*4.1. Comparison of Different Sound Source Detection*

In this section, using LES databases obtained in Section 2 for the weakly compressible flow field around NACA0012 airfoil under the conditions of *Re* = <sup>2</sup> × 105, *<sup>M</sup>* = 0.0875, and *<sup>α</sup>* = <sup>9</sup>◦, results of the sound field due to the flow field are discussed. The relationship between our proposed sound source model (34) and the classical sound source models by Lighthill [11] and Powell [19] is discussed through the comparison of the distribution of sound source terms around the airfoil using our LES database, as shown in Figure 5. Figure 5a,b show the instantaneous and cross-sectional profiles of the well-know classical sound source models of ∇ · (∇ · **T**) by Lighthill [11] and ∇ · (*ω* × *u*) by Powell [19], respectively; Figure 5c,d show the instantaneous and cross-sectional profiles of our derived sound source of Equation (34) for <sup>−</sup>*ρ*<sup>0</sup> *<sup>D</sup> Dt*(∇ · **<sup>u</sup>**) and *<sup>ρ</sup>*0(∇ · **<sup>u</sup>**)2, respectively. Because the experimentally estimated sound source which was caused by the separation bubble was confirmed near the leading-edge, we focused on that region. In Figure 5a–c, the distributions of ∇ · (∇ · **T**), ∇ · (*<sup>ω</sup>* <sup>×</sup> *<sup>u</sup>*), and <sup>−</sup>*ρ*<sup>0</sup> *<sup>D</sup> Dt*(∇ · **u**) are locally similar, near the leading edge. Pairs of positive and negative patterns are observed near the leading edge in the suction side, and intense regions correspond to the experimentally estimated sound source region. But, the distribution of *<sup>ρ</sup>*0(∇ · **<sup>u</sup>**)<sup>2</sup> differs from the other terms and has a relatively very small value. Therefore, from Figure 5, it is considered that *<sup>ρ</sup>*0(∇ · **<sup>u</sup>**)<sup>2</sup> does not nearly contribute to the sound field, while *<sup>ρ</sup>*<sup>∞</sup> *<sup>D</sup> Dt*(∇ · **u**) plays an important role in generation of sound.

According to the theory of vortex sound by Powell [19] and Howe [22], the main sound source of the aerodynamic noise is related to the behavior of vortex. Thus, the comparison of the behavior of the spanwise vorticity *<sup>ω</sup><sup>z</sup>* and the main sound source <sup>−</sup>*ρ*<sup>0</sup> *<sup>D</sup> Dt*(∇ · **u**) was conducted. Figure 6 shows time evolution of instantaneous and cross-sectional profiles of <sup>−</sup>*ρ*<sup>0</sup> *<sup>D</sup> Dt*(∇ · **u**) and *ω<sup>z</sup>* near the leading edge. The significant distribution of *ω<sup>z</sup>* exists in the suction side, and its region is similar to the distribution region of <sup>−</sup>*ρ*<sup>0</sup> *<sup>D</sup> Dt*(∇ · **u**), apart from the area of strong compressibility. Moreover, the period of moving of <sup>−</sup>*ρ*<sup>0</sup> *<sup>D</sup> Dt*(∇ · **<sup>u</sup>**) corresponds to that of *<sup>ω</sup>z*: its value is 2.3 × <sup>10</sup><sup>−</sup>4. From results that time evolution of <sup>−</sup>*ρ*<sup>0</sup> *<sup>D</sup> Dt*(∇ · **u**) is similar to the time evolution of the unsteady vortex *ω<sup>z</sup>* near the leading edge, and the distribution of <sup>−</sup>*ρ*<sup>0</sup> *<sup>D</sup> Dt*(∇ · **u**) is similar to the distribution of *ρ*0∇ · (*ω* × *u*), it is considered that our sound source model might reproduce the vortex sound appropriately.

We compared sound pressure levels (SPL) for different acoustic models using our LES database and the published results from computation and measurement in the far-field; see Figure 7. The sound pressure in the far-field for our acoustic model was obtained by Equation (37) and converted to SPL by Equation (38). It is necessary to perform volume integration when determining the sound pressure in our sound source model. And the range of volume integration is determined by reference to the distribution of *ρ*0∇ · [(*u* · ∇)*u*] in Figure 5 and in consideration of the calculation costs. In Figure 7, the SPL profile calculated from the Lighthill–Curle's equation [18] employing our LES database was computed through following equation

$$p\_a'(\mathbf{x}, t) = \frac{\rho\_0 \mathbf{x}\_i}{4\pi \varepsilon c\_0 |\mathbf{x}|^2} \frac{\partial}{\partial t} \int n\_j p' \delta\_{ij} \left(\mathbf{y}, t - \frac{|\mathbf{x}|}{c\_0}\right) \nabla Y\_i(\mathbf{y}) d^3 y\_\prime \tag{39}$$

where *p*, is the sound pressure, **x** the observation point, **y** the sound source point, *nj* the component of the outward pointing unit normal of the surface, and *c*<sup>0</sup> the speed of sound. Finally, the SPL is converted by Equation (38). In order to correspond to the experimental conditions of Miyazawa et al. [31] in the prediction of sound pressure, for the computed SPL, the representative velocity in the main flow direction is set to *U*<sup>0</sup> = 30 × *m*/*s*, the chord length is *C* = 0.1*m*, and the observation point is 1*m* away from the leading edge in the direction normal to the streamwise direction.

(**a**)

(**b**)

(**c**)

(**d**)

**Figure 5.** Instantaneous, cross-sectional profile of the sound source terms around NACA0012 airfoil at *<sup>M</sup>* <sup>=</sup> 0.0875: (**a**) ∇ · (∇ · **<sup>T</sup>**); (**b**) ∇ · (*<sup>ω</sup>* <sup>×</sup> *<sup>u</sup>*); (**c**) <sup>−</sup>*ρ*<sup>0</sup> *<sup>D</sup> Dt*(∇ · **<sup>u</sup>**); (**d**) *<sup>ρ</sup>*0(∇ · **<sup>u</sup>**)2.

**Figure 6.** Time evolution of instantaneous and cross-sectional profiles of <sup>−</sup>*ρ*<sup>0</sup> *<sup>D</sup> Dt*(∇ · **u**) and *ω<sup>z</sup>* near the leading edge at *<sup>M</sup>* <sup>=</sup> 0.0875: (**a**,**c**,**e**) denote <sup>−</sup>*ρ*<sup>0</sup> *<sup>D</sup> Dt*(∇ · **u**) at different moments, where the time interval is 3.5 <sup>×</sup> <sup>10</sup>−<sup>2</sup> s; (**b**,**d**,**f**) denote *<sup>ω</sup><sup>z</sup>* at different moments corresponding to the times of (**a**,**c**,**e**), respectively.

Overall, the SPL profiles by <sup>−</sup>*ρ*<sup>0</sup> *<sup>D</sup> Dt*(∇ · **u**) and the Lighthill–Curle's method obtained by our LES do not agree well with the experimental data. However, the SPL profile obtained by the calculation of Miyazawa et al. [31] does not agree with their experimental data either. The reason for the discrepancies between the experimental data and the results obtained by the numerical calculations may be lack of resolution in the simulation of the flow field, or the compact assumption that the sound source area is regarded as a point source in sound pressure prediction. Thus to verify our sound source model, we focused on comparing the SPL profile calculated from the Lighthill–Curle's method which

is widely used for sound pressure prediction, and the SPL profile obtained by our sound source model <sup>−</sup>*ρ*<sup>0</sup> *<sup>D</sup> Dt*(∇ · **u**). Both SPL profiles were in reasonable agreement, especially in high-frequency regions. On the other hand, in both cases, the peak value existed at about 4300 Hz. This frequency is in good agreement with the period of the moving of positive and negative patterns near the leading edge in the suction side confirmed in Figure 5c. The computational cost to use <sup>−</sup>*ρ*<sup>0</sup> *<sup>D</sup> Dt*(∇ · **u**) as a sound source increases approximately 40% against the Lighthill–Curle analogy, due to the volume integral required in our method, while the Lighthill–Curle formulation needs only surface integral. But our method enables the understanding of the relationship between the behavior of the sound source and the sound generation. Moreover, our method might be applicable to higher Mach numbers.

**Figure 7.** Sound pressure level at *M* = 0.0875.
