J.
Fluid
Mech.
(1996).
col.
32
I.
pp.
59-86
Copyright
0
1996
Cambridge
University Press
59
Experimental
studies
of
vortex
disconnection
and
connection
at
a free
surface
By
MORTEZA
GHARIB
AND
ALEXANDER
WEIGAND
Graduate
Aeronautical
Laboratories,
California
tnstitute
of
Technology,
Pasadena,
CA
91125,
USA
(Received
14
August
1995
and
in
revised
form
22
March
1996)
An
experimental study
is presented
that
examines
the
interaction
of
a vortex
ring with
a free
surface.
The main
objective of
this study
is to
identify
the
physical
mechanisms
that are
responsible
for
the
self-disconnection
of
vortex
filaments
in
the
near-surface
region
and
the subsequent connection
of
disconnected vortex
elements
to
the
free
surface.
The understanding
of those
mechanisms
is essential
for
the
identification
and
estimation
of
the
appropriate
spatial
and
temporal
scales
of
the
disconnection
and
connection
process.
In
this
regard,
the velocity
and
vorticity
fields
of
an
obliquely
approaching laminar
vortex ring with
a Reynolds
number
of
1150
were
mapped
by
using Digital
Particle
Image
Velocimetry
(DPIV).
The
evolution
of
the
near-surface
vorticity
field
indicates
that
the connection
process
starts
in
the
side regions of
the
approaching
vortex ring where
surface-normal vorticity already
exists
in
the
bulk.
A
local
strain
rate analysis
was
conducted to support
this
conclusion. Disconnection in
the near-surface
tip region
of
the
vortex ring
occurs
because of
the removal
of surface-
parallel vorticity
by
the
viscous
flux
of vorticity
through
the
surface.
Temporal
and
spatial
mapping
of
the vorticity
field
at
the surface
and
in
the
perpendicular plane
of
symmetry shows
that
the
viscous flux is
balanced
by
a local
deceleration
of
the
flow
at
the
surface.
It
is
found
that
the
observed timescales of
the
disconnection
and
connection
process scale with
the near-surface vorticity
gradient rather than
with
the
core
diameter
of
the
vortex
ring.
1.
Introduction
Surface shear
flows,
such
as
the wake
of
an
island
or a
ship,
and
flows
generated
by
under-surface
current
present intriguing features.
The
persistence of
some
of
the
observed
features
are
uniquely puzzling
in
that
they
do
not
obey
any known
decay
laws
of
fully
submerged
flows.
Typical ship
wakes
and
surface
currents
in
the
ocean
possess
Reynolds
numbers
in
excess
of
lo8
which, in
combination
with
the
complex
nature
of
these
turbulent
flows,
makes their
full simulation
difficult,
if not
impossible,
to
perform.
One interesting
feature during the
formation
and
evolution
of surface shear
flows
is
the
dominant
role
of sub-surface
vortices
and
large-scale
vortical
structures.
In
this
regard,
studies
of
the
free-surface
interaction
of vortices
such
a vortex
rings
and
vortex
pairs
are
critical
in
the understanding
of
the
main features of surface
shear
flows.
Therefore, the
research scene in
free-surface turbulence
is dominated
by
efforts
that
focus
on
the
understanding
of
the
interaction
of
elemental vortex
flows with free
surfaces.
Vortex rings
and
vortex
pairs
have
received
much
attention
as
the
primary
constituents
of
any
generic
shear
flow. The
problem
of
a vortex
ring
or
a vortex
pair
approaching a boundary
with
a free-slip
or
no-slip
condition (whether
it is
solid
or
60
M.
Gharib
and
A.
Weigand
deformable)
is
challenging,
since it involves
unsteady three-dimensional vortex
interactions. The pioneering work
of
Sarpkaya
&
Henderson
(1989,
and
subsequent
important
studies
of
Ohring
&
Lugt
(1991)
and Dommermuth
(1992)
on
the vortex-
pair problem,
and
the work
by
Kwon
(1989)
and
Bernal
&
Kwon
(1989)
on
the vortex-
ring problem opened
a
new
era
in
our
understanding
of
vortex/free-surface
interactions.
In
those studies, vortex trajectories
and
surface signatures such as scars
and
striations
have
been
investigated extensively.
However, the connection
of
vortex
filaments
to a
free surface had
not
received
serious
attention
until the experiments
of
Bernal
&
Kwon
(1989)
and
the analytical studies
of
Lugt
(1987)
who elegantly
explained the
proper
kinematic conditions
at the surface.
Bernal
&
Kwon made the
first
clear
demonstration
of
the early stages
that
a vortex ring undergoes as it
rises
toward
a
free surface. Their observations raised many
important fundamental
questions
regarding the interaction
with
the free surface
which
is drastically
different
from
the
case
of
a vortex ring interacting
obliquely with
a solid
wall
(Lim 1989;
Chu,
Wang
&
Hseih
1993).
However,
Bernal
&
Kwon
did
not
clarify
the
issue
of
whether the apex
of
the vortex ring
simply
tilts
and
connects
to
the free surface,
or
whether
it disconnects
and
subsequently connects its free ends
to
the surface.
In
their experiments, the
effect
of
the angle
of
approach
was
found
to
be
significant
regarding the connection
timescale,
while
the Reynolds number
of
the vortex ring did
not
play
an important
role.
They argued
that
the
viscous
diffusion timescale
overpredicts the connection
timescale
and,
therefore, assumed
an
inviscid
scaling based
on
the circulation
and core
diameter
of
the vortex ring. However, the
latter
resulted in
the underprediction
of
the connection
timescale (see
figure
14
of
Kwon
1989).
Therefore, the main question is:
What
are
the
proper
physical
parameters
that
describe the connection
timescale
for
a given
angle
of
approach
of
a vortex ring?
In
this paper,
we
focus
on
the mechanisms
that
lead
to
the disconnection
and
surface
connection
of
a laminar vortex ring as it approaches the
free
surface
at a shallow angle.
Our
objective
is
to
present
a clear picture
of
the stages
that
are
involved
in the early
disconnection
and
subsequent connection
process
by
using
results
of
velocity-
and
vorticity-field
measurements. Obviously, as has
been shown
by
Bernal
et
al.
(1989),
the
physical
state
of
the
free
surface
is
a point
of
concern
in
terms
of
the
presence
of
surfactants.
We
address
these
concerns
by
conducting
two
sets
of
experiments
where
a vortex ring approaches
a very
clean
and
a
semi-clean
water surface.
In
our
investigations, Digital Particle Image
Velocimetry
(DPIV)
was
employed
to
map
the
spatial
and
temporal evolution
of
the
velocity
and
vorticity
fields
during the interaction
with the free surface.
The
mapping process
was
carried
out
(in some
cases
simultaneously)
very close
to
the
free
surface
and
at
various two-dimensional cross-
sections
in
the
flow.
2.
Experimental
setup and procedures
Experiments
were
conducted
in
a water
tank
using
a mechanical vortex-ring
generator.
As
the schematic
in figure
1
shows, vortex rings
of
diameter
D,
circulation
T,
and
propagation
velocity
U,
were
generated
by
a piston
that
pushes fluid
out
of
a
sharp-edged cylindrical
nozzle
with
an
inner
orifice
diameter
of
Do
=
3.0
cm.
The
non-
deformed water surface
coincides with
the (x, z)-plane
where
the positive
z-axis
points
toward
the reader.
The
centreline
of
the vortex-ring
generator
(i.e. the
x’-axis)
is
inclined
to
the water surface
at
an
angle
a,
and
the origin
x’
=
0
is
located
at x
=
0
and
y
=
-h,
where
h
designates the submergence
depth
of
the generator.
To
minimize
surface-contamination
effects
and
surface-tension
differences between
Vortex
disconnection and connection
at
a, free
surface
61
Free
surface
yt
x
FIGURE
1.
General
schematic
of
the experimental
setup.
the water
in
the
bulk
and
at
the
surface, careful steps
were
taken
to
keep
the water
and
its
surface clean.
Besides using de-ionized
water
and
working in a
clean
environment
(i.e.
in a
class
100000
clean
room,
wearing
non-contaminating
gloves,
cleaning all
parts
in contact
with water
using
ethyl alcohol,
and
operating
a UV
filter
to
prevent bacterial
growth
in
the
water), a
constantly operating
skimmer
and
a
vacuum-operated suction
device were using
to
remove
the aging
water
surface. Surface-tension measurements
were
performed before
and
after
each
experimental
run
using
a ring
tensiometer
(Fisher Surface
Tensiomat,
Mod.
21).
An IBM
PC
provided
precise
timing
and
synchronization
of different
events (time
resolution better
than
s)
including vortex-ring
generation,
dye injection
for
flow
visualization,
and
initialization
of
measurement
processes.
As
figure
2
(a)
shows,
an
off-
axis
shadowgraph setup
with
a parallel
beam
diameter
of
8
in.
was used
for
the
visualization
of
the
vortex-ring/free-surface
interaction
and
the
resulting surface-
deformation
fields.
Regarding Digital Particle
Image
Velocimetry
(DPIV),
only
those
aspects
of
the
measurement technique
that
are
relevant
to
the
present work
are
described
in this
paper.
(For
a more
detailed discussion, the reader
is referred
to
Willert
&
Gharib
1991
.)
DPIV
measures
the
two-dimensional displacement-vector
field
of
particles
that
are
suspended
in
the
flow
and
illuminated
by
a thin,
pulsed sheet of laser
light. A
video
camera
is
positioned
normal
to
the
illuminated measurement plane
and
records a
sequence
of
particle-image
fields.
The
exposure
times
and
time
difference between successive image
exposures
are
synchronized
and
controlled
by
a camera controller
and
a shutter.
The
latter
prevents
streaking
particle
images
and
limits
the
maximum displacement
of
particles
in
the
imaging plane.
The
recorded
image sequence
is
stored
on
an
analog
video disk
and
subsequently
digitized
by
a frame grabber.
By
cross-correlating spatial
sub-samples (windows)
of
two
successive video images,
the
average
local
displacement
vector
of
the
particles
contained
in
the correlation
window
is estimated. Moving
the
correlation
window over
the
entire image, the displacement-vector
field is
obtained.
The
latter
is
divided by
the time
difference between
two
successive
image exposures
yielding
the
velocity-vector field.
In the
present experiments,
we
used single
and
quasi-simultaneous
double-plane
DPIV
measurements in
order
to
map
the
velocity fields close
to
the
free
surface
and
in
the
plane
of
symmetry
z
=
0.
Figure
2(b)
shows
the
experimental
setup
of
the
quasi-
simultaneous
measurement technique
that
uses
two separate
DPIV
cameras, two
62
M.
Gharib
and
A.
Weigand
(a)
(Q
Frame
~
112
of
DPIV
camera
Free
surface
Alternating
camera
Scanner
mirror
Shutter
Laser
beam
Frame
___
I
I
I
I
I
Shutter control
and
illumination
of
DPIV camera
nrr
ni
Time
t
I
I
I
I
I
I
I
I
1
I
I
Sequence
of
I
scanner
motion
*
Time
t
FIGURE
2.
(a)
Shadowgraph
setup.
(b)
DPIV
system for
the
quasi-simultaneous
velocity-field
measurement in two perpendicular measurement planes.
(c)
Timing sequence
and
synchronization
of
the
video
camera, shutter,
and
laser scanner.
optical setups for the illuminating laser-light sheets,
and a
laser-scanning
device.
The
scanner
oscillates
the
laser beam (green line
of
an
Argon/Ion
laser,
h
=
514.5
nm)
between
the
two
light-sheet optics
and
alternates the illumination
of
the horizontal
(y
=
-0.1
cm)
and
vertical
(z
=
0)
measurement plane
in
synchronization with
the
DPIV
cameras.
The
light sheets
are
formed
by
using
cylindrical
lenses
with
a focal
length
off=
0.64 cm,
while
their thickness
of
0.1
cm
is adjusted
by
a lens with
a focal
length
off=
100
cm.
The
scanning light-sheet technique
was
implemented
by
placing
a rotating mirror
at
the focal
point
of
a scanner
lens
(f=
30.48
cm,
F#
=
3)
such
that
the
rotation
of
the
laser beam
is transformed
into a
plane-parallel
motion.
The
stepper-motor
drive
of
the
rotating mirror
is synchronized
to
the frame marker
of
one
video
camera
and
allows
for a
rapid oscillatory
beam
displacement
over
a distance
of
up
to
6.0cm
with
a
scanning
rate
of
20
ms/scan.
As
the
pulse sequence
in
figure
2
(c)
shows,
the
DPIV
cameras
and
their frame read-
out
are
phase-locked
to
each
other,
while
the illumination
sequence
is controlled
by
a
synchronized
shutter.
The individual
DPIV
measurements
are
phase-shifted
by
two
Vortex
disconizection
und connection
ut
a
free
surface
63
video
frames
(66
ms) resulting
in
the quasi-simultaneous
and
alternating
measurement
of
the
velocity fields in
the horizontal
and
vertical
plane.
In
the DPIV
measurements, neutrally
buoyant
silver-coated
glass
spheres
with
an
average
diameter
of
14
k
5
pm
were
used
as
seeding
particles.
The
exposure
times
and
time difference between
exposures
of
the
DPIV
cameras
were
tPl
=
teZ
=
4
x
lop3
s
and
At
=
7
x
lo-"
s,
respectively.
The
video images were digitized with
a resolution of
768
x
480 pixels,
and
processed with
a window wide of
32
x
32
pixels
and
a step
size
of
8
x
8 pixels (75
YO
window
overlap).
The
data
processing results in
a
field
measurement
of
96
x
60
velocity
vectors
and,
for
a typical
field
of
view
of
11
x
8
cm,
in a
spatial
wavelength
resolution
of
0.46
x
0.54
cm.
Since
the
location
of the cross-correlation
peak
can
be resolved with
a sub-pixel
accuracy
of
approximately
0.01
pixel (Willert
&
Gharib
1991)
the
maximum uncertainty
based
on the
local velocity
and
vorticity
magnitude
is
C
1
YO
and
-t
3
%,
respectively.
3.
Results
3.
I.
Initial
conditions
Prior to the
investigation
of
the
free-surface
interaction,
the
physical
properties
of
the
laminar
vortex
rings were investigated in
the
fully
submerged
case
by
means
of
flow
visualization
and
Digital Particle Image
Velocimetry
(Weigand
&
Gharib
1995).
In
the
present work
the
oblique interaction
with
the
free surface
was
studied
by
using vortex
rings with
a diameter
of
D
=
3.0
cm
and
a Reynolds
number
of
Re
=
1150
where
the
Reynolds
number
is defined
by
the
ratio
of
the
circulation
rand
the kinematic
viscosity
1'
(i.e.
Re
=
T/v).
As
a result of
the
DPIV
measurements, the
core
size of
the
vortex
rings was
found
to
be
relatively large with
an
initial
core
to
vortex-ring diameter
ratio
of approximately
0.6. In
all flow
cases,
the
submergence
depth
and
inclination angle
of
the
vortex-ring
generator are
h
=
3.0
cm
(one
nozzle
diameter)
and
cx
=
7",
respectively.
The
resulting
Froude
and
Weber
numbers
are
Fr
=
r/(gD3)'"
z
0.07
and
We
=
pP/(nD)
z
0.6,
using
the measured surface tension
of
D.1.-water
v
=
72.120.5
dyn cm-'
(1
dyn
=
lo-"
N),
the
density of
water
p
=
1000
kg m-3,
and
the
gravitational
acceleration
g
=
9.81
m
s-l.
In order
to
be
able
to map
the
velocity
field
close
to
the
free
surface, the Reynolds
and Froude
numbers
were chosen
to
be
relatively
small such
that
only
small surface
deformations
are
generated
(i.e. deformations
of
the
order
of
magnitude
of
the laser-sheet thickness).
In
the
following
sections,
the
results
of two distinct
flow
cases
are
presented.
In
the
first, referred
to
as the
'clean'
case,
extraordinary care
was
taken
to
keep
the
water
surface
clean
and
uncontaminated,
while,
in the second,
referred
to
as
the
'
semi-clean'
case,
slight
surface
contamination
was
present. However, surface-tension
meas-
urements
using
a ring
tensiometer
with
a nominal accuracy
of
CO.1
dyn
cm-l
and
an
experimentally determined repeatability
of
C0.5
dyn
cm-l
did
not
reveal
the
degree of
contamination.
Therefore,
the
distinction
between
the
clean
and
semi-clean
flow
cases
is based on differences in
their surface-interaction
and
connection
behaviour,
and on
differences in
the
formation
of
Reynolds
ridges
(Scott
1982).
In
contrast
to the
semi-
clean
case,
the
formation
of
Reynolds
ridges due to
surface-tension differences was
not
observed in
the
clean
case.
3.2.
Results
of
flow
risua/izurion
To
obtain
a qualitative
understanding
of
the
flow
at
the
free
surface,
the shadowgraph
technique
was
used
(Weigand 1995).
In order
to
improve the quality
of
the
shadowgraph
images,
the
visualization
experiments
were
conducted
at
higher
Froude
M.
Ghavih and
A.
Weigand
FIGURE
3(a).
For
caption
see
facing
page
Vortex
tlisconnection
and
c'onnection
ut
a
fret
S~N~UCP
65
FIGURE
3.
Shadowgraph
images
of
thc
surface
deformation during the
vortex-ring
interaction
with
(a)
clean
surface,
(h)
contaminated
surface.
Re
=
5000
and
Fr
=
0.2
(direction
of
the
flow
is
from
left
to
right.
all
frames
are
the same
scale,
and
time
is indicated
in
each frame).
66
M.
Gharib
and
A.
Weigand
numbers
with
Fr
=
0.2.
However,
the
DPIV
measurements
were
conducted
with
the
aforementioned
Froude
number
of
Fr
zz
0.07
in
order
to
avoid
strong
surface
deformations.
As the
shadowgraph
sequence
in
figure
3(a)
indicates,
the
clean
interaction
is
initially
characterized
by
an
elliptical
surface
deformation
followed
by
the appearance
of
two distinct
and
round
surface depressions
that
enlarge
with
time.
As
was
reported
by
Bernal
&
Kwon
(1989),
later
stages
involve
the fold-up
and
splitting
of
the
lower
part
of
the
vortex
ring which results
in
the
formation
of
two
half-vortex
rings
that are
connected
to
the
surface. Similar
to
the observations
of
Bernal
&
Kwon,
the
symmetric
surface depressions
appear
to
be
associated with
the connection
of
the vortex ring,
i.e.
the
opening
of the upper portion
of
the vortex-ring.
In
this
paper,
we
focus
on
the
early
but
important
stages
of
the vortex-ring
connection. In
order
to
relate
these
observations
to
the connection
process, the evolution
of
the
velocity
and
vorticity
fields
as
the
surface
and
in planes
normal
to
it are
discussed
next.
3.3.
Free-surface
velocity
field
In
the
clean
and
semi-clean cases,
DPIV
provided
many
velocity fields of
the
vortex
ring
during the
course
of
its
interaction
with
the
free
surface.
This enabled
us
to
identify
some
important
milestones
in the
interaction
process
which
are
referred
to
as
stages.
Based
on
the
evolution
of
the
surface-velocity
field,
those stages
were
observed
to
be
common
to
both
clean
and
semi-clean cases.
However,
the reader should
be
cautioned
that,
in
the
case
of
extremely
contaminated
surfaces with
more than
5dyncm-'
difference in
surface tension, the vortex
connection
and
the
observed stages
might
not
occur
(Gharib
1994;
Willert
&
Gharib
1995).
An
example
of
such a
highly
contaminated
surface
is
shown
in
the shadowgraph
images of
figure
3
(b).
In
this
case,
the
observed stages
are
drastically
different
from
the
flow-visualization
results
of
the
clean
case.
In
comparison
to
figure
3(a)
the
most
obvious
difference evolves
during
the
final
stages
where
the
surface
connection
of
the
lower
part
of
the
vortex
ring
and
the formation
of
two
symmetric half-vortex
rings were
not
observed.
Stage
I
in figure 4
shows
the
initial surface
motion
induced
by
the
irrotational
velocity
field
of
the
vortex ring.
This
early stage
is characterized
by
the appearance
of
two stagnation
regions
(marked
by
S1
and
S2)
that
are
located
on
the
centreline
at
x1
=
8.3
cm
and
x,
z
6.1
cm,
respectively.
The
flow
in
the
centre
region accelerates
from
S1
in
the
negative
x-direction
and
decelerates
towards
S2.
Between
Stages
I
and
111,
the
surface-flow
pattern
resembles
that
of
a doublet. This
pattern
results
from the
oblique
approach
of
the
vortex ring
toward the
surface
and
is
a
manifestation
of
the
induced
velocity
field
of
the
small
upper
arc-portion
of
the
vortex ring. While
S2
moves
downstream towards
S1,
the two stagnation
regions
merge
and
form
a larger
region
with
an
overall reduction in
velocity
magnitude.
From
Stage
I11
to
IV,
a qualitative
change
in
the
flow
behaviour
occurs in
the form
of
a
flow
reversal
between
S1
and
S2.
The
flow
reversal changes
the doublet
into
a
dipole
pattern
by
Stage
IV.
Subsequently, a
strong
forward
current
on
the
centre-line
enhances
the dipole
pattern
which
results
in
the appearance
of
two
counter-rotating
elongated regions
at
Stage
V.
Stage
VI
shows
the
formation
of two
identifiable
circular
surface-flow
patterns
and
the completion
of
the connection
process.
Figure
5
shows
simultaneous
velocity-vector fields
for
the
clean
case,
obtained
in
the
plane
of
symmetry
z
=
0
and at
the
free
surface.
The
initial
formation
of
the
stagnation
regions
can
be clearly seen
to
correspond
to
the
sub-surface induced
velocity
field
of
the
approaching
tip
of
the
vortex
ring (figure
5cz-c).
However,
during
the
flow
reversal
Vortex
disconnection
and
connection
ut
u
free
.surface
67
3
1
t=43s
0:
s2.
-1
i
-2
3
I
2
o
cm
s-'
-3
8
9
10
11
5
61
8
9
10
11
5
6
1
3
2
I
h
-0
E
N
1
-I
-2
4
I11
(-41s
Y3
-1
1
1
-2
t
=
5.0
s
10
11
5
67
8
9
10
I1
5
61
8
9
1
12.0
cm
s
'
-3
2
1
0
-1
-2
-3
5
61
8
9
10
11
9
10
I1
12
13
14
15
16
17
x
(cm)
x
(cm)
FIGURE
4.
Surface
velocity
fields
due
to
the interaction
of
the
vortex
ring
with
a clean surface.