# Noise Reduction Mechanisms of an Airfoil with Trailing Edge Serrations at Low Mach Number
## 2 Computational Methodology
### Acoustic Equations  
The compressible continuity equation and Navier-Stokes equation are given as:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho u_{j})}{\partial x_{j}}=0, \tag{20}\]  
\[\frac{\partial(\rho u_{i})}{\partial t}+\frac{\partial}{\partial x_{j}}(\rho u _{i}u_{j})+\frac{\partial p}{\partial x_{i}}=\frac{\partial\sigma_{ij}}{ \partial x_{j}}, \tag{21}\]  
where  
\[\sigma_{ij}=2\mu\left(S_{ij}-\frac{1}{3}\delta_{ij}S_{ij}\right),\quad S_{ij} =\frac{1}{2}\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial u_{j}} {\partial x_{i}}\right). \tag{22}\]  
The density and pressure of flow field can be decomposed as the mean component and perturbation component:  
\[\rho=\rho_{0}+\rho,\quad p=p_{0}+p \tag{23}\]  
where due to low Mach number  
\[\rho_{0}\gg\rho^{\prime},\quad p_{0}\gg p^{\prime}. \tag{24}\]  
By multiplying \(u_{i}\) by Equation (20) and \(\rho^{\prime}\) by Equation (21), taking the divergence of the sum, subtracting time derivation of the mass conservation law in Equation (20) and adding \(-c_{0}^{2}\frac{\partial^{2}\rho}{\partial x_{i}\partial x_{j}}\) with respect to the equation both sides, we finally obtain an approximation wave equation with the velocity divergence as the sound source:  
\[\frac{\partial^{2}\rho}{\partial t^{2}}-c_{0}^{2}\frac{\partial^{2}\rho}{ \partial x_{i}\partial x_{i}}=\rho_{0}\
abla\cdot[(\mathbf{u}\cdot\
abla)\mathbf{u}]. \tag{25}\]Our wave equation (Equation (25)) with the velocity divergence \(\rho_{0}\
abla\cdot[(\mathbf{u}\cdot\
abla)\mathbf{u}]\) as the sound source is similar to [PERSON]'s wave equation [44] with the vorticity \(\rho_{0}\
abla\cdot(\mathbf{\omega}\times\mathbf{u})\) as the sound source. Obviously, two wave equations both take the effect of the compressibility into account.  
Applying \(\rho_{0}\
abla\cdot[(\mathbf{u}\cdot\
abla)\mathbf{u}]\) in Equation (25) as sound source to predict sound field, the sound pressure in the far field can be obtained by solving Equation (25) using the compact [PERSON]'s function [45]. Assuming that the object is stationary, the velocity at the boundary \(S\) of the object is 0, and the observation point \(x\) is sufficiently far from the sound source area \(y\), Equation (25) can be expanded as follows by using [PERSON]'s formula:  
\[p_{a}(\mathbf{x},t)=-\rho_{0}\iint[(\mathbf{u}\cdot\
abla)\mathbf{u}]\cdot\
abla G(\mathbf{x}, \mathbf{y},t-\mathbf{\tau})d^{3}\mathbf{y}d\mathbf{\tau}, \tag{26}\]  
where \(p_{a}\) means pressure sound and \(G\) is the compact Green function, which can be expressed as  
\[G(\mathbf{x},\mathbf{y},t-\mathbf{\tau})=\frac{1}{4\pi|\mathbf{x}|}\delta\left(t-\mathbf{\tau} \frac{|\mathbf{x}|}{c_{0}}\right)+\frac{\mathbf{x}\cdot\mathbf{Y}}{4\pi c_{0}|\mathbf{x}|^{2} }\frac{\partial}{\partial t}\delta\left(t-\mathbf{\tau}-\frac{|\mathbf{x}|}{c_{0}} \right). \tag{27}\]  
Here, \(\mathbf{Y}\) is Kirchhoff vector and can be solved by \(\
abla^{2}\mathbf{Y}_{i}=0\). Substituting Equation (27) into Equation (26) leads to  
\[p_{a}(\mathbf{x},t)=-\frac{\rho_{0}x_{i}}{4\pi c_{0}|\mathbf{x}|^{2}}\frac{\partial}{ \partial t}\int[(\mathbf{u}\cdot\
abla)\mathbf{u}]\left(\mathbf{y},t-\frac{|\mathbf{x}|}{c_{0} }\right)\
abla\mathbf{Y}_{i}(\mathbf{y})d^{3}\mathbf{y}. \tag{28}\]  
From Equation (28), Fourier transform is performed to obtain \(p_{a}\). Using the obtained \(p_{a}\), the sound pressure level (SPL) is determined by the following equation:  
\[SPL(dB)=10\times log\left(\frac{p_{a}^{2}}{p_{b}^{2}}\right), \tag{29}\]  
where \(P_{b}\) represents the reference pressure, and its magnitude is \(P_{b}=2\times 10^{-5}Pa\).