Non-dimensional Model Parameters

The parameters of the dynamical vocal fold model shown in Figure 3.1 and related by dimensional equations of motion (3.1), (3.2), and (3.3) are gathered in Table 3.1. These are subsequently combined with the geometric parameters and the uid parameters to obtain dimensionless groups such as mass ratios and frequency ratios which enable an overall characterization and summary of the vocal fold model developed. Model parameters are summarized in Table 3.4 below.

From these dimensional parameters, characteristic scales are selected and gathered in Table 3.5. Neutral glottal width is selected as the characteristic length scale because it ostensibly determines the glottal ow; since Vs is constant and volumetric ow rate is the product of velocity and cross-sectional area, Ang = hngldetermines nominal glottal ow. The velocity scale is the vortex advection speed in absence of a free stream, Vc = Γ/a. The pressure scale is proportional to a dynamic pressure determined by the characteristic velocity scale.

Finally, characteristic time is the ratio of characteristic length and velocity.

Table 3.5: Characteristic scales.

dimension characteristic scale interpretation

L l_c= h_ng neutral glottal width, characterizes intraglottal ow LT⁻¹ V_c= Γ/a characteristic velocity, due to vortex conguration M L⁻¹T⁻² Pc= ρΓ²/a² a dynamic pressure

T t_c= ah_ng/Γ characteristic time determined by lc/V_c

The dimensional equations of motion may be transformed via the change of variables that time is replaced with the ratio of dimensional time and characteristic time, t^∗ = t/t_c,

Table 3.4: Summary of model parameters.

parameter description

M body mass

m cover mass

I_c cover moment of inertia

K body translational stiness

k cover translational stiness

κ cover torsional stiness

B body mass translational damping

b cover translational damping

B_c torsional damping of cover h_s subglottal width of larynx A_s subglottal sectional area

h_e epilaryngeal width of larynx A_e epilaryngeal sectional area

tV F vocal fold thickness

l vocal fold length

l_node/t_{V F} nodal position as fraction of vocal fold thickness

A_ng neutral glottal area

h_ng neutral glottal channel width θ_ng neutral glottal convergence angle

ρ uid density

PL lung pressure

P_s subglottal pressure

P_e epilaryngeal pressure

Q volumetric ow rate

V_s subglottal velocity

Γ circulation due to vortex

a intravortex spacing

and similarly that displacements are scaled with characteristic length as y^∗ = y/l_c, via chain rule for the substitutions, the dimensionless form of (3.1) is then

mΓ²

the right hand side of which follows by observing that the aerodynamic loading term Fain (3.1) may be written as a mean pressure over the projected area of the medial surface of the vocal fold,

F_a= P l t_{V F} = P^∗ρΓ² a²l t_{V F} .

So, by dividing through by the coecient of the dimensionless pressure, ρ^Γ_a²2ltV F, we obtain the dimensionless form of (3.1) given by

m The coecients of the terms in (3.30) determine dimensionless groups, a mass ratio, a damping ratio, and a stiness ratio. To obtain a frequency ratio, if one recalls that the natural frequency of a simple spring-mass system is ωn=pk/m, and that, after dividing the dierential equation through by the coecient of the second-order term, the coecient of the zeroth-order term is the square of the natural frequency, one obtains a frequency

ratio as follows _kh

So that a characteristic frequency is given by f_c= Γ

2π a h_ng (3.31)

however, notice that this is simply the reciprocal of the characteristic time. This yields an interpretation of the characteristic time that it is a characteristic period.

A similar argument may be applied to (3.3). However, despite that θ is dimensionless, upon dividing through by Ic,

θ + 2ζω¨ _nθ + ω˙ _n²θ = Ta

I_c (3.32)

in which the damping ratio and natural frequency are given by ζ = B_c

equation (3.32) remains dimensional. However, as before, with t^∗ = t/t_c, application of chain rule yields

which, upon substitution into (3.32) yields Γ²

a²h_ng² d²θ

dt^∗2 +2ζω_nΓ ahng

dθ

dt^∗ + ω_n²θ = T_a Ic

These frequency ratios, mass ratios, and various other useful dimensionless groups, are summarized in Table 3.6.

Table 3.6: Dimensionless groups dimensionless group characterizes

m

ρ l tV Fhng ratio of plate mass to uid mass

b

Γ

aρ l tV F damping ratio

k ρ l tV F Γ2

a2hng

stiness ratio

ωn Γ a hng

frequency ratio

a/hs geometry of vortex distribution

Γ/a

Vs relative strength of vortex

4 | Results

The simulation model developed in Chapter 3, with steady and unsteady versions of Bernoulli's equation employed to obtain the pressure eld, is validated against the bar-plate body-cover model of Titze [99], which employs a 1-D steady potential ow model of the glottal ow. This validation is discussed in Section 4.1. In order to address the ques-tion of this thesis, whether intraglottal vortices inuence the rate of closing of the vocal folds, the vortex advection scheme is employed, and advecting vortex pairs are introduced at some upstream position and allowed to advect into the glottis. In Section 4.2, the re-sults of the study of the inuence of advecting vortices are discussed. The 2-D glottal ow model employing trailing edge ow separation together with the unsteady Bernoulli equa-tion to determine the pressure eld in the presence of the advecting vortices is employed for the simulations. The advecting vortices account for the intraglottal vortices which are shed from the medial surface of the vocal folds downstream of ow separation in actual glottal ow. The simulation was performed with a trailing edge separation to ensure that the pressure variation due to the presence of advecting vortices would be impressed upon the medial surfaces of the vocal folds in a natural way regardless of their axial position.

Finally, Section 4.3 presents a discussion of the limitations of the model, whereas mention of possible approaches to overcome the limitations of the model or to extend it shall be postponed until Chapter 5.