# Recent Advances in Acoustics of Transitional Airfoils with Feedback-Loop Interactions: A Review
## 4 Discussion of AFL Frequency Selection Mechanism and Airfoil Tonal Acoustic Structure  
The clustering of equally spaced tones observed in AFL-dominated airfoil acoustic spectra (e.g., in Figure 13) are described by the feedback-loop relationship proposed by [PERSON] and [PERSON] [10] in mathematical interpretation of the fact that the convected  
Figure 19: Linear stability analysis (LSA) results (correlated with numerical predictions of laminar separation bubble (LSB) zones) showing instability chordwise amplification on suction and pressure sides for \(Re=144,000\) (a) and \(Re=288,000\) (b) [18]. Figure 18: Separation regions with increasing Reynolds number [18]. BL instabilities must be in phase with coupled acoustic waves at the trailing edge. In application to the AFL scenario suggested by [PERSON] [7] in Figure 2, the resulting formula is,  
\[f_{n}\ \ \frac{L}{U_{c}}+\frac{L}{c_{0}-U_{c}}\ \ =(n+\frac{1}{2}) \tag{11}\]  
where the tonal frequency \(f_{n}\) is related to the modal number \(n\), distance \(L\) between the instability wave inception point (typically, the point of maximum velocity on the surface), the wave convection speed \(U_{c}\), and the speed of sound \(c_{0}\). However, while Equation (11) may help interpret the frequency spacing \(\Delta\,f\) in the observed cluster of peaks, it does not explain the actual selection of frequencies in the spectrum. Several studies employed LSA analysis as the primary tool for identifying the AFL frequency selection mechanism and the relationship between the frequencies of the peak radiated acoustic tone and the most amplified instability modes [12; 16; 17; 18]. [PERSON] and [PERSON] [12] developed the method based on that proposed by [PERSON] et al. [11] and [PERSON] et al. [43] in order to calculate amplification of T-S instability wave on airfoil surface. They employed XFOIL's [44] mean-flow airfoil solutions to solve Orr-Sommerfeld equation [39], and further combined results with the AFL Equation (11) of [PERSON] [10]. The predicted frequencies agreed reasonably well with experimental data of [PERSON] et al. [1]. However, [12] also suggested that due to the nonlinear nature of the AFL mechanism, their model cannot relate the level of the produced acoustic tones to instability amplification over the surface. As discussed further, [15] and [17] later proposed and validated such model based on [PERSON]'s TE noise theory using experimental results and numerical predictions, respectively. Based on their DNS case study of NACA-0012 airfoil in the tone-producing regime with AOA of 2\({}^{\circ}\), \(M\) = 0.1 and \(Re\) = 200,000, [PERSON] et al. [16] developed the LSA analysis to match the radiated peak tonal frequency with that of the most amplified instability mode on the pressure side, but not on the suction side. They proposed a complex dual-feedback-loop scenario of multi-tonal acoustic response involving interaction and modulation between pressure and suction side AFLs. However, as has been later experimentally proven using BL trippings (e.g., [14; 15]), each side can independently produce the acoustic tonal structure characteristic of AFL interactions. The DNS analysis of [PERSON] [16] was further extended by [PERSON] and [PERSON] [17] for several AOAs between 0\({}^{\circ}\) and 2\({}^{\circ}\). When LSA was applied to time-averaged boundary layer profiles, the tonal noise occurred at a frequency significantly lower than that of the most convectively amplified instability wave. Furthermore, with increasing AOA, the most convectively unstable frequency increased significantly while the vortex shedding frequency changed only slightly. The analysis thus revealed that the vortex shedding and acoustic tone observed were not caused directly by convective instability growth, in contrast to the findings in [11; 16; 43].