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between the measured signals s i (t) and the model given
in equation (2). In practice, such a parametric estimate of
amplitude and frequency modulations is very robust with
respect to the experimental noise. The estimation is done
in several steps:
1. As the time scale of α(t) (of order d/u
Particle Seeding Issues For practical applications, the
particle seeding density has to be adjusted in order to
be low enough so that one does not observe events with
two particles at the same time in the measurement vol-
ume but high enough to observe several trajectories per
second. For a fully turbulent flow with Reynolds num-
ber at the Taylor microscale, Re λ
5ms)is
very large as compared to 1 /δf = 0.01 ms, it is removed
with high-pass filtering at several kilohertz.
2. To obtain an absolute definition of the local fre-
quencythroughtheevolutionof thephase φ(t) thefiltered
signal s (t) istransformedintoananalyticalcomplexsignal
x (t) : This is done using the Hilbert transform HT
600, a collection of
15,000 trajectories with mean duration 20 Kolmogorov
times ( τ η ) is enough to converge velocity statistics, accel-
eration statistics, and acceleration autocorrelation func-
tion. In the case of acceleration autocorrelation C aa (τ ) =
s ]
a 2
(t)
of the measured signal with the definition x (t) = s (t) +
i HT
[
, one gets access to an estimate of the
local kinetic energy dissipation because the integral of
the positive part of th e curve is very close to the dissi-
pation time τ η = ν/ [ Volk et al. , 2011]. As shown in
Figure 15.7 (top), there is a good rescaling between the
acceleration autocorrelation functions when time is mea-
sured in τ η units for fully developed turbulent flows of
water (with kinematic viscosity 10 6 m 2 /s and dissipa-
tion scale 19 μ m) and water-glycerol mixtures (kinematic
viscosity 8
a(t)a(t + τ)
/
s ]
(t) . The amplitude and frequency of the signal are
then α(t) =
[
and f p (t) = δf + dφ/dt , respectively.
3. The complex signal x (t) is then demodulated from
the carrying frequency by multiplication by exp(−2iπδf (
x (t)
·
t) . The real part of such a demodulated complex signal
for a typical burst is displayed in Figure 15.6 (middle).
4. For a fast and precise measurement of the modu-
lation frequency, an approximated maximum likelihood
(AML) method is coupled with a Kalman filter so as
to perform a parametric estimation of the instantaneous
amplitude and frequency. In a moving window of dura-
tion δT , centered at time t , it assumes that the signal is
made of a modulated complex exponential plus Gaus-
sian noise n(t) and compares the measured signal x (t) to
the functional form
2 iπδf
10 6 m 2 /s, dissipation scale 90 μ m). These
two curves also show that particles with diameters D =5 η
(the dissipative scale) can still be considered tracers of
the flow. As shown in 15.7 (bottom), this is no longer the
case for larger particles for which one observes a decrease
of particle acceleration variance following a power law
×
(D/η) 2 / 3 . This decrease of acceleration
variance goes together with an increase of the particle
acceleration autocorrelation time [ Qureshi et al. , 2007;
Brown et al. , 2009; Volk et al. , 2011].
a 2 D
a tracer
/
i 2 π t
0
ν(t ) dt +
z (t) = A(t) exp
[
]
+ n(t) ,
(15.3)
where A(t) and ν(t) are the unknown amplitude and fre-
quency to be estimated and ψ a constant phase originat-
ing from the initial particle position in the measurement
volume. As an output, one obtains for each trajectory an
estimate of the frequency ν(t) = u
15.3. ACOUSTIC TECHNIQUES
Whenever a sound wave encounters an obstacle or inho-
mogeneity along its propagation path, it is deflected from
its original course, a phenomenon called acoustic scatter-
ing. The scatterer can be either a material obstacle or a
physical inhomogeneity such as temperature or velocity
gradient, which creates a contrast of acoustic impedance
and influences the propagation of sound. The properties
of the scattered acoustic wave depends upon the frequency
of the incident wave, the shape and size of the obstacles,
as well as their velocity. It is possible to take advantage of
these scattering features to probe the dynamics of fluids.
We detail here two recent techniques based on acous-
tic scattering: (i) Lagrangian acoustic tracking, which
exploits the Doppler shift of the wave scattered by mov-
ing particles, and (ii) acoustical measurements of vorticity,
which exploits the scattering properties of eddies in a
fluid (with no need of seeding the flow). These techniques
are intrinsically related to acoustic scattering properties
and differ from more classical ones based on echo and
time-of-flight measurements.
(t)/a , amplitude A(t) ,
and a confidence criterion h(t) which measures the qual-
ity of the estimation at each time step. This is done for
each trajectory on a personal computer using Matlab to
obtain a collection of trajectories to be further used to
compute Lagrangian statistics of the flow.
Initially designed for acoustical Doppler measurements
(described in the next section), the demodulation tech-
nique proved to be fast and accurate enough to perform
Lagrangian ELDV measurements with typical time reso-
lution of 10 μ s and a sampling rate of 300 kHz. This repre-
sents the highest sampling rate ever used for Lagrangian
measurements in high-Reynolds-number flows. It is par-
ticularly adapted to investigate flows with rapid and
intense multiscale swirling structures. Therefore, although
this technique has never been used (to our knowledge)
in experiments with geophysical motivations, it is very
likely to be a good candidate for the investigation of most
extreme atmospheric events.
 
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