# Examining the Quasi-Steady Airflow Assumption in Irregular Vocal Fold Vibration
## 3 Results
### Momentum Budget Analysis  
A comprehensive momentum budget analysis was conducted for each case to evaluate the relative importance of unsteady acceleration in glottal flow. This approach facilitates the comparison of each term, namely the unsteady term, convective term, pressure term, and shear term in the Navier-Stokes momentum equation, along with their variations throughout the cycle. The analysis aims to provide information on the dynamic interplay of these terms, shed light on their respective contributions to glottal flow dynamics, and reveal important instances where the unsteady term plays an important role. The Navier-Stokes momentum equation was nondimensionalized as follows:  
\[\frac{\partial\overset{\rightarrow}{V}}{\partial t}+(\overset{\rightarrow}{V }.\overset{\rightarrow}{\
abla})\overset{\rightarrow}{V}=-\overset{ \rightarrow}{\
abla}P+\frac{1}{Re}\overset{\rightarrow}{V}, \tag{6}\]  
where \(\overset{\overset{\rightarrow}{\partial V}}{\partial t}\) is the unsteady acceleration term, \((\overset{\rightarrow}{V}.\overset{\rightarrow}{\
abla})\overset{ \rightarrow}{V}\) is the convection term, and \(\overset{\rightarrow}{\
abla}P\) and \(\frac{1}{Re}\overset{\rightarrow}{V}\overset{\rightarrow}{V}\) are the pressure and shear stress terms, respectively. To quantify and compare the magnitude of different terms, the glottis was taken as the control volume (CV), and a volume integral was performed for the convection, pressure, and shear stress term within the control volume. By utilizing the divergence theorem, the volume integrals were  
Figure 6: RMSE of glottal wall pressure between the quasi-steady/pseudo-static and dynamic case over one vibration cycle. converted to the surface integrals over the boundary of the glottis, which were evaluated in the flow direction (y-direction) as follows:  
\[\iiint_{CV}\left(\overset{\rightarrow}{V}.\overset{\rightarrow}{\
abla}\right) \overset{\rightarrow}{V}dV=\iint_{\partial V}\overset{\rightarrow}{V}\left( \overset{\rightarrow}{V}.\overset{\rightarrow}{n}dS\right)=-\sum V_{1i}.Q_{1i }+\sum V_{2i}.Q_{2i}, \tag{7}\]  
\[\iiint_{CV}\overset{\rightarrow}{\
abla}PdV=\iint_{\partial V}P\overset{ \rightarrow}{n}dS=-\sum P_{1i}.A_{1i}+\sum P_{2i}.A_{2i}+\sum P_{wi}.A_{wi}, \tag{8}\]  
\[\iiint_{CV}\frac{1}{Re}\overset{\rightarrow}{\
abla}\overset{ \rightarrow}{V}dV=\frac{1}{Re}\iint_{\partial V}(\
abla V_{x},\ \ \ \
abla V_{y},\ \
abla V_{z})\overset{ \rightarrow}{n}dS=\frac{1}{Re}\left[\sum\left(\frac{\partial V}{\partial x}ydz \right)_{i}+\sum\left(\frac{\partial V}{\partial z}dxdy\right)_{i}\right]\!, \tag{9}\]  
where the subscripts 1, 2, and \(w\) denote the glottal inlet, outlet, and wall, respectively, and the subscript \(i\) (1, 2, 3, \(\ldots\)) denotes the grid number through the corresponding area. Because the simulation results were not output at every time step, the volume integral of the unsteady acceleration term was obtained through balancing the momentum equation:  
\[\iiint_{CV}\frac{\partial V}{\partial t}dV=-\iiint_{CV}\left(\overset{ \rightarrow}{V}.\overset{\rightarrow}{\
abla}\right)\overset{ \rightarrow}{V}dV-\iiint_{CV}\overset{\rightarrow}{\
abla}PdV+\iiint_{CV} \frac{1}{Re}\overset{\rightarrow}{V}dV. \tag{10}\]  
Figure 7 depicts the variation in the convection, pressure, shear, and unsteady terms over a single vibration cycle in each dynamic case. At both 100 Hz and 500 Hz, the unsteady term exhibited a magnitude comparable to that of the convection and pressure terms. Throughout both the flow acceleration (unsteady term is positive) and deceleration (unsteady term is negative) stages, the magnitude of the unsteady term initially rose, reached a peak roughly halfway through the stage, and then declined. The term was nearly zero only briefly during the transition between flow acceleration and deceleration, as well as during early opening and near-closed phases. In most of the glottal cycle, the unsteady term was non-negligible at both frequencies.