INSTITUTE OF PHYSICS PUBLISHING
MEASUREMENT SCIENCE AND TECHNOLOGY
Meas. Sci. Technol. 13 (2002) 1020–1028
PII: S0957-0233(02)31771-5
Simulation of particle trajectories in a
vortex-induced flow: application to
seed-dependent flow measurement
techniques
A Lecuona, U Ruiz-Rivas1 and J Nogueira
Escuela Politécnica Superior, Universidad Carlos III de Madrid, Leganés, Spain
E-mail: ulpiano@ing.uc3m.es
Received 10 December 2001, in final form 19 April 2002, accepted for
publication 23 April 2002
Published 20 June 2002
Online at stacks.iop.org/MST/13/1020
Abstract
The trajectories of heavy particles (ρparticle /ρfluid 1) are simulated in a
two-dimensional free vortex flow. The results show that heavy particles,
even with small diameters, cannot properly trace the fluid and develop a
centrifugal motion. This behaviour leads to a rapid depletion of particles in
the vortex core, which, in seed-dependent measurements such as particle
image velocimetry (PIV) or laser Doppler velocimetry, produces a marked
increase in measurement errors. Some examples are given and the evolution
of the local concentration of particles is simulated. Synthetic images have
been generated using the information about its motion in a two-dimensional
free vortex flow. A wing-tip vortex is simulated, showing good agreement
with experimental results present in the literature. Standard PIV
measurements have been performed over the synthetic images, showing the
effect of core depletion of particles on the incidence of erroneous
measurements.
Keywords: fluid flow velocity, PIV, wing-tip vortices
(Some figures in this article are in colour only in the electronic version)
1. Introduction
The dynamics of a particle embedded in a fluid flow has
gathered constant attention over the last two decades. The
precise determination of the particle motion is of paramount
importance over a wide range of particle-laden flows that can
be found in nature and it is also of importance in engineering
applications. Moreover, the study of particle motion is relevant
to the implementation of flow measurement techniques such
as laser Doppler velocimetry (LDV) and particle image
velocimetry (PIV). These techniques rely on the motion of
seeded particles, and therefore their reliability is based on the
capability of the particles to accurately follow the flow.
1
Address for correspondence: Avda de la Universidad 30, 28911 Leganés,
Spain.
0957-0233/02/071020+09$30.00
© 2002 IOP Publishing Ltd
In PIV, particles seeded in the flow are illuminated with
two pulses of a pulsed laser sheet, separated by a small
time interval. The particle reflections are focused on a CCD
sensor, giving two separate images per particle, with a small
displacement between them. Both particle tracking and image
correlation techniques are used to measure the displacement
between the two images. Therefore, individual particle images
are required for these methods. The CCD pixel size provides
a lower limit for the particle image size. Moreover, the light
scattering cross section of the particle decreases rapidly with
particle size, giving a lower limit for the particle size. On the
other hand, a smaller particle will follow the flow better and
a higher number of particles can be obtained per interrogation
area. These opposing requirements must be studied in order to
know, among other issues, when a PIV measurement is prone
Printed in the UK
1020
Simulation of particle trajectories in a vortex-induced flow
to fail. If, for any reason, particles concentrate on a region of
the imaged flow, migrating from other regions, measurement
difficulties can also arise, not only because of the particle
concentration inhomogeneities, but also for the lack of particles
in other regions.
LDV relies in the assumption of homogeneous fluid
seeding and on a perfect flow tracing by the seeding
particles. Any deviation from these assumptions leads to
non-recoverable measurement errors. Smaller particles better
follow the flow, but could pass undetected by the apparatus.
Therefore, the circumstances over which the measurement
might fail must be described.
In this paper, we will analyse the dynamics of a tracer
particle in a vortical flow. Flows with strong vortices are also
widely present in nature and engineering applications (i.e. the
Karman Vortex Street, aircraft wing-tip vortices, the mixing
layer between parallel streams, etc).
We will consider a small spherical particle embedded in
a two-dimensional, isolated vortex. The particle diameters
will vary around 1 µm and the density ratio between fluid and
particle is of the order of 10−3 . This is typical for tracers such
as oil drops or solid particles in airflows.
Throughout this paper we will consider that the particle
motion does not affect the fluid flow, an assumption based
on the small particle dimensions and the moderate number
of particles embedded in the flow. This is, of course, a
general assumption taken in the generality of seed-dependent
measurement techniques. Also, the separation between
particles is commonly several orders of magnitude larger than
the particle diameter, so that each particle can be considered as
isolated. This means that the diffusion of particles, collisions
between particles and particle breaking processes will not be
considered. Finally, for the case of liquid seeds, the vapour
pressure of the oils used for seeding is so low at the usual
working conditions that evaporation is negligible.
2. Equation of motion
The general form of the equation of motion for a small rigid
sphere was proposed by Maxey and Riley (1982) and is
generally accepted. It considered the effect of five forces acting
over the particle: the steady state drag force, the gravitational
force, the added (or virtual) mass effect, the fluid acceleration
at the particle location (pressure gradient) and the Basset force
(considering the time history of the particle). This equation of
motion is then given by the formula
dv ∗
18µc
1 2 ∗2 ∗
ρc Du∗
∗
∗
=
u
−
v
+
∇
u
D
+
dt ∗
D 2 ρp
24
ρp Dt ∗
ρc
1 ρc d
1 2 ∗2 ∗
∗
∗
+
+ g∗ 1 −
u
−
v
+
∇
u
D
ρp
2 ρp dt ∗
40
t∗
1 2 ∗2 ∗
d
9 ρc ν c
∗
∗
u
−
v
+
∇
u
D
+
D ρp π 0 dt ∗
24
× (t ∗ − τ )−0.5 dτ
(1)
where D is the particle diameter, µ, ν and ρ are, respectively,
the dynamic and kinematic viscosity and the mass density
(denoted p or c to distinguish between particle and continuum
fluid), g is the acceleration vector of gravity, u is the velocity
vector of the flow at the particle location and v is the particle
velocity vector. The operator d/dt denotes time derivatives
following the particle and the operator D/Dt is a time
derivative following the fluid. The ∇ 2 u terms are the Faxen
correction for the non-uniformity of the flow. The asterisk (∗)
over a symbol indicates that it is a dimensional quantity, while
its dimensionless form will be written without it.
The vortex characteristic radius Ro and velocity Uo
define, respectively, the length and velocity scales of the flow.
Therefore, we can introduce the dimensionless variables:
x=
x∗
Ro
u=
u∗
Uo
t=
t ∗ Uo
.
Ro
(2)
In dimensionless form, equation (1) becomes
dv
1
ε
1 2 2
Du
=
u−v+
δ ∇ u +
dt
St (1 − ε/2)
24
1 − ε/2 Dt
1 − 3ε/2
1 2 2
ε/2 d
u−v+
+g
+
δ ∇ u
1 − ε/2
1 − ε/2 dt
40
t 1 2 2
ε
d
3
u−v+
δ ∇ u
+
1 − ε/2 2π St 0 dt
24
× (t − τ )−0.5 dτ.
(3)
Four dimensionless parameters appear in equation (3): a
Stokes number, a density ratio, a length scale ratio between
the particle dimensions and the characteristic flow scale (the
vortex radius) and a gravity coefficient:
St =
Uo D 2
18ενc Ro
D
δ=
Ro
ε=
ρc
ρp + ρc /2
Ro g ∗
g=
Uo2
(4)
where the Stokes number (St) is the viscous time divided by the
characteristic time scale of the vortex flow, Ro /Uo . Note that
the density ratio has been defined taking into account the effect
of the added mass. The Faxen corrections in equation (2) can
be neglected for a tracer particle in an airflow, as δ ∼ O(10−5 ).
Lasheras and Tio (1994) have carried out the asymptotic
study of the resulting equation, using a modified Rankine
vortex to model the flow field. Their modification of the
Rankine vortex is performed to avoid the discontinuity of the
derivative in r = 1. In terms of a cylindrical coordinate
system, the (dimensionless) fluid velocity and vorticity for this
axisymmetric vortex are given by
u=
2r
eθ
1 + r2
ω=
4
ez .
(1 + r 2 )2
(5)
In figure 1 we show the velocity and the vorticity fields.
The asymptotic analysis of Lasheras and Tio was carried
out for three different regions of the flow field, defined by
the order of magnitude of the position: x ∼ O(St), O(1)
and O(St −1 ). Let us examine the orders of the different
parameters involved in the dynamics of a tracing particle
in a large-scale airflow. The particle diameters for a good
tracer are D ∼ O(10−6 ) m. Air tracers are commonly
oil drops or solid particles (Al2 O3 or similar), for which
ρp ∼ O(103 ) kg m−3 . The vortex parameters (Uo , Ro )
can be assumed to be ∼O(10) m s−1 and ∼O(10−1 ) m,
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A Lecuona et al
(Dimensionless) Azimuthal Velocity, u
1
4
0.8
3
0.6
2
0.4
1
0.2
0
(Dimensionless) Axial Vorticity, w
5
1.2
0
(Dimensionless) Radial Coordinate, r
Figure 1. The velocity and vorticity fields of a modified Rankine vortex.
(Dimensionless) Radial Velocity /
Stokes Number, v r /St
1.2
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
(Dimensionless) Azimuthal Velocity,
vθ
1.2
1.4
0
0
0.5
1
1.5
2
2.5
3
3.5
4
(Dimensionless) Radial Coordinate, r
Figure 2. Components of the velocity of a rigid spherical particle embedded in a modified Rankine vortex. The error bars (indistinguishable
for the azimuthal component) show the error margin allowed in the asymptotic study. The radial velocity is shown divided by the Stokes
number.
respectively (for example, for an aircraft model in a large wind
tunnel). With such conditions, the dimensionless parameters
of equation (4) are
St ∼ O(10−3 )
ε ∼ O(10−3 )
δ ∼ O(10−5 )
g ∼ O(10−2 ).
1022
2
3
u (r)
ε−1 − θ
− gr (θ ) + O(St 3/2 )
2
r
(7)
vθ (r, θ ) = uθ (r) − gθ (θ )St 23 ε − 1 + O(St 3/2 ).
vr (r, θ ) = St
With these values it seems reasonable to disregard the
asymptotic analysis for x ∼ O(St) and O(St −1 ) in a practical
evaluation of seeding dynamics. Therefore, the only region
of interest is x ∼ O(1), where, according to the asymptotic
analysis of Lasheras and Tio, the particle velocity is given by
v = u + St ( 23 ε − 1)(u · ∇ u − g ) + O(St 3/2 ).
Now, let us separate the two components of the particle
velocity in cylindrical coordinates:
(6)
Consider the equation for the azimuthal particle velocity, vθ .
The terms on the right-hand side of the equation have the
following orders of magnitude: uθ ∼ O(1), the effect of
gravity is of order O(10−2 )×O(10−3 ) ∼ O(10−5 ) and the error
assumed in the asymptotic analysis is O(10−4 ). Therefore the
tracer particles can be considered to follow quite accurately
the azimuthal fluid motion. On the other hand, if we consider
the equation for the radial particle velocity vr , it shows that
the centrifugal term is of order O(10−3 ), the effect of gravity
is of order O(10−5 ) and the error assumed in the asymptotic
Simulation of particle trajectories in a vortex-induced flow
(Dimensionless) Radial Coordinate, r
10
1
0.1
0.01
0.001
0.1
1
10
100
(Dimensionless) Time · Stokes Number, t·St
Figure 4. Radial motion of the particle versus time.
Figure 3. Trajectory of a rigid spherical particle embedded in a
modified Rankine vortex.
analysis is O(10−4 ). Note that here, as the fluid flow radial
velocity is zero, the centrifugal term cannot be neglected and
the tracer particles will experience a radial motion. Hence, the
particle velocity can be calculated as a function only of the
radial coordinate from the following equations:
2 3
u (r)
ε−1 − θ
+ O(10−4 )
vr (r) = St
2
r
(8)
vθ (r) = uθ (r) + O(10−4 ).
vθ (r, θ) = 2r − gθ (θ)St ( 23 ε − 1) + O(St 5/2 ).
t d (r = 2)
3.5
3
2.5
2
t d (r = 1)
t d (r = 0.5)
1.5
1
0.5
t d (r = 0.25)
0
0.001
0.01
0.1
1
(Dimensionless) Initial Radial Position, r o
In figure 2 we plot the two components field (for
ε = 10−3 ).
In the following we will consider that the particles will
range over a region of interest that extends from r ∼ O(St)
to O(10). As has been said before, Lasheras and Tio
(1994) defined three regions in their asymptotic analysis of
the equation of motion of a heavy particle embedded in a
flow: x ∼ O(St), O(1) and O(St −1 ). Being in our case
St ∼ O(10−3 ), the outer region is of no importance in our
study, but we should consider the inner region. Nevertheless,
the analysis of Lasheras and Tio shows that, for this inner
region,
vr (r, θ) = St ( 23 ε − 1)(−4r − gr (θ )) + O(St 5/2 )
(Dimensionless) Time · Stokes Number, t d ·St
4
(9)
Equations (9) and (7) are quite similar, as, for r ∼ O(St),
equation (5) gives uθ ∼ 2r and u2θ /r ∼ 4r within the error of
the analysis. The difference here is that in equations (9) the
gravity effects are much more important than in equations (7).
In equations (9), the radial velocity of the particle depends on
4rSt ∼ O(10−6 ) and gr St ∼ O(10−5 ), while the azimuthal
velocity of the particle depends on 2r ∼ O(10−3 ) and gθ St ∼
O(10−5 ). This means that the azimuthal velocity of the particle
is still well defined by the local azimuthal flow velocity, while
the effect of gravity is of paramount importance in the radial
velocity of the particle. Nevertheless, this second conclusion
needs further attention. If we consider just the effect of gravity
in the radial velocity of the particle, the radial motion of the
Figure 5. Time for a particle to escape from the vortex core, td .
particle becomes periodic, moving inwards for 0 < θ < π
and outwards for π < θ < 2π (with the same instant radial
velocity in each position and identical time lapse to advance in
both semi-periods). Hence, although the effect of gravity in the
motion of the particle will produce larger displacements than
the centrifugal effect (4r), it is this last one that is responsible
for the ultimate tendency to escape the vortex core. Therefore,
in the calculations that follow, the gravity term will not be
considered, as it will not produce any global motion but just a
periodic motion of small amplitude.
Finally, some error is due to the variation of the term (4r)
during the motion of the particle on a cycle. This error can be
estimated considering the ratio between radial and azimuthal
velocities in equations (9): vr /vθ ∼ O(10−2 ) and therefore the
change in radial position during a cycle will be r ∼ O(10−5 ),
being r ∼ O(10−3 ). Hence, the radial velocity error is
within the error of the asymptotic analysis and equations (9)
(neglecting the gravity terms) become
vr (r) = St ( 23 ε − 1)(−4r) + O(10−7 )
vθ (r) = 2r + O(10−7 ).
(10)
Note that these last equations are equivalent to
equations (8), but with a smaller error. The use of equations (8)
to obtain the global displacement can therefore be accepted in
the region between r ∼ O(St) and O(1).
1023
A Lecuona et al
1.6
Particle Concentration (arbitrary units)
Particle Concentration (arbitrary units)
1.6
1.4
1.2
1
0.8
0.07
0.17
0.6
0.27
0.37
0.4
0.47
0.57
0.2
0.77
1.4
1.2
1
0.8
0.01
0.03
0.04
0.06
0.13
0.24
0.38
0.58
1.00
0.6
0.4
0.2
1.00
0
0
0
0.5
1
1.5
2
2.5
3
3.5
4
(Dimensionless) Radial Coordinate, r
a
0
0.5
1
1.5
2
2.5
3
3.5
4
(Dimensionless) Radial Coordinate, r
b
Figure 6. Concentration C of particles as a function of radial location and for different times (in s) in a Rankine vortex with Uo = 25 m s−1
and Ro = 0.05 m for particles with ε = 10−3 and diameters: (a) 0.5 µm; (b) 1.5 µm.
3. Results
The analysis mentioned seems to be relevant to the
implementation and normal use of the measurement techniques
that rely on the visualization of a tracer, such as PIV or
LDV. The relevant fact is not that the velocity obtained by
looking at the tracer does not coincide with the fluid velocity.
This error is small in the azimuthal velocity, which is generally
the important magnitude. Also, the vorticity calculation is
not affected by the particle’s inability to trace the flow, as the
radial velocity is a function of the radial coordinate and thus is
irrotational. The problem is the possibility of finding large
regions in the vortex core where no particle remains, thus
rendering the measurement impossible. Some experimental
results (for example, Kompenhans and Stanislas (2000)) have
shown that the inner zones of strong vortices get depleted of
tracers, making the measurement very difficult or impossible.
In the following we will try to state the major characteristics
of this phenomenon.
According to equation (8), a good tracer embedded in a
Rankine vortex will deviate from the flow motion in the radial
direction. Therefore, the particle will follow a spiral path
instead of following the circular streamlines of the flow. This
behaviour can be observed in figure 3.
From equation (8) we can also obtain the radial position
of the particle versus time. This has to be implemented
avoiding the singularity in the vortex centre (a particle located
in the centre of the vortex will not move from its equilibrium
position). We have considered an initial position (t = 0) at
r = St. Figure 4 shows a plot of the dimensionless quantities.
Logarithmic scale is used to show the behaviour for small radii.
We can thus calculate the time that a tracer needs to
depart from the vortex core, td . Figure 5 shows, using
dimensionless variables, the time that a particle, located in a
certain initial position, needs to arrive at certain characteristic
radial positions (0.25Ro , 0.5Ro , Ro and 2Ro ). For example, a
vortex of Uo = 25 m s−1 and Ro = 0.05 m seeded with oil
drops of 1.5 µm of diameter (St = 3.4×10−3 ) will have its core
(r < Ro ) practically depleted of particles in approximately
0.3 s, which seems a small elapsed time. These values are
typical for an Airbus model with a wingspan of 1 m tested in
a wind tunnel, flying at 60 m s−1 . The time of depletion will
represent a travelling distance of the order of 20 wingspans.
1024
More detailed information can be given. The evolution
in time and space of the concentration of particles can be
obtained solving the continuity equation for the radial particle
concentration C = dN/(2π r drl) for an axisymmetric flow:
∂C 1 ∂(rCvr )
= 0.
+
r
∂r
∂t
(11)
We have considered a uniform distribution of particles as initial
condition (C = 1 in arbitrary units). The boundary condition
should state that the solution is finite at the vortex axis.
For simplicity, we have defined C = 2π rlC. Thus,
equation (11) becomes
∂C ∂(C vr )
+
=0
∂t
∂r
(12)
with an initial condition of the form C = 2π rl and a boundary
condition of the form C (r = 0, t) = 0. With such conditions,
equation (12) can be easily integrated numerically. We
have performed this integration using the MatLab ODE 45
Integrator for the time derivative and a first-order central finite
difference scheme for the spatial derivative. The MatLab
DE 45 Integrator uses a Runge–Kutta–Fehlberg algorithm (an
adaptive stepsize integrator of fourth order).
Figure 6 shows the concentration (C) evolution for tracer
particles of diameters 0.5 and 1.5 µm in the vortex already
mentioned. The smaller particles show a slower tendency to
be depleted from the vortex core. As the residence time goes
inversely with the Stokes number, which goes with D 2 , the
time behaviour of the two types of particles differ by a factor
of 9.
Note that, in addition to the reduction of particle
concentration in the vortex core, there is an increase at higher
radii: this effect can also damage the acquisition of good
measurements in such zones.
All this information has been implemented in a computer
program that shows the evolution of seeding particles in a
PIV recording as time increases. Figure 7 shows several
images obtained for a Rankine vortex seeded with 1.5 µm
particles.
Note that, even for small times the vortex
core is strongly depleted of particles, thus rendering the
measurement impossible using any seed-dependent technique.
The accumulation of particles in an annular region outside the
Simulation of particle trajectories in a vortex-induced flow
t = 0s
t = 0.1s
t = 0.3s
t = 0.7s
t = 1.0s
Figure 7. Evolution of tracer particles of 1 µm of diameter in a Rankine vortex (indicated by the black circle) with Uo = 25 m s−1 and
Ro = 0.05 m and for particles with ε = 10−3 .
1.E+07
Experimental Data
Rossin Rambler
Gamma
Log Normal
1.E+06
PDF
vortex core can be observed in the images for larger times.
This effect may induce a bias error in the measurements.
The parameters involved in this behaviour are, as has
been shown, the vortex parameters, the fluid viscosity and
the particle diameter and density. The first three are given
by the experiment and generally they cannot be changed,
while the particle density cannot be changed by any large
amount. Therefore, to avoid this problem, the use of
smaller particles seems customary. Nevertheless, the use of
small particles has its drawbacks, as the ratio between the
wavelength of the illuminating light and the particle diameter
decreases, thus reducing its light scattering efficiency (besides
that, its geometrical cross section is reduced). Another
problem is that the usual equipment for seeding oil particles
in air experiences difficulties at producing large quantities of
particles of diameters smaller than 1 µm, as stated by Kähler
et al (2001).
Nevertheless, the seeding is not monodispersed and
particles smaller than 1 µm are seeded in the flow. To
take this effect into account, we have considered a particle
generator that produces particles with the distribution shown
in figure 8, obtained using a modified Laskin nozzle (Kähler
et al 2001). The distribution can be modelled using a log–
normal, Rossin-Rambler or gamma distribution. Here we
have chosen a gamma distribution because it provides better
fits for both the smaller and the larger particles, as can be
seen in the graph. (Note that the experimental measurement
for the smaller diameter might be over-dimensioned, carrying
information of the smallest particles, due to the problems
existing in measuring such tiny particles.)
1.E+05
1.E+04
1.E-07
1.E-06
1.E-05
Particle Size (m)
Figure 8. Particle size distribution of a typical oil droplet generator.
The process to obtain the images in figure 7 can be repeated
with the information on the particle size distribution. In figure 9
we have plotted the results.
A comparison between the images of figures 7 and 9
shows, for the larger times, a less dramatic depletion of the
core in the second case. But for smaller times the effect is the
opposite. The generator distribution for figure 9 is centred on
the particle diameter used for figure 7. Then, for small times,
when the particles of figure 7 have not got the time to leave the
vortex, the higher-diameter particles of the distribution used
for the results in figure 9 would have left the centre. For larger
1025
A Lecuona et al
t = 0s
t = 0.1s
t = 0.7s
t = 0.3s
t = 1.0s
Figure 9. Evolution of tracer particles from the data shown in figure 8 (mean diameter 1 µm) in a Rankine vortex with Uo = 25 m s−1 and
Ro = 0.05 m and for particles with ε = 10−3 .
times, we observe the smaller particles of the generator in
the zone where, in figure 7, the particles have already been
depleted.
Nevertheless, the use of smaller particles to soften the
depletion effect is questionable. The scattering cross section
of a particle goes as D 2 and thus the intensity of the image
of the particle decreases very rapidly, reducing the signal to
noise ratio. Moreover, when the size of the particle image
is smaller than the pixel size, peak locking appears and
increases the measurement uncertainty. Finally, when the ratio
(particle diameter)/(light wavelength) reaches πD/λ < 1,
the scattering mode changes to Rayleigh scattering and the
scattering cross section decreases even more rapidly.
The images of figures 7 and 9 have been obtained, for
clarity purposes, with low particle density (the mean distance
between particles is around 4 pixels), while PIV images
can reach far larger densities (mean particle distance around
1 pixel).
In order to approach real conditions, random background
noise (with a mean value of 5% of the image range) has been
added to the images.
As a check, we have compared our synthetic images with
an experimental image obtained in a large-scale aerodynamic
facility. The image corresponds to the core of a wing-tip vortex
generated in the DNW wind tunnel with a fixed model of
half a wing of an airplane, corresponding to a wingspan of
3.5 m. More information can be found in Kompenhans and
Stanislas (2000). The comparison is shown in figure 10. The
images here are not the negatives, as in figures 7 and 9, for
1026
clarity purposes. No attempt has been made to precisely match
the real image appearance with the synthetic one, as there are
many influencing factors, such as the optics and CCD response
to the laser scattered light. The more relevant parameters are
the particle concentration and the average particle image size,
which is 1.5 pixels in the synthetic images.
We have applied a standard correlation PIV algorithm over
synthetic images, similar to the one shown in figure 10. This
algorithm uses a three-point Gaussian peak fitting (in the two
directions) to detect the maximum correlation, a 32 × 32 pixel
interrogation window and a 75% window overlapping. No
window shift has been applied, so that a slight peak locking
must be present. Figure 11 shows the results as compared with
the exact solution given by equation (5). Case (a) corresponds
to an image with homogeneous seeding, in order to compare it
with case (b), where the image was obtained considering a time
lapse of 0.002 s for the depletion process. The PIV synthetic
images have been obtained with a time interval of 10−5 s to
avoid PIV malfunctions due to the high vorticity. Nevertheless,
this is still present near the vortex axis in case (a). The obvious
erroneous vectors in case (b) are shown as zero velocity instead
of its out-of-range value in the horizontal axis.
The results in figure 11 show that the depletion process
produces a clear increment in the incidence of erroneous
vectors in the vortex core. In this region, seeding particles
are few and of small diameter. Therefore, the signal to noise
ratio in the zone will be low and erroneous vectors appear, even
in the case when there would be some seeding particles present
in the zone.
Simulation of particle trajectories in a vortex-induced flow
Ro
Synthetic image
Experimental image (Kompenhans and Stanislas 2000)
Figure 10. Comparison of a synthetic image, obtained by computing the particle trajectories with equation (8) and an experimental image
obtained for a wing-tip vortex in a large wind tunnel. The images are 200 × 200 pixels (26 mm × 26 mm). The vortex model was defined by
a radius Ro = 9 mm (shown on the left side) and a maximum velocity Uo = 60 m s−1 .
70
(Dimensional) Azimuthal Velocity (m/s)
(Dimensional) Azimuthal Velocity (m/s)
70
60
50
40
30
20
10
0
60
50
40
30
20
10
0
0
0.5
1
1.5
2
2.5
0
0.5
1
1.5
(Dimensionless) Radial Coordinate, r
(Dimensionless) Radial Coordinate, r
a) Homogeneous seeding (t = 0s)
b) t = 0.002s
.5
Figure 11. Azimuthal velocities (in m s−1 ) obtained applying PIV to synthetic images for (a) homogeneous seeding and (b) vortex-induced
inhomogeneous seeding.
4. Conclusions
The evolution of heavy particles in a flow has been studied and
applied to seeding particles for LDV or PIV measurements in
airflows. The results show that strong vortices are depleted
of seeding particles, while an increase in particles occurs
simultaneously in an annular zone outside the vortex core. The
effect is less rigorous for smaller particle diameters (td ∼ D 2 ).
Particles around 1 µm will be depleted from the core of a
strong vortex (for example, an aircraft wing-tip vortex) in less
than 1 s.
The effect is studied and its main characteristics are
pointed out. The use of poly-dispersed seeding seems to
mitigate the problem, as a considerable amount of particles
have diameters smaller than 1 µm. Nevertheless, particles
with diameters smaller than a micron have other well known
drawbacks, and relying on them for PIV measurements seems
problematic, with the available laser power and image sensor
resolution.
The basic equations that run these processes have been
implemented in a program that synthesizes images for PIV
measurements. We have used a standard correlation PIV
algorithm, and it shows that the process of seeding depletion
has a direct effect in the incidence of erroneous vectors.
PIV techniques that use image distortion, such as LFC–PIV
(Nogueira et al 1999), could considerably improve the correct
vector yield.
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A Lecuona et al
Acknowledgments
References
This work has been partially funded by the Spanish Research
Agency grant DGICYT TAP96-1808-CE, PB95-0150-CO202 and under the EUROPIV 2 project (A joint program
to improve PIV performance for industry and research)
which is a collaboration between LML URA CNRS 1441,
DASSAULT AVIATION, DASA, ITAP, CIRA, DLR, ISL,
NLR, ONERA, DNW and the Universities of Delft, Madrid
(Carlos III), Oldenburg, Rome, Rouen (CORIA URA CNRS
230), St Etienne (TSI URA CNRS 842) and Zaragoza. The
project is managed by LML URA CNRS 1441 and is funded
by the CEC under the IMT initiative (contract no: GRD11999-10835).
The authors would also like to thank Dr Carlos Martı́nezBazán and Mr Javier Rodrı́guez Rodrı́guez for useful
comments and discussion.
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