Oscillations, Waves, and Interactions - GWDG

98 D. Ronneberger et al.
an increase of the total radiated sound power nor an increase of the static pressure
drop is effected by the synchronization.
The sound pressure that originates from the mainly studied resonator section and
that causes the spectral peak A 1 in Fig. 2 exhibits some properties which might be
due to an absolute instability and which have already been mentioned in connection
with Fig. 3. With all the studied configurations consisting of at least six of the
cavities shown in Fig. 1 a particular flow velocity U jump is found at which the peak
frequency jumps from ca. 1200 Hz to ca. 1260 Hz. At somewhat lower flow velocities
the peak width decreases with increasing flow velocity while the increase of the peak
amplitude is steeper than at other flow velocities. In this range of flow velocities
also a strong change of the distribution of the pressure amplitudes is found which
finally is very similar to the distribution of a sine wave. In addition the pressure
oscillation can be synchronized with incident sound [35]. So the acoustic transmission
coefficient becomes particularly large at these flow velocities (see Fig. 4 for 16 cavities
with U jump/c ≈ 0.35). However while U jump strongly depends on the number of
cavities and on the geometry of the duct cross-section the ‘jump’-frequency does not.
Furthermore the strong oscillation of the pressure becomes weaker or even breaks
off when the flow velocity exceeds U jump. These latter facts do not belong to the
obvious concomitants of an absolute instability.
4.2 Effect of shear stress
4.2.1 Turbulent Stokes layer and its effect on the boundary condition at the wall
The essential simplifying assumptions which have been made in the previously discussed
calculations are, except for the axial homogeneity of the mean flow, (i) uniform
mean flow over the cross-section, (ii) homogeneous locally reacting compliance
of the wall, and (iii) disregarding of shear stress. (i) The introduction of an adequate
approximation of the real velocity profile has indeed an appreciable effect on
the dispersion relations (not shown here), however the general dependency on the
frequency and on the flow velocity remains unchanged. (ii) The wavelength of the
observed instability wave is much larger than the spacing of the cavities which makes
it unlikely that the inhomogeneity of the wall plays an essential role in the phenomena
to be explained here. In fact the sound amplification has been observed to increase
when the ratio between the width of the cavities and the wavelength of the instability
wave is decreased [26]. So the inhomogeneity is obviously not a precondition of the
considered instability. (iii) Finally the possible effect of shear stress remains to be
studied more extensively.
In the flow to be considered the turbulent mixing and the generation of Reynolds
stress are still effective very close to the wall here because the permeability of the
wall allows the turbulent eddies to pass through the wall. So the axial momentum of
the eddies is very efficiently transported to the wall which means that the Reynolds
shear stress τ ′′ = −ρu ′ v ′ including the wall shear stress τ w are much higher (by
nearly one order of magnitude) than with an impervious smooth wall. Nevertheless
the actual transfer to the wall is effected by viscosity within a very thin layer. Within
this layer the mean shear rate dU/dy = τ w/µ must be extremely high so that the flow