The application of a 3D PTV algorithm to a mixed convection flow
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Experiments in Fluids 33 (2002) 603–611
DOI 10.1007/s00348-002-0513-9
The application of a 3D PTV algorithm to a mixed convection flow
R.N. Kieft, K.R.A.M. Schreel, G.A.J. van der Plas, C.C.M. Rindt
603
Abstract A 3D particle-tracking velocimetry (PTV) algo- PTV techniques, individual particles in subsequent images
rithm is applied to the wake flow behind a heated cylinder. are tracked, whereas in PIV techniques the average dis-
The method is tested in advance with respect to its accu- placement is determined within a segment of an image
racy and performance. In the accuracy tests, its capability (interrogation area).
to locate particles in 3D space is tested. It appears that the In 2D techniques, the flow is illuminated with a thin
algorithm can determine the particle position with an ac- light sheet and only the velocity components within this
curacy of less than 0.5 camera pixels, equivalent to sheet can be evaluated. Although a few methods exist for
0.3 mm in the present test situation. The performance tests analysing 3D velocities in a point (3D laser Doppler ane-
show that for particles located in a 2D plane, the algorithm mometry) or plane (3D stereo-PIV), only a fully 3D tech-
can track the particles with a vector yield reaching 100%, nique based on the illumination of a flow volume rather
which means that a velocity vector can be determined for than a flow sheet will give the information needed to
almost all particles detected. The calculated velocity vec- construct the instantaneous 3D velocity fields. Presently, a
tors for this situation have a standard deviation of less few different methods can be applied (Hinsch 1995). The
than 1%. The performance is also tested on a mixed first method is an extension of the 3D stereo-PIV tech-
convection flow behind a heated cylinder in which the 2D nique. In this method, velocity information in a volume is
flow transits into a 3D flow. As there is no exact solution of obtained by acquiring several slices of the flow field using
such a flow available, the 3D PTV results are compared a scanning technique (Rockwell et al. 1993; Brücker 1995).
with visualisation results. The results show that the 3D The second method is holographic PIV (Zhang et al. 1997),
PTV method can capture the main features of the 3D in which the spatial distribution of particles is recorded
transition of the 2D vortex street. holographically. The holograms are analysed afterwards by
sampling the hologram in slices. A very high spatial res-
1 olution can be obtained in this way, but the experimental
Introduction requirements are stringent and the analysis time con-
Using tracer particles for fluid velocity measurements is a suming. The third method is a 3D extension of the parti-
well-established technique. Tracer particles are seeded in a cle-tracking method. Instead of tracking particles in a thin
fluid and illuminated within a defined area. The images of light sheet, the particles are now tracked in an illuminated
the moving particles in this area can be recorded and volume. The obtained 3D particle trajectories can be used
processed. Currently, several techniques based on particle to calculate the 3D velocity field. Because the actual path of
visualisation have been developed to measure the velocity. the particles is analysed, particle-tracking techniques are
When the defined area is a thin light sheet, techniques generally more accurate than PIV-based techniques
such as 2D particle-tracking velocimetry (2D PTV) and (Cowen and Monismith 1997). Besides, the trajectories of
particle image velocimetry (2D PIV) can be applied. In the particles are analysed in a flow volume, which enables,
in contrast to scanning techniques, the construction of
instantaneous velocity fields. The technique of 3D particle
tracking was introduced by Chang and Taterson (1983)
Received: 15 September 2001 / Accepted: 13 June 2002 and further developed by (among others) Racca and
Published online: 31 July 2002 Dewey (1988) and Maas et al. (1993). In most of these
Springer-Verlag 2002
investigations, a 3D particle localisation algorithm is used,
based on a so-called epipolar-lines method. In this meth-
R.N. Kieft, K.R.A.M. Schreel, G.A.J. van der Plas, C.C.M. Rindt (&)
Energy Technology Division, od, several transformations between the physical 3D do-
Department of Mechanical Engineering, main and the camera images are needed, resulting in long
Eindhoven University of Technology, computational times (for details, see Maas 1996).
P.O. Box 513, 5600 MB Eindhoven, The Netherlands In the present investigation, the performance of a 3D
E-mail: C.C.M.Rindt@wtb.tue.nl
particle localisation and tracking technique is studied and
This work is part of the research programme of the Netherlands tested with respect to its accuracy. Both for the calibration
Foundation for Fundamental Research on Matters (FOM), which is
financially supported by the Netherlands Organisation for Scientific
and the accuracy tests, synthesised data is created by
Research (NWO). The authors would like to take the opportunity to traversing a precisely manufactured 2D grid through the
thank the technical staff of the Energy Technology section for their measuring volume. For the performance test, the fluid flow
support. behind a heated cylinder is measured.In Sect. 2 a thorough discussion of the algorithm is PTV algorithm (as described in van der Plas et al. 1999 and
presented showing the details of the 3D localisation pro- Bastiaans et al. 2001). First, the captured images are
cedure, the calibration method and the matching algo- dynamically thresholded (Dynamic Thresholding). Then,
rithm. The testing problem is discussed in Sect. 3, while within each image, the 2D representation of a particle is
the results concerning accuracy and the performance of detected (Blob detection). From the particles located in the
the matching algorithm are discussed in Sect. 4. The re- three cameras, a 3D position can be deduced (Mapping to
sults of the fluid flow measurements are presented in the Lines of Possible Position and 3D localisation). As soon as
Sect. 5. The article ends with a brief discussion and some the 3D position of the particles is known, the procedures
conclusions. for matching and path storage are almost equivalent to the
ones used in a 2D algorithm. In the following sections, the
604
2 3D localisation of the particles and the 3D calibration
Methodology procedure, which are quite different from their 2D repre-
In 3D PTV methods, at least two synchronized cameras sentatives, are discussed extensively. The other parts of the
need to be used. Only then can stereo images required to algorithm are discussed briefly in appendix A.
determine the 3D position of the particles (comparable to
the human eye) be obtained. With two cameras, the pos- 2.1
sibility exists that, especially for high seeding densities, Localisation
particles are hiding behind each other. As it is the aim to The main idea behind the 3D localisation of particles is
track particles as long as possible, the effect of particle that under normal circumstances a particle in a camera
‘hide and seek’ needs to be minimized. Therefore, a third image will have a position in the world coordinate system
synchronized camera is applied which focuses at the same somewhere along the particle projection line (Fig. 2a). The
volume as the other two cameras (Fig. 1a). A third camera ‘normal-circumstances’ criterion means basically that no
also reduces the ambiguity occurring during the 3D abrupt change in refractive index of the medium is allowed
localisation. (no ‘mirror’ effects) and that possible changes are known
From the obtained images, the 3D position of the par- and stable or predictable in time. This is not a serious
ticles can be determined using a 3D localisation algorithm restriction since the applicability of seeding techniques for
(Fig. 1b). The algorithm functions comparable to the 2D flow visualisation in general satisfies this criterion. Note
that this also holds for different refractive indices (e.g. a
Fig. 1a, b. 3D PTV setup and algorithm. a Basic 3D PTV configu- Fig. 2a, b. Principles of the 3D localisation. a Particle projection.
ration. b Algorithm scheme for 3D PTV b Method of crossing linesglass–water interface), or for deforming media (e.g. lens- dependent on the calibration data, the data also deter-
like deformations in the glass wall), without affecting the mines whether 3D particles can be found or not.
above statement. In the case of a medium with a uniform To obtain the calibration data, a calibration system is
refractive index, the lines of possible positions will be developed, allowing to position a precisely manufactured
straight. This property is used for the construction of a 2D grid on well-known positions inside the measuring
transformation between pixel coordinates of the particle volume (Fig. 3). The calibration grid consists out of a
images and the particle projection line in world coordi- blackened square copper-bronze foil (200·200·0.1 mm)
nates. When this transformation is known for all cameras, glued onto a flattened opaque glass plate. In this foil, pin-
the detected blobs in the images can be transformed to a holes with a diameter of 0.1 mm are etched with a spacing
set of lines of possible particle positions (Fig. 2b). The of 5 mm. The grid can be positioned with an accuracy of
points in space where the lines from the different cameras ±5 lm. By illuminating the grid from behind with a diffuse
605
cross are the possible positions of the particles. This 3D light source, the regular pattern of pin-holes is visualised
localisation method, based on crossing projection lines, and captured by all three cameras. For each grid position,
only requires the construction of these lines (Yamamoto the (x,y,z)-positions of the detected pin-holes are exactly
et al. 1993). In the so-called epipolar-line method, as used known. This information is used to determine which
by various researchers, the virtual images of the particle (x,y,z)-position is projected on each camera pixel. Use is
projection lines belonging to cameras 1 and 2 need to be made of nz different z-positions, meaning that each camera
constructed in camera 3. This demands an additional pixel corresponds to nz (x,y,z)-positions. By applying a
back-transformation from world coordinates to camera least square fit through these points, the coefficients of the
coordinates, causing an increase of the computational straight-unique line are established for every cluster of
effort. 25 camera pixels. Because blob detection is performed at
The ‘crossing-line method’ allows using only two sub-pixel accuracy, a continuous function, describing the
cameras. However, in high-seeding-density flows, acci- line coefficients within a camera image, is needed.
dental crossing of two lines is quite common, resulting in Therefore, a 2D nf -th-order polynomial (with nf,c-th cross-
multiple crossings and consequently in ambiguities in order terms) is fitted through the coefficients belonging to
determining the 3D particle position. Besides, particles can each pixel. For this particular experiment, it turns out that
also hide behind one another, resulting in no crossings at for nf ‡3 and nf,c ‡3, the error in z-position becomes
all. The use of three cameras is therefore almost indis- smaller than the accuracy in the traversing system.
pensable. The chances of accidental crossings of three lines
through one point are relatively small. 2.3
A blob, detected in each camera image, is now repre- Experimental device
sented by three spatial lines crossing somewhere in the The setup consists of a system in which three synchro-
measuring volume. Due to optical disturbances, random nized cameras are integrated (Fig. 1a). These three b/w
camera noise and errors in the calibration data, an exact CCIR video cameras (JAI 1021, Copenhagen, Denmark) are
crossing of the three lines is unlikely. Therefore, a line connected to a RGB frame grabber (PCI frame grabber
crossing is detected when the minimum distance between with AM-CLR module, ITI, Indianapolis, Ind.). Each
the three lines is smaller than a certain critical value Dc. camera is connected to a different color input of the frame
Depending on the quality of the calibration data and grabber and the digitised images are stored as RGB files.
camera/lens characteristics, Dc should be set to an optimal Synchronization of the cameras is obtained by using the
value. In general, too large a value results in an increase of sink signal from one of the cameras as triggering signal for
crossing possibilities, especially for experiments where a the other ones. At the processing stage, the separate
high seeding density is used. A very small value, in turn, camera images are then extracted again from the stored
results in no crossings at all. In the present study, this RGB images. In this way, a cost-effective setup is realized
approaching distance was set to 0.01 mm. with relatively easy synchronization and storage. The
frame grabber is installed in a Pentium-class personal
2.2
Calibration
For the transformation of the detected blobs in the camera
images to lines in world coordinates, in principle a
mathematical model can be developed based on the mea-
sured geometry of the experimental setup and camera
characteristics. The derivation of the transformation pa-
rameters in this model, however, would be a time-con-
suming task. Furthermore, some kind of check of the
transformation has to be performed with in-situ objects. In
the present study, in-situ calibration is the preferred
technique to determine the transformation.
The transformation parameters are determined by tra-
versing a well-defined object through the measuring vol-
ume. The calibration process should be performed very
accurately. Not only the accuracy of the 3D position is fully Fig. 3. Calibration configurationcomputer with a 40-GB RAID0 volume (also called stripe system which is connected to the construction which
set). The RAID0 volume is created by using the stripe set carries the cylinder.
features of Windows NT on ten 4-Gb hard disks, con- By towing the cylinder through the water tank, the flow
nected in groups of five to two UW-SCSI controllers. This behind the cylinder is created. When fixing the camera to
allows for a minimal sustained data rate of 30 Mb/s, which the setup instead of the translation system, the shed
is enough for real-time hard-disk recording with these structures remain within the interrogation volume.
cameras. The recording software is Eye-Image calculator Therefore, these structures can be followed for a long
(IO Industries, London, Ontario, Canada). Since the period of time. This allows us to monitor the transition
cameras are interlaced, the lighting source cannot be process of the shed structures into unstable structures
pulsed. Therefore, during the application experiments we where heat effects dominate (Kieft et al. 1999).
used a cw Ar+ laser (Spectra Physics 2016, Mountain View,
606
Calif., 6 W all lines) which provides enough intensity to
4
illuminate a typical volume of 10·10·10 cm3. For seeding,
Accuracy and performance
50-lm hollow-glass particles are used, which are dispersed
The development of the present 3D PTV code can be
in the flow about 1.5 h before the actual experiment.
thought of as in two parts, the particle-tracking routines
and the 3D localisation routines. The integral accuracy of
3 the present 3D PTV method can then be seen as a com-
Experimental test problem bination of the 3D localisation accuracy together with the
The application on which the 3D PTV code is tested
particle-tracking accuracy. In the present article, the
concerns a mixed convection flow behind a heated cylin-
analysis of the accuracy and performance is focused on the
der (Fig. 4). In this problem, a heated cylinder is cooled by
3D localisation. The particle-tracking routine is more or
a convection flow. The determining dimensional parame-
less independent of whether it is used in two or three
ters are then the Reynolds number, ReD=U0D/m, and the
dimensions. For more details about the accuracy of the
Richardson number, RiD=GrD/Re2D=Dgb(T1–T0)/(U20),
tracking routine, the reader is referred to Bastiaans et al.
with b the thermal expansion coefficient and m the kine-
(2001) in which the accuracy of this part of the method is
matic viscosity and the other parameters as defined in
discussed.
Fig. 4. For small heat addition (small RiD), a 2D von
Kärmän vortex street is found, whereas for moderate heat
addition, the 2D stable vortex street becomes disturbed 4.1
and a transition to a 3D flow field takes place (Kieft et al. Accuracy in 3D localisation
1999). The performance of the method is first tested for its ca-
The experiments were done in a water tank facility. In pability to localise well-known positioned markers some-
this setup, the heated cylinder (D=8.5 mm, L=495 mm) is where in the calibrated measuring domain. To that end,
towed through the tank rather than being exposed to a the calibration grid is placed at several known z-positions
forced main flow. The specific dimensions of the water in this volume. The positions of the pin-holes are now
tank are length·width·height=500·50·75 cm. The main reconstructed and compared with the known positions. A
advantage of this device is a minimal creation of boundary measure of the average error i in the i-th component of
layers and an almost uniform inflow velocity distribution the located pin-hole position vector is the standard devi-
(Anagnostopoulos and Gerrard 1978). To obtain the de- ation in the located position with respect to the known
sired cylinder wall temperature, an electric rod heater is position. This can be defined according to
used with a maximum heat density of 8.0 W/cm2. The sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Pnd 2ffi
temperature of the cylinder is kept constant in time by k¼1 x i;k ð known Þ x i;k ð located Þ
controlling the heat input with the help of the measured ei ¼ nd 1
ð1Þ
wall temperature. The translation of the construction is
obtained by an electric motor which is corrected for its with xi,k(known) the i-th vector component of the known
variation in rotational speed by means of a closed circuit, position vector of the k-th pin-hole, xi,k(located) the i-th
resulting in a variation of the rotational speed of less than component of position vector as calculated by the 3D lo-
0.2%. The motor is coupled to the drive wheel by using a calisation algorithm and nd the total number of located
1:100 gear. An almost inelastic fibre-based tape is looped pin-holes.
around the drive and the idle wheel of the translation It turns out that the error in the located positions i is
dependent on the camera configuration (Fig. 5a). The
camera configuration is determined by the camera viewing
direction and the optical path lw through the water. The
camera viewing directions a1,2,3 are defined as the angles
between the optical axis of the cameras and the z-direction
(central line in Fig. 5a) after refraction by the air/test-
section interface.
The dependency of i on these variables is investigated
by three sets of experiments, where in the first set, the
camera viewing angles are a1=a2=a3=a. In the second set,
Fig. 4. Definition of the mixed-convection problem a1a25 and a3 is varied, while in the third set, lw isdeformation of the images at the air/water interface when
viewing with an angle.
For the second set (denoted by a triangle and scaled
with the upper axis, Fig. 5b), in which only a3 was varied,
an increasing a3 also results in a decrease of z, but less
strongly than for the first set. Furthermore, at a3=25, the
accuracy of the z-position is still improving, while for a
varying a, this improvement seems to diminish for a>20.
The influence of the optical path length is determined
for a13 and by varying the position of the measuring
volume within the test section (denoted by a square and
607
scaled with the middle axis, Fig. 5b). By doing so, it turns
out that this variation has no detectable effect on the ac-
curacy in z-position. Only a slight improvement can be
observed if the measuring volume is positioned closer to
the air/water interface.
Considering the camera positions, the ideal configura-
tion is an orthogonal setting, resulting in a set of images
with the smallest dependency. From the results one can
conclude that the smallest error is observed for viewing
angles larger than 20 with regard to the central line. No
significant improvements in accuracy can be gained by a
further increase of this angle. The remaining error in
z-position z is approximately 0.003 cm. Comparing this
accuracy with the accuracy in x- and y-position, it is found
out that z is about three to four times larger than x,y. This
accuracy is similar to values found by other researchers
(Sata and Kasagi 1992).
4.2
Tracking performance
The 3D tracking of particles is tested for the optimal
camera configuration (a20, lw150 mm). In this test,
the calibration grid is translated through the measurement
Fig. 5a, b. Performance variables and their influence on the domain with a constant step size Dz=0.1 cm. As in this test
z-position. a Camera configuration. b Dependency of the accuracy in all particles are situated in a plane, no ambiguities during
z-position on the camera configuration. Three tests were performed the localisation (occurrence of two crossing possibilities)
(see text for definitions) or the phenomenon of ‘hiding’ particles occur. Further-
more, only about 200 white dots are viewed by the cam-
eras, corresponding to a low seeding density. At every
varied. During these experiments, the accuracy of the in- position of the grid, three independent camera images are
stalled angle was estimated to be 1 and the accuracy of the captured, from which the pin-hole positions are deter-
measured optical length was 0.5 mm. mined. Doing this for a sequence of ten steps, a tracking
For all three sets, it was found that x and y are more or run is simulated where a virtual velocity can be detected
less independent of the varied variable and equal to about between two subsequent frames. The accuracy of the
10 lm. The error in z-position z, in turn, strongly varied matching in z-direction can then be determined by plot-
as a function of the chosen parameters (Fig. 5b). For a ting the distribution of all found z-velocities during the
changing a (depicted as the black circles and scaled ac- entire tracking run (Fig. 6). These z-velocities show only a
cording to the bottom axis), the strongest variation in z small standard deviation of about 1.2% of the mean value.
can be observed. For increasing a, z decreases rapidly for In a real flow, the total number of localised particles is
8position behind the cylinder, 3D phenomena will arise.
The 2D vortex tubes shed from the cylinder will then show
a 3D transition, eventually leading to a collapse of the
vortex street. This transition is analysed here by consid-
ering the z-component of the vorticity vector xz (Fig. 7-
a,c,e). The evolution of xz is interpreted using the
visualisation results.
The results show indeed that the flow remains 2D over a
certain period after the vortex structures are shed. The
isovorticity surfaces (here xz=–0.3 and xz=0.3, Fig. 7a)
appear as more or less axisymmetric vortex tubes, which
608
are parallel to the cylinder axis. The 2D character of the
flow can also be seen in the dye visualisation results
(Fig. 7b). A regular pattern of dye streaks is observed up to
x/D609
Fig. 7a–f. Isovorticity surfaces
[xz=–0.3 (shaded) and xz=+0.3
(grid)] as determined by 3D PTV
(left) and visualisation results
(right) for ReD=75 and RiD=1.3
measuring volume is at least 15. For viewing angles above • In the 3D localisation algorithm used, only the first
25, no significant improvement in localisation accuracy found crossing possibility is used. Other crossing
was found. possibilities are not taken into account and this
From the 3D PTV results it is concluded that the information is lost. During the particle matching, an
method is capable of detecting a thermally induced tran- erroneous crossing may be chosen. This will result in
sition of a stable vortex street. In these experiments, about no matching with previously located particles and
500 vectors were constructed, allowing the calculation of therefore cause a decrease in the velocity yield. At this
the vorticity distribution. The 3D PTV results closely re- point, the algorithm can be improved by using all
semble the results observed by visualisation experiments. crossing possibilities for matching. The incorrect
Although not presented in this article, the results also crossings then do not match with other particles and
showed that, for an increased seeding density, the number automatically drop out during the matching
of resulting vectors drops significantly with respect to the procedure.
found blobs. In comparison to other reported results • For a high seeding density, the possibility exists that two
(Maas 1996), this decrease is quite common and is mainly or more particles are located on one particle projection
caused by the strong increase of ambiguities for increasing line (hiding particles, Fig. 2b). This results in only one
seeding density. In order to obtain a higher vector yield, particle image in the camera frame. Within the present
the present algorithm can be improved at least with algorithm, every 2D localised blob can only be used
respect to the following points. once, resulting in a loss of 3D particles. Allowing mul-tiple usage of a 2D blob will overcome this problem and After this validation step, the particle positions are
consequently result in an increased number of 3D determined with sub-pixel accuracy by using the
localised particles. grey-value-weighted centre of gravity (volume centroid) of
the segmented blob.
The above-suggested improvements are currently
being implemented in the 3D PTV code. 3D localisation
The localization method is thoroughly described in the
main text of the article.
Appendix: Description of the interrogation procedure
Matching
Dynamic thresholding
610 Every particle in frame f is matched with a candidate
In order to accurately determine the position of a par-
particle in frame f+1, where the candidate particles are
ticle, the background light (due to the reflections and the
defined as all particles within a given maximum matching
non-uniform intensity profile of a light sheet) is re-
distance from the particle in frame f. From all candidate
moved. A simple and fast algorithm was chosen: a
particles, the one which is positioned closest to the po-
square min-max subtraction filter. The filter leaves the
sition of the particle in frame f+1 is matched (Matching).
intensity profile undisturbed while removing non-uni-
The matching algorithm is improved by using an esti-
formities of its filter size. The filter consists of three
mation for the particle position in frame f. This estimated
basic operations:
position is provided by a prediction algorithm (Predic-
• Step 1: Min. Each pixel in a copy from the original image tion). In the 3D PTV algorithm, the prediction is deduced
is replaced by the minimum value in a square filter from the flow calculated in the frame set f–1. An im-
window. portant parameter for matching is the maximum
• Step 2: Max. Each pixel in the Min-filtered image is then matching distance. As the cameras operate at 25 Hz and
replaced by the maximum value in a square filter win- the average flow velocity is about 1 cm/s, the maximum
dow. matching distance was set to 0.2 cm (equivalent to
• Step 3: Sub. The result of step 2 is then subtracted from 10 pixels).
the original image.
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