## 1 Introduction
Therefore, it is important to understand the onset condition of cavitation in the propeller design process for developing low-noise propellers at high-speed operations. Primarily, tip-vortex cavitation is known to occur earlier than others such as sheet cavitation as the operation speed increases, and the accurate prediction of the tip-vortex cavitation noise is crucial for designing low-noise propellers [13; 14]. Conversely, the prediction of CIS for full-scale vehicles is of considerable significance to developers and manufacturers. In practice, a small-scale laboratory test is typically performed for the analysis of tip-vortex cavitation in a design stage. Subsequently, the scaling law is employed to extrapolate the small-scale result for the full-scale application. However, there are some cases where the classical scaling law based on the McCormick formula should be corrected [15; 16]. The correction is required due to real flow effects, such as nuclei distribution and dynamics and bubble/flow interaction and unsteadiness, which are generally termed as scale-effects. Based on a study by [PERSON] [16], several experimental studies have been conducted to extrapolate the model-scale results in full-scale results by varying the McCormick exponent [17; 18]. Furthermore, a proper criterion for determining the onset of cavitation inception is not clear. Two representative measures correspond to acoustic and optical criteria. The former is the variation of sound pressure level, and the latter involves a bubble size larger than a threshold. In any case, developing a high-fidelity numerical method that can accurately and efficiently analyze cavitation initiation and noise is critical for evaluating CIS at the design stage. The numerical methods for predicting cavitation flows can be classified into two: one is based on Eulerian approaches that the other is based on Eulerian-Lagrangian approaches. The Eulerian approaches are further classified into two: one is based on the Reynolds-averaged Navier-Stokes (RANS) solvers, and the other is based on the high-order solvers such as Large Eddy Simulation (LES)and Direct Numerical Simulation (DNS) methods, both of which are combined with cavitation models. [PERSON] et al. [19] investigated the effects of viscous flux in using the preconditioning and dual-time stepping methods with a homogeneous mixture model on numerical prediction of the sheet cavitation and noise of hydrofoils. [PERSON] et al. [20; 21] performed two-phase flow simulations using the Chimera grid method on massage passing interface (MPI) parallelization. [PERSON] et al. [22] used the Sauer cavitation model and volume of fluid (VOF) technique to predict tip-vortex cavitation of a NACA 66\({}_{2}\)-415 wing. In general, the RANS simulation results [19; 20; 21; 22] were in reasonable agreement with those that were experimentally observed when the cavitation is strong and large, such as sheet and cloud cavitation. However, it is challenging to simulate the tip vortex cavitation using the RANS solver alone because the tip-vortex core center where the cavitation bubble starts to develop belongs to the micro-scale phenomenon. It is reported that the much higher grid resolution is required to capture the vortex core [23]. Besides, a turbulence model contaminates the numerical solution of the vortex core due to excessive turbulence kinetic energy and dissipation [24]. Recently, to overcome the difficulty in resolving the tip-vortex flow, the LES and DNS simulations are employed [25; 26; 27], which require much numerical cost and still have difficulty for the prediction of cavitation noise. Furthermore, in the case of the Eulerian approaches, cavitation inception is determined by only using the vapor pressure. Here, it is assumed that different phases are homogeneously distributed in the given control volume. In this respect, Eulerian methods exhibit two limitations in simulating the cavitation inception.