#
Numerical Simulation of the Flow in a Kaplan Turbine Model during Transient Operation from the Best Efficiency Point to Part Load^{ †}

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

_{runner}, where f

_{runner}represents the runner rotational frequency. It is therefore necessary to accurately predict the RVR formation in order to avoid the unfavorable operation of hydraulic turbines. The analysis of the vortex formation and its mitigation may allow for the development of effective countermeasures. Both experimental and numerical techniques have been used for such investigations.

## 2. Materials and Methods

#### 2.1. Test Case

_{GV}= 0.859°/s and the total time of the load variation is Δt = 7.57 s. The main parameters for the BEP and PL operating points are presented in Table 1.

#### 2.2. Numerical Model

#### 2.1.1. Analysis Domain

#### 2.1.2. Mesh

^{6}. The mesh quality criteria are presented in Table 2.

^{+}values. The size of the guide vane mesh is comparable to the previous version, but the runner domain mesh is refined.

^{+}values presented in Table 2 are, in some areas, larger than the recommended value of 1; therefore, the automatic wall function is locally employed in the CFX simulations [36]. This automatic wall treatment switches between the wall function approach and the low Re approach (k-ω) depending on the grid spacing near the wall [37]. A very fine mesh, that meets the y

^{+}<1 criterion, considerably increases the mesh size, the computational demands and the total simulation time.

_{c}, y

_{c}) shown in Figure 5. The location of the guide vane blade (x

_{t}, y

_{t}) is calculated at each time step using coordinates relative to the previous mesh position (x

_{t}

_{−1}, y

_{t}

_{−1}):

_{t}and y

_{t}are the coordinates of the guide vane blade at the current time step, x

_{t}

_{−1}and y

_{t}

_{−1}are the coordinates of the guide vane blade at the previous time step and x

_{c}and y

_{c}are the coordinates of the center of rotation.

_{GV}), related to the initial position, is calculated at each time step (dt) as:

#### 2.1.3. Boundary Conditions

#### 2.1.4. Time Step Sensitivity Analysis

## 3. Results and Discussion

#### 3.1. Validation with Experimental Velocity Profiles

^{+}values). Therefore, the simulations show a recirculation area formed at the inlet of the draft tube, even during BEP operation. The different time steps have a marginal effect on the runner and no effect on the draft tube as the velocity profiles overlap on a single profile.

#### 3.2. Inlet Boundary Conditions

#### 3.2.1. Main Parameters

_{GV}= 6.5° and PL operation is reached.

_{Q}) is specified at the inlet of the guide vane domain according to the experiment. Using the total constant pressure as the inlet boundary condition leads to an underestimation of the flow rate (Q

_{P}) by approximatively 3% during PL operation. The influence of the different boundary conditions over the variation of the parameters becomes visible around t = 4 s. The results confirm that the head losses are overestimated by the turbulence model. The justification for this overestimation is that the head losses are predominantly due to turbulent kinetic energy dissipation [23], hence being overestimated by URANS models. Another possible cause may be the wall function used in the numerical simulations for y

^{+}> 1 [39]. The turbulent fluctuations are overpredicted, and the energy losses are large, especially near the solid boundaries. However, resolving the boundary layers up to y

^{+}= 1 would result in a too large number of hexahedral cells to perform calculation in a reasonable amount of time.

#### 3.2.2. Pressure Oscillations

_{runner}.

_{runner}= 11.61 Hz. All simulations show amplitude peaks at approximately the same dimensionless frequencies. Their amplitude is, however, strongly influenced by the time step size.

_{runner}= 0.19) and the asynchronous mode of the RVR (f/f

_{runner}= 0.81) on the runner, corresponding to the two largest amplitude peaks captured in Figure 12a. The lower peaks represent the harmonics of the two modes. Figure 12b shows that the monitor point located on the draft tube wall, in the stationary frame of reference, still captures the same frequency of 0.19 × f

_{runner}and its harmonics. The results confirm the experimental values reported in [6,31].

#### 3.3. Rotating Vortex Rope

#### 3.3.1. Flow Structure

#### 3.3.2. Spectral Analysis

_{inf}= 0.15 × f

_{runner}and f

_{sup}= 0.25 × f

_{runner}(Figure 15c).

_{runner}[6]. The pressure fluctuations caused by the RVR rotating mode have a frequency of 0.8 × f

_{runner}in the rotating frame of reference, i.e., the runner.

_{pl}= 0.2 × f

_{runner}. The spectrogram shows that, in the runner rotating domain, the rotating component appears 1 s later than the plunging component at the dimensionless frequency of f

_{rot}= 0.8 × f

_{runner}.

_{runner}in both cases. However, the time of appearance is not clear. The resolution of the spectrograms is, of course, influenced by the total number of samples and the sampling frequency corresponding to the simulation time step.

#### 3.3.3. Decomposition of the Rotating Vortex Rope

_{ax}*) monitored during the transient simulation is presented in Figure 18 starting from the beginning of the guide vane closure (t = 1.3 s). The velocity was recorded using 39 monitor points defined along the diameter of the draft tube cone (Section I and I

_{ex}, Figure 5). The velocity values are normalized with respect to the bulk velocity at this section, calculated from the flow rate. The contour plot starts at t = 1.3 s, corresponding to the end of the BEP steady-state simulation (Figure 18a).

_{GV}= 22.9°) the stagnation region is developing along the centerline of the draft tube. Just below the runner hub, the velocity values are negative. The RVR is already formed at t = 6.6 s and the bulk flow is visibly pushed towards the walls. By the end of the transient operation, at t = 8.87 s, the RVR is wrapped around the low velocity areas.

## 4. Conclusions

_{runner}) and its harmonics were captured by all monitor points. The plunging and rotating modes of the pressure fluctuations induced by the RVR were also visible in the spectrograms. The plunging mode component was captured by the numerical simulation approximately 4 s after the start of the guide vane closure at f

_{pl}= 0.2 × f

_{runner}. The spectrogram of the pressure monitored on the runner blade showed that the rotating component appeared 1 s later at the dimensionless frequency of f

_{rot}= 0.8 × f

_{runner}. The numerical results were similar to the values determined experimentally [6].

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Banshwar, A.; Sharma, N.K.; Sood, Y.R.; Shrivastava, R. Market based procurement of energy and ancillary services from renewable energy sources in deregulated environment. Renew. Energy
**2017**, 101, 1390–1400. [Google Scholar] [CrossRef] - Amiri, K.; Cervantes, M.J.; Mulu, B. Experimental investigation of the hydraulic loads on the runner of a Kaplan turbine model and the corresponding prototype. J. Hydraul. Res.
**2015**, 53, 452–465. [Google Scholar] [CrossRef] - Goyal, R.; Cervantes, M.J.; Gandhi, B.K. Vortex rope formation in a high head model Francis turbine. J. Fluids Eng.
**2017**, 139, 041102. [Google Scholar] [CrossRef] - Houde, S.; Fraser, R.; Ciocan, G.; Deschênes, C. Experimental study of the pressure fluctuations on propeller turbine runner blades: Part 2, transient conditions. IOP Conf. Ser. Earth Environ. Sci.
**2012**, 15, 062061. [Google Scholar] [CrossRef] [Green Version] - Chirag, T.; Cervantes, M.J.; Bhupendrakumar, G.; Dahlhaug, O.G. Pressure measurements on a high-head Francis turbine during load acceptance and rejection. J. Hydraul. Res.
**2014**, 52, 283–297. [Google Scholar] [CrossRef] - Amiri, K.; Mulu, B.; Raisee, M.; Cervantes, M.J. Unsteady pressure measurements on the runner of a Kaplan turbine during load acceptance and load rejection. J. Hydraul. Res.
**2016**, 54, 1–18. [Google Scholar] [CrossRef] - Ciocan, G.D.; Iliescu, M.S.; Vu, T.C.; Nennemann, B.; Avellan, F. Experimental study and numerical simulation of the FLINDT draft tube rotating vortex. J. Fluids Eng.
**2006**, 129, 146–158. [Google Scholar] [CrossRef] - Favrel, A.; Müller, A.; Landry, C.; Yamamoto, K.; Avellan, F. Study of the vortex-induced pressure excitation source in a Francis turbine draft tube by particle image velocimetry. Exp. Fluids
**2015**, 56. [Google Scholar] [CrossRef] - Lipej, A.; Čelič, D.; Tartinville, B.; Mezine, M.; Hirsch, C. Reduction of CPU time for CFD analysis of hydraulic machinery development process. IOP Conf. Ser. Earth Environ. Sci.
**2012**, 15, 62011. [Google Scholar] [CrossRef] [Green Version] - Židonis, A.; Aggidis, G. State of the art in numerical modelling of Pelton turbines. Renew. Sustain. Energy Rev.
**2015**, 45, 135–144. [Google Scholar] [CrossRef] - Trivedi, C.; Cervantes, M.J. State of the art in numerical simulation of high head Francis turbines. Renew. Energy Environ. Sustain.
**2016**, 1, 20. [Google Scholar] [CrossRef] [Green Version] - Denton, J.D. Some limitations of turbomachinery CFD. In Proceedings of the ASME Turbo Expo 2010: Power for Land, Sea, and Air, Glasgow, UK, 14–18 June 2010; pp. 735–745. [Google Scholar] [CrossRef]
- Tang, T. Moving mesh computations for computational fluid dynamics. Contemp. Math.
**2005**, 383, 141–173. [Google Scholar] [CrossRef] [Green Version] - Matsushima, K.; Murayama, M.; Nakahashi, K. Unstructured dynamic mesh for large movement and deformation. In Proceedings of the 40th AIAA Aerospace Sciences Meeting & Exhibit, Reno, NV, USA, 14–17 January 2002. [Google Scholar] [CrossRef]
- Perot, J.B.; Nallapati, R. A moving unstructured staggered mesh method for the simulation of incompressible free-surface flows. J. Comput. Phys.
**2003**, 184, 192–214. [Google Scholar] [CrossRef] - Liu, J.; Liu, S.; Yuekun, S.; Yulin, W.; Leqin, W. Three dimensional flow simulation of load rejection of a prototype pump-turbine. Eng. Comput.
**2013**, 29, 417–426. [Google Scholar] [CrossRef] - Kolšek, T.; Duhovnik, J.; Bergant, A. Simulation of unsteady flow and runner rotation during shut-down of an axial water turbine. J. Hydraul. Res.
**2006**, 44, 129–137. [Google Scholar] [CrossRef] - Li, J.; Yu, J.; Wu, Y. 3D unsteady turbulent simulations of transients of the Francis turbine. IOP Conf. Ser. Earth Environ. Sci.
**2010**, 12, 012001. [Google Scholar] [CrossRef] - Mössinger, P.; Jester-Zürker, R.; Jung, A. Francis-99: Transient CFD simulation of load changes and turbine shutdown in a model sized high-head Francis turbine. J. Phys. Conf. Ser.
**2017**, 782, 12001. [Google Scholar] [CrossRef] [Green Version] - Gagnon, J.-M.; Flemming, F.; Qian, R.; Deschênes, C.; Coulson, S. Experimental and numerical investigations of inlet boundary conditions for a propeller turbine draft tube. In Proceedings of the ASME 2010 3rd Joint US-European Fluids Engineering Summer Meeting, Montreal, QC, Canada, 1–5 August 2010. [Google Scholar] [CrossRef]
- Nilsson, H.; Cervantes, M.J. Effects of inlet boundary conditions, on the computed flow in the Turbine-99 draft tube, using OpenFOAM and CFX. IOP Conf. Ser. Earth Environ. Sci.
**2012**, 15, 32002. [Google Scholar] [CrossRef] [Green Version] - Nicolle, J.; Giroux, A.M.; Morissette, J.F. CFD configurations for hydraulic turbine startup. IOP Conf. Ser. Earth Environ. Sci.
**2014**, 22, 32021. [Google Scholar] [CrossRef] - Wilhelm, S.; Balarac, G.; Métais, O.; Ségoufin, C. Analysis of head losses in a turbine draft tube by means of 3D unsteady simulations. Flow Turbul. Combust.
**2016**, 97, 1255–1280. [Google Scholar] [CrossRef] [Green Version] - Nicolet, C.; Arpe, J.; Avellan, F. Identification and modelling of pressure fluctuations of a Francis turbine scale model at part load operation. In Proceedings of the IAHR 22nd Symposium on Hydraulic Machinery and Systems, Stockholm, Sweden, 29 June–2 July 2004. [Google Scholar]
- Liu, S.; Li, S.; Wu, Y. Pressure fluctuation prediction of a model Kaplan turbine by unsteady turbulent flow simulation. J. Fluids Eng.
**2009**, 131, 101102. [Google Scholar] [CrossRef] - Wu, Y.; Liu, S.; Dou, H.-S.; Wu, S.; Chen, T. Numerical prediction and similarity study of pressure fluctuation in a prototype Kaplan turbine and the model turbine. Comput. Fluids
**2011**, 56, 128–142. [Google Scholar] [CrossRef] - Ruprecht, A.; Helmrich, T.; Aschenbrenner, T.; Scherer, T. Simulation of vortex rope in a turbine draft tube. In Proceedings of the Hydraulic Machinery and Systems 21st IAHR Symposium, Lausanne, Switzerland, 9–12 September 2002. [Google Scholar]
- Maddahian, R.; Cervantes, M.J.; Sotoudeh, N. Numerical investigation of the flow structure in a Kaplan draft tube at part load. IOP Conf. Ser. Earth Environ. Sci.
**2016**, 49, 22008. [Google Scholar] [CrossRef] [Green Version] - Iovănel, R.G.; Bucur, D.M.; Dunca, G.; Cervantes, M.J. Numerical analysis of a Kaplan turbine model during transient operation. IOP Conf. Ser. Earth Environ. Sci.
**2019**, 240, 022046. [Google Scholar] [CrossRef] - Mulu, B.; Jonsson, P.; Cervantes, M.J. Experimental investigation of a Kaplan draft tube—Part I: Best efficiency point. Appl. Energy
**2012**, 93, 695–706. [Google Scholar] [CrossRef] - Jonsson, P.; Mulu, B.; Cervantes, M.J. Experimental investigation of a Kaplan draft tube—Part II: Off-design conditions. Appl. Energy
**2012**, 94, 71–83. [Google Scholar] [CrossRef] - Amiri, K.; Mulu, B.; Cervantes, M.J. Experimental investigation of the interblade flow in a Kaplan runner at several operating points using Laser Doppler Anemometry. J. Fluids Eng.
**2015**, 138, 021106. [Google Scholar] [CrossRef] - Melot, M.; Nennemann, B.; Désy, N. Draft tube pressure pulsation predictions in Francis turbines with transient Computational Fluid Dynamics methodology. IOP Conf. Ser. Earth Environ. Sci.
**2014**, 22, 32002. [Google Scholar] [CrossRef] [Green Version] - Alfonsi, G. Reynolds-averaged Navier–Stokes equations for turbulence modeling. Appl. Mech. Rev.
**2009**, 62, 040802. [Google Scholar] [CrossRef] - Mulu, B.G.; Cervantes, M.J.; Devals, C.; Vu, T.C.; Guibault, F. Simulation-based investigation of unsteady flow in near-hub region of a Kaplan Turbine with experimental comparison. Eng. Appl. Comput. Fluid Mech.
**2015**, 9, 1–18. [Google Scholar] [CrossRef] [Green Version] - Menter, F.; Carregal-Ferreira, J.; Esch, T.; Konno, B. The SST turbulence model with improved wall treatment for heat transfer predictions in gas turbines. In Proceedings of the International Gas Turbine Congress, Tokyo, Japan, 2–7 November 2003. [Google Scholar]
- Ansys 16.2 Documentation CFX Modelling guide 4.2. Modelling Flow Near the Wall. 2014. Available online: http://http://read.pudn.com/downloads500/ebook/2077964/cfx_mod.pdf (accessed on 12 June 2020).
- Trivedi, C.; Cervantes, M.J.; Dahlhaug, O.G. Experimental and numerical studies of a high-head Francis turbine: A review of the Francis-99 test case. Energies
**2016**, 9, 74. [Google Scholar] [CrossRef] [Green Version] - Nicolle, J.; Cupillard, S. Prediction of dynamic blade loading of the Francis-99 turbine. J. Phys. Conf. Ser.
**2015**, 579, 012001. [Google Scholar] [CrossRef] [Green Version] - Iovanel, R.G. Numerical Investigation of the Flow in a Kaplan Turbine. Ph.D. Thesis, Politehnica University of Bucharest, Bucharest, Romania, 2018. [Google Scholar]
- Nishi, M.; Kubota, T.; Matsunaga, S. Surging characteristics of conical and elbow type draft tubes. In Proceedings of the 12th IAHR Symposium on Hydraulic Machinery, Equipment, and Cavitation, Stirling, Scotland, 27–30 August 1984. [Google Scholar]

**Figure 1.**Pressure sensors location on two consecutive runner blades. (

**a**) PS1 to PS6 on the pressure side. (

**b**) SS1 to SS6 on the suction side.

**Figure 2.**Pressure sensors location on the wall of the draft tube cone. (

**a**) The positions are marked by blue dots. (

**b**) Circumferential positions.

**Figure 3.**Locations of the velocity monitor points along the blue lines: Section RB (between the runner blades), Section RC (runner cone below the runner blades), Sections numbered I to III (in the draft tube cone), and Section I

_{ex}(the extension of Section I).

**Figure 4.**Computational domains and interfaces: 1. Guide vane and stay vane domain. 2. Guide vane–Runner interface. 3. Runner domain. 4. Runner–Draft tube interface. 5. Draft tube domain.

**Figure 5.**Spatial discretization of the guide vane domain at the beginning of the transient operation (BEP) and the end of the transient operation PL).

**Figure 6.**Experimental and simulated axial (V

_{ax}*) and tangential (V

_{tan}*) velocity at the BEP, in the runner blade channel (Section RB I) and below the runner hub (Section DT I). The dotted vertical lines represent the hub wall and cone.

**Figure 7.**Experimental and simulated axial (V

_{ax}*) and tangential (V

_{tan}*) velocity at PL, in the runner blade channel (Section RB I) and below the runner hub (Section DT I). The dotted vertical lines represent the hub wall and cone.

**Figure 8.**Time-dependent variation of the guide vane angle and discharge (Q) during the load variation. The dotted vertical lines represent the start and the end of the guide vane closure.

**Figure 9.**Time-dependent variation of the guide vane angle and head (H) during the load variation. The dotted vertical lines represent the start and the end of the guide vane closure.

**Figure 10.**Pressure–time variation (monitor point PS1 in the runner domain). The dotted horizontal line represents the mean pressure measured at the BEP. The dotted vertical lines represent the start and the end of the guide vane closure, respectively.

**Figure 11.**Pressure–time variation (monitor point 3c in the draft tube domain). The dotted horizontal line represents the mean pressure measured at the BEP. The dotted vertical lines represent the start and the end of the guide vane closure.

**Figure 12.**Amplitude spectra of the numerical pressure signal recorded throughout the load variation and PL operation. (

**a**) Monitor point PS1 near the leading edge, on the pressure side of the blade. (

**b**) Monitor point 3c on the draft tube cone wall.

**Figure 13.**Axial velocity (V

_{ax}) in the draft tube at (

**a**) the BEP and (

**b**) PL. Results of the InletTotalPressure simulation.

**Figure 14.**Iso-pressure contour of the rotating vortex rope (RVR) in the draft tube of a Kaplan turbine model during PL operation. Tangential velocity contours are presented for Sections I, II and III (see Figure 3).

**Figure 15.**(

**a**) RVR components—raw signal. (

**b**) Savitzky–Golay filtered pressure signal.

**(c)**Truncated pressure signal. The y-scales are different for the three plots.

**Figure 16.**Spectrograms of the pressure monitor PS1 (runner blade, pressure side). (

**a**) Full spectrum. (

**b**) Low-frequency spectrum. The black solid line represents the guide vanes angle with the y-scale to the right. The dotted vertical lines represent the start and the end of the guide vane closure.

**Figure 17.**Spectrograms of the RVR plunging mode (

**a**) and rotating mode (

**b**). The black solid line represents the guide vanes angle with the y-scale to the right. The dotted vertical lines represent the start and the end of the guide vane closure.

**Figure 18.**Normalized axial velocity at Section I. (

**a**) and detail around the end of the guide vane closure (

**b**). The black horizontal line represents the ending of the transient operation.

**Figure 19.**Axial projection of the velocity vector field during load rejection, t = 1.3 s at the BEP (the beginning of the transient operation), t = 4.2 s and t = 6.6 s (during the guide vane closure), t = 8.87 s (the end of the transient operation), t = 16 s at PL.

Operating Point | BEP | PL |
---|---|---|

α_{GV} (°) | 26.5 | 20 |

Q (m^{3}/s) | 0.69 | 0.62 |

H (m) | 7.5 | 7.5 |

Domain | No. of Elements (×10^{6}) | Minimum Angle (°) | Expansion Factor (–) | Aspect Ratio (–) | y^{+}(min/avg/max) (–) |
---|---|---|---|---|---|

Guide vane | 0.34 | 20.2 | 16 | 58 | 2.70/15.6/27.1 |

Runner | 9.96 | 16.8 | 48 | 668 | 0.90/15.7/67.0 |

Draft tube | 3.66 | 30.5 | 9 | 7635 | 0.02/1.12/4.40 |

Case | Time Step dt [s] | Corresponding Runner Rotation dθ (°) |
---|---|---|

1 | 0.001195 | 5 |

2 | 0.014579 | 61 |

3 | 0.028919 | 121 |

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**MDPI and ACS Style**

Iovănel, R.G.; Dunca, G.; Bucur, D.M.; Cervantes, M.J.
Numerical Simulation of the Flow in a Kaplan Turbine Model during Transient Operation from the Best Efficiency Point to Part Load. *Energies* **2020**, *13*, 3129.
https://doi.org/10.3390/en13123129

**AMA Style**

Iovănel RG, Dunca G, Bucur DM, Cervantes MJ.
Numerical Simulation of the Flow in a Kaplan Turbine Model during Transient Operation from the Best Efficiency Point to Part Load. *Energies*. 2020; 13(12):3129.
https://doi.org/10.3390/en13123129

**Chicago/Turabian Style**

Iovănel, Raluca G., Georgiana Dunca, Diana M. Bucur, and Michel J. Cervantes.
2020. "Numerical Simulation of the Flow in a Kaplan Turbine Model during Transient Operation from the Best Efficiency Point to Part Load" *Energies* 13, no. 12: 3129.
https://doi.org/10.3390/en13123129