measurement of laminar, transitional and turbulent pipe

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.


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.


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 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 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).


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+ (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.


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

<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 *(

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

 /

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

 /



(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

 /



(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).’

References

Born M, Wolf E (1975) Principles of optics. Pergamon Press,Oxford

Christensen K (2004) The influence of peak-locking errors on theturbulence statistics computed from PIV ensembles. ExpFluids 36:484–497

Coudert S, Schon J (2001) Back-projection algorithm withmisalignment corrections for 2D-3C stereoscopic PIV. MeasSci Technol 12:1371–1381

Draad A (1998) Laminar-turbulent transition in pipe flow for Newtonian and non- N ewtonian fluids. PhD thesis, DelftUniverstity of Technology

Draad A, Nieuwstadt F (1998) The earth’s rotation and laminarpipe flow. J Fluid Mech 361:297–308

Draad A, Kuiken G, Nieuwstadt F (1998) Laminar-turbulenttransition in pipe flow for N ewtonian and non- N ewtonianfluids. J Fluid Mech 377:267–312

Eggels J, Unger F, Weiss M, Westerweel J, Adrian R, FriedrichR, Nieuwstadt F (1994) Fully developed pipe flow: acomparison between direct numerical simulations andexperiments. J Fluid Mech 268:175–209

Faisst H, Eckhardt B (2003) Traveling waves in pipe flow. PhysRev Lett 91:224502

Foucaut J, Miliat B, Perenne N, Stanislas M (2004) Character-ization of different PIV algorithms using the EUROPIVsyntetic image generator and real images from a turbulentboundary layer. In: Stanislas M et al (eds) Particle imagevelocimetry: recent improvements. Proceedings of the EU-ROPIV 2 workshop on particle image velocimetry, Zar-agoza, Spain, March 31–April 1 2003, Springer, BerlinHeidelberg New York

Hoeven Jvd (2000) Observations in the turbulent boundary-layer. PhD thesis, Delft Universtity of Technology

Hof B, Doorne C, Westerweel J, Nieuwstadt F, Faist H,Eckhardt B, Wedin H, Kerswell R, Waleffe F (2004)Experimental observation of nonlinear traveling waves inturbulent pipe flow. Science 305:1594–1598

Hutchins N, Hambleton W, Marusic I (2005) Inclined cross-stream stereo particle image velocimetry measurements inturbulent boundary layers. J Fluid Mech 541:21–54

Jimenez (1994) Resolution requirements for velocity gradients inturbulence. Annu Res Briefs (CTR), pp 357–364

Keane R, Adrian R (1993) Theory of cross-correlation of PIVimages, In: Nieuwstadt F (ed) Flow visualization and imageanalysis. Kluwer, pp 1–12

Kim J, Moin P, Moser R (1987) Turbulence statistics in fullydeveloped channel flow at low reynolds number. J FluidMech 177:133–166

Lawson N, Wu J (1997) Three-dimensional particle imagevelocimetry: error analysis of stereoscopic techniques. MeasSci Technol 8:894–900

Marassi M, Castellini P, Pinotti M, Scalise L (2004) Cardiac valveprosthesis flow performances measured by second and third-stereo particle image velocimetry. Exp Fluids 36:176–186

Matsuda T, Sakakibara J (2005) On the vortical structure in around jet. Phys Fluids 17

Meyer K, Westerweel J (2000) Advection velocities of flowstructures estimated from particle image velocimetry mea-surements in a pipe. Exp Fluids 29:S237–S247

Fig. 31 3D visualization ofthe iso-contours ofstreamwise vorticity (±3.5 s–1)in a puff, withz� ¼ ðt0 � tÞU=D

278 Exp Fluids (2007) 42:259–279

123

Prasad A (2000) Stereoscopic particle image velocimetry. ExpFluids 29:103–116

Prasad A, Adrian R (1993) Stereoscopic particle image veloc-imetry applied to liquid flows. Exp Fluids 15:49–60

Raffel M, Willert C, Kompenhans J (1998) Particle imagevelocimetry. Springer, Berlin Heidelberg New York

Robinson S (1991) Coherent motions in the turbulent boundarylayer. Annu Rev Fluid Mech 23:601–639

Scarano F (2002) Iterative image deformation methods in PIV.Meas Sci Technol 13:R1–R19

Schlichting H (1955) Boundary-layer theory. McGraw-Hill BookCompany

Schmid P, Henningson D (1994) Optimal energy density growthin Hagen–Poiseuille flow. J Fluid Mech 277:197–225

Soloff S, Adrian R, Liu Z (1997) Distortion compensation forgeneralised stereoscopic particle image velocimetry. MeasSci Technol 8:1441–1454

Taylor GI (1938) The spectrum of turbulence. Proc R Soc LondA 132:476–490

Trefethen L, Trefethen A, Reddy S, Discroll T (1993) Hydro-dynamic stability without eigenvalues. Science 261:578–584

van Doorne C (2004) Stereoscopic PIV on transition in pipe flow.PhD thesis, Delft Universtity of Technology

van Doorne C, Westerweel J, Nieuwstadt F (2006) On the spatialevolution of a wall-imposed periodic disturbance in pipe Poiseuille flow at Re = 3,000. Part 2: experiments, submittedto J Fluid Mech

van Doorne C, Hof B, Lindken R, Westerweel J, Dierksheide U(2003) Time resolved stereoscopic- PIV in pipe flow.Visualising 3D flow structures. Proceedings of fifth interna-tional symposium on particle image velocimetry. Busan,South-Korea, September 22–24

Waleffe F (1997) On a self-sustaining process in shear flows. PhysFluids 9:883–899

Wedin H, Kerswell R (2004) Exact coherent structures in pipeflow: traveling wave solutions. J Fluid Mech 508:333–371

Westerweel J (1993) Digital particle image velocimetry. Theoryand application. PhD thesis, Delft Universtity of Technology

Westerweel J (2000a) Theoretical analysis of the measurementprecision in particle image velocimetry. Exp Fluids 29(Sup-plemantary): S3–S12

Westerweel J (2000b) Effect of sensor geometry on the perfor-mance of PIV interrogation. In: Adrian RJ (ed) Lasertechniques applied to fluid mechanics. Springer, BerlinHeidelberg New York, pp 37–55

Westerweel J, Draad A (1996) Measurement of temporal andspatial evolution of transitional pipe flow with PIV, devel-opments in laser techniques and fluid mechanics. Eighthinternational symposium, Lisbon, Portugal, pp 311–324

Westerweel J, Nieuwstadt FTM (1991) Performance tests on 3-dimensional velocity measurements with a two-cameradigital particle-image velocimeter. In: Dybbs A, GhorashiB (eds) Laser anemometry—advances and applications.ASME, New York

Westerweel J, van Oord J (2000) Stereoscopic PIV measure-ments in a turbulent boundary layer. In: Stanislas M,Kompenhans J, Westerweel J (eds) Particle Image Veloc-imetry: progress towards industrial application. Kluwer,Dordrecht

Wieneke B (2005) Stereo PIV using self-calibration on particleimages. Exp Fluids 39:267–280

Willert C (1997) Stereoscopic digital particle image velocimetryfor application in wind tunnel flows. Meas Sci Technol8:1465–1479

Willert C, Gharib M (1991) Digital particle image velocimetry.Exp Fluids 10(4):181–193

Wygnanski I, Sokolov M, Friedman D (1975) On transition in apipe. Part 2 the equilibrium puff. J Fluid Mech 69:283–304

Exp Fluids (2007) 42:259–279 279

123

measurement of laminar, transitional and turbulent pipe

Documents