## 2 CFD Model Setup and Validation
### Grid Generation and Validation Study
The Grid Convergence Index (GCI) described in Celik [14] is introduced to evaluate the discretization error for the thrust and torque of the propeller under two representative loading conditions: J = 0.1 and J = 0.5. GCI is a discretization error estimation method based on Richardson extrapolation and is calculated in the following way:  
Figure 4: Mesh distribution: (**a**) surface mesh on duct, propeller, and hub; (**b**) volume mesh. \[e_{21}=\left|\frac{\varphi_{1}-\varphi_{2}}{\varphi_{1}}\right| \tag{9}\]  
Figure 3:0  
where \(e_{21}\) is the relative error between the result from the coarse and the medium mesh, \(\varphi_{1}\) and \(\varphi_{2}\) are the results from the coarse and the medium mesh, respectively, 1.25 is the safety factor that is adopted when three or more meshes are used, \(r_{21}\) is the refinement ratio between the coarse and the fine mesh, and \(p\) is the order of numerical accuracy, which is 2 in this study. For the GCI between the medium and the fine mesh, the same procedures can be followed. The results show that for both thrust and torque the GCI values of the fine mesh are lower than that of the medium mesh, which demonstrates that the numerical uncertainty becomes smaller when the mesh is refined. The highest numerical uncertainty is reported within 5.23% and 1.95% for the medium mesh and fine mesh, respectively. Based on the mesh sensitivity study, the fine mesh is used as the final mesh for the present study. After the grid independence study, the numerical results of the ducted propeller are also compared with experimental data by [PERSON] [11] to validate the CFD model. Figure 5 provides an overview of the comparison between the numerical results from the fine mesh and experimental measurements under different advance coefficients. The former is presented in lines and the latter in dots. The curves show that the thrusts on the propeller and duct are predicted well by the simulations. However, the propeller torque is over predicted, causing a discrepancy with the experimental data. The difference in efficiency is within 5% for all the advance coefficients and a similar discrepancy is found in the research of [PERSON] [15]. The discrepancy can be attributed to experimental errors during measurements, physical modeling errors as the geometry is generated from the coordinates provided in [PERSON] [11], and the errors caused by the turbulence model. Since the SST \(k\)-\(\omega\) model assumes a fully turbulent boundary layer, which may not be the case on certain parts of the propeller surface as the rotational speed is rather small, this may be the reason for the exaggerated torque prediction. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Mesh**} & \multirow{2}{*}{\
\begin{tabular}{c} **Total Cells** \\ **(Million)** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Refinement** \\ **Ratio r** \\ \end{tabular} } & \multicolumn{2}{c}{**J = 0.1**} & \multicolumn{2}{c}{**J = 0.5**} \\ \cline{4-7}  & & & & & **GCI (\%)** & & **GCI (\%)** \\ \cline{1-1}  & & & & **Ktp/Kq** & & **Ktp/Kq** & & **Ktp/Kq** \\ \hline Coarse & 5.02 & - & 0.257/0.0456 & - & 0.170/0.0334 & - \\ Medium & 9.79 & 1.25 & 0.262/0.0465 & 4.32/4.39 & 0.174/0.0340 & 5.23/3.19 \\ Fine & 19.6 & 1.26 & 0.263/0.0461 & 0.86/1.95 & 0.175/0.0337 & 1.22/1.88 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Discretization error estimation using the GCI.