J. Fluid Mech. (1998), vol. 360, pp. 121–140. Printed in the United Kingdom
c
1998 Cambridge University Press
121
A universal time scale for vortex ring formation
By MORTEZA GHARIB
1
, EDMOND RAMBOD
1
AND
KARIM SHARIFF
2
1
Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena,
CA 91125, USA
2
NASA Ames Research Center, Moffett Field, CA 94035, USA
(Received 20 December 1996 and in revised form 10 November 1997)
The formation of vortex rings generated through impulsively started jets is studied
experimentally. Utilizing a piston/cylinder arrangement in a water tank, the velocity
and vorticity fields of vortex rings are obtained using digital particle image velocimetry
(DPIV) for a wide range of piston stroke to diameter (L/D) ratios. The results indicate
that the flow field generated by large L/D consists of a leading vortex ring followed
by a trailing jet. The vorticity field of the leading vortex ring formed is disconnected
from that of the trailing jet. On the other hand, flow fields generated by small stroke
ratios show only a single vortex ring. The transition between these two distinct states
is observed to occur at a stroke ratio of approximately 4, which, in this paper, is
referred to as the ‘formation number’. In all cases, the maximum circulation that
a vortex ring can attain during its formation is reached at this non-dimensional
time or formation number. The universality of this number was tested by generating
vortex rings with different jet exit diameters and boundaries, as well as with various
non-impulsive piston velocities. It is shown that the ‘formation number’ lies in the
range of 3.6–4.5 for a broad range of flow conditions. An explanation is provided
for the existence of the formation number based on the Kelvin–Benjamin variational
principle for steady axis-touching vortex rings. It is shown that based on the measured
impulse, circulation and energy of the observed vortex rings, the Kelvin–Benjamin
principle correctly predicts the range of observed formation numbers.
1. Introduction
Vortex rings are a particularly fascinating fluid mechanical phenomenon. From
starting jets to volcanic eruptions or the propulsive action of some aquatic creatures,
as well as the discharge of blood from the left atrium to the left ventricular cavity in
the human heart, vortex rings (or puffs) can be identified as the main flow feature.
The generation, formation, and evolution of vortex rings have been the subject of
numerous experimental, analytical and numerical studies. The reviews of Shariff &
Leonard (1987), and Lim & Nickels (1995) describe much of the current understanding
of vortex ring phenomena as well as some unresolved issues.
In the laboratory, vortex rings can be generated by the motion of a piston pushing
a column of fluid of length L through an orifice or nozzle of diameter D. This results
in the separation of a boundary layer at the edge of the orifice or nozzle and its
subsequent spiral roll-up. The main focus of vortex ring studies in the past has been
to describe the evolution of the ring’s size, position, and circulation. For example,
Maxworthy (1977), Didden (1979), Auerbach (1987a), Glezer (1988), and Glezer &
122 M. Gharib, E. Rambod and K. Shariff
Coles (1990) studied some of the fundamental aspects of vortex ring formation as well
as vortex ring trajectory and evolution. Glezer (1988) considered a few different piston
velocities as a function of time known as the ‘velocity program’. Didden’s (1979) work
has been very popular among vortex ring researchers, since it provides a clear picture
of the role of internal and external boundary layers in the formation process and
circulation of the vortex ring. Utilizing similarity theory, Saffman (1978) and Pullin
(1979) obtained expressions for the vortex ring trajectory, circulation and its vorticity
distribution. Weigand & Gharib (1997) revisited the vortex ring problem using digital
particle image velocimetry (DPIV). They showed that vortex rings generated by a
piston/cylinder arrangement possess a Gaussian vorticity distribution in their core
region. James & Madnia (1996) present a numerical study of vortex ring formation
for different generator configurations. They concluded that the total circulation and
impulse in the flow field of the ring are approximately the same for nozzles with and
without a vertical wall at the nozzle exit plane.
The piston/cylinder arrangement has been extensively used to address the problem
of vortex ring generation. However, except for the investigations of Baird, Wairegi
& Loo (1977) and Glezer (1988), all of the available experimental, analytical and
numerical investigations of vortex rings use small stroke ratios (L/D). Glezer’s (1988)
work focused mainly on mapping the boundaries for the laminar to turbulent transi-
tion as a function of L/D, while Baird et al.’s work addresses the role of the impulse
of the vortex ring in the formation process and presents some flow visualization
observations, which we will discuss later in this paper. Therefore, to the best of our
knowledge, the flow behaviour of vortex rings that can be generated with large stroke
ratios (L/D) has not been examined before.
In particular, let us consider the question of the largest circulation that a vortex
ring can achieve by increasing L/D, keeping the average piston velocity fixed. In
general, the vorticity flux provided by the separated shear layer is the main source of
vorticity for the forming vortex ring (Didden 1979). Therefore, a termination of the
piston motion inhibits the flow of shear layer vorticity and thus its accumulation in
the core region of the vortex ring. In this case, we should expect the circulation in
the vortex formed to be approximately equal to the discharged circulation from the
nozzle or orifice. In the limit of large L/D, the question arises about the existence
of a limiting process which would inhibit the vortex ring from evolving (growing)
indefinitely while still being fed by the vorticity emanating from the tube. In other
words, for a given geometry, is there an upper limit to the maximum circulation that
a vortex ring can acquire?
The purpose of this paper is to address this question. To do this, we must observe
some of the global features of the vortex ring formation such as its velocity and
the vorticity fields from which vortex ring circulation as a function of L/D can be
obtained. We will present experimental evidence to prove the existence of such a
limiting process.
2. Experimental setup
Figure 1 shows a schematic of the experimental setup. Experiments were conducted
in a water tank using a constant-head tank in conjunction with a computer-controlled
flow monitoring valve. Vortex rings are generated by allowing the flow from the
constant-head tank to drive a piston that pushes fluid out of a sharp-edged cylindrical
nozzle into the surrounding fluid. The x-axis coincides with the centreline of the vortex
ring generator, and the nozzle-exit plane is located in the plane x =0.
A universal time scale for vortex ring formation 123
Dump tank
Overflow
Pump
Constant-
head tank
Control
valve
Computer controller
Ultrasonic
flow meter
Water tank
D
L
α
Figure 1. General schematic of vortex ring generator.
The main cylindrical nozzle had an inner diameter (D) of 2.54 cm. The outer
contour of the cylindrical nozzle was shaped to form a wedge with a tip angle of
α =20
◦
and a length of 1.5 cm. In order to study the effect of the exit geometry,
α was increased to 90
◦
by simply mounting a vertical plate at the nozzle exit plane.
In order to generate smaller vortex rings, a smaller nozzle with an inner diameter of
1.63 cm and an L/D of 12 was partially inserted inside the main nozzle. With this
configuration, the main nozzle’s piston action can be used to drive the fluid through
the smaller nozzle.
A computer-controlled variable-area valve controlled the flow rate from a constant-
head tank which in turn controlled the velocity program of the piston. An ultrasonic
flow meter (Transonic Systems, Inc., Model T-208) was used to monitor the fluid
volume displaced by the piston motion. The overall length of the cylinder limited the
maximum stroke of the piston to (L/D)
max
= 15 and the maximum acceleration and
deceleration to |a|
max
≈ 250 cm s
−2
. Figure 2 presents typical piston velocities versus
time for one case of impulsive motion and two cases of different ramp profiles. The
computer control provides precise timing and synchronization of various events with
a time resolution of approximately 10
−3
s. These events include, for example, vortex
ring generation and initialization of measurement processes, such as ultrasonic flow
metering and DPIV.
Fluorescent dye as a fluid marker, in conjunction with a laser light sheet, was used
to make the vortex ring visible. For the purposes of DPIV, the flow was seeded with
neutrally buoyant silver-coated glass spheres with an average diameter of 14 ±5 µm
and illuminated by a sheet of laser light with a thickness of approximately 0.1 cm.
The technique of DPIV (Willert & Gharib 1991), was implemented to map the
flow field. DPIV measures the two-dimensional displacement-vector field of particles
suspended in the flow and illuminated by a thin pulsed sheet of laser light. The
present experiment used a high-speed version of DPIV that is described in detail by
Weigand & Gharib (1997).
The imaging video camera was positioned normal to the measurement plane and
recorded image sequences of particle fields with spatial resolution of 768 ×480 pixels.
With a typical field of view of 11 × 8 cm, the spatial resolution is 0.23 × 0.23 cm,
and the uncertainty in the velocity and vorticity measurement is ±1% and ±3%,
respectively.
124 M. Gharib, E. Rambod and K. Shariff
20
15
10
5
0 0.5 1.0 1.5 2.0 2.5 3.0
Time (s)
Impulse Fast ramp Slow ramp
Piston velocity (cm s
–1
)
Figure 2. Piston velocity vs. time for three different acceleration conditions.
3. Parameters governing the vortex ring’s circulation
For a given geometry, the circulation of a vortex ring (Γ ) depends on the history
of the piston velocity u
p
(t), nozzle or orifice diameter (D), kinematic viscosity (ν) and
discharge time (t). The piston stroke (L) is a derived parameter related to u
p
(t)by
L=
R
t
0
u
p
(t)dt.
We introduce
¯
U
p
(the running mean of the piston velocity,
¯
U
p
=(1/t
R
t
0
u
p
dt)as
the suitable velocity scale. The aforementioned set of dimensional parameters can
then be reduced to the non-dimensional piston velocity history U
p
(t)/
¯
U
p
and to
two non-dimensional parameters Γ/
¯
U
p
D and
¯
U
p
t/D. This non-dimensional time is
equivalent to the ratio of length to diameter of the ejected fluid column (stroke
ratio), i.e. L/D =
¯
U
p
t/D, and will be referred to as the ‘formation time’ in this
paper.
Using the boundary layer assumption, we can show that the vorticity flux from
the nozzle is approximately equal to U
2
m
/2 where U
m
is the maximum velocity within
the cylinder at the exit plane. For the slug model, which assumes a uniform profile,
we have U
2
p
/2=U
2
m
/2. However, this assumption will not be valid for large stroke
ratios (L/D), where the velocity profile in the pipe would show acceleration in the
central region caused by the growth of the boundary layer region (Didden 1979).
In this case, one needs to obtain a time history of U
2
m
and use it to normalize
instantaneous circulation values. For this reason, formulas suggested by Didden, in
which circulation is non-dimensionalized by U
p
D, do not predict the circulation value
for large L/D values correctly (Shariff & Leonard 1992). Glezer (1988) suggested using
the kinematic viscosity (ν) to normalize circulation (Γ )asΓ/ν, which can also be
considered the Reynolds number of the vortex rings. In the limits of a high Reynolds
number, kinematic viscosity should not play a role in the vortex ring’s formation
process. In this paper, we chose to present the circulation data in its dimensional
form.
A universal time scale for vortex ring formation 125
(a)
(b)
(c)
Figure 3. Visualization of vortex rings at X/D ≈ 9for(a)L
m
/D =2,Re ≈ Γ/ν ≈ 2800;
(b) L
m
/D =3.8, Re ≈ 6000; and (c) L
m
/D =14.5. Picture is taken at
¯
U
p
t/D = L/D = 8. All three
cases were generated by an impulsive piston velocity depicted in figure 2.
4. Flow visualization
Figures 3(a), 3(b) and 3(c) show three vortex rings generated by three different
maximum stroke ratios (L
m
/D). The vortex rings shown in these pictures are at an
approximate axial position of X ≈ 9D from the nozzle exit. In figure 3(a), L
m
/D =2,
while in figure 3(b), L
m
/D ≈ 3.8. For the case in figure 3(c), the piston was passing
through the position L/D ≈ 8 at the time the picture was taken. The piston motion
was only stopped later at L
m
/D =14.5. In all cases, vortices were generated with
similar impulsive piston motion to that depicted in figure 2.
One striking feature in these pictures is the existence of a trailing jet of fluid behind
the leading vortex ring in figure 3(c) and lack of it in figures 3(a) and 3(b). It appears
that in figures 3(a) and 3(b) almost all of the discharged fluid has been entrained
into the vortex ring. However, for the case in figure 3(c), the vortex ring shows a
clear separation from the active trailing jet-like region behind it. It is apparent that
the formation of the vortex ring has been completed and the vorticity is no longer
entrained from the shear layer region of the trailing jet. It is interesting to note that the
size of the leading vortex ring in figure 3(c) is approximately the same as that of the
vortex ring in figure 3(b) and is larger than that depicted in figure 3(a). Considering
that the pictures are taken at the same downstream position of X ≈ 9D, this variation
