Evaluation of aerodynamic characteristics of a coupled fluid-structure system using generalized Bernoulli’s principle: An application to vocal folds vibration

Abstract

In this work we explore the aerodynamics flow characteristics of a coupled fluid-structure interaction system using a generalized Bernoulli equation derived directly from the Cauchy momentum equations. Unlike the conventional Bernoulli equation where incompressible, inviscid, and steady flow conditions are assumed, this generalized Bernoulli equation includes the contributions from compressibility, viscous, and unsteadiness, which could be essential in defining aerodynamic characteristics. The application of the derived Bernoulli’s principle is on a fully-coupled fluid-structure interaction simulation of the vocal folds vibration. The coupled system is simulated using the immersed finite element method where compressible Navier-Stokes equations are used to describe the air and an elastic pliable structure to describe the vocal fold. The vibration of the vocal fold works to open and close the glottal flow. The aerodynamics flow characteristics are evaluated using the derived Bernoulli’s principles for a vibration cycle in a carefully partitioned control volume based on the moving structure. The results agree very well to experimental observations, which validate the strategy and its use in other types of flow characteristics that involve coupled fluid-structure interactions.

1. INTRODUCTION

In classical fluid mechanics, the Bernoulli equation is applied to ideal fluids where the fluids are assumed to be incompressible, inviscid, and steady.^{(1, 2)} Based on these assumptions, the Bernoulli equation is derived based on the simplified Cauchy momentum equation, or the balance of momentum, for the ideal fluids along a streamline. Local form of the Bernoulli equation evaluates the amount of mechanical energy stored at a particular point along the streamline. The mechanical energy includes three terms: kinetic (related to the flow velocity), potential (related to the body force), and pressure (related to pressure) energies. The decomposition of the total energy would reveal a more detailed contribution to the overall flow characteristics. For the past few decades, Bernoulli’s principle has been widely accepted in fluid mechanics as a way to perform mechanical energy accounting.

However, ideal fluids are not ideal for practical problems. In reality, fluids can be compressible, viscous, and unsteady. The Bernoulli’s analysis can no longer neglect those terms from the original momentum equation. This problem is accentuated when the fluid interacts with a moving and deformable structure where the unsteadiness comes from the inherent fluid dynamics and its interactions with the moving structure. An example of such complex coupled system is the airflow in the glottis.

Laryngeal airflow and its interactions with the vocal fold tissues are very important aspects of the phonation process. The laryngeal flow, driven by lung pressure, is responsible for pushing the vocal folds in such patterns that ensures a positive net energy influx into the structures over the oscillation cycles⁽³⁾ in order to maintain the vocal folds’ sustained vibration.^{(4, 5)} The vocal folds structures, which can voluntarily constrict the glottis, in return regulates the laryngeal airflow by altering the glottal cross-sectional area as a result of the inflicted vibration, as shown in Figure 1. The laryngeal airflow drives vocal folds vibrations via fluid-structure interaction, and is the carrying medium of acoustic propagation. The fluctuations in the laryngeal airflow, along with several other factors such as turbulence in the glottal airstream and sudden onset of phonation,⁽⁶⁾ contribute to sound generation. Among them, drag force on the glottal flow by the vocal fold structures and volume fluctuation in the glottis have been attributed as the primary and the secondary sound sources in human phonation process,⁽⁷⁾ to which the fluid-structure interaction between the laryngeal airflow and the vocal folds is accountable. Therefore, the quantification and evaluation of the fluid-structure interaction in the larynx are important in characterizing laryngeal airflow and vocal fold vibration mechanism.

Fig. 1.

A schematic drawing of the glottis opening and closing process.

The self-sustained vibration can be explained as a self-repeating physical process where the energy in the fluid-structure interaction system must be properly transferred and balanced between the vibrating structure and its surrounding fluid. One study shows that compressibility of fluid, or medium volume change is the key factor for a self-sustained vibration.⁽⁸⁾ It is widely accepted that net mechanical energy transfer from the fluid flow to the interacting structure is necessary for self-sustained vibration, in order to compensate for the energy loss due to damping in the structure.^{(3, 9–11)} Furthermore, this net gain of energy from the flow is used as an indication of the onset of flutter^(12–14) to analyze the energy transfer between airflow and vocal folds and concluded that there is indeed positive net energy transferred into the vocal folds over a full-cycle period. To fully understand the self-sustained vibration process including that of the vocal folds, it is instrumental to investigate all the physical contributions without making too many simplified assumptions.

An initial work that evaluated the aerodynamic characteristics in a glottal flow was proposed in Ref. [15] by decomposing the terms into unsteady and convective flow accelerations. Based on the momentum equation along the symmetry streamline and various assumptions, this work derived an energy balance equation describing glottal flow, with terms representing unsteady and convective accelerations, correction terms due to separation point motion and vocal fold wall motion, and friction. The derivation could be reduced to a familiar form of unsteady Bernoulli equation⁽¹⁶⁾ if the effects of motions of vocal fold wall and separation point were neglected. This derivation, listing all the terms due to their causes, not only made it feasible to compare the magnitudes of different terms and related effects, but also brought the aerodynamic analysis a step closer towards the understanding the self-sustained vibration in a fluid-structure interaction system.

However, the simplification in a flow without the actual representation of the vocal fold lacks the fundamental physics of a coupled system. In reality, vocal folds vibration and deformability are an important piece of inputs that contribute to the aerodynamics. In a coupled system, the aerodynamics of the flow responds to the movement or displacement of the structure while the behavior of the structure responds to the surrounding aerodynamics. Therefore, an evaluation based on a more physical simulation result is necessary, particularly to the application where the flow and the solid behaviors are so tightly related. In this work, a generalized Bernoulli’s principle is derived along with physical interpretation of the derived contribution terms pertaining to the laryngeal flow and vocal folds vibration. The defined regions and streamline in a control volume are carefully defined for the evaluation of the flow characteristics near the glottis based on a moving vocal folds. The simulation results in a coupled fluid-structure framework are shown to demonstrate the validity of the chosen numerical technique. Finally, the flow is evaluated using the proposed generalized Bernoulli’s principle.

The paper proceeds as follows. In Section 2.2, the derivations of the generalized Bernoulli equation is first given in detail, with explanation of the roles and contributions of different terms in those expressions. The numerical simulation method and setup are introduced in Section 3. In Section 4, evaluations of the derived terms are performed based on the simulation results, and full analysis are discussed to advance the understanding of the complex laryngeal flows. Finally, conclusions are drawn with a brief discussion on relevant future work.

2. DERIVATIONS OF GENERALIZED BERNOULLI EQUATION IN A MOVING CONTROL VOLUME

2.1. Defining Streamline and Control Volume

A Bernoulli’s analysis requires the identification of a streamline where the Bernoulli head and loss terms are tracked. However, it is generally difficult to identify and track a streamline in a laryngeal flow that remains in place on which a meaningful Bernoulli analysis can be performed. The airflow in the supraglottal region (downstream of the glottis) is widely observed to form an orifice-modulated jet,⁽¹⁷⁾ whose jet deflection angle changes stochastically over oscillation cycles, both in experiments^(17–21) and numerical simulations.^(22–24) Therefore, to provide the possibility of tracing a streamline, symmetry is enforced on the centerline where only half of the larynx is modeled, which is also a common practice seen in numerical simulations.^{(25, 26)} For this reason the aforementioned half-space model is used here for the numerical simulation and for Bernoulli and control volume analyses.

It was proposed that two different flow regions exist in the glottis:

1. at glottal entry and throughout the glottis where the flow is non-turbulent and

2. at glottal exit where the flow becomes turbulent and energy losses are substantial.

This division of the flow region suggested that a jet flow existed compactly near the centerline in the supraglottal region.^{(27, 28)} To be consistent with these observations, the control volume defined in this study is in a volume near the vocal folds, which is chosen as a volume con-fined by an entrance plane S_entrance located at the upstream extremum of the vocal fold and an exit plane S_exit located at a short distance downstream of the structure, as shown in Figure 2. The vocal tract wall and the vocal fold surface are denoted as S_wall and S_VF respectively. Since a line of symmetry line is imposed, the control volume is also bounded by the centerline S_centerline. The volume is split into two domains, as suggested in Ref. [28], the contraction region V_contraction (upstream of the glottis opening) and the jet region V_jet (downstream of the glottis opening). They are separated by the vocal fold as it contacts the centerline and close off the glottis, and separated by a transverse plane S_separation at the longitudinal location of the separation point when the glottis is open. The separation point here is conveniently defined as the most upstream location of minimum glottal constriction, since the geometric feature is easy to locate numerically and is found to be close to the actual separation point.⁽²⁸⁾ Similarly, the streamline or the symmetry centerline S_centerline is also divided into segments S_{centerline-contraction} and S_{centerline-jet} respectively. With a defined streamline, a control volume and its boundaries, the following derivations based on governing laws yield the physical terms that would assist in the full understanding of the aerodynamic behavior in a fluid-structure interaction system. Noting that even though the terms are defined using the geometry of the vocal folds and vocal tract, the approach is general to be used in any geometrical set up.

Fig. 2.

Definitions of streamline and control volume in the glottal channel at the extrema of vocal fold deformations, when (a) the glottis is open; (b) the glottis is closed.

2.2. Generalized Bernoulli Equation

Bernoulli equation, when based on the momentum equation without any assumptions, is able to show kinetic and compressional internal energy changes in the airflow along a streamline, and airflow mechanics such as flow density variations, unsteadiness and frictional losses. When the terms are evaluated in the contraction and the jet regions, it also reveals the distinct characteristics of the airflow in these regions.

To derive a generalized Bernoulli equation, we start with integrating the Cauchy momentum equation along a streamline segment. The momentum equation, neglecting body force, for a compressible, Newtonian, viscous fluid can be written in its index form (Einstein summation convention is applied unless explicitly stated). All the indices i and j throughout the derivation are the physical dimensions where i, j = 1, 2, 3:

$${(\rho u_i)}_{,t}+{(\rho u_i{u}_j)}_{,j}=-p_{,i}+{\tau }_{ij,j}$$ (1)

which after expansion of the left-hand side becomes:

$$(\rho u_{i,t}+u_j{u}_{i,j})+({\rho }_{,t}+{\rho }_{,j}{u}_j+\rho u_{j,j})u_i=-p_{,i}+{\tau }_{ij,j}$$ (2)

The second term on the left hand side of Eq. (2) is zero because of the continuity equation stipulates that ρ_,t + (ρu_j)_{, j} = 0, therefore an equivalent form of the momentum equation is:

$$(\rho u_{i,t}+u_j{u}_{i,j})=-p_{,i}+{\tau }_{ij,j}$$ (3)

where the deviatoric stress tensor τ_ij = μ(u_{i, j} + u_{j, i} − (2/3)u_{k, k}δ_ij), δ_ij is the Kronecker Delta, and μ is the dynamic viscosity. Its derivative is then:

$${\tau }_{ij,j}=\mu \left(u_{i,jj}+u_{j,ij}-\frac{2}{3}{u}_{k,kj}{\delta }_{ij}\right)=\mu \left(u_{i,jj}+\frac{1}{3}{u}_{j,ji}\right)$$ (4)

The convective acceleration in Eq. (3) can be expressed with velocity and vorticity:

$$u_j{u}_{i,j}=\frac{1}{2}{(u_j{u}_j)}_{,i}-{\epsilon }_{\mathit{ijk}}{u}_j{\omega }_k$$ (5)

where ε_ijk is the permutation tensor, also known as the Levi-Civita symbol, and ω = ∇ × u is the vorticity.

Bernoulli equation can be obtained by integrating the momentum Eq. (3) with definitions in Eqs. (4) and (5) along a streamline, s, with its differential vector in the streamline direction ds = ŝds, or in its index form ds_i = ŝ_ids, where ds = |ds| is the differential vector length and ŝ is the unit vector indicating the streamline direction:

$$\int \frac{\partial u_i}{\partial t}{\widehat{s}}_{i}ds+\int \frac{\partial }{\partial x_i}\left(\frac{1}{2}{u}_j{u}_j\right){\widehat{s}}_{i}ds-\int {\epsilon }_{\mathit{ijk}}{u}_j{\omega }_k{\widehat{s}}_{i}ds=-\int \frac{1}{\rho }\frac{\partial p}{\partial x_i}{\widehat{s}}_{i}ds+\int \frac{\mu }{\rho }\left(\frac{{\partial }^2{u}_i}{\partial x_j\partial x_j}+\frac{1}{3}\frac{{\partial }^2{u}_j}{\partial x_j\partial x_i}\right){\widehat{s}}_{i}ds$$ (6)

It can be interpreted that in Eq. (6) the dot products with ŝ are essentially the projections of the terms onto the streamline direction. Which yields the following properties:

$$u_i=u{\widehat{s}}_i\Rightarrow u_i{\widehat{s}}_i=u{\widehat{s}}_i{\widehat{s}}_i=u\Rightarrow u_i{\widehat{s}}_{i}ds=\mathit{uds}$$ (7)

where u = |u| is the magnitude the velocity. Note that a generalization of Eq. (7) indicates that for a vector γ, if and only if it is parallel to the streamline, i.e., γ = γŝ where γ = |γ|, would the property γ·ŝ = γŝ·ŝ = γ hold true and thus γ·ŝds = γds. For this reason, an assumption for unsteady flows is made to maintain simplicity, that the streamline does not change its position over time, which conveniently holds true for a centerline on which symmetry is enforced:

$$\frac{\partial u_i}{\partial t}{\widehat{s}}_i=\frac{\partial u}{\partial t}{\widehat{s}}_i{\widehat{s}}_i=\frac{\partial u}{\partial t},\text{if}\frac{\partial \mathbf{u}}{\partial t}\Vert \mathbf{u}$$ (8)

Similarly, for any arbitrary scalar β, e.g., pressure p, its spatial derivative dot the streamline unit vector ŝ_i is:

$$\frac{\partial \beta }{\partial x_i}{\widehat{s}}_i=\frac{\partial \beta }{\partial s}\Rightarrow \frac{\partial \beta }{\partial x_i}{\widehat{s}}_{i}ds=\frac{\partial \beta }{\partial s}ds$$ (9)

Lastly, the third term in Eq. (6) can be evaluated as ε_ijku_j ω_kŝ_i = 0 since (u × ω) × u = 0 which also means (u × ω) × ŝ = 0.

Finally, integrating Eq. (6) along a streamline from point s₁ to point s₂, we have

$${\int }_{s_1}^{s_2}\frac{\partial u}{\partial t}ds+{\int }_{s_1}^{s_2}\frac{\partial }{\partial s}\left(\frac{1}{2}{u}^2\right)ds+{\int }_{s_1}^{s_2}\frac{\partial }{\partial s}\left(\frac{p}{\rho }\right)ds=-{\int }_{s_1}^{s_2}\frac{p}{{\rho }^2}\frac{\partial \rho }{\partial s}ds+\mu {\int }_{s_1}^{s_2}\frac{1}{\rho }\left[\frac{{\partial }^2{u}_i}{\partial x_j\partial x_j}{\widehat{s}}_{i}ds+\frac{1}{3}\frac{\partial }{\partial s}\left(\frac{\partial u_j}{\partial x_j}\right)\right]ds$$ (10)

Re-arranging this equation then yields a generalized Bernoulli equation for unsteady, compressible, viscous flow:

$$\underset{(\mathrm{i})E_{\text{con}}}{\underbrace{{\left(\frac{1}{2}{u}^2\right)|}_{s_1}^{s_2}}}+\underset{(\text{ii})E_{\text{pre}}}{\underbrace{{\left(\frac{p}{\rho }\right)|}_{s_1}^{s_2}}}=-\underset{(\text{iii})A_{\text{uns}}}{\underbrace{{\int }_{s_1}^{s_2}\frac{\partial u}{\partial t}ds}}-\underset{(\text{iv})E_{\text{den}}}{\underbrace{{\int }_{s_1}^{s_2}\frac{p}{{\rho }^2}\frac{\partial \rho }{\partial s}ds}}-\underset{(\mathrm{v})E_{\text{vis}}}{\underbrace{\left(-\mu {\int }_{s_1}^{s_2}\frac{1}{\rho }\left[\frac{{\partial }^2{u}_i}{\partial x_j\partial x_j}{\widehat{s}}_i+\frac{1}{3}\frac{\partial }{\partial s}\left(\frac{\partial u_j}{\partial x_j}\right)\right]ds\right)}}$$ (11)

where:

1. the convective acceleration head E_con can be interpreted as the change in kinetic energy per unit mass along the streamline between points s₁ and s₂;

2. the pressure head E_pre indicates the change in internal energy per unit mass due to compression and rarefaction;

3. the unsteady acceleration head loss A_uns is the head loss due to flow unsteadiness;

4. the density head loss E_den is the head loss due to density variation;

5. the viscous head loss E_vis is the head loss due to viscosity.

The combined left-hand-side terms, ${((1/2)u^2+(p/\rho ))\mid }_{s_1}^{s_2}$, can be identified as change in the total head from point s₁ to point s₂ along a streamline, which is often seen in the standard incompressible and inviscid Bernoulli equation for steady flows, where the change in the total head is zero. The right-hand-side terms are the factors that change the total head, i.e., head loss terms along the streamline.

3. SIMULATION METHOD AND SETUP

Once the essential Bernoulli terms are identified, either experimental or simulation results can be evaluated or post-processed. In this study, the evaluation of the Bernoulli terms is based on a fully-coupled fluid-structure interaction simulation.

Over the past two decades, a number of numerical models are built to facilitate the understanding of the self-sustained vocal fold vibration, which span from multi-mass models where the vocal folds are modeled as mass-spring-damper system^(29–32) to continuum models where the vocal folds are modeled using realistic physical representation. With the dramatic increase in computational capabilities and efficiencies in the recent years, more continuum vocal fold models have been developed using 2-D and 3-D finite elements, some of which are coupled with Bernoulli equation for fluid to study the mechanisms behind the self-sustained oscillation process.^(33–37) A recent review dealing with validation issue^{(38, 39)} when the underlying problem is viewed as a fluid-structure interaction can be found in Ref. [40].

In this work, the simulation of the laryngeal flow and vocal folds vibration involving their fluid-structure interaction that produces a self-sustained oscillation is performed with a fluid-structure interaction algorithm, the modified Immersed Finite Element Method (mIFEM),^(41–43) in which the laryngeal air is modeled as a compressible isentropic fluid with the Navier-Stokes equations and the vocal folds as linear visco-elastic solid. The two components, Navier-Stokes equations for air and solid mechanics for the vocal fold, are fully coupled. The original Immersed Finite Element Method^(44–48) was developed for solutions of fluid and deformable structure interaction problems. It is efficient as it is a non-boundary-fitted technique where no mesh update or re-meshing is necessary. However, the original formulation cannot handle large density disparities between fluid and the solid,⁽⁴⁹⁾ and does not directly solve for solid dynamics which tend to over-estimate the solid deformation, especially for high-Reynolds-number flows. To compensate for these deficiencies, the mIFEM was developed to more accurately handle complicated fluid-structure interactions where the solid behavior is dominant. The mIFEM has been validated and verified in several studies, and is shown to be very robust and accurate in fluid-structure interaction simulations.^(41–43) Particularly, with the inclusion of fluid compressibility, a self-sustained vocal fold vibration is easily obtained with no artificially imposed numerical treatment. The detailed algorithm, methodology, simulation setup and validation can be found in our previous work.⁽⁵⁰⁾

The model setup for the numerical simulation is shown in Figure 3. The numerical model setup follows a vocal fold acoustic experimental setup designed by Campo and Krane,⁽⁵¹⁾ which is a simplified 2-layer (body and cover) real-size model. The total length of the vocal tract is 31.514 cm, while its width is 1.397 cm in the half-space model. A constant and fixed pressure of 1300 Pa is applied to the inlet boundary on the left, and a stress-free boundary condition is applied on the outflow boundary on the right. The vocal tract length is long enough to ensure that the exit boundary would not produce any numerical reflections that can impact the flow in the upstream. No-slip boundary conditions are applied to the side walls, while the symmetry boundary condition is applied on the vocal tract centerline. The vocal fold characteristics such as dimensions and properties vary in a large range from humans to animals. Experimentalists continue to refine these measurements using more advanced tools and methods.^{(52, 53)} In this study, the vocal fold structure has a roughly triangular shape, and is consisted of a body layer and a cover layer as introduced in Ref. [51]. Both layers have the same density of ρ_body = ρ_cover = 1.0013 g/cm³, damping coefficient of η = 10 Poise, and Poisson’s ratio of ν = 0.3. The elastic moduli for the two parts, however, are different. The cover layer is typically softer than the body. Studies have measured the Young’s moduli for the tissues to be in the range between 2.7 kPa and 110 kPa.^(54–56) Our study has the body elastic modulus to be 4 times higher than that of the cover, where E_cover = 10 kPa and E_body = 40 kPa. A summary of the simulation setup that includes the geometry, material properties and boundary conditions for the vocal fold (solid) and the air in the vocal tract (fluid) is listed in Table I.

Fig. 3.

Numerical setup of vocal fold and laryngeal flow.

Table I.

Simulation setup.

	Vocal tract (isentropic air)	Vocal fold (2-layer, linear visco-elastic material)
Size/shape	Length = 31.514 cm Height = 1.397 cm (half space)	Body base length = 1.505 cm Body base height = 1.397 cm Cover layer thickness = 0.15 cm Convergent angle = 10°
Material properties	Density ρ0 = 1.3 × 10−3 g/cm3 at T = 273 K Viscosity ν = 1.8×10−4 g/cm · s	Density ρbody = ρcover = 1.0013 g/cm3 Damping coefficient η = 10 Poise Young’s modulus: Ebody = 40 kPa, Ecover = 10 kPa
Boundary conditions	Pin = 1300 Pa No-slip on vocal fold walls No-slip on bottom of the vocal tract Symmetry on vocal tract centerline	Fixed at vocal fold base

The computing resource to perform these computations is the IBM Blue Gene Q system located in the Center of Computation Innovations at Rensselaer Polytechnic Institute, with each computing node consisting of a 16-core 1.6 GHz A2 processor and 16 GB of DDR3 memory. The source code is efficiently parallelized with MPI, please refer to Ref. [57] for the computing efficiency studies. For each run, 1024 cores are utilized, which takes approximately 40 hours to complete 10,000 time steps of simulation. With the time step we choose at 0.0001 s, it is 50–60 cycles of the vocal folds vibration, which is more than sufficient number of cycles to reach a steady-state vibration.

4. RESULTS AND DISCUSSION

4.1. Simulation Results

To demonstrate that the simulation reaches and sustains steady-state vibration, Figure 4 shows the time history of the gap size h, measured between the vocal fold surface and the centerline. The vocal fold vibration reaches its steady state about 0.12 s after initiation, with a frequency of 234 Hz. Due to the periodicity of the laryngeal airflow and the vocal fold vibration, a single period t/T = 0.1646 s~0.1689 s, demarcated with the red dashed lines in Figure 4, is used for the ensuing analyses.

Fig. 4.

Time history of gap size between vocal fold surface and the centerline. The vocal fold vibration reaches its steady state after about 0.12 s. The red dashed lines demarcate the cycle during which the ensuing analyses are conducted.

The snapshots of the transient laryngeal jet flow and vocal fold deformation within a vibration cycle from opening, to a fully-opened, then to a closed position after reaching steady-state vibration are shown in Figure 5. The glottal jet is clearly captured in each frame as it propagates downstream. A new jet forms at the beginning of the next vibrational cycle as the vocal fold is pushed open from a closed position.

Fig. 5.

Snapshots of the laryngeal jet (colored by vorticity) during one vocal fold vibration cycle. (a) t/T = 0.28, glottis starts to open; (b) t/T = 0.45, glottis further opens; (c) t/T = 0.62, glottis is fully opened; (d) t/T = 0.72, glottis is closing; (e) t/T = 0.83, glottis is closed.

Figure 6(a) shows the vocal fold surface shape over one cycle after reaching steady-state vibration, where the third axis is time. The vocal fold vibration shows strong periodicity across the entire surface. Figure 6(b) shows the projection of the preceding figure onto the xy-plane, where the dashed and dash-dot curves are the extreme positions of the vocal fold when it closes the glottis and when it deforms towards the superior direction respectively, while those in green are the traces of points on the vocal fold surface. The vocal fold deforms in such a way that the upper portion and the lower portion generally move in opposite directions. Most points on the structure surface vibrate in slim traces in the shape of sickles, while some points near the tip of the vocal fold follow traces in the shape of lemniscate.

Fig. 6.

Vocal fold deformation over time.

4.2. Bernoulli Analysis

The following is to perform numerical post-processing of the simulation results using Bernoulli analysis derived in Section 2.2. All the terms are evaluated independently. The total residual is first computed, which are found to be within 0.3% in magnitude compared to the other computed terms. This demonstrates that the error caused by the post analysis is insignificant and well within computational error tolerances. Because of the strong periodicity of the vocal fold vibration as well as the laryngeal flow, all the Bernoulli terms are plotted against the normalized time t/T and are shown over a single period.

Since the laryngeal flow dynamics are substantially different in the subglottal and supraglottal regions as discussed previously, the Bernoulli’s analysis separates the streamline into the contraction region and jet region as indicated in Figure 2.

In the contraction region, among the five derived Bernoulli terms, the convective acceleration head E_con and the pressure head E_pre are significant, dwarfing the other three terms, shown in Figure 7. This indicates that the predominant dynamics in the contraction region is the conversion of compressional internal energy to kinetic energy, as the air flows from higher-pressure upstream region to lower-pressure region near the separation point. This gain of kinetic energy is due to convective acceleration. The convective acceleration term is positive when the glottis is open, which suggests as the laryngeal flow goes through the glottis, the velocity at the separation point tends to be much larger than that at a location upstream of the vocal fold structure; this is because the tapering shape of the channel in the contraction region facilitates velocity increases as the air squeezes through narrowing cross sections, and correspondingly pressure drop is significant in the contraction region. The only interval during which the unsteady acceleration head loss is much smaller compared to the convective acceleration head is when the glottis is open to its maximum extent, roughly from t/T = 0.5 to t/T = 0.7. The unsteady acceleration head loss is positive during flow acceleration when the glottis opens, and negative during flow deceleration when the glottis closes, as the glottis regulates the laryngeal flow. The unsteady acceleration head loss can be non-zero even when the glottis is closed, due to ongoing vocal fold deformation.

Fig. 7.

Bernoulli terms in the contraction region during one oscillation cycle.

The flow dynamics is very different in the jet region, as shown in Figure 8. The most significant terms are the convection acceleration head E_con, which correlates with the velocity difference between the separation point and the exit plane, and the unsteady acceleration head loss A_uns. The pressure head E_pre does not diverge far from zero suggesting that the pressure drop is insignificant in the jet region. The unsteady acceleration head loss also shows flow acceleration as the glottis opens and flow deceleration as the glottis closes; however, a flow acceleration is also seen from t/T = 0.6 to 0.8 when the glottis is closing. The acceleration terms show lags in their onsets after the glottis starts to open, and remain non-zero for a significant time after the glottis closes, as the glottal jet takes time to convect. Since the two acceleration terms are commensurate in magnitude, the flow in the jet region is hence considered inherently unsteady. This result further supports the analysis of the flow dynamics in the subglottal and supraglottal regions, as estimated in Refs. [27, 28] using a mass-spring-damper system, where the unsteadiness is an inherently significant factor in the supraglottlal region.

Fig. 8.

Bernoulli terms in the jet region during one oscillation cycle.

Combining the two regions, Figure 9 provides an overall flow dynamics in the system at any given time. During the closed glottis period (t/T = 0 to 0.2), flow is completely shut off and there is no mechanical energy exchange. During the period when the glottis is open (t/T = 0.2 to 0.8), the whole glottis is driven by the pressure, this pressure causes the unsteadiness in the flow that pushes the vocal folds to open. The convective acceleration in the glottis has a slightly delayed response to the pressure input, starting at t/T = 0.4. During the glottis closing period (t/T = 0.8 to 1.0), the convective acceleration is almost fully compensated by the flow unsteadiness.

Fig. 9.

Bernoulli terms in the combined region during one oscillation cycle.

5. CONCLUSIONS AND OUTLOOK

In this work, a generalized Bernoulli equation is derived with each term identified with distinct physical meaning. The generalized Bernoulli equation identifies the contributions of total head and the head loss terms, and is used to characterize the airflow in the region upstream of the vocal fold structure whose predominant feature is the contracting channel width and in the region downstream of the structure where the glottal jet is formed. Analysis with Bernoulli’s principle shows that the flow dynamics in the contraction region and the jet region are inherently different. The contraction region, where pressure drop drives air into the tapering channel through the glottis, is characterized with conversion of compressional internal energy to kinetic energy, and the unsteady acceleration term is minor but yet non-negligible. In the jet region, where pressure drop is not as pronounced, is dominated by the convective acceleration head and the unsteady acceleration head loss, which are comparable in magnitude, therefore revealing that the flow in the jet region is inherently unsteady.

It is worth noting that the evaluations shown in the current work are based on numerical simulation with a specific model, therefore serves as a representation. Nonetheless, the equations derived here are generic and derived from first principles of physics, hence is applicable in both 2D and 3D cases, and can be modified to study and analyzevarious coupled flow-structure phenomena and flow-induced vibrations such as wind energy farming applications. Towards this end, it is also important to understand that Bernoulli’s principle only evaluates the flow characteristics along a streamline, it may not represent the entire flow characteristics in the control volume. A control volume analysis would reveal more in detail, which may provide other physical information that is otherwise missed in the Bernoulli’s analysis. The authors are preparing a manuscript to address this issue in a separate paper.