## 3 Aeroacoustic Model
### Engine Acoustic Source Model
#### 3.1.2 Acoustic RSI Noise Source Model  
As mentioned, only the tonal RSI noise mechanism is considered in the present study. The physical background of the mechanism is illustrated in Figure 3: due to wall adhesion, a turbulent boundary layer is formed on the pressure and suction side of the rotor blade. When passing the rotor blade trailing edge, these boundary layers merge and form a wake shear flow, which propagates towards the stator vane leading edge. The mean velocity deficit of the wake causes a periodic fluctuation of the inflow incidence angle at the stator vane leading edge and therefore a periodic fluctuation of the lift force. This fluctuation of the lift force results in tonal noise at the blade passing frequency (BPF) and its higher harmonics. The focus on this noise source is justified by the fact that the rotor is operating at subsonic conditions as shown in Table 1. Thus, no shock noise is generated. Other tonal self-noise sources, such as steady drag, steady lift, and thickness noise, are cut-off for a subsonic ducted rotor [13]. The turbulence, contained in the wakes, additionally excites broadband noise at the stator vane leading edge. However, in the context of this work, only the tonal RSI mechanism is considered. Note that the turbulent wakes are extracted from the numerical simulation at the mixing plane. Using the assumption of stationary and circumferential uniform rotor wakes, the propagation of the wakes towards the stator leading edge is performed according to [PERSON] [12]. Following Moreau [9], the modal amplitudes, consisting of the azimuthal and radial mode orders \(m\) and \(n\), of the tonal RSI noise are given by  
\[A_{mn}^{\pm}=V\int g_{mn}(\omega)\cdot\mathcal{R}(\omega)\cdot\Psi(\omega) \cdot\zeta(\tilde{\omega})\cdot\exp(ik_{x}x_{LE}+im\theta_{LE})\cdot\mathrm{d }r_{s} \tag{1}\]  
In Equation (1), the parameter \(V\) denotes the stator vane count, \(g_{mn}\) is the Green's function for an infinitely long annular channel, and \(k_{x}\) the axial acoustic wave number. \begin{table}
\begin{tabular}{c c} \hline \hline
**Variable** & **Value** \\ \hline Mach flight & 0.78 \\ Mach rotor tip & 0.60 \\ Altitude & 10,363.2 m \\ Engine Mass Flow & 99.45 kg/s \\ Engine Rotational Speed & 2500 RPM \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Aerodynamic and engine conditions. Figure 3: Illustration of Rotor–Stator Interaction (RSI) noise mechanism. Note that the subscript \(LE\) denotes the stator vane leading edge position and the subscript \(s\) the position of the source. This function is given by [9]  
\[g_{mn}(x,r,r_{s},\omega)=\frac{J_{m}(k_{r}r)+Q_{mn}Y_{m}(k_{r}r)}{\sqrt{F_{mn}}} \exp(-ik_{x}^{\pm}x)\cdot\frac{i}{4\pi R}\cdot\frac{J_{m}(k_{r}r_{s})+Q_{mn}Y_{ m}(k_{r}r)}{kRa_{mn}\sqrt{F_{mn}}} \tag{2}\]  
With the acoustic wave number in radial direction \(k_{r}\), radial position \(r\), the Bessel function of first and second kind \(J_{m}\) and \(Y_{m}\), the cut-on factor \(\alpha_{mn}\), and the outer radius \(R\) of the annular channel. Note that the quantity \(Q_{mn}\) models the impact of the hub body and is zero in case of its absence. The quantity \(F_{mn}\) is a normalization factor [9; 12; 14]. The term \(\mathcal{R}=ik_{n}c\) in Equation (1) models the radiation efficiency of the sound source from the blade surface. Note that \(k_{n}\) is the acoustic wave number in normal direction to the blade surface and \(c\) the chord length of the blade. For the acoustic modeling, all blades and vanes are assumed to be flat plates [9]. The parameter \(\Psi(\omega)\) models the dimensionless source distribution along the chord and is given by  
\[\Psi(\omega)=\frac{1}{c}\int_{l=0}^{c}h_{L}(l)\exp(-ik_{l}l)\mathrm{d}l, \tag{3}\]  
with the acoustic wave number in direction of the blade chord \(k_{l}\) and \(h_{L}(l)\) the non-dimensional distribution of the blade loading along the chord with the maximum load at the leading edge of the blade [9].