## 3 Pressure Data Analysis
### Pressure Pulsations Decomposition  
[PERSON] [46] identifies two types of pressure fluctuations associated with the draft tube surge. First one is the asynchronous (rotating) type of the pressure fluctuation being associated to the precession of the helical vortex. The asynchronous pressure fluctuation is trapped in the discharge cone, according to [PERSON] et al. [47]. The other fluctuation is synchronous (plunging) one. This fluctuations travel as standing waves along to the hydraulic passage being dangerous in operation. Several theories have been advanced to account for synchronous pressure fluctuations [48; 49]. [PERSON] et al. [21] suggested that the interaction between the vortex rope and the draft tube elbow is the source of the synchronous fluctuations. [PERSON] and [PERSON] [50] have performed a phase analysis for two simultaneously acquired pressure signals on the discharge cone of a Francis turbine to discriminate between rotating (asynchronous) and plunging (synchronous) oscillations, respectively. In their investigations, the synchronous (plunging) component was lower in amplitude than the asynchronous (rotating) component. However, the synchronous (plunging) pressure pulsations are accepted as the  
Figure 9: Comparison of the dimensionless equivalent amplitudes obtained for following cases: swirling flow with vortex rope (long dashed line), axial water jet method with 10% discharge (dashed line) and the diaphragm control method with shutter area of 50% (d = 0.113 m) (solid line).  
driving mechanism behind most operational problems related to draft tube surging, although there are times when simple vibration and noise caused by the asynchronous (rotating) pulsations can be serious.  
Synchronous (plunging) pressure fluctuations can produce variations in head, discharge, torque, and output power known in the literature as a power swing [13]. Equations (5) and (6) are applied for two unsteady signals acquired on the same level to discriminate between asynchronous and synchronous components, respectively:  
\[\mathrm{P(t)=}\frac{\mathrm{ST_{1}+}\mathrm{ST_{2}}}{2} \tag{5}\]  
\[\mathrm{R(t)=ST_{1}-}\mathrm{P} \tag{6}\]  
As expected, the equivalent amplitudes associated with the rotating component (RC) are larger than the plunging component (PC) for the case without the diaphragm (Figure 10). This means the helical structure of the vortex rope is well developed. The plunging component (PC) becomes dominant regarding the rotating component (RC) when the diaphragm with a shutter area ratio larger than 50% throttles the cone outlet area. In these cases, the flow structure in the cone changes as the cross-section closes, revealing a bubble type for the case with an area ratio of 70%. In addition, the mitigation of pressure pulsations by up to 45% for all diaphragm cases highlighted in the previous section implies a significant decrease of the rotating component (up to 80-90%) correlated with a slight increase (10% to 50%) of the plunging component compared with the case without a diaphragm (note that we investigated the diaphragm method on a surrogate model of swirl generator). However, the goal was to mitigate the rotating part associated to the vortex rope. This conclusion supports our previous recommendation about selecting a diaphragm with a shutter area ratio from 40% to 60% [34].  
Figure 10: Rotating—RC (asynchronous) and plunging—PC (synchronous) components along to the cone wall for the cases without the diaphragm and with diaphragm area ratios of 50% (d = 0.113 m), 60% (d = 0.1 m) and 70% (d = 0.088 m), respectively.