Oscillations, Waves, and Interactions - GWDG

94 D. Ronneberger et al.
(wavenumber) x (pipe radius)
12
10
8
6
4
2
0
−2
−4
−6
−8
U
U
U
0.6 0.8 1.0 1.2 1.4
frequency [kHz]
Figure 13. Dispersion relations of a m = 0 mode in a lined duct with dimensions according
to Fig. 1. The flow velocity of the uniform compressible mean flow has been varied between
0.1 ≤ U/c ≤ 0.3 in steps of 0.01. The real (green) and the imaginary (red) parts of the
wavenumber are shown in the upper and the lower part of the Figure, respectively. The
experimental data (colored, from Ref. [29]) and the corresponding theoretical values (black)
refer to U/c = 0.17(+), 0.20(△), 0.23(◦), 0.25(⊓⊔). The (blue) circled cross and the dashed
lines describe a bifurcation (see text).
by circles which happen to appear at equivalent points of the dispersion relations,
characterized e. g. by the maximum of ℜ{α}. Likewise it can only be speculated
that the structure found in the theoretical dispersion relations at frequencies around
1000 Hz might be related to the increase of the pressure drop that is achieved by the
excitation of the antisymmetric mode at similar frequencies (see Fig. 12).
The instability denoted by C in Fig. 10 and marked by the circles in Fig. 14 obviously
does not rely on the compressibility of the air. The dynamics behind this
instability is very similar to the Kelvin-Helmholtz instability which results from a
balance of the accelerating forces that the fluid elements experience on both the
sides of shear layers due to the unsteady deflection of the shear layer. In the present
case the fluid flow in the interior of the duct behaves like an oscillating jet which
forces the surrounding fluid in the cavities to oscillate in the azimuthal direction. So
the accelerating forces are proportional to ˆη0 ω 2 in the cavities and proportional to
ˆηU(ω − αU) 2 in the flow. The sum of these forces has to cancel out which yields
α = (ω/U)(1 ∓ iκη) wherein κη = � ˆη0/ˆηU depends on the geometric details of the