## 2 Models and Methods
### Mathematical Model
Frowcs Williams-Hawkings equation:  
\[\rho_{p}\left(\frac{\partial u_{x}}{\partial t}+u_{x}\frac{\partial u_{x}}{ \partial x}+u_{y}\frac{\partial u_{x}}{\partial y}+u_{z}\frac{\partial u_{x}}{ \partial z}\right) =-\frac{\partial p}{\partial x}+\mu_{p}\left(\frac{\partial^{2}u_ {x}}{\partial x^{2}}+\frac{\partial^{2}u_{x}}{\partial y^{2}}+\frac{\partial^{ 2}u_{x}}{\partial z^{2}}\right) \tag{2}\]9where:  
\[\rho_{p}\left(\frac{\partial u_{y}}{\partial t}+u_{x}\frac{\partial u_{y}}{ \partial x}+u_{y}\frac{\partial u_{y}}{\partial y}+u_{z}\frac{\partial u_{y}}{ \partial z}\right) =-\frac{\partial p}{\partial y}+\mu_{p}\left(\frac{\partial^{2}u _{y}}{\partial x^{2}}+\frac{\partial^{2}u_{y}}{\partial y^{2}}+\frac{\partial^ {2}u_{y}}{\partial z^{2}}\right) \tag{3}\]0 \[\rho_{p}\left(\frac{\partial u_{y}}{\partial t}+u_{x}\frac{\partial u_{y}}{ \partial x}+u_{y}\frac{\partial u_{y}}{\partial y}+u_{z}\frac{\partial u_{y}}{ \partial z}\right) =-\frac{\partial p}{\partial y}+\mu_{p}\left(\frac{\partial^{2}u _{y}}{\partial x^{2}}+\frac{\partial^{2}u_{y}}{\partial y^{2}}+\frac{\partial^ {2}u_{y}}{\partial z^{2}}\right) \tag{3}\]1 \[\rho_{p}\left(\frac{\partial u_{y}}{\partial t}+u_{x}\frac{\partial u_{y}}{ \partial x}+u_{y}\frac{\partial u_{y}}{\partial y}+u_{z}\frac{\partial u_{y}}{ \partial z}\right) =-\frac{\partial p}{\partial y}+\mu_{p}\left(\frac{\partial^{2}u _{y}}{\partial x^{2}}+\frac{\partial^{2}u_{y}}{\partial y^{2}}+\frac{\partial^ {2}u_{y}}{\partial z^{2}}\right) \tag{3}\]2 \[\rho_{p}\left(\frac{\partial u_{y}}{\partial t}+u_{x}\frac{\partial u_{y}}{ \partial x}+u_{y}\frac{\partial u_{y}}{\partial y}+u_{z}\frac{\partial u_{y}}{ \partial z}\right) =-\frac{\partial p}{\partial y}+\mu_{p}\left(\frac{\partial^{2}u _{y}}{\partial x^{2}}+\frac{\partial^{2}u_{y}}{\partial y^{2}}+\frac{\partial^ {2}u_{y}}{\partial z^{2}}\right) \tag{3}\]3  
\(u_{i}\)--fluid velocity component in the \(x_{i}\) direction  
\(u_{n}\)--fluid velocity component normal to the surface \(f\)  
\(v_{i}\)--surface velocity component in the \(x_{i}\) direction  
\(v_{n}\)--surface velocity component normal to the surface \(f\)  
\(T_{ij}\)--Lighthill turbulence stress tensor  
\(P_{ij}\)--compressible stress tensor  
\(H(f)\)--Heaviside function  
\(\delta(f)\)--Dirac delta function  
\(\delta_{ij}\)--Kronecker delta  
\(c_{0}\)--speed of the sound  
\(\rho^{\prime}\)--density fluctuation. Because the flow is incompressible it is possible to assume \(p^{\prime}=c_{0}^{2}\left(\rho-\rho_{0}\right)=c_{0}\rho^{\prime}\). Furthermore, the velocity components normal to the surface are \(u_{n}=0\) i \(v_{n}=0\). Using the above assumptions, the Ffowcs Williams-Hawkings equations simplify to Equations (13)-(15). \[\rho_{p}\left(\frac{\partial u_{y}}{\partial t}+u_{x}\frac{\partial u_{y}}{ \partial x}+u_{y}\frac{\partial u_{y}}{\partial y}+u_{z}\frac{\partial u_{y}}{ \partial z}\right) =-\frac{\partial p}{\partial y}+\mu_{p}\left(\frac{\partial^{2}u _{y}}{\partial x^{2}}+\frac{\partial^{2}u_{y}}{\partial y^{2}}+\frac{\partial^ {2}u_{y}}{\partial z^{2}}\right) \tag{3}\]4  
\[\rho_{p}\left(\frac{\partial u_{y}}{\partial t}+u_{x}\frac{\partial u_{y}}{ \partial x}+u_{y}\frac{\partial u_{y}}{\partial y}+u_{z}\frac{\partial u_{y}}{ \partial z}\right) =-\frac{\partial p}{\partial y}+\mu_{p}\left(\frac{\partial^{2}u _{y}}{\partial x^{2}}+\frac{\partial^{2}u_{y}}{\partial y^{2}}+\frac{\partial^ {2}u_{y}}{\partial z^{2}}\right) \tag{3}\]5  
\[\rho_{p}\left(\frac{\partial u_{y}}{\partial t}+u_{x}\frac{\partial u_{y}}{ \partial x}+u_{y}\frac{\partial u_{y}}{\partial y}+u_{z}\frac{\partial u_{y}}{ \partial z}\right) =-\frac{\partial p}{\partial y}+\mu_{p}\left(\frac{\partial^{2}u _{y}}{\partial x^{2}}+\frac{\partial^{2}u_{y}}{\partial y^{2}}+\frac{\partial^ {2}u_{y}}{\partial z^{2}}\right) \tag{3}\]6  
As a result from the calculations, the pressure fluctuation \(p^{\prime}\) is received as a function of time. The pressure fluctuation is used to calculate the sound pressure level \(L_{sp}\) according to Equation (16).