AIAA-2001-0696
APPLICATION OF WAVELET VECTOR MULTI-RESOLUTION
TECHNIQUE TO PIV MEASUREMENTS
Hui LI
Department of Mechanical Engineering
Kagoshima University
1-21-40, Korimoto, Kagoshima 890-0065
JAPAN
e-mail: li@mech.kagoshima-u.ac.jp
Hui HU
Turbulent Mixing and Unsteady Aerodynamics Lab.
A22, Research Complex Engineering
Michigan State University
East Lansing, Michigan48824
USA
Toshio KOBAYASHI, Tetsuo SAGA, Nobuyuki TANIGUCHI
Institute of Industrial Science
University of Tokyo
7-22-1 Roppongi, Minato-Ku, Tokyo 106-8558
JAPAN
ABSTRACT
A wavelet-based vector multi-resolution technique
was applied to analyze the three-dimensional
measurement results of a high-resolution stereoscopic
PIV system in this paper. The instantaneous threedimensional flow structures in the near field of lobed
jet mixing flow were successfully decomposed into
large- and small-scale structures based on the wavelet
vector multi-resolution analysis. It is found that as
increasing the downstream distance, the large- and
small-scale streamwise vortices and the higher smallscale w velocity component first grow up and appear
around the trailing edge of the lobed nozzle, and then
they decay rapidly and appear in the center region of
jet.
INTRODUCTION
The turbulent jet exhibited complex structures with
a wide range of coexisting scales and a variety of
shapes in the dynamics and its physics of mixing
process is important in the engineering. It is well
known fact that the streamwise vortices generated in a
jet flow, in additional to the azimuthal (or ring type)
vortices, have been found to mix fluid streams even
Copyright©2000 by the American Institute of
Aeronautics and Astronautics, Inc. All rights reserved.
more efficiently. The streamwise vortices in jet mixing
flows can be generated by many methods. One of
methods is to use a lobed nozzle to generate the largescale streamwise vortices, which has been considered
to be one promising method for the jet mixing
enhancement.
During the last couple of years the development of
PIV techniques has made it possible to provide more
detailed information on flow structure, such as the
instantaneous values of various flow quantities, as
well as their distribution and transient variation.
Recently, Hu et al. (1), (2) (2000) employed twodimensional and stereoscopic PIV system to measure
the near flow field of a lobed jet mixing flow. The
characteristics of the mixing process in a lobed jet
mixing flow compared with a conventional circular jet
flow were discussed based on the two and threedimensional PIV measurement results. Despite the
usefulness of information were obtained by examining
the measured instantaneous flow fields and the timemean 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 turbulent
flow data. This technique allows to track turbulent
structures in terms of time and scale and extracts new
information on turbulence structures (Li, 1998; Li et
al., 1999) (3), (4). To extract the instantaneous multiscale turbulent structures from the two-dimensional
1
American Institute of Aeronautics and Astronautics
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 wavelet vector multiresolution
analysis can be summarized in two steps:
(1) Compute the wavelet coefficients of vector data
based on the discrete wavelet transform of Eq. (1).
(2) Inverse wavelet transform of Eq. (2) is applied to
wavelet coefficients at each wavelet level, and
vector components are determined at each level or
scale.
The wavelet vector multi-resolution analysis may
process fewer data by selecting the relevant details
that are necessary to perform an extraction of the
multi-scale structures, and decompose the vector data
in both Fourier and physical spaces. The technique is
unique in terms of its capability to separate turbulence
structures of different scales.
PIV measurement results in a lobed jet mixing flow,
Li et al. (5) (2000) developed a new signal processing
technique, i.e. wavelet vector multi-resolution
analysis. For the highly three-dimensional flow fields
like lobed jet mixing flows, however, the analysis of
two-dimensional measurement results may not be able
to reveal its three-dimensional feature successfully.
The aim of this paper is to apply the wavelet vector
multi-resolution technique to analyze the threedimensional measurement results of a high-resolution
stereoscopic PIV system for revealing the
instantaneous three-dimensional multi-scale turbulent
structures in the near field of lobed jet mixing flow.
WAVELET VECTOR MULTI-RESOLUTION
TECHNIQUE
H
For a two-dimensional vector field f (x1 , x 2 ) and a
wavelet basis Ψm1 , n1 ; m 2 , n 2 (x1 , x2 ) the two-dimensional
discrete wavelet transform is defined by
Wf m , n ; m , n =
1 1 2 2
åå
i
j
(
L
f ( x1i , x2j )Ψm1 , n1 ; m 2 , n 2 x1i , x2j
(1)
The reconstruction of the original vector field can
be achieved by
H
f ( x1 , x 2 )
=
ååååWf
m1,n1;m2 ,n2 Ψm1,n1;m2 ,n2
(x1 , x2 )
m1 m2 n1 n2
(2)
The
two-dimensional
wavelet
basis,
Ψm1 , n1 ; m 2 , n 2 (x1 , x2 ) , is simply to take the tensor
product functions generated by two one-dimensional
bases as
Ψm1,n1;m2 ,n2 ( x1 , x 2 )
=2
−(m1+ m2 ) 2
(
ψ2
−m1
)(
x1 − n1 ψ 2
− m2
x 2 − n2
)
.
)
EXPERIMENTAL SET-UP AND
STEREOSCOPIC PIV SYSTEM
A test lobed nozzle with six lobes, as shown in
Fig.1, is used in the present study. The width of each
lobe is 6 mm and the height of each lobe is 15 mm (H
= 15 mm). The inner and outer penetration angles of
the lobed structures are θin=220 andθout=140
respectively. The equivalent nozzle diameter is
designed to be D = 40 mm. The z-axis is taken as the
direction of the main stream; the x-y plane is
perpendicular to the z-axis and is taken as the cross
plane of the lobed jet. u, v and w are defined as the
velocity components in x, y and z directions,
respectively.
Figure 2 shows the air jet experimental set-up used
in the present study. A centrifugal compressor was
used to supply air jet flows. A cylindrical plenum
Lobe trough
(3)
X
Lobe peak
Lobeside
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, BattleLemarie basis, Baylkin basis, and spline basis, etc..
Y
Lobe height
H=15
Fig.1 The test lobed nozzle
2
American Institute of Aeronautics and Astronautics
Centrifugal compressor
Test nozzle
Cylindrical
Convergent connection
plenum chamber
Two-dimensional translation mechanism
Fig.2 The air jet experimental setup
optics
Host computer
Laser sheet
Double-pulsed Nd:YAG Laser
Sy n ch r on izer
650mm
250
Lobed nozzle
650mm
250
Measurement region
80mm by 80mm
high-resolution
CCD cameras
Fig.3 The schematic of the stereoscopic PIV system
chamber with honeycomb structures was used to settle
the airflow. Through a convergent connection
(convergent ratio is about 50:1), the airflow is
exhausted from the test nozzles.
The velocity of the air jet exhausting from the test
nozzle can be adjusted and the core jet velocity (U0)
was set at about 20 m/s in the present study. The
Reynolds number of the jet flow is about 60,000 based
on the equivalent nozzle diameter (D) and the core jet
velocity.
Figure 3 shows the schematic of the stereoscopic
PIV system used in the present study. The objective jet
mixing flows were illuminated by a double-pulsed
Nd:YAG laser set (New Wave 50 mJ/pulse) with the
laser sheet thickness being about 2 mm. The doublepulsed Nd:YAG laser set can supply the pulsed laser
at the frequency of 15 Hz. The time interval between
the two-pulsed illuminations was settled as 30 µs .
Two high-resolution (1K by 1K) cross-correlation
CCD cameras (TSI PIVCAM10-30) were used to do
stereoscopic PIV image recording. The two CCD
cameras were arranged in an angular displacement
configuration to get a big overlapped view. In order to
have the measurement field focused on the image
planes perfectly, tilt-axis mounts were installed
between the camera bodies and lenses, the lenses and
camera bodies were adjusted to satisfy the
scheimpflug condition. In the present study, the
3
American Institute of Aeronautics and Astronautics
30
0
-10
-20
-30
-40
-30
Instantaneous Three-Dimensional Multi-Scale
Velocity Fields
In order to gain insight into the multi-scale flow
structures, the wavelet vector multi-resolution analysis
is applied to the three-dimensional measurement
results of PIV. In the present study, the measured three
velocity components of 64x64 are used. The
H
instantaneous velocity vector u (x, y ) is first
decomposed into three wavelet levels and three
H
velocity vector compositions ui (x, y ) within different
scale ranges are produced based on the wavelet vector
multi-resolution analysis. The velocity vector
composition of wavelet level 1, which corresponds to
the central scale of 8 mm, is employed to describe the
large-scale flow structure. Then the sum of velocity
vector compositions of wavelet levels 2 and 3, which
corresponds to the central scale range of 2-4 mm,
constructs the smaller flow structure. Of course, the
H
measured velocity vector u (x, y ) can be written as the
H
sum of velocity vector compositions ui (x, y ) , i.e.
(4)
0
10
20
30
40
5 m/s
W m/s
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
Y mm
10
0
-10
-20
-30
-40
-30
-20
-10
0
10
20
30
40
X mm
Fig. 4 (b) The instantaneous velocity field at the
central scale of 8 mm in the cross plane of z/D =
0.5
30
5 m/s
20
W m/s
4
3
2
1
0
-1
-2
-3
-4
-5
10
Y mm
å
-10
Fig. 4 (a) The instantaneous velocity field of the
stereoscopic PIV measurement results in the cross
plane of z/D = 0.5
20
H
u i ( x, y ) ,
-20
X mm
30
3
W m/s
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
10
RESULTS AND DISCUSSION
H
u ( x, y ) =
5 m/s
20
Y mm
distance between the illuminating laser sheet and
image recording plane of the CCD camera is about
650 mm, and the angle between the view axes of the
two cameras is about 500. For such arrangement, the
size of the overlapped view of the two image
recording cameras for stereoscopic PIV measurement
is about 80 mm by 80 mm.
The two-dimensional particle image displacements
in every image planes were calculated separately by
using Hierarchical Recursive PIV (HR-PIV) software
(6)
(Hu et al. 2000). The Hierarchical Recursive PIV
software is based on a hierarchical recursive process
of conventional spatial correlation operation with
offsetting of the displacement estimated by the former
iteration step and hierarchical reduction of the
interrogation window size and search distance in the
next iteration step.
0
-10
i =1
-20
Figure 4 (a) shows an instantaneous velocity vectors
of the stereoscopic PIV measurement in the cross
plane ((x, y)-plane view) overlapping on the
corresponding the contours of w velocity component
at the downstream location of z/D = 0.5. The falsecolors have been assigned to the value of w velocity
-30
-40
-30
-20
-10
0
10
20
30
40
X mm
Fig. 4 (c) The instantaneous velocity field at the
central scale of 2-4 mm in the cross plane of z/D =
0.5
4
American Institute of Aeronautics and Astronautics
30
5 m/s
W m/s
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
20
Y mm
10
0
-10
-20
-30
-40
-30
-20
-10
0
10
20
30
40
X mm
Fig. 5 (a) The instantaneous velocity field of the
stereoscopic PIV measurement results in the cross
plane of z/D = 1.5
30
5 m/s
W m/s
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
20
Y mm
10
0
-10
-20
-30
-40
-30
-20
-10
0
10
20
30
40
X mm
Fig. 5 (b) The instantaneous velocity field at the
central scale of 8 mm in the cross plane of z/D =
1.5
30
5 m/s
20
W m/s
4
3
2
1
0
-1
-2
-3
-4
-5
10
Y mm
component, and the highest concentration is displayed
as read and the lowest as a blue. This is the original
data before the wavelet decomposition. The irregular
flow structures that imply a multi-scale structure can
be observed. The irregular large-scale streamwise
vortices can be seen to be in the same cofiguration as
the trailing edge geometry of the lobed nozzle. The
highest velocity region of w velocity component exists
in the center region of the jet flow.
The analysis results of the instantaneous velocity
vectors of the stereoscopic PIV measurement
(Fig.4(a)) based on the wavelet vector multi-resolution
technique are shown in Fig.4 (b) and (c), in which the
two different scale componets of instantaneous
velocity field can be seen. Figure 4 (b) displays the
large-scale structures with a central scale of 8 mm. Six
pairs of large-scale streamwise vortices can be clearly
seen around the edge position of the lobed nozzle,
although this location is the initial region of the largescale streamwise vortices generated by the lobed
nozzle. These vortices corresponded quite well to the
irregular vortices appeared in Fig.4 (a). This
agreement provides a validation for the present data
analysis technique. The flow structures with a central
scale range of 2-4 mm can be shown in Fig.4 (c). A
number of the smaller-scale vortices appeared around
the edge position of the lobed nozzle. By comparing
the Fig.4 (b), it is found that these each smaller-scale
vortices are contained in the large-scale streamwise
vortices. The higher w velocity component within the
central scale range of 2-4 mm is also identified clearly
at the same position. Note that such structure cannot
be extracted by traditional techniques.
Figure 5 (a) displays the stereoscopic PIV
measurement results of the cross plane at the z/D =
1.5. The geometry of the lobed nozzle can be
identified from the instantaneous velocity vector field.
The instantaneous velocity field is found to become
more complex than that in the upstream cross plane of
z/D = 1.0. The core jet flow is found to diffuse to the
ambient flow substantially, and the size of the higher
velocity region in the center of the jet flow becomes
smaller compared with Fig. 4(a). The large-scale
structure with a central scale of 8 mm, as shown in
Fig.5 (b), can be extracted based on the wavelet vector
multiresolution analysis. Several pairs of large-scale
streamwise vortices can be clearly seen at the position
of lobe. The vector plot of the cross stream flow
shows that the streamwise vortices have spread
outward. Figure 5 (c) shows the flow structures with a
central scale range of 2-4 mm. The smaller-scale
vortices almost distribut in the whole measured field.
Espeically, they are much more active at the position
of the trailing edge geometry of the lobed nozzle.
0
-10
-20
-30
-40
-30
-20
-10
0
10
20
30
40
X mm
Fig. 5 (c) The instantaneous velocity field at the
central scale of 2-4 mm in the cross plane of z/D =
1.5
5
American Institute of Aeronautics and Astronautics
ω zi =
D ∂vi ∂ui
(
),
−
U 0 ∂x
∂y
(5)
where i stands for the scale.
Figure 7 (a) shows the distribution of the measured
instantaneous streamwise vorticity at z/D = 0.5. The
false-colors have been assigned to the vorticity values,
and the highest concentration is displayed as red and
the lowest as blue, and the positive and negative
vorticities are simultaneously denoted by solid and
dashed lines, respectively. The alternative positive and
5 m/s
W m/s
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
20
Y mm
10
0
-10
-20
-30
-40
-30
-20
-10
0
10
20
30
40
X mm
Fig. 6 (a) The instantaneous velocity field of the
stereoscopic PIV measurement results in the cross
plane of z/D = 4.0
30
5 m/s
W m/s
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
20
10
Y mm
Instantaneous Multi-Scale Streamwise Vorticity
The above velocity vector plots have shown the
existence of very strong multi-scale cross-stream in
the lobed jet mixing. The core jet flow expends
outward along the lobes and ambient flow ejects
inward in the lobe troughs, which result in the
generation of large-scale streamwise vortices.
Therefore, a pair of counter rotating streamwise
vortices in the lobed jet mixing flow can be generated
for each lobe.
In order to study the evolution of multi-scale
streamwise vortices quantitatively, the instantaneous
streamwise vorticity was calculated based on the
velocity data obtained by the stereoscopic PIV
measurement and the wavelet vector multi-resolution
analysis. The normalized instantaneous component of
streamwise vorticity, ω zi , at scale i can be defined in
terms of the derivatives of the instantaneous velocity
components viz.
30
0
-10
-20
-30
-40
-30
-20
-10
0
10
20
30
40
X mm
Fig. 6 (b) The instantaneous velocity field at the
central scale of 8 mm in the cross plane of z/D =
4.0
30
5 m/s
20
W m/s
4
3
2
1
0
-1
-2
-3
-4
-5
10
Y mm
Some of them exist in the large-scale streamwise
vortices and others exist independently. The region of
higher w velocity component is found to be the same
cofiguration as the trailing edge geometry of the lobed
nozzle.
At further downstream location of z/D = 4, the
stereoscopic PIV measurement results are shown in
Fig.6 (a). The geometry of the lobed nozzle almost
cannot be identified from the instantaneous velocity
field. The region and magnitude of the higher w
velocity component in the center of the jet flow are
found to decrease rapidly due to the intensive mixing
of the core jet flow with ambient flow. Figure 6 (b)
gives a clear picture of the large-scale structure with a
central scale of 8 mm. The large-scale streamwise
vortices can be observed in the range of the equivalent
circular nozzle diameter. From the flow structures
with a central scale range of 2-4 mm, as shown in
Fig.6 (c), many active smaller-scale vortices and the
region of the higher w velocity component are
identified in the center region of the jet.
0
-10
-20
-30
-40
-30
-20
-10
0
10
20
30
40
X mm
Fig. 6 (c) The instantaneous velocity field at the
central scale of 2-4 mm in the cross plane of z/D =
4.0
6
American Institute of Aeronautics and Astronautics
30
30
0.5
-0.5
Y mm
10
0.5 0.5
0.5
0.5
-0.5 0.5
-1.5
-2.5 -1.5
0
-1.5
0.5
-0.5
-0.5
-0.5
0.5
0.5
-40
-30
1.5
-20
0.5
-0.5
-0.5
0
-0.5
0.5
-0.5
-10
0.5
0.5
1.5
0.5 2.5
-20
-30
1.5
1.5
-0.5
2.5
0.50.5
1.5
-1.5
-0.5
0.5
0.5
-0.5
-0.5
-0.5
-0.5
1.5 0.5
-10
Vorticity
4.5
3.5
2.5
1.5
0.5
-0.5
-1.5
-2.5
-3.5
-4.5
0.5
-1.5
0.5
0.5 -1.5
0.5
0.5
-0.5
-0.5
0.5
0.5
-1.5
-2.5
3.5
-2.5
0.5
20
-0.5
0.5
-0.5
10
-1.5-1.5
-2.5
-1.5
-0.5
-1.5
0.5
2.5
-3.5
-1.5 -0.5
0
1.5
1.5
0.5
0.5
-1.5
-0.5
-0.5
-2.5 2.5
-4.5
-0.5
0.5
-0.5
-30
-0.5
-1.5
1.5
-0.5
1.5
1.5
-1.5
-0.5
2.5
0.5
0.5
0.5
-0.5
-1.5
-0.5 -0.5
-20
-0.5
-2.5
-0.5
-0.5
-0.5
-0.5
-0.5
-0.5
0.5
-0.5
-3.5
-2.5
-1.5
Vorticity
4.5
3.5
2.5
1.5
0.5
-0.5
-1.5
-2.5
-3.5
-4.5
-0.5
0.5 -1.5
-10
-0.5
-0.5
-0.5
1.5
-0.5
-0.5 -0.5
0.5
-0.5
10
-1.5
-2.5
1.5
20
Y mm
0.5
-3.5
-0.5
-0.5 -0.5
-1.5
0.5
-0.5
20
30
-40
-30
40
-20
-10
0
10
20
30
40
X mm
X mm
Fig.7 (a) The instantaneous streamwise vorticity
distributions of the stereoscopic PIV measurement
results in the cross plane of z/D = 0.5
Fig.8 (a) The instantaneous streamwise vorticity
distributions of the stereoscopic PIV measurement
results in the cross plane of z/D = 1.5
30
30
0.5
-0.5
Y mm
10
0.5 0.5
-0.5
0
-0.5
-0.5
0.5 -0.5
0.5
0.5
-0.5
1.5
0.5
-0.5
-0.5
-0.5
-0.5
-0.5
0.5
-0.5
0.5
1.5
-0.5
0.5
-0.5
-0.5
0.5 -0.5
10
-1.5-1.5
-2.5
-0.5
0.5 -0.5
-2.5
-2.5 -0.5
0
-0.5
-0.5
-0.5
-1.5
-1.5
-0.5
-0.5 1.5
-2.5
0.5
-1.5
-0.5
1.5
-0.5
-0.5
-0.5
1.5
-20
-10
0
10
20
30
-40
-30
40
Fig.7 (b) The instantaneous streamwise vorticity
distributions at the central scale of 8 mm in the cross
plane of z/D = 0.5
-1.5
-1.5 -0.5
0.5
0.5
0.5
-0.5
-3.5
-0.5 -0.5
-0.5
-0.5
1.5
-20
-10
0
10
20
30
40
X mm
Fig.8 (b) The instantaneous streamwise vorticity
distributions at the central scale of 8 mm in the
cross plane of z/D = 1.5
30
30
0.7
10
0
0
0 0
0
0
0
0.7
-0.7
0.7
0
0
0
-2.1
0.7
0.7
0.7
-0.7
0.7
0
-0.7
0
0
-0.7
0
0
00
-0.7
0
0
0
0
0.7
0.7
0
0
0
0
0
0
-1.4
0.7 0.7
-20
0
0
0
0.7
-0.7
-0.7
-0.7
-0.7
-0.70
0
0 -0.7
-10
0
-0.7
0
1.4
-0.7
0
20
0.7
0
0
0
Vorticity
3.5
2.8
2.1
1.4
0.7
0
-0.7
-1.4
-2.1
-2.8
-3.5
10
1.4
-0.7
0.7
0 -0.7
0.7
-1.4
-0.7
0
0.7
0
0
0.7
0
0.7
-2.1 1.4
-2.1
1.4
0
0
0
0
0
0
-0.7
00
0
0
0
-0.7
0.7
0
0
0
-0.7
0.7
-0.7
0
0.7
1.4
0-0.7
0
0
0.7
-20
-30
-0.7
0.7
0
-0.7 0.7
-10
0
0
-0.7
-0.7
0
0
0
0
0
0
-0.7
0
20
Y mm
0.7
Y mm
-0.5
-0.5
X mm
-30
0.5
-0.5
-0.5
-40
-30
-0.5
2.5
1.5
-20
-30
-1.5
-1.5
-0.5
-0.5
-0.5
-0.5
-0.5
-1.5
-0.5
-3.5
-2.5
-0.5
-0.5
Vorticity
4.5
3.5
2.5
1.5
0.5
-0.5
-1.5
-2.5
-3.5
-4.5
-0.5
0.5 -2.5
-10
0.5
-0.5
-0.5
1.5
-0.5
-0.5
0.5
0.5
-0.5
-2.5
1.5
20
-0.5
0.5
2.5
-0.5 2.5
-20
-30
-0.5
-0.5
2.5
1.50.5
1.5
-1.5
0.5
1.5
1.5 0.5
-10
0.50.5
0.5
-0.5
Vorticity
4.5
3.5
2.5
1.5
0.5
-0.5
-1.5
-2.5
-3.5
-4.5
0.5
0.5
-2.5
-0.5
-0.5 0.5
-1.5
-3.5 -2.5
0.5
0.5
-1.5
-2.5
2.5
-1.5
0.5
20
-0.5
-0.5
Y mm
-0.5
0.7
0
0
0
0.7 0
0
Vorticity
3.5
2.8
2.1
1.4
0.7
0
-0.7
-1.4
-2.1
-2.8
-3.5
-0.7
-0.7
0
0
-40
-30
-40
-30
-20
-10
0
10
20
30
-20
-10
X mm
Fig.7 (c) The instantaneous streamwise vorticity
distributions at the central scale of 2-4 mm in the
cross plane of z/D = 0.5
0
10
20
30
40
X mm
40
Fig.8 (c) The instantaneous streamwise vorticity
distributions at the central scale of 2-4 mm in the
cross plane of z/D = 1.5
negative peaks can be clearly seen around the edge
7
American Institute of Aeronautics and Astronautics
30
-0.5
0.5
0.5
Y mm
10
0.5 2.5
-0.5
0.5
0.5
0.5
-0.5 -1.5
0.5
-0.5 -0.5
0
1.5
3.5
-0.5
-0.5
1.5
-0.5
0.5
-0.5 0.5
-1.5
2.5
-0.5
-0.5
0.5
0.5
0.5
0.5
0.5
-0.5
-1.5
-1.5
-0.5
0.5
-0.5
0.5
1.5
2.5
-0.5 -0.5
1.5
-0.5 -0.5
-20
-30
-0.5
-1.5
-0.5
0.5
0.5
0.5
-2.5
1.5
-1.5
0.5
-0.5
-0.5
Vorticity
4.5
3.5
2.5
1.5
0.5
-0.5
-1.5
-2.5
-3.5
-4.5
-0.5
-0.5 0.5
-10
0.5
-1.5
-0.5
-0.5
-0.5
0.5
20
0.5
0.5
-0.5
0.5
-40
-30
-20
-10
0
10
20
30
40
X mm
Fig.9 (a) The instantaneous streamwise vorticity
distributions of the stereoscopic PIV measurement
results in the cross plane of z/D = 4.0
30
-0.5
0.5
0.5
Y mm
10
0.5 1.5
-0.5
-1.5
1.5
-1.5 -1.5
0.5
-0.5 -0.5
0
-0.5
1.5
-0.5
0.5
-1.5
-0.5
0.5
1.5
-1.5
0.5
-0.5
-0.5
0.5
-0.5
1.5
-1.5
0.5
0.5
-0.5
0.5
1.5
1.5
0.5 -0.5
-20
-30
-1.5
-1.5
0.5
1.5
0.5
-1.5
-3.5
-0.5 0.5
-10
Vorticity
4.5
3.5
2.5
1.5
0.5
-0.5
-1.5
-2.5
-3.5
-4.5
-0.5
-1.5
-0.5
-0.5
0.5
0.5 0.5
-0.5
1.5
1.5
-0.5
3.5
-0.5
-1.5
-0.5
0.5
-0.5
0.5
20
-0.5
-1.5 -0.5
1.5
0.5
-0.5
0.5
-40
-30
-20
-10
0
10
20
30
40
X mm
Fig.9 (b) The instantaneous streamwise vorticity
distributions at the central scale of 8 mm in the
cross plane of z/D = 4.0
30
0
0
0
10
0.7
0.7
0
0
0
0.7
0
0
0
-10
0 0
0
1.4
-0.7
1.4
-0.7 0.7
-0.7
0
0.7
0
0
0
-0.7
0
-0.7
00
1.4
0.7
0
0.7
-1.4
0
0 00
0
0
0
0
-0.7
0
0
0
00
0
0
0
0
0.7
0
0.7
0.7
-0.7 -0.7
-20
-30
0.7
0
-0.7
-0.7
0
0
20
Y mm
negative peaks can be clearly seen around the edge
position of the equivalent circular nozzle diameter,
which indicates pairs of streamwise vortices. But it is
difficulty to identify the smaller-scale structures using
the measured instantaneous vorticity. Figure 7 (b) and
(c) provides information on the distribution of multiscale vortices in the lobed mixing turbulent jet. The
pairs of large-scale streamwise vortices that
correspond to vortices appeared in Fig.7 (a) can be
clearly observed in Fig.7 (b). Figure 7 (c) shows the
distribution of the smaller-scale vorticity with a central
scale range of 2-4 mm. The alternative positive and
negative peaks can be clearly seen around the position
of lobe, which indicates pairs of the smaller-scale
streamwise vortices. It maybe implies the existence of
horseshoe vortical structures.
As increasing the downstream distance to z/D = 1.5,
as shown in Fig.8 (a), the distribution of the measured
instantaneous streamwise vorticity exhibits many
alternative positive and negative peaks at the position
of lobe. They imply the multi-scale pairs of
streamwise vortices. However, Figure 8(b) only
displays the distribution of large-scale streamwise
vorticity. Several pairs of large-scale streamwise
vortices can be clearly seen at the position of lobe.
The distribution of the smaller-scale vorticity with a
central scale range of 2-4 mm is shown in Fig.7 (c).
As indicated in the above velocity vector plot, many
positive and negative peaks that imply the smallerscale vortices appear in the whole measured field.
At further downstream location of z/D = 4, from the
distribution of the measured instantaneous streamwise
vorticity in Fig.9 (a), many positive and negative
peaks mainly distribute in the center region of the jet.
From the results of the wavelet multi-resolution
analysis, as shown in Fig.9 (b) and (c), it is found that
both the large-scale streamwise vortices and smallerscale streamwise vortices are concentrated in the
center region of jet. The maximum vorticity value of
these streamwise vortices is found be decreased when
compared with that at z/D = 1.5.
From the above discuss at different downstream
cross planes, it can be seen that as increasing the
downstream distance, the size and strength of the
large-scale streamwise vortices generated by the lobed
nozzle first grow up and appear at the position of lobe.
Then they decay rapidly and only appear in the center
region of jet.
0
Vorticity
3.5
2.8
2.1
1.4
0.7
0
-0.7
-1.4
-2.1
-2.8
-3.5
0 0.7
0
0
0
-40
-30
-20
-10
0
10
20
30
40
X mm
CONCLUSIONS
In order to extract the three-dimensional multi-scale
structures features of the lobed jet mixing flow, the
Fig.9 (c) The instantaneous streamwise vorticity
distributions at the central scale of 2-4 mm in the
cross plane of z/D = 4.0
8
American Institute of Aeronautics and Astronautics
wavelet vector multi-resolution technique was applied
to analyze the three-dimensional measurement results
of a high-resolution stereoscopic PIV system in this
paper. The following main results are summarized.
(1) The instantaneous three-dimensional flow
structures were successfully decomposed into largeand small-scale structures based on the wavelet
vector multi-resolution analysis.
(2) The pairs of large- and small-scale streamwise
vortices and the higher values of the small-scale w
velocity component were found around the position
of lobe at the location of z/D=0.5.
(3) The pairs of large-scale streamwise vortices can
be clearly observed at the position of lobe and the
small-scale vortices appear in the whole measured
field when increasing the downstream distance to
z/D = 1.5. The higher values of the small-scale w
velocity component distribute around the trailing
edge of the lobed nozzle.
(4) The large- and small-scale streamwise vortices
and the higher small-scale w velocity component
appear in the center region of jet at further
downstream location of z/D = 4.
(2)
(3)
(4)
(5)
(6)
REFERENCES
(1) Hu, H., Saga, T. and Kobayashi, T., “Research on
the Vortical and Turbulent Structures in the
Lobed Jet Flow by Using LIF and PIV”,
Measurement Science and Technology, Vol.11
(2000), pp.698-711.
Hu, H., Saga, T., Kobayashi, T. and Taniguchi,
N., “Stereoscopic PIV Measurement of a Lobed
Jet Mixing Flow”, Developments in Laser
Techniques and Applications to Fluid Mechanics,
R. J. Adrian et al. (Eds.), Springer-Verlag (2001).
Li H., “Identification of Coherent Structure in
Turbulent Shear Flow with Wavelet Correlation
Analysis”, ASME Journal of Fluids Engineering,
Vol.120 (1998), No.4, pp.778-785.
Li H., Takei, M., Ochi, M., Saito, Y. and Horii,
K., “Application of Two-dimensional Orthogonal
Wavelets to Multiresolution Image Analysis of a
Turbulent Jet”, Transactions of the Japan Society
for Aeronautical and Space Sciences, Vol.42
(1999), No.137, pp.120-127.
Li H., Hu, H., Saga, T., Kobayashi, T. and
Taniguchi, N., “Extraction of Multi-scale
Turbulent Structure from PIV Results based on
Wavelet Vector Multiresolution Technique”,
Proceedings of the 9th International Symposium
on Flow Visualization, UK (2000), No.383, pp.19.
Hu, H., Saga, T., Kobayashi, T., Taniguchi, N.
and Segawa S., “The Spatial Resolution
Improvement of PIV Result by Using Hierarchical
Recursive Operation”, Journal of Visualization,
No.3 Vol. 2 (2000).
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American Institute of Aeronautics and Astronautics