# Numerical Investigation of Tonal Trailing-Edge Noise Radiated by Low Reynolds Number Airfoils
## 3 Flow Configuration and Comparison of 2D vs. 3D Numerical Approaches
### Spectral Analysis of Surface and Far-Field Pressure Data Samples  
In addition to the statistical parameters, spectral analysis was conducted for the NACA-0012 airfoil. The numerical predictions consist of a Fourier analysis on 214 pressure samples over a time signal of 0.487 s, whereas experimental measurements were sampled over several seconds. Figure 5 compares the computed and measured spectra for various monitor points along the NACA-0012 surface (Figure 1) as well as the far-field, which is located 12.5 chords directly above the TE. Along the airfoil surface (near-field), the results show a strong correlation between the numerical predictions and measurements in predicting the main tonal signature, which occurred at 610 Hz. However, the overall results reveal that the 3D ILES provides a stronger correlation with the experiment in capturing the harmonics of the main tones, as shown in the experiment, as well as accurately predicting the peak frequency near the trailing edge at 0.945c. It should be noted that the overall broadband level is slightly reduced in the 3D ILES simulations, further improving the comparison with the experiments.  
In the far-field, a Ffowcs-Williams and Hawkings (FWH) approach discussed in application to the current problem by [PERSON] et al. [20] was utilized to generate the numerical spectra, with corrections made to the levels by accounting for the periodic boundary condition along the span, as suggested by [PERSON] et al. [21]. As predicted, 3D ILES shows better agreement with the experimental measurements by continuing the trend of lower broadband levels and a near perfect match between 500 and 1500 Hz, compared to the 2D results. Further evaluation of the far-field spectra elucidates the fact that higher broadband levels shown in the numerical surface spectra result from significant hydrodynamic effects dominating the surface pressure fluctuations that may not be fully captured in measurements. As the pressure signal propagates to the far-field, the unsteady hydrodynamic disturbances quickly dissipate and only the acoustic field dominated by tonal noise components is preserved. Overall, the numerical results demonstrate both broadband levels and peak frequencies correlate well with experimental measurements for frequencies near the primary tone.  
The results shown in this section demonstrate the validity of utilizing 2D analysis by revealing a good agreement with 3D simulations and experimental measurements. The idea of employing 2D simulations for this work stems from the analysis of [PERSON] et al. [5], who argued that 2D simulations are capable of capturing the physics of the AFL due to its inherent two-dimensional nature. In their numerical analysis, 2D simulations were used and performed well in capturing the peak tones exhibited by the airfoil, and in certain cases, captured the neighboring frequencies as well. Therefore, results in the subsequent  
Figure 4: Comparison of 2D (blue) and 3D ILES (magenta) vs. measured (red) suction-side boundary layer profiles. Averaged tangential U-velocity, NACA-0012 at a 0\({}^{\circ}\) AoA and U = 16 m/s.  
sections are obtained mostly using a two-dimensional mesh, except for special cases where 2D analysis would actually lead to contradictory or misleading results, as discussed further below. In Section 4, such results are complemented with the linear stability analysis (applied in the streamwise direction) to further characterize and link the behavior and transition of viscous instabilities with LSB dynamics.  
Figure 5: Comparison of 2D (blue) and 3D ILES (magenta) vs. measured (red) airfoil surface and far-field pressure spectra. NACA-0012 U = 16 m/s. R/\(c\) represents the distance from the airfoil centerline and \(\theta\) represents the angle measured from the chordline (counter-clockwise).