# Dynamic Analysis of Cavitation Tip Vortex of Pump-Jet Propeller Based on DES
## 3 Cavitation and Vortex Dynamics Analysis of Pump-Jet Propeller
### Slight cavitation at \(J=1.02\)  
#### 3.2.1 Pressure fluctuation  
In this work, non-dimensional parameter \(C_{p}\) is used to represent pressure pulsation, which is computed by:  
\[C_{p}=\frac{p-\overline{p}}{1/2\rho v^{2}} \tag{17}\]  
Figure 8: Comparison of propulsion efficiency of pump-jet propeller before and after cavitation at different speeds.  
Figure 7: Diagram of the monitoring points positioning (**a**) rotor blades (**b**) location of monitor points.  
where \(C_{p}\) is pressure coefficient, \(p\) is the local pressure, \(\overline{p}\) is the time average pressure, \(v\) is the incoming velocity. By calculation, the advance ratio \(J\) is equal to 1.02 and the cavitation number \(\sigma\) is equal to 3.98 for both non-cavitation and cavitation scenarios.  
Figure 9 shows the time domain diagram of pressure pulsation before and after cavitation at TLV monitoring points. It can be found that when there is no cavitation, the peak and bottom of the five monitoring points of TLV appear almost at the same time, and the number of peak and bottom are the same showing evident regularity. In the time domain diagram after cavitation, LP1 and LP2 fluctuate disorderly with less evident regularity compared to LP4 and LP5, and the degree of disorder is less at LP3. It indicates that the slight cavitation has affected the middle position of the tip. By observing the fluctuations of LP4 and LP5, it can be found that the peak pressure coefficient after cavitation is twice high compared to that without cavitation. Figure 9b shows the secondary peak, which indicates that cavitation in the middle of the tip affects the flow field near the trailing edge of the blade.  
Figure 10 presents frequency domain diagram of pressure fluctuation before and after cavitation at TLV monitoring points. The pressure fluctuations in Figure 10 are extremely distinctive before and after cavitation. The horizontal axis in this figure represents the multiples of the shaft frequency (SFM), and the value of the shaft frequency equals the rotational speed in r/s. It can be seen from Figure 10a that prior to the cavitation, the main frequency of the five monitoring points is 9 times the shaft frequency and the secondary frequency is four times the shaft frequency, which might be caused by the interaction between rotor blades and stator. After cavitation, LP1 is located in the core area of cavitation, so its pulsation has no apparent regularity. For LP2-LP5, their main and secondary main frequencies are still nine times and four times of the shaft frequency, respectively. Nevertheless, compared with the case without cavitation, the fluctuation components increase after cavitation, and low-frequency and high-frequency fluctuations also occur at the same time.  
Figure 11 provides the results of pressure and vorticity changes before and after cavitation at TLV monitoring points. As can be seen from Figure 11, regardless of the existence of the cavitation, the pressure of the TLV core increases and the vorticity decreases from the tip inlet to the outlet. This is due to the fact that the TLV is in the developing stage at the tip leading edge, so the internal pressure of the vortex core is low and the vorticity is large. When it gradually decays and dissipates, the internal pressure of the vortex core gradually returns to the value of the mainstream and the vorticity will decrease accordingly. When cavitation occurs, because of the low-pressure area exists around vortex cores, the pressure inside the vortex core will further decrease. Furthermore, cavitation bubbles will  
Figure 9: Time domain diagram of pressure pulsation before and after cavitation at TLV monitoring points (**a**) non-cavitation (**b**) cavitation.  
increase the pressure gradient in the tip cap, which may cause the velocity curl to come up, and then the vorticity enlarges.