9TH. INTERNATIONAL SYMPOSIUM ON FLOW VISUALIZATION, 2000
EXTRACTION OF MULTI-SCALE TURBULENT
STRUCTURE FROM PIV RESULTS BASED ON
WAVELET VECTOR MULTIRESOLUTION
TECHNIQUE
Hui LI 1, Hui HU 2, Toshio KOBAYASHI 2, Tetsuo SAGA 2, Nobuyuki TANIGUCHI2
Keywords: orthogonal vector wavelet transform, particle image velocimetry,
multiresolution analysis, multi-scale vortex, turbulent jet
ABSTRACT
The wavelet-based vector multiresolution technique was proposed and applied to the post-processing of PIV
results for identifying the multi-scale turbulent structures. By analyzing the instantaneous streamlines and
vorticity fields of various scales the important information on the multi-scale flow structures was obtained.
The large-scale vortices or coherent structures with a central scale of a=15.36mm were easily extracted from
PIV measurement results. The intermediate- and small-scale vortices embedded within the large-scale
vortices were separated and visualized.
1 INTRODUCTION
The turbulent jet exhibited a complex structure with a wide range of coexisting scales and a variety
of shapes in the dynamics, and its coherent structures were responsible for most of the momentum,
mass and heat transfer. During the last couple of years the development of PIV techniques has
made it possible to provide more detailed information on the flow structures, such as the
instantaneous values of various flow quantities, as well as their distribution and transient variation.
Despite the usefulness of information were obtained by examining the measured instantaneous
fields and the time-mean turbulent quantities, further information on the instantaneous multi-scale
structures has not yet been clarified.
In the past decade, there has been a growing interest in the use of wavelet analysis for the
turbulent flow data [1]-[6]. This technique allows to track turbulent structures in terms of time and
scale and extracts new information on turbulence structures. To gain deeper insight into the multiscale structures and coherent structures in the turbulent flows, it is an important to analyze the
instantaneous full flow field. Recently, Li et al. [7], [8] firstly applied the two-dimensional
orthogonal wavelets to the turbulent images for the identification of the multiresolution turbulent
structures. However, few investigates concerned the extraction of multi-scale turbulent structures
from PIV data.
In this paper the wavelet vector multiresolution technique was developed and applied to the
Author(s):
1
Department of Mechanical Engineering
Kagoshima University, Japan.
2
2nd Department, Institute of Industrial Science
University of Tokyo, Japan.
Corresponding author: Hui Li
E-mail: li@mech.kagoshima-u.ac.jp
Paper number 383
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Hui LI et al.
post-processing of PIV data for revealing and visualizing the instantaneous multi-scale turbulent
structures.
2 WAVELET VECTOR MULTIRESOLUTION TECHNIQUE
!
For a two-dimensional vector field f (x1 , x 2 ) and a wavelet basis Ψ m1 ,n1 ;m2 ,n2 (x1 , x2 ) the twodimensional discrete wavelet transform is defined by
Wf
m1 ,n1 ;m2 ,n2
=
∑∑
i
(
)
"
f ( x1i , x 2j )Ψ m1 ,n1 ;m2 ,n2 x1i , x 2j .
(1)
j
The reconstruction of the original vector field can be achieved by using
!
f (x1 , x 2 ) =
∑∑∑∑
m1
m2
n1
Wf
m1 ,n1 ;m2 ,n2Ψ m1 ,n1 ;m2 ,n2
(x1 , x2 ) .
(2)
n2
The two-dimensional wavelet basis, Ψ m1 ,n1 ;m2 ,n2 (x1 , x2 ), is simply to take the tensor product
functions generated by two one-dimensional bases as
(
)(
)
Ψ m1 ,n1 ;m2 ,n2 (x1 , x2 ) = 2 −(m1+m2 ) 2ψ 2 − m1 x1 − n1 ψ 2 − m2 x2 − n2 .
(3)
The oldest example of a function ψ (x ) for which the ψ m ,n (x ) constitutes an orthogonal basis
is the Haar function, constructed long before the term “wavelet” was coined. In the last ten years,
various orthogonal wavelet bases have been constructed, for example, Meyer basis, Daubechies
basis, Coifman basis, Battle-Lemarie basis, Baylkin basis, and spline basis, etc.. They provide
excellent localization properties both in physical space and frequency space. In this study we use
the Daubechies basis with index N=20, which is not only orthonormal, but also have smoothness
and compact support, to analyze the flow image.
The procedure of the vector multiresolution analysis can be summarized in two steps:
(1) The wavelet coefficients Wf
m1 ,n1 ;m2 ,n2
, representing vector field in orthonormal wavelets basis,
are computed using the discrete wavelet transform of Eq. (1).
(2) Inverse wavelet transform of Eq. (2) is applied to wavelet coefficients at each wavelet level,
and components of vector field are obtained at each level or scale.
3 EXPERIMENTAL PROCEDURE
The experiment was carried out in liquid-phase turbulent-jet flows. The PIV measurement was
supplied by a double-pulsed Nd:YAG Laser at the frequency of 10 Hz and power of 200 mJ/pulse.
The time interval between the two pulses can be adjustable, which is about 2 to 5 ms for the present
study. The PIV images of transverse sections at downstream position z/d=12 (jet-nozzle diameter d
is 2.54mm) were captured by a 1008x1016 pixels Cross-Correlation CCD array camera. The spatial
resolution of the PIV images for the present research case is about 120 µm pixel . The images were
divided into 8 by 8 pixel interrogation windows, and 50% overlap grids were employed. The
Hierarchical Recursive PIV method was used in the present study for obtaining about 225x225 velocity
vectors. The post-processing procedures which including sub-pixel interpolation and spurious velocity
9th International Symposium on Flow Visualization, Heriot-Watt University, Edinburgh, 2000
Editors G M Carlomagno and I Grant.
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EXTRACTION OF MULTI-SCALE TURBULENT STRUCTURE FROM PIV RESULTS BASED
ON WAVELET VECTOR MULTIRESOLUTION TECHNIQUE
deletion were used to improve the accuracy of the PIV results. More details have been given in Hu et
al. [9].
1000
800
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0
0
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1000
X
Fig.1. Original instantaneous streamlines of PIV results in a lobed mixing jet at Re=3000
1000
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1000
X
(a) a=15.36mm
Fig.2. Multiresolution instantaneous streamlines of PIV results in a lobed mixing jet at Re=3000
9th International Symposium on Flow Visualization, Heriot-Watt University, Edinburgh, 2000
Editors G M Carlomagno and I Grant.
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Hui LI et al.
1000
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0
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X
(b) a=7.68mm
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X
(c) a=3.84mm
Fig.2. Multiresolution instantaneous streamlines of PIV results in a lobed mixing jet at Re=3000
4 RESULTS AND DISCUSSION
Figure 1 showed an original instantaneous streamlines of a cross plane at downstream position
z/d=12 in a lobed mixing jet with Re=3000. The irregular flow structures that imply a multi-scale
9th International Symposium on Flow Visualization, Heriot-Watt University, Edinburgh, 2000
Editors G M Carlomagno and I Grant.
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EXTRACTION OF MULTI-SCALE TURBULENT STRUCTURE FROM PIV RESULTS BASED
ON WAVELET VECTOR MULTIRESOLUTION TECHNIQUE
1000
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Y
600
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200
0
0
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600
800
1000
X
Fig.3. Original instantaneous vorticity of PIV results in a lobed mixing jet at Re=3000
structure can be observed. The irregular large-scale vortices, i.e. coherent structure, can be seen at
the position of lobe. For identifying the multi-scale flow structures, the velocity vector filed
obtained from the PIV measurement was decomposed into five components with different wavelet
levels or broader scales based on the vector multiresolution analysis. The corresponding
components of the streamlines were computed from the components of the velocity vector. Three
components of instantaneous streamlines were shown in Fig.2. Figure 2 (a) exhibited clearly the
large-scale structures with a central scale of a=15.36mm. The large-scale vortices, i.e. coherent
structure, can be clearly seen at the position of lobe. These vortices corresponded quite well to the
vortices in Fig.1. This agreement provides a validation for the present data analysis technique. The
flow structures with a central scale of a=7.68mm can be shown in Fig.2 (b). A number of
intermediate-scale vortices appeared at the position of lobe, and existed in the large-scale vortices
of a=15.36mm (in Fig.2 (a)). It is clear that such structure cannot be extracted by traditional
techniques. Figure 2 (c) showed the flow structures with a central scale of a=3.84mm. A number of
smaller-scale vortices were identifiable. These vortices seem to occur all over the position of lobe.
For even smaller-scale flow structures, it is difficulty to describe such flow structures using the
instantaneous streamlines. We adopted the vorticity to discuss the vortical structures in the
following session.
Figure 3 illustrated a vorticity contour obtained from the original instantaneous PIV
velocities, which corresponded to the instantaneous streamlines of Fig.1. The positive and negative
vorticities were denoted by solid and dashed lines, respectively. Only a number of smaller peaks in
the vorticity field that represented the smaller-scale vortices can be observed in Fig.3.
In order to extract the distribution of the multi-scale vorticity field, the components of the
velocity vector obtained using the wavelet vector multiresolution technique were employed to
compute the vorticity field. The components of vorticity contour with five broader scales were
displayed in Fig.4, and provided information on the distribution of multi-scale vortices in a lobed
mixing turbulent jet. Several alternative larger positive and negative peaks can be clearly seen at
the position of lobe in Fig.4 (a), and indicated pairs of large-scale streamwise vortices with a
central scale of a=15.36mm. They are the uppermost and coherent structures. These larger positive
and negative peaks consistent with the large-scale vortices observed in the instantaneous
streamlines of Fig.2 (a).
9th International Symposium on Flow Visualization, Heriot-Watt University, Edinburgh, 2000
Editors G M Carlomagno and I Grant.
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1000
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0
0
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1000
X
(a) a=15.36mm
1000
800
Y
600
400
200
0
0
200
400
600
800
1000
X
(b) a=7.68mm
Fig. 4. Multiresolution instantaneous vorticity of PIV results in a lobed mixing jet at Re=3000
In the vorticity component of scale a=7.68mm (Fig.4 (b)), several pairs of positive and
negative peaks appeared at the position of each lobe. They implied pairs of intermediate-scale
streamwise vortices. By comparing the Fig.4 (a), it was found that each pair of intermediate-scale
streamwise vortices was contained in a larger-scale vortex. With decreasing the scale to a=3.84mm,
as shown in Fig.4(c), a number of stronger positive or negative peaks mainly distributed in
9th International Symposium on Flow Visualization, Heriot-Watt University, Edinburgh, 2000
Editors G M Carlomagno and I Grant.
383-6
EXTRACTION OF MULTI-SCALE TURBULENT STRUCTURE FROM PIV RESULTS BASED
ON WAVELET VECTOR MULTIRESOLUTION TECHNIQUE
1000
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0
0
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X
(c) a=3.84mm
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0
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1000
X
(d) a=1.92mm
Fig.4. Multiresolution instantaneous vorticity of PIV results in a lobed mixing jet at Re=3000
neighbor of lobe, and correspond to the vortices at this scale range. When further decreasing the
scale to a=1.92 and 0.96mm, as shown in Fig.4 (d) and (e), a clear distribution of a number of
smaller positive or negative peaks can be observed in the interior of the flow. This means that the
smaller-scale vortices exist in the whole flow field. Note that it is difficulty to identify such multiscale flow structures by analyzing the original vector field of PIV results.
9th International Symposium on Flow Visualization, Heriot-Watt University, Edinburgh, 2000
Editors G M Carlomagno and I Grant.
383-7
Hui LI et al.
1000
800
Y
600
400
200
0
0
200
400
600
800
1000
X
(e) a=0.96mm
Fig.4. Multiresolution instantaneous vorticity of PIV results in a lobed mixing jet at Re=3000
Although preliminary, the results presented here demonstrated that the wavelet vector
multiresolution technique could be used effectively for decomposition and analysis of multi-scale
turbulent structures.
5 CONCLUSION
In this paper the wavelet-based vector multiresolution technique was developed and applied to PIV
data. The components of instantaneous streamlines and vorticity were obtained, and important
information on the multi-scale flow structures was provided. The large-scale vortices or coherent
structures with a central scale of a=15.36mm were extracted from PIV measurement results. The
intermediate- and small-scale vortices embedded within the large-scale vortices were visualized.
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9th International Symposium on Flow Visualization, Heriot-Watt University, Edinburgh, 2000
Editors G M Carlomagno and I Grant.
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EXTRACTION OF MULTI-SCALE TURBULENT STRUCTURE FROM PIV RESULTS BASED
ON WAVELET VECTOR MULTIRESOLUTION TECHNIQUE
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9th International Symposium on Flow Visualization, Heriot-Watt University, Edinburgh, 2000
Editors G M Carlomagno and I Grant.
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