*4.2. Comparison of Different Mach Numbers*

In this subsection, using the same computational parameters such as Reynolds number *Re* = <sup>2</sup> × <sup>10</sup><sup>5</sup> and the angle of attack *<sup>α</sup>* = <sup>9</sup>◦ in Section 2, we conducted a LES of the turbulent flow field around the NACA0012 airfoil at Mach number *M* = 0.15. In order to verify if our acoustic method could reproduce the behavior of sound source due to the variation of Mach number, and, investigate the influence of the increase of Mach number on the acoustic field, the results of the sound field in case of *M* = 0.15 are compared with that in case of *M* = 0.0875. Figure 8 shows the instantaneous distribution of ∇ · *u* around NACA0012 airfoil colored by |∇ · *u*| ≤ 0.1 at Mach numbers 0.0875 and 0.15. In the case of *M* = 0.15, the remarkable distribution of ∇ · *u* was observed near the leading edge in the suction side, while a relatively small distribution was shown in case of *M* = 0.0875. From this observation, the compressibility effect due to the increase of Mach number was represented by ∇ · *u*, and thus the change of the behavior of sound source was reproduced.

(**b**) **Figure 8.** Instantaneous, cross-sectional profiles of ∇ · *u* around NACA0012 airfoil at different Mach numbers: (**a**) *M* = 0.0875; (**b**) *M* = 0.15.

Figure <sup>9</sup> shows the instantaneous distribution of <sup>−</sup>*ρ*<sup>0</sup> *<sup>D</sup> Dt*(∇ · **u**) near the leading edge at Mach numbers *<sup>M</sup>* <sup>=</sup> 0.0875 and 0.15. The noticeable difference of the distribution for <sup>−</sup>*ρ*<sup>0</sup> *<sup>D</sup> Dt*(∇ · **u**) between Figure 9a,b was not observed in the immediate vicinity of the leading edge. However, in comparison with the case of *<sup>M</sup>* <sup>=</sup> 0.0875, the clear patterns of <sup>−</sup>*ρ*<sup>0</sup> *<sup>D</sup> Dt*(∇ · **u**) were observed in the circle region in case of *M* = 0.15. From this observation, our acoustic theory was proven to be able to reproduce the influence of Mach number on the sound field.

**Figure 9.** Instantaneous, cross-sectional profiles of <sup>−</sup>*ρ*<sup>0</sup> *<sup>D</sup> Dt*(∇ · **u**) near the leading edge: (**a**) *M* = 0.0875; (**b**) *M* = 0.15.

Figure 10 compares the SPL at Mach number *M* = 0.15 and 0.0875 measured at point 1 m from the leading edge in the direction normal to the streamwise direction. Hutcheson et al. [26] measured the aerodynamic noise generated from the flow field around the NACA0015 airfoil under the condition of the angle of attack *α* = 10◦, and the Mach number at *M* = 0.09, 0.11, and 0.127. Their investigation shows that the position of the peak frequency of SPL tends to move from the high-frequency region to the low-frequency region as the Mach number increases. In Figure 10, the maximum peak value obtained by *M* = 0.15 is observed in a low frequency region, as it was against the profile of SPL obtained by *M* = 0.0875. In other words, the tendency of SPL due to the increase of Mach number is reproduced by our sound source model.

**Figure 10.** Sound pressure level obtained by using *ρ*0∇ · [(*u* · ∇)*u*] at *M* = 0.0875 and 0.15.
