# Modal Decomposition of the Precessing Vortex Core in a Hydro Turbine Model
## 4 Results and Discussion
### Symmetric/Antisymmetric POD at Various Flow Regimes  
According to the decomposed pressure fluctuations in Figure 3, the mode \(m=2\) has a higher amplitude than is expected for the first harmonic of the PVC. This indicates that the \(m=2\) mode is driven by an individual mechanism. To shed more light on this, we intend to track the \(m=1\) and \(m=2\) instabilities as a function of the operating condition. For this purpose, we apply a modified POD to the symmetric/antisymmetric decomposed velocity fluctuations, which allows for a clear separation, even at conditions where the modes are relatively weak.  
To extract the antisymmetric and symmetric POD structures, the recorded PIV snapshot sequence is decomposed into antisymmetric and symmetric parts, according to Equation (6). The resulting spatial modes are shown in Figure 6a-d and compared to the classic POD. It is clearly seen that the first mode pair of the classical POD is equivalent to the first mode pair of the antisymmetric POD. Considering Figure 6c,d, it is shown that the second mode pair of the classical POD decomposition (3 rd and 4 th) is equivalent to the first mode pair of the symmetrical POD. The relative energy content of the POD modes computed from the antisymmetric and symmetric parts of the velocity fluctuations likewise feature a gap between the first mode pair and the remaining modes (Figure 6e). Additionally, the phase portrait of the first two modes in Figure 6f indicates an oscillatory motion. The phase portrait of Figure 6g indicates a synchronization between \(a_{1}^{antisymm}\) and \(a_{1}^{symm}\). The fact that \(m=2\) is double the frequency of \(m=1\) is also proven by the eight-like shape of the Lissajous figure (Figure 6g). All these observations suggest that the first and  
Figure 5: POD analysis for 0.5\(Q_{c}\): TKE content of POD modes (**a**), Lissajous figures of \(a_{1-4}\) (**b–e**), and the POD mode shapes (**f**).  
second antisymmetric modes represent the global mode \(m=1\) and can be associated with the PVC. In turn, the first and second symmetric modes represent the mode \(m=2\).  
Figure 6: Comparative POD analysis of the flow snapshots decomposed in symmetric/antisymmetric parts (right column) and without decomposition (left column) for all three components of velocity fields: 1 st and 2 nd modes of classical POD analysis (**a**); 3 rd and 4 th modes of classical POD analysis (**c**); 1 st and 2 nd modes of antisymmetric POD (**b**), 1 st and 2 nd modes of symmetric POD (**d**), TKE content of POD modes (**e**), Lissajous figure of \(a_{1}^{antisymmetry}(a_{2}^{antisymmetry})\) (**f**) and Lissajous figure of \(a_{1}^{symmt}(a_{1}^{antisymmetry})\) (**g**). Flow rate is \(0.5Q_{c}\)  
Thus, the POD analysis based on the decomposed velocity snapshots allows distinguishing between modes with azimuthal wave numbers \(m=1\) and \(m=2\) by consideration of the antisymmetric and symmetric parts of the velocity fluctuations, respectively. This is particularly important when considering flow regimes where vortex dynamics are less dominant and covered by turbulent noise. In this case, classical POD will not well be able to clearly distinguish between \(m=1\) and \(m=2\) due to the mixing of the modes.