# A Computational Study on the Aeroacoustics of a Multi-Rotor Unmanned Aerial System
## 2 Materials and Methods
### Grid Verification Study  
Grid verification study is conducted for the fully appended CRC-3 advancing at \(U_{0}=16\) m/s with a pitch angle of \(\theta=0^{\circ}\) and propellers rotating at RPM \(=12,000\). For this purpose, coarse and fine grid systems are generated, where the average cell size of the aforementioned grid, which can be considered as a medium grid, is increased/reduced by a factor of \(r_{G}=\sqrt{2}\) in any direction to generate the coarse/fine grid system. The coarse and fine grids consist of about 3 million and 22 million elements, respectively. Grid uncertainty is estimated using the factor of safety method [45] by identifying the convergence type of the simulation using the coarse (\(S_{3}\)), medium (\(S_{2}\)), and fine grid solutions (\(S_{1}\)):  
\[R_{G}=\frac{e_{21}}{e_{32}}=\frac{S_{2}-S_{1}}{S_{3}-S_{2}}=\left\{ \begin{array} []{cl}0<R_{G}<1&:\text{Monotonic Convergence}\\ -1<R_{G}<0&:\text{Oscillatory Convergence}\\ R_{G}>1&:\text{Monotonic Divergence}\\ R_{G}<-1&:\text{Oscillatory Divergence}\end{array} \right. \tag{2}\]  
When monotonic convergence is achieved, the grid uncertainty is computed as:  
\[U_{G}=\left\{ \begin{array}{cl}\left[9.6(1-CF)^{2}+1.1\right]\left|\delta \right|&\left|1-CF\right|<0.125\\ \left[2|1-CF|+1\right]\left|\delta\right|&\left|1-CF\right|\geq 0.125\end{array} \right. \tag{3}\]  
where  
\[CF=\frac{r^{P_{G}}-1}{r^{P_{H}}-1} \tag{4}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Case Label** & **Propeller** & **Wing** \\ \hline C1 & 2.22 & 2.62 \\ C2 & 1.67 & 1.60 \\ C3 & 2.08 & 2.37 \\ C4 & 2.09 & 2.23 \\ C5 & 2.46 & 1.72 \\ C6 & 2.00 & 1.57 \\ C7 & 1.75 & 1.30 \\ C8 & 0.57 & 1.01 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Average \(y^{+}\) values on propeller and wing for all cases. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Variable** & **Far-Field** & **Propeller** & **Wing** & **Other Surfaces** \\ \hline \(u\) & Uniform & No-slip & No-slip & No-slip \\ \(p\) & \(\partial p\partial n=0\) & \(\partial p\partial n=0\) & \(\partial p\partial n=0\) & \(\partial p\partial n=0\) \\ \(k\) & 0 & 0 & 0 & Wall function \\ \(Omega\) & 0.02 & \((6.0v)\left(\beta y^{2}\right)\) & \((6.0v)\left(\beta y^{2}\right)\) & Wall function \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Boundary conditions. where \(p_{th}\) is the theoretical order of accuracy (\(p_{th}=2\)) and \(p_{G}\) is the numerical order of accuracy, defined as:  
\[p_{G}=-\ln(R_{G})/\ln(r_{G}) \tag{5}\]  
In addition, the error estimation, \(\delta\) is computed as:  
\[\delta=\frac{\epsilon_{21}}{r_{G}^{2}-1} \tag{6}\]  
When oscillatory convergence is achieved, the range of variation in the solutions is computed as the level of uncertainty. The obtained uncertainty values are normalized by the solution for the fine grid \(S_{1}\). Figure 5 shows the predicted time histories of thrust coefficient \(C_{T}\) for propeller and lift coefficient \(C_{l}\) on the wing as well as directivity of the overall sound pressure level (OASPL) for all of the grids. OASPL is calculated for the first 3 harmonics using Equation (7):  
\[\text{OASPL}=10 log_{10}\left\{\sum_{1}^{3}10^{\frac{SP_{t}}{10}}\right\} \tag{7}\]  
It should be noted that grid study is conducted for a half-domain case with a symmetric boundary condition passing through the center of the geometry and normal to the \(z\) direction, to decrease computational cost, as the full CRC-3 case is symmetric with respect to the described plane. Mean parameters are computed and listed in Table 6, which indicate monotonic convergence for \(C_{T}\), \(C_{l}\) and average OASPL with \(R_{G}=0.47\), \(R_{G}=0.68\) and \(R_{G}=0.48\), respectively. The average of the numerical accuracy is about 2.24, close to the theoretical accuracy of the second order discretization scheme used for the spatial terms in governing equations.