Experimental investigation
of flows at low Reynolds number using
optical techniques
Christian J. Kähler
Institut für Strömungsmechanik der TU Braunschweig
Bienroder Weg 3, 38106 Braunschweig, Germany
E-mail: c.kaehler@tu-bs.de
RTO / AVT-104 VKI LECTURE SERIES
LOW REYNOLDS NUMBER AERODYNAMICS ON AIRCRAFT
including applications in EMERGING UAV TECHNOLOGY
24–28 November 2003
Contents
1 Introduction
1.1 Laser Two Focus Velocimetry . . . . . . .
1.2 Laser Doppler Velocimetry . . . . . . . . .
1.3 Doppler Global Velocimetry . . . . . . . .
1.4 3D Particle Tracking Velocimetry (3D PTV)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
2 Particle Image Velocimetry
2.1 Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Tracer particles . . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Recording of the particle images . . . . . . . . . . . . . .
2.2 Evaluation of image pairs . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Particle image density, loss of pairs and velocity gradients
2.2.2 Signal-peak detection and displacement determination . .
2.3 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Optical flow control . . . . . . . . . . . . . . . . . . . .
2.3.2 Laminar separation bubble with turbulent re-attachment .
2.3.3 Time-resolved PIV study of flow separation at low Re . .
2.3.4 Phase-locked PIV investigation of oscillating airfoils . . .
3 Stereo-scopic Particle Image Velocimetry
3.1 Principles . . . . . . . . . . . . . . . . . . . . . .
3.1.1 Error analysis . . . . . . . . . . . . . . . .
3.1.2 Scheimpflug condition . . . . . . . . . . .
3.2 Evaluation of stereo-scopic recordings . . . . . . .
3.2.1 Determination of the mapping function . .
3.2.2 Image warping . . . . . . . . . . . . . . .
3.2.3 Vector field warping . . . . . . . . . . . .
3.3 Calibration validation . . . . . . . . . . . . . . . .
3.4 Application: Coherent structures in boundary layers
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
2
2
3
4
5
.
.
.
.
.
.
.
.
.
.
.
6
6
7
9
10
13
14
15
15
17
19
21
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
24
25
26
28
29
29
31
32
32
35
4 Multiplane Stereo Particle Image Velocimetry
4.1 Principles . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Four-pulse-laser System . . . . . . . . . . . . . . . . . .
4.2.1 Performance of spatial light-sheet separation . . .
4.2.2 Generation and controlling of the timing sequence
4.3 Modes of Operation . . . . . . . . . . . . . . . . . . . . .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
38
38
40
41
42
43
1
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Chapter 1
Introduction
Despite the technological progress in optical- and microelectronic devices, that build the basis
for state-of-the-art measurement techniques, the experimental investigation of aerodynamic
flows at low Reynolds number is still a challenging task. First, because the flows are usually
highly unsteady and three-dimensional but also because of the large dynamic range of the
fluid mechanical variables and their sensitivity to small disturbances. In the following, various
measurement concepts will be outlined and their potential and limitations for the investigation
of low Reynolds number flows is discussed. However, because of the mentioned sensitivity of
flows to small disturbances, only non-intrusive optical techniques will be discussed.
1.1 Laser Two Focus Velocimetry
The simplest optical technique applied today in fluid mechanics is called Laser Two Focus
Velocimetry (L2F). This technique was proposed by Thompson in 1968 and successfully developed and applied by Schodl in 1974. This method estimates the velocity of small tracer
particles that passes in a certain time two light boundaries, each approximately 10 µm in diameter and 200 µm apart from each other in the direction of the velocity component of interest.
A schematic setup is shown in figure 1.1.
Rochonprism
Mirror
HeNe-Laser
Lens
Rotator
Photodiode
Photodiode
Lens
1
2
Filter Apertur
Flowdirection
F IGURE 1.1: Schematic setup of a Laser Two Focus Velocimeter.
A single linearly polarized laser beam is spitted by means of a Rochonprism into two
spatially separated beams of equal intensity. After the separation, both beams pass a pair of
2
lenses to generate two focal points in the measurement domain. When a particle passes the
focal points at time t and t
t ∆t, the scattered light is accumulated and collimated with
an additional lens system and afterwards filtered with an aperture and band-pass filter. Is the
displacement of the focal points known, a time accurate recording of the light pulses with
a pair of photodiodes allows to determine the first order approximation of a single velocity
component of the particle motion and also the direction of the flow can be determined with this
arrangement. By rotating the prism also the other velocity components can be measured. Is the
particle concentration sufficiently high, the temporal variation of a single velocity component
can be determined at a single point. However, the principle of the technique requires that the
separation of the particles inside the fluid must be large relative to the measurement volume
for reliable velocity measurements. This limits the temporal resolution of the system.
1.2 Laser Doppler Velocimetry
The simultaneous determination of two or three velocity components is possible with a Laser
Doppler Velocimetry (LDV) system that was proposed in 1964 by Yeh and Cummins. In
contrast to the L2F system, were just the intensity of the laser radiation is essential for the velocity measurement, the LDV technique makes also use of the coherence of the light. Figure
1.2 reveals the principle setup for a one component system. In the displayed configuration, the
Photodiode
2ϕ
Filter
Lens
Mirror
Lens
Braggcell
Flow
HeNe-Laser
Beamsplitter
Mirror
opt. comp.
Flowdirection
F IGURE 1.2: Schematic setup of a Laser Doppler Velocimeter.
monochromatic beam emitted by the laser is at first separated into two beams of equal intensity by means of a beam-splitter cube and afterwards superimposed at the location were the
flow velocity is of interest. When each beam is described by a plane electro-magnetic wave,
the superposition of both beams results in a stationary interference pattern and the distance
between the intensity maxima is given by:
∆x
λ
2 sin φ
(1.1)
Thus the separation depends only on the wavelength λ of the electro-magnetic wave and the
angle between the two propagation directions. When a particle passes the interference pattern
with a constant velocity, it is periodically illuminated and the scattered light pulses can be
3
recorded by using a photodiode and analyzed with a counter or spectral analyzer as illustrated
in figure 1.3 to determine the frequency ν of the pulse sequence. When ν, λ and φ are know, the
velocity component of the particle normal to the intensity pattern can be determined according
to:
λν
U ∆xν
(1.2)
2 sin φ
Signal
conditioning
Bandpassfilter
Photodiode
t
Frequency
analysis
t
t
νD
ν
F IGURE 1.3: Diagram of the Laser Doppler Velocimeter signal-analysis procedure.
Due to the fact that the direction of the particle motion can not be determined with a
stationary interference pattern, a Bragg-cell is usually introduced to generate a small frequency
shift of one beam. In effect the interference pattern moves with a fixed velocity given by:
λνS
2 sin φ
(1.3)
2U sin φ νS
λ
(1.4)
US
This means that also a particle at rest (U 0) will periodically scatter light whereby the
scattering frequency νD is directly related to the frequency shift νS according to:
νD
However, when the motion of the interference pattern is above the motion of the particles the
direction of the particle motion can be uniquely determined because the frequency generated
by the particle motion is added in one case (U 0 νD νS ) and subtracted from the natural
frequency of the interference pattern in the other case (U 0
νD νS ). This is the well
known Doppler effect. The LDV technique can measure simultaneously two velocity components of the particle in the measurement volume when another pair of crossing beams is
rotated by 90 relative to the pair of beams shown in the drawing and the light scattered by
the particles from the two orthogonal interference patterns can be uniquely separated. This
is possible either by using different frequency shifts νS or orthogonal states of polarization
or different wavelength λ of the laser radiation. The basic drawback for investigations at low
Reynolds number is the fact that this technique is only a single point measurement technique.
1.3 Doppler Global Velocimetry
To overcome the limitations of the single-point measurement probes the so called Doppler
Global Velocimetry (DGV) was invented by Komine in 1990. This technique illuminates the
4
particles within a two dimensional domain by using a frequency stabilized light-sheet and the
scattered light by the particles within the sheet is recorded simultaneously with a pair of CCD
camera whereby one camera is located behind a iodine-cell. The iodine-cell causes an intenLight-sheet (ν0 )
Mirror
T (ν0 −νD)
CCD 1
Transmission
Lens
Iodine-cell
CCD 2
Beam-splitter
~ 1GHz
T (ν0 )
T (ν0 +νD)
Velocitycomponent
ν0 −νD
ν0
ν0 +νD
Frequency
F IGURE 1.4: Left: Setup of a Doppler Global Velocimeter. Right: Absorption line of Iodine.
sity shift that is directly related to the frequency shift of the Doppler signal, see right drawing
of figure 1.4. When the intensity distribution behind the iodine-cell is properly normalized
with the intensity field measured with the reference camera, the spatial distribution of the velocity component at half-angle between the propagation direction of the laser light and the
observation direction can be measured. By using more laser-sheets with different propagation
directions and more cameras, the technique can be extended to a 2D3C technique (two spatial
dimensions, three velocity components). However, in contrast to LDV and L2F, today only the
mean values of the velocity can be determined with DGV. This is the main drawback for applications where in-stationary effects dominate the flow phenomena. In addition this technique
is not well suited for low Reynolds number applications because the accurate determination
of the intensity shift requires a strong frequency shift due to the absorption-line of iodine, the
frequency drift of the laser and the camera noise.This means that small velocity variations can
not be resolved today.
1.4 3D Particle Tracking Velocimetry (3D PTV)
The only technique that is able to determine the time dependent distribution of all velocity
components in the three dimensional space is the so called 3D Particle Tracking Velocimetry.
This technique was successfully applied by Virant and Dracos in 1997. Principally, the flow of
interest is homogeneously seeded with small tracer particles and illuminated with a white light
source and the light, scattered by the particles, is recorded simultaneously by means of four
CCD cameras which observe the particles from different directions. By using photogrammetric methods it is possible to identify identical particles on the four images and afterwards their
position in the flow. So it becomes possible to trace each particle in time. However, to avoid
particles hidden behind other ones, this method requires that the particle concentration is very
low and the flow must be slow with state of the art illumination and camera technologies.
5
Chapter 2
Particle Image Velocimetry
2.1 Principles
In the last two decades a multi-point technique, called Particle Image Velocimetry (PIV),
was developed that is one of the most powerful tools for the investigation of complex flow
fields today. The Particle Image Velocimetry is a non intrusive technique for measuring the
spatial distribution of the velocity within a single plane inside the flow, indirectly via the
displacement of moving particle groups within a certain time, see figure 2.1 and [32]. For
this purpose the flow is homogeneously seeded with appropriate tracer particles whereby the
concentration of the particles must be well adjusted with regard to the finest flow structures. In
addition, the deviation of the particle velocity up from the real flow motion u must be negligible
compared to the uncertainty of the imaging and recording system and to the uncertainty due
to the evaluation procedure1 . After the required seeding concentration has been obtained, a
desired plane inside the flow is illuminated twice by a thin laser light-sheet. The light scattered
by the tracer particles at time t and t in the direction of the recording optics is stored on
individual frames of a single frame transfer CCD camera, whose optical axis is perpendicular
to the light-sheet. It is obvious that the particle displacement ∆x must be small relative to
the finest
scales, as only phenomena that occur over a time interval which is longer
flow
than ∆t t t and that have a spatial extent larger than the absolute displacement ∆x can be
resolved, but the particle image displacement ∆X , on the other hand, must be large for accurate
measurements. The local displacement of the double exposed particles is finally determined
from the two single exposed recordings by means of spatial cross-correlation techniques and
afterwards divided by ∆t and the magnification factor M of the imaging system to calculate
the first order approximation of the velocity field2 .
t ∆t
u t dt
t
t ∆t
up t dt
∆x
t
∆x ∆t
∆X u
M∆t
(2.1)
This brief overview already implies that the complexity of the technique arises basically
from the technical components involved and their mutual dependence on each other and less on
the principles of the technique itself. In terms of accuracy, for example, the particles should be
1 The
2
symbol u indicate a three component vector and u one single velocity component of u.
In this imaging configuration the projection of the velocity field in the light-sheet plane is determined.
6
Mirror
Light sheet optics
Double pulse Laser
Light sheet
Illuminated
tracer particles
Flow with
tracer particles
Particle image positions at t
Particle image positions at t
y
Imaging optics
x
t
Flow direction
Image plane
t
F IGURE 2.1: Schematic set-up of particle image velocimetry system. A desired plane inside the flow is
illuminated twice by a thin laser light-sheet and the scattered light emerging from the homogeneously
distributed particles in the direction of the imaging optics is recorded.
sufficiently small and their density should exactly match the density of the surrounding fluid.
Unfortunately, this is not often feasible for a desired field of view and a given laser power, light
sheet thickness, transparency of the fluid, imaging optics and sensitivity of the digital camera,
as the scattering intensity decreases rapidly with decreasing particle diameter. Decreasing the
light-sheet width or thickness may partially help but the size of the largest resolvable scales
will decrease as well and the three dimensionality of the flow may cause further problems as
will be explained later. To use a powerful laser seems to be the appropriate solution but beside
the costs, strong reflections from model surfaces or undesirable disturbances of the flow due
to acoustic excitation or thermal response of the flow have to be taken into account. An
intensified camera could be applied as well but a reduced spatial resolution and an increased
noise level must be accepted. Alternatively, the evaluation procedure could be adapted but
then the accuracy of the velocity estimation and the validity of the well established principles
may become questionable. Furthermore, it cannot be recommended to analyze data where the
desired fluid mechanical information, hidden in the particle image displacement, cannot be
uniquely determined. So increasing the particle size may be the appropriate solution at the
end, but how this can be achieved is unknown. Anyway, in order to make the right decision
while setting up and aligning the experiment, a clear understanding of the basic components
is important.
2.1.1 Tracer particles
The starting point and basic problem associated with all tracer based optical velocity measurement techniques in fluid mechanics is the reproducible generation of sufficiently monodisperse
particles with an appropriate size, shape and density such that they follow the macroscopic
flow motion faithfully without disturbing the flow or fluid properties. In the past, air-operated
7
100
Cumulative percentage
Frequency distribution
2
1
0
1
50
d=0.5, p=0.5, n=4
d=1.0, p=0.5, n=4
d=2.0, p=0.5, n=4
d=1.0, p=0.5, n=12
d=1.0, p=0.5, n=4, Imp
0
10
0
2
Particle size [µm]
6
8
10
100
Cumulative percentage
2
Frequency distribution
4
Particle size [µm]
1
0
1
10
Particle size [µm]
50
d=0.5, p=1.0, n=4
d=1.0, p=1.0, n=4
d=2.0, p=1.0, n=4
d=1.0, p=1.0, n=12
d=1.0, p=1.0, n=4, Imp
0
0
2
4
6
8
10
Particle size [µm]
F IGURE 2.2: Volumetric particle size distribution for various hole diameters (d 0 5 1 2 mm), pressure (p 0 5 and 1 bar), number of holes per nozzle (n 4 and 12) and impactor.
aerosol atomizers have been widely and successfully used as these devices easily produce the
particle concentrations required for high resolution measurements in air [29]. However, the
operating conditions of these devices must be carefully controlled because the particle size
distribution may strongly deviate from the desired distribution if the liquid level in the generator changes, the pressure varies, or the nozzle holes are contaminated for any reason [6, 19].
This may cause serious problems, especially for investigations with strong vortices or transonic flows with shocks, as the accuracy of all tracer-based velocity measurement techniques,
such as laser Doppler anemometry (LDA), particle image velocimetry (PIV) or Doppler global
velocimetry (DGV), is ultimately determined by the particle dynamics. Figure 2.2 shows the
volumetric particle size distributions as a function of the atomizer nozzle outlet diameter d,
the external pressure p, the number of outlets per nozzle n and the effect of an impactor behind the output of the seeding generator where large particles are rejected. The upper left
plot of figure 2.2 implies that shortly above the minimum pressure required for the production
of particles, only the impactor is well suited to keep the particle size distribution sufficiently
narrow and the mean particle diameter sufficiently small for accurate flow investigations. Unfortunately, the particle concentration produced is quite low when only a single four-outlet
8
nozzle is used. The effect of the impactor can be clearly seen by comparing the dot-dashed
line with the dotted graph. By increasing the pressure from 0.5 to 1 bar, the exit velocity of the
air jet reaches
the speed
and the performance of the two high mass flux nozzles with
of sound
(d 2 n 4) and (d 1 n 12) becomes comparable with the desired impactor distribution
but the number of particles produced is several orders of magnitude higher. The other nozzles,
in contrast, need much higher pressure before the shoulder disappears and the mean diameter
becomes about 1 µm.
Generally, it can be stated from the graphs that the bandwidth and mean diameter of the
distribution decrease to a final distribution if the diameter d, pressure p, and the number of
nozzle outlets n are increased, or when an impactor is used alternatively, see [19, 17] for
details. However, it has to be taken into account that a decreasing particle diameter reduces
strongly the light scattered by the particles as shown in figure 2.3 for a spherical oil particles of
different size in air. To overcome this problem powerful lasers are required for the illumination
of the tracer particles or sensitive cameras for the registration of the particle images.
Light
0o
10
3
10
10
5
7
10
Light
180o0o
180o
F IGURE 2.3: Light intensity scattered by spherical oil particles of different size in air (left: d p 1 µm,
right: dp 10 µm), illuminated from left with a plane monochromatic wave front (same intensity
scale). The complex spatial intensity distribution, with a maximum in forward direction, results from
the interference between the reflected, refracted and diffracted wave front.
2.1.2 Recording of the particle images
The registration, storage and read-out of the individual particle images is another key element in PIV, beside the particle dynamics, because the accuracy of the technique strongly depends on the precision with which the image displacement can be related to particle locations
and their respective particle displacements [2, 42]. In the recording process the continuous
intensity distribution of the particle images is transformed into a discrete signal of limited
bandwidth as shown in figure 2.4. When the discretization of the image-signal matches the
minimum sampling rate, the frequency contents of the original signal can be reconstructed in
principle without any losses and thus the particle location as well. This is the contents of the
sampling theorem which states that a bandwidth-limited signal can be perfectly reconstructed
from its discrete samples when the sampling rate of the signal is at least twice the signal
bandwidth. Due to the noise introduced by inhomogeneous illumination or absorption by the
surrounding fluid, optical aberration, the evaluation technique and the common peak-fit routines for sub-pixel accuracy, for example, the principal resolution given by the aperture, focal
length, wavelength of the monochromatic light and the magnification of the imaging system
9
Intensitätsverteilung
Integration
überdistribution
die aktiven of tracer particlesDiskretisierte
F IGURE
2.4: From left auf
to right: Continuous
intensity
in the image-plane.
der Bildebene
Bereiche der CCDIntensitätsverteilung
Integration of the photons registered by the light
sensitive area of a CCD sensor with fill factor smaller
Elemente
one. Discretised intensity distribution in memory of computer.
never need to be fully resolved in practice. This allows the use of digital recording systems
which are easy to handle and the time consuming focusing, film change, developing and scanning procedure, for the photographic recording methods utilized some years ago, are no longer
required [41, 43]. But it should be emphasized that beside the loss-of-information due to the
discretization of the spatial coordinate with a limited resolution and beside undesired artefacts
like Moire or Mach-band effects, a possible modification of the incoming intensity signal, due
to the grey-level quantization, nonlinearities in the pixel response or during the read-out of the
images or AD-conversion, amplification and transportation via long connections, may reduce
further the performance of the measurement technique. As a digital camera is a complex system this bias may be introduced in many different ways and its strength strongly depends on
the CCD (charge coupled device) architecture and pixel (picture element) characteristics. To
detect possible CCD errors in the recordings and to understand their consequences for PIV, a
deeper understanding of the CCD principles is required. However, this is beyond the scope of
this paper.
2.2 Evaluation of image pairs
In order to extract the displacement information from the two single exposed grey-level patterns acquired at t and t , both images are usually evaluated by means of statistical evaluation techniques, for two reasons. First, individual particle image pairs cannot be identified
sufficiently reliable due to the high seeding concentration required to resolve the small flow
scales. Secondly, the statistical evaluation is less sensitive to noise and discretization effects
as outlined in the previous section. For the evaluation, the whole image is sampled with
an
appropriate step-size, typically half the linear dimension of the grey-level
sample I x y . For
each sampling location a two-dimensional grey-level sample I x y of certain size and shape
is extracted from the source image
as indicated in figure 2.1, and it is cross-correlated with
the corresponding sample I x y from the search recording according to the following equation3 [23, 43].
3 As
each single grey-level sample back-projected in the object space and multiplied by the effective lightsheet thickness, given by the intensity distribution of the light-sheet itself, the particle size distribution, the imaging optics and sensitivity of the CCD camera, represent a fluid volume, the cross-correlation can be considered
as weighed volume average.
10
RII x y
n
i
m
∑ ∑
nj
I i j I i
x j
y
(2.2)
m
The cross-correlation formula produces one cross-correlation value R II x y for each possible
displacement between the two samples by calculating the sum of the products of all overlapping pixel intensities so that the output of this calculation can be displayed in the form of a
two dimensional cross-correlation plane, see figure 2.5 and 2.6. Due to the symmetry of the
cross-correlation function the template I x y can be selected in either of the two images, but
the sign of the displacement has to be reversed when the template originates from the image
recorded at t ∆t. The cross-correlation formula is usually properly normalized so that its
Shift (∆x = 0, ∆y = 0)
Shift (∆x =-2, ∆y = 2)
Shift (∆ x = 2, ∆y = 2)
Shift (∆ x =-1, ∆y =-1)
Shift (∆ x = 1, ∆y =-2)
+2
∆y 0
-2
0
-2
+2
∆x
Cross-correlation plane
F IGURE 2.5: Example of the formation of the correlation plane by direct cross-correlation: here a
4 4 pixel template (grey-square) is correlated with a 8 8 pixel size sample to produce a 5 5 pixel
cross-correlation plane.
amplitude is invariant under scale changes in the amplitude of I and I and thus not affected by
variations in the particle image number, size and intensity. This allows to quantify the quality
of the signal which is of great importance when aligning the experimental components, such
as light-sheet and cameras, with respect to each other.
n
cII x y
i
n
i
∑
∑
m
nj
∑
m
m
nj
∑
I i j
I I i
m
I i j
I
n
2
i
∑
x j
m
nj
∑
m
Ii
y
x j
I
y
I
2
(2.3)
I denotes the mean intensity value
of the search window calculated over the portion which
coincides perfectly with the I i j and I is the corresponding mean intensity for the template.
It is obvious that I has to be calculated for each possible shift to guarantee that 0 cII 1
(negative values are usually not considered as intensities are always positive and the mean
11
RD
F IGURE 2.6: Spatial distribution of the the cross-correlation coefficient calculated over five randomly located particle image pairs of varying intensity moving homogeneously ∆x 1 89 pixel ∆y
1 89 pixel. The displacement correlation peak R D is surrounded by random noise due to other pairing
possibilities. Whereas RD is proportional to the number of paired particle images as only these add
to the signal strength, the height of the noise peaks depends on the intensity of the images and on the
organization of the underlying pattern.
value is close to zero for moderate image densities). As a direct implementation of this formula
in the spatial domain becomes quite time intense, especially for large correlation windows, the
computation of the correlation is usually performed in the frequency domain via fast Fourier
transformation (FFT) based algorithms. They are fast as symmetry properties are taken into
account but artifacts like aliasing, circular effects and limitations in the size and shape of
the interrogation window may lower the quality of the velocity estimation and restrict the
performance of the method for particular applications.
As the cross-correlation approach is based on the idea that the location of the signal peak
yields the information about the displacement of the image pattern, the conditions for the
correctness of this assumption need to be analyzed in detail. Due to the fact that this is mathematically not tractable when a single realization of a particle image pattern is considered,
the pattern is thought of as a single realization of a random process and it is assumed that
this particular pattern represents the entire random process to which it belongs 4 . Under this
assumption, it can be shown by analyzing all possible tracer patterns for a single flow field
realization that the particle images must be exactly identical in size, shape and intensity, homogeneously distributed and the structure of the pattern must be invariant under spatial transformations to ensure that the particle image displacement is directly related to the location of
the correlation maximum as assumed in PIV evaluation [1, 23, 41]. In practice, of course,
the required conditions are only approximately realized, so that experimental, numerical and
analytical investigations are required to analyze the effects of non ideal conditions introduced
by inhomogeneities in the particle image distribution, variations in the particle image sizes,
shape and intensity, image sampling, grey-level quantization as well as non-linearities in the
pixel response or light sensitivity, shot noise, charge leakage, dark-current noise in the CCD
and electronic noise in the CCD, camera and frame grabber, gradients and so on.
4 For
ergodic processes it is possible to derive the desired statistical information about the entire random
process from an appropriate analysis of a single arbitrary sample function.
12
2.2.1 Particle image density, loss of pairs and velocity gradients
In the previous sections it has always been assumed that the signal peak R D x y is uniquely
defined, reliable to detect and not affected by random noise superimposed on the signal peak.
In order to obtain this ideal conditions in any real experiment, the following relation has to be
taken into account while setting up the experiment [1, 23, 41].
RD x y
NI FI FO F∆
(2.4)
NI represents the so called image density which is actually the number of particle image pairs
in the interrogation domain which contribute to the signal strength. FI and FO denote the inplane and out-of-plane loss of pairs as a result of the finite size of the measurement volume
and F∆ accounts for the loss of correlation due to gradients within the measurement volume.
NI
FI
FO
F∆
∆zo
Dx Dy
M2
∆x 1
Dx ∆z 1
∆z0 M ∆u ∆t
Dy
N
∆y NI
1
Dy
(2.5)
(2.6)
(2.7)
(2.8)
Equation (2.4) states that the amplitude of the cross-correlation peak increases with increas
ing image density NI N and decreases with increasing displacement, no matter in which
direction or if the displacement of the particle images is not constant. In order to optimize
the performance of the experiment, the number of paired particle images is of primary importance as only these add to the signal strength. In addition the number of correct pairings with
respect to incorrect combinations has to be optimized for increasing the signal-to-noise ratio
and thus the detectability of the signal. This can be achieved either by changing the seeding
concentration, the magnification of the imaging system or simply by increasing the light-sheet
thickness, see equation (2.5). In plane loss of pairs, caused by particles entering or leaving the
measurement volume during the illuminations, due to the motion of the fluid, can be reduced
by decreasing ∆t or eliminated when the search window contains the corresponding particle
image pattern selected with the template. This can be achieved either by correlating differently
sized samples as described before or by using window-shifting in connection with multi-pass
interrogation techniques [40, 43]. Using this approach two equally sized samples, separated
by the local particle image shift are cross-correlated after the local shift has been determined
within a first evaluation. Since the correlation windows are symmetrically shifted with respect to each other and relative to the exact measurement position, it becomes necessary to
determine where the footprint of the vector should be located (center or corner). This is especially important when different evaluation methods are compared because the particle pattern
employed for the evaluation may be different for the same pixel coordinates of the displacement vector [37]. Out-of-plane loss-of-pairs can be compensated by shifting the light-sheet in
the direction of the mean particle displacement, as proposed in [23], or by using multi-lightsheet arrangements when the out-of-plane motion is not uniform, see chapter 4. The effect
of in-plane gradients can be reduced by increasing the magnification of the imaging system,
13
by decreasing the time-separation between the two illuminations or by using properly shaped
interrogation windows whose linear dimension is reduced in the direction of the gradient.
2.2.2 Signal-peak detection and displacement determination
To increase the accuracy in determining the location of the displacement peak from 0 5 pixel
to sub-pixel accuracy, an analytical function is fitted to the highest correlation peak by using
the adjacent correlation values [43]. Usually two one-dimensional Gaussian fits along the two
coordinates through the highest correlation coefficient are applied. The motivation for this
particular function is based on the fact that under ideal imaging conditions the shape of the
signal peak is of Gaussian shape, like a diffraction limited image of a particle, and close to 3
pixels in each spatial direction for realistic applications of PIV.
f x
C0 exp
x0
x
k
2
(2.9)
x0 indicates the exact location of the maximum and C0 k are coefficients of no direct interest.
Using this expression for the main and the adjacent correlation values and the fact that the
first derivative of this expression must be zero, the position can be estimated with sub-pixel
accuracy. Unfortunately, differences from this ideal Gaussian shape are normal due to the discretization, electronic noise, velocity gradients and optical aberrations and introduce artefacts
which reduce the performance of this peak fit and lead to systematic measurement errors. Better results can be achieved by fitting a two dimensional Gaussian function to a larger number of
points or by using other sophisticated analysis methods that have been proposed in the past to
optimize the evaluation procedure with regard to accuracy and spatial resolution and to overcome the limitations associated with the required image density and the tolerable gradients.
For details the interested reader may consult the existing literature [8, 10, 12, 28, 37]. The
excellent performance of the 3-point Gaussian fit when the location of the correlation maximum is exactly symmetrical relating to a pixel stimulated Lecordier et al [28] to make use
of the accuracy by transforming the measured image in such a way that the measured velocity
becomes zero after the transformation, see section 3.2.2. To increase the spatial resolution
without increasing the probability that a random correlation peak will exceed the height of
the displacement peak and deteriorate the velocity measurement, it is possible to calculate the
correlation twice with a small spatial separation in such a way that the image displacement
is identical but the noise uncorrelated [10]. Once this has been done, the signal peak can be
clearly detected by calculating a correlation of the correlation (the uncorrelated noise will vanish), but again it should be taken into account that evaluation schemes based on few particle
images are highly sensitive to noise that is introduced by the characteristics of digital cameras. In order to enhance the signal quality when strong in-plane gradients are present, Huang
et al [12] has proposed to transform the image in such a way that the deformation vanishes
after the mapping, see also [8]. This approach works fine as long as the flow under consideration possesses only in-plane gradients (∂u ∂x, ∂u ∂y and ∂v ∂x, ∂v ∂y). In case of strong
out-of-plane gradients (∂u ∂z) this and most of the other sophisticated methods will fail, as
can be easily realized by considering a turbulent boundary layer experiment with a light-sheet
parallel to the flat plate. Under these conditions the apparent particle image gradients caused
by particles from different layers cannot be compensated because of arising instabilities in the
image analysis.
14
2.3 Application
2.3.1 Optical flow control
The active control of laminar and turbulent flows with dynamical actuators is of great scientific and technological interest in nearly any field of fluid mechanics. On one hand it becomes
possible to generate artifical flow structures whose properties and significance for the turbulent mixing, or their interaction with other flow structures, can be examined. On the other
hand, these devices suppress flow separation on profiles or increase the performance of flow
engines. Today, most of the well established actuator concepts are based on pneumatic and
micro-mechanic basis. However, owing to their limited dynamic range their potential seems
to be limited from the present point of view, see Gad-el-Hak (2001). For this reason, the performance of an optical actuator was examined that allows to excite the flow non-intrusively
with nearly any pulse-width and repetition-rate, see [20] for details.
The experiment was performed in the small wind-tunnel of the Institute of Fluid Mechanics located at the Technical University of Braunschweig. This facility has an open test section,
940 mm in length, with a circular nozzle, 500 mm in diameter, and a 4:1 contraction ratio.
A 470 mm long, 248 mm wide and 10 mm thick perspex plate with an elliptical leading edge
(6:1 aspect ratio) and a blunt trailing edge was horizontally installed between two transparent
side-plates with a slightly divergent orientation. Perspex is usually a well suited material for
laser-induced excitation for two reasons. First, due to the small heat capacity a strong heating
of the surface is promoted before the interaction with the flow takes place. Secondly, the low
heat conductivity ensures that the heat is not transfered inside the model and thus lost for the
control of the flow. However, due to the low absorption and melting temperature (80–130 Celsius) the surface was protected with a 10 mm 10 mm 0 5 mm silicon inlay that was flush
embedded inside the model, x 305 mm behind the leading edge. The loss of radiation due to
reflection was minimized by cauterizing the polished silicon surface. The flow-field investigation was done with a frequency doubled Nd:YAG double-pulse
laser system (Quantel Brillant)
with the following properties: Pulse energy 167 mJ at λ 532 nm; Beam-pointing stability
5 %; Beam-divergence 0.43 mrad; Direction stability 100 µrad; Pulse-width 4 2 ns. The
laser beam was formed into a light-sheet, 0 5 mm in thickness, to illuminate the tracer particles. The light scattered by the particles was recorded by means of a Peltier cooled CCD
camera (FlowMaster 3S) with a Sigma 105/2.8 Macro Ex lens. The observation distance was
approximately 240 mm and the field of view measures 16 45 13 1 mm2 . The evaluation of
the recordings was achieved with an 2nd order accurate multi-grid method with 32 32 pixel
interrogation windows and 50 % overlap. The corresponding resolution
was 0 2 mm in phys
ical space and the dynamic range 0 3 ∆x 16 2 pixel for ∆t 15 µs time delay between
the illuminations. Based on the assumption that the resolution of the evaluation method is
around 0 05 pixel, see [37] for details, velocity fluctuations with a magnitude of 0 03 m/s can
be resolved at 10 m/s free-stream velocity. The generation of the thermal disturbance was
performed with a second Nd:YAG double-pulse laser-system whose beams were delivered simultaneously. The beams were focused onto the silicon inlet by means of a converging lens.
The diameter of the focal point was approximately 0 2 mm and the maximum energy-density
about 3.5 J/mm3 . In a first series of experiments the flow quality along the flat plate was examined at 5 locations and compared with the Blasius-solution. The rectangles A to E in figure
2.7 display the size and location of the observation area and the solid line indicate the mean
15
stream-wise velocity at the edge of the laminar boundary layer, calculated with XFoil 6.9.
Ue / U∞
1.15
A
1.1
B
C
D
E
1.05
F IGURE 2.7: Location and size of the measurement domain and ratio between the velocity Ue
at the boundary layer edge and the free stream
velocity U∞ along the flat plate.
1
0
100
200
x [mm]
300
400
The numerical flow simulation implies that the acceleration effects, caused by the leading and
trailing edge of the plate, are below 2% at the location where the PIV measurements take
place. For the determination of the mean velocity profiles 200 statistical independent velocity
fields were measured for each location and averaged pointwise and along the x-coordinate.
This is possible because the growth of the laminar boundary layer within the field of view is
below the spatial resolution. The graphs in the left image of figure 2.8 agree nicely with the
2
2
window A
window B
window C
window D
window E
Blasius-profile
1
1.5
y/δ
y/δ
1.5
0.5
0
U∞ = 10 m/s
U∞ = 11 m/s
U∞ = 12 m/s
U∞ = 13 m/s
U∞ = 14 m/s
U∞ = 15 m/s
U∞ = 16 m/s
U∞ = 17 m/s
Blasius-profile
1
0.5
0
0.25
0.5
0.75
0
1
u / Ue
0
0.25
0.5
0.75
1
u / Ue
F IGURE 2.8: Dependence of the normalized boundary-layer profiles from the stream-wise position at
14 m/s (left) and from the free-stream velocity at position E (right)
calculated Blasius-profile, only the results obtained at E deviate because the flow is already in
a transitional state at this position. A variation of the free-stream velocity shows that this effect
becomes visible for U∞ 12 m/s and vanish at 15 m/s after reaching the turbulent flow state.
This can be concluded from the invariance of the normalized velocity profiles for u 15 m/s.
Based on this results it was decided to perform the main flow investigations at 10 m/s because
here, the generation and development of the laser-induced flow structure in a laminar boundary
layer, that is stable against small disturbances, is of interest. To examine the effectiveness of
the laser-induced disturbance for the generation of turbulent flow structures within a stable
laminar boundary layer along a flat plate without stream-wise pressure gradient, the laser beam
was focused at 10 m/s onto the silicon inlay located at 305 mm down-stream of the leading
edge of the plate, and the effect on the flow was analyzed at position E according to figure 2.7.
Figure
2.9 shows the spatial distribution of the stream-wise velocity fluctuations measured
∆t 9, 10, 12 and 16 milliseconds after the laser-induced excitation of the flow. The upper left
16
y [mm]
12
∆ u [m/s]
t = 10ms
1.2
0.75
9
9
6
6
0.3
-0.15
3
3
-0.6
0
12
y [mm]
12
t = 9ms
thickness of the
undisturbed boundary layer
-1.05
-1.5
380
384
388
380
392
12
t = 12ms
9
9
6
6
3
3
380
384
388
384
388
392
392
388
392
t = 16ms
380
x [mm]
384
x [mm]
F IGURE 2.9: Convection of the laser-induced flow structure through the measurement domain located
80 mm behind the position where the thermal disturbance is introduced
representation indicate the shape and size of the down-stream region of the generated structure
after entering the observation area. The thickness of this flow region is three times larger than
the laminar boundary layer, diagrammed by the horizontal line at y 3 5 mm, and the graylevel scale indicate that the mean velocity is up to 15% lower in this region relative to the
undisturbed flow. However, with decreasing wall distance, the velocity difference change its
sign and reaches values up to 12% and more, see parts of the turbulent structure which reach
the observation area later. This investigation demonstrates clearly that an artifical turbulent
flow structure with characteristic properties can be non-intrusively generated with a focused
pulse-laser beam by thermal excitation of a solid surface. The excitation is possible with
nearly any pulse-length and repetition rates and the shape can be both point- or line-like but
also 2D excitation areas are possible with sufficient laser power.
2.3.2 Laminar separation bubble with turbulent re-attachment
Another characteristic aerodynamical phenomena at low Reynolds number is the separation
of the flow on the suction side of a thin airfoil at large angle of attack. This process is very
complex in its details and still subject of fundamental research. However, here only mean
properties will be outlined to demonstrate that such problems can be successfully investigated
with the PIV technique. For this investigation, a 200 mm long and 280 mm wide SD7003airfoil, made of glass-fiber reinforced Plastic, was installed between the side-walls of a small
water tunnel and
the flow-field
on the suction
side of the airfoil was examined with a PIV
system at U∞ 0 3 m/s, Re 6 104 and α 8 angle of attack. Of primary interest was the
17
SD7003-Profil,
α = 8°, U∞ = 0,3 m/s, Re = 60000
l = 202,6 mm
| u | / U∞

y [mm]
2.0
1.9
1.8
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
25
20
15
10
20
25
30
35
40
45
x [mm]
F IGURE 2.10: High resolution measurements of the mean velocity field of a laminar separation bubble
with turbulent re-attachment on the suction side of a SD7003 airfoil at α 8 . The location of the
x-coordinate indicates the distance from the leading edge of the airfoil.
examination of the size, shape and internal structure of the laminar separation bubbles but also
the region where the transition process takes place and the intensity of the Reynolds shearstresses was of interest. The basic problem associated with this investigation was the small
size of the laminar separation bubble and the low up-stream velocity close to the wall. Figure
2.10 shows the non-dimensional magnitude of the velocity field (average over 500 samples)
measured with a FlowMaster 3S camera along with a Sigma 105 mm macro-lens. The illumination of the tracer particles (glass hollows, 20 µm in diameter on average) was achieved
with a frequency doubled Nd:YAG laser. The stream-lines indicate
clearly the existence of a
separation bubble on the suction side of the SD7003-airfoil at α 8 angle of attack. The distribution of the Reynolds shear-stresses component, displayed in figure 2.11, indicates that the
bubble is laminar to a large extend. However, inside the strong shear-layer an irregular fluctuation of the velocity takes place due to the transition process. With increasing distance from
the leading-edge of the airfoil the turbulence level increases and the stream-lines indicate the
re-attachment of the flow. Problems arise close to the surface of the airfoil because of light reflections. This difficulty can be solved by using transparent airfoils whose index of diffraction
is nearly identical with the surrounding fluid or by using digital masking techniques. Another
problem is the low particle image displacement close to the wall. This problem can be partially
solved by using the time-resolved PIV technique (TR-PIV).
18
SD7003-Profil,
α = 8°, U∞ = 0,3 m/s, Re = 60000
l = 202,6 mm

2
u´v´/ U∞
y [mm]
0.022
0.017
0.011
0.005
-0.000
-0.006
-0.012
-0.017
-0.023
-0.029
-0.034
-0.040
-0.046
-0.051
-0.057
-0.063
-0.068
-0.074
-0.080
-0.085
-0.091
25
20
15
10
20
25
30
35
40
45
x [mm]
F IGURE 2.11: High resolution measurements of the Reynolds shear-stresses.
2.3.3 Time-resolved PIV study of flow separation at low Re
The Particle Image Velocimetry equippment, described in the previous sections, can usually
not be applied for time resolved flow investigations because of the low repetition rate of flashlamp pumped solid state lasers and the low frame rate of high resolution CCD cameras. However, there are diode-pumped solid state lasers available with repetition rates up to 50 kHz and
high resolution cameras with CMOS-technology that can aquire 2000 images per second with
1 MPixel resolution and up to 100 kHz at reduced image size. Unfortunately, the performance
of the components differs strongly from the components applied in the previous applications.
The output energy of typical diode pumped lasers at 10 kHz for example is less than 5% of the
laser energy usually applied in PIV and the sensitivity of the high-speed CMOS cameras is far
away from state-of-the art CCD cameras. Another drawback associated with the high-speed
cameras is the strong under-sampling of the particle due to the large pixel size relative to the
CCD pixel dimensions. This has a great impact on the accuracy of the time-resolved Particle
Image Velocimety, because the sub-pixel routine described in section 2.2.2 works only well
when the particle image size is 2–3 pixel, see [31]. On the other hand, it becomes possible
to optimize the evaluation with regard to accuracy, for example, or to increase the dynamic
range, by using more sophisticated evaluation schemes that take advantage from the analysis
of long image sequences [18]. For instance, a general limitation of conventional PIV is the
fact, that the time interval between the illuminations is not well suited for the determination of
all flow velocities inside the light-sheet. In case of a flow with a large dynamic, as displayed
in section 2.3.2, it is principally impossible to obtain reliable measurements everywhere. This
becomes immediately evident because when the particle image displacement in the outer flow
is 10 Pixel, for example, the correct displacement inside the bubble can not be determined,
19
Vortex
Laminar separation bubble
wavy structure
25
50
75
x [mm]
F IGURE 2.12: Flow visualization performed with a CMOS-camera in long exposer mode and an Ar laser at the location of the separation bubble at α 8 .
because the measurement error is large relative to the shift of the particle images. Increasing
the delay between the illuminations would solve this problem for the flow inside the bubble,
but in this case the correct displacement in the free flow can not be determined because of the
large displacement or due to three dimensional effects (loss of pairs). However, when a time
sequence is available it is possible to calculate the cross-correlation for image pairs acquired
symmetrically around a time t. This results in a set of velocity fields whereby the relative measurement error decreases with increasing temporal separation between the images consulted
for the calculation of the cross correlation. In regions where the separation becomes to large
for a reliable determination of the displacement, a spurious vector will be detected. Thus it
is clear that from the set of velocity fields that characterize the flow properties at time t one
optimized velocity field can be composed by applying particular criteria. The outcome of this
approach is a velocity field for each time t where all velocity measurements are optimized with
regard to a criteria and it becomes possible to determine with high accuracy simultaneously
the flow in the outer region and the region were the separation takes place.
The principal value of the time-resolved PIV technique will be demonstrated for the two
dimensional flow problem already discussed in section 2.3.2. The average flow field of the
laminar separation bubble does not give any information about the dynamic of the bubble
because these details are smeared out after calculating the mean flow. For this experiment
a high speed CMOS-camera (Photron FastCam Ultima APX) with 50 mm Nikon lens was
applied in combination with an cw Argon-Ion laser with 6 Watt power. The long-exposed
image, shown in figure 2.12, reveals a single sample from a series of images taken with a
frame rate of 60 Hz and 1 60 s integration time. The field of view is approximately 30 mm as
can be seen from the scale at the lower edge of the image. The image indicate that only the first
60% of the bubble are laminar and it can be also seen that the flow structure in the remaining
part is more complex. Clearly visible are two vortex like structures with a clockwise rotation.
20
From the analysis of the total image sequence it can be concluded that the upstream vortex is
not stable and disappears frequently. The other vortex on the other hand seems to be stable,
but the vorticity and the position of the vortex is not constant. However, this motion does not
affect the total size of the separation bubble to a large extent and also the laminar part of the
bubble is not biased due to this unsteady effect. The turbulent wake of the bubble that was
already visualized in figure 2.11, shows a wavy behavior and the largest amplitude can be
observed in the middle of the boundary layer. Although further experiments are required to
examine the fine structure of the wave-like motion, this example indicate nicely the value of
the TR-PIV technique.
2.3.4 Phase-locked PIV investigation of oscillating airfoils
For the efficient actuation of micro air vehicles (MAV) one promising concept is the harmonic
oscillation of the airfoil, as applied by birds and insects, for example. As the flow dynamics
is quite complex, detailed time-resolved measurements would be required to find the optimal
operating parameter. However, as the flow is periodic, a phase-locked PIV setup was applied to
examine the flow phenomena around an oscillating SD7003-airfoil with high spatial resolution
and low measurement error. Of primary interest was the effects introduced by the different
amplitude, frequency and angle of attack. For the generation of the movement, a special
device was designed and developed that allows periodic motion up to 10 Hz with an amplitude
of 40 mm. The experiments were performed with a conventional PIV system by using a high
resolution CCD-camera (FlowMaster 3S) in combination with a double-pulse Nd:YAG laser.
For the synchronization of the equipment a light barrier was mounted at the rotating wheel,
that forces the airfoil to oscillate, and the TTL-signal coming from this photodiode was used
to trigger a frequency generator that was introduced for the generation of the phase shift. The
output signal of this device was plugged to the external trigger port of a 16 channel sequencer
board that was required to trigger the flash-lamps and Q-switch of the laser and the CCDcameras.
For this investigation, the amplitude of the oscillating SD7003-airfoil
was set to
z 20 mm, the frequency to f 1 Hz and the angle of attack to α 0 and α 4 . The
non-dimensional
frequency,
based on a cord length of l 200 mm and a free-stream velocity
of U∞ 0 3 m/s, was k π f l U∞ 2 1. To obtain an accurate estimate of the flow velocities
with a high spatial resolution, the flow fields displayed in figure 2.13 and 2.14 where recorded
with four cameras located next to each other. The total size of the measurement domain was
412 mm 83 mm and the total number of velocity vectors for each recording is 20480. The
period of the oscillation was sampled with 72 points with a resolution of 5 . For each phase
angle a total number of 50 velocity field were recorded to determine the ensemble averaged
behavior of the flow with sufficient convergence. The phase angle of the selected results,
shown in figure 2.13 and 2.14, is altered in steps of 45 over the full period of the oscillation.
Clearly visible in both cases is the movement of the stagnation point and the period generation
of the starting vortex at the trailing edge of the airfoil and its movement in the wake of the
airfoil. However, dynamical flow separation can be only observed in figure 2.14, see [31] for
further information.
21
F IGURE 2.13: Velocity fields measured at α
0 , in steps of ϕ
22
45 phase angle starting at ϕ
0
F IGURE 2.14: Velocity fields measured at α
4 , in steps of ϕ
23
45 phase angle starting at ϕ
0
Chapter 3
Stereo-scopic Particle Image Velocimetry
Conventional PIV as described in the previous sections yields reliable results as long as the
flow under investigation is two-dimensional and parallel to the light-sheet. For investigations
with a velocity component being normal to the light-sheet, either due to three dimensionality,
or the orientation of the light-sheet with respect to the main flow direction, the out-of-plane
velocity component remains unknown whereas the in-plane components are biased due to the
perspective error.
F IGURE 3.1: Projectionerror as a function of the
image location and out-ofplane displacement.
As this error is directly proportional to the viewing angle, according to figure 3.1, the object distance must be increased to minimise the error while keeping the field of view. As this
approach requires long focal length lenses it is obvious that this approach is not satisfactory.
To overcome this constraint completely and to obtain the out-of-plane velocity component, a
stereoscopic observation arrangement has to be applied, which will be outlined in the following sections.
24
3.1 Principles
Using the stereoscopic recording technique, the images of tracer particles are recorded simultaneously from two different viewing directions, and the correct displacement (without
perspective error) of the particle ensembles are reconstructed by using the proper equations.
The basic recording arrangements can be classified either with respect to the camera position
relative to the light-sheet or with respect to the propagation direction of the light-sheet plane
according to figure 3.2. The left drawing reveals a configuration where both cameras are located on the same side of the light-sheet. As a consequence, this recording arrangement can be
operated in forward/backward or ninety degree configuration when the propagation direction
of the light-sheet plane is considered. An alternative arrangement is shown in the right drawing
of the same figure. In this case the cameras are separated by the light-sheet plane so that the
pure forward, backward and ninety degree arrangements are possible. As long as only the intensity of the scattered light is considered, the most efficient light-sheet camera-configuration
is the purely forward scattering set-up, according to the Mie scattering diagram in figure 2.3,
followed by the forward/backward configuration, purely backward and finally ninety degree
case. This may change when the state of polarisation has to be taken into account beside the
intensity (this will be further analysed in chapter 4 and the experimental parts of this thesis).
F IGURE 3.2: Stereoscopic recording arrangements. The orientation of the principle observation ray
(oblique lines) relative to the light sheet (dark plane) or the propagation direction of the light-sheet
plane (indicated by the arrows) can be used to define various recording configurations with different
properties. The observation direction from one light-sheet side (left drawing) allows forward/backward
and ninety degree imaging and from opposite sides (right drawing) forward, backward and ninety
degree.
Beside the above classification, it is common practice to differentiate between the stereoscopic recording approaches regarding to the field distortions into translation and angular displacement methods. In case of the translation method the light-sheet plane, the main plane of
the lens and the image plane are parallel in relation to each other. As a result, the magnification
factor is constant across the field of view (e.g. the image of a regular grid appears undistorted)
and the image analysis varies only slightly from the analysis outlined in the previous chapter,
see [16] for details. The drawback, on the other hand, is the decreased performance of this arrangement for increasing stereo opening angles. This happens because of optical aberrations
and the decrease of the modulation transfer function towards the edges of the field of view
and because of the limited overlap of the observation areas of both cameras when the CCD
sensor is not shifted with respect to the optical axis, see [16]. For these reasons the angular
displacement method is usually applied where the light-sheet plane and the main plane of the
25
Observation plane
Observation plane
Lens plane
Image plane
F IGURE 3.3: Stereo-scopic imaging configurations. Left: translation method. Right: angular displacement method.
lens intersect in a common line1 . In this configuration the magnification factor varies across
the field of view and typical distortions appear as indicated in figure 3.4 for both camera-lightsheet arrangements shown in figure 3.2. The size, shape and location of the dark areas indicate
the image of a rectangular area in the object space as a function of the camera arrangement.
The difficulties associated with this effect and possible solutions will be further analysed in
section 3.2.
F IGURE 3.4: Linear field distortions of a regular grid due to the oblique observation directions for two
angular displacement camera arrangements. Left: both cameras are located on the same side of the
light-sheet according to the left drawing in figure 3.2). Right: cameras are separated by the light sheet
(right drawing in figure 3.2).
3.1.1 Error analysis
It is obvious from our experience with the visual system of the human being, that the measurement error for the out-of-plane component and the accuracy of the perspective correction
depend on the opening angle between the two cameras. Lawson and Wu have derived from
geometrical considerations that for a symmetrical translation arrangement the relative out-of1
The translation imaging configuration can be seen as a special case of the angular-displacement arrangement
with the line of intersection between the image plane, the main plane of the lens and the object plane at infinity.
26
plane error σ∆z σ∆x as a function of the off-axis position x is given by
σ∆z σ∆x
x d0
1
2
h d0
(3.1)
2
where d0 denotes the object distance and 2h indicates the shortest distance between the lenses [27].
The upper left graph of figure 3.5 shows the variation of this relative out-of-plane error as
a function of the off-axis position for various x d0 given by equation 3.1. The slope of
the graphs implies that the relative measurement error can be significantly improved within
0 x d0 0 1 by increasing the opening angle between the observation directions whereas
for larger x d0 this effect becomes weaker.
Translation
Angular displacement
12
12
h / d0 = 0.1
h / d0 = 0.2
h / d0 = 0.3
h / d0 = 0.4
h / d0 = 0.5
10
8
σ∆z / σ∆x
σ∆z / σ∆x
8
6
6
4
4
2
2
0
0
0.1
0.2
0.3
0.4
α=5
α = 10
α = 15
α = 20
α = 25
α = 30
α = 35
α = 40
α = 45
10
0
0.5
0
0.1
0.2
12
12
10
10
8
8
6
4
2
2
0
0.2
0.4
0.4
0.5
30
40
50
6
4
0
0.3
x / d0
σ∆z / σ∆x
σ∆z / σ∆x
x / d0
0.6
0.8
1
h / d0
0
0
10
20
α
F IGURE 3.5: Upper graphs: Variation of this relative out-of-plane error as a function of the off-axis
position for the translation method (left column) and the angular displacement arrangement (right column). Lower graphs: Dependence of the error for the principal observation rays from the viewing
angle.
For the centre of the field of view (x 0) the relative out-of-plane error as a function of the
opening angle reduces to equation 3.2. This dependence, which gives the appropriate opening
27
angle between the principle rays for a desired out-of-plane error, is shown in the lower left
plot of figure 3.5.2
1
σ∆z (3.2)
σ∆x d0 h
In case of the angular displacement arrangement the variation of the relative out-of-plane
error as a function of the x coordinate is given by expression 3.3. Compared with the translation set-up, the dependence is rather weak as can be seen by comparing the upper plots of
figure 3.5.
cos2 α
σ∆z σ∆x
1
2
sin α
2
sin α 1
x
d0 x
d0 2
2
cos2 α
(3.3)
cos2 α
1
2
By substituting x 0 in equation 3.3 it turns out that the relative out-of-plane error at the
optical axis is the reciprocal
of the tangents of the off-axis angle α, according to the following
equation, and for α 45 (opening angle θ 90 ) the out-of-plane error σ∆z becomes comparable with the in-plane error σ∆x at the centre of the field of view, see lower right plot of
figure 3.5.
σ∆z σ∆x
1
tan α
(3.4)
3.1.2 Scheimpflug condition
Unfortunately, large opening angles are often not feasible by using
standard
equipment
due to
the limited depth of focus. For a typical configuration with f 5 6 M 0 5 and λ 532 nm,
for example, the depth of focus is only δz 0 35 mm. To overcome these difficulties, the
conditions which improve the imaging when the object plane is tilted relative to the main plane
of the lens, will be briefly derived.
Assuming that the main-plane of the lens intersects with the object- and image-planes
at P1 and P2 respectively according to figure 3.6, it follows from geometrical considerations
that the distance
from the centre
of the lens to
the points of intersection can be expressed as
OP1 Z0 Y Z and OP2 z0 y z. Y Z denotes the coordinates of a non-axial objectpoint (measured
from the intersection of the optical axis of the lens with the object plane)
and y z are the corresponding
image coordinates. Using the definition for the transversal
magnification MT y Y z0 Z0 and the relation z ML Z MT2 Z for the longitudinal magnification, it turns out that the two points of intersection coincide.
OP2
zo y z
M T Zo M T Y ZMT2
2 As
Zo Y OP1
Z
(3.5)
the performance of the stereoscopic approach decreases when the object distance is large with respect
to the lens separation, the visual system of the human being changes from the stereoscopic approach to the
interpretation of the perspective (relative magnitude of known objects and their position relative to each other remote objects appear higher than close ones), shadow sizes, image contrast (absorption of light increases with
increasing distance), degree of accommodation and other aids.
28
pa
p
A
a
Z
y
z
-Y
A
-Z
zo
o
F IGURE 3.6: Scheimpflug condition: The image-, object- and main plane of the lens need to intersect
in a common line for ideal imaging (P1 P2 ).
Thus the image-, object- and main plane of the lens need to intersect in a common line
for ideal imaging. Although this relation was proven theoretically by the French mathematician Girard Desargues (1591–1661), and experimentally in 1901 by Jules Carpentier
(Patent GB 1139/1901), this condition is named after the Austrian aerial cartographer Theodor
Scheimpflug (1865–1911), who has derived this relations from the optical laws in his British
Patent (GB 1196/1904) from 1904. However, it should be noted that this Scheimpflug condition is only a necessary condition which requires in addition that the object distance d o is
larger than the focal length f of the imaging system. For practical purposes this condition
becomes significant for L δz cos α where α denotes the angle between the optical axis of the
lens and the light-sheet plane, δz is the depth of focus and L is the size of the field of view.
For small aperture or long focal length imaging, the adjustment of the image plane can be neglected for a wide range of opening angles between the two cameras in stereo-scopic imaging
configuration.
3.2 Evaluation of stereo-scopic recordings
The advantage of the angular-displacement technique with respect to the translation method
is that optical aberrations as well as intensity losses, due to the decrease of the modulationtransfer function towards the edges of the field of view, can be neglected. The inherent drawback, on the other side, is the characteristic variation of the magnification factor across the
field of view due to the oblique viewing direction. Beside a variation of the spatial resolution
across the field of view and a varying particle image density (number of particle image pairs
per unit area) additional errors have to be taken into account, especially when both cameras
are located on one side of the light-sheet. In this case, the size of each of a pair of measurement volumes considered for the calculation of the third velocity component is inversely
proportional to each other according to figure 3.4.
3.2.1 Determination of the mapping function
For conventional PIV (single light-sheet single camera configuration) the variation of the magnification factor over the field of view is usually negligible because the image-plane and the
29
main plane of the imaging system are parallel to the light-sheet. To ensure that the interrogation windows from each of a pair of stereoscopic recordings correspond to the same flow
region and also, that the magnification is constant for all positions within the image, the distortions along with possible differences in the field of view have to be carefully determined
before the line-by-line evaluation procedure, described in section 2.2, can be applied. For
this process images of a regular calibration grid (or dot pattern) with a known line-spacing
are usually acquired prior to the measurements while the calibration target must be perfectly
aligned with the centre of the light-sheet planes in order to avoid systematic measurement errors, compare section 3.3. According to the procedure proposed in [44] each image is inverted
and cross-correlated with an appropriate correlation mask (+ in the case of a calibration grid
and when a dot pattern is used) in order to determine the coordinates of the line crossing
with sub-pixel accuracy3 . After this step the imaging function between the image- and object
plane can be changed from a discrete into a continuous representation by means of fitting a
standard least squares surface to each of the image-object point sets, so that the first order
projection matrix can be calculated for each observation direction along with the translation,
rotation and magnification factor, [44].
a11 x a12 y a13
X
(3.6)
a31 x a32 y 1
a21 x a22 y a23
(3.7)
Y
a31 x a32 y 1
In order to take into account aberrations of higher order and other non-linear distortions,
the second order projection in form of equation (3.10) has to be applied. For this purpose
the coefficients of the first order projection matrix have to be used as an initial estimate to a
Levenberg-Marquart non-linear least squares fitting algorithm.
X
Y
a33
a11 x
a31 x
a21 x
a31 x
1
a12 y
a32 y
a22 y
a32 y
a14 x2
a34 x2
a24 x2
a34 x2
a13
a33
a23
a33
a15 y2
a35 y2
a25 y2
a35 y2
a16 xy
a36 xy
a26 xy
a36 xy
(3.8)
(3.9)
As both sets of transformation equations describe just a mapping between two planar domains without any three-dimensional information, the location of the image with respect to the
object has to be known in addition. This can be done either directly, by measuring the exact
camera positions relative to the centre of the field of view, or indirectly by using the following
set of equations, see [24] for mathematical details or [35] for the applicability in PIV.
X
Y
a11 x
a31 x
a21 x
a31 x
a12 y
a32 y
a22 y
a32 y
3 As
a13 z
a33 z
a23 z
a33 z
a14
a34
a24
a34
(3.10)
(3.11)
this method is quite time consuming, Hough transformation methods are usually applied in this thesis to
find the coordinates of the line crossings, see [7].
30
The appearance of the z-coordinate requires that the calibration procedure has to be repeated for different z locations in order to determine all unknown coefficients. This is especially useful when the position of the cameras is not accessible or for applications in water,
where the air-glass-water interface has to be taken into account.
3.2.2 Image warping
Once the reconstruction coefficients have been properly determined, the transformation equations can be applied to deform each acquired, single-exposed image in such a way that the
magnification factor is constant across the back-projected image and the field of view is identical for all acquired images. Using this technique optical parameters such as the focal length
and the magnification factor never need to be determined, and non-linear distortions introduced by the transparent test-section wall or other optical components in a non-collimated
beam can be accounted for, when the transformation equations are extended to higher order.
Furthermore, the simplified stereo-equations can be applied to calculate one three-component
displacement field from a pair of two-component sets as the magnification factor is constant.
∆x2 tan α1 ∆x1 tan α2
(3.12)
∆X
tan α2 tan α1
∆y2 tan β1 ∆y1 tan β2
∆Y
(3.13)
tan β2 tan β1
∆x1 ∆x2
(3.14)
∆Z
tan α2 tan α1
As the grey-level values are only defined at integer pixel locations within the image, a spatial
transformation based on equation (3.6) to (3.11) causes a mapping into locations for which
no grey levels are defined. Thus, it becomes necessary to infer what the correct grey-level
values at those locations should be, based on the grey-level values at integer pixel coordinate
locations.
To validate the performance of different image interpolation methods, two particle image
fields, created from one measured (or simulated) single
exposed grey-level pattern by using
the transformation
(a11 0 999, a22 1, other coefficients zero) for the first
coefficients
image and (a11 1, a22 1, other coefficients zero) for the second image, where crosscorrelated. As the true displacement linearly varies by two pixels over a range of 2000 pixels
(reference line in figure 3.7) a correlation between the two back-projected images will yield
information about artefacts due to non-properly chosen sub-pixel increments (number of grid
points per pixel considered for calculating the grey-value in the back-projected image) or due
to interpolation procedures which have been chosen improperly. The simplest schemes based
on nearest-neighbour approach are easy to implement but undesirable artefacts like distortions
can be hardly avoided and systematic errors appear as indicated in figure 3.7.
The so called de-sampling method yield better results, provided each pixel is subdivided
in at least four sub-pixel or the bilinear interpolation where the grey levels of the four nearest
integral neighbours of a non integral coordinate are consulted to determine the appropriate
value. More sophisticated approaches like fitting a sin x x type surface through a much larger
number of neighbours (cubic convolution interpolation method) yield much smoother results
but at the cost of computational time.
31
Experimental
2
1.5
1.5
1
Nearest neighbour
2 × 2 Desampling
4 × 4 Desampling
8 × 8 Desampling
16×16 Desampling
Bilinear interpol.
Reference
0.5
0
0
250
500
750
∆x [pixel]
∆x [pixel]
Numerical
2
1
0.5
1000
x [pixel]
0
Reference
16 × 16 Desampling
Bilinear interpolation
0
250
500
750
1000
x [pixel]
F IGURE 3.7: Systematic deviation of the measured displacement ∆x for different interpolation
schemes. A linearly varying particle displacement appears as a step function after the image de-warping
has been performed whereby the step-size decreases with increasing performance of the image interpolation scheme.
3.2.3 Vector field warping
As the proper de-warping in the image space is time consuming and requires a redistribution
of the original images, which does not increase confidence4 , the de-warping can be performed
in the vector space as soon as the conventional line-by-line interrogation procedure has been
applied. Using this approach the equally spaced grid points in the object space are transformed
into the image space according to the transformation equations (3.6) and (3.7), and the velocity
at each image grid point is calculated by means of linear interpolation. In order to minimise or
avoid the interpolation procedure, the acquired images can either be evaluated with a smaller
step-size or interrogated at the proper positions given by the grid points in the object space.
The inherent drawback of this approach is the fact that the magnification factor varies over
the field of view as well as the measurement error and the detectability [22, 23]. Furthermore,
if the camera system is symmetrical and located on one side of the light-sheet (left image
in figure 3.4), the interrogation windows from each of a pair of stereoscopic images backprojection in the physical space may differ substantially in size, which means that in effect
different flow volumes are considered for the calculation of the third velocity component 5 .
The significance of this statement will be considered in the section 3.3.
3.3 Calibration validation
To ensure that the interrogation windows from each of a pair of stereoscopic images correspond to the same region of flow, the properties of the imaging must be constant, when the
4 An
initially circular particle image for example may appear elliptically after the transformation has been
performed. This artifact can bias or lower the principal measurement accuracy.
5 In the case of a symmetrical setup with a light-sheet between the two cameras (dashed configuration in figure
3.2 and right image in figure 3.4) the fluid elements considered for the calculation of the third velocity component
are equal in size but the spatial resolution remains a function of the image location.
32
measurements take place, and identical with the calibration condition. Unfortunately, interrupting the experiment and entering the test section of a wind-tunnel for the calibration procedure may lead to different boundary conditions, and mechanical or thermal variation during
the experiment can be hardly avoided, especially in industrial environments. To demonstrate
the effect of non properly aligned calibration grids or poorly performed evaluation, a pure 2
dimensional flow with a Rankine vortex was simulated, see upper right image of figure 3.8,
with a tangential particle image displacement of 4 pixel and the effect of an artificial translation was examined according to equation 3.14. The 512 512 pixel2 images were analysed
with 32 32 pixel2 interrogation windows, each of them containing approximately 25 particles
generated at random positions within the Gaussian shaped light-sheet of finite thickness. The
size of the particle images defined by the e 2 diameter is 2 4 pixel and their intensity by the position within the light-sheet. The curvature of the particle trajectory is taken into account, such
that particle images close to the core of the vortex actually orbit the core at the same radius
but centrifugal forces are not considered. Before the analysis has been performed the simulated particle image pattern has been duplicated and properly shifted in opposite directions by
choosing the values of the coefficient a13 and a23 and the sub-pixel increment properly. The
opening angle between both observation directions is 90 and the object distance is assumed
to be large so that the perspective error is negligible.
x16_avg.sm
−untitled−
−untitled−
F IGURE 3.8: Spatial distribution of the out-of-plane displacement introduced by non properly aligned
calibration grid or poorly performed evaluation (average over 20 fields). The line spacing in the contour plots is incremented in steps of 0 1 pixel and different line styles indicate different out-of-plane
directions.
The range of the out-of-plane displacements as a function of the in-plane shift depends on
both, magnitude and direction and can be quite large with respect to all other error sources in
PIV according to table 3.1. Figure 3.8 shows the spatial distribution of the error. The shift is
33
In-plane shift [pixel] Out-of-plane error [pixel]
∆x 8 ∆y 0
0 520 εZ 0 482
∆x 16 ∆y 0
0 688 εZ 0 705
∆x 32 ∆y 0
1 049 εZ 1 109
∆x 8 ∆y 8
0 576 εZ 1 147
∆x 16 ∆y 16
0 824 εZ 1 526
∆x 32 ∆y 32
1 120 εZ 2 076
∆x 0 ∆y 8
0 436 εZ 1 076
∆x 0 ∆y 16
0 602 εZ 1 611
∆x 0 ∆y 32
0 969 εZ 2 049
TABLE 3.1: Artificial out-of-plane displacements εZ as a function of the magnitude and direction of
the in-plane misalignment. Maximum in-plane displacement: 4.2 pixel
∆x 16 pixel for the upper left image, ∆y 16 pixel for the lower right and 16 pixel in each
direction for the lower left sample. A surprising result is the completely different symmetry as
only the displacements in x-direction account for the out-of-plane component. This explains
the different εZ for identical shifts in orthogonal spatial direction, see table 3.1.
Under real flow conditions this artefact is usually less visible especially for complex flows,
due to the superimposed fine scale flow structure and noise, so that care must be taken before
analysing the results in terms of fluid-mechanics. To validate the boundary conditions a crosscorrelation between the de-warped single exposed images from the camera pair in stereoscopic
imaging configuration, acquired at the same instant of time, has to be calculated in order to
obtain the local misalignment between the images. As the illuminated particles are almost
identical, especially for pure forward, backward and 90 degree observation direction6 , and
their different image position has been corrected by the applied warping, based on the analysis
of the calibration grid, the output must be a zero-displacement field. Any serious misalignment
due to translation, rotation or deformation (magnification change of one image) will become
obvious in the displacement field and can be corrected by calculating the mapping function between both samples and combining the coefficients of this transformation with the coefficients
determined with the calibration grid analysis [16]. This non-intrusive approach, based on the
particle image fields itself, can be applied to each acquired image pair to guarantee that everything involved in the measurement is unaffected by wind tunnel vibrations, thermal distortions
e.g. for the duration of the measurement [16]. This becomes important in noisy environments
or for long acquisition times. Unfortunately, an equal displacement of both cameras can not
be detected by using this approach, except if a mark is present in both images such as a bright
model reflection or a tiny dark shadow within the light-sheet.
6 Due
to the finite aperture of the imaging system this approach is also feasible for non symmetrical observation directions or forward/backward configurations but the quality is slightly reduced due to possible unpaired
particle images.
34
3.4 Application: Coherent structures in boundary layers
In this section, the application of the stereo-scopic PIV to turbulent flows will be outlined for
two reasons. First, most of the real flows are turbulent. Second, turbulent flows are highly
instationary and three dimensional with a large range of scales. The statistical description of
turbulence, in form of the Reynolds equation in boundary layer approximation, implies that a
strong correlation between the velocity fluctuations parallel to the mean flow and those in the
direction normal to the wall is of primary importance for the turbulent transport of mass and
momentum in wall-bounded flows. Since all vortices which do not have their axes of rotation
exactly perpendicular to the wall will induce such transport, a number of partially contradicting vortical models have been proposed in the past to explain qualitatively the outflow of low
momentum fluid by means of inclined vortex loops, referred as hairpin, horseshoe, U-shaped
or lambda vortices in the literature. The idea, which intends to explain the complex turbulent
motion in terms of relatively simple vortical structures is the basic purpose of the concept of
coherent flow structures. As order can always be observed when the convection terms in the
equation of motion are dominant over the production, diffusion and dissipation terms, there
is no doubt about the existence of organized structures. However, there is a general controversy about which ones are fundamental and which ones are only secondary, which ones are
dominant and which ones are irrelevant. To examine the physical process associated with
the production and transport of turbulence in wall bounded flows, a fully developed turbulent
boundary
layer flow
along a flat plate was investigated in stream-wise span-wise planes at
20 and Reθ 7800 by using a high resolution stereo-scopic PIV system. Of primary
y
interest was the structural features of the coherent flow structures, such as their average size
and shape, but also their intensity, dynamics and mutal interaction.
The experimental investigation was performed
with a Nd:YAG laser system with approximately 255 mJ output energy per pulse at λ 532 nm and the aquisition of the data was
done with a pair of high resolution PCO cameras with 1280 pixel by 1024 pixel resolution.
The mean observation distance was 840 mm and the opening angle between the left and right
camera systems was 94 4 to resolve the out-of-plane motion properly. This is important as
the out-of-plane
component is required to calculate the dominant Reynolds stress component
τturb
ρuv. For the displacement estimation with sub-pixel accuracy, a two-dimensional
Gaussian peak-fit routine was applied and by applying the following set of band-pass and
gradient filters (1 ∆x 12 pixel; 4 ∆y 4 pixel and ∆xi ∆xi 1 5 pixel), the number of correct measurements was on average above 99 9 %. The tracer particles generated for
this experiment were delivered from a smoke generator which produces high concentrations
of monodisperse poly-ethylene-glycol particles with a mean diameter of 2 µm, see [19]. To
obtain ideal conditions for accurate PIV measurements, the closed circuit wind-tunnel was
completely seeded and continuously operated until no seeding inhomogeneities could be observed at all.
To illuminate the complexity of the turbulent motion two typical instantaneous velocity
fields are shown in figure 3.9 and 3.10 with large uv 2 intensities. However, it should
be noted that these structures do not occur frequently [17]. The
lower image of figure 3.9
shows a representative instantaneous flow field measured at y
20 whereby the local mean
velocity of 1 4 m/s was subtracted in order to display the fluctuating velocities u (stream-wise)
and w (span-wise) clearly. The mean flow direction is from left to right and the contours
behind the velocity vectors indicates the out-of-plane motion (solid line: v 0, dashed line:
35
300
ejection
250
lifting streak
200
X
+
150
100
lifting streak
lifting streak
50
0
0
50
100
150
200
250
0
50
100
150
200
250
300
350
400
450
500
550
300
350
400
450
500
550
300
250
y [???]
x [???]
200
X
+
150
100
50
0
Z
+
F IGURE 3.9: Distribution of the Reynolds stress component uv (top) and velocity vector field (bottom).
Solid line: v 0, uv 0. Dashed line: v 0, uv 0. The white lines in the lower image indicate the
location of the low speed streaks.
??]
x [???]
v 0). Predominant structures are the elongated low-speed fluid regions which are convecting downstream with approximately half the local mean velocity. It can be seen that the u
component of the velocity is frequently low (u 0) in regions where the fluid moves away
from the wall (v 0) and the stream-wise extend of the wall-normal motion is quite short
with respect to total streaks length as stated before. This illustrates that in general no strong
stream-wise vortices flank the low speed regions. Only a weak vortex-like flow motion can
be seen as the lifting of low speed regions into higher momentum flows must be compensated
36
300
ejection
250
ejection
ejection
200
X
+
150
100
ejection
lifting streak
50
sweep
0
0
50
100
150
200
250
0
50
100
150
200
250
300
350
400
450
500
550
300
350
400
450
500
550
300
250
y [???]
x [???]
200
X
+
150
100
50
0
Z
+
F IGURE 3.10: Distribution of the Reynolds stress component uv (top) and velocity vector field (bottom). Solid line: v 0, uv 0. Dashed line: v 0, uv 0. The black and white circles in the lower
image indicate counter rotating vortex pairs.
??]
x [???]
due to continuity. However, it seems that these vortices are only a secondary motion which
is produced locally. The upper image of figure 3.9 shows the spatial distribution of the instantaneous Reynolds stress component uv to highlight the importance of low-speed streaks
for the production of turbulence. Figure 3.10 does not reveal any extended streak pattern but
the intensity of the Reynolds stress component is very large as can be estimate from the upper
image. However, by visual inspection of the velocity field it can be seen that the regions of
strong production is associated with counter rotating vortex pairs, indicated by the circles.
37
Chapter 4
Multiplane Stereo Particle Image
Velocimetry
The Stereo-scopic Particle Image Velocimetry (PIV) described in the previous chapter has
become a widely used technique for investigations where the spatial distribution of the velocity is needed to understand the flow physics. Unfortunately, the distribution of the velocity
within one single plane, captured at one instant in time, does not yield always the information required to answer fluid-mechanical questions. In case of turbulent flows, for example,
space-time correlations together with the spectrum of the fluctuations are important in the statistical theory of turbulence as there is a link between these quantities and the decomposition
of the flow field into a series of harmonic modes. For the identification of the most energetic velocity structures, the proper orthogonal decomposition is a well suited technique but
in general it requires spatial and temporal information about the flow field. The eigenvalues of
the velocity gradient tensor are also increasingly used to describe various flows because these
quantities play a role in the theory of dynamical systems and topology. The vorticity vector
(all components) is another important quantity due to its Galilean invariance as well as the acceleration in its Lagrangian and Eulerian form in order to study the formation and interaction
processes of moving flow structures. To retrieve the desired information about the spatiotemporal flow unsteadiness other sophisticated imaging techniques have been developed such
as the phase-conjugate holographic system, the scanning light-sheet PIV, or the 3D particle
tracking technique [3, 5, 39]. Despite the high level of development of each cited technique
the information about the spatial-temporal flow unsteadiness remains limited and their implementation remains challenging due to their complexity, costs and manpower demand [11].
To overcome these limitations a stereoscopic PIV based technique has been developed which
is well suited to determine many fluid-mechanical quantities with high accuracy and spatial
resolution [18]. This technique is reliable, robust and easy to handle. Furthermore it is based
on standard PIV equipment and evaluation procedures so that available PIV systems can be
easily extended. In the following sections this technique will be referred as multiplane stereo
PIV.
4.1 Principles
The multiplane stereo PIV system consists of a four-pulse laser system delivering orthogonally
polarised light, two pairs of high resolution progressive scan CCD cameras in an angular
38
imaging configuration with Scheimpflug correction, two high reflectivity mirrors and a pair
of polarising beam-splitter cubes, each consisting of two cemented right-angle prisms with an
appropriate interference coating in between, see figure 4.1.
After the illumination of the tracer particles with orthogonally linearly polarised light, the
polarising beam splitter cube (7) separates the incident wave-front scattered from the particles
into two parts. The light which passes straight through the cube emerges linearly polarised
with the plane of the electric field vector parallel to the plane of incidence defined for the
multi-layer film (p-polarised). The light emerging from the cube at right-angles to the incident wave front (having been reflected inside the cube at the dielectric multi-layer film) is
oriented orthogonally to the plane of incidence (s-polarised). The separation based on polarisation works perfectly as long as the radius of the spherical particles is comparable to the
wavelength of the laser light and the observation direction is properly aligned to the direction
of the polarisation vector [4]. In order to avoid contrast reduction in the image plane due
to background light emerging from the opposite beam-splitter-camera system, one surface of
each cube is covered with an absorbing material (8).
α1 α2
6
5
7
8
3
4
2
s-polarized light
p-polarized light
1
F IGURE 4.1: Schematic setup of the recording system. 1-4 digital cameras, 5 lens, 6 mirror, 7 polarising beam-splitter cube with dielectric coating between the two right-angle prisms, 8 absorbing material,
α opening angle.
Before entering the lens the s-polarised light coming from the polarising beam-splitter
cube is reflected to achieve identical orientation of the image plane for all cameras. This
simplifies the matching of the four observation areas and saves computing time during the
evaluation of the images. The polarisation characteristics of these mirrors is not important as
only the intensity of the scattered light is needed after the polarising beam-splitter cube has
been passed.
The orthogonally linearly polarised light emerging from the polarising beam-splitter cube
is finally recorded by means of high resolution CCD cameras in an angular imaging configuration. Due to the oblique viewing direction and the limited depth of focus of the lens, the
image plane, the main plane of the lens and the object plane need to intersect in a common line
(Scheimpflug condition) according to section 3.1.2, in order to obtain focused particles everywhere within the image plane. Therefore, each camera lens must be connected to a specially
designed one-axis tilt-adapter (Scheimpflug-adapter). The axis of rotation should coincide
with the centreline of the CCD sensor to ensure that all particle images along this axis remain
39
in focus under rotation. This simplifies the installation of the system and the focusing process
since the image location of the centreline in object space and the opening angle between corresponding camera pairs remain constant under Scheimpflug adjustment. For magnification
and field of view adjustments (necessary for maximising the amount of stereo information) all
Scheimpflug-adapters should be mounted on a two-axis linear translation stage which allows
high precision translations by thumb screws. In order to simplify the adjustment procedure
without restricting the flexibility of the system the left and right recording systems can be
connected to different base-plates with individual rotation stage.
4.2 Four-pulse-laser System
For the illumination of the tracer particles the beams of four independent laser-oscillators
need to be combined in such a way that the linearly polarised light-sheets can be positioned
independently with respect to each other. This can be easily and precisely done by the fourpulse system shown in figure 4.2. The lasers consist of a Neodymium-Yttrium-AluminumGranat rod embedded in an unstable resonator with a variable reflectivity output mirror (the
transmission decreases from the centre to the edge to eliminate the intensity maxima of higher
orders introduced by the unstable resonator) and a white light flash-lamp for the excitation
of the crystal-atoms. The monochromatic unpolarised radiation spontaneously emitted by the
laser material (1) will be linearly polarised when it emerges from the dielectric Glan-Laser
polariser (6) and circularly polarised behind the λ 4 retardation plate (5). The direction of
propagation of the circularly polarised beam is reversed by reflection in a mirror (2) and thus
the sense of the circular polarisation as well (due to the 180 phase shift induced by the mirror)
before the second transformation into linearly polarised light. Thus, the linearly polarised
wave of the reflected beam, as it finally emerges from its second path through the retarder
(5), is orthogonal to the incident linearly polarised wave and can be rejected by means of
the dielectric polariser (6) constructed from an air-spaced right-angle prism pair with parallel
optical axis. In this configuration the polariser λ 4 plate combination acts as an isolator (or
closed switch) when the angle between the plane of linear polarisation and the crystalline
optical axis equals exactly either 45 (for other angles the transformation is from linear to
elliptical polarisation due to the different amplitudes of the ordinary and extraordinary beam
behind the retardation plate). For the stimulated emission of radiation the electro-optical block
(4) can be made equivalent to a λ 4 plate by appropriate choice of block length, bias voltage
and suitable orientation with respect to the quarter-wave plate such that the beam can pass the
Glan-Laser polariser when the population inversion reaches its maximum. Using this optical
arrangement the laser power can be increased by reducing the pulse duration and the output
energy of the laser is adjustable by changing the time delay between the flash-lamp and the
Pockels cell bias voltage.
In figure 4.2 the linearly orthogonally polarised infra-red beams released by the resonator
pair are combined using a dielectric polariser at the Brewster angle (8a) which reflects the
p-polarised light partially at every dielectric interface within the multi-layer coating while
transmitting the s-polarised light with almost no reflection. A retardation plate behind the
Brewster window (5) transforms both linearly polarised beams into circularly polarised light
before they enter a properly cut and temperature stabilised highly doped KD*P crystal (10) for
polarisation selection and generation of the second harmonic (532 nm) from the fundamental
wavelength (1064 nm). In order to separate the two wavelengths, a high energy harmonic
40
4
2
7
1
5 6
3
12
12
7b
11
5
8a
12
10
9
11 8c
8b
F IGURE 4.2: Four-pulse four frequency doppler laser system. 1 Pump cavity, 2 Full reflective mirror,
3 Partially transmitting mirror, 4 Pockels cell, 5 λ 4 retardation plate, 6 Glan-Laser polariser, 7 Mirror,
8 Dielectric polariser, 9 Dichroic mirror, 10 Frequency doubler crystal with phase angle adjustment, 11
λ 2 retardation plate, 12 Beam dump.
separator (9) is used consisting of a specially coated substrate which reflects the harmonic
Rmax at 532 nm) and transmits the fundamental (Tmax at 1064 nm) wave. As the linearly
polarised light emerging from the frequency doublers possesses the same state of polarisation,
a λ 2 retardation plate (11) has to be inserted before the superposition of the four beams by
means of another dielectric polariser (8c) can take place. As the orientation of the frequency
doubler (10) affects the efficiency of the second harmonic generation and may change the
angle between the incoming and outgoing beam, problems such as different output energy
may occur when using a system where two beams pass the same frequency doubler crystal.
4.2.1 Performance of spatial light-sheet separation
The appropriate method of adjusting the displacement between the orthogonally polarised
light-sheets depends on the desired distance. Small separations between the orthogonally
polarised light-sheet pairs (up to a few millimetres) can be generated by a simple rotation of
mirror 8c in the re-combination optics around the axis perpendicular to the laser-beam plane.
This is possible as the divergence between the s- and p-polarised beams is negligible when
the separation is small with respect to the distance from the measurement position to mirror
8c. A translation of mirror 8c on the other hand cannot be applied in order to separate the
orthogonal polarised beams as the focal lens in the light-sheet optics will superimpose parallel
rays at the focal point. The relative separation between the beams can be easily controlled by
means of a target located at the measurement position which is slightly tilted with respect to
the light-sheets in order to increase the resolution.
For a wider range of light-sheet spacings (up to a few cm) and independent positioning of
both beam-pairs, it is useful to remove mirror 7b along with the beam dump (12) such that
two spatially separated laser beams with orthogonally polarised radiation emerge. Using two
separate light-sheet-optics (one for each polarisation) each with a 45 mirror behind the last
lens, all positions are possible by moving the mirrors. Once calibrated, the actual position of
each pair of light-sheets is given by a micrometer scale. For the conservation of the polarisation the mirror needs a dielectric broad-band coating optimised for 532 nm and 45 angle
41
of incidence to provide a 99 % reflectance in both s- and p-planes also when the angle is not
exactly 45 . In addition, a high damage threshold ( 10 J/cm2 for 15 ns pulse duration) is
necessary to minimise the probability of laser induced damage. Due to the thermal sensitivity
of the laser material and the unstable resonator, the previously mentioned adjustments should
be performed under thermal equilibrium conditions.
4.2.2 Generation and controlling of the timing sequence
To trigger all necessary components quickly and easily a 16 channel sequencer-board with an
output frequency between 0.01 Hz and 1 MHz can be applied. Figure 4.3 shows the userinterface for the generation and operation of the appropriate timing-sequence.
F IGURE 4.3: Window-interface to select and control the laser- and camera-timing
The first two sliders located at the top of the window control the pulse separation for
both laser pairs separately whereas the third one determines the time-interval between the first
pulses of each laser pair. Thus, all pulse combinations described in section 4.3 for multipleplane recording can be easily generated. Slider 4 to 7 from the top specify the delay between
the flash-lamps and the Pockels-cells in order to adjust the output energy of each laser individually and with slider 8 the time for the read-out of the first camera images can be adjusted
with respect to the first laser pulse. In the centre menu in the lower half of figure 4.3 the specifications of the laser-system (Quantel, BMI) can be chosen as well as the mode of operation
(high or low energy, flash-lamps only to keep the laser material at the appropriate temperature,
laser off). All other possibilities like cameras only (for field of view adjustments) or cameras
together with only one oscillator (for checking the separation by means of the polarisation)
can be selected in the lower left menu by a mouse click on the right bottom. After selecting
42
the appropriate set of parameters the sequence has to be sent to the electronic board and can
be activated using the start bottom at the lower right of figure 4.3 and afterwards terminated
with the neighbouring stop bottom. This user-friendly interface enables the user to perform
all alignments and calibrations within reasonable time.
4.3 Modes of Operation
Once installed the multiplane stereo system is well suited to determine different fluid-mechanical
quantities simply
the time sequence or light-sheet position. For constant pulse
by changing
separation (∆t t2 t1 t3 t2 t4 t3 ) and overlapping light-sheets, a time sequence of
three velocity fields can be measured at any repetition rate by cross-correlating the first acquired grey-level distribution with the second, the second with the third and the third with the
grey-level distribution from the last illumination,
4.4. By increasing the time delay
see figure
between the second and third illumination (∆t t2 t1 t4 t3 t3 t2 ) the first order estimation of the acceleration field in its Lagrangian and Eulerian form can be calculated in order
to study the dynamic behaviour and the interaction processes of moving flow structures [15].
The Lagrangian acceleration of a moving fluid-element is defined as the temporal derivative
of the velocity and can be simply implemented as a difference quotient
ui j t
ai j t
(4.1)
t
t
whereby
t and t are defined
as the mean between the corresponding illuminations e.g. t
t4 t3 2 respectively and ui j denote the velocity of a fluid volume at any
t2 t1 2 and t
time t (the indices i j denote
the discrete location in the measurement plane). Equation (4.1)
states that in case that ui j t is continuous in t t t and differentiable in t
t t there
exists a t in the specified time-interval for which the acceleration is given by expression 4.1.
The exact time t is unknown except for the case of a linear variation of the velocity during the
time interval. In case of the Eulerian acceleration
ui j t
∂u x t a x t
u x t ∇u x t
∂t
the first term on the right hand side can be approximated by
(4.2)
∂u x t ∂t
u x t
t
u x t
t
with the fluid velocities u x t and u x t measured at x and at times t and t respectively.
The nonlinear term on the right hand side of equation 4.2 can be approximated to:
u x t
∇u x t
∂ui x t
1
∂ui x t
uj x t
uj x t
ei
∑
2ij
∂x j
∂x j
where ei with (i 1 2 3 indicates the three unit vectors. By fitting each component of the
spatial velocity gradient at a given point to its two adjacent velocity vectors along the axis of
the given component the partial velocity gradients can be determined from a pair of velocity
43
Intensity
F IGURE 4.4: Timing diagram for the temporally
separated determination of
all three velocity components. Different shading of
the light-sheet profile indicates different states of polarisation.
t
t1
z
t2
t3
t4
fields. The two products containing the partial velocity gradients are determined from each of
the velocity fields and are averaged together, for details see [15].
Besides of measuring the acceleration the temporal behaviour of the moving flow structures can be estimated as well which yields information about the formation and decay of flow
structures. Furthermore, three dimensional space-time correlations can be
simply
measured
by increasing the time
delay between
a pair of images being acquired (∆t t 2 t1 t4 t3
n
t3 t2 ) with ui x y
∑i 1 ui x y t n.
m
Ri j ∆x ∆y ∆t
n
∑ u
∑
n
i
i 1j 1
u j
ui u j
i 1j 1
m
∑ ∑ ui
ui
2
m
∑
∑ u
n
ui
i
i 1j 1
(4.3)
2
When the light-sheet pairs with equal polarisation are spatially separated as indicated in
figure 4.5 further important information about the flow field can be achieved. For small separations (or partially overlapping light-sheets) the multiplane stereo PIV technique allows the
determination of all three components of the vorticity vector – PIV and Stereo-PIV offer only
one component – or the divergence of the velocity field, according to equations 4.4 and 4.5.
rot u
∇
u
∂w
∂y
∂v
ex
∂z
∂u
∂z
∂w
ey
∂x
∂v
∂x
∂u
e
∂y z
(4.4)
∂u ∂v ∂w
(4.5)
∂x ∂y ∂z
This
requires that the differently polarised light-pulses are fired simultaneously (t 1 t3 and
t2 t4 ) as shown in figure 4.5.
The same data can be analysed in terms of critical point theory which displays mathematically the important features of a given set of first order differential equations without having
the exact solutions. This analysis, which allows vortex identification to be made more reliable,
involves locating certain critical points, linearising the equation in its vicinity and examining
the topological features of the solution trajectories. This leads to the classification of possible
critical points which may be displayed in a (P,Q,R)-diagram [36].
For calculating gradients normal to the light-sheets (i.e. ∂ui ∂z with i 1 2 3) it is obvious
that either the forward- or backward-difference scheme has to be applied as other extrapolation
techniques require three or more grid points in each spatial direction. This is not a disadvantage (especially in terms of spatial resolution) as long as the spacing between the light-sheets
divu
∇ u
44
Intensity
∆t
∆ z
z
t1
t
F IGURE 4.5: Timing diagram for the simultaneous
determination of all three
velocity components in
spatially separated planes
t2
is sufficiently large so that the data entering in the formula is not correlated. For the in-plane
derivatives (e.g. ∂ui ∂x and ∂ui ∂y with i 1 2 3) the three or even five point difference
scheme is more suited especially for 50 percent interrogation window overlap analysis in order to reduce the truncation or random errors in the values of the velocity function [43]. By
increasing the spacing ∆z between the light-sheet pairs and varying the time delay between
the second and third illumination, all components of the four dimensional spatio-temporal
correlation tensor Ri j can be measured according to
m
Ri j ∆x ∆y ∆z ∆t
n
ui u j
i 1j 1
m
∑
∑ u
n
i 1j 1
i
∑ ∑ ui
ui
2
m
∑
u j
∑ u
n
i 1j 1
i
ui
(4.6)
2
Furthermore
the
direction
of
a
vortex
which
crosses
these
planes
can
be
determined
for
∆z
0
and ∆t 0 (this is especially valuable for aircraft wake vortex investigations) and many other
quantities not mentioned here which are helpful for the understanding of fluid-mechanical
problems. For applications the reader should consult the original publications, see [18, 21, 17,
30, 13] for example.
45
Bibliography
[1] A DRIAN RJ (1988): Statistical properties of particle image velocimetry measurements
in turbulent flow. Laser Anemometry in Fluid Mechanics III. pp 115–129
[2] A DRIAN RJ (1995): Limiting resolution of particle image velocimetry for turbulent flow.
Advances in Turbulence Research, Proc. 2nd Turbulence Research Assoc. Conf. Pohang
Inst. Tech. pp. 1–19
[3] BARNHART DH, A DRIAN RJ, PAPEN , GC (1994): Phase-conjugate holographic system for high-resolution particle image velocimetry. Appl. Opt. 33, 7159–7170
[4] B ORN M, W OLF E (1985) Principles of Optics, Press, 6th edition
[5] B R ÜCKER C (1997) 3D scanning PIV applied to an air flow in a motored engine using
digital high-speed video. Meas. Sci. Technol. 8 1480–92
[6] E CHOLS WH, YOUNG YA (1963) Studies of portable air-operated aerosol generators.
NRL Report 5929, Naval Research Laboratory, Washington D.C.
[7] Ehrenfried K. (2002) Processing calibration-grid images using Hough transformation.
Meas. Sci. Technol. 13, 975–983
[8] F INCHAM A, D ELERCE G (1999) Advanced optimization of Correlation Imaging Velocimetry algorithms. 3rd International Workshop on PIV, University of California, Santa
Barbara, USA, Sept. 16-18
[9] G AD - EL -H AK M (2001) Flow Control: The Future, Journal of Aircraft 38
[10] H ART DP (2000) PIV Error Correction. Selected papers from the 9th Int. Symp. on
Appl. of Laser Techn. to Fluid Mech., Lisbon, Portugal, June 13-16, Springer
[11] H INSCH K (1995) Three-dimensional particle velocimetry. Meas. Sci. Tech. 6, 742–753
[12] H UANG HT, F IEDLER HE, WANG JJ (1993) Limitation and improvement of PIV. Experiments in Fluids. 15, 168–174 and 263–273
[13] H U H, S ATA T, KOBAYASHI T, TANIGUCHI N, YASUKI M (2001) Dual-plane stereoscopic particle image velocimetry: system set-up and its application on a lobed jet mixing flow. Exp. Fluids. 31, 277–293
[14] ISO 13320-1 (1999) Particle size analysis – Laser diffraction methods. First edition,
1999-11-01, Reference number: ISO 13320-1:1999(E)
46
[15] JAKOBSEN ML, D EWHIRST TP, G REATED CA (1997) Particle image velocimetry for
predictions of acceleration force within fluid flows. Meas. Sci. Technol. 8 1502–16
[16] K ÄHLER CJ, A DRIAN RJ, W ILLERT CE (1998) Turbulent boundary layer investigations with conventional and stereoscopic PIV. 9th Int. Symp. on Appl. of Laser Techn.
to Fluid Mech., Lisbon Portugal, paper 11.1, June 13–16
[17] K ÄHLER CJ (2003) Investigation of the spatio-temporal flow structure in the bufferregion of a turb. boundary layer by means of multiplane stereo PIV. Exp. Fluids. 34
[18] K ÄHLER CJ, KOMPENHANS J (2000) Fundamentals of Multiple Plane Stereo PIV. Exp.
Fluids. [Suppl.], S70–S77
[19] K ÄHLER CJ, S AMMLER B, KOMPENHANS J (2002) Generation and Control of Particle
size distributions for Optical Velocity Measurement Techniques in Fluid Mechanics. Exp.
Fluids. 33, 736–742
[20] K ÄHLER CJ, S CHOLZ U (2003) Investigation of laser-induced flow structures with timeresolved PIV, BOS and IR technology. 5th Int. Symp. on Particle Image Velocimety,
Busan, Korea, September 22–24, Paper 3223
[21] K ÄHLER CJ, S TANISLAS M, D EWHIRST TP, C ARLIER J (2001) Investigation of the
spatio-temporal flow structure in the log-law region of a turbulent boundary layer by
means of multi-plane stereo particle image velocimetry. Selected Papers from the 10th
Int. Symp. on Appl. of Laser Techn. to Fluid Mech., Lisbon, Portugal, July, Springer
[22] K EANE RD, A DRIAN RJ (1990) Optimization of particle image velocimeters. Part I:
Double pulsed systems. Meas. Sci. Technol. 1, 1202–1215
[23] K EANE RD, A DRIAN RJ (1992) Theory of cross-correlation analysis of PIV images.
Appl. Sci. Res. 49, 191–215
[24] K LEIN F (1925) Elementarmathematik vom höheren Standpunkte aus. Bd. 2, 3rd edition,
Springer, Berlin
[25] KOMINE H (1990) System for measuring velocity field of fluid flow utilising laserSoppler spectral image converter. US Patent 4 919 536
[26] L ASKIN S (1948) Submerged Aerosol Unit, AEC Project Quaterly Report UR-38, Univ.
of Rochester.
[27] L AWSON NJ, W U J (1997) Three dimensional particle image velocimetry: error analysis of stereoscopic techniques. Meas. Sci. Technol. 8, 894–900
[28] L ECORDIER B,L ECORDIER JC, T RINITE M (1999) Iterative sub-pixel algorithm for
the cross-correlation PIV measurements. 3rd International Workshop on PIV, University
of California, Santa Barbara, USA, Sept. 16–18
[29] M ELLING A (1997) Tracer particles and seeding for particle image velocimetry. Meas.
Sci. Technol. 8 261–304
47
[30] M ULLIN JA, DAHM W (2002) Highly-resolved three dimensional velocity measurements via Dual-Plane Stereo Particle Image Velocimetry (DSPIV) in turbulent flows.
AIAA Paper No. 2002-0290
[31] N ERGER D, K ÄHLER CJ, R ADESPIEL R (2003)
Zeitaufgelöste PIV-Messungen
an einem schwingenden SD7003-Profil bei Re
6 104 . Lasermethoden in der
Strömungsmechanik, GALA e.V., Braunschweig, Germany
[32] R AFFEL M, W ILLERT CE, KOMPENHANS J (1998) Particle Image Velocimetry – A
practical Guide. Springer Verlag
[33] RONNEBERGER O, R AFFEL M, KOMPENHANS J (1998) Advanced evaluation algorithms for standard and dual plane particle image velocimetry. 9th Int. Symp. on Appl.
of Laser Techn. to Fluid Mech., Lisbon Portugal, Paper 10.1, June 13–16
[34] S CHODL R (1974) A laser fual beam method for flow measurements in turbomachines.
ASME Paper 74-GT-157
[35] S OLOFF SM,A DRIAN RJ, Z IU Z-C (1997) Distortion compensation for generalized
stereoscopic particle image velocimetry. Meas. Sci. Technol. 8 1441–54
[36] P ERRY AE (1984) A study of degenerate and nondegenerate critical points if threedimensional flow fields. Deutsche Forschungs- und Versuchsanstalt f”ur Luft- und Raumfahrt, Research Report DFVLR-FB 84-36.
[37] S TANISLAS M, O KAMOTO K, K ÄHLER CJ (2003) Main results of the First International PIV Challenge. Meas. Sci. Technol. 14, R1–R27
[38] T HOMPSON DH (1968) A tracer-particle fluid velocity meter incorporating a laser. J.
Phys. E: J. Sci. instrum. 1 929–932
[39] V IRANT M, D RACOS T (1997) 3D PTV and its application on Lagrangian motion.
Meas. Sci. Technol. 8 1539–52
[40] W ESTERWEEL J, DABIRI D, G HARIB M (1995) Noise reduction by image shifting in
DPIV. Flow Visualization VII. 688–694
[41] W ESTERWEEL J (1997) Fundamentals of digital particle image velocimetry. Meas. Sci.
Technol. 8 1379–92
[42] W ESTERWEEL J (1998) Effect of sensor geometry on the performance of PIV interrogation. 9th Int. Symp. on Appl. of Laser Techn. to Fluid Mech., Lisbon Portugal, Paper 1.2,
June 13–16
[43] W ILLERT CE, G HARIB M (1991) Digital particle image velocimetry. Exp. Fluids. 10,
181–193
[44] W ILLERT CE (1997) Stereoscopic digital particle image velocimetry for application in
wind tunnel flows. Meas. Sci. Technol. 8 1465–79
[45] Y EH Y, C UMMINS HZ (1964) Localized flow measurements with an He-Ne laser spectrometer. Appl. Phys. Lett. 4 176–178
48