12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

SINGLE-PIXEL PIV MEASUREMENTS OF HIGH-RE TURBULENT PIPE
FLOW

Jerry Westerweel
Laboratory for Aero & Hydrodynamics
Delft University of Technology
Mekelweg 2, 2628 CD Delft, Netherlands
j.westerweel@tudelft.nl

Sudarshan SridharanDept. Process & Energy, Faculty 3ME
Delft University of Technology
Mekelweg 2, 2628 CD Delft, Netherlandss.sridharan@tudelft.nl

Gosse Oldenziel
Deltares & Delft University of Technology
P.O. Box 177, 2600 MH Delft, The Netherlands
gosse.oldenziel@deltares.nl

ABSTRACTIn this paper we present PIV measurements of turbulent pipe flow at Reynolds numbers between 3.4×105 and6.9×105 . We apply a so-called ‘single-pixel correlation’ thatyields a superior spatial resolution (Westerweel et al., 2004).
We use the location and shape of the averaged correlation peakto obtain the mean velocity and normal and Reynolds stresses(Scharnowski et al., 2012). A novel aspect of the single-pixel
correlation approach is the extension to determine the spatial correlation of the velocity fluctuations. In this paper we
present the results for Re = 4.98×105 , corresponding to a shear
Reynolds number Reτ = 10.3×103 , with a spatial resolution of∆y+ = 18.

The shape of the displacement-correlation peak represents
the local probability-density function for the displacement thatis convoluted with the particle-image self-correlation (Westerweel, 2008). This makes it possible to estimate also the normal stresses and Reynolds stress from the measured shape ofthe correlation peak (Scharnowski et al., 2012). However, tofully describe the turbulent flow statistics, it is necessary toalso measure the local length scales. In this paper, we present
a further extension to estimate the local spatial correlation of
the velocity fluctuations that is based on the single-pixel algorithm.

INTRODUCTION
One of the main limitations in obtaining experimental
data of wall turbulence at very high Reynolds number is thelimited spatial resolution, especially in the near-wall region.
Intrusive measurement methods, such as Pitot tube and hotwire anemometry, are limited by the dimensions of the probe.
Therefore, there has been a drive to design smaller measurement probes (Vallikivi et al., 2011). An alternative is to change
the scale of the facility relative to the typical dimensions of theprobe (Vinuesa et al., 2016). Particle image velocimetry (PIV)
is a non-intrusive measurement method where commonly
the instantaneous flow velocity field is estimated in smallinterrogation regions (Adrian & Westerweel, 2011). The finite
dimensions of the interrogation windows, typically 32×32
pixels, limit also the spatial resolution in the near wall region.
Using an optical system with a high magnification Willert
et al. (2017) obtained detailed measurements in the near-wallregion of the CICLoPE facility, although a mirror needed to be
introduced into the facility to accommodate the measurements.

0.02
0.018
0.0005

0.016

0.04

0.00020.014

0.00010.00005
0.000020.00001

0.012

0.03
1

0.01
105

106

0.02

2
0.01
3

103

104

105

106

107

Figure 1. The friction factor c f for pipe flow as a function of
the bulk Reynolds number ReD . The blue dots represent thepresent data in the Alpha Loop. The numbered lines represent:(1) the friction factor for Poiseuille flow (c f = 64/ReD ); (2) the
friction law for a smooth wall (Zagarola & Smits, 1998); and
1/4(3) the Blasius friction law (c f = 0.316/ReD ). Open symbols represent experimental data (den Toonder & Nieuwstadt,
1997; McKeon et al., 2004), and closed symbols numerical
data (Eggels et al., 1994; Wu & Moin, 2008; Ahn et al., 2015).
The inset shows the present data along with lines for different
values of the relative roughness ks /D according to the correlation of Colebrook (1939).

Another approach is to reduce the spatial resolution of
the PIV measurement to a single pixel. This approach was
first described for microfluidic applications (Westerweel et al.,2004), and that was later also applied to measure the mean
velocity profile in the near-wall region of a turbulent boundary
layer (Kähler et al., 2006).

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
EXPERIMENTAL SET-UP
We demonstrate this approach by measuring the turbulent
flow in the ‘Alpha Loop’ at Deltares (Delft, The Netherlands).
This facility is a closed-loop water-filled pipe with a length of
320 m and a diameter of 206 mm. The flow loop is intended
for industrial-scale testing of multiphase flows, and is madeof a smooth steel pipe. Measurements of were done at bulk
Reynolds numbers ReD (= Ub D/ν) between 3.4×105 and6.9×105 (where Ub is the bulk velocity, D is the pipe diameter,and ν the kinematic viscosity of the fluid). In Figure 1 weshow the friction coefficients based on the measured pressure
gradient and measured bulk flow rate. The inset of Fig. 1
shows that the pipe has a small roughness of ks /D = 3×10−5 ,corresponding to ks = 6.2 µm.
For the PIV measurements the pipe was fitted with a transparent test section (made of PMMA) enclosed in a rectangular transparant water-filled box that can withstand the operating pressure (3 bar) and avoids serious light refraction fromthe curved pipe wall; a special arrangement of slits reduced
internal reflections of the light sheet from the recorded images (Sridharan, 2018). The flow was seeded with 10 µm
diameter tracer particles (Sphericel 110P8, Potters Inductries
Inc.). The flow was illuminated with a 0.8 mm thick lightsheet generated from the beam of a double-pulsed frequencydoubled dual Nd:YAG laser with a light wavelength of 532 nm.
Images were recorded using a CCD camera (SensiCam QE,
1376×1040 pixels with 6.45 µm pixel pitch) equipped with a
50 mm lens (Micro Nikkor) using a f /4 aperture stop. The image magnification was M0 =0.035, ensuring ample focal depth
and diffraction-limited particle images of 5.4 µm diameter.
The recorded images were pre-processed using a 5×5-px minmax filter (Adrian & Westerweel, 2011) to reduce reflections
that are visible in the image and normalize the image contrast(see Figure 2); this reduces the number of spurious vectors (seebelow) by a factor of 4-6. Any remnants of the internal reflections that occur as thin horizontal lines near the pipe walls,
were removed using a Fourier filter, similar to the method forremoving striations in LIF images (Westerweel et al., 2011);
see Fig. 2.

...

...

Figure 2. Image processing: (a) raw image; (b) contrast normalization by means of a 5×5-pixel min-max filter (Adrian &Westerweel, 2011); (c) detail near the lower pipe wall in image(b), and: (d) removal of remaining reflections in (c) by meansof a Fourier filter (Westerweel et al., 2011).

u’2 1/2 v’2 1/2v’2

u’v’ =

u’2

images 1a, 2a, 3a, ...

images 1b, 2b, 3b, ...

Figure 3. Principle of single-pixel PIV. For a set of image
pairs the intensity at a given pixel location in the first exposure
is correlated over a small region in the second exposure. The
velocity is determined from the location of the displacementcorrelation peak. The shape and orientation of the correlation
peak yields hu0 2 i, hv0 2 i, and hu0 v0 i.

SINGLE-PIXEL CORRELATION

recording time-resolved flows. Hence, this approach is very
attractive for stationary turbulent flows where a high spatial
resolution is required.

The conventional PIV method sub-divides each image
pair in small interrogation windows, of typically 32×32
pixels, and determines the local instantaneous particle-image
displacement by computing the spatial correlation in typically 32×32-pixel interrogation windows, and identifying
the location of the displacement-correlation peak in eachinterrogation window; the velocity is obtained by dividing
the particle-image displacement by the exposure time-delay
between the two laser light pulses that illuminate the flow
(Adrian & Westerweel, 2011).

When the correlation is averaged over a sufficientnumber of frame pairs, it is also possible to extract thenormal and Reynolds stresses from the shape and orientation of the displacement-correlation peak (Scharnowski
et al., 2012). The correlation peak represents the local
time-averaged probability-density function of the particleimage displacements, convoluted with the particle-image
self-correlation (Westerweel, 2008; Adrian & Westerweel,2011). This self-correlation can be obtained from computing
the auto-correlation of the recorded images. Then a general two-dimensional Gaussian distribution is fitted to thedisplacement-correlation peak, which is corrected for the
convolution with the particle-image self-correlation. Finally,
the normal and Reynolds stresses are obtained from thesecond-order moments of the remaining peak; see Figure 3(Scharnowski et al., 2012).

Alternatively, the particle-image displacement can befound by correlating a single pixel in the first image framewith pixels in a small domain in the second frame, andsumming the correlations over many frame pairs; see Figure 3.
When one uses 1024 frames, in principle the same amount of
information is processed as for a single 32×32-pixel (=1024
pixels) interrogation window. However, the gain is that the
spatial resolution is now determined by the dimension of asingle pixel, rather than the dimension of the interrogation
window; one thus improves the spatial resolution by morethan an order of magnitude, at the cost of losing the ability ofIn this paper we introduce an extension to the single-pixel
approach, and that is the evaluation of spatial correlation of2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

2.5

U [m/s]

2.0
1.5
1.0
0.5
0.0

20

40

60
80
100
radial position [mm]

96

98

100

102

Figure 5. (left) Measured mean axial velocity profile for the
lower half of the pipe. Black dots represent the conventional
PIV with 32×32-pixel interrogation windows; red dots are thesingle-pixel correlation results. (right) Detail for the 10 mmregion near the wall.

Figure 4. The estimation of the joint probability function
f (u1 , u2 |x1 , x2 ) by the summation of single-pixel correlations
at two positions separated by a distance ∆x. The summation iscarried out over all pixels along any homogeneous direction of
the flow (here the axial direction for turbulent pipe flow) andover all recorded frames.

30(log y + +2.0)/0.405
25

the velocity fluctuations. This is normally done by correlating the velocity fluctuations in two points. For a flow with ahomogeneous direction, such as the axial direction in a turbulent pipe flow, the correlation is only a function of the distance∆x = x2 − x1 . Alternatively, the correlation can be found fromthe joint moment of the (estimated) joint probability density
function:

U+

20
15
10
5

Ruu

(∆x) = u0 (x)u0 (x + ∆x) =ZZu1 u2 f (u1 , u2 |x1 , x2 )du1 du2 ,

0
10

(1)
with: u1 = u0 (x), u2 = u0 (x + ∆x), and: ∆x = x2 − x1 . Given
that each single-pixel correlation for an individual frame pair isan estimate of the probability density function (pdf) f (u, v|x),
we multiply the pdf’s at two distinct positions, and thus
obtain an estimate of the joint probability density function
f (u1 , v1 , u2 , v2 |x1 , x2 ). We now simplify this by considering a
projection of the joint pdf for only one velocity component, i.e.
f (u1 , u2 |x1 , x2 ). By averaging over a substantially large number of pixel pairs (i) in a large set of frame pairs, and making
use of any homogeneous direction in the flow (here the axial
flow direction), we can obtain a converged result from which
we can estimate the spatial correlation from the joint momentof the (estimated) probability density function:
f (u1 , u2 |x1 , x2 ) ∼= ∑ f (i) (u1 |x1 ) × f (i) (u2 |x2 ),

100

1000

10000

y+

Figure 6. The mean axial velocity profile in wall units. Symbols correspond to this in Fig. 5. The first reliable measurement for conventional PIV (black dots) occurs at y+ = 484 andfor single-pixel PIV at y+ = 56.

The 32×32-pixel interrogation result contains on averageless than 1.8% spurious vectors (67 outliers per image pairyielding 3,763 vectors each, with 45 and 96 as the 5- and 95percentiles, respectively); this indicates that there is sufficientseeding and negligible loss-of-correlation due to out-of-planemotion.
The spatial resolution of the conventional PIV is givenby the dimensions of the 32×32-pixel interrogation window.
For the measurement presented here this corresponds to a
resolution of ∆y+ ∼= 600. Evidently, this is too crude to
make measurements in the near-wall region. The single-pixel
approach gives a measurement at each (radial) pixel location,and thus gives a spatial resolution of ∆y+ = 19. This would
in principle be sufficient to determine the flow profiles inthe entire log-region. Figure 5 shows the measured velocity
profiles with both the conventional and the single-pixel PIV
approaches. The conventional PIV was performed over theentire flow, while the single-pixel results were only done forthe near wall flow region, and ensuring a sufficient overlap
with the conventional PIV data. Due to interference withremaining internal reflections, a few of the single-pixel results
deviate with respect to the overall result. Figure 6 shows thesame data, but now in a semi-log plot, which illustrates thehigh level of accuracy of the single-pixel PIV results andthe extension of the range in the log-region by a decade inwall units. The profile coincides with the log profile with avon-Kármán constant of k = 0.405.

(2)i

as illustrated in Figure 4. Evidently, this approach only works
when a particle image is present in x1 and another particle image is present in x2 . This is why a large number of pixel pairs
needs to be evaluated; by making use of a homogeneous direction in the flow—here the axial direction of the turbulent
pipe flow—and summing over a large number of frames, aconverged result is obtained. In the present case we have 9000
frame pairs and 900 pixels along an image line, yielding a totalof more that 8×106 pixel pairs for each separation ∆x.

RESULTSFor each Reynolds number a total of 104 double-frame
images were recorded. In this paper we present the results
obtained at ReD = 4.98×105 using a subset of 9000 image
pairs. The corresponding shear Reynolds number Reτ =10.3×103 , with: Reτ = uτ R/ν, where uτ is the wall friction
velocity, and R = 21 D the pipe radius.

Figure 7 shows the results for the normal stresses and312th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

R uu( x) 10 3 [mm2/s2]

0.0

u’v’+

-0.2
-0.4
-0.6
-0.8
-1.0

6
4
2
0
0.0

0.1

u’+ , v’+3

0.3
0.4 0.00
x/D

0.02

0.04

0.06

Figure 9. (left) The spatial correlation Ruu (∆x) of the axial velocity fluctuations at the pipe centerline. Symbols correspond to those in Fig. 5. (right) Detail of Ruu (∆x) for∆x/D < 0.075.

2

1

0
-0.5

0.2

-0.4

-0.3

-0.2

-0.1

pixel results match the conventional 32×32-px PIV data thatis available for y+ > 103 only. The original single-pixel PIV
data show some strong fluctuations for 250 < y+ < 103 (thegray region in Fig. 8). This can be attributed to the reflections
that occur close to the pipe wall; see Fig. 2. By expanding
the number of modes that is suppressed in the Fourier filter forimages prior to the single-pixel correlation, the effect of thesereflections could be substantially suppressed. The results show
an acceptable agreement with the Superpipe and CICLoPE
data to within an error margin that corresponds to variationsin the particle-image displacement of 0.078 px (corresponding

0.0

r/ D
Figure 7. The root-mean-square turbulent fluctuation in theaxial (u0 + ) and radial (v0 + ) directions, and Reynolds stress+−u0 v0 , normalized by the wall friction velocity uτ . Symbols
correspond to those in Fig. 5.

+

to 14 mum displacement in the flow), and the plateau in u0 2
that occurs at these high Reynolds numbers can be resolved.
In Figure 9 we present the results for the spatial correlation Ruu of the axial velocity fluctuations at the pipe centreline. The single-pixel result was normalised to match the tailof the spatial correlation as determined from conventional PIV
data. The difference towards zero offset would indicate theresult of the spatial filtering of the conventional PIV analysis.
This also demonstrates the potential for single-pixel PIV to estimate the local Taylor micro-scale and dissipation rate for thistype of flows from these high-resolution PIV measurements.
In Figure 10 the full spatial correlation coefficient ρuu (∆x) isshown for both the single-pixel PIV results and the conventional 32×32-px PIV results. By taking the Fourier transformof ρuu (∆x) we find the longitudinal spectrum Fuu (κ) at thecenterline. The single-pixel PIV result extends the range inwavenumber by more than a decade (owing to the 16 times
higher resolution compared to the conventional 32×32-px PIVdata), but also becomes noisy at the higher wave numbers. By
using binning over equal sections along the logarithmic axis
(red dots in the bottom graph of Fig. 10) a consistent result
is obtained for κD < 103 , whereas the conventional PIV data
appears to give consistent results for κD < 150. This demonstrates the potential of single-pixel PIV to provide informationof the spatial structure of the turbulence.

+

Figure 8. The streamwise normal stress u0 2 as a functionof the distance from the wall for Reτ = 10.3×103 (ReD =4.98×103 ). The red circles represent the 32×32-px PIV data,
the light red dots the raw unfiltered single-pixel PIV data, andthe bright red dots the median filtered single-pixel PIV data.
The grey area indicates where the single pixel data is influenced by reflections. The dash-dotted green curve represents
the data of Willert et al. (2017) for Reτ = 11.7×103 and thedashed blue curve represents the data of Hultmark et al. (2013)
for Reτ = 10.5×103 .

the Reynolds stress. The conventional PIV method using
32×32-pixel interrogation windows shows erroneous results
for the radial normal stress and the Reynolds stress for|r|/D > 0.43, which corresponds to a distance y+ < 103 fromthe wall. The single-pixel PIV results appears to give reliable
results when approaching the wall, although the Reynolds
stress results appear to give a higher noise level.

CONCLUSION
We present high-resolution PIV results (ultimately limited by the pixel size) of turbulent pipe flow, using a singlepixel PIV approach. This improves the spatial resolution bymore that an order of magnitude over conventional PIV methods, while still recording the data over the full pipe diameter.
This is applied to an industrial-style pipe flow facility, at shear
Reynolds numbers between Reτ = 7.2×103 and 13.9×103 .
The results presented in this paper are for the measurement
at a shear Reynolds number of Reτ = 10.3×103 . Conventional
PIV using 32×32-pixel interrogation windows only covers the+

In Figure 8 the normalized normal stress u0 2 is plotted
as a function of the distance from the wall y+ . The data iscompared with data obtained by Hultmark et al. (2013) at Reτ= 10.5×103 in the Princeton Superpipe and by Willert et al.
(2017) at Reτ = 11.7×103 in the CICLoPE facility. The single-

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
the capabilities of single-pixel PIV for turbulence measurements of high-Re wall turbulence, without recourse to specialized equipment or dedicated flow facilities. Using cameras
with a larger format would make it possible to resolve the turbulence statistics even closer to the wall.

1.2
1

x)

0.8

uu

(

0.6
0.4
0.2
0
-0.2
-0.8

REFERENCES
-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Adrian, R. J. & Westerweel, J. 2011 Particle Image Velocimetry. Cambridge University Press.
Ahn, J. Lee, J. H., Lee, J., Kang, J. & Sung, H. J. 2015 Direct
numerical simulation of a 30R long turbulent pipe flow at
Reτ = 3008. Phys. Fluids 27, 065110.
Colebrook, C. F. 1939 Turbulent flow in pipes, with particularreference tothe transition region between smooth and roughpipe laws. J. Inst. C. E. 11, 133–156.
Eggels, J. G. M., Unger, F., Weiss, M. H., Westerweel, J.,
Adrian, R. J., Friedrich, R. & Nieuwstadt, F. T. M. 1994
Fully developed turbulent pipe flow: a comparison between
direct numerical simulation and experiment. J. Fluid Mech.
268, 175–209.
Hultmark, M., Vallikivi, M., Bailey, S. C. C. & Smits, A. J.
2013 Logarithmic scaling of turbulence in smooth- andrough-wall pipe flow. J. Fluid Mech. 728, 376–395.
Kähler, C. J., Scholz, U. & Ortmanns, J. 2006 Wall-shearstress and near-wall turbulence measurements up to single
pixel resolution by means of long-distance micro-PIV. Exp.
Fluids 41, 327–341.
McKeon, B. J., Li, J., Jiang, W., Morrison, J. F. & Smits, A. J.
2004 Further observations on the mean velocity distribution
in fully developed pipe flow. J. Fluid Mech. 501, 135–147.
Scharnowski, S., Hain, R. & Kähler, C. J. 2012 Reynolds
stress estimation up to single-pixel resolution using PIVmeasurements. Exp. Fluids 52, 985–1002.
Sridharan, S. 2018 Towards high resolution particle image velocimetry. Master’s thesis, Technische Universiteit Delft.
den Toonder, J. M. J. & Nieuwstadt, F. T. M. 1997 Reynolds
number effects in a turbulent pipe flow for low to moderate
Re. Phys. Fluids 9, 3398–3409.
Vallikivi, M., Hultmark, M., Bailey, S. C. C. & Smits, A. J.
2011 Turbulence measurements in pipe flow using a nanoscale thermal anemometry probe. Exp. Fluids 51, 1521–
1527.
Vinuesa, R., Duncan, R. D. & Nagib, H. M. 2016 Alternativeinterpretation of the Superpipe data and motivation for CICLoPE: the effect of a decreasing viscous length scale. Eur.
J. Mech. B-Fluid 58, 109–116.
Westerweel, J. 2008 On velocity gradients in PIV interrogation. Exp. Fluids 44, 831–842.
Westerweel, J., Geelhoed, P. F. & Lindken, R. 2004 Singlepixel resolution ensemble correlation for micro-PIV applications. Exp. Fluids 37, 375–384.
Westerweel, J., Petracci, A., Delfos, R. & Hunt, J. C. R. 2011
Characteristics of the turbulent/non-turbulent interface of a
non-isothermal jet. Phil. Trans. R. Soc. A 369, 723–737.
Willert, C. E., Soria, J., Stanislas, M., Klinner, J., Amili, O.,
Eisfelder, M., Cuvier, C., Bellani, G., Fiorini, T. & Talamelli, A. 2017 Near-wall statistics of a turbulent pipe flowat shear Reynolds numbers up to 40 000. J. Fluid Mech. 826,
R5.
Wu, X. & Moin, P. 2008 A direct numerical simulation study
on the mean velocity characteristics in turbulent pipe flow.
J. Fluid Mech. 608, 81–112.
Zagarola, M. V. & Smits, A. J. 1998 Mean-flow scaling of turbulent pipe flow. J. Fluid Mech. 373, 33–79.

x/D

103
102

Fuu ( )

101

-5/3

100
10-1
10-2
-3

10

101

102

103

D

Figure 10. (top) The full logitudinal spatial correlation coefficient ρuu (∆x) for the axial velocity fluctuations at the pipe
centerline. Blue dots: single-pixel correlation results; black
line: same data after applying 16-px moving average filter;cyan dots: 32×32-pixel PIV result. (bottom) The corresponding spectrum. Blue lines: single-pixel result; red dots: binned
data of single-pixel result; + symbols: 32×32-pixel PIV result.

wake region of the turbulent pipe flow, with the interrogation
region closest and non-overlapping to the wall residing at theedge of the log-layer. The single-pixel PIV result is obtainedby processing 104 image pairs. Making use of the homogeneous axial direction in turbulent pipe flow, 8×106 pixel pairs
are processed and resulting in a 5×5-pixel correlation peak foreach radial position. Sub-pixel interpolation yields the mean
velocity profile and makes it possible to resolve the full loglayer down to the buffer layer.
From the shape of the single-pixel correlation peak also
the axial and radial normal stresses and the Reynolds stresscan be extracted. The accuracy of the single-pixel PIV result
is demonstrated by resolving the plateau in the axial normal
stress that forms into an outer peak at higher Reynolds numbers. However, it appears that the result is sensitive to thepresence of reflections in the image. These reflections arise
from the laser light sheet that enters the optical box and innerand outer transparant pipe walls within the optical box. In thepresent application conventional non-fluorescent tracer particles were used; the interference of these reflections could be
avoided by the use of fluorescent tracer particles in combination with an appropriate optical filter.
A novel aspect of the work is the extension of single-pixel
PIV from single-point to two-point flow statistics, demonstrated by a high-resolution result for the spatial correlationof the axial velocity fluctuations. This makes it possible to obtain a reliable estimate of the turbulent spectral density for theaxial velocity fluctuations.
These results were obtained with a standard PIV systemwith a modest resolution of 1,000 pixels. This demonstrates

5