## 2 Numerical Methods
### Initial Nuclei Distribution and Critical Pressure  
Nuclei distributed in an undissolved state of water grow into a bubble, i.e., cavitation occurs when the pressure around nuclei is lower than the critical pressure of nuclei [29; 31]. The pressure value below which nuclei begin to grow into bubbles is termed as the critical pressure. The nuclei distribution and critical pressure vary with nucleus size and are the main properties of the water quality [17]. Nuclei distribution in the inlet flow should be determined for the prediction of cavitation inception. Nuclei distribution varies with water type (seawater or freshwater), sea area, water temperature, and seasons. Cavitation is more likely to occur in seawater than in freshwater because seawater includes more impurities, such as plankton, gas bubble, and dust, and has more nuclei than freshwater. [PERSON] et al. [31] measured the nuclei distribution in different areas of Pacific seawater. [PERSON] [40] compared the nuclei distribution between seawater and freshwater and reported that the seawater environment could be approximated by changing the air content rate in the freshwater. It was reported that the number of nuclei in seawater is about ten times higher than that in freshwater. The nuclei distribution of the freshwater used in the experiment is reproduced by multiplying the reported seawater condition by 0.1. The number density of initial nuclei used for the simulations is shown in Figure 5, which is a modified version of the number concentration density distribution of nuclei reported by [PERSON] et al. [31]. [PERSON] et al. showed that the holographic detection method was a more reliable technique. Besides, the nuclei distribution for the range of 10-25 \(\upmu\)m was measured by using a sampled water of small volume, 1 mL, while the nuclei larger than 25 \(\upmu\)m were measured by using a sampled water of large volume, 150 mL. Therefore, the nuclei larger than 25 \(\upmu\)m are used in the present study. A similar range of initial nuclei sizes was used in the previous studies of [PERSON] et al. [37], [PERSON] et al. [35; 41], and [PERSON] [40]. It should be noted that a parametric study with the varying number density of nuclei is also conducted to investigate the effects of the initial nuclei distribution on the predicted cavitation inception and noise. Figure 5: Number of nuclei per unit volume as a function of the particle radius in the ocean. When the pressure around nuclei is low enough to reach critical pressure, the nuclei grow into bubbles, and cavitation occurs. [PERSON] [28] described this behavior of nuclei in terms of intermolecular forces from a microscopic point of view. Figure 6 shows the variation in potential \(\varPhi\) of intermolecular force based on the distance between molecules, \(x\). Furthermore, slope \(\frac{\partial\varPhi}{\partial x}\) is the intermolecular force. Intermolecular force is attractive at distances where the slope is positive, while it is repulsive at distances where the slope is negative. The so-called surface tension originates from the intermolecular force. Subsequently, position \(x_{0}\) represents an equilibrium state where there is no intermolecular force, i.e., no surface tension. The initial nuclei are assumed to be in an equilibrium where external pressure is equal to the internal pressure of the nuclei. When the external pressure decreases, the volume of the nuclei increases, and intermolecular distance on the nuclei surface increases. Hence, the attraction force is applied. Therefore, for nuclei to grow into cavitation bubbles, the external pressure must be sufficiently lowered based on the difference between vapor pressure and additional force due to surface tension. Typically, the equilibrium distance \(x_{0}\) of the liquid is extremely small, approximately \(10^{-10}\) m. If the cavitation shape is relatively large, similar to that of sheet cavitation, then the effect of surface tension can be neglected.