Oscillations, Waves, and Interactions - GWDG

Sound absorption, sound amplification, and flow control in ducts 87
The conspicuous low-frequency peaks (C) in Fig. 10(a) the mid-frequencies of which
are proportional to the flow velocity and which are absent in Fig. 2(b), are obviously
caused by an antisymmetric pressure distribution according to Figs. 10(b) and (c).
With the assumption that this pressure distribution is due to a single mode |m| we
infer from the measured coherence functions that cos(m·180 o ) < 0 and cos(m·120 o ) <
0, according to Eq. (2), so that |m| = 1, or |m| = 5, 7, · · · . In Figs. 11(b) and (c)
this mode manifests itself by the broad dips at low frequencies. There are some
indications (Sect. 4.1.2) that the considered mode is a hydrodynamic mode which
cannot exist but with non-zero mean flow.
The effect of the higher-order modes on the pressure drop was investigated only
with |m| = 1. In order to excite this mode, three loudspeakers with an azimuthal
spacing of 120 o and driven at 120 o phase difference were connected to the first cavity
of the resonator section. Also the axisymmetric mode m = 0 can be excited by this
means, namely by driving the loudspeakers with identical phases. First of all it turns
out that these acoustically excited modes, particularly m = 1, reach the backmost
cavity only within certain frequency bands. For frequencies up to ca. 2 kHz, these
bands more or less coincide with the frequency bands within which the coherence
function (∆ϕ = 180 o , Fig. 10(b)) is negative, respectively positive when the m = 0
mode is propagated through the resonator section. There is however some overlap
between the respective pass-bands, thus, e. g., both the modes propagate within the
band denoted by C in Fig. 10.
As expected, the static pressure can be noticeably influenced by the m = 1 mode
only within the pass-bands of this mode. This is shown in Fig. 12 for various flow
velocities. With low flow velocities the low-frequency hydrodynamic mode C is more
effective than the mode which propagates above the first acoustical cut-on frequency
of the |m| = 1 modes, however for U/c > 0.2 the pressure drop is higher with this
‘acoustical’ |m| = 1 mode. Interestingly, this latter mode is effective in practically
the same freqency range as the m = 0 mode the effects of which have been denoted
by A and have been described in the previous Section 2.1. A comparison between
the Figures 5 and 12 shows that nearly the same acoustically induced pressure drop
is achieved with both these modes. A closer inspection reveals that some kind of
interference seems to occur between the modes around 1.2 kHz: the pressure drop is
high with the m = 1 mode when it is low with the m = 0 mode and vice versa.
3 Physical mechanisms
3.1 Interaction between the mean and the fluctuating parts of the flow
3.1.1 Momentum transport by flow oscillations and stability of the mean flow
In order to study the physical mechanisms behind the sound amplification and the
acoustical control of the static pressure, we describe the interaction between the
mean and the fluctuating parts of the flow in the common way: the flow velocity
u = (u, v, w) t , the pressure p, and the density of mass ρ are decomposed into mean
and fluctuating parts, u = u + u ′ , p = p + p ′ , ρ = ρ + ρ ′ , and are substituted in the
Navier Stokes equation (conservation of momentum). Then the same type of average,