Vortex sound of the flute

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Vortex sound of the flute

• An Overview

Andreas Bamberger
Phys. Institute of the Albert-Ludwig-University
Freiburg
2
What do we know about the excitation mechanism in
flutes?

• After basic papers by Cremer ('50's) pressure
  drive (Coltman 1968) vs. volume flow drive
  (Fletcher 1976)
• Vortex sound was stated by Powell in 1964
• Basically jet-edge system with acoustic feedback
  mechanism serves as base
• Howe (1975) derived an expression for power based
  on the coriolis force on a moving vortex together
  within acoustic field of the resonator,
• Experimental technique for quantitative results
  Particle Image Velocimetry (PIV)
• Quantitative velocity determination with high
  resolution imaging properties
• this investigation is done with the flute for the
  following reason The high register has rather
  low content of higher partials (few ). The
  higher harmonics are in principle tractable, but
  complicate things.

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What is the vortex sound?

• the source terms follows from the stagnation
  enthalpy B and results in a vortex source term
  with the vorticity ? rot U and speed U (Powell
  1964, Howe 1975)
• no viscous terms, small Mach numbers (v/c ltlt1),
  and long wave length approximation (consequence
  integrate source terms over the volume with their
  proper phases, coherency)
• given the Greens function at the remote listener
  results in a fictitious velocity potential at
  the source which in turn interact with the
  coriolis force (Howes analogon)

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Vortex sound

• one is reminded on the Magnus effect lift of a
  vortex moving with a velocity U (see also Lorentz
  force)
• the power P
• where T is the periode. The acoustical
  field is rotation free (Ugrad ) i.e. vortex
  crosses equi-potential lines and delivers work.
• Some examples ventillator noise, flapping noise
  of landing helicopters, noise generation of high
  lift wing (Airbus)

gains pot. energy
force density velocity
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A recent result using Howes analogon

• For numerical simulation of aeolian noise from
  the pantograph of Shinkansen train the authors
  Takaishi at al. 2006 use Howes formulation
  taking care of the proper description of the
  velocity potential within a truncated volume. The
  sound is evaluated with a fictitious velocity
  potential.
• The listener being far away should not hear a
  part the Karman vortex street outside the volume
  considered (realistically would have dissipated
  at a short distance from the obstacle).

flow
source region
(fictitious) rotation free flow
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Why did it raise my interest?

• Being familiar with nuclear physics Weisskopf
  Unit is the electromagnetic analogon to the
  sound sources (deals with the life time of
  ?-transitions in nuclei with a single nucleon
  being bound in a harmonic potential)
• Comparable with respect to the long wave length
  approximation used
• Coherency of the sources has to be considered
• (also different types of transitions (e.g E2 and
  M1) are coherently superimposed)

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How did I got acquainted with PIV

• Playing flute since some time I was interested in
  vocal tract resonances
• I wanted to observe my own vocal tract while I
  was playing
• MMR Since it was only possible with wood I took
  a recorder, clearly seen resonance with a mike
  attached to a tube
• The effect appears in higher partials using a
  mock-up system with an end blown flute, but less
  pronounced with a Boehm flute
• The then used jet diagnostics was hot wire
  anemometry
• Particle Image Velocimetry (PIV) is a
  quantitative tool for investigation of fluid
  flow

investigation region
U
?x
air loaded with seeding droplets exposed to a
pair of laser shot with a time difference ?t
U ?x/?t
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Program

• Experimental technique
• Assessment of the jet vorticity
• Creation of the possibly rotation-free
  acoustic velocity field
• Extract source terms by spatial integration and
  time wise integration
• Verify the result by an absolute SPL measurement
• Make an estimation of the coriolis force extract
  the phase with respect to the acoustic field
• Remarks
• Desired requirement is the unobstructed
  observation of the jet for a full cycle
• Controlled geometry for the mouth - labium system

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Overview of the experimentregularly operated by
students in practice courses
pressure gauge
endoscope
seeding generator
CCD-camera with double shutter
microfone
flute
flute and mouth replica
speaker
double Nd-YAG laser head
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Setup for Endo-piv
y
x
z
as viewed by the camera
FLUE thickness d1mm

• the endoscope allows to have the first lens flush
  with chimney of the embouchure,
• the window provides the light sheet inside the
  head joint from below onto the labium

FLUTE
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Procedure of the measurement

• flute operated at 1100 Hz at various pressures
• double pictures taken with typ. 2-10 µs laser
  pulse distance
• 5 pictures for averaging (there are small eddies
  varying from shot to shot) taken at 16 points
  along a period,
• endoscopic PIV requires to correct for nonlinear
  mapping of the fish eye lens

• cross correlation for displacement ?x of the
  seeding droplets, apply velocity filter and
  averaging over some grid area
• same sequence for the acoustic field measurement,
  jet switched off, containment of seeding realized
  with a bag

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Measurement of the jet at 20 m/s
Period of 850 µs sampled with 16 Phases
phase 3
upper lip lower lip
LABIUM

FLUE
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Creation the acoustic field
Remark Obvious to realize for open ended
resonators
resonator corresponds to the fingering of D5
horn weakly coupled to the flute
position of the microphone (phase determination
and as reference for the far field power
determination)
speaker with horn driven by sine wave generator
set to the same frequency
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velocity fields at 20,0 m/s
jet acoustics
velocity field with external excitation
acoustics
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jet moving up (phase 6)
acoustic field at the same phase
jet moving up
field up
vorticityof the jet
power density
-?(?xU)vac
?rot U
spatially integreated power has a positive value
positive net value because of curvature
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Phase advanced by 180
jet moving down and acoustic field downwards
vorticity of the jet
power density
?rot U
-?(?xU)vac
?rot U
integrated power positive again
acoustic phase switches sign
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20,0 m/s vorticity and power density
clear vortex shedding but adds little to the
final result
The integration over both shear layers at every
pase results in a net positive power! The main
point is that there is a net vorticiy across the
jet changing sign with the phase, and the
acoustic field does it also. We are locking at a
large eddy.
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Models and interpretation by discrete vortices

• Jet edge system, see Holger et al. 1968

thinn jet
thick jet

convection speed
Raley instability
2 shear layers

net vorticity large eddy
treated by Meissner 2002

resonator tube
this experiment
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Phase dependence of the instantaneous power
double peak push pull action
24 m/s
36 m/s
Note Negative values might occur depending on
the geometry of the jet labium position
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Concentration of the power density near the
labium and spanwise distribution
Pvort/?x
piece wise integration in strips of 0.5 mm
The effective spanwise width of the jet is
dermined by a scan Wz 6 mm A check was carried
out for jet homogeneity along z- axis by rotating
camera and light by 90.
gt
jet
in this region acoustic field is highest
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Power determination by integration over the
volume of 1cm3 at the embouchure

• the range of investigated jet velocities 19 37
  m/s
• input power 26 to 200 mW
• power range a factor 20
• comparison with measured free field power
  measured with the same setup in a reverberation
  room at the Fraunhofer Institute/Stuttgart

of reference microphone at the resonator
The remarkable result is that there is just a
factor 2 between the two methods
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Efficiency of acoustic power of the flute

• the integration over the width results in a near
  cancellation of the vorticities of the upper and
  lower shear layers at the flue exit
• the interaction with the standing wave of the
  resonator turn the jet towards the labium into a
  large structure eddy and results in an energy
  producing vortex sound source (there is an order
  of magnitude reduction in the vorticity as the
  integration over the width is done)
• input power and efficiency with Aflue cross
  section of the flue

The value is already known from other
investigations. Some of the power is spent for
the excitation of the standing wave, and some
part as a flow past the embouchure.
ISMA2007, 10.9.2007
A. Bamberger, Phys. Inst. of University Freiburg
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Interpretation of the vortex sound as pressure
gradient across the jet (Coltman 1965)

• vector algebra gives us
• power
• last term in our case vanishes
• for finite step the first term corresponds to a
  pressure difference for a volume V?yS and with
  vac to the power. This holds also for our case of
  large scale eddy.
• the dynamic pressure difference is the driving
  force for the acoustic field

0
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Phase shift determination between coriolis force
and acoustic field
This set of measurement is done at a fixed
geometry over a jet velocity range between 19 and
28 m/s

• Measured phase shift with respect to the
  acoustic. field
• transverse position of the jet
• net rot(UJet), integration over a relevant strip
  near the labium
• net coriolis force (y component only)

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Estimation of the coriolis force
phase of upwards moving jet
rotUj ? in color code
vertical component of ?ac
phase shift
Coriolis force density is a double layer
vertikal components are integrated over (main
component when acoustic field is considered)
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Phase shift for the fundamental component ofthe
coriolis force with respect to the acoustic field
plot as a function of the Strouhal number
net coriolis force
this experiment
-25ltFlt-57 with 19ltUjetlt27.5 m/s !
red line is in agreement with data by Coltman
displayed for comparison
Conclusion The phase changes with Ujet , the
slope being similar!
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Discussion of the results

• Over blowing condition of the flute eases the
  evaluation. For lower pitch like 560, or 256 Hz
  the contribution of higher partials hampers the
  evaluation however, this regime is
  esthetically not unimportant, see organ pipes
  which use to have a rich spectrum.
• There might be a concern that there are vortices
  near the labium at a scale which not resolved by
  this investigation (Fabre, B. et al.,(1996). This
  cannot be excluded at a scale of 0.14 mm.
  Nevertheless the vortex power versus the free
  field sound pressure level does not indicate a
  saturation.
• The acoustic field near the labium at high jet
  velocities reaches 5 - 6 m/s. It is not excluded
  that there is a vortex shedding at the high end
  of the investigated jet velocities (Disselhorst
  etal.,1980). However, as it demonstrated in this
  experiment the sound sources are spatially
  distributes over some range before the jet
  actually reached the tip of the labium its of
  minor importance.

tip of the labium
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Discussion of the results cont.

• The overestimation of factor 2 might be an effect
  of radiation at the foot which would represent
  the end correction radiation. Interference
  pattern of the two sources are well known.
• In spite of the characteristics of the vortex
  sound sources as a pressure gradient across the
  jet, the volume flow might play an equally
  sizable role.
• In this respect it is interesting to note that
  the thickness d of the jet never explicitly
  enters in this evaluation. In second order it
  matters as there is a gradient of field vac.
  Flute players try to economize the total flow at
  a given SPL. Systematic experiments with
  different d as a parameter would be interesting
  Is the net vorticity affected ? What is the
  volume flow?

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Estimates of the power as a function of the jet
velocity
n in this experiment somewhat higher 5 - 6
main result of this investigation
This is measured for a given d what happens
with d1/2 mm at the same pres ?
exp.
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Conclusion

• vortex sound power is positive averaged over a
  cycle
• also instant power tends to be positive, 2
  maxima/period for symmetric configurations, in
  general also absorbing part may occur
• The power of vortex source is up to a factor 2
  higher than power determined in the far field
• Net coriolis force and acoustic field has phase
  difference which tends to affect the power as the
  jet speed increases
• the power as a function of the suggests scaling
  laws being consistent with both the pac and
  pvort.
• What happens when the thickness of the jet d is
  changed?