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OCR for page 479
Vortex Ring Interaction with a Free Surface
M. Song, N. Kachman, I. Kwon, L. Bernal, G. Tryggvason
(The University of Michigan, USA)
ABSTRACT
The results of numerical and experimental studies on
the interaction of vortex rings with a free surface are
presented. New results are reported on the interaction of a
large vortex ring with a clean surface at normal incidence.
The early stages of the interaction are well described by a
simple axisymmetric vortex filament model. Transition to
a fully three-dimensional state is observed at later stages of
the interaction. Surface waves are generated at high
Froude numbers by these three-dimensional motions.
Results are also presented on the interaction of vortex
rings with clean and contaminated free surfaces at inclined
incidence. The phenomenon of vortex lines breaking and
attachment to the free surface is documented. It is shown
that small amounts of surface active agents greatly alter the
interaction at inclined incidence. The effect differ
depending on the local topology of the vortex lines. A
Reynolds ridge and secondary vorticity generation are
observed during the interaction with a contaminated
surface.
INTRODUCTION
The disturbance on a free surface created by a
moving ship is composed of several superimposed and
sometimes interacting phenomena. The most dramatic and
best understood is the surface wave pattern generated,
generally referred to as the Kelvin wake. Not only is the
Kelvin wake the more visible mark left by the ship, it
contributes also significantly to the drag of the ship, and is
therefore of direct economic significance. Although it is,
of course, well known that the ship also has a large
viscous wake that is turbulent and can last for a long time,
traditionally the turbulent wake has only been of interest as
it directly relates to the drag of the ship, and in most such
considerations the effect of the free surface can be
neglected. Furthermore, for the purpose of drag estimation
the focus is on the turbulent wake near the ship. It is only
recently that the far wake of the ship has generated
interest. The motivation is remote sensing technology. In
order to process the signal and to determine what is being
detected, as well as to be able to reduce the detectability of
ships, it is necessary to understand the detailed
mechanisms of generation of surface signatures of ship
wakes.
The surface signature of unsteady vertical motions
below a free surface has recently been the subject of
several investigations. Sarpkaya & Henderson ~
experimented with a delta wing moved below a free
surface. The wing was set at a negative angle of attack, so
that the trailing vortices moved upward to the free surface.
As the vortices approached the surface a pair of long and
narrow marks were observed on the free surface that
appear to be directly related to the trailing vortices. These
marks, called scars by Sarpkaya, are parallel to the
direction of motion and moved outward with the vortices.
The scars were accompanied by other features called
"striations", perpendicular to the line of motion. A
somewhat different setup, two-dimensional vortex pair
was investigated by Willmarth et al.2 and by Sarpkaya et
al.3 The surface signature of the pairs is similar to the
trailing vortices, but the mean motion is now strictly two-
dimensional. These experiments were motivated primarily
by a desire to understand the surface signature of ship
wakes, and the focus was mainly on the large scale
motion.
Several numerical studies of this problem have
followed the experimental work. These have mostly
assumed an inviscid, two-dimensional motion.
Tryggvason4 presents a brief numerical study of surface
deformation due to the roll-up of a submerged vortex sheet
using a boundary integral/vortex method. Sarpkaya et ala
and TelsteS use a similar technique to follow the motion of
a vortex pair toward a free surface. A finite difference
simulation of the point vortex problem have been reported
by Marcus6 who also discusses linearized aspects of the
problem. More realistic vortex structure are used by
Willmarth et a[2 who simulate the formation of a vortex
pair from an initially flat vortex sheet and the subsequent
vortex motion and free surface deformation. A brief
479
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comparison of experimental and computational results is
contained in Reference 2. A thorough discussion of both
vortex collision as well as the formation of vortices from a
shear layer, and the resulting surface signature, is given
by Yu and Tryggvason7, who simulated a large number of
cases, and, in particular, explored the limits of high and
low Froude numbers.
It should be noted that the signal from the ocean
surface that is received by remote radar sensors is directly
related to the presence of relatively short waves. It is
therefore the surface roughness that is observed directly.
Generally the large features of more interest such as
changes in the ocean depth or currents and the wakes of
ships can only be inferred through their modulation of the
free surface roughness. For interpretation of these data it
is therefore necessary to understand the interplay between
the large and small scale features of the flow and their role
in the generation of short surface waves. An example of
such interplay is the relation between the scars and
striations in the vortex wake problem studied by Sarpkaya
and coworkersl'3 and more recently by Hirsa.8 Also of
primary interest in this regard is the role of smaller scale
motions in the turbulent wake in the generation of short
waves.
To examine the surface response to a turbulent
subsurface flow Bernal and Madnia9 studied a turbulent jet
parallel to the surface located a few diameters from the
surface. Near the jet exit, vortex rings are generated that
produce some surface deformations and waves, but the
most dramatic signature is produced when these vortex
rings open and reconnect with the free surface. To explore
this mechanism in more detail, Bernal and Kwon1O and
Kwonll experimented with a single ring moving parallel
and at inclined incidence relative to the surface,
respectively.
The above discussed investigations were not
concerned with the effects of surface contaminants as
such, but it appears that some of those results were
influenced by the fact that a free surface is hardly ever
hydrodynamically clean. Earlier, Daviesl2 discussed the
damping of turbulent eddies at a free surface, and Davies
and Driscolll3 experimented with ejecting pulses of
colored water to a free surface, specifically addressing the
rate of surface renewal and the effect of surface
contamination. They found that the spreading of the
colored water at the surface is reduced considerably for
contaminated surfaces. However, their visualization
technique did not allow for a clear explanation of the
mechanism responsible for this behavior. Experiments on
the collision of two-dimensional vortex pairs with a free
surface were reported by Barker and Crowl4 whose main
interest was in vortex collision with a rigid surface. The
motivation for their experiments was the observed
rebounding of aircraft trailing vortices from rigid surfaces.
This rebounding of a vortex pair from a solid surface is
due to the separation of the ground boundary layer and
subsequent formation of secondary vortices. Therefore it
is not expected that rebounding will occur if the rigid
surface is replaced by a stress free surface. However,
Barker and Crowl4 observed rebounding in their free
surface experiments, just as the rigid surface case, and
suggested that this rebounding might be due to inviscid
effects effects such as the deformation of the vortex cores.
Saffmanis refuted this suggestion, and showed that for
inviscid flow and a flat boundary rebounding can not
occur. He suggested that the behavior might be due to
surface tension effects. Peace and Rileyl6 performed
numerical simulations of the Navier-Stokes equations for a
two-dimensional vortex pair colliding with a flat no-slip
and stress-free surface, and concluded that even for a
stress-free boundary viscous effects could cause
rebounding. However, even though their calculations
clearly show rebounding, those are for rather low
Reynolds numbers, and with increasing Reynolds
number, the rebounding decreased significantly. Their
results can therefore not explain the behavior in the Baker
and Chow experiments, which were conducted at a much
higher Reynolds number.
The explanation for rebounding from a free surface
is clear from recent experiments by Bernal, Hirsa, Kwon
and Willmarthl7 who investigated the collision of both
vortex rings and pairs with a free surface. They observed
that the cleanness of the surface lead to considerable
differences in the vortex motion itself. For very clean
surfaces sufficiently weak vortices were deflected outward
in a manner similar to what inviscid theory predicts (if the
surface deforms some rebounding is predicted but most
experiments have been conducted under conditions where
surface deformation is minimal), but for contaminated
surfaces the behavior was more like vortices encountering
a rigid wall where secondary vorticity from the wall
boundary layer is pulled away by the primary vortex that
then rebounds as a result of its interaction with the wall
vorticity. Detailed observations using Laser Induced
Fluorescence (LIF) flow visualization lead Bernal et all7
to conclude that the surface motion induced by the vortex
generated an uneven distribution of contaminant that in
turn caused shear stress at the surface, generating
secondary vorticity. This vorticity rolls-up into a
secondary vortex which results in the rebounding of the
primary vortex. This generation and roll-up of secondary
vorticity and its subsequent interaction with the primary
vortex appears to be the leading effect of the surface
contaminants.
Although rebound can usually be associated with
viscous effects Dahm, Scheil and Tryggvasonl8 have
shown that a weak vortex colliding with a weak density
interface can engulf a portion of the interface containing
baroclinically generated vorticity which then causes the
primary vortex to rebound in completely inviscid
simulations. Yu and Tryggvason7 also show that a
deformable surface can lead to rebounding. However this
occurs at much higher Froude number than in the
experiments.
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Observations of contaminated free surfaces have
been reported on numerous occasions for over a century.
One important phenomenon is the Reynolds ridge which
appears on the boundary between the contaminated and
clean surface regions when the surface flow is stopped by
a barrier. This flow configuration develops when vortices
collide with a free surface as shown by Hirsa8. The
upwelling generated by the vortices pushes the
contaminated surface water to the side, thereby
compressing the contaminated layer. The surface above
the vortices is cleaner and is separated from the
contaminated surface by a Reynolds ridge. For a
Thorough discussion of the Reynolds ridge with historical
perspective see Scottl9. We should note that the
occurrence of a Reynolds ridge, although often observed
when separation takes place, is not directly related to the
generation of secondary vortices. Indeed, a Reynolds
ridge is easily generated in the absence of separation (see
Scottl9) and separation can take place without the
formation of a Reynolds ridge.
In what follows an overview of recent results on the
interaction of vortex rings with the free surface is
presented. The flow geometry is shown schematically in
Figure 1. Results on the interaction at normal incidence,
Figure la, are presented first. Next the results of Kwon1 1
for inclined incidence are briefly reviewed. Finally the
results of recent experiments on the effect of surface
contamination on the interaction at inclined incidence are
presented and discussed.
INTERACTION AT NORMAL INCIDENCE
The interaction of a vortex ring with the free surface
at normal incidence is perhaps the simplest flow geometry
involving the interaction of a vertical flow with the free
surface. This type of interaction has been studied in detail
in a recent experimental and numerical investigation by
Song, Bernal and Tryggvason.20 One objective of the
investigation was to study the interaction at a scale
substantially larger than previous experiments which were
conducted in small water tank facilities.l°~ll
A large vortex ring generator with a nozzle exit
diameter of 10 cm was used in this study. The general
design and operating characteristics are similar to the
smaller scale vortex ring generators. The experiments
were conducted in Tow Tank facility at the
Hydrodynamics Laboratories of the University of
Michigan. The water surface was cleaned by a continuous
surface current. The current was interrupted and the flow
motion allowed to dissipate before each vortex ring test.
Because of these precautions the water surface is believed
to have been free from contaminants on these tests.
Hot film velocity measurements along the axis of the
flow were used to determine the vortex ring formation
characteristics. The flow field during the interaction was
LLl
(a)
ma_
Figure 1. Schematic diagram of flow geometry.
a) Normal incidence. b) Inclined incidence.
characterized by flow visualization of the underwater flow
using the Hydrogen bubble technique and of the free
surface using the shadowgraph technique. The surface
signature during the interaction was also characterized by
measurements of the free surface elevation using a
capacitance probe.
The numerical simulations were conducted using a
vortex/bounda~y-integral method. A single vortex filament
with finite core size was found adequate for the objectives
of the study. Comparison with experiments was made by
adjusting the vortex ring circulation and core parameter to
obtain the same circulation and initial propagation speed as
in the tests. For additional details on the experimental
setup and numerical simulations the reader is referred to
Song et al.20
Flow visualization
Typical flow visualization results of the interaction at
normal incidence are shown in Figure 2. Figure 2a are
visualizations of the surface deformation and underwater
flow at a Froude number F/(g R3)1l2 = 0.252 where His
the circulation of the vortex ring, g is the gravitational
acceleration and R is the radius of the ring before
interaction. The corresponding Reynolds number was
F/v=15,000. Figure 2b are similar visualizations at a
481
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(a)
(b)
Figure 2. Flow visualization of the interaction of a vortex ring with a free surface at normal
Incidence. Top is a shadowgraph image of the surface, Bottom, underwater flow
visualized by hydrogen bubbles. (a) Froude number 0.252. (b) Froude number
Froude number of 0.988. The Reynolds number was
64,700. In both cases it is shown the shadowgraph image
of the surface on top and the side view image obtained
using the hydrogen bubble technique at the bottom. Both
images were obtained on the same realization of the flow.
The bubbles in the side view pictures tend to migrate
towards the core of the vortices due to their low density.
Note that on the side view the mirror image of the
underwater flow is observed caused by total reflection of
scattered light on the water surface.
The flow visualization results at low Froude
number, Figure 2a, show a highly coherent axisymmetric
surface pattern. The side view picture also shows a
coherent axisymmetric core. The surface signature
consists of a dark band bounded by two bright regions
these features indicate a local depression of the surface.
This surface depression is located above the vortex core
and moves with it as it stretches outward due to the
velocity field induced by the image vorticity above the
surface. There were no surface waves generated during
this process. At very low Froude number these features
were observed during the entire interaction process.
482
Figure 2b shows flow visualization pictures at a
higher Froude number as indicated. The shadowgraph
visualization of the surface shows small scale three-
dimensional structures superposed on a axisymmetric dark
band associated with the core. Surface waves are
generated by these small scale three-dimensional motions.
These waves coalesce to form an axisymmetric wave front
propagating away from the interaction region. The
visualization of the core in the side view picture show
small scale three-dimensional distortions of the core
associated with the surface features. The development of
the small scale three-dimensional features was found to
occur rapidly, through an instability process. Before the
instability the surface signature and general flow
appearance was similar to the case shown in Figure 2a.
After transition the flow features discussed in relation to
Figure 2b appeared.
OCR for page 483
Surface signature and subsurface flow
The measured and calculated trajectories of the
vortex cores at a Froude number of 0.25S and 0.988 are
shown in Figure 3. The measured and calculated results
are in good agreement. There is no rebounding of the
vortex core in the low Froude number case. This is
expected since contaminants were not allowed to
accumulate on the surface. At the larger Froude number
the calculations show a small rebounding of the core. The
measurements on the other hand are not accurate enough
to confirm this result. An interesting observation shown
by the data in Figure 3 is that the vortices attain a different
final depth after the interaction for the two cases
presented. Studies of many numerical simulations of the
flow revealed that the final depth of the vortex core is
controlled primarily by the core size parameter. For the
cases shown in Figure 3 the core size parameter Rle,
where e is the core radius, was 4.9 for the low Froude
number case and 2.7 for the high Froude number case.
~ .o
o.o
-1 .0
z/R
-2 .0
-3.0
-4.0
-~.0
~ o
Ids- _
pa
d.
l . 1 , 1 1 1 , 1
0.0 1.0 2.0 3.0
r/R
4.0 5.0 6.0
Figure 3. Vortex core trajectories for normal incidence.
Froude number 0.252: Solid line calculated
results, circles measurements. Froude number
0.988: Dashed line calculated results, squares
measurements.
Several parameters can be used to characterize the
surface signature during the interaction of a vortex ring
with the free surface. Perhaps the simplest measure of the
strength of the interaction is the surface elevation at the
center. Figure 4 is a plot of the normalized maximum
elevation at the center, hlR, as a function of Froude
number, F/(g R3J]/, for all the cases tested in the
h/R
.1
1
r/(g R3 )1/2
.01
/
laboratory. The straight line in this plot has slope 2, the
expected scaling behavior derived from considerations of
momentum balance of the vertical flow and the surface
deformation.
.001 ~ 1._ ~ _d A ~ _
10
Figure 4. Normalized maximum surface elevation at the
centerline as a function of Froude number. Only
experimental results are presented.
The surface signature during the interaction was
obtained from the numerical simulations. Figure 5 is a plot
of the instantaneous surface shape for the case Fr= 0.252
at nondimensional time tF/R2 = 14.16. Note that the
vertical axis is stretched by a factor of ten compared to the
horizontal axis. The surface features in this plot are in
good qualitative agreement with the shadowgraph flow
visualization picture of the surface shown in Figure 2a.
The dark band in the photograph corresponds to the
surface depression at r/R a 3.0. The bright regions next to
the dark band in the photograph correspond to the surface
rise at either side of the depression.
More detailed comparisons between the measured
and calculated results is presented in Figures 6 and 7 for
Froude numbers of 0.252 and 0.988 respectively. In both
Figures we plot various measures of the position of the
vortices and associated surface features as a function of
time. The solid lines are the calculated location of the
vortex core. The open circles are the measured location of
the cores determined from the hydrogen bubbles flow
visualization. The broken line is the radial location of the
maximum surface depression from the numerical
simulations. The cross symbols are the measured location
of the dark band in the shadowgraph images.
483
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0.3
0.2 _
0.1 _
z/R
o.o
-o. 1
-0.2
-0 . 3
-0.4 _
-0.5 , I . 1 , I , I , I
0.0 1.0 2.0 3.0 4.0 5.0 6.0
r/R
Figure 5. Calculated free surface shape for Froude
number 0.252.
The results at low Froude number, Figure 6, show
good agreement between the numerical calculation and the
measurements. The agreement is excellent for the distance
6.0
4.0
r/R
to the free surface. The calculation show that the location
of the maximum depression moves outwards ahead of the
core. The calculated speed of this outward motion is in
good agreement with the measured value. Although there
are discrepancies in the absolute location of the core. At
Later times, however, (~/R2 ~ 15) there appears to be a
change in the measured core trajectory.
The results at high Froude number, Figure 7, show
significant discrepancies between the calculated and
measured results. Again there is excellent agreement
between the calculations and the measurements for the
distance to the free surface. The evolution of the radial
location of the core is in good agreement up to tFIR2~ 12.
After that time the measurements show a slower outward
motion of the vortex core compared to the calculations.
The reason for this discrepancy is the three-dimensional
instability observed in the flow visualization study. Before
the instability the outward motion of the core results from
the induced velocity field caused by the image vorticity
above the free surface. This induced motion is well
characterized by the axisymmetric vortex filament model
used in the numerical calculations. As three-dimensional
motions develop after the instability, vortex lines break
and connect to the free surface, a phenomenon discussed
in more detail later, thus destroying the axial coherence of
the vorticity distribution. These vortex patches attached to
the free surface determine the speed of the outward motion
of the core after the three-dimensional instability.
A/
i+++
/+~+
+
r/R
6.0
4.0 _
2.0 r ~1 2.0 _
~o~
0.0
-2 .0
-4.0
,/
To
l I · I , 1
0.0 10.0 20.0 30.0 40.0
rUR2
Figure 6. Temporal evolution of various flow parameters
at a Froude number of 0.252. Solid line
calculated core location. Dashed line radial
location of minimum surface depression.
Circles measured core location. Crosses radial
location of dark band.
0.0
-2 .0
z/R
+ ~
_--P
a/
,~
~+++
lo:- 40! + +
~ O ~ ~ 00
it
-4.0 .
0.0 10.0
I . I . 1 . 1
20.0 30.0 40.0
r dR2
Figure 7. Temporal evolution of various flow parameters
at a Froude number of 0.988. For key to the
symbols see caption of Figure 6.
484
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INTERACTION AT INCLINED INC1:DENCE
Investigations on the interaction of a turbulent jet
with the free surfaced revealed the development of
characteristics features on the free surface consisting of
dark dimples associated with vortex lines terminating at
the free surface. It was also observed that vertical
structures in the near field of the jet were directly
responsible for the formation of the dimples. Bernal and
Kwon10 experimented with vortex rings moving parallel
1~
to the surface. These experiments demonstrated that
surface dimples result from vortex lines breaking and
reconnecting with the free surface. To further examine this
phenomenon a systematic investigation on the interaction
of a vortex ring, at an inclined incidence to the free
surface, was undertaken (see Figure lb).
_
_
(a) (b) (c)
Figure 8. Flow visualization pictures of the interaction of a vortex ring with a clean free
surface at a Reynolds number of S,OOO and incidence angle of 20°.
Shadowgraph visualizations of the surface are shown on top and Cross-section
views at the bottom. In each photograph the vortex motion is from right to left.
(a) rtlR2 = 18. (b) rtlR2 = 26. (c) rtlR2 = 43.
485
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In considering these flow processes at a free surface
the role of surface contamination is an important
parameter. The flow visualization results by Bernal,
Hirsa, Kwon and Willmarthl7 showed that vorticity
generation at the free surface can alter the evolution of the
subsurface flow. The study of this interaction in the more
complicated case of inclined incidence was another
motivation for the present study.
The experiments were conducted in the free surface
water tank described by Bernal and Madnia9. Several
modifications were added to help control contamination of
the free surface. A stand-up drain pipe was installed, and a
continuous current of water was allowed to flow into the
facility to remove surface contaminants before they could
have accumulated on the free surface. During the actual
tests the water current was temporarily halted and the
remaining turbulence in the tank was allowed to dissipate
before generating the vortex rings. This procedure resulted
in highly reproducible results for a clean free surface.
A vortex ring generator having a 2.5 cm nozzle exit
diameter was used in all the experiments at inclined
incidence. A short duration water pulse is allowed to flow
out of the generator to produce the vortex rings. A
pressurized tank and a solenoid valve are used to control
the speed and duration of the water pulse. A detailed study
was conducted to determine the operating characteristics of
the vortex ring generator. These as well other information
on the operation of the vortex ring generator can be found
in Kwonll. The flow field and surface signature was
documented by Laser Induced Fluorescence (LIF) flow
visualization of the symmetry plane and by shadowgraph
visualization of the free surface.
Interaction with a clean free surface
Shown in Figure 8 are flow visualization pictures of
the interaction at an incidence angle of 20°. On top are the
shadowgraph images of the free surfaces, at the bottom
are LIP cross-sections through the symmetry plane. The
vortex ring motion is from left to right. The flow
conditions were Reynolds number Fly = 5,000 and the
Froude number Fl~gR3~1/2 =0.81. These photographs
show the three stages of the interaction process. The first
photographs at l~tlR2 = 18, Figure 8a, show the upper
vortex core interacting with the free surface which causes
a surface depression. The second set of photographs
obtained at l~tlR2= 26, Figure 8b, show a single vortex
core in the cross-section while the surface signature shows
two dimples on the surface at either side of the symmetry
plane. These features indicate that the vortex lines in the
ring have opened-up and are now attached to the surface
at the dimples. The shadowgraph image also shows that
surface waves were generated during the vortex line
breaking and reconnection process. At a later time (Figure
8c, FtlR2= 43) the flow visualization of the cross-section
shows little evidence of the vortex ring core while the
surface shadowgraph shows four dimples on the surface.
The top pair propagates away from the lower pair in time.
Each individual pair of dimples represents the surface
signature of a half vortex ring with vortex lines beginning
and terminating at the free surface.
Flow visualization studies were conducted to
determine the effect of Reynolds number and incidence
angle on the interaction. The results are summarized in
Figure 9. Vortex line breaking and reconnection of the top
and bottom parts of the ring were found for a large range
of Reynolds numbers (from 2,000 to 7,000) and incidence
angles in the range between 10° and 30°. At a Reynolds
number of 5,000 the vortex line breaking and reconnection
of the lower part of the core was observed for angles up to
45°. Small variations were observed within this range of
parameters in the sense that the half-vortex rings would
propagate at different angles after the interaction, and in
some cases they will move along converging paths,
interact with each other again and form a single open ring.
~ Lower Core Vortex Breakdown
em Lower Core Vortex Reconnection
50
40
30
a
20
10
1
l
1] 1111] 1~ i I
2 4 6 8 10
F/v (x10-3)
Figure 9. Observed interaction outcome as a function of
vortex ring Reynolds number and incidence
angle.
For vortex rings formed at conditions outside this
region, the vortex lines of the upper core break and
reconnect to the surface as in the other cases. But when
the bottom core reached the surface the vertical region
broke down into smaller scale vortical structures.
486
OCR for page 487
The vortex line breaking and reconnection process
can be quantified be a characteristic time, tr. This
reconnection time was defined in the experiments as the
elapsed time between the time when the vortex outline in
the LIF image first reaches the surface and the time when
the dimples on the surface are first observed.
Measurement were conducted of the reconnection time of
the upper vortex core at several conditions. The results are
shown in Figure 10 where the reconnection time
nondimensionalized by the circulation and the core
diameter, 8, is plotted as a function of the Reynolds
number for different incidence angles. The results show
that the normalized reconnection time is independent of the
Reynolds number, suggesting that the breaking of vortex
lines is by and large an inviscid process. There is a
systematic reduction of the normalized reconnection time
as the incidence angle is increased suggesting a strong
dependence on the details of the vortex line topology as
they approach the surface.
20
s
. . . .
r tr/ I; 2 ~ ~ a - 30°
10~ ~~c`=45'~
u
1000 r/v
Figure 10. Effect of vortex ring Reynolds number and
incidence angle on vortex line reconnection
~ ~i]
'
~ .~: ~ ~ ~ ~ ~ ~ ~]
_,:
(a)
EM
. ~,., ~, ~, ,.,.~.,.,., ,. ~_
_ ~ ~ ~'1~
~ ~""~ ~ ~ ~ ~ ~
E'_ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
_
~ 11~
. ~ ~ ~
· __
'.. ~ ~ ~ _
<."""-' ~ ~ ~ ~ ~ ~'~
'' ~ ~ ~_
(C)
Figure 11. Flow visualization photographs of the interaction of a vortex ring with a
contaminated surface at a Reynolds number of 5,000, incidence angle of 20°
and surface pressure of 12.7 dynes/cm. Shadowgraph visualizations of the
surface are shown on top and Cross-section views at the bottom. In each
photograph the vortex motion is from left to right. (a) I~tIR2 = 42. (b) 1~tIR2 =
53. (c) Ft/R2 = 65.
487
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The flow visualization at a nondimensional time
Ft/R= 53 shown in Figure 1 lb reveals the formation of a
pair of counter rotating vortices trailing behind the lower
core. This small pair of vortices is the remanence of the
upper vortex core and the opposite sign vorticity generated
by surface tension effects at the contaminated surface. The
shadowgraph image of the surface shows the dimples due
to the reconnection of the top vortex core with the surface
and the waves generated in the process of reconnection.
The shadowgraph image also shows a distinct Reynolds
ridge moving ahead of the reconnected vortex cores.
At a later time, l~t/R2 = 65 Figure lie, the cross
section view still shows evidence of the counter rotating
vortex pair at the surface which has now been entrained by
the lower vortex core. Also a large secondary vortex has
formed ahead of the lower vortex core. The induced
velocity field of the secondary vortex tends to stop the
motion of the lower vortex core towards the surface. The
surface signature in this case show well defined circular
dimples as well as a highly distorted Reynolds ridge.
These flow visualization results illustrate the effect
that surface active agents have on the dynamics of vorticity
near a free surface. The effect vary depending on the local
details of the flow. For the upper core the surfactant does
not suppress the vortex line breaking and reconnection
process. This is consistent with the observed Reynolds
number independence of the reconnection time. In contrast
the lower core dynamics is strongly influenced by the
presence of surface active agents. As the lower core region
approaches the free surface secondary vorticity generated
at the surface by surfactant action accumulates in a
secondary vortex. The velocity field induced by the
secondary vortex causes rebounding of the core and
prevents vortex line breaking at the surface.
ACKNOWLEDGEMENTS
This research was sponsored by the Office of Naval
Research, Contract no. N000184-86-K-0684 under the
U.R.I. Program for Ship Hydrodynamics.
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W.W., "On the interaction of vortex ring and pairs with a
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DISCUSSION
Daniel Marcus
Lawrence Livermore National Laboratory, USA
What is the effect of changing the ratio of core radius to ring radius
on the reconnection event?
AUTHORS' REPLY
We have only limited amount of data on the effect of core radius or
ring radius on the reconnection event. The data presented in Figure
10 shows that the vortex reconnection time when nondimensionalized
by core parameters is approximately independent of the Reynolds
number. For these data, the change in Reynolds number is
accompanied by a change of the ratio of core radius to ring radius.
Kwon (1989) estimates the values of this parameter for the cases
plotted in Figure 10. The values changed from a minimum value of
0.23 at the low Reynolds number case to a maximum value of 0.39
at the largest Reynolds number case. On the basis of this evidence,
it appears that the nondimensional reconnection time is also
independent of the ratio of core radius to ring radius.
DISCUSSION
Theodore Y. Wu
California Institute of Technology, USA
The authors deserve our warm thanks for the very enlightening report
on their investigations of this fascinating subject. In view of
Helmholz's theorem that no vortex filaments can terminate in the
midst of a fluid (or fluids), I wonder if the authors have investigated
the vertical flow induced in the air by the submerged vortices that
have opened up at the interface. I further wonder if such more
complete vortex systems may have a bearing on the secondary
vortices sequentially generated in the course of the vortex bifurcation.
AUTHORS' REPLY
We would like to thank Professor Wu for his kind remarks. Yes,
Helmholz's theorem requires that vortex lines with a component
normal to the free surface continue in the air above the free surface.
We have not attempted to measure or in any way characterize the air
motion above the free surface during the interaction. Elucidation of
the topology of the vortex lines in air is a challenging problem. The
Reynolds number for the air flow is one order of magnitude lower
than for the water flow and consequently the air vortices are expected
to dissapate more rapidly. In any case, these induced vortices in air
or other weak air flow disturbances above the surface do not
significantly influence the vortex reconnection process because of the
large density ratio. Free surface dynamics, including contamination
effects, dominate the interaction process.
DISCUSSION
Richard Yue
Massachusetts Institute of Technology, USA
In your work, the induced velocity on the surface is a known function
(of space and time) since the influence of the free surface on the
vortex pair is not considered. This is consistent with the assumption
of deep submergence. In this context it seem that the present
problem can be addressed by simply considering the dispersion
relationship in terms of the known surface advection which can be
applied directly to a wave spectrum.
AUTHORS' REPLY
Although the induced velocity on the surface is a known function of
space and time, a solution method based solely on the spectral
variation due to this known surface current requires an additional
assumption. This assumption is that the variation in surface current
be slowly varying in time and space with respect to the length and
time scales of the ambient waves. The ambient wavelengths here is
allowed to be of the same order of magnitude as the vortex pair
separation and depth. The problem is therefore transient in nature
rather than quasi-steady as would be assumed in the direct
modification of the dispersion relationship. The transient problem is
handled in a straightforward manner using the simulation technique.
In the limit of small waves under the influence of large scale vortices,
such as in the ship example at the end of the paper, the direct method
suggested will likely give reasonable agreement.
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Representative terms from entire chapter:
vortex ring
