# On the Vaneless Space Vortex Structures in a Kaplan Turbine Model Operating at Speed No Load
## 3 Results  
### Identification of the VSV Structures  
A VSV structure is a hydraulic phenomenon originated in the vaneless space which is formed by a certain number of vortices, \(z_{D}\), ideally distributed equidistant to each other. The entire structure precesses at a particular frequency, \(f_{0}\), resulting in all the vortices following a circular orbit around the rotor. Based on the work of [PERSON] et al. [28], the frequency of a particular VSV structure, \(f_{S}\), detected directly with an off-board sensor depends on the number of vortices of the structure, \(z_{D}\), and its precession frequency, \(f_{0}\):  
\[f_{S}=z_{D}\cdot f_{0} \tag{5}\]  
The frequency detected with an on-board sensor, \(f_{R}\), depends on \(z_{D}\), \(f_{0}\) and \(f_{n}\), as shown in Equation (6), if the turbine rotates in the same direction as the precession of the VSV structure:  
\[f_{R}=z_{D}\cdot(f_{n}-f_{0}) \tag{6}\]  
But if the turbine rotates in opposite direction to the precession of the VSV structure, the frequency detected with an on-board sensor, \(f_{R}\), would depend on \(z_{D}\), \(f_{0}\) and \(f_{n}\) as follows:  
\[f_{R}=z_{D}\cdot(f_{n}+f_{0}) \tag{7}\]  
Combining Equations (5) and (6) or Equations (5) and (7), \(z_{D}\) can be isolated to obtain Equation (8) or Equation (9), respectively. Hence, \(z_{D}\) can be determined only knowing \(f_{S}\), \(f_{R}\) and \(f_{n}\). \[f_{R}+f_{S}=z_{D}\cdot f_{n} \tag{8}\]  
\[f_{R}-f_{S}=z_{D}\cdot f_{n} \tag{9}\]  
Figure 3 presents the spectra of the guide vane torque (TGV), the pressure (PVS\({}_{\rm A}\)), the blade strain (SB\({}_{\rm R}\)) and the shaft torque strain (SS\({}_{\rm T}\)) for \(\alpha\) equal to 3.98\({}^{\circ}\). The TGV and PVS\({}_{\rm A}\) off-board spectra show two frequency peaks, each one corresponding to a particular VSV structure. Similarly, the SB\({}_{\rm R}\) and SS\({}_{\rm T}\) on-board spectra also show two frequency peaks, corresponding to the same particular structures when detected from a rotating frame of reference. The frequency peaks measured off-board, \(f_{S}\), and on-board, \(f_{R}\), can then be combined according to Equation (8) to calculate the number of vortices \(z_{D}\) forming the VSV structure. For instance, considering the first frequency peaks at \(f_{S}^{*}=1.25\) and \(f_{R}^{*}=0.77\), \(z_{D}\) equates to 2. Then, considering the second frequency peaks at \(f_{S}^{*}=1.71\) and at \(f_{R}^{*}=1.28\), \(z_{D}\) equates to 3. These results indicate that the first frequency peak is associated with a VSV structure formed by two vortices and the second peak with a VSV structure formed by three vortices which precess in the same direction as the turbine rotation. Throughout this article, the VSV structures with two vortices and three vortices will be named as 2V and 3V, respectively. Figure 4 presents the TGV and SB\({}_{\rm R}\) spectra showing the frequency peaks corresponding to 2V and 3V at all SNL conditions. Although the frequencies \(f_{S}\) and \(f_{R}\) corresponding to 2V and 3V change with the SNL condition, their combination, according to Equation (8), always results in constant values of \(z_{D}=2\) and \(z_{D}=3\), respectively. In addition to these dominant 2V and 3V structures, two additional frequency peaks with significantly lower amplitudes can also be distinguished in the spectra shown in Figure 4. For the frequency peaks highlighted in orange, \(z_{D}\) equates to 4, and the structure will be referred to as 4V. For the frequency peaks highlighted in purple, inducing low strains on the blades, \(z_{D}\) equates 6, and the structure will be referred to as 6V\({}_{\rm GV}\). Similarly, for the frequency peaks highlighted in green, inducing low torques on the guide vanes, again \(z_{D}\) equates 6, and the structure will be referred to as 6V\({}_{\rm B}\). Surprisingly, these two structures present the same number of vortices. Nevertheless, the 6V\({}_{\rm GV}\) presents higher amplitudes than the 6V\({}_{\rm B}\) in the TGV spectra, suggesting that it occurs closer to the guide vanes.