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Numerical Simulation and Validation of a High Head Model Francis Turbine at Part Load Operating Condition

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Hydraulic turbines are operated over an extended operating range to meet the real time electricity demand. Turbines operated at part load have flow parameters not matching the designed ones. This results in unstable flow conditions in the runner and draft tube developing low frequency and high amplitude pressure pulsations. The unsteady pressure pulsations affect the dynamic stability of the turbine and cause additional fatigue. The work presented in this paper discusses the flow field investigation of a high head model Francis turbine at part load: 50% of the rated load. Numerical simulation of the complete turbine has been performed. Unsteady pressure pulsations in the vaneless space, runner, and draft tube are investigated and validated with available experimental data. Detailed analysis of the rotor stator interaction and draft tube flow field are performed and discussed. The analysis shows the presence of a rotating vortex rope in the draft tube at the frequency of 0.3 times of the runner rotational frequency. The frequency of the vortex rope precession, which causes severe fluctuations and vibrations in the draft tube, is predicted within 3.9% of the experimental measured value. The vortex rope results pressure pulsations propagating in the system whose frequency is also perceive in the runner and upstream the runner.

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Best efficiency point


Diameter of runner, m


Guide vane’s opening, degree


Fast Fourier transform


Observed frequency, Hz

fn :

Runner rotational frequency, Hz


Normalised frequency, minus

frh :

Rheingans (vortex rope) frequency, Hz ≡ f/3.6


9.821465 m/s2, as tested and measured at NTNU


Head, m


Sampling length


Runner speed, rev/s

nED :

Speed factor [−], \({\text{n}}_{\text{ED}} { = }\frac{\text{nD}}{{\sqrt {{\text{gH}}_{\text{M}} } }}\)

ns :

Specific speed [−], \({\text{n}}_{\text{s}} = \frac{{\left( {{\text{n}}_{\text{P}} \frac{\uppi }{180}} \right)\sqrt {{\text{Q}}_{\text{P}} } }}{{\left( {2{\text{gH}}_{\text{P}} } \right)^{{\frac{3}{4}}} }}\)


Pressure difference across the turbine, Pa

\({\tilde{\text{p}}}\) :

Acquired pressure signal, kPa

\({\bar{\text{p}}}\) :

Mean pressure, kPa

\({\text{p}}_{{}}^{ *}\) :

Fluctuating pressure, kPa


Pressure, kPa, harmonic order (1, 2,…)


Power, MW


Flow rate, m3/s−1

qED :

Discharge factor [−], \({\text{q}}_{\text{ED}} { = }\frac{\text{Q}}{{{\text{D}}^{2} \sqrt {{\text{gH}}_{\text{M}} } }}\)


Runner inlet radius, m


Guide vane’s opening, degree


Rotor stator interactions


Rotating vortex rope


Time step


Turbulence kinetic energy


Time, s




Discrete quantity

\(\overline{\text{X}}\) :

Average value


Wavelength, m


Angular vane/blade position, degree


Angular velocity, rad/s

ηh :

Hydraulic efficiency, %


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Correspondence to Bhupendra Kumar Gandhi.

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Goyal, R., Trivedi, C., Kumar Gandhi, B. et al. Numerical Simulation and Validation of a High Head Model Francis Turbine at Part Load Operating Condition. J. Inst. Eng. India Ser. C 99, 557–570 (2018).

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