prototype. Mech. Syst. Signal Process.
**2022**, 169, 108666. [Google Scholar] [CrossRef] - Kalantar, M.; Mousavi, S.M.G. Dynamic behavior of a stand-alone hybrid power generation system of wind turbine, microturbine, solar array and battery storage. Appl. Energy
**2010**, 87, 3051–3064. [Google Scholar] [CrossRef] - Zhu, D.; Tao, R.; Xiao, R.; Pan, L. Solving the runner blade crack problem for a Francis hydro-turbine operating under condition-complexity. Renew. Energy
**2020**, 149, 298–320. [Google Scholar] [CrossRef] - Li, P.; Xiao, R.; Tao, R. Study of vortex rope based on flow energy dissipation and vortex identification. Renew. Energy
**2022**, 198, 1065–1081. [Google Scholar] [CrossRef] - Zhu, D.; Yan, W.; Guang, W.; Wang, Z.; Tao, R. Influence of guide vane opening on the runaway stability of a pump-turbine used for hydropower and ocean power. J. Mar. Sci. Eng.
**2023**, 11, 1218. [Google Scholar] [CrossRef] - Tamura, Y.; Tani, K.; Okamoto, N. Experimental and numerical investigation of unsteady behavior of cavitating vortices in draft tube of low specific speed Francis turbine. IOP Conf. Ser. Earth Environ. Sci.
**2014**, 22, 032011. [Google Scholar] [CrossRef] - Arthur, F.; Muller, A.; Landry, C.; Yamamoto, K.; Avellan, F. LDV survey of cavitation and resonance effect on the precessing vortex rope dynamics in the draft tube of Francis turbines. Exp. Fluids
**2016**, 57, 168. [Google Scholar] - Zhang, J.; Appiah, D.; Zhang, F.; Yuan, S.; Gu, Y.; Asomani, S.N. Experimental and numerical investigations on pressure pulsation in a pump mode operation of a pump as turbine. Energy Sci. Eng.
**2019**, 7, 1264–1279. [Google Scholar] [CrossRef] - Bosioc, A.I.; Susan-Resiga, R.; Muntean, S.; Tanasa, C. Unsteady pressure analysis of a swirling flow with vortex rope and axial water injection in a discharge cone. J. Fluids Eng.
**2012**, 134, 669–679. [Google Scholar] [CrossRef] - Geng, C.; Li, Y.; Tsujimoto, Y.; Nishi, M.; Luo, X. Pressure oscillations with ultra-low frequency induced by vortical flow inside Francis turbine draft tubes. Sustain. Energy Technol. Assess.
**2022**, 51, 101908. [Google Scholar] [CrossRef] - Liu, S.; Shao, Q.; Yang, J.; Wu, Y.; Dai, J. Unsteady turbulent simulation of Three Gorges hydraulic turbine and analysis of pressure in the whole passage. J. Hydroelectr. Eng.
**2004**, 23, 97–101. [Google Scholar] - Liao, W.; Ji, J.; Lu, P.; Luo, X. Unsteady flow analysis of francis turbine. Chin. J. Mech. Eng.
**2009**, 45, 134–140. [Google Scholar] [CrossRef] - Zhang, Y.; Liu, S.; Wu, Y. Detailed simulation and analysis of pressure pulsation in Francis turbine. J. Hydroelectr. Eng.
**2009**, 28, 183–186. [Google Scholar] - Pasche, S.F.; Gallaire, F.; Avellan, F. Origin of the synchronous pressure pulsations in the draft tube of Francis turbines operating at part load conditions. J. Fluids Struct.
**2019**, 86, 13–33. [Google Scholar] [CrossRef] - Kim, H.H.; Rakibuzzaman, M.; Kim, K.; Suh, S.-H. Flow and fast fourier transform analyses for tip clearance effect in an operating kaplan turbine. Energies
**2019**, 12, 264. [Google Scholar] [CrossRef] - Wu, Y.; Guang, W.; Tao, R.; Liu, J. Dynamic mode structure analysis of the near-wake region of a Savonius-type hydrokinetic turbine. Ocean Eng.
**2023**, 282, 114965. [Google Scholar] [CrossRef] - Tian, L.; Wu, J. Fault diagnosis and analysis of submersible sewage pump based on zoom-fft. Mach. Des. Res.
**2018**, 34, 171–174. [Google Scholar] - Wang, W.; Tai, G.; Shen, J. Experimental investigation on pressure pulsation characteristics of a mixed-flow pump as turbine at turbine and runaway conditions. J. Energy Storage
**2022**, 55 Pt C, 105562. [Google Scholar] [CrossRef] - Tang, Z.; Wang, M.; Ouyang, T.; Fei, C. A wind turbine bearing fault diagnosis method based on fused depth features in time–frequency domain. Energy Rep.
**2022**, 8, 12727–12739. [Google Scholar] [CrossRef] - Jin, F.; Tao, R.; Lu, Z.; Xiao, R. A spatially distributed network for tracking the pulsation signal of flow field based on CFD simulation: Method and a case study. Fractal Fract.
**2021**, 5, 181. [Google Scholar] [CrossRef] - Lu, Z.; Tao, R.; Jin, F.; Li, P.; Xiao, R.; Liu, W. The Temporal-Spatial Features of Pressure Pulsation in the Diffusers of a Large-Scale Vaned-Voluted Centrifugal Pump. Machines
**2021**, 9, 266. [Google Scholar] [CrossRef] - Pope, S.B. Turbulent Flows; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
- Terentiev, L. The Turbulence Closure Model Based on Linear Anisotropy Invariant Analysis; VDM Verlag: Saarbrucken, Germany, 2008. [Google Scholar]
- Menter, F.R. Two-Equation Eddy-Viscosity Turbulence Models for Engineering Applications. AIAA J.
**1994**, 32, 1598–1605. [Google Scholar] [CrossRef] - Celik, I.B.; Ghia, U.; Roache, P.J.; Freitas, C.J.; Coloman, H.; Raad, P.E. Procedure for estimation and reporting of uncertainty due to discretization in CFD applications. J. Fluids Eng.
**2008**, 130, 78001. [Google Scholar]

**Figure 9.**Time domain diagram and corresponding spectrum of P1 pressure pulsation signal based on VMD decomposition under different operating conditions.

**Figure 10.**Time domain diagram and corresponding spectrum of P2 pressure pulsation signal based on VMD decomposition under different operating conditions.

**Figure 11.**Time domain diagram and corresponding spectrum of P3 pressure pulsation signal based on VMD decomposition under different operating conditions.

**Figure 12.**Time domain diagram and corresponding spectrum of P4 pressure pulsation signal based on VMD decomposition under different operating conditions.

**Figure 13.**Pulsation amplitude of runner frequency f

_{r}on all four monitoring points of the left tilt and right tilt of the draft tube.

Component | Mesh Number | Range of y+ | Average of y+ |
---|---|---|---|

Volute | 209,260 | 11–230 | 53 |

Stay vane | 558,230 | 2–165 | 38 |

Guide vane | 623,824 | 2–165 | 45 |

Runner | 897,680 | 1.5–211 | 31 |

Draft tube | 464,692 | 16–253 | 88 |

Total | 2,753,686 | — | — |

Left Tilt | Right Tilt | |||||||
---|---|---|---|---|---|---|---|---|

P1 | P2 | P3 | P4 | P1 | P2 | P3 | P4 | |

IMF1 (Hz) | 137.10 | 141.41 | 139.33 | 138.67 | 87.70 | 78.28 | 102.43 | 71.15 |

IMF2 (Hz) | 81.36 | 99.89 | 93.67 | 78.72 | 42.30 | 42.06 | 45.16 | 40.25 |

IMF3 (Hz) | 6.24 | 46.85 | 41.29 | 4.67 | 5.51 | 5.78 | 20.28 | 2.58 |

IMF4 (Hz) | 2.89 | 2.77 | 2.27 | 2.35 | 2.60 | 2.80 | 2.75 | 1.34 |

IMF5 (Hz) | 1.41 | 1.38 | 1.33 | 1.33 | 1.35 | 1.40 | 1.37 | 0.34 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, T.; Hu, Z.; Liu, X.; Lu, J.; Song, X.; Zhu, D.; Wang, Z.
Comparison of Pressure Pulsation Characteristics of Francis Turbine with Different Draft Tube Arrangement Direction. *Water* **2023**, *15*, 4028.
https://doi.org/10.3390/w15224028

**AMA Style**

Zhang T, Hu Z, Liu X, Lu J, Song X, Zhu D, Wang Z.
Comparison of Pressure Pulsation Characteristics of Francis Turbine with Different Draft Tube Arrangement Direction. *Water*. 2023; 15(22):4028.
https://doi.org/10.3390/w15224028

**Chicago/Turabian Style**

Zhang, Tao, Zilong Hu, Xinjun Liu, Jiahao Lu, Xijie Song, Di Zhu, and Zhengwei Wang.
2023. "Comparison of Pressure Pulsation Characteristics of Francis Turbine with Different Draft Tube Arrangement Direction" *Water* 15, no. 22: 4028.
https://doi.org/10.3390/w15224028