Oscillations, Waves, and Interactions - GWDG

Sound absorption, sound amplification, and flow control in ducts 97
The condition (ii) is frequently violated, in particular, if free shear layers with finite
thickness are approximated by infinitesimally thin vortex layers (then in many cases
a hydrodynamic mode is encountered with ℑ{Ω(α)} → ∞ for α → ∞ + 0i. This
problem has been investigated by Jones and Morgan [38] and later by Crighton and
Leppington [39], and the Briggs criterion has been modified by the latter authors.
They propose to trace α = inv{Ω n}(ω) while running on a quarter arc of a circle in the
first quadrant of the ω-plane, i. e. from i|ω| to |ω| with |ω| kept constant. Rienstra and
Peake [16] have compared these two criteria for the wave propagation in resonatingly
lined circular ducts which are comparable to our case, except for the much higher
quality factor of our resonators. The authors find that the causality direction indeed
differs between the two criteria in several cases. However the authors do not discuss
the existence of branch points of the dispersion relations which cause a violation of
the condition (i). We find a great number of branch points with ℑ{ωvgr0} > 0 not
depending on whether a flat velocity profile or a more realistic representation of the
turbulent flow in the lined duct is assumed [35]. We even suspect that all the modes
in the forth quadrant of the α-plane are connected via branch points.
4.1.5 Search for experimental indications of absolute instability
The existence of branch points with ℑ{ωvgr0} > 0 in the theoretical dispersion relations
raises the question whether there are experimental observations that hint at
absolute instabilities of the flow and have been overlooked up to now. In most cases
a steady, however absolutely unstable flow assumes a spatially and temporally periodic
state the frequency and wavenumber of which is not far from ℜ{ωvgr0} and
ℜ{αvgr0} – a salient example is the vortex street in the wakes of bluff bodies [40]. An
axisymmetric oscillatory state in the resonator section should radiate sound into the
rigid pipes connected to the resonator section, and the distribution of the radiated
pressure amplitudes should be similar to the distribution of a sine wave; by contrast
turbulent fluctuations that have passed a narrow-band filter or a narrow-band
amplifier are normally distributed.
Brandes [29,30] indeed finds narrow peaks in the pressure spectra which are radiated
into the pipe upstream of the lined duct section. While the peak denoted by
D in Fig. 2(c) reflects only a weak example of this kind of instability, much stronger
peaks have been found with higher resonance frequencies (2934 Hz in most of Brandes’
experiments). The signals that result from this instability have all the mentioned
properties of self-excited oscillations. However the frequency of the oscillation is inversely
proportional to the the spacing of the cavities. So it is very likely that the
oscillations are generated by a feedback loop rather than by an absolute instability. It
is remarkable that also narrow-band sound amplification is observed at the frequencies
of these oscillations; the amplified sound is radiated in the direction against the
flow [25], in contrast to the sound amplification shown in Fig. 4. Brandes explains this
kind of sound amplification by partial synchronization of the self-exited oscillation
with the incident sound. Thereby the amplitude of the synchronized sound radiation
is proportional to the much smaller amplitude of the incident sound wave. However
with high incident sound amplitudes, the self-excited oscillation is completely synchronized,
and it is even possible to considerably change its frequency. Yet, neither