# Examining the Quasi-Steady Airflow Assumption in Irregular Vocal Fold Vibration

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{−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. Kucinschi and Scherer [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.

## 2. Methods

#### 2.1. Glottal Shapes and Movement

_{0R}as a function of anterior–posterior ($y$) and inferior–superior ($z$) directions:

_{0R}(y,z) = (1 − y/L) [ξ

_{0R2}+ (ξ

_{0R1}− ξ

_{0R2}− 4ξ

_{BR}z/T) (1 − z/T)],

_{0R1}and ξ

_{0R2}are, respectively, the inferior and superior glottal half-widths at the vocal process, and ξ

_{BR}is a surface bulging parameter, which controls the vertical curvature of the medial surface. The subscript R denotes the right medial surface. The same equation with a subscript $L$ exists for the pre-phonatory glottal width of the left medial surface (ξ

_{0L}).

_{R}(y,z,t) = ξ

_{mR}sin(mπy/L) [sinωt − n(ω/c)(z − z

_{mR}) cosωt],

_{mR}is the modal displacement amplitude, ω is the angular frequency, c is the mucosal wave speed, and z

_{mR}is the inflection point for the vertical half wavelength. An equivalent equation exists for the modal displacement of the left medial surface (${\xi}_{L}$). The overall three-dimensional glottal shape at any moment of time was obtained by superimposing the modal displacements on the pre-phonatory shape of the medial surfaces:

_{0R}(y,z) + ξ

_{0L}(y,z) + ξ

_{R}(y,z,t) + ξ

_{L}(y,z,t).

_{m}was 0.1 $\mathrm{cm}$, and the inflection point z

_{m}was defined as $T\times (0.6-0.02{\xi}_{B})$.

#### 2.2. Airflow Simulation

#### 2.3. Simulation Setup and Case Summary

^{−6}s and 1.016 × 10

^{−6}s were used for the case of 100 Hz and 500 Hz vibration frequency, respectively. Such time steps were chosen for (1) outputting the results at the specified 16 phases and (2) complying with the Courant–Friedrichs–Lewy condition. The same time steps were used for pseudo-static and quasi-steady simulations. For a few cases that experienced numerical convergence problems, the time step was appropriately reduced. Time independence was achieved for all three types of simulations. For each dynamic simulation, glottal airflow was simulated for two cycles, reaching steady cycles based on the observation that cycle-to-cycle flow variations were very small. For each pseudo-static or quasi-steady simulation, glottal airflow was calculated until a sustained steady state was reached. The simulations were run on the XSEDE COMET cluster, which used Intel’s Xeon Processor E5-2600 v3 family. One hundred and twenty-eight processors were used for each dynamic or pseudo-static simulation, and sixty-four processors were used for each quasi-steady simulation. The computational time of the dynamic simulations ranged from 40 h to 240 h, depending on the vibration frequency and number of grids. The average computational time for a pseudo-static or quasi-steady simulation was around 20 h.

## 3. Results

#### 3.1. Glottal Flow Rate Comparison

#### 3.2. Strouhal Number and Velocity Ratio Analysis

#### 3.3. Errors of Important Aerodynamics and Sound Spectrum Parameters

#### 3.4. Errors of Glottal Pressure

#### 3.5. Momentum Budget Analysis

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Ishizaka, K.; Flanagan, J.L. Synthesis of voiced sounds from a two-mass model of the vocal cords. Bell Syst. Tech. J.
**1972**, 51, 1233–1268. [Google Scholar] [CrossRef] - Van den Berg, J. Myoelastic-aerodynamic theory of voice production. J. Speech Hear. Res.
**1958**, 1, 227–244. [Google Scholar] [CrossRef] [PubMed] - Titze, I.R. The Myoelastic Aerodynamic Theory of Phonation; National Center for Voice and Speech: Iowa City, IA, USA, 2006. [Google Scholar]
- Scherer, R.C.; Guo, C.G. Laryngeal modeling: Translaryngeal pressure for a model with many glottal shapes. In Proceedings of the First International Conference on Spoken Language Processing, Kobe, Japan, 18–22 November 1990. [Google Scholar]
- Scherer, R.C.; Shinwari, D.; De Witt, K.J.; Zhang, C.; Kucinschi, B.R.; Afjeh, A.A. Intraglottal pressure profiles for a symmetric and oblique glottis with a divergence angle of 10 degrees. J. Acoust. Soc. Am.
**2001**, 109, 1616–1630. [Google Scholar] [CrossRef] - Story, B.H.; Titze, I.R. Voice simulation with a body-cover model of the vocal folds. J. Acoust. Soc. Am.
**1995**, 97, 1249–1260. [Google Scholar] [CrossRef] [PubMed] - Zhang, Z. Cause-effect relationship between vocal fold physiology and voice production in a three-dimensional phonation model. J. Acoust. Soc. Am.
**2016**, 139, 1493–1507. [Google Scholar] [CrossRef] [PubMed] - Zhang, Z. Effect of vocal fold stiffness on voice production in a three-dimensional body-cover phonation model. J. Acoust. Soc. Am.
**2017**, 142, 2311–2321. [Google Scholar] [CrossRef] [PubMed] - Mongeau, L.; Franchek, N.; Coker, C.H.; Kubli, R.A. Characteristics of a pulsating jet through a small modulated orifice, with application to voice production. J. Acoust. Soc. Am.
**1997**, 102, 1121–1133. [Google Scholar] [CrossRef] [PubMed] - Zhang, Z.; Mongeau, L.; Frankel, S.H. Experimental verification of the quasi-steady approximation for aerodynamic sound generation by pulsating jets in tubes. J. Acoust. Soc. Am.
**2002**, 112, 1652–1663. [Google Scholar] [CrossRef] - Pelorson, X. On the meaning and accuracy of the pressure-flow technique to determine constriction areas within the vocal tract. Speech Commun.
**2001**, 35, 179–190. [Google Scholar] [CrossRef] - Vilain, C.E.; Pelorson, X.; Fraysse, C.; Deverge, M.; Hirschberg, A.; Willems, J. Experimental validation of a quasi-steady theory for the flow through the glottis. J. Sound Vib.
**2004**, 276, 475–490. [Google Scholar] [CrossRef] - Krane, M.H.; Wei, T. Theoretical assessment of unsteady aerodynamic effects in phonation. J. Acoust. Soc. Am.
**2006**, 120, 1578–1588. [Google Scholar] [CrossRef] [PubMed] - Krane, M.H.; Barry, M.; Wei, T. Dynamics of temporal variations in phonatory flow. J. Acoust. Soc. Am.
**2010**, 128, 372–383. [Google Scholar] [CrossRef] [PubMed] - Švec, J.G.; Horáček, J.; Šram, F.; Veselý, J. Resonance properties of the vocal folds: In vivo laryngoscopic investigation of the externally excited laryngeal vibrations. J. Acoust. Soc. Am.
**2000**, 108, 1397–1407. [Google Scholar] [CrossRef] [PubMed] - Neubauer, J.; Mergell, P.; Eysholdt, U.; Herzel, H. Spatio-temporal analysis of irregular vocal fold oscillations: Biphonation due to desynchronization of spatial modes. J. Acoust. Soc. Am.
**2001**, 110, 3179–3192. [Google Scholar] [CrossRef] [PubMed] - Triep, M.; Brücker, C. Three-dimensional nature of the glottal jet. J. Acoust. Soc. Am.
**2010**, 127, 1537–1547. [Google Scholar] [CrossRef] [PubMed] - Kucinschi, B.R.; Scherer, R.C.; DeWitt, K.J.; Ng, T.T. An experimental analysis of the pressures and flows within a driven mechanical model of phonation. J. Acoust. Soc. Am.
**2006**, 119, 3011–3021. [Google Scholar] [CrossRef] - Deverge, M.; Pelorson, X.; Vilain, C.; Lagrée, P.Y.; Chentouf, F.; Willems, J.; Hirschberg, A. Influence of collision on the flow through in-vitro rigid models of the vocal folds. J. Acoust. Soc. Am.
**2003**, 114, 3354–3362. [Google Scholar] [CrossRef] - Motie-Shirazi, M.; Zañartu, M.; Peterson, S.D.; Erath, B.D. Intraglottal aerodynamic pressure and energy transfer in a self-oscillating synthetic model of the vocal folds. medRxiv
**2020**. [Google Scholar] [CrossRef] - Ringenberg, H.; Rogers, D.; Wei, N.; Krane, M.; Wei, T. Phase-averaged and cycle-to-cycle analysis of jet dynamics in a scaled up vocal-fold model. J. Fluid Mech.
**2021**, 918, A44. [Google Scholar] [CrossRef] - Wei, N.; Haworth, A.; Ringenberg, H.; Krane, M.; Wei, T. Phase-averaged, frequency dependence of jet dynamics in a scaled up vocal fold model with full and incomplete closure. Phys. Rev. Fluids
**2022**, 7, 123102. [Google Scholar] [CrossRef] - Kanaya, M.; Tokuda, I.T.; Bouvet, A.; Van Hirtum, A.; Pelorson, X. Experimental study on the quasi-steady approximation of glottal flows. J. Acoust. Soc. Am.
**2022**, 151, 3129–3139. [Google Scholar] - Smith, S.L.; Titze, I.R. Vocal fold contact patterns based on normal modes of vibration. J. Biomech.
**2018**, 73, 177–184. [Google Scholar] [CrossRef] [PubMed] - Mittal, R.; Dong, H.; Bozkurttas, M.; Najjar, F.M.; Vargas, A.; Von Loebbecke, A. A versatile sharp interface immersed boundary method for incompressible flows with complex boundaries. J. Comput. Phys.
**2008**, 227, 4825–4852. [Google Scholar] [CrossRef] [PubMed] - Zheng, X.; Xue, Q.; Mittal, R.; Beilamowicz, S. A coupled sharp-interface immersed boundary-finite-element method for flow-structure interaction with application to human phonation. J. Biomech. Eng.
**2010**, 132, 111003. [Google Scholar] [CrossRef] - Xi, J.; Si, X.A.; Dong, H.; Zhong, H. Effects of glottis motion on airflow and energy expenditure in a human upper airway model. Eur. J. Mech.-B/Fluids
**2018**, 72, 23–37. [Google Scholar] [CrossRef] - Saldías, M.; Laukkanen, A.M.; Guzmán, M.; Miranda, G.; Stoney, J.; Alku, P.; Sundberg, J. The vocal tract in loud twang-like singing while producing high and low pitches. J. Voice
**2020**, 35, e1–e807. [Google Scholar] [CrossRef]

**Figure 2.**Contact patterns of the sixteen shapes of the (1,0)–(2,1) and (1,1)–(2,0) sets of glottal wall motions. For each phase, the contact area (black) and open glottis (white) are marked on the rectangular medial surface.

**Figure 3.**Computational domain and vocal fold model: (

**a**) setup with supraglottic vocal tract. (

**b**) Setup without supraglottic vocal tract. The representative glottal shape corresponds to the (1,0)–(2,1) set at t/T = 0.0000.

**Figure 4.**Flow rate waveforms of the three simulations and percent errors of flow rate between quasi-steady/pseudo-static and dynamic simulation over one vibration cycle.

**Figure 5.**${v}_{g}/{v}_{v}$ versus t/T during the open phase, with the flow rate errors being presented in the form of error bars. Maximum values of ${v}_{g}/{v}_{v}$ for the (1,0)–(2,1) set at 100 Hz are much larger than 500. They are limited to 500 for a better view of other ${v}_{g}/{v}_{v}$ values.

**Figure 6.**RMSE of glottal wall pressure between the quasi-steady/pseudo-static and dynamic case over one vibration cycle.

**Figure 7.**Variation in convection, pressure, shear, and unsteady term over one vibration cycle in each dynamic case.

**Figure 8.**Comparison of convection, pressure, and shear term between the dynamic and quasi-steady/pseudo-static cases.

Study | VF Geometry | Frequency Investigated | Vocal Tract | Approach |
---|---|---|---|---|

[9] | Convergent | 10~120 Hz | No | Experimental |

[11] | Uniform, circular, and divergent | 142 Hz | Yes | Theoretical and experimental |

[10] | Convergent, straight, and divergent | 70~120 Hz | Yes | Experimental |

[19] | Straight, rounded, and Gaussian | 45~315 Hz | No | Theoretical and experimental |

[12] | Straight and rounded | 38~540 Hz | No | Theoretical and experimental |

[18] | Time-dependent | 28.1~112.5 Hz | Yes | Experimental and numerical |

[13] | Not applicable | 100 Hz | Yes | Theoretical |

[14] | Half cylinder | 30~126 Hz | Yes | Theoretical and experimental |

[20] | Reproduced from the coronal section and stained slices of the VFs | 160 Hz | Yes | Experimental |

[21,22] | Half cylinder | 52.5~97.5 Hz | Yes | Theoretical and experimental |

[23] | M5 | 112.5 Hz | No | Theoretical and experimental |

100 Hz | 500 Hz | |||
---|---|---|---|---|

Without Vocal Tract | With Vocal Tract | Without Vocal Tract | With Vocal Tract | |

(1,0)–(2,1) | 0.032 | 0.031 | 0.154 | 0.165 |

(1,1)–(2,0) | 0.031 | 0.030 | 0.145 | 0.145 |

**Table 3.**Errors of important aerodynamic and sound spectrum parameters between quasi-steady/pseudo-static and dynamic cases. Phase shift is calculated as the phase difference in the peak flow between the quasi-steady/pseudo-static and dynamic cases. H1–H2 and H1–H4 are the spectral amplitude differences between the first two harmonics and the first and fourth harmonics, respectively. Spectral slope is determined using linear regression in MATLAB to fit the first 20 harmonics of each case.

Without Vocal Tract | With Vocal Tract | ||||
---|---|---|---|---|---|

Frequency | Parameters | (1,0)–(2,1) Quasi/Pseudo | (1,1)–(2,0) Quasi/Pseudo | (1,0)–(2,1) Quasi/Pseudo | (1,1)–(2,0) Quasi/Pseudo |

Peak flow (%) | −1.18/−2.18 | −1.18/−3.28 | −0.33/2.51 | 3.49/5.35 | |

Mean flow (%) | −2.24/−3.94 | −3.36/−1.74 | −0.61/−0.39 | −2.14/2.61 | |

100 Hz | Phase shift (degree) | −11.34/−20.37 | −8.82/−13.93 | −25.48/−27.83 | −24.89/−28.04 |

MFDR (%) | 0.49/−16.55 | −9.51/−20.48 | −13.83/−22.94 | −13.19/−15.81 | |

H1–H2 (%) | −5.84/−9.23 | −9.53/−0.20 | −0.11/−7.65 | −10.72/−1.31 | |

H1–H4 (%) | −7.19/−5.87 | 8.54/11.47 | −17.07/−13.15 | 8.76/10.09 | |

Spectral slope (%) | 12.30/14.75 | 27.64/44.44 | 8.75/13.00 | 14.78/6.05 | |

Peak flow (%) | 0.09/1.46 | 2.44/−6.79 | 28.12/32.95 | 37.71/29.57 | |

Mean flow (%) | −5.58/−10.26 | −11.01/−8.56 | 14.07/10.68 | 4.46/10.75 | |

500 Hz | Phase shift (degree) | −21.27/−44.63 | −27.62/−34.76 | −59.11/−77.93 | −61.89/−66.04 |

MFDR (%) | 0.79/−34.05 | −2.18/−25.48 | 12.84/−16.15 | 17.69/−3.63 | |

H1–H2 (%) | −31.45/−43.76 | −42.60/−21.55 | −26.88/−41.37 | −50.66/−32.46 | |

H1–H4 (%) | −21.90/−14.58 | −19.46/−23.71 | −2.43/8.23 | −15.88/−19.58 | |

Spectral slope (%) | 11.75/36.04 | 9.09/26.84 | 12.03/36.99 | 6.18/5.44 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wang, X.; Zheng, X.; Titze, I.R.; Palaparthi, A.; Xue, Q.
Examining the Quasi-Steady Airflow Assumption in Irregular Vocal Fold Vibration. *Appl. Sci.* **2023**, *13*, 12691.
https://doi.org/10.3390/app132312691

**AMA Style**

Wang X, Zheng X, Titze IR, Palaparthi A, Xue Q.
Examining the Quasi-Steady Airflow Assumption in Irregular Vocal Fold Vibration. *Applied Sciences*. 2023; 13(23):12691.
https://doi.org/10.3390/app132312691

**Chicago/Turabian Style**

Wang, Xiaojian, Xudong Zheng, Ingo R. Titze, Anil Palaparthi, and Qian Xue.
2023. "Examining the Quasi-Steady Airflow Assumption in Irregular Vocal Fold Vibration" *Applied Sciences* 13, no. 23: 12691.
https://doi.org/10.3390/app132312691