RESEARCH ARTICLE
Measurement of laminar, transitional and turbulent pipe flowusing Stereoscopic-PIV
C. W. H. van Doorne Æ J. Westerweel
Received: 26 January 2006 / Revised: 14 November 2006 / Accepted: 14 November 2006 / Published online: 8 December 2006� Springer-Verlag 2006
Abstract Stereoscopic particle image velocimetry
(SPIV) is applied to measure the instantaneous three
component velocity field of pipe flow over the full
circular cross-section of the pipe. The light sheet is
oriented perpendicular to the main flow direction, and
therefore the flow structures are advected through
the measurement plane by the mean flow. Applying
Taylor’s hypothesis, the 3D flow field is reconstructed
from the sequence of recorded vector fields. The large
out-of-plane motion in this configuration puts a strong
constraint on the recorded particle displacements,
which limits the measurement accuracy. The light sheet
thickness becomes an important parameter that
determines the balance between the spatial resolution
and signal to noise ratio. It is further demonstrated that
so-called registration errors, which result from a small
misalignment between the laser light sheet and
the calibration target, easily become the predominant
error in SPIV measurements. Measurements in laminar
and turbulent pipe flow are compared to well estab-
lished direct numerical simulations, and the accuracy
of the instantaneous velocity vectors is found to be
better than 1% of the mean axial velocity. This is
sufficient to resolve the secondary flow patterns in
transitional pipe flow, which are an order of magnitude
smaller than the mean flow.
1 Introduction
Transition from laminar to turbulent flow in a straight
pipe is one of the oldest and most fundamental prob-
lems of fluid mechanics. Various transition scenarios
have been suggested, such as the linear transient
growth of initially small perturbations (Trefethen et al.
1993; Schmid and Henningson et al. 1994), the estab-
lishment of a self-sustained non-linear process (Wale-
ffe 1997), and the organization of the flow around a few
dominant exact solutions of the Navier–Stokes equa-
tions (Faisst and Eckhardt 2003; Wedin and Kerswell
2004). In all these transition models the appearance
and development of streamwise vortices and low-speed
streaks plays a crucial role. In order to capture these
structures with stereoscopic PIV measurements, we
have applied a light sheet that is oriented perpendic-
ular to the mean flow direction, which allows us to
measure the velocity over the entire circular cross
section of the pipe. A further advantage of this orien-
tation of the light sheet is that the flow structures are
advected by the mean flow through the measurement
plane. For time-resolved measurements, the quasi-
instantaneous 3D flow field can therefore be recon-
structed from the sequence of recorded vector fields by
application of Taylor’s hypothesis (Taylor 1938).
The large out-of-plane motion (or cross-flow) in this
configuration puts a strong constraint on the recorded
particle displacements, which limits the measurement
accuracy. Especially for the measurement of the
C. W. H. van Doorne (&) � J. WesterweelJ.M. Burgers Center for Fluid Dynamics,Laboratory for Aero- and Hydrodynamics,Delft University of Technology,Leeghwaterstraat 21, 2628 CA Delft, The Netherlandse-mail: [email protected]
Present Address:C. W. H. van DoorneShell Global Solutions International BV,P.O. Box 38000, 1030 BN Amsterdam, The Netherlands
123
Exp Fluids (2007) 42:259–279
DOI 10.1007/s00348-006-0235-5
in-plane motions, i.e., the secondary flow pattern which
is an order of magnitude smaller than the mean flow, a
careful optimization of the measurements is required.
The signal-to-noise (S/N) ratio, defined as the velocity
fluctuations divided by the PIV measurement noise,
can be improved by increasing the light sheet thickness;
this, however, will decrease the spatial resolution.
Hence, the S/N-ratio, and the spatial resolution are
strongly correlated and impose conflicting demands on
the experimental parameters. Measurements behind a
heart valve mounted in a pipe (Marassi et al. 2004),
show that it is not trivial to balance these contradicting
requirements. Other SPIV measurements with large
cross-flow were performed by Matsuda and Sakakibara
(2005) in a turbulent jet and by Hutchins et al. (2005)
in a turbulent boundary layer.
For the calculation of the three components of the
velocity vector the velocity projections observed by the
two cameras are mapped (dewarped) from the image
plane onto the measurement plane. Errors in this
mapping procedure can lead to a mismatch of the two
dewarped vector fields, the so-called misregistration.
This means that velocity information from different
regions in the measurement plane is combined leading
to errors in the 3C-velocity field. This error source was
recognized by Willert (1997), Coudert and Schon
(2001), and Wieneke (2005), but the effect on the
measurement accuracy was never quantified based on
fluid mechanical data.
Our research is a continuation of the work by Draad
and Nieuwstadt (1998) and Westerweel and Draad
(1996), who considered a jet-like disturbance in lami-
nar pipe flow and reconstructed the flow in the mid-
plane of a turbulent slug by combining a sequence of
PIV data fields. For SPIV in general and with large
cross-flow in particular, small experimental details
have a large effect on the obtained measurement
accuracy. The main purpose of this paper is therefore
to give a detailed and systematic description of the
measurement accuracy for SPIV with large out-of-
plane motion. After a brief discussion of the principles
of stereoscopic-PIV in Sect. 2, the experimental setup
and calibration procedure are presented in Sect. 3. The
evaluation of the vector fields from the recorded PIV
images is explained in Sect. 4, and forms the basis for
the discussion of the measurement uncertainty in
Sect. 5. In Sects. 6 and 7 the measurement accuracy is
further investigated on the basis of measurements in
laminar and turbulent flow. An example of the 3D flow
structures measured during the transition from laminar
to turbulent pipe flow is given in Sect. 8. Finally, the
main results and conclusions are summarized in
Sect. 9.
2 Principles of stereoscopic-PIV
With stereoscopic-PIV we can measure all three com-
ponents (3C) of the velocity in the plane of the laser
light sheet. SPIV uses two cameras that look from
different directions to the light sheet and each camera
measures the particle displacement perpendicular to its
viewing direction. We obtain thus two different pro-
jections of the velocity, one from each camera, and the
complete velocity vector can therefore be recon-
structed. This is illustrated in Fig. 1, where we use a
local coordinate system that does not need to coincide
with the orientation of the light sheet. The x- and z-
axes lay in the plane defined by the two cameras and
the measurement point, and the x-axis divides the an-
gle (2a) between the two cameras in two equal halves.
The y1- and y2-axes of cameras 1 and 2 respectively are
further chosen to be parallel to the y-axis of the above
defined coordinate system. For a paraxial approxima-
tion, which is valid when the particle displacements are
much smaller than the distance to the camera lenses,
the reconstruction formula for the 3C displacement
vector (Dx, Dy, Dz) is given by:
Dx ¼ Dx1 � Dx2
2 sinðaÞ
Dy ¼ Dy1 þ Dy2
2
Dz ¼ �Dx1 � Dx2
2 cosðaÞ
ð1Þ
z∆
x∆1
∆x 2
x 2
x1
x∆1 x∆ 2
x∆
x
zα
α
α
camera 1 camera 2
Fig. 1 Illustration of the principle of SPIV. A particle displace-ment Dz is observed by two cameras. Cameras 1 and 2 measurethe projected displacements Dx1 and Dx2, from which the realdisplacement Dz can be reconstructed if the projection angle isknown. The inset shows the projection of a particle displacementDx
260 Exp Fluids (2007) 42:259–279
123
In practice, first the 2C vector fields of the particle
displacements observed by each camera are evaluated
by standard PIV correlation methods. For the calcu-
lation of the 3C-vector fields, the two 2C-vector fields
must be mapped (dewarped) from the image planes
onto the measurement plane of the light sheet and
interpolated on a common grid. Then the displacement
vectors from both cameras are combined to calculate
(reconstruct) the three components of the particle
displacement. The dewarping and reconstruction can in
principle be based on the exact knowledge of the
geometry of the setup, but most often they are based
on a calibration procedure. In Sect. 4 we describe the
applied dewarping, reconstruction and the 3D-cali-
bration procedure. For a more extensive description of
the principles of SPIV we refer to Prasad (2000), Raffel
et al. (1998), Soloff et al. (1997), and Willert (1997).
3 Experimental setup
The SPIV system and the laboratory coordinate system
are shown in Fig. 2, and an overview of the most
relevant parameters is presented in Table 1.
For our measurements we use a pipe with an inner
diameter of 40 mm and a total length of 28 m. The
working fluid is water, and due to a well designed con-
traction and thermal isolation of the pipe, the flow can
be kept laminar up to Re = 60,000 (Draad et al. 1998).
For the turbulent flow measurements at Re = 5,300 the
flow was tripped at the inlet of the pipe. All measure-
ments were carried out at 26 m from the inlet.
The PIV images are recorded with two dual-frame
cameras (Kodak-ES-1.0), which operate at 15 Hz. The
images have a resolution of 1,008 · 1,008 pixels and an
eight-bit dynamic range. We use 50 mm camera lenses
(Nikon micro–Nikkor), which have a minimum f-
number of 1.8. The two cameras look at angles of +45
and –45� to the light sheet and satisfy the Scheimpflug
condition (Prasad 2000).
According to Mie theory (Born and Wolf 1975),
small particles scatter more light in the forward
direction of the illuminating light source. Therefore,
the cameras are placed on opposite sides of the light
sheet and look (under 45�) in the direction of the light
source (Willert 1997).
The system is filled with tap water that has passed
through a filter to remove particles larger than 10 lm.
Nearly neutrally buoyant hollow glass spheres of
10 lm (Sphericel) are added to the water to increase
the particle number density (�5 gr of particles per m3
water). A substantial part of the particles remains in
suspension for many hours, also without constant
mixing of a turbulent flow. This makes it possible to
measure in both laminar and turbulent flows.
The flow is illuminated by a dual cavity frequency-
doubled pulsed Nd:YAG laser with a maximum energy
of 200 mJ/pulse (Spectra Physics PIV-200). In principle
it is not necessary to use such a strong laser. When the
aperture of the camera lens is opened completely
(f-number 1.8), PIV images of good quality can already
be recorded with 10 mJ/pulse. For our measurements
the f-number was 5.6. The light sheet is formed with
two lenses, and a mirror on a micro traverse is used to
adjust the position of the light sheet.
In order to minimize optical deformation of the
PIV-images, a rectangular test section filled with water
surrounds the pipe and two water-prisms are attached
to the window (Prasad 2000; Westerweel and van Oord
2000). Inside the test section the perspex pipe is re-
placed by a thin glass tube with a wall thickness of 1.6
and a 40 mm inner diameter.
Two black-painted aluminum screens inside the test
section are used to reduce the background illumination
and to enhance the contrast of the PIV-images. The
screens are held in position and can also be moved with
small magnets on the outside of the test section. The
screens form a slit orthogonal to the pipe, which is used
to align the light sheet in the vertical direction (parallel
to the y-axis). The horizontal alignment is based on the
reflections of the light sheet from the pipe.
The calibration grid is a 2 mm spaced lattice of dots
with a diameter of 0.5 mm printed on a transparent
sheet and glued between two 0.5 mm thick glass plates
(Fig. 3). The grid is kept in position by a cylindrical
transparent sheet (0.1 mm thick), which is mounted on a
plastic rod that precisely fits into the pipe. The cylin-
drical sheet forms a solid support for the grid and allows
= 40 mm
(1) (2) (3) (4)
(6)(5)
45
x
zy
(7) (8) (9)
Fig. 2 Schematic (top view) of the stereoscopic-PIV system. 1Nd: YAG-laser, 2 lenses, 3 mirror on a micro-traverse, 4traversing direction of the mirror, 5 test section, 6 anti-reflectionscreen, 7 CCD camera on Scheimpflug mount, 8 light sheet, 9water prism
Exp Fluids (2007) 42:259–279 261
123
for optical access from both sides of the grid without any
noticeable optical distortion. To enter the calibration
grid in the pipe, an open tank is placed behind the test
section (Fig. 3). For the first set of calibration images the
grid is placed in the plane of the light sheet, and a second
set of calibration images is made after the calibration
grid has been displaced 0.5 mm in the downstream
direction. The calibration grid can be translated with an
accuracy of 10 lm. An example of a calibration image
recorded by camera 1 is shown in Fig. 4(a). In the upper
and lower parts of the calibration image the grid points
deviate slightly from straight lines, which can be
attributed to the refraction by the round glass tube.
4 Vector evaluation
The evaluation of the 3C-vector fields from the PIV-
images is performed with commercial PIV-software
(DaVis 6.2, LaVision), and an overview of the relevant
parameters is presented in Table 2. First, for each
camera the particle displacements are evaluated in
exactly the same way as for standard (2C-) PIV (Raffel
et al. 1998; Foucaut et al. 2004). The subsequent de-
warping, interpolation and recombination of the vector
fields are discussed below.
4.1 Dewarping
After the computation of the two 2C-vector fields from
each camera, the displacement vectors are mapped
(dewarped) from the image plane to the measurement
plane. The mapping function ðMÞ can for example be
based on a camera pinhole model (Wieneke 2005), but
other approaches have been proposed as well (Prasad
2000).
The perspective image deformation can be corrected
by a second-order polynomial that maps straight lines
onto straight lines (Prasad 2000; Prasad and Adrian
1993; Westerweel and van Oord 2000). We used a
Table 1 Overview of the relevant experimental parameters forlaminar flow measurements at Re = 3,000
Pipe Diameter 40 mmLength 28 mMaterial GlassWall thickness 1.6 mm
Flow Fluid WaterRe 3.0 · 103
Seeding Type Hollow glass sphereSpecific weight 1 gr/cm3
Diameter 10 lmConcentration 5 gr/m3
Lightsheet
laser type Nd:YAGMaximum energy 200 mJ/pulseWave length 532 nmPulse duration 6 nsThickness 1.5 mm
Camera type CCDResolution 1,008 · 1,008 pxPixel size 10 lmDiscretization 8 bitRepetition rate 15 HzLens focal length 50 mm
Imaging f-number 5.6Diffraction limit 9 lmSignal level 200/256Viewing angles ±45�Image magnification 0.220 ±0.014a
Viewing area 40 · 57 mm2
Exposure time-delay 2 msMaximum particle
displacement8 px
PIVanalysis
Reconstruction method Three-dimensionalcalibration
Interrogation area (IA) 32 · 32 pxOverlap IA 50%Approximate resolution 1.5 · 1.5 · 1.5 mm3
a Nominal magnification on the optical axis and the variationover the image in the x-direction
test section
water tank
side view
z
y
−x
calibration gridholder
micro traverse
fixed support
cam1 cam2
top view
y
x
z
Fig. 3 Left Photo of thecalibration grid and its holder.Right An open water tank isplaced behind the test sectionto insert the calibration gridinto the pipe
262 Exp Fluids (2007) 42:259–279
123
third-order polynomial mapping function to account
for additional aberrations by the round glass tube. The
coefficients of the mapping function are estimated with
a least-square method from the coordinates of the real
position of the markers on the calibration grid and the
observed location in the calibration images.
Figure 4b shows the dewarped image of the cali-
bration image in Fig. 4a. A regular grid of white dots,
2 mm apart, is superimposed on the dewarped cali-
bration image. An ideal mapping function would pro-
ject all markers from the calibration image onto the
position of the white dots. The mapping function is
found to be within 1 px accurate over almost the entire
image. Only in the upper and lower part of the image,
where the glass pipe causes an additional deformation
of the calibration image, the mapping function be-
comes less accurate. In this region, the maximum error
of the mapping function is about 5 px, which corre-
sponds to 0.27 mm in physical space. The effect of this
error on the measurement accuracy is discussed in
Sect. 5.2.
4.2 Interpolation
The 2C-vector fields from both cameras are calculated
on a rectangular grid in the image planes. After the 2C-
vector fields are mapped from the image planes to the
measurement plane, the vectors lay on two different
non-uniform grids. The two 2C-vector fields must
therefore be interpolated on a common grid in the
Fig. 4 Left View of camera 1on the calibration grid when itis placed in the measurementplane. The deviation of thedots from straight lines in theupper and lower parts of theimage (see enlargement) isdue to refraction by the roundglass tube. Right Dewarpedimage of the calibration gridon which a regular lattice ofwhite points is superimposed.The deviation of the blackcalibration dots from thewhite dots (see enlargement)reveals a small error (5 px atmost) in the mapping functionin the upper and lower part ofthe image
Table 2 Parameters of theSPIV-analysis for the laminarflow measurements
Image enhancement Sliding-min-max-filter (Meyer and Westerweel 2000)Filter length = 3 px
First passPIV-correlation IA = 32 · 32 px
Three point Gaussian peak fitDewarping Third order polynomial mapping functionInterpolationRecombination 3D-calibration method (Soloff et al. 1997)Vector validation Median test, remove spurious vectors,
Try to replace with second to fourth correlation peakInterpolate missing vectors Average of four neighborsFilter vector fields 3 · 3 smoothing
Second passPIV-correlation IA = 32 · 32 px
Three point Gaussian peak fitWindow shifting with an integer pixel displacement
Dewarp/interpolate/recombine See first passVector validation See first pass
Exp Fluids (2007) 42:259–279 263
123
measurement plane, before the reconstruction of the
3C-vectors can take place (Prasad 2000; Westerweel
and Nieuwstadt 1991). A drawback of this interpola-
tion is that errors will spread, as a single spurious
vectors can affect several interpolated vectors around
it. In principle, the interpolation could be avoided if
the evaluation of the two 2C-vector fields would be
performed on two different grids that would be map-
ped directly onto a common grid in the measurement
plane (this, however, was not possible within the ap-
plied software).
4.3 Recombination
For the combination of the two interpolated 2C-vector
fields into a single 3C-vector field we make use of the
3D-calibration method proposed by Soloff et al. (1997).
In the calibration procedure the calibration grid is first
placed in the measurement plane and a calibration-im-
age is recorded for each camera. These images are used
to derive the 2D-mapping function ðMÞ; which projects
the measurement plane on the image planes. Then the
calibration grid is traversed in the out-of-plane direction
(in our case 0.5 mm in the downstream direction), and a
second calibration image is recorded for each camera.
For each camera it is in principle possible to derive a 3D-
mapping function (F) from the calibration images. The
linear approximation for the projection of the real par-
ticle displacement (Dx, Dy, Dz) onto the particle-image
displacements (Dx1, Dy1) and (Dx2, Dy2) recorded by the
two cameras can be written as:
Dx1
Dy1
Dx2
Dy2
2664
3775 ¼
@Fx1
@x@Fx1
@y@Fx1
@z@Fy1
@x
@Fy1
@y
@Fy1
@z@Fx2
@x@Fx2
@y@Fx2
@z@Fy2
@x
@Fy2
@y
@Fy2
@z
266664
377775
x;y;z¼0ð Þ
�DxDyDz
24
35 ð2Þ
For the calculation of the 3C-vector from the two 2C
particle-image displacements Eq. (2) must be inverted.
The expression provides four equations for three un-
knowns and can be solved in a least-square sense.
For the practical implementation of Eq. (2), the
derivatives of the 3D-mapping function (F) in the x-
and y-direction (in the measurement plane z = 0) are
obtained from the 2D mapping function @F@x ¼ @M
@x and@F@y ¼ @M
@y : The partial derivatives in the z-direction are
obtained from the difference between the calibration
images when the calibration grid is displaced over a
small distance Dz in the z-direction. If Dx1 is the cor-
responding displacement of a grid point as observed by
camera one, then: @Fx1
@z ¼Dx1
Dz (the other derivatives in
the z-direction are obtained in a similar manner).
5 Measurement accuracy
In this section the errors related to each step of the
vector evaluation will be estimated. The two 2C-vector
fields calculated for each camera contain the usual
errors found in standard 2C-PIV, which are the cor-
relation-noise, bias and peak-locking. Errors in the
mapping from the image plane to the measurement
plane can lead to a mismatch (misregistration) of the
two dewarped vector fields. This means that velocity
information from different regions in the illuminated
plane will be combined, which leads to further errors in
the 3C-vectors, as explained in Sect. 5.2. Finally, in
Sect. 5.3, the accuracy of the recombination of the two
2C vector fields into the 3C vector field is investigated.
Besides the uncertainty of the velocity, other
important measures for the performance of a PIV-
system are the signal-to-noise (S/N) ratio, the fraction
of spurious vectors, the spatial resolution, and the
temporal resolutions of the measurements. Some
values typically found in the literature have been
summarized in Table 3, together with the control
parameters that have to be tuned to optimize the PIV
measurements. The relation between these control
parameters and the measurement accuracy is subject of
extensive research (Keane and Adrian 1993; Wester-
weel 2000a; Scarano 2002; Foucaut et al. 2004).
5.1 Correlation noise
Correlation noise generally forms the major error
source in standard 2C-PIV measurements. When
sufficient particle-pairs are present in an interrogation
area and the velocity gradients are not too large
(see Table 3), the RMS of the measured particle-image
displacements rcorr � 0.1 px (Westerweel 2000a;
Foucaut et al. 2004).
The noise level of the 3C-vector fields follows from
the error propagation in the recombination equations
(Lawson and Wu 1997). If we assume that the viewing
angle (a = 45�) and the magnification are approxi-
mately constant over the entire field of view and
identical for both cameras, it follows from Eq. (1) that:
Dx � �Dx1 þ Dx2ffiffiffi2p
Dy � Dy1 þ Dy2
2
Dz � Dx1 þ Dx2ffiffiffi2p
ð3Þ
If we further assume that the correlation noise (rcorr) is
the same for all 2C-particle image displacements, it
follows from Eq. (3) that:
264 Exp Fluids (2007) 42:259–279
123
RMSðDxÞ �Mrcorr
RMSðDyÞ �M1
2
ffiffiffi2p
rcorr
RMSðDzÞ �Mrcorr
ð4Þ
The correlation-noise of the x- and z-components of
the 3C velocity vector are thus of the same order as the
correlation-noise in the 2C-vector fields. The noise
level of the y-component is smaller, because it is the
average of two independent measurements.
5.2 Velocity errors due to misregistration
When we performed our first measurements in laminar
pipe flow, it became clear that even for a very small
misalignment of the light sheet and the calibration grid,
the measurement uncertainty was dominated by errors
in the velocity due to misregistration. This error source
is therefore investigated in detail here.
In the SPIV-analysis the 2C-vector fields from both
cameras are mapped (dewarped) from the image
planes to the calibration plane. Errors in the mapping
procedure can lead to a mismatch of the two dewarped
vector fields, the so-called misregistration. This means
that velocity information from different regions in the
illuminated plane (measurement plane) is combined
for the calculation of the 3C-vectors, leading to errors
in the 3C-velocity field. The error in the position of the
dewarped vectors (dxi) can also be interpreted as an
error in the 2C-velocity vectors, which can be
approximated by the local velocity gradient times the
registration error, e.g. dux1 = (¶ux1/¶xi) dxi. Substitu-
tion of this error in the recombination Eq. (1) reveals
the error in the 3C-velocity vector:
du � max1
sinðaÞ;1
cosðaÞ
� �@u
@xidxi ð5Þ
Obviously, the larger the spatial velocity gradients
(¶u/¶xi), the larger the velocity errors (du) due to
misregistration (dxi). Further it follows that for very
large and very small angles of the cameras (a), the
SPIV becomes more sensitive to registration errors
(Lawson and Wu 1997). If we require that the velocity
errors due to misregistration must be smaller than the
correlation noise (e), which is of the order of 0.1 px,
and we assume the angle between the two cameras to
be of the order of 90�, then it follows from Eq. (5) that
we should satisfy the following relation:
@U
@x
��������Dtdx .M � � ð6Þ
the misregistration can result from two different
effects:
1 errors in the 2D mapping function ðMÞ which is
derived from the calibration images, and
2 the misalignment of the laser light sheet and the
calibration plane.
For the first case, Fig. 4b reveals that in our
experiment small errors (of maximum 5 px) occur in
the mapping of the upper and lower parts of the image.
However, due to the symmetric configuration of the
two cameras, which are located on opposite sides of the
Table 3 The measures (with typical values) and control parameters (with recommended values) that determine the quality of theSPIV-measurements
Quality measures Control parameters
S/N-ratio (100–600) Laser-pulse delay-time (Dt)Light sheet thickness
Correlation-noise (rcorr~ 0.1 px) Particle image densityNI ‡ 10, where NI is the mean number of particle images in an interrogation area.
Particle loss due to out-of-plane-motionFO ‡ 0.75, where FO is the fraction of particles that is present in both images.
Velocity gradientM |DU| Dt / ds £ 0.5, where M (px/m) is the image magnification, DU (m/s) is the
velocity difference within one interrogation area, and ds (px) the mean particle imagediameter.
Systematic errors- Bias- Peak-locking Mean particle image diameter (2 px £ ds £ 4 px)- Velocity errors due to misregistration Registration accuracy
@U@x
�� ��Dtdx\ 0.1 px, where @U@x is the velocity gradient and dx is the registration error.
Number of spurious vectors (less than 5%)Spatial resolution (60 · 60 vectors)Temporal resolution (8-500 Hz)
Exp Fluids (2007) 42:259–279 265
123
light sheet, the errors in the mapping functions are
equal for both cameras, and the two dewarped cali-
bration-images fall exactly on top of each other. For
the applied configuration, the mapping errors therefore
do not result in misregistration. However, in the upper
and lower part of the images the vectors will be cal-
culated at a slightly wrong position. In case that the
two cameras would be positioned on the same side of
the light sheet, the mapping error would lead to a
maximum registration error of about 10 px (or
0.5 mm), and this could lead to considerable errors in
the velocity estimation as will be shown below.
For the second case, Fig. 5 illustrates the registration
errors caused by a misalignment of the light sheet and
the calibration plane. The 2C-vectors from both cam-
eras, which are measured at different locations in the
light sheet, are mapped onto the same location in the
calibration-plane and are recombined to find a single
3C-vector. For a viewing angle of 45� for both cameras,
half the misregistration (dx) is equal to the misalign-
ment of the light sheet (dz). For the simple 2D shear
flow displayed in the figure, it follows from geometric
considerations that
dux ¼@uz
@xdz: ð7Þ
The gradient in the z-component of the velocity leads
in this case to an error in the x-component of the
velocity.
5.2.1 Measurement of the velocity error due
to misregistration in laminar pipe flow
In laminar pipe flow the in-plane velocity field is zero,
and therefore the vector field shown in Fig. 6 is a direct
measurement of the velocity errors due to the mis-
registration. From Eq. (7) and the gradient of the
parabolic velocity profile (¶uz/¶x = –2 uc x/R2), the
relative error in the in-plane velocity can be predicted
by
dux
uc¼ �2
x
R
dz
R:ð8Þ
The velocity error due to misregistration is thus inde-
pendent from the y-coordinate and is zero for x = 0.
The average of the x-component of the velocity in the
rectangle shown in Fig. 6 is plotted as function of the
misalignment between the light sheet and the calibra-
tion plane in Fig. 7. The misalignment of the light sheet
was varied in small steps by moving the mirror on a
micro-traverse back and forth (Fig. 2). The prediction
by Eq. (8) for x/R = 0.9 is found to be in agreement
with the measurements. From Fig. 7 it can further be
concluded that in order to measure the in-plane
velocity with a precision of 1% of the centerline
velocity, the alignment of the laser sheet and calibra-
tion plane should be better than 0.1 mm. In turbulent
flow the velocity gradients are even larger than in
laminar flow, and the required alignment precision
becomes even more stringent. However, due to the
turbulent motions the instantaneous in-plane velocity
is non-zero and the velocity errors due to misregistra-
tion are not so easily recognized in the instantaneous
velocity fields.
5.2.2 Minimization of the registration error
For our measurements, the proper registration was
obtained by a very accurate alignment (within 0.1 mm)
of the light sheet and the calibration plane. The
vpiv
verr
v0
dxdvz δx
δx
δxvpiv
cam1 cam2
calib
atio
n−pl
ane
y z
x
light
she
et
α
v2
vc2
vc1
vc2
vc1
=45deg αδ
(x ,z )0 0
z
α
α
v1
v0
Fig. 5 Illustration of thevelocity error (verr) and theregistration error (2dx) inshear flow due to amisalignment (dz) of the lightsheet and the plane of thecalibration target
266 Exp Fluids (2007) 42:259–279
123
alignment was based on the minimization of the
velocity errors that were measured for x/R = 0.9 and
–0.9 in laminar pipe flow. Although this procedure
proved to give very satisfactory results, it has a few
disadvantages. First of all, the procedure is quite time
consuming and complicated, which makes it liable to
errors. Furthermore, when the SPIV is to be applied to
other flow configurations, it is very unlikely that a well
known laminar flow can be imposed to verify the
registration by a direct measurement of the velocity
errors.
It was pointed out by Willert (1997) that the mis-
registration can also be found by cross-correlation of
the dewarped PIV images of the two cameras. For
SPIV measurements with a rather thick light sheet,
however, the cross-correlation is very noisy when
determined from a single image-pair. Wieneke (2005)
reports that the accuracy can be improved when the
cross-correlation planes of several PIV-images are
averaged.
Figure 8 shows the (slow) convergence of the aver-
age cross-correlation peak calculated from an increas-
ing number of dewarped PIV images. At least 400
images are required to obtain a reliable representation
of the cross-correlation function. The undulations show
that the profile of the light sheet is not quite Gaussian.
The width of the correlation function is about two
times the light sheet thickness, because the viewing
angle of the two cameras is ±45�. The maximum of the
correlation is located about 3 px from the center, which
corresponds to about 0.15 mm. For a symmetric cor-
relation function this would correspond to the regis-
tration error, which is about two times the
misalignment of the light sheet. In Fig. 9, the correla-
tion function is calculated at different positions in the
measurement plane. The position of the correlation
peak is slightly different for each position. This shows
that the registration error is not constant over the
measurement plane, which is caused by a small tilt (of
the order of 0.5�) between the calibration plane and
the laser light sheet.
These results show that, when sufficient images are
available, also for SPIV measurements with a rather
thick light sheet, it is possible to calculate the average
cross-correlation between the dewarped PIV images.
−20 −10 0 10 20−20
−15
−10
−5
0
5
10
15
20
x (mm)
y (m
m)
Fig. 6 Registration error in laminar pipe flow. Only 1/4 of thevectors is displayed
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4
−0.04
−0.02
0
0.02
0.04
0.06
0.08
0.1
mirror posistion (mm)
u x / u zm
ax
measurementprediction at r/R=0.9
Fig. 7 Average registration error in the rectangle shown inFig. 6 as a function of the difference in the positions of the lightsheet and the calibration plane
−50 −25 0 25 50
0
0.2
0.4
0.6
0.8
1number of frames
x (px)
cros
s−co
rrel
atio
n
96040010020
Fig. 8 Cross-correlation of the dewarped particle images ofcameras 1 and 2. The correlation functions are determined for aninterrogation area of 128 · 128 px in the middle of the image,and averaged over a varying number of image pairs (x = 10 pxcorresponds to 0.5 mm)
Exp Fluids (2007) 42:259–279 267
123
From the position of the correlation peak, or the
position of the center of mass, a disparity vector map
and a correction of the mapping functions can be
derived. Implementation of this method proposed by
Wieneke (2005), avoids velocity errors due to misreg-
istration and relaxes the requirement for a perfect
alignment of the calibration plane and the laser light
sheet.
5.3 Accuracy of the recombination equation
The largest uncertainty in the recombination Eq. (2)
arises from the uncertainty in the displacement of the
calibration grid when it is traversed 0.5 mm in the
downstream direction during the calibration proce-
dure. The accuracy of the micro-traverse that moves
the grid is estimated at 10 lm. The displacement is thus
known with a relative uncertainty of about 2%.
Suppose that there is an error in the displacement of
the calibration grid of 2%. In that case all coefficients
in the third column of the matrix in Eq. (2) also have
an error of 2%, which in turn results in an error in the
reconstructed downstream particle displacement Dz,
and the downstream velocity component of also 2%.
The in-plane velocity components, however, are not
affected. In order to verify the accuracy of the
streamwise velocity, the flow rate is calculated by
integration of the streamwise velocity over the entire
cross-section of the pipe and compared with the flow
rate indicated by the flow meter (see also Sect. 7.2).
The errors in first two columns of the matrix in
Eq. (2) are related to the errors in the 2D mapping
function ðMÞ: In Sect. 4.1 the accuracy of the mapping
function was found to be better than 1 px over almost
the entire image. It follows that the error in the
derivatives of the mapping function is smaller than
1 px/4 mm = 0.05/4 mm = 1% in the center of the
image and about 4 px/4 mm = 4% in the upper and
lower parts of the image. These systematic errors
introduced by the 2D-mapping function are thus of the
same order of magnitude as the correlation noise,
which is also about 1% of the mean flow rate.
5.4 S/N and spatial resolution
In this section we will investigate the influence of the
light sheet thickness on the spatial resolution and
signal-to-noise (S/N) ratio. A thick light sheet leads to
a reduced spatial resolution, not only in the direction
normal to the light sheet, i.e. the z-direction, but also in
the x-direction. This is related to the (45�) viewing
angle of the cameras, and is illustrated in Fig. 10. All
particles in the indicated volume in the light sheet are
projected in the same region of the image. Suppose this
region is an interrogation area, the velocity estimated
by the PIV is then the spatial average of the velocity in
the indicated volume. With the cameras under 45� the
probe size in the x-direction is therefore approximately
the light sheet thickness plus the width of the interro-
gation area. The resolution in the y-direction is not
influenced by the light sheet thickness and is deter-
mined by the width of the interrogation area only.
On the other hand, when the mean flow is perpen-
dicular to the light sheet plane, a thin light sheet leads
to a low signal-to-noise ratio. This is related to the so-
called one-quarter-rule (the second control parameter
in Table 3). If the loss of particle pairs is less than 25%,
then the correlation-noise (the RMS of the mean par-
ticle displacement) is of the order of 0.1 px (Foucaut
et al. 2004; Westerweel 2000a). In our SPIV-setup the
flow is perpendicular to the light sheet and the maxi-
mum particle displacement is thus limited to 1/4 of
the light sheet thickness. For a thin light sheet the
−50 −25 0 25 50
0
0.2
0.4
0.6
0.8
1
x=15 (c1/max(c
1))
x=0 (c2/max(c
2))
x=−15 (c3/max(c
3))
x=0 (c2/max(c
1))
x=−15 (c3/max(c
1))
x (px)
cros
s−co
rrel
atio
n
Fig. 9 Cross-correlation function for three different positions ofthe interrogation area
x
z
Fig. 10 For a thick light sheet the non-orthogonal viewing of thecamera causes spatial filtering of the velocity estimate in thein-plane direction
268 Exp Fluids (2007) 42:259–279
123
maximum attainable particle displacement is small,
while the noise level remains fixed around 0.1 px. This
leads thus to a small signal-to-noise level for a thin
light sheet.
Therefore the optimal light sheet thickness is a trade
off between the spatial resolution and the signal-to-
noise level. In turbulent flow and transition measure-
ments the velocity fluctuations are a few percent of the
bulk velocity. In order to capture these fluctuations the
measurement accuracy should therefore be of the
order of 1% of the mean flow speed. For a noise level
of 0.1 px, this means that the mean displacement of the
particles should be at least 10 px. We have just argued
that the particle displacement should be smaller or
equal to 1/4 of the light sheet thickness, and the min-
imum light sheet thickness would thus correspond to
40 px, which would be 2.3 mm in reality.
The required spatial resolution of the SPIV-system
is determined by the size of the smallest structures in
the flow. For turbulent flow, the friction velocity
ðu� ¼ UB
ffiffiffiffiffiffiffiffif=2
pÞ and the viscous length scale (y0
+ =
m/u*) can be calculated from the Fanning friction
factor (f) which follows from the Blasius-friction-law
f = 0.079 Re1/4 (Schlichting 1955). For a Reynolds
number of 5,300 we find f = 0.00926, u* = 9.0 mm/s,
and y0+ = 0.11 mm. The smallest structures in the flow
measure at least several times the viscous length scale.
A resolution of 5 y0+, that is 0.55 mm, seems therefore
sufficient to resolve the turbulent flow field. This is
further supported by the results of Jimenez (1994),
which show that up to the 6-order-moments of the
velocity can still be estimated with a high precision
from measurements performed with a probe size of
three times the viscous length scale.
The light sheet thickness should thus be chosen
somewhere in between 2.3 and 0.55 mm. As demon-
strated in the following two sections, good results in
laminar and turbulent flow were obtained for a
light sheet thickness close to 1.5 mm and for observed
particle displacements up to 8 px.
6 Laminar flow measurements
Laminar pipe flow proved to be a critical test case for
the investigation of the measurement accuracy of the
SPIV system. The measurements are performed at
Re = 3,000, for which the centerline velocity is
approximately 150 mm/s. The mean flow and the RMS
of the velocity fluctuations are calculated from 100
independent vector fields. We refer to Tables 1 and 2
for an overview of all the experimental parameters and
the parameters used for the vector evaluation.
6.1 Mean velocity profile
A 3D view of the mean streamwise velocity profile is
shown in Fig. 11. The laminar profile is extremely
sensitive to small disturbances, e.g. due to thermal
convection or a minor misalignments of the pipe seg-
ments; It was shown that even the Coriolis force affects
the laminar velocity profile (Draad and Nieuwstadt
1998). This leads to an asymmetry of the laminar
velocity profile at Re = 3,000, as revealed in Fig. 12.
6.2 Velocity errors due to misregistration
The measurement accuracy can be found directly from
the in-plane velocity components. The mean of ux over
the 100 vectors is shown in Fig. 13, and it reveals a bias
from zero which is smaller than 1% of the maximum
streamwise velocity in every point. The small magni-
tude of this bias, which is an almost random function of
location, shows that the registration error has been
eliminated by the proper alignment of the light sheet
and the calibration plane (see also Sect. 5.2). In Fig. 14
we show the measurement of Æux æ when the light sheet
is misaligned on purpose and displaced 0.3 mm in the
downstream direction, which is 20% of the light sheet
thickness. We now find a large velocity error due to the
misregistration, which attains a maximum value of
about 0.4 px near the wall.
6.3 Correlation noise
Because the laminar flow is stationary, the RMS of the
velocity fluctuations immediately reveals the PIV-cor-
relation noise (see Sect. 5.1). The RMS of ux is shown
in Fig. 15. The noise level is 0.05 px at the center of the
−20
−10
0
10
20 −20−10
010
20
0
50
100
150
Y (mm)
X (mm)
<uz>
(m
m/s
)
Fig. 11 Three-dimensional view of the streamwise velocitymeasured in laminar pipe flow
Exp Fluids (2007) 42:259–279 269
123
pipe. When the wall is approached the noise level in-
creases gradually and reaches a maximum value of
about 0.18 px close to the wall, which is explained by
the increased velocity gradient toward the wall.
It is recommended that the difference in the particle
displacements within one interrogation area should be
kept smaller than about one half of the particle image
diameter: M |D| Dt / ds £ 0.5 (fourth control parameter
in Table 3U). The velocity gradient of the laminar flow
close to the wall is approximately duz / dr = –2Uc/R.
This velocity gradient results in a gradient of the pro-
jected velocities observed by the cameras, i.e.
dux1=dr � 1ffiffi2p duz=dr for camera one. If the maximum
particle displacement is substituted for the centerline
velocity (M Uc Dt = 9 px) and the size of the inter-
rogation area for dr (dr/R = 32 px / 350 px), then the
difference of the particle displacements across an
interrogation area is found to be M |DU| Dt ~1.1 px.
The mean particle image diameter (ds) is about 2 px,
and it follows that M |DU| Dt / ds ~ 0.5. Monte–Carlo
simulations by Foucaut et al. (2004) showed that for
this shear rate the RMS of the estimated particle dis-
placement is approximately 0.14 px, which is not to far
from the observed value of 0.18 px in Fig. 15.
6.4 Spurious vectors
When the vector fields are evaluated from the raw
PIV-images, i.e. no image enhancement is applied, all
vectors in the central region of the pipe are valid.
However, in the near-wall region quite a few spurious
vectors are found. This is due to the cumulative effect
of several unfavorable factors in the near-wall region.
First of all, the glass tube results in a dark region in
the PIV-images, which appears as a bright region in the
inverted image, indicated by No. 1 in Fig. 16. This re-
duces the correlation when an interrogation area partly
overlaps the wall, which leads to more noise and an
increased chance to find a spurious vector. The second
problem in the near-wall region is the fouling of the
tube due to seeding particles that stick to the wall.
When many particles are attached to the wall, they
form a continuous and bright curved line in the PIV
images (No. 2 in Fig. 16). This line correlates very well
along its own direction and therefore most of the
spurious vectors point parallel to the wall. Immobile
particle images can also appear at a small distance
from the wall (No. 4 in Fig. 16). In this case a very
large particle is attached to the wall at some distance
from the laser light sheet. The particle scatters indirect
laser light in the direction of the camera, and due to the
large opening angle of the cameras it appears at some
distance from the wall in the PIV image. Small scrat-
ches in the glass tube have the same effect (No.3 in
Fig. 16). It is thus very important to keep the wall of
the tube clean and free from scratches during the
experiments. The number of spurious vectors in the
near-wall region is further increased by the large
velocity gradient close to the wall, which was discussed
in the previous section.
In an attempt to decrease the noise level and the
amount of spurious vectors in the near-wall region,
part of the PIV images was masked. At the wall of the
tube and outside the tube the gray level was set to zero.
However, this resulted in even more spurious vectors
that pointed parallel to the edge of the mask, i.e.,
parallel to the wall of the tube. This is caused by the
non-uniform background in the masked image. When
the value of the masked pixels was set to the mean gray
value inside the flow, or to the value of the mean
−0.5 0 0.50
0.2
0.4
0.6
0.8
1
x / D , y / D
<u z>
/ u zm
ax
<uz> on x−axis
<uz> on y−axis
0
1
2
3
4
5
6
7
8
disp
lace
men
t (p
ix)
Fig. 12 Horizontal and vertical profiles of the streamwisevelocity in laminar pipe flow, Re = 3,000
−0.5 0 0.5−0.02
−0.01
0
0.01
0.02
x / D , y / D
<u x>
/ u zm
ax
<ux> on x−axis
<ux> on y−axis
−0.1
0
0.1
disp
lace
men
t (p
ix)
Fig. 13 Horizontal and vertical profiles of the mean horizontalin-plane velocity component in laminar pipe flow
270 Exp Fluids (2007) 42:259–279
123
background inside the flow, this did not lead to any
significant improvement.
A renormalization of the PIV images with a min-
max-filter (Meyer and Westerweel 2000; Westerweel
1993) with a filter length of 3 px removed practically all
spurious vectors in the near-wall region. The large
improvement in the near-wall region is probably re-
lated to the fact that the lines visible in the original
image, which correspond to the glass tube and particles
attached to the wall, are hardly visible in the filtered
image. As a result the correlation along the wall is
effectively reduced. After application of the renor-
malization filter there are typically about ten spurious
vectors in the entire flow domain. The whole cross-
section of the pipe contains about 1,400 vectors, so that
the fraction of spurious vectors is approximately 0.7%.
7 Turbulent flow measurements
Measurements were performed in fully developed tur-
bulent pipe flow at Re = 5,300. For this Reynolds
number several data sets are available that can be used
for comparison, e.g., the results of a direct numerical
simulation (DNS) by Eggels et al. (1994). The experi-
mental parameters are identical to those used for the
laminar flow measurements (Table 1), except for the
above mentioned Re number and the exposure delay
time, which was 2.5 ms. In the second pass of the vector
evaluation of the turbulent flow the interrogation area is
16 · 16 px, after which the vector fields are smoothed
with a 3 · 3 averaging filter. These modifications of the
PIV-analysis are clarified in the following section. All
the statistics presented in this chapter are calculated
from 900 statistically independent vector fields.
−0.5 0 0.5−0.05
0
0.05
x / D
<ux>
/ uzm
ax
on X
−axi
s
alligned0.3 mm misaligned
−0.4
−0.2
0
0.2
0.4
disp
lace
men
t (p
ix)
Fig. 14 Effect of the registration error on the measurement ofthe mean in-plane velocity Æuxæ in laminar pipe flow
−0.5 0 0.50
0.01
0.02
0.03
x / D , y / D
rms(
u′x)
/ uzm
ax
rms(ux) on x−axis
rms(ux) on y−axis
0
0.05
0.1
0.15
0.2
0.25
rms(
dis
plac
emen
t )
(pix
)
Fig. 15 Noise level of the horizontal in-plane velocity compo-nent in laminar pipe flow
(2)(3)
(1)
(4)
16x16 32x32 64x64
x px
y px
150 200 250 300 350 400
150
200
250
300
350
400
Fig. 16 Left Inverted PIVimage; 1 glass tube, 2 foulingof the wall by tracer particles,3 scratches, 4 large particleattached to the wall. Rightdetail of the PIV image
Exp Fluids (2007) 42:259–279 271
123
For turbulent pipe flow it is common to normalize
all statistics with the wall friction velocity u* (Schlich-
ting 1955). It is important to use an accurate value for
u*; errors in the estimated value of u* can lead to
overall systematic errors in the normalized turbulence
statistics (Kim et al. 1987; Eggels et al. 1994; Wester-
weel et al. 1996). Nonetheless, despite the rigorous
validity of Blasius’ friction law (which is accepted to
hold for 104 < Re < 105), for a Reynolds number as
low as 5,300 one can expect small deviations, and one
should be careful in using this expression for Reynolds
numbers below 104 without further consideration. We
therefore apply two methods for estimating u*: one is
based on Blasius’ semi-empirical law and denoted as
u*(1), while the second one is based on an extrapolation
of the measured total shear stress to the wall and is
denoted by u*(2).
7.1 Correlation noise
For the turbulent flow the velocity gradients in the
near-wall region are much larger than for the laminar
flow. This leads to an increased noise level near the
wall, and the PIV-analysis was adapted to deal with
these large velocity gradients.
The velocity gradient at the wall estimated from the
Blasius friction law is Duz/Dr � 0.25f Re Ub/R. This
velocity gradient results in a gradient of the projected
velocities observed by the cameras, i.e. Dux1=Dr �1ffiffi2p Duz=Dr: After substitution of the mean axial particle
displacement for the bulk velocity (M Ub Dt� 6 px), the
size of the interrogation area for Dr (Dr/R = 32 px/350
px), the Reynolds number (Re = 5,300), the Fanning-
friction-factor (f = 0.00926), and the particle image
diameter (ds � 2 px), we find for the difference of the
particle displacements across an interrogation area of
32 · 32 px close to the wall M |DU|Dt/ds � 2.2. From
Monte Carlo simulations by Foucaut et al. (2004) it
follows that the RMS of the correlation noise is larger
than 0.8 px for such large velocity gradients. This noise
level is too high to resolve the turbulent velocity fluc-
tuations in the near-wall region. Therefore the interro-
gation area was reduced to 16 · 16 px, which reduced
the difference in the particle displacements across an
interrogation area to M |DU|Dt/ds � 1.1, for which the
RMS of the noise level is about 0.3 px (Foucaut et al.
2004). For a PIV analysis with 16 · 16 px interrogation
area and 50% overlap the spacing of the vectors is
0.46 mm. This is in fact an oversampling of the velocity
field, because the spatial resolution is limited by the light
sheet thickness to about 1.5 mm (Sect. 5.4). It is there-
fore acceptable to reduce the noise level further by
application of a 3 · 3 smoothing of the vector fields,
after which the RMS of the noise becomes approxi-
mately 0.1 px. In the central part of the flow domain the
velocity gradients are much smaller and the noise level
can be expected to be smaller than 0.1 px.
In Fig. 17 we show an arbitrarily chosen vector plot
of the in-plane velocity components. Small-scale and
large-scale structures can be observed. The vector field
is very smooth and the fraction of valid vectors is larger
than 99%.
For fluctuating signals it is possible to investigate the
noise from the auto-correlation function. The correla-
tion noise is in principle uncorrelated between two
different PIV vectors, except for the direct neighbors
due to the 3 · 3 smoothing of the velocity field. The
noise may therefore result in a sharp peak at the
maximum of a correlation function and the height of
this peak corresponds to the noise variance. However,
all the correlation functions in Figs. 29 and 30 are very
smooth around the maximum. It is therefore concluded
that the correlation noise must be very small.
7.2 Mean axial velocity
Figure 18 shows the flow rate as function of time. This
flow rate is obtained by integration of the axial velocity
over the cross-section of the pipe. The flow rate indi-
cated by the flow meter was 604 liters per hour, which
corresponds to a bulk velocity of 133.5 mm/s; this is in
very good agreement with the average value of
134 mm/s obtained from the SPIV measurements. This
confirms that the displacement of the calibration grid
has been sufficiently accurate during the 3D calibration
procedure, because an error in the displacement results
in a linearly proportional error in the estimation of the
axial velocities (Sect. 5.3) The figure also shows that
the flow rate is kept constant during the observation
period, which is important for the measurement of the
turbulence statistics.
The mean axial velocity profile is shown in Fig. 19.
The agreement between the results of the SPIV and
the DNS by Eggels et al. (1994) is very good. Error
bars, representing the 95% reliability interval due to
the sampling error, have been plotted for r/D � 0.1,
0.2, 0.3, 0.4, 0.45 and 0.48, but most of them are not
visible because they are smaller than the symbols. The
friction velocity u*(1) = 9.24 mm/s, used to normalize
the measurements, is calculated from the Fanning
friction factor and the bulk velocity: u�ð1Þ ¼ Ub
ffiffiffiffiffiffiffiffif=2
p:
7.3 Turbulence statistics
Figure 20 illustrates the turbulent intensities and
Fig. 21 displays the viscous stress, the Reynolds stress
272 Exp Fluids (2007) 42:259–279
123
and the total stress profiles. The agreement between
the SPIV and the DNS data is again very good, and the
only significant deviation occurs in the RMS(ur) for
r/D ‡ 0.45. Most of the error bars, which have been
plotted for r/D � 0.1, 0.2, 0.3, 0.4, 0.45 and 0.48, are
smaller than the symbols.
We note that the friction velocity u*(2) used to nor-
malize the turbulence intensities has been obtained by
extrapolation of the linear profile of the measured total
stress, i.e. the sum of the viscous and Reynolds stress,
to the wall. The Reynolds stress vanishes at small
scales and therefore would be less influenced by (pos-
sibly unresolved) small scale turbulence. The resulting
value is found to be u*(2) = 8.75 mm/s, which is
appreciably smaller than the friction velocity
u*(1) = 9.24 mm/s used for the renormalization of the
mean velocity profile. We conjecture that this differ-
ence would be caused by the rather coarse spatial
resolution of the SPIV measurements, which acts as a
low pass filter on the data. This will be discussed in
more detailed in the next Sect. 7.4.
7.4 Spatial resolution
The underestimation of the turbulent energy, the
Reynolds-stress and u*(2), which was found in the
previous section, is due to the limited spatial resolu-
tion, which is inherent to PIV measurements. A PIV
velocity vector is determined from the mean displace-
ment of the particles in the corresponding interroga-
tion area. The estimated velocity is therefore the
spatial average of the velocity in the volume deter-
mined by the dimensions of the interrogation area and
the light sheet thickness.
In order to obtain a first estimation of the effect of
the limited spatial resolution on the turbulent mea-
surements, we have filtered the spectra of the axial and
radial velocity components from the DNS by Eggels
et al. (1994). These two spectra are shown together
with a k–5/3 model in Fig. 22. The velocity spectra from
the DNS are calculated from a time series at the center
of the pipe and the frequency is converted to a wave
number with the mean velocity. The k–5/3 spectrum is
modeled with a constant exponent (slope in a log–log
−20 −15 −10 −5 0 5 10 15 20
−20
−15
−10
−5
0
5
10
15
20
Fig. 17 Example of the instantaneous turbulent flow field in across-section of the pipe. Displayed resolution is 8 · 8 px, or0.5 · 0.5 mm2
200 400 600 8000
50
100
150
time (s)
mea
n flo
w r
ate
(mm
/s)
500 525 550 575 600132
133
134
135
136
Fig. 18 Flow rate measured by integration of uz over the cross-section of the pipe
0 0.1 0.2 0.3 0.4 0.50
5
10
15
20
r / D
<uz>
/ u*(
1)
050100150
0
20
40
60
80
100
120
140
160
180
y+<u
z> (m
m/s
)
Fig. 19 Mean streamwise velocity profile
Exp Fluids (2007) 42:259–279 273
123
plot) from the integral length scale L � 0.25
D = 10 mm, down to the Kolmogorov length scale
g = (m3/e)1/4 = 0.17 mm, where m is the kinematic vis-
cosity and e = f Ub3/R is the energy dissipation.
In Table 4 the energy losses related to the filtering
of the spectra are shown for two different filters and
two different filter lengths. The sinc2-function is the
filter that corresponds to the calculation of a uniform
average of the velocity in an interrogation area (Willert
and Gharib 1991), and it seems therefore the most
appropriate description of the PIV method. The cut-off
filter, which corresponds to a non-uniform averaging of
the velocity field (weighted with a sinc function), is
shown to demonstrate the large influence of the shape
of the filter on the energy loss. The (Gaussian) distri-
bution of the light intensity over the light sheet thick-
ness will for instance lead to a non-uniform averaging
of the velocity in this direction.
The energy loss observed in our experiments is
about 10% for all velocity components, i.e. (u*(2)/
u*(1))2 = 0.9. The energy losses predicted in Table 4
vary from 0.1 to 47%, depending on the spectrum, the
filter and the filter length. One would expect the filter
length to be around 1.8 mm (32 px) and the sinc2-filter
to be the best representation of the spatial filtering, but
the energy losses for this combination are beyond any
realistic value (47 and 24% for the DNS-spectra). Ei-
ther the filtering of the PIV is overestimated, or the
spectra of the DNS are incorrect and contain too much
energy at high wave numbers.
Based on the large energy losses predicted in Ta-
ble 4, we conclude that a significant underprediction of
the turbulent energy and Reynolds stress has to be
expected from the spatial filtering related to the SPIV
technique. However, a more precise analysis needs to
be performed to be able to make an accurate predic-
tion. For example, the higher turbulence levels found
in the DNS results may possibly be related to the fact
that the DNS is computed for a periodic pipe section
with a prescribed pressure gradient (Eggels et al.
1994), so that the total mass flow rate can fluctuate
slightly. This would slightly increase the observed tur-
bulence levels in the DNS data and contribute to the
observed differences between the experimental and
numerical results.
7.5 Bias
In Fig. 23 we show the measured mean in-plane
velocities, which should in principle be zero in fully
0 0.1 0.2 0.3 0.4 0.50
0.5
1
1.5
2
2.5
3
r/D
rms(
u′)
/ u*(
2)
rms(ur)
rms(ut)
rms(uz′)
050100150
0
5
10
15
20
25
y+
rms(
u′)
(mm
/s)
Fig. 20 Turbulence intensities
0 0.1 0.2 0.3 0.4 0.50
0.2
0.4
0.6
0.8
1
r/D
norm
alis
ed s
tres
s
<ur u
z′>
ν d<uz> / dr
total stress
050100150
0
10
20
30
40
50
60
70
80
y+
stre
ss (
mm
2 /s2 )
Fig. 21 Stress profiles in turbulent pipe flow
0 10 20 30 40 500
0.05
0.1
0.15
0.2
k D
norm
aliz
ed p
ower
spe
ctra
l den
sity
E(k
) Fuzu
z DNS
Furu
r DNS
k−5/3 model
sinc2(k ∆x) filtercut−off filter
0
0.2
0.4
0.6
0.8
1
filte
rs
Fig. 22 The normalized spectral density and normalized spatialfilters; see text for explanation
274 Exp Fluids (2007) 42:259–279
123
developed turbulent pipe flow. The deviations from
zero are all smaller than 1% of the maximum stream-
wise velocity, which is similar to the accuracy obtained
for the bias in the laminar flow measurements
(Fig. 13).
The turbulence stress components Æur uhæ and Æuh uzæare expected to be zero as well. The measured values
are displayed in Fig. 24, and the small deviations from
zero can be explained by the statistical fluctuations
that arise from the finite length of the measurement
sequence.
7.6 Peak locking
Most PIV measurements suffer from some degree of
peak locking (Westerweel 2000b). Significant peak
locking can affect the statistics of the histogram,
such as the mean and the RMS (Christensen 2004).
Figure 25 shows several histograms of the particle
displacement from camera 1 (ux1 and uy1) before the
reconstruction of the 3C-vector field. A small amount
of peak-locking is visible for ux1 in the proximity of the
wall. In the center of the flow, the peak locking results
in some weak fluctuations in the slope of the histo-
grams.
7.7 Velocity errors due to misregistration
The effect of velocity errors due to misregistration on
the statistics of turbulence measurements is investi-
gated for the case that the light sheet is misaligned on
purpose and displaced 0.3 mm in the downstream
direction. Figure 26 shows the mean in-plane velocity
component < ux > along the x-axis. The error is seen to
be large in the near wall region were the velocity
gradient is large as well. As for the laminar flow
measurements (Sect. 5.2), the errors due to misregis-
tration are negligible for the other velocity components
(uy and uz). Because the velocity error of Æuxæ is large
only on the left and right side of the tube, it is only the
RMS of ur that is affected by the misregistration. This
can be seen in Fig. 27. The effect on the Reynolds
stress is shown in Fig. 28.
7.8 Spatial correlations
Figures 29 and 30 show auto-correlation functions of
the streamwise velocity fluctuations at different radial
positions. First the 2D-correlation, from a particular
grid point in the xy-plane to all the other grid points of
the vector field was calculated. The correlation func-
tions shown in Figs. 29 and 30 are the cross-sections in
azimuthal and radial direction through this 2D-corre-
lation function. To improve the statistical convergence,
the 2D-correlation function was calculated at eight
different azimuthal positions and then averaged.
The negative correlation in the azimuthal direction,
which corresponds to the spanwise direction in
boundary layer flow and channel flow, can be attrib-
uted to the presence of low and high speed streaks in
Table 4 Relative energy loss (in%), for different 1D-spectra andfor different filter settings, which simulate the spatial filtering ofthe PIV technique
Filter Filter length DNS Furur DNS Fuzuz k–5/3
Cut-off 1.8 mm (32 px) 14 4 7Sinc2 1.8 mm (32 px) 47 24 17Cut-off 0.9 mm (16 px) 0.4 0.1 4Sinc2 0.9 mm (16 px) 24 10 10
−0.5 0 0.5
−0.2
0
0.2
x/D or y/D
norm
alis
ed m
ean
in−
plan
e ve
loci
ties
<ux>/u
*(2) on X−axis
<uy>/u
*(2) on X−axis
<ux>/u
*(2) on Y−axis
<uy>/u
*(2) on Y−axis
Fig. 23 Mean in-plane velocity profiles
0 0.1 0.2 0.3 0.4 0.5−0.05
−0.025
0
0.025
0.05
r/D
norm
alis
ed s
tres
s<u
r u
t>
<ut u
z′>
020406080100120140160
−0.4
−0.2
0
0.2
0.4
y+
stre
ss (
103 m
m2 /s
2 )
Fig. 24 Turbulent stress components Æur uhæ and Æuh uzæ in(mm2/s2) and normalised by u*(2)
2
Exp Fluids (2007) 42:259–279 275
123
the buffer layer. The spanwise position of the minimum
in the correlation function corresponds to the mean
separation between the low and high speed streaks. In
literature, the streak spacing is defined as two times the
position of the minimum in the correlation function.
From Fig. 29 it follows that very close to the wall
(y+ = 9) the mean streak spacing is found to be about
100 wall units. The streak spacing increases with the
distance from the wall, to about 200 wall units halfway
to the center (y+ = 90). These observations are in
agreement with the values reported in the literature
(Robinson 1991; Van der Hoeven 2000).
In Fig. 30 we show the correlation function in the
radial direction. In the center of the pipe, the axial
velocity is negatively correlated with the flow around
it. This is a consequence of the conservation of mass,
which dictates that the flow rate is constant everywhere
in the pipe. It is further found that for r1/D ‡ 0.38 there
is no correlation between points on opposite sides of
the center of the pipe.
8 Quasi-3D flow structure
In this section we give an example of the 3D flow
structures that were observed in the transition from
laminar to turbulent flow in a pipe. As explained
before, the light sheet of the SPIV system is perpen-
dicular to the mean flow direction, and therefore the
flow structures are advected by the mean flow through
the measurement plane. Applying Taylor’s hypothesis,
the quasi-instantaneous 3D flow field can be recovered
−9 −8 −7 −6 −5 −4 −3 −2 −1 0 1 20
0.2
0.4
0.6
0.8
1
uy1
1
53
ux1 1
2
3
4
5 6
displacement (px)
prob
abili
ty d
ensi
ty (
1/px
)
1: r/R<0.22: 0.2<r/R<0.43: 0.4<r/R<0.64: 0.6<r/R<0.85: 0.8<r/R<0.956: 0.9<r/R<0.95
Fig. 25 Pdf’s of ux1 and uy1 measured by camera 1
−0.5 0 0.5−1.5
−1
−0.5
0
0.5
1
1.5
x / D
<u x>
/ u *(
2)
on x
−ax
is
aligned0.3 mm misaligned
Fig. 26 Effect of the registration error on the measurement ofthe mean in-plane velocity Æuxæ in turbulent flow
0 0.1 0.2 0.3 0.4 0.50
0.2
0.4
0.6
0.8
1
r / D
rms(
u) /
u *(2)
rms(uθ)
rms(ur)
DNSaligned0.3 mm misaligned
Fig. 27 Effect of the registration error on the measurement ofthe turbulent intensities in turbulent flow
0 0.1 0.2 0.3 0.4 0.50
0.2
0.4
0.6
0.8
1
r / D
<u z u
r> /
u2 *(2)
DNSaligned0.3 mm misaligned
Fig. 28 Effect of the registration error on the measurement ofthe Reynolds-stress in turbulent flow
276 Exp Fluids (2007) 42:259–279
123
from a time-resolved measurement sequence. We as-
sume a single constant advection velocity for the
structures in the flow, for which we take the bulk
velocity U and this, although not quite correct, will at
least provide us with a good qualitative impression of
the 3D structure of the flow. The normalized stream-
wise distance follows from z� ¼ ðt0 � tÞU=D:
A puff, which is a turbulent spot at low Reynolds
number (Wygnanski et al. 1975), was created by
injection of a strong jet for short duration ð1D=UÞtrough a (1 mm) hole into fully developed laminar pipe
flow at Re = 2,000. The SPIV measurements were
made 150 D downstream of the injection point. For
these measurements the cameras and laser were re-
placed by much faster components (Van Doorne et al.
2003), and time-resolved measurements were made at
62.5 Hz.
In Fig. 31 we show a 3D visualization of the iso-
surfaces of the streamwise vorticity, which was evalu-
ated with the circulation method (Raffel et al. 1998).
The numerous streamwise vortices form a complicated
structure and reveal the internal organization of the
flow. A more detailed investigation showed that a
quasi-periodic formation of strong hair-pin like vorti-
ces at the upstream end of the puff result in a contin-
uous transition from laminar to turbulent flow at this
location (Van Doorne 2004). In the interior of the
turbulent region transients of traveling waves were
observed (Hof et al. 2004). A related discussion on the
flow structure and transition scenario induced by a
periodic injection and extraction of fluid from the wall
is given by Van Doorne et al. (2006).
9 Summary and conclusions
Stereoscopic-PIV was applied to measure the instan-
taneous velocity field over the entire circular cross-
section of a pipe. The system is based on an angular
displacement of 45� of the two cameras and a 3D cal-
ibration based reconstruction method proposed by
Soloff et al. (1997). A special calibration grid was made
to fit into the pipe and give the two cameras, which
stand on either side of the light sheet, a clear view on
both sides of the grid.
It was expected that the large out-of-plane motion of
the tracer particles in the light sheet would limit the
accuracy of the measurements, which was therefore
investigated in great detail for laminar and turbulent
flow.
The laminar flow measurements revealed the
importance of a precise alignment of the light sheet
with respect to the calibration plane. Misalignments as
small as 0.1 mm will lead to large errors due to mis-
registration. Although the velocity error due to mis-
registration was described by several authors before, it
was never properly quantified. We explain the origin of
this error and predict its effect and magnitude. We
validate this based on a comparison of the measured
and theoretical properties of laminar and turbulent
pipe flow.
After alignment, the laminar velocity profile and
turbulent statistics were reproduced with very high
accuracy. The noise level of individual vectors was
smaller than 0.1 px units, which corresponds to 1% of
0 50 100 150 200 250
−0.2
0
0.2
0.4
0.6
0.8
1
(θ1−θ
2) r u* / ν
<u z′(θ
1,r)
u z′(θ2,r
)> /
<u z′(θ
1,r)2 >
(1)
(1) r/D=0.125 ; y+=131(2) =0.25 ; =88(3) =0.375 ; =44(4) =0.425 ; =26(5) =0.475 ; =9
(5)
Fig. 29 Correlation of the streamwise velocity fluctuations in theazimuthal direction at different distances from the wall
−0.5 0 0.5
0
0.2
0.4
0.6
0.8
1
r2 / D
<u z′(r
1) u z′(r
2)> /
<u z′(r
1)2 >
(1) (2) (3)(1) r1/D=0
(2) =0.125(3) =0.25(4) =0.375(5) =0.425(6) =0.475
04080120160
y+
Fig. 30 Correlation of the streamwise velocity fluctuations in theradial direction at different distances from the wall
Exp Fluids (2007) 42:259–279 277
123
the maximum streamwise velocity. This demonstrates
that the measurements were able to determine the
secondary fluid motion in the plane perpendicular to
the pipe axis, which are an order of magnitude smaller
than the axial fluid motion, and we found that artifacts
due to the misalignment of the calibration targer are
much smaller than the velocity variations. This dem-
onstrates the applicability of SPIV to flows with large
out-of-plane motion.
Applying Taylor’s hypothesis, the quasi-instanta-
neous 3D flow field can be recovered from a time-re-
solved measurement sequence. These measurements,
which are the first of this kind in pipe flow, have en-
abled us to derive several quantities which until now
could only be obtained from numerical simulations.
These are:
1 the full 3D structure of vortices and streaks in the
flow, which are visualized by perspective viewing of
the iso-surfaces of the fluctuations of the stream-
wise velocity and the vorticity;
2 the fluxes of e.g. the mass, momentum and kinetic
energy, which can be obtained from integration of
the velocity field over the cross-section of the pipe.
As an example we have presented the 3D structure
of the streamwise vortices in a turbulent puff. This type
of structural information is extremely valuable in
understanding transition in a pipe (Van Doorne 2004),
and the approach can easily be extended to other flow
configurations as well.
Acknowledgments This work is part of the research pro-gramme of the ‘Stichting voor Fundamenteel Onderzoek derMaterie (FOM)’, which is financially supported by the ‘Neder-landse Organisatie voor Wetenschappelijk Onderzoek (NWO).’
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