5.1. Frequency and Reynolds number dependence

At the outset, it is worth commenting on the frequency range covered in this study. The actual range of frequencies examined was just under a factor of two, 52.5 Hz ≤ f_life ≤ 97.5 Hz. Since the maximum frequency, f_life = 97.5 Hz lies within the fundamental frequency range of adult males, this study has physiological relevance in its own right. Note, however, that the lowest Reynolds number case corresponds to f* = 0.0511, which is almost twice the f* = 0.0261 value of the highest-frequency case at Re = 7200. As such, this study provides insights into frequencies beyond the nominal range examined.

Frequency dependencies of phase-averaged kinematic and dynamic quantities have been highlighted throughout the preceding section. It was shown in § 4.2 that as frequency increases, flow accelerates through the glottis after the vocal folds start to close. This acceleration appears as an increasingly pronounced hump in the maximum jet velocity around t/2T_o = 0.3 in figure 5, and a leaning of the volumetric flow time traces to the right with increasing f* in figure 6. In the phase-averaged pressure traces in § 4.3, figures 7 and 8, supraglottal pressure minima decrease in magnitude with increasing frequency and the location of the peaks shifts toward t/2T_o = 0.5, i.e. when the vocal folds close.

In terms of the dynamics, it was shown in figures 9 and 10 in § 4.4 that the transglottal pressure force and the vocal-fold drag increases in magnitude with increasing frequency. The maximum streamwise momentum flux, presented in figure 11, decreased and occurred later in the cycle with increasing frequency; this was correlated with maxima in volume flow rate. Finally, inertia effects became more pronounced with increasing f*, specifically when the vocal folds first started opening and when they shut. However, even at the highest f*, the magnitude of the inertia forces was an order of magnitude smaller than the vocal-fold drag.

Discerning Reynolds number effects in this study is less straightforward because varying flow speed affects both Re and f* even when the oscillation frequency is held constant. From a phase-averaged perspective, frequency effects appear to dominate Reynolds number effects. For example, for a fixed f*, one would expect that transglottal pressure would increase with increasing Reynolds number because transglottal pressure increases with flow speed. However, figure 9(b) shows the opposite; the effects of increasing flow speed are much more consistent with f* trends observed in figure 9(a) than they are with what one would expect to see if Reynolds number were truly an independent variable. Still, Reynolds number effects become more evident when looking at cycle-to-cycle variations and will be examined in greater detail in § 5.3.

5.2. The dynamics of glottal jets

A key contribution of this paper to the human phonation literature is the ability to examine the entire momentum balance across the complete oscillation cycle. In this section, focus is on the phase-averaged balance. Cycle-to-cycle variations are addressed in the following sub-section.

To begin, it is important to revisit the foundational premise of this paper. As discussed in § 1, the goal of this study is to identify the flow features that contribute most to phonatory sound production. Because vocal-fold drag has been identified as the source of sound, these features may be ranked in terms of how they contribute to vocal-fold drag. For this, the reader is referred back to the control volume shown in figure 2 containing the fluid moving between the vocal folds.

For this analysis, the first term in (1.1) is reformulated using mass conservation, to break out the bulk unsteady acceleration and acceleration due to wall motion, and the pressure force integral into driving pressure force and vocal-fold drag, yielding

(5.1)

Here, V_xj is the glottal volume between axial location x and x_d, the axial location of the control-volume outflow face. The orders of magnitude of these terms are

(5.2a)\begin{gather}{\rho _0}\frac{\textrm{d}}{{\textrm{d}t}}\int_{{x_u}}^{{x_d}} {{u_j}{S_j}\,\textrm{d}x} \sim {\rho _0}\frac{{{u_j}{h_{max}}{L_G}}}{{{T_a}}}L,\end{gather}

(5.2b)\begin{gather}{\rho _0}\frac{\textrm{d}}{{\textrm{d}t}}\int_{{x_u}}^{{x_d}} {{{\dot{V}}_{xd}}\,\textrm{d}x} \sim {\rho _0}{L^2}{L_G}\frac{{{h_{max}}}}{{T_o^2}},\end{gather}

(5.2c)

(5.2d)\begin{gather}({P_u} - P)S\sim \Delta PS\sim {\rho _0}u_j^2S,\end{gather}

(5.2e)

where T_a is a time scale describing the duration of the sharp rise at the beginning of the cycle, and the rapid fall of jet velocity, u_j, at the control-volume exit, as seen in figure 5. In addition, L is the streamwise extent of the laryngeal control volume, L_G is the depth into the page of the glottis, h_max is the maximum glottal width and S_j ~ h_maxL_G is the cross-sectional area of the jet at the control-volume exit face. Following Krane & Wei (Reference Krane and Wei2006)

(5.3)\begin{equation}{\tau _{ij}} \sim {\rho _0}u_j^2{\left( {\frac{{{u_j}L}}{\nu }} \right)^{ - 1/2}}.\end{equation}

Note that a scaling for vocal-fold drag, D, has not been chosen. Rather, this is to be a result from this analysis. From Krane & Wei (Reference Krane and Wei2006) and Krane et al. (Reference Krane, Barry and Wei2010), the Bernoulli's equation may be scaled using $\Delta PS\sim {r_o}u_j^2S$.

These estimates then yield ratios of the terms in (5.1)

(5.4a)\begin{gather}\frac{\displaystyle{{\rho _0}\frac{\textrm{d}}{{\textrm{d}t}}\int_{{x_u}}^{{x_d}} {{\dot{V}}_{xd}}\,\textrm{d}x}}{\displaystyle{{\rho _0}\frac{\textrm{d}}{{\textrm{d}t}}\int_{{x_u}}^{{x_d}} {u_j}{S_j}\,\textrm{d}x\; \; }}\sim {f^\ast }\frac{{{T_a}}}{{{T_o}}}\sim \frac{{{Q_w}}}{{{Q_j}}}\frac{{{T_a}}}{{{T_o}}},\end{gather}

(5.4b)

(5.4c)

(5.4d)

(5.4e)

(5.4f)

where ${v_W}\sim h/{T_o}$ is the vocal-fold wall velocity scale. The most important takeaway from this analysis is that the acceleration terms are at least an order of magnitude smaller than the driving pressure force. From (

) and (

), it is clear that the bulk unsteady acceleration and wall acceleration terms are smaller than the convective acceleration by factors of f*T_o/T_a and f*², respectively. From figure 5, T_a/2T_o ≈ 0.15, so T_o/T_a ≈ 3.3. This analysis, (5.4a)–(

) also shows that the ratio of Q_w, the volume flow displaced by vocal-fold motion, to Q_j, the glottal volume throughflow, is proportional to the reduced frequency of vibration, f*, consistent with the results of Deverge et al. (Reference Deverge, Pelorson, Vilain, Lagrée, Chentoug, Willems and Hirschberg2003) and Krane & Wei (Reference Krane and Wei2006), where ${Q_w}\sim {v_W}{L_G}L$ was used. The convective acceleration is smaller than the driving pressure force by a factor of S_j/S ~ O(10⁻¹), as shown in (

).

The friction force is also smaller than convective acceleration by a factor of $Re_L^{ - 1/2}(L/{h_{max}})$. Thus, while $Re_L^{ - 1/2}\sim O({10^{ - 1}})$ for the flows studied here, the friction force is in general much smaller than the convective acceleration, but can exceed it for small h, i.e. when the glottis is nearly closed. Thus, when the glottis is open, the leading term after the driving pressure force appears to be the convective acceleration, followed by the unsteady acceleration, and the friction force, in descending order. When the vocal folds are nearly closed, however, the convective acceleration is smaller, and the unsteady acceleration and friction can compete with it for predominance among the second-order terms, as shown by (5.4f). As a result, the integral momentum equation reduces to

(5.5)\begin{equation}D \approx ({p_u}-{p_d})S.\end{equation}

This is consistent with estimates from measurement shown in figure 13. More about the relative sizes of the unsteady and convective acceleration terms is discussed in § 5.4.

Figure 13.

Time traces of the phase-averaged integral streamwise integral momentum balance for (a) Re = 7200; f* = 0.0141, (b) Re = 7200; f* = 0.0261 and (c) Re = 3650; f* = 0.0511. Dotted vertical lines indicate the times, t/2T_o = 0.25 and 0.5, when the vocal folds are fully opened and when they return to fully closed.

Traces of four of the five terms in (1.1), transglottal pressure force, vocal-fold drag, unsteady inertia force and net momentum flux, are shown in figure 13 for three cases: the lowest frequency, Re = 7200 and f_life = 52.5 Hz, lowest Reynolds number, Re = 3650 and f_life = 97.5 Hz, and the case common to the variable frequency and variable flow speed studies, Re = 7200 and f_life = 97.5 Hz. Based on the discussion in § 5.1, these three cases were selected because they include the extremes of the f* parameter range of this study. It was shown in § 4.4 that the viscous shear forces are negligible. Also note that the signs of each term are consistent with the convention that a positive force points in the positive x-direction.

One of the most prominent and important features of figure 13 is the correlation between the transglottal driving pressure force and vocal-fold drag, already highlighted in the presentation of figures 9 and 10, and (5.5). The salient point is that transglottal pressure force can indeed serve as a surrogate for the vocal-fold drag. This directly supports the foundational work of Zhang et al. (Reference Zhang, Mongeau and Frankel2002a,Reference Zhang, Mongeau and Frankelb), Krane (Reference Krane2005), Howe & McGowan (Reference Howe and McGowan2007) and McPhail et al. (Reference McPhail, Campo and Krane2019). Closer examination of those two particular terms, however, reveals subtle differences. For example, the magnitudes of vocal-fold drag tend to be smaller than the driving pressure force. Indeed, (5.5) suggests that the driving pressure force is equal to the drag plus small corrections, due to the acceleration terms, which do not exceed order of magnitude $\Delta P{S_j} \ll \Delta PS$. In addition, the vocal-fold drag traces seem to be slightly more rounded than the pressure force traces. This will be explored further in subsequent paragraphs.

The dependence of the momentum balance on reduced frequency, highlighted in (5.4), is evident in the three plots comprising figure 13. As shown earlier, with increasing f*, the jet accelerates after the vocal folds reverse motion and start to close. This gives rise to an increasing transglottal pressure force accompanied by increasing vocal-fold drag. In the highest f* case, figure 13(c), the transglottal pressure and vocal-fold drag waveforms have much more defined peaks and much larger amplitudes, reaching their largest magnitudes around t/2T_o = 0.5, corresponding to when the vocal folds close.

It was pointed out in § 1.3 that it is problematic to attempt to extrapolate an entire dynamic model around the observation of a single kinematic phenomenon. This is where the simultaneous measures of kinematics, e.g. figures 4–6, and dynamics, figure 13 are so important. From this comprehensive perspective, it can be seen that the effect of f* on transglottal pressure force is due to the acceleration of the jet as the vocal folds begin to close; this has already been proposed in § 4.3. This would become more pronounced at f* increases because there would be comparatively less time for a quasi-steady jet to form after the folds open and before they begin to close. This is supported by the fact that the unsteady inertia term, although still small, becomes more significant with increasing f* while the momentum flux term decreases with increasing f*.

It is interesting to observe that the inertia term adds to the momentum flux while the vocal folds are opening and subtracts while the vocal folds close. This is simply because the flow accelerates when the vocal folds open and decelerates when they close. The important point is that the unsteady inertia does not cancel the momentum flux. That is, the current experiments show that the left-hand side of the integral momentum equation is not, in fact, zero for glottal jet flows.

To examine this point further, time traces of the total acceleration (i.e. left-hand side of the streamwise integral momentum equation, or the sum of the unsteady inertia and momentum flux terms) were plotted along with the net force acting on the control volume (i.e. the sum of the driving transglottal pressure force and vocal-fold drag, or the right-hand side of the momentum equation). A sample plot for the Re = 7200 and lowest-frequency, f_life = 52.5 Hz, case is shown in figure 14; this was computed from the data shown in figure 13(a).

Figure 14.

Comparison of total acceleration (unsteady inertia plus momentum flux) with the net streamwise force (driving pressure plus vocal-fold drag) for Re = 7200 and f* = 0.0141. Dotted vertical lines indicate the times, t/2T_o = 0.25 and 0.5, when the vocal folds are fully opened and when they return to fully closed.

Before exploring the implications of this plot, it is important to recognize the dc difference of 0.1 between the driving transglottal pressure force and the vocal-fold drag. From a broader perspective, this difference is actually relatively small as the two terms have maximum magnitudes of the order of 1.5. Further note that there are only two pressure taps positioned within the curved portion of the vocal-fold models, at (x − x_u)/(x_d − x_u) = 0.25 and 0.75. As such, one can expect that the contribution to drag in that part of the model may not be fully resolved; this can account for the failure of the two force terms to cancel when the vocal folds are completely closed.

The key point of figure 14, then, is the alignment of total acceleration and net force. Both curves reach a maximum around the time when the vocal folds are fully opened and both have a maximum value above 0.2. Because these values are comparatively small, the conclusion that the vocal-fold drag serves as a surrogate for the transglottal pressure force remains unchanged. This is, however, a new and interesting finding in the context of understanding the detailed dynamics, which includes both pressure and momentum. Because the source of voiced sound, i.e. vocal-fold drag, is the sum of the transglottal pressure force, momentum flux and unsteady inertia, the current measurements indicate that momentum flux and inertia may be large enough to affect sound quality and perception. While subjective parameters such as quality and perception lie beyond the scope and capability of the current work, a fluid dynamics framework for examining these ideas is developed further in the following sub-section.

5.3. Cycle-to-cycle variations

In an earlier work, Sherman et al. (Reference Sherman, Lambert, Krane and Wei2019) examined cycle-to-cycle variations in a duct with an oscillating constriction over a range, 0.012 ≤ f* ≤ 0.048 at Re = 8000. In that study, cycle-to-cycle variations in jet direction, volume flow rate as well as integral momentum equation terms were observed. Variations were most pronounced for f* = 0.036 where the jet switched directions almost every cycle. Further, it appeared that every time the jet turned to the left, the volume flow rate was approximately twice that as when the jet turned to the right.

Informed by this previous study and the seemingly bifurcating phase-averaged glottal jet in figure 4, was clear that cycle-to-cycle variations were likely prominent in this flow as well. That this is the case is seen in figure 15 which contains instantaneous DPIV vector fields from successive oscillation cycles at t/2T_o = 0.22. This time corresponds to when the vocal folds are still opening and the gap is almost fully open, h = 0.88h_max. Vector fields were taken for the last eight (of twenty-four) oscillations for Re = 7200 and f_life = 97.5 Hz. As in figure 4, flow is left to right, but the full vector resolution is shown. The vocal-fold models are again masked in white.

Figure 15.

Instantaneous velocity vector fields from eight consecutive oscillation cycles at t/2T_o ≈ 0.22 for Re = 7200, f_life = 97.5 Hz. Flow is left to right with the full vector resolution shown. Observe the strong cycle-to-cycle variation in the glottal jet.

Figure 15 clearly shows cycle-to-cycle variations in terms of both jet strength as well as direction. Note that the jets either deflected to the right of the centreline or were straight. In other runs (that were not shown), the jet deflected to the left of the centreline, accounting for the seeming jet bifurcation observed in the phase-averaged vector fields shown in figure 4. As discussed in § 1, this type of behaviour has been observed for many different vocal-fold geometries and oscillation frequencies. Further, based on the work of Sherman et al. (Reference Sherman, Lambert, Krane and Wei2019), it appears that this switching phenomenon becomes established in one to two oscillation cycles. As such, this appears to be an inherent feature of flows of this type and not a start-up transient. That this degree of variability occurs in a simple model geometry like this is interesting in its own right and warrants further investigation.

The critical question, however, is whether these variations are meaningful in terms of voiced sound generation. Here again the power of dynamic analysis comes to the fore. Time traces of volume flow rate, transglottal pressure and net streamwise momentum flux are shown in figure 16 for the eight oscillation cycles shown in figure 15.

Figure 16.

Time traces of (a) gap opening, (b) transglottal pressure (–$\cdot$–) and momentum flux (—) and (c) volume flow rate for eight successive oscillations for the Re = 7200; f_life = 97.5 Hz; f* = 0.0261 case. Observe the cycle-to-cycle variations and the correlation between transglottal pressure and corresponding phase-locked vector fields in figure 15.

Figure 16(b) shows time traces of transglottal pressure and net momentum flux for the eight consecutive oscillations. Strong cycle-to-cycle variations are clearly visible particularly in the transglottal pressure traces. A notable example is that the cycle-to-cycle variations in transglottal pressure exactly correlate to the jet strength observed in the vector fields comprising figure 15. The first oscillation in this sequence has the strongest jet, figure 15(a), as well as the largest transglottal pressure. Over the next four cycles, the transglottal pressure peak values successively decrease, figure 16(b), while the jet strength correspondingly diminishes, as seen in figures 15(b)–15(e). Figure 15(f) shows that the sixth oscillation has a stronger jet which correlates with the increased transglottal pressure.

The jets are successively weaker in the final two oscillations, figures 15(g) and 15(h), as are the corresponding transglottal pressures, figure 16(b). Examination of the time trace of net momentum flux for the eight oscillation cycles shows the same correlation although the traces are noisier and the magnitudes of the flux term are approximately 20 % of the transglottal pressure.

While there does not appear to be a strong correlation between jet direction and transglottal pressure or momentum flux, there does appear to be a relationship between jet direction and volume flow rate. Four of the cycles with the highest volume flow rates, i.e. the second, third, seventh and eighth in figure 15, are also the oscillations where the jet deflects to the right of the centreline. For the three middle cycles with the lowest volume flow rate, i.e. the fourth, fifth and sixth, the jet is straight. Irrespective of whether jet direction and volume flow rate are correlated, cycle-to-cycle variation in volume flow rate is clearly visible.

The significance of this observation lies in the fact that transglottal pressure (as a surrogate for the vocal-fold drag) and volume flow rate are both directly related to acoustic source strength, cf. Fant (Reference Fant1970), McGowan (Reference McGowan1988) and Hirschberg (Reference Hirschberg1992). Since the cycle-to-cycle variations are quite significant in these two quantities, they will likely directly influence the sound produced. Further, it seems that some form of cycle-to-cycle variation is broadly present in flows of this type; the work of Sherman et al. (Reference Sherman, Lambert, Krane and Wei2019) and a number of studies described in Mittal et al. (Reference Mittal, Erath and Plesniak2013) covering a range of geometries and oscillatory motions all report what Mittal et al. (Reference Mittal, Erath and Plesniak2013) refer to as ‘cycle-to-cycle asymmetries’. While voice quality and colour are perceptual measures, and beyond the scope of this work, it is nevertheless hypothesized that these inherent variations could be unique to each individual and play a role in identifying her/his voice. From these results, it appears that both jet strength and direction may play a role in the acoustics of phonation.

The relative influence of Reynolds number and reduced frequency on cycle-to-cycle variations can be seen in sample time traces of transglottal pressure shown in figure 17. Two cases are shown including the lowest Reynolds number case, Re = 3650 and f* = 0.0511, in figure 17(a), and the highest Reynolds number case, Re = 8100 and f* = 0.0129, in figure 17(b). These represent the highest and lowest reduced frequency cases, respectively.

Figure 17.

Time traces of transglottal pressure for successive oscillations at (a) Re = 3650; f* = 0.0511 and (b) Re = 8100; f* = 0.0129. Note that the physical frequency for both cases is identical, f_life = 97.5 Hz and the ordinates of the two plots are displaced but both cover 70 Pa.

The obvious difference between the two plots comprising figure 17 is the pressure values. These are dimensional plots with pressure in units of Pascals. It is intuitively obvious that the mean pressure level and amplitude of the fluctuations should be smaller at the lower Reynolds number. Recall, however, that the lowest Reynolds number case, i.e. the case with the highest reduced frequency case, showed the largest non-dimensional amplitude, as seen in figure 8(c).

It appears therefore that Reynolds number effects are most noticeable in dimensional data whereas non-dimensionalization appears to draw out frequency effects. The dimensional plots in figure 17 show that increasing the flow speed is necessarily accompanied by increasing transglottal pressures, both in the mean and fluctuation amplitudes. It follows further that the overall energy levels are higher at higher speeds, implying that the acoustic energy levels are correspondingly higher. This is consistent with the idea that one requires greater energy to speak louder.

While not all of the individual oscillation traces are shown in this paper, cycle-to-cycle variations occur over the entire parameter space covered in this study. However, the degree of variation is not the same for every case. If the envelope of maximum peaks is considered, assuming for the sake of illustration that there is some sort of beating behaviour, the number of oscillations in a ‘beat period’ seems to vary from one case to another. It is longest in the Re = 3650 and f* = 0.0511 case, figure 17(a) and shortest in the Re = 7200 and f* = 0.0261 case, figure 16(b). The highest Reynolds number case, figure 17(b), appears to have an intermediate ‘beat period’. Similarly, if one looks at the envelope of minimum peaks, those minima in figure 17(a) are essentially constant. In contrast, the minima for the transglottal pressure trace in figure 16(b) are the most variable and even irregular. Whether there is, in fact, some sort of dynamic resonance behaviour is an interesting subject for future investigation.

It is perhaps of interest to consider the duct as a bluff body in the water tunnel with a characteristic transverse dimension of 28 cm. For the Re = 7200 cases, the flow speed outside the duct (on either side and underneath) is ~20 cm s⁻¹. Assuming a Strouhal number of around 0.2, the corresponding vortex shedding period for the duct would be around 7 s. Recalling that the oscillation period is 15 s ≤ 2T_o ≤ 28 s, vortex shedding from the duct, if it were to occur and if it were strong enough, would fall in the range of harmonics of the vocal-fold oscillation frequency. It is therefore within the realm of possibility that there may be some sort of resonance between the duct shedding frequency and the glottal jet. It is noted, however, that there was no evidence of any such vortex shedding from the duct. Further, it would be extremely difficult to imagine that the cycle-to-cycle variations at the Reynolds numbers could be caused by duct vortex shedding where the flow speed outside the duct is only ~11 cm s⁻¹.

Unfortunately, given that experimental runs were restricted by camera memory limitations, it is not possible to explore this variability at this time. However, this frequency and Reynolds number interdependence is consistent with observations in Sherman et al. (Reference Sherman, Lambert, Krane and Wei2019). In that study, up to sixteen cycles were captured in a single run as there was no delay time between each oscillation. For that study, cycle-to-cycle variations were most prominent at the third highest of the four frequencies examined. In this study, the strongest and most frequent variations appear to occur at the highest frequency but at the third highest Reynolds number.

The final questions to address involve the source of the cycle-to-cycle variations and the reason for the dependence on Reynolds number and frequency. Before beginning this discussion, it is important to note that cycle-to-cycle variations have been observed by numerous researchers in a wide variety of facilities, air and water, and with very different vocal-fold models. So at the outset, one can rule out pump or model vibrations, bulk flow around the glottal duct model or other apparatus induced disturbances. For this specific study, all of these were addressed in § 2.

Returning to the previously posed questions about source and dependencies of the cycle-to-cycle variations, then, it is possible that these questions might actually be addressed by a single explanation. A clue can be extracted from the two instantaneous velocity-field plots shown in figure 18. Both vector fields are taken from the Re = 7200, f_life = 97.5 Hz case for the first oscillation presented in figures 15 and 16. The vector field in figure 18(a) was captured at time, t/2T_o = 0.004, at the very beginning of the entire experimental run. The second velocity field in figure 18(b) shows the flow at t/2T_o = 0.95 just before the start of the second cycle of the sequence. Every other vector is shown in these fields and the vector lengths have been magnified five times relative to those shown in figure 15. In addition, spurious vectors generated inside the vocal folds have not been masked.

Figure 18.

Instantaneous DPVI vector fields taken during the first oscillation in a sequence at: (a) t/2T_o = 0.004 and (b) t/2T_o = 0.95 for Re = 7200; f_life = 97.5 Hz. Note that the vector lengths in both fields are five times longer than in figure 15.

The key point of these two vector fields is that the flow downstream of the vocal folds is quiescent only at the very beginning of each experimental run before the vocal folds ever move. It can be seen in figure 18(a), then, that before they open for the first time, there is no flow downstream of the vocal folds. After that first oscillation, however, after the vocal folds have opened, closed, and remained closed for almost half the oscillation period, the flow has not fully returned to quiescence. There appears to be a large, albeit weak, clockwise circulation centred downstream of the field of view. This appears as a coherent field of vectors pointed down and to the left in the upper right quadrant of figure 18(b). This weak circulation is believed to have been responsible for deflecting the jet in the subsequent oscillation, cf. figure 15(b), to the right. In general, it is hypothesized that the residual motions after the vocal folds closed are the perturbations that cause cycle-to-cycle variations.

It is likely that each successive glottal jet will be highly sensitive to initial conditions. As such, it would be difficult to predict a priori the strength and direction of each jet. With that in mind, however, there are some general comments that can be made regarding frequency and Reynolds number dependence of cycle-to-cycle variations.

The first is that cycle-to-cycle variations should become less pronounced with increasing delay time between the close of the vocal folds and their next opening. This is simply because the longer delay provides more time for the supraglottal flow to dissipate. Consequently, it would be expected that the variations would occur more often and grow stronger with increasing oscillation frequency at a fixed Reynolds number. Similarly, a shorter delay time between the closing and successive opening of the vocal folds, e.g. 80 % versus 120 % of T_o, would enhance these variations.

Increasing the Reynolds number could also increase cycle-to-cycle variations. The rationale for this hypothesis is that the increased momentum of the glottal jet induces greater mixing in the supraglottal region which, in turn, creates a more unsteady environment for the formation of the successive jet. At the same time, pressure levels increase with increasing Reynolds number. For example, while the percentage cycle-to-cycle variations in transglottal pressure are the same in figures 17(a) and 17(b), there would obviously be far greater acoustic energy produced at the higher Reynolds number.

But there is a counter-effect. For a fixed dimensional frequency, increasing the Reynolds number simultaneously decreases the reduced frequency. As such, there is a competition of effects as increased mixing from the higher momentum jet is counterbalanced by relatively longer settling times afforded by the lower reduced frequency. This may explain why cycle-to-cycle variations for the highest, Re = 8100, case are not as strong as for the next lowest, Re = 7200, case.

5.4. Revisiting the quasi-steady flow assumption

As noted in § 1.5, the quasi-steady character of glottal jets is widely accepted but has not yet been fully demonstrated. First, it is not clear that the jets are quasi-steady when the vocal folds begin to open and when they shut. Second, the bulk of the work on this subject has been done at lower frequencies equivalent to the adult male voice. In this sub-section, quasi-steadiness will be revisited in the context of both the integral momentum equation and the unsteady Bernoulli equation, using estimates for terms of these equations and data presented earlier.

As discussed in Krane et al. (Reference Krane, Barry and Wei2010), in the context of the Bernoulli equation, a time-varying flow can be considered quasi-steady when

(5.6)\begin{equation}\frac{\displaystyle{{\rho _o}\int_{{x_u}}^{{x_d}} \frac{{\partial u}}{{\partial t}}\,\textrm{d}x}}{{\dfrac{{{\rho _o}}}{2}(u_d^2 - \; u_u^2)}}\; \ll 1.\end{equation}

The analogous comparison for the integral momentum equation is given by

(5.7)

A comparison of the numerator and denominator of (5.7) is given in figures 11–13, and these terms of the integral momentum equation are estimated in (5.4). What is noticeable, especially in figure 13, is that, as f* increases, the unsteady acceleration increases in size, relative to the convective acceleration. This is especially true during the early part of the opening phase (0 < t/2T_o < 0.15), where the convective acceleration is identically zero, because the glottal jet has not yet pierced the control-volume exit plane. During the closing phase, when the convective acceleration is non-zero, the unsteady acceleration can be nearly as large, especially at larger f*. The unsteady acceleration reaches its peak just after the convective acceleration, at t/2T_o ≈ 0.35. In figure 13, the unsteady acceleration is non-negligible for the remainder of the time the glottis is open, for the larger two f* cases shown. The combination of developing jet flow early in the cycle and large unsteady deceleration late in the cycle suggests the flow is quasi-steady, at most, for the interval 0.15 < t/2T_o < 0.35.

Another way to address the question of whether time-varying glottal flow is quasi-steady is to use the present data to compute terms of the Bernoulli equation. The unsteady Bernoulli equation contains the traditional balance of static and dynamic pressures with an additional temporal derivative term on the left-hand side of the equation, as well as a term representing the effect of friction (Deverge et al. (Reference Deverge, Pelorson, Vilain, Lagrée, Chentoug, Willems and Hirschberg2003), Vilain et al. (Reference Vilain, Pelorson, Fraysse, Deverge, Hirschberg and Willems2004), Krane & Wei (Reference Krane and Wei2006), Krane et al. (Reference Krane, Barry and Wei2010). Therefore, if glottal flows are quasi-steady, the unsteady term should be negligible and the static and dynamic pressures should balance, to within the magnitude of the friction term. On a phase diagram, with transglottal pressure plotted as a function of dynamic pressure, a quasi-steady glottal jet should appear on a straight line with slope equal to unity. The exceptions would be when the folds just begin to open, when the jet pinches off at closure and, of course, when the vocal folds are closed. According to the works of Mongeau et al. (Reference Mongeau, Francheck, Coker and Kubli1997), Deverge et al. (Reference Deverge, Pelorson, Vilain, Lagrée, Chentoug, Willems and Hirschberg2003), Vilain et al. (Reference Vilain, Pelorson, Fraysse, Deverge, Hirschberg and Willems2004), Krane & Wei (Reference Krane and Wei2006) and Krane et al. (Reference Krane, Barry and Wei2010), it is expected that unsteadiness is relevant for some interval after glottal opening, and just prior to glottal closure, and that friction is likely relevant during these same intervals.

To test this hypothesis, phase plots were generated showing the transglottal pressure plotted as a function of dynamic pressure. These are shown in figure 19 for the four different frequencies at Re = 7200, figure 19(a), and for the different flow speeds at f_life = 97.5 Hz, figure 19(b). The dynamic pressure coefficient was calculated as one half the square the maximum exit plane jet velocity, u_j, multiplied by the fluid density and non-dimensionalized by (P_uo − P_do). For reference, time traces of the dynamics pressure, non-dimensionalized consistent with all other plots, are presented in figure 20 and corresponding plots of transglottal pressure appear in figure 9. Every point on each plot in figure 19, then, shows the phase-averaged transglottal and dynamic pressures at a particular instant in time. Red lines with unity slope have been added to represent the idealized quasi-steady flow condition.

Figure 19.

Phase plots of transglottal pressure vs. dynamic pressure for (a) the four oscillation frequencies at Re = 7200, and (b) the four different flow speeds at f_life = 97.5 Hz. For reference, time stamps have been included for the overlapping case, Re = 7200; f* = 0.261, in both (a,b).

Figure 20.

Time traces of the dynamic pressure coefficient across the vocal-fold exit plate for: (a) the four oscillation frequencies at Re = 7200, and (b) the four different flow speeds at the highest frequency, f_life = 105 Hz.

A cursory look at figure 19 clearly shows that significant portions of the plots are not coincident with the red lines. In fact, there are only relatively small parts of the plots where the transglottal pressure and dynamic pressure are even close to being equal.

One should keep in mind that time is not evenly distributed around a phase plot. For ease of reference, time stamps have been superimposed on figure 19(b) for the Re = 7200 case. The start of the cycle, t/2T_o = 0, appears on the left edge of the plots where the transglottal pressure is unity, (P_u − P_d)/(P_uo − P_do) = 1, and dynamic pressure is zero. The moment when the vocal folds close, t/2T_o = 0.5, again appears on the left edge of the plots where the dynamic pressure is zero. But at that time, (P_u − P_d)/(P_uo − P_do) ≈ 1.5. Thus, the first half of the phase-averaged oscillation cycle, in which the vocal folds open and close, 0 ≤ t/2T_o ≤ 0.5, appears as a counter-clockwise circuit starting and ending on the ordinate. The second half of the cycle, 0.5 ≤ t/2T_o ≤ 1.0, resides along the vertical axis from (P_u − P_d)/(P_uo − P_do) = 1.5 down to (P_u − P_d)/(P_uo − P_do) = 1. This is the part of the cycle where the vocal folds remain closed and the transglottal pressure returns back toward steady state, without fully reaching it.

With the exception of the lowest Reynolds number case, i.e. where f* is the largest, the portions of the phase plots in which the transglottal and dynamic pressures lie in the vicinity of the red lines occur between the time just before the vocal folds are fully open to when they are approximately 40 % closed, on 0.24 ≤ t/2T_o ≤ ~0.34. This represents approximately 20 % of the time the vocal folds are open and corresponds to the vector fields shown in figures 4(d) and 4(e). Observe that this time period corresponds to the part of the oscillation cycle where the momentum flux is largest, as seen in figure 11, and is consistent with the observations given the relative size of integral momentum equation acceleration terms seen in figures 11–13, and the results of Krane et al. (Reference Krane, Barry and Wei2010).

It should be pointed out that extrapolating the degree of quasi-steadiness in the present rigid vocal-fold model, to real phonation should be done with care. The shape of the real vocal fold is more channel-like in that the contact between the folds takes place over a much longer streamwise extent than in our model. The shape of the glottis in the current model might, then, understate the importance of unsteady and friction effects when the folds are just open and are just about to close. Evidence can be drawn from the direct empirical comparison of flow through the gap between semicircle model vocal folds with flow through a more channel-like glottis formed by rectangular-shaped vocal folds (Deverge et al. (Reference Deverge, Pelorson, Vilain, Lagrée, Chentoug, Willems and Hirschberg2003).

Having said that, it was previously noted regarding figure 19, that even for the current model in which unsteadiness might be attenuated, the phase plot only touches the red line for an instant at the highest reduced frequency. This observation, taken together with the clear trends shown in figure 11 and (5.4) imply that the phonatory airflow in adult female or child voices, or for soprano singing, may less quasi-steady than the adult male voice. Accordingly, the quasi-steady assumption might be worth revisiting in general. Additionally, most of the sound produced during phonation is generated just before the folds close, at which time the quasi-steady assumption does not apply. In this regard, the relevance of this assumption is also an open question.