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Three-dimensional simulation of the flute
using the Lattice Boltzmann Method

The three-dimensional MRT lattice Boltzmann method (LBM) [1] was used to study the sound generation in flue pipes like flutes, recorders or flue organ pipes. LBM simulates the dynamics of an ideal, isothermal gas at low Mach numbers and automatically incorporates acoustics.

A small, stopped pipe with recorder-like dimensions of the mouth was used as a first test object:

dimensions of the pipe: 60 x 10 x 10 mm
dimensions of the flue: 15 x 7 x 1.5 mm
distance flue exit – labium: 7 mm
inclination of the labium: 14°

To stabilize the simulation numerically a viscosity had to be chosen six times higher than air. Therefore the speed of the air jet also had to be six times higher (35 m/s) to get a jet similar to that in a recorder.


Transient and stationary oscillations

Air density:
red: inside, near the stopped end of the pipe: 160 dB SPL
blue: outside, 5 cm away from the mouth of the flute: 130 dB

The rising time of the jet was chosen quite fast. The velocity of the air flow in the flue builds up quickly and the jet is formed. Shortly after that a stationary oscillation is established in the flute.

Motion of the air jet during start up and stationary oscillations (video)

The air jet in the flute is visualized by an isovelocity surface. During the animation there is a "zoom" into the core of the jet:
starting at 30% of the maximum jet speed (blue) to 60% of the maximum jet speed (red).
(This video has 3.5 MB and needs a

Sound spectrum

Sound spectra
red: inside, at the stopped end of the flute,
blue: outside, 5 cm away from the mouth of the flute

The spectra of the radiated sound (blue) and the sound of the standing wave inside the pipe (red) were normalized with respect to their fundamental in order to be comparable. Higher harmonics are radiated more efficiently and in the pipe a standing wave is formed with a very strong fundamental. The jet feeds the standing wave with a spectrum containing all harmonics. Therefore the sound also contains even harmonics. The stopped pipe does not support them but damp them highly.

Sound production: Dependency on the shape of the labium

Is the lattice Boltzmann method able to simulate small geometry changes? To test this the simulations of a flute with a sharp labium was compared with a flute with a round labium. The sharp labium produces some stronger higher harmonics than the round labium. But this effect is not very pronounced probably because the pipe resonance is rather dominated by the fundamental.


sharp labium                        round labium

Normalized spectra of the radiated sound:
red: sharp labium, blue: round labium

Interaction of the flow field of the jet with the sound field of the standing wave in the flute

The air jet can be regarded as to consist of two parallel shear layers. They form a vortical flow field that interacts with the sound field of the standing wave in the mouth of the flute. According to the vortex sound theory of Howe and Doak in every period of oscillation both rotational energy of the vortical field is converted into sound - sound is produced - and the other way round sound is absorbed by converting sound into vorticity.

Sound production and absorption by flow-acoustic intercation:
red: regions acting as sound source, blue: egions acting as sound sink.
(This video has 9 MB and needs a divx-codec.)


Energy transfer averaged over one period
regions acting as sound source, blue: egions acting as sound sink.
There is a surplus of acoustical power that is radiated.


Publications (7)

Kühnelt, Helmut (2007)
"Vortex sound in recorder- and flute-like instruments: Numerical simulation and analysis,"
in Proceedings of the International Symposium on Musical Acoustics, ISMA 2007 (Institut d'Estudis Catalans, Barcelona, Spain).
Kühnelt, Helmut (2005)
"Simulation and analysis of the flow-acoustic interactions in the mouth of flute-like instruments,"
in Forum Acusticum Budapest 2005, 4th European Congress on Acoustics (Budapest, Hungary) p. 417-422.
Kühnelt, Helmut (2004)
"Simulating the sound generation in flutes and flue pipes with the Lattice-Boltzmann-Method,"
in Proceedings of the International Symposium on Musical Acoustics, ISMA 2004 (Nara) p. 251-254.
Kühnelt, Helmut (2003)
"Simulating the mechanism of sound generation in flutes using the lattice Boltzmann method,"
in Proceedings of the Stockholm Music Acoustics Conference 2003, SMAC 03. Volume I, edited by Roberto Bresin (Royal Swedish Academy of Music, Stockholm (SE)) 1, p. 401-404.
Kausel, Wilfried and Kühnelt, Helmut (2003)
"Modelling Wave and Fluid Propagation in Wind Instruments - Methods and Applications,"
in Proceedings of the First Congress of AAAA and Third Congress of Slovenian Acoustical Society (Mirko Cudina, Portoroz) 2003, p. 335-348.
d'Humières, D.; Ginzburg, I.; Krafczyk, M.; Lallemand, P., and Luo, L. S. (2002)
"Multiple-relaxation-time lattice Boltzmann models in three-dimensions,"
Philosophical Transactions of Royal Society of London A 360(1792), 437-451.
Kühnelt, Helmut (2002)
Simulation dreidimensionaler Schallfelder in Flöteninstrumenten mittels Lattice-Boltzmann-Methode
Institut f. Wiener Klangstil, University of Music and Performing Arts.
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