## 3 Computational Setup
### Serrated Leading-Edge Geometries  
There are various leading-edge treatments for noise mitigation available in the literature [25; 26; 27]. In this work, regular and irregular serrated trailing edge geometries are tested. The designs with regular serrated leading edges are designed so that the chord length is extended half of the wave length, ensuring a similar wetted area to that of the standard design (STD). New airfoils are designed according to Ref. [11]. The serrated leading-edge airfoil's chord length in relation to the spanwise coordinate 'z' is calculated by Equation (10) where \(c\), \(A\), and \(W\) are the chord length, amplitude, and wave length. In the nose region, the coordinates of the standard leading-edge airfoil undergo adjustments based on Equation (11). Specifically, the x coordinates near the nose undergo stretching  
Figure 1: General view of the computational domain and locations of measurement points and lines.  
or contraction in accordance with the spanwise chord length of the serrated leading-edge airfoil, while the coordinates towards the rear remain unaltered [11]. So, \(y\) coordinates of the airfoil remain the same, while the new \(x\) coordinate (\(x_{new}\)) is calculated according to the new chord length (\(c(z)\)). For instance, for the airfoil with an amplitude of \(A=0.1c\), the minimum and maximum chord lengths will be 0.95c and 1.05c, respectively. There will also be airfoil sections with chord lengths between these values, whose chord lengths can be calculated according to their span position using Equation (10). After calculating the chord lengths, the \(x_{new}\) location is determined using Equation (11). For instance, the new \(x\) coordinate where \(y=0\) for the airfoil with the chord length of \(1.05c\) will be \(x=1.05\), while it will be \(x=0.95\) for the airfoil with the chord length \(0.95c\):  
\[c(z)=c+\frac{A}{2}\cos(\frac{2\pi}{W}z) \tag{10}\]  
\[\begin{cases}x_{\text{new}}&=\begin{cases}\frac{x_{\text{old}}}{x_{\text{max}} }\left[x_{\text{max}}+(c(z)-c)\right]-\left[c(z)-c\right]&x_{\text{old}}&<x_{ \text{max}}\\ x_{\text{old}}&x_{\text{old}}&\geq x_{\text{max}}\\ y_{\text{new}}&=y_{\text{old}}\end{cases}\end{cases} \tag{11}\]  
In Figure 2, various airfoils utilized in this study are illustrated. Leading edges with regular serrations maintain consistent amplitudes across the span, whereas for the irregular design, amplitudes vary with every second serration, either being increased or reduced by half of the amplitude. The main serration lengths are chosen as 0.025c (A2W1) and 0.05c (A1W1), and the wavelengths between two serration tips are \(\lambda_{S1}=0.075c\) and \(\lambda_{S2}=0.0375c\) for the regular designs. In the case of irregular designs, the maximum chord lengths are 1.075c (A3W2), 1.0375c (A4W2), 1.05c (A5W2), and 1.025c (A6W2), while the minimum chord lengths are 1.05c, 1.025c, 1.025c, and 1.0125c, respectively. Illustrations of the regular (left) and irregular (right) serrated leading-edge airfoils are presented in Figure 3.  
Figure 2: Illustration of some airfoil geometries: The A1W1 configuration features regular serrations with a 0.05c amplitude, producing chord lengths between 0.95c and 1.05c, while the A2W1 configuration has regular serrations with a 0.025c amplitude, yielding chord lengths between 0.975c and 1.025c. The irregular A5W2 and A6W2 designs alternate serration amplitudes by \(\pm 50\%\) of their base values (0.05c and 0.025c respectively), creating asymmetric chord distributions.