Advances in Hydroinformatics pp 1069-1083 | Cite as

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

## Abstract

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.

## Keywords

Homogeneous model Unsteady analysis ANSYS CFX## Nomenclature

*C*_{p}Pressure coefficient (−)

- \(\vec{C}\)
Velocity vector (m s

^{−1})*C*_{z}Axial velocity component (m s

^{−1})*C*_{u}Circonferential velocity component (m s

^{−1})*C*_{r}Radial velocity component (m s

^{−1})*D*Runner diameter (m)

*E*Specific hydraulic energy (J kg

^{−1})*k*Turbulent kinetic energy (m

^{2}s^{−2})*n*Runner rotating frequency (s

^{−1})- \(n_{\text{ED}} = \frac{nD}{{E^{0.5} }}\)
Speed factor (−)

- NPSE
Net Positive Suction Head [12] (−)

*p*Pressure (Pa)

- \(\overline{p}\)
Spatially averaged pressure in a specific section (Pa)

*p*_{v}Saturated vapor pressure (Pa)

*Q*Flow discharge (m

^{3}s^{−1})- \(Q_{\text{ED}} = \frac{Q}{{D^{2} E^{0.5} }}\)
Discharge factor (−)

*R*Radius (m)

- \(R_{{\overline{1} }}\)
Runner radius at the runner outlet (m)

*r*_{g}Gas volume fraction (−)

*r*_{nuc}Volume fraction of the nucleation sites (−)

*R*_{nuc}Radius of the nucleation sites (m)

*S*Swirl number (−)

*T*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})*α*_{L}Liquid volume fraction (−)

*μ*Molecular dynamic viscosity (kg m

^{−1}s^{−1})*μ*_{t}Turbulent dynamic viscosity (kg m

^{−1}s^{−1})*υ*Specific speed (−)

*ρ*Density (kg m

^{−3})*ρ*_{g}Gas density (kg m

^{−3})*ρ*_{f}Liquid density (kg m

^{−3})- \(\sigma = \frac{\text{NPSE}}{E}\)
Thoma number (−)

*σ*_{u}Cavitation number based on the runner peripheral velocity (−)

*ω*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})

## Notes

### Acknowledgements

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).

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