# Numerical Investigation of Tonal Trailing-Edge Noise Radiated by Low Reynolds Number Airfoils
## 2 Experimental and Numerical Approaches
### Numerical Approach
#### 2.2.2 Numerical Implementation  
The computational domains employed in this work is a 1281 \(\times\) 789 O-mesh generated about a NACA-0012 airfoil with a no-slip, adiabatic wall with 4 th-order extrapolation ([PERSON] et al. (2015, 2016)). At 100 chords away from the airfoil, a freestream condition is applied to the far-field with the grid rapidly stretching towards the boundary to ensure effective elimination of spurious reflections achieved in conjunction with the low-pass spatial filtering ([PERSON] and [PERSON] (2011)). The mesh is carefully clustered near the airfoil surface to achieve the wall-normal and wall-tangent mesh sizes of \(\Delta\)y/c = 2.5 \(\times\) 10\({}^{-5}\) and \(\Delta\)x/c = 0.5 \(\times\) 10\({}^{-3}\), where c is the airfoil chord length. In terms of the wall units, yw\({}^{+}\)/c = 3.13 \(\times\) 10\({}^{-5}\) is estimated for the characteristic flow condition, with \(\Lambda_{\infty}\) = 0.0465 and \(Re_{c}\) = 1.4 \(\times\) 10\({}^{5}\). Such a grid refinement corresponds to the non-dimensional values of \(\Delta\)y\({}^{+}\)\(\approx\) 1 and \(\Delta\)x\({}^{+}\) = 20, with 12 grid points clustered in the region 0 \(<\) y\({}^{+}\) \(<\) 10. For 3D simulations, such grid parameters correspond to a high-resolution LES according to estimates in [PERSON] et al. [17] (p. 209). Such a grid is also finer compared to the mesh employed in the DNS study by [PERSON] et al. [5] conducted using a mesh with \(\Delta\)y/c = 3.8 \(\times\) 10\({}^{-4}\) and \(\Delta\)x/c = 6 \(\times\) 10\({}^{-3}\). Note that the use of 2D analysis (much less computationally intensive, which is particularly critical for this parametric study) could be justified based on the assumption that, though inherently unsteady, the investigated flow regimes remain primarily laminar (with possible separation zones) and exhibit transitional features. Nevertheless, the next section will present a thorough comparison of the two approaches for the selected cases.  
The current 3D ILES analysis utilizes a NACA-0012 with a spanwise extension of 0.1c, and periodic conditions applied at the span tip planes. Overall, the comparative analysis conducted in the current work employs a matrix of 2D and 3D case studies using the grid configurations shown in Table 1. The baseline 2D setup employs a non-dimensional time step of 9 \(\times\) 10\({}^{-5}\) corresponding to a physical time step of 0.675 \(\times\) 10\({}^{-6}\) s, whereas the 3D simulations require a time step that is 4x finer, a 0.0225 \(\times\) 10\({}^{-5}\) non-dimensionalized time or 0.16875 \(\times\) 10\({}^{-6}\) s, to accommodate the fine grid spacing along the span. In all simulations, the solutions were first marched forward in time for 20 characteristics cycles to ensure transient processes have dissipated and a quasi-steady state is reached. The pressure signals were then recorded for over 65 cycles (720,000 iterations); hence, for the baseline setup, we collected the data sample for 0.487 s with a sampling rate of 33.6 kHz, achieving a frequency resolution of \(\Delta\)f = 2.05 Hz.