Geoscience Reference
In-Depth Information
p
scat
(
k
,
ν
)
40
p
inc
20
n
q
scat
k
0
-20
(
k
0
,
ν
0
)
θ
scat
-
k
0
Ω(
r
,
t
)
n
0
-40
0
20
40
60
80
100
120
140
160
180
Figure 15.13.
Typical implementation of acoustic scattering for
probing the vorticity field of a flow. An ultrasonic emitter gen-
erates a plane acoustic wave in the direction
θ
scat
n
at the frequency
ν
0
. A receiver listens to the acoustic wave scattered in a given
direction
Figure 15.14.
Angular factor
L(θ
scat
)
.From
Poulain et al.
[2004].
r
. The relative amplitude of the scattered acoustic
pressure to the incident acoustic pressure can be related to the
spectral distribution of the vorticity field
(
x
,
t)
according to
It is important to stress that this measurement is local in
Fourier space, meaning that for a given scattering vector,
only the mode of vorticity at wave number
relation 15.6. From
Poulain et al.
[2004].
q
scat
is actually
measured. Hence the measurement is naturally global in
space and the corresponding spectral mode is character-
ized across the entire measurement volume. Though the
scattering structures of the flow are tracked as they move
across the flow (this results for instance in a Doppler shift
of the scattered acoustic wave), the recorded signal rep-
resents a coherent average of all structures at the probed
scale simultaneously present in the measurement volume,
and no information is extracted from individual scatterers.
As a consequence this technique is not properly speak-
ing of Lagrangian type, although the instrumentation is
almost identical to that described in the previous section
on Lagrangian acoustic tracking.
An important point to be considered is the angular
factor
L(θ
scat
)
present in equation (15.6). Figure 15.14
shows the dependency of
L(θ
scat
)
with the scattering angle
θ
scat
as calculated by Lund et al. It shows a quadrupolar-
like radiation pattern which diverges at zero angle (Born
approximation fails in this limit) and vanishes for scat-
tering angles
θ
scat
=90
◦
and
θ
scat
= 180
◦
(back-scattering
situation). Those two specific scattering angles are to be
avoided for the vorticity measurement. On the contrary,
they are optimal for particle tracking as no signal is then
recorded from scattering effects of the fluid itself, and only
the particles seeding the flow will be seen. This explains
the back-scattering configuration chosen for the acoustic
particle tracking described in the previous section.
qualitatively understood as the fact that each vortex in
the flow acts as a scatterer which radiates a sound wave
as it is perturbed by the incident impinging acoustic
wave. The global scattered wave, results from the coherent
average over the scatterer distribution. Figure 15.13 illus-
trates a typical acoustic scattering configuration which
can be used to probe the vorticity of a flow. Lund et al.
have shown that the acoustic pressure amplitude and the
Fourier transform of the vorticity field are related as
p
scat
k
,
t
=
L(θ
scat
)
˜
−
k
0
,
t
p
inc
k
0
,
t
,
(15.6)
q
scat
=
k
⊥
where
k
0
and
k
are the vector wave numbers of the incom-
ing and scattered acoustic waves, respectively,
θ
scat
is the
scattering angle,
p
inc
and
p
scat
are the complex pressure
amplitudes of the incoming and scattered acoustic wave,
respectively,
L(θ
scat
)
is an angular factor which will be
discussed further below, and
˜
is the component of
the space Fourier transform of the vorticity perpendic-
ular to the scattering plane defined by the vector wave
numbers of the incident and scattered acoustic waves (see
Figure 15.13):
⊥
⊥
q
,
t
=
n
·
r
,
t
e
−
iq
·
r
d
3
˜
n
0
×
r
. (15.7)
Equation (15.6) therefore shows that the amplitude of
the scattered wave gives a direct measurement of one
Fourier mode of the vorticity component
. Interest-
ingly, the Fourier mode at which the vorticity field is being
probed is directly selected by the imposed scattering vector
15.3.2.2. Experimental Implementation and Typical
Results.
Experimental evidence of ultrasonic scattering
by vortical structures in a flow has first been given by
Baudet et al.
[1991] in the canonical configuration of
the von Kármán vortex street behind a cylinder at low
Reynolds number. Since then, the technique has been
ported to turbulent flows at moderate Reynolds number
(in a turbulent jet of air [
Poulain et al.
, 2004]) and at
high Reynolds number (in a cryogenic turbulent jet of
gaseous helium [
Bezaguet et al.
, 2002;
Pietropinto et al.
,
⊥
q
scat
=
k
−
k
0
. Hence, it is possible to reconstruct the com-
plete vorticity spectra by spanning the explored scattering
vector, and this can be done either by changing the angular
position of the acoustic receiver or by changing the oper-
ating acoustic frequency (as
q
scat
4
πν
0
/c
sin
(θ
scat
)
,
−
assuming the Doppler shift
ν
ν
0
remains small com-
pared to
ν
0
).
