## 2 Numerical Methods
### Lagrangian Approach Based on Discrete Properties
#### 2.2.1 Initial Nuclei Distribution  
Nuclei distribution varies according to water temperature, seasons, sea areas, and water types (seawater or freshwater). Experimentally, it is also dependent on the air content rate in the cavitation tunnel. Therefore, it is difficult to establish quantitative standards, but it is possible to grossly characterize the distribution of nuclei between seawater and freshwater from the available literature. [PERSON] et al. [19] measured the number concentration density of nuclei as a function of particle radius in the two sea regions of Santa Catalina Island and southwest of Los Angeles by using holographic and Coulter counter measurement techniques. Figure 4 presents the number concentration density of nuclei obtained from the data for Santa Catalina Island seawater in the reference [19]. Because the experiment for cavitation inception of the scaled body is performed in the cavitation tunnel using freshwater, the nuclei distribution needs to be adjusted to represent freshwater quality. [PERSON] [20] showed that the number of nuclei in freshwater is about ten times less than that in the seawater. In this study, the distribution of number concentration density in Santa Catalina Island seawater is used for the nuclei of radius between 30 to 100 \(\upmu\)m, as shown in Figure 4b. However, the number of nuclei is adjusted to reflect the freshwater condition. In a flow field where the large-scale cavitation structure occurs due to a decrease in the cavitation number, the molecules on the bubble surface are sufficiently far away for one another; thus, the molecular force is negligible. Then, the critical pressure being a function of the only temperature equals the vapor pressure, which is why the cavitation flows, such as sheet and cloud cavitation, have been successfully predicted with the Eulerian methods based on the RANS solver combined with the homogeneous mixture model [34; 35; 36; 37; 38; 39; 40]. However, the Eulerian method has difficulty in accounting for the water quality effect because the critical pressure varies with the initial nuclei size. The scale of nuclei is generally in a range of micrometers; thus, the molecular force cannot be ignored. The molecular force between particles can be described in terms of potential energy that is a function of the distance between the particles: the molecular force is proportional to the gradient of potential energy. If the distance between molecules is too close, the potential energy increases in the direction of pushing each other. In contrast, if the distance is far, the potential energy increases in the direction of pulling each other. There is an equilibrium position between close and far, where the repulsion and attraction forces are in equilibrium. When the distance is too far away, the potential energy is constant; thus, the molecular force becomes negligible [30]. The critical pressure via which the initial nuclei size affects the cavitation inception is written in the following form [41]:  
\[p_{cr}=p_{v}-\frac{4\gamma}{3R_{0}}\Bigg{[}3\Bigg{(}1+\frac{p_{0}-p_{v}}{ \frac{2\gamma}{R_{0}}}\Bigg{)}\Bigg{]}^{-\frac{1}{2}}. \tag{11}\]  
where \(\gamma\) and \(p_{0}\) denote the surface tension and the freestream pressure, respectively. It can be found from Equation (11) that the smaller the initial nuclei radius (\(R_{0}\)), the lower the critical pressure (\(p_{cr}\)). The fact that the ambient pressure around a nucleus must be lower than the critical pressure for the nucleus to grow into the cavitation bubble implies that the smaller the nuclei radius is, the more difficult it is for it to become a bubble. The critical pressure versus the radius of a nucleus is presented in Table 2. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \(R_{0}\) (\(\mu\)m) & 30 & 40 & 50 & 60 & 70 & 80 & 90 & 100 \\ \hline \(P_{cr}\) (\(Pa\)) & 726 & 924 & 1030 & 1095 & 1137 & 1167 & 1189 & 1206 \\ \hline \end{tabular}
\end{table} 
Table 2: Critical pressure for each nuclei size.