Interspeech 2024
1-5 September 2024, Kos, Greece

Measurement and simulation of pressure losses due to airflow in vocal tract
models
Peter Birkholz, Patrick Häsner
Institute of Acoustics and Speech Communication, TU Dresden, Germany
peter.birkholz@tu-dresden.de

the length l remains constant, then ∆Pv ∝ A−3 . Fulcher et
al. [7] proposed a modification to the viscous term, which better fitted their own data from flow experiments with a physical
replica of the glottis, namely

λ
w
12µ
∆Pv = ∆x 3
U,
(2)
lw
wref

Abstract
We propose a unified model for viscous and kinetic energy
losses in a discrete tube model of the vocal system including
the glottis. In this model, a lossless Bernoulli flow is assumed
at each transition between two tube sections if the downstream
section has a smaller diameter than the upstream section, and
otherwise the recovery of a fixed fraction of the dynamic pressure. For viscous losses, we propose a general equation according to which the pressure drop within a tube section is inversely
proportional to a certain power of its cross-sectional area. The
parameters of the model were adjusted to reproduce the results
of measurements with physical replicas of the glottis and the
vocal tract. The best agreement with the experimental data was
achieved when 29% of the dynamic pressure were recovered at
tube expansions, and when the viscous losses were proportional
to the tube area to the power of -2.9. These results may improve
articulatory speech synthesis.
Index Terms: one-dimensional flow model, flow resistance

where wref = 0.01 cm and λ = 0.41. With this modification
the viscous pressure loss becomes proportional to A−2.59 .
The term ∆Pk in Eq. (1) represents the loss of kinetic energy, where % is the density of the air, and kent and kexit describe the effects responsible for energy losses at the glottal
entrance and the effect of pressure recovery at the glottal exit,
respectively. The case kent = 1 and kexit = 0 would correspond to a lossless conversion of static to dynamic pressure at
the entrance of the glottis according to Bernoulli’s equation, and
a loss of all dynamic pressure at the exit of the glottis. Potential energy losses during the conversion from static to dynamic
pressure can be considered by setting kent > 1. Furthermore, a
certain percentage of dynamic pressure is usually recovered at
the glottal exit, which can be considered by setting kexit > 0.
The coefficients are frequently set to the values obtained by van
den Berg et al. [6], namely kent = 1.375 and kexit = 0.5.
However, there is no consensus on these values. For example,
Ishizaka et al. [8] proposed to calculate kexit based on the conservation of momentum at the glottal exit, which makes kexit a
function of the cross-sectional area of the glottis and the area
immediately downstream from the glottis. The resulting coefficient is typically in the order of 0.05 to 0.4. With respect to the
entrance coefficient, Pelorson et al. [9, 10] argue that kent = 1
instead of 1.375 because the human glottis has a smooth entrance at which no significant flow separation should occur.
In the vocal tract above the glottis, viscous losses are often
modeled as distributed pressure losses using Poiseuille’s equation, which describes the pressure drop across a cylindrical tube
section of length ∆x and cross-sectional area A under the assumption of a fully developed laminar flow [11, 12]:

1. Introduction
Articulatory speech synthesizers mostly use one-dimensional
aerodynamic-acoustic simulations of the vocal tract, because
they are much faster than two-dimensional or three-dimensional
simulations and physically valid up to frequencies of about
4 kHz [1, 2, 3, 4]. An important element for realistic simulations is the modeling of energy losses. For voiced speech
sounds they determine the bandwidths of the formants, and for
unvoiced sounds they determine the airflow rate for a certain
subglottal pressure, which in turn affects the power spectrum of
turbulent noise sources [5].
Currently, the energy loss at the glottis is usually modeled
differently than losses further downstream in the vocal tract.
With regard to the glottis, the transglottal pressure drop ∆P
is often modeled as
12µ
%
∆P = ∆x 3 U + (kent − kexit ) 2 U 2
lw
2A
| {z } |
{z
}
∆Pv

(1)

∆Pk

∆Pv = ∆x

based on a study by van den Berg et al. [6]. The term ∆Pv
represents the viscous pressure loss of a fully developed laminar flow through a rectangular duct (the glottis), where ∆x is
the length of the duct (axial glottal length), l is the long side of
the rectangle (anterior-posterior glottal length), w is the short
side of the rectangle (glottal width), µ is the viscosity of the air,
and U is the volume velocity through the glottis. The area of
the glottis is A = lw, hence the viscous pressure loss can be
rewritten as ∆Pv = ∆x12µl2 /A3 . When we assume that the
glottal area changes mainly due to changes of its width w, and

8µπ
U.
A2

(3)

In contrast to ∆Pv in Eqs. (1) and (2), here ∆Pv ∝ A−2 .
A supraglottal kinetic energy loss is usually only considered locally at the site of a critical constriction with the area Ac
(e.g. in fricatives) as
∆Pk = kc ·

%
U 2,
2A2c

(4)

where kc is an empirical coefficient [13, 1]. However, using
a lumped loss at constrictions can cause acoustic artifacts in

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10.21437/Interspeech.2024-1970

Cylinder

/a/

//

/e/

/i/

/o/

Laptop

/u/

l = 16 mm

Trachea

Pressure
transducer

Vocal tract

Data
acquisition

Single-board
computer

w = 4 mm

Figure 1: Back row: Tube models of the vocal tract. The mouth
opening is at the upper end. Front row, from left to right: Glottis
models with glottal widths of 1, 2, 4, 8, and 16 mm.

Lung

Pressure
tap

Glottis
Larynx

Blower

Airflow
sensor

Trachea

Figure 2: Measurement setup. The right side shows a cutaway
view of the vocal tract for /a/, the glottis replica (“Larynx”),
and the upper part of the trachea.

aerodynamic-acoustic simulations of running speech, where the
constriction sites change dynamically [14].
The above overview shows that there are several ways of
modeling the viscous and kinetic energy losses in the vocal
tract. However, there is little experimental evidence for onedimensional flow models of the vocal tract [15], especially
when they include both the glottis and the supraglottal cavities.
The goal of the present study was to propose a unified approach
to modeling viscous and kinetic energy losses in the vocal tract,
which is suitable to be included in one-dimensional models for
articulatory speech synthesis. In this model, not only the viscous losses but also the kinetic energy losses are distributed
(as opposed to localized losses in previous studies). The model
has three parameters that were adjusted to reproduce flow data
obtained with physical replicas of the vocal tract including the
glottis.

1.7 cm. Further details about these models can be found in [16].
The glottis models consist of three regions: a short cylindrical part with a diameter of 1.8 cm that represents the upper part of the trachea, a 7 mm long section that represents the
conus elasticus where the cross-section tapers towards that of
the actual glottis, which is represented as a rectangular duct
(with slightly rounded edges) with a length of 3 mm (glottal
depth), a long side (glottal length) of 16 mm and a short side
of the glottal width w. We made five glottis models with glottal widths of 1, 2, 4, 8, and 16 mm. For the width of 16 mm,
the glottal area became a full circle with a diameter of 16 mm,
as shown in Figure 1. Glottal widths of w = 1 mm and
w = 2 mm roughly represent intermediate and maximal areas
during phonation, w = 4 mm and w = 8 mm correspond to
glottal areas used for voiceless fricatives [5], and w = 16 mm
corresponds to an area of 201 mm2 typical for breathing [17].
The area functions and 3D-printable STL files of all models are
provided in the supplemental material1 .
The glottis and vocal tract replicas were used in the measurement setup shown in Figure 2. Here, air was fed from a
radial blower (U71HX-024KX-6 by micronel, Tagelswangen,
Switzerland) through a 2 m long tube into a box representing
the lungs and from there through a model of the trachea. The
glottis and vocal tract models were mounted on the upper end
of the trachea, as shown in the cutaway view at the right side
of the figure. A pressure transducer (DMU4, Kalinsky Sensor
Elektronik, Erfurt, Germany) was used to capture the subglottal
pressure at the upper end of the trachea, and an airflow sensor (AWM720-P1, Honeywell, Charlotte, North Carolina) was
used to capture the flow rate at the entrance to the “lung box”.
The sensor values were digitized by a data acquisition device
(DataTranslation 9837C, Norton, MA) with a sampling rate of
48 kHz and 24 bits quantization per channel and sent to a laptop. The laptop was running a custom software to display the
data and control the power of the radial blower.
For each combination of a glottis model and a vocal tract
model, the blower power was adjusted sequentially to generate 7 fluid power levels: 100, 200, 400, 700, 1100, 1600, and

2. Method
2.1. Measurements with physical tube models
To obtain reference data for the proposed theoretical model (see
Sec. 2.2), we performed flow experiments with physical replicas
of the glottis and the vocal tract. Glottis models with different
glottal areas were combined with tube models of the vocal tract
for different vowels, and the pressure drop across these models
was measured for multiple airflow rates.
A photograph of all vocal tract and glottis models is shown
in Figure 1. They were designed with the software Autodesk
Inventor (version 2019) and 3D-printed on an Ultimaker 5S
printer using the material polylactic acid. The vocal tract models comprise a cylinder as well as straight axisymmetric tubes
with area functions of the vowels /a, E, e, i, o, u/. The area
functions were exported from the articulatory speech synthesizer VocalTractLab 2.3 (www.vocaltractlab.de) based
on the respective vocal tract shapes of the standard speaker
model. The tube models were chosen because they cover a
range of conditions with different numbers of constrictions and
constrictions of different lengths. For example, the models for
/e/ and /i/ have a single relatively long constriction in the anterior part, while the models for /o/ and /u/ have two shorter
constrictions in the middle and at the anterior end. The smallest
cross-sectional areas of these models are in the range of 0.25
to 0.36 cm2 . The diameter at the glottal end is 1.7 cm for all
models, which tapers to a diameter of about 0.68 cm (area of
0.36 cm2 ) at the upper end of the epilaryngeal tube. The cylinder model has a length of 17 cm and a uniform diameter of

1 https://www.vocaltractlab.de/index.php?page=
birkholz-supplements

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6

a)

Mouth opening

Glottis

Area in cm

2

5
Glottis

4
3
2

Oral cavity

Pressure in Pa

1
0
b) 300

R1
+
Psub
-

200

Q1

R2

Q2

...

QN-2 RN-1 QN-1

RN

U

100

Figure 4: Discrete tube model and fluidic network.

0
-2

0

2

4

6
8
10
Position in cm

12

14

16

between these sections, namely

Figure 3: a) Continuous (black) and discrete (red) area functions for the glottis model (glottal width of 2 mm) concatenated
with the tube model for /a/. b) Simulated distribution of the
static pressure for a flow rate of 500 cm3 /s.



2200 mW. At each power level, the pressure and airflow rate
were measured. The measurements were carried out three times
on different days and averaged over the repetitions. Since the
results for the glottal widths of 8 mm and 16 mm were nearly
identical (less than 5% deviation for all measurements), the data
with the 16 mm glottis were excluded from further analysis.
2.2. Theoretical model
Our approach to model viscous and kinetic losses in a uniform
way assumes that the vocal tract (including the glottis and subglottis) is represented as a sequence of short abutting cylindrical
tube sections [18]. The lengths and cross-sectional areas of the
tube sections are given in terms of a discrete area function. As
an example, the red curve in Figure 3a shows the discrete area
function that was derived from the combined continuous area
function (black curve) of the physical replicas of the glottis and
the vocal tract for /a/. As proposed in [19], the areas of the
discrete sections assume the minimum of the continuous area
function in the regions of the sections.
As shown in Section 1, previous approaches to model the
viscous pressure loss ∆Pv over a short tube section of length
∆x have in common that they are proportional to the airflow
rate U , to ∆x and to a (negative) power of the cross-sectional
area. However, the proportionality factor and the area exponent
differ between the equations. Here we propose the following
general equation for the viscous loss over a tube section i,
∆Pv,i = li Rref · (Aref /Ai )α U,

kc · (%/2)(1/A2i+1 − 1/A2i )U 2
ke · (%/2)(1/A2i+1 − 1/A2i )U 2

for Ai+1 < Ai
otherwise,
(6)
where the proportionality factors kc and ke account for potential
losses at tube contractions and expansions, respectively. With
this approach, the total kinetic pressure change across a tube
contraction or expansion that extends over multiple tube sections (sum of the involved ∆Pk,i ) depends only on the initial
and final cross-sectional areas of the expansion or contraction.
This makes the results independent of the discretization step
size of the area function.
A summation of all ∆Pk,i for the whole tube furthermore
shows that the overall kinetic pressure loss is essentially a function of the difference (kc − ke ) of the coefficients, so that their
individual values cannot be independently determined from our
experimental data. Hence, following [9] we set kc = 1 in this
study, assuming a lossless conversion from static to dynamic
pressure at tube contractions. This leaves ke as the only parameter to be determined for the kinetic energy losses.
With these considerations, the pressure changes in the vocal
tract can be represented in terms of the fluidic network shown
in Figure 4. Here, Psub is the subglottal pressure, N is the number of tube sections, Rn are linear resistors that cause a pressure change proportional to U according to Eq. (5), and Qn are
non-linear resistors that cause a pressure change proportional
to U 2 according to Eq. (6). Note that QN = 0, because there
is no pressure recovery at the tube exit [20]. The network allows to determine the complete pressure distribution along the
tube axis for steady flows and can be readily combined with
the transmission-line circuit representation of the vocal tract to
model both aerodynamics and acoustics [18, 14].
∆Pk,i =

(5)

2.3. Determination of model parameters

where li and Ai are the length and cross-sectional area of the
tube section, and Rref , Aref and α are free parameters. Aref was
fixed to 1 cm2 , so that Rref can be interpreted as the flow resistance (ratio of pressure drop to volume flow) per unit length for
an area of 1 cm2 , and α models the variation of the resistance
with the area. Both Rref and α were adjusted to fit the simulations to our experimental data (see Section 2.3).
Inspired by [14], kinetic energy losses are distributed along
the entire vocal system and depend on whether the tube is
locally contracting or expanding. Specifically, the pressure
change at each transition between two tube sections i and i + 1
is assumed to be proportional to that of a lossless Bernoulli flow

To find values for the three model parameters Rref , α and ke
that minimize the deviation between measurements and simulations, we performed a complete grid search in the intervals
Rref ∈ [800; 2.6 · 106 ] Pa·s/m4 , α ∈ [2; 3], and ke ∈ [0; 0.8].
For each setting of the parameters, the subglottal pressure was
simulated for all airflow rates and tube models used in the experiments in Section 2.1 based on the discretized area functions.
The overall deviation between simulations and measurements
for a certain parameter setting was then calculated as the root
mean square percentage error E [21] between simulated and
measured subglottal pressure values across all tube models and
flow rates.

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Pressure Psub in Pa

2500 Glottis only
2000 (no vocal tract)
1500
1000
500
0

Uniform cylinder

/a/

Pressure Psub in Pa

2500 //
2000
1500
1000
500
0

/e/

/i/

Pressure Psub in Pa

2500 /o/
2000
1500
1000
500
0
0

/u/

0

500
1000
Flow U in cm3/s

1500

0

500
1000
Flow U in cm3/s

500
1000
Flow U in cm3/s

1500

1500

Flow resistance per unit length in Pa∙s/m4

Figure 5: Measured flow-pressure points (circles) and simulated flow-pressure curves with optimized parameters. The black, red, green,
and blue curves were obtained for glottal widths of 1 mm, 2 mm, 4 mm, and 8 mm, respectively.

1012

10

With regard to viscous losses, we obtained an area exponent of 2.9, which is close to the value of 3 assumed in many
studies for the glottis, and higher than the value of 2.59 obtained by Fulcher et al. [7] or the value of 2 in Poiseuille’s law.
Figure 6 compares the flow resistance as a function of the tube
area between the proposed Eq. (5) with the optimized parameters (red curve), Eq. (2) proposed for the glottis by Fulcher et
al. (light gray curve), the viscous resistance for the glottis in
Eq. (1) (gray curve), and Poiseuille’s law (black curve). This
shows that Poiseuille’s law, which is often used to model viscous losses in the vocal tract, strongly underestimates the losses
for small cross-sectional areas. This in turn leads to an underestimation of formant bandwidths in acoustic simulations [22].

Propsed
Glottis (Fulcher)
Glottis (van den Berg)
Poiseuille's law

11

10

10

109
8

10

107
106
105
10

4

10-2

10-1
Area in cm2

100

Figure 6: Variation of the flow resistance per unit length as a
function of the cross-sectional area for different equations.

The fluidic network shown in Figure 4 and the optimized
model parameters allow to calculate the pressure distribution
along the tube axis for any given area function and airflow rate.
As an example, Figure 3b shows the pressure distribution for the
glottis with a width of 2 mm connected to the vocal tract for /a/
at a flow rate of 500 cm3 /s. For the three tube contractions and
expansions in the posterior part, the associated pressure drops
and (partial) recoveries can be clearly seen. In future studies,
it would be interesting to compare such pressure profiles with
distributed pressure measurements along the whole tube axis.

3. Results and discussion
The parameter values that minimize the deviation between the
measured and simulated subglottal pressure values across all
models and flow rates are Rref = 347, 922 Pa·s/m4 , α = 2.9
and ke = 0.29 with an error of E = 14.9%. Figure 5 shows
the measured flow-pressure pairs as circles along with the simulated flow-pressure curves. Different glottal widths are represented by different colors (black for 1 mm, red for 2 mm, green
for 4 mm, and blue for 8 mm), while each panel represents one
vocal tract model (including the absence of a vocal tract in the
top left panel). The simulations approximate the measured data
points reasonably well.
With regard to the kinetic energy losses, the difference of
the contraction and expansion factors is kc − ke = 0.71 in our
model. This value is somewhat lower than 0.875 as determined
by van den Berg for the glottis [6], and lower than the value of
1.0 assumed by Birkholz et al. [14]. Hence, the overall kinetic
energy losses in the proposed model are lower than in these earlier models.

Finally, we would like to compare the error of 14.9% between measurements and simulations for the optimized model
parameters with other choices for the model parameters. If we
model viscous losses with Poiseuille’s law and set ke = 0 for
tube expansions as in [14], the error increases to 20.9%. When
the viscous loss according to Poiseuille’s law is combined with
ke = 0.29, the error becomes 28.0%, and when the optimized
viscous resistance is combined with ke = 0, the error becomes
31.2%. This shows that the parameters for both viscous and kinetic losses must be set correctly to achieve optimum agreement
with the experimental data.

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4. Acknowledgements
We thank Mario Fleischer for useful discussions about this
study.

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M. Pichelin, G. Caillibotte, and A. Giovanni, “Realistic glottal
motion and airflow rate during human breathing,” Medical Engineering & Physics, vol. 37, no. 9, pp. 829–839, 2015.

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