# Noise Reduction Mechanisms of an Airfoil with Trailing Edge Serrations at Low Mach Number
## 4 Grid Resolution Study and Validation of Numerical Method  
Hereafter, the data were collected by time-averaging and spatial averaging in the spanwise direction. Before discussing the results of the numerical investigation, we first examine the dependence of the intensity of fluctuation of the pressure coefficient on the grid resolution, as shown in Figure 3. Here, the pressure coefficient \(C_{p}\) is defined using the freestream pressure \(p_{0}\), that is, \(C_{p}=(p-p_{0})/\frac{1}{2}\rho U_{0}^{2}\). Then, \(C_{p\,rms}\) means the root-mean-square of \(C_{p}\). Since the resolution in the spanwise direction most affects the turbulence statistics, the grid points in the spanwise direction were set as \(N_{z}=20,60,100\) for a fixed spanwise direction length, as shown Straight-1, -2, and -3 cases in Table 1. In Figure 3, when the grid resolution in the spanwise direction is low, i.e., \(N_{z}=20\), the peak value of \(C_{p\,rms}\) is overestimated compared to the other cases. However, in the case of \(N_{z}=60\) and more grid points \(N_{z}=100\), there is almost no difference in the peak value of \(C_{p\,rms}\). Therefore, it was shown that the peak value of \(C_{p\,rms}\) does not depend on the grid resolution if the grid resolution that can reproduce the anisotropy of the wall turbulence properly is ensured.  
To demonstrate the validity of the numerical method, we compared the results of the present numerical model with the experimental and computational data obtained by [PERSON] et al. [48] and [PERSON] et al. [47], respectively. As for \(C_{p}\) in Figure 4, the results of our model and the experimental data are in good agreement. There is almost no difference in the distribution of \(C_{p}\) on the airfoil due to the change in the Mach number. Comparing the results of the present model at Mach number \(M=0\) with the results of [PERSON] et al., overall good agreement can be seen apart from differences near the leading edge of the airfoil. This difference is believed to be due to the difference of the SGS model. With regard to \(C_{prms}\) on the suction surface side of the airfoil in Figure 5, the location of the peak value of \(C_{p\,rms}\) agrees well with the experimental data regardless of the Mach number, but there is a big difference in the peak value corresponding to the change of Mach number. In the case of \(M=0.0875\), which is the same setup as in the experiment, the agreement of the profile of \(C_{p\,rms}\) between the simulation and experiment is quite good at all locations. In the case of \(M=0\), the profile of \(C_{p\,rms}\) agrees well with the experimental data from the airfoil center to the trailing edge, while the peak value of \(C_{p\,rms}\) near the leading edge is overestimated compared to the experimental data. The results of [PERSON] et al. show the same tendency as that of \(M=0\), although they adopted a different SGS model for a different  
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Case** & \(Ma\) (\(\times 10^{-2}\)) & \(Re\) (\(\times 10^{5}\)) & **Angle of Attack** & \(N_{x}\times N_{y}\times N_{z}\) & \(\Delta_{x}^{+},\Delta_{y}^{+},\Delta_{z}^{+}\) \\ \hline Straight-1 & 8.75 & 2 & \(9^{\circ}\) & \(1600\times 160\times 60\) & \(40,12,40\) \\ Straight-2 & 8.75 & 2 & \(9^{\circ}\) & \(1600\times 160\times 20\) & \(40,12,80\)  
case. Thus, we believe that the overestimation of \(C_{prms}\) observed near the leading edge in the cases of \(M=0\) and [PERSON] et al. is caused by not considering the effect of compressibility near the leading edge of the airfoil. These observations therefore mean that the effect of compressibility cannot be ignored in the vicinity of an airfoil even under low Mach number conditions and the present numerical method is suitable for considering the weak compressibility at low Mach number and then in reasonable agreement with the experimental data.