5th International Symposium on Particle Image Velocimetry
Busan, Korea, September 22-24, 2003
PIV’03 Paper 3002
A. Lecuona, J. Nogueira P. A. Rodríguez and A. Acosta
Abstract Particle image velocimetry (PIV) is now applied with confidence in industrial facilities like large wind tunnels, yielding data not possible before. Many research and development programs have obtained benefits out of this measurement technique (Heliflow, Apian, C-wake, among others in Europe). 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. The requirements in them usually force the researcher to take measurements near or even out of the limits of conventional PIV operating envelope. This means large velocity gradients, small spatial wavelength flow structures, background light, reflections, etc. Experts are familiarized to deal with these situations and to evaluate when and where to push PIV limits. This is generally done within a conscious knowledge of the reduction in the confidence in the measurements taken under these conditions, often only at certain locations of the flow field.
The difficulties in these situations make that additional knowledge is always welcomed, particularly on the behavior of the PIV technique as well as on improvements in such areas as advanced PIV processing methods and hardware. Our work has been focused on two of these aspects: (i) The behavior of conventional PIV when the conventional limits of the processing algorithm are near or surpassed and (ii) Which kind of advanced methods can reduce the main sources of error that are characterized in this paper.
Here the focus is put on the description of vortex flows, not only for its relevance but also as a test case. In addition to that, the conclusions are applicable to other high shear flow structures. Group locking, a major source of error, is introduced, modeled, and metrologically characterized. This means a concept for additional evaluation of the processing PIV algorithms.
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. Also, they indicate situations in which PIV images already obtained in past experiments can be reprocessed with the new advanced method in order to extract more information and with a higher accuracy.
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.
The limits that define the working envelope of conventional PIV processing have been progressively defined in the open literature.
A. Lecuona, J. Nogueira, P. A. Rodríguez, A. Acosta
Dep. of Thermal and Fluids Engineering, Universidad Carlos III de Madrid, Spain.
Correspondence to:
Prof. Antonio Lecuona, Dep. of Thermal and Fluids Engineering, Universidad Carlos III de Madrid. c/ Butarque 15, 28911, Leganés, Spain.
E-mail: lecuona@ing.uc3m.es
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PIV’03
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 > ∼ 4 F . 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/( πδ 2 ) 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 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 · δ 2 ) ≈ 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
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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: v
θ
=
2
1 + (
(
/
/
0
0
) 2
)
; v r
= 0
(1)
U
0
is the maximum tangential velocity, located at the vortex radius R
0
. The vortex circulation is Γ = 4 π R
0
U
0
. 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, ω p
, corresponds to:
p
= 4 U
0
/ R
0
(2)
It diminishes to quickly reach ω p
/4 at a distance R
0
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 = 4 R
0
.
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 > ∼ 5.6
F as a first approximation. For F = 32 pixels this limit corresponds to R
0
~ 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
θ p
ω p
~ 0.06
∆ t
= 1.7 deg. between PIV pulses ( θ = 90 ω / π ).
-1 . This means a rotation of the flow in the core of the vortex of
R
0
With these orders of magnitude in mind, several tests on synthetic images were arranged. They were run with
≤ 60 pixels and for two different peak vorticities: ω p
= 0.06
∆ t -1 and a large value, ω p
= 0.35
∆ t -1 ( θ p
= 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 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.
ω
−
(
= v
( x v y −
− 1,1
+ 4
+ v
4 v x 0,1 y 1,0
+
+ u
1,1 v
1,1
) (
12
) (
12 v
∆
ω x v
∆
ω y
+ 4
+ v
4 v x − y − 1,0
+ v
+ v x − y − 1,1
)
)
− (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, R
0
and ω p
, 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 R
0
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.
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0.8
0.6
0.4
True peak vorticity,
ω p
, normalized
1.4
1.2
Case
ω p
= 0.35 [1/
∆ t]
1
Correct peak vorticity normalized
1.4
Case
ω p
= 0.35 [1/
∆ t]
1.2
1
0.8
0.6
0.4
True peak vorticity,
ω p
, normalized
Correct peak vorticity normalized
0.2
Case
ω p
= 0.06 [1/
∆ t]
0.2
Case
ω p
= 0.06
0
0 10 20 30 40 50 60 7 0
0
0 10 20 30 40 50 60 7 0
Vortex core radius (modified Rankine)
a
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 ∆ respectively stand for low and high vorticity.
ω
Vortex core radius (modified Rankine) b
= 16 pixels. Circles and triangles
• 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 < R
0
< 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 < R
0
< 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
R
0
= 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 R
0
> 30 pixels. 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 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, window, U
0
∆ t ,< 1 pixel for the low vorticity case and U
0
∆ t
U
0
∆ t , were much smaller than the interrogation
,< 6 pixels for the high vorticity case.
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 ω p
= 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
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PIV’03 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. v
0
∆ t
d √ 2 v
1
∆ t
d √ 2
Zone with particles contributing to v
0
peak
Zone with particles contributing to v
1
peak
Figure 2.
Explanation of why group locking can mean a bias different from the “most frequent” displacement when the diameter of the particles is not null.
In this figure the “most frequent” displacement is v
0
∆ 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 v
0
, the measured velocity in absence of noise would be a smaller value towards v
1
= v
0
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
4
2
0
-2
8
6
-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.
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PIV’03
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 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 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
180000
114000
170000
112000
160000
110000 2.0
2.9
150000
1.5
108000
2.1
140000
106000
1.0
1.3
130000
0.5
104000 0.5
120000
102000 110000
0.0
-0.3
100000
4.5 4.0
3.5 3.0
2.5 2.0
1.5 1.0
Axis Y displacement [pixels]
0.5 0.0
-0.5 -1.0
-1.9
-2.7
-3.5
-1.1
Axis X displacement
-1.5 -2.0
-2.5
a
100000
2.0
1.5
1.0
-0.5
-1.0
Axis X displacement
[pixels]
0.5
-1.5
[pixels]
0.0
Axis Y displacement [pixels]
-0.5
-1.0
-2.0
-1.5
-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 R
0
= 45 pixels and particles of d = 2 pixels. a) plot corresponding to ω p
= 0.35
∆ t -1 . Presents the peak at v y v y
∆ t = 1.99 pixels when the displacement corresponding to the center of its interrogation window is
∆ t = 1.36 pixels. b) plot corresponding to ω p
= 0.06 ∆ t -1 . Presents the peak at v displacement corresponding to the center of its interrogation window is v y y
∆ t = 0.16 pixels when the
∆ 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. The other is the less spread peak fro 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.
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PIV’03
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.
8
6
4
8
6
4
2 2
0 0
10
-2 -2
8
-4 -4
6
4 -6 -6 a
2
0
-2
-4
-6
-8
-10
-10 -8 -6 -4 -2 0 2 4 x[grid nodes]
6 8 10 b
) 
-8
10
8
6
4
2
0
-2
-8 -6 -4 -2 0 2 x[grid nodes]
4 6 8 c
-8
10
8
6
4
2
0
-2
-8 -6 -4 -2 0 2 x[grid nodes]
4 6 8
-4
-6
-4
-6
-8 -8 d
-10
-10 -8 -6 -4 -2 0 2 x[grid nodes]
4 6 8 10 e
-10
-10 -8 -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,
R
0
= 45 pixels in all cases. Results from: a) exact processing indicating the grid nodes. b) prediction based on group locking model for the case ω peaks has been arbitrarily selected. c) conventional PIV processing for the case ω
ω p
= 0.06 ∆ t -1 , ∆ = 8 pixels. p
= 0.35
∆ t -1 . Asymmetries are due to multiple peaks presence. In these cases, one of the prediction based on group locking model for the case ω p
= 0.06
∆ t -1 p
= 0.35
∆ t -1 , ∆ = 8 pixels. d)
. e) conventional PIV processing for the case
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 U
0
∆ t it has been 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.
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PIV’03
0.8
0.6
0.4
True peak vorticity,
ω p
, normalized
1.4
1.2
Case
ω p
= 0.35 [1/
∆ t]
1
Correct peak vorticity normalized
1.4
Case
ω p
= 0.35 [1/
∆ t]
1.2
1
0.8
0.6
0.4
True peak vorticity,
ω p
, normalized
Correct peak vorticity normalized
0.2
Case
ω p
= 0.06 [1/
∆ t]
0.2
Case
ω p
= 0.06
0
0 10 20 30 40 50 60 7 0
0
0 10 20 30 40 50 60 7 0
Vortex core radius (modified Rankine)
a
Vortex core radius (modified Rankine) 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 red thin dashed line corresponds to the peak summit before discretisation.
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 R
0
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 red thin dashed 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 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.
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
8

PIV’03 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 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 different advanced methods. Other cases within the same vortex flow test exercise had dense seeding, with only the spatial resolution ( R
0 vorticity ( ω p
~ 1.9
∆ t results of case B003 from the pivchallenge 2001 data base are depicted in figure 7.
∼ 18 pixels) and the high
-1 ; θ p
~ 55 deg.) as difficulties. This corresponds to the difficulties studied in this paper. 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, R
0
~ 18 pixels, ω 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. p
~ 1.9 ∆ t -1 . a) vortex
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 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.
9

PIV’03
1.2
1
0.8
0.6
0.4
0.2
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 LFC-PIV the image deformation iterations lead to a situation where the displacements to be measured after each iteration are theoretically smaller all along the 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.
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 8
6
4
2
0
-2
-4
-6
6
4
2
0
-2
-4
-6
6
4
2
0
-2
-4
-6 a -8
-8 -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
-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,
R
0
= 45 pixels in all cases. Results from: a) exact processing, indicating the grid nodes . b) LFC-PIV processing for the case of figure 5c, ω p
= 0.35
∆ t -1 . c) LFC-PIV processing for the case of figure 5d, ω p
= 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
small errors in the displacement produce large distances to the matching in-the-window flow field location. In order
< 12% down to R
0
∆ t -1 ). Consequently, 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
= 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 ω p
= 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 True peak vorticity,
ω p
, normalized
1.4
1.4
Correct peak vorticity normalized
1.2
1
0.8
0.6
0.4
0.2
Correct peak vorticity normalized
0
0 10 20 30 40 50 60 7 0
0
0 10 20 30 40 50 60 7 0
Vortex core radius (modified Rankine)
a
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 ω corresponding to ∆
ω p
= 1.2
∆ t -1
Vortex core radius (modified Rankine) b
has been added and plotted with squares. a) plot
= 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 the vortex core. The improvements obtained by an image deformation method over
10

PIV’03 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 R
0
= 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 state-of-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
Figure 11.
Comparison of application of conventional PIV and LFC-PIV to a real image from an industrial facility. b
Dark grey means solid objects; light grey 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.
11

PIV’03
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 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 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).
Vor[1/dt]: -0.18 -0.15 -0.13 -0.10 -0.08 -0.05 -0.03 0.00 0.03 0.05 0.08 0.10 0.13 0.15 0.18
Vor[1/dt]: -0.18 -0.15 -0.13 -0.10 -0.08 -0.05 -0.03 0.00 0.03 0.05 0.08 0.10 0.13 0.15 0.18
Vor[1/dt]: -0.18 -0.15 -0.13 -0.10 -0.08 -0.05 -0.03 0.00 0.03 0.05 0.08 0.10 0.13 0.15 0.18
620
610
600
590
580
570
670
660
650
640
630
620
610
600
590
580
570
670
660
650
640
630
620
610
600
590
580
570
670
660
650
640
630
560
1100
X[pixels]
1150 1200
a
560
1100
X[pixels]
1150 1200 b
560
1100
X [pixels]
1150 1200 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 as well for particles of diameter d ≈ 2 pixels. This model allows to drive some conclusions relevant to the industrial operators of PIV systems.
Group locking not only can increase the bias of the measurement 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 R
0
∼ 4 F , where F is the window side length, implying this a spatial wavelengths up to λ ∼ 8 F . This is a relevant limitation when designing a PIV experiment. This confirms that the maximum peak vorticity that a conventional PIV method can evaluate is in the order of ω p corresponds to a rotation of θ p
= 1.7 deg.
= 0.06
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 R
0
> 4 F .
∆ t -1 , which
12

PIV’03
Advanced methods, such as LFC-PIV, can perform well at 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 ω p
= 1.2
∆ t -1 ( θ p
= 34 deg.) with errors < 12% for R
0
≥ 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 these methods use. This method 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 ω p
= 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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