## 1 Introduction  
Hydropower is the main renewable source of energy generation in the world with 3.7% of the total final energy consumption [1]. In order to increase the part of the hydropower in the energy mix, efforts are made to develop micro-hydropower [2], to recover energy from existing water networks [3], or to improve the efficiency of the existing power plants during refurbishment [4]. Beyond these efforts and due to the development and integration of renewable energy resources, hydraulic turbines and pump-turbines are key technical components for load shifting and frequency control of the grid [5, 6]. However, in order to act as a grid stabilizer, the hydraulic machines have to extend their operating range [7]. Computational Fluid Dynamics (CFD) is one of the main tools used to investigate the dynamics of the flow in various hydraulic turbines at both stable and unstable operating points [8, 9]. Most of the numerical investigations are carried out by solving the Unsteady Reynolds-Averaged Navier-Stokes (URANS ) equations. The Reynolds stresses are often modeled using [PERSON]'s assumption, which requires computing the eddy viscosity, usually calculated using a two-equation turbulence model. Such an approach has been used to compute full head Francis turbines at the Best Efficiency Point (BEP) including also full load [10; 11; 12; 13], partial load [14; 15; 13], and Speed-No-Load (SNL) [16; 17; 18] operating points or transition from pump to turbine mode [19], start-up [20], or the S-shaped region of pump-turbines [21]. The Francis-99 workshop, whose summary is available in [22; 23], gave an overview of the state of the art for the simulation of high head Francis turbines at design and off-design operating points. Regarding the turbulence modeling, the authors mentioned that the BEP is well predicted by both URANS and hybrid turbulence models. The part load operating point is more challenging, and results show a larger discrepancy compared to the measurements. This discrepancy is explained by the influence of the mesh near the wall boundaries. Hybrid turbulence models improve the results at part load operating conditions compared to the standard URANS models. Regarding more precisely the SNL operating conditions, several authors put in evidence the inability of standard two-equation turbulence models to capture the pressure fluctuations [17; 18]. Indeed, these models are built to compute steady flows in absence of streamline curvature, stagnation points and large vortices, which are characteristic of the flow at SNL. Therefore, in order to overcome the limitations of the standard turbulence models, Scale-Adaptive Simulation (SAS) [17] based on the Shear Stress Transport (SST) \(k-\omega\) model or Large Eddy Simulation (LES) [18] has been carried out. In both cases, the content in pressure fluctuations is enhanced and shows an improved agreement with the experimental measurements. LES in a complete hydraulic machine requires a very fine mesh with around 100 million nodes [18; 24]. In order to avoid the use of such large meshes, which require large Central Processing Unit (CPU) resources, some authors intended to use hybrid RANS/LES models. Hybrid RANS/LES models are designed to adjust the level of the eddy viscosity in relation to the grid spacing, at least in the framework of Detached Eddy Simulation (DES) [25; 26]. [PERSON] et al. [27] used the Improved Delayed-Detached Eddy Simulation (IDDES) model to simulate the flow in a Francis pump turbine with a mesh containing 20 million elements. Compared to the results provided by the SST \(k-\omega\) turbulence model, the IDDES model improved the prediction of the standard deviation of the velocity components and of the turbulent kinetic energy in the draft tube cone. [PERSON] et al. computed the part load and full load operating conditions in a reduced domain of the Francis-99 turbine using a DES SST \(k-\omega\) model and a mesh with only six million control volumes [28]. The results show an improvement of the accuracy regarding the r.m.s.