Geoscience Reference
In-Depth Information
a pixel taking into account the magnification of the pro-
jective system). Thus the combination of three sensors of
10 3
y
10 3 pixels provides an effective spatial resolution
of four decades in 3D. Adding a fourth camera is then
essentially interesting to increase the number of particles
actually followed. Indeed, ambiguous situations where a
particle hides another one in the line of sight of a cam-
era may occur. These ambiguities can be lifted by adding
a fourth camera at a different angle in order to maximize
the number of particles which are seen at any time by at
least three cameras.
×
0.9 mm
Velocity (m/s)
0.8
x
0
z
6.3 mm
Trajectory Reconnections Lagrangian tracking consists
in reconnecting particle trajectories between successive
time steps. This requires to identify at time t + 1 particles
already detected at time t . Lagrangian tracking algorithms
are generally based on the minimization of a given cost
function. The simplest algorithm, called nearest neighbor ,
simply consists in connecting a particle (let us say particle
j ) whose position at time t +1is
10 0
ρ p
m
ρ p = 1.06, d p = 220 μ m
ρ p
= 1.06,
d p
= 26
μ
10 -1
= 2.5,
d p
=138
μ
m
10 -2
x j (t +1 ) to the parti-
cle i whose position at time t minimizes the cost function
φ ij =
10 -3
. This simple algorithm is accu-
rate only if the interframe displacement is significantly
less than the average interparticle separation. It is there-
fore generally limited to relatively diluted configurations.
In higher seeding density situations, more sophisticated
algorithms are required. As shown by Ouellette et al., one
robust algorithm consists in defining a cost function φ ij
based on four consecutive images. Qualitatively, it is based
on a smoothest acceleration criterion. Quantitatively it is
implemented as follows: Assume trajectories have been
reconnected up to time step t ; the velocity of the parti-
cles is estimated from positions
x i (t)
x j (t +1 )
10 -4
-15
-10
-5
0
5
10
15
a + = a /< a
2 > 1/2
Figure 15.4. Top: Example of high-resolution reconstruction of
particle trajectories in experiment shown in Figure 15.2. The
small spheres mark every other measured position of the par-
ticles and are separated by 0.074 ms; the large spheres mark
every 30th position. Color indicates the velocity of the particle
along its trajectory. From Bourgoin et al. [2006]. Bottom: One
component acceleration statistics measured in a von Kármán
swirling flow seeded with particles of different size and density.
From Xu and Bodenschatz [2008]. For color detail, please see
color plate section.
1 ) , which
allows one to estimate their probable position at time t +1,
˜
x i (t) and
x i (t
x i (t +1 ) ; then the particle acceleration is estimated from
x i (t) and ˜
x i (t
x i (t +1 ) and this propagates at time
t + 2 an estimation ˜
1 ) ,
x i (t +2 ) of the position for each par-
ticle in the vicinity of ˜
x i (t +1 ) ; then the most probable
trajectory is the one which minimizes the cost function
φ ij =
˜
x i (t +2 )
x j (t +2 )
.
twice to access particle acceleration. Optical tracking of
small particles has shown the highly intermittent dynam-
ics of such fluid tracers in turbulent flows. This is revealed
for instance by measurements of acceleration statistics
in von Kármán swirling flows (shown in Figure 15.4b),
which exhibit highly non-Gaussian fluctuations corre-
sponding to events of very high acceleration occurring
with a probability order of magnitude larger than what
would be expected for a normal random process with
equivalent variance.
15.2.1.3. Example of 3D Optical Tracking. Figure
15.4a shows an example of tracking of pairs of particles
by Bourgoin et al. [2006] in the high-Reynolds-number
experiment previously shown in Figure 15.2. Figure
15.4a only shows two trajectories, but hundreds of such
trajectories are simultaneously reconstructed. This allows
a rapid statistical convergence of particle displacement,
velocity, and acceleration statistics. Such data can be used
to investigate different properties of the flow. In the study
by Bourgoin et al. separation statistics are investigated in
order to address the longstanding question of turbulent
super diffusion. But time-resolved trajectories can be dif-
ferentiated with time, once to obtain particle velocity and
15.2.2. Extended Laser Doppler Velocimetry
As already mentioned, particle tracking is very demand-
ing in terms of acquisition frequency, which needs to be
 
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