# Noise Reduction Mechanisms of an Airfoil with Trailing Edge Serrations at Low Mach Number
## 6 Effect of Serrations on the Sound Field  
Figure 12 shows the distribution of sound source \(\rho_{0}\
abla\cdot[(\mathbf{u}\cdot\
abla)\mathbf{u}]\) around the baseline NACA0012 airfoil and the trailing edge serrated airfoil. The value of \(\rho_{0}\
abla\cdot[(\mathbf{u}\cdot\
abla)\mathbf{u}]\) is slightly smaller near the trailing edge in the case of the serrated airfoil (section \(B^{\prime}B\)) than that in the case of the baseline NACA0012 airfoil. At the root of the serrations (section \(A^{\prime}A\)), a relatively strong distribution of \(\rho_{0}\
abla\cdot[(\mathbf{u}\cdot\
abla)\mathbf{u}]\) is confirmed in comparison with that at the section \(B^{\prime}B\), while in the wake away from the trailing edge of \(A^{\prime}A\) there is no strong distribution of sound source compared to in the wake of section \(B^{\prime}B\). This is believed to be due to the reduction of fluctuation components in the vertical cross-section of the mainstream direction, which is thought to be due to the development of a strong main flow direction vortex near the wake of section \(A^{\prime}A\) compared to near the wake of section \(B^{\prime}B\) in Figures 9 and 10. Figures 13 and 14 show the instantaneous distribution of the spanwise direction vorticity \(\omega_{z}\) around the baseline airfoil and the profile of averaged friction coefficient on the suction side of NACA0012 airfoil, respectively. In Figure 12a, it is shown that the strong sound source near the trailing edge is distributed at the location of \(x/C=0.82\), which can be confirmed by the distribution of the spanwise direction vorticity \(\omega_{z}\) (see Figure 13). It is interesting that the strong sound source location of \(x/C=0.82\) is in good agreement with the position where the separated boundary layer reattaches at the trailing edge of the airfoil (see Figure 14). From this observation, it can be said that it is effective to reduce the strong sound source around an airfoil due to the reattachment of the separated boundary layer by using serration before reattaching the separated boundary layer. Figure 11: Profile of the pressure fluctuation near the trailing edge: (a) NACA0012 airfoil: and (b) serrated airfoil. Figure 14: Averaged friction coefficients on the suction side of NACA0012 airfoil. Figure 12: Instantaneous distribution of \(\rho_{0}\
abla\cdot[(\mathbf{u}\cdot\
abla)\mathbf{u}]\) around airfoils, colored by \(|\rho_{0}\
abla\cdot[(\mathbf{u}\cdot\
abla)\mathbf{u}]|\leq~{}8000\): (**a**) NACA0012 airfoil; and (**b**) serrated airfoil. The sound source, as shown in Figure 12, was used to calculate the sound pressure. Figure 15 compares the sound pressure level (SPL) measured at a point 0.5 m from the trailing edge in the direction normal to the mainstream velocity for the \(x-y\) plane. The SPL obtained by the Lighthill-Curle method [45] using our LES database is compared. In addition, the experiment results by [PERSON] et al. [49] measured at same condition are also shown for comparison. In Figure 15, the SPL profile obtained from the flow field around the baseline NACA0012 airfoil is in reasonable agreement with the experimental data between 1000 and 3000 Hz. Especially, the location of the peak SPL around 1800 Hz seen in the experimental data are well reproduced, while the value of the peak SPL seen in the measurement is slightly underestimated. For the SPL profile obtained from the flow field around the serrated airfoil, the peak value near 1800 Hz disappears and the peak value is seen around 2000 Hz. However, it can be seen that the peak value for the serrated airfoil is smaller than the value for the baseline airfoil, specifically 83 dB for the peak value of baseline airfoil and 91 dB for the peak value of serrated airfoil. On the other hand, the overall sound pressure level (OASPL) obtained by summing up for each frequency band is determined by the following equation:  
\[\text{OASPL(dB)}=10\times log\left[\int(\frac{p_{a}^{2}}{p_{b}^{2}})df\right], \tag{31}\]  
where \(f\) means frequency.