## 2 Numerical and Experimental Methodologies
### Fowcs-Williams and Hawkings (FW-H) Acoustics Model  
For predictions of near- to far-field acoustic noise, the FW-H acoustics model based on the [PERSON]'s acoustic analogy provide a feasible alternative to the direct aeroacoustic simulation. The FW-H equation is essentially an inhomogeneous wave equation that can be derived by manipulating the continuity equation and the N-S equations. The FW-H formulation adopts the most general form of [PERSON]'s acoustic analogy, and is capable of forecast sound generated by equivalent acoustic sources such as monopole, dipole, and quadrupole. The FW-H equation [15; 16; 17] can be summarized as:  
\[\frac{1}{a_{0}^{2}}\frac{\partial^{2}\dot{p}}{\partial t^{2}}-\
abla^{2}\dot {p}=\frac{\partial^{2}}{\partial x_{i}\partial x_{j}}\Big{\{}T_{ij}H(f)\Big{\}} -\frac{\partial}{\partial x_{i}}\Big{\{}\big{[}P_{ij}n_{j}+\rho u_{i}(u_{n}-v _{n})\big{]}\dot{\delta}(f)\Big{\}}+\frac{\partial}{\partial t}\{[\rho_{0}v_ {n}+\rho(u_{n}-v_{n})]\dot{\delta}(f)\} \tag{9}\]  
where, \(u_{i}\), \(u_{n}\), \(v_{i}\) and \(v_{n}\) are fluid velocity component in the \(x_{i}\) direction, fluid velocity component normal to the surface \(f=0\), surface velocity components in the \(x_{i}\) direction, surface velocity component normal to the surface, respectively. Moreover, \(\delta(f)\), \(\delta(h)\), \(n_{j}\), \(a_{0}\) and \(T_{ij}\) Dirac delta function, Heaviside function, unit normal vector indicating the exterior region, far-field sound speed and Lighthill stress tensor, respectively. \(\dot{p}\) is \(p-p_{0}\), which denotes the sound pressure at the far field. \(f=0\) corresponds to a mathematical surface introduced to include the exterior flow problem (\(f>0\)) in a borderless space. The surface (\(f=0\)) indicates the source surface, can be coincident with an impermeable body surface or a permeable surface off the body surface. Namely, the impermeable and permeable body surface refer to the wall of the body and the fluid region around the body, respectively. The Lighthill stress tensor is defined as:  
\[T_{ij}=\rho u_{i}u_{j}+P_{ij}-a_{0}^{2}(\rho-\rho_{0})\delta_{ij} \tag{10}\]  
Here, \(P_{ij}\) is the compressive stress tensor and the free-stream quantities are represented by subscript 0. The Equation (9) can be integrated analytically under the premise of the free-space flow without any objects between the acoustic sources and the receivers. The complete solution is composed of surface integrals of the first and second terms, and volume integrals of third term on the right in the Equation (11). However, the contribution of the quadrupole source in the third term is disregarded in the present large eddy simulations because its effect becomes small when the flow is low subsonic.  
\[\dot{p}\Big{(}\overrightarrow{x},t\Big{)}=\dot{p}_{T}\Big{(} \overrightarrow{x},t\Big{)}+\dot{p}_{L}\Big{(}\overrightarrow{x},t\Big{)}+ +\dot{p}_{Q}\Big{(}\overrightarrow{x},t\Big{)} \tag{11}\]  
\[\begin{array}{rl}4\pi\dot{p}_{T}\Big{(}\overrightarrow{x},t\Big{)}&=\int_ {f=0}\Bigg{[}\frac{\rho_{Q}\big{(}\dot{u}_{L}+U_{g}\big{)}}{r(1-M_{\rm r})^{2} }\Bigg{]}dS\\ &+\int_{f=0}\Bigg{[}\frac{\rho_{Q}\big{(}\dot{u}_{L}+U_{g}\big{)}}{r^{2}(1-M_{ \rm r})^{3}}\Bigg{]}dS\\ \end{array} \tag{12}\]  
\[\begin{array}{rl}4\pi\dot{p}_{L}\Big{(}\overrightarrow{x},t\Big{)}=\frac{ 1}{a_{0}}&\int_{f=0}\Bigg{[}\frac{\dot{u}_{r}}{r(1-M_{\rm r})^{2}}\Bigg{]}dS\\ &+\int_{f=0}\Bigg{[}\frac{L_{r}-U_{M}}{r^{2}(1-M_{\rm r})^{2}}\Bigg{]}dS\\ \end{array}+\frac{1}{a_{0}}\int_{f=0}\Bigg{[}\frac{L_{r}\Big{[}r\dot{M}_{r}+a_ {0}\big{(}M_{\rm r}-M^{2}\big{)}\Big{]}}{r^{2}(1-M_{\rm r})^{3}}\Bigg{]}dS \end{array} \tag{13}\]  
Here, the square brackets in Equations (12) and (13) mean that the kernels of the integrals are calculated at the corresponding retarded times (\(\tau\)), receiver time (t), and distance to the receiver (r). The FW-H formulation in ANSYS Fluent can handle rotating surfaces as well as stationary surfaces. A complete description of the model is available in [15; 16; 17].