## 2 Integrated Model of Turbofan Engine and Jet Noise
### Semi-Empirical Model of Jet Noise
The above proportional relationship of unit volume can be simplified as:  
\[I\propto\frac{\rho_{s}^{2}v^{8}}{L\rho_{0}c_{0}^{5}x^{2}} \tag{7}\]  
The sound power of unit volume is proportional to:  
\[W\propto\frac{\rho_{s}^{2}v^{8}}{L\rho_{0}c_{0}^{5}} \tag{8}\]  
For the jet flow field of turbofan engines, the volume of a ring sheet in the flow field is  
\[V\propto Dy_{1}dy \tag{9}\]  
where \(D\) is the equivalent diameter of nozzle area and \(y_{1}\) is the distance from the nozzle. Considering that \(L\) is positively correlated with \(y_{1}\), the sound power of a ring sheet \(W^{\prime}\) is proportional to:  
\[W^{\prime}\propto\frac{\rho_{s}^{2}v^{8}D}{\rho_{0}c_{0}^{5}} \tag{10}\]The length of the mixing zone of the jet is about \(4D\), so the total sound power of the jet is proportional to:  
\[P(x,t)\approx\frac{\rho_{s}}{4\pi c_{0}^{2}x^{3}}V_{e}\frac{\partial^{2}u_{r}^{ 2}\left(y,t-\frac{|x|}{c_{0}}\right)}{\partial t^{2}} \tag{2}\]0  
When the distance between the receiving point and sound source is \(r\), the sound pressure is proportional to:  
\[P(x,t)\approx\frac{\rho_{s}}{4\pi c_{0}^{2}x^{3}}V_{e}\frac{\partial^{2}u_{r}^{ 2}\left(y,t-\frac{|x|}{c_{0}}\right)}{\partial t^{2}} \tag{2}\]1  
Then, the sound pressure level generated by the jet flow at the far-field can be simplified as:  
\[P(x,t)\approx\frac{\rho_{s}}{4\pi c_{0}^{2}x^{3}}V_{e}\frac{\partial^{2}u_{r}^{ 2}\left(y,t-\frac{|x|}{c_{0}}\right)}{\partial t^{2}} \tag{2}\]2  
where \(M,\ N,\ W,\ K\) are constants, \(v\) is the velocity of jet flow, \(\rho_{s}\) represents the average density in the turbulent region, \(A\) is the outlet area, \(r\) is the distance between the sound source and receiving point, and \(f(c_{0},\rho_{0})\) is calculated by local density \(\rho_{0}\) and sonic speed \(c_{0}\). According to the different flow field characteristics, the jet flow field of a turbofan engine with a separate exhaust is divided into several zones for analysis and calculation, which is shown in \[P(x,t)\approx\frac{\rho_{s}}{4\pi c_{0}^{2}x^{3}}V_{e}\frac{\partial^{2}u_{r}^{ 2}\left(y,t-\frac{|x|}{c_{0}}\right)}{\partial t^{2}} \tag{2}\]3 There are four main zones in the figure, namely, the jet core zone, full mixing zone, external shear layer, and internal shear layer. Therefore, the sound pressure level of four zones is calculated by Equation (13), and the total sound pressure level is obtained through the logarithmic superposition of sound pressure level of four parts, which is shown in Equation (14). \[P(x,t)\approx\frac{\rho_{s}}{4\pi c_{0}^{2}x^{3}}V_{e}\frac{\partial^{2}u_{r}^{ 2}\left(y,t-\frac{|x|}{c_{0}}\right)}{\partial t^{2}} \tag{2}\]4  
where \(SPL\) is the total sound pressure level, \(SPL_{ex}\) is the sound pressure level of external shear layer, \(SPL_{in}\) is the sound pressure level of internal shear layer, and \(SPL_{cor}\) is the sound pressure level of the jet core zone. According to Ref [26], NASA carried out a full-size jet noise measurement experiment on a separate exhaust nozzle, with an inner diameter of 0.7 m and an outer diameter of 1.37 m. The experimental object is similar to the object studied in this paper. The experimental data can be used to verify the jet noise model in this paper. Therefore, a simulation test was carried out to verify the calculation accuracy of the semi-empirical model. By regulating the Mach number in the mixed zone, the simulation conditions were similar to the experiment conditions. The measuring radius of the sound field is 50 m. The comparison results between the simulation and experiment are shown in \[P(x,t)\approx\frac{\rho_{s}}{4\pi c_{0}^{2}x^{3}}V_{e}\frac{\partial^{2}u_{r}^{ 2}\left(y,t-\frac{|x|}{c_{0}}\right)}{\partial t^{2}} \tag{2}\]5 The difference between the simulation and experiment at each observation angle is within the range of 5 dB, and the change trend is basically consistent.