Flow separation in a computational oscillating vocal fold model

Flow separation in a computational oscillating vocal fold model Fariborz Alipoura) Department of Speech Pathology and Audiology, The University of Iowa, Iowa City, Iowa 52242

Ronald C. Scherer Department of Speech Pathology and Audiology, The University of Iowa, Iowa City, Iowa 52242 and Department of Communication Disorders, Bowling Green State University, Bowling Green, Ohio 43403

共Received 1 October 2003; revised 9 June 2004; accepted 15 June 2004兲 A finite-volume computational model that solves the time-dependent glottal airflow within a forced-oscillation model of the glottis was employed to study glottal flow separation. Tracheal input velocity was independently controlled with a sinusoidally varying parabolic velocity profile. Control parameters included flow rate 共Reynolds number兲, oscillation frequency and amplitude of the vocal folds, and the phase difference between the superior and inferior glottal margins. Results for static divergent glottal shapes suggest that velocity increase caused glottal separation to move downstream, but reduction in velocity increase and velocity decrease moved the separation upstream. At the fixed frequency, an increase of amplitude of the glottal walls moved the separation further downstream during glottal closing. Increase of Reynolds number caused the flow separation to move upstream in the glottis. The flow separation cross-sectional ratio ranged from approximately 1.1 to 1.9 共average of 1.47兲 for the divergent shapes. Results suggest that there may be a strong interaction of rate of change of airflow, inertia, and wall movement. Flow separation appeared to be ‘‘delayed’’ during the vibratory cycle, leading to movement of the separation point upstream of the glottal end only after a significant divergent angle was reached, and to persist upstream into the convergent phase of the cycle. © 2004 Acoustical Society of America. 关DOI: 10.1121/1.1779274兴 PACS numbers: 43.70.Aj, 43.70.Bk 关AL兴

I. INTRODUCTION

The mechanics of human phonation involves the coupling of the vibration of the vocal folds with the airflow through the glottis. The pressurized air from the lungs and the biomechanical characteristics of the vocal folds drive the vocal folds into oscillation. The vocal folds modulate the flow of air from the trachea, turning the airflow into cyclic pulsatile jet flow exiting the glottis. During the closing portion of the phonatory cycle when the glottal walls take on a diverging shape, flow separation is predicted to occur somewhere along the glottal walls 共Guo and Scherer, 1993; Alipour et al., 1996a, b; Pelorson et al., 1994; Scherer et al., 2001; Shinwari et al., 2003; Hofmans et al., 2003; Kucinschi et al., in review兲. It is generally accepted that the flow separation point in a divergent duct is a primary factor in the determination of the pressures of that duct. Flow separation also has been found to be a positive factor in the lowering of the phonation threshold pressure that helps make it easier to create self-oscillation of the vocal folds 共Lucero, 1998兲. The theoretical notion of the Bernoulli equation that relates the pressure, area, and volume velocity at each crosssectional area of a physical duct and the quasi-steady approximation of the glottal flow may no longer hold once the flow has separated from the walls. It seems highly likely that the effective glottal 共flow兲 duct, then, becomes nearly uniform for a short distance following the separation point. If this held, then one might assume that the wall pressures past a兲

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the separation point would equal the pressure in the downstream reservoir just past the glottal exit. This is the basic assumption used in some phonatory models 共Titze, 1988; Pelorson et al., 1994; Story and Titze, 1995; Lous et al., 1998兲. However, it is now known that the pressure just downstream of the point of separation may continue to rise until reaching a plateau value 共Senoo and Nishi, 1977; Scherer et al., 2001; Alipour and Scherer, 2002; Shinwari et al., 2003兲. Flow separation from a wall of the glottis takes place when the skin friction or shear stress on the wall approaches zero. The shear stress, a frictional effect exerted by the layers of the fluid on each other, is proportional to the viscosity and the derivative of the flow velocity across the layers 共normal to the wall兲. The pulsatile nature of the glottal flow causes the shear stress distribution to change with time in contrast to that found with constant flow through static models. Since the peak of the shear stress occurs at locations of steep velocity gradients and wall bends, an oscillatory flow with tissue movement may generate high peak values on the walls. When separation occurs, a slightly lowered pressure region is generated near the flow separation point and the air downstream of the separation point between the glottal wall and the jet flow will circulate at a lower rate of flow. This flow separation may influence the driving pressures on the glottal walls. In a static geometry, flow separation can easily be predicted through the solution of the Navier–Stokes equations. It has been reported that Reynolds number, divergence angle, pressure gradient, and flow deceleration are major determinants of the flow separation location 共Senoo and Nishi, 1977; Ojha et al., 1989兲.

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Pulsatile flow with flow separation has received attention in numerous areas, such as in arterial stenosis. For example, the ultrasonic Doppler shift has been used for the analysis of moving flow separation in arterial stenoses 共Tamura and Fronek, 1990; Siouffi et al., 1998兲. However, an accurate measurement of the wall shear stress is very difficult and an estimated error of 20%–50% can be expected for pulsatile flows 共Deplano and Siouffi, 1999兲. Therefore, in general, computational models may aid in determining important aspects of flow separation. Numerical modeling has been used to investigate pulsatile flow in models of heart valves, carotid arterial bifurcation, and arterial stenosis, where data on oscillatory shear stress and flow separation have been reported 共e.g., Perktold et al., 1998; Zhao et al., 2000; Long et al., 2001; Lai et al., 2002兲. Glottal flow is more complicated than blood flow in arterial stenoses. Both the angle and diameter of the glottis continually change during the open portion of the vibratory cycle, and flow oscillates on the order of 100 times faster. Also, glottal walls oscillate with large amplitudes, and the possible glottal closure within each cycle adds further complication to the analysis and prediction of flow separation locations. In terms of nondimensional parameters, the Strouhal number reflects unsteadiness. The Strouhal numbers for phonation are approximately two to ten times larger than for blood flow 共Alipour et al., 1995; Hofmans et al., 2003; Tutty, 1992兲. Pelorson et al. 共1994兲 reported a quasi-steady model of flow separation that allowed for a moving separation point as the vocal folds oscillated. They based their model on a combination of the unsteady Bernoulli equation and the von Karman equation using the Polhausen cubic polynomial approximation 共White, 1974兲. They reported that at the beginning of the closing cycle, the flow separation point moves upstream and during the last part of the closing 共prior to closure兲 moves downstream. Zhao et al. 共2002兲 showed moving separation points in their computational model of forced vocal fold motion, but did not quantify the separation point location. Alipour et al. 共1996a兲 and Alipour and Scherer 共1998兲 used computational fluid dynamics and forced oscillatory glottal models and reported asymmetric flow separation within the glottis, as did Shinwari et al. 共2003兲. In the study here, we present a forced-oscillation glottal model in which the locations of the separation points are calculated by solving the time-dependent Navier–Stokes equations and are tracked for a variety of static and vibratory conditions. The model predicts the location of the separation point as a function of the amplitude of glottal motion, flow rate, and phase difference between the inferior and superior glottal margins, conditions that have not been pursued in earlier studies. The outcome of the study should lend conceptual guidance to the notion of the location of glottal flow separation points for phonation. II. METHODS

A two-dimensional CFD computer code using a pressure-based finite volume method, developed at the Iowa Institute of Hydraulic Research 共University of Iowa兲, was used for the numerical modeling 共acoustic effect was not included兲. Lai and Przekwas 共1994兲, Lai et al. 共1995兲, and J. Acoust. Soc. Am., Vol. 116, No. 3, September 2004

FIG. 1. Design of the oscillating vocal folds and portions of the grids.

Lai and Alipour 共2002兲 have described the details of this numerical model. In this method the computational mesh moves freely to conform to the wall motion and provides a solution for the flow with a moving boundary. An unstructured grid technique was adopted in order to use a multiblock-based mesh to represent the geometry of the glottis. A. Design of the flow channel

The channel height was 2.5 cm with a 5-cm inlet duct length and a 50-cm outlet duct length. The vocal fold model had an equilibrium height of 1.15 cm with independent amplitudes ranging from 0 to 0.9 mm. An upstream symmetric flow condition was assumed and only one-half of the channel was discretized for computation. Thus, the glottis moved symmetrically about the midline, and the flow through the larynx was also symmetric. The computational grid was divided into three blocks. Block 1 共inlet兲 had 21⫻46 grid points, block 2 共glottis兲 had 31⫻26 grid points, and block 3 共outlet兲 had 76⫻46 grid points 共see Fig. 1兲. B. Design of the oscillating glottal walls and the specification of the tracheal flow

The glottal surface was modeled with two sinusoidal curves connected with a tangent straight line. The heights H i and H s defined the inferior and superior glottal openings, respectively 共Fig. 1兲. The glottal walls oscillated 共were ‘‘forced’’兲 sinusoidally at the inferior and superior margins. The phase between the two locations and the amplitude of each location were controlled. The time variations of the inferior and superior glottal heights were H i ⫽H 0 ⫹A i cos共 2 ␲ f t⫹ ␾ 兲 ,

共1兲

H s ⫽H 0 ⫹A s cos共 2 ␲ f t 兲 ,

共2兲

where H 0 was the equilibrium height value of 1.15 cm from the lateral wall for both the inferior and superior glottal locations, A i and A s were the amplitudes of oscillation at the inferior and superior glottal openings, respectively, t was time, f was the frequency of oscillation, and ␾ was the phase lead of the inferior opening relative to the superior opening. At time zero, the glottal diameter of the superior margin was smallest. There was no completely ‘‘closed’’ portion to the vibratory cycle.

F. Alipour and R. C. Scherer: Flow separation in computational glottis

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The flow was not governed by a constant transglottal pressure, but was independently controlled. The flow was synchronized in a sinusoidal fashion with the motion of the glottal walls. The inlet 共tracheal兲 mean velocity (U T ) at the X⫽0 location was set such that at the beginning of the cycle, which occurred when the superior margin was at its minimal diameter, the velocity was zero. The inlet velocity was in phase with the diameter of the upper glottal margin, viz., U T ⫽0.5U c 共 1⫺cos 2 ␲ f t 兲 ,

共3兲

where U c was the reference maximum center velocity derived from the prescribed Reynolds number, and Re ⫽UcHT /␯, where H T was the upstream height and ␯ was the kinematic viscosity. C. Basic simulation procedure

For a particular Re value, U c was determined and A i , A s , and ␾ were chosen. This permitted the calculation of the dynamic H i and H s values, as well as the flow U T , and the calculation of the pressure and velocity fields for the first time step. At every time step of the simulation, the new glottal wall position was calculated and a new mesh was generated from the previous mesh. The pressure and velocity fields of the previous time step were used as an initial guess in the iteration method. Computations were performed on a Silicon Graphics workstation. A converged solution provided the velocity and pressure distribution in the computational domain for each time interval. The distributions were stored on a computer disk for later analysis. The wall shear stress for every time step was calculated from the velocity distributions. Using a polynomial interpolation technique within the MATLAB package, the location of the flow separation point was calculated for each time step. The separation point was that axial location where the velocity gradient was zero (du/dy⫽0) 关MRS criterion discussed in Schlichting and Gersten 共2000兲兴, whereby for moving walls the additional criterion u⫽0 is imposed, yielded changes of the separation points within the distance between nodes, and therefore was deemed insignificant for this study. III. RESULTS A. Basic characteristics of the mean cross-sectional velocities and transglottal pressure

An example of the kinds of signals obtained from this glottal model is shown in Fig. 2共a兲, and a sequence of glottal shape change is shown in Fig. 2共b兲. This figure shows two cycles of the glottal inlet mean velocity, glottal outlet mean velocity, glottal outlet volume velocity, tracheal velocity, transglottal pressure, and the glottal angle 共positive values for glottal convergence, negative for divergence兲. Two phase values, 60° and 90°, with the inferior glottis leading the superior, are shown, while other parameters were held constant 共Reynolds number Re⫽1000, fundamental frequency Fo⫽100 Hz, and amplitudes of A i ⫽0.75 mm, A s ⫽0.90 mm). The patterns seen in Fig. 2 resulted from the interaction of the prescribed flow and wall motion. The mean crosssectional inlet and outlet velocities, as well as the outlet vol-

FIG. 2. 共a兲 Glottal and tracheal velocities, angle, and transglottal pressure for the phase lag values of 60° and 90°, and Re⫽1000, Fo⫽100 Hz, A i ⫽0.75 mm, A s ⫽0.90 mm. 共b兲 One cycle of motion of the vocal folds for the 60° phase case. The minimal diameter is 0.2 mm. The time between each frame number is 0.5 ms. Pressure units are in cm H2 O 共1 cm H2 O ⫽98.1 Pa).

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ume velocities, are skewed to the right relative to the tracheal velocity. This skewing is consistent with human volume velocity signals. The skewing here appears to be due to the changing shape of the glottis and related to the resulting displacement flows. The maximum glottal velocity peaks occur while both inferior and superior glottal margins are moving medially, and the glottal angle is divergent, even though the tracheal velocity is decreasing. The inward wall movement creates a displacement flow in the downstream direction that increases the glottal velocities and delays the peak glottal velocities relative to the tracheal velocity. We note that the volume velocity through the glottis and the glottal inlet velocity primarily have positive 共downstream兲 going flow except for a portion of the cycle during which the glottis is opening. The reversal in direction of the flow 共toward upstream兲 is due to the outward movement of the glottis 共especially the inferior section兲 and the corresponding decreasing 共and independent兲 tracheal flow. The outward wall motion, decreasing upstream flow, and incompressibility also create negative transglottal pressures; the absence of a dc flow prevents the maintenance of a positive subglottal pressure. The greatest effect of these conditions is on the glottal outlet velocity, where relatively large negative velocities are seen during glottal opening. It is noted that in the normal larynx, the flow decreases because of greater glottal flow resistance as the glottal diameter reduces during the glottal closing gesture, but here the flow is independently reduced 共and has no dc component兲. This model does not mimic certain features of normal phonation, such as the usual lack of negative volume flows from the glottis and the modulated but positive transglottal pressure 共e.g., see Cranen and Boves, 1985兲, but this model should tend to emphasize the effect of the wall movement on the flow behavior, and will therefore be highly sensitive to the creation of flow separation from the glottal walls. It is assumed that the range of flow separation locations may therefore be greater than expected in normal phonation, and thus this experiment should ‘‘bracket’’ that expected range. Figure 2 also indicates the effect of changing the phase of motion on the inferior and superior portions of the glottis. This effect is to shift the peak of the mean and volume velocities more to the right 共later in time兲 for the 60° phase condition compared to the 90° phase due to timing of the upper and lower diameters.

FIG. 3. Velocity vectors within a divergent glottis for Re⫽1000, Fo ⫽100 Hz, A i ⫽0.0 mm, A s ⫽0.90 mm, and ␾ ⫽60°. The length of the small arrows indicates the magnitude of the velocities. Velocities that are not shown near the vocal fold surface are negative and point upstream. The dashed line represents an approximation to the effective ‘‘flow tube.’’

plotting routine to show them. The line in the figure designates where the velocity was zero, and is the lateral ‘‘wall’’ or boundary of the ‘‘flow tube.’’ Figure 4 shows a few selected velocity profiles from Fig. 3 in an overlaid format with the grid numbers that refer to their locations in Fig. 3. At grid number 11, which is at the minimum glottal area, the velocity has the largest peak value and the largest velocity gradient on the wall ( ␦ u/ ␦ y) 共and consequently the wall shear stress has a large positive value, see below兲. Further downstream, this velocity gradient on the wall decreases and eventually passes through zero and becomes negative. Once the shear stress becomes negative, the flow reverses. Profiles 17, 20, and 24 show reverse flow below a certain height 共negative velocity vectors兲. Flow separation occurs at the location of the zero shear value.

B. Velocity profiles

The velocity profile across the glottis at any cross section will change as the glottis moves and as the tracheal input velocity changes. Figure 3 shows a snapshot of velocity profiles within the glottis for the nominal conditions (Re ⫽1000, Fo⫽100 Hz, A i ⫽0.0 mm, A s ⫽0.90 mm, and ␾ ⫽60°) and a divergent glottal shape. The velocity magnitudes increase at the minimum cross-sectional area. The velocity vectors are almost flat for most of the channel and vanish on the wall due to the no slip condition. However, downstream of approximately cross section 15, the velocity vectors do not appear for a certain distance away from the wall. The velocities that are not shown are negative and point upstream, and do not appear because of the inability of the J. Acoust. Soc. Am., Vol. 116, No. 3, September 2004

FIG. 4. Glottal velocity profiles at a few selected x-grid locations 共from Fig. 3兲.


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FIG. 5. Variations of wall pressures 共Pw兲 and center pressures 共Pc兲 along with the wall shear stress in a portion of the modeled glottis with flow separation. Also shown is an outline of the glottal surface and the separation point. The conditions are the same as in Figs. 3 and 4. Pressure units are in cm H2 O 共1 cm H2 O⫽98.1 Pa).

Figure 5 demonstrates the variation of centerline pressure, wall pressure, and wall shear stress near a small portion of the glottal wall in the region of the minimum diameter for the same case as in Figs. 3 and 4. The shear stress has a large peak value at the location of the maximum curvature. The wall and centerline pressures are almost identical everywhere except in the region of high shear stress. The vertical line at zero shear stress intercepts the curves at the flow separation location 共approximately X⫽6.1). After the separation point, the discrepancy between these two pressures disappears 共and is shown to rise slightly, not stay constant兲. This discussion introduces the topic of flow separation. The following section will show how the separation point changes as phonatory parameters are altered in this model.

FIG. 6. Effect of the glottal angle on separation point locations: 共a兲 inferior glottis is fixed and superior margin oscillate, and 共b兲 superior glottis is fixed and inferior margin oscillates. Otherwise, the conditions were nominal (Fo ⫽100 Hz, H o ⫽1.15 cm, A i ⫽0.75 mm, A s ⫽0.90 mm, and phasing of 60° between the lower and upper glottal margin兲.

In this section, locations of the flow separation point for divergent glottal shapes are studied relative to motion of the inferior glottis, the superior glottis, and Reynolds number. For the following discussion, the nominal phonatory condition was Fo⫽100 Hz, H 0 ⫽1.15 cm, A i ⫽0.75 mm, A s ⫽0.90 mm, and phasing of 60° between the lower and upper glottal margin. Figure 6 shows the effects of glottal angle on flow separation location. In the upper graph 共a兲 the inferior margin is fixed and the superior margin oscillates with an amplitude of 0.9 mm. The lower graph 共b兲 shows the inferior margins oscillating with an amplitude of 0.75 mm with the superior margin fixed. The separation point was tracked over the duration of a phonatory cycle as the inferior or superior amplitude was varied, other parameters having the default values. The symbols indicate the separation point locations for each instant of the oscillation. In both graphs, the glottal angle

was a major determinant of the separation point locations. As this angle increased, the separation point moved upstream after a relatively small angle was surpassed. To show the effect of glottal wall amplitude, a case of a divergent glottis with a constant 5° half angle was modeled. The constant angle means that the inferior and superior margins were moved as a unit 共zero phase difference between the upper and lower margins兲, preserving the angle, but allowing glottal motion. In Fig. 7 three panels are displayed. The top panel 共a兲 shows glottal inlet mean velocities for two oscillation amplitudes. The solid curve shows the inlet mean velocity for the static glottis, the flow being symmetric in time. The increase of the amplitude to 0.25 mm 共dashed curve兲 creates a skewing in the flow due to the wall motion 共motion inward displaces flow in the downstream direction, skewing the flow兲. The second panel 共b兲 shows flow separation points for the static glottis at time instants of 15, 16, 17, and 18 ms. These separation points are on the decreasing portion of the flow, as can be seen from the first panel. The second panel indicates that for this static divergence, greater flow yielded separation points more downstream in the glottis, the opposite of the usual finding for the static divergent glottis 共Shinwari et al., 2003兲, thus suggesting that changing flow rates may yield different results than for constant flow rates for a static glottis. The third panel of Fig. 7共c兲 includes four solid curves,

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C. Flow separation locations


FIG. 7. Effects of the oscillation amplitude on the flow separation location in a 5° half-angle divergent glottis; 共a兲 one cycle of glottal inlet velocities, 共b兲 flow separation points for the static glottis, and 共c兲 flow separation points for a 0.25 mm amplitude. ‘‘SP’’ stands for the separation point character, and ‘‘time’’ refers to the time of the first panel. Note that the times in panel 共c兲 have been augmented by 1 ms to show the similar range of values as in 共b兲.

which display wall locations and separation points at time instants of 16, 17, 18, and 19 ms. A comparison of the times and separation points for the second and third panels indicates that the inward motion of the vocal folds moves the separation point downstream 共while still preserving the finding that lower flows yield more upstream separation locations兲. The effect of increasing flow 共Reynolds number兲 on flow separation is shown in Fig. 8. The top panel 共a兲 shows one cycle of the glottal inlet mean velocities for Reynolds numbers of 500 and 1000. The amplitude of 0.5 mm 共again for the zero phase glottis of 5° half angle兲 is associated with more skewing for the smaller Reynolds number 共solid curve兲. The middle panel 共b兲 shows three frames of wall motion at Reynolds number of 500 and their corresponding separation points for time instants of 17, 18, and 19 ms. Similarly, three wall positions and the separation points for the Reynolds number 1000 are shown in the third panel 共c兲. It appears that as the Reynolds number increases, the flow separation locations move upstream, which is the typical expectation. The amount of the shift is not the same for each of the moments in time, however; the greatest movement was for the separation point at 18 ms, the separation point not near the glottal end points. Once the flow separates, the ‘‘flow tube,’’ that is, the effective spatial region within which the air flows through the glottis, has a larger cross-sectional area than the miniJ. Acoust. Soc. Am., Vol. 116, No. 3, September 2004

FIG. 8. Effects of the Reynolds number on the flow separation locations in a 5° half-angle divergent glottis: 共a兲 one cycle of glottal inlet velocities, 共b兲 flow separation points for Re⫽500, and 共c兲 flow separation points for Re ⫽1000.

mum glottal area 共the line cutting through the velocity profiles in Fig. 3 connects points where the flow velocity equals zero and gives the approximate effective boundary of the ‘‘flow tube’’兲. The ratio of the glottal cross-sectional area at the flow separation location and the minimal glottal crosssectional area, called the separation cross-sectional ratio, is plotted in Fig. 9共a兲 for the 5° 共zero phase兲 divergent glottal model for amplitudes of 0, 0.25, 0.50, and 0.75 mm. There is a trend for increasing ratio values from about 1–1.2 to approximately 1.5–2.0 in the first 25% of the cycle. This is the portion of the cycle in which the tracheal flow is accelerating 共up to the inflection point of the velocity curve兲. From 25% to 50% of the cycle, the ratio value decreases to about 1.4 – 1.5 or so. This phase of the tracheal flow is when it is decreasing its rate of increase toward its maximum flow, suggesting that this slowing of the velocity increase creates a condition in which the static or outward wall movement pulls the separation point upstream. From 50% to 80%, the ratio value takes different paths depending upon the amplitude. For the zero amplitude 共static兲 case, as shown in Fig. 7, the value of ratio continues to decrease as the flow decreases, meaning that the separation point continues to move upstream on the wall. When the amplitude is nonzero, the wall motion inward deflects 共squeezes兲 the flow, pushing the separation point more downstream, the largest effect being shown by the largest amplitude 共0.75 mm兲, moving the crosssectional value to approximately 2. From about 80% to 100%, the tracheal flow goes to zero, and the amplitude cases show a fast drop of the ratio value toward 1.0. Figure


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for divergent glottal angles only. The range of values shown, approximately 1.1 to 1.9, is representative of the data from this study. The overall average was 1.47 (sd⫽0.23), and is slightly larger than adopted in the voice literature 关for example, 1.3 by Lucero 共1999兲兴. Figure 10共b兲 shows glottal half-angle versus the percent of the cycle for the four cases of oscillation of Fig. 10共a兲. Thus, the beginning of the cycle is at 0%, and ends at 100%. The figure indicates that the different cases have different beginning glottal angles, and different angles at each percentage of the cycle. Figure 10共c兲 gives the separation locations versus the glottal angle for the cases in Fig. 10共b兲. Figures 10共a兲–共c兲 are to be considered in establishing further insight into the dynamic change of the separation point locations. These figures indicate the sensitivity of this model to both the flow and wall motion, and yield results that suggest delays in the movement of the separation point relative to what might be expected from steady flow in diffusers. For example, for the four cases in Fig. 10共c兲, three of them have their cycles start with a convergent shape, but with separation points upstream of the glottal exit 共6.5 cm兲 location 共note that the effective glottal entrance is at 6.0 cm兲. The glottal angle at which flow separation begins to move upstream, away from the exit of the glottis, varies, not happening until the angle reaches a divergent 6°, 3.5°, and 8° half-angle for cases 1, 3, and 4, respectively, and essentially never for case 2, where separation only occurs for nearly uniform and convergent shaping near the end of the cycle. Furthermore, cases 2 and 4 end their cycle with a convergent angle but with separation still upstream of the glottal exit. These findings that movement of the separation point upstream away from the exit begins to occur well into the divergent angles, and that upstream separation can persist into the convergent angle portion of the cycle, strongly suggests inertial forces may act to retard the effects of the flow after the wall movement has occurred. These findings need further, more complete investigation with more realistic models of phonation. IV. DISCUSSION

9共b兲 shows the separation locations of the same data shown in Fig. 9共a兲. For the static condition 共zero amplitude兲, the flow separation location is governed by the flow rate as shown in Fig. 7共a兲. After a period of stationary condition, the deceleration portion of the cycle causes the separation point to move steadily upstream. Increase of the amplitude makes the stationary period longer. Separation phenomena are shown for a variety of amplitudes in Fig. 10. Figure 10共a兲 is a series of four cases for which the Reynolds number was 1000, frequency was 100 Hz, and the amplitudes of the inferior and superior glottal margins were varied as shown on the graph. The values are

Because this CFD forced dynamic model of phonation controls tracheal flow and wall motion, rather than lung pressure, it may be highly sensitive regarding the resulting locations for the flow separation points in the glottis. The results indicate that increasing the Reynolds number 共or flow兲 tends to move the separation points upstream, as predicted by empirical studies of diffusers and steady flows. However, surprisingly, the direction of movement of flow separation locations appears to be opposite to expectations 共or confounded兲 regarding the dynamic flow changes during a cycle of glottal oscillation. This is shown in a rather pure sense for the divergent glottis, static and moving, with zero phase between the inferior and superior margins, Fig. 9共a兲. For this condition, while the flow rate increases over the first 25% of the cycle, the separation point moves downstream, which is opposite to steady flow expectations 共which would be movement of the separation point upstream兲. After the tracheal flow moves past its inflection point 25% into the cycle, the separation points move upstream during static and outward

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FIG. 9. Effects of amplitude on 共a兲 the separation cross section ratio and 共b兲 the flow separation point location in a 5° half-angle divergent model with zero phase lead between inferior and superior glottal margins. The tracheal flow and glottal excursion reach their maximum when the cycle is at 50%.


FIG. 10. 共a兲 Separation cross-sectional ratio for the divergent angles for four cases of differing inferior, superior amplitudes. 共b兲 Glottal half-angle for the same four cases of differing inferior, superior amplitudes. 共c兲 Location of flow separation points 共separation location relative to Fig. 1 x axis兲 relative to glottal half-angle for the four cases. The location 6.0 cm is at the glottal entrance 共A in Fig. 1兲, and location 6.5 cm is the effective glottal exit 共B in Fig. 1兲.

movement of the glottis 共when the flow is still increasing兲, which is consistent with steady flow expectations, but perhaps not because the flow is increasing, but because the rate of increase is decreasing. Past the 50% cycle point, when the flow begins to decrease, for the static duct case, the separation point continues to move upstream, contrary to steady flow expectations. However, when there is wall movement, Figs. 9共a兲 and b suggest a relatively strong influence of the wall movement: the larger the amplitude 共and therefore the greater the rate of movement of the glottal walls inward兲, the J. Acoust. Soc. Am., Vol. 116, No. 3, September 2004

more the separation points are moved in the downstream direction because of the ‘‘squeeze’’ given to the airflow. These results suggest a strong interplay between the rate of change of the flow through the glottis 共relative to the 100-Hz period兲 and the motion of the walls. The phenomenon of having the separation point move further downstream in the glottal duct as the velocity increases, and then having a change toward upstream near the inflection point during the velocity increase, is consistent with the phenomenon of the delay in creating the Coanda or


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skewing effect in the glottis for accelerated flows as shown by Hofmans et al. 共2003兲. That is, their data show that crosssectional pressure differences, which would reflect a change in status of the separation points in the glottis, occur approximately at or just after the velocity 共or tracheal pressure兲 increase inflection point, similar to the computational findings here for the static divergent glottis with unsteady flow. Because of the interest in the cross-sectional ratio between the glottal duct diameter 共area兲 at the separation location and the minimum glottal diameter 共area兲, the ratio was mapped for a wide range of conditions 共Fig. 9兲. Those data indicate that the range of this ratio value was from approximately 1.1 to 1.9, with an average of 1.47 (sd⫽0.27). This number does, however, vary with condition. The findings for the cases with dynamic glottal angle and diameter suggest that the separation point location is a delayed phenomenon relative to the motion of the vocal folds; when divergent angles begin, the flow may stay attached at the glottal exit, to begin to move up the divergent glottis for larger divergent angles, and then retain separation upstream of the glottis exit even as the glottis takes on the convergent shape. Pelorson et al. 共1994兲 also show changing separation point locations over a wide range of the glottal duct, but their two-mass model results can not be directly compared here because of the lack of glottal angle information. Even though the computational model presented here may exaggerate this phenomenon of separation point movement due to strong inertial influence, this needs to be hypothesized to exist in more realistic flow-induced oscillation biophysical model. The shear stress along the glottal walls is strongest at the location of the greatest curvature of the glottis, which would be at the entrance of the glottis for the time when the glottis has a divergent shape. This stress would act tangentially on the vocal folds in addition to the normal air pressures, and thus would add to the forces tending to move the tissue. Although the shear stresses are relatively small, they need to be considered in a more complete modeling process.

共4兲共5兲共6兲

共7兲

Reynolds number of the upstream flow兲 tends to move flow separation points further upstream, consistent with steady flow diffusers. Larger divergent glottal angles tend to move the separation point upstream, consistent with steady flow diffusers. Inward motion of the divergent glottal walls tends to move the separation point downstream 共via ‘‘squeezing’’ the flow兲 and skew the peak velocities. Mean velocity increase tends to move the separation point downstream in the diverging glottis, and mean velocity decrease can move the separation point upstream, contrary to steady flow diffuser behavior. The rapid change of airflow through the glottis appears to have a strong influence on the flow separation locations, most likely due to an inertial effect of the air. There appears to be a significant delay in establishing separation point locations during vocal fold motion with changing glottal angle, compared to the separation points expected during steady flow, static diffuser conditions.

Results here suggest that the rate of change of the flow may be important, as well as whether the flow is increasing or decreasing. These aspects need to be studied in more detail. Specifically, what conditions create 共and control兲 reversals of separation location movement relative to expectations from steady flow diffuser results? This model did not include the influence of the airflow 共air pressure兲 on the tissue per se, and therefore a parametric study of flow separation within a fluid-surface interaction model is pertinent. ACKNOWLEDGMENTS

The authors would like to thank Dr. Yong G. Lai and Iowa Institute of Hydraulic Research for assistance in adapting their CFD code for this project. This work was supported by NIDCD Grant No. DC03566-05.

共1兲 The separation cross-sectional ratio varied from about 1.1 to 1.9 in this study, with an average of 1.47. 共2兲 Decreasing the inferior-superior phase from 90° to 60° tends to delay the flow and velocity peaks more 共greater skewing to the right兲. 共3兲 Increasing the average flow rate 共in this study, the peak

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V. CONCLUSIONS

This study used a computational fluid dynamics model of phonation with the special characteristics of forced dynamic motion of the vocal folds and a prescribed tracheal flow that varied sinusoidally. The focus on the location of flow separation points during divergent glottal shaping was justified relative to the common application of their location and influence on designating pressures in the glottis. Thus, a general study of the influences on the separation point movement might lend further insight into phonation modeling requirements. Results suggest the following.


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