## 3 Results and Discussion
### Baseline Case
#### 3.1.2 Baseline Case: Acoustics  
Figure 5a shows the acoustic pressure contour at a specific time instant, illustrating the overall distribution of acoustic pressure. A probe placed 5 cm from the mouth, as indicated in Figure 5a, was used to record the acoustic signals. The phase-averaged acoustic pressure recorded at this location is shown in Figure 5b. Focusing on t/T = 0.7, the pressure rises sharply, reaching a peak with a maximum value of approximately 2 Pa. This peak corresponds to the moment when the vocal folds are closing and the volumetric flow rate deceleration is maximum, which occurs around t/T = 0.6. The phase difference of around 0.1 is due to the time required by the sound to travel from glottis to the probe. Following this, a second peak occurs around t/T = 0.15, though lower in magnitude than the first. The amplitude of the oscillations continues diminishing after the second peak, \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Computed Value** & **Typical Range [38]** \\ \hline \(f_{0}(\mathrm{Hz})\) & 154 & 65–260 \\ \(\tau_{s}\) & 1.32 & 1.1–3.4 \\ \(\tau_{o}\) & 0.69 & 0.4–0.7 \\ \(q_{peak}\left(\frac{\mathrm{mL}}{\mathrm{s}}\right)\) & 388 & 200–580 \\ \(q_{mean}\left(\frac{\mathrm{mL}}{\mathrm{s}}\right)\) & 198 & 110–220 \\ \(\mathrm{MFDR}\left(\frac{\mathrm{L}}{\mathrm{s}^{2}}\right)\) & 481 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Normal human phonation quantities computed from the realistic geometry simulation and their typical physiological ranges [38]. \(f_{0}\) is the fundamental frequency, \(\tau_{s}\) is the skewness quotient, \(\tau_{o}\) is the open quotient, \(q_{peak}\) is the peak glottal flow rate, \(q_{mean}\) is the mean glottal flow rate and MFDR is the maximum flow declination rate. reflecting the damping effect in the supraglottal tract. Overall, the curve captures the typical oscillatory pattern of acoustic pressure during phonation, with noticeable damping as the energy of the sound waves dissipates towards in the cycle [39; 40; 41]. In our previous study utilizing simplified geometries [24], the acoustic pressure also exhibited approximately three oscillations per vibration cycle. However, in the current study, the amplitude of the acoustic pressure peaks decreases more significantly within the same cycle. This discrepancy in acoustic pressure amplitude can be attributed to the differing boundary conditions used in the two studies. In the previous work, a total reflection boundary condition was applied at the mouth, causing all incident waves to be reflected, resulting in higher amplitudes. In contrast, the present study incorporates the full mouth shape, along with a box domain outside the mouth, allowing for partial dissipation of the incident waves at the mouth opening. This partial dissipation leads to lower amplitudes in the second and third peaks of the acoustic pressure wave observed in the current simulation. To analyze the acoustic characteristics of the vocal tract, we calculated the formant frequencies using a method based on the source-filter mechanism [42], as introduced by [PERSON] and [PERSON] [26]. This method defines a transfer function (TF) that represents the formant frequencies:  
\[\text{TF}(f)=\frac{\Delta p(f)}{[\partial Q/\partial t](f)} \tag{5}\]  
where these variables are in the frequency domain. Figure 5c shows the formant frequencies obtained using the TF method, alongside the corresponding values for the same supraglottal tract reported in the experiment study by [PERSON] et al. [33] and LES study by [PERSON] et al. [43] for comparison. The Fourier transform spectrum of the acoustic pressure from our simulation is also overlaid to facilitate discussion. The first four formants from the TF method are F1 = 510 Hz, F2 = 818 Hz, F3 = 2186 Hz, and F4 = 3441 Hz. These formants correspond to distinct peaks in the Fourier transfer spectrum. The LES study by [PERSON] et al. [43] reported F1 = 480 Hz and F2 = 1170 Hz, while the experiment study by [PERSON] et al.