## 2 Experimental and Numerical Method
### Numerical Modeling  
In this study, computation was performed using commercial CFD software, STAR-CCM+ v13.06. Implicitly unsteady, incompressible flow was selected as the physical model for computations. The continuity equation and Reynolds average Navier-Stokes (RANS)  
\begin{table}
\begin{tabular}{c c c c c} \hline \(J_{A}\) & RPS & U(m/s) & \(C_{j}\) & \(Re\) \\ \hline
0.1 & & 0.2 & 0.0/0.00429 & 2.471 \(\times\) 10\({}^{5}\) \\
0.2 & & 0.4 & 0.0/0.00428 & 2.479 \(\times\) 10\  
equations were used as the governing equations and can be expressed in the form of integral equations, as shown in Equations (3) and (4). \[\frac{d}{dt}\int_{\Omega}\rho d\Omega+\int_{\mathrm{S}}\rho u_{i}n_{i}dS=0 \tag{3}\]  
\[\frac{d}{dt}\int_{\Omega}\rho d\Omega+\int_{\mathrm{S}}\rho u_{i}n_{i}dS=\int_ {\mathrm{S}}(\tau_{ij}n_{j}-pn_{i})dS+\int_{\Omega}\rho b_{i}d\Omega \tag{4}\]  
\(u_{i}\) and \(b_{i}\) represent the tensors of velocity and volume, respectively. \(p\) is the pressure and \(\tau_{ij}\) is defined in Equation (5) as the effective stress due to viscosity and turbulence. \[\tau_{ij}=\mu_{e}\Bigg{[}\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{ \partial u_{j}}{\partial x_{i}}\right)-\frac{2}{3}\delta_{ij}\frac{\partial u_ {k}}{\partial x_{k}}\Bigg{]} \tag{5}\]  
Model-scale propellers cannot use turbulence simulators in resistance tests, so turbulent and laminar flow areas coexist. As a result, it is difficult to analyze a fully turbulent model, so there are many studies that apply a transition model. [PERSON] et al. [19] used a fully turbulent model and transition model for the \(SST\)\(k-\omega\) model and compared them with paint-test results. It was confirmed that the flow pattern was improved when the transition model was used. After that, [PERSON] et al. [20] analyzed the turbulence behavior of the turbulence model and the transition model for several Reynolds number. [PERSON] et al. [21] and [PERSON] et al. [22] showed that the result of the transition model was more similar to the experiment by comparing the POW test result corresponding to the Reynolds number of the transition region and the computation result using the fully turbulent model and the transition model. Considering the Reynolds number of the experimental conditions performed in this study, it corresponds to the transition region. Considering the results of previous studies, it is judged that the calculation of this study is appropriate to use the transition model. Therefore, the computations were performed by additionally setting a transition model, and the \(SST\)\(k-\omega\) model and gamma transition model were used. Figure 4 shows the domain and boundary conditions used in the computation. The inlet condition is set to a velocity inlet, and the outlet condition is set to a pressure outlet. Except for the inlet and outlet conditions, all of the surfaces are far enough away from the propeller so as not to affect the calculation, and a slip-wall boundary condition was applied. In the same method as in experimental conditions, the modeling was performed so that the water for jet injection could flow into the propeller through the shaft. A mass flow inlet boundary condition was given to the end of the propeller shaft, and the mass flow rate measured in the experiment was set equally in computations. Figure 5 shows the grid system used in the numerical simulations of this study. A Self-propelled method where the propeller and grid around the propeller rotate together was used for propeller rotation and the timestep of the simulations was set to the time of the propeller rotating 2\({}^{\circ}\). Figure 4: Computation domain and boundary conditions. A grid dependency test was conducted to determine the reliability of the numerical analysis. A total of 4 grid sizes were considered, and each grid size is shown in Table 3. The test was operated at an advance coefficient of 0.1 and the convergence of \(K_{T}\) and \(10K_{Q}\) was compared.