Advanced PIV algorithms with Image Distortion
Validation and Comparison using Synthetic
Images of Turbulent Flow
B. Lecordier1 and M. Trinité
CORIA – UMR CNRS 6614, Université de Rouen, BP 12, F-76801 Saint Etienne
du Rouvray Cedex (France)
Abstract
In the present paper, two advanced PIV algorithms are described and compared
with conventional cross-correlation sub-pixel PIV evaluation techniques. The first
algorithm is an iterative continuous windows shift technique (CWS). The second,
based on the same correlation and peak-finding techniques, includes an image
distortion module to improve measurement of velocity gradients. PIV algorithms
are validated and compared using synthetic images, onto which Direct Numerical
Simulation (DNS) adds the motion of tracer particles to simulate a turbulent flow
with homogeneous and isotropic properties. PIV algorithms are compared and
their limitations studied in terms of velocity fluctuations, vorticity field, spectrum
or other relevant turbulence parameters. The effect of the out-of-plane motion in
3D turbulent flows and the resulting uncertainties are also investigated.
1 Introduction
Nowadays, Particle Image Velocimetry (PIV) is a well-developed measurement
technique, which is widely used for fundamental research and industrial applications. In the past five years, in order to improve the accuracy of velocity measurements, numerous advanced PIV algorithms have been proposed. Nevertheless,
up to now, the intrinsic limitations of these algorithms have not always been well
established, especially their limitations for investigating turbulent flows in terms
of scales, energy and spectrum. Various experimental studies using fundamental
flows such as grid turbulence, wind tunnel or pipe flow have been used to try to
evaluate, in a real configuration, how well the PIV technique can measure turbulence quantities from PIV maps. Unfortunately, it is always difficult to conclude
which advanced PIV treatment is the most accurate or the most appropriate to
measure a given flow characteristic. Indeed, as quantities such as the energy spectrum or scales of turbulence are only known with experimental uncertainties, the
comparison of PIV algorithms is not straightforward. In addition, from experi1
Corresponding author: Bertrand.Lecordier@coria.fr - http://www.coria.fr
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mental validations, it is quite difficult to evaluate which main experimental settings (seeding density, laser, particle size…) are the most important for the accuracy of the PIV technique.
The primary objective of this paper is to compare the standard PIV treatments
proposed in the literature with advanced PIV algorithms. These comparisons
should lead to a better understanding of the intrinsic limitations of the different
standard and advanced PIV algorithms, and especially their limitations in terms of
measurement of turbulence properties. The comparisons are performed from a
fully digital approach based on synthetic images of a flow field with particles. The
main idea consists in producing realistic synthetic images of particles within
known turbulent velocity fields. To this end direct numerical simulations of multiphase flows have been used to generate digital turbulent flow fields and particle
fields with known homogeneous and isotropic properties (Lecordier et al. 2001).
The synthetic images are produced thanks to the Synthetic Image Generator
(S.I.G.) developed in the framework of the EUROPIV II European project. A detailed description of the SIG is given elsewhere in this book by Lecordier et al.
In the present paper, three different PIV algorithms are described and compared
in terms of their capacity to evaluate turbulence properties: a conventional crosscorrelation method with sub-pixel accuracy (CPIV), a continuous window shift
technique using a predictor/corrector iterative process (CWS) and an original algorithm including an image distortion module (MDPIV). The section below describes the 3 different PIV algorithms with more attention to the approach utilising
the image distortion technique. In the two following sections, the results of the
comparison of the PIV algorithms from synthetic images will be presented.
Fig. 1 Principle of different interrogation window management.
The second section evaluates the effect of the in-plane particle motion on the
PIV accuracy while that the out-of-plane particle motion will be investigated in
third section. The main conclusions are summarized in the final section.
2 Description of the Different PIV Algorithms
The next three sub-sections contain a description of the different PIV algorithms
compared in the present paper. We shall consider 3 different PIV treatments:
Advanced Algoritms 117
•
•
•
Conventional sub-pixel PIV treatment (CPIV).
Continuous window shift technique (CWS).
Multi-grid shift technique with image distortion (MDPIV).
In the framework of the EUROPIV II project, the developments of the CORIA
have been mainly focused on the last two PIV algorithms with the demanding task
of combining a high order image distortion module with a multi-grid PIV algorithm.
2.1 Conventional sub-pixel PIV method (CPIV)
This algorithm consists in cross-correlating small corresponding window samples
at the same location in the two successive images of particles (cf. figure 1). The
normalised 2D correlation signal is obtained by using the Fast Fourier Transform
(FFT) technique and the peak location is determined with sub-pixel accuracy. In
the present work, the standard 2D Gaussian peak interpolation method has been
used (Lourenco and Krothapalli, 1995). A detailed description of the conventional
cross-correlation PIV method is beyond the scope of this paper and details can be
found elsewhere in references (Willert and Gharib, 1991; Huang et al., 1993).
The conventional PIV approach has several drawbacks and one of the sources
of uncertainty is introduced by the in-plane particle displacement. To overcome
this problem, iterative discrete offset of the interrogation windows can be adopted
(cf. figure 1) (Scarano and Riethmuller, 1999). This increases the dynamic range,
and allows the size of the interrogation windows to be reduced, thus improving the
spatial resolution.
2.2 Sub-pixel iterative approach (CWS) (Lecordier 1997, 1999)
The main idea of this PIV evaluation consists in an iterative velocity measurement
based on a predictor/corrector approach, which tends progressively towards the
measurement of zero displacement, the most accurate detectable displacement by
the cross- correlation algorithm. To this end, our algorithm introduces an iterative
shifting of the interrogation windows to reduce in-plane particle motion effects,
but contrary to the discrete windows offset technique, the windows are shifted in
fractions of a pixel. Thus, after a few iterative loops, the correlation peak is centred on the origin of the correlation plane and so the measured displacement is
nearly equal to zero. The displacement required to determine the local velocity
vector is given by the displacement of the shifted windows. The iterative loop is
stopped when the “maximum” possible resolution is reached. The second originality of our treatment is the alignment of the interrogation windows with the local
direction of the particle displacement (cf. Fig. 1). So, during the treatment, the
window size can be reduced in the direction perpendicular to the displacement to
increase the spatial resolution and reduce the effects of velocity gradients (cf. Fig.
2).
The tricky part of our treatment is the sub-pixel translation and rotation of the
interrogation windows (Fig. 3). The centres of the first and second interrogation
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windows are symmetrically shifted in the two images of the PIV recording and the
magnitude of shift is obtained from the previously determined particle displacement. Next, the interrogation windows are rotated in order to align the windows
with the predicted direction of the particle displacement.
Fig. 2 Principle of the translation and rotation of an interrogation window.
As presented in Fig. 3, each pixel (xp,yp) of the rotated window is then interpolated from 25 neighbouring pixels sampled in the digital images of the PIV recording. The grey level of pixel value results from 5 horizontal and 1 vertical
polynomial interpolations of degree 4. Each horizontal interpolation is performed
from 5 horizontal pixels and they are used to estimate 5 grey levels at xp location
in the vertical direction. Next, these 5 new values are interpolated to compute the
grey level at (xp,yp) location in the rotated window.
Fig. 3 Principle of the measurement of a uniform velocity gradient by using rotated interrogation windows.
Advanced Algoritms 119
It must be emphasised that this method is independent of the interpolation directions and that 5 vertical interpolations followed by 1 horizontal interpolation
lead to the same result. A simpler image interpolation method may be used without dramatically influencing the velocity measurements. However, in a few
situations such as an alignment of particle displacement with the pixel mesh, velocity uncertainties can increase significantly. To avoid the divergence of the
computation between successive loops, the estimated velocity field is validated
thanks to 3 different methods based on: the minimum signal-to-noise ratio of the
correlation signal, the vector magnitude and neighbourhood comparisons with a
median filter. The spurious vectors are interpolated using the DAM method (see
next section) and the result is used as the next estimated velocity field.
It can be noticed that our sub-pixel iterative method does not dramatically increase the computer load. The computation time is only 3 times longer than the
conventional PIV approach. Indeed, the number of loops needed to measure residual displacements smaller than 0.05 pixel is generally smaller than 3.
2.3 Multi-grid continuous window shift technique with image
distortion (MDPIV)
As described in the previous section, the CWS treatment leads to an increase of
the dynamic range (window translation) of the displacement measurements, reduces the in-plane particle motion and the “peak-locking” effects (Westerweel J.,
1998; Lecordier et al. 1999, 2001) and in a few cases improves the measurement
of the velocity gradient. Nevertheless, for large velocity gradients and small scales
encountered in turbulent flows, the previous approaches present some limitations.
In order to improve the possibility of measuring large velocity gradients, in the
framework of the EUROPIV II project, we have combined our CWS algorithm
with an image distortion technique. Up to now, this approach, originally introduced by Huang et al., 1993, was inapplicable to large datasets due to very long
computing times. However, within the past few years, the rapid improvement of
computer performances has led us to reconsider the advantages of image distortion
methods, even when large datasets of PIV recording have to be analysed.
The main idea of our image distortion method consists in cancelling, as far as
possible, the particle displacement in the two images of each PIV recording. This
iterative process of deformation is started (step n=0) by a first estimation of the
velocity (u0,v0 ) field from the initial images I 0 and J 0 . Next, two deformed images I1 and J1 (step n=1) are obtained by taking into account the first predicted
velocity field in the initial images. A new velocity field, called corrector (u1c,v1c ) ,
is then computed from the deformed images and added to the initial velocity field
(u0,v0 ) . The next steps consist in a successive iterative computation of a new corrector followed by a distortion of the initial images. The iterative process is
stopped when parameters such as mean, fluctuation and maximum velocity magnitude of the corrector velocity (unc,vnc ) field become smaller than defined values.
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Fig. 4 Evolution of the corrector velocity fields at 6 steps of the deformation.
Examples of the evolution of successive corrector velocity fields during image
distortions are presented in Fig. 4. The initial step 0 (top-left plot) corresponds to
the result of the CWS algorithm from initial images I 0 and J 0 . During the iterative
process of deformation, the size and intensity of residual structures on deformed
images decrease. In the last step (bottom right plot), the corrector velocity field is
nearly equal to zero and so indicates that the deformed images perfectly match
each other. The final velocity field is then obtained from the equations:
i =n
un(x, y)=u0(x, y)+∑uic(x, y)
i =1
i =n
vn(x, y)=v0(x, y)+∑vic(x, y)
i =1
At the step n, to deform the initial images I 0 and J 0 according to the velocity
field (un,vn) , each pixel in the deformed images I n +1 and J n +1 is relocated using
the expressions:
u (i, j) vn(i, j) 
I n +1(i, j)= I 0(x p, y p)= I0 i − n
, j−
2 
2

u (i, j) vn(i, j) 
J n +1(i, j)= J 0(xq, yq)= J 0 i + n
, j+
2 
2

Advanced Algoritms 121
where un(i, j) and vn(i, j) are the velocities at the pixel location (i, j) . The expressions I 0(x p, y p) and J 0(x p, y p) correspond to image intensity interpolated at
the locations (x p, y p) and (xq, yq) and are obtained using the interpolation
scheme described in section 2.2. In (x p, y p) and (xq, yq) expressions, the velocity
[
]
field un(i, j),vn(i, j) can be outside of the velocity mesh. In this case, the velocity field is interpolated by using a diffuse approximation method (DAM) of the
third order (Prax and Sadat, 1996; Hamel et al., 2000). This high order approximation, able to take into account the effect of flow rotation and dilatation, permits
us to cancel the velocity gradients on the images without a direct calculation of
velocity derivatives.
It can be noticed that the accuracy of velocity interpolation is a more critical
point than the image interpolation procedure. An unsuitable interpolation scheme
introduces significant errors during the determination of corrector velocity fields
due to oscillation phenomena, additional noises or over-estimation of the intensities of vortices.
To evaluate the accuracy of different velocity interpolation schemes, a fixed
number of vectors has been randomly removed from a known velocity field and
next, each removed vector has been interpolated and compared to the initial value.
The relative errors of two interpolation schemes for u and v are plotted in Fig. 5.
For an inverse-distance weighting method with Gaussian kernel (left distribution),
the relative error reaches 4% whatever the velocity, and is around 20 times
larger than for the 3rd order DAM (right distribution). In addition, the DAM being
based on a statistical approach, it is not very sensitive to noise and so adapted to
experimental data. In addition this interpolation provides accurate estimations of
flow derivatives up to the second order.
Fig. 5 Relative error between vectors removed from known velocity field (30 %) and interpolated vectors. Interpolation method: inverse-distance weighting interpolation method
with Gaussian kernel (left) and DAM of 3rd order (right).
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Between two successive image deformations, the corrector velocity fields are
obtained from our CWS algorithm previously described. During the image deformation process, additional noise can be added if the corrector velocity fields contain spurious vectors. Thus, in order to avoid additional noise and propagation of
errors, at each step, the predictor velocity field is validated and spurious vectors
are replaced by using the diffuse interpolation scheme. This result of velocity
validation is also used as an input to adjust a multi-grid technique in our CWS
computation. Indeed, in order to reach the final solution as fast as possible and
then reduce the number of deformations, at each step, the interrogation window
sizes are locally adjusted according to the validation of the intermediate velocity
field. The size is reduced when the previous vector is validated and it is increased
if the vector is considered as spurious. A process of dichotomy fixes the magnitude of the size variation in order to minimise the number of loops. The iterative
computation is stopped when the window size distribution has converged. Other
multi-grid techniques are based on different parameters such as the local velocity
gradients or flow curvature (Scarano, 2002).
3 Validation of the PIV algorithms using Direct Numerical
Simulation and Synthetic Images
3.1 Introduction
In order to evaluate the accuracy and the intrinsic limitations of each PIV algorithm, we have used synthetic images of flow fields with tracer particles, which
simulate turbulent flow conditions. The synthetic images are obtained from the
common Synthetic Image Generator (S.I.G.) developed within the EUROPIV-2
project (contract G4RD-CT-2000-00190). A detailed description of the SIG is
given elsewhere in this book by Lecordier et al. The flow condition imposed on
the synthetic images is a homogeneous and isotropic turbulent flow without mean
velocity. The description of how such images are produced by using Direct Numerical Simulation of two-phase flow may be found in the reference (Lecordier et
al., 2001). The main idea consists in generating synthetic images from successive
2D or 3D particle fields produced thanks to a Direct Numerical Simulation of
multi-phase flows. The successive locations of each particle are used to generate
couples of synthetic images.
In the present work, the parameters of the simulation have been maintained
constant and are summarized in Table. 1. These flow parameters are quite close to
real flow conditions, but due to the limited size of computer memory they could
not be used to initialise a 3D simulation. Nevertheless, we will explain later how
to combine two 2D DNS with these parameters to simulate a pseudo 3D situation
in order to evaluate the effect of out-of-plane particle motion on the velocity
measurement.
Advanced Algoritms 123
Table 1. Properties of generated turbulent flows utilising the Direct Numerical Simulation.
u’
lint
Size
kt
η
ε
lλ
Npart
Mesh
Reλ
[m2/s2] [m2/s3] [m/s]
[mm]
[m]
[mm] [mm]
2
1024
0.1x0.1
1.12
31.8
1.06
6.44
1.66
0.1
200 32.105
The parameters required to generate synthetic images have been adjusted to
produce realistic images with properties similar to those recorded by CIRA partner
in its large wind tunnel. These parameters have been determined by comparing
synthetic and real image properties in terms of mean and RMS grey levels and
width and dispersion of the self-correlation signal. The analysis of this selfcorrelation signal is important in adjusting the size of the particles and thus simulating complex effects such as “peak-locking”. The laser sheet has a Gaussian
shape and a thickness of 800 µm defined at 10e-2 of intensity profile. The image
size is 1024x1024 with a magnification factor of 0.0977 mm/pixel, which leads to
a size of the Kolmogorov scale close to one pixel. The velocities are in a range of
± 3 m/s or ± 4.6 pixels.
3.2 Effect of image deformation on the measurement of velocity
gradients
In Fig. 6 an example of velocity measurements from a couple of synthetic images
is presented. In this example, the two velocity fields have been computed with the
same interrogation window size: 32x32 pixels (≈ 3x3 mm2). The result of the
CWS computation (left plot) presents a few spurious vectors, localized near the
centre of vortices, where intense velocity gradients are encountered. From the
same images, the result obtained by the MDPIV algorithm (right plot) does not
exhibit spurious vectors in these areas. From this very simple observation, it is
clear that, at equivalent sizes of interrogation window, the image deformation
technique is less sensitive to velocity gradients than the conventional PIV technique.
CWS (32x32 Pixels)
MDCPIV (32x32 Pixels)
Fig. 6 Effect of the image deformation technique to improve the measurement of velocity
gradients.
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Fig. 7 Comparison of imposed vorticity field (left) with the standard PIV (CPIV 32x32 centre) and the advanced PIV algorithm with image deformation (MDPIV - right).
In order to look closer at the effect of velocity gradients on the velocity measurement, a second study has been focused on the local measurement of the vorticity field. In Fig. 7 an example of a comparison of the vorticity measurement for
the CPIV and MDPIV algorithms is presented. The CWS results are not reported
in Fig. 7 because similar results to the CPIV algorithm have been obtained. The
vorticity field on the left corresponds to the simulation and the black curve is one
horizontal profile. From these results, the low-pass filtering effect of the conventional PIV technique (CPIV – centre plot) is clearly observed. The mean level of
vorticity is well estimated but the maximum of the intensity of the smallest flow
structures is underestimated. This effect is clearly observed by comparing the imposed profile and the measured profile (red curve). On the contrary, the low-pass
filtering effect is less pronounced for the PIV treatment including the image distortion module (MDPIV - right plot). In that case, the vorticity field is close to the
imposed values and the two profiles are nearly identical except very close to the
maximum of vorticity levels, where slight differences can be observed.
Fig. 8 Distribution of the velocity component (v) for 3 PIV algorithms.
The difference of behaviour between CPIV and MDPIV has been observed
from numerous simulations with different turbulence intensities. This suggests to
us that MDPIV including the image deformation technique has more potential
than standard CPIV for resolving strong velocity gradients. In the next section, we
will analyse whether the improvements introduced by the deformation also lead to
an improvement in the measurement of turbulence properties.
Advanced Algoritms 125
3.3 Measurements of turbulence properties
The measurement of turbulence properties from PIV recording has been investigated from 20 synthetic images produced from 20 independent simulations, initialised with the same turbulence properties.
In Fig. 8 the reconstruction of the velocity distribution (u component) for 3 PIV
algorithms is presented. The imposed distribution is plotted in red. The 3 PIV distributions are always well centred around zero and thus show that the mean velocity is well estimated. For the CPIV technique (left distribution), the “peaklocking” effect can be observed, as was the case for the real images of CIRA. This
effect is not visible from CWS and MDPIV algorithms. That improvement is the
result of the iterative and continuous window shift technique, (Lecordier et al.
1999, 2001). On the other hand, for the CPIV and CWS techniques, the distributions present an overestimation close to zero and an underestimation at the distribution tails. For the MDPIV algorithm, the distribution (right plot) is very close to
the simulation and thus, should lead to a better estimation of the statistical parameters of turbulence. This remark has been confirmed by the determination of
the kinetic energy of turbulence. Indeed, in Fig. 9 the kinetic energy and the dissipation rates for different algorithms and interrogation window size are presented.
In these plots, the red bars represent the simulation. It must be emphasized that
these values are smaller than the imposed values shown in Table 1 because for the
comparison, the DNS results are analysed with the same methods as the PIV velocity field. In addition, the analysis is performed with the PIV mesh, which has a
grid distance 4 times larger than the initial DNS mesh, and thus reduces the magnitudes of the spatial derivatives.
Fig. 9 Kinetic energy and dissipation rate measurements from CWS and MDPIV
algorithms with different interrogation window sizes.
For the kinetic energy (left plot) and the dissipation rate (right plot), the effect of
the interrogation window size is clearly observed. For the CWS technique, the PIV
results get closer and closer to the imposed values as the window size decreases.
Nevertheless, the size could not be smaller than 16x16 pixels without significantly
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increasing the number of spurious vectors (≈20%). On the other hand, by using the
image deformation (MDPIV), the size can be decreased up to 8x8 pixels without
introducing spurious vectors. This leads to a better estimation of the turbulence
properties, even though low noise addition slightly overshoots the imposed
values. In Fig. 10 the turbulence spectrum obtained from 20 velocity fields is
presented. The continuous line corresponds to the simulation. The CPIV method is
not reported in this plot. Indeed, the results of CPIV in terms of scales are very
similar to those of the CWS technique.
Fig. 10 Iterative PIV (CWS) compared to the algorithm with image deformation (MDPIV)
for the evaluation of turbulence spectrum.
From these results, relevant information about the accuracy of the measurement
of turbulent scales, cut-off frequency and noise level can be extracted. For the low
frequencies, all the algorithms have the same behaviour and slightly underestimate
the spectrum of simulation.
This effect is due to the low number of velocity fields, which does not allow a
sufficient statistic for the large scales. On the other hand, the number of small
scales in each velocity field being much higher than that of large scales, the statistic is sufficient for the high frequencies. In Fig. 10, for all the PIV evaluation
methods, the cut-off frequency linked to the algorithm itself is always smaller than
the one due to computation noise (rise of spectrum). This point is very important
and it allows us to evaluate the maximum expectable frequency accessible from a
given algorithm. The cut-off frequency of the CWS technique is always lower
than that of the MDPIV technique. Nevertheless, the results in Fig. 10, clearly
show that whatever the algorithm, the cut-off frequency is always strongly linked
to the size of the interrogation window. Indeed, the cut-off frequencies are always
very close to the theoretical values (dashed lines in Fig. 10) defined by Foucault et
al. (2002) and equal to 2.8/∆, where ∆ is the size of the interrogation window. An
important advantage of the deformation is to progressively cancel the displace-
Advanced Algoritms 127
ment on the images, thus allowing the size of the interrogation window to be reduced much more than with the CPIV and CWS methods, and this without dramatically increasing the number of spurious vectors.
Fig. 11 (u,v,w) imposed from two 2D direct numerical simulations.
4 Influence of the 3rd Velocity Component on Velocity
Measurements in a Turbulent Flow
4.1 Introduction
The previous results have clearly shown that the image distortion module induces
significant improvements for the measurement of large velocity gradients (Fig. 6
and Fig. 7) and the detection of small scales of turbulence (Fig. 10). Nevertheless,
in real 3D turbulent flows, the 3rd component of the velocity (perpendicular to the
laser sheet) can strongly influence the accuracy of in-plane velocity measurements
and can even prevent the determination of the in-plane velocity component. In order to investigate the effect of the 3rd component, the best way will be to use 3D
direct numerical simulation and then generate synthetic images from 3D particle
fields. However, at the present time, the size of computer memory restricts the
computation domain to very small volumes, and the energy which could be injected in the simulation is always much lower than in real experimental conditions. To overcome this problem, an original approach based on two simultaneous
2D simulations has been developed. The first 2D simulation provides the (u,v)
components in the centre plane of the laser sheet and the second simulation is used
to impose a realistic w component and normal velocity gradients. In Fig. 11, an
example of the imposed velocity components (u,v,w) is presented. In our approach, 3D turbulence properties such as velocity correlations are not necessarily
simulated as accurately as with a real 3D direct numerical simulation. Nevertheless, the main advantage of our approach is to enable us to use large computation
meshes (up to 4096x4096) to generate velocity fields with high turbulence intensities, having a more realistic w component than when either a simple analytic
Function (uniform, sinusoidal…) or random distributions are used. In addition,
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the intensity and maximum magnitude of the w velocity component being independent of the (u,v) components, the effect of the w can be investigated for a
large range of flow conditions. In the present work, the in-plane velocity fields
have the same properties as those used in the previous section (see Table 1). To
study the effect of the 3rd velocity component, 4 different cases summarised in Table 2 have been considered. Case C1 corresponds to the previous simulations
(w’=0 m/s) and case C3 imposes isotropic properties of turbulence (w’/u’=1).
Cases C2 and C4 are intermediate. For the 4 situations, the mean velocity w is
equal to zero. In Table 2, the parameter Ls and ∆t correspond respectively to the
laser sheet thickness (800 µm) and the time between images (150 µs).
Table 2. Properties of 4 turbulent flows to imposed w component on synthetic images in
order to study the out-of plane effect.
Case
C1
C2
C3
C4
w’
[m/s]
0.00
0.53
1.06
1.27
w’/u’
0.0
0.5
1.0
1.2
ktz
[m2/s2]
0.00
0.28
1.12
1.61
εz
[m2/s3]
0.00
7.95
31.8
45.8
ηz
[mm]
0.00
0.14
0.10
0.91
w’.∆t/Ls
0.00
0.10
0.20
0.24
4.2 Results
In Fig. 12, the relative error of the u and v component of the velocity for the 4
cases is presented. The error is normalised by the in-plane velocity fluctuation u’.
In these plots, no correlation between the errors of the u and v components is observed. Nevertheless, the dispersion of the errors increases significantly with increase of the w component. For w’=0, the RMS of error is limited to 0.1 and it
reaches 0.23 in case C4. Clearly, the w component has a strong influence on the
accuracy of the in-plane velocity measurement. To study it more precisely, in Fig.
13 the correlation plot of the w against the relative error of u is presented. In these
plots a normalised parameter w.∆t/Ls is also indicated, which represents the ratio of
the displacement along z to the laser sheet thickness. As is shown by plots in Fig.
13, for low values of the w component, the relative error is uniformly distributed.
It corresponds to a range of ± 0.2 in terms of normalised displacement, namely
z displacement smaller than 20% of the laser sheet thickness. For larger z displacements, we can notice a significant increase of the dispersion of the relative
error. In Fig. 14 the RMS of the relative error versus the w component and the
normalised displacement is presented. As observed previously, the RMS of u dispersion is minimum for the z displacement close to zero but increases four-fold as
soon as the normalised displacement exceeds ± 0.4.
The previous observations are not without repercussions on the accuracy of the
measurement of turbulent quantities. Indeed, in Fig. 15, representing the turbulence spectrum of cases C1, C2, C3 and C4, the effect of the out-of-plane motion
is clearly observed. For the first two cases, C1 and C2, the spectrum is well resolved and the cut-off frequency is nearly equal to the theoretical value k8, im-
Advanced Algoritms 129
posed by the size of the interrogation window of 8x8 pixels. On the other hand, for
case C3, the cut-off frequency due to the increase of the spectrum is observed long
before the theoretical value k8 and it is close to the k16 value The difference
between the cut-off frequency and its theoretical value is even larger for case C4,
for which the cut- off frequency is close to k64, even though the computation has
been performed with 8x8 pixels interrogation window size.
Fig. 12 Correlation of the normalised relative error of the u and v velocity components.
Fig. 13 Relative error of u versus w and the normalised z displacement.
For cases C3 and C4, the rise of the spectrum is typically due to noise in the
velocity field, despite the velocity fields being free of spurious vectors. The previous observations have significant consequences for the measurement of turbulent
flows. Indeed, from an ideal situation of synthetic images, it has been shown that
the measurement uncertainties in turbulent flows increase very quickly when particle displacements exceed 20% of the laser sheet thickness. In real experiments, it
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is often tricky to adjust ∆t to force the z displacement in this range, in particular if
the main flow is along z.
Fig. 14 RMS of the relative error versus w and the normalised displacement.
Fig. 15. Effect of the out-of-plane particle motion on the turbulence spectrum.
Advanced Algoritms 131
5. Conclusions
An original PIV algorithm, including an image distortion technique has been proposed and described. It has been validated and compared with two other PIV
methods thanks to synthetic images generated from direct numerical simulation,
on which turbulent flows with homogeneous and isotropic properties have been
imposed.
These PIV algorithms have been compared and their limitations investigated in
terms of velocity fluctuations, dissipation, vorticity field and spectrum. The distortion technique has shown the potential to improve greatly the accuracy of
measurement in turbulent flows as well as to access much steeper velocity gradients than standard PIV algorithms. From the study of the turbulence spectrum, it
has been shown that the cut-off frequency of all PIV algorithms is always strongly
linked to the size of the interrogation window. Nevertheless, the main advantage
of the deformation technique is to reduce interrogation window sizes and to reach
sizes, which cannot be obtained with the other PIV evaluation techniques.
In the last part, the effect of the out-of-plane particle motion on the turbulent
velocity measurement has been investigated thanks to an original use of two simultaneous 2D DNS. It has been shown that for z displacements exceeding 20%
of the laser sheet thickness, significant uncertainties and noises are added to the
in-plane velocity measurement. These bias sources have considerable incidences
on the decreasing of the cut-off frequency.
Acknowledgements
This work has been performed under the EUROPIV2 project. EUROPIV2 (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 and the universities of Delft, Madrid, 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 European Union within the 5th framework (Contract N°: G4RD-CT-2000-00190).
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