# Examining the Quasi-Steady Airflow Assumption in Irregular Vocal Fold Vibration
## 3 Results  
### Glottal Flow Rate Comparison  
In Figure 4, we compare the glottal flow rates over one vibration cycle across different simulation setups for each of the two sets of glottal shapes, including dynamic, pseudo-static, and quasi-steady simulations. The flow rate waveforms for the quasi-steady and pseudo-static simulations were obtained using the \"spline\" function in MATLAB to interpolate over the 16 discrete flow rate points. To assess the accuracy of the pseudo-static and quasi-steady assumptions, we calculated the percent errors of the flow rate by comparing and normalizing the values obtained from these simulations to those obtained from the dynamic simulations under corresponding configurations. At a frequency of 100 Hz, the flow rate waveforms of the dynamic, quasi-steady, and pseudo-static simulations for the (1,0)-(2,1) glottal shape set showed significant similarities when the vocal tract was not considered. The cycle-averaged absolute percent errors for the quasi-steady and pseudo-static simulations were 6.5% and 7.2%, respectively. The errors were primarily due to a slight phase difference during flow deceleration. However, including the vocal tract led to larger  
Figure 3: Computational domain and vocal fold model: (**a**) setup with supraglottic vocal tract. (**b**) Setup without supraglottic vocal tract. The representative glottal shape corresponds to the (1,0)–(2,1) set at t/T = 0.0000.  
phase differences, causing the quasi-steady and pseudo-static simulations to deviate from the dynamic cases, with cycle-averaged absolute percent errors increasing to 14.7% and 15.0%, respectively.  
At the higher frequency of 500 Hz, significant departures of flow rate from those of the dynamic case were observed, with cycle-averaged absolute percent errors of 24.1% and 22.2% for the quasi-steady and pseudo-static simulations, respectively, when the vocal tract was not considered. When the vocal tract was included, the errors nearly doubled, with cycle-averaged absolute percent errors of 46.1% and 44.5% for the quasi-steady and pseudo-static simulations, respectively. These data indicate that flow rate errors nearly tripled as the vibration frequency increased to 500 Hz, including the fact that the vocal tract in the simulation nearly doubled the errors. Similar observations were made for the flow rate waveforms of the (1,1)-(2,0) set, with cycle-averaged absolute percent errors ranging from 8.0% to 41.2% for the quasi-steady and pseudo-static simulations compared to the dynamic case. The errors were nearly identical for the two different sets of glottal shapes, indicating that the accuracy of flow rate calculation by quasi-steady or pseudo-static approximation was insensitive to glottal shapes.  
Figure 4 also shows that the flow rate error varies during a vibration cycle for all the cases, with the error increasing and then decreasing during glottis opening and closing. The highest errors were observed in the middle of opening and closing, corresponding to peak flow acceleration and deceleration. This variation in errors was more prominent at the 500 Hz frequency. The nonuniform error levels suggest that the importance of unsteady effects varies within one vibration cycle. Furthermore, at 100 Hz frequency, the flow rate waveforms of quasi-steady and pseudo-static cases almost overlap, indicating that glottal wall motion has a negligible impact on flow rates at low vibration frequency. However, at 500 Hz frequency, discrepancies between the two flow rate waveforms are apparent, particularly during flow deceleration. Generally, the quasi-steady cases display higher cycle-averaged errors than the pseudo-static cases, indicating that incorporating glottal wall motion in the model can improve the accuracy of flow rate prediction at high vibration frequency.  
Figure 4: Flow rate waveforms of the three simulations and percent errors of flow rate between quasi-steady/pseudo-static and dynamic simulation over one vibration cycle.