Exp Fluids (2008) 44:235–247
DOI 10.1007/s00348-007-0397-9
RESEARCH ARTICLE
Measurement of ambient fluid entrainment during laminar vortex
ring formation
Ali B. Olcay Æ Paul S. Krueger
Received: 11 December 2006 / Revised: 14 September 2007 / Accepted: 14 September 2007 / Published online: 14 October 2007
Springer-Verlag 2007
Abstract Planar laser induced fluorescence (PLIF) and
digital particle image velocimetry (DPIV) combined with
Lagrangian coherent structure (LCS) techniques are utilized to measure ambient fluid entrainment during laminar
vortex ring formation and relate it to the total entrained
volume after formation is complete. Vortex rings are
generated mechanically with a piston-cylinder mechanism
for a jet Reynolds number of 1,000, stroke ratios of 0.5, 1.0
and 2.0, and three velocity programs (Trapezoidal, triangular negative and positive sloping velocity programs). The
quantitative observations of PLIF agree with both the total
ring volume and entrainment rate measurements obtained
from the DPIV/LCS hybrid method for the jet Reynolds
number of 1,000, trapezoidal velocity program and stroke
ratio of 2.0 case. In addition to increased entrainment at
smaller stroke ratios observed by others, the PLIF results
also show that a velocity program utilizing rapid jet initiation and termination enhances ambient fluid entrainment.
The observed trends in entrainment rate and final entrained
fluid fraction are explained in terms of the vortex roll-up
process during vortex ring formation.
A. B. Olcay
Department of General Engineering,
University of Wisconsin-Platteville,
1 University Plaza, Platteville, WI 53818, USA
e-mail: [email protected]
P. S. Krueger (&)
Department of Mechanical Engineering,
Southern Methodist University, P.O. Box 750337,
Dallas, TX 75275, USA
e-mail: [email protected]
1 Introduction
Transient ejection of a jet from a nozzle is a common flow
configuration which engenders the formation of a vortex
ring. During the jet ejection, the shear layer which separates at the nozzle lip rolls up and entrains some of the
ambient fluid into the forming vortex ring as described by
Didden (1979). Consequently, both ejected fluid which
comes from inside the cylinder and ambient fluid which is
pulled from the vicinity of the nozzle must be accelerated
as the ring forms. The convective nature of the entrainment
process is directly relevant for a wide variety of problems
including cooling of a CPU unit (Kercher et al. 2003),
extinguishing oil well fires at places where bringing manpower and technology can be very expensive (Akhmetov
et al. 1980), mixing two different fluids, and transferring
mass from one location to another. Hence, a detailed
understanding of the entrainment mechanics in transient
jets could be applied to enhance or lessen entrainment
effects in a variety of applications.
Most work on entrainment in vortex rings to date has
considered formed or steady vortex rings. In this state, a
closed streamline encircles the vortex ring in a frame of
reference moving at the vortex ring velocity. Thus, the fluid
transported within the ring in the vortex ‘‘bubble’’ is clearly
defined as described by Shariff and Leonard (1992).
Maxworthy (1972) conducted experiments using dye
visualization and hydrogen bubble techniques to study the
diffusion of vorticity, which results in entrainment after the
vortex bubble is formed. He concluded that while some of
the vorticity diffuses by pulling irrotational fluid inside the
vortex bubble making vortex bubble larger in size, some
stays behind the traveling vortex ring forming a wake.
Maxworthy (1977) also performed some experiments to
study the vortex ring formation as well as evolution of the
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236
vortex rings. He noted the effect of Reynolds number on
the formation process, but he did not comment about its
effect on entrainment. Fabris and Liepmann (1997) analyzed vortex ring formation, and they concluded that there
are three distinct regions in a steady vortex ring, namely
the core region where rotational flow is present, an intermediate region where irrotational flow exists in the form of
ejected and entrained fluid, and finally an external region
where potential flow encloses the moving vortex bubble.
Dabiri and Gharib (2004) investigated entrainment using
steady bulk counter-flow to hold the rings in the field of
view and streamlines obtained from DPIV to identify the
bubble volume. They observed the rings contained up to
65% entrained fluid volume when they were completely
formed. It was also observed that entrainment fraction
could be increased by using a smaller stroke ratio. A diffusive fluid entrainment model was developed by relating
the ratio of entrained fluid flux (i.e., time rate of change of
entrained fluid volume) to the total fluid flux in the dissipation region behind the formed ring with the ratio of
ambient fluid energy loss rate by viscous dissipation to
ambient fluid energy entering the dissipation region. These
ratios where expressed in terms of the vortex ring’s governing parameters (e.g., velocity, volume of the vortex ring,
and diameter of vortex ring generator).
For formed rings experiencing periodic forcing, the
unstable manifold of the forward stagnation point no longer
coincides with the stable manifold of the rear stagnation
point and intersections between these manifolds identify
‘‘lobes’’ of fluid which can cross the ring boundary
(entrainment or detrainment) as the flow evolves. Shariff
et al. (2006) studied this process for numerically generated,
time-periodic vortex rings using dynamical systems theory.
By monitoring evolution of the lobes and changing oscillation amplitude of the periodic disturbance, they showed
that the exchanged volume can be increased when a higher
oscillation amplitude is used. They also noted the quantitative similarity between their results and detrainment from
experimentally generated turbulent rings.
Shadden et al. (2006) studied empirically generated
vortex rings and observed entrainment and detrainment by
lobe dynamics for nominally steady (i.e., quasi-steady),
aperiodic flows. In this case, the stable and unstable manifolds delineating the vortex boundary were identified as
ridges in the finite-time Lyapunov exponent (FTLE) field
obtained from digital particle image velocimetry (DPIV)
data of the velocity field. They also computed the bubble
volume identified by the ridges [called Lagrangian coherent structures (LCSs)] and compared it with that
determined from streamlines.
Although all of these studies highlight various features
of entrainment during the steady (or quasi-steady) phase of
laminar vortex ring motion when a closed vortex bubble
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Exp Fluids (2008) 44:235–247
has formed, none of them address the process of entrainment during initial ring formation and roll-up. Yet, the bulk
of the entrained fluid in a steady vortex ring is acquired
during the formation process (Dabiri and Gharib 2004).
Indeed, Auerbach (1991) made the distinction between
convective entrainment during shear layer roll-up phase
and diffusive entrainment after the vortex ring is formed.
While he was unable to provide quantitative measure of
ambient fluid entrainment during vortex ring formation, he
concluded that depending on the formation details as much
as 40% of the fluid carried with a steady ring can be
ambient fluid.
The objective of this study is to measure ambient fluid
entrainment during laminar vortex ring formation, and
evaluate the effect of vortex ring formation parameters on
the entrainment process. Since there is no closed volume
associated with the vortex during the formation process
(i.e., prior to achieving a nearly constant translational
velocity), we instead focus on the rate at which fluid is
entrained into the forming vortex spiral. Explicit identification of the entrance to the spiral will be given in
Sect. 3.3. The experimental observations are made using
planar laser induced fluorescence (PLIF) and DPIV. Using
dye as a Lagrangian marker of the ejected fluid, PLIF
allows direct observation of the vortex spiral during the
formation process. DPIV combined with LCS techniques
gives analogous results, but also includes velocity information allowing a more detailed analysis. Both data sets
(PLIF and DPIV) can be used to deduce the size of the
vortex bubble once formed, indicating the overall entrainment. Using these techniques, the present investigation
studies the evolution of entrainment during ring formation
under the influence of different jet velocity programs and
ejected jet length-to-diameter ratios (L/D).
2 Experimental setup and techniques
A schematic of the experimental apparatus is given in
Fig. 1. The experimental apparatus consisted of a piston–
cylinder mechanism for generating the vortex rings, a water
tank, and a pressurized tank to drive the piston. The water
tank was 61 cm wide, 61 cm deep and 244 cm long. The
walls of the tank were 1.27 cm thick glass for flow visualization purposes. The piston and cylinder of the vortex
ring generator were made from high-impact strength PVC
rod and clear PVC schedule-40 pipe, respectively. The
cylinder’s inner diameter (D) was 3.73 cm. A critical
parameter for vortex ring formation was the length-todiameter ratio (L/D), defined as the ratio of the total piston
displacement (during jet ejection) to the piston diameter.
The outer surface of cylinder nozzle was machined to have
a wedge with a tip angle (a) of 7 to ensure clean flow
Exp Fluids (2008) 44:235–247
237
separation at the nozzle exit plane. The piston–cylinder
mechanism was connected to a 30-gallon tank which was
pressurized to 15 psig (103 kPa gage) to actuate the piston
during measurements. A proportional solenoid valve
(SD8202, ASCO Valve Inc.) and an inline ultrasonic flow
rate probe (ME19PXN, Transonic Systems Inc.) were used
to control and measure the volumetric flow rate, respectively. The piston velocity was determined from the ratio of
volumetric flow rate to the piston area. An in-house-code
programmed in Labview (National Instruments) provided
feedback control of the flow rate allowing the piston to
follow an arbitrary velocity program.
To generate the vortex rings, the piston was commanded
to execute finite duration jet pulses. Three different jet
velocity programs were considered: trapezoidal, triangular
positive sloping (PS), and triangular negative sloping (NS).
Examples of all three are shown in Fig. 2. Jet Reynolds
number (ReJ) is calculated based on the piston’s maximum
velocity (UM), piston diameter (D) and the fluid viscosity
(m), namely,
ReJ ¼
UM D
:
m
ð1Þ
In Fig. 2 solid lines show the commanded velocity, and
hollow triangles indicate the measured piston velocity. As
seen from the plots, there is a very good agreement
between commanded and measured velocities. Repeatability of velocity programs was better than 10% in ReJ and
5% in L/D. While acceleration and deceleration periods
were chosen to last 0.1tp (tp being the pulse duration) for a
trapezoidal velocity program, triangular PS and triangular
NS velocity programs commanded the piston with 0.9tp
and 0.1tp in the acceleration phase, 0.1tp and 0.9tp during
the deceleration phase, respectively. The triangular PS and
Fig. 1 Schematic of
experimental apparatus
Pressurized
air
NS cases introduce the effects of non-impulsive jet initiation and termination, respectively.
In order to study the effects of piston velocity program
and L/D ratio, a number of experiments were performed. A
summary of tests used in this study along with the utilized
techniques are given in Table 1.
For PLIF measurements a cover plate similar to that
used by Johari (1995) was initially placed over the end of
the cylinder (see Fig. 1). A hole in the plate allowed dyed
fluid to be drawn into the cylinder while the plate was in
place ensuring the fluid in the cylinder and the ambient
fluid were initially separate. Just before the test began the
cover plate was drawn up vertically. Velocity of the cover
plate was 0.49 ± 0.10 cm/s. This velocity was small
enough to cause minimal dye disturbance prior to tests.
When the plate was clear, the piston was actuated. Using
this procedure, only dyed fluid was in the cylinder when the
jet was initiated and therefore the dye may be considered as
a Lagrangian marker for the ejected fluid. Fluorescein dye
at 4.7 · 10–7 M was used as the fluid marker. The Schmidt
number for fluorescein was about 1,000 as suggested by
Green (1995), so the dye tracked the fluid motion, but not
vorticity diffusion.
An Argon ion laser (Innova 70-2, Coherent Inc.) was
used to illuminate the dye. A 0.15 cm thick laser sheet was
obtained using a cylindrical lens. A black and white 8-bit
digital CCD camera (UP-1830, Uniq Vision Inc.) was
placed perpendicular to the laser sheet to record the flow
evolution at 30 Hz. The recorded field had a 1,024
· 1,024 pixels spatial resolution with a 4.65D · 4.65D
field of view.
Digital particle image velocimetry (DPIV) was used to
obtain velocity field data for the vortex ring formation
process. The DPIV system consisted of a pair of frequency
doubled, pulsed Nd:YAG lasers (Vlite200, LABest Inc.),
Pressure
regulator
Pressurized
tank
Proportional
solenoid valve
Cover plate
and lifting
mechanism
Flowrate meter
Laser sheet
Water tank
Water
Piston
α
Up(t)
Feedback control
through Labview
D
Vortex ring
generator
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238
Exp Fluids (2008) 44:235–247
(b) 1.2
1
1
0.8
0.8
Up(t) / UM
(a) 1.2
Up(t) / UM
Fig. 2 Typical piston velocity
programs for ReJ = 1,000,
L/D = 2.0. a Trapezoidal
velocity program (tP = 2.77 s),
b triangular positive sloping
(tP = 4.98 s), and c triangular
negative sloping (tP = 4.98 s)
velocity programs
0.6
0.6
0.4
0.4
Commanded velocity
Measured velocity
0.2
Commanded velocity
Measured velocity
0.2
0
0
0
0.2
0.4
0.6
0.8
1
0
0.2
(c)
0.4
0.6
0.8
1
t/tp
t/tp
1.2
1
Up(t) / UM
0.8
0.6
0.4
Commanded velocity
Measured velocity
0.2
0
0
0.2
0.4
0.6
0.8
1
t/tp
Table 1 Table of the tested cases
L/D
ReJ
Velocity program
Flow analysis technique
0.5, 1.0
1,000
Trapezoidal
PLIF
2.0
1,000
Trapezoidal
PLIF, Streamline, and LCS
0.5, 1.0
1,000
Triangular NS
PLIF and LCS
2.0
1,000
Triangular NS
PLIF and LCS
0.5, 1.0
1,000
Triangular PS
PLIF
2.0
1,000
Triangular PS
PLIF and LCS
optics to transform the laser beam into a 0.1 cm thick laser
sheet, and a delay generator (555 Pulse Delay Generator,
Berkeley Nucleonics Corporation) for synchronizing the
laser and the camera. The particles used to seed the flow
were 15–20 lm diameter neutrally buoyant silver-coated
hollow glass spheres (SH400S20, Potters Industries Inc.).
The obtained 1,024 · 1,024 pixels particle images were
recorded, paired and processed using Pixel Flow (FG
Group LLC.), which uses a cross-correlation algorithm
similar to the one described by Willert and Gharib (1991).
A laser pulse separation of 22.7 ms was used, giving
maximum particle displacements of 7–8 pixels for a
3.0D · 3.0D field of view. With 32 · 32 pixel interrogation windows at 50% (16 pixels) overlap, the spatial
123
resolution of the resulting vector fields was 0.094D ·
0.094D. To improve the accuracy, the data were processed
a second time with a window-shifting algorithm as
described by Westerwheel et al. (1997). The uncertainty of
velocity measurements was 0.04 pixels as stated by
Westerwheel et al. (1997), which was less than 1% for the
majority of the measured velocity field.
The DPIV data were also used to obtain Lagrangian
information about the flow as expressed using the finite
time Lyapunov exponent (FTLE). Details of this approach
may be found in Shadden et al. (2005); however, a brief
overview will be presented here for completeness. The
equation describing the trajectory of a fluid particle at
position x0 at time t0 may be expressed as
_ t0 ; x0 Þ ¼ Vðxðt; t0 ; x0 Þ; tÞ
xðt;
ð2Þ
where xðt0 ; t0 ; x0 Þ ¼ x0 :
The right side of Eq. (2) can be attained from the DPIV
velocity field data. The solution to (2) is a flow map
ð/tt00 þT ðx0 ÞÞ describing the position at time t = t0 + T of the
fluid particle initially at x0 at time t0 namely;
/tt00 þT ðx0 Þ ¼ xðt0 þ T; t0 ; x0 Þ:
ð3Þ
Then the finite time Lyapunov exponent (FTLE) is defined as
t*=0.5
t*=1.0
t*=1.5
t*=2.0
t*=0.5
t*=1.0
t*=1.5
t*=2.0
t*=0.5
t*=1.0
t*=1.5
t*=2.0
Triangle (PS)
Velocity Program
Triangle (NS)
Velocity Program
Fig. 3 PLIF flow visualization
vortex rings generated by
trapezoidal, triangular NS and
PS velocity programs for ReJ =
1,000 and L/D = 2.0. Gray and
black pixels represent the
ejected and ambient fluid,
respectively
239
Trapezoidal Velocity
program
Exp Fluids (2008) 44:235–247
rTt0 ðxÞ 1 pffiffiffiffiffiffiffiffiffi
ln kmax
jT j
ð4Þ
where kmax is the maximum eigenvalue of
r/tt00 þT ðxÞ
r/tt00 þT ðxÞ
placed in the domain and advected using the given velocity
field. For the present investigation, computation of r was
performed with a uniform grid of 0.0067D resolution to
produce clear ridges for both attracting and repelling LCS.
ð5Þ
and ()* denotes the adjoint operation. It can be shown
(Shadden 2005) that the separation of particles advected by
T
the flow is proportional to ert0 ðxÞjT j to highest order. Hence,
the FTLE is roughly a measure of the maximum expansion
rate of particle pairs advected by the flow.
Lagrangian coherent structures (LCS) are defined as the
ridges in the FTLE field. Shadden et al. (2005) show that
the flux across a LCS scales like jT1 j and thus, for large |T| a
LCS can be treated as a material line or transport barrier in
the flow. Additionally, LCS obtained by forward (T [ 0)
and backward (T \ 0) time integration recovers the stable
(also called repelling LCS) and unstable manifolds (also
called attracting LCS), respectively, surrounding a vortex
ring. Since LCSs behave like material lines, they can
identify the vortex bubble boundaries. This was illustrated
by Shadden et al. (2006), who combined the attracting and
repelling LCS to study entrainment of a formed ring. Since
the formulation applies equally well to unsteady flows, it
can be used to identify the vortex boundary during ring
formation and hence, is a useful tool for studying
entrainment during this phase of ring evolution as well.
LCSs in this study were calculated by using a software
package called ManGen developed by Francois Lekien and
Chad Coulliette in 2001 (http://www.lekien.com/*
francois/software/mangen/). ManGen provided the FTLE
field by computing Eq. (4) for a grid of massless particles
3 Results
3.1 Qualitative observations
Figure 3 illustrates vortex ring formation from a trapezoidal, triangular NS and PS velocity programs for ReJ of
1,000 and L/D = 2.0. All the rings in Fig. 3 travel
from left
to right, and t* in the figures is defined as t tp : In these
figures, gray pixels represent the ejected fluid coming from
inside the cylinder, and black pixels represent ambient fluid
which was initially outside the nozzle. During jet ejection
(0 £ t* £ 1), entrainment is apparent through the growing
black spiral, but the volume of entrained fluid is clearly
much less than the ejected fluid.
Comparison of the spiral formation among the velocity
programs shows distinct differences. Since trapezoidal and
triangular NS velocity programs have a similar start up
acceleration, both produce a long tightly wound spiral
during the initial jet ejection. In contrast, the initial spiral
for the PS case is not wound tightly because the slow jet
initiation provides much less vorticity initially yielding
fewer spiral loops, and the width of each loop is larger. In
general, trapezoidal and triangular NS velocity programs
have steeper start up accelerations than the triangular PS
velocity programs. This steep start up acceleration is
responsible for high velocity gradients at the inner nozzle
wall, which produce stronger vorticity at start up. The
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240
(b) 1.25
1
8
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
0.75
r/D
0.5
0.25
0
-0.25
-0.5
-0.75
-1
-1.25
0
0.5
1
1.5
2
2.5
(c) 1.25
1
ω (1/s)
8
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
0.75
0.5
0.25
0
-0.25
-0.5
-0.75
-1
-1.25
0
1
ω (1/s)
ω (1/s)
8
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
0.75
0.5
r/D
1.25
r/D
(a)
Exp Fluids (2008) 44:235–247
0.25
0
-0.25
-0.5
-0.75
-1
-1.25
0.5
1
x/D
1.5
x/D
2
2.5
0
0.5
1
1.5
2
2.5
x/D
Fig. 4 Contour plots of vorticity with the stagnation streamline indicated by dashed lines. a–c are at t* = 0.72, 1.08, and 1.81, respectively
result is strong Biot-Savart induction, a tightly wound
spiral and, as will be shown later, high ambient fluid
entrainment. Additionally, while the vortex rings obtained
with trapezoidal and triangular NS velocity programs leave
behind a noticeable quantity of ejected fluid, rings generated by triangular PS velocity program pull in nearly all the
ejected fluid. This results in proportionally more ambient
fluid within the ring for the former case. Finally, it is noted
that most of the entrainment occurs after the piston has
stopped (see Fig. 3 for t*[ 1.0). Once the piston stops, the
ejected fluid boundary is no longer held out by the jet and it
may contract under the influence of the ring vorticity,
making a larger area available for entrainment of ambient
fluid. The vorticity created during shear layer roll up drives
the ambient fluid entrainment not only during formation
phase but also after piston has stopped. These observations
agree with Didden’s (1979) results.
Qualitative observations of the ring formation are also
obtained using DPIV by computing the streamfunction in a
frame of reference moving with the vortex ring. The
velocity field is converted to this reference frame by subtracting the ring velocity based on the position of the peak
ring vorticity. To obtain accurate ring velocity measurements, a Gaussian fit of the vortex core is used to obtain
subgrid estimates of vortex location and a third order
polynomial fit of the results is used for computing velocity.
Then, Stoke’s streamfunction is obtained by solving the
governing equation
1 o2 W
o 1 oW
¼ xh
þ
r ox2
or r or
ð6Þ
with a second order accurate finite difference method using
the vorticity field obtained from DPIV data and the velocity
data as boundary conditions.
Figure 4 displays DPIV measurements of vortex ring
evolution for ReJ = 1,000 and L/D = 2.0. The dashed line in
Fig. 4 represents the stagnation streamline which identifies
the boundary of the vortex bubble. Figure 4a shows that, in
the field of view, the W = 0 streamline does not reach r = 0
123
on the backside of the ring while the jet is on. Although
data was collected only for x [ 0, we can conclude that the
W = 0 streamline does not reach r = 0 for x \ 0 while the
jet is on since instantaneous streamlines cannot intersect.
Thus, mass is still entering the ring in Fig. 4a. The W = 0
streamline may, however, close on the outer annulus of the
nozzle. Once the piston has stopped moving, a stagnation
point develops behind the ring, forming a closed bubble as
shown in Fig. 4b, c.
The vorticity distribution is also given in Fig. 4. During
jet ejection the vortex core is small and moves slightly
outward, above the piston radius (i.e., 0.5D). As the ring
continues to form (Fig. 4b), the bounding streamline
obtains fore-aft symmetry. During further evolution of the
vortex ring, vorticity starts to diffuse out of the vortex
bubble (Fig. 4c) causing enlargement of the vortex bubble
volume as mentioned by Maxworthy (1972). The movement of the vorticity core during ring formation determines
the area where induced ambient fluid entrainment takes
place.
Once the DPIV velocity vector fields are obtained from
experiments, FTLE fields are calculated using ManGen
(see Fig. 5a). It is noted that high FTLE values (i.e., ridges)
illustrate the LCS (also called attracting LCS since T \ 0)
as stated by Shadden et al. (2006). During post-processing,
a threshold is applied to FTLE fields to identify the LCS
and locate Dr to be used for volume and entrainment calculations, respectively (see Fig. 5b). This is discussed
further in Sect. 3.3.
The time evolution of the LCS for a trapezoidal velocity
program for ReJ = 1,000 and L/D = 2.0 is shown in Fig. 6.
The repelling and attracting LCSs in Fig. 6a–c were
obtained with forward integration (i.e., T [ 0) and backward integration (i.e., T \ 0), respectively. Combining
these repelling and attracting LCSs generate a closed
transport barrier defining the vortex bubble as described by
Shadden et al. (2006). The extended back side can be
observed in Fig. 6a, b representing the mass that will
comprise the final vortex bubble. This unique property of
LCS determines the volume associated with the vortex ring
Exp Fluids (2008) 44:235–247
241
Fig. 5 a shows the color contour plots of FTLE field at |T| = 5.01 s(|T|/tp = 1.81); b illustrates the LCS after thresholding. It is noted that ambient
fluid entrainment occurs through Dr
1.25
(a)
1
ω (1/s)
8
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
r/D
0.5
0.25
0
-0.25
-0.5
-0.75
-0.25
-0.5
-0.75
-1
-1
-1.25
0.5
1
1.5
2
2.5
Primary
vortex
0
-1.25
0
8
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
0.5
0.25
0
0.5
1
8
7
6
5
4
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
0.5
0.25
0
-0.25
-0.5
-0.75
-1.25
1.5
x/D
ω (1/s)
0.75
-1
Stopping vortex
x/D
(c)
1
ω (1/s)
0.75
r/D
0.75
1.25
(b)
1
r/D
1.25
2
2.5
0
0.5
1
1.5
2
2.5
x/D
Fig. 6 Vortex evolution observed from LCS. The solid line is the repelling LCS and the dashed line is the attracting LCS. a–c illustrates
evolution of the vortex bubble at t* of 0.72 (T = 1.99 s), 1.08 (T = 2.99 s) and 1.81 (T = 5.01 s), respectively
before it is completely formed. It is also noted that the LCS
in Fig. 6b delineates the primary vortex and the stopping
vortex as observed by Didden (1979). Since the stopping
vortex does not enter into the forming ring, the LCS
verifies this flow is outside of the vortex bubble. Lastly,
Fig. 6c illustrates the LCS after the ring is completely
formed. The LCS representing boundary of the vortex
bubble demonstrates near fore-aft symmetry once ring
formation is complete (in agreement with Fig. 4c).
3.2 Entrainment for a trapezoidal velocity program
For a quantitative assessment of the entrainment, we first
focus on the trapezoidal velocity program as a canonical
case frequently studied in the literature. Figure 7 illustrates
PLIF of a vortex ring just after formation is complete for
ReJ = 1,000 and L/D = 2.0. To find the boundary of the
vortex bubble ðoXÞ from such data, an edge detection
algorithm was applied to the images after thresholding.
While the front edge of the moving vortex bubble can be
clearly observed with the PLIF technique, the rear edge of
the vortex bubble cannot be identified by this technique.
Therefore, front-back symmetry is assumed using the upper
and lower boundary of the vortex bubble to define the
mirror axis as indicated in Fig. 7. As noted earlier, this
assumption applies well for a formed ring (Fig. 4b, c).
Integrating over the hatched region in Fig. 7 and assuming
axisymmetry gives the vortex bubble volume (VB).
Although the uncertainty of the volume calculation
depends on the location of mirror axis, threshold value for
edge detection, and image quality, uncertainty analysis
shows that overall uncertainty in VB falls below 7%.
Once the vortex bubble boundary is determined, the
volume of ejected fluid within this volume (VEJ) is calculated by integrating the volume of the gray pixels within
oX (again assuming axisymmetry). This approach accurately obtains the amount of ejected fluid which remains in
the vortex bubble at the end of a piston stroke since it does
not consider ejected fluid not entrained into the vortex
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Exp Fluids (2008) 44:235–247
1.6
1.4
1.2
V B / V EJ
1
0.8
0.6
Volume by PLIF
Volume by streamlines
Volume by LCS
Ejected Volumeinring (PLIF)
0.4
Fig. 7 The components of a moving vortex bubble obtained by PLIF
for ReJ = 1,000 and L/D = 2.0
0.2
0
0
Mirror
axis
0.75
1
1.25
1.5
1.75
2
2.25
2.5
Fig. 9 Vortex bubble volume calculation using PLIF, DPIV and LCS
data (ReJ = 1,000, L/D = 2.0)
B
Fig. 8 Illustration of the rear edge obtained in PLIF images assuming
fore-aft symmetry
bubble (see PLIF data for trapezoidal and triangular NS
when t* ‡ 1.5 in Fig. 3). Therefore, once the ejected fluid
in the vortex bubble is known, volume of entrained
ambient fluid (VE) can be calculated as
ð7Þ
Dye diffusion in the vortex core, however, leads to ambiguity in identification of ejected versus entrained fluid in
the core. The related uncertainty in VEJ is less than 4.2%
for t* ‡ 0.5.
The preceding analysis employing fore-aft symmetry to
identify the boundary of a formed ring may be readily
extended to a forming ring. During jet ejection, fore-aft
symmetry is clearly violated near the centerline because
there is no stagnation point on the back side of the ring, but
near the forming spiral such symmetry holds approximately. This is illustrated in Fig. 8 where the boundary
obtained from the front of the ring is reflected about the
mirror axis to compare with the back side of the ring. The
lack of symmetry on the back side near the axis is of no
consequence for computing VE, however, since the ejected
123
0.5
t / tP
A
VE ðtÞ ¼ VB ðtÞ VEJ ðtÞ:
0.25
fluid in this region is subtracted out. Thus, VE(t) computed
with this procedure, in essence, identifies the fluid
entrained into the spiral across the arc AB in Fig. 8. Specifically, dVdtE computed from these results should give an
accurate measurement of the rate at which ambient fluid is
entrained into the ring during formation. This statement
will be justified experimentally in Sect. 3.3.
Results of the PLIF volume calculations for ReJ = 1,000,
L/D = 2.0, and a trapezoidal velocity program are shown in
Fig. 9. ‘‘Volume by PLIF’’ refers the volume calculation
performed on hatched region given in Fig. 7, and ejected
volume in the ring is the volume of gray pixels (i.e., ejected
fluid) in the hatched region of Fig. 7. The results show that
the computed VB and VEJ increase at nearly identical rates
during formation so that VE is small during this phase.
Once the piston stops, ejected fluid entry slows dramatically since any ejected fluid left at the vicinity of the nozzle
does not have enough momentum to catch the vortex
bubble. Nevertheless, the vortex bubble volume continues
to increase after the jet stops until it reaches an asymptotic
value of VB/VEJ = 1.25. This final increase in the volume is
mostly due to the ambient fluid entrainment. Therefore, the
vortex ring velocity (see Fig. 10) decelerates in this region
(between t* of 1.0 and 1.5) since momentum initially
supplied by the piston needs to be shared with this additional mass, namely the entrained ambient fluid. This
indicates most of the ambient fluid is entrained during the
impulse-preserving phase of motion after the jet stops (as
observed qualitatively in Sect. 3.1).
The PLIF data shows a nearly constant bubble volume
after ring formation is complete, verifying that the ring is
formed since its shape and volume remain unchanged. In
reality, the vortex bubble continues to increase slowly due
Exp Fluids (2008) 44:235–247
243
bubble volume for 0.7 £ t* £ 1.4, even though the PLIF
data indicate the ring is not formed until t* [ 1.5. This is
because the nature of the LCS as transport barriers allows
them to identify the fluid volume eventually to appear in
the ellipsoidal volume of the formed vortex ring bubble,
even before the bubble is formed (see Fig. 6a, b). The LCS
data for t* [ 1.4 illustrates an increase in vortex bubble
volume in agreement with the streamfunction volume
calculation to within experimental uncertainty.
As with the streamfunction data, the LCS data agree
with the volume obtained from the PLIF results to within
experimental uncertainty, confirming the validity of the
fore-aft symmetry assumption used in computing the
bubble volume from the PLIF images, at least in the case of
a completely formed ring.
0.4
0.35
0.3
Wr / UM
0.25
0.2
0.15
0.1
0.05
0
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
2.25
t / tP
Fig. 10 Vortex ring velocity obtained from polynomial fit of vortex
peak locations (ReJ = 1,000, L/D = 2.0)
to the vorticity diffusion as mentioned by Dabiri and
Gharib (2004) and Maxworthy (1972). This is not apparent
in the PLIF data due to slow dye diffusion (high Schmidt
number). As a consequence, the PLIF data indicates volume obtained during formation only and not as a result of
subsequent vorticity diffusion. The streamfunction volume
calculation is also given in Fig. 9 and verifies the volume
increase due to vorticity diffusion. Streamfunction volume
calculations are only obtained after a vortex bubble is
formed so that a closed streamline (see Fig. 4b, c) is
obtained.
Using the stagnation streamline defining the vortex
bubble as shown in Fig. 4, VB can also be computed once
the ring is formed. As before, axisymmetry is assumed in
the volume calculation. The major source of uncertainty
comes from the vortex ring velocity calculation and is less
than 12%. The streamfuction volume calculation agrees
with the PLIF results to within the experimental error just
after ring formation is complete (1.35 £ t* £ 2.0). As time
proceeds, the vortex bubble volume obtained from the
streamfunction starts to increase, reflecting the effect of
vorticity diffusion as the ring advects downstream. Since
vortex bubble volume increases by this additional mass, we
can see that vortex ring slows down after t* of 1.75 (see
Fig. 10).
The filled diamonds in Fig. 9 represent the vortex bubble volume calculation from the LCS technique. These
values were obtained by integrating the volume inside the
LCS boundaries (see Fig. 6) assuming axisymmetry. The
uncertainty in these volume calculations comes mainly
from threshold values applied for repelling and attracting
LCSs and is less than 3%. The LCS data indicate a constant
3.3 Entrainment rate
The rate of fluid entrainment into the vortex ring (QE) is
defined as
QE dVE
dt
ð8Þ
where VE is the volume of ambient fluid in the vortex ring
spiral. This can be estimated from the PLIF data, but the
PLIF data relies on the assumption of front-back symmetry
during ring formation. The LCS data, on the other hand,
can obtain QE directly. Specifically, we consider the volume flow rate into the entrance gap of the vortex spiral as
identified by the attracting LCS. For convenience, we
consider the gap Dr = ro – ri identified along the line
connecting vortex cores as shown in Fig. 11a. This is
analogous to entrance of the vortex spiral identified in the
PLIF data as can be seen by comparing Fig. 11a, b.
Although taken from different runs, the location and the
magnitude of Dr for the LCS and PLIF data show reasonable agreement. In particular, the entrance surface area
(p(r2o – r2i )) in LCS and PLIF is calculated to be 3.22 and
3.75 cm2, respectively. The percent difference is comparable to the repeatability of Reynolds number and L/D.
Both Shariff et al. (2006) and Shadden et al. (2006)
used attracting and repelling LCSs for an already formed
vortex ring to investigate the entrainment/detrainment
between irrotational fluid outside the vortex ring and the
fluid in the vortex ring. Here, however, we are interested in
the flow of fluid into the developing spiral. While the
attracting LCS identifies the spiral (i.e., unstable manifold
which separates ejected fluid from entrained fluid in the
vortex ring), the repelling LCS does not. Thus, we only use
the attracting LCS in this analysis. With the spiral entrance
identified, QE is computed as
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Exp Fluids (2008) 44:235–247
Fig. 11 Entrance surface area
through which ambient fluid is
entrained is shown for both LCS
data in a with |T| = 5.01 s
(|T|/tp = 1.81) and PLIF data in
b with t* = 1.81. Reference
horizontal lines based on the
LCS data are given for
comparison purposes
QE ðtÞ ¼ 2p
Zro
ðuðrÞ Wr Þrdr
ð9Þ
where u(r) is obtained from DPIV and Wr is the ring
velocity. Although Dr considered here is different from the
spiral entrance identified in the PLIF results (AB in Fig. 8),
that should not affect the comparison of QE between the
methods since the fluid entering AB also passes through
Dr.
The comparison of QE obtained from LCS/DPIV and
PLIF is given in Fig. 12. The average slope obtained from
PLIF data for t* £ 0.5 and t* ‡ 1.0 demonstrates reasonable agreement with LCS/DPIV results for the same
interval. However, PLIF results during 0.5 \ t* \ 1.0 were
not able to capture gradual increase of entrainment
obtained from LCS/DPIV. The disagreement is a combination of the fact that the PLIF result is an average value
and uncertainty in VE obtained by PLIF due to dye diffusion in the vortex core. Nevertheless, the simplistic
approach used in PLIF results during ring formation give
good quantitative measurements of entrainment rates during and after jet ejection, even though the transition
between the two rates appears more abrupt in the PLIF data
than is actually indicated by the LCS/DPIV data.
The high entrainment rate after the jet stops agrees with
the previous observation that most of the ambient fluid is
entrained during the impulse preserving phase of motion.
As indicated in Fig. 13, a key factor in the increased QE is
the dramatic increase in Dr after jet termination. From
123
QE / (0.25pD2PUM )
ri
t / tp
Fig. 12 Entrainment rate comparison between PLIF and LCS/DPIV
results (ReJ = 1,000 and L/D = 2.0)
Eq. (9), however, the flow velocity plays a role as well. In
particular, note that time-accurate LCS results show a
descending trend for 0.9 £ t* £ 1.5. The peak near t* = 0.9
is reasonable since piston velocity program starts to slow
down after this point. This causes reduction in velocity, but
the Dr is almost constant between t* of 1 and 1.5. Consequently, the entrainment rate starts to diminish until Dr
starts to rise after t* of 1.5. This is observed as the LCS
entrainment rate starts to increase again at late time (i.e.,
t* [ 1.5).
Exp Fluids (2008) 44:235–247
245
1.2
0.2
1
0.15
0.8
Γ / ΓΜ
Dr / D
0.25
0.1
0.6
0.4
0.05
Trapezoidal
Triangle (NS)
Triangle (PS)
0.2
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
t / tP
Fig. 13 Variation of Dr
D for ReJ = 1,000 and L/D = 2.0 (The
uncertainty of Dr
D is less than 5%)
0.2
0.18
0.16
QE / (0.25pD2PUM )
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
t / tp
Fig. 15 Vortex ring circulation comparison for trapezoidal, triangular NS, and PS velocity programs Rfor RReJ = 1,000 and L/D = 2.0.
1 1
Circulation is calculated from C ¼ 0 0 xh dxdr and CM refers the
maximum circulation obtained from trapezoidal velocity program.
Uncertainty of the circulation calculations is less than 1% of the
maximum circulation
0.22
0.14
0.12
0.1
0.08
0.06
Trapezoidal
Triangular NS
Triangular PS
0.04
0.02
0
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
t / tP
Fig. 14 PLIF entrainment rate (calculated from the average slope of
PLIF entrained volume data) for trapezoidal, triangular NS and PS
velocity programs at ReJ = 1,000 and L/D = 2.0
3.4 Quantitative comparison between trapezoidal
and triangular velocity programs
The quantitative validity of the PLIF results having been
confirmed, only PLIF entrainment results will be used here
for brevity. The effect of velocity program on entrainment
rate obtained from PLIF data can be seen in Fig. 14. Since
trapezoidal and triangular NS velocity programs exhibit
similar acceleration behavior at the start up, nearly the
same entrainment rates are obtained during jet initiation
(0 £ t* £ 1.0). Similarly, during the momentum conserving
period (1.0 £ t* £ 1.5), near identical entrainment rates are
calculated for trapezoidal and triangle PS velocity
programs since these programs behave similarly at the
deceleration phase.
For rapidly accelerating velocity programs, both the
vorticity in the forming spiral and the entrainment area are
key parameters. First, Fig. 15 shows that the fast acceleration cases (i.e., trapezoidal and triangular NS) have higher
circulation during jet ejection, which produces a higher
entrainment velocity (due to the Biot-Savart induction)
compared to the triangular PS velocity program. Second,
since rapidly accelerated velocity programs cause tighter
spirals for 0 £ t* £ 1.0, a larger Dr is available for
entrainment as shown in Fig. 16.
For rapidly decelerating velocity programs, on the other
hand, the primary effect is on the area through which fluid
is entrained (Dr), as determined by the effect of the stopping vortex. Figure 16 shows that Dr rapidly increases for
both the trapezoidal and triangular PS cases following jet
termination, arriving at a final value of nearly twice that of
the triangular NS case. This is contrasted with the total
circulation, CM, which differs by only about 30% between
these three cases.
Finally, to understand the effect of L/D on total
entrained fluid volume, the entrainment fraction
ði.e., gent VVEB Þ is plotted in Fig. 17 below. When ReJ is
fixed, the piston is required to reach the commanded
velocity in a shorter time as L/D is decreased. This results
in a higher gent as L/D is decreased for all the velocity
programs given in Fig. 17 as a more compact vortex core is
generated at initiation for higher initial jet acceleration. On
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Exp Fluids (2008) 44:235–247
0.18
0.225
0.2
Triangular NS
Triangular PS
0.175
0.16
0.15
Dr F / D
Dr / D
0.14
0.125
0.1
0.12
0.075
0.05
0.1
0.025
0
0
0.25
0.5
0.75
1
1.25
1.5
1.75
2
t / tP
0.5
Trapezoidal
Triangle (NS)
Triangle (PS)
0.45
0.4
ηent
0.35
0.3
0.25
0.2
0.15
0.5
1
1.5
2
2.5
L/D
Fig. 17 Entrainment fraction ði.e.; gent VVEB Þ comparison for vortex
rings at ReJ = 1,000
the other hand, the proximity of the ring to the nozzle when
the jet terminates and the strength of the stopping vortex
determine the final size of Dr (DrF) following jet termination. When L/D is decreased causing a stronger stopping
vortex, DrF/D increases as illustrated in Fig. 18. This also
contributes to the larger gent at small L/D.
4 Conclusions
Ambient fluid entrainment during vortex ring formation
due to Biot-Savart induction was investigated from a
123
0
0.5
1
1.5
2
2.5
L/D
Fig. 16 Dr comparison between triangular NS and PS velocity
programs for ReJ = 1,000 and L/D = 2.0. The dashed line shows the Dr
D
obtained from trapezoidal velocity program as a reference
0
0.08
Fig. 18 DrF/D variation with respect to L/D for ReJ = 1,000 and the
Triangular NS velocity program
piston–cylinder mechanism. PLIF and DPIV combined
with LCS methods were utilized first to identify the vortex
bubble and then to compute the ambient fluid entrainment.
The piston velocity programs and L/D ratio were changed
in order to study the effects of these parameters on
entrainment in the forming vortex ring.
PLIF method gives entrainment during ring formation
using the assumption of the fore-aft symmetry. This
assumption is justified for a formed ring. During ring formation it is also justified for calculation of VE due to
approximate fore-aft symmetry of the vortex spiral. The
DPIV streamline method accurately provides the vortex
bubble shape via the stagnation streamline once the vortex
is formed. Indeed, the streamfunction shows bubble growth
by diffusion; however, it can only be used after the ring is
formed. The DPIV/LCS method provides more detailed
information than either method, but requires significantly
more data processing. The LCS results confirmed the
conclusions drawn from PLIF using the assumption of foreart symmetry.
Studying the PLIF results in detail revealed several key
factors affecting entrainment during vortex ring formation.
First, the effect of the velocity program on entrainment rate
is determined primarily by the magnitude of jet acceleration during initiation and termination as shown in Fig. 14.
While high initial accelerations such as a trapezoidal or a
triangular NS velocity program can enhance the initial
entrainment rate by as much as 50% compared to a triangular PS velocity program during jet ejection, high
terminal deceleration like a trapezoidal velocity program or
a triangular PS velocity program can increase the final
entrainment rate up to only 10% compared to a triangular
Exp Fluids (2008) 44:235–247
NS velocity program. Second, L/D has a strong effect on
entrainment as well. As L/D is lowered from 2.0 to 0.5, the
piston must be accelerated at a higher rate causing higher
vorticity in the forming spiral. Also, when L/D is reduced
from 2.0 to 0.5, DrF/D increases by as much as 80%. This
is due to the rapid stop of the piston which generates a
stronger stopping vortex and creates a larger area for
ambient fluid entrainment. The net effect of these trends is
an increase of up to 67% in the final entrainment fraction as
L/D is reduced from 2.0 to 0.5.
These observations provide insight into enhancing
ambient fluid entrainment during vortex ring formation.
Specifically, a trapezoidal velocity program with low L/D
ratio is an ideal candidate since this program benefits from
an impulsive jet initiation as well as a rapid jet termination.
Acknowledgments This material is based upon work supported by
the National Science Foundation under Grant No. 0347958.
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