# CFD Design Performance Analysis for a High-Speed Propeller
## 4 Results and Discussion
### Velocity on propeller
The results are presented in three plots representing the residuals of mass and momentum conservation, turbulence parameters (SST k-\(\omega\) model), and wall boundary treatment, respectively. All monitored variables exhibited a consistent and monotonic decrease in root mean square (RMS) residuals, with no signs of numerical oscillation or divergence. The mass conservation residual (P-Mass) reached values below \(10^{-5}\), while momentum components and turbulence parameters decreased below the \(10^{-3}\) threshold, indicating a well-converged flow field. Additionally, the wall scale residual remained well below \(10^{-4}\) confirming the adequate resolution of the near-wall region--crucial for the accurate modeling of momentum and heat transfer near solid boundaries. Therefore, it can be concluded that the solution is numerically converged and physically stable, validating the robustness of the simulation setup and ensuring the reliability of the computed results for flow and associated phenomena analysis. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Fine Mesh** &
\begin{tabular}{c} **Default-Mesh** \\ **Utilized in This Study** \\ \end{tabular} & **Coarse Mesh** \\ \hline Force [N] & 1024.6 & 1020 & 1005 \\ Torque [Nm] & 45.82 & 43.23 & 40.47 \\ Max speed [m/s] & 51.45 & 51.30 & 51.46 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Numerical results for the extreme speed values of 6 [m/s] considered in the analysis. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** &
\begin{tabular}{c} **Between Default and Fine Mesh** \\ **[\%]** \\ \end{tabular} & **Between Default and Coarse Mesh** \\ \hline Force [KN] & 0.5 & 1.5 \\ Torque [Nm] & 6.0 & 6.8 \\ Max speed [m/s] & 0.3 & 0.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The percentages results for the extreme speed values of 6 [m/s] considered in the analysis. Figure 11: Turbulence convergence diagram for v = 2 [m/s] and 3600 rpm. Figure 10: Momentum and mass convergence diagram for v = 2 [m/s] and 3600 rpm. Based on the convergence Figures 13-15 for the case with an inlet velocity of 6 m/s and a rotational speed of 3000 RPM, it can be concluded that the numerical solution is stable and reliable. The RMS residuals for mass and momentum conservation (U, V, W) showed a consistent decrease, reaching values below \(10^{-5}\) and \(10^{-4}\) respectively, which indicates an excellent convergence of the flow field. Similarly, the residuals for the SST turbulence model parameters (k and \(\omega\)) continuously decreased throughout the simulation, and the RMS residuals for the SST turbulence model parameters are also shown in Figure 16.  
Figure 12: Wall and boundary convergence diagram for v = 2 [m/s] and 3600 rpm. Figure 13: Momentum and mass convergence diagram for v =6 [m/s] and 3000 rpm. stabilizing at values below the \(10^{-3}\) threshold, which confirms an adequate resolution of the turbulent phenomena. The rapid convergence of the wall scale residual below \(10^{-4}\) also validates the quality of the computational mesh and the numerical treatment applied at the solid boundaries, which is essential for accurate modeling. Therefore, the absence of oscillations and the monotonic decreasing trend of all residuals confirm the robustness of the simulation setup and the reliability of the obtained results.