## 2 Integrated Model of Turbofan Engine and Jet Noise  
### Semi-Empirical Model of Jet Noise  
A real-time model of the jet noise of turbofan engines with a separate exhaust is the basis of active jet noise control. To meet the real-time requirements of active noise control, a semi-empirical model was chosen for the calculation of the jet noise. There are three main semi-empirical calculation methods: STNJET, ST2 JET, and SAE. STNJET and ST2 JET are suitable for far-field noise, and SAE is suitable for calculating near-field noise. As the measurable jet noise is far-field noise, SAE cannot ensure good calculation accuracy for the jet noise of turbofan engines. The application ranges between STNJET and ST2 JET are different. The subjects of STNJET are usually single or coaxial double flow nozzles, which are limited by the fact that jet speed in the inner duct is faster than that in the outer duct. Compared with STNJET, the application ranges of ST2 JET include convergent or convergent-expanding nozzles, and single or coaxial double flow nozzles; the nozzle of turbofan engines is convergent or convergent-expanding. Besides, ST2 JET has higher calculation accuracy, especially for high bypass turbofan engines [24]. Therefore, this paper selected ST2 JET to build a semi-empirical model of the jet noise of turbofan engines. The magnitude of noise is usually described by sound power \(W\), sound intensity \(I\), and sound pressure \(P\). The sound power \(W\) reflects the total sound energy radiated by the sound source per unit time. The square of sound pressure \(P\) is proportional to sound intensity \(I\). The sensitivity of the auditory nerve to sound pressure is directly proportional to its logarithm. Therefore, sound pressure level \(SPL\) is usually used to measure the size of sound pressure \(P\), which is defined as:  
\[SPL=10\lg\frac{P^{2}}{P_{ref}^{2}} \tag{1}\]where \(P_{ref}\) is the threshold of sound pressure, which can be recognized by the human ear, and \(P_{ref}=2\times 10^{-5}\)Pa. The noise source of jet noise of aero-engines is a small area centered on the nozzle. The type of jet noise is a quadrupole noise source. For the sound field generated by the turbulent vortex, the sound pressure at the far-field receiving point is:  
\[P(x,t)\approx\frac{\rho_{s}}{4\pi c_{0}^{2}x^{3}}V_{e}\frac{\partial^{2}u_{r}^{ 2}\left(y,t-\frac{|x|}{c_{0}}\right)}{\partial t^{2}} \tag{2}\]  
where \(x\) is the position vector of the receiving point, \(y\) is the position vector of the turbulent vortex, \(\rho_{s}\) is the average density of gas in the turbulent region of the jet, \(V_{e}\) is the volume of the turbulent vortex, \(c_{0}\) is the local speed of sound, \(u_{r}\) is the pulsation speed of the vortex, and \(t\) is the time variable. There is no correlation between the pulsations of different vortices. The mean square value of sound pressure at the receiving point in the far-field is the linear superposition of the mean square value of sound pressure generated by each vortex, namely:  
\[\overline{P(x)}^{2}=\left(\frac{\rho_{s}}{4\pi c_{0}^{2}x}\right)^{2}\overline {\left(\frac{\partial^{2}u_{r}^{2}}{\partial t^{2}}\right)^{2}}\sum_{1}^{N}V _{e}^{2} \tag{3}\]  
where \(N\) is the number of vortices. The sound intensity \(I(x)\) is:  
\[I(x)=\frac{\overline{P(x)}^{2}}{\rho_{0}c_{0}}=\frac{\rho_{s}^{2}}{16\pi^{2} \rho_{0}c_{0}^{5}x^{2}}\overline{\left(\frac{\partial^{2}u_{r}^{2}}{\partial t ^{2}}\right)^{2}}\sum_{1}^{N}V_{e}^{2} \tag{4}\]  
The turbulent pulsation of jet flow field is monochromatic. The frequency is assumed as \(w\). Then, the following proportional relationship can be obtained:  
\[I(x)\propto\frac{V_{e}^{2}\rho_{s}^{2}w^{4}u_{r}^{4}}{\rho_{0}c_{0}^{5}x^{2}} \tag{5}\]  
According to [25], it can be known that:  
\[w\propto\frac{v}{L},\ u_{r}\propto v,\ V_{e}\propto L^{3} \tag{6}\]  
where \(v\) is the outlet velocity of jet and \(L\) is the longitudinal spatial scale of turbulent vortex.