## 1 Introduction  
Recently, there has been an increase in demand for propulsion systems with higher hydrodynamic performance and lower underwater-radiated noise, as environmental issues are gaining more attention in addition to the traditional military necessity. The underwater-radiated noise from underwater propellers can be grossly categorized into two parts: one is non-cavitation noise, and the other is cavitation noise. It is well known that, when the cavitation occurs on the propeller, the tonal and broadband noise increase rapidly in the wide frequency range. The sheet cavitation is of periodic characteristics at the rotating frequency, and the tip vortex cavitation (TVC) is of statistically ergodic characteristics in a broadband frequency range [1]. Therefore, the reliable and efficient numerical methods which can be used to predict and analyze the cavitation inception and noise are essential for developing low-noise and high-performance underwater propellers, in practice by increasing the so-called cavitation inception speed (CIS). There have been a lot of previous experimental studies to predict the CIS of underwater propellers. [PERSON] et al. [2] experimentally showed that cavitation might be caused by protrusions such as bolts in a full-scale ship and increased the cavitation inception speed by streamlining the propeller surface, including bolts. [PERSON] et al. [3] tried to delay the TVC inception using the additional structure of twisted thread on the propeller blade tip. [PERSON] et al. [4] evaluated the propulsion efficiency and cavitation performance of propellers of a container ship and improved the cavitation performance by decreasing the expanded area ratio and increasing the average pitch. The thru-holes near the leading edge of model propellers were recently applied to delay the cavitation inception by flowing the fluid into holes in a uniform flow condition [5]. [PERSON] et al. [6] tried to investigate the quantitative correlations between cavitation patterns and pressure field variation near a propeller. The cavitation patterns recorded using a high-speed camera in a cavitation tunnel were compared with the measured pressure in the flow field near the propeller when the non-uniform wake was incident on the propeller. The empirical formula using volume variation of cavitation of a propeller was suggested to predict pressure fluctuation at a location apart from the propeller. [PERSON] et al. [7] and [PERSON] et al. [8] evaluated the cavitation inception speed using the detection of envelope modulation on noise (DEMON) analysis. The acceleration signal on the hull near the propeller was used because it is difficult to directly measure the acoustic signal. DEMON analysis, however, can identify specific components only in a particular frequency band. To resolve this limitation and quantitatively determine CIS, [PERSON] et al. [9] extracted the cyclo-stationarity from the acceleration signal on the hull. However, the experimental method is challenging to apply in a design stage, and the scale effect for extending the experimental results obtained for the scaled models to the full-scale product is not straightforward. The numerical methods widely used to predict the cavitation inception of wings and propellers are the Eulerian approaches consisting of the Reynolds-averaged Navier-Stokes (RANS) solvers and the large-eddy simulation (LES) techniques. The potential theory was frequently used in a design stage for underwater-propellers due to its cost-effectiveness. [PERSON] et al. [10; 11] predicted the tonal noise caused by sheet cavitation using cavitation volume variation obtained from the potential-based flow simulation. These results were validated through the comparison with the measured data. [PERSON] et al. [12] and [PERSON] et al.