## 4 Results
### Integrated Variables  
The time-history of the efficiency, the dimensionless volume of vapour in the computational domain, the torque factor \(T_{ED}\) and the speed factor \(n_{ED}\) are plotted on Figure 4 (the experimental values for the \(T_{ED}\) and \(n_{ED}\) are added in the caption of the figure). The divergence of the simulation _Case 1_ just after the collapse of the cavitation volume is clearly put in evidence. On the contrary, simulations _Case 2_ and _Case 3_ do not show a complete collapse of the cavitation volume even if a time oscillation of the different quantities is observed. The first cycle seems to be shortened by increasing the requirement on the time integration since it is shorter for _Case 2_ compared to _Case 1_ and the shortest for _Case 3_.  
Regarding the efficiency, the torque factor and the speed factor, the differences between the simulations are around 1.5%, whereas, for the volume of vapour, simulation _Case 1_ predicts a volume that is at least four times higher. Compared to the measurements, the \(T_{ED}\) and \(n_{ED}\) are overestimated respectively by approximately 2.5% and 3.5%. Simulations _Case 2_ and _Case 3_ have a rather similar behaviour even if a time shift is observed due to a faster first cycle for _Case 3_. Furthermore, due to the smaller time step imposed in _Case 3_, fluctuations at higher frequencies are observed in all the signals.  
For simulation _Case 2_ between the seventh and eleventh runner revolution, the slow variation of the efficiency, the torque factor, and the speed factors disappear, whereas a fluctuation at a low frequency is still observed for the volume of vapour. By splitting the time history of the volume of vapour between the volume in the runner and draft tube domains (see Figure 5), it is noticeable that between the seventh and eleventh runner revolution, the volume of vapour in the draft tube is close to zero and then increases again after the tenth runner revolution. Therefore, the low frequency seems to be related to the volume of vapour in the draft tube, which means with the presence of a cavitating vortex rope. This point seems to be in agreement with the experimental investigations [1].  
Figure 3: Instantaneous iso-surface of the liquid volume fraction \(\alpha_{L}=0.9\). Downstream view (top) and side view (bottom).  
A Fourier analysis of the efficiency signal is shown in Figure 6 on the left. The frequency is divided by the runner frequency \(f_{n}\). The modes put in evidence are the ones associated with the blade passage frequency \(f_{b}=16f_{n}\) and the first and fifth harmonic. Both _Case 2_ and _Case 3_ capture the fifth harmonic, which corresponds to the rotor/stator interaction since \(5f_{b}=4f_{gv}=80f_{n}\). A Fourier analysis of the volume of vapour has also been carried out even if only the simulation _Case 2_ has achieved enough cycles to provide an estimation of the low frequency characteristic of the time evolution of the volume of vapour. By focusing only on _Case 2_, a low frequency around \(0.3f_{n}\) is put in evidence, which is closed to \(0.25f_{n}\), the value observed on the torque during the measurements [1].  
Figure 4: Time-history of the efficiency, the dimensionless volume of vapour, the torque factor \(T_{ED}\) (measured value: 0.127) and the speed factor \(n_{ED}\) (measured value: 0.288).  
Figure 5: Time-history of the dimensionless volume of vapour in the runner (left) and in the draft tube (right).