Oscillations, Waves, and Interactions - GWDG

Sound absorption, sound amplification, and flow control in ducts 95
(wavenumber) x (pipe radius)
2
0
−2
−4
U
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4
frequency [kHz]
Figure 14. Like Fig. 13, but for the m = 1 mode, 0.05 ≤ U/c ≤ 0.35 in steps of 0.05. The
circles mark the frequencies that correspond to the peak C in Fig. 10(a).
fluid oscillation. Comparing this result with the dispersion relations in Fig. 14 yields
κη = 1 in the low frequency limit.
4.1.3 Bifurcations and absolute instabilities
Besides the discrepancies between the theoretical and experimental results the dispersion
relations exhibit unexpected branch points which particularly show up with
ℑ{α}. Two of these are marked by circled crosses in Figs. 13 and 14. The existence
of such branch points has serious consequences for the stability of the flow because
they may be associated with absolute instabilities. In homogeneous, absolutely unstable
flow a locally excited perturbation temporally grows and finally spreads over
the whole flow region. By contrast, a convective instability which is supposed to
cause the spatial growth of the pressure fluctuations in the resonator section, results
in amplifying perturbations that propagate away from the source; so after the source
has been switched off the perturbation decays at fixed locations.
Whether an absolute instability exists is commonly investigated by means of the
group velocity dΩ n/dα. If for any mode n a complex wavenumber αvgr0 is found
such that dΩ n/dα = 0 for α = αvgr0, a wave packet with the complex mid-frequency
ωvgr0 = Ω n(αvgr0) will stay where it is. It then depends on the sign of ℑ{ωvgr0}
whether the amplitude of this in principle infinitely extended wave package grows
(ℑ{ωvgr0} > 0 ⇒ absolute instability) or decays. It is to be annotated that
the group velocity, particularly if it is not real, does not allow any conclusion about
the direction of ‘propagation’ of the mode, i. e. about the question, whether the
mode contributes to the wave field upstream or downstream of the source. This so
called causality problem has to be resolved (Sect. 4.1.4) if no absolute instability is
encountered and if a given mode shall be checked for convective instability. This is
the case if the mode grows in the causality direction.
The relation between absolute instability and the existence of a branch point in
the dispersion relation becomes obvious when the dispersion relation is expanded in