# Modal Decomposition of the Precessing Vortex Core in a Hydro Turbine Model
## 4 Results and Discussion
### Classic POD at Part-Load Regime  
We now focus on the modal decomposition of the velocity fluctuations to understand the influence of the PVC for the case of the highest level of pressure fluctuations (flow rate of \(0.5Q_{c}\)). For this purpose, the classical POD, as described in Section 3.2, is applied to the recorded PIV snapshot sequence. The eigenvalue spectrum of the POD modes is shown in Figure 5a. The POD spectrum shows that the first two leading modes contain 52% of the total TKE, while the second two modes contain approximately 12%. The energy contained in the first two modes is expected to be associated with the unsteady dynamics of the PVC, as was previously observed in other swirling flows [39; 52].  
To elucidate the temporal correlation of the identified coherent structures in the first four POD modes, the phase portraits of the corresponding temporal amplitudes \(a_{i}\) are investigated (Figure 5b-e). Considering the phase portrait of \(a_{1}\) and \(a_{2}\) (Figure 5b), it is clearly seen that the modes have a \(\pi/2\) phase shift and describe a quasi-periodic process [39]. The comparison of the values of \(a_{1}(a_{3})\) and \(a_{1}(a_{4})\) reveals that the second pair of modes is synchronized with the first pair and oscillates at the double frequency, as indicated by the shape of the Lissajous figure (Figure 5c,d). In addition, modes \(a_{3}\) and \(a_{4}\) have a \(\pi/2\) phase shift between each other and describe an oscillatory process too. This analysis clearly shows that the flow dynamics are dominated by four temporarily synchronized modes. The first and second pairs of the POD modes combine to two individual oscillatory motions that are synchronized, where the first rotates at half the frequency of the second.  
The shape of the leading four POD modes are shown in Figure 5f for the transversal velocity component (\(x\)-component). All values are normalized with respect to the maximum value. The first two POD modes look typical for the PVC possessing a symmetrical distribution with respect to the \(z\)-axis [39; 51]. The next two mode shapes (3 rd and 4 th) have an antisymmetric distribution. Generally, this fact corresponds to even/odd azimuthal wave numbers. While the first two modes most likely represent a \(m=1\) mode, the next two modes (3 rd and 4 th) represent a \(m=2\) mode, considering the symmetry properties of the mode shapes. It is also evident that the antisymmetric mode pair has a non-vanishing radial velocity component on the jet axis (\(x/D=0\)). According to [51; 53], only a mode with  
Figure 4: Mean axial (**a**,**d**,**g**) and mean tangential (**b**,**e**,**h**) components of velocity, and TKE distributions (**c**,**f**,**i**) for \(0.3Q_{c}\), \(0.5Q_{c}\), and \(Q_{c}\) cases.  
an azimuthal wave number \(m=1\) can have this property. Moreover, it is most likely that the 3 rd and 4 th POD modes correspond to an azimuthal mode \(m=2\), as it has the double frequency of the \(m=1\) mode (see Lissajous figures on Figure 5c,d). From the pressure spectra, we also know of the existence of the \(m=2\) mode at double the frequency of the \(m=1\) mode for this operating regime (Figure 3, 0.5\(Q_{c}\)).