# Numerical Investigation of Tonal Trailing-Edge Noise Radiated by Low Reynolds Number Airfoils
## 4 Results of the Parametric AFL Sensitivity Study
### Variation in Angle of Attack for Re\({}_{c}\) = 180,000: Tone-Producing Regimes
Figure 8: Instability chordwise amplification at peak total frequency vs. statistical parameters for \(\alpha=0^{\circ}\). Figure 7: Far-field acoustic spectra for cases with tonal peaks. shown due to the lack of LSB on the pressure side and the subsequent suppression of instability amplification. Additionally, the predicted modal amplification at the frequency of the acoustic tone is plotted for comparison. Although the peak frequency of 6375 Hz theoretically reaches a much higher amplification within the LSB, as predicted by the stability analysis (compared to the observed acoustic tonal peak of 1209 Hz), its growth quickly saturates at 20% chord with the flow reattachment, and the instability is likely to be damped out before reaching the TE. In contrast, the acoustic frequency mode, although significantly lower in amplitude, is sustained and grows in amplitude over the length of the airfoil. Overall, the results clearly illustrate that the rapid growth of disturbances is associated with the LSB and the presence of the separation zone, which features an attached and separated region. In the attached regions, the instability waves develop as slowly-growing Tollmien-Schlichting (T-S) waves associated with the BL viscous effects. In the separated regions, the instability switches to the fast-growing Kelvin-Helmholtz (K-H) waves associated with the velocity gradients in the detached shear layers. This switch from T-S to K-H instabilities is critical and associated with the presence of the LSB and the ability of the convected mode to sustain the required amplitude before the instability mode is scattered into acoustic waves at the TE. For the considered range of the tone-producing AoAs, Figure 10 compares the LST-predicted frequencies corresponding to the instability waves with maximum amplification (dashed blue lines) against the peak frequencies of calculated acoustic spectra (dashed black lines). In addition, the solid green line is intended to indicate the airfoil surface location where each frequency mode reaches its maximum. Note that for \(\alpha=0^{\circ}\), the predicted frequency of the most amplified instability mode agrees well with the observed far-field frequency of 1550 Hz, showing a discrepancy of only 3.2%. Figure 9: Instability chordwise amplification at peak tonal frequency vs. statistical parameters for \(\alpha=6^{\circ}\). For other cases, however, a substantial difference is seen, and is also observed between the predicted peak amplifications on the suction and pressure surfaces. At \(\alpha=4^{\circ}\), the absence of the LSB on the pressure side of the airfoil results in weak instability growth, and similarly for \(\alpha=6^{\circ}\) due to the favorable pressure gradient; thus, only the suction side is shown. Overall, these results are indicative of the increase in angle of incidence driving the discrepancy between the peak LST-predicted and acoustic tonal frequencies. Figure 11 supports this conclusion and shows the vortex shedding frequency agrees with the observed acoustic peak tone within 2% error at a low AoA. These results correspond to those of [PERSON] et al. (2016) and [PERSON] and [PERSON] (2017) who performed their stability analysis based on DNS simulations of NACA-0012 airfoil at \(Re_{c}=\) 100,000 at \(0^{\circ}\) to \(5^{\circ}\) and  
Figure 10: Instability mode amplifications (blue line), location of the modal peaks (green line), and the peak acoustic frequency (dashed black line) for \(\alpha=0^{\circ}\), \(2^{\circ}\), \(4^{\circ}\), and \(6^{\circ}\). found that the acoustic tone was identical to the vortex shedding frequency but was of lower frequency than the most amplified frequency predicted by LST. It should be noted that the tonal amplitudes are significantly higher on the suction surface, thus indicating instabilities on this side are the dominant source of pressure fluctuation growth shown earlier in Figures 8 and 9.