# Noise Reduction Mechanisms of an Airfoil with Trailing Edge Serrations at Low Mach Number
## 3 Computational Setup and Test Case  
The object of calculation was the three-dimensional flow around baseline NACA0012 airfoil and the NACA0012 airfoil with the trailing edge serrations. The schematics of the serrated airfoils from top view, center cross section and side cross section are shown in Figure 1. The serration wavelengths and the serration depths were \(\lambda/C=0.2\) and \(h/C=0.1\), respectively. The airfoil thickness was changed starting at 80% the chord length from the leading edge. The computational domain and boundary conditions for NACA0012 airfoil and serrated airfoils are shown in Figure 2. A Cartesian coordinate system was used to define \(x\) in the mainstream direction,  
Figure 1: Sketch configuration of the serrated airfoils. \(z\) in spanwise direction and \(y\) in the vertical direction (perpendicular to \(x\) and \(y\)). The boundary-fitted grid of C-type was applied in the \(x-y\) plane. Actually, all computations were conducted on a general curvilinear coordinate system (\(\xi\),\(\eta\),\(\zeta\)), in which \(\xi\) means the direction following the mainstream surface of the airfoil, \(\eta\) is the direction away from the surface of airfoil, and \(\zeta\) is the same as the direction of \(z\). The computational domain size was defined as follows: the diameter of a half-circle of C-type grid is 8\(C\); and the length of the wake side and the spanwise side are 8\(C\) and 0.5\(C\), respectively, where \(C\) is chord length. As shown in Figure 2, the inflow was a uniform stream without disturbance. Thus, the turbulence was developed in the boundary layer around the airfoil after the transition. The outflow boundary condition was defined as the convective boundary condition. In the spanwise direction, the periodic boundary condition was used. The gradients of variables in the \(\zeta\) direction at the top and bottom boundary were assumed to be zero. The nonslip boundary condition was applied at the surface of the airfoil. To remove the reflection of pressure waves, for pressure, a non-reflective boundary condition by [PERSON] and Kajishima [46] was used in the boundary condition of inflow, outflow, top, and bottom. A second-order central finite-difference discretization scheme was used for the diffusion terms in the equation of motion, and the QUICK method for the convective terms. The QUICK method used here is to reduce the numerical instability resulting from the grid arrangement based on the general curvilinear coordinate system in high Reynolds number flow. As shown in Section 2.2, the fractional method was selected for coupling the continuity equation and the pressure field. For the time advancement, the second-order accuracy Adams-Bashforth method was used for the convective term and viscous term in the Navier-Stokes equation. For the transport equation of the SGS kinetic energy, the donor cell method was employed as spatial discretization scheme, and the second-order accuracy Adams-Bashforth method was utilized to the convective, production, dissipation and diffusive terms. The initialization data of \(K_{gsg}\) was solved from \(k_{gsg}=\left(\
u_{gsg}/C_{v}\tilde{\Delta}\right)^{2}\) using the results of \(\
u_{gsg}\) from LES with Vreman model. Computational parameters of all possible cases conducted in this study are summarized in Table 1. First, straight and serrated in the first column of Table 1 mean the NACA 0012 airfoil with straight trailing edge and serrated trailing edge. To validate numerical method of this study, we conducted LES of the compressible flow around NACA0012 airfoil with the angle of attack, \(9^{\circ}\); the Reynolds number based on the chord length and the mainstream velocity, \(2\times 10^{5}\); and the Mach number, \(8.75\times 10^{-2}\), which matches the computational setup of [PERSON] et al. [47] and the experimental setup of [PERSON] et al. [48]. To exclude the effect of Mach number on the numerical method, we performed simulations of Mach numbers 0, 0.01 and 0.0875 under the same Reynolds numbers, angle of attack and mesh arrangement.