###### Abstract  
The flow in a Francis turbine at full load is characterised by the development of an axial vortex rope in the draft tube. The vortex rope often promotes cavitation if the turbine is operated at a sufficiently low Thoma number. Furthermore, the vortex rope can evolve from a stable to an unstable behaviour. For CFD, such a flow is a challenge since it requires solving an unsteady cavitating flow including rotor/stator interfaces. Usually, the numerical investigations focus on the cavitation model or the turbulence model. In the present works, attention is paid to the strategy used for the time integration. The vortex rope considered is an unstable cavitating one that develops downstream the runner. The vortex rope shows a periodic behaviour characterized by the development of the vortex rope followed by a strong collapse leading to the shedding of bubbles from the runner area. Three unsteady RANS simulations are performed using the ANSYS CFX 17.2 software. The turbulence and cavitation models are, respectively, the SST and Zwart models. Regarding the time integration, a second order backward scheme is used except for the transport equation for the liquid volume fraction, for which a first order backward scheme is used. The simulations differ by the time step and the number of internal loops per time step. One simulation is carried out with a time step equal to one degree of revolution per time step and five internal loops. A second simulation used the same time step but 15 internal loops. The third simulations used three internal loops and an adaptive time step computed based on a maximum CFL lower than 2. The results show an influence of the time integration strategy on the cavitation volume time history both in the runner and in the draft tube with a risk of divergence of the solution if a standard set up is used. v 2020 20 January 2021 2  
the amount of cavitation is still small and the cavitating vortex rope is stable, whereas, by decreasing further the Thoma number, the vortex rope enters in a self-oscillating mode [1]. On one hand, various studies have been performed to better understand the self-oscillating behaviour of the cavitating vortex rope based on mathematical approaches [2] or 1D analyses [3], experimental investigations [4], 3D CFD [5] and coupling between 1D-3D numerical simulations [6; 7]. On the other hand, several investigations look at identifying some key parameters (mainly the mass flow gain factor) useful for the calibration and the development of 1D hydro-acoustic models [8; 9] using either experimental measurements [10; 11] or CFD simulations [12; 13; 14]. Among the 3D CFD studies, the set up for the time integration is not often discussed since it is rare that the mean and the maximum Courant-Friedrichs-Lewy (CFL) numbers or the number of internal loops per time step are provided. However, due to the large range of values that can be taken by the speed of sound in a vapour/liquid mixture and the large time density gradient that can occur in a cell, the time integration requires attention. Usually, in order to improve the accuracy of a simulation at each time step, two ways are available: decreasing the time step or increasing the number of internal loops per time step (if such an option is available). The latter choice can provoke some instabilities [15; 16]. In addition, by considering the Ansys CFX software, which is a coupled solver, the equation for the volume fraction is treated in a segregated manner and, in case of the simulation of a rotating turbomachinery, only a first order time scheme is available. Based on these considerations, this paper investigates the influence of the time resolution set up used to compute an unstable cavitating vortex rope at full load using the Ansys CFX software. The sensitivity to the time integration set up is often disregarded due the large CPU effort required and authors focus rather on grid sensitivity. In addition, sensitivity analysis is often carried out at BEP.