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OCR for page 521
Measurement and Computations of Vortex Pair Interaction
with a Clean or Contaminated Free Surface
A. Hirsa, G. Tryggvason, I. Abdollahi-Alibeik, W.W. Willmarth
(The University of Michigan, USA)
Observations of surface deformations produced
by the interaction of trailing vortices with a free
surface, behind a submerged delta wing at
negative angle of attack, were described by
Sarpkaya and Henderson (1985~. The flow
field during the laminar interaction with the free
surface is studied experimentally using a pair of
vertically oriented, computer controlled,
counter rotating flaps to generate laminar vortex
pairs with the same Reynolds number and
Froude number as the trailing vortices.
Numerical computations in two-dimensions,
flow visualization (laser induced fluorescence
and shadowgraph) and particle image
velocimetry (PIV) with known surfactants on
the surface show that surface contamination has
a significant influence on the flow field and
surface motion during the interaction. New
vorticity produced on either side of the vortex
pair when the surface is contaminated initially
forms a Reynolds ridge, (the surface signature
at the leading edge of a subsurface boundary
layer), and then the new vorticity beneath the
contaminated surface rolls up to form secondary
vortices outboard of the original vortices and
with opposite sign. The secondary vortices
cause the original vortex pair to rebound from
the free surface.
INTRODUCTION
Synthetic aperture radar images made at high
altitudes show that a ship produces a long, narrow
signature on the surface in the wake that is detectable long
after the passage of the ship (Munk et al. 1987~. Since
the wake directly behind a ship contains vertical
structures produced by the hull and the propeller blades
which interact with the surface, it appears that an
explanation of the surface signature in the wake will
require a fundamental understanding of the interaction of
vorticity with the free surface.
When a pair of counter rotating vortices in an ideal
fluid approach a flat surface the velocity field due to the
image vortices will cause them to diverge (Lamb 1932~.
Since the motion of each vortex is a result of the velocity
induced by the other vortex as well as the image vortices,
the diverging vortices ultimately move along straight lines
parallel to the flat surface. When the viscosity of the fluid
A. Hirsa Department of Mechanical Engineering,
Mechanics, Rensselaer PolytechnicInstitute,Troy NY
is considered, the flow field will be different owing to
the diffusion of the initial vorticity and the additional no
slip boundary condition at the flat surface. The observed
surface features produced by the wake of a submerged
delta wing at a negative angle of attack, as originally
described by Sarpkaya and Henderson (1985), are
sketched in Figure 1.
Delta Wing
Trailing Vortices Propagating at a Negative
Toward the Free Surface and Angle of Attack
Carrying with them some Fluid
\ / ii~Vonex Sheet
\ ,~ /~ Roli-Up
\/
_~~
BY Z ~Striations
1< /; - \Scars Slightly Deformed
X ~c~ O Free Surface
.~ - ~~3-Dimensional Structures
including Vortex Lines
Terminatulg at the Surface
Figure 1 Surface deformations resulting from the trailing
vortices interacting with a free surface.
521
The subsurface flow field associated with many of the
surface features that they described are not yet
understood. There are also numerous other experimental
and theoretical studies of the interaction of vortex pairs
with solid and free surfaces. A survey of the literature for
many of these studies has been made by Hirsa (1990~.
From the literature it is clear that the interaction of
vorticity with a free surface is a very complex problem.
The flow may or may not be turbulent and one must
consider the Reynolds number. In addition, the
momentum of the vortex pair may be great enough to
cause vertical displacement of the free surface with
rebounding of the vortex pair during the interaction, as
Tryggvason has shown through numerical computations
[see Willmarth et al. (1989~] and one must consider the
Froude number. Finally, it has become apparent (as will
be demonstrated in this paper) that an understanding of
the interaction of vortex pairs with a free surface at low
Aeronautical Engineering and
12180 U.S.A.
G. Tryggvason and J. Abdollahi-Alibeik Department of Mechanical Engineering and
Applied Mechanics, The University of Michigan, Ann Arbor MI 48109 U.S.A.
W. W. Willmarth Department of Aerospace Engineering, The University of Michigan,
Ann Arbor MI 48109 U.S.A.
OCR for page 522
Froude and Reynolds numbers requires that one also
must consider the effect of a dimensionless number
representing the ratio of the viscous force tangential to the
free surface to the tangential surface force exerted by a
surface film with a gradient in concentration. When
surface contaminating films are not present on the free
surface (which in nature is a very rare condition) the
basic interaction, at low Froude numbers and for a
laminar flow, is approximately similar to the flow that
would be observed for an inviscid fluid with a flat
surface. However, clean surfaces are a rarity and
varying amounts of surface film contaminants are usually
present on the surface. When present, as will be shown,
surface contaminants can completely change the nature of
the flow field during the later stages of the interaction of
vortex pairs with the surface.
Initial experiments were made with a small delta
wing, similar to the one used by Sarpkaya and
Henderson (1985), which was towed beneath and
parallel to the surface at negative angle of attack. The
flow beneath the surface was observed using a
fluorescent dye (Fluorescein) injected at the trailing edge
of the wing and illuminated from below the surface by a
thin sheet of laser light normal to the surface and to the
towing direction. These flow visualization experiments
were in general agreement with the results described by
Sarpkaya and Henderson (1985~.
In addition to observations of the wake cross-
section, we observed the motion of the surface with the
aid of particles placed on the surface after the wing had
passed. We found that as the vortex pair approached the
surface the particles~on the center line of the wake were
swept to either side by the upwelling of fluid carried with
the vortex pair. We also discovered that the surface
motion was not consistent because the velocity and
amount of surface motion observed on either side of the
wake was dependent upon the type of particles used as
passive markers. In one case, punched circles (chaff)
from oiled, paper computer tape placed on the surface
after the wing were completely motionless. However, a
few pieces of chaff that were completely wet and had
sunk slightly below the surface were observed to be
moving outward from the wake center line while the drier
pieces of chaff just above them and floating on the surface
were stationary.
After these initial observations, the status of the
project was discussed with personnel of the Naval
Research Laboratory and the Office of Naval Research.
We were alerted to the possibility that surfactants could be
present in the Ann Arbor city water used to fill our small
towing tank. Jack Kaiser of the Naval Research
Laboratory offered to measure the surface tension of our
water and found severe surface contamination of the Ann
Arbor tap water. He suggested that we investigate
methods to clean the water in our tow tank water and
referred us to the papers of Scott (1975 and 19821. Scott
(1975) describes methods to prepare clean water for fluid
mechanical experiments with an uncontaminated surface.
In the other paper, Scott ( 1982), experimental
measurements are described of the boundary layer flow
beneath a contaminated surface in which the convection of
surface contaminants is blocked by a barrier which
penetrates below the surface. A steady state boundary
layer is formed as a result of a balance between viscous
shear forces in the viscous flow beneath the surface and
the force produced by gradients in the surface tension
which are a result of surface active agent concentration
gradients in the blocked film of surface active agent. At
the leading edge of the contaminated surface, where the
surface tension begins to decrease, there is a rapid
variation in height of the surface which appears as an
easily observable ridge. The ridge is formed when the
upstream flow, with high surface tension, first encounters
the contaminated surface which has a considerably lower
surface tension. The upstream fluid is rapidly decelerated
as a result of the upstream "pull" of the clean oncoming
flow with a rapid increase and then decrease in surface
elevation.
At this initial stage of the investigation it was
apparent that a study of the interaction of vorticity with a
free surface would require both experimental
measurements of the flow field and numerical
computations of the velocity, vorticity and displacement
of the fluid particles in the wake flow field. Experimental
measurements of the flow field in the wake of the delta
wing would require many measurements for a large
number of runs in the tow tank facility.
To reduce the time and labor required for the
experimental and numerical investigation of the wake
flow field, the flow in the wake was approximated by a
two-dimensional vortex pair propagating toward and
interacting with a free surface. In the experiments the
flow field of a laminar vortex pair was produced by a
vortex pair generator designed for this investigation. In
the numerical computations the time dependent, two-
dimensional flow resulting when two line vortices were
placed at an initial position beneath the surface was
calculated. The line vortices were approximated by two
blobs of vorticity, with opposite sign. Both a clean free
surface and a free surface contaminated by various known
amounts of surface active agents were studied in the
experiments and in the numerical computations.
In the paper we first present a description of the
flow visualization experiments and the quantitative
measurements that we have made which serve to outline
many fundamental aspects of the problem. This is
followed by a description of the numerical computations
for vortex blobs interacting with clean and contaminated
free surfaces at low Froude and Reynolds numbers. The
numerical computations allow the study, at little cost in
time and labor, of the effect of many different parameters
on the flow variables during the interaction of vorticity
with the free surface.
EXPERIMENTS
1) Vortex pair Generation
The vortex pair generator used for the experiments
was designed to produce a two-dimensional pair of
counter rotating vortices that propagate upward toward
the free surface. A pair of initially vertical flaps, see
Figure 2, which were driven by a computer controlled
stepping motor were rotated toward each other to produce
a pair of vortices without any measurable effect on the
free surface caused by the flap motion. The velocity
induced by the voriicity in the vortex pair causes the pair
to propagate upwards as the flaps close. The flow field
produced by the flaps was found to be uniform along the
span. This vortex generation scheme also produced little
interference between the flaps and the newly formed
vortex pair. As the vortices form, the inward rotation of
the flaps allows the vorticity to roll up and entrain
additional fluid as required for the formation of the Kelvin
oval associated with the vortex pair. The Kelvin oval is
the oval region bounded by a closed streamline around the
vortex pair which can be observed in frame of reference
moving with the vortex pair. The triangle formed by the
flays after they are closed had an apex angel of less than
74 . See Hirsa (1990) for further information on the
522
OCR for page 523
vortex pair generator.
: ~
Lmage Vortices
:~x Free Surface
1
h
Vortex Pair
A \
// \\
// \\
// \\
// \\
//\\
'of ~b~
Mylar Sheet
Kelvin Oval
Counter Rotating,
Stainless Steel Flap
/
Figure 2 Schematic of the vortex pair approaching a free
surface.
The dimensions of the vortex pair generator were
chosen so that the vortex pair would have a Reynolds
number, Re, and Froude number, Fr, matching the
vortex pair produced in the wake of the delta wing. The
Reynolds number of the vortex pair is defined by:
Re
r
v
where r is the circulation about one of the vortices and v
is the kinematic viscosity. The Froude number of the
vortex pair is defined by:
r
Fr-, (2)
where g is the gravitational acceleration and SO is the
vortex pair separation when the vortices are far beneath
the surface.
2) Flow Field of the Vortex Pair
Observations, using laser induced fluorescence
(LIF), of a cross-section of the flow field when vortex
pairs interact with the free surface were made for a
number of different strength vortex pairs generated with
the flaps and for various degrees of surface
contamination. The dye used was fluorescein which was
injected with a syringe fitted with a slender plastic tube
into the water between and around the flaps before flap
motion was initiated. The laser light sheet was less than 1
mm thick and illuminated the flow field above the flaps
from one side, see Hirsa (1990) for further information.
- As described in the introduction, our initial
measurements with the delta wing and a contaminated
surface demonstrated conclusively that the motion of the
surface was inhibited if the surface was contaminated.
During the experiments with the vortex pair generated
with the flaps the free surface with an area of
approximately one half square meter above the vortex
generator was cleaned for at least one hour by means of a
drain pipe and a fan blowing air toward the drain prior to
the first experiment each day. After each vortex pair was
generated, the free surface was cleaned first for a
minimum of 5 minutes by surface draining with the fan
on, then the fan was turned off and the surface drain was
continued for at least another 5 minutes. The surface
drain was then stopped and the free surface was
undisturbed for a minimum of 5 minutes to allow the
surface currents to completely decay before the next run.
To illustrate the typical flow phenomena observed
for low Froude number vortex pairs of various strengths
a set of (LIF) photographs of a the flow produced for a
typical vortex pair (Re=12,400 and Fr=0.217)
interacting with the free surface with and without
surfactant added on the surface is shown in Figure 3.
The photographs on the left, labeled (A), are for a
relatively clean free surface with no oleyl alcohol added.
The photographs on the right, labeled (B), are for an
initially clean free surface with 1.06 x 10-7 (cm3/cm2) of
oleyl alcohol spread on the surface before the vortex pairs
were generated. Hirsa (1990) contains a complete
description of the method used to determine the surface
concentration of the oleyl alcohol . From the
concentration and using the state relationship for oleyl
alcohol, see Hirsa (1990) Figure 4.5, the initial surface
pressure ~ - 60 - 6 (the difference between the surface
tension, GO, of the clean and the surface tension, 6, of
the contaminated free surface was approximately 2.5
dynes/cm).
For the vortex pairs shown in Figure 3, the tips
of the flaps (when vertical) were at a depth of 21 (cm).
The average velocity and spacing of the vortices when the
pairs were well below the surface was vp0 = 2.36
(cm/see) and SO = 6.93 (cm/sec). A dimensionless time
I* was defined as:
:* = ' 'i (3)
60 / vp O
3 where t is the time since the start of the flap motion and,
t1 = 7.55 (sec), is the time elapsed from the start of the
523
OCR for page 524
flap motion until the time the vortices are at a depth equal
to the vortex pair separation after roll-up. Thus, SO /
vpO, is the time it takes the vortex pair to propagate a
distance equal to their separation, when they are far from
the free surface. The circulation, A, and the Froude
number, Fr, for the vortex pairs produced in the
experiments was estimated using an inviscid point vortex
model. The model treats each of the vortices as a point
vortex and consists of two point vortices and the two
image vortices associated with the original vortices when
they are in the proximity of a free surface. Lamb (~932)
describes the trajectory of a vortex pair approaching a flat
wall in inviscid fluid. Using the trajectory equations and
observations of the position, spacing and velocity of the
vortex pairs, the apparent circulation of the vortex pairs
was estimated. This is equivalent to assuming an inviscid
vortex pair with Fr=O, approaching a free surface.
~ ~ ~ PA ~ ; ~:~ ~ ~ (x-~`x*~ Surfs ~ ~ ~ ~ ~ (~3 ~ ~
~- . ~ ~it. I
:_ _ it:
~:~ ~ ~ ~:~ ~ ~ ~
~(i) em
'I
__ I,
:: - :~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
::::::: ~ _ _ t~
:~:(j^,) ~
'I ~ ~'~
it__
:~::: i: : :: ~ ~ i: ~ ~
~ ~:~ _ ~
~ ~11 ~ on
~:~ ~ ~2 ~
~ ~ n
~ ~ (iY) ~
~ ~ ~ ~ __ ~ ~ '__ ~ ~
~ :~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~-7 0
~ ~ : ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ _ _ ~ ~ ~
: ~ :__ ~ ~ __ ~
~ :1_ (V) ~
__' ~ i: ~
: _ ~ ~ -__
:- i: :~ ~ .: ~ ~ ~ :
: ~ :: :~ :: it: it: :~ : i: i: ~ ::~ :: ::::: ~ :: : : :~ :~ ~ ~ ~ ~ ~ ~ ~ ~ : ~ ::
_~e
b ~ __
~:~ _
Figure 3 LIF photograph of a vortex pair interacting
with a free surface with: (A) no surfactant added & (B)
with oleyl alcohol added with surface pressure of 2.5
(dynes/cm); initial vortex pair separation bo=6.93 (cm)
and propagation speed vpo=2.36 (cm/sec), Re=12,400
and Fr=0.217 (scale: 1 cm on the photograph = 8.4 cm in
the flow field). The dimensionless times are: (i) I* =
0.59, (ii) I* =0.93, (iii)* = 1.27, (iv)* = 1.61, (v)
I* = 1.95, and (vi) I* =2.29.
Figure 3 (i) shows the vortex pair approaching the
free surface. The dimensionless time I* is 0.59. The
vortices have started to move apart and no difference is
apparent between the case without oleyl alcohol (A), and
the case with oleyl alcohol (B). The subsequent
photographs, (ii) to (vi), show the vortex pair at equal I*
intervals of 0.34. The mirror image that appears on the
upper portion of each photograph is due to total internal
reflection at the free surface. Location of the free surface
was determined in each photograph by drawing a bisector
through the symmetrical image.
There is little visible difference in the flow field
between the case (A), and case (B) up to the time
~*=1.61. At ~*=1.61, Figure 3 (iv) shows a marked
difference between the two cases. In the case where oleyl
alcohol is added, the streamlines near the free surface
appear to diverge from the free surface whereas in the
case with no oleyl alcohol the streamlines near the free
surface stay close to the surface. By ~*=1.95, Figure 3
(v) shows that a secondary vortex is being formed for the
case (B), but in case (A) no secondary vorticity is
formed. In the last Figure 3 (vi), a pair of secondary
vortices are visible in the case (B) with oleyl alcohol
which drastically alter the path of the primary vortices as
they begin to rebound from the free surface. The case
without oleyl alcohol shows no sign of secondary
vorticity formation or rebounding of the primary vortices.
Free Surface
Secondary Vortices V
, ,_
,~-'
~` \.\
1 1 1
--Free Surface - no surfactant added
- " w/ Oleyl alcohol, ~0.3
- 1.1
1 .. ·1 x=9.0
-2.5
.... .......
Solid Wall
x/bO
Figure 4 Average trajectory of the apparent center of the
(right) vortex during the interaction of the vortex pair with
a solid wall and a free surface with various amounts of
oleyl alcohol whose surface pressure is denoted by it.
Re=12,400 and Fr=0.217. A tick mark on the primary
vortex trajectory indicates the position of the primary
vortex at the beginning of the secondary vortex trajectory.
The time at the beginning and the end of each primary and
secondary vortex trajectory is as follows:
no surfactant; -1.55 5 I* < 2.19, not formed:
= 0.3; -1.55 S I* < 2.19, 1.85 < I* < 2.19:
= 1.1; -1.55 < I* 5 1.85, 1.51 S I* S 1.85:
= 9.0; -1.55 S I* 5 1.51, Turbulent:
Solid Wall, -1.55 S I* 5 1.85; 1.17 5 I* 5 1.85.
524
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3) Vortex Pair Trajectories
Figure 4 shows the trajectories of the apparent
center of the right vortex for various surface conditions
for vortex pairs with, Re=12,400 and Fr=0.217. Also
plotted, are the apparent center of the secondary vortex
which is produced outboard of the right vortex. The five
primary vortex trajectories overlap in the beginning, but
diverge at later times during the interaction. The primary
vortex trajectories all begin at ~*= -1.55. The time at the
end of each primary vortex trajectory is given in the
caption along with the time at the beginning and the end
of secondary vortex trajectories. The secondary vortex
for the case with the largest surface pressure, 9.0
(dynes/cm), rapidly became turbulent and no distinct
vortex center could be observed. A tick mark on each of
the primary vortex trajectories indicates the position of the
primary vortex at the time of formation of the secondary
vortex. Turbulence in the primary vortex was first
observed at time, ~*= 2.19 (+0.24) for all the surface
conditions tested.
The trajectories plotted in Figure 4 show that
when oleyl alcohol is present on the free surface, the
trajectory of the primary vortex departs from the trajectory
for the case with no surfactant. From these trajectories it
is apparent that the greater the surface pressure, the
greater the amount of rebounding of the primary vortex
from the free surface. When the surface pressure is as
high a 9.0 (dynes/cm), the vortex trajectory is very
similar to that for vortex interaction with a solid wall.
4) Particle Image Velocimetry Measurements
In order to provide quantitative data for the
phenomena observed and results obtained using flow
visualization the entire velocity field in the cross section
was measured for a vortex pair at Re=12,400 and
Fr=0.217 using a particle image velocimetry (PIV)
technique. The method we used is one of the variations
of double-pulse velocimetry used in the past fifteen years,
as described in Stetson (1975), Lauterborn and Vogel
(1984), and Dudderar et al. (1986~. From a double-
exposed transparent photograph of a seeded flow, the
velocity at each point in the flow is deduced by measuring
the displacement of the seeding particles during the time
between two exposures. In the present study, a pulsed
copper vapor laser was used to illuminate a thin cross-
section of the vortex pair flow field. The Young's fringe
method was chosen to determine the particle
displacements and direction of displacement. The
magnitude of the velocity was determined from the
displacement divided by the time between the two
exposures. The flow direction was obtained by
inspection of the velocity magnitude data. The system
and method we used to interrogate double exposure
photographs of the particle images was developed by L.
P. Bernal and is described by Kwon (1989~. This system
consists of a Helium-Neon laser, a set of mirrors, a
transform lens, a beam stopper, a video camera with a
pair of imaging lenses and a Gould model ED 5000 image
analysis system controlled by a Zeos model 286 personal
computer. Some time was spent learning how to seed the
flow. After a number of trials the seeding particles
selected for PIV measurements were 12 to 50 microns
particles of titanium dioxide in the rutile crystalline form
and 50 to 100 micron diameter micro-balloons. Micro-
balloons are hollow glass bubbles with specific gravity of
less than one which tend to migrate inward toward the
core of the vortices while the much heavier titanium
dioxide particles migrate away from the vortex core. The
fluid in the vicinity of the flaps was seeded with both
glass micro-balloons and titanium dioxide particles and
titanium dioxide particles were also continuously
deposited on the free surface.
Figure 5 shows the velocity vector field of the
vortex pair for Re=12,400 and Fr=0.217 as it approaches
a clean surface at time ~*= 0.49.
It} (cm/see) 1 1
5 (cm'
Figure 5 Vortex pair velocity field during the interaction
with a free surface with no surfactant added; ~*= 0.49,
Re=12,400 and Fr=0.217.
The non-zero velocities on the top of the figure show that
the free surface is adequately clean and free to move. The
lack of data near the center of the vortices is primarily due
to the limits in the spatial resolution of the interrogation
system. Inadequate seeding is responsible for the
missing data near the lower part of the figure. The
dynamic range of the interrogation system limits the
lowest velocity that could be measured. The flow on the
plane of symmetry at the free surface resembles a
stagnation region of very low velocity. As a result, the
spacing between the Young's fringes is too large to
measure. PIV images at later times with a clean surface
were also obtained and there was no sign of the
development of secondary vorticity near the free surface.
From the PIV data the contours of constant
vorticity in the primary vortices were determined. The
circulation for each side of the vortex pair was also
calculated using a line integral around a square contour
surrounding the vortex core. The magnitude of
circulation for each vortex was found to be 145.9
(cm2/sec). A few of the vorticity contour lines extended
outside the domain of the calculation. This implies that
the total circulation for each vortex is slightly more than
the 145.9 (cm2/sec) measured. In contrast, the apparent
circulation, i.e. the circulation based on the measured
propagation speed and spacing of the vortex pair and
using a point vortex model, was found to be 124
(cm2/sec).
In marked contrast to the flow with a clean free
surface, Figure 6 shows the velocity vector field of the
left primary vortex and the secondary vortex developed
outboard of the left primary vortex at a time ~*= 1.54,
when the free surface is contaminated with oleyl alcohol
with a surface pressure of 9.0 (dynes/cm).
525
OCR for page 526
1() (cm/see)
~ ~ ~ . . . . . . ~ . I . .
'' ;~
, ~ ,,
,, ,, ~ ~ ~ ~ ~ If ~ J ~ ..........
/ J J J I ~ J I
J ~ I ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ _ '''
_ ~ ~ ~ ~ \ ~ ~ ~ ~ ~ ~ ~,, ,
1 1
2 (cm)
Figure 6 Velocity field for the left vortex interacting with
a free surface with oleyl alcohol with ~=9.0 (dynes/cm);
the right edge of the figure is 7.7 (cm, actual) from the
centerline; ~*=1.54, Re=12,400 and Fr=0.217.
The circulation around the secondary vortices in this PIV
vector plot was calculated and found to be approximately
-45 (cm2/sec), opposite to and approximately 30% of the
circulation in the primary vortex. This clearly illustrates
the fact that surface contamination can lead to the
generation of new vorticity during interaction of vorticity
with the free surface.
5) Free Surface Signatures of a Vortex Pair
The Froude number for the vortex pairs in these
experiments was not large enough to cause appreciable
wave generation when the vortex pairs interact with the
free surface. However, a variety of slight surface
deformations were observed that were associated with the
velocity and pressure field of the primary, secondary and
any other vorticity produced by the vortex pair. These
surface deformations were visually observed using the
shadowgraph method. The shadowgraph system used in
the experiments with the vortex pair generator was very
simple, a mercury vapor light source beneath the glass
bottom tank containing the vortex pair generator and a
white cloth screen 6 X 2 feet mounted 3 feet above the
free surface. The image on the screen was recorded on
video tape using the video camera and recorder. The
shadowgraph effect is a result of the refraction of light by
the free surface. When an approximately parallel beam of
light passes upward from water to air normal to the
surface, a depression of the surface will result in the light
rays diverging and an upward displacement of the free
surface will cause the light rays to converge. The screen
placed above the free surface will then show surface
depressions as darker regions and elevations of surface as
brighter regions.
The vortex pair with Re=18,700 and Fr=0.277
produced a surface signature which from our
observations of the signatures of various strength vortex
pairs (at low Froude number) contains the typical
phenomena observed with weaker and stronger vortex
pairs before transition to turbulent flow occurs. The
surface signature for the right half of a clean surface
above a vortex pair is shown in Figure 7.
526
_.
_
~ 1
-
8~
~3
IB8l 8
Figure 7 Shadowgraph view (of the right half) of the
free surface during the interaction of a vortex pair with a
free surface with no surfactant added; Re=18,700 and
Fr=0.277 (scale: 1 cm on the photograph = 10.0 cm on
the surface).
The first observable features are the striations that were
first described by Sarpkaya and Henderson (1985~.
Using LIF and other techniques to observe the flow
beneath the surface we have determined that the striations
are caused by spanwise vorticity which is stretched in the
upwelling flow field of the primary vortex pair, see Hirsa
(1990~. A paper describing and summarizing these
observations of the flow field of the striations is in
preparation. The first photograph in Figure 7, (i), is at
time ~*=0.79. The striations, observed to produce
narrow dark strips of surface depression, are clearly
visible in this photograph. A surface depression which is
wider than and normal to the striation depressions can be
observed directly above the right hand primary vortex.
Just outboard of this depression the initial formation of a
Reynolds ridge can be observed (an undulating, narrow,
line bright on the left and dark on the right) which
becomes stronger in the following photographs, (ii)
through (iv). The photograph (ii), taken at ~*=1.16,
shows the striations and the Reynolds ridge as well as a
surface dimple, a dark region at the top of the
photograph. This dimple is caused by a vortex line
terminating at the free surface. A similar dimple occurred
on the bottom but does not appear in this photograph.
The vortex induced at the end wall (see Yamada and
Honda 1989) is thought to be responsible for this
phenomenon. The third photograph (iii), taken at
~ *= 1.90, shows that the Reynolds ridge has been
convected to the right. A few striations are still visible in
this photograph. Adjacent to the large dimple near the top
of the photograph, a series of smaller dimples appear in
this picture. The last photograph (iv), shows the
Reynolds ~idge which by this time has moved very much
to the right. The photograph was taken at ~*=2.63.
There are some last remnants of the striations visible to
the left of the Reynolds ridge. The dimples are more
numerous by this time. It should be noted that the small
dimples all appear on the outboard, i.e. on the
contaminated side of the Reynolds ridge.
The surface deformations on the ~ight side of the
same vortex pair formed when a small amount of oleyl
alcohol, surface pressure of 0.3 (dynes/cm), is present
on the free surface are shown in Figure 8.
OCR for page 527
~ -
hi=
At=
~ -
- ~ ~
- ~ ~
- -
- -
-
- -
:
-
:~
Figure 8 Shadowgraph view (of the right half) of the
free surface during the interaction of a vortex pair with a
free surface with oleyl alcohol with ~=0.3 (dynes/cm);
Re=18,700 and Fr=0.277 (scale: 1 cm on the photograph
= 10.0 cm on the surface).
The first photograph, (i), at ~*=0.79, shows the
striations, an intense, narrow bright and dark wavy line,
the Reynolds ridge, and two wide depressions (which
will be referred to as scars) parallel to the Reynolds ridge
and to the primary vortex. The weaker depression is
formed to the left of the Reynolds ridge (above the
primary vortex) and a stronger one just to the right of the
Reynolds ridge (above the secondary vortex). The next
photograph, (ii), at ~*=1.16, shows that the Reynolds
ridge has moved slightly to the right. The striations are
still visible in this photograph. A series of surface
deformations, primarily depressions, are visible between
the two scars. The next photograph, (iii), shows the
effect of three-dimensional structures along with vortex
lines terminating at the free surface (dimples) to the right
outboard) of the Reynolds ridge. The last photograph,
(iv), at ~*=2.63 again shows the surface signatures of
the three-dimensional structures which have been
onvected to the right. The Reynolds ridge, which did not
move very much since the last photograph appears to be
less intense at this time.
Figure 9 is for the same vortex pair ~ Re=18,700
and Fr=0.277) as the previous two figures but with a
higher concentration of oleyl alcohol. For this case,
enough oleyl alcohol was spread on the surface to saturate
it so that the surface pressure was equal to the saturation
pressure which for oleyl alcohol is approximately 31.5
(dynes/cm). The striations and the scars are visible in this
photograph. The photograph shows that no Reynolds
ridge is formed. The scar above the secondary vortex
observed out board of the primary vortex is more intense
than the previous cases.
NUMERICAL COMPUTATIONS
1) Problem Formulation and Numerical Method
The flow is assumed to be viscous, and confined
to two dimensions. In addition to the assumption of two-
dimensionality, the major limitation is that the free
surface is assumed to remain flat for all tinges. This limits
the results presented here to relatively low Froude
numbers. However, these are the cases most frequently
Figure g Shadowgraph view of the free surface showing
the scar during the interaction of a vortex pair with the
free surface with ~=31 (dynes/cm); ~*= 0.79, F= 187
(cm2/sec) (Re=18,700), Fr=0.277; (scale: 1 cm on the
photograph = 4.0 cm on the surface).
studied experimentally, and since the surface
deformations are observed to be small the linutation is not
as severe as might be thought. In order to avoid any
arbitrary modeling of inflow and outflow boundaries we
simply take the flow domain to be periodic in the
horizontal direction, and to have a flat full slip bottom.
The effects of this limited domain size is discussed below
in section 21. The flow is governed by the Navier-Stokes
equation, which in vorticity form can be written,
Ocl,/Ot + J(~r~c~) = He V2(c'~) (4)
where Jo= fa~lay'~a~lax'- ra~lax1fa~lay', and a
Poisson equation relating the stream function to the
vorticity
V2~= ~ (5)
Here the the Reynolds number is defined as Re = F/v.
The free surface boundary condition is very
important for the present investigation. Surface
contaminants are known to have an effect on the motion
of vortices. In a previous investigation Davies (1966)
described the damping of turbulent eddies at a free surface
and later Davies and Driscoll (1974) experimented with
ejecting pulses of colored water towards a free surface,
specifically addressing the rate of surface renewal and
they found that the spreading of colored water at the free
surface is reduced considerably for contaminated
surfaces. Their simple visualization technique did not
allow a clear explanation of the mechanism responsible
for this behavior. The explanation for rebounding from a
free surface is clear from recent experiments of Bernal et.
al. (1989) who investigated collision of both vortex rings
and two-dimensional vortex pairs with a free surface.
They observed (as did Davies and Driscoll) that surface
contamination led to considerable differences in the vortex
motion itself. Using LIF for flow visualization they
found that the surface motion induced by vorticity
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approaching a contaminated free surface generated an
uneven distribution of surface contaminant that in turn
caused a shear stress at the surface, thereby generating
vortieity with sign opposite to the initial vortieity. The
generation of vortieity at a contaminated surface appears
to be a primary effect of the surface contaminants since
for contaminated surfaces the behavior is similar to
vortices encountering a rigid wall.
For the numerical computations the boundary
condition at the surface requires knowledge of the surface
contaminant which is assumed to be conserved, leading
to a hyperbolic conservation equation
De/Ot +akuSc~lax = 0 (6)
Here us is the horizontal velocity at the surface. Notice
that since the surface divergence of us is in general not
zero and depends on e, this equation allows for the
possibility of "contamination shocks" (that is the
development of a discontinuity in e). The surface
contaminant affects the flow field through shear stresses
induced by variations in the surface tension. At the free
surface, surface tension gradients induce a shear given by
= Rolex
(7)
Since the surface is flat the vortieity at the surface is ces=
Ou/Oy. The surface tension depends on the amount of
contaminant, c>= Ares, and the boundary conditions for
the vortieity, at the surface, is therefore
ce = p-I(36/Oc~Oc/Ox (8)
The quantity ~ = co (36/Oc) is usually called the elasticity
of the surface. If the contamination, e, is
nondimensionalized by its initial value, en, and the
vortieity is as before, we end up with the boundary
condition COS = cOe/ax in nondimensional units, where
C = [L/~] en (96/Oe) (9)
The flow is therefore governed by the parameters Re and
C as well as the initial vortieity configuration. The
dimensionless variables used for the computations were,
the dimensionless time = t F/L2, and the dimensionless
distances, x/L and y/L. Where ~ is the circulation and L
is the half width of the computational box. To solve these
equations numerically we have used a rather standard
finite difference approximations. Equation (4) is
integrated by an explicit second order predietor-eorreetor
method in time, and the spatial diseretization is done with
second order eentered-differenees. For the Jacobian,
J(x,y), Arakaw's conservative stencil is used. The
Poisson equation is solved with a fast solver (HWCRT
form FISHPACK). For the contaminant we also use a
second order predictor corrector in time, and second order
differences in space. For stability an artificial viscosity
term is added on the right hand side of (6) with viscosity
that is small everywhere except where the contaminant
value changes rapidly. The surface velocity is found by a
one sided, second order differentiation of the stream
function. Several of our results have been checked for
convergence by repeating the calculation using a different
resolution.
2) Results and Parameter Studies
Most of our computations have been done for the ease of
a two dimensional vortex pair colliding head on with the
top surface. Since the problem is symmetric about the
centerline it is sufficient to calculate only one of the
vortices and use symmetry boundary conditions. The
central question that we are addressing is how the
contaminants on the surface affect the evolution of the
vortieity, and how it differs from the ease when the free
surface is clean. In Figure 10, for Re = 2000, we show
the evolution of the flow produced by two blobs of
primary vortieity with opposite sign, with an initial
spatial distribution proportional to r exp(-ar2) where r is
the distance from the center of each blob. The boundary
conditions on the top surface are, (a) a stress free
boundary (also called a full slip boundary) and (b) a
contaminated top surface with C =2. The right hand
vortex is initially half way between the top and bottom
boundary, and the first frame is at the time the motion
begins. There is no boundary layer for the full slip
surface. A slight boundary layer (not visible at the initial
time, t = tip is formed beneath the contaminated surface
an instant after the motion begins. In the second frame
the upward motion of the vortex has ended, and, due to
the image vortieity above the free surface, it is now
moving outward. The boundary layer in (b) has grown
considerably, and it is clear that separation is about to
take place. In the third frame the vortex in (a) continues
its outward motion along the full slip boundaries, but in
(b) the boundary layer has separated and formed a
secondary vortex that deflects the path of the primary
vortex away from the surface. This evolution continues
in the fourth frame, the vortex in (a) moves out along the
wall, but in (b) the primary vortex has moved further
away from the wall under the influence of the secondary
vortex. At the same time the stronger primary vortex
swings the secondary vortex around so it is now almost
below the primary one, and thus induces an inward
motion. Viscosity now has visible effect on the evolution,
the maximum vorticity of both the single vortex in (a), as
well as in (b) has decreased compared with the previous
frames. In the last frame the vortex in (a) has
encountered the outer boundaries of the computational
box, and is starting to move downward along the outer
wall, and in (b) the primary vortex is actually moving
upward again es well asinward. Perhaps the most
striking feature of the above sequence is the similarity
between the results for the contaminated surface case (b)
and results of calculations (not shown in this paper) for
the case of a "no slip" rigid wall . The flow field,
primary vortex motion and secondary vortex motion for
case (b) and the "no slip" wall case are very similar.
The rebounding of the primary vortex for the
contaminated top surface in (b) is obviously due to the
uneven distribution of the contaminant produced after the
motion begins. This distribution is shown in Figure 11,
at times corresponding to those in Figure 10. In (a) the
contaminant is passive, and is simply advected with the
flow and this does not lead to any shear stresses on the
fluid at the boundaries, as mentioned above. As the
vortex collides with the surface the contaminant is swept
outward, depleting the region between and above the
vortices of contaminant and accumulating it outward of
the vortices. This contaminant peak is then pushed
outward. Since the computational box is of finite width,
the contaminant eventually reaches large values at the
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OCR for page 529
(a)
(b)
(a)
(b)
-
,~,
,~,
t3
t4
l~ts~
.
Figure 10 Computational results showing contours of
constant vorticity on the right side of a vortex pair
approaching the free surface for a clean surface (a),
C = 0 and a contaminated surface (b), C = 2. Re =
2000. The dimensionless time is zero at the top of the
Figure and increases in increments of 0.7 dimensionless
. .
time units.
t3
l
lll
c: tip t'4 tats
Figure 11 Contamination profiles for the computations
shown in Figure 10. Full slip case is (a) and the C = 2
case is (b). The time between the profiles is the same as
in Figure 10 and the profiles all become steeper with time
except for the last profile for the case C = 2.
l Ill
. ~
t'2 arts
outer side of the box in the down welling region.
Although the finite box size obviously has effects on the
final profile, the maximum contamination peak increases
rapidly even before the side effects become significant,
since the outward velocity decreases outboard of the
vortex. In the second frame, (b), C = 2 as in Figure 10.
Now the uneven contaminant distribution creates shear
stresses on the top surface that opposes the outward
motion due to the vortices. This balance---outward
motion due to the vortices, and inward motion due to the
uneven contaminant distribution---eventually slows down
the spreading of the clean region above the vortices. The
shear stresses due to the contaminants create vorticity that
eventually separates and causes the primary vortex to
rebound. As the vortices rebound their effect on the
surface diminishes, and the contamination "shock" that
separates the clean and contaminated surface starts to
move inward again. In Figure 11 at time is the inward
motion has just started. The large accumulation of
contaminants, seen for the "full slip" case does not take
place in the contaminated surface case although the
contamination profile behind the shock equilibrates with
time. We have made computations for larger values of C
and find that the restoring effect of the contaminants is
much stronger. As a result for C = 10, only a small
clean region forms on the surface between the primary
vortices. The vortices then move outboard of the shock,
and as they pass under and rebound the "hole" closes
rapidly. For the case with C = 50 the vortices only cause
a small initial dimple in the contamination profile which
disappears rapidly as the vortices move outward.
Perhaps the most noticeable feature of the contamination
profiles is how different they are when compared to the
similarity of the vorticity distributions which the
contamination has created. Except for the completely
stress free boundary a secondary vortex is formed and the
primary vortex rebounds, even though in some cases a
clean region is formed, and in others the contaminant
distribution is hardly changed at all. The only difference
in the vorticity distribution is that the boundary layer at
the top starts further away from the center when a clean
region is formed.
The above runs have all been done in a relatively
small computational domain. To assess the influence of
the boundaries on the evolution we have repeated one of
s29
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the runs with, C = 2, in a domain that is twice as wide.
The vorticity distribution appeared almost identical, and
only for the latest times were there any significant
differences in the contamination profiles. The primary
difference was that the value of the contamination
concentration was slightly higher behind the shock for the
shorter box, and as a result the shock moved slightly
faster to close the hole in the contamination profile after
the vortices had rebounded. We therefore feel rather
confident that boundary effects are of minimal
. .
significance.
(a)
~ _-----_ ~_
l
(b)
i'
Figure 12 The paths of the primary and secondary
vortices. In (a), the full slip case with no secondary
vortex is a dotted line, the solid line is for C = 2 and the
dashed line is the solid wall case. The two large radius
curves are the secondary vortex paths. In (by, the
primary and secondary vortex paths for increasing
Reynolds numbers of See, 1000, 2000, & 4000 are
shown for contaminated surface with, C = 2. There are
no secondary vortices for Re = 500 and 1000. For Re =
2000 and 4000 the secondary vortex paths have a large
radius of curvature as the Reynolds number increases.
To efficiently display the similarities and
differences in the flow fields caused by the surface
boundary condition and/or Reynolds number, we plot in
Figure 12 (a), the path of the primary vortices and
secondary vortices, at Re = 2000, for the no-stress
case, for a contaminated surface with C = 2 and for the
rigid boundary case. For the no-stress case there is no
secondary vortex and no rebounding. The slight
downturn in the primary vortex trajectory is caused by the
proximity of the right hand boundary. For the
contaminated surface and for the rigid wall there is a
secondary vortex and there is rebounding of the primary
vortex.
In Figure 12 (b) the paths of the primary and
secondary vortices are shown for a contaminated surface
with C = 2 at various Reynolds numbers of 500, 1000,
2000, and 4000. At the higher Reynolds numbers the
diffusion of vorticity is less rapid in comparison to
convection. This results in greater outboard motion of the
primary and secondary vortices as the Reynolds number
Increases.
CONCLUSIONS
· Experiments and numerical computations for
vorticity interaction with a free surface show good
qualitative agreement.
.
Surface contamination, in the experiments and
computations, was found to have a strong influence on
the nature of the interaction of the vortex pair (or the
mailing vortices) in both the surface deformations and the
flow field below the surface.
· During interaction of a vortex pair (or trailing
vortices) with a contaminated free surface, shear stress
produced at the free surface causes the production of
vorticity. This vorticity can roll-up into a pair of
secondary vortices with sign opposite to the adjacent
primary vortex. This alters the trajectory of the original
vortices.
· A Reynolds ridge is formed as a result of the
interaction of vortex pair (or the trailing vortices) with a
slightly contaminated free surface.
· The strong scar (surface depression) was found
to be caused by secondary vortices formed when a
surfactant was present on the surface.
· The circulation of the secondary vortex produced
as a result of the vortex pair interacting with a free surface
with surfactant was found to be about one third of the
circulation of the primary vortex.
REFERENCES
Davies, J. T., 1966 The effect of surface films in
damping eddies at a free surface of a turbulent liquid.
Proc. Royal Soc. London A 290, 515-526.
Davies, J. T. & Driscoll, J. P., 1974 Eddies at free
surfaces, simulated by pulses of water. Industrial and
Engr. Chemistry Fundamentals 13, 105-109.
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Dudderar, T. D., Meynart, R., & Simpkins, P. G., 1986
Laser speckle velocimetry. The Tenth U. S. National
Congress of Appl. Mech., The Univ. of Texas at
Austin.
Hirsa, A., 1990 An Experimental Investigation of
Vortex Pair Interaction with a Clean or Contaminated
Free Surface. PhD. Thesis, Dept. of Aerospace Engr.
The University of Michigan.
Kwon, J. T., 1989 Experimental study of vortex ring
interaction with a free surface. PhD. Thesis, Dept. of
Aerospace Engr. The University of Michigan.
Lamb, H., 1932 Hvdrodynamics 6th ed. Cambridge
University Press.
Lauterborn, W., & Vogel, A., 1984 Modern optical
techniques in fluid mechanics. Ann. Rev. Fluid Mech.
16, 223-244.
Sarpkaya, T. & Henderson Jr., D. O., 1985 Free
surface scars and striations due to trailing vortices
generated by a submerged lifting surface. AIAA paper
no. 85-0445, AIAA 23rd Aerospace sciences meeting,
Jan. '85 Reno, Nevada.
Scott, J. C., 1982 Flow beneath a stagnant film on
water: the Reynolds Ridge. J. Fluid Mech. 116, 283-
296.
Scott, J. C., 1975 The preparation of water for surface
clean fluid mechanics. J. Fluid Mech. 69 pt. 2, 339-
351.
Stetson, K. A., 1975 A review of speckle photography
and interferometry. Optical Engr. 14 (5), 482-489.
Willmarth, W. W., Tryggvason, G., Hirsa, A., & Yu,
D., 1989 Vortex pair generation and interaction with a
free surface. Phys. Fluids A 1~2), 170-172.
Yamada, H. & Honda, Y., 1989 Wall vortex induced by
and moving with a confined vortex pair. Phys. Fluids
A 1 (7), 128()-1282.
DISCUSSION
Targut Sarpkaya
Naval Postgraduate School, USA
Contrary to authors' arguments, contamination does not change the
physics of the phenomena. Authors' use of low speeds and scales in
their model overly accentuated the effect of surface tension and hence
the Weber number. They could have minimized the said effects at
relatively larger Froude numbers and thereby gained a clearer
understanding of the physics of the phenomenon. As I have shown
in my papers, variations in contamination at higher Froude numbers
(relatively lower numbers) did not alter the fundamental character of
the scars and striations. In ocean environments, the vortex motion
and the turbulent wake is such that the relative significance of Weber
number is not exaggerated.
AUTHORS' REPLY
In the near-wake of large, high speed ships, it is possible to
encounter very energetic, high Froude number vortices which try to
Leap out" of the surface. For such high Froude number vortices, it
is true that the effect of surface contamination might not be as great
as it is for vortices with lower Froude number. On the other hand,
in the far-wake of ships, which are observed in the SAR images, the
turbulence is decaying and the Froude number for the eddies is
relatively small and therefore surface contamination plays an
important role in the interaction of the eddies with the free surface.
DISCUSSION
Hyong-Tae Kim
The University of Iowa, USA (Korea)
1. Besides the secondary vortices identified in your measurement of
the velocity for the case of the contaminated surface, could you really
resolve the free-surface boundary layer in the measurement?
2. Could you tell how this vortex pair model is related with the
persistent trace on the ocean surface of the ship wake?
AUTHORS' REPLY
1. The measurements which we made using PIV do not resolve the
velocity within the free surface boundary layer. For that information
we have to rely on our results from full Navier-Stokes simulations.
Information on the finer scales of flow obtained from the
computations should be reliable since the agreement on large scale
comparisons to the laboratory measurements is very good and the
flow is relatively laminar.
2. Although there are vortex pair-like structures in the wake of ships
(e.g., bilge vortices and propeller vortices), an exact comparison
between the vortex pair flow and the actual ship wake is not possible
and in fact is not intended. The vortex pair offers a simple flow
which can be studied in order to provide insight into the nonlinear
ship wake problem. For example, the mass transport to the surface
by a vortex pair and the important effects of surface contamination on
this transport process shows the role surfactants can play in the wake
of a ship in the ocean.
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Representative terms from entire chapter:
vortex pair
