## 2 Numerical Methods
### Background Hydrodynamic Flow Simulations Using RANS Solver  
In this study, an elliptic wing with NACA 16-020 cross-section [32] is utilized for the investigation of wing-tip vortex cavitation and noise. The wing of the same geometry is also experimentally investigated. The detailed shape of the targeted wing, dimensions of the test section of the water tunnel, and applied boundary conditions are depicted in Figure 2. The chord and height of the wing are 0.5 m, and the angle of attack is set at 8\({}^{\circ}\). The dimensions of the entire computational domain are determined based on the actual size of the test section of the large cavitation tunnel (LCT) in Korea Research Institute of Ships and Ocean Engineering (KRISO). The related boundary conditions are set as identical to those used in a previous study [18]. The boundary conditions of inlet velocity and outlet pressure are set at 9 m/s and 49.885 kPa, respectively. The vapor pressure is 1.304 kPa, which corresponds to a cavitation number of 1.2. Figure 2: (**a**) NACA 16-020 wing geometry; (**b**) computational domain, boundary conditions, and targeted wing. The hydrodynamic flow field around the wing is predicted by numerically solving the three-dimensional steady incompressible RANS equations. The semi implicit method for pressure linked equations (SIMPLE) scheme, where the continuity and momentum equations are sequentially solved, is used to satisfy the mass conservation. The gravity effect is applied along the \(z\)-coordinate direction. Fluent v19.2 tool is used to perform the numerical RANS simulations described above. A wing-tip vortex begins to form in the lowest pressure region around the wing-tip and develops in the downstream direction. It becomes partially laminar inside its vortex core due to the extremely low turbulent kinetic energy when it is fully established [22]. However, it is difficult to numerically resolve the vortex structure due to the excessive turbulent viscosity of the turbulence model. In this study, five turbulence models, namely, standard k-\(\varepsilon\), realizable k-\(\varepsilon\), re-normalization group (RNG) k-\(\varepsilon\), k-\(\omega\) SST, and the Reynolds stress model (RSM), are considered. The second-order accuracy is retained for numerical differencing schemes, and thus a grid-refinement study is performed to find an appropriate grid size near the vortex structure. Four types of grids are used, and their detailed information is listed in Table 1. The computational domains are divided into three regions: tunnel, wing-near, and vortex zones, as shown in Figure 3. Furthermore, the identical grid-resolution is retained in the tunnel and wing-near zones for all the four grid types. However, the resolution in the vortex zone is varied. The dependence of numerical solutions on grid-resolution is analyzed in terms of the variation of vortex structure along the downstream direction, especially the tangential velocity distribution around the vortex core and vortex core-radius. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Grid Refinement**} \\ \hline \multirow{2}{*}{**Domain**} & **Maximum** & **Number of Cells** & **Total Number of Cells** \\  & **Cell Size (m)** & **(Vortex Domain Only)** & **(Tunnel + Wing-Near + Vortex)** \\ \hline Tunnel & \(\sim\)0.1 & - & - \\ \hline Wing near & \(\sim\)0.0028 & - & - \\ \hline \multirow{4}{*}{Vortex} & \(\sim\)0.0052 & \multirow{2}{*}{8 M *} & \multirow{2}{*}{21.7 M} \\  & (Test) & & \\ \cline{1-1} \cline{3-4}  & \(\sim\)0.0014 & & \\ \cline{1-1} \cline{3-4}  & (Coarse) & & \\ \cline{1-1} \cline{3-4}  & (Middle) & & \\ \cline{1-1} \cline{3-4}  & \(\sim\)0.00035 & \multirow{2}{*}{83.2 M} & \multirow{2}{*}{116.5 M} \\  & (Dense) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Grid refinement in vortex domain. Figure 3: Magnified computational domains for (**a**) test grids and (**b**) coarse, middle, and dense grids domain. To identify the location of the vortex core line, the preliminary RANS simulation is performed with the test grids.