Oscillations, Waves, and Interactions - GWDG

102 D. Ronneberger et al.
Nevertheless, with regard to the approximations that have been made in the previous
section, it is still open to question whether the gap between experiment and
theory can be closed by taking the shear stress into account. In fact more than one of
these assumptions turn out to be problematic. Even the very first assumption that
the velocity can be decomposed into two non-interacting parts which are determined
by the pressure and by the shear stress, respectively, cannot be maintained with low
phase velocities and small ratios ℑ{α}/ℜ{α}. In such cases a so called critical layer
yc exists with ˇω(yc) ≈ 0 so that the right-hand side of the Rayleigh equation (4) as
well as the integrand in Eq. (15) are nearly singular there. This unphysical singularity
causes, among others, the discontinuity of | ˆ δS|/| ˆ δw| when the real α-axis is passed
(see Fig. 15). In reality the shear stress adjusts itself such that the numerator in
Eq. (14) becomes small together with ˇω. Also is the Stokes layer thickness not small,
i. e. |ℜ{α}|δS = O(1) with the wavenumbers observed in the experiment nor is the
WKBJ approximation unquestionable, close to the wall.
So it appears to be necessary to solve the full problem before a final answer to
the question can be expected whether or not the observed instability waves and their
dependency on the essential parameters can be described by the wave propagation in a
homogeneous environment (i. e. that all parameters are assumed to be independent of
x). This leads to a fourth order differential equation if eddy-viscosity is assumed, and
the algorithm for the solution of the respective eigenvalue problem has to be enhanced
for this purpose. Yet it is questionable whether the response of the turbulence can
indeed be described by an eddy-viscosity in view of the fact that the turbulence is
by itself subject to flow instability so that its state can be anticipated to be far from
equilibrium. Beyond that, regarding the bifurcations of the dispersion relations and
the associated absolute instabilities, it is suspected that it is indispensable to take
the spatial development of the flow into account. This might lead to the conclusion
that the distance within which the flow is subject to a certain absolute instability
is too short for the development of a global instability [52], however that the wave
propagation is nevertheless governed by the absolute instability.
5 Summary and concluding remarks
A strong convective instability has been found with a short section of a flow duct
which was provided with a resonating lining. Sound waves that propagate in the
direction of the mean flow as well as turbulent pressure fluctuations are amplified
by this instability, and the pressure drop along the lined duct section is drastically
increased due to the power consumption by the excited amplifying instability wave.
While the sound radiation from the rear end of the lined duct section into the trailing
rigid pipe is possible only with the fundamental axially symmetric mode (m = 0), the
increase of the pressure drop is achieved also with an antisymmetric mode (m = 1)
nearly as efficiently as with the (m = 0) mode. This is in favour of technical applications
of the phenomenon because the mostly undesired radiation of the controlling
sound is avoided in this way. One such application, namely the active suppression of
low-frequency sound transmission through the lined duct section, has already been
demonstrated. The sound-induced pressure gradient can be increased by reducing