# Numerical Investigation of Tonal Trailing-Edge Noise Radiated by Low Reynolds Number Airfoils
## 4 Results of the Parametric AFL Sensitivity Study
### Acoustic Radiation Frequency Structure
\begin{table}
\begin{tabular}{c} \hline \hline \(\mathbf{U_{\infty}}\) **(m/s), \(Re_{c}\)** \\ \hline
16 (128,000), 19 (152,000), 21 (168,000), 25 (200,000) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Mean flow condition. Figure 23: Airfoil surface pressure spectra at selected monitor points, U = 16 m/s. Figure 22: Instantaneous Z-Vorticity contour: (**a**) U = 16 m/s; (**b**) U = 19 m/s; (**c**) U = 21 m/s; (**d**) U = 25 m/s. As the mean flow velocity is increased to 19 m/s (\(Re_{c}\) = 152,000) (Figure 24), the peak tone shifts rightward in the spectrum and occurs at a higher frequency of 746 Hz (while maintaining \(St\) -0.2). Again, correlating this value with [PERSON]'s formula (14) provides a good match and reveals that the increase from 16 m/s (\(Re_{c}\) = 128,000) to 19 m/s (\(Re_{c}\) = 152,000) would result in a shift from 574 Hz to 729 Hz. This confirms that the peak tones do in fact scale with \(\sim\)\(U^{1.5}\), thus corresponding to the airfoil's natural shedding frequency. Additionally, the increase in the flow velocity leads to neighboring tones with higher amplitudes that are more prominent and easily distinguished in the spectrum. Further increase in mean flow velocity to 21 m/s (\(Re_{c}\) = 168,000) and 25 m/s (\(Re_{c}\) = 200,000) shows a similar trend. Figures 25 and 26 demonstrate the shift of the primary tones to higher frequencies, in agreement with [PERSON], all the considered cases exhibit a similar trend, with the spectral pressure levels peaking at the TE. On the other hand, the case of 25 m/s (\(Re_{c}\) = 200,000) (Figure 26) reveals that while the highest amplitude tone is observed at point A at 1199 Hz, correlating well with shedding frequency (14), and the probes at points B, C, and D show a peak tone at 1281 Hz. As suggested based on the LST analysis, this could be linked to the amplification of various instability modes occurring along the airfoil surface. Perhaps the peak shedding tone of 1199 Hz observed at the TE reduces in amplitude as it propagates upstream and interacts with probes B, C, and D. At the same time, it could be that the secondary peak of 1281 Hz results from a mode that is more amplified in the regions close to points B, C, and D but appears weaker at the TE, where the interaction of the saturated instability modes with the primary shedding mechanism dominates. It is also worth noting that increasing the flow velocity from 16 m/s (\(Re_{c}\) = 128,000) to 25 m/s (\(Re_{c}\) = 200,000) directly affects the amplitudes of the neighboring tones surrounding the peak, and as a result, leads to a very pronounced peak. This comparison is clearly illustrated in Figures 24 and 26 in which the former shows a very low amplitude with nearly suppressed neighboring tones, while the latter shows clearly defined tones. This observation correlates well with the Z-vorticity contours in Figure 22 by visually inspecting the vortices scattering in the wake and correlating the amplification of the neighboring tones with the amplitudes of the vortical structures developing along the airfoil surface. It is evident that at 16 m/s (\(Re_{c}\) = 128,000) the vorticity contours are not as saturated compared to 25 m/s (\(Re_{c}\) = 200,000), which is indicative of stronger vortex cores. These stronger vortex cores will, in turn, produce stronger acoustic waves and thus show higher amplitudes of the predicted spectral peaks. Results obtained from the spectral analysis are now analyzed in order to reconstruct the ladder-type tonal frequency structure originally obtained by [PERSON] et al. [1]. Figure 27 shows such a reconstruction using the numerical data presented in Figures 23-26, in comparison with the shedding-tone formula (14) and experimental results of [PERSON] et al. [31]. Note that \(f_{s}\) is graphically determined by selecting a single highest amplitude peak tone recorded at the four monitor points along the airfoil (typically at the TE). The remaining neighboring tones, \(f_{n}\), are obtained by selecting the equally spaced tones surrounding the peak tone.