# Scalability of Mach Number Effects on Noise Emitted by Side-by-Side Propellers
## 2 Numerical Tools
### The Aerodynamic Tool  
The aerodynamic solver exploits a Boundary Element Method (BEM)to solve the boundary integral formulation presented in [25]. Let us consider an incompressible, potential flow for which the velocity field, \(\mathbf{v}\), can be expressed as \(\mathbf{v}=\
abla\varphi\), where \(\varphi\) denotes the velocity potential given by the sum of an incident potential, \(\varphi_{I}\), and a scattered one, \(\varphi_{S}\) (i.e., \(\varphi=\varphi_{I}+\varphi_{S}\)). The scattered potential is given by sources and doublets distributed over the body surface, \(S_{B}\), and by the doublets distribution over the near wake, \(S_{W}^{N}\), namely the portion of the wake close to the emitting trailing edge. The incident potential is instead generated by the doublets distribution over the far wake, \(S_{W_{S}}\) (i.e., the near wake complementary wake regions) [25]. This splitting of the wake surface is such that only the far wake may impact bodies. The incident and scattered potentials are discontinuous across \(S_{W}^{F}\) and \(S_{W}^{N}\), respectively. Furthermore, the scattered potential equation is [25]  
\[\varphi_{S}(\mathbf{x},t)=\int_{S_{B}}\biggl{[}G(\mathbf{v}_{n}-\mathbf{u}_{n})-\varphi_{S }\frac{\partial G}{\partial n}\biggr{]}dS(\mathbf{y})-\int_{S_{W}^{N}}\Delta \varphi_{S}\,\frac{\partial G}{\partial n}\,dS(\mathbf{y}) \tag{1}\]  
where \(G=-1/4\pi\,||\mathbf{y}-\mathbf{x}||\) is the 3D free-space Green function, whereas \(\Delta\varphi_{S}\) is the jump of the scattered potential across the wake surface, which comes from the time history of scattered potential discontinuity at the corresponding body trailing edge exploiting [PERSON]'s condition [29,30]. Furthermore, from the impermeability of the body surface \(S_{B}\), \(\mathbf{v}_{n}=\mathbf{v}_{B}\cdot\mathbf{n}\), with \(\mathbf{v}_{B}\) denoting the body velocity and \(\mathbf{n}\) the unit normal vector to the body surface, oriented towards the fluid domain, while \(\mathbf{u}_{n}=\mathbf{u}_{I}\cdot\mathbf{n}\), where \(\mathbf{u}_{I}\) is the far-wake induced velocity.  
The numerical solution of Equation (1) is obtained through a zeroth-order boundary element method, which discretizes both the body and the near-wake surfaces into quadrilateral panels. Then, the scattered potential, \(\varphi_{S}\), its normal derivative, \(\mathbf{v}_{n}\) and its jump across the wake, \(\Delta\varphi_{S}\), are assumed uniformly distributed on each discretization panel (with the value matching that at the panel centroid), and the far-wake induced velocity, \(\mathbf{u}_{I}\), is evaluated by exploiting the equivalence between the distribution of surface doublets and vortices, through the Biot-Savart law applied to vortices shaped as the wake panels contour. This formulation becomes singular in strong blade-wake interaction conditions (i.e., once the wake vortices impact a body). The Rankine vortex model is used to remove this singularity and ensure a regular induced velocity field even in severe blade-vortex interactions [25]. The wake shape evolution is obtained through a free-wake algorithm, which moves the vertices of the wake panels as per the velocity field induced by the bodies and their wakes. Equation (1) implies that the incident potential influences the scattered one through the induced-velocity, while the scattered potential influences the incident one through its trailing-edge jump that is transported by the wake material points and defines the intensity of the far-wake vortices [25]. Once the scattered potential and the far-wake-induced velocity are known, the Bernoulli theorem allows the evaluation of the pressure field on the bodies [31] from whichthe body loads can be easily evaluated. Note that the Prandtl-Glauert transformation, applied to the pressure field, is used to include the compressibility effects to the aerodynamic formulation.