# Modal Decomposition of the Precessing Vortex Core in a Hydro Turbine Model
## 1 Introduction  
Hydropower represents a reliable and cost-effective renewable energy source with the capacity of flexible operation over a wide range of loads that allows to meet varying energy demands. However, if the turbine operates at part-load regime, a substantial amount of swirl remains in the flow behind the runner, which produces a negative effect on the turbine operation. When the swirl intensity exceeds a critical value, the vortex core undergoes breakdown [1]. This then triggers strong hydrodynamic oscillations, which are known as the precessing vortex core (PVC) [2, 3]. This phenomenon is also well known as the vortex rope, and it has been observed in many hydraulic machines from the 1950s. The PVC itself generates synchronous pressure oscillations (pressure surges) in the draft tube of the hydro turbine, which may resonate with the entire hydraulic system, causing serious damage [4, 5]. Understanding the primary physical mechanisms responsible for the PVC appearance, as well as the effect of inlet flow and channel geometry, is of practical interest for further modeling and control of the PVC and was the focus of many previous studies (for detailed references, see review [6]). Nevertheless, the relationship between the velocity distributions and pressure fluctuations generated by the PVC is still not fully understood. Modern computational fluid dynamics (CFD) packages are powerful tools for the investigation of the complex three-dimensional unsteady flows in hydro turbines [7; 8]. Lately, CFD has been used extensively to investigate the PVC occurrence in the draft tube of hydro turbines [9]. However, such numerical simulations require high computational resources and sufficiently long recordings to identify unsteady flow features. Moreover, there is a need for the verification of the numerical results with experimental data [10]. Experiments focusing on the PVC in large-scale hydro turbine prototypes are very expensive. An alternative approach is to carry out the experiments in laboratory-scale models [11]. Therefore, experimental studies focusing on the PVC appearance in part-load regimes are widely represented in the literature [12; 13; 14; 15; 16; 17]. Another possibility is to use simplified swirl flow generators instead of the full geometrical similitude of the water passage (the spiral case, the stay vanes, the guide vanes, and the runner). [PERSON] et al. [18; 19] have shown that laboratory-scaled vane-based swirlers designed by applying an inverse quasi-three-dimensional method can reproduce a predetermined mean velocity distribution that mimics the real inlet flow behind a hydro turbine runner. This approach has been proven as a powerful means of bringing new insights on the vortex breakdown and PVC dynamics in hydro turbines. Examples are vortex reconnection occurring at part load regimes [20; 21; 22; 23; 24] and the development of new promising methods for PVC mitigation [25; 26; 27; 28; 29]. Upon further simplifying the test rig design, the experiments may be carried out in air instead of water to investigate the PVC dynamics in more detail [30; 31; 32; 33]. The quantitative investigations of the PVC usually require velocity measurements using non-intrusive techniques, such as laser doppler anemometry (LDA) and particle image velocimetry (PIV). Many researchers employed PIV to investigate the flow in hydraulic turbines during off-design operating conditions. The overview of different applications of PIV techniques in the context of hydro turbines can be found in [34; 35]. Proper orthogonal decomposition (POD) is also used to extract the spatial structure and the energy of the PVC in the context of model hydro turbines [36; 37]. To characterize the dominant frequency and amplitude of the pressure fluctuations induced by the PVC, pressure sensors installed into the draft tube cone are used [14]. Thereby, the decomposition of pressure fluctuations into asynchronous and synchronous modes are used to characterize the pressure fluctuations induced by the PVC [38].