## 2 Numerical Analysis
### Grid System  
This study utilizes an unstructured grid system, which is divided into inlet, rotor, and outlet regions as plotted in Figure 1a. The inlet and outlet regions simulate the external spaces, which must be large enough to avoid the boundary effect and ensure the validity of the simulation outcome. At first, the incoming airstream enters the fan inlet to flow into the rotating rotor. It follows that energy transfer to the fluid via contact with the blade surface occurs inside the blade passages. Clearly, due to the abrupt energy increase, the physical phenomenon becomes very complex after the incoming flow moves into the fan rotor. Therefore, as shown in Figure 1b, most of the grid points are placed in this rotor region to obtain the detailed information of the flow field.  
Figure 1: Grid system of the stand fan. (**a**) Overall mesh; (**b**) rotor mesh; (**c**) wall \(y^{+}\) over the rotor surface.  
Moreover, it is essential and instructive to check the grid quality around the rotor surface to ensure the appropriate numerical model via the usual wall \(Y_{\text{plus}}\) (\(y^{+}\)) in the CFD community [8]. \(Y_{\text{plus}}\) is the dimensionless distance from the wall and is based on the distance from the wall to the first node and the wall shear stress. It is a non-dimensional number similar to local Reynolds number, determining whether the influences in the wall-adjacent cells are laminar or turbulent, hence indicating the part of the turbulent boundary layer that they resolve.  
\[y^{+}=\frac{u_{z_{air}}y}{v_{air}}. \tag{6}\]  
In previous references [12; 13], the range of \(30<y^{+}<300\) is suggested for a fully turbulent portion, which is similar to the flow pattern associated with the fan rotor. Thus, for the stand fan, Figure 1c illustrates the distribution of wall \(y^{+}\) over the fan rotor, which is around 20-60 for most of the blade surfaces. In addition, to ensure the reliable CFD outcome, the enhanced wall function is utilized on the near-wall treatment in this work.  
In addition, the grid dependence study is executed on the stand fan to identify the appropriate grid system. Table 1 lists the calculated flow rate for several grid systems with different grid numbers. Apparently, the results of the 4.4 M-grid and 5.7 M-grid systems are almost identical and sufficient to yield the detailed flow characteristics associated with the stand fan. Consequently, this study utilizes the 4.4 M-grid system to perform the related CFD simulations. Also, Figure 2 presents the velocity distribution for these grid systems. Obviously, the velocity distribution becomes stable for a grid number higher than 4.4 million. Noticeably, the calculation is reached steadily around the convergence of \(10^{-3}\) while the convergence criterion of this CFD work is set at \(10^{-5}\).  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Case** & **Mesh Number (Millions)** & **Volume Flow Rate (CFM)** \\ \hline
1 & 3.09 & 2341 \\
2 & 3.63 & 2331 \\
3 & 4.40 & 2289 \\
4 & 5.73 & 2288 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Calculated flow rates for the grid dependence study.  
Figure 2: Simulated velocity distributions for the grid dependence study. (**a**) 3.09 million; (**b**) 3.63 million; (**c**) 4.40 million; (**d**) 5.73 million.