## 1 Introduction
The hybrid methods where the Eulerian methods are coupled with the Lagrangian approach have been developed to simulate the cavitation flow as an alternative. These methods estimate cavitating flow by tracking the variation of nuclei distributed in the inflow region of the associated hydrodynamic flow field. Compared to the conventional Eulerian approaches, the Lagrangian approach has the advantage that it can give a simple and straightforward but reasonably comprehensive account of the cavitation inception from the bubble dynamics at a much reduced numerical cost. In the conventional computational fluid dynamics (CFD) methods such as the RANS solver and LES techniques, the cavitation is described using vapor volume fraction in the fluid under the assumption of a homogeneous mixture. However, there is intrinsic ambiguity in determining the cavitation inception by using the vapor volume fraction. If a low volume fraction value, such as vapor volume fraction \(\alpha_{v}=0.1\), is used to identify the cavitation flow at the interface between the cavitation and the surrounding fluid, the cavitation may be excessively over-predicted. If a high volume fraction value, such as \(\alpha_{v}=0.9\), is applied, the cavitation may be underestimated. There is no generally accepted criterion about which value of volume fraction is physically proper for determining the cavitation inception, especially at the beginning of cavitation. Besides, because the cavitation generation criterion depends solely on the vapor pressure, which is a function of temperature, water quality effects, such as the number density and the detailed distribution of nuclei, are not considered. To resolve this issue, some studies used modified cavitation models considering water quality effects by defining the number of nuclei per unit volume [15]. However, it is assumed that all nuclei have the same size. This means that it does not reflect the distribution of nuclei with various sizes. Regarding the nuclei distribution, [PERSON] et al. [19] characterized the quality of seawater in the specific region of the Pacific Ocean in terms of the number concentration density distribution as a function of radius. [PERSON] [20] compared the cavitation trend in terms of cavitation inception due to the nuclei distribution by applying the different air content rate in the freshwater in the cavitation tunnel. It was confirmed from this result that nuclei distribution in the freshwater could be approximately 1/10 times that of seawater. Because the Eulerian-Lagrangian coupling method can consider nuclei distribution directly in the flow field, cavitation inception can be defined quantitatively by using bubble properties, such as the bubble radius, the number of cavitation bubbles, and acoustic signal by the bubble. Additionally, water quality effects can be directly reflected by varying the initial distribution of the nuclei of different sizes. [PERSON] et al. [21] investigated the TVC behavior of the wing having the NACA0015 cross-section using the Eulerian-Lagrangian approaches. The vortex model was used to reconstruct the tip vortex. The dissipation vortex model (DVM) was introduced to consider the vortex dissipation effect during its convection. The cavitation phenomena were simulated using the modified Rayleigh-Plesset equation for homogeneously distributed nuclei. Cavitation inception was analyzed using the cavitation bubble radius and the number of peak values in the time-history of acoustic pressure. [PERSON] et al. [22; 23; 24] and [PERSON] et al. [25] have developed the 3 DYNAFS solver, which is an Eulerian-Lagrangian two-way coupling solver that can simulate both micro and macro variations in cavitation flow. This method can simulate bubble deformations, such as growth, elongation, separation, and collapse, during the transition process between cavitation patterns.