## 3 Aeroacoustic Model
### Propagation Model  
The aeroacoustics propagation model is based on the numerical solution of the acoustic analogy equation proposed by [PERSON] in 1952 [15; 16], and more developed in the succeeding works of [PERSON] [17], [PERSON] [18], and [PERSON] [19]. The frequency domain counterpart of the aforementioned analogy equations leads to a convected Helmholtz equation which results in an integral equation numerical handled with a Boundary Element Method (BEM). The convected Helmholtz equation reads  
\[\bigtriangledown^{2}h(x,x_{q})-(-ik+\mathbf{M}\cdot\
abla)^{2}h(x,x_{q})=q(x_{ q}) \tag{8}\]  
where \(h\) represents the acoustic pressure \(p\) or the scalar velocity potential \(\varphi\), \(q\) the source term, \(k=2\pi/\lambda\) the wave number, and \(\mathbf{M}=v/c\) is the Mach number of the fluid with uniform velocity \(v\). In the potential formulation, the acoustic pressure \(p\) reads  
\[p(x)=\rho c(ik\varphi(x)-\mathbf{M}\cdot\
abla\varphi(x)) \tag{9}\]As extensively reported in several works [20; 21; 22], exploiting the free-field Green's function, the boundary integral formulation of Equation (8) reads  
\[\begin{split} Ch(x,x_{q})=\int_{V}g(x,y)q(y,x_{q})dV(y)+\oint_{S}B_{ 1}(x,x_{s})h(x_{s},x_{q})dS(x_{s})\\ +\oint_{S}B_{2}(x,x_{s})\frac{\partial h}{\partial n}(x_{s},x_{q} )dS(x_{s})\end{split} \tag{10}\]  
where \(x\), \(x_{q}\), and \(x_{s}\) are the position of the observer, the sources, and the points on the surface \(S\), respectively; \(C\) is a coefficient that depends on the location of the point \(x\) (it is 1 in the flow, \(1/2\) on the boundary surface and 0 inside the surface); and the functions \(B_{1}(x,x_{s})\) and \(B_{2}(x,x_{s})\) are given by  
\[\begin{split} B_{1}(x,x_{s})&=(2 ikM_{n}g+[M_{n} \mathbf{M}-n]\cdot\
abla g+\
abla_{tg}\cdot[gM_{n}M_{tg}])\\ B_{2}(x,x_{s})&=(1-M_{n}^{2})g\end{split} \tag{11}\]  
where the tangential divergence \(\
abla_{tg}\cdot[gM_{n}M_{tg}]=M_{n}\Big{[}M_{i}\frac{\partial g}{\partial t} +M_{s}\frac{\partial g}{\partial s}\Big{]}\), involves the directional derivatives and the Mach number components along the \(s\) and \(t\) directions lying in the plane orthogonal to the normal unit vector defined positive pointing into the fluid, and \(g\) is the convected free-field green function as expressed in [23]:  
\[g(x,x_{s})=\frac{1}{4\pi R^{*}}e^{ik(R^{*}+M\cdot(x-x_{s}))/\beta^{2}} \tag{12}\]  
where  
\[\begin{split} R^{*}&=\sqrt{\beta^{2}|x-x_{s}|^{2}+( M\cdot(x-x_{s}))^{2}}\\ \beta&=\sqrt{1-M^{2}}\end{split} \tag{13}\]  
The \(\int_{V}g(x,y)q(y,x_{q})dV(y)\) term of Equation (10) represents the incident field generated at an observer point, \(x\), by a a generic source, \(q\), located in \(x_{q}\). This can be reviewed as a generic incident solution, \(h_{inc}(x,x_{q})\), provided by an external solver. Assuming this generalized incident field, the Equation (10) can be written as  
\[\begin{split} Ch(x,x_{q})=h_{inc}(x,x_{q})+\oint_{S}B_{1}(x,x_{ s})h(x_{s},x_{q})dS(x_{s})\\ +\oint_{S}B_{2}(x,x_{s})\frac{\partial h}{\partial n}(x_{s},x_{q} )dS(x_{s})\end{split} \tag{14}\]  
that allows to solve a generalized scattering problem.  
ACO-FAM is an ACOustic FAST Multipole code developed at CIRA's department of computational acoustics [24]. ACO-FAM implements the aforementioned integral boundary element formulation for the solution of interior and exterior acoustic problems (radiation and scattering) and solves the inverse problem through a collocation Boundary Element Method (BEM) accelerated with an Adaptive black-box Fast Multipole approach (AbbFMM).