# Prediction and Control of Broadband Noise Associated with Advanced Air Mobility--A Review
## 3 Methods for Prediction of BBN
### BBN Prediction Models
Historically, there has been quite extensive work in the field of airfoils, with a few researchers applying them to rotating blades. The most prominent analytical theory was proposed by [PERSON] [22] and could be derived for rotor TE noise by [PERSON] and [PERSON] [23]. They derived an airfoil in rectilinear motion:  
\[S_{pp}(x,\omega)=\left(\frac{K_{x}Mzb}{2\pi\sigma^{2}}\right)^{2}\frac{\pi s }{2U_{c}}\big{|}L\big{(}x,K_{x},\lambda,K_{y}\big{)}\big{|}^{2}\Phi_{qq}( \lambda,K_{y}) \tag{9}\]  
\[|L|=\int_{-c}^{0}g\big{(}x_{0},K_{x},\lambda,K_{y}\big{)}e^{-i\mu x_{0}(M-x/ \sigma)}\frac{dx_{0}}{b} \tag{10}\]  
\[G_{TBL-TE,s}(f)=\frac{\delta_{s}^{*}M^{5}L\overline{D}_{h}}{r_{e}^{2}}\cdot H _{s}\left(\frac{f\delta_{s}^{*}}{U},M,\text{Re}_{c}\right) \tag{2}\]0  
\[G_{TBL-TE,s}(f)=\frac{\delta_{s}^{*}M^{5}L\overline{D}_{h}}{r_{e}^{2}}\cdot H _{s}\left(\frac{f\delta_{s}^{*}}{U},M,\text{Re}_{c}\right) \tag{2}\]1  
where \(s\) is the airfoil span and \(\Phi_{qq}\) is the wavenumber spectrum produced by the turbulence. The effective lift function is \(L\). Transforming Equation (9) to rotating and introducing the Doppler-shifted frequency \(\omega/\omega_{0}\), the azimuthally averaged spectrum is then as follows:  
\[G_{TBL-TE,s}(f)=\frac{\delta_{s}^{*}M^{5}L\overline{D}_{h}}{r_{e}^{2}}\cdot H _{s}\left(\frac{f\delta_{s}^{*}}{U},M,\text{Re}_{c}\right) \tag{2}\]2  
[PERSON] et al. [24] proposed a new formulation of trailing-edge noise for rotating blades. They gave the instantaneous spectrum, as shown in Equation (14), and time-averaged PSD, as shown in Equation (15), and validated with [PERSON] and [PERSON] [23]. \[G_{TBL-TE,s}(f)=\frac{\delta_{s}^{*}M^{5}L\overline{D}_{h}}{r_{e}^{2}}\cdot H _{s}\left(\frac{f\delta_{s}^{*}}{U},M,\text{Re}_{c}\right) \tag{2}\]3\[G_{TBL-TE,s}(f)=\frac{\delta_{s}^{*}M^{5}L\overline{D}_{h}}{r_{e}^{2}}\cdot H _{s}\left(\frac{f\delta_{s}^{*}}{U},M,\text{Re}_{c}\right) \tag{2}\]4  
The analytical model is a physical-based frequency domain approach, which helps in understanding the mechanisms of noise generation and allows for detailed analysis across different frequencies. However, the mathematical rigor of the analytical model introduces a level of complexity that can be challenging to implement. Its applicability is primarily limited to trailing-edge noise, and the derivation of its mathematical equations necessitates several assumptions and simplifications, such as the adoption of linear acoustics and the assumption of steady flow conditions. These assumptions may constrain the model's accuracy. Moreover, it lacks the accuracy necessary for low-frequency BBN. Time domain model: BBN can be predicted in the time domain by solving the Ffowcs-Williams and Hawkings (FW-H) equation [25]. At a low Mach number, the quadrupole noise is negligible; the noise is dominated by the thickness noise and loading noise. The BBN is the unsteady loading noise and takes the form of a dipole, which can be solved by the loading part of [PERSON]'s 1A Formulation [26]:  
\[G_{TBL-TE,s}(f)=\frac{\delta_{s}^{*}M^{5}L\overline{D}_{h}}{r_{e}^{2}}\cdot H _{s}\left(\frac{f\delta_{s}^{*}}{U},M,\text{Re}_{c}\right) \tag{2}\]5  
where \(M\) is the surface motion Mach number, r is the distance between the source and observer, and \(L\) is the loading term that is obtained from simulations. \(L_{r}\) and \(M_{r}\) represent the source time derivatives. The subscript \(r\) denotes a dot product of the vector with the unit vector in the radiation direction. The term \(L_{M}=L_{i}M_{i}\). The formulation above assumes a control surface defined by \(f=0\). More recently, [PERSON] and [PERSON] [27; 28] devised a new time domain Formulation 1B for broadband noise predictions, which is a solution of FW-H. Their method demonstrates that time domain approaches could be effectively employed for predicting broadband noise of a propeller/rotor. However, Formulation 1B has certain limitations that do not strictly meet the geometry, kinematics, validation, and observer requirements.