# Impact of the Sub-Grid Scale Turbulence Model in Aeroacoustic Simulation of Human Voice

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## Abstract

**:**

## 1. Introduction

## 2. CFD Model of Incompressible Flow in the Larynx

#### 2.1. Mathematical Model

#### 2.1.1. Smagorinsky SGS Model

#### 2.1.2. One-Equation SGS Model

#### 2.1.3. Wall-Adapting Local Eddy-Viscosity SGS Model

#### 2.2. Boundary Conditions

#### 2.3. CFD Geometry and Mesh

#### 2.4. Discretization and Numerical Solution

#### 2.5. CFD Results

## 3. Computational Aeroacoustic (CAA) Model of Human Phonation

- i.
- A monopole source term due to the motion of vocal folds (the term is zero, when the walls are fixed and also the monopole source term at inlet is often omitted due to a non-reflecting boundary condition).
- ii.
- A dipole source term due to the unsteady force exerted by the surface of the vocal folds onto the fluid.
- iii.
- A quadrupole sound term due to the unsteady flow inside the vocal tract. See [2] for more details.

#### 3.1. Mathematical Model

#### 3.1.1. Acoustic Perturbation Equations (APEs)

- The velocity field is purely solenoidal, that is, $\nabla \xb7{\mathit{U}}^{\mathrm{i}c}=0$,
- The density ${\rho}_{0}$ is constant, that is, $\nabla {\rho}_{0}=0$ and $\partial {\rho}_{0}/\partial t=0$,
- The acoustic field is irrotational, that is, $\nabla \times {\mathit{U}}^{\mathrm{a}}=0$.

#### 3.1.2. Perturbed Convective Wave Equation (PCWE)

#### 3.2. Geometry, Mesh and Numerical Solution

#### 3.3. CAA Results

#### 3.3.1. Acoustic Sources

#### 3.3.2. Wave Propagation

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**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 ${\mathsf{\Gamma}}_{front}$ and ${\mathsf{\Gamma}}_{back}$.

**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.

**Figure 4.**Velocity fields [$\mathrm{m}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$] in mid–sagittal plane in three time instants.

**Figure 5.**Vorticity fields $\left|\mathbf{\omega}\right|$ in mid–coronal plane in range (0, 30,000) [${\mathrm{s}}^{-1}$].

**Figure 6.**Turbulent viscosity ${\nu}_{t}$ [${\mathrm{m}}^{2}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-1}$] in mid–coronal plane.

**Figure 7.**Geometry, mesh and probe location for computational aeroacoustic (CAA) simulations—vocal tracts [u:] (

**top**) and [i:] (

**bottom**). Red—larynx, purple—vocal tract, green—PML.

**Figure 8.**Spatial distribution of sound sources [$\mathrm{kg}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-6}\phantom{\rule{0.166667em}{0ex}}{\mathrm{s}}^{-2}$] in mid–coronal plane at four frequencies (as a result of Fast Fourier Transform).

**Figure 9.**Acoustic sound spectra from the numerical simulation of vocalization of [u:] (

**left**) and [i:] (

**right**) at monitoring point “MIC 1”.

**Table 1.**Boundary conditions for the filtered flow velocity $\overline{\mathit{U}}$ and static pressure $\overline{p}$. The symbol ${n}_{i}$ is the unit outer normal and $h(\mathit{x},t)$ is the prescribed displacement of the vocal folds.

Boundary | $\overline{\mathit{U}}\phantom{\rule{4pt}{0ex}}\left[\mathbf{m}\phantom{\rule{4pt}{0ex}}{\mathbf{s}}^{-1}\right]$ | $\overline{\mathit{p}}\phantom{\rule{4pt}{0ex}}\left[\mathbf{Pa}\right]$ |
---|---|---|

Inlet ${\mathsf{\Gamma}}_{in}$ | from flux, ${\overline{U}}_{i}{n}_{i}<0$ | 350 |

0, ${\overline{U}}_{i}{n}_{i}>0$ | ||

Outlet ${\mathsf{\Gamma}}_{out}$ | $\nabla \left(\overline{\mathit{U}}\right)\xb7\mathit{n}=0$, ${\overline{U}}_{i}{n}_{i}>0$ | 0 |

$\overline{\mathit{U}}=0$, ${\overline{U}}_{i}{n}_{i}<0$ | ||

Vocal folds ${\mathsf{\Gamma}}_{bVF}$, ${\mathsf{\Gamma}}_{uVF}$ | ${\overline{U}}_{2}=\frac{\partial}{\partial t}h(\mathit{x},t)$ | $\nabla \left(\overline{p}\right)\xb7\mathit{n}=0$ |

${\overline{U}}_{1}={\overline{U}}_{3}=0$ | ||

Fixed walls ${\mathsf{\Gamma}}_{wall}$ | $\overline{\mathit{U}}=0$ | $\nabla \left(\overline{p}\right)\xb7\mathit{n}=0$ |

Symbol | Meaning | Time [s] |
---|---|---|

${t}_{N}$ | closed divergent | 0.1900 |

${t}_{C}$ | closed convergent | 0.1927 |

${t}_{O}$ | open glottis | 0.1963 |

Case | Turb. Modelling | SGS Model | Walltime |
---|---|---|---|

LAM | laminar | - | 27 days |

OE | LES | One-Equation | 34 days |

WALE | LES | WALE | 37 days |

**Table 4.**Sound pressure levels [dB] at probe MIC 1 for aeroacoustic simulations (vowel u: and i:), based on CFD simulations with the laminar, One-Equation and Wall-Adapting Local Eddy-viscosity (WALE) sub-grid scale (SGS) models. Values at ${f}_{o}=100\phantom{\rule{3.33333pt}{0ex}}\mathrm{Hz}$, ${f}_{1}=200\phantom{\rule{3.33333pt}{0ex}}\mathrm{Hz}$, ${f}_{2}=300\phantom{\rule{3.33333pt}{0ex}}\mathrm{Hz}$, non-harmonic frequency $1235\phantom{\rule{3.33333pt}{0ex}}\mathrm{Hz}$ and formant frequencies.

Case | ${\mathit{L}}_{{\mathit{f}}_{\mathit{o}}}$ | ${\mathit{L}}_{{\mathit{f}}_{1}}$ | ${\mathit{L}}_{{\mathit{f}}_{2}}$ | ${\mathit{L}}_{1235}$ | ${\mathit{L}}_{\mathit{F}1}^{\mathit{u}:}$ | ${\mathit{L}}_{\mathit{F}2}^{\mathit{u}:}$ | ${\mathit{L}}_{\mathit{F}3}^{\mathit{u}:}$ |

LAM-u | 44.05 | 57.48 | 55.20 | 33.12 | 34.25 | 57.31 | 45.49 |

OE-u | 38.34 | 55.13 | 47.79 | 15.33 | 28.94 | 40.88 | 35.76 |

WALE-u | 44.06 | 56.86 | 53.52 | 20.03 | 33.42 | 48.29 | 42.15 |

Case | ${\mathit{L}}_{{\mathit{f}}_{\mathit{o}}}$ | ${\mathit{L}}_{{\mathit{f}}_{1}}$ | ${\mathit{L}}_{{\mathit{f}}_{2}}$ | ${\mathit{L}}_{1235}$ | ${\mathit{L}}_{\mathit{F}1}^{\mathit{i}:}$ | ${\mathit{L}}_{\mathit{F}2}^{\mathit{i}:}$ | ${\mathit{L}}_{\mathit{F}3}^{\mathit{i}:}$ |

LAM-i | 52.68 | 53.31 | 51.52 | 28.99 | 32.46 | 34.62 | 56.02 |

OE-i | 42.07 | 57.76 | 46.08 | 15.24 | 28.49 | 29.70 | 43.95 |

WALE-i | 47.69 | 59.45 | 51.88 | 19.86 | 34.13 | 35.96 | 58.77 |

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**MDPI and ACS Style**

Lasota, M.; Šidlof, P.; Kaltenbacher, M.; Schoder, S.
Impact of the Sub-Grid Scale Turbulence Model in Aeroacoustic Simulation of Human Voice. *Appl. Sci.* **2021**, *11*, 1970.
https://doi.org/10.3390/app11041970

**AMA Style**

Lasota M, Šidlof P, Kaltenbacher M, Schoder S.
Impact of the Sub-Grid Scale Turbulence Model in Aeroacoustic Simulation of Human Voice. *Applied Sciences*. 2021; 11(4):1970.
https://doi.org/10.3390/app11041970

**Chicago/Turabian Style**

Lasota, Martin, Petr Šidlof, Manfred Kaltenbacher, and Stefan Schoder.
2021. "Impact of the Sub-Grid Scale Turbulence Model in Aeroacoustic Simulation of Human Voice" *Applied Sciences* 11, no. 4: 1970.
https://doi.org/10.3390/app11041970