PIV Measurements of Turbulence Decay behind a
Grid.
G. Avallone1, F. De Gregorio2, G.P.Romano1
1
2
- Università di Roma “La Sapienza”, Via Eudossiana 18, Roma, Italy
- Centro Italiano Ricerche Aerospaziali, Via Maiorise, Capua (CE), Italy
Abstract
The scope of this work is to evaluate the capability of PIV to measure small-scale
properties of a turbulent flow. To this aim, due to the available theoretical and
empirical results, homogeneous and isotropic turbulence was selected for the investigations.
The influence of different recording parameters and different evaluation methods used in PIV on mean velocity and turbulent statistics is evaluated, and in particular the turbulence decay is investigated.
A study on the minimum number of images necessary to obtain satisfactory
turbulence statistics is also performed. The relationship between this minimum
number and the turbulence level of the flow is shown.
Wave-number spectra have been also computed.
1 Introduction
The capability of PIV to measure turbulent flows has been evaluated investigating
the case of homogeneous and isotropic turbulence. This particular kind of turbulence has been chosen because it allows to extrapolate some empirical relations,
and has already been studied in previous works, which provide valuable comparison for PIV data assessment [1-5]. Homogeneous and isotropic turbulence is obtained by inserting a grid in the wind tunnel test chamber.
According to the theory [1], three regions of turbulence decay are identified behind a grid. The first is the developing region, near the grid, where the rod wakes
merge, the flow is inhomogeneous and anisotropic with production of turbulent
kinetic energy. In the second region, the flow is nearly homogeneous, isotropic
and locally isotropic; the energy transfer from one wave-number to another is significant. The third region is where viscous effects act directly on the large energy
containing scales.
For the second region, Gad-El-Hak and Corrsin [4] proposed the turbulence decay law
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n
2
1
U 02
 x 
 x  U0
=
B
= B2  
;


1
2
2
u'
 M  w'
M 
n2
(2.1)
The first equation of 2.1 is related to the longitudinal turbulence component decay (u’), and the second is referred to the lateral component (w’). In these relations, Uo is the mean flow velocity, M is the grid mesh, and x is the investigated
axis for turbulence decay. The values of the parameters B1, n1, B2, n2, used in
these equations, are empirically determined. Gad-El-Hak and Corrsin [4] summarised the values of the parameters obtained by several authors in different grid turbulence works. For the longitudinal turbulence decay, typical values for the parameter n1 are include between 1.13 and 1.37, while for B1 values between 13 and
25 are found. The values of the parameters of the lateral turbulence decay are 1.14
< n2 < 1.35 and 14 < B2 < 35.
Comte-Bellot and Corrsin [5] fixed the beginning of the homogeneous and isotropic region at 50 meshes downstream the grid, while Gad-El-Hak and Corrsin
[4] determined the beginning of the second turbulence decay region at 30 meshes
downstream the grid.
In order to evaluate the degree of homogeneity of the flow, Mohamed and La
Rue [3] measured the variation, along a vertical axis, of the homogeneity parameter
(u’-u’0)/u’0
(2.2)
where u’, the standard deviation of the longitudinal component of velocity, is
the absolute turbulence level of the flow, and u’0 is the turbulence level measured
at the middle of the investigated axis. For an homogeneous flow, a value of zero
for the homogeneity parameter is expected.
To evaluate the degree of isotropy of the flow, Mohamed and La Rue [3] suggested to measure the velocity skewness S(u), keeping in mind that for an isotropic flow S(u)=0. Another method to evaluate the isotropy of the flow is presented by Comte-Bellot and Corrsin [5] and by Uberoi and Wallis [6]. They
measure the ratio between the longitudinal an the lateral turbulence level which,
for an isotropic flow, is: u’/w’=1. Comte Bellot and Corrsin show that, using a
straight test chamber, values of u’/w’ in the range between 1.1 and 1.4 are obtained, depending on the flow Reynolds number and on the grid characteristics,
while u’/w’=1 may be obtained by using a convergent duct downstream the grid.
In the inertial subrange, the Kolmogorov law for the energy spectrum holds:
E(K)=Aε2/3K-5/3
(2.3)
where ε is the mean turbulent kinetic energy dissipation rate. The slope like k-5/3
is typical for homogenous and isotropic turbulence [2].
Turbulence and PTV 265
2 Experimental Set-up
The tests have been conducted in the CIRA low speed wind tunnel (CT-1). The
CT-1 is an Eiffel type open circuit wind tunnel, with the following main characteristics: 5 m/s to 55 m/s velocity range, 16:1 nozzle contraction ratio and 0.1%
maximum value of turbulence level at 50 m/s flow speed. The test chamber is 600
mm long, 302 mm wide and 302 mm high. It allows 100% of optical access being
built of Plexiglas and it is equipped with a 3D traversing system (for HW and
pressure probes).
In order to generate an homogenous and isotropic turbulent flow, a round rods
(d=1,2 mm diameter) grid with mesh size M=4,5 mm has been mounted at the
wind tunnel nozzle exit, in front of the test chamber entrance. The grid is characterized by a solidity ratio σ=0.45; the solidity ratio being defined as :
σ=d/M(2-d/M)
(3.1)
Comte-Bellot and Corrsin [5] suggest grids with solidity ratio lower than 0.5, in
order to avoid the instability of the flow.
The PIV system consists of a laser source and of a digital camera for the images
acquisition.
The Nd:YAG pulse laser was mounted on the frame prepared near the test
chamber of the wind tunnel. The laser is composed of two resonator heads, with
second harmonic generators providing a pulse energy of 300 mJ at 532 nm length
wave.
The camera is a PCO SensiCam with a resolution of 1024 by 1280 pixels. The
grey level resolution is 12 bit.
The camera has been equipped with different lenses with focal lengths of
60mm, 100mm and 180 mm to provide different magnification factors and better
space resolution. As seeding material, the Di(2.EthylHexyl) Sebacate oil was used.
The camera and the laser heads were synchronised by a dedicated electronic.
The complete recording and synchronisation system was located in the control
room.
3 Measurement Description
The flow field immediately downstream the grid has been investigated by Hot
Wire (HW), Laser Doppler Velocimetry (LDV) and Particle Image Velocimetry
(PIV) [11].
In this zone, HW, LDV and PIV measurements have been performed along the
main flow direction but at 4 different transverse positions, indicated as R0, R1, R2
and R3 in Fig. 1. This was done to investigate the grid wake in the first region,
where the influence of the grid is significant. The R0 axis is located in the middle
of the vertical bar, R1 is at the bar crossing, R2 is in the middle of a mesh cell and
R3 is in the middle of the horizontal bar.
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The reference system has been fixed with the origin located on the intersection
between the axis R0 and the grid plane, laying on the WT centre line. The X axes
is parallel to the main flow direction, the Z axes, laying in the grid plane, is directed vertically and orthogonal to the X axes and the Y axes is oriented following
the right hand rule.
The first zone, up to 26 meshes (M) downstream the grid, has been investigated
by PIV using different lenses (focal length 60mm, 100mm and 180mm), in order
to evaluate the space resolution influence on PIV measurements, and different delay times in the range between 5 and 70µs .
Fig. 1. The investigated axis.
To ensure the homogeneity and isotropy of the turbulent flow, the flow field
has been investigated up to 120 meshes downstream the grid using only the PIV
system. A single lens, with focal length of 100 mm, has been used, providing a
field of view (FOV) of about 80*65mm2, in such a way eight different zones,
slightly overlapped, were necessary to cover all the region of interest. In each
zone, images were acquired with different time interval between laser pulses (Dt)
in a range between 45µs -110µs. The grid turbulence has been investigated for two
values of the Reynolds number: ReM=3000 and ReM=6000, corresponding to a free
stream velocity of respectively 10 and 20m/s.
In order to obtain satisfactory statistical results, at least 1000 images have been
acquired for each investigated zone and configuration.
4 Data analysis
The PIV images have been evaluated using two methods: the standard crosscorrelation method, with interrogation windows of 322 pixels and the Hart correla-
Turbulence and PTV 267
tion method [8], with same window sizes in order to evaluated the effect of Hart
method on the turbulence measurements. The interrogation windows have been
overlapped 50% providing a matrix of 79x63 displacement vectors. To improve
the signal/noise ratio a multi pass algorithm has been used, in which several interactions are used to shift the interrogation window with respect to each other in accordance with the local displacement vector [7]. For all the analysed flow fields,
the energy wave number spectrum of the flow was computed from the spatial
auto-correlation coefficient of the velocity along the stream wise direction; the
evaluation is performed by averaging several (about 1000) instantaneous PIV
fields, and all the rows in each image (about 63), at each separation distance
(ranging from 0 to 80 mm).
5 Results
The mean velocity behaviour, obtained along the R2 axis from PIV with different
spatial resolution, is presented in Fig. 2. The graph presents the velocity normalised with the Pitot probe velocity versus the position downstream of the grid, normalised with the grid mesh size. Since the R2 axis is at the middle of the mesh, an
acceleration in the first points downstream the grid, and a successive deceleration
of the flow is expected. The right behaviour, according to the HW measurements,
is evaluated especially with the 1.44 mm-1, but also with the 1.23 mm-1 resolution
(FOV of 11 and 13 grid mesh), but not with a 0.6 mm-1 resolution (FOV of 26 grid
mesh).
2.5
space resol.=1.23 1/mm
space resol.=1.44 1/mm
spase resol.=0.60 1/mm
Hot Wire
Vx/Vpitot
2
1.5
1
0.5
0
0
5
X/M
10
15
Fig. 2. Effect of the spatial resolution on the velocity behaviour evaluation.
The lowest space resolution, corresponds to an interrogation windows of about
3.3mm2. This window covers both the central region of the mesh interested by the
velocity acceleration, but also the surrounding region affected by the wakes shed-
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ding of the grid (with reverse flow and slow motion). So the displacement obtained is the results of the average on the above-mentioned zones. That is the reason for the behaviour observed in Fig. 2 for these spatial resolutions.
In order to reach a compromise between the reliability of the measurements
and the field size, the measurements have been carried out with the 1.23 mm-1
resolution (using a 100 mm Zeiss lens).
2.5
Vx/Vpitot
2
R0 axis
R1 axis
R2 axis
R3 axis
1.5
1
0.5
0
-0.5
0
2
4
6
8
10
12
14
X/M
Fig. 3. Effect of the axis position on the grid rods on velocity measurements.
In Fig. 3, the influence of the axis position with respect to the grid rods is presented. Along the R1 axis, located at the rod crossing a low velocity (almost zero
behind the grid) and a successive acceleration is measured. Along the R3 axis, located at the middle of the horizontal bar the velocity behaviour close the grid
shows a negative values, indicating a reverse flow in the wake. Downstream, the
velocity increases reaching unit values. Similar velocity behaviour was expected
along the R0 axis, located at the middle of the vertical bar, but a different result is
encountered. Immediately behind the grid the normalised velocity presents a value
of about 2.2. Farther downstream a deceleration is observed. This is probably an
effect of a misalignment of the light sheet with respect to the vertical rod. Along
the R2 axis, located at the middle of the mesh, the expected velocity behaviour,
with high velocity at the first points and a successive deceleration, is assessed.
In Fig. 4, the different behaviour of flow turbulence level obtained along the
different axis, is presented. As for the velocity, the transverse position behind the
grid influences the level of turbulence only up to about 7-8 mesh behind the grid.
Farther, the four curves show the same behaviour.
Turbulence and PTV 269
25
R0 axis, Dt=10E-6 s
R1 axis, Dt=10E-6 s
R2 axis, Dt=10E-6 s
R3 axis, Dt=10E-6 s
LTRx %
20
15
10
5
0
0
2
4
6
X/M
8
10
12
14
Fig. 4. Effect of the axis position on turbulence measurements.
In Fig. 5, the influence of the delay time on longitudinal turbulence measurements is shown. The turbulence level of the flow is evaluated by the expression
(VPitot/u’), for the longitudinal component, and (VPitot/w’) for the lateral component;
VPitot is the measured upstream velocity in the test chamber.
A good agreement is obtained between HW and the PIV with Dt=5 µs, for the
first 14 meshes downstream of the grid. This good agreement is extended up to 20
and 23 meshes for a delay time equal to 10 and 15 µs, while for Dt =70 µs, the
PIV curve follows the HW for more than 26 meshes. Therefore the capability of
the PIV technique to measure turbulence fluctuations at increasing distances from
the grid, is here subjected to the use of a large delay time.
(Vpitot/u')2
2000
Dt=5 E-6 s
Dt=10 E-6 s
Dt=15 E-6 s
Dt=70 E-6 s
Hot Wire
1500
1000
500
0
0
5
10
15
X/M
20
25
Fig. 5. Effect of delay time Dt on turbulence measurements.
30
(Vpitot/u')2
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1600
Dt= 70E-6 s - Ha rt
Dt= 70E-6 s - no Ha rt
1200
Hot W ire da ta
800
400
0
0
5
10
15
X /M
20
25
30
Fig. 6. Hart correlation influence on turbulence.
In Fig. 6, the comparison of the turbulence level behaviour evaluated with standard correlation and with Hart correlation for a delay time of 70 µs versus the position behind the grid, is presented. The turbulence level evaluated by HW measurements is also reported. Using Hart correlation, a lower turbulence level, with
respect to the one obtained with standard cross-correlation, is obtained. The PIV
curve “no Hart” shows a good agreement with the HW data, so all the acquired
images have been processed without Hart algorithm.
In Fig. 7, the flow homogeneity degree, evaluated by means of the method presented by Mohamed and La Rue [3], is presented. The parameter (2.2), measured
at different distances from the grid and for different Reynolds number of the flow
is evaluated; a perfect flow homogeneity would attain a value equal to zero.
Z/M
4
3
2
1
0
-1
-2
-3
-4
-0.15
X/M=15.15
X/M=41,79
X/M=106,08
-0.1
-0.05
0
(u'-u'0)/u'0
0.05
0.1
0.15
Fig. 7. Flow homogeneity degree at different distances downstream the grid.
Mohamed and La Rue obtained a maximum value of 0.04 (reference dashed
line in the Fig. 7) at 11 meshes downstream the grid for ReM=6000 and
ReM=14000, by means of Hot Wire measurements. The reported PIV curves, obtained for ReM=3000 at different distances from the grid, show values of the ho-
Turbulence and PTV 271
mogeneity parameter included in the range ± 0.05, so the flow may be considered
almost homogeneous since 20 meshes downstream of the grid.
1.5
skew(u) - Dt=90E-6 s
skew(u) - Dt=110E-6 s
skew(u) - Dt=10E-6 s
skew(u) hot wire
skewness(U)
1
0.5
0
-0.5
-1
0
30
60
X/M
90
120
Fig. 8. Flow degree of isotropy evaluated with the skewness of velocity.
In Figs. 8 and 9, the isotropy degree of the flow is discussed with two different
methods. In the first figure the velocity skewness obtained with different Dt is reported together with the one obtained with HW measurements. The PIV curves
show a fluctuation around the zero value, typical of isotropic flow, since 4 mesh
downstream the grid.
2.5
Reference 1.2
u'/w' - Dt=90E-6 s
u'/w' - Dt=110E-6 s
u'/w' - Dt=10E-6 s
Hot Wire
(u'/w')
2
1.5
1
0.5
0
30
60
X/M
90
120
Fig. 9. Flow degree of isotropy evaluated with the ratio between longitudinal and lateral
turbulence level.
In Fig. 9 the isotropy degree is evaluated by means of the ratio u’/w’. Since 12
meshes downstream the grid, fluctuations are limited around the value 1.2. These
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fluctuations are similar to the one expected for an isotropic flow realised in a
straight wind tunnel test chamber.
According to the two different methods presented, the flow may be considered
almost isotropic since 12 meshes downstream the grid.
In Fig. 10, the PIV longitudinal turbulence decay, obtained using delay time of
90µs and 110 µs, and the Hot Wire curve, are reported.
(Vpitot/u')2
10000
1000
(U/u')^2 - Dt=90E-6 s - Re=3000
(U/u')^2 - Dt=110E-6 s - Re=3000
Extrapolated Hot Wire data
Refer. decay - Re=3000 (n1=1.26; B1=18)
100
0
20
40
60
X/M
80
100
120
Fig. 10. Longitudinal turbulence decay for ReM = 3,000.
The PIV data are interpolated by means of the first of (2.1), using the appropriate parameters B1 and n1; this curve, reported in the figure as reference decay,
shows a very good agreement with the HW curve. For the parameters the values
B1=18 and n1=1.26 are obtained. They are included in the ranges individuated by
many authors [4] (1.13 < n1 < 1.37 and 13 < B1 < 25).
(Vpitot/w')
2
10000
1000
(U/w')^2 - Dt=90E-6 s - Re=3000
(U/w')^2 - Dt=110E-6 s - Re=3000
Extrapolated Hot Wire data
Refer. decay - Re=3000 (n2=1.28; B2=25)
100
0
20
40
60
X/M
80
Fig. 11. Lateral turbulence decay for ReM = 3,000.
100
120
Turbulence and PTV 273
The same curves presented in fig.10 are reported, for the lateral turbulence, in
Fig. 11, and the reference curve is obtained using the appropriate parameters in the
second equation of (2.1). The values obtained for the parameters are 25 for B2 and
1.28 for n2, and also these values are included in the ranges 1.14 < n2 < 1.35 and 14
< B2 < 35, [4].
log (Vpitot/u')2
10000
1000
(U/u')^2 - Dt=45E-6 s - Re=6000
Refer. decay - Re=6000 (n1=1.22;
100
0
40
80
120
X/M
Fig. 12. Longitudinal turbulence decay for ReM = 6,000.
log (Vpitot/w') 2
10000
1000
(U/w')^2 - Dt=45E-6 s - Re=6000
Refer. decay - Re=6000 (n2=1.15; B2=32)
100
0
20
40
60
X/M
80
100
120
Fig. 13. Lateral turbulence decay for ReM = 6,000.
In the Figs. 12 and 13, the PIV data and the respective reference curves, obtained by the equations (2.1), are presented for the longitudinal and lateral turbulence decay of the flow characterized by Reynolds number 6,000.
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For the decaying parameters, the following values are obtained: B1=15, n1=1.22,
B2=32, n2=1.15. All these values are included in the respective literature ranges,
and the reference curve shows a very good agreement with the PIV curve.
log E1(K)
0.1
U - X/M=0
U - X/M=12
U - X/M=35
U - X/M=68
slope -5/3
0.01
0.001
0.0001
10
100
log (K) - m-1
1000
Fig. 14. Longitudinal energy spectra at different distances downstream the grid.
In Fig. 14, the longitudinal energy spectrum, at Re=3000, is presented for different distances from the grid. A trend to increasing slope is clear when considering the energy spectra at 0, 12 and 35 meshes downstream the grid, while no appreciable slope difference is observed between the curves at 35 and 68 meshes
from the grid.
log E2(K)
0.1
V - X/M=0
V - X/M=12
V - X/M=35
V - X/M=68
slope -5/3
0.01
0.001
0.0001
10
100
log (K) - m -1
1000
Fig. 15. Lateral energy spectra at different distances downstream the grid.
The same behaviour is shown by the lateral spectrum curves, presented in Fig.
15. The behaviour of the different spectra slopes indicates that the turbulence con-
Turbulence and PTV 275
mean error %
dition of homogeneity and isotropy is reached at a distance between 13 and 35
meshes downstream the grid. After 35 meshes, the spectra show a constant slope
of -5/3, in agreement with the Kolmogorov law (2.3).
16
14
12
10
8
6
4
2
0
100
Dt=70E-6 s
Dt=90E-6 s
Dt=110E-6 s
Dt=45E-6 s - Re=6000
300
500
700
number of analized images
900
Fig. 16. Mean error on longitudinal turbulence level evaluation for different number of averaged images and for different Dt.
All the PIV measurements previously presented, have been realized analysing
1000 images for each investigated zone and for each analysis configuration. In order to define the minimum number of images necessary for evaluating reliable
turbulence statistics, the mean percentage error, defined as the difference between
the turbulence data obtained processing 1000 images and the turbulence obtained
varying the number of samples (6.1), are reported in Fig. 16. Data, obtained for
different delay times, are shown as well.
Mean error = (u’Nsample - u’1000)/u’1000
(6.1)
The least error is obtained when using a delay time of 90 µs; analysing 300 images, a difference lower than 2% is evaluated with respect to the results obtained
analysing 1000 images. A difference lower than 4% is obtained analysing 600 images independently from the time interval.
6 Conclusions
In this work, grid turbulence has been investigated by means of HW, LDV and
PIV measurements. Conditions of homogeneity and isotropy of the turbulent flow
have been obtained from about 10 meshes and have been verified by means of the
PIV system.
The capability of PIV to measure small-scale properties of turbulent flow has
been demonstrated, up to 120 mesh downstream the grid.
The PIV data showed a fairly good agreement with the HW results.
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Using robust and fast analysis software a dynamic range of 100 has been obtained measuring velocity fluctuation of about 0.1 m/s with mean velocity value of
20 m/s.
The influence of the delay time, i.e. the pixel displacement, has been evaluated
on the velocity statistics. To obtain reliable measurements of turbulence fluctuations at increasing distances from the grid, an increasing time interval between the
two PIV images must be selected. To this end, a compromise should be taken by
considering a delay time as long as possible avoiding particles to move out of the
measurement plane.
The effect of the space resolution has been also investigated by varying the lens
magnification factor and consequently the dimension of the acquired field of view.
The optimal selection depends on the phenomenon under investigation and on the
effective turbulence scale.
The longitudinal and lateral turbulence decay laws have been determined, and
the derived parameters are in agreement with literature data.
The energy spectrum has been evaluated as well. A wave number of about 650
mm-1 has been resolved with the adopted space resolution. The slope of the spectrum is in agreement with the Kolmogorov –5/3 law valid in the inertial sub-range.
A link between the number of processed images and the reliability of the turbulence measurements has been investigated; showing a minimum number of 600
images is sufficient in order to obtain satisfactory statistics independently from the
time interval used.
Acknowledgement
This work has been partially funded by European Commission in the frame
work of the EUROPIV 2 research project (contract number G4RD-CT200000190). The authors wish to acknowledge the valuable contributions enabling the
success of this investigation by dr. A. Marino for the support in the first recording
campaign.
References
1.
2.
3.
4.
5.
6.
Monin S, Yaglom AM (1971) Statistical fluid mechanics, vol II. MIT
Hinze JO (1997) Turbulence. McGraw-Hill
Mohamed, La Rue JC (1990) The decay power low in grid-generated turbulence. Journal of Fluid Mechanics
Gad-El-Hak M, Corrsin S (1974) Measurements of the nearly isotropic turbulence behind a uniform jet grid. Journal of Fluid Mechanics
Comte-Bellot G, Corrsin S (1966) The use of a contraction to improve the isotropy of
grid-generated turbulence. Journal of Fluid Mechanics
Uberoi MS, Wallis S (1966) Small axisymmetric contraction of grid turbulence. Journal of Fluid Mechanics
Turbulence and PTV 277
7.
Willert C., Raffel M., Kompenhans J.: Particle Image Velocimetry, Book series: 'Experimental Fluid Mechanics', Eds. R.J. Adrian, M. Gharib, W. Merzkirch, D. Rockwell, J.H. Whitelaw, ISBN 3-540-63683-8, Springer-Verlag, 255 pages, 1998.
8. Hart DP (1998) PIV error correction. Applications of Laser Anemometry to Fluid Mechanics, 9th Intern. Symp. Lisbon
9. Westerweel J, Dabiri D, Gharib M (1997) The effect of a discrete window offset on the
accuracy of cross correlation analysis of digital PIV recordings. Experiments in Fluids
10. Westerweel J (1997) Fundamentals of digital Particle Image Velocimetry. Measurement Science and Technology
11. Tino N., De Gregorio F., Diliberto D., Esposito B. “Experimental Comparison of
Anemometric Measurement Techniques in 2D Flow Field”, RAE 2001 Congress, 910/4/2001, London.