# Examining the Quasi-Steady Airflow Assumption in Irregular Vocal Fold Vibration
By comparing the glottal pressure predicted by the model with the unsteady measurements on straight and rounded rigid replicas of the vocal folds, they reported that the QSFA does not stand when the glottis is almost closed. To address this limitation, [PERSON] and [PERSON] [13] conducted a theoretical study to assess the order of magnitude of the unsteady and convective acceleration terms in the momentum equation, which relates the transglottal pressure to the volume flow. The effects of glottal wall motion, friction, and the movement of the separation point were considered in the equation. They showed that glottal flow dynamics could be divided into an unsteady-effect-dominated flow-initiation/shutoff interval and a convective-acceleration-dominated quasi-steady interval, based on the relative importance of each term. Subsequently, [PERSON] et al. [14] combined the theoretical work with experimentally measured flow velocity data and calculated the waveshapes of unsteady and convective accelerations. Flow velocity measurements were conducted in a scaled-up vocal fold model using digital particle image velocimetry (DPIV), and unsteady and convective acceleration terms were calculated along the centerline of the flow between the vocal folds. Their results indicated that although glottal jet inertia was nearly zero during the middle 40% of the open phase, unsteady acceleration still dominated the flow for the remainder of the cycle. A summary of previous studies on the QSFA is displayed in Table 1. Due to the challenges of experimentally implementing the complex geometry of the vocal folds, almost all studies used simplified geometries. However, under pathological vocal fold vibrations, the glottal shape can be asymmetric and involve longitudinal wave propagations [15; 16]. Since different glottal shapes can generate significantly different glottal flow dynamics [5; 17], there remains a question as to whether the previous conclusions still stand when the complex geometry of the vocal folds is considered. In addition, the frequency investigated by the preceding studies concentrates in the range of 10-160 Hz. The nondimensional inequality condition of \(S_{t}\ll 1\) suggests that the validity of QSFA is frequency dependent. The \(S_{t}\) for glottal flow in normal speech phonation is approximately \(10^{-2}\), estimated based on a frequency range of 100-200 Hz. However, higher vibration frequencies, such as those encountered in singing, can lead to larger \(S_{t}\), which, coupled with an increase in the oscillation speed of vocal folds, may undermine the justification of QSFA. Therefore, the dependence of the validity of QSFA on frequency also needs to be explored. Another factor that affects the validity of QSFA is the air inertia effect in the vocal tract. [PERSON] and [PERSON] [18] showed that the inertance of air in the trachea and vocal tract generates a dynamic pressure at the glottal inlet and outlet, which interacts with the unsteady glottal flow and can cause the flow rate and transglottal pressure to vary with frequency. Thus, considering the air inertia effects in the validation of QSFA may yield different results. The purpose of this study is to utilize numerical methods to explore the limitations of QSFA in human phonation, taking into account factors such as phonation frequency, air inertance in the vocal tract, and irregular glottal shapes. To isolate the effects of flow unsteadiness and vocal fold vibration, three simulation types were conducted. The first simulation type, dynamic simulation, involved solving the unsteady Navier-Stokes equations with prescribed vocal fold motion, including glottal wall shapes and velocity at each time step. This simulation type considered both unsteady flow and vocal fold motion effects. The second simulation type, pseudo-static simulation, computed the steady solution of the Navier-Stokes equations for each glottal wall shape with the corresponding glottal wall velocity. This simulation type considered the effect of glottal wall motion but not the effects of unsteady flow.