Lattice Boltzmann method (LBM) has a potential to simulate airfoil self-noise with low Mach number flow including turbulent flow and aerodynamic feedback loops. In this study, the computational techniques concerning LBM were developed toward direct numerical simulation of aeroacoustic fields with low Mach number. For applications of multi-scale phenomena such as flow and acoustic fields, multi-scale model was introduced, which enables to use locally refined grids. The grids were efficiently arranged using the Building-Cube Method (BCM) by dividing the computational domain into multiple blocks with various grid sizes. Furthermore, the zonal DNS and LES approach was adopted to suppress the numerical instability in the region of coarse grids. The grid dependency of the results provided by the present numerical method was investigated by two-dimensional simulations of flow fields around a NACA0012 airfoil using four different grids. Furthermore, a three-dimensional simulation of flow around a NACA0018 airfoil with moderate Reynolds number was conducted. The computational results were compared and have a good agreement with the experimental ones. The present method can simulate flow around airfoil with moderate Reynolds number involving the laminar-to-turbulent transition.