## 6 Conclusions  
The influence of the time resolution set up on the simulation of an unstable cavitating rope has been performed using the Ansys CFX software. A standard set up _Case 1_ usually used for the URANS computation of a Francis turbine is not able to capture the flow due to the crash of the simulation after the first collapse of the cavitating vortex rope. By increasing the number of internal loops from 5 to 15, the simulation _Case 2_ is able to capture several cycles of the vortex rope with a dampened amplitude compared to the experiment but with a frequency close to the measured one. It has also been noticed that the low frequency observed on the torque or the efficiency seems to be related to the presence of a cavitating vortex rope in the draft tube since, in its absence, the low frequency disappears. The simulation _Case 3_ carried out with an imposed maximum CFL number lower than 2, i.e., with a smaller time step, and only three internal loops does not provide a dynamic behaviour very different than _Case 2_ even if a longer simulation could show differences.  
Surprisingly, the simulation _Case 1_ is the only one that gives a duration and an amplitude of the pressure peak in agreement with the experiment. Nevertheless, the simulation stops when the vortex rope collapse due to a solver crashes and no successive cycles are computed. On the contrary, the simulations _Case 2_ and _Case 3_ do not show any growth or collapse of the cavitating vortex rope. This difference is explained by the fact that the simulation _Case 1_ does not respect the mass conservation equation at each time step contrary to the two other simulations.  
To compute an unstable vortex rope, it is therefore necessary to take care of the time integration set up by increasing at least the number of internal loops. To improve the accuracy of the simulation, the U-RANS simulation needs, in addition, to be coupled with a 1D hydroacoustic model allowing for adjusting the boundary conditions at the inlet and outlet boundaries of the computational domain [6,7].  
Figure 12: Time history of the left- and right- hand terms of Equation (2) for each case.  
**Author Contributions:** [PERSON] contributed to carrying out the simulations and the post-treatment of the CFD results. He is the main writer of the paper. [PERSON] and [PERSON] performed the experimental work and provide the experimental results used in this paper. They also contribute to the redaction of the paper. [PERSON] and [PERSON] were responsible for the supervision, project administration, and the funding acquisition. All authors have read and agreed to the published version of the manuscript.  
**Funding:** 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)  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** The data presented in this study could be available on request from the corresponding author. The data are not publicly available due to NDA.  
**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.