Oscillations, Waves, and Interactions - GWDG

Sound absorption, sound amplification, and flow control in ducts 93
vortex layer, a jump condition for the wall-normal velocity component can be derived.
In a wave-like field according to Eq. (3)
ˆv⊥ = ˇω
ω ˆvw
is obtained wherein ˆv⊥ and ˆvw are the wall-normal velocity components of the wall
and the fluid elements, respectively, and ˇω = ω−αU is the Doppler-shifted frequency
in the frame of reference of the fluid elements. So while the boundary condition at
the wall is a local relation between the wall-normal velocity and the pressure for zero
mean flow, it becomes non-local due to the dependency on α in the presence of mean
flow.
4.1.2 Dispersion relations
In view of the sound amplification in the resonator section Großer [34,36] has studied
the potential of various common approaches to the theoretical treatment of the
wave propagation in lined ducts. Unfortunately none of these approaches results in
dispersion relations that at least approximately exhibit the observed dependencies
on the frequency and the flow velocity. One such approach will be discussed in the
following in order to identify the immanent problems which still have to be resolved.
In the Figures 13 and 14 both, the real and the imaginary part of the wavenumber are
shown as functions of the frequency for an axisymmetric (m = 0) and an azimuthal
mode (m = 1), respectively. These modes have some properties in common with
hydrodynamic instability modes: they travel in the direction of flow (except for a
certain frequency interval for m = 1) while their amplitudes grow, and they exist
only with superimposed mean flow, i. e., with U → 0 ⇒ α → ∞ which however
is found to be valid only at low frequencies. In the respective eigenvalue problem,
uniform compressible mean flow in a circular pipe has been considered. The acoustic
admittance of the wall can be reliably computed from the geometrical parameters
of the resonator cavities [29], and the jump condition (11) has been used to fix the
boundary condition.
Some data points obtained from the pressure distribution in the lined duct section
(Sect. 2.1.2 and Ref. [29]) are displayed in Fig. 13, and the respective theoretical values
are correspondingly marked. While, with U/c = const = 0.23, the measured phase
constant ℜ{α} strongly increases as a function of the frequency and the spatial
growth rate −ℑ{α} exhibits a clear maximum at 1077 Hz, the theoretical values
only slightly depend on the frequency above the resonance frequency (840 Hz), and
even worse, ℜ{α} has the opposite slope. When in contrast the flow velocity is
varied and the data points are taken at the frequencies of maximum spatial growth
of the instability wave (Fig. 6), then the general trend is reproduced for ℜ{α} but
not for ℑ{α}. Also the dispersion relations shown in Fig. 14 for m = 1 are only
marginally compatible with the observations. After all, the wavenumber depends
nearly exclusively on ω/U at low frequencies, and this kind of dependency has also
been observed with the spectral peaks and the associated coherence of the pressure
fluctuations in the backmost cavity (see the peak marked by C in Fig. 10). The
theoretical wavenumbers at the respective peak frequencies are marked in Fig. 14
(11)