## 2 Experimental Setup  
### Swirling Flow Apparatus and Experimental Setup  
The experimental investigations are performed on the test rig available at Politehnica University Timisoara. The hydraulic circuit presented in Figure 2 contains the swirl apparatus with two parts: (i) the swirl generator and (ii) the convergent-divergent test section. The swirl generator delivers a swirling flow configuration similar to one determined in a Francis runner operated at 70% partial discharge [31]. At this regime, the vortex rope generates the largest pressure fluctuations. Figure 1: Meridian view of the elbow draft tube (cone, elbow and part of the diffuser) together with the adjustable diaphragm at the cone outlet. Three openings of the adjustable diaphragm associated to different operating points are detailed. Figure 2: Sketch of the test rig with the main elements of the hydraulic circuit (**a**). Detail of the test section and swirl generator (**b**). Several details about the swirl generator are available in two benchmark cases: (i) the Timisoara Swirl Generator test case available at the Turbomachinery Special Interest Group (TSIG) [30]; and (ii) the AC6-14 test case included in the ERCOFTAC database [32]. The swirl generator has an upstream annular section with stationary and rotating blades for generating swirling flow (Figure 2). It has three components: the ogive, the guide vanes, and the free runner. The ogive with four leaned struts has the role to sustain the swirl generator. The main part of the swirl generator is the free runner. Its main purpose is to redistribute the total pressure at the entrance of the cone. The free runner induces an excess of energy near the shroud and a deficit of energy near the hub. Therefore, it acts as a turbine at the hub and as a pump at the shroud having a vanishing total torque. The 10-bladed runner spins freely and ensures the designed output flow configuration. The cone angle of 17\({}^{\circ}\) selected in this case corresponds to modern hydraulic turbines. We performed the unsteady pressure measurements on the cone wall using a set of eight fast response transducers. The transducers are displaced on four levels (labeled L0, L1, L2 and L3) along to the cone. The first level (L0) is located in the throat of the test section. The next three levels are located at 50 mm, 100 mm and 150 mm downstream the throat section. The pressure transducers with an accuracy of \(\pm\)0.13% within a full range of \(\pm\)100 kPa are flush-installed on the cone wall. This experimental setup allows us to examine the swirling flows with a single vortex rope as we visualize it on our test rig. We select the time interval of 32 s at a sampling rate of 256 samples per second for data acquisition based on several investigations. At least 10 data sets were acquired for each flow regime to ensure data reliability. The standard deviation \(\sigma\) is defined according to Equation (1) being around \(\pm\)1% for all measured values. \[\sigma=\sqrt{\frac{\sum\left(\rho_{\text{i}}-\overline{\rho}\right)^{2}}{\text {N}-1}} \tag{1}\]  
We have achieved the measurements for a discharge of Q = 30 L/s corresponding to a Reynolds number of \(3.8\times 10^{5}\). The discharge correspond to a flow rate coefficient of q = 0.23 [33]. A flow meter with an accuracy of \(\pm\)0.15% within the full range is installed on the test rig, Figure 2. The water temperature from 20 \({}^{\circ}\)C to 24 \({}^{\circ}\)C is determined during our measurements with no cooling system installed on the test rig. As a result, the uncertainties given by the temperature variations are negligible because of the water volume which is large enough (larger than 4 m\({}^{3}\)). The entire hydraulic circuit was completely filled with water with no volume of air trapped inside. We perform the experimental investigations considering over-pressure conditions. Only non-cavitating vortex rope was examined in our investigations. Then, a constant value is selected for the water density. The method developed by [PERSON] et al.