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OCR for page 503
Scarred and Striated Signature
of a Vortex Pair on the Free Surface
T. Sarpkaya, P.B. Suthon (Naval Postgraduate School, USA)
ABsTRAcr
This paper describes a numerical and
experimental study of three-dimensional
flow structures (striations, whirls, and scars)
resulting from the interaction of a pair of
ascending vortices with the free surface.
The characteristics of the flow features at
the scar-striation interface (a constellation
of whirls or coherent vertical structures,
thought to be the boundaries of the dark
narrow radar images!, are investigated
through the use of an infra-red camera, flow
visualization, an image analysis system, and
the vortex-element method. The results
have shown that the striations are a conse-
quence of the short wavelength instability,
inherent to the vortex pair itself, and the
whirls result from the interaction of stria-
tions with the surface vorticity Dice = 2qK,
twice the tangential velocity times the curva-
ture of the intersection of the free surface
with the plane normal to it). The whirl-
merging leads to an up-cascading process in
size and energy in fewer vortices while the
total energy decreases rapidly.
INTRODUCTION
Just a narrow patch of darkness,
bounded by two bright lines, provides the
impetus for this investigation partly because
it is seen in the synthetic aperture radar
(S1\R) images of a ship's wake, partly because
it extends many miles directly in the ship's
track, partly because the reasons for its exis-
tence have not yet been explained, and
partly because the surface footprints of sub-
surface phenomena can give trace of the
generating bodies. This is the basis of the
current intense interest in the interaction of
internal waves, wakes, and vortices with the
free surface.
A few facts are known about the SAR
images: Their physics is elusive; they are by
no means easily accessible to precise mea-
surement; they are not related, at least
directly, to the Kelvin wake; they do not
reflect the incident electro-magnetic waves
back to the source (negative spectral pertur-
bation); and they can bifurcate. Various pro-
posals have been advanced to provide a fea-
sible explanation of the dark band: Interac-
tion of the wake of a vortex pair with the
free surface; turbulence and surface mean
flow resulting from the ship's motion; redis-
tribution of surface impurities by large-scale
vertical motions (as in Langmuir circulations
[11 and Reynolds ridges t211; entrained air in
the wake; bubble scavenging of surface and
subsurface surfactant materials; interaction
of Kelvin waves, ambient waves, and momen-
tum wake; generation of vorticity-retaining
inverse bubbles and drops by a Kelvin-
Helmholtz instability [3l, just to name a few
of the existing proposals. Each model at-
tempts to provide a more feasible explana-
tion of the dark narrow band seen in the
SAR images.
As far as the footprints of a vortex pair on
the free surface are concerned, the original
observations leading to the present study
may be summarized as follows. When a
trailing vortex pair, generated by a lifting
foil, rises toward the free surface, with or
without mutual induction instability and/or
vortex breakdown, the vortices and/or the
crude vortex rings give rise to surface dis-
turbances (whirls, scars, and striations).
These were first reported by Sarpkaya and
Henderson [4, 51 and Sarpkaya [6, 71. The
striations are essentially three-dimensional
free-surface disturbances (which appear as
ridges as shown in Figs. 1 and 2], normal to
the direction of the motion of the lifting
Prof. Turgut Sarpkaya and LT Peter B. Suthon, Mechanical Engineering, Code: ME,
Naval Postgraduate School, Monterey, California 93943-5000 U.S.A.
sol
OCR for page 504
surface, and come into existence when the
vortex pair is at a distance equal to about one
initial vortex separation from the free sur-
face. The scars are small free-surface
depressions, comprised of many randomly
distributed whirls, and come into existence
towards the end of the pure striation phase,
as in Fig. 2, and when the vortices are at a
distance equal to about sixty percent of the
initial vortex separation from the free sur-
face (Fig. 31. When the vortices migrate
large distances upward, the vortex pair
usually undergoes both short-wavelength
and long-wavelength sinusoidal instability
(Fig. 4) and often breaks up into isolated
rings (Fig. Al, (for additional details see, e.g.,
Sarpkaya t811.
Fig. 3 Evolution of scars and whirls
Fig. 1 Appearance of striations
Fig. 2 Transition from striations to scars
Sarpkaya and Henderson's [51 and
Sarpkaya's t61 theoretical models of the scar
cross-section created by the trailing vortices
was based on the classical solution of Lamb
t9l, assuming the vortices to be two-dimen-
sional and the free surface to be a rigid
plane. For small Froude numbers Fr (=
VO/~O, where VO is the initial mutual
induction velocity of the vortex pair), the
vortices follow the simple path described by
Lamb's potential-flow solution, the free sur-
face remains fairly flat, and each scar front
Fig. 4 Onset of long wave instability
. fig
~ id_
_
Fig. 5 Single inclined rising ring
approximately coincides with the stagnation
point on the Kelvin oval, formed by one of
the pair of the trailing vortices and its
image, as shown by Sarpkaya and Henderson
t4-71. For Froude numbers larger than about
0.15, not only the deformation of the free
surface but also the nonlinear interaction
504
OCR for page 505
between the said deformation and the
motion of the vortices become significant.
The vortices follow Lamb's solution only
during the early stages of their rise. Subse-
quently, they exhibit paths of varying
degrees of complexity, depending on Fr and
the Reynolds number Re (see, e.g., Fig. 6 for
Fr = 0.6 and Re ~ 4000, reproduced from
Leeker's thesis t1011. Here the vortex paths
rise vertically upward and, instead of moving
away from the center, move initially toward
the center line as the vortex is drawn up
into the domed area underneath the free
surface (Fig. 71. Occasionally, the path of the
approximate center of a vortex forms a loop
as seen in Fig. 6. It is not clear whether this
is due to three-dimensional instabilities
along the axis of the vortex, or due to the
formation of the "wall vortex", observed by
Yamada and Honda [111, or due to the suc-
cessive 'rebounding' of the decaying turbu-
lent vortex (for rebounding in general, see,
e.g., Peace and Riley [1211.
Even though it was fully realized at the
outset that the problem ultimately to be
solved is the understanding of the three-
dimensional nature of the phenomenon, the
relative ease of the two dimensional coun-
terpart has attracted the immediate atten-
tion of experimentalists and numerical anal-
ysists alike (e.g., Sarpkaya et al. [131 Dahm et
al. [141, Willmarth et al. [15], Marcus [16],
Marcus and Berger [ 17], Tryggvason 118],
Telste [191) and Ohring and Lugt [20], just to
name a few). Most of the numerical simula-
tions dealt with the inviscid, two-dimen-
sional interaction between a pair of counter-
rotating line vortices and a free, initially pla-
nar, surface. In these calculations the criti-
cal time at which the numerical instability
manifests itself does not correspond to the
instability of the free surface or to its
extremum position. The calculations of
Ohring and Lugt are for a two-dimensional
laminar flow at relatively small Reynolds
numbers (Re = VO b O / v = 10 and 501.
Recently, Marcus [211 calculated the defor-
mation of a density interface for Fr = 1.125,
Re = 500, and a density ratio of 5:1 using a
second-order finite-difference projection
method for the full variable-density (i.e.,
without invoking the Boussinesq approxima-
tion). The resolution of fine structure in the
flow is obtained through the use of higher-
order Godunov methods in evaluating the
nonlinear advective terms. Figure 8 shows
sample vorticity and density plots
(reproduced here with permission from the
original color plots).
Even though some insight has been
gained through the use of two-dimensional
r
5
Fig. 6 Sample vortex-center paths
Fig. 7 Sample scar cross sections
viscous- or inviscid-flow numerical models,
the results are not likely to lead to the
understanding of the physics of the dark
narrow images. Observations and measure-
ments strongly suggest that the three
sos
OCR for page 506
Fig. 8 Vorticity and density contours, (from
Marcus t21] with permission)
dimensionality of the phenomenon is essen-
tial to the existence and longevity of the
scars and striations.
The three-dimensional instability of an
initially parallel vortex pair has attracted
great attention because of its importance in
the understanding of the demise mecha-
nisms of aircraft trailing vortices. Crow [221
was the first to show that both symmetric
and asymmetric modes of instability will
develop on the vortices due to the mutual
inductance of the sinusoidally perturbed
pair. Figure 9, adopted from Widnall [231,
shows Crow's stability diagram for the vortex
pair. The shaded areas show the stability
regions. The solid curve shows the wave
1n
\/b~
4
2
o
0 0.1 0.2 0.3 0.4 0.5 0.6
rctbO
Fig. 9 Stability diagram for a vortex pair
lengths of maximum amplification for each
rc/bo. The upper curve shows the most
unstable long wave and the lower curve
shows the most unstable short wave. The
preferred mode of instability at the longest
wavelength is 8.6bo. The wave number ,B for
the short wavelength instability varies from
about 4 to 17, corresponding to wavelengths
of ~ = 0.37bo to 1~57bo.
It is in view of the foregoing that it was
decided to generate nearly two-dimensional
vortex pairs at relatively large Froude num-
bers, rather than inclined trailing vortices at
relatively small Froude numbers, and to con-
struct a numerical model of the three-
dimensional surface structures. Such an
effort is complicated by the difficulty and
expense of quantitative measurements, im-
age analysis, and high-resolution numerical
simulations. Some of the experimental diff~-
culties stem from the occurrence of
Helmholtz instability during the roll-up of
the vortex sheets, the short-wavelength and
the lon~wavelength instabilities of the vor-
tex pair, turbulent diffusion of the trailing
vortices, and the unknown (and perhaps
unknowable) physical condition of the liquid
surface. Some of the numerical problems
stem from the difficulty of prescribing the
initial conditions, including the free-surface
characteristics, the three-dimensional
nature of the resulting surface disturbances,
and the stochastic nature of the whirl con-
stellation.
EXPERIMENTAL PROCEDURES
Experiments were conducted in three
different water basins. The first was a long
towing tank. It was used previously for the
exploration of the characteristics of scars
and striations resulting from the trailing
vortices generated by lifting bodies
(Sarpkaya and Johnson [241, Sarpkaya et al.
[131) The second, a multipurpose basin, was
used for the study of the scar cross section
in a two-dimensional mode (Sarpkaya et al.
~ 131) either through the use of counter-
rotating plates or through the use of a
streamlined nozzle, mounted on top of a
piston chamber. The third water basin was
constructed partway through the current
investigation when it was realized that the
existing basins were insufficient for the type
and scope of experiments desired. It was
built three times wider than the second tank
and allowed the generation of two-
dimensional Kelvin ovals (with aspect ratios
as large as 18) and the study of two
counter rotating stationary circulations at
prescribed depths below the free surface,
through the use of two counter-rotating
506
OCR for page 507
cylinders of identical diameter and angular
velocity.
The test tanks were covered with ply-
wood and the water was recirculated con-
tinuously prior to each experiment in order
to protect the free surface from airborn con-
taminants. Nevertheless, the tanks may have
had a contaminated free surface with a sur-
face viscosity larger than the corresponding
bulk viscosity. The average value and/or the
spatial distribution of the surface tension
ultimately produced are not easy to quantify.
The surface tension always exists at a free
surface (due to intermolecular cohesive
forces) regardless of whether the interface
is contaminated or not by agents foreign to
either fluid. Surface tension can affect the
motion of the free surface partly by imposing
a tangential pressure gradient as a result of
spatial variations of the curvature of the air-
water interface and partly by applying a tan-
gential stress to the liquid beneath as a
result of the variations of the surface tension
along the interface, brought about by the
gradients of surfactant concentration and
temperature. The effects of viscosity, curva-
ture, surface-velocity, and surface-tension
appear to play significant roles in the cre-
ation of the surface vorticity and the evolu-
tion of surface signatures.
Experimental data were collected using
the same equipment regardless of which
basin the experiments were conducted in.
Hero video-recorder systems were used, one
with a black-and-white video camera and the
other with a color video camera. The output
signal of each camera was sent to a screen
date/timer before it is recorder on a video-
tape. The black-and-white camera was used
primarily with the shadowgraph technique
(for optimum contrast and clarity of the
shadows of the surface disturbances), and
the color camera was used primarily for the
recording of the striations, visualized
through the use of various florescent dyes
(introduced into the Kelvin oval at the nozzle
exit, not onto the free surfacel. In addition,
two still cameras were used for the record-
ing of streaklines through the use of surface
markers. Thermal-imaging was conducted
using an infra-red camera which sent a video
signal to a date/timer and then to a video-
cassette recorder
The video-screen images of whirls, scars,
and striations were tracked through the use
of a digital image processing system and the
data were transferred to a computer for flow
structure and motion recognition in whirl
constellation {for additional details, see
Leeker [ 1011. The stored data could be
manipulated to generate detailed informa
tion on whirl centers, number of whirls per
reference area, angular velocities as function
of whirl radius, whirl-migration velocity,
whirl-whirl interaction (fusing and cancella-
tion, etc). All of the experiments cited
above were repeated at least twice to assure
that the results could be reproduced within
reasonable experimental errors. The exper-
imental data corresponding to the earlier
stages of the motion were used to prescribe
the initial conditions in the numerical calcu-
lations. The data at later times were used
for comparison with the numerical predic-
tions at the corresponding times.
NUMERICAL SIMULATION
Following extensive observations and
measurements, the vortex dynamics or the
vortex-element method (see, e.g., Sarpkaya
[251) was used to simulate the phenomenon.
In doing so, the most important flow fea-
tures to be reproduced by this or any other
model were identified and a numerical code,
with very little or no sensitivity to the per-
turbations in the input parameters, was
developed. The model is idealized enough
for simple calculation but realistic enough to
be interesting.
The initial separation ho of the vortex
pair was chosen as the reference length; the
mutual-induction velocity VO of the vortex
pair as the reference velocity; bo/Vo as the
reference time, and the initial strength F0 of
a vortex as the reference vortex strength.
The equations describing the motion were
non-dimensionalized through the use of the
characteristic parameters ho, VO, bo/Vo' and
IN where VO = F0 /~2~bol.
The numerical model simulates the
mutual interaction of the whirls on the free
surface, beginning with their creation,
through the use of the vortex-element
method. The results of the previous investi-
Cations (Sarpkaya et al. t61), and the experi-
mental results obtained in this investigation
helped to define the parameters necessary
for the construction of the numerical model.
As shown in Fig. l 0, the position of the
inboard edge of the scar band is denoted by
s, the width of the scar band by w, the aver-
age spacing of whirls in the longitudinal
direction (a measure of whirl population
density per unit length) by c, and the hori-
zontal component of the convection velocity
of one of the vortices in the original vortex
pair by Vb. It has been shown previously that
the scars are slaved to and transported by
the vortices [4-71.
507
OCR for page 508
low = (Rnd3Fm)(-l)Rnd4 (4)
v,, v,,
so
Fig. 10 Schematic of scars and striations
A right-handed coordinate system has
been defined in the complex plane where
the real x-axis is parallel to the scars and the
y-axis is perpendicular to the scar bands.
Ideally, the length of the scar band should
extend from -so to +~. However, this is not
numerically possible. Instead, a finite scar
length L, defined by L= c iNw] and extend-
ing from (-L/2, y) to (L/2, y), is used. Nw is
the number of whirls in each scar band.
The whirl strength.is given by
Ew = 2xrCVt; = 2~r2o ~ 1 ~
where rc is the core radius and the tangen-
tial velocity at the core boundary is Vt = rc m.
The normalized, time- averaged, angular
velocity ce of each whirl core, for a large
number of whirls, was determined from the
image analysis and was found to vary from
about 0.6 to 1.2, the majority of the data
falling closer to unity. The core radius was
then calculated from
rc =
where now ce is assigned approximately the
size- and time-averaged value of unity.
The numerical analysis began by ran-
domly placing the whirls into two parallel
scar bands,
z = [RndiL- 24+iRnd2 w (3)
where z is the position of the whirl center
and Rnd1 and Rnd2 are random numbers
(independently seeded) with a uniform dis-
tribution from 0.0 to 1.0, inclusive. Each
whirl is also assigned a vortex strength in a
random manner,
where Ew is the strength of a whirl, I-m is
the mean absolute value of the strengths of
all whirls, Ends is a random number from a
standard normal distribution (with a mean of
Em and a standard deviation one-quarter the
expected range of the whirl strengths), and
Rn d 4 is a random integer from 1 to 10,
inclusive.
It has been customary to amalgamate two
or more vortices into a single vortex, placed
at their center of vorticity, (see, e.g.,
Sarpkaya [251), when their cores touch, or
are less than a prescribed critical distance £,
or overlap by a prescribed amount. Like-
signed whirls merge into a larger whirl.
Oppositely signed whirls are merged into a
smaller whirl, to mimic the cancellation of
oppositely signed vorticity (thought to be the
major mechanism of enhanced energy dissi-
pation in turbulent flow). If the resulting
whirl strength is below a prescribed mini-
mum (ymin), the whirl is removed from the
scar. The amalgamation process does not
conserve total vorticity nor is the linear or
angular momentum conserved due to the
merging of oppositely-signed whirls and the
removal of weak whirls. Nevertheless, the
conservation of these quantities is not con-
sidered important for a number of reasons.
First, the purpose of the simulation is not an
exact treatment of the viscous diffusion
although the removal of very small whirls do
in fact accomplish this purpose indirectly.
Second, the random nature of the distribu-
tion has far greater effect on the mutual
interaction of the whirls than the occasional
amalgamation of the whirls. Third, the cal-
culations with different ranges of the funda-
mental parameters, such as the whirl den-
sity, minimum survival strength (7min),
2 ~amalgamation distance, etc., have shown that
the fundamental nature of the randomly dis
tributed vortices within a narrow band is to
self-limit the amalgamation process and to
reduce the effect of the imposed perturba
tions (for sensitivity analysis) on the result
ing characteristics of the scar band.
For an incremental time step At, the
mutual induction velocity of each whirl is
calculated using a cutoff scheme similar to
that first proposed by Rosenhead [261
Az=u-iv=~21~ z ~(~z-z,,~'+i32)
s08
OCR for page 509
The cutoff parameter ~ was taken to be pro-
portional to the sum of the core sizes on the
basis of past experience [251. Each whirl is
moved to a new position, after the calcula-
tion of the induction velocity, using
t + At Zt + he + in + Vb:~At (6)
The whirl-energy density or the specific
energy in the system is estimated through
the use of
E ~ Abut Ok °' ~ skulk (7)
The diffusion of vorticity due to viscosity
is introduced artificially by reducing the
strength of each whirl either through the
use of a Gaussian vorticity distribution or
through the use of a simple percentage,
(e.g., 0.5 percent per time step), in a man-
ner similar to that done previously by others
(for details see, e.g., Sarpkaya [2511. The
artificial reduction of circulation is justified
on the grounds that it accounts for the
three-dimensional deformation of vortex fil-
aments, for the cross-diffusion of oppositely-
signed vorticity, and for the observed fact
that the strength of the vortices continues to
decrease with time or downstream distance.
Numerical experiments did not show a sig-
nificant dependence on the type and magni-;
tude of the circulation reduction (0.5 % to
1% per time step).
A few additional features of the model
need to be explained. These concern the
creation of the whirls, sensitivity analysis,
and the selection of the numerical parame-
ters (c, so, w, Fm. 6, c, Nw, At, and Amino
Whirls were created at a particular time in
the initial stages of the scar formation in
accordance with the observed characteris-
tics of the whirl population. The sensitivity
of the predictions of the later stages of the
scar motion to the parameters involved in
the simulation was examined with great
care. Accordingly, every significant parame-
ter involved in the calculations was varied
within reasonable limits. For example, c was
varied by 100 percent and w was varied from
zero to 0.4. The other parameters, such as
Em, 6. £, Nw, At, Amine were varied by at
least 100 percent.. The initial inboard posi-
tion of the scar front so was kept at the
experimentally-observed value of 1.6 at the
corresponding time and was not varied fur-
ther since it naturally increased as a function
of time and since it influenced only indi-
rectly the mutual interaction of the two scar
strips. Several runs were made with identi
cal sets of parameters, changing only the
seeding of the random numbers involved, in
order to ascertain that the insensitivity of
the results and basic conclusions to the vari-
ation of the parameters selected was not a
consequence of the use of a particular set of
random numbers. In fact, the randomization
was performed by the computer using the
system clock time at the commencement of
each run. Several plots were created at
regular intervals during the execution of the
numerical simulation, carried out to times
corresponding to the disappearance of the
surface structures, in order to compare the
predicted results with those obtained exper-
imentally.
DISCUSSION OF RESULTS
Physical Experiments
A careful frame-by-frame analysis of the
video recordings of literally hundreds of test
runs has shown that the sloping of the vor-
tex pair relative to the free surface or the
generation of vortices by a lifting surface is
not necessary for the creation of scars and
striations. Nearly identical surface signa-
tures can be produced through the use of an
initially horizontal vortex pair. Figure 11
and many others like it have shown clearly
that the three-dimensional instability lead-
ing to striations do not originate at the free
Fig. 11 Inception of short wavelength in
stability on a Kelvin oval below the
free surface
son.
OCR for page 510
surface but rather on the Kelvin oval, long
before the latter 'sees' the free surface. Sub-
sequently, the free surface interacts with
and is modified by the instability brought to
it by the Kelvin oval (see Figs. 12-15), i.e.,
the striations (nearly uniform corrugations,
at least at their inception} are a manifesta-
tion of the short wavelength instability of the
vortex pair. The flow features on the scar
front (Figs. 13 -15) are clearly identifiable
and show the creation of whirls. Figure 16
shows the whirls and the whirl pairs
(Kelvin-oval like structures at the free sur-
facel, resulting from a pair of trailing vor-
tices. The only contribution of the sloping of
Fig. 12 Rise of corrugated vortex dome
the trailing vortices is the divergence of the
scar lines (the V-shaped footprints) and the
fact that the striations come into existence
sequentially rather than simultaneously.
As noted previously, the wave number ~
for the short wavelength instability varies
from about 4 to 17, corresponding to wave-
lengths of ~ = 0.37bo to 1 .57bo . The wave-
length ~ of the striations has been deduced
Fig. 14 Intensification of scars
Fig. 13 Formation of scars and whirls
Fig. 15 Later stages of scar formation
silo
OCR for page 511
Fig. 16 Whirl formation in scars
from numerous runs as a function of the ini-
tial spacing of the main vortex pair. The his-
tograms of the striation wavelength have
shown that \/bo varies from about 0.75 to
1.25, depending primarily on the initial roll-
up of the vortex sheets as dictated by the
Froude number. As far as a vortex pair in an
infinite homogeneous medium is concerned,
the striational instability corresponds to the
short wavelength instability (see Fig. 91.
The interaction of a 'corrugated' Kelvin
oval with the free surface under the influ-
ence of gravitational, centrifugal, and surface
tension effects is anything but simple. The
surface acquires space and time dependent
curvature and velocity and produces vorticity
even in the absence of surface tension or
surface contamination. This vorticity is Ace =
2qK, twice the tangential surface velocity
times the surface curvature [271. The sur-
face velocities are induced partly by the
motion of scars and striations and partly by
the vortex pair. The surface vorticity is
largest at the scar-striation intersection
because this is where the curvature and the
surface velocities are largest. Consequently,
the scars are the regions where the surface
vorticity is most likely to be converted into
whirls with vertical axes. Thus, what begins
as a short wavelength instability quickly
degenerates into a far more complex three-
dimensional free-surface phenomenon (Figs.
14-15) with its own source and mechanism
of vorticity generation. The unsteady motion
of the striations (particularly those of the
ends where strongest surface-tension con-
centration and whipping action occurs)
Fig. 17 An additional sample of Kelvin oval
free-surface interaction
Fig. 18 Vortex dome for Fr = 1.125
Fig. 19 Rapid increase of the striations
OCR for page 512
provide the stroke necessary to concentrate
the surface vorticity into whirls. The
hypothesis of vorticity transport to the free
surface from the main vortex pair is not
necessary simply because the free surface
acquires its own curvature and surface veloc-
ity even if it were initially flat and free of
vorticity. Figures 17 through 19 show addi-
tional samples of the evolution of Kelvin oval,
striations, and scars at a Froude number
close to unity.
Experiments with rotating cylinders
have shown that even at fairly low angular
velocities and at fairly large depths of
immersion (as much as 1.5D below the free
surfacel, the rotation of the cylinders give
rise to striations and whirls. Figure 20
shows the evolution of a whirl and its neigh-
borhood during a time interval of 0.25 sec-
onds. It is clear that neither a trailing vor-
tex pair nor a nearly two-dimensional Kelvin
oval, neither a contaminated free surface nor
a Reynolds ridge is necessary to create scars
and striations. The mere presence of two
counter-rotating circulations near a free
surface (with curvature and velocity] is suf-
ficient to create all the phenomena observed
in other scar experiments. There is, obvi-
ously, much more to be explored. The only
purpose of reference to rotating-cylinder
experiments at this time is to point out that
conjectures regarding surface contamination
and Reynolds ridge are not necessary.
There are, obviously, some fundamental dif-
ferences as well as strong similarities
between the classical Taylor instability, the
striational instability, resulting from the
counter-rotation of two cylinder, and the
three-dimensional instability observed in the
actual vortex experiments. These will be
explored in some detail in future studies.
The sequence of events emerging from
the experimental observations and mea-
surements may be summarized as follows:
Uorten pair and Keluin offal;
Short rueuelength instability;
Interaction with the free surface;
Formation of striations, feeding to:
3-D curvature, Surface velocity, and;
Dividing and painug of striations;
Surface uorticity, mostly near the seers;
Grauitational' centrifugal and surface
tension effects;
Formation of whirls, whirl pairing, and
cascading of uorticity;
Self-limiting amalgamation process;
As the Kelvin oval rises, the spiralling
vortex sheets in each vortex undergo
Helmholtz and, subsequently, Rayleigh
instability and, eventually, degenerate into
turbulent motion. The distance travelled by
the oval more or less determines the
sequence of events. In the present experi-
ments the said distance to the free surface
has been kept below 6bo in order to obtain a
clearly defined oval near the free surface.
The vorticity is initially confined to the
Kelvin oval and the motion outside it is irro-
tational. With the passage of time, the vor-
ticity diffuses over a wider area and some
vorticity gets annihilated in the overlapping
regions of oppositely-signed vorticity. The
distance between the vortices during the
initial period of rise of the Kelvin oval
remains fairly constant but the core radius
and, hence, rc/bo increases due to diffusion.
As the short wavelength instability begins to
grow, the initial value of \/bo in Figure 9 is
in the order of unity. Had the cores
touched, \/bo would have been as large as
2.45 (point B in Figure 91. However, the
vortex cores do not grow to such large sizes,
not at least during the short period of
migration of the Kelvin oval toward the free
surface. Once the vortices begin to diverge,
- - - - Fig 20 Evolution of a single whirl in 025 s.
Dissipation of Whirls.
si2
OCR for page 513
As in the case of parallel line vortices
approaching a rigid plane, the distance
between them increases. This decreases the
mutual interaction of the vortices and the
relative wavelength \/bo decreases. It is not.
therefore, surprising to see that the stria-
tions multiply quickly and concentrate in
the regions directly above the vortices and
adjacent to the scars (see Figs. 13-151. The
dividing and pairing of the striations are
accompanied by the stroking and meander-
ing of the ends (a fishtail-like motion) of the
striations. It is this action that is thought to
be responsible for the reconstitution of the
surface vorticity into numerous, randomly-
sized, randomly-shaped, and randomly-dis-
tributed whirls.
A number of whirls in close proximity to
each other may give rise to a number of
complex interactions, such as, amalgama-
tion, annihilation (at least partially), or
pairing for a brief period and then re-sepa-
ration. Thus, it is clear than the interaction
of three-dimensional whirls, penetrating
only a short distance into the fluid, is not a
simple matter and needs further study
through the inclusion of viscous effects. The
amalgamation as well as annihilation is par-
ticularly strong during the formation period
of the scars. The amalgamation amounts to
cascading of circulation into larger vortices
and hence to their longer life-span. It is this
process and the generation of surface vortic-
ity that are thought to be responsible for the
longevity of the SAR images. Evidently,
some sort of regeneration process is neces-
sary without violating the principles of con-
servation of energy and circulation. One
could interpret this whirl-growth phe-
nomenon as an indication of an inverted
energy cascade. However, the total internal
energy decreases in spite of the growth in
size of a number of whirls. This completes
the observed as well as perceived evolution
of the surface signatures. It will be interest-
ing to discover as to how the scar band,
comprised of whirls, tend to increase the
radar return in order to provide bright lines
in the SAR images.
Numerical Experiments
Figures 21a through 21c show at So/bo =
1.6 (where time is taken to be ~ = 0 I, the
position of the whirls, the streamlines (with
respect to a coordinate system moving with
the scars), and the streamlines (with
respect to a fixed coordinate system). The
size of the whirls is drawn proportional to
the square root of their strength. A solid
circle denotes clockwise circulation and a
hollow circle indicates counter-clockwise
circulation. Figures 22a through 22c show
~ o~oO. ~ Oo°04 to <~ - .-9 .~ °8 ~ ~o~9c
30~0~30~° ~.o'~°~0~^a4
Fig. 21a Initial whirl distribution on scars
Fig. 21b Streamlines at ~ = 0 (comoving
with the scars)
Fig. 21c Streamlines at ~ = 0 (relative to a
fixed coordinate system}
the corresponding plots at ~ = 1.6. Thus, it
is seen that the whirls amalgamate as time
increases. Figures 21c and 22c show that
5~3
OCR for page 514
the wide region between the scar bands is
fairly calm whereas the scar bands are a con-
stellation of depressions created by the
whirls. Additional facts emerging from
these figures are as follows: The width of
the scar band increases naturally partly
because of the transport velocity imposed on
them by the main vortex pair and partly
because of the mutual-induction of the
whirls. It is rather remarkable that the band
essentially retains its identity (eventually, a
long wave instability is seen to develop).
What is more remarkable is the fact that the
calculated mean scar-separation, for the
example shown here, is within 3 percent of
that measured at the corresponding times.
O me; 0 · O · C a; O · ~ , O ; O O ~ O O .. c 0 c
of ,OO. · O O .o. O LIMO ..e
Fig. 22a Predicted whirl distribution on
scars at I= 1.6
Figure 23 shows a sample plot of the path
lines of a number of particles initially situ-
ated along a line on the scar.
Typical whirl strength distribution is
shown in Fig. 24. Clearly, the total strength
of the whirl system decreases with time.
However, the amalgamation process does
lead to a set of larger whirls. The increase
of the number of large structures slows
down or stops after a short time period. In
other words, the whirl system reaches an
equilibrium.
Fig. 22c Predicted streamlines at ~ = 1.6
(relative to a fixed coordinate
system)
Fig. 22b Predicted Streamlines at ~ = 1.6
(comoving with the scars)
Fig. 23 Marker path lines on scars at I= 2
514
OCR for page 515
So!
80-
Ul
O 70
00
> 60
0 40
1~ 30-
ID
~ ~-
Z 10
~0
In 80
U
70
o lo
30
a
Z 10
~4 1.2 jig ~05 flu ~
instability as seen in Fig. 22a. At no time did
the scar fronts exhibit a V-shaped diver-
gence as observed in the SAR images. This
was entirely expected because it is the con-
tinuous creation and upwash of the trailing
vortices that give rise to the V-shaped scar
bands with an included angle of 2a Where a
is arctantVO/U)~. It was deemed necessary
to apply the numerical model to the trailing
vortex case with a few minor differences,
partly to further substantiate the predictions
of the model and partly to observe the simi-
larities or differences between the mea-
surements and calculations. Figure 25
Fig. 24 Whirl strength distribution
Clearly, the whirl-whirl interaction
should contribute most significantly to the
evolution of the initial distribution, and such
events should occur only if the number of
whirls is high enough. In the present calcu-
lations the whirl population has been dou-
bled several times to explore this very ques-
tion. Furthermore, numerical experiments
were carried out with different random-
number seedings to ascertain that the
results concerning the energy-density dis-
tribution and the cascading of the energy did
not depend on either the number of the
whirls or on their statistical distribution. It
has been found that the population density
and the number of random samplings are
sufficiently large to arrive at statistically
meaningful conclusions. The time variations
of the distributions, therefore, allow one to
estimate whether whirl-whirl interactions
are important. The results presented above
show that the shift in the size distribution
toward larger structures and the concentra-
tion of energy in these structures are an
important ingredient of the scar formation
and scar life-span.
The numerically-simulated scars were
intentionally started parallel, with no further
intrusion into their evolution in succeeding
steps. As described above, the scars evolved,
remaining essentially parallel, with the
superposition of a long wavelength sinusoidal
\\\\~\\\\\\\\\\\\\\\\\~\~\~
Fig. 25 V-shaped scar streamlines at I= 0
Removing with the scars)
shows the instantaneous streamlines at ~ =
0, as they would be seen by an observer
moving with the scars, and Fig. 26 shows at
~ = 0-1.2, the instantaneous streamlines as
they would be seen by an observer fixed to
the coordinate system. Finally, Figures 27a
and 27b show a comparison of the experi-
mental and numerical results. One half of
these figures represent the numerical
results and the other half the experimental
results. A comparison of the two halves
show the remarkable similarity between the
observations and predictions at the corre-
sponding times.
V. CONCLUSIONS
The numerical and experimental results
presented herein warranted the following con-
clusions:
1. The sloping of the vortex pair (as in
the case of trailing vortices) is not necessary
to produce surface signatures or footprints
of vortex wakes in the form of scars and stri
s~s
OCR for page 516
Fig. 26 Growth of a V-shaped scar in time
and space
516
ations. An initially two-dimensional vortex
pair yields similar structures.
2. A fully submerged vortex pair is sub-
ject to both long- and short-wavelength instabili-
ties. The latter is particularly prevalent when
the former is suppressed. The experiments
have revealed their existence and the role
played by them in the generation of striations.
3. The short-wavelength instability has a
wavelength in the order of unity, compared
with the initial spacing of the main vortex pair.
The wavelength decreases as the vortex spacing
increases.
O
O. o.
0.
. °d. · '
~ ·.
· me ~
·0
O ~
~ O me
Too .
· ~
~ O
O.
~ .
_0. -
~ O
· .e
c
c, I,
.o~o
.0~; ~
·0 ~
08;
,O
~0
o ~
me
· ~
0,
Cue
o
Fig. 27a Comparison of measured and pre-
dicted V-shaped scars (No. 1)
OCR for page 517
o o.
ego
· ~
-
oou.
-a -
b;O
_ .
O
a_
I_.
· O
o
~o.o
to
.o.~; ~
ooO.
·0 ~
OB ;
0~ .
O;
e
0
'a-o
·e
· ~
° ~
O ~ ~
do ~ 0
.oO
04e
o
· c
Jo
C,O.O
~ 0 Cot.
Oc-~;
.
0 _ ~C>
Fig. 27b Comparison of measured and pre-
dicted V-shaped scars (No 2)
4. The short wavelength instability leads
to striations, surface curvature, surface veloc-
ity, and surface vorticity. The strongest sur-
face-tension concentration occurs near the tips
of the striations. Consequently, these are the
regions where the new whirls are created
through the use of the surface vorticity. This
type of whirl formation appears to be unique to
the scar generation.
5. The scar and striation generation may
be affected by the degree of surface contamina-
tion. However, the changes in surface tension
and surface elasticity brought about by contam-
ination, beyond that due to intermolecular
cohesive forces, are neither necessary nor suff;-
cient for the creation of striations and scars.
The most indispensable ingredients for the gen-
eration of whirls are the viscosity of fluid, gen-
eration of surface vorticity due to surface curva-
ture, and the unstable motion of the ends of
whirls. Also, it is equally important to note that
the so-called Reynolds ridge is akin to but not
related to the scar formation. The Reynolds
ridges do not give rise to whirls.
6. The numerical simulation of the phe-
nomenon through the use of vortex dynamics or
vortex element method has shown that all of the
fundamental characteristics of the scar evolu-
tion (e.g., the preservation of scar-band width,
whirl distribution, whirl-whirl interaction,
energy cascading, mutual annihilation of
whirls, self-limiting amalgamation, etc.) are
faithfully reproduced.
7. Among the numerous mechanisms
proposed to explain the physics of the SAR
images, the hypothesis of the interaction of a
vortex pair with the free surface emerges as the
most viable one in view of the observations and
measurements made and the conclusions
arrived at in this investigation.
8. The above conclusions are further
indicated by the experiments carried out with
two rotating cylinders. The analogy between
the two types of circulatory motions need to be
further explored, partly because such experi-
ments will provide an excellent opportunity to
discover many new and interesting vortex pat-
terns and partly because they will lead to the
closer examination of the similarities between
the striations, whirls, and scars generated by
the two types of circulations.
ACKNOVVLEDGMENTS
The authors wish to express their sin-
cere appreciation to the Office of Naval
Research and the Naval Postgraduate School
for the support of the investigation. The
project was monitored by Dr. Edwin P. Rood.
REFERENCES
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3. Furey, R. J., "Hydrodynamic Stability and
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OCR for page 518
4. Sarpkaya, T. and Henderson, D. O.,
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5. Sarpkaya, T., and Henderson, D.,
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9. Lamb, H. (Sir), Hydrodynamics, Dover
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10. Leeker, R. E., Jr., "Free Surface Scars
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13. Sarpkaya, T., Elnitsky, J., and Leeker, R.
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14. Dahm, W. J. A., Scheil, C. M.,
Tryggvason, G., "Dynamics of Vortex Inter-
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16. Marcus, D. L., "The Interaction Between
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18. Tryggvason, G., "Deformation of a Free
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sl8
OCR for page 519
DISCUSSION
Owen M. Griffin
Naval Research Laboratory, USA
In your introductory remarks you noted that the surface disturbances
and effects are caused by shed vortices which accompany the passage
of a submerged body. There is equal, if not more, interest in the
wake of a surface ship in terms of the persistent synthetic aperture
radar (SAR) signature produced by the ship's passage. Can you
comment upon the possible importance of the vortex-free surface
interactions discussed in your paper as they might apply to the
remotely sensed surface ship wake?
AUTHORS' REPLY
As I have noted in the Introduction of the written version of the
paper, "Just a narrow patch of darkness, bounded by two bright lines,
provides the impetus for this investigation partly because it is seen in
the synthetic aperture radar (SAR) images of a ship's wake, partly
because it extends many miles directly in the ship's track, ..... Dr.
Griffin is of course correct in reinforcing this fact. My introductory
remarks at the oral presentation of the paper were confined to
ascending heterostrophic vortices, generated by submerged bodies,
primarily because of the mechanism with which the vortices were
created in our experiments. I should have pointed out that a well-
known example of such a SAR image is that of the wake of the
surface ship USS Quapaw.
As far as the possible importance of the vortex/free-surface
interactions discussed in the paper to the understanding of the
mechanisms leading to the SAR images of surface ship wakes is
concerned, the current state of the understanding of either phenomena
does not allow one to explain the physics of what relationship could
scars and striations have with the SAR images. Among the many
proposals made, one that appeals this writer most is the interaction of
the vertical fluid motions generated by the boundary layers and
propellers of the ship with the free surface. However, such an
interaction is not as simple and as relatively clean as that of ascending
vortices because of intense turbulence (patches and parcels of vorticity
of many scales and intensities) and air-water mixture accompanying
the ship's wake. The development of a U-shaped vortex wrapping
around the outside of the main vortex core (discussed in 1985 in
author's Refl6]), restructuring of vorticity in the wake, the reverse
energy cascading, and the self-limiting growth of the surface whirls
(all discussed in the present paper) may go long ways towards
establishing a relationship between the scars and striations generated
by ascending heterostrophic vortices and the SAR images of ship
wakes. It is also possible that the real fall-out benefits of the
investigation will be in yet-unthought-of areas of experimental and
computational fluid dynamics. The investigation has just begun.
519
OCR for page 520
Representative terms from entire chapter:
vortex pair
