3D-PIV Measurements near the Wall.
D. Calluaud, L. David
Laboratoire d'Etudes Aérodynamiques (UMR 6609-CNRS), Boulevard Marie et
Pierre Curie, Téléport 2, B.P. 30179, 86960 FUTUROSCOPE Cedex, France,
Laurent.David@univ-poitiers.fr
Abstract
Difficulties are encountered for the study of the flows near walls by stereoscopic
PIV. A 3D-calibration method outside the illuminated zone is proposed, using the
pinhole camera model. The validation of this calibration is tested for elementary
micro-displacements of a sheet of sandpaper. 3D-measurements of the flow
around a surface mounted block for a section near the wall are realized and compared with 2D-PIV measurements obtained in perpendicular sections.
1 Introduction
Stereoscopic PIV techniques are generally classified into one of three reconstruction categories according to the calibration method used Prasad (2000). For 2D
calibration-based reconstruction, a thick target of precisely known depth is placed
in the illuminated plane and for the 3D calibration method, this target is moved inside the plane in the out-plane direction. In addition, for these types of calibration,
Willert (1997) has shown the influence of laser-sheet misalignment with the target. Coudert and Schon (2001) have also suggested systematic corrections of the
investigated plane equation based on the vector field calculated by correlation in
the object reference between the projected particle images from the two cameras.
Near the wall some difficulties appear for the calibration, and the displacement
of the target in the laser sheet becomes impossible. Also, a 3D calibration method
outside the illuminated zone is proposed, using the pinhole camera model. The assumption of linear reconstruction matrices is first verified for the experimental
configuration. The validation of this calibration is then tested for elementary micro-displacements of a sandpaper sheet. Afterwards, this calibration technique is
applied to measure the early stages of the flow around a surface mounted block for
a section near the wall. The 3D velocity measurements are finally compared with
2D-PIV measurements obtained in perpendicular sections for the same dimensionless times.
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2 Camera Model and Stereoscopic Reconstruction.
The pinhole camera model gives some analytical relations between the object and
images references. The general transformation for a camera between the image coordinates (xi, yi) and the corresponding point (X, Y, Z) in the object reference is
written:
 xi 
y  = M
 i
 1 
X 
Y 
⋅ 
Z 
 
1
(1)
In this case the matrix M is a composition of several transformations (projections, rotations, translations). The optical distortions due to an optical nonalignment, a non-linearity of the lenses, and/or an optical refraction of windows,
dioptres, and other optical elements of the experiment are not taken into account in
the model, and are minimized with the geometry of the apparatus. In this way, the
transformations between the different references stay linear and simpler to resolve.
The system of two camera matrices is able to represent each point location in
space and not just in the limited volume described by the target displacement.
For the stereoscopic reconstruction, the two velocity fields are calculated at
first by cross correlation for each camera and interpolated on a matching grid. Afterwards each reconstruction of the three velocity components is found by solving
an over-determined linear system of four equations and three unknowns by a least
squares method for the vector origin located in the centre plane of the laser sheet.
This reconstruction method is close to that suggested by Soloff et al. (1997) and is
explained in detail by Calluaud and David (2002).
3. Experimental Set-up
Experiments were carried out on an apparatus especially set up to study the first
stages of hydrodynamic flow evolution around impulsively started obstacles. The
block mounted on a plate is towed in a vertical octagonal water tank
(100×100×150 cm3 capacity) and guided during its fall. The tank (made of altuglass) is filled with water seeded by hollow glass particles (15 µm diameter). Before an experiment, the surface-mounted block carriage is linked to the fixed
frame by an electromagnet until the fluid comes to rest. At the initial time, the carriage is released. After a short initial period of acceleration, a constant downward
velocity U0 is reached under the combined action of gravity and a damping dashpot. In this study, the block has a square section of D2=60×60 mm2, and a height of
H=18 mm and is placed at 40 mm to the leading edge of the plate (Fig. 1). Its ve-
Stereoscopic PIV 409
locity is about 1.6 cm/s, defining the Reynolds number Re = U 0 ⋅ D = 1000 where
υ
ν is the kinematic viscosity of water. Flow development is sequentially recorded at
regular dimensionless time-steps ∆t*=1 (t*=t.U0/D; 0 ≤ t* ≤ 12) by two cameras.
Fig. 1. Characteristic dimensions
Fig. 2. Top view of the water
tank.
To reduce the optical aberations due to the air-altuglass-water dioptre, the CCD
cameras are placed at angle of 90° to the tank faces. The viewing angles of each
camera are fixed at 45° and -45° respectively to the Z-axis (Fig. 2) and Scheimpflug arrangements are used to obtain entirely focused images.
The 2D velocity fields are calculated by cross correlation and local adaptive
window shifts with the Dantec software. The matching step and the reconstruction
are calculated by our algorithm.
4 Camera Model Validation
To validate the linear camera model in this application, the hypothesis of small
distortions has to be verified. For the 3D calibration, the target is moved in the
out-plane direction in 0.5 mm steps from the section Z=7.5 mm to the section
Z=11.5 mm and recorded by the two cameras. The matrix coefficients for the two
cameras are then calculated knowing the target point location and their projections
in the image references.
The errors introduced by the localization technique for the target point projected on the images have been quantified and are smaller than 0.15 pixels. The
real co-ordinates of the target points are calculated by reconstruction from the
matrices and the target points images co-ordinates (x1, y1) and (x2, y2). Thus, we
can estimate the global errors introduced by the calibration while defining the amplitudes, the averages and the RMS of the differences between the real coordinates (XR, YR, ZR) and the reconstructed co-ordinates (XM, YM, ZM) of the target
points.
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 Eλi = λMi − λRi

 EX, EY, E Z = E Xi2 + EYi2 + E Zi2
i

N

 Eλ = 1 ∑ Eλi
N i=1


2
N
1
E ' =
E
λi
(N − 1) ∑
 λ
i =1
(2)
With λ=X, Y, Z and N the number of localized points.
To represent the local errors between the linear model and the reconstructed
points from the calibration, contour plots of the errors in each direction are represented. In this way, the Fig.s 3-a, 3-b and 3-c show that there is no spatial relation
of the errors in the three directions. The larger error is localized for one point at
X=0 mm, Y=80 mm and Z=-7mm. For the other points, the norm is under 0.1 mm.
The mean value of the differences is lower than 0.04 mm and the RMS value stays
within the range between -0.05 mm and 0.05 mm. Also a second order or more
polynomial approximation is not necessary due to the degree of accuracy of the
point detection.
Finally, to confirm the linear camera model approach for this application, the
dispersions between the projected points on the same line of the target and the
nearest line defined by the points are evaluated. The mean value of these dispersions gives the order of accuracy of the point detection and is found to be 0.13
pixels. In conclusion, the linear model accurately describes the volume swept by
the target displacement and is always in good agreement close to the boundaries.
This model is applied for measurements outside the 3D calibration area.
5 Results
For the validation of the reconstruction, based on a 3D calibration outside the zone
of interest, two different tests are carried out. At first, a sandpaper sheet placed in
the water tank at the same location as the illuminated plane for the flow measurements is moved in the three directions with micro-translation tables. Then, the reconstruction is applied to the measurements of the flow around a surface mounted
block in the first stages of its establishment.
Stereoscopic PIV 411
(a)
(b)
(c)
Fig. 3. Variations of the error norm between the real and the reconstructed point coordinates.
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5.1 Validation by displacements of a sandpaper sheet
A sandpaper sheet has been moved by micro- translation tables in the in-plane and
out-plane direction with regular steps in order to evaluate, by reconstruction, the
real displacement of the particles fixed in the paper. The advantage of the sandpaper in this case compared to a resin block is the fact that the signal to noise ratio is
not influenced by the loss of particle pairs. Although the accuracy of the microtable is given to be about 1/100 mm, a simple test of different displacements in the
three directions with a return to the original location has been carried out. The reconstructed displacements between the initial and final positions yield an average
of the differences of about 0.04 mm and a root mean square value of 0.005 mm.
Below this level of translation, the accuracy of the method is therefore difficult to
extract; the errors are shared between displacement table errors, reconstruction errors and others.
The global error in each direction has been represented by the ratio of the norm
of the differences NEE*c :
NEE * c / xyz =
∑λ
i
∑
N
(3)
(∆xRi − ∆x) 2 + (∆yRi − ∆y ) 2 + (∆z Ri − ∆z ) 2
N
with c=x, y or z and λi = ∆cRi − ∆c the difference between the reconstructed and
imposed displacements.
7
7
6
6
6
5
5
5
NEE* z/xyz [%]
NEE* x/xyz [%]
NEE*y/xyz [%]
7
4
4
4
3
3
3
2
2
1
1
0
0
0
1
2
∆z [mm]
3
Centred calibration
2
1
0
1
2
∆z [mm]
0
3
[%]
6
NEE* z/xyz
5
4
xyz
NEE* y/xyz
1
2
∆z [mm]
3
Decentred
calibration
6
NEE* x/xyz
7
0
NEE* x/xyz
NEE* y/xyz
NEE* z/xyz
Fig. 4. Percentage of the norm of the differences between reconstructed and imposed displacements for the 3 directions.
For a fixed XY displacement (∆x=2.5 mm, ∆y=0 mm) and different Z displacements (0 mm ≤ Z ≤ 3 mm), Fig. 4 presents a comparison of the errors from a
calibration centred around the zone of the sandpaper displacements with those
Stereoscopic PIV 413
from a calibration decentred by 7 mm in the Z direction. The ratios of the norm
between the results provided by the two calibrations are lower than 2%.
In regard to the scatter plots of the differences between the reconstructed and
imposed displacements (Fig. 5), we notice that the difference distributions have
the same shapes. Furthermore, we can defined some quantities εx=k. Ex’, εy=k. Ey’
and εz=k. Ez’ (with k=2.5 and Ex’, Ey’ and Ez’ the RMS values of the differences
between the reconstructed and imposed displacements).
0.04
0.04
0.04
0.02
0.02
0.02
0
∆z R,i - ∆z i
0.06
∆zR,i - ∆zi
0.06
∆yR,i - ∆yi
0.06
0
0
-0.02
-0.02
-0.02
-0.04
-0.04
-0.04
-0.06
-0.06
-0.04
-0.06
-0.06
-0.02
-0.02 0.040
0-0.04 0.02
∆xR,i - ∆xi
0.0
0.06
∆xR,i - ∆xi
-0.06
-0.06
-0.02
0.04
0.04
0.04
0.02
0.02
0.02
∆z R,i - ∆zi
0.06
∆zR,i - ∆z i
0.06
∆yR,i - ∆yi
0.06
0
0
-0.02
-0.02
-0.04
-0.04
-0.04
-0.04
-0.06
-0.06
-0.02
-0.02 0.040
0-0.04 0.02
∆xR,i - ∆xi
0.0
0.06
∆xR,i - ∆xi
0.02
0.06
∆xR,i - ∆xi
0.04
0.06
(a)
0
-0.02
-0.06
-0.06
-0.02 0.040
0-0.04 0.02
∆xR,i - ∆xi
-0.06
-0.06
-0.02
-0.02 0.040
0-0.04 0.02
∆xR,i - ∆xi
0.02
0.06
∆xR,i - ∆xi
0.04
0.06
(b)
Fig. 5. Scatter plots of the differences ∆xR,i-∆x, ∆yR,i-∆y et ∆zR,i-∆z for the sandpaper displacement (∆x=2.5 mm, ∆y=0 mm et ∆z=2 mm). (a) Centred calibration, (b) Decentred
Calibration.
For each calibration, the values εx, εy, and εz allow to defined windows which
correspond to 98 % of the dots (Fig. 5). The value εx varies between 0.041 mm and
0.054 mm according to the calibration method. Also, for the decentred calibration
the errors increase slightly in the X direction whilst decreasing in the Z direction.
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5.2 Validation on flow measurements
This technique of calibration outside the illuminated section is applied to the
measurements of the early stages of the establishment of the flow around a surface-mounted block. The third dimensionless times are presented in the Fig. 6 for
the section Z=2.5 mm. The laser sheet thickness is of the order of 4 mm and the
acquisition rates are adapted to have homogenous measurements in the three directions.
Fig. 6. First steps of the flow around a surface-mounted block (Z=2.5mm).
A Z-axis vortex is generated at each of the side faces of the block from the start
of the establishment of the flow. These two vortices are increasing with time to
form a nearly attached wake in this section. In the XZ plane, a Y-axis starting
vortex appears and is transported to the far field. This vortex is represented in Fig.
6 by areas of negative and positive W components which induce the roll-up of this
type of vortex. To compare these three instantaneous velocity components with
conventional PIV measurements, two XZ flow sections are investigated for
Y=10mm and 20 mm. The dimensionless U and W velocity profiles are drawn and
compared along the X-axis for the Z=2.5 mm position. Since it is not the same experiment and errors in the time of the first acquisition could exist, instantaneous
Stereoscopic PIV 415
measurements by the both methods are compared. The amplitudes and variations
of the U and W velocity components are very close and localize at the same abscissa (Fig. 7). However, the 2D measurements seem less accurate in this case,
and the velocity components fluctuate more because of the presence of reflections
near the wall which tend to decrease the signal to noise ratio.
t*=2.5
t*=3
Fig. 7. Velocity profiles by 2D and 3D PIV for Y=20 mm, t*=2.5 and 3.
6 Conclusion
To allow velocity measurements close to the wall, a 3D-calibration method outside the illuminated zone has been proposed using a pinhole camera model. The
assumptions of a linear model have been verified for the experimental configuration with two cameras placed at 45° relative to a normal to the plate. The optical
aberrations and the radial distortions have been reduced by the octagonal shape of
the tank and by Scheimpflug arrangements. The measurements outside the calibration volume have been tested with displacements of a sandpaper sheet and have
shown very small differences between results obtained by centred and decentred
calibration. This type of measurement with a calibration outside the illuminated
section has been carried out to study the early stages of the establishment of the
flow around a surface mounted block for a section near the plate. The results are in
agreement with measurements coming from 2D-PIV for orthogonal sections and
show the 3D topology and the evolution of the flow from its initial conditions.
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References
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29, pp 103-116
Soloff SM; Adrian RJ; Liu ZC (1997) “Distortion compensation for generalized stereoscopic particle image velocimetry.” Measurement Science and Technology , vol. 8, pp
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Willert C (1997) “Stereoscopic digital particle image velocimetry for application in wind
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