U-RANS Simulations and PIV Measurements of a Self-excited Cavitation Vortex Rope in a Francis Turbine

  • Jean DecaixEmail author
  • Andres Müller
  • Arthur Favrel
  • François Avellan
  • Cécile Münch
Conference paper
Part of the Springer Water book series (SPWA)


In the course of the massive penetration of alternative renewable energies, the stabilization of the electrical power network significantly relies on the off-design operation of turbines and pump-turbines in hydropower plants. The occurrence of cavitation is, however, a common phenomenon at such operating conditions, often leading to critical flow instabilities, which undercut the grid stabilizing capacity of the power plant. In order to predict and extend the stable operating range of hydraulic machines, a better understanding of the cavitating flows and mainly of the transition between stable and unstable flow regimes is required. In the case of Francis turbines operating at full load, an axisymmetric cavitating vortex rope develops at the outlet runner in the draft tube. The cavity may enter self-oscillation, with violent periodic pressure pulsations propagating throughout the entire hydraulic system. The flow fluctuations lead to dangerous electrical power swings and mechanical vibrations through a fluid-structure coupling across the runner, imposing an inconvenient and costly restriction of the operating range. The paper deals with a numerical and experimental investigation of the transition from a stable to an unstable operating point on a reduced scale model of a Francis turbine at full load. Unsteady homogeneous two-phase RANS simulations are carried out using the ANSYS CFX solver. Cavitation is modelled using the Zwart’s model that required solving an additional transport equation for the void fraction. Turbulence is solved using the SST k-ω model. Simulations are compared with the experimental measurements and some insights are provided for a first comprehensive analysis of the transition between the stable and unstable states.


Homogeneous model Unsteady analysis ANSYS CFX 



Pressure coefficient (−)


Velocity vector (m s−1)


Axial velocity component (m s−1)


Circonferential velocity component (m s−1)


Radial velocity component (m s−1)


Runner diameter (m)


Specific hydraulic energy (J kg−1)


Turbulent kinetic energy (m2 s−2)


Runner rotating frequency (s−1)

\(n_{\text{ED}} = \frac{nD}{{E^{0.5} }}\)

Speed factor (−)


Net Positive Suction Head [12] (−)


Pressure (Pa)


Spatially averaged pressure in a specific section (Pa)


Saturated vapor pressure (Pa)


Flow discharge (m3 s−1)

\(Q_{\text{ED}} = \frac{Q}{{D^{2} E^{0.5} }}\)

Discharge factor (−)


Radius (m)

\(R_{{\overline{1} }}\)

Runner radius at the runner outlet (m)


Gas volume fraction (−)


Volume fraction of the nucleation sites (−)


Radius of the nucleation sites (m)


Swirl number (−)


Torque (N m)

\(T_{\text{ED}} = \frac{T}{{\rho D^{3} E}}\)

Torque factor (−)

\(U_{{\overline{\text{I}} }}\)

Rotating velocity at the outer diameter at the runner outlet (m s−1)


Liquid volume fraction (−)


Molecular dynamic viscosity (kg m−1 s−1)


Turbulent dynamic viscosity (kg m−1 s−1)


Specific speed (−)


Density (kg m−3)


Gas density (kg m−3)


Liquid density (kg m−3)

\(\sigma = \frac{\text{NPSE}}{E}\)

Thoma number (−)


Cavitation number based on the runner peripheral velocity (−)


Runner rotating speed (rad s−1)

\(\overline{\overline{\tau }}\)

Viscous stresses (kg m−1 s−2)

\(\overline{\overline{{\tau_{\text{t}} }}}\)

Turbulent stresses (kg m−1 s−2)



The research leading to the results published in this paper is part of the HYPERBOLE research project, granted by the European Commission (ERC/FP7-ENERGY-2013-1-Grant 608532).


  1. 1.
    Favrel, A., Müller, A., Landry, C., Yamamoto, K., & Avellan, F. (2015). Study of the vortex-induced pressure excitation source in a Francis turbine draft tube by particle image velocimetry. Experiments in Fluids, 56(12), 215.CrossRefGoogle Scholar
  2. 2.
    Favrel, A., Müller, A., Landry, C., Yamamoto, K., & Avellan, F. (2016). LDV survey of cavitation and resonance effect on the precessing vortex rope dynamics in the draft tube of Francis turbines. Experiments in Fluids, 57(11), 168.CrossRefGoogle Scholar
  3. 3.
    Koutnik, J., Nicolet, C., Schohl G., & Avellan, F. (2006). Over-load surge event in a pumped-storage power plant. In 23rd IAHR Symposium on Hydraulic Machinery and Systems (Vol. 1, pp. 1–15).Google Scholar
  4. 4.
    Müller, A., Favrel, A., Landry, C., & Avellan, F. (2017). Fluid-structure interaction mechanisms leading to dangerous power swings in Francis turbines at full load. Journal of Fluids and Structures, 69, 56–71.CrossRefGoogle Scholar
  5. 5.
    Panov, L. V., Chirkov, D. V., Cherny, S. G., Pylev, I. M., & Sotnikov, A. A. (2012). Numerical simulation of steady cavitating flow of viscous fluid in a Francis hydroturbine. Thermophysics and Aeromechanics, 19(3), 415–427.CrossRefGoogle Scholar
  6. 6.
    Dörfler, P. K., Keller, M., & Braun, O. (2010). Francis full load surge mechanism identified by unsteady 2-phase CFD. In IOP Conference Series: Earth and Environmental Science (Vol. 12(1), p. 012026.Google Scholar
  7. 7.
    Flemming, F., Foust, J., Koutnik, J., & Richard, K. F. (2009). Overload surge investigation using CFD data. International Journal of Fluid Machinery and Systems, 2(4), 315–323.Google Scholar
  8. 8.
    Mössinger, P., Conrad, P., & Jung, A. (2014). Transient two-phase CFD simulation of overload pressure pulsation in a prototype sized Francis turbine considering the waterway dynamics. In IOP Conference Series: Earth and Environmental Science (Vol. 22, p. 032033).Google Scholar
  9. 9.
    Chirkov, D., Panov, L., Cherny, S., & Pylev, I (2014). Numerical simulation of full load surge in Francis turbines based on three-dimensional cavitating flow model. In IOP Conference Series: Earth and Environmental Science (Vol. 22, p. 032036).Google Scholar
  10. 10.
    Panov, L. V., Chirkov, D. V., Cherny, S. G., & Pylev, I. M. (2014). Numerical simulation of pulsation processes in hydraulic turbine based on 3D model of cavitating flow. Thermophysics and Aeromechanics, 21(1), 31–43.CrossRefGoogle Scholar
  11. 11.
    Alligné, S., Decaix, J., Müller, A., Nicolet, C., Avellan, F., & Münch, C. (2016). RANS computations for identification of 1-D cavitation model parameters: Application to full load cavitation vortex rope. In IOP Conference Series: Earth and Environmental Science (Vol. 49, p. 082014).Google Scholar
  12. 12.
    IEC standards. (1999). 60193: Hydraulic turbines, storage pumps and pump-turbines—model acceptance tests (2nd ed.). International Electrotechnic Commission.Google Scholar
  13. 13.
    Brennen, C. E. (1995). Cavitation and bubble dynamics. New York: Oxford University Press.zbMATHGoogle Scholar
  14. 14.
    Menter, F. R. (1993). Zonal two equation k–ω turbulence models for aerodynamic flows. In AIAA 93–2906, 24th Fluid Dynamics Conference.Google Scholar
  15. 15.
    Zwart, P., Gerber, A., & Belamri, T. (2004). A two-phase flow model for predicting cavitation dynamics. In Fifth International Conference on Multiphase Flow (Vol. 152).Google Scholar
  16. 16.
    Plesset, M. S. (1949). The dynamics of cavitation bubbles. Journal of Applied Mechanics, 16, 277–282.Google Scholar
  17. 17.
    Bakir, F., Rey, R., Gerber, A. G., Belamri, T., & Hutchinson, B. (2004). Numerical and experimental investigations of the cavitating behavior of an inducer. The International Journal of Rotating Machinery, 10(1), 15–25.CrossRefGoogle Scholar
  18. 18.
    Wack, J., & Riedelbauch, S. (2015). Numerical simulations of the cavitation phenomena in a Francis turbine at deep part load conditions. Journal of Physics: Conference Series, 656, 012074.Google Scholar
  19. 19.
    Müller, A., Favrel, A., Landry, C., Yamamoto, K., & Avellan F. (2014). On the physical mechanisms governing self-excited pressure surge in Francis turbines. In IOP Conference Series: Earth and Environmental Science (Vol. 22, p. 032034).Google Scholar
  20. 20.
    Gupta, A. K., Lilley, D. G., & Syred, S. (1984). Swirl flows. Tunbridge Wells, Kent, England: Abacus Press.Google Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  • Jean Decaix
    • 1
    Email author
  • Andres Müller
    • 2
  • Arthur Favrel
    • 2
  • François Avellan
    • 2
  • Cécile Münch
    • 1
  1. 1.University of Applied Sciences and Arts Western Switzerland ValaisSionSwitzerland
  2. 2.Ecole Polytechnique Fédérale de LausanneLausanneSwitzerland

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