ERCOFTAC Bulletin - Centre Acoustique

20log10(Jm(kD/2sintheta)/Jm(kD/2sintheta)), dB
10
0
-10
-20
-30
-40
-50
0 1 2 3 4 5
(m)
St=0.2
St=0.5
St=0.8
St=1.1
Figure 1: Relative efficiencies of azimuthal mode m compared
to the axisymmetric case, considering θ = 30 ◦ ,
M = 0.9 and r = D/2.
expect considerable sound radiation from structures in
the flow with significant azimuthal coherence. For higher
frequencies the differences become gradually lower, and
m = 0 and 1 have similar efficiencies at a Strouhal number
around 1. For lower Mach numbers, the dominance of
mode 0 is accentuated, as seen in Figure (2) for M = 0.6.
20log10(Jm(kD/2sintheta)/Jm(kD/2sintheta)), dB
10
0
-10
-20
-30
-40
-50
0 1 2 3 4 5
(m)
St=0.2
St=0.5
St=0.8
St=1.1
Figure 2: Relative efficiencies of azimuthal mode m compared
to the axisymmetric case, considering θ = 30 ◦ ,
M = 0.6 and r = D/2.
If we are interested at the peak directions and frequencies
for subsonic jet noise, which correspond to low angles
and low Strouhal numbers, we can thus concentrate on
the axisymmetric mode. Moreover, if kr sin θ ≪ 1, we
can make a further approximation by taking J0(kr sin θ)
to be 1, and the axisymmetric source is equivalent to a
line distribution of quadrupoles. The far-field pressure is
given by
×
p(x, m = 0, ω) = − k2 cos 2 θe −ikx
e −ikz cos θ dz
2x
T11(z, r, m = 0, ω)rdr. (3)
A further interesting result is given by Crow[10, 7],
who assumed a line distribution consisting of a hydrodynamic
wave with an envelope given by a Gaussian,
T11(z, r, m, ω)rdr = 2ρUũe −ikHz e − z2
L 2 , (4)
to model the amplification, saturation and decay of an instability
wave (or wavepacket). For this model the sound
radiation is given by
p(x, m = 0, ω) = − ρ0UũM 2 c (kHD) 2L √ π cos2 θ
e
8x
−ikx
×e − L2k 2
(1−Mc cos θ)2
H
4 . (5)
The exponential dependence of the sound field on θ,
present in Eq. (5), has been labelled superdirectivity[9, 6].
This dependence is significant if the axial extent of the
wavepacket, which is related to L, is comparable to the
acoustic wavelength.
This is in contrast with the results for compact sources.
A compact eddy has a directivity (for the pressure field)
given by the factors cos 2 θ, sin 2 θ or sin θ cos θ depending
on the orientation of the quadrupole axes. The superdirective
field of Eq. (5) is due to the interference in
an extensive source between the regions of positive and
negative sign.
The theoretical results presented so far show that if
such axisymmetric wavepackets, with a high axial extent,
are present in a jet, we should expect to measure such
superdirectivity in the acoustic field. The next section
presents an experimental investigation of this issue.
3 Experimental observations in
the acoustic field
The decomposition of the experimental acoustic field into
azimuthal Fourier modes can be done either by performing
correlations[15, 20] or by using a microphone array[2],
and, in a linear context such as any linearised acoustic
analogy, we can expect that the lower azimuthal modes
will correspond to sound radiation by structures with
high azimuthal coherence.
The cited studies[15, 20, 2] have shown that only a
small number of azimuthal modes are present in the
acoustic field. This is also the case for the near pressure
field[37]. Figure (3) reproduces results from Cavalieri et
al.[6]. The results show that most of the sound radiation
at low axial angles is axisymmetric, the higher modes
having lower contributions to the OASPL.
OASPL (dB)
100
95
90
85
Total
Mode 0
Mode 1
Mode 2
80
20 30 40 50 60 70 80 90
theta (deg)
Figure 3: Directivity for a M = 0.6 jet.
ERCOFTAC Bulletin 90 35