Numerical Approach for Aerodynamics around a tone hole of woodwind
instruments: an example solving moving boundary problems with
topologically change
Taizo Kobayashi(1) , Daiki Wakasa(2) , Sho Iwagami(2) , Toshiya Takami(3) , Kin’ya Takahashi(2)(1) Kyushu University, Japan, tkoba@cc.kyushu-u.ac.jp(2) Kyushu Institute of Technology, Japan, takahasi@mse.kyutech.ac.jp(3) Oita University, Japan, takami-toshiya@oita-u.ac.jp

Abstract
In this paper, we discuss the numerical reproducibility of the compressible fluid behavior around a tone holeof woodwind instruments by using compressible Large Eddy Simulation (LES). In particular, we focus on thesituation that the tone hole is opened and closed with moving a pad above the tone hole, which is regarded asa moving boundary problem with topology change. Our numerical model is based on a 2D analog of Keefe’s
experimental model, which has a tone hole placed at the center of the resonance tube. To reproduce the opening
and closing the tone hole, the pad is moved continuously. Actually, the position of the pad was continuously
changed in the order of "open - close - open". Our numerical results agree rather well with Keefe’s experimental
results. We solved the moving boundary problem with topology change under the situation of acoustics of fluidstructure interaction, and reproduced transitions between resonance and non-resonance states. In addition, thedetails of complex vortices near the tone holes clarified by this numerical calculation will be reported.
Keywords: Flue instruments, Tone hole, Moving boundary conditions, Numerical simulation

1

INTRODUCTION

The transient phenomena around tone holes of woodwind musical instrument have been studied as a long standing problems in the field of musical acoustics[1, 2, 3, 4, 5, 6, 7, 8]. The main difficulty in studying the functionof tone holes comes from the highly complicated interaction around compressible fluid dynamics (non-linear dynamics), aerodynamic sound as a linear wave motion and a moving boundary problem with topology change.
In order to achieve a rigorous study on it, we had developed an compressible LES solver resolving the moving
boundary problem[9]. Then we had studied the function of a tone hole by using a 2D model of a small
recorder-like-instrument with a tone hole[10]. In that study, the pad above a tone hole is set to a fixed position,that is, the distance between the pad and the tone hole is taken at fixed values, 0, 0.5, 1, 2, 3, 5, 10, 20mm, andthe change of pitch depending on the position of the pad is observed. The results is consistent with Keefe’s
experimental results. In the next study, we had developed numerical models in order to manipulate moving
mesh without topology change[11].
In this paper, we give a short review of recent results about this series of study. We reproduce numericallythe compressible fluid behavior around a tone hole of woodwind instruments by using compressible LES. We
focus on the situation that the tone hole is opened and closed with moving a pad above the tone hole, whichis regarded as a moving boundary problem with topology change. In order to achieve seamless numericalcalculation of moving boundary problem with topology change, we develop compressible LES solver and amethod to move and remap the mesh structure. Numerical results are consistent with Keefe’s experimental
results, and error rates of pressure between mesh remapping with/without topology change are in the sameorder ∼ 10−9 .

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2

MODEL AND NUMERICAL METHOD

We continuously use the same model which is a 1/4 scaled 2D analog of Keefe’s experimental model (seeFigure.1 left), which has a tone hole placed at the center of the resonance tube[11]. To reproduce the opening
and closing the tone hole, the pad is moved continuously.

h=1.6
U

d=10

dgap=0~1

tha=6.6
L=137.5

Figure 1. (left) Dimensions of the model. (right) typical mesh structure. “dgap” is the distance between the padand the top of the tone hole. The unit is mm.
We employ a compressible Large Eddy Simulation (LES) with the one-equation sub-grid-scale (SGS) model[12].
In practice, we have developed a extended compressible LES solver “rhoPimpleMyDyMFoam” which incluldes
a function of “DynamicMesh” for the numerical calculation of moving mesh. This solver “rhoPimpleMyDyMFoam” is based on “rhoPimpleFoam” in the open source software, OpenFOAM ver.4.1. The pressure and temperature at the rest are taken as p0 = 105 Pa and T0 = 300K, respectively. The smallest mesh size around thetone hole is ∆x = 0.17mm and the time step of the numerical integration is ∆t = 1.0 × 10−7 s. The number ofmesh cells is 24125. To excite an acoustic wave in the resonance tube at a given frequency, the velocity “U” at
the left end of the tube is changed periodically. The amplitude of U is taken as |U| = 0.024m/s in the steadystate. The frequencies of U are taken at 1269Hz and 2538Hz as the resonance frequencies of fundamental modeand 2nd higher mode, respectively.

dgap=1mm

dgap=0.8mm

dgap=0.6

dgap=0.4mm

dgap=0.2mm

closedFigure 2. Mesh structures around the pad and tone hole. The distance between the pad and the top of the tone
hole is 1 to 0mm.
To numerically solves the moving boundary problem with topology change, we use a combination of “Dy-6484

namicMesh” and “mapFields”, which controls the pad movement and the mesh manipulation. In practice, we
configure moving velocity of mesh in the configuration file “pointMotionU” and “cellMotionU”. In order torelieve the distortion of mesh structure, we use “mapFields” every interval of 0.2mm for the pad movement.
We configure this mesh structure remapping in the configuration file “mapFieldsDict”.

(a)

1

(1)

(2) (3) (4) (5)(6)

Distance: Pad <-> Tone hole [mm]0.1Velocity of Pad [m/s]

0.05

0

Closed-0.05(7) (8) (9)(10) (11)(2) (3) (4) (5)(6)(7) (8) (9)(10)

(1)

(11)

0.8

0.6

Closed

0.4

0.2

0

-0.1
0.045

(b)

0.05

0.055

0.06

0.065

0.07

0.075

0.08

0.085

0.045

0.05

0.055

0.06

Time [s]

0.065

0.07

0.075

0.08

0.085

Time [s]Figure 3. Time evolution of the pad movement. (a) the velocity, (b) “dgap”: the distance between the pad andthe top of tone hole
The actual pad movement is shown in Figure 3. The quantity “dgap” as a function of time must be continuously
differentiable in order to avoid shock waves. And furthermore, at the point (dgap= 0mm) where the topologicalchange of the mesh-boundary arises, the velocity of the pad as a function of time must be also continuously
differentiable. The maximum velocity of the pad is set to 0.1m/s.
Table 1. Mesh dimensions and corresponding time of them
Region
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)

Time[s]0∼0.053
0.053∼0.055
0.055∼0.057
0.057∼0.059
0.059∼0.06198
0.06198∼0.07
0.07∼0.073
0.073∼0.075
0.075∼0.077
0.077∼0.079
0.079∼0.082

Time of mapping mesh[s]
0.053
0.055
0.057
0.059
0.06198
0.07
0.073
0.075
0.077
0.079

Default “dgap”
1mm
0.8mm
0.6mm
0.4mm
0.2mm
0mm
0.2mm
0.4mm
0.6mm
0.8mm
1mm

Table 1 shows the mesh dimensions and the time of mesh structure remapping.

3

NUMERICAL RESULT

3.1 Resonance conditions
Our numerical model has a tone hole placed at the center of the resonance tube. Thus, in the case of fundamental frequency resonance mode, stable oscillation can be obtained irrespective of the position of pad, becauseof the node of standing wave of pressure is placed at the center of resonance tube at which the tone hole is also

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placed. On the other hand, in the case of 2nd higher mode, due to an antinode of standing wave of pressure isplaced at the center of resonance tube, stable oscillation is prohibited at the condition of opening tone hole.
3.2 Numerical result of fundamental mode (1269Hz)
Numerical results of fundamental mode are shown in Figure 4 and 5.

open to close

closedclose to open

Figure 4. Snapshots of spatial distributions of pressure p (upper line) and magnitude of velocity U (lower line).

100600100600100400100400100400100200100000Pressure [Pa]100600Pressure [Pa]Pressure [Pa]Figure 4 shows snapshots of spatial distributions of pressure p (upper line) and magnitude of velocity U (lowerline) in a steady state. Because of the antinode of standing wave of velocity at the center of resonance tube,
the velocity around the pad and tone hole is relatively active.

100200100000100000

99800

99800

99800

99600

99600

99600

0.05

0.055

0.06

0.065

0.07

0.075

0.08

0.05

0.055

0.06

Time [s]

0.07

0.075

0.08

0.05

-0.5

-1

0.5

0

-0.5

-1

0.06

0.065

Time [s]

0.06

0.07

0.075

0.08

0.05

0.065

0.07

0.075

0.08

0.07

0.075

0.08

1

Particle Velocity [m/s]Particle Velocity [m/s]

0

0.055

0.055

Time [s]

1

0.5

0.05

0.065

Time [s]1Particle Velocity [m/s]

100200

0.5

0

-0.5

-1

0.055

0.06

0.065

0.07

0.075

0.08

0.05

0.055

0.06

Time [s]0.065Time [s]sampling position: “Left”, “Center”, “Right” side in the pipe
Figure 5. Time evolution of pressure p(upper line) and horizontal particle velocity Ux (lower line) oscillations.
Each line color corresponds to each mesh dimension in Table 1.
Figure 5 shows time evolution of pressure p (upper line) and horizontal particle velocity Ux (lower line) oscillations. The sampling points of p and Ux are as follows: All sampling points are on the center line of the tube.

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“Left” is the point at 1mm right from the left end of the tube. “Center” is the point at the center of the tube.
“Right” is the point at 1mm left from the right end of the tube.
As shown in Figure 5, our numerical method for moving mesh with topology change works very well, during
the position of the pad is continuously changed in the order of "open - close - open". Amplitudes of p andUx are large at the antinode positions and small at the node positions, i.e. behaviors of p and Ux obey theresonance condition of fundamental mode.
3.3 Numerical result of second higher mode (2538Hz)
Numerical results of fundamental mode are shown in Figure 6 and 7.

open to close

closedclose to open

Figure 6. Snapshots of spatial distributions of pressure p (upper line) and magnitude velocity U (lower line).
Figure 6 shows snapshots of spatial distributions of pressure p (upper line) and magnitude of velocity U (lowerline) in a steady state. Because of the resonance condition, i.e. node of standing wave of velocity at the centerof resonance tube, the velocity around the pad and tone hole is inactive.
Figure 7 shows time evolution of pressure p (upper line) and horizontal particle velocity Ux (lower line) oscillations. The sampling points of p and Ux are as follows: All sampling points are on the center line of the tube.
“Middle-Left” is the point at 34.375mm right from the left end of the tube. This point is a node of pressure in
the 2nd higher mode condition. “Center” is the point at the center of the tube. “Model-Right” is the point at
34.375mm left from the right end of the tube. This point is also a node of pressure in the 2nd higher modecondition.
As shown in Figure 7, our numerical method for moving mesh with topology change also works very well,during the position of the pad is continuously changed in the order of "open - close - open". Amplitudes of pand Ux are large at the antinode condition and small at the node condition, i.e. behaviors of p and Ux obey theresonance condition of 2nd higher mode. In this case, transitions between resonance and non-resonance states
are observed clearly, when the pad is closing and opening.
3.4 Error estimations of mesh manipulation
Table 2 shows errors of pressure at the “Center” sampling point. In the case of “increase mesh (0.8mm →1mm)”, error is bigger than the case of “decrease mesh (1mm → 0.8mm)”. We assume that this increasing oferror rate is caused by a higher order complement of physical quantities when they are mapped from on small
mesh onto large one.
Other error rates are in the same order ∼ 10−9 . Thus, the function of moving boundary calculation with topology change in our numerical model is ensured.

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100300

100200100200100200100100100000

99900

99800
0.05

Pressure [Pa]100300Pressure [Pa]Pressure [Pa]100300100100100000

99900

0.055

0.06

0.065

0.07

0.075

99800
0.05

0.08

0.055

0.06

0.07

0.075

99800
0.05

0.08

0

-0.2

-0.4

0.2

0

-0.2

-0.4

0.065

0.06

0.07

0.075

0.08

0.05

Time [s]

0.065

0.07

0.075

0.08

0.07

0.075

0.08

0.4

Particle Velocity [m/s]

0.2

0.06

0.055

Time [s]0.4Particle Velocity [m/s]Particle Velocity [m/s]0.065Time [s]

0.4

0.055

10000099900

Time [s]0.05100100

0.2

0

-0.2

-0.4

0.055

0.06

0.065

0.07

0.075

0.08

0.05

0.055

0.06

Time [s]0.065Time [s]sampling position: “Middle-Left”, “Center”, “Middle-Right” side in the pipe
Figure 7. Time evolution of pressure p (upper line) and horizontal particle velocity Ux (lower line) oscillations.
Each line color corresponds to each mesh dimension in Table 1.
Table 2. Errors of pressure at the “Center” sampling point
Mesh manipulation
decrease mesh (1mm → 0.8mm)increase mesh (0.8mm → 1mm)
topology change (Open → Close)
topology change (Close → open)

4

Default mesh dimension
1mm(Before)
0.8mm(After)
0.8mm(Before)
1mm(After)
0.2mm(Before)
0mm(After)
0mm(Before)
0.2mm(After)

Pressure [Pa]100004.3572
100004.3581
100003.5042
100003.5021
100077.2729
100077.2735
100120.6458
100120.6459

Error rate
9.0 × 10−9−2.1 × 10−8
6.0 × 10−9
1.0 × 10−9

CONCLUSIONSIn this paper, we study the numerical reproducibility of the compressible fluid behavior around a tone hole ofwoodwind instruments by using compressible LES. In particular, we focus on the situation that the tone hole isopened and closed with moving a pad above the tone hole, which is regarded as a moving boundary problem
with topology change.
Our numerical model is a 2D analog of 1/4 scaled Keefe’s experimental model, which has a tone hole placedat the center of the resonance tube. For seamless numerical calculation of moving boundary problem with
topology change, we have also developed our own compressible LES solver “rhoPimpleMyDyMFoam” on an
open source software OpenFOAM ver.4.1. This solver is applied to our numerical tone hole model and ensured
that the functions of mesh manipulation as moving boundary problems with/without topology change work very
well. From the view point of numerical reproducibility, the most important result is as follows. The transitions
between resonance and non-resonance arise for the calculation of the second mode, when the position of the pad
is continuously changed in the order of "open - close - open". They are consistent with Keefe’s experimental
results. Error rates of pressure at the mesh manipulation is also consistent with this reproducibility.

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It is necessary to check that the same results occurs for a 3D model, and reproduce the pitch change in the
opening and closing the tone hole of more practical woodwind instrument model.

ACKNOWLEDGEMENTS
The present work was supported by JST CREST Grant Number JPMJCR1501, japan, 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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Fluid Mech. 44 1-25 (2011).
[3] D. Keefe, “Theory of the single woodwind tone hole”, J.Acoust.Soc.Am. 72, 676–687 (1982).
[4] D. Keefe, “Experiments on the single woodwind tone hole”, J.Acoust.Soc.Am. 72, 688–699 (1982).
[5] D. Keefe, “Woodwind air column models”, J.Acoust.Soc.Am. 88, 35–51 (1990).
[6] C.J. Nederveen, J.K.M. Jansen, R.R. van Hassel, “Corrections for woodwind tone-hole calculations”, Acustica · acta acustica, 84, 957–966 (1998).
[7] V. Dubos, J. Kergomard, A. Khettabi, J.-P. Dalmont, D.H. Keefe, C.J. Nederveen, “Theory of sound propagation in a duct with a branched tube using modal decomposition”, Acustica · acta acustica, 85, 153–169(1999).
[8] J.-P. Dalmont, C. J. Nederveen, V. Dubos, S. Ollivier, V. Mśerette, E. teSligte, “Experimental determinationof the equivalent circuit of an open side hole: linear and non linear behaviour”, Acta Acustica united with
Acustica, 88, 567–575 (2002).
[9] Y. Ito, T. Kobayashi, K. Takahashi, T. Takami, A. Nishida, M. Aoyagi, “Reproduction of Transitions among
Notes on an Air-reed Musical Instrument with Compressible LES Combined with Moving Boundary Technique”, in Proceedings of Open Source CFD International Conference 2011, Volume Session V, A 1 (OSCIC2011) (2011)[10] T. Iwasaki, T. Kobayashi, K. Takahashi, T. Takami, A. Nishida, M. Aoyagi, “Numerical study on thefunction of tone holes of a recorder like instrument from the viewpoint of the aerodynamic sound theory”,
Proc. Mtgs. Acoust. 19, 035024 (2013)[11] T. Kobayashi, H. Matsuda, T. Takami, K. Takahashi, “Numerical study for the function of moving pad ontone hole acoustics”, The Journal of the Acoustical Society of America 140(4):3037-3037, (2016)[12] C. Wagner, T. Hüttl, and P. Sagaut, eds. Large-Eddy Simulation for Acoustics, (Cambridge Univ. Press, New
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