Oscillations, Waves, and Interactions - GWDG

90 D. Ronneberger et al.
3.2 Qualitative explanation of the observed phenomena
With these basic physical mechanisms that have been summarized in the previous
sections we can speculate about the physics behind the phenomena that have been
described in Sect. 2. To begin with, the development of the turbulent flow in the
resonator section will be considered in the presence of the hypothetical convective
instability. Obviously part of the turbulent fluctuations of the incoming flow is amplified
by the instability according to the spectra shown in Figures 2 and 10. The
radial distribution of the Reynolds shear stress which is necessarily involved in the
amplification of the fluctuations (Eq. (5)) is expected to differ from the distribution
in the fully developed channel flow (Eq. (8)). This difference inevitably leads to an
alteration of the velocity profile with the consequence that the stability of the flow
and particularly the dispersion relation of the considered instability wave depends
on the axial coordinate. Besides the development of the mean flow profile also the
considerable nonlinear interaction between the growing fluctuations contributes to
the change of the dispersion relation. This interaction will be described by means of
the oscillating component of the Reynolds shear stress in Sect. 4.2.1.
The axial growth rate of acoustically excited small-amplitude instability waves has
indeed been found to depend considerably on the travelling distance (Sect. 2.1.2).
This indicates that a significant readjustment of the velocity profile already occurs
within the short lined duct section – its length L has been varied between 1.8R ≤
L ≤ 7.2R in the experiments. Apart from the flow instability it is expected that the
permeability of the wall causes a sudden increase of the wall shear stress when the
flow enters the lined duct section (to be discussed in more detail in Sect. 4.2.1). This
is confirmed by the pressure drop along the lined duct section which is closely related
to the wall shear stress according to Eq. (6): also with short resonator sections the
pressure drop increases by nearly one order of magnitude (compared to impermeable
smooth wall). So it can be assumed that the readjustment of the mean flow profile
starts at the wall and immediately at the leading edge of the lined duct section.
With regard to the acoustically induced increase of the pressure drop we consider
the effect of the instability wave that is excited by the irradiated sound. Both, the
kinetic energy of the fluctuating part of the flow and the Reynolds shear stress will
increase in proportion to the square of the amplitude of the instability wave since
the turbulent fluctuation and the acoustically excited oscillation are uncorrelated.
With low sound amplitudes the acoustical part of the Reynolds shear stress is small
compared to the turbulent part. So the mean flow remains unaffected, and the
propagation of the acoustically excited instability wave is exclusively governed by
the developing flow which has been described in the previous paragraph. Only when
the acoustical and the turbulent parts of the Reynolds stress become comparable, the
flow will be affected and both, the static pressure drop and the acoustic transmission
coefficient will depend on the sound amplitude. In fact, the thresholds at which the
dependency on the sound amplitude becomes noticeable, are fairly identical for the
two effects.
The pressure drop can be calculated by integration of Eq. (6) over the length L
of the resonator section, provided the second term on the right-hand side of the
equation, i. e. the effect of alteration of the velocity profile can be disregarded. This