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°
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.
Results
Transient und 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
SPL
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
divx-codec.)
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
red:
regions acting as sound source,
blue:
egions acting as sound sink.
There is a surplus of acoustical power that is radiated.
[1] D. d'Humières, I. Ginzburg, M. Krafczyk, P. Lallemand, and L.-S. Luo, Multiple-relaxation-time lattice Boltzmann models in three-dimensions, Philosophical Transactions of Royal Society of London A 360(1792):437-451 (March 2002).
05.02.2005 - contact: Helmut Kuehnelt
