# Numerical Investigation of Tonal Trailing-Edge Noise Radiated by Low Reynolds Number Airfoils
## 4 Results of the Parametric AFL Sensitivity Study
### Linear Stability Analysis
The code solves a set of linearized Navier-Stokes equations presented in the form  
\[\begin{array}{l}\Gamma\frac{\partial\phi}{\partial t}+\frac{A}{h_{1}}\frac{ \partial\phi}{\partial x}+B\frac{\partial\phi}{\partial y}+\frac{C}{h_{3}} \frac{\partial\phi}{\partial Z}+D\phi=\\ \frac{1}{R_{0}}\left(\frac{V_{xx}}{h_{1}^{2}}\frac{\partial^{2}\phi}{\partial x ^{2}}+\frac{V_{yy}}{h_{1}}\frac{\partial^{2}\phi}{\partial x\partial y}+V_{yy }\frac{\partial^{2}\phi}{\partial y^{for the flow-divergence effects in predicting the instability mode evolution. Substituting (9), the following set of equations are solved for the instability modal growth rates, \(-\alpha_{i}\), with the no-slip wall boundary conditions applied at the airfoil surface and Dirichlet boundary condition applied in the free stream,  
\[\left(B+\frac{i\beta V_{yz}}{h_{3}R_{0}}-\frac{i\alpha V_{xy}}{h_{1}R_{0}} \right)\frac{d\psi}{dy}+\left(D-i\omega\Gamma+\frac{i\alpha A}{h_{1}}+\frac{a^ {2}V_{xx}}{h_{1}^{2}R_{0}}-\frac{i\beta V_{xz}}{h_{3}R_{0}}+\frac{i\beta C}{h_ {3}}+\frac{\beta^{2}V_{zz}}{h_{3}^{2}R_{0}}\right)\psi=\frac{V_{yy}}{R_{0}} \frac{d^{2}\psi}{dy^{2}} \tag{11}\]  
In LASTRACT, the equations are discretized via a 1 st-order scheme in the streamwise direction and 4 th-order central difference scheme in the wall-normal direction. The solution is obtained in a two-step process. In the first step, the viscous terms are neglected to recast the equation in the linear form that can be solved as an eigenvalue problem. The obtained global eigenvalue spectrum generally contains all the discrete modes as well as the continuous spectrum. Once the global eigenvalues are obtained, a local eigenvalue search is performed using the results from the global search as a starting point for the viscous solution using the iterative [PERSON]'s method applied to the governing equations. Note that \(N\)-factor in Equation (9) is essentially the normalized accumulated growth rate of the modal amplitudes integrated over a specified streamwise distance along the BL. The local growth rates of the instability modes thus indicate the slopes of the corresponding \(N\)-factor curves. On the other hand, the root-mean-square (RMS) of the flow disturbance pressure, \(p_{RMS}\), was determined from a high-fidelity numerical analysis based on a sample of \(M\) selected tonal frequency modes,  
\[p_{RMS}(x)=\sqrt{\frac{1}{M^{2}}{\sum_{m}}|p_{m}(x)|^{2}} \tag{12}\]  
and the slopes of the \(p_{RMS}\) curves may be easily related to the weighted modal growth rates,  
\[dp_{RMS}(x)/dx=\frac{1}{M}\frac{\sum\limits_{m}\left(-\alpha_{im}\right)|p_{m }(x)|}{\sqrt{\frac{\sum\limits_{m}|p_{m}(x)|^{2}}}} \tag{13}\]  
The connection between the RMS and \(N\)-factor curves for the disturbance solution thus becomes evident and has been demonstrated in our previous study on the effect of upstream turbulence on AFL suppression ([PERSON] et al. [24]). This feature is employed below to match and elucidate the results obtained from the ILES and LST analyses in the parametric AFL sensitivity study.