**5. Conclusions**

Aerodynamic noise from wind turbine blades is one of the major hindrances for the widespread use of large-scale wind turbines to generated green energy. Generally, the weakly compressible turbulent flows around large-scale wind turbine blades that induce the aerodynamic noise, are typical, high Reynolds number flows at low Mach numbers. In order to accurately guide wind turbine blade manufacturers to optimize the blade geometry for aerodynamic noise reduction, an acoustic model that not only understands the relationship between the behavior of sound source and the sound generation but accounts for the compressibility effect, is required. To that end, in this study, a new acoustic theory was proposed, in which we rearranged the continuity and Navier–Stokes equations as a wave equation with a lump of source terms, including the material derivative and square of the velocity divergence. These source terms were used for sound source detection and the estimation of the far-field sound. Our acoustic model was applied to low Mach number, weakly compressible turbulent flows around NACA0012 airfoil. For the computation of flow fields and considering the weak compressibility in flow fields, a LES with the dynamic Smagorinsky subgrid-scale model [28,29] and the CIP-combined unified numerical procedure method [30] was conducted. The reproduced turbulent flow around NACA0012 airfoil was in good agreement with the experimental data. For the estimation of acoustic fields, different acoustic models were performed using our LES database. The distribution of the sources obtained from our acoustic model was compared with the classical sound source models, such as Lighthill [11] and Powell [19] in the case of very low fluctuation of density. The investigation suggested that the derived material derivative of the velocity divergence plays a dominant role as a sound source, and the distribution of our derived source model is consistent with that of the classical sound models. The sound pressure level predicted based on the above-mentioned LES and our newly derived acoustic model was compared with the SPLs calculated from the Lighthill–Curle's equation [18] employing our LES database and the experimental data by Miyazawa et al. [31]. The results showed that the SPL

from our acoustic model was in reasonable agreement with the experimental data. The influence of the increase of Mach number on the acoustic field was investigated. From the observation, our acoustic source model was verified to be capable of treating the influence of Mach numbers on the acoustic field. As a result, noise prediction of the large-scale wind turbine blade using our acoustic source model is more accurate at a low Mach number, and further, can more accurately guide wind turbine blade manufactures to optimize the blade geometry for aerodynamic noise reduction. At this stage, further validation of sound rated by comparisons with experimental data is necessary. But we believe our proposal contributes to the development of computational aeroacoustics for applications in low Mach number turbulent flows, such as large-scale wind turbine blades.

**Author Contributions:** H.T. and Y.L. conceived the original ideal; H.T. wrote and edited the manuscript; and Y.L. and X.L. supervised the study.

**Funding:** This research was funded by a project of the Chinese National Natural Science Foundation (grant number: 51575220), a project of the Key Scientific and Technological Project of Jilin Province (grant numbers: 20160519008JH and 20170204073GX), and a project of National Key R&D Program of China (grant number: 2016YFB0101402).

**Acknowledgments:** The authors thank Takeo Kajishima for his help with the numerical method proposed and simulations conducted.

**Conflicts of Interest:** The authors declare no conflict of interest.
