It is known that vortex rings formed by large stroke ratios (in a piston/cylinder arrangement) pinch off from their generating jets at a fairly constant universal time scale. In this paper we show that the hypothesis that at the pinch off the translational velocity of the ring equals the jet flow velocity near the ring is equivalent to the recently proposed idea based on a variational principle by Kelvin and Benjamin that the pinch off occurs when the apparatus is no longer able to deliver energy at a rate required for steady vortex ring existence. A formula for the propagation velocity of a thick vortex ring is also proposed and compared with available experimental data and empirical relations

Gharib, Morteza

Shusser, Michael

English

Caltech Authors - Main

PHYSICS OF FLUIDS VOLUME 12, NUMBER 3 MARCH 2000Energy and velocity of a forming vortex ring
Michael Shusser and Morteza Gharib
Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, California 91125
~Received 4 May 1999; accepted 15 November 1999!
It is known that vortex rings formed by large stroke ratios~in a piston/cylinder arrangement! pinch
off from their generating jets at a fairly constant universal time scale. In this paper we show that the
hypothesis that at the pinch off the translational velocity of the ring equals the jet flow velocity near
the ring is equivalent to the recently proposed idea based on a variational principle by Kelvin and
Benjamin that the pinch off occurs when the apparatus is no longer able to deliver energy at a rate
required for steady vortex ring existence. A formula for the propagation velocity of a thick vortex
ring is also proposed and compared with available experimental data and empirical relations.
© 2000 American Institute of Physics.@S1070-6631~00!00703-0#f
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asI. INTRODUCTION
Recently Gharibet al.,1 investigating the phenomenon o
vortex ring formation in a piston/cylinder arrangement, de
onstrated the existence of a universal time scale for vo
ring formation. This time scale~the ‘‘formation number’’! is
the time when the vortex ring reaches its maximum circu
tion and energy, which physically coincide with the pinch o
of the vortex ring in terms of its entraiment, velocity, an
vorticity fields from its generating axisymmetric jet.
Gharibet al.1 showed that the existence of the formati
number was based on the Kelvin–Benjamin variational p
ciple for steady axis-touching vortex rings.2 It follows from
this principle that the pinch off occurs when the apparatu
no longer able to deliver energy at a rate compatible with
requirement that a steady translating vortex ring have m
mum energy with respect to impulse-preserving iso-vorti
perturbations.
Mohseni and Gharib4 used this idea to predict the for
mation number analytically by considering the dimensionl
energy of a vortex ring. It follows from their analysis that th
translational velocity of the ringW50.5UP , whereUP is the
piston velocity. The predictions of the model were in reas
able agreement with experiment, though the authors men
that in practice the translational velocity of a vortex ring
higher.
An alternative approach was used by Shusser
Gharib,3 who modeled inviscid vortex ring formation in
nozzle flow generator by using a hypothesis that a vor
ring completes its formation and pinches off from its gen
ating axisymmetric jet when the translational velocity of t
ring becomes equal to the jet flow velocity near the ring. T
flow in the jet was approximated as a one-dimensional ro
jet of a variable cross section, its diameter being equal to
nozzle diameter near the nozzle and to the diameter of
vortex ring in the immediate vicinity of the ring~see Fig. 1!.
Relating the jet flow velocity to the piston velocity by co
servation of mass and choosing the thin vortex ring appro
mation for the radius and the velocity of the vortex ring t
authors were able to reasonably estimate the formation n6181070-6631/2000/12(3)/618/4/$17.00
Downloaded 11 Jan 2006 to 131.215.240.9. Redistribution subject to AIP-
x
-
-
is
e
i-
l
s
-
on
d
x
-
e
d
e
e
i-
-
ber. This theory gave good predictions for a broad range
flow conditions.
While the Kelvin–Benjamin principle offers an elega
theory for the mere existence of a limiting process, it fails
suggest a process for the dynamics of pinch off. In this pa
we will try to show that the kinematic model suggested
Shusser and Gharib3 is the physical manifestation of th
Kelvin–Benjamin principle.
II. VORTEX RING FORMATION
We start by considering the formation of a vortex ring
a piston/cylinder arrangement. In order to concentrate on
main physical factors, we first study the basic case of a c
stant piston velocity~an impulse velocity program! and as-
sume a uniform velocity profile within the cylinder an
across the ring generating axisymmetric jet. Time-depend
piston velocities are considered by Shusser and Gharib.3 We
also adopt the slug flow approximation1,4
G5 12LUP , ~1!
I 5 14pD
2rLUP , ~2!
E5 18pD
2rLUP
2 . ~3!
HereE is the energy,I is the impulse,G is the circulation,r
is the density of the fluid,D is the diameter of the piston,L
is the piston stroke andUP is the piston velocity. The advan
tages and limitations of this approach are discus
elsewhere.5,6
To apply the Kelvin–Benjamin variational principle, on
needs to compare the energy provided by the pistonE, as
given by~3!, with the energy required for steady vortex rin
ER . The formation process will not be complete as long
E.ER . We now show that this will result in a limitation on
the value of a dimensionless coefficienta defined by Gharib
et al.1 as
a5
E
ArIG3
. ~4!
Generally, a vortex ring can be described by its radiusR,
radius of the corea, circulationG, and the distribution of the
circulation within the core.5,6 Mohseni and Gharib~Ref. 4, p.© 2000 American Institute of Physics
 license or copyright, see http://pof.aip.org/pof/copyright.jsp
b
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619Phys. Fluids, Vol. 12, No. 3, March 2000 Energy and velocity of a forming vortex ring2436! pointed out that the experimental results provided
Gharibet al.1 for different piston velocity programs and ex
conditions suggest that the limiting formation number is n
very sensitive to the vorticity distribution. We will therefor
assume that the functional form of the circulation distrib
tion ~but not the absolute values! does not change during th
formation process. Then the properties of the ring are de
mined by choosing the values of three independent par
eters. These can be taken kinematically asR, a, and the trans-
lation velocityW or dynamically asE/r, I /r, G.
On the other hand, these three quantities are not ent
independent during the formation. Experimental eviden
~see Figs. 4.4.1, 4.4.3, 4.4.4 of Ref. 6! has clearly demon-
strated that after the initial period of growth the vortex ri
radius R remains constant during the formation. This
caused probably by the inability of the cylindrical vorte
sheet having a radius ofD/2 near the nozzle exit to increas
this radius substantially during the roll up.
The fact that the ring radius remains constant is a c
straint that limits the number of independent parameter
two, which can be taken asI /r andG. Therefore the energy
of a steady vortex ring is a function of these two paramet
ER
r
5 f S Ir ,G D . ~5!
Only one dimensionless combination can be obtain
from these quantities and hence according to thePi theorem7
ER
ArIG3
5const. ~6!
Therefore the energy of a steady vortex ring
ER5AArIG3, ~7!
whereA is an unknown constant.
Substituting~7! into the requirementE>ER and using
~4!, we obtain
a>A. ~8!
A is the limit value ofa that was found experimentall
by Gharibet al.1 as 0.33. One sees that the formation proc
continues as long as the value ofa exceedsA. To illustrate
the evolution of the jet and ring energies during vortex ri
formation we define dimensionless energy
FIG. 1. Vortex ring formation in a piston/cylinder arrangement as mode
by Shusser and Gharib~Ref. 3!.Downloaded 11 Jan 2006 to 131.215.240.9. Redistribution subject to AIPy
t
-
r-
-
ly
e
-
to
s,
d
s
End5
4E
rpD3UP
2 . ~9!
In Fig. 2 we plottedEnd for the jet and the ring as a
function of the formation numberL/D. The energy provided
by the piston was calculated using the slug model~3!, while
for the ring energy we used~7! with A50.33. One sees tha
initially the jet energy is larger but the ring energy grow
faster. The pinch off is predicted atL/D'3.8, in agreement
with the results of Gharibet al.1
Next, we estimate the thickness of the formed vort
ring using the experimental value ofa. Following Mohseni
and Gharib4 we approximate the ring as a member of No
bury’s family of vortex rings8 and utilize Fraenkel’s9 second-
order formulas for these vortices:
E5
1
2
rRG2F ln 8e2 74 1 38 e2 ln 8e G , ~10!
I 5rpGR2~11 34e
2!. ~11!
Heree5a/R, i.e., the dimensionless core radius.
Substituting~10! and ~11! into ~4! we obtain
a5
ln
8
e
2
7
4
1
3
8
e2 ln
8
e
2ApA11 3
4
e2
. ~12!
Then froma50.33 followse'0.50. It should be noted
that Mohseni and Gharib,4 considering the propagation ve
locity of a vortex ring, arrived at a lower value ofe'0.3. On
the other hand, the dependence of propagation velocity oe
is relatively weak. Indeed, the propagation velocityW can be
calculated as9
W5
G
4pR F ln 8e2 14 1 3e
2
8 S 542 ln 8e D G . ~13!
Eliminating the vortex ring radiusR from ~11! and ~13!
one obtains
d
FIG. 2. Vortex ring and jet energy. license or copyright, see http://pof.aip.org/pof/copyright.jsp
e
y
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-
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7.2
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ity
rib
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620 Phys. Fluids, Vol. 12, No. 3, March 2000 M. Shusser and M. GharibW5BArG3
pI
. ~14!
Here
B5
1
4
A11 3
4
e2F ln 8e2 14 1 3e
2
8 S 542 ln 8e D G . ~15!
One sees from~15! that whene50.3, B'0.7658 and
when e50.5, B'0.6483. Thus the dependence is inde
weak.
We now use~14! to compare the vortex ring velocit
with the jet flow velocity near the ring, as done by Shus
and Gharib.3
Shusser and Gharib3 pointed out that the jet velocity dif
fers from the piston velocity due to the inapplicability of th
slug model in the immediate vicinity of the ring, as shown
Fig. 1. We denote the ratio of jet to piston velocitiesN with-
out specifying its value at this stage. Then according to Sh
ser and Gharib3 the formation process continues as long a
W<NUP . ~16!
From ~2! and ~3! it follows that
UP5
2E
I
. ~17!
Substituting~14! and ~17! into ~16! one obtains
E>
B
2N
ArG3I
p
. ~18!
That is,
a>
B
2NAp
. ~19!
To compare the approaches quantitatively, one must
timate the value ofN. Following Shusser and Gharib3 we
assume that in the vicinity of the ring the radius of the ge
erating jet increases to that of the ringR. Then in the one-
dimensional approximation~see Fig. 1! it follows immedi-
ately from conservation of mass that
N5
D2
4R2
. ~20!
Substituting the slug-model formulas~1! and ~2! into
~11! one can calculate the ring radius as
R5
D
A2~11 34e2!
. ~21!
Therefore from~20!,
N5 12~11
3
4e
2!. ~22!
It follows from ~15! and ~22! that for e50.50,N'0.59
and B50.6483. For comparison, Mohseni and Gharib4
analysis predictsN50.5. Substituting the obtained values
N andB into the right-hand side of~19! we obtain the value
of 0.31, which is very close to the experimental value of 0
previously obtained by Gharibet al.1Downloaded 11 Jan 2006 to 131.215.240.9. Redistribution subject to AIPd
r
s-
s-
-
3
We see that both dynamical~based on the Kelvin–
Benjamin variational principle! and kinematic~based on
comparison of the velocities! approaches results in the sam
limitation on the value ofa ~8! and ~19!, respectively.
III. VORTEX RING VELOCITY
The fact that the piston velocity is larger than the flo
velocity in the ring generating jet (N,1) can be verified by
using numerical calculations of Rosenfeldt al.10 Though in
a real flow the velocity is not uniform in the cross sectio
the physically important part is the velocity in the bounda
layer because it is where the vortex ring vorticity is creat
One can see from Fig. 13 of Rosenfeldet al.10 that this ve-
locity, which is initially the highest in the cross section an
slightly higher than the piston velocity, decreases with tim
and can become substantially smaller than the piston ve
ity. For example, we can retrieve from Fig. 13~b! of Ref. 10
that the initial peak velocity decreases to approximately 1
cm/s at later times. For the piston velocity of 30 cm/s,
reported by Rosenfeldet al. ~Ref. 10, p. 301!, this corre-
sponds toN50.57, which is in good agreement with th
value of 0.59 obtained in the present paper.
It is interesting to note that one can use~19! to utilize the
experimental value ofN to calculateB, in order to obtain a
formula for the propagation velocity of thick vortex ring
For example, if we use the vortex ring propagation veloc
at the pinch off that can be retrieved from Fig. 11 of Gha
et al.,1 we will obtain thatN'0.4575, which by substituting
~19! will correspond toB50.5352. Therefore the following
relation for the propagation velocity of a vortex ring can
recommended:
W50.5352ArG3
pI
. ~23!
This formula is supposed to be good close to the pin
off, i.e., for thick vortex rings. On the other hand, Gu¨hler and
Sallet11 proposed the following empiric relationship for
vortex ring propagation velocity:
W
UP
5
31p
4p
2
31p
3p2
D
L
. ~24!
FIG. 3. Vortex ring velocity. license or copyright, see http://pof.aip.org/pof/copyright.jsp
o
s
th
ex
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in
on
ir
621Phys. Fluids, Vol. 12, No. 3, March 2000 Energy and velocity of a forming vortex ringIn Fig. 3 we compared relations~23! and~24! with three
experimental values calculated from Fig. 11 of Gharibet al.1
These values correspond to the range of 2<L/D<3.8 cov-
ering the second part of the formation process.
It is seen from Fig. 3 that though~23! was obtained
using the pinch off velocity its predictions are very good n
only for L/D53.8, but also forL/D53. Even for as low
values asL/D52 the accuracy of the prediction remain
fair. On the other hand, Gu¨hler and Sallet’s11 formula ~24! is
good only close to the pinch off because it overpredicts
velocity for shorter piston strokes.
One can conclude that Eq.~23! is a good approximation
for the propagation velocity of a thick vortex ring.
1M. Gharib, E. Rambod, and K. Shariff, ‘‘A universal time scale for vort
ring formation,’’ J. Fluid Mech.360, 121 ~1998!.
2T. B. Benjamin, ‘‘The alliance of practical and analytical insights into t
non-linear problems of fluid mechanics,’’ inApplication of Methods ofDownloaded 11 Jan 2006 to 131.215.240.9. Redistribution subject to AIPt
e
Functional Analysis to Problems in Mechanics, edited by P. Germain and
B. Nayroles@Lect. Notes Math.503,8 ~1976!#.
3M. Shusser and M. Gharib, ‘‘A new model for inviscid vortex ring form
tion,’’ 30th AIAA Fluid Dynamics Conference, Norfolk, 28 June–1 Ju
1999.
4K. Mohseni and M. Gharib, ‘‘A model for universal time scale of vorte
ring formation,’’ Phys. Fluids10, 2436~1998!.
5K. Shariff and A. Leonard, ‘‘Vortex rings,’’ Annu. Rev. Fluid Mech.24,
235 ~1992!.
6T. T. Lim and T. B. Nickels, ‘‘Vortex rings,’’ inFluid Vortices, edited by
S. I. Green~Kluwer, Dordrecht, 1995!.
7G. Birkhoff, Hydrodynamics ~Princeton University Press, Princeton
1960!.
8J. Norbury, ‘‘A family of steady vortex rings,’’ J. Fluid Mech.57, 417
~1973!.
9L. E. Fraenkel, ‘‘Examples of steady vortex rings of small cross-section
an ideal fluid,’’ J. Fluid Mech.51, 119 ~1972!.
10M. Rosenfeld, E. Rambod, and M. Gharib, ‘‘Circulation and formati
number of laminar vortex rings,’’ J. Fluid Mech.376, 297 ~1998!.
11M. Gühler and D. W. Sallet, ‘‘The formation of vortex rings and the
initial motion,’’ Z. Flugwiss. Weltraumforsch.3, 109 ~1979!. license or copyright, see http://pof.aip.org/pof/copyright.jsp


Energy and velocity of a forming vortex ring

PHYSICS OF FLUIDS VOLUME 12, NUMBER 3 MARCH 2000Energy and velocity of a forming vortex ring
Michael Shusser and Morteza Gharib
Graduate Aeronautical Laboratories, California Institute of Technology, Pasadena, California 91125
~Received 4 May 1999; accepted 15 November 1999!
It is known that vortex rings formed by large stroke ratios ~in a piston/cylinder arrangement! pinch
off from their generating jets at a fairly constant universal time scale. In this paper we show that the
hypothesis that at the pinch off the translational velocity of the ring equals the jet flow velocity near
the ring is equivalent to the recently proposed idea based on a variational principle by Kelvin and
Benjamin that the pinch off occurs when the apparatus is no longer able to deliver energy at a rate
required for steady vortex ring existence. A formula for the propagation velocity of a thick vortex
ring is also proposed and compared with available experimental data and empirical relations.
© 2000 American Institute of Physics. @S1070-6631~00!00703-0#I. INTRODUCTION
Recently Gharib et al.,1 investigating the phenomenon of
vortex ring formation in a piston/cylinder arrangement, dem-
onstrated the existence of a universal time scale for vortex
ring formation. This time scale ~the ‘‘formation number’’! is
the time when the vortex ring reaches its maximum circula-
tion and energy, which physically coincide with the pinch off
of the vortex ring in terms of its entraiment, velocity, and
vorticity fields from its generating axisymmetric jet.
Gharib et al.1 showed that the existence of the formation
number was based on the Kelvin–Benjamin variational prin-
ciple for steady axis-touching vortex rings.2 It follows from
this principle that the pinch off occurs when the apparatus is
no longer able to deliver energy at a rate compatible with the
requirement that a steady translating vortex ring have maxi-
mum energy with respect to impulse-preserving iso-vortical
perturbations.
Mohseni and Gharib4 used this idea to predict the for-
mation number analytically by considering the dimensionless
energy of a vortex ring. It follows from their analysis that the
translational velocity of the ring W50.5UP , where UP is the
piston velocity. The predictions of the model were in reason-
able agreement with experiment, though the authors mention
that in practice the translational velocity of a vortex ring is
higher.
An alternative approach was used by Shusser and
Gharib,3 who modeled inviscid vortex ring formation in a
nozzle flow generator by using a hypothesis that a vortex
ring completes its formation and pinches off from its gener-
ating axisymmetric jet when the translational velocity of the
ring becomes equal to the jet flow velocity near the ring. The
flow in the jet was approximated as a one-dimensional round
jet of a variable cross section, its diameter being equal to the
nozzle diameter near the nozzle and to the diameter of the
vortex ring in the immediate vicinity of the ring ~see Fig. 1!.
Relating the jet flow velocity to the piston velocity by con-
servation of mass and choosing the thin vortex ring approxi-
mation for the radius and the velocity of the vortex ring the
authors were able to reasonably estimate the formation num-6181070-6631/2000/12(3)/618/4/$17.00
Downloaded 11 Jan 2006 to 131.215.240.9. Redistribution subject tober. This theory gave good predictions for a broad range of
flow conditions.
While the Kelvin–Benjamin principle offers an elegant
theory for the mere existence of a limiting process, it fails to
suggest a process for the dynamics of pinch off. In this paper
we will try to show that the kinematic model suggested by
Shusser and Gharib3 is the physical manifestation of the
Kelvin–Benjamin principle.
II. VORTEX RING FORMATION
We start by considering the formation of a vortex ring in
a piston/cylinder arrangement. In order to concentrate on the
main physical factors, we first study the basic case of a con-
stant piston velocity ~an impulse velocity program! and as-
sume a uniform velocity profile within the cylinder and
across the ring generating axisymmetric jet. Time-dependent
piston velocities are considered by Shusser and Gharib.3 We
also adopt the slug flow approximation1,4
G5 12LUP , ~1!
I5 14pD2rLUP , ~2!
E5 18pD2rLUP
2
. ~3!
Here E is the energy, I is the impulse, G is the circulation, r
is the density of the fluid, D is the diameter of the piston, L
is the piston stroke and UP is the piston velocity. The advan-
tages and limitations of this approach are discussed
elsewhere.5,6
To apply the Kelvin–Benjamin variational principle, one
needs to compare the energy provided by the piston E, as
given by ~3!, with the energy required for steady vortex ring
ER . The formation process will not be complete as long as
E.ER . We now show that this will result in a limitation on
the value of a dimensionless coefficient a defined by Gharib
et al.1 as
a5
E
ArIG3
. ~4!
Generally, a vortex ring can be described by its radius R,
radius of the core a, circulation G, and the distribution of the
circulation within the core.5,6 Mohseni and Gharib ~Ref. 4, p.© 2000 American Institute of Physics
 AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp
619Phys. Fluids, Vol. 12, No. 3, March 2000 Energy and velocity of a forming vortex ring2436! pointed out that the experimental results provided by
Gharib et al.1 for different piston velocity programs and exit
conditions suggest that the limiting formation number is not
very sensitive to the vorticity distribution. We will therefore
assume that the functional form of the circulation distribu-
tion ~but not the absolute values! does not change during the
formation process. Then the properties of the ring are deter-
mined by choosing the values of three independent param-
eters. These can be taken kinematically as R, a, and the trans-
lation velocity W or dynamically as E/r , I/r , G.
On the other hand, these three quantities are not entirely
independent during the formation. Experimental evidence
~see Figs. 4.4.1, 4.4.3, 4.4.4 of Ref. 6! has clearly demon-
strated that after the initial period of growth the vortex ring
radius R remains constant during the formation. This is
caused probably by the inability of the cylindrical vortex
sheet having a radius of D/2 near the nozzle exit to increase
this radius substantially during the roll up.
The fact that the ring radius remains constant is a con-
straint that limits the number of independent parameters to
two, which can be taken as I/r and G. Therefore the energy
of a steady vortex ring is a function of these two parameters,
ER
r
5 f S Ir ,G D . ~5!
Only one dimensionless combination can be obtained
from these quantities and hence according to the Pi theorem7
ER
ArIG3
5const. ~6!
Therefore the energy of a steady vortex ring
ER5AArIG3, ~7!
where A is an unknown constant.
Substituting ~7! into the requirement E>ER and using
~4!, we obtain
a>A . ~8!
A is the limit value of a that was found experimentally
by Gharib et al.1 as 0.33. One sees that the formation process
continues as long as the value of a exceeds A. To illustrate
the evolution of the jet and ring energies during vortex ring
formation we define dimensionless energy
FIG. 1. Vortex ring formation in a piston/cylinder arrangement as modeled
by Shusser and Gharib ~Ref. 3!.Downloaded 11 Jan 2006 to 131.215.240.9. Redistribution subject toEnd5
4E
rpD3UP
2 . ~9!
In Fig. 2 we plotted End for the jet and the ring as a
function of the formation number L/D . The energy provided
by the piston was calculated using the slug model ~3!, while
for the ring energy we used ~7! with A50.33. One sees that
initially the jet energy is larger but the ring energy grows
faster. The pinch off is predicted at L/D’3.8, in agreement
with the results of Gharib et al.1
Next, we estimate the thickness of the formed vortex
ring using the experimental value of a. Following Mohseni
and Gharib4 we approximate the ring as a member of Nor-
bury’s family of vortex rings8 and utilize Fraenkel’s9 second-
order formulas for these vortices:
E5
1
2 rRG
2F ln 8e2 74 1 38 e2 ln 8e G , ~10!
I5rpGR2~11 34e2!. ~11!
Here e5a/R , i.e., the dimensionless core radius.
Substituting ~10! and ~11! into ~4! we obtain
a5
ln
8
e
2
7
4
1
3
8
e2 ln
8
e
2ApA11 3
4
e2
. ~12!
Then from a50.33 follows e’0.50. It should be noted
that Mohseni and Gharib,4 considering the propagation ve-
locity of a vortex ring, arrived at a lower value of e’0.3. On
the other hand, the dependence of propagation velocity on e
is relatively weak. Indeed, the propagation velocity W can be
calculated as9
W5
G
4pR F ln 8e2 14 1 3e
2
8 S 542ln 8e D G . ~13!
Eliminating the vortex ring radius R from ~11! and ~13!
one obtains
FIG. 2. Vortex ring and jet energy. AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp
620 Phys. Fluids, Vol. 12, No. 3, March 2000 M. Shusser and M. GharibW5BArG3
pI . ~14!
Here
B5
1
4 A11
3
4 e
2F ln 8e2 14 1 3e
2
8 S 542ln 8e D G . ~15!
One sees from ~15! that when e50.3, B’0.7658 and
when e50.5, B’0.6483. Thus the dependence is indeed
weak.
We now use ~14! to compare the vortex ring velocity
with the jet flow velocity near the ring, as done by Shusser
and Gharib.3
Shusser and Gharib3 pointed out that the jet velocity dif-
fers from the piston velocity due to the inapplicability of the
slug model in the immediate vicinity of the ring, as shown in
Fig. 1. We denote the ratio of jet to piston velocities N with-
out specifying its value at this stage. Then according to Shus-
ser and Gharib3 the formation process continues as long as
W<NUP . ~16!
From ~2! and ~3! it follows that
UP5
2E
I . ~17!
Substituting ~14! and ~17! into ~16! one obtains
E>
B
2N A
rG3I
p
. ~18!
That is,
a>
B
2NAp
. ~19!
To compare the approaches quantitatively, one must es-
timate the value of N. Following Shusser and Gharib3 we
assume that in the vicinity of the ring the radius of the gen-
erating jet increases to that of the ring R. Then in the one-
dimensional approximation ~see Fig. 1! it follows immedi-
ately from conservation of mass that
N5
D2
4R2 . ~20!
Substituting the slug-model formulas ~1! and ~2! into
~11! one can calculate the ring radius as
R5
D
A2~11 34e2!
. ~21!
Therefore from ~20!,
N5 12~11 34e2!. ~22!
It follows from ~15! and ~22! that for e50.50, N’0.59
and B50.6483. For comparison, Mohseni and Gharib’s4
analysis predicts N50.5. Substituting the obtained values of
N and B into the right-hand side of ~19! we obtain the value
of 0.31, which is very close to the experimental value of 0.33
previously obtained by Gharib et al.1Downloaded 11 Jan 2006 to 131.215.240.9. Redistribution subject toWe see that both dynamical ~based on the Kelvin–
Benjamin variational principle! and kinematic ~based on
comparison of the velocities! approaches results in the same
limitation on the value of a ~8! and ~19!, respectively.
III. VORTEX RING VELOCITY
The fact that the piston velocity is larger than the flow
velocity in the ring generating jet (N,1) can be verified by
using numerical calculations of Rosenfeld et al.10 Though in
a real flow the velocity is not uniform in the cross section,
the physically important part is the velocity in the boundary
layer because it is where the vortex ring vorticity is created.
One can see from Fig. 13 of Rosenfeld et al.10 that this ve-
locity, which is initially the highest in the cross section and
slightly higher than the piston velocity, decreases with time
and can become substantially smaller than the piston veloc-
ity. For example, we can retrieve from Fig. 13~b! of Ref. 10
that the initial peak velocity decreases to approximately 17.2
cm/s at later times. For the piston velocity of 30 cm/s, as
reported by Rosenfeld et al. ~Ref. 10, p. 301!, this corre-
sponds to N50.57, which is in good agreement with the
value of 0.59 obtained in the present paper.
It is interesting to note that one can use ~19! to utilize the
experimental value of N to calculate B, in order to obtain a
formula for the propagation velocity of thick vortex rings.
For example, if we use the vortex ring propagation velocity
at the pinch off that can be retrieved from Fig. 11 of Gharib
et al.,1 we will obtain that N’0.4575, which by substituting
~19! will correspond to B50.5352. Therefore the following
relation for the propagation velocity of a vortex ring can be
recommended:
W50.5352ArG3
pI . ~23!
This formula is supposed to be good close to the pinch
off, i.e., for thick vortex rings. On the other hand, Gu¨hler and
Sallet11 proposed the following empiric relationship for a
vortex ring propagation velocity:
W
UP
5
31p
4p 2
31p
3p2
D
L . ~24!
FIG. 3. Vortex ring velocity. AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp
621Phys. Fluids, Vol. 12, No. 3, March 2000 Energy and velocity of a forming vortex ringIn Fig. 3 we compared relations ~23! and ~24! with three
experimental values calculated from Fig. 11 of Gharib et al.1
These values correspond to the range of 2<L/D<3.8 cov-
ering the second part of the formation process.
It is seen from Fig. 3 that though ~23! was obtained
using the pinch off velocity its predictions are very good not
only for L/D53.8, but also for L/D53. Even for as low
values as L/D52 the accuracy of the prediction remains
fair. On the other hand, Gu¨hler and Sallet’s11 formula ~24! is
good only close to the pinch off because it overpredicts the
velocity for shorter piston strokes.
One can conclude that Eq. ~23! is a good approximation
for the propagation velocity of a thick vortex ring.
1M. Gharib, E. Rambod, and K. Shariff, ‘‘A universal time scale for vortex
ring formation,’’ J. Fluid Mech. 360, 121 ~1998!.
2T. B. Benjamin, ‘‘The alliance of practical and analytical insights into the
non-linear problems of fluid mechanics,’’ in Application of Methods ofDownloaded 11 Jan 2006 to 131.215.240.9. Redistribution subject toFunctional Analysis to Problems in Mechanics, edited by P. Germain and
B. Nayroles @Lect. Notes Math. 503, 8 ~1976!#.
3M. Shusser and M. Gharib, ‘‘A new model for inviscid vortex ring forma-
tion,’’ 30th AIAA Fluid Dynamics Conference, Norfolk, 28 June–1 July
1999.
4K. Mohseni and M. Gharib, ‘‘A model for universal time scale of vortex
ring formation,’’ Phys. Fluids 10, 2436 ~1998!.
5K. Shariff and A. Leonard, ‘‘Vortex rings,’’ Annu. Rev. Fluid Mech. 24,
235 ~1992!.
6T. T. Lim and T. B. Nickels, ‘‘Vortex rings,’’ in Fluid Vortices, edited by
S. I. Green ~Kluwer, Dordrecht, 1995!.
7G. Birkhoff, Hydrodynamics ~Princeton University Press, Princeton,
1960!.
8J. Norbury, ‘‘A family of steady vortex rings,’’ J. Fluid Mech. 57, 417
~1973!.
9L. E. Fraenkel, ‘‘Examples of steady vortex rings of small cross-section in
an ideal fluid,’’ J. Fluid Mech. 51, 119 ~1972!.
10M. Rosenfeld, E. Rambod, and M. Gharib, ‘‘Circulation and formation
number of laminar vortex rings,’’ J. Fluid Mech. 376, 297 ~1998!.
11M. Gu¨hler and D. W. Sallet, ‘‘The formation of vortex rings and their
initial motion,’’ Z. Flugwiss. Weltraumforsch. 3, 109 ~1979!. AIP license or copyright, see http://pof.aip.org/pof/copyright.jsp


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Energy and velocity of a forming vortex ring

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