# Effect of Subglottic Stenosis on Expiratory Sound Using Direct Noise Calculation
## 2 Methods
### Solution Independence  
To ensure grid independence, a multi-step mesh refinement approach was employed. Initially, a coarse non-uniform unstructured mesh was designed to resolve the basic flow features, and, subsequently, the mesh was refined by decreasing the grid interval size by 25% for each successive mesh. Mesh independence was checked for a geometric configuration (\(o=0.4\) cm, \(d=3.0\) cm, and \(g=0.4\) cm) that exhibited the highest sound pressure level in the preliminary simulation results. A total of five meshes were tested at flowrates of 300 mL/s and 1000 mL/s, with the total number of cells ranging from 17 K to 105 K. Figure 2a shows the overall sound pressure level (SPL, see definition in Section 3) versus the number of grid cells. Figure 2b shows a comparison of the acoustic spectra for both flowrates between  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{3}{*}{Solver options} & \multirow{3}{*}{Solver type} & \multicolumn{3}{c}{Spatial discretization} & \multirow{3}{*}{Time integration} \\ \cline{3-3} \cline{5-5}  & & Pressure & Others & \\ \cline{2-5}  & & Pressure based & 2 nd order & QUICK & 2 nd order \\ \hline \multirow{4}{*}{Air properties} & \multirow{2}{*}{Viscosity} & Thermal & \multirow{2}{*}{Specific heat} & \multirow{2}{*}{Molecular weight} \\  & & conductivity & & \\ \cline{2-5}  & 1.7894 \(\times\) 10\({}^{-5}\) & 0.0242 & 1006.43 & 28.966 \\  & kg/(m s) & W/(m K) & J/(kg K) & kg/kmol \\ \hline \multirow{3}{*}{Domain parameters} & \multirow{3}{*}{Trachea length} & \multirow{3}{*}{Vocal tract length} & \multirow{3}{*}{Tract diameter} & \multirow{3}{*}{Environmental space radius} \\  & & & & \\ \cline{1-1} \cline{3-3} \cline{5-5}  & & 10.0 cm & 15.0 cm & 2.0 cm & 2.0 m \\ \hline \hline \end{tabular}
\end{table}
Table 1: Simulation setup summary.  
the two densest meshes. Since the focus of this study was the tonal sound, quantitative comparison was made for the SPL and the primary frequency. The data are summarized in Table 2, with errors in the primary frequency being below 2% and errors in the SPL below 4 dB. The results indicate that the second densest mesh with 73 K cells can predict the location and amplitude of the peaks in the spectrum with adequate accuracy compared to the densest mesh. The minimum grid size for the 73 K mesh was 0.12 mm. The grid was not further refined due to concerns regarding computational speed. It will be shown in the validation case that such grid size can accurately reproduce the experimental results. Moreover, the smaller mesh was favored to considerably reduce the computational cost as the wall time scaled linearly with grid size. The simulation was run using 6 cores on a 3.4 GHz Intel Xeon (Skylake) CPU. For the adopted grid size (73 K cells), each time step took about 3.8 s to converge, and it took about 24 h to run the simulation to 110 ms.  
To ensure that the simulation produced statistically steady results, the spectra of the 1000 mL/s case were computed using two different time durations: 60-110 ms and 140-250 ms. As shown in Figure 2c, the two spectra were found to be nearly identical, indicating that the simulation reached statistical steadiness at up to 110 ms. Therefore, most of the cases in the parametric space discussed below were simulated for 110 ms, and an interval of 60-110 ms was used for analysis. However, in cases where the flowrate was very low, resulting in a low sound frequency, the simulation was extended to 220 ms to provide sufficient data for Fourier analysis.