# 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 \(J=1.02\)
#### 3.2.2 Evolution of TLV  
Figure 12 shows the evolution of tip vortex in a rotating period. The TLV and TSV shown are based on the \(Q\)-criterion ISO-surface. From the figure, it can be seen that after 1/8 cycle of rotor rotation, TSV has begun to merge into TLV. By 2/8 cycle, TSV has fully merged and then both TLV and TSV begin to dissipate. Because of the relative motion between blade tip and end wall, there is induced vortex formed near the blade suction surface. TLV is generated at 5/8 cycles and the induced vortices are separated from TSV. As the rotor rotates, the induced vortices gradually converge towards TLV. After 1/3 cycles, the induced vortices are fully integrated with TLV. It should be noted that the main frequency in the figure is equal to the passing frequency of the stator. Figure 11: Pressure and vorticity changes before and after cavitation at TLV monitoring points. Figure 10: Frequency domain diagram of pressure fluctuation before and after cavitation at TLV monitoring points (**a**) non-cavitation (**b**) cavitation. Figure 13 shows periodic evolution of vortices under cavitation at \(J=1.02\). As shown in Figure 13, at T\({}_{0}\) without the cavitation, the length of the vortices extends to the trailing edge of the blade. At 1/8 cycle, the non-cavitation TSV has begun to merge into the TLV, while at cavitation conditions, the two vortices have completely merged at 2/8 cycle, which indicates that the merging time of TSV to the TLV under cavitation is between 1/8 cycle and 2/8 cycle. The cavitation can delay the merging process of vortices. This conclusion can be proved at 5/8 cycles: TSV has already started to separate the induced vortices which began to move towards TLV in non-cavitation condition. However, there are almost no induced vortices for the cavitation until 6/8 cycles. Therefore, by comparing the vortex shape evolution before and after cavitation, it can be deduced that cavitation will make TLV extend to the trailing edge of blade, while delaying the fusion between TSV and TLV. Figure 12: Periodic evolution of vortices under non-cavitation at \(J=1.02\), \(Q=0.01\). Figure 13: Periodic evolution of vortices under cavitation at \(J=1.02\), \(Q=0.01\). Figure 14 presents the Z-direction component contour of stretching term at two adjacent times before and after cavitation. The stretching term comes from the vorticity transport equation which is introduced as follows:  
\[\frac{d\omega}{dt}=(\omega\cdot\
abla)V-\omega(\
abla\cdot V)-\
abla\left(\frac{1 }{\rho}\right)\times\
abla p\ +\ \
u\
abla^{2}\omega \tag{18}\]  
where \(\omega\) is the vorticity, \(\rho\) is the local fluid density, \(p\) is the local pressure, \(V\) is the flow velocity, and \(\
u\) is the dynamic viscosity. The definition of vorticity \(\omega\) is:  
\[\omega\ =\
abla\times V \tag{19}\]  
In Equation (17), the stretching term \((\omega\cdot\
abla)V\) on the right-hand side describes the stretching or tilting of vorticity due to the flow velocity gradients. The expansion term \(\omega(\
abla\cdot V)\) describes stretching of vorticity due to flow compressibility. The term \(\
abla\left(\frac{1}{\rho}\right)\times\
abla p\) is the baroclinic term. It accounts for the changes in the vorticity due to the intersection of density and pressure surfaces. The viscous dissipation term, \(\
u\
abla^{2}\omega\), accounts for the diffusion of vorticity due to the viscous effects [50; 51]. Because the fluid flow medium used in this paper is incompressible, the baroclinic and expansion terms are zero. Moreover, since the viscous dissipation effect can be neglected in high Reynolds number fluid, only the stretching term is analyzed in this paper. It can be found that the stretching component value in the core region of TLV is positive while the surrounding is negative. The stretching value is positive in TSV core region, and the positive distribution extends to TLV core region.