## 1 Introduction
Subsequently, the team performed numerical calculations for the cavitation inception of this ducted propeller with the minimum pressure coefficient method, and found that the calculated incipient cavitation number and inception position differed greatly from the experimental results [21; 22; 23; 24]. It further indicates that the rationality of making a prediction for propeller TVC inception with the minimum pressure coefficient method needs further discussion. Moreover, the minimum pressure coefficient method is heavily dependent on the local grid resolution at the tip vortex, and grid quality and refinement influence the prediction accuracy directly [24; 25]. Therefore, it is necessary to develop a new numerical prediction method for TVC inception so as to accurately predict and investigate propeller TVC inception. In order to overcome the limitations of the above methods, many researchers have set about modeling microbubbles. In recent years, the numerical prediction method for cavitation inception with the combination of the Lagrangian framework and the Eulerian framework based on bubble dynamics theory has developed rapidly. [PERSON] [26] was the first to undertake research into nuclei and observed the generation of nuclei in a narrow duct and the collapse of nuclei in a downstream high-pressure region. The theoretical analysis of nuclei, originally carried out by [PERSON] [27], is the foundation of modern research on cavitation and bubble dynamics. [PERSON] solved the flow problem of an infinite flow field that has a spherical cavity with a vacuum or constant internal pressure based on the law of conservation of energy and proposed an ordinary differential equation (i.e., Rayleigh equation) that neglected the compressibility, surface tension and viscosity to describe the evolution process of bubbles. [PERSON] [28] derived the well-known Rayleigh-Plesset equation (i.e., R-P equation) by incorporating the surface tension and viscous effect into the [PERSON] model and replacing the constant internal pressure of liquid with the ambient pressure of the actual flow varying with time, promoting the development of bubble dynamics greatly. Subsequently, [PERSON] et al. [29] considered the compressibility of liquid on the basis of the R-P equation and derived the Herring-Trilling equation by modifying the R-P equation. [PERSON] [30] introduced the Kirkwood-Bethe hypothesis [31] and put forward a bubble dynamics equation that took into account the compressibility and dealt with the effect of liquid compressibility more successfully. However, this model is only applicable to the operating condition with a high drive pressure, and it is difficult to obtain an analytical solution in a complex flow field directly [32]. [PERSON] [33,34], [PERSON] and [PERSON] [35], [PERSON] [36] and [PERSON] [37,38] considered the compressibility of liquid while reckoning in the surface tension and viscous effect of liquid, which further improved the bubble dynamics theory. With the continuous improvements of relevant theories, bubble dynamics has been applied to a wide range of physical scenarios, such as gas diffusion, heat transfer, and the magnetic field effect. The most representative one is the SAP spherical bubble model, proposed by [PERSON] and [PERSON] that is based on single bubble dynamics, which was applied to the prediction of propeller cavitation inception [39, 40, 41, 42, 43, 44, 45, 46]. This model solves for flow field trajectories and boundary conditions with the Eulerian equation, which is followed by releasing microbubbles in the corresponding characteristic region of a flow field, and obtains the growth and movement states of bubbles by solving the Rayleigh-Plesset equation and Lagrangian equation, thus implementing an accurate prediction of TVC inception.