Anti-phase synchronization between the oscillation in the pipe and that in thefoot of a flue organ pipe
Kin’ya TAKAHASHI(1) , Sho IWAGAMI(1) , Shuhei TATEISHI(1) , Genki TSUTSUMI(1) ,Taizo KOBAHASHI(2) , Toshiya
TAKAMI(3)(1) Kyushu Institute of Technology, Japan, takahasi@mse.kyutech.ac.jp(2) Kyushu University, Japan, tkoba@cc.kyushu-u.ac.jp(3) Oita University, Japan, takami-toshiya@oita-u.ac.jp

Abstract
In the recent work (Acoust. Sci. & Tech. vol.40, No.1 pp.29-39 (2019)), we found with the numerical studyof a 2D flue organ pipe model that the foot works as a Helmholtz resonator. When the frequency of acoustic
oscillation in the pipe is higher than the resonance frequency of the Helmholtz resonator by almost the full-widthat half-maximum, the most stable oscillation is observed, and the oscillation in the foot is anti-synchronizedwith that in the pipe. However, if the Helmholtz resonance frequency is nearly equal to the acoustic oscillation
frequency, the oscillations in the pipe and foot become rather unstable and the oscillation in the foot lags behindthat in the pipe by nearly π /2. In this paper, we study the phase relations among the pressure oscillation in
the foot, that in the pipe, and the hydrodynamic oscillation of the jet, which are the key to understanding themechanism of stabilizing and destabilizing oscillations of the flue organ pipe.
Keywords: Flue Organ Pipe, Foot, Anti-phase synchronization

1

INTRODUCTION

The sounding mechanism of flue organ pipes has been studied by many authors in the field of musical acoustics[1,2, 3, 4, 5, 6, 7, 8]. The major difficulty in studying flue organ pipes from the view of Physics comes fromthe fact that the sound source of flue organ pipes, the edge tone, is aerodynamic sound that is caused by theunsteady motion of a fluid flow with non-zero vorticities[1, 9, 10]. For studying flue organ pipes experimentallyand numerically, one has to pay special attention to the interaction between the acoustic oscillation in the pipeand hydrodynamic oscillation of the jet. The numerical method with compressible fluid solvers, which allows
us to simultaneously calculate the jet motion and acoustic oscillation in the pipe, has been recently developed
by several authors[11, 12, 13, 14].
In a recent study[15], we found with the numerical study of a 2D flue organ pipe that the foot works as aHelmholtz resonator and plays an important role in stabilizing the acoustic oscillation in the pipe. Namely,when the frequency of acoustic oscillation in the pipe is higher than the resonance frequency of the Helmholtz
resonator by almost its full-width at half-maximum, the most stable oscillation is observed, and the oscillationin the foot is anti-synchronized with that in the pipe. However, if the Helmholtz resonance frequency is nearly
equal to the acoustic oscillation frequency, the oscillations in the pipe and foot become rather unstable and theoscillation in the foot lags behind that in the pipe by nearly π /2.
In this paper, we give a short review of this study. Furthermore, to clarify the mechanism of stabilizing anddestabilizing oscillations, we consider the interactions among the acoustic and hydrodynamic oscillations. Actually, we study the phase relations among the pressure oscillation in the foot, that in the pipe, and the hydrodynamic oscillation of the jet. The volume flow supplied through the flue to the pipe periodically changes owing
to the pressure oscillation in the foot. In order to sustain and stabilize the oscillation in the pipe, the volume
flow supplied to the pipe becomes the maximum at a certain time in the period of oscillation, otherwise theoscillation is destabilized, and the foot with an appropriate volume of the body, namely an adjusted Helmholtz
resonator, assists this process.

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2

MODEL AND NUMERICAL METHOD

We use the same model studied by the recent work[15] (see Fig.1 (a)), which is a 2D analog of the flue organ
pipe studied by Ségoufin et al.[8]. The model consists of an inlet with a length of 50mm and a height of 3mm,
a foot, a flue with a length of 3mm and a height of 1mm and a pipe with a length of 141.5mm and a heightof 20mm. The volume labeled ’Out side’ indicates the outside space with the external atmosphere. Our model
has a closed pipe while the instrument studied by Ségoufin et al.[8] has an open pipe with a length of 283mm.
Thus, they have almost the same fundamental pitch around 500Hz, when they are driven by a jet. The footconsists of the left rectangular part and the right channel part connecting with the flue, where the lower channel
block below the channel is a quarter circle. The length of the rectangular part is 60mm for the model in Fig.1.
Figure 2 shows the dimensions of the mouth opening, flue and channel block. The flue has chamfers at theright end, which stabilize the jet motion and acoustic oscillation[8, 15].
Let us call the 2D model shown in Fig.1 the Reference model. To investigate the role of the foot, considering
how the geometry and volume of the foot influence the jet motion and acoustic oscillation, we treat three othermodels. Models called the Short and Long models have the rectangular parts of the foot with a length of 30mm
and 120mm, respectively. We also investigate a model without the foot and inlet called the Non-foot model.
For the numerical calculation, we adopt a compressible LES (Large Eddy Simulation) with the one-equation
sub-grid-scale (SGS) model [11, 16]. In practice, we use a compressible LES solver ’rhoPisoFoam’ in the open
source software, OpenFOAM ver.2.2.2[15]. The pressure and temperature at the rest are taken as p0 = 105 Pa
and T0 = 300K, respectively. The smallest mesh size around the mouth opening is ∆x = 0.1mm and the time
step of the numerical integration is ∆t = 5.0 × 10−8 s. For the Reference model, the number of mesh cells is120165. The mean jet velocity at the flue exit is taken as V = 6m/s in the steady state, at which a stableoscillation is observed for the Reference model. Since the height of the inlet is 3 mm, which is three times aslarge as that of the flue, the flow velocity at the left end of the inlet should be set as one-third of the desired
jet velocity, namely 2m/s. In practice, the flow velocity at the inlet is gradually increased to reach the desired
value at t = 2ms. Note that for the Non-foot model, the air flow is directly injected from the left end of theflue. The inlet boundary condition is taken for the left end of the inlet except for the Non-foot model, forwhich the left end of the flue is an inlet. The outlet boundary condition is taken for the right, left and top
walls of the volume labeled ’Out side’ in Fig.1 (a). The pressure oscillations are observed at the center of theright end of the pipe and at the upper-left point of the rectangular part of the foot. The jet velocity is observedat the center of the flue exit.
(a)(b)
Out side
Flue
Inlet

Foot

Pipe

Figure 1. Snapshots of spatial distributions of pressure and velocity for the Reference Model. (a) Pressure
distribution. (b) Velocity distribution.

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lf

l
he

w cf
hf

hcb

R

hp
chamfer

Figure 2. Dimensions of the edge, flue with chamfers and foot channel: h p = 20mm, l = 4mm, he = 1.2mm,
θ = 20◦ , wc f = 0.71mm, h f = 1mm, l f = 3mm, hcb = 20.8mm, R = 19.8mm.

3

NUMERICAL RESULT

3.1 Pressure oscillations in the pipe and footFigure 1 (a) and (b) show pressure and velocity distributions in the steady state for the Reference model. In thesteady state, pressure oscillations are well sustained in both pipe and foot, but they seem to be in anti-phasesynchronization. Indeed, at the moment shown in Fig. 1 (a), the pressure takes the minimum and maximum
values in the pipe and foot, respectively, while in another half period, the opposite distributions are observedin the pipe and foot. As shown in Fig. 1 (b), the oscillating jet collides with the edge emitting the edge tone,
which drives the resonance pipe.
Figure 3 (a) shows the time evolution of the pressure oscillation in the pipe for the Reference model comparedwith that for the Non-foot model. The pressure oscillation for the Reference model is more stable and larger inamplitude than that for the Non-foot model. This means that the interaction between the pipe and foot assists
the acoustic oscillation in the pipe to be stabilized with large amplitude. Figure 3 (b) shows the time evolutionof the pressure oscillation in the pipe compared with that in the foot for the Reference model. In the steadystate, the pressure oscillations in the pipe and foot have the same frequency at fa = 482Hz, but they are inanti-phase synchronization.
As shown in Fig.4 (a), for the Short model, the pressure oscillation in the pipe at fa = 481Hz is more unstable
and smaller in amplitude than that for the Reference model. On the other hand, the pressure oscillation in thefoot is in the same order as that for the Reference model but is more unstable. The pressure oscillations in thepipe and foot seem to strongly correlate to each other and, most of the time, the pressure oscillation in the footlags behind that in the pipe by nearly π /2.
As shown in Fig.4 (b), for the Long model, the pressure oscillation in the pipe at fa = 479Hz is slightly unstable
and is slightly smaller in amplitude than that for the Reference model. However, the pressure oscillation in thefoot is much smaller in amplitude than that for the Reference model and irregularly fluctuates in first stage ofthe time evolution, though the oscillations in the pipe and foot seem to fall into anti-phase synchronization in
the steady state.
Figure 5 shows the horizontal velocity at the center of the flue exit vx for the Reference, Short and Long
models. The behavior of vx shows the same tendency as the pressure oscillation in the pipe for all the models.
That is, for the Reference model, vx is the most stable and the largest in amplitude among the three models.
On the other hand, for the Short model, vx is the most unstable and the smallest in amplitude among thethree models. For the Long model, vx is slightly unstable and slightly smaller in amplitude than that for theReference model. The phase relation between vx and the pressure oscillation in the pipe will be discussed in
the next subsection.

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(a)

(b)
100200

Reference pipe
Non foot pipe

100100Pressure [Pa]Pressure [Pa]100200100000

Reference pipe
Reference foot100100100000

482Hz
99900
0

0.01

0.02

0.03

Time [s]

0.04

0.05

99900

0

0.01

0.02
0.03
Time [s]

0.04

0.05

Figure 3. Pressure oscillations for the Reference model. (a) Pressure oscillation in the pipe compared with thatfor the Non-foot model. (b) Pressure oscillations in the pipe and foot.
3.2 The role of the foot and relative phases among the acoustic and hydrodynamic oscillations
In the recent study[15], we found that the function of the foot depends on its volume and it acts as a Helmholtz
resonator. To consider the properties of the foot as a Helmholtz resonator, we use the foot model without
the pipe, which is formed by the flue and foot connected to the inlet (see Fig.6 (a)). The frequency of theHelmholtz resonator fH is normally given by
!
c
S
fH =
,
(1)
2π VH L
where c is the speed of sound in air, S is the cross section of the neck, L is the length of the neck and VH
is the volume of air in the resonator’s body. For the 2D model, S and VH are the height of the neck and the2D area of the foot, respectively[15]. However, to obtain the resonance frequency of the foot, we hardly use
Eq.(1), because there is no method to determine the boundary between the resonator’s body and neck for acomplex-shaped resonator like the foot. Here, we use the compressible LES to obtain the resonance frequencyof the foot separated from the pipe. To obtain the frequency response of the foot, the flow velocity at the inlet
is periodically changed as Uin = U0 sin ω t, where U0 = 1m/s.
Figure 6 (a) shows a snapshot of the pressure for the Reference model without the pipe at f = 380Hz, at which
the amplitude of the pressure oscillation takes the maximum value. Figure 6 (b) shows the frequency responseof the amplitude of the pressure oscillation in the foot. A resonance peak is observed at f = 380Hz. However,the frequency of the acoustic oscillation at fa = 482Hz is higher than the Helmholtz resonance frequency at
fH = 380Hz by nearly the value of the full-width at half maximum (FWHM).
Table 1 shows the Helmholtz frequency fH compared with the acoustic frequency fa for the Reference, Shortand Long models. When fa − fH ≥ FWHM, anti-phase synchronization occurs. If fa ≈ fH , the pressure oscillation in the foot lags behind that in the pipe by nearly π /2. From this fact, we can presume that the acoustic
oscillation in the pipe drives the pressure oscillation in the foot[15]. Actually, the theory of forced harmonic
oscillators (TFHO) well explains the change of phase difference between the pressure oscillations in the pipeand foot with change of fH .
Table 2 shows relative phases among the oscillations of pipe, foot and horizontal jet velocity at the flue exit vx ,where, for example, values in the column labeled ’pipe - foot’ indicate how much the oscillation in the pipe
lead that in the foot in phase. Roughly speaking, the relative phase of the oscillation in the pipe from that in
the foot take values at θr ≈ π for the Reference and Long models and at θr ≈ π /2 for the Short model. But,

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(a)

(b)
100200

Short pipe
Short foot100100Pressure [Pa]Pressure [Pa]100200100000Long pipeLong foot100100100000

481Hz
99900

0

0.01

0.02
0.03
Time [s]

0.04

479Hz
99900

0.05

0

0.01

0.02
0.03
Time [s]

0.04

0.05

Figure 4. Pressure oscillations in the pipe and foot. (a) Short model. (b) Long model.
14

Reference
Short
Long

12

vx [m/s]

10
8
6
4

vx

2
0

0

0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05
Time [s]Figure 5. Horizontal velocity at the center of the flue exit vx .
more precisely, it increases with fa − fH and is close to π for the Long model. For the Short mode, it is lessthan π /2, since fa is slightly less than fH by 14Hz. Such a change of the relative phase can be explained with
TFHO. TFHO also explains the decrease in amplitude of the oscillation in the foot with increasing fa − fH ,when fa − fH ≥ FWHM. However, TFHO cannot explain the destabilization of oscillations in the pipe and foot,when fa ≈ fH . The strong interactions among the oscillation in the foot, the jet motion and the oscillation in
the pipe should destabilize themselves.
To consider the interactions among the acoustic and hydrodynamic oscillations, we check relative phases among
them as shown in Table 2. As shown in the columns labeled ’foot - jet velocity vy ’ and ’jet velocity vx pipe’, the oscillation of vx lags behind the oscillation in the foot and leads that in the pipe for all models.
The relative phase of the pressure oscillation in the foot from the oscillation of the jet velocity vx decreases
with increasing fa − fH . On the other hand, the relative phase of the oscillation of the jet velocity vx from thepressure oscillation in the pipe is almost fixed in a small range (1.4 < θr < 1.5) for the Reference and Long
models, for which stable oscillations are observed, while it takes a much different value as θr = 2.301 ≈ 3π /4for the Short model, for which the oscillation is destabilized.
The change of vx indicates the change of the supply of the volume flow to the pipe. Therefore, to sustain

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and stabilize the oscillation in the pipe, the volume flow supplied through the flue to the pipe becomes the
maximum at a certain time in the period of oscillation, otherwise the oscillation is destabilized, and the footwith an appropriate volume of the body, namely an adjusted Helmholtz resonator, assists this process.
(a)

(b)
600

Foot
Inlet

Acoustic Pressure [Pa]Out Side

500

FWHM

400

Acoustic oscillation

300

200
300

FWHM

320

340

360

380

400

420

440

460

480

500

Frequency [Hz]Figure 6. Helmholtz resonator (foot) of the Reference model. (a) Snapshot of the pressure distribution at
f = 380Hz. (b) Frequency response of the Helmholtz resonator of the Reference model.

Table 1. Helmholtz frequency fH compared with the frequency of the acoustic oscillation fa for the representative models.
Reference
Short
Long
fH [Hz]
380±5
495±5
265±5
fa [Hz]482
481
479
fa − fH
FWHM-14 [Hz]
2FWHM

Table 2. Relative phases among the pressure oscillation in the pipe, that in the foot and the oscillation ofhorizontal jet velocity at the flue exit vx .
Modelpipe - footfoot - pipefoot - jet velocity vx jet velocity vx - pipe
Reference
2.857 ! π
3.426 " π1.955 " π /2
1.471 ≈ π /2
Short
1.215 ! π /2 5.068 " 3π /22.767 " 3π /42.301 ≈ 3π /4Long
3.229 ≈ π
3.054 ≈ π
1.629 ≈ π /2
1.425 ≈ π /2

4

CONCLUSIONSIn this paper, we studied the problem of how the geometry of the foot influences the acoustic oscillation in thepipe by using numerical simulations of the 2D flue organ pipe model with three different shaped foots. As aresult, it was found that the function of the foot depends on its volume and it acts as a Helmholtz resonator.
We gave a conjecture that the acoustic oscillation in the pipe drives the oscillation in the Helmholtz resonator.
The theory of forced harmonic oscillators (TFHO) well explains the change of phase difference between thepressure oscillations in the pipe and foot with change of the Helmholtz frequency fH .
For the Reference model, where the frequency of the oscillation in the pipe fa is higher than that of theHelmholtz resonator fH by almost the resonator’s FWHM, the most stable oscillation is observed among the5570

three models and anti-phase synchronization between the oscillations in the pipe and foot occurs. For theLong model with fa − fH ≈ 2FWHM, anti-phase synchronization still occurs and the oscillation in the pipe isstable, but the oscillation in the foot is very small in amplitude accompanied with small irregular fluctuations.
The results for the Reference and Long models are almost explained with TFHO. For the Short model withfa ≈ fH , the oscillation in the pipe leads that in the foot in phase by nearly π /2, which can be explained with
TFHO. However, both oscillations in the pipe and foot are destabilized by interactions among the acoustic andhydrodynamic oscillations, which can not be explained with TFHO.
To consider the interactions among the acoustic and hydrodynamic oscillations, we checked relative phases
among them. As a result, the relative phase of the jet velocity vx from the pressure oscillation in the pipe isalmost fixed in a small range (1.4 < θr < 1.5) for the Reference and Long models, for which stable oscillations
are observed, while it takes a much different value as θr = 2.301 ≈ 3π /4 for the Short model, for which
the oscillation is destabilized. This fact means that to sustain and stabilize the oscillation in the pipe, thevolume flow supplied through the flue to the pipe should take the maximum at a certain time in the periodof oscillation, otherwise the oscillation is destabilized, and the foot with an appropriate volume of the body,
namely an adjusted Helmholtz resonator, assists this process. To unravel the detailed mechanism of stabilizing
and destabilizing oscillations, we need more precise knowledge of the jet motion disturbed by the oscillationsin the foot and the pipe[1, 17]. This task is left for future work. It should also be checked by simulations of
3D models and experiments.

ACKNOWLEDGEMENTS
The present work was supported by Grant-in-Aid for Scientific Research (C) Nos. 16K05477 and 19K03655
from the Japan Society for the Promotion of Science (JSPS) and “Joint Usage/Research Center for Interdisciplinary Large-Scale Information Infrastructures” and “High Performance Computing Infrastructure” in Japan(Project IDs: jh180007-MDH and jh190010-MDH).

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