## 2 CFD Model of Incompressible Flow in the Larynx
### CFD Results  
The current study reports on the results of three CFD simulations using different turbulence modeling approaches, which are summarized in Table 3. The laminar case \"LAM\" used no turbulence model. \"OE\" and \"WALE\" are LES simulations with the One-Equation and WALE SGS models, respectively. All the simulations were run for 20 periods of vocal fold oscillations, that is, \(t=0.02\) s. The CFD simulations provide the filtered velocity and pressure fields \(\overline{U}\), \(\overline{p}\). For simplicity, the overbars are dropped in the following presentation. Figure 2 shows the flow rate and glottal opening during the last four simulated cycles of vocal fold oscillation. The predicted peak flow rate in the laminar case is, for the same boundary conditions, higher than in the One-Equation and WALE SGS models by 16.76% and 5.26%, respectively. This is caused by the different values of the SGS viscosity, which add to the molecular viscosity and limit the flow rate through the glottal constriction. The laminar model does not capture the influence of small-scale turbulence, which corresponds to \(\
u_{t}=0\). The WALE SGS model and the One-Equation SGS model compute with non-zero SGS viscosity, with the latter one significantly higher due to the already mentioned deficiency of the One-Equation model, which overestimates the turbulent viscosity near the vocal fold surfaces (see also Figure 6). The flow rate does not reach zero value, since the vocal folds do not fully close, corresponding physiologically to breathy phonation. The velocity and pressure distributions along the laryngeal mid-line are plotted in Figure 3 in three distinctive times (see Table 2). The high-velocity glottal flow corresponds to a low static pressure due to the Bernoulli effect. The three simulations give almost identical results in the subglottal space, where the flow is laminar, but differ in the supraglottal volume where turbulent fluctuations are present. Figure 4, plotting the velocity fields in a sagittal (x-z) plane, shows the structure of the expa \begin{table}
\begin{tabular}{c c c c} \hline Case & Turb. Modelling & SGS Model & Waltime \\ \hline LAM & laminar & - & 27 days \\ OE & LES & One-Equation & 34 days \\ WALE & LES & WALE & 37 days \\ \hline \end{tabular}
\end{table} 
Table 3: Overview of the CFD simulations. Figure 2: Flow rates during four oscillation cycles and glottal surface. The visualizations of the vorticity fields \(\mathbf{\omega}=r^{\prime}\quad\mathbf{U}\), which are commonly used for characterizing turbulence in cases with no entrainment rotation, are shown in mid-coronal (x-y) plane in Figure 5. They reveal the shear layers, where vortices are shed as a consequence of Kelvin-Helmholtz instability. The vortices may undergo successive  
Figure 4: Velocity fields [m s \({}^{1}\)] in mid–sagittal plane in three time instants. Figure 3: Velocity magnitude (**left**) and pressure distribution (**right**) along the glottal mid–line in three time instants \(t_{N}\) (**top**), \(t_{C}\) (**middle**) and \(t_{O}\) (**bottom**). Gray background denotes the region of the moving vocal folds. instabilities, leading to a direct kinetic-energy cascade towards the small scales. The supraglottal jet deflects stochastically towards either of the ventricular folds. This behavior is not a consequence of the SGS model, it is caused by the bistability of the flow in this symmetric geometry (see e.g., [46; 47]). The effect of the unresolved turbulent sub-grid scales on the resolved velocity and pressure fields is carried by the turbulent SGS viscosity \(\
u_{t}\) (see Equations (10) and (15)). Figure 6 clearly shows that the turbulent viscosity predicted by the One-Equation model is very high in regions of pure shear, especially within glottis. At certain phases of the vocal fold motion, \(\
u_{t}\) reaches values three times higher than the molecular viscosity \(\
u=1.58~{}~{}10~{}~{}^{5}\) m\({}^{2}\) s \(~{}~{}^{1}\).