14th International Symposium on Particle Image Velocimetry – ISPIV 2021
August 1–5, 2021

Noise canceling to compute second order statistic from
SPIV

J-M. Foucaut1∗, R. Yuvaraj1, C. Cuvier1

1 Univ. Lille, CNRS, ONERA, Arts et Métiers ParisTech, Centrale Lille, UMR9014-LMFL-Laboratoire de
Mécanique des Fluides de Lille-Kampé de Fériet, F-59000, Lille, France, Lille, France

∗ jean-marc.foucaut@centralelille.fr

Abstract
Stereoscopic PIV (SPIV) is performed near the wall in a turbulent boundary layer at high Reynolds numbers
(Reτ = 2272, 3840) in the streamwise-wallnormal plane. A novel method for denoising the statistics is pro-
posed by the use to two independent SPIV systems to capture the same field of view, with the objective being
that the random noise associated with the velocity fluctuations are not correlated between two independent
PIV systems. The derivatives in the third (spanwise) direction are obtained by axisymmetry assumptions
[George and Hussein (1991)] and continuity equation. The statistics are in agreement with that of DNS at
comparable Reynolds number, from y+ = 25 for lower Reτ experiment and from y+ = 40 for higher Reτ

experiment.

1 Introduction
Stereoscopic PIV (SPIV) is a well recognized experimental method to study turbulent flow. Many re-
searchers also use this method to compute statistics of the flow such as mean velocity, Reynolds stress
tensor, probability density function or spectra [Adrian et al. (2000), Foucaut et al. (2011), Herpin et al.
(2013)]. SPIV allows the measurement of the three components of the velocity in a plane with an accuracy
of about 1-2% (0.1 pixel). For a turbulent flow, it opens up the unique capability of studying the organization
of the turbulence. Generally, due to the limited spatial resolution of PIV, the smallest scales of the flow that
can be investigated is limited. The PIV random noise will also affect the statistics such as variance. In addi-
tion there is also the noise amplification that occurs when derivatives are computed [Foucaut and Stanislas
(2002)]. Nevertheless, the velocity gradient is necessary to determine the vorticity, the shear, the dissipation
and specific criteria allowing the detection of vortices such as the Q criterion which are useful tools to study
the flow organization.

In the present contribution we focus on statistics. The proposition here is to build an experiment of time
resolved SPIV in the near wall region of a turbulent boundary layer with a very good spatial and temporal
resolution. This experiment is carried out in the LMFL boundary layer wind tunnel. It is setup with two
SPIV system in order to be able to characterize and to get rid of the noise. The measured velocity is con-
sidered ui = uiT + uiN where uiT is the true value and uiN is the noise (random error). The variance, then
computed by 〈uiui〉 = 〈uiT uiT 〉+ 〈uiN uiN 〉 where 〈〉 corresponds to averaging in time and homogeneous direc-
tion (in streamwise direction for the present experiment), is affected by the variance of the noise and thus it
could be overestimated. The solution is to use 2 independent PIV systems to measure the same velocity at
the same point. In that case the variance is computed as

ui1 = uiT +uiN1
(1)

ui2 = uiT +uiN2
(2)

〈ui1ui2〉= 〈uiT uiT 〉 (3)
where the numbers 1 and 2 correspond to each PIV system. The variance is free of noise because the noise
of the two systems are not correlated. This allows the computation of denoised statistics and an estimation
of the RMS value of the noise by

σ
2
uiN

= 〈ui1ui1〉−〈ui1ui2〉 (4)



2 Experimental setup

Figure 1: Schematic of the SPIV experimental setup

The measurement was done at 19.2 m from the inlet of the test section. At this station, the boundary layer
thickness is of 0.273 m and 0.243 m for the Reynolds numbers Reτ of 2220 and 3840 for infinite velocities
of 3m/s and 6m/s, respectively. Figure 1 shows the two sets of Stereoscopic PIV (2D- 3C) employing 4
Miro cameras to capture a field of view illuminated by a laser light sheet of 60.4 × 18.4 mm2 (503+×153+)
for Reτ = 2220 and 29.6 × 18.4 mm2 (493+× 306+) for Reτ = 3840, both along the streamwise x and
wall-normal direction y. The seeding for the experiment is 1 µm-size water-ethylene glycol mixture which
is fully seeded in the wind tunnel. The cameras and the Laser are synchronized by LaVision’s High Speed
Controller. The chip-size of the camera were reduced at 1280× 512 pixels to achieve 4.5 kHz PIV frequency
and 640×512 pixels corresponding to 7.5 kHz for the lowest and highest Reynolds numbers, respectively
and the lasers are triggered at same time with 10 mJ/pulse. The images were acquired in a series of 5
images burst mode, up to a total of 1803 sets of time series images in order to offer possibility to make
statistics on time derivative. This time-resolved SPIV employed here is capable of producing a 2D-3C
velocity field in a XY-plane, which provides all the derivatives in x- and y- directions and time. With the
help of continuity equation, it is possible to obtain the ∂w/∂z term. The experiment is calibrated by using an
accurate calibration plate which has crosses at known places and are spaced equally by 1 mm distance, and
this is in-turn captured by the cameras in 11 spanwise positions. The Soloff method is used for the three-
component reconstruction. This calibration procedure is followed by self-calibration [Wieneke (2008)].
Special care is taken to compute the velocity vectors at the same point for the two SPIV system. To ensure
that the velocities are computed at the same point the mesh is generated in the object plane and projected
in the planes of the 4 cameras. The acquired data are then processed using a modified version of MatPIV
toolbox at LMFL. A multi-pass cross-correlation PIV analysis is used with deformation at the last pass and
with a final interrogation window size of 18 × 24 px which corresponds to 0.97 × 0.97 mm2 for the system
1 and 0.88 × 0.95 mm2 for the system 2, which are of the same order of magnitude corresponding to about
8+ for Reτ = 2220, and 16+ for Reτ = 3840. The small variation come from a little difference of angles and
magnifications between the two systems.

Figure 2 shows the mean velocity profile in wall-units in SPIV experiment at the two Reτ, and is then
compared with DNS of TBL at Reτ = 1989 [Borrell et al. (2013)] and TCF at Reτ = 4200 [Hoyas and Jiménez
(2008)]. The ranges of accessible wall-distances from the SPIV experiment are 10.25≤ y+ ≤ 157.13 (green
vertical line) and 20.19≤ y+ ≤ 309.6 (red vertical line), for the lower and higher Reτ datasets respectively.
The mean velocity (u+) of the SPIV experimental dataset at both the Reynolds numbers shows a good
agreement that of the DNS datasets.

Figure 3 shows the comparison of variances computed from individual systems and denoised with that of
DNS datasets at nearest Reτ. It is observed that the statistics of both the individual PIV systems and denoised
version from Reτ = 2220 follows that of the DNS at Reτ = 1989 [Borrell et al. (2013)]. However, as the wall



100 101 102 103

0

10

20

y+

u+
PIV (Reτ = 2220) PIV (Reτ = 3840) DNS (Reτ = 1989)
DNS (Reτ = 4200)

Figure 2: Comparison of average streamwise velocity normalised with inner co-ordinates u+ from the
present PIV experiments at Reτ = 2220 and 3840 with DNS datasets of TCF at Reτ = 1989,3000,4200.
The green vertical line shows the range of wall-distances for PIV experiment with Reτ = 2220, and the red
vertical line shows the range of wall-distances for PIV experiment with Reτ = 3840

is approached one can observe that only the denoised statistics tend to follow that of DNS down to y+ = 10.
This result shows the advantages of using two SPIV systems with the order of spatial resolution in present
experiment for accurate measurement near the wall. The SPIV with higher Reτ = 3840 does approximately
follow that of DNS of TCF at Reτ = 4200 [Hoyas and Jiménez (2008)] for u′, v′ and u′v′, and there is some
discrepancy for w′. It is clear that the noise is very low in the present experiment, with the exception for the
w component it is difficult to distinguish the noise.

Figure 4 gives a field of an estimation of RMS value of the noise associated with the two systems for the
streamwise component in pixels. It is observed that with the exception on the left side of the fields and near
the wall, the noise is of the order of 0.06-0.08 px. The higher value on the left side is probably linked to a
problem of vector validation at the border.

Figure 5 shows the same parameter but averaged along x for both SPIV systems in pixels for Reτ = 2220.
The noise of the two systems stay under that 0.06 px which is remarkable. It is compared with the model of
George and Stanislas (2020) (denoted as ‘Estimation George’) which is able to estimate the noise by taking
into account the pixelisation error and the fluctuation of particle positions in the interrogation window due
to the local turbulence. This noise from ‘Estimation George’ is given by :

σth =

√
1
N

(
〈u′21 〉vol +

∆2

12

)
W1(Uδt) (5)

where the first part of the right hand side of the equation correponds to the velocity fluctuation inside in-
terrogation volume and the second part corresponds to the pixelisation error (∆ = 0.5) and N is the number
of particles in the interrogation window. It is also compared with an estimation from spectral analysis as in
Foucaut et al. (2004) at distances of 45+ and 100+. The noise from George and Stanislas (2020) is smaller
compared to the noise from the present SPIV experiment, which therefore shows that other sources of noise
needs to be accounted in addition to the pixelisation error and fluctuation of particle positions. This model
does not take into account other source of noise such as isolated particles or gradient in the interrogation
window. The estimation from the spectra [Foucaut et al. (2004)] is approximately of the same order as the
noise estimated from the experiments. The spectra of the two systems and the subsequent noise estimation
from them are very similar and of the same order as predicted by Equation 4.

Figure 6 shows the dissipation which is composed by a sum of variances of spatial derivative of velocity,



101 102

4

6

8

10

y+

u′
2
+

101 102

0

0.5

1

y+

v′
2
+

101 102

1.5

2

2.5

3

y+

w
′2
+

101 102

−1

−0.5

0

y+

u′
v′
+

Figure 3: Comparison of variances and covariances of streamwise, wall-normal and spanwise veloc-
ity components in the present PIV experiments at Reτ = 2220 and 3840 with DNS of TCF datasets at
Reτ = 1989,3000 and 4200

200 100 0 100 200
x +

50
100
150

y
+

PIV system 1

200 100 0 100 200
x +

50
100
150

PIV system 2

Figure 4: Noise associated with SPIV sys1 (left) and sys2 (right) in pixels

obtained with individual system and denoised by the two systems. The derivatives are computed by a Least
Squared Method [Foucaut and Stanislas (2002)] which is based on second-order central difference scheme
and is optimized to not amplify the noise. The missing variance of derivatives are obtained by the assump-
tion of axisymmetry (see George and Hussein (1991)). We can see the noise which is very small except
close to the wall, and the effect of filtering for the highest Reynolds Number. This allows to evidence the
filtering effect of PIV for the highest Reynolds number [Atkinson et al. (2014)] which is mainly visible in
the dissipation rate. The assumption of axisymmetry works down to y+ = 25 for the SPIV experiment at



Figure 5: Noise associated with SPIV sys1, sys2 in pixels, compared with the noise estimation of George
and Stanislas (2020), and that of estimation from spectra Foucaut et al. (2004)

101 102

0

5 ·10−2

0.1

0.15

0.2

y+

D
+

PIV sys1 (Reτ = 2220) PIV sys1 (Reτ = 3840) 15ν( du
dx )

2 (Reτ = 2220)
PIV sys2 (Reτ = 2220) PIV sys2 (Reτ = 3840) 15ν( du

dx )
2 (Reτ = 3840)

PIV denoised(Reτ = 2220) PIV denoised(Reτ = 3840) DNS−T BL(Reτ = 1989)

Figure 6: Comparison of dissipation computed from individual PIV systems 1, 2 and denoised at Reτ = 2220
and 3840 with DNS of TBL datasets at Reτ = 1989. Vertical green line shows the limit of axisymmetric
assumption in the computation of dissipation

Reτ = 2220 which follows the explanation in Foucaut et al. (2020).

Figure 7 shows the effect of denoise on the streamwise derivatives of two velocity components obtained from

SPIV experiment at Reτ = 2220. It is observed for
(

∂u′
∂x

)2
+

, the difference between the individual systems

and that of denoised version are significant as the wall is approached, when y+ ≤ 25. And for
(

∂v′
∂x

)2
+

, the
difference between the denoised and individual system is small despite the denoised value being lower than
that predicted by individual systems. So this method of denoising has the ablity to obtain accurate statistics
especially near the wall.



101 102
0

0.5

1

1.5

·10−2

(
∂u′
∂x

)2
+

101 102

0.5

1

1.5

2
·10−3

(
∂v′
∂x

)2
+

PIV sys1 PIV sys2 PIV denoised

Figure 7: Effect of denoise on the derivatives of streamwise and wall-normal velocities at Reτ = 2220

3 Conclusion
A method to denoise the statistics of turbulent flow is proposed in the present paper. This method is based
on the use of two independant PIV systems. It allows the computation of the noise level of the order of 0.06
px. The derivative tends to amplify the noise (see Foucaut and Stanislas (2002)). This method allows also
to denoise variances of velocity derivatives such as the terms of the dissipation rate. By using two different
spatial resolutions we evidence that if the interrogation window size is of the order of 8 px it is possible to
compute these dissipation with a very good accuracy. If the interrogation window size increases (doubled
here) the filtering effect of PIV is strongly observed in this parameter. If the interrogation window size is
smaller than 4 Kolmogorov length scale which is about 2 wall units, the dissipation will be computed with
a good accuracy.

Acknowledgements
This work was carried out within the framework of the CNRS Research Foundation on Ground Transport and
Mobility, in articulation with the ELSAT2020 project supported by the European Community, the French
Ministry of Higher Education and Research, and the Hauts de France Regional Council.

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	Introduction
	Experimental setup
	Conclusion