## 3 Aerodynamic Noise Analysis of the Panel  
### Incompressible Numerical Simulation  
#### 3.1.1 Convective Component of FSP  
(a) Corcos Model  
There are several empirical wavenumber-frequency surface pressure models that can be used to extract the convective component. The most commonly used and widely accepted model is the Corcos model [5]. The model supposes that the waves travelling streamwise are independent of the waves travelling across to the stream. So streamwise decay and cross stream decay can be gained separately. Another assumption is that the spatial correlation of fluctuating surface pressure is homogenous within a given flow zone and the WFS over that zone is approximately equal to the product of space-averaged pressure spectral density and the spatial correlation. The Corcos model can be expressed as follows  
\[\varPhi_{p}\Big{(}r_{x},r_{y},\omega\Big{)}=\varPhi(\omega)e^{-\alpha_{x}| \omega r_{x}|/U_{c}}e^{-\alpha_{y}|\omega r_{y}|/U_{c}}e^{i\omega r_{y}/U_{c}} \tag{1}\]  
Figure 8: Velocity vector diagram for \(Y=0\) plane. Figure 9: Distribution of sound sources on the side mirror and Curle surface of the panel. where, \(\alpha_{x}\) and \(\alpha_{y}\) represent the decay of surface pressure coherence in streamwise and cross flow direction. \(\Phi(\omega)\) is the space-averaged pressure PSD. They are chosen to yield the best agreement with the experiment data. After the two-dimensional wavenumber transform of Equation (1), one finds [19]:  
\[\Phi_{p}\Big{(}k_{x},k_{y},\omega\Big{)}=\Phi(\omega)4\alpha_{x}\alpha_{y}/[ \alpha_{x}^{2}+U_{c}^{2}k_{x}^{2}/\omega^{2}][\alpha_{y}^{2}+\Big{(}U_{c}k_{y} /\omega-1\Big{)}^{2}] \tag{2}\]  
Firstly, the area calculated is divided into several small areas. For each area, the cross-spectrum between any two nodes in the direction of airflow and the direction of vertical airflow is obtained, and the cross-spectrum matrix is formed. Because the number of nodes is very large, the cross-spectral matrix of each small area is averaged to reduce the computational complexity. Then the cross-spectra of these small regions are averaged to obtain the spatial correlation function \(\mathrm{R}\Big{(}r_{x},r_{y},\omega\Big{)}\). After obtaining the phase of the spatial correlation function, the wavenumber \(k_{x}\) along the direction of the airflow is obtained using the following Equation (3). The attenuation coefficients along the airflow direction and across airflow direction are obtained by using the following Equations (4) and (5). \[Phase=R(r_{x},0,\omega)=\frac{\omega r_{x}}{U_{c}}=k_{c}r_{x} \tag{3}\]  
\[\alpha_{x}=-ln|R(r_{x},0,\omega)|\frac{U_{c}}{|\omega r_{x}|} \tag{4}\]  
\[\alpha_{y}=-ln|R\Big{(}0,r_{y},\omega\Big{)}|\frac{U_{c}}{|\omega r_{y}|} \tag{5}\]  
Four parameters of the Corcos model (convective velocity \(U_{c}\), attenuation coefficient \(\alpha_{x}\) along the airflow direction, attenuation coefficient \(\alpha_{y}\) in the vertical airflow direction, and spatial average pressure spectrum \(\Phi(\omega)\)) are obtained. Figure 10 shows the spatial average pressure obtained by means of the Corcos model. The wavenumber in the X direction is \(k_{x}=\omega/U_{c}\), \(U_{c}\approx 0.55U\). \(U\) is the airflow velocity. The dissipation coefficients in X and Y directions are 0.1 and 0.72. In Figure 10, we can see that the SPL on the surface of the panel decreases rapidly from 0 to 1000 Hz, and gently above 1000 Hz. Most of the energy of the convective component is concentrated in the middle and low frequencies. When the Corcos model is used to extract the convective component, the panel is required to build a statistical energy analysis (SEA) model. [PERSON] [20] proposed to calculate the local input power of the SEA model by using Equation (6)--concerning the properties and fluid parameters of the SEA structural subsystem. \[\Pi_{int}(\omega)=2\Phi(\omega)\Delta\omega\frac{\pi A}{\rho_{s}}n_{1}(\omega )J^{2}(\omega) \tag{6}\]  
Figure 10: Spatial average pressure obtained by means of the Corcos model.