## 3 Cavitating Jet
### Structure of Cavitating Jet
Figure 7: Schematic diagram of the development of cavitation cloud in cavitating jet: (**a**) 1 st cloud produced by the 1 st jet core; (**b**) 2 nd jet core passing through the 1 st cloud; (**c**) 2 nd cloud produced by the 2 nd jet core and 3 rd jet core passing through the 1 st cloud; (**d**) 3 rd cloud produced by the 3 rd jet core and 4 th jet core passing through the 1 st and 2 nd clouds; (**e**) 4 th cloud produced by the 4 th jet core, and 5 th cloud produced by 5 th jet core passing through the 1 st clouds. swirl directly hit the impinging surface, the pressure around cavitation would be the pressure in the shear layer around the jet, and it gradually increases with the shedding of cavitation, as the jet speed decreases from the distance from the nozzle. On the other hand, the pressure of the cloud cavitation, which impinges the surface, increases at the impinging jet center suddenly and decreases in the ring vortex cavitation, then increases again. To consider the mechanism of the ring erosion occurring on the impinging surface, Figure 10 shows a schematic diagram of the local cavitation number on the surface [72]. The impinging pressure profile on the surface has a maximum at the jet center, \(p_{max}\), and it changes with the injection pressure and cavitation number. The \(b\) in Figure 10 is a kind of jet width defined by the flow velocity [72]. The cavitation number of the cavitating jet, \(\sigma\), is defined by the injection pressure, i.e., the upstream pressure of the nozzle \(p_{1}\), the downstream pressure of the nozzle \(p_{2}\) and the vapor pressure of water \(p_{v}\) as follows [2], and is simplified as Equation (1) and as \(p_{1}>>p_{2}>>p_{v}\). \[\sigma=\ \frac{p_{2}-p_{v}}{p_{1}-p_{2}}\ \approx\ \frac{p_{2}}{p_{1}} \tag{1}\]  
In Figure 10, the local cavitation number \(\sigma_{L}\) is defined by Equation (2) and is proportional to the ratio of \(p-p_{v}\) and \(\frac{1}{2}\)\(\rho\)\(v_{max}\)2. Footnote 2: The pressure profile is defined by Equation (2) and is proportional to the ratio of \(p-p_{v}\) and \(\frac{1}{2}\)\(\rho\)\(v_{max}\)2 [72]. \[\sigma_{L}\ \propto\ \frac{p-p_{v}}{\frac{1}{2}\ \rho\ {v_{max}}^{2}}\ \propto\ \frac{p-p_{v}}{p_{max}-p_{2}}\ f(r) \tag{2}\]  
Here, \(p\) and \(v_{max}\) are the pressure and the maximum flow velocity on the impinging surface. The \(v_{max}\) has a maximum at a certain distance from the jet center, \(r\), as it is zero at the jet center and at a further point from the jet center as shown in the lower figure of Figure 10. As \(v_{max}\) is mainly determined by the pressure difference, \(p_{max}-p_{2}\), \(\sigma_{L}\) is described by a function of \(f(r)\), as shown in the right-hand term of Equation (2). When the ring vortex cavitation sheds on the surface, the cavitation develops the \(\partial\sigma_{L}\partial r<0\) region, and it collapses at \(\partial\sigma_{L}\partial r\approx 0\). This is why the ring treatment area is obtained by the impinging cavitating jet. The detail of the pressure distribution on the impinging surface was shown in references [72; 73]. Figure 9: Schematic diagram of cavitating jet: (**a**) swirl; (**b**) vortex, cloud and ring cavitation.