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RESEARCH ARTICLE
Population, characteristics and kinematics of vortices
in a confined rectangular jet with a co-flow
B. Kong •M. G. Olsen •R. O. Fox •
J. C. Hill
Received: 30 March 2010 / Revised: 3 September 2010 / Accepted: 11 October 2010 / Published online: 20 November 2010
ÓSpringer-Verlag 2010
Abstract Vortex behavior and characteristics in a con-
fined rectangular jet with a co-flow were examined using
vortex swirling strength as a defining characteristic. On the
left side of the jet, the positively (counterclockwise)
rotating vortices are dominant, while negatively rotating
vortices are dominant on the right side of the jet. The
characteristics of vortices, such as population density,
average size and strength, and deviation velocity, were
calculated and analyzed in both the cross-stream direction
and the streamwise direction. In the near-field of the jet, the
population density, average size and strength of the dom-
inant direction vortices show high values on both sides of
the center stream with a small number of counter-rotating
vortices produced in the small wake regions close to jet
outlet. As the flow develops, the wake regions disappear,
these count-rotating vortices also disappear, and the pop-
ulation of the dominant direction vortices increase and
spread in the jet. The mean size and strength of the vortices
decrease monotonically with streamwise coordinate. The
signs of vortex deviation velocity indicate the vortices
transfer low momentum to high-velocity region and high
momentum to the low velocity region. The developing
trends of these characteristics were also identified by
tracing vortices using time-resolved particle image veloc-
imetry data. Both the mean tracked vortex strength and size
decrease with increasing downstream distance overall. At
the locations of the left peak of turbulent kinetic energy,
the two-point spatial cross-correlation of swirling strength
with velocity fluctuation and concentration fluctuation
were calculated. All the correlation fields contain one
positively correlated region and one negatively correlated
region although the orientations of the correlation fields
varied, due to the flow transitioning from wake, to jet, to
channel flow. Finally, linear stochastic estimation was used
to calculate conditional structures. The large-scale struc-
tures in the velocity field revealed by linear stochastic
estimation are spindle-shaped with a titling stream-wise
major axis.
1 Introduction
The study of coherent structures in turbulent shear flows is
of great importance due to the structures’ significant con-
tribution to fluid entrainment and the transfer of mass,
momentum and heat. The properties of these structures,
such as population, size, circulation, and energy, can be
useful in the understanding of turbulence and property
transport and can aid in the development of more efficient
and more environmentally benign applications. Moreover,
detailed measurements of the behavior and characteristics
of large-scale structures can be used to validate the pre-
dictions of computer modeling techniques, such as large
eddy simulation, where the largest scales of the turbulence
are resolved. The objective of the work presented here is to
use a vortex identification method to analyze the experi-
mental data collected in a confined rectangular liquid jet
with a co-flow with regard to the population, size, kine-
matics and characteristics of vortices within the jet.
In the chemical process industry, liquid jets usually are
placed in a confined space, as in a mixer or a reactor, and
are commonly applied within a co-flow or cross-flow to
improve mixing and chemical reaction between two or
more fluids, like a concentric pipeline mixer or a coaxial jet
mixer (Lu et al. 1997). A typical geometry for this type of
B. Kong M. G. Olsen (&)R. O. Fox J. C. Hill
Department of Mechanical Engineering,
Iowa State University, Ames, IA 50010, USA
e-mail: mgolsen@iastate.edu
123
Exp Fluids (2011) 50:1473–1493
DOI 10.1007/s00348-010-0995-9
mixer is an axisymmetric jet with a co-flow or cross-flow.
However, cylindrical reactors pose challenges in the lab-
oratory with laser techniques such as particle image ve-
locimetry (PIV) due to the curvature and associated image
distortion in the measurements. An alternative approach is
to use a rectangular configuration (Feng et al. 2005), which
can provide much better measurement accuracy and also a
much simpler geometry to conduct CFD simulations. When
the aspect ratio of a rectangular jet nozzle (w/h, where
wand hare the long and short sides of the nozzle) is not
large enough to eliminate the 3D effect, the jet usually can
no longer be considered a planar jet, but instead a rectan-
gular jet (Deo et al. 2007a). For example, Pope (2000)
pointed out that the aspect ratio of a plane nozzle must be
significantly large, typically 50 or more, to ensure that the
flow is statistically two-dimensional and free from the
effects of sidewalls. For a liquid phase jet, especially when
designed to study the mixing process in a high Schmidt
number environment, it is not practical to build and study a
high aspect ratio planar jet mixer. The aspect ratio of the jet
in the current research is five, which indeed is a rectangular
jet. But according to Deo et al. (2007b), the sidewalls of
the confinement can significantly limit the development of
the jet in the spanwise direction, and make the 2-D region
extend a longer distance than a rectangular jet without
sidewalls. In this study, the farthest observation down-
stream location was X/d=30, and thus the measured flow
fields are more comparable to planar jets, instead of most
studies of low- aspect-ratio rectangular jets in which the
focus is mainly on a free jet, in which the 3-D vortex ring
structure can break-down faster than in axisymmetric jets
due to the presence of sharp corners (Grinstein 2001;
Gutmark and Grinstein 1999). Although the confined
rectangular jet with a co-flow is not common in turbulent
jet research, it provides a simple and well-defined geometry
suitable for both experimental measurements and numeri-
cal simulations, especially for PIV experiment and 3-D
CFD simulations.
The jet flapping phenomenon first reported by Golds-
chmidt and Bradshaw (1973) is one of the first indications
of large-scale structure in the planar turbulent jet. How-
ever, Oler and Goldschmidt (1982) suggested that corre-
lation results indicating the flapping motion can also be
explained by the presence of an antisymmetric array of
counter-rotating spanwise vortices. Antonia et al. (1983)
also performed correlation measurements, which support
this concept and showed that the apparent flapping could
indeed be explained in terms of the passage of vortical
structures past the fixed probe pair and was not associated
with bulk lateral displacement of the jet. The existence of a
large-scale structural array has also been demonstrated
later by the velocity fluctuation two-point correlation fields
obtained in different jet facilities by Mumford (1982),
Antonia et al. (1986) and Thomas and Brehob (1986). Such
large-scale structural arrays propagate at approximately
60% of the local centerline mean velocity, estimated by
Goldschmidt et al. (1981), Antonia et al. (1983) and Tho-
mas and Brehob (1986). Recently, Gordeyev and Thomas
(2000) used proper orthogonal decomposition (POD) to
analyze experimental data, also suggesting that the pres-
ence of planar structures aligned in the spanwise direction
as well as three-dimensional structures with asymmetrical
shape in the cross-stream direction and pseudo-periodically
distributed in the spanwise direction.
Most of the earlier coherent structure investigations on
planar jets were focused on the large structures, because
they were conducted using point-wise velocity measure-
ments and then analyzed with a correlation technique.
In the past two decades, particle image velocitimetry
(PIV) has become a popular experimental technique for
turbulence studies. Since PIV provides instantaneous
two-dimensional velocity field data, it is well suited for
visualizing and identifying vortical structures. Methods of
analyzing and interpreting these spatially resolved velocity
field data were discussed in Adrian (1999) and Adrian et al.
(2000a). Prime examples of using PIV to study coherent
structure are provided by Christensen and Adrian (2001,
2002a,b) in turbulent boundary layers and Agrawal and
Prasad (2002a,b,2003)in turbulent axisymmetric jets. Of
particular interest to the present study, Agrawal and Prasad
(2003) and Chhabra et al. (2006) reported results on vor-
tices present in the axial plane of a self-similar turbulent
axisymmetric jet by using a high-pass filter and vortex
extraction method based on velocity quadrant and angular
variance. These studies include measurements of the vortex
characteristics, such as the vortex population, energy,
vorticity, and rms of velocity fluctuations. Their studies
also show that the centers of a larger vortices spin faster
than centers of a smaller vortices. After normalization,
vorticity results for different vortex radii collapse upon
each other. In contrast, the similar studies of small-scale
vortices in planar jets and rectangular jets are lacking.
Besides the geometry of the jet, another new element of
current study is the presence of co-flow. Although the vast
majority of turbulent jet research has been focused on jets
discharged into ambient fluids, a jet with a co-flow is more
commonly seen in industrial applications. One of the ear-
liest investigations of jets with a co-flow is the analytical
study done by Abramovich et al. (1984). He showed the
presence of three essential regions in co-flow jets: the
initial, principal and transition areas. When the flow is
confined, the process of the co-flow driven by the jet is
modified and the mixing process depends strongly not only
on the velocity ratio but also on the interaction between the
boundary layer, the mixing layer and the main flow
(Gazzah 2010). Gutmark and Wygnanski (1976) found that
1474 Exp Fluids (2011) 50:1473–1493
123
a jet exhausting into a slow-moving co-flowing stream is
narrower than a comparable jet exhausting into quiescent
surroundings. Curtet (1958) was interested by recirculation
phenomena generated by a considerable pressure gradient,
and proposed a parameter of similarity called the parameter
of CrayaCurtet, which was formulated by Steward and
Guruz (1977). For a CrayaCurtet parameter value greater
than 0.8, recirculation can be avoided. Because of its great
practical importance, an increasing attention has been
given to the jet development and mixing in jets with a co-
flow (Bradbury and Riley 1967; Nickels and Perry 1996;
Chu et al. 1999; Benayad et al. 2001; Enjalbert et al.
2009). Another study of interest is the experimental
investigation of an axisymmetric jet discharging in a co-
flowing air stream by Antonia and Bilger (2006), which
shows that the far jet flow may be strongly dependent on
the nozzle injection conditions. Despite all of this previous
work, the behavior of turbulent vortices in such a flow
remains relatively unstudied. Thus, in the work that fol-
lows, the characteristics of vortices in a confined rectan-
gular turbulent jet with a co-flow have been investigated.
2 Experimental facility and methodology
The flow facility used in the experiments presented here is
shown in Fig. 1. The measurements are carried out in a
Plexiglas test section with a rectangular cross-section
measuring 60 mm by 100 mm and with an overall length of
1 m. There are three streams separated by two splitter plates,
each emitting from its own flow conditioning section con-
sisting of a packed bed and turbulence reducing screens and
a 16:1 contraction section. The slope of the surface of the
splitter plates is 3°along the side channels and 1°along the
center channel, and the thickness of the tips of the splitter
plates are less than 0.5 mm. Two different flow cases as
defined by different bulk flow Reynolds numbers of 20,000
and 50,000 (based on average velocity and hydraulic
diameter) were studied. The volumetric flow rates for the
two flow cases are listed in the Table 1.
Experimental data from simultaneous PIV/PLIF exper-
iments and high-speed PIV experiments were collected and
analyzed. The simultaneous PIV/PLIF technique has been
used to study many turbulent mixing problems, such as
Westerweel et al. (2002,2005) in a turbulent jet.
Approximately 24 g of hollow glass spheres (Spherical,
Potters Industries, Inc.) were added to the water reservoir
with total volume of 3,500 l in both the regular and high-
speed PIV measurements. The nominal diameter of the
seed particles was 11.7 lm and the density of the particles
was 1.1 g/cm
3
. The 0.5-mm-thick laser sheet used in high-
speed PIV experiments was produced by a Quantronix
Darwin-Duo dual oscillator, single head Nd:YLF CW laser,
which passes through the centerline of the test section in
the z-direction. A Photron ultima APX-RS high-speed 10-
bit CMOS camera was used to capture particle images. The
image magnification of the CMOS camera was 0.27, and
the numerical aperture was 8. The image capturing fre-
quency was 125 Hz for the Re =20K case and 250 Hz for
the Re =50K case. A multi-pass cross-correlation tech-
nique with decreasing window sizes was used to compute
the velocity field. The final interrogation spot size mea-
sured 16 pixels by 16 pixels, corresponding to 1.02 mm on
each side. With 50% overlap between interrogation win-
dows, the velocity vector spacing was 0.51 mm in both the
x- and y-directions. The time interval between two laser
pulses was 1 ms in the low flow rate case, and 600 ls in the
high flow rate case. Since the memory capacity of CMOS
camera is limited, 1,024 PIV image pairs can be collected
in each run. At each observed location, the experiment was
repeated 20 times, thus a total of 20,480 PIV realizations
were collected and analyzed at each locations in both cases.
The optical set up for the PIV/PLIF experiments is
shown in Fig. 2. In the simultaneous PIV/PLIF measure-
ments, illumination was provided by a New Wave
Research Gemini PIV laser. PIV and PLIF images were
obtained using two 12-bit LaVision Flowmaster 3S CCD
cameras. The image magnification of the two CCD cameras
Fig. 1 Photograph and schematic of the confined planar jet test
section
Table 1 Flow rates of two cases
Reynolds number Flow rate (l/s)
Center stream Outer stream
20K 0.8 0.4
50K 2.0 1.0
Exp Fluids (2011) 50:1473–1493 1475
123
was 0.12, and the numerical aperture was 8 for PIV and 5.6
for PLIF. A dichroic mirror (Q545LP, Chroma Technology
Corp.) was placed at an angle of 45°to the laser sheet to
separate the light paths and direct them to either the PIV or
the PLIF camera. The PLIF camera lens was filtered with a
long-pass optical filter (E560LP, Chroma Technology
Corp.), and the PIV camera lens was filtered with a narrow-
band-pass optical filter (Z532/10X, Chroma Technology
Corp.) The seeding method, time interval between two
laser pulses and cross-correlation technique in PIV mea-
surement in the simultaneous PIV/PLIF experiments were
the same as in high-speed PIV measurement. The fluores-
cent dye Rhodamine 6G was used as a passive scalar in
PLIF. In the center stream, the source concentration of
Rhodamine 6G was 45 lg/l, while the other two streams
were pure water. The in-plane spatial resolution of the
PLIF measurements in the present study was actually
limited by the flow area imaged per pixel, which was
approximately 56 lm. The simultaneous PIV/PLIF data at
high Reynolds number are a subset of measurements
reported by Feng et al. (2007b). Since the experimental
apparatus and procedure have been described in detail
elsewhere (Feng et al. 2005,2007b), the reader is directed
to the literature for further information. At each observed
location, 3,250 simultaneous velocity and concentration
realizations were analyzed in high Re cases and 2,500
realizations in low Re case. The simultaneous PIV/PLIF
experimental data for the low Re case are available at two
locations, X/d=4.5 and X/d=7.5, where Xis down-
stream distance and dis the initial jet width, 20 mm.
The smallest Kolmogorov scale in the flow field can be
estimated based on the exit width of the jet and the tur-
bulent kinetic energy at the tips of the two splitter plates.
Although the smallest turbulent scales of the flow cannot
be fully resolved, second-order quantities such as velocity
fluctuations and characteristics of large-scale structures can
be measured accurately. As in Prasad et al. (1992), the
random error in the PIV measurements was estimated as
one-tenth of the effective particle image diameter. The
measurement resolution and uncertainty of the two cases
are listed in the Table 2.
To assistant in the understanding of the flow configu-
ration of current investigation, Fig. 3gives the profiles of
the mean of streamwise velocity, U, for both Reynolds
number cases at 3 different downstream locations, X/d=1,
4.5 and 15. After normalization by DU, which is the
average velocity difference between the center stream and
two side streams (DU¼0:2 m/s for Re = 20K and DU¼
0:5 m/s for Re =50K), the profiles of these two flow cases
are very similar. Near the channel inlet, at X/d=1, two
small wake regions on both sides of the tips of the splitter
plates can be observed in the velocity profile. These result
from the boundary layers formed on the splitter plates. At
the further downstream location, X/d=4.5, the two wake
regions disappear and are replaced by two mixing layers
regions that quickly grow together, resulting in the poten-
tial core in the center jet disappearing for both cases. At
X/d=15, the potential cores in the outer streams also
disappear, and the flow continues its development toward
channel flow because of the confinement by the two walls
in cross-stream direction. The wake regions in the near-
field of a jet and confinement configuration of the jet are
rather uncommon in turbulent jet research Again, as
mentioned in the introduction, the design of the jet con-
figuration is determined by the practical considerations of
simulating flow conditions in an industrial reactor.
3 Swirling strength vortex-ID method
Several methods have been suggested and implemented by
researchers to identify vortices in instantaneous two-
dimensional velocity fields, such as those obtained by PIV.
Adrian et al. (2000a,b) described a number of vortex
Table 2 PIV spatial resolution and measurement uncertainty
Reynolds
number
Spatial resolution (g:
local Kolmogorov
scale)
Uncertainty
Center
stream
(%)
Outer
stream
(%)
High speed 20K 5.4g±1.2 ±2.5
PIV 50K 6.7g±0.8 ±1.6
Simultaneous 20K 4.6g±1.7 ±3.4
PIV/PLIF 50K 6.0g±1.3 ±2.7
Fig. 2 Schematic of the optical setup for the combined PIV and PLIF
experiments
1476 Exp Fluids (2011) 50:1473–1493
123
eduction techniques and compared their effectiveness in
identifying vortices. They found that swirling strength
provides a reliable means of extracting the small-scale
vortices, including those which are not visible in velocity
decompositions. Vollmers (2001) also validated and com-
pared several coherent structure eduction methods. He also
concluded that the swirling strength is the best indicator for
the presence of vortices in turbulent flow. Thus, the authors
have chosen a vortex extraction technique based on
swirling strength to analyze the turbulent vortices in a
confined rectangular turbulent jet in this paper.
The concept of swirling strength, k
ci
, was described by
Adrian (1999) as the imaginary part of the complex con-
jugate eigenvalues (k
cr
±k
ci
) of the local velocity gradient
tensor. Physically, k
ci
-1
represents the period required for a
fluid particle to swirl once about the k
cr
-axis (Piomelli et al.
1996; Adrian 1999). Thus, a non-zero k
ci
indicates a local
swirling motion, and spatially connected regions of non-
zero k
ci
represent vortices (Wu and Christensen 2006).
Tomkins and Adrian (2003) multiplied k
ci
by the sign of the
local vorticity to capture the direction of rotation at each
location in the flow field. This modified form of swirling
strength can be written as (Wu and Christensen 2006)
Kciðx;yÞ¼kci ðx;yÞxzðx;yÞ
jxzðx;yÞj ð1Þ
This method has two great advantages. First, it is frame
independent, meaning a priori information of the bulk
motion of the vortex core is not necessary. Second, it only
reveals the regions with vortices, which means the flow
regions with high vorticity but no local swirling motion,
such as shear layers, do not produce non-zero k
ci
. However,
like other vortex extraction methods based on the velocity
gradient, the method does have one important drawback
when applied to PIV data. Due to the sensitivity of velocity
gradient to noise, this type of method is often not applicable
to experimental data if no smoothing has been applied
(Vollmers 2001), since the method may falsely identify
measurement noise as vortices. However, too much velocity
smoothing may eliminate the small vortices. Thus, in
addition to carefully choosing experiment parameters to
reduce the measurement uncertainty and applying slight
smoothing, we utilized a threshold of jKci j1:5Krms
ci (Wu
and Christensen 2006) to limit the influence of experiment
noise, which only affects the edges of clusters and does not
alter the populations appreciably. Further information about
this method can be found in the references mentioned above.
4 Vortex characteristics analysis
4.1 Instantaneous structures
Two examples of the instantaneous swirling strength field
at the near-field 0 \X/d\2.5 and far-field 6.5 \X/d\9
of the jet at Re =20K are shown in Fig. 4to illustrate the
effectiveness of the swirling strength as a vortex indicator.
It is contrasted with the mean-bulk Galilean decomposed
velocity field in Fig. 4a1, a2, which reveals only the vor-
tices moving at their respective spatial convective veloci-
ties. The local Galilean decomposed velocity fields are
shown in Fig. 4b1, b2, where the local convective veloci-
ties have been subtracted from each vortex center. The
background contours are the clusters of Kci after filtering
with a universal threshold of jKci j1:5Krms
ci . Where the
clusters of filtered Kci identified vortices, all of the local
Galilean decomposed velocity vectors on Fig. 4b1, b2
display a clear swirling motion. This demonstrates the
effectiveness of using the swirling strength as a vortex
identification method in the present flow field. Due to
−1.5 −1 −0.5 0 0.5 1 1.5
0
0.5
1
1.5
2
2.5
Mean of Stream wise velocity
Re = 20K
X/d
<U> / ΔU (ΔU = 0.2 m/s)
−1.5 −1 −0.5 0 0.5 11.5
0
0.5
1
1.5
2
2.5
Re = 50K
X/d
<U> / ΔU (ΔU = 0.5 m/s)
Fig. 3 The mean of streamwise
velocity. Filled circle:
X/d=1, open square:
X/d=4.5, asterisk:X/d=15
Exp Fluids (2011) 50:1473–1493 1477
123
spatial resolution limitations, clusters of Kci with fewer
than three grid points across their span in both the xand
ydirections are not considered.
Comparing the definition of swirling strength above
with Robinson’s definition of a vortex, ‘‘A vortex exists
when instantaneous streamlines mapped onto a plane nor-
mal to the vortex core exhibit a roughly circular or spiral
pattern, when viewed from a reference frame moving with
the center of the vortex core’’ (Robinson 1991), the edges
of swirling strength clusters do not necessarily accurately
define the very edge of a vortex. By its definition, the
swirling strength is the indicator of how strong the swirling
motion of a particular local fluid particle is, or how fast the
particle rotates. Therefore, the swirling strength clusters are
only the parts of vortices that only include fluid particles
with distinguishable strong swirling motions. For those
vortices with weak rotation at the vortex edge, the filtering
process performed to eliminate the effect of the measure-
ment noise will also eliminate the swirling strength sig-
natures of these regions. A small fluid particle with close to
zero swirling strength, with no rotation or swirling motions
itself, could be part of large-scale coherent motion.
Therefore, the swirling strength clusters identified from the
instantaneous velocity fields are only defined as vortex
cores, or the highly rotating part of a vortex. As Agrawal
and Prasad (2003) argued, vortices in turbulent flows range
in size from the integral length scale down to the Kol-
mogorov scale, and there are always small-scale vortices
embedded in large-scale structures. Large-scale vortices
account for most of the turbulent energy, while the small
Y/d
X/d
-1 0 1
6.5
7
7.5
8
8.5
9
50
40
30
20
10
0
-10
-20
-30
-40
-50
Y/d
X/d
-1 0 1
6.5
7
7.5
8
8.5
9
Y/d
X/d
-1.5 -1 -0.5 0 0.5 1 1.5
0
0.5
1
1.5
2
50
40
30
20
10
0
-10
-20
-30
-40
-50
Y/d
X/d
-1.5 -1 -0.5 0 0.5 1 1.5
0
0.5
1
1.5
2
(a1)
(b1)
(a2)
(b2)
Fig. 4 Example of vortex identification in an instantaneous PIV
velocity field in near-field 0\X/d\2.5 and far-field 6.5 \X/d\9
of the jet, at Re =20K. aThe Galilean decomposition of this
instantaneous velocity field with 0.7U
c
, where U
c
is the mean velocity
of the center stream at each Xlocation. bLocal Galilean decompo-
sition of vortices with the velocity at each vortex core. The contours
of Kci ¼1:5Krms
ci are also shown in the background of the figure aand
b
1478 Exp Fluids (2011) 50:1473–1493
123
scales carry almost the entire vorticity. These swirling
strength clusters are those small-scale vortices in their
argument.
Figure 4shows that on the left side of the jet, the pos-
itively (counterclockwise) rotating vortices are dominant,
while negatively rotating vortices are dominant on the right
side of the jet. One may notice from the instantaneous
fields of vortices that in both the near-field and the far-field
of the jet, the distribution of the vortices seems random and
does not display as the large-scale structural array in planar
jets suggested by the Mumford (1982), Antonia et al.
(1986), nor the roller and helical modes revealed by low-
pass filtering axisymmetric jets (Agrawal and Prasad
2002a). The velocity fluctuation correlation reported by
Feng et al. (2007a) actually shows, in this jet, there are the
chain of large coherent structures in the stream-wise
direction, as previous researchers suggested in planar jets.
As mentioned earlier, the large-scale coherent motions
visualized by the correlation technique are not the swirling
motions of the small-scale vortices in the individual
instantaneous velocity field. Also, notice that besides those
dominant positively rotating vortices, there are also some
negatively rotating vortices on the left side of the jet. The
counter-rotating vortices are due to the small wake regions
downstream of the splitter plates. In the far-field of the jet,
the instantaneous vortex fields bear some resemblance to
the small-scale vortex field revealed by a high-pass filter in
the far-field of an axisymmetric jet (Agrawal and Prasad
2002b,2003), where small-scale vortices spread across the
jet body. Spatial correlation calculations with the swirling
strength were also performed in the present study, like the
calculations done by Christensen and Adrian (2001), who
found a strong position and angle preference between the
vortices in wall turbulence. The results here show that the
vortices in entire body of this jet have no strong spatial
correlations with other vortices at all, confirming the high
level of randomness observed in the instantaneous fields. In
addition to the vortices in the jet region, there are also some
vortices in the two boundary layers near the walls, which
are not of interest in the current study.
4.2 Vortex population
To identify the population trends of the vortices, a similar
vortex definition as Wu and Christensen (2006) is used to
define the population density of vortices, PpðnÞ.PpðnÞis
herein defined as the local ensemble-averaged number of
detectable positively rotating or negatively rotating vorti-
ces whose centers reside at a given PIV grid node. Figure 5
shows PpðnÞcross-streamwise profiles at 3 downstream
locations for the two different Reynolds numbers
−1.5 −1 −0.5 0 0.5 1 1.5
0
0.5
1
Y/d
Per Realization
Re = 20K X/d = 1
−1.5 −1 −0.5 0 0.5 11.5
0
0.5
1
Y/d
Per Realization
Re = 50K X/d = 1
−1.5 −1 −0.5 0 0.5 1 1.5
0
0.5
1
Y/d
Per Realization
Re = 20K X/d = 4.5
−1.5 −1 −0.5 0 0.5 11.5
0
0.5
1Re = 50K X/d = 4.5
Y/d
Per Realization
−1.5 −1 −0.5 0 0.5 1 1.5
0
0.5
1
Y/d
Per Realization
Re = 20K X/d = 15
−1.5 −1 −0.5 0 0.5 11.5
0
0.5
1
Y/d
Per Realization
Re = 50K X/d = 15
Fig. 5 Local vortex population
density of positively and
negatively rotating vortices at
different streamwise locations.
Open circle: positively rotating
vortices, dot: negatively rotating
vortices
Exp Fluids (2011) 50:1473–1493 1479
123
investigated. To minimize scatter in the profiles, PpðnÞis
presented as the vortex population over a half jet width
(d/2 =1 cm) long area in the streamwise direction. The
plots presented in Fig. 5demonstrate that the distributions
of the positively and negatively rotating vortices are nearly
axially symmetric at all three downstream locations in both
two cases.
These profiles confirm the observations from the
instantaneous vortices fields above. At X/d=1 in the near-
field of the jet, the population density profiles of the
dominant direction vortices and counter-rotating vortices
both show clear peaks, at the locations on either side of the
center of the wake region, according to the streamwise
velocity mean profile in Fig. 3. The population of the
counter-rotating vortices almost disappears completely at
X/d=4.5, as the wake regions decay and are replaced by
mixing layer-like region population profiles. The popula-
tion profiles of the dominant vortices also broaden with
increasing downstream distance. These vortices are ini-
tially confined to the high shear, mixing layer regions with
few vortices existing in the free streams, but as the flow
develops toward channel flow at the farthest downstream
locations, the distribution of the vortices spreads through-
out the entire width of the reactor. At X/d=15, the pop-
ulation density profiles approach the vortex distribution of
a channel flow. One may notice the remarkable difference
between the results here and the small-scale vortex distri-
bution results in far-field of a axisymmetric free jet in
Agrawal and Prasad (2002b), which shows that the prob-
ability of finding a vortex is uniform up to the edge of the
jet and a substantial number of both clockwise and counter-
clockwise rotating eddies exist on both sides of the jet axis.
In the far-field of the jet, the two walls on the spanwise
direction produce the new source of the turbulence, which
reaching into the center of the jet as the flow develops. This
causes the small-scale vortices distributed as in a channel
flow, instead of a free jet.
Also notice that the highest vortex population actually
occurs in a very small region of the wall boundary layers,
which is not a surprise because of the higher velocity
gradient in the wall region. As mentioned earlier, the
population here only includes the vortices whose core
(swirling strength cluster) diameters are larger than 13.8g
in the low Re case and 33.9gin the high Re case.
By adding the number of positively rotating vortices
together between the center of the center stream Y/d =0
and the center of the left side stream Y/d =-1 in this
1-cm-long region, the streamwise population changing
trend of detectable positively rotating vortex can be more
easily observed, as shown in Fig. 6. The number of the
vortices in the same size region per velocity realization at
different downstream locations initially increased and then
decreased. This is reasonable because right after the flow
enters the test section, the high velocity gradients produce
many vortices. At the same time, the decay and dissipation
of these vortices also occurs. As the flow progresses further
downstream, the potential cores disappear, and fewer vor-
tices are produced. The vortex dissipation process contin-
ues, though, and later in the far-field of the jet it eventually
overwhelms the jet vortex production. Also notice the
population is much higher in the high Reynolds number
case than in the low Reynolds case, specially in the far-
field of the jet, where the flow approaches a fully devel-
oped channel flow and the boundary layers on the walls
play a bigger role in the vortex production than the shear
layer caused by the jet. Figure 6also shows that the
development toward channel flow is more rapid for the low
Re case, which of course makes sense, since the entrance
length in a channel or pipe increases with increasing
Reynolds number (Sadri and Floryan 2002).
4.3 Size and strength of vortex cores
In addition to the vortex population analysis presented
earlier, other properties of the identified vortices can be
obtained from the experimental data. The results of the size
of vortex core are presented in a similar manner to the
vortex populations. Figure 7shows the mean vortex core
size profiles across the channel, and Fig. 8gives stream-
wise profiles. The values of mean vortex core size appear
to be rather small compared to the jet width. Also the
smallest size of swirling strength cluster we can identify is
d
2
/400 for both investigated Reynolds numbers.
0 5 10 15 20 25 30
6
7
8
9
10
11
12
13
X/d
Per Realization
Re = 20K
Re = 50K
Fig. 6 Mean populations of positively rotating vortex in the half jet
width (1 cm) streamwise long region between the centers of the
center of center stream (Y/d=0) and the center of left side stream
(Y/d=-1). Open circle:Re =50K, filled square:Re =20K
1480 Exp Fluids (2011) 50:1473–1493
123
The cross-stream profiles in Fig. 7show that on both
sides of the jet where the dominant direction vortex pop-
ulation is highest, the average vortex size is also the largest,
especially at X/d=1. Although the population density of
the vortices in the wall boundary layers is highest, the size
of the vortices in the boundary layer is small. Figure 8
shows that the average vortex core size between the center
of the center stream and the left stream monotonically
decreases with increasing downstream distance. This might
initially seem contrary to what was reported in Feng et al.
(2007b). In the Re = 50K case, the size of the coherent
structures was observed linearly growing almost from
X/d=1 through at least X/d=15, which would be
expected for jet flow. Glancing back at the instantaneous
vortex field in Fig. 4, the areas of the identified swirling
strength clusters in the far-field are indeed smaller than in
the near-field. In the near-field of the jet, the current vortex
areas are about two orders of magnitude smaller than in the
previous report; in the far-field of the jet, they are almost
four order of magnitude smaller than in the previous report.
Again, this can be explained based on the difference
between the two measurement techniques. In Fig. 6, from
X/d=1toX/d=7.5, the populations of vortex cores
increase by at least 40%, while the mean sizes only
decrease by 15%. Therefore, the overall swirling area in the
jet grows by approximately 20%. However, as the jet
progresses downstream, there are more small structures
with increasing downstream distance, as previously men-
tioned. The increase in the number of small structures
brings the mean size values down. When the flow becomes
fully developed channel flow, the mean structure size will
remain constant beyond this point. This explains why the
vortex core size becomes smaller with increasing down-
stream distance and eventually stabilizes as the flow fully
−1.5 −1 −0.5 0 0.5 1 1.5
0
0.005
0.01
0.015
Y/d
Size/ d2
Re = 20K X/d =1
−1.5 −1 −0.5 0 0.5 1 1.5
0
0.005
0.01
0.015
Y/d
Size/ d2
Re = 50K X/d =1
−1.5 −1 −0.5 0 0.5 1 1.5
0
0.005
0.01
0.015
Y/d
Size/ d2
Re = 20K X/d = 7.5
−1.5 −1 −0.5 0 0.5 1 1.5
0
0.005
0.01
0.015
Y/d
Size/ d2
Re = 50K X/d = 7.5
−1.5 −1 −0.5 0 0.5 1 1.5
0
0.005
0.01
0.015
Y/d
Size/ d2
Re = 20K X/d = 15
−1.5 −1 −0.5 0 0.5 1 1.5
0
0.005
0.01
0.015
Y/d
Size/d2
Re = 50K X/d = 15
Fig. 7 Profiles of the size of
vortex cores. Open circle:
positively rotating vortices, dot:
negatively rotating vortices
0 5 10 15 20 25 30
0.0085
0.009
0.0095
0.01
0.0105
0.011
0.0115
0.012
X/d
Size/ d2
Re = 20K
Re = 50K
Fig. 8 Mean vortex size in the half jet width (1 cm) streamwise long
region between the centers of the center of center stream (Y/d=0)
and the center of left side stream (Y/d=-1). Open circle:
Re =50K, filled square:Re =20K
Exp Fluids (2011) 50:1473–1493 1481
123
develops. This is to be expected as the confined jet flow
transitions from a jet-like flow to a channel-like flow,
which would have the vortex size independent of down-
stream location.
The strength of the vortex cores also can be determined.
Note that inside the swirling strength clusters, the strength
value is not evenly distributed. The highest swirling
strength value of the vortex core is used to characterize the
vortex, since it gives the maximum rotation speed of the
given vortex, according to the swirling strength definition.
Figure 9shows cross-stream profiles of the average peak
value of the vortex core and Fig. 10 gives streamwise
profiles. All these profiles were normalized using DU=d.In
Fig. 9, it can be observed that the cross-stream profiles of
the maximum strength of vortex core show very similar
characteristics as the plot of the size of vortex core. The
center of the shear region of the jet has the maximum
vortex population density, the largest vortex size, and the
strongest swirling motion. Figure 10 shows that as the flow
develops, the strength of the vortices decreases, then
eventually stabilizes, just as the vortex size. Comparing the
two Reynolds number cases, the values for the Re =20K
case are higher than the Re =50K case after normaliza-
tion, especially at X/d=1. However, the difference
between the two Reynolds numbers becomes smaller and
smaller as the flow progresses downstream.
The cross-stream and streamwise profiles of both vortex
size and vortex strength all indicate that the high Reynolds
number case develops more slowly than the low Reynolds
number case, just as the profiles of vortex population
density show.
−1.5 −1 −0.5 0 0.5 1 1.5
−10
−5
0
5
10
Y/d
Λcid /ΔU
Re = 20K X/d =1
−1.5 −1 −0.5 0 0.5 1 1.5
−5
0
5
Y/d
Λcid /ΔU
Re = 50K X/d =1
−1.5 −1 −0.5 0 0.5 1 1.5
−4
−2
0
2
4
Y/d
Λcid /ΔU
Re = 20K X/d = 7.5
−1.5 −1 −0.5 0 0.5 1 1.5
−4
−2
0
2
4
Y/d
Λcid /ΔU
Re = 50K X/d = 7.5
−1.5 −1 −0.5 0 0.5 1 1.5
−4
−2
0
2
4
Y/d
Λcid /ΔU
Re = 20K X/d = 15
−1.5 −1 −0.5 0 0.5 1 1.5
−4
−2
0
2
4
Y/d
Λcid /ΔU
Re = 50K X/d = 15
Fig. 9 Profiles of the peak
swirling strength value of vortex
cores. Open circle: positively
rotating vortices, dot: negatively
rotating vortices
0 5 10 15 20 25 30
2
2.5
3
3.5
4
4.5
5
5.5
6
X/d
Λcid /ΔU
Re = 20K
Re = 50K
Fig. 10 Mean the peak swirling strength value of the vortex cores in
a half jet width (1 cm) streamwise long region between the centers of
the center of center stream (Y/d=0) and the center of left side stream
(Y/d=-1). Open circle:Re =50K, filled square:Re =20K
1482 Exp Fluids (2011) 50:1473–1493
123
4.4 Deviation velocity of vortex cores
The instantaneous velocity field data can also lend insight
into the motion of the vortices. The velocity of vortex cores
can be mapped back to the velocity field once the vortex is
identified. By subtracting the mean velocity from the
velocity values of the vortex cores, what we define as
the deviation velocity of the vortex can be obtained, i.e., the
deviation of the instantaneous vortex convection velocity
from the local mean velocity. Figures 11 and 12 show
profiles of the average cross-stream (V) and streamwise
(U) deviation velocities cross the channel at three down-
stream locations, which all are normalized by DU.
Figure 11 shows that the mean deviation velocity Vis
zero at the location of the population density peak.
Considering the left side of the jet, on the right side of
population density peak, vortices tend to move toward the
center of the center stream, and on the left side, the
vortices tend to move toward the center of the left side
(or outer) stream, which corresponds to the spreading of
the shear layers at the center regions of the jet. Particu-
larly, at X/d=1, the deviation velocity Vis maximum at
the very edge of the potential core of the free streams.
Since there are virtually no vortices in the potential core,
the average deviation velocity there is zero. In addition,
the normalized values of vortex deviation velocity Vare
higher in the low Reynolds number case than in the high
Reynolds number case.
The mean deviation velocity Uis also zero at the
location of the population density peak as shown in Fig. 12.
At X/d=1, toward the jet center, the values are negative,
and toward the side streams, the values are positive, which
corresponds to one of the roles of vortices in the flow:
transferring low momentum fluid to the high-velocity
region and transferring high momentum fluid to the low
velocity region. However, at further downstream locations,
X/d=4.5 and X/d=15, the profiles of mean deviation
velocity Uof the counter-rotating vortices overlap with the
profiles of dominant rotating direction vortices. Once
again, glancing back at the instantaneous vortex field in the
far-field of the jet in Fig. 4, these small number of counter-
rotating vortices are blending with the dominant direction
vortices.
The probability density function of vortex deviation
velocity was also calculated at three downstream locations,
X/d=1, X/d=4.5 and X/d=15 for the Re =20K case,
and compared to the PDF of the velocity fluctuations of the
overall flow, which are shown in Figs. 13,14 and 15.At
each downstream location, three observation points were
used, one is at the location of the peaks of positively rotating
vortex population density (Y/d =-0.45 at X/d=1,
Y/d=-0.42 at X/d=4.5), one is on the left side and one is
−1.5 −1 −0.5 0 0.5 1 1.5
−0.1
0
0.1
Y/d
Vλ/ΔU
Re = 20K X/d =1
−1.5 −1 −0.5 0 0.5 11.5
−0.1
−0.05
0
0.05
0.1
Y/d
Vλ/ΔU
Re = 50K X/d =1
−1.5 −1 −0.5 0 0.5 1 1.5
−0.1
−0.05
0
0.05
0.1
Y/d
Vλ/ΔU
Re = 20K X/d = 7.5
−1.5 −1 −0.5 0 0.5 11.5
−0.1
−0.05
0
0.05
0.1
Y/d
Vλ/ΔU
Re = 50K X/d = 7.5
−1.5 −1 −0.5 0 0.5 1 1.5
−0.05
0
0.05
Y/d
Vλ/ΔU
Re = 20K X/d = 15
−1.5 −1 −0.5 0 0.5 11.5
−0.05
0
0.05
Y/d
Vλ/ΔU
Re = 50K X/d = 15
Fig. 11 Vortex deviation
velocity in the cross-stream
direction. Open circle:
positively rotating vortices, dot:
negatively rotating vortices
Exp Fluids (2011) 50:1473–1493 1483
123
on the right side where the middle points of the population
density peak declining slopes are. In each case, all the
distributions of the deviation velocity appear narrower than
the velocity fluctuation distribution. One reason to explain
this is the swirling strength vortex identification method
reveals the core of the vortex. It is actually the part of the
−1.5 −1 −0.5 0 0.5 1 1.5
−0.2
−0.1
0
0.1
0.2
Y/d
Uλ/ΔU
Re = 20K X/d =1
−1.5 −1 −0.5 0 0.5 1 1.5
−0.1
−0.05
0
0.05
0.1
Y/d
Uλ/ΔU
Re = 50K X/d =1
−1.5 −1 −0.5 0 0.5 1 1.5
−0.1
0
0.1
Y/d
Uλ/ΔU
Re = 20K X/d = 7.5
−1.5 −1 −0.5 0 0.5 1 1.5
−0.1
0
0.1
Y/d
Uλ/ΔU
Re = 50K X/d = 7.5
−1.5 −1 −0.5 0 0.5 1 1.5
−0.1
−0.05
0
0.05
0.1
Y/d
Uλ/ΔU
Re = 20K X/d = 15
−1.5 −1 −0.5 0 0.5 1 1.5
−0.1
−0.05
0
0.05
0.1
Y/d
Uλ/ΔU
Re = 50K X/d = 15
Fig. 12 Vortex deviation
velocity in the streamwise
direction. Open circle:
positively rotating vortices, dot:
negatively rotating vortices
−0.5 0 0.5
0
0.02
0.04
0.06
0.08
Value
Distribution
V/ΔU, Y/d = −0.57
−0.5 0 0.5
0
0.02
0.04
0.06
0.08
Value
Distribution
V/ΔU, Y/d = −0.45
−0.5 00.5
0
0.02
0.04
0.06
0.08
Value
Distribution
V/ΔU, Y/d = −0.31
−0.5 0 0.5
0
0.02
0.04
0.06
0.08
Value
Distribution
U/ΔU, Y/d = −0.57
−0.5 0 0.5
0
0.02
0.04
0.06
0.08
Value
Distribution
U/ΔU, Y/d = −0.45
−0.5 00.5
0
0.02
0.04
0.06
0.08
Value
Distribution
U/ΔU, Y/d = −0.31
Fig. 13 Vortex deviation
velocity distribution, at X/
d=1, Re =20K. Times:
velocity fluctuation distribution,
open circle: vortex deviation
velocity distribution
1484 Exp Fluids (2011) 50:1473–1493
123
vortex close to the vortex edge that generates high velocity
fluctuations, where the swirling motion could be very weak.
Another possible reason is the vortex identification method
used here does have spatial limitations, and vortices that are
too small to be detected could play some role in the flow
unsteadiness. It is also possible that some of the fluctuations
−0.5 0 0.5
0
0.02
0.04
0.06
0.08
Value
Distribution
VΔU, Y/d = −0.72
−0.5 0 0.5
0
0.02
0.04
0.06
0.08
Value
Distribution
V/ΔU, Y/d = −0.42
−0.5 0 0.5
0
0.02
0.04
0.06
0.08
Value
Distribution
V/ΔU, Y/d = −0.16
−0.5 0 0.5
0
0.02
0.04
0.06
0.08
Value
Distribution
U/ΔU, Y/d = −0.72
−0.5 0 0.5
0
0.02
0.04
0.06
0.08
Value
Distribution
U/ΔU, Y/d = −0.42
−0.5 0 0.5
0
0.02
0.04
0.06
0.08
Value
Distribution
U/ΔU, Y/d = −0.16
Fig. 14 Vortex deviation
velocity distribution, at X/
d=4.5, Re =20K. Times:
velocity fluctuation distribution,
open circle: vortex deviation
velocity distribution
−0.5 0 0.5
0
0.02
0.04
0.06
0.08
Value
Distribution
V/ΔU, Y/d = −0.82
−0.5 0 0.5
0
0.02
0.04
0.06
0.08
Value
Distribution
V/ΔU, Y/d = −0.52
−0.5 0 0.5
0
0.02
0.04
0.06
0.08
Value
Distribution
V/ΔU, Y/d = −0.21
−0.5 0 0.5
0
0.02
0.04
0.06
0.08
Value
Distribution
V/ΔU, Y/d = −0.82
−0.5 0 0.5
0
0.02
0.04
0.06
0.08
Value
Distribution
V/ΔU, Y/d = −0.52
−0.5 0 0.5
0
0.02
0.04
0.06
0.08
Value
Distribution
V/ΔU, Y/d = −0.21
Fig. 15 Vortex deviation
velocity distribution, at X/
d= 15, Re =20K. Times:
velocity fluctuation distribution,
open circle: vortex deviation
velocity distribution
Exp Fluids (2011) 50:1473–1493 1485
123
are due to other unsteadiness or large-scale coherent
motions in the jet flow, such as stream-wise jet pulsing or
cross-stream flapping of the jet.
Similar to the mean vortex deviation velocity profiles, at
the location of the population density peak, the mean of
both the vortex deviation velocities are zero. However, the
mean of velocity fluctuations are also close to zero at this
location. Interestingly, on both sides of the peak location,
the distribution peaks of vortex deviation velocity appear
on the opposite side of zero as the distribution peaks of
velocity fluctuation. It makes sense that the velocity fluc-
tuation close to the jet center is negative, because low
momentum fluid is transferring into the jet center (where
the mean velocity is high); the velocity fluctuation close to
the outer stream center (where the mean velocity is low) is
positive, because high momentum fluid is transferring in.
Notice also that compared to the distributions at X/d=1,
as downstream distance increases, the vortex deviation
velocity distributions become closer to the distribution of
the velocity fluctuation (an exception to this is at X/d=4.5
and Y/d =-0.16, where the streamwise velocity fluctua-
tion distribution has a much higher peak, which is simply
because the location is very close to the jet center). As the
flow progresses further downstream, the vortex deviation
velocity should become more similar to the distribution
shape of the velocity fluctuations, as the mean vortex
deviation velocity profiles indicate.
5 Vortex tracking analysis
From the high speed PIV data, the trajectories of individual
vortices can also be determined. After identifying a par-
ticular vortex in a pair of velocity realizations, the velocity
of the swirling strength cluster can be found from the
change in the vortex position between the two frames.
Repeating this frame by frame over the high speed PIV
data set, the location of the vortex in each frame can be
estimated. The vortex on the next frame which has the
maximum overlap with the estimated location is considered
the same vortex as the selected vortex from the previous
frame. Figure 16 shows two trajectories of a positively
rotating vortex on the left side of the jet and a negatively
rotating vortex on the right side of the jet to demonstrate
the effectiveness of this tracking method. Only vortices that
newly appear at the bottom of the velocity fields were
traced to maximize tracking time because the length of
observation window is limited. Once a vortex has been
identified in a series of velocity realizations at different
time steps, how the size and strength of the vortex changes
with time can be determined. Because the vortex property
development of one given vortex could be rather noisy, to
obtain the main characteristics of vortex development,
averaging was performed for certain vortices which trav-
eled through the current watching window, t0:2 s for
Re =20K and t0:08 s for Re =50K. Those vortices
which disappeared during this time frame are not included.
The mean vortex size and strength are shown in Figs. 17
and 18, using observation windows beginning at X/d=0,
X/d=3.5 and X/d=7.5. Since the experimental data at
these three locations were collected at different times, the
vortices that were tracked in one window definitely could
not appear in the other windows. The mean tracked vortex
maximum strength decreases with increasing downstream
distance after the flow enters the channel, the same
development as we observed in the previous section.
However, the mean tracked vortex size increases slightly
after the vortices enter the reactor, and then decreases. The
vortices that were tracked close to the jet outlet must have
already existed when the flow entered the test section. They
are likely those well organized vortices reported in wall
turbulence (Adrian et al. 2000b), and continue to develop
for a while before the changing of the flow field affects
them. When these vortices meet the counter-rotating vor-
tices on the other side of the splitter plates, and when
vortices begin to be produced by the mean shear in the jet,
the properties of the vortex core will change. They can be
weakened or pushed aside by the counter-rotating vortices.
Also, the swirling strength values in the swirling strength
clusters could be redistributed, as some parts of the rotating
fluid particles can spin off from the cluster, a phenomenon
observed in some instantaneous vortices fields from the
high speed PIV data.
Also compared to the mean vortex size and strength
profiles, at X/d=1 and X/d=4.5, presented in Figs. 7and
Y/d
X/d
Re = 20K
−1.5 −1 −0.5 0 0.5 1 1.5
0
0.5
1
1.5
2
2.5
−60
−40
−20
0
20
40
60
Fig. 16 The trajectories of a positively rotating vortex and a
negatively rotating vortex, at the jet outlet, Re =20K. The
background contours are the plots of the swirling strength clusters
at several different time steps
1486 Exp Fluids (2011) 50:1473–1493
123
9, all the values calculated by tracing individual vortices
using high speed PIV data are higher. This is because the
averaging here is actually conditional averaging as
mentioned earlier. Those vortices chosen here have rela-
tively long life spans, a probable consequence of their
being larger and stronger than the ‘‘average’’ vortex.
0.05 0.1 0.15
2
3
4
5
6
7
Λcid /ΔU
Starting at X/d = 0
0.05 0.1 0.15
2
3
4
5
6
7
X/d = 3.5
0.05 0.1 0.15
2
3
4
5
6
7
X/d = 15
0.04 0.06
2
3
4
5
6
7
time (sec)
Λcid /ΔU
Starting at X/d = 0
0.04 0.06
2
3
4
5
6
7
time (sec)
X/d = 3.5
0.04 0.06
2
3
4
5
6
7
time (sec)
X/d = 15
Re = 50K
Re = 20K
Fig. 17 The mean tracked
vortex maximum strength
starting at three different
downstream locations
0.05 0.1 0.15
0.015
0.02
0.025
0.03
Size / d2
Starting at X/d = 0
0.05 0.1 0.15
0.015
0.02
0.025
0.03 X/d = 3.5
0.05 0.1 0.15
0.015
0.02
0.025
0.03 X/d = 15
0.04 0.06
0.02
0.025
0.03
time (sec)
Size / d2
Starting at X/d = 0
0.04 0.06
0.015
0.02
0.025
0.03
time (sec)
X/d = 3.5
0.04 0.06
0.015
0.02
0.025
0.03
time (sec)
X/d = 15
Re = 50K
Re = 20K
Fig. 18 The mean tracked
vortex size starting at three
different downstream locations
Exp Fluids (2011) 50:1473–1493 1487
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6 Spatial correlation functions and linear stochastic
estimation
6.1 Two-point spatial correlations
Normalized by the rms of swirling strength and some flow
property, r, the two point spatial cross correlation of
swirling strength and the fluctuation of rcan be defined as
Rkr0ðX;Y;x;yÞ¼ hkciðX;YÞr0ðx;yÞi
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
hk2
ciðX;YÞihr02ðx;yÞi
qð2Þ
where (X,Y) and (x,y) are the coordinates of the basis point
(i.e. the point about which the correlation is calculated) and
an arbitrary point in the flow field, respectively. In the
present study, locations along the left peak of turbulent
kinetic energy at different downstream locations were
chosen as basis points. This location also corresponds to
the peak of the population density of positively rotating
vortices. The spatial correlations of Rku0and Rkv0for basis
points located at four downstream locations for both Re
cases are presented in Figs. 19 and 20. The streamwise
flow direction is from the bottom to the top of the figures.
The cross correlation fields of swirling strength and
fluctuations of the two velocity components exhibit ‘‘but-
terfly’’ like shapes. The right wing of Rku0contains positive
values and the left wing contains negative values. The
symmetry line of this ‘‘butterfly’’ shape in Rku0, called ‘‘the
axis’’ of these correlation fields hereafter, is oriented ver-
tically in the streamwise direction, which the contour line
of Rku0=0 overlaps in the region of the basis point. The
symmetry line of Rkv0is oriented horizontally with negative
values downstream of the basis point and positive values
upstream of the basis point. Notice at X/d=1, the axis of
Rku0is not oriented vertically as it is at the other down-
stream locations. Instead, the axis tilts about 30°away from
the jet center, which is caused by high velocity gradients in
this narrow wake region. Once the wake region disappears,
the tilting of the axis becomes less significant due to
smaller and smaller velocity gradients. There is a second
Y/d
X/d
-1 -0.8 -0.6 -0.4 -0.2 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Y/d
X/d
-1 -0.8 -0.6 -0.4 -0.2 0
3.8
4
4.2
4.4
4.6
4.8
5
5.2
Y/d
X/d
-1 -0.8 -0.6 -0.4 -0.2 0
6.8
7
7.2
7.4
7.6
7.8
8
8.2
Y/d
X/d
-1 -0.8 -0.6 -0.4 -0.2 0
14.2
14.4
14.6
14.8
15
15.2
15.4
15.6
15.8
Y/d
X/d
-1 -0.8 -0.6 -0.4 -0.2 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Y/d
X/d
-1 -0.8 -0.6 -0.4 -0.2 0
3.8
4
4.2
4.4
4.6
4.8
5
5.2
Y/d
X/d
-1 -0.8 -0.6 -0.4 -0.2 0
6.8
7
7.2
7.4
7.6
7.8
8
8.2
Y/d
X/d
-1 -0.8 -0.6 -0.4 -0.2 0
14.2
14.4
14.6
14.8
15
15.2
15.4
15.6
15.8
Fig. 19 Two-point cross correlation Rkciu0(first row), Rkci v0(second row). At X/d=1, X/d=4.5, X/d=7.5 and X/d=15, Re =50K. The black
dot indicates the location of swirling strength basis point
1488 Exp Fluids (2011) 50:1473–1493
123
positively correlated region on the left side of the negative
peaks, and a second negatively correlated region on the
right side of the positive peaks in the Rku0contours, espe-
cially in the low Re case, at X/d=1. Also, there is a
weakly correlated negative region upstream of the strong
positive region close to the basis point and a weak positive
region downstream of the strong negative region in Rkv0.
Assuming the vortex at the basis point is accompanied by
vortices upstream and downstream of the basis point
location, these two correlation fields indicate locations of
accompanying vortices. Also the correlation in the region
far from the basis point is actually very weak (that is close
to zero), which makes the location of the Rku0=0 contour
somewhat random. This agrees with the observation from
the instantaneous vortex fields, in Fig. 4. Also, as the flow
progresses downstream, the correlation areas of both
Rku0=0 and Rkv0=0 grow larger. Notice the peak values
of these correlations in Fig. 22. The peak values of Rku0are
approximately 0.2 for both cases close to the jet inlet. The
peak values of Rkv0begin at approximately 0.4 near jet inlet
in both cases and decrease with downstream distance,
approaching a value of 0.25 for both Reynolds number
cases after the flow passes beyond X/d=12.
Figure 21 shows Rkci/0, the cross-corrlation of swirling
strength with the concentration fluctuation, for Re =50K
and Re =20K, respectively. Note that simultaneous PIV/
PLIF data were only collected for two downstream loca-
tions for Re =20K, compared to six locations for
Re =50K. There are two correlated areas in the contours;
one positively correlated region downstream of the basis
point and one negatively correlated region upstream. This
shape suggests that the positively rotating (counterclock-
wise) vortices bring high concentration fluid (i.e., positive
concentration fluctuations) from the center stream to the
side stream downstream of the vortex core and bring the
low concentration fluid from the side stream to the center
stream upstream of the core. These two correlation areas
are not only comparable in size but also in absolute peak
values. As mentioned before, the locations chosen here also
are the locations of the peak of positive vortex population
density, which means there are predominately positively
rotating vortices centered at this location. The contour lines
Y/d
X/d
-1 -0.8 -0.6 -0.4 -0.2 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Y/d
X/d
-1 -0.8 -0.6 -0.4 -0.2 0
3.8
4
4.2
4.4
4.6
4.8
5
5.2
Y/d
X/d
-1 -0.8 -0.6 -0.4 -0.2 0
6.8
7
7.2
7.4
7.6
7.8
8
8.2
Y/d
X/d
-1 -0.8 -0.6 -0.4 -0.2 0
14.2
14.4
14.6
14.8
15
15.2
15.4
15.6
15.8
Y/d
X/d
-1 -0.8 -0.6 -0.4 -0.2 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Y/d
X/d
-1 -0.8 -0.6 -0.4 -0.2 0
3.8
4
4.2
4.4
4.6
4.8
5
5.2
Y/d
X/d
-1 -0.8 -0.6 -0.4 -0.2 0
6.8
7
7.2
7.4
7.6
7.8
8
8.2
Y/d
X/d
-1 -0.8 -0.6 -0.4 -0.2 0
14.2
14.4
14.6
14.8
15
15.2
15.4
15.6
15.8
Fig. 20 As Fig. 19 but at Re =20K
Exp Fluids (2011) 50:1473–1493 1489
123
of Rkci/0¼0:1 and Rkci /0¼0:1 are nearly symmetrical
about the basis point, except at the location X/d=1. As
previously stated, because the axis of positively rotating
vortices close to the jet outlet is obliquely oriented, positive
Rkci/0is located in the second quadrant instead of the first
quadrant. Considering these two regions together provides
an explanation for the elliptic shape of the R/0/0with the
major axis inclined at 45°, reported in a previous paper
(Feng et al. 2007a). Also, considering the location of the
correlation peaks with respect to the basis point, the results
are consistent with the location of the vortex core in linear
stochastic estimates of the velocity field given the event of
/0(x
o
)=?2/0
rms
(x
o
) (Feng et al. 2007a). Notice also that
with increasing downstream distance, the size of the cor-
relation area becomes steadily larger. However, the peak
values of Rkci/0, shown in Fig. 22, also first increase, then
decrease in the high Re case. Compared to the high Re
case, the correlation area in the low Re case is considerably
larger and the peak values are lower. Also, unlike the high
Re case, the negative region not only occupies most of the
third quadrant but also the fourth quadrant, with the
contour line of Rkci/0¼0 inclined at approximately 30°
with respect to the xdirection.
6.2 Linear stochastic estimation
One useful tool to interpret spatial correlation data is
linear stochastic estimation, in which conditional aver-
ages are calculated from measured correlation fields
(Adrian 1994; Olsen and Dutton 2002,2003) Although
conditional averages can be calculated directly from an
experiment dataset, this requires either a very large
ensemble size or averaging over a large flow region
instead of a particular location in the flow, due to the
low population density of the vortices. Derived directly
from the two point spatial correlation, LSE can give the
typical underlying flow structures more precisely with a
smaller ensemble size.
Letting k
ci
(X,Y) be the swirling strength value at loca-
tion (X, Y), the linear stochastic estimate of the velocity
fluctuation u
i
0(x,y) over the entire flied given the condition
k
ci
(X,Y) is,
Y/d
X/d
-1 -0.8 -0.6 -0.4 -0.2 0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
Y/d
X/d
-1 -0.8 -0.6 -0.4 -0.2 0
3.8
4
4.2
4.4
4.6
4.8
5
5.2
Y/d
X/d
-1 -0.8 -0.6 -0.4 -0.2 0
6.8
7
7.2
7.4
7.6
7.8
8
8.2
Y/d
X/d
-1 -0.8 -0.6 -0.4 -0.2 0
14.2
14.4
14.6
14.8
15
15.2
15.4
15.6
15.8
Y/d
X/d
-1 -0.8 -0.6 -0.4 -0.2 0
3.8
4
4.2
4.4
4.6
4.8
5
5.2
Y/d
X/d
-1 -0.8 -0.6 -0.4 -0.2 0
6.8
7
7.2
7.4
7.6
7.8
8
8.2
Fig. 21 Two-point cross-correlation Rkci/0.AtX/d=1, X/d=4.5, X/d=7.5 and X/d=15, for Re =50K (first row) and at X/d=4.5, X/
d=7.5, for Re =20K (second row)
1490 Exp Fluids (2011) 50:1473–1493
123
hu0
iðx;yÞjkciðX;YÞi Likci ðX;YÞð3Þ
The linear coefficient, L
i
, can be determined in the
following way,
Li¼hu0
iðx;yÞkciðX;YÞi
hkciðX;YÞkci ðX;YÞi ð4Þ
where hu0
iðx;yÞkciðX;YÞi is the (unnormalized) spatial
correlation of u
i
0and k
ci
.
As Christensen and Adrian (2001) argued, since the
event of the conditional average is a single scalar value, the
character of the conditionally averaged results remains the
same for all values, just the magnitudes are simply
amplified and attenuated. Thus, the thresholding of the k
ci
is not necessary for just examining the fluctuation field. But
to see the real typical structure in the flow field, the esti-
mated fluctuation fields need to be added with the mean
field. Thus, the thresholding of the event will only change
the size of the structure. Here we chose the event of kci ¼
2krms
ci in order to get the features of typical structure in the
flow.
Adding the mean field and subtracting U(x
o
) from each
vector results in Fig. 23, which shows the typical velocity
field of one vortex at different downstream locations. All
the roller structures revealed by the linear scholastic esti-
mation are spindle-shaped with a stream-wise major axis.
The tilting of the major axis becomes much more visible
here than in the spatial correlation fields. The high
momentum of the jet pushes the upper part of the roller
structure outwards, and the low momentum fluid was
entrained inwards to the center of the jet by the lower part
of the structure. In the near-field of the jet, there are two
vortices seen adjacent to the vortex at the base point in both
upstream and downstream direction. As the flow progresses
downstream, the roller structures grow larger. Although the
sizes of the vortex cores decrease, the large-scale coherent
motions of the flow, which the swirling motions at the base
point are part of, are indeed growing. Comparing the two
Reynolds number case, the roller structures for Re =20K
are larger than those in the Re =50K case.
7 Conclusions
A vortex identification method based on swirling strength
was employed to analyze the properties of vortices in a
confined rectangular jet. Experimental data from simulta-
neous PIV/PLIF experiments and high speed PIV experi-
ments were used in this analysis. Swirling strength fields
were computed from velocity fields, and then filtered with
a universal threshold of jKcij1:5Krms
ci . By identifying
clusters of filtered Kci, vortex structures were identified.
Instantaneous swirling strength field data indicate that
positively (counterclockwise) rotating vortices are domi-
nant on the left side of the jet and negatively (clockwise)
rotating vortices are dominant on the right side. The pop-
ulation density, average size and strength, deviation
velocity of vortices were calculated and analyzed, in both
the cross-stream direction and the streamwise direction. In
the region close to the channel inlet, the population density,
average size and strength all show high values on both
sides of the center stream. There are some counter-rotating
vortices next to the dominant direction vortices that are
indicative of a wake region formed downstream of the
splitter plate tips by the boundary layers that form on both
sides of the splitter plates. At the further downstream
location, X/d=3.5, the wakes disappear, as do most of the
counter-rotating vortices. As the flow develops toward
channel flow at the farthest downstream locations, the
distribution of the vortices spreads throughout the entire
reactor. The mean size and strength of the vortices decrease
continuously downstream from the channel inlet. The mean
vortex deviation velocity in both the Xand Ydirections are
zero at the location of the population density peak. The
signs of vortex deviation velocity Vindicate the vortices
move from the high vortex population region to the low
vortex population region. The signs of mean deviation
velocity Uare negative on the side near the jet center and
positive on the side of near center of the outer stream,
which indicate vortices transfer low momentum fluid to
high-velocity region and transfer high momentum fluid to
the low velocity region.
The development trends of vortex size and strength were
also identified by tracking vortices using high speed PIV
experimental data. Both the average tracked vortex
strength and size decrease with increasing downstream
Fig. 22 Average of positive and negative peak values of cross
correlations. (open square:Rku0;open triangle:Rkv0;times:Rk/0;solid
line:Re ¼20K;dotted line:Re ¼50K)
Exp Fluids (2011) 50:1473–1493 1491
123
distance overall. However the average tracked vortex size
increases before it starts to decrease in the area close to the
jet inlet.
Two point spatial cross-correlations of swirling strength
with velocity fluctuations and concentration fluctuations
were calculated at the location of the left peak of turbulent
kinetic energy. The cross-correlation fields of swirling
strength and fluctuations of the two velocity components
exhibit a ‘‘butterfly’’ like shape. The right wing of Rku0
contains positive values and the left wing contains negative
values. The axis of Rku0, the contour line of Rku0=0, is
oriented vertically in the streamwise direction, slightly
tilted toward the outer stream. With increasing downstream
distance, the angle of orientation of the Rku0axis becomes
smaller. Also at the X/d=1.0 downstream locations for
both Reynolds number cases, there are a weak negatively
correlated regions upstream of the strong positively cor-
related region close to the basis point and a weak posi-
tively correlated region downstream of the strong
negatively correlated region in Rkv0. This indicates that a
vortex at the basis point is usually accompanied by at least
one counter-rotating vortex. The axis of Rkv0is aligned
with the cross-stream direction with negative values
downstream of the basis point and positive values
upstream of the basis point. There are also two correlated
areas in the Rk/0correlation field; one positively correlated
region downstream of the basis point and one negative
upstream of the basis point, indicating that the positively
rotating vortices bring high concentration field from the
center stream to the side stream downstream of the vortex
core and bring the low concentration field from the side
stream to the center stream upstream of the core. Finally,
linear stochastic estimation was used to calculate condi-
tional structures. The estimation was based on the swirling
strength values at chosen locations in the flow. The large-
scale structures in the velocity field revealed by linear
stochastic estimation are spindle-shaped with a titling
stream-wise major axis.
Y/d
X/d
-1 -0.8 -0.6 -0.4 -0.2 0
0.5
1
1.5
0.5 m/s
Y/d
X/d
-1 -0.8 -0.6 -0.4 -0.2 0
3.8
4
4.2
4.4
4.6
4.8
5
5.2
0.5 m/s
Y/d
X/d
-1 -0.8 -0.6 -0.4 -0.2 0
6.6
6.8
7
7.2
7.4
7.6
7.8
8
8.2
0.5 m/s
Y/d
X/d
-1 -0.8 -0.6 -0.4 -0.2 0
14.2
14.4
14.6
14.8
15
15.2
15.4
15.6
15.8
0.5 m/s
Y/d
X/d
-1 -0.8 -0.6 -0.4 -0.2 0
0.5
1
1.5
0.5 m/s
Y/d
X/d
-1 -0.8 -0.6 -0.4 -0.2 0
3.8
4
4.2
4.4
4.6
4.8
5
5.2
0.5 m/s
Y/d
X/d
-1 -0.8 -0.6 -0.4 -0.2
6.8
7
7.2
7.4
7.6
7.8
8
8.2
0.5 m/s
Y/d
X/d
-1 -0.8 -0.6 -0.4 -0.2
14.2
14.4
14.6
14.8
15
15.2
15.4
15.6
0.5 m/s
Fig. 23 Linear stochastic estimation of velocity field with U(x
o
) subtracted from each vector. At X/d=1, X/d=4.5, X/d=7.5 and X/d=15,
Re =50K (first row), Re =20K (second row)
1492 Exp Fluids (2011) 50:1473–1493
123
Acknowledgments This work was supported by the National Sci-
ence Foundation through Grants CTS-9985678 and CTS-0336435.
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