Fifth Australian Conference on Laser Diagnostics in Fluid Mechanics and Combustion
The University of Western Australia
3-4 December 2008
Interaction Between a Vortex Ring and a Uniform Cross-Flow
E. R. Hassan, R. M. Kelso and P. V Lanspeary
School of Mechanical Engineering, The University of Adelaide, S.A. 5005, AUSTRALIA
[email protected]
ABSTRACT
This paper describes features in the structure of a laminar vortex ring ejected vertically from an elevated pipe into a uniform
horizontal cross flow. Measurements are obtained by particleimage velocimetry (PIV) and flow-visualisation images are obtained by planar-laser-induced fluorescence (PLIF). Cross flow
produces a shear layer over the open end of the elevated pipe.
As a vortex ring is ejected, it lifts the cross-flow shear layer
away from the elevated pipe. As the sides of the shear layer
stretch and convect in the direction of the cross flow, they form
a pair of counter-rotating “columnar” vortices. The columnar
vortices link the rear of the pipe to the middle of the vortex
ring.
Optics
Cross−Flow
y Laser −Light
Sheet
Elevated
Pipe
z
x
CCD Camera
1. INTRODUCTION
This is an experimental study of a laminar vortex ring which
is ejected from vertical elevated pipe into a horizontal uniform
flow. Hassan et al. [3] have observed the growth and decay
of the vortex-ring circulation and its interaction with the shear
layer which is produced by the cross flow over the elevated pipe.
In the central x-y plane (see figure 1) there are two principal
cross sections of the vortex ring. The effect of the shear layer is
to increase the circulation of one vortex-ring cross section and
to decrease the other.
In this paper, we learn more about the interaction between a
vortex ring and a cross flow by measuring and inspecting the
evolution of the vorticity field in several planes perpendicular
to the cross-flow velocity vector.
2. APPARATUS
Experiments are conducted in a closed-return water channel
with a working section of 500 mm × 500 mm × 2000 mm.
A vortex ring is generated by impulsively translating a piston within a cylinder for a short period. In this context “impulsively translating” means that the ideal piston velocity is a
square-wave function of time. A stepper motor driving a lead
screw advances the piston in a controlled and repeatable manner. Fluid from the cylinder flows through a hose into an elevated pipe. The elevated pipe has a diameter of D=25 mm and
an exit plane which is 2.5D above the false floor of the water
channel. The axis of the elevated pipe is perpendicular to direction of the cross-flow, and the outside of the pipe is tapered at
an angle of 7o to a sharp edge at the tip.
3. PIV METHOD
Velocity components are measured by two-component Particle
Image Velocimetry (PIV). The flow is seeded with 8 µ m meandiameter hollow glass spheres with a specific gravity of 1.1.
Illumination is provided by a dual-cavity Quantel Brilliant B
Nd:YAG laser. The wavelength of the laser light is 532 nm and
TM
Quantel Brilliant−B
Nd:YAG Laser
False Floor
Mirror
Figure 1: Optical and geometric arrangement of the experi-
ment.
Location of
laser sheet
x-y plane, z/D = 0
z-y plane, x/D = 0
z-y plane, x/D = 0.5
z-y plane, x/D = 1.0
Number of
Realisations
58
9
5
5
Table 1: Number of realisations for each laser-sheet position.
the duration of each laser pulse is approximately 5 ns. An optics train reshapes the laser beam into a light sheet of thickness
1.5 mm. The PIV images are recorded by a Kodak Megaplus
ES 1.0 CCD camera with an array of 1008 pixels × 1018 pixels
and a Nikon 70-300 mm zoom lens. The field of view is approximately 90 mm × 90 mm giving a spatial resolution of 11 pixels/mm. PIV image pairs are collected at a rate of 10 Hz. A
Stanford DG535 pulse-delay generator provides trigger pulses
to the Q-switch in each laser cavity. XCAP-Plus software collects the images and stores them on computer hard disk as uncompressed 16-bit TIFF files. PIV velocity vectors are obtained
by cross correlation of the image pairs. PIV interrogation windows have a size of 32 pixels × 32 pixels and an overlap of
50%. Correlation peaks are detected using a multi-pass double
correlation, least squares Gaussian fit.
Anomalous velocity vectors (outliers) are detected by the universal median technique of Westerweel and Scarano [5]. This
algorithm is extended so that it also replaces outliers with a
weighted average of valid surrounding vectors. For all results
presented here, the number of outliers in any one velocityvector field is less than 3% of the total number of vectors. An
adaptive Gaussian window interpolator filters noise from the
raw vector field. The estimated maximum error in the velocity
measurements is 8%. This error is evaluated for the worst case
conditions where the velocity gradient is largest. Elsewhere the
error is significantly smaller.
Figure 1 is a schematic diagram showing the optical and geo-
To ensure statistical reliability, data collection is repeated a
number of times for each laser-sheet position. The number
of realisations is given in Table 1. A custom-built electronic
controller synchronises the start of vortex generation with the
next available laser pulse and enables the camera trigger. This
ensures that the n-th PIV image pair in any one realisation
is synchronised with the n-th image pair in all other realisations. Therefore, data collected from different realisations can
be compared and ensemble averaged.
In addition to the PIV, we use planar laser-induced fluorescence
(PLIF) in the x-y plane at z/D = 0 as a method of flow visualisation. Fluid in the pipe is marked with Rhodamine dye. Although
the flow is illuminated only in the x-y plane at z/D = 0, there is
sufficient randomly scattered light to observe features adjacent
to but not actually in the laser sheet.
4. FLOW CONDITIONS
A vortex ring is generated by impulsively translating a piston
within a cylinder. The duration of piston motion is τ = 1second.
The instantaneous spatially-averaged pipe exit velocity, U(t),
can be integrated over time
R to obtain the total length of the displaced fluid slug, L0 = 0∞ U(t)dt ≈ 40 mm. The ratio of the
total displaced slug length to the duration of piston motion,
U0 = L0 /τ ≈ 40 mm/s, is a vortex-ring velocity scale. The duration of piston motion corresponds to a nondimensional time
of e
t =tU0 /D = 1.6. The Reynolds number of the vortex ring is
most appropriately defined in terms of the circulation of the vortex ring, ReΓ =Γ/ν. The vortex slug model described in Shariff
and Leonard [4] provides a convenient estimate of the circulation, Γslug so that
1 2
U τ
Γslug
≈ 2 0 .
ν
ν
3.5
−4
−2
0
2
3.5
(a)
3
2.5
2.5
2
2
y/D
3
1.5
1.5
1
1
0.5
0.5
−1
0
x/D
1
4
6
(b)
−1
0
1
2
x/D
e , of the vortex ring,
Figure 2: Nondimensional vorticity field, ω
at e
t =5.6, (a) in the absence of a cross-flow and (b) in a R≈3.6
left-to-right cross flow.
5. DATA PROCESSING
The out-of-plane vorticity field, ω, at each time step is estimated
from the in-plane velocity components. The hybrid compactRichardson extrapolation (CR4) vorticity-estimation scheme of
Etebari & Vlachos [1] is a noise-optimised, fourth-order scheme
which simultaneously reduces both the random error and the
bias error. This vorticity estimation scheme is applied to all the
data. Applying the error propagation analysis of Fouras & Soria
[2] gives maximum random and bias errors which are 7.3% and
5.9% respectively of the peak vorticity value. The maximum total error in vorticity is therefore 9.4% of the peak vorticity value.
e =ωD/U0 .
Vorticity is presented in the nondimensional form, ω
6. PRIMARY STRUCTURE OF VORTEX RING
In the absence of a cross flow, an axisymmetric vortex ring
forms as the piston pushes fluid from the elevated pipe, and the
vortex ring moves away along the axis of symmetry of the pipe.
Figure 2 shows the nondimensional vorticity field of the vortex ring at nondimensional time e
t =5.6 both in the absence of
a cross-flow and in a R≈3.6 cross-flow. Two cross-sections of
the vortex ring are visible in figure 2(a). The cross-section on
the left has positive vorticity and, for convenience, is referred
to as the positive core. The cross-section on the right has negative vorticity and is referred to as the negative core. The cores
initially grow in time but remain equal in size; they remain sym-
(1)
For current data ReΓ ≈ 800. The equivalent piston stroke length
is L ≈1.6D. The cross-flow velocity, U∞ =11 mm/s corresponds
to a cross-flow Reynolds number of Re∞ =U∞ D/ν ≈ 270. The
velocity ratio, R =U0 /U∞ ≈ 3.6, the nondimensional slug length
L/D and Reynolds number ReΓ determine the characteristics of
the flow.
Measurements made with a cross-flow only (in the absence of
a vortex ring) at Re∞ ≈ 450 show that, slightly upstream of the
pipe, the Blasius boundary layer on the floor of the water channel has a thickness of δ(x) ≈ 0.8D. The boundary layer thickness is much smaller than the height of the elevated pipe. Also,
in the absence of a vortex ring, we observe a separated shear
layer at the top of the elevated pipe. This is referred to as the
“cross-flow shear layer”.
1.5
e = Γ/U0 D
Nondimensional Circulation Γ
ReΓ =
−6
y/D
metric arrangement of the experiment. PIV measurements are
obtained first in the plane defined by the axis of the elevated
pipe and the direction of the cross-flow. This is the x-y plane.
For this same flow, PIV data is also obtained at three streamwise
locations, x/D = 0, 0.5 and 1.0 in the y-z plane. To provide a
line-of-sight to the CCD camera, a mirror is placed 15 diameters downstream of the pipe axis at an angle of 45o from the
cross-flow. Flow visualization shows that the mirror has no adverse effects on the flow in the vicinity of the pipe. The time
delay between the first and second images in each pair is determined by the desire to optimise dynamic velocity range and by
the need to avoid loss of correlation due to out-of-plane motion.
For PIV in the x-y plane, the time delay is 10 ms. For PIV in
the z-y plane, we use a longer time delay of 17 ms.
No Cross Flow
1.0
0.5
R=3.6
Piston stops at
e
t = 1.6
0.0
−0.5
No Cross Flow
−1.0
−1.5
−2.0
0
R=3.6
1
2
3
4
5
6
7
t = tU0 /D
Nondimensional Time e
8
Figure 3: Growth and decay of ensemble-averaged circulation
in the absence of a cross-flow and in a R≈3.6 cross flow.
1
1
0
1
2
−1
0
x/D
3
D
y/D
y/D
1
2
−1
1
3
D
2
2
−1
0
z/D=0
3
3
N
−1
A
2
y/D
F
0
1
2
−1
0
z/D=0
3
1
2
−1
3
3
N
0
3
3
P
N
0
1
1
2
−1
0
3
N
y/D
y/D
−1
A
0
1
−1
3
2
2
N
P
−1
3
NO DATA
x/D=1.0
e
t = 7.4
3
e
t = 7.4
N
P
2
1
G
P
−1
0
x/D
−1
3
2
1
1
−1
3
NO DATA
2
1
3
NO DATA
G
F
G
−1
0
x/D=1.0
e
t = 9.0
3
e
t = 9.0
N
P
2
1
2
G
G
1
F
−1
0
1
x/D
2
−1
0
1
2
−1
x/D
0
z/D
1
1
z/D
x/D=0.5
e
t = 9.0
1
1
0
z/D
x/D=0
e
t = 9.0
2
0
z/D
z/D=0
e
t = 9.0
y/D
y/D
2
x/D
z/D=0
3
1
N
2
1
G
y/D
2
y/D
1
y/D
0
1
z/D
F
−1
0
x/D=0.5
e
t = 7.4
F
F
1
z/D
1
1
2
e
t = 5.8
G
0
x/D=0
e
t = 7.4
1
1
z/D
y/D
3
y/D
y/D
2
z/D=0
B
2
2
1
x/D
z/D=0
e
t = 7.4 D
1
1
0
x/D=1.0
e
t = 5.8
G
x/D
3
2
N
z/D
y/D
0
−1
x/D=0.5
1
1
B
P
2
z/D
y/D
−1
y/D
y/D
y/D
A
1
2
e
t = 4.6
1
−1
3
1
G
P
2
0
x/D=1.0
2
1
e
t = 5.8
N
z/D
x/D=0.5
x/D=0
e
t = 5.8
−1
e
t = 4.6
z/D
z/D=0
e
t = 5.8
1
1
x/D
P
F
0
G
x/D
2
z/D
B
−1
e
t = 3.7
1
−1
1
1
3
N
1
P
1
x/D=1.0
P
F
0
e
t = 4.6
2
0
z/D
2
x/D=0
3
−1
e
t = 3.7
z/D
y/D
y/D
y/D
2
1
1
D
2
1
1
e
t = 4.6
D
0
x/D=0.5
P
z/D=0
e
t = 4.6
N
−1
2
x/D
2
z/D
F
1
x/D=1.0
e
t = 1.6
1
1
1
x/D
3
0
e
t = 3.7
A
0
1
x/D=0
e
t = 3.7
1
−1
2
z/D
z/D=0
e
t = 3.7
2
N
x/D
z/D=0
3
P
3
P
y/D
−1
2
(e)
x/D=0.5
e
t = 1.6
y/D
C
3
y/D
2
(d)
x/D=0
e
t = 1.6
y/D
1
3
y/D
2
(c)
z/D=0
e
t = 1.6
y/D
3
y/D
y/D
3
(b)
z/D=0
e
t = 1.6
y/D
(a)
−1
0
z/D
1
−1
F
0
1
z/D
Figure 4: Temporal development of the vortex ring in cross flow. Column (a) shows flow visualisation in the x-y plane and columns
e , in each of the four laser-sheet planes (Table 1). In the x-y plane, the
(b)–(e) show the ensemble-averaged out-of-plane vorticity, ω
cross-flow is from left to right and, in the z-y planes, the cross flow is out of the page. The colormap for the vorticity field is the same
as that in figure 2.
metrically disposed about the axis of the pipe. Remnants of the
stopping vortex, which is generated by cessation of piston motion, are visible at the pipe exit.
With the addition of a uniform cross flow, we still observe the
development of the two vortex cores but many of the details
are different. The self-induced motion of the vortex ring is augmented by convection in the direction of the cross flow as shown
by comparing figure 2(b) with figure 2(a). Also, the vortex ring
tilts as it moves into the cross-flow; the line between vortex
cores tilts at an angle of up to 23 degrees from the cross flow
direction. On the upstream side (with respect to the cross flow)
of the vortex ring, vorticity of the shear layer and vorticity of
the positive core are of opposite sign. As shown in Figure 3,
diffusion leads to cancellation of vorticity and a reduction in
positive-core circulation. On the downstream side of the vortex ring, shear-layer vorticity has the same sign as vorticity of
the vortex ring. The result, (Figure 3), is a larger circulation
on the downstream side of the ring. Details of the circulation
measurements are available in Hassan et al. [3].
7. OBSERVATIONS OF SECONDARY STRUCTURE
Figure 4 shows the temporal development of a vortex ring in
cross flow. Column (a) in this figure shows PLIF flow visualisation in the x-y plane at z/D = 0. Adjusting contrast levels
in these images improves the visibility of features which are
near the laser sheet. Soon after the vortex ring forms, unmarked
fluid from the downstream side of the pipe is ingested by the
vortex ring. From time e
t ≈ 3.7 to e
t ≈ 9.0, we observe an ap-
proximately vertical “columnar” feature (labelled “A”) joining
the vortex ring to the elevated pipe. In these images, the columnar feature is faint because it is outside the PLIF laser sheet.
Flow visualisation in the x-z plane confirms that the columnar
feature in the PLIF images is in fact a pair of counter-rotating
vortices.
Stretching of the columnar vortices is clearly visible as time
progresses and the vortex ring moves away from the pipe. As
indicated by label “B” in figure 4(a), dye-marked fluid in the
columnar vortex is carried into the laser sheet between the two
principal vortex cores. Roll-up of the vortex ring induces the
top part of the columnar vortex into the centre of the vortex
ring. The bottom of the columnar vortex remains attached to
the elevated pipe.
Columns (b)–(e) of figure 4 show the ensemble-averaged outof-plane vorticity in each of the four laser-sheet planes. In the
x-y plane, the cross-flow direction is from left to right and, in the
z-y planes, the cross-flow direction is out of the page. All features in the ensemble averages are visible in the vorticity fields
of individual realisations.
The thin feature labelled “C” at time t˜ = 1.6 in column (b) is vorticity of the cross-flow shear layer. As the vortex ring forms, the
cross-flow shear layer vorticity is rapidly ingested by the negative vortex core. Inspection of subsequent frames shows that
features labelled “D” (t˜ = 3.7, 4.6), despite apprearances, are unrelated to the cross-flow shear layer.
In figure 4(c)–(e) the two principal vortex cores (labelled “P”
and “N”) are easily identified as they move through the lasersheet positions. The vorticity fields also contain a number
smaller, weaker features. As the vortex ring rolls up, it draws
in (induces) fluid from the wake region of the elevated pipe.
The vorticity of this induced flow is labelled “F” in figure 4(c)–
(e). Features “G” are cross sections of the columnar vortices.
They are visible because the vortex axes are tilted away from
the vertical and therefore have non-zero x-components of vorticity. Unlabelled features remain unexplained at this stage of
the investigation.
8. ROLE OF CROSS FLOW IN DEVELOPMENT OF
SECONDARY STRUCTURE
Cross flow produces a boundary layer on the surfaces of the flat
plate and elevated pipe. Before the process of generating a vortex ring begins, there is also a shear layer at the exit plane of the
pipe and a separated wake downstream of the pipe. The boundary layer and the exit-plane shear layer are shown in figure 5(a)
as a “canopy-like” structure of vortex lines.
Figures 5(b) and (c) show how the vortex ring lifts the shearlayer canopy of vortex lines away from the exit plane of the
elevated pipe. Roll-up of the vortex ring induces motion of the
nearby canopy vortex lines. Canopy vortex lines gather underneath and then move up through the middle of the ring. Vortex lines in the boundary layer are convected in the direction of
the cross flow U∞ so that vortex lines forming the sides of the
canopy are gathered towards the rear (downstream side) of the
pipe. As the vortex ring moves away from the pipe, vortex lines
joining the ring to the boundary layer stretch and form a pair of
well-defined columnar vortices.
In effect, the bottom of each columnar vortex remains anchored
to the downstream side of the pipe-exit plane. The top of each
columnar vortex is drawn into the middle of the vortex ring.
As the ring is convected in the direction of the cross flow, the
vortex
ring
shear layer
"canopy"
U
U
a) t1 <0
before generating
the vortex ring
b) t 2>0>t 1
columnar
vortices
y
z
x
U
c) t 3 > t 2
Figure 5: Development of “columnar-vortex structure” from
the cross-flow shear layer for a vortex ring generated at t= 0.
columnar vortex (and its vorticity field) are tilted. When viewed
in the x-y plane of figure 4, the direction of tilting is clockwise.
In figure 4(d–e) these tilted cross sections of the columnar vortices are visible as regions labelled “G”.
9. CONCLUSIONS
In the absence of a vortex ring there is a cross-flow shear layer
at the top of the elevated pipe. As a vortex ring is ejected, it lifts
the cross-flow shear layer away from the pipe. Roll-up of the
vortex ring pushes the nearest shear-layer fluid underneath and
then up through the middle of the vortex ring. As the sides of
the shear-layer simultaneously
• stretch along the trajectory of the vortex ring and
• convect in the direction of the cross flow,
they form a pair of counter-rotating columnar vortices. The
columnar vortices link the rear of the pipe to the middle of the
vortex ring.
REFERENCES
[1] Etebari, A. and Vlachos, P., Improvements on the accuracy
of dervative estimation from DPIV velocity measurments.,
Experiments in Fluids, 39, 2005, 1040–1050.
[2] Fouras, A. and Soria, J., Accuracy of out-of-plane vorticity measurements derived from in-plane velocity field data,
Experiments in Fluids, 25, 1998, 409–430.
[3] Hassan, E., Kelso, R. and Lanspeary, P., The effect of a
uniform cross-flow on the circulation of vortex ring, in
Proceedings of the Sixteenth Australasian Fluid Mechanics Conference, Gold Coast, Australia, The University of
Queensland, 2007.
[4] Shariff, K. and Leonard, A., Vortex rings, Annual Review
of Fluid Mechanics, 24, 1992, 235–279.
[5] Westerweel, J. and Scarano, F., Universal outlier detection
for PIV data, Experiments in Fluids, 39, 2005, 1096–1100.