# A Computational Study on the Aeroacoustics of a Multi-Rotor Unmanned Aerial System
## 2 Materials and Methods
### Computational Domain, Grids, and Boundary Conditions  
For all of the simulations discussed in Section 2.4, the computational domain is a sphere with a diameter of \(18D_{p}\), with the geometry located at center of the domain, as depicted in Figure 3. The grid of fully appended CRC-3 consists of five blocks, four circular blocks for four propellers, and one block for the rest of the computational domain. The sliding mesh technique is employed for rotating the propeller block in the computational domain. The CRC-3 grid has 16 million tetrahedral volume elements, of which around 80% of grid elements are packed near CRC-3 within a small sphere of radius \(2D_{p}\). To capture the acoustic waves, the grid is designed to resolve the first three harmonics of the acoustic wave. Since the 1 st BPF is 200 Hz in simulations, the maximum frequency required to be resolved was assigned as 600 Hz with a wavelength of \(\lambda=c/f=0.57\) m, with \(c=343\) m/s for the standard speed of sound in air. Therefore, the maximum grid size was assigned 0.055 m in the region surrounding the geometry and microphones, which gives at least 10 grid points in acoustic wavelength, as recommended by [PERSON] [44]. It  
\begin{table}
\begin{tabular}{c c c c} \hline
**Case** & **RPM** & \(\mathbf{U}_{0}\) (m/s) & \(\mathbf{\theta}\) (**deg**) \\ \hline C1 & 6000 & 7 & 0 \\ C2 & 6000 & 7 & 15 \\ C3 & 6000 & 7 & 35 \\ C4 & 6000 & 7 & 55 \\ C5 & 6000 & 7 & 75 \\ C6 & 6000 & 4 & 15 \\ C7 & 6000 & 13 & 15 \\ C8 & 12,000 & 7 & 15 \\ \hline \end{tabular}
\end{table}
Table 2: Test matrix.  
Figure 2: Microphone positions: (**a**) circular array; (**b**) on the CRC-3 body.  
should be noted that although the grid is not designed to spatially resolve higher than third harmonics, those harmonics may still be compared between cases to discuss trends.  
Regarding the temporal discretization, the designed time step resulted in no more than 0.54 degree of rotation per time step for propellers for all RPM values, corresponding to time steps of \(1.5\times 10^{-5}\) s and \(0.75\times 10^{-5}\) s for cases with \(\text{RPM}=6000\) and \(\text{RPM}=12,000\), respectively. Since the shortest period should be discretized with 10 to 20 steps through time [44], the aforementioned time-step values are small enough to temporally resolve noises up to 16 th harmonic.  
Figure 4 shows grid distribution on the surface of CRC-3 and on a slice in the computational domain and Table 3 lists more details of the grid distribution. On propeller and wing surfaces, the grid was designed with an average \(y^{+}<2.6\) to resolve the turbulent boundary layer, while on other solid surfaces (fuselage and rest of appendages) larger grid spacing along with wall-functions were used to minimize the computational cost for modeling the turbulent boundary layer. The average values of \(y^{+}\) for propeller and wing surfaces for all CRC-3 cases are provided in Table 4. Each side of the propeller's blade consists of 80 grid elements on the chord and 220 on the span, resulting in a total of 64 K cells on the propeller surface. The wing's surface poses 300 K grid elements with 600 on its chord and 1100 on its span. Finally, there are 44 K cells on the fuselage, with 94 in the vertical direction and 86 in the horizontal direction.  
Figure 4: Grid representation of CRC-3.  
Figure 3: Computational domain for CRC-3 simulations.  
The boundary conditions consist of uniform velocity with zero pressure gradient on the far-field boundary and no-slip walls on the solid surfaces. Details of boundary conditions are provided in Table 5.