# On the Vaneless Space Vortex Structures in a Kaplan Turbine Model Operating at Speed No Load
## 3 Results
### Induced Responses of the 2V and 3V Structures  
Figure 7 shows the amplitude of the 2V and 3V frequency peaks detected in the SB\({}_{\text{R}}\), SS\({}_{\text{T}}\) and TGV spectra. A first observation is that the amplitude evolutions resemble those observed on the pressure signals, which confirms that they are induced by the same phenomena detected from different locations. Additionally, it must be noted that although most sensors do not detect the 2V at 7.35\({}^{\circ}\) and the 3V at 7.86\({}^{\circ}\) and 8.13\({}^{\circ}\), the SB\({}_{\text{R}}\) and TGV spectra show small peaks confirming the coexistence of both structures at all SNL conditions and reinforcing the hypothesis that they are not independent. Among all strain measurements, the blade ones (SB\({}_{\text{R}}\)) present higher values than the shaft torque ones (SS\({}_{\text{T}}\)) with a maximum of 0.55 \(\upmu\)m/m induced by the 2V at 8.13\({}^{\circ}\) and 0.9 \(\upmu\)m/m by the 3V at 7.35\({}^{\circ}\). Similarly, the TGV measurements also present a maximum of 0.16 Nm induced by 2V at 7.86\({}^{\circ}\) and 0.21 Nm by 3V at 7.35\({}^{\circ}\). It must be noted that the maximum induced responses by 3V present higher values than 2V, unlike as observed in the pressures.  
Equation (11) [28; 36] permits to calculate the frequency of a VSV structure, \(f_{\text{S}\_{R}}\), when measuring its induced vibration transmitted from the rotating rotor with an off-board sensor, using \(f_{\text{S}\_{R}}\), \(k\), \(Z_{\text{R}}\), \(f_{n}\) and \(j\):  
\[f_{\text{S}\_{R}}=k\cdot Z_{\text{R}}\cdot f_{n}-j\cdot f_{\text{S}} \tag{11}\]  
Table 3 presents the values of \(f_{\text{S}\_{R}}\) for 2V and 3V calculated using Equation (11) when \(\alpha\) is 2.6\({}^{\circ}\), 3.98\({}^{\circ}\), 6.8\({}^{\circ}\) and 7.35\({}^{\circ}\).  
Figure 7: Amplitudes of the frequency peaks of 2V and 3V on (**a**) SB\({}_{\text{R}}\), (**b**) SS\({}_{\text{T}}\) and (**c**) TGV, at all SNL conditions.  
Figure 8a,b present the spectra of the vibration velocity measured on the draft tube cone (A0y) and the amplitudes of the peaks induced by 3V at 2.6\({}^{\circ}\), 3.98\({}^{\circ}\), 6.8\({}^{\circ}\) and 7.35\({}^{\circ}\), respectively. It must be noticed that 2V is not detected by A0y, suggesting that 2V is not well transmitted to the turbine supporting structure. However, 3V is detected at the same frequencies listed in Table 3. The amplitude evolution of the 3V peak correlates with those previously found for the pressures, strains and torques, as presented in Figures 6b, 7a, 7b and 7c, respectively. In this case, the frequency peaks associated with 3V present lower amplitudes than other sources of excitation, such as the mechanical unbalance at \(f^{*}=1\) and their harmonics, as shown in Figure 8a. These results differ from the vibrations measured in a Kaplan turbine prototype by [PERSON] et al. [28] showing high peaks dominated by the VSV structures but agrees with the results found by [PERSON] et al. [33] concluding that the vibrations derived from VSV structures have low intensity compared with other sources of excitation.