Sent to Measurement Science And Technology Journal
PIV Evaluation Algorithms for Industrial Applications
A Lecuona, J Nogueira, P A Rodríguez and A Acosta
Dep. of Thermal and Fluids Engineering, Universidad Carlos III de Madrid.
c/ Butarque 15, 28911, Leganés, Madrid, Spain.
Abstract
Particle image velocimetry (PIV) is now applied with confidence in industrial facilities
like large wind tunnels, yielding data not possible before. Despite the difficulties that
arise in such environments, the conventional PIV methods can provide high quality
data, especially when dealing with spatial and temporal slowly varying values of the
flow magnitudes.
Obtaining highly spatially resolved velocity fields is still challenging due to the
inherent difficulties on these industrial facilities, such as large velocity gradients,
background light and reflections.
This work has been focused on: (i) The behavior of conventional PIV when the
conventional limits of the processing algorithm are approached or surpassed and (ii)
Which kind of advanced methods can reduce the main sources of error that are
characterized in this paper. The focus is put on the description of vortex flows.
Group locking, a major source of error, is introduced, modeled, and metrologically
characterized.
The part of the study devoted to advanced methods deals with one that already has
shown to be of profit in images from industrial facilities, Local Field Correction PIV
(LFCPIV). This is a robust and precise method, able to obtain a high yield of
measurements in these environments, without the need of external adjustment to each
particular situation. To illustrate this point, some examples of processing of real images
are given.
The conclusions of this work suggest guidelines about the error figures when
measuring flows with embedded vortex using conventional and advanced methods in
industrial facilities.
Keywords: PIV, advanced algorithms, LFCPIV, error, vortex, group locking, wind tunnels, spatial resolution.
The limits that define the working envelope of
conventional PIV processing have been progressively
defined in the open literature.
Willert and Gharib (1991), among others, established
the spatial resolution of the conventional PIV. In that
work, it can be observed that to describe a certain single
harmonic spatial wavelength, λ, the side length of the
interrogation window, F, is limited. As an example, 10%
error in the description of a single 1D harmonic
fluctuation of the velocity demands λ > ∼4F. Obviously,
in every method the correct description of these
structures requires also small enough grid distances
∆ < ~ λ/2 and small enough mean distances between
particles δ < ~ λ/2, being the seeding surface density
4/(πδ2) particles per pixel area (ppp). In what follows we
will consider only square interrogation windows.
Another interesting limit in a conventional PIV
processing is the maximum allowable velocity gradients,
∆v. The unit used here for this variable is
[pixel/(∆tpixel)] that is [∆t-1], where ∆t is the time delay
between the two PIV laser pulses. Based on extensive
1. Introduction: conventional PIV limitations and
common scenario in some industrial facilities
Particle image velocimetry (PIV) in its usual conception
based on cross-correlation algorithms and digital image
acquisition has become a grown-up reliable technique.
The implementation of PIV can follow some design
variations. Here we will refer to conventional PIV as a
single pass methodology that does not use window
offset, window weighting, and neither image
deformation. The two last techniques are used in a
combined way in some implementations of what has
been called advanced PIV methods. Now it is current
practice to perform a two-step or even a three processing
steps method, in which the displacement resulting from
a step is used for the window offset in the following
step. This technique when used in this paper will be
indicated. Test performed using the correct window
offset in synthetic images showed no significant
difference on the specific errors analyzed in this paper.
Section 4 develops this topic.
1
Monte Carlo simulations, Keane and Adrian (1993)
found an empirical limit ∆v∆t < ~0.03, being this in
order to assure accurate correlation peak location and
simultaneously high peaks. They raised another limit
based on the particle diameter, d. This limit corresponds
to ∆v∆tF < d, being this in order to assure single
correlation peak built-up (Lecuona et al. 2002b and
Nogueira et al. 2002). This last limit is generally less
restrictive than the previous one unless for very large F.
Some recently identified limits that are interesting for
this work, are the ones that deal with the absolute size of
the particles images. Westerweel (1998) states that the
diameter of the particles should be d ≥ 2 pixels, in order
to avoid loss of information caused by peak locking.
Furthermore Westerweel et al. (1997) indicate that
errors are minimized for small particle diameters. This
results in an optimum particle diameter d ≈ 2 pixels.
Operating inside these limits leads to accurate
measurements, especially in well-controlled experiments
with low noise in the images.
Once the limits of conventional PIV are raised, it is
the time to evaluate the scenario in industrial facilities
such as wind tunnels. It is common that these facilities
are large, expensive to run and with limited optical
access. One consequence is that the PIV images often
cover rather large flow areas. In this situation and with
modest optical magnifications, the diameter of the
seeding particles defined by geometrical optics is
smaller than their Airy disc at the sensor surface. The
result is an image of fairly monodisperse particle images
with a diameter corresponding to the Airy disc. With a
proper selection of the f-number of the lenses and
enough light for illumination, this diameter can
reasonably approximate 2 pixels. This means a good
situation regarding the limits of PIV related to peak
locking. The small particle diameter allows a high
particle density, meaning this high information content.
The situation is not so favorable in respect to the
other limits of conventional PIV: velocity gradients and
spatial resolution. The industrial PIV images usually try
to depict complex high Reynolds number flows. In this
context covering large areas leads to situations in which
the limits on velocity gradients and small flow features,
where viscosity dominates, are surpassed. This way
local accuracy is sacrificed in the sake of other
advantages, like global description of flow features
and/or high velocimetry data rate.
It is common that these small λ features incorporate
large velocity gradients. This, together with other
sources of noise, lead to a considerable difficulty in
using window sizes smaller than F = 32 pixels in an
industrial facility.
To shed some light in the characterization of the
performances of conventional PIV when pushed this
way out of the operating limits, synthetic images of
vortex flows have been processed with the common
F = 32. Results are discussed in section 2.
The motivation of this study is two-fold. Not only can
it illustrate about conventional PIV but also on the
critical first steps of advanced PIV methods, which rely
on previous processing to compensate for image
distortion and offset, so that after a certain number of
iterative steps the processing converges into the
conventional working envelope, thus being able to yield
accurate measurements.
Synthetic images were purposely designed to test the
performance of the methods in respect to the spatial
wavelength content and velocity gradient. They contain
no noise except for the spatial discretisation of the
simulated image sensor and the usual effective 8 bit gray
level sampling. The mean distance between the
randomly located particle images is small, δ = 2 pixels,
i.e. 4/(π·δ2) ≈ 0.3 ppp. The e-2 diameter of all the
Gaussian particle images is d = 2 pixels, simulating the
small size dispersion previously commented. The
Gaussian shape of the particle images was integrated
with unity fill factor over each square pixel surface
(Westerweel 1998). Constant intensity profile was
selected for the light sheet, so that each particle can
equally contribute to the correlation. Where particles
overlap, the corresponding intensities are added. The
associated particle surface density is on the edge where
speckle starts to appear, but this phenomenon will not be
taken into account in this paper. The rationale behind the
high density chosen is to reduce the error coming from a
finite number of particles inside the interrogation
window, and this way more clearly reveal the nature of
group locking. On the other hand, the resulting images
look very similar to real ones obtained in wind tunnels,
being the particle surface density similar. Grey level
saturation resulted to be statistically insignificant. No
out of plane displacement was considered. This kind of
images has been already used by the authors in some
cases like Lecuona et al. (2002a).
The performance of conventional PIV is obtained in
section 2. The following section 3 analyzes the origin of
the errors and allows understanding in section 4 why
some types of advanced methods can avoid the sources
of error. In section 5 these methods are tested in a
typical, but difficult, image taken in an industrial facility
from a main European laboratory. In this section the
unsuitability of windows with F < 32 for the difficult
part of the flow field is shown as evident.
2. Trespassing the conventional PIV limits: out of the
working envelope
At first sight, the limits for conventional PIV that are
commented in the previous section may seem too
conservative. Analyzed in detail, they show to be rather
realistic. In this section synthetic images of
representative modified Rankine vortex are processed to
find out to which degree these limits are realistic. The
velocity axis-symmetrical flow field in each image
follows the expression:
vθ =
2
2U 0 ( r / R0 )
1 + ( r / R0 )
2
; vr = 0
(1)
prediction of the peak vorticity. This will be performed
here by means of a well known circulation type filter,
shown in expression (3). Discussion about different
derivative calculation schemes can be found in Lecuona
et al. (1998) and Acosta et al. (2000), among others.
U0 is the maximum tangential velocity, located at the
vortex radius R0. The vortex circulation is Γ = 4πR0U0.
The parameter selected in this analysis to compare
performances is the vorticity (curl of the velocity), a
basic flow magnitude for viscous structures. At the
center of the vortex, the peak vorticity, ωp, corresponds
to:
ωp = 4U0/R0
(v
ω=
(2)
(v
−
It diminishes to quickly reach ωp/4 at a distance R0 of
the center of the vortex.
Particularizing the limits of the above referred
authors to the processing of the vortex flow by
conventional PIV, the following conclusions can be
drawn:
• The core of a single vortex can be closely
reproduced by 2D single frequency harmonic field
of λ = 4R0. Moreover, the response to a 2D single
harmonic field can be coarsely approximated by the
square of the normalized amplitude response (0 ÷ 1)
to a 1D one. Consequently, to locate the error in the
order of 10% (2D normalized amplitude response
0.9) the 1D response should be around 0.95 (5% of
error). This leads to λ > ∼5.6F as a first
approximation. For F = 32 pixels this limit
corresponds to R0 ~ 45 pixels.
• Considering the limit ∆v∆t < ~0.03 as independent
for both spatial coordinates, x and y, the maximum
vorticity in the core of a vortex results to be
ωp ~ 0.06∆t-1. This means a rotation of the flow in
the core of the vortex of θp = 1.7 deg. between PIV
pulses (θ = 90ω/π).
With these orders of magnitude in mind, several tests
on synthetic images were arranged. They were run with
R0 ≤ 60 pixels and for two different peak vorticities:
ωp = 0.06∆t-1 and a large value, ωp = 0.35∆t-1 (θp = 10
deg.).
Vorticity from PIV data is generally evaluated
calculating the mean vorticity around the point of
interest, here the center of the vortex, thus giving a
y1,−1
+ 4v y1,0 + v1,1 ) − ( v y −1,−1 + 4v y −1,0 + v y −1,1 )
12∆ω
x −1,1
+ 4v x 0,1 + u1,1 ) − ( v x −1,−1 + 4v x 0,−1 + v x1,−1 )
12 ∆ω
True peak vorticity, ω p, normalized
Norm. resp. = Meas. Vort. / Correct peak Vort.
Norm. resp. = Meas. Vort. / Correct peak Vort.
True peak vorticity, ω p, normalized
Case ω p = 0.35 [1/∆ t]
1
0.8
0.6
Correct peak vorticity normalized
0.4
0.2
(3)
i, and j are the index of the 3 by 3 neighborhood of
velocity vectors v, separated a calculation grid node
distance ∆ω. This calculation has been performed for
∆ω = 8 pixels and ∆ω = 16 pixels, giving an idea of the
accuracy of measurements of the peak vorticity with
data at 8 and 16 pixels of distance from the vortex
center. For each pair of values, R0 and ωp, 15 different
synthetic images were processed in order to represent
the statistical dispersion of the processing. The results
are depicted in figure 1, where each point represents a
run. In that figure the thick line represents the vorticity
that a correct PIV processing would give, calculated
using expression (3) and expression (1), and was taken
as a reference for normalization. The rationale behind
this is that a finite difference vorticity algorithm such as
(3) cannot reconstruct the true peak, but typically a
lower one, owing to its low-pass effect. There is
difference between this and the true peak vorticity,
expression (2), also depicted in the figure with a thin
line. This parameter is maintained constant when R0
changes, but looks changing in the figure, owing to the
normalization. Separation from datum 1 for the
normalized amplitude response must be interpreted as an
error. Systematic error (bias), in addition to random
error (dispersion) is evident in figure 1.
1.4
1.2
−
Case ω p = 0.06 [1/∆ t]
Case ω p = 0.35 [1/∆ t]
1.4
1.2
1
0.8
0.6
Correct peak vorticity normalized
0.4
0.2
Case ω p = 0.06 [1/∆ t]
0
0
0
10
20
30
40
50
60
0
70
Vortex core radius (modified Rankine)
10
20
30
40
50
60
70
Vortex core radius (modified Rankine)
a
b
Figure 1. Normalized response of conventional PIV algorithm on a vortex flow when trespassing the working envelope. a) plot
corresponding to ∆ω = 8 pixels. b) plot corresponding to ∆ω = 16 pixels. Circles and triangles respectively stand for low and high
vorticity.
3
•
interrogation window, U0∆t,< 1 pixel for the low
vorticity case and U0∆t,< 6 pixels for the high vorticity
case.
For the low vorticity case, it can be observed that
conventional PIV behaves as a low-pass filter that
tends to reach reasonable accuracy on vorticity, in
the order of 30%, for: 45 pixels < R0 < 60 pixels.
This is in agreement with the previously
commented PIV limits on spatial resolution. It
shows that the values given for these limits are quite
realistic, needing F < ∼λ/8 to obtain an acceptable
error ∼ 15%, because variations are in both spatial
coordinates. The dispersion of the measurement
found with these data is ∼ 15% for: 45
pixels < R0 < 60 pixels.
• For the large vorticity case the dispersion of the
measurements is larger, making this kind of
processing unsuitable for accurate measurements. It
can be observed that either ∆ω = 8 pixels or ∆ω = 16
pixels give large errors and large dispersion. The
tendency of the measurement average is not good
even
when
approaching
R0 = 60
pixels,
overestimating the peak vorticity. Lets remark that
for ∆ω = 8 pixels, there is one out of the fifteen
measurements in the range 0.5 to 0.8 of normalized
response for all the tests performed with R0 > 30
pixels, even being most of the normalized responses
larger than 1.
Even in the extreme cases, the corresponding
displacements were checked to be within the range of
actual displacement covered by the interrogation
window. This means that they can be attributed to
existing measurements rather than outliers. Additionally,
they were always in the correct vortex circulation
direction. In any case, whether they can be considered
outliers or not is always questionable, as it depends on
the outlier detection algorithm. In a real environment
they would seem even more reliable due to the
contribution of noise and the unknown underlying flow
field.
The reason for the underestimation of peak vorticity
is a group locking to smaller displacements as they are
more similar in magnitude from particle to particle than
larger displacements, thus they build a higher correlation
peak.
Overestimation of peak vorticity, appearing for the
larger vortex radiuses, is caused by group locking to
displacements larger than in the center of the
interrogation window, caused by their smaller value of
∂Vθ / ∂r .
3. Source of error: not the signal to noise ratio but
the built-in group locking effect
It is important to discriminate whether the errors
observed in the previous section are due to a low signal
to noise ratio (s/n) or to a conventional PIV built-in
error. Moreover, the way of modifying the PIV method
to obtain better results will be different if the loss in
performance when surpassing the limits of the operating
envelope is due to either: (i) loss in s/n ratio or (ii)
correct identification of biased peaks. In the first case,
with particularly low noise setups the measurement
would improve. In the second one, only refinements in
the processing algorithm or methodology of its
application would lead to improvement.
As commented in the introduction, the synthetic
images generated for these tests were specifically
designed so that the noise content is low. Vortex
radiuses of 60 pixels are of a reasonable size in terms of
operating envelope. Even the case of ωp = 0.35∆t-1 the
processing here described did not give outliers. All of
this increases the confidence on the hypothesis that the
error is coming from built-in errors. To check this
hypothesis the way the different displacements within
the interrogation window interact has been studied.
The processing in conventional correlation PIV loses
the information of where each particle is located in the
interrogation window, just it superposes the individual
correlation contributions. Quite often this is addressed as
a bias towards the “most frequent” displacement. This is
so because particles add their contribution with no
privilege between them. In some way this source of
error has been addressed already (Adrian 1988),
although for a different purpose, and has not been yet
characterized. Actually, it will be shown below that the
bias is away from the “most frequent” displacement.
This source of error is characterized for providing a
displacement corresponding to the group of particles that
contribute to the highest correlation peak, or at least
biased towards it, instead of the one corresponding to
the center of the window. In consequence, it will be
referred to as “group locking”.
To clarify concepts, it is good to make a first sketch
devoted to explaining why the bias is not exactly the
“most frequent displacement”. To do this, let’s focus on
a simple situation like a 1D single harmonic sinusoidal
displacement field. In this situation the correlation peak
of a certain displacement, v∆t, is built with contributions
by all the particles with a displacement < v∆t±d/√2, if
we assume Gaussian particles just for simplicity. This
leads to situations like the one depicted in figure 2.
Larger particle diameter will result in a smaller
overestimation.
The case ∆ω = 8 pixels implies 75% window
overlapping. This higher than recommended value is
justified for the objective of the paper is just to illustrate
the mechanisms of error growth in images in the absence
of noise, and not the design rules for a PIV processor for
real images, which usually contain noise.
In all the cases depicted in figure 1, the maximum
displacements, U0∆t, were much smaller than the
4
v0∆t
v1∆t
d√2
d√2
Zone with particles
contributing to v1 peak
Zone with particles
contributing to v0 peak
Figure 2. Explanation of why group locking can mean a bias different from the “most frequent” displacement when the diameter
of the monodisperse particle images is not null.
In this figure the “most frequent” displacement is
v0∆t, where the slope of the displacement spatial
distribution vanishes. Even if the center of the
interrogation window is located in the position
corresponding to this velocity v0, the measured velocity
in absence of noise would be a smaller value towards
v1 = v0-d/(∆t√2), due to the presence of more particles
contributing to its correlation peak. This means a bias
away from the “most frequent” displacement. This effect
can be observed in figure 3, where a synthetic couple of
images with the properties described in the introduction
and depicting a single 1D harmonic field with λ = 64
pixels has been processed by the above described
conventional PIV using windows of F = 32 pixels. The
result in figure 3 shows coherence with the group
locking concept. It can be argued that peak locking
contribution could increase the separation from the most
frequent displacement. This would be so only in the case
of a group locking departure from the most frequent
displacement > 0.5 pixels.
A non linear response of PIV is evident in this case,
introducing high spatial frequencies evidenced by the
square shape of the resulting displacement distribution.
The resulting high window overlapping in this case
allows fully revealing the phenomenon. In a practical
method with less overlapping the introduction of high
frequencies associated to group locking would be
different in each case, depending on the degree of
overlapping and the spatial phase where the
measurement is taken, not mentioning the effect of other
PIV parameters.
10
Displacement (pixels)
8
6
4
2
0
-2
-4
-6
-8
-10
32
48
64
80
96
112
128
Location (pixels)
Figure 3. Results of a conventional PIV processing of a
synthetic image containing a 1D single harmonic displacement
field, depicting the effect commented in figure 2. Continuous
line denotes real displacements, circles stand for results with
F = 32 pixels.
If a single constant gradient is present in the
interrogation window, group locking will not appear, but
a degenerate situation where an elongated peak is
formed, like a wall. The number of particles contributing
to each displacement is equal in theory, giving
correlation peaks of the same height. Owing to the finite
number of particles, the height of this elongated peak
will not be constant along it. As a result, the peak
finding algorithm will detect a random peak on the wall.
Repeating with a interrogation window a random error
will appear on the measurement ensemble, but not a
bias. This indicates that velocity gradients (first spatial
derivative) are not the underlying explanation for group
locking bias, but the second spatial derivative (Lecuona
et al. 2002b).
With a more complex displacement distribution, the
appearance of secondary peaks caused by group locking
will be possible. This will happen when there are
particle displacement grouping separated enough to
form two geometrically resolvable peaks and separated
more than 2 pixel. For smaller displacements
differences, a deformed, or biased peak will result.
There are ways of determining whether the errors
depicted in figure 1 are coming from group locking or
not. The path followed in the analysis starts by
considering a continuous and uniform distribution of
overlapping Gaussian particles all along the
5
instead of the real non uniform contribution, which is
due to the finite number of particles and thus different in
shape for each image. The ratio between the peak and its
average level depends on the seeding density of the PIV
image. Here the case of δ = 2 pixels has been computed
to match the synthetic images. Different results arise for
the cases of low and high vorticity, as can be observed
in figure 4.
118000
200000
116000
190000
114000
180000
112000
110000
2.9
108000
2.1
106000
1.3
104000
0.5
102000
100000
Arbitrary units
Arbitrary units
interrogation window, although the reasoning would be
valid for non Gaussian particle. Then, the correlation of
each particle with itself from the couple of PIV images
is calculated. This gives a Gaussian peak of diameter
d√2 at the location of the corresponding displacement in
the correlation domain. All these peaks add up, giving a
correlation peak where the unavoidable noise coming
from the correlation between different particles has been
neglected. In this model this is added as a constant level
-0.3
Axis X
-1.1
4.5 4.0
-1.9 displacement
3.5 3.0
2.5 2.0
[pixels]
1.5 1.0
-2.7
0.5 0.0
-0.5 -1.0
-3.5
Axis Y displacement [pixels]
-1.5 -2.0
-2.5
a
170000
160000
2.0
150000
1.5
140000
1.0
130000
0.5
120000
0.0
110000
100000
-0.5
2.0
1.5
1.0
-1.0
0.5
0.0
Axis Y displacement [pixels]
-1.5
-0.5
-1.0
-1.5
Axis X
displacement
[pixels]
-2.0
-2.0
b
Figure 4. Biased correlation peaks due to group locking. Both correspond to the point 8 pixels to the right of the center of a
vortex with R0 = 45 pixels and particles of d = 2 pixels. a) plot corresponding to ωp = 0.35∆t-1. Presents the peak at vy∆t = 1.99
pixels when the displacement corresponding to the center of its interrogation window is vy∆t = 1.36 pixels. b) plot corresponding
to ωp = 0.06 ∆t-1. Presents the peak at vy∆t = 0.16 pixels when the displacement corresponding to the center of its interrogation
window is vy∆t = 0.23 pixels.
In the real correlation a finite number of particles
induce noise coming from the correlation between
different particles. This, together with other sources of
error introduce some dispersion in the final location of
the upper part of the correlation peak. The plots depicted
in figure 4 give three reasons why the uncertainty in
peak vorticity is larger in the case of large vorticity, as
was observed in figure 1.
• The first two of these reasons are related to the peak
itself. One is the fact that a larger number of
particles build up the peak on the case of low
vorticity, giving a higher similarity with the real
peak build-up using PIV processing. The other is
the less spread peak for the low vorticity case, even
normalizing with the magnitude of the displacement
that corresponds to the peak. This reduces the
possibility of deviation of the measurement.
• The third reason has to deal with the fact that eight
different displacements are used for each vorticity
calculation of those depicted in figure 1. The peak
on the large vorticity case is clearly non
symmetrical. In our case the asymmetry is more
probable towards smaller displacements than
towards larger ones. Consequently, it is less
probable that the different deviations of the eight
vectors cancel each other.
These plots explain not only the dispersion of the
measurements but the bias too. Due to the effect of
group locking in these peaks, rarely the peak maximum
matches the displacement corresponding to the center of
the window. Instead, they occur at a different
displacement within the interrogation window.
To compare this deviation model against the real
case, the correlation peaks like the ones depicted in
figure 4 have been pixel discretized and then located
with a usual three points Gaussian peak fitting
algorithm. Figure 5 gives examples of the location of the
resulting displacement in the analytical flow field that
pure group locking yields in comparison with the
displacements that the conventional PIV processing of
figure 1 yields.
6
Y[grid nodes]
8
8
6
6
4
4
2
2
Y[grid nodes]
Y[grid nodes]
10
8
0
-2
0
-2
6
-4
-4
4
-6
-6
2
-8
b
0
-2
-8
-8
-6
-4
-2
0
2
4
x[grid nodes]
6
c
8
-8
-6
-10
-8
-4
-2
0
2
x[grid nodes]
4
6
8
-4
10
10
8
8
6
6
4
4
-6
-8
-6
-4
-2
0
2
4
x[grid nodes]
6
8
10
Y[grid nodes]
a
-10
d
Y[grid nodes]
-8
-10
2
0
-2
2
0
-2
-4
-4
-6
-6
-8
-8
-10
-10
-10
-8
-6
-4
-2
0
2
x[grid nodes]
4
6
8
10
e
-6
-4
-2
0
2
x[grid nodes]
4
6
8
10
Figure 5. Locations in the vortex flow field of the displacements corresponding to interrogation windows with center on the
nodes of a grid with 8 pixels of spacing. The discontinuous circle represent the radius of the vortex, R0 = 45 pixels in all cases.
Results from: a) exact processing indicating the grid nodes. b) prediction based on group locking model for the case
ωp = 0.35∆t-1. Asymmetries are due to multiple peaks presence. In these cases, one of the peaks has been arbitrarily selected. c)
conventional PIV processing for the case ωp = 0.35∆t-1, ∆ = 8 pixels. d) prediction based on group locking model for the case
ωp = 0.06∆t-1. e) conventional PIV processing for the case ωp = 0.06 ∆t-1, ∆ = 8 pixels.
plotted at the vortex radius where the maximum
displacement is located.
The square shape of the interrogation window imply
the appearance of the symmetry axis that can be
appreciated on the plot.
As it has been previously commented, in
conventional PIV processing, the finite number of
particles within an interrogation window produces
statistical differences between the prediction with an
infinite number of particles and the reality. Nevertheless,
the magnitude and tendency of the deviations are
correctly predicted by the model using an infinite
number of particles. One further check is to use this
model to predict the measurements in figure 1. The
results obtained are plotted in figure 6.
The similarity between model and PIV processing
confirms the group locking argument, indicating that
under the circumstances imposed, large deviations on
the peak location occur, especially near the vortex radius
where a change in behavior is also noticeable. This
change is caused by the grouping of particle correlation
peaks near the maximum vortex velocity, where the first
radial derivative of the velocity vanishes. Deviations are
larger for the low vorticity case owing to the smaller
gradients on the flow field.
In figure 5 all the displacements are within those
present in the flow field ± d/2. In those cases that the
resulting displacement is larger than U0∆t it has been
7
True peak vorticity, ω p, normalized
1.2
Norm. resp. = Meas. Vort. / Correct peak Vort.
Norm. resp. = Meas. Vort. / Correct peak Vort.
True peak vorticity, ω p, normalized
1.4
Case ω p = 0.35 [1/∆ t]
1
0.8
0.6
Correct peak vorticity normalized
0.4
0.2
Case ω p = 0.06 [1/∆ t]
0
Case ω p = 0.35 [1/∆ t]
1.4
1.2
1
0.8
0.6
Correct peak vorticity normalized
0.4
0.2
Case ω p = 0.06 [1/∆ t]
0
0
10
20
30
40
50
60
70
0
Vortex core radius (modified Rankine)
10
20
30
40
50
60
70
Vortex core radius (modified Rankine)
a
b
Figure 6. Prediction of the normalized response of the conventional PIV algorithm on a vortex when trespassing the working
envelope. Same code for symbols as figure 1. a) plot corresponding to ∆ω = 8 pixels. b) plot corresponding to ∆ω = 16 pixels. The
predictions are depicted by a thick dashed line. A dotted line depicts the vorticity for displacements corresponding to the mid way
between the location of the correlation peak top and its center of mass. This gives an approximation on the expected bias and
dispersion magnitude. The dash and dots thin line corresponds to the peak summit before discretisation.
locking error taking as a reference the flow field at the
same point, like in figure 3, and not a local average,
always related to the parameters of the averaging
process.
It can be observed that the dispersion in the case of
high vorticity is toward smaller vorticity, as predicted
from the peak shape in figure 4. Taking this into
account, the line depicting the vorticity for
displacements corresponding to the mid way between
the location of the correlation peak top and its center of
mass has been included in the graph. Although this is
only one of the three sources of dispersion commented
when analyzing figure 4, it gives a reasonable
prediction. Only in the cases of low R0 and ∆ω = 16 this
prediction of dispersion is not adequate. This is so
because in those cases the differences in the
displacements evaluated are in the order of 0.03 pixels.
In this situation, the depicted difference comes from the
peak-fitting uncertainty and not from a real difference in
location of the center of mass and the top of the peak.
As some questions regarding the effect of different
peak fitting algorithms can be raised, the dash and dots
thin line indicates results obtained using the peak
summit location before discretisation, although no peak
fitting algorithm would be able to fully recover this
information. It is of importance that such an algorithm
would not reduce the error significantly.
These results again supports that group locking is a
built-in drawback, responsible for errors that limit the
applicability of methods based on the straight
application of conventional PIV algorithms. To avoid
this error, modifications in the conventional method
seems appropriate or alternatively modifications on the
basic algorithm. Such a method or algorithm, free of this
built-in error, would provide the possibility of extracting
additional information from PIV images, at the same
time expanding some of the most relevant limits of
conventional PIV processing.
Up to now the reference for comparison has been the
displacement in the center of the interrogation window.
The average displacement along the interrogation
window could be an alternative. The selection of this
reference responds to the aim of describing the group
4. Advanced methods: avoidance of the group
locking effect
This section focuses on considerations about which kind
of advanced methods can avoid the effect of group
locking in experiments oriented towards the industrial
sector.
Weighting of the interrogation window can modify
the measurement response of PIV algorithm, including
the effect of group locking. The problem that arises is
that it generally reduces the s/n ratio; in this sense it is
like reducing the size of the interrogation window. This
is a considerable drawback in industrial environments. If
weighting is applied in a way that does not increase the
occurrence of outliers, increasing the window size for
example, the group locking can be modified but not
avoided. The coupling between effective size of the
interrogation window and window weighting prevents
this option, when applied alone, to be considered as a
way of avoiding group locking.
Regarding methods that improve the s/n ratio, one of
them that is usually applied in industrial environments is
the offset of the interrogation window in the basic
algorithm. Due to the reduction of the in-plane loss of
particles the s/n ratio increases. Unfortunately, the offset
of the interrogation window does not improve the
rejection of group locking. The information of the
location of the particles is still lost. Several test runs on
the synthetic images of figure 1 did not show
improvement.
In order to check the suitability of other advanced
methods, some results from literature can be mentioned.
In particular, the “pivchallenge 2001” edition
(www.pivchallenge.org) showed the state of the art in
8
different advanced methods. Other cases within the
same vortex flow test exercise had dense seeding, with
only the spatial resolution (R0 ∼ 18 pixels) and the high
vorticity (ωp ~ 1.9∆t-1; θp ~ 55 deg.) as difficulties. This
corresponds to the difficulties studied in this paper. The
results of case B003 from the pivchallenge 2001 data
base are depicted in figure 7.
that moment. One of the tests of that international
collaboration was focused on synthetic images of vortex
flows. In order to test the response of the methods to
different conditions, some cases of this same test
included seeding density and CCD saturation as sources
of noise, for example case B002. These effects limit the
information contained in the image in a non reversible
way, reducing performance differences between the
a
b
c
d
e
f
g
h
Figure 7. Example of application of advanced methods from Pivchallenge 2001. The case corresponds to a synthetic PIV couple
of images of the core of a small vortex with high vorticity, R0 ~ 18 pixels, ωp ~ 1.9 ∆t-1. a) vortex displacements vectors to be
measured. b) error vectors after processing the images with LFC-PIV. c) d) e) f) g) and h) error vectors after processing the
images with other participants advanced methods.
et al. 2002a). The good behavior of the LFC-PIV
technique when processing industrial PIV images has
been presented in Nogueira et al. (2003). The following
section will give a further example.
Moreover, image deformation like this one are
advanced methods that increase the s/n ratio (Nogueira
el al. 2001.).
Returning to group locking, in a method like LFCPIV the image deformation iterations lead to a situation
where the displacements to be measured after each
iteration are theoretically smaller all along the
Besides the high quality of the data obtained by all
the groups, this figure remarks the suitability of Local
Field Correction PIV (LFC-PIV) to obtain good
descriptions of the velocity field even near the vortex
core. This technique, already applied by the authors in
the past (Nogueira et al. 2001), is based on iterative
image deformation, including subpixel offset, and
proprietary weighting on square windows of F = 64
pixels. The combination of the weighting and the large
size of the window gives an effective window size of
about 32 pixels relative to outlier occurrence (Lecuona
9
With this idea in mind, a test has been performed
over the synthetic images used for figure 5. The results
are plotted in figure 8.
8
8
6
6
6
4
4
4
2
2
2
0
-2
Y[grid nodes]
8
Y[grid nodes]
Y[grid nodes]
interrogation window. This leads to a progressive
reduction of the differences in displacements and thus a
reduction on the effect of group locking in the
processing. Consequently, it seems a good candidate to
check the possibility in avoiding group locking.
0
-2
0
-2
-4
-4
-4
-6
-6
-6
-8
-8
-8
a
-6
-4
-2
0
2
x[grid nodes]
4
6
8
b
-8
-8
-6
-4
-2
0
2
x[grid nodes]
4
6
8
c
-8
-6
-4
-2
0
2
x[grid nodes]
4
6
8
Figure 8. Locations in the vortex flow field of the displacements corresponding to interrogation windows with center on the
nodes of a grid with 8 pixels of spacing. The discontinuous circle represents the radius of the vortex, R0 = 45 pixels in all cases.
Results from: a) exact processing, indicating the grid nodes . b) LFC-PIV processing for the case of figure 5c, ωp = 0.35∆t-1. c)
LFC-PIV processing for the case of figure 5d, ωp = 0.06∆t-1.
It can be observed that the deviations have been
reduced substantially, particularly near the vortex center.
In the case of low vorticity, some more error than in the
high vorticity case can be observed in the outer part of
the flow field. Let’s remark that for this case, out of the
vortex radius the gradients are very small (< 0.01∆t-1).
Consequently, small errors in the displacement produce
large distances to the matching in-the-window flow field
location. In order to evaluate the importance of those
errors as well as the ability of LFC-PIV to avoid group
locking, the cases of figure 1 were processed using this
technique. The results are plotted in figure 9. It can be
observed that the bias is < 12% down to R0 = 15 pixels.
They also present a small dispersion, < 12%, down to
that radius.
In addition to the cases of figure 1, a set of images
with a very high peak vorticity ωp = 1.2∆t-1 has been
included in figure 9, where the results from only two
couples of images are depicted for each radius. The
performance of the LFC-PIV method for those cases is
very similar to that of the cases with smaller vorticities,
making it difficult to differentiate the symbols. This
illustrates the large dynamic range that can be described
by the use of this method.
True peak vorticity, ω p, normalized
Norm. resp. = Meas. Vort. / Correct peak Vort.
Norm. resp. = Meas. Vort. / Correct peak Vort.
True peak vorticity, ω p, normalized
1.4
1.2
1
0.8
0.6
Correct peak vorticity normalized
0.4
0.2
0
1.4
1.2
1
0.8
0.6
Correct peak vorticity normalized
0.4
0.2
0
0
10
20
30
40
50
60
70
0
Vortex core radius (modified Rankine)
10
20
30
40
50
60
70
Vortex core radius (modified Rankine)
a
b
Figure 9. Normalized response of the LFC-PIV method on a vortex flow. Same code for symbols as figure 1 and 5. A set of
cases with very high peak vorticity ωp = 1.2∆t-1 has been added and plotted with squares. a) plot corresponding to ∆ω = 8 pixels.
b) plot corresponding to ∆ω = 16 pixels.
It should be remarked that all the cases have been
calculated with deformation based on a grid spacing
∆ = 8 pixels. This is so for both implementations of the
finite difference scheme in expression (2), that is, for
∆ω = 8 pixels and for ∆ω = 16 pixels. This has been
performed this way because the aim is to check the
performance of the method at different distances from
10
the vortex core. The improvements obtained by an
image deformation method over conventional PIV are
reduced for coarse deformations with large grid spacing.
For example, with deformation based on a grid with
∆ = 16 pixels the error grows slightly over 10% with
R0 = 30 pixels. This is like that because LFC-PIV is
based on a bilinear interpolation of the displacement
between grid nodes. Second order interpolation would
probably yield results more similar to the displacement
at the grid node, owing to the more accurate description
of the local displacements.
5. Application to real images
The previous study has been performed on synthetic PIV
images. These images are a powerful tool that allows to
expose particular details an reveal the underlying
mechanisms. Nevertheless, the genuine objective of the
work here reported is real images. Even though the flow
field in a real image is unknown, thus precluding to offer
a measurement error figure, tests are customary for
checking the coherence between the results obtained
from synthetic images and those from real ones obtained
in an industrial facility.
In this section the tests on real images focus on two
different topics:
• Suitability of the LFC-PIV method. This is
important if this method is proposed as a way of
avoiding errors coming from group locking.
• Relevance of the errors coming from group locking
in comparison to other sources of error.
In order to check the first topic, a couple of PIV
images from a measurement campaign carried out by
DLR within the Europiv2 consortium was selected. The
test campaign focused on the 2D low Mach number
aerodynamics around a wing profile in high lift
configuration. Figure 10 gives a sketch of the
configuration and the position chosen for the image
analyzed in this section. The image can be observed in
Lecuona et al. (2003). This couple of images is a
especially difficult case where typical commercial stateof-the-art software had problems in correctly evaluating
the flow field. The software used for comparison
implements conventional PIV plus integer window
offset, using ∆ = 16 pixels. The window offset was
implemented by means of a three steps multigrid
scheme. The purpose of the first two steps is to locally
determine an accurate value for integer window offset
for the third one. No image deformation was used, and
window sizes F were 128, 64 and 32 pixels in the three
successive steps. Between steps a validation algorithm
was implemented, rejecting vectors that differed more
than 45% of the local median of a 3 by 3 neighborhood.
The resulting vorticity field is plotted in figure 11a.
Figure 10. Location of the PIV image analyzed in this section. DLR ref: 456_18_c3_0020.b16.
a
b
Figure 11. Comparison of application of conventional PIV and LFC-PIV to a real image from an industrial facility. Hatched
pattern means solid objects; cross-hatched pattern means places where reflections and shadows suppress all data. a) Vorticity plot
obtained from conventional PIV data. b) Vorticity plot obtained from LFC-PIV data.
On the same PIV image pair, LFC-PIV was applied
giving the results presented in figure 11b. The Von
Kârman vortex street in the wake of the slat is clearly
visible and also there is much less spurious results than
with conventional PIV. They are easily detected because
11
of the inverse vorticity structure appearing at both sides
of the spurious vector.
Easier cases were also analyzed, but although the real
flow field is not known, the results were always
apparently better when LFC-PIV was applied (Nogueira
et al. 2003).
In addition to the accuracy improvement obtained,
this example supports the robustness of the LFC-PIV
method when processing images coming from
industrial-like setups.
The second subject of this section is checking the
relevance of group locking in a real image. For this task
the region marked with a square in figure 11 has been
selected. LFC-PIV gave good data inside it and the
analysis, above offered, using synthetic images indicates
that the results can be expected to be free of group
locking. Consequently, it has been supposed that these
data are a good representation of the real flow field.
Based on this data and by bilinear interpolation between
grid nodes, a prediction of the flow field measurement
incorporating the bias of group locking has been
obtained using the method described in section 3. Data
Vor[1/dt]: -0.30 -0.26 -0.21 -0.17 -0.13 -0.09 -0.04 0.00 0.04 0.09 0.13 0.17 0.21 0.26 0.30
obtained processing with the described conventional
PIV resulted to be closer to the predicted biased
measurement in 214 vectors out of the 289 total vectors.
The other 75 vectors were closer to the supposed real
flow field. This result is coherent with the dispersion
around the prediction already presented in figure 6. The
vorticity plots for the LFC-PIV measurement, the
prediction of the biased measurement and the
measurement of a conventional PIV are detailed in
figure 12. Figure 12c presents a better look than the
corresponding square in figure 11a because a correct
mean displacement of 20 pixels has been subtracted
previously to the conventional PIV processing. When a
three-step multigrid scheme is implemented not always
the correct offset is found (Nogueira et al. 2002). It has
been checked that the result coming from conventional
PIV processing is not a simple low-pass filtering of the
correct
results.
Vor[1/dt]: -0.30 -0.26 -0.21 -0.17 -0.13 -0.09 -0.04 0.00 0.04 0.09 0.13 0.17 0.21 0.26 0.30
Vor[1/dt]: -0.30 -0.26 -0.21 -0.17 -0.13 -0.09 -0.04 0.00 0.04 0.09 0.13 0.17 0.21 0.26 0.30
670
660
660
660
650
650
650
640
640
640
630
630
630
Y[pixels]
620
Y[pixels]
670
Y[pixels]
670
620
610
620
610
610
600
600
600
590
590
590
580
580
580
570
570
560
570
560
1100
1150
X[pixels]
a
1200
560
1100
1150
X[pixels]
b
1200
1100
1150
1200
X[pixels]
c
Figure 12. Comparison of different processing methods on a part of the real image indicated in figure 11. a) Vorticity plot
obtained from LFC-PIV data. b) Vorticity plot obtained from the same LFC-PIV data, but biased by the group locking prediction.
c) Vorticity plot obtained from conventional PIV data.
The root mean square of the differences and the
maximum difference between the LFC-PIV vorticity
data and the conventional PIV vorticity data are three
times larger than the difference between the
conventional PIV vorticity data and the group locking
biased data coming from LFC-PIV processing. This
again supports the hypothesis that group locking is a
significant source of error in the real measurement
performed.
6. Conclusions
An evaluation has been performed on the error of PIV
processing produced by the differences in the particle
displacements within the interrogation window. It shows
that when the spatial displacement distribution includes
fraction of particles with similar displacements, the
resulting individual correlation peak grouping can
deform the ensemble correlation peak and even form a
high enough secondary peak, causing significant error.
This has been named group locking.
A model for group locking has been developed,
showing a reasonable description of its effects in
synthetic images and on real images from vortex flows
as well, for particles of diameter d ≈ 2 pixels. This
model allows to drive some conclusions relevant to the
industrial operators of PIV systems.
12
Group locking not only can increase the bias of the
measurement in a non-linear way but also can contribute
to deteriorate the repeatability of the measurement.
Group locking can signify a substantial source of
error when measuring vorticity in the core of vortex
flows. Its effect is clearly visible for vortex of diameter
up to R0∼ 4F, where F is the window side length,
implying this a spatial wavelengths up to λ ∼ 8F. This is
a relevant limitation when designing a PIV experiment.
This confirms that the maximum peak vorticity that a
conventional PIV method can evaluate with the
specified quality is in the order of ωp = 0.06∆t-1, which
corresponds to a rotation of θp = 1.7 deg.
When evaluating higher peak vorticities in a vortex ,
conventional PIV processing, even using a well
established low-pass circulation vorticity calculation
algorithms can overestimate the peak, even for R0 > 4F.
Advanced methods, such as LFC-PIV, can perform
well with substantially smaller spatial wavelengths,
showing a potential for high accuracy. It has been shown
that it is able to give correct measurements for peak
vorticities as high as ωp = 1.2∆t-1 (θp = 34 deg.) with
errors < 12% for R0 ≥ 15 pixels, when the vortex center
coincides with a grid point. This is primarily due to the
avoidance of the group locking effect by means of a non
diverging iterative image deformation technique that this
method uses. LFCPIV offers a high robustness against
noise and other image difficulties in industrial images
and also offers a large dynamic range to evaluate
vorticity, being able to accurately measure up to
ωp = 1.2 ∆t-1 with no external adjustment.
Acknowledgements
This work has been partially funded by the Spanish
Research Agency (DGCYT) grant AMB1999-0211,
project DPI2000-1839-CE, Feder 1FD97-1256 in
collaboration with the Universidad de Las Palmas de
Gran Canaria, Departamento de Ingeniería de Procesos,
grupo de Energía y Medio Ambiente, prof. Austín
Macías, project DPI2002-02453 “Técnicas Avanzadas
de Velocimetría por Imagen de Partículas (PIV)
Aplicadas a Flujos de Interés Industrial” and under the
EUROPIV 2 project (A JOINT PROGRAM TO
IMPROVE PIV PERFORMANCE FOR INDUSTRY
AND RESEARCH) 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), Zaragoza. The project is
managed by LML URA CNRS 1441 and is funded by
the CEC under the IMT initiative (CONTRACT N°:
GRD1-1999-10835)
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