## 2 CFD Model of Incompressible Flow in the Larynx  
### Mathematical Model  
Large Eddy Simulation is a mathematical concept for modeling turbulent flows, which deals with flow structures carrying most kinetic energy, that is, spatially organized large scales. These consist of two main categories: coherent structures and coherent vortices of recognizable shape [31]. In the numerical implementation, the characteristic length \(\Delta\), defining a cutoff between resolved scales and modeled sub-grid scales, is usually given by the mesh grid spacing [32].  
In the LES concept, any flow variable \(f(\mathbf{x},t)\), where \(\mathbf{x}=(x_{1},x_{2},x_{3})\) is the spatial coordinate and \(t\) time, may be decomposed as  
\[f(\mathbf{x},t)=\overline{f}(\mathbf{x},t)+f^{\ell}(\mathbf{x},t), \tag{1}\]  
where \(\overline{f}(\mathbf{x},t)=G(\mathbf{x})\quad f(\mathbf{x},t)=\int G(\mathbf{r},\mathbf{x},\Delta )\,f(\mathbf{x}\quad\mathbf{r},t)\,d\mathbf{r}\) is the large-scale component, obtained by spatial filtering, and \(f^{\ell}(\mathbf{x},t)\) is the small, sub-grid scale contribution. The convolution introduced above contains a filter function \(G\) separating spatial scales \(j\mathbf{r}/\)\(\Delta/2\) from \(j\mathbf{r}/j>\Delta/2\). In current simulations, a top-hat filter is used, which is a common choice in low-order finite volume methods. The effect of sub-grid scale (SGS) contributions on the large flow scales relies on the assumption of isotropic (non-directional) small-scale turbulence and is modeled.  
The continuity and momentum equations for the incompressible fluid flow, with LES filtering applied, can be written as  
\[\frac{\partial\overline{U}_{i}}{\partial x_{i}}=0, \tag{2}\]  
\[\frac{\partial\overline{U}_{i}}{\partial t}+\frac{\partial}{\partial x_{j}}( \overline{U_{i}U_{j}})=\quad\frac{1}{\rho}\frac{\partial\overline{p}}{ \partial x_{i}}+\
u\frac{\partial^{2}\overline{U}_{i}}{\partial x_{j}\partial x _{j}}, \tag{3}\]where \(\overline{U}_{i}\) is the filtered velocity, \(\overline{p}\) represents the filtered static pressure and \(\
u=1.58\)\(10\)\(5\)\(\mathrm{m}^{2}\)s \(1\) is the kinematic molecular viscosity. The term \(\overline{U_{i}U_{j}}\) is the dyadic product and cannot be expressed directly [33]. Modification of the momentum Equation (3) by \(+\frac{\partial}{\partial x_{j}}(\overline{U_{i}U_{j}})\) yields  
\[\frac{\partial\overline{U}_{i}}{\partial t}+\frac{\partial}{\partial x_{j}}( \overline{U}_{i}\overline{U}_{j})=\quad\frac{1}{\rho}\frac{\partial\overline {p}}{\partial x_{j}}+\
u\frac{\partial^{2}\overline{U}_{i}}{\partial x_{j} \partial x_{j}}\quad\frac{\partial\tau_{ij}}{\partial x_{j}}. \tag{4}\]  
The new term on the right-hand side of (4) is the divergence of the sub-grid scale (SGS) stress tensor  
\[\tau_{ij}=\overline{U_{i}U_{j}}\quad\overline{U}_{j}\overline{U}_{j}=\quad( \overline{u^{\prime}_{i}u^{\prime}_{j}}+\overline{U_{i}u^{\prime}_{j}}+ \overline{U_{i}\overline{U}_{j}}+\overline{U_{i}\overline{U}_{j}}\quad \overline{U}_{i}\overline{U}_{j}), \tag{5}\]  
where the individual tensors are \(\overline{u^{\prime}_{i}u^{\prime}_{j}}\) the Reynolds-stress-like term, \((\overline{\overline{U_{i}u^{\prime}_{j}}}+\overline{u^{\prime}_{i}\overline {U}_{j}})\) the [PERSON] term [34] and \((\overline{\overline{U_{i}}\overline{U_{j}}}+\overline{U_{i}\overline{U}_{j}})\) the Leonard term [35]. The SGS stress tensor is left to be modeled to close the set of equations. Compared to the Reynolds stresses in RANS, the SGS stresses carry much less of the turbulent energy, and so the accuracy of the model is less important.