## 2 Materials and Methods
### Finite Volume Flow Solver  
The conservation equations of mass, momentum, and energy are given in non-dimensional form  
\[\frac{\partial\rho}{\partial t}+\mathbf{\
abla}\cdot(\rho\mathbf{u})=0 \tag{1}\]  
\[\frac{\partial\rho\mathbf{u}}{\partial t}+\mathbf{\
abla}\cdot(\rho\mathbf{u}\mathbf{u}+p\mathbf{ I})+\frac{\tau}{Re_{0}})=0 \tag{2}\]  
\[\frac{\partial\rho E}{\partial t}+\mathbf{\
abla}\cdot\left((\rho E+p)\mathbf{u}+\frac {1}{Re_{0}}(\mathbf{\tau}\mathbf{u}+\mathbf{q})\right)=0 \tag{3}\]  
The quantity \(\rho\) represents the fluid density, \(\mathbf{u}=(u,v,w)^{T}\) the velocity vector, and \(E\) the total specific energy, while \(I\) is the identity matrix. The definition of the total specific energy for a perfect gas is  
\[\rho E=\frac{p}{\gamma-1}+\frac{1}{2}\rho(\mathbf{u}\cdot\mathbf{u}) \tag{4}\]  
where \(p\) is the pressure and \(\gamma\) is the heat capacity ratio. The stagnation state denoted by the subscript \((_{0})\) is used for non-dimensionalization. The Reynolds number \(Re_{0}\) is defined \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{\#} & \multirow{2}{*}{\begin{tabular}{c} **Rod** \\ **Diam.** \\ **(mm)** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Gap** \\ **(mm)** \\ \end{tabular} } & \multirow{2}{*}{**Exp./CFD**} & \multirow{2}{*}{\begin{tabular}{c} **Flow** \\ **Speed** \\ **(m/s)** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **Airfoil Geometry** \\ **Mod. and/or** \\ **Additional Note** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Acoustic** \\ **Method** \\ \end{tabular} } \\ \hline
1 & 10, 16 & 100 & Exp. & \
\begin{tabular}{c} 30.5, 61, \\ 72, 115 \\ \end{tabular} &
\begin{tabular}{c} No, benchmark \\ study \\ \end{tabular} & Exp. & [1] \\
2 & 6 & 220 & Exp. & 15 & No &
\begin{tabular}{c} Curle’s \\ acc. an. \\ \end{tabular} & [2] \\
3 & 10 & 100 & CFD & 72 & No &
\begin{tabular}{c} FWH 1 \\ \end{tabular} & [3] \\
4 & 10 & 100 & Exp. \& CFD & 72 & No & Exp. & [5] \\
5 & 10 & 20–100 & CFD & 72 & No &
\begin{tabular}{c} FWH \\ \end{tabular} & [6] \\
6 & 5, 16 & 86, 124 & Exp. \& CFD & 26 & No &
\begin{tabular}{c} FWH \\ \end{tabular} & [7] \\
7 & 10 & 100 & CFD & 72 & Yes, LE serrations &
\begin{tabular}{c} FWH \\ \end{tabular} & [8] \\
8 & 10, 15, 20 & 150 & Exp. & 72 & Yes, LE serrations &
\begin{tabular}{c} Exp. \\ \end{tabular} & [9] \\
9 & 10 & 100 & CFD & 40 & Yes, LE serrations &
\begin{tabular}{c} Generalized \\ Lighthill Eq. \\ \end{tabular} & [10] \\
10 & 10 & 100 & CFD & 72 & Yes, LE serrations &
\begin{tabular}{c} FWH \\ \end{tabular} & [11] \\
11 & 10 & 100 & CFD & 72 & Yes, LE porous-serrations &
\begin{tabular}{c} FWH \\ \end{tabular} & [12] \\
12 & 15 & 150 & CFD & 40 & Yes, LE porous-serrations &
\begin{tabular}{c} FWH \\ \end{tabular} & [13] \\
13 & 10 & 100 & CFD & 72 & \
\begin{tabular}{c} Yes, regular and irregular \\ LE serrations \\ \end{tabular} &
\begin{tabular}{c} APE-4 2 \\ \end{tabular} & Present study \\ \hline \hline \end{tabular}  
* Flowcs Williams-Hawkings; 2 Acoustic Perturbation Equations.  
\end{table} 
Table 1: Some of the previous rod– airfoil studies. \(Re_{0}=\frac{\partial\omega\circ\Omega L}{\mu_{0}}\), with \(L\) being a reference length, and \(\rho_{0}\), \(c_{0}\), and \(\mu_{0}\) being the density, the speed of sound, and the dynamic viscosity at stagnation. The stress tensor reads  
\[\tau=-2\mu S+\frac{2}{3}\mu(\mathbf{\
abla}\cdot\mathbf{u})\mathbf{I} \tag{5}\]  
where \(\mathbf{S}=[\mathbf{\
abla}\mathbf{u}+(\mathbf{\
abla}\mathbf{u})^{\mathsf{T}}]/2\) is the rate of strain tensor, and the dynamic viscosity \(\upmu\) is determined by [PERSON]'s law. The vector of heat conduction \(q\) is determined by [PERSON]'s law  
\[\mathbf{q}=-\frac{k}{\Pr(\gamma-1)}\mathbf{\
abla}T \tag{6}\]  
where \(T\) is the static temperature. The Prandtl number is defined by \(\Pr=\mu_{0}c_{p}/k_{0}\) with the specific heat at constant pressure \(c_{p}\). For a constant Prandtl number, the thermal conductivity k can be obtained through the relation \(k(T)=\mu(T)\).