## 2 CFD Model of Incompressible Flow in the Larynx
### CFD Geometry and Mesh  
The computational domain for the CFD simulation represents a simplified model of the human larynx with a rectangular cross-section, consisting of a short subglottal channel, glottal constriction formed by the vocal folds, ventricles, further contraction by the false vocal folds and straight supraglottal channel (see Figure 1). The geometry of the vocal folds is based on the M5 parametric shape by [PERSON] et al. [38]. The false vocal folds were specified according to data by [PERSON] et al. [39]. The dimensions and more details on the geometry can be found in [18]. In wall-bounded flows, the presence of solid walls fundamentally influences the flow dynamics, turbulence generation and transport in the near-wall regions due to large \begin{table}
\begin{tabular}{c c c} \hline Boundary & \(\overline{\mathbf{U}}\) [m s\({}^{-1}\)] & \(\overline{p}\) [Pa] \\ \hline Inlet \(\Gamma_{in}\) & from flux, \(\overline{U}_{i}n_{i}<0\) & 350 \\  & \(0\), \(\overline{U}_{i}n_{i}>0\) & \\ \hline Outlet \(\Gamma_{out}\) & \(\Gamma\) (\(\overline{\mathbf{U}}\)) \(n=0\), \(\overline{U}_{i}n_{i}>0\) & 0 \\  & \(\overline{\mathbf{U}}=0\), \(\overline{U}_{i}n_{i}<0\) & \\ \hline Vocal folds \(\Gamma_{bVF}\), \(\Gamma_{uVF}\) & \(\overline{U}_{2}=\frac{\partial}{\partial t}h(x,t)\) & \(\Gamma\) (\(\overline{p}\)) \(\mathbf{n}=0\) \\  & \(\overline{U}_{1}=\overline{U}_{3}=0\) & \\ \hline Fixed walls \(\Gamma_{wall}\) & \(\overline{\mathbf{U}}=0\) & \(\Gamma\) (\(\overline{p}\)) \(\mathbf{n}=0\) \\ \hline \end{tabular}
\end{table} 
Table 1: Boundary conditions for the filtered flow velocity \(\overline{\mathbf{U}}\) and static pressure \(\overline{p}\). The symbol \(n_{i}\) is the unit outer normal and \(h(x,t)\) is the prescribed displacement of the vocal folds. Figure 1: CFD computational domain in mid–coronal section and details of the CFD mesh near the vocal and ventricular folds. The z-normal boundaries are denoted \(\Gamma_{front}\) and \(\Gamma_{back}\). viscous stresses. The accuracy of the numerical simulation is thus closely related to the grid resolution near the fixed walls. According to the classification by [PERSON] [40], LES of wall-bounded flows can be classified as Large-Eddy Simulation with Near-Wall Resolution (LES-NWR) with a grid sufficiently fine to resolve 80% of the turbulent energy in the boundary layer, and Large-Eddy Simulation with Near-Wall Modeling (LES-NWM), which employs a modeling approach similar to RANS in the near-wall region. For these simulations, an important parameter is the wall unit  
\[y^{+}\,=\,\frac{u_{\tau}y}{\
u}\,, \tag{16}\]  
where \(u_{\tau}=\sqrt{\frac{L_{W}}{\rho}}\) is the friction velocity, \(\tau_{W}=\mu_{eff}\left(\frac{\partial U}{\partial y}\right)\Big{|}_{y=0}\) is the wall shear stress, \(\mu_{eff}=(\mu+\mu_{t})\) is the effective dynamic viscosity and \(y\) the dimensional distance in normal direction from the wall. The wall unit \(y^{+}\), commonly referred to as \"\(y\) plus\", is used as the dimensionless wall-normal distance. Using the same normalization, \(x^{+}\) and and \(z^{+}\) denote the dimensionless streamwise and spanwise distances. Wall units are also commonly used to indicate LES adequacy. According to [41; 42], in LES-NWR the theoretical limits for the grid spacing adjacent to the wall are 50 \(\Delta x^{+}\) 150, \(\Delta y^{+}\,<1\) and 15 \(\Delta z^{+}\) 40, with at least 3-5 gridpoints between \(0<y^{+}<10\). The computational mesh in the current CFD simulation is block-structured in order to capture well the boundary layer and consists of 2.1 M hexahedral elements. The mesh deforms in time due to vocal fold oscillation. The grid resolution in wall units was evaluated in three distinct time instants, corresponding to maximum opening of the vocal folds (\(t_{O}\)), maximum closure during the divergent phase (\(t_{N}\)) and maximum closure during the convergent phase (\(t_{C}\)), see Table 2 and Figure 2. On the boundary \(\Gamma_{bVF}\) at time \(t_{C}\) these values were evaluated: \(Y^{+}_{avg}=1.77\), \(z^{+}\,=14\) and \(x^{+}\,=8\).