# Recent Advances in Acoustics of Transitional Airfoils with Feedback-Loop Interactions: A Review
## 3 Laminar Separation Bubble and Acoustic Feedback Loop Process
### Numerical Approaches
#### 3.2.3 Numerical Implementation  
All numerical approaches (both DNS and LES) must use wall-resolved analysis to accurately predict all details of the AFL process, including BL instabilities inception and development before, through, and after LSB, followed by their TE scattering into acoustic waves and propagation of the latter to the point of instabilities inception occurring though BL receptivity process. Such requirement determines a need for extremely high-resolution meshes and correspondingly demanding computational resources. For instance, the fourth-order accurate DNS computations of [PERSON] and [PERSON] [17] involved 3125 \(\times\) 729 C-mesh grid, yielding approximately 2.3 \(\times\) 10\({}^{6}\) grid-points in total. The sixth-order accurate ILES computations of [PERSON] et al. [18] used 1281 \(\times\) 789 O-mesh, with the grid 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\({}^{-3}\). In terms of the wall units \(y_{\mathrm{w}}\)\({}^{\prime}\)/c = 3.13 \(\times\) 10\({}^{-5}\) estimated for the characteristic flow condition with M = 0.0465 and \(Re\) = 1.4 \(\times\) 10\({}^{5}\), such 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 [38]. Such grid is also finer compared to the mesh employed in 2D DNS study by [PERSON] et al. [16] that was conducted using the mesh with \(\Delta\)y/c = 3.8 \(\times\) 10\({}^{-4}\) and \(\Delta\)x/c = 6 \(\times\) 10\({}^{-3}\). It should be noted that while the use of 2D analysis typically employed in DNS studies could be justified based on the hypothesis of the essentially two-dimensional AFL process, the comparative analysis of 2D vs. 3D approaches conducted in [18] indicates that 2D approach may become misleading. Indeed, the LSB development terminates with the turbulent reattachment occurring close enough to TE but still sufficiently upstream so that the effect of spanwise turbulent redistribution has an impact both on the amplitude and the phase of the vortical structures at TE and the resulting scattered acoustic waves. While such effects will be addressed further below, it should be noted that by comparison,rate of the modal amplitudes integrated over a specified streamwise distance along the boundary layer. The local growth rates of the instability modes thus indicate the slopes of the corresponding \(N\)-factor curves. On the other hand, e.g., for the root mean square (RMS) of the flow disturbance pressure \(P_{RMS}\) determined from the high-fidelity numerical analysis based on a sample of \(M\) selected tonal frequency modes as,  
\[P_{RMS}(x)=\frac{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  
band. Figure 12 (bottom plot) illustrates the result of the method applied to the pressure signal extracted from the near-field simulations [14] (that produced the sound spectrum in Figure 11) using 15 bands with 50% overlap. The amplitudes of the present tones vary by color from deep blue to dark red. They appear to fluctuate with time although the main tonal frequency content remains unchanged. [PERSON] et al. [16] also realized the presence of the random component in the computed signal but proved the consistency of the primary tonal spectrum using the related periodogram method based on averaging the results of the Fourier transform applied to each time band. The first successful comparison of the predicted sound pressure spectrum against experimental results was presented by [PERSON] et al. [18] in the follow-up to the study [14], obtained using the same computational IIES approach.