# Modal Decomposition of the Precessing Vortex Core in a Hydro Turbine Model
## 4 Results and Discussion
### Identification of Stability  
In the final step, we apply the decomposed POD approach to the PIV data acquired at a range of flow rates to track the onset of the \(m=1\) and \(m=2\) instability. Figure 7a summarizes the results. The left ordinate axis represents the energy content of the leading mode pair in terms of \(E=(\lambda_{1}+\lambda_{2})/V_{0}^{2}\) calculated from the antisymmetric (\(m=1\)) and symmetric (\(m=2\)) parts of the velocity fluctuations. The right ordinate axis represents the level of the dominant peak of the PSD spectra of the pressure data (refer to Figure 3). The abscissa axis represents the flow rate, with \(Q/Q_{c}=\) representing the BEP.  
Comparing the results from the POD and PSD, we observe a similar qualitative trend for the \(m=1\) mode. For the \(m=2\) mode, we observe quite a few deviations. The differences are most likely due to the different spatial locations of the PIV domain and cross section A (Figure 1a), where the acoustic sensors are mounted. Further, the difference is also due to the fact that we compare kinetic energy and pressure, which are two different quantities. The impact of mode \(m=2\) on the energy content is significant for both cases in the part load regime (between \(0.5Q_{c}\) and \(0.55Q_{c}\)). It again shows that with the contribution of \(m=2\) the energy content should not be considered as a higher harmonic of PVC.  
The evaluated swirl number of the flows is plotted in Figure 7b as a function of \(Q\). The swirl number is calculated based on the mean velocity distributions, as given in Appendix A. In addition, the normalized frequencies of the fluctuations of the \(m=1\) and \(m=2\) modes are shown as a function of \(Q/Q_{c}\) (right axis). The frequencies remain rather constant with the changing flow rate, with a slight minimum at the point of maximal  
Figure 7: (**a**) Kinetic energy \(E\) and the wall pressure amplitudes of the \(m=1\) and \(m=2\) modes as a function of flow rate. The inset shows the linear fit to determine the bifurcation point of both modes. (**b**) Swirl number and mode frequencies as a function of flow rate.  
registered pressure fluctuations (\(0.5Q_{c}\)). It is further seen that the frequency of \(m=2\) is approximately twice the frequency of \(m=1\).  
Finally, we would like to evaluate if the \(m=1\) and \(m=2\) modes originate from two individual stability mechanisms. According to [1; 54], the observed single-helical PVC is the result of a self-excited global mode due to a supercritical Hopf bifurcation. Close to the bifurcation point, the limit-cycle amplitude of such a mode is well represented by the Landau equation, which implies that with the swirl number as the control parameter, the limit-cycle amplitude is proportional to \(\sqrt{S-S_{cr}}\). Upon consideration of \(E=(\lambda_{1}+\lambda_{2})/V_{0}^{2}\) as the squared saturation amplitude, we should expect this quantity to linearly depend on the swirl number for the supercritical Hopf bifurcation type.  
The zoomed insert of Figure 7a shows the energy of the leading POD mode pair of the \(m=1\) and \(m=2\) modes as a function of the swirl number. The linear dependence is clearly seen. A linear fit of the saturation amplitudes further yields a critical swirl number of \(S_{cr}^{m=1}=0.47\) and, \(S_{cr}^{m=2}=0.61\), for the two modes, respectively. The linear dependence we observe in this figure, as well as the two different bifurcation points, clearly show that the two modes correspond to two individual global instabilities.