Oscillations, Waves, and Interactions - GWDG

86 D. Ronneberger et al.
R / L ⋅ ∆p ac / p dyn
0.06
0.05
0.04
0.03
0.02
0.01
0
U/c = 0.1
0.15
0.2
0.25
0.2 0.4 0.6 0.8 1 1.2 1.4
frequency [kHz]
Figure 12. Increase of the pressure drop along the lined duct section effected by acoustical
excitation of the m = 1 mode, as a function of the frequency and for various flow velocities.
This hypothesis is supported by the power cross-spectral density between locations
the azimuthal coordinates of which differ by ∆ϕ = 180 o and by ∆ϕ = 120 o ,
respectively. First of all, the imaginary parts of the cross-spectra are expected to
vanish, since otherwise one of the two directions of rotation of the constituent modes
(m positive or negative) would be preferred. Nevertheless, in a few cases significant
imaginary parts of the power cross-spectral density are encountered, the origin of
which was not pursued, however. Instead the imaginary parts are simply ignored
in the following. If the modes ˆpm exp(imϕ) contained in the pressure field ˆp(ϕ) are
incoherent (what is to be expected) we obtain
ℜ〈ˆp(ϕ1)ˆp ∗ (ϕ2)〉 = 〈|ˆp0| 2 〉 +
0.3
∞�
(〈|ˆpm| 2 〉 + 〈|ˆp−m| 2 〉) cos[m(ϕ1 − ϕ2)] (2)
m=1
wherein ˆp(ϕ) is the Fourier transform of one possible pressure signal measured at the
azimuthal location ϕ, and 〈· · ·〉 is the average over all possible realizations. So all
the modes with identical order |m| contribute to the real part of the cross-spectrum
in proportion to their mean square amplitudes.
The real part of the coherence function, i. e. the real part of the ratio between
the power cross-spectral and (auto-)spectral densities is plotted as a function of the
flow velocity in Figs. 10(b) and (c) for ∆ϕ = 180 o and ∆ϕ = 120 o , respectively. Also
the average of the coherence function over the flow velocity (U/c = 0.13 · · · 0.25) was
computed and was depicted in Figs. 11(b) and (c) together with coherence functions
computed according to Eq. (2) with the assumption that the pressure field consists of
one single mode namely the one which belongs to the nearest lower cut-on frequency,
in each case. Obviously this assumption is not too bad, at least for low-order modes.