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OCR for page 491
Submerged Voriex Pair Influence
on Ambient Free Surface Waves
S. Fish (David Taylor Research Center, USA)
C. van Kerczek (University of Maryland, USA)
ABSTRACT
This paper examines how free surface gravity
waves travelling over a fluid are modified by the
presence of a submerged pair of vortex singularities.
Greater attention has recently been focused on the
problem of vortex flows near a free surface due to its
possible importance ill determining the influence of
vertical ship wakes on radar intakes of the ocean
surface. Although much research has concentrated on
studying vortices near a still free surface, a theory for
vertical influence on ambient free surface waves has
not been addressed. Initial steps in uncovering the
subtleties of the ambient wave modification by
submerged vortices are taken utilizing first order free
surface boundary conditions to demonstrate the
importance of the vortexinduced surface current in
the interaction of vortices with ambient waves. A
parametric study is then conducted showing that the
relative strength of the vortices and their geometric
configurations are more important in characterizing
the forth of surface modification than the relative
phase or steepness of the ambient waves. And
finally, analysis of simulation data is performed
indicating how a vortex pair ship wake model can
contribute to the observed dark centerline wakes in
radar images of ships at sea.
INTRODUCTION
Research interest in the behavior of vortex
structures near a free surface has recently been
revived by the detection of dark regions in radar
images of the sea surface. The significance of these
long narrow dark streaks is their coincidence with the
centerline wake regions of ships travelling on the sea
surface at the time of radar exposure. For this reason,
they are generally referred to as dark centerlines.
Examples and details on the radar operation and
image processing can be found in Lyden, et all. and
Peltzer et al . Although the cause of the dark
centerlines is not well understood, there is widespread
agreement that the lower energy radar return in this
region is a result of the elimination of surface waves
491
normally responsible for constructive interference of
the reflected radar signal. Constructive interference
occurs when the component of the surface
wavenumber vector in the "look" direction of the
radar satisfies the following expression (Wrights:
kw = 2 kr sin(~) (1)
where: kW = 27~/\w = surface wavenumber
k:r = 2~/\r = radar wavenumber
= wavelength
= vertical incidence angle of the radar
(normal to the free surface: 0=0°)
Waves satisfying this criterion, termed Bragg
scattering waves, are typically wind generated, and
form the background signal return level in SAR
images. Dark centerlines in the radar images of
moving ships at sea suggest the reduction or
elimination of these Bragg waves in the ship wake.
The elimination of these Bragg waves and the
long term maintenance of a Braggfree region in the
wake is the subject of intense research in several
hydrodynamic fields, from surfactant distribution to
nonlinear ship wave generation. It is likely that many
of these hydrodynamic processes are combining to
produce the dark centerlines and that their relative
importance depends on the ambient environmental
conditions. The process being studied here is the
effect of the large scale rotations generated in the ship
wake on the ambient waves present in the background
of the radar image. These longitudinal vortexlike
notions in the ship wake have been measured
(Lindenmuth4) and calculated (Griffin et al.5) for a
variety of ships and are illustrated in figure 1. The
origin of this vorticity is the overall boundary layer of
the ship and occurs in slight variations regardless of
propulsion configuration. The weaker persistence of
these vortices in the wake Cabot, by itself, account
for the length of the dark centerline but may
contribute to both the initial Bragg wave elimination
and help sustain the other processes (such as
surfactant redistribution) in the far wake. The
presence of ambient waves in the real ocean
OCR for page 491
environment therefore indicates the need to determine
how waves may be modified by the wake itself. This
work will concentrate on wave modification by the
large scale wake vorticity just described.
hi
VORTICES
Fig. 1 Ship Wake Vortex Model with Ambient
Waves
Prior studies of vortexfree surface interactions
focused on the dynamics of an initially flat free
surface under the influence of the vortex pair. This
problem is relevant only under the limited conditions
of a glassy smooth sea surface. The more common
windgenerated ambient wave background is of
concern here. In this case, the important phenomenon
is the modification of the ambient Bragg scattering
waves by vortices of relatively weak strength. The
term "weak" here refers to circulation strength and
speed of the vortex relative to the gravitational forces
on the free surface. Weak vortices are representative
of the large scale flow measurements in the far wake,
and result in minimal vertical surface deflections, and
no wave breaking. It will be shown that a first order
formulation of the free surface boundary condition
retains the dominant nonlinear surface characteristics
without the additional computational difficulties
associated with nonlinear surface modelling.
Linearization of the free surface also allows
simulation of simple monochromatic ambient waves.
The alteration of these simple waveforms by
submerged vortices can therefore be easily identified
both visually and using spectral analysis. Because
Bragg waves for typical radar wavelengths are much
shorter than the depth of the modelled wake vortices,
their inviscid influence on the vortex paths is
negligible and will be ignored in this study. The
resulting paths of the vortex pair correspond to the
paths for a flat, rigid free surface. Simulations of
the two dimensional vortex pair near a free surface
with monochromatic ambient waves are used here to
determine which parameters of the governing
equations are dominant. The Froude number and
initial vortex positions are shown to play the most
important roles in modifying ambient waves. Other
parameters, such as the ambient wave steepness and
relative phase that result from the analytic study as
independent quantities, play secondary roles, as might
be expected from linear theory.
In addition, spectral analysis is used in a form
which extracts characteristics in the surface similar to
the radar imaging process. This analysis shows that
submerged vortices can give images with darker
center regions where the ambient waves undergo the
modification. This analysis will be performed on a
vortex pair with parameters derived from typical ship
operating characteristics.
PROBLEM FORMULATION AND SOLUTION
METHOD
In developing a solution technique for the
vortex/ambient wave interaction problem, one would
like to start with the simplest model possible while
retaining the most significant flow interaction terms.
The model used here follows the classical
assumptions of inviscid, incompressible flow without
surface tension. In addition, the three dimensional
ship wake will be considered as a quasitwo
dimensional (2D) flow. See figure 2. This
modelling hypothesis assumes that velocity gradients
in the axial direction are sufficiently small to permit
2D flow realizations in a plane perpendicular to the
ship track to be integrated in time to represent the
three dimensional problem. The constant ship speed
may then be used to transform time data into axial
distance bellied the ship.
9
2A 1 ? ANAL
waves D Nisi
~ .
~ J
r
Fig. 2 2Di~nensional Flow Model
Because simple sinusoidal ambient waves are
desired for the interaction problem, a linearized
approach is formulated with the small perturbation
parameter, c, defined as follows:
is= r
where D is vortex depth, and g the gravitational
acceleration. Although ~ represents the traditional
492
(2)
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form of Froude number governing the flow when no
ambient waves are present, the use of "Froude
number" in this paper will refer to another parameter,
defined later, which the reader will find more intuitive
for cases including ambient waves. Nevertheless, the
value of £ will be kept small since this is
representative of the "weak" large scale vertical
motion in the ship wake.
Following a standard perturbation theory and
using a zeroth order solution of the vortex near a rigid
surface (corresponding to ~ approaching 0), one can
derive the following boundary conditions for the free
surface. (See Fish or Fish and von Kerczek7 for
details.)
Dynamic:
Kinematic:
where:
~ '+°StP X + Tl =0 (3)
(I y+ll '+¢oX11 Xll¢Oyy=o (4)
=40~f 1
=~0+~q 1
It should be noted here that these equations include
the first order Taylor series expansion of each term
about its evaluation on the still water level
(corresponding to the rigid surface boundary). Note
also that the O.S¢x term in tl~e dynamic boundary
condition and the Foxed term in the kinematic
boundary condition represent the influence of induced
surface currents on the modification of ambient
waves. In particular, the t0~q,: term provides a direct
coupling of the vortex induced surface current and the
ambient wave slope. These terms are excluded in the
classical free surface linearizations utilizing the
quiescent zeroth order base state.
For the problem here, the ambient waves are
propagated from the side of the 2D domain as shown
in figure 2. These waves represent ambient waves
moving transverse to the ship track. Because of Me
quasi2D nature of the ship wake model, modelling
waves propagating in directions much different from
this lateral condition should not be assumed practical.
The solution method used to march through
the 2D computations utilizes the widely used
Boundary Integral Technique (BIT) of Trygvasson8,
Telste9 and others. The BIT utilizes the Cauchy
Integral Theorem to describe potential flows such as
this one by the values of the potential and stream
function on the boundaries of the fluid domain.
r ~ dz+iaO~`zO t,=0 USE
where:

= velocity potential
= stream function
a0 = exterior angle between boundary
tangents on either side of zO
The boundary is discretized for numerical integration
assuming a continuous linear variation in potential
and stream function over each element. In the present
study, the boundaries are composed of the free
surface, two vertical side walls and a flat bottom. At
each time step Cauchy's Integral Theorem is used to
determine the equilibrium unknown {i or yri on a
boundary element in which the corresponding Hi or (i
is prescribed. This is achieved by solving the set of
simultaneous equations resulting from the application
of equation 5 to the node point, ZOi, of each element
of the boundary.
The prescribed values of by and ~ for He next
time step are determined as follows. First, the (i
distribution over the free surface elements is obtained
by integrating the dynamic free surface boundary
condition.
((t+At) = t(~+~°.Stox+loxt 1 x+ ~ ~
Hi on the free surface is also updated by integrating
the kinematic boundary condition.
11~+~) =11~+~§ y~0xTl X+llto'y) (7)
Then the new distribution of Wall on the side and
bottom boundaries is prescribed by the undisturbed
deep water ambient wave stream function.
Wall+ = gaff) Cos(eXt+At)k~i) (8)
where:
k = 27~/\ ~ = ~
A comparison of calculation methods for the
surface disturbance caused by a single vortex moving
at constant speed below the free surface is given in
figure 3. One can see that the method used here
compares very well with the fully nonlinear
calculations of Telste9 while calculations utilizing the
classical zero flow base state for linearization are in
error.
In determining the relative importance of the
additional terms in equations 3 and 4 with respect to
the classical linearized form, equivalent simulations
were conducted with each term deleted. It was found
in this study that the terms containing ¢0~`,
representing the induced surface current from the
vortices were significant contributors to the solution.
The 1140',~, value remained small. In addition, the
cross term +0x¢1~ provides a direct interaction term
between the ambient waves (whose potential will be
contained in ~ 1 and the submerged vortices).
Without this term calculations were almost equivalent
493
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to the superposition of independent calculations of the
vortices and waves. Figures 4a and 4b show the
influence of the induced current terms on the ambient
wave modification. This figure shows free surface
profiles at discrete time intervals corresponding to the
period of the ambient wave, T. which is propagating
frown left to right. A counter rotating pair is
positioned in the center region as shown in figure 2.
The details of the flow parameters will be described
further in the next section.
3 ~
 Clossicol Uneorbed
 Alternate Uneorized
Nonlineor ~elate)
2.0 l is 1.0 0.5 o.o
X/S
o.s 1.0 1 S 2.0
Fig. 3 Free Surface Profiles Above Single Vortex
£ = 0.5
Cal _
~ _
c ~
~ .= ~
! A,~ ~ ~ rem
. ~ ~ Am, ~ ~ ~,
_ _ _ _ _ _ 7_ ,_
, ~ ~
! of of
! of
.
.
! ACE' ~
! ~' an
, ~ Elf
.
! of ~
. ~ ~

.
. ~ .
. ~ ~
_ ~ ~ ~f ~ ~
' ~ 5 ~ 3  2  1 O 1 2 3 ~ 5 ~ 7
X/}
Fig. 4a Surface Profiles with Vortex Pair: Classical
Linearization
_%
cat
o
u
vet
c
·o
a
·c
Nb~
~~W^~

I `~
`~ IT
NOT
7 ~ 5 ~ 3  2  1 0 1 2 3 ~ 5 ~ 7
X/x
12T
Fig. 4b Surface Profiles with Vortex Pair: First Order
Theory with Induced Current Terms
NONDIMENSIONAL PARAMETER
IDENTIFICATION
When ambient waves are included in the
vortex/free surface problem, the characteristic length
scale should correspond to the ambient wavelength.
This wavelength is of primary concern since
eventually it will be this scale which determines the
character of radar return from the free surface. The
problem therefore is nondimensionalized as follows:
(where ' indicates a dimensional quantity)
x=x 'I y=y'~ M~'~
t=t,\2w ¢=¢,~
(9)
The governing inputs to the problem are A, g, A, S.
D, and \. Dimensional analysis reduces these six
variables to four parameters. The choice of parameter
definitions must be made with care to allow
meaningful conclusions to be drawn. The ambient
waves, for instance, are described by their steepness,
Am.
Since the paths taken by the vortices will be
defined by their "rigid wall" paths and therefore are
known apriori, one choice for a parameter would be
the ratio of tile initial separation distance to initial
depth, S/D. Small values of S/D indicate that the
vortex pair is initiated at greater depth than
separation, and will initially propagate upwards
toward the free surface. Large values of S/D
represent a vortex pair which is widely separated and
will move parallel to the free surface under the
primary influence of their image vortices above the
free surface. One then needs to define either S/\ or
D/\ as the other parameter relating the vortex
geometric configuration to the ambient wavelength.
Most of the cases studied here are for small values of
S/D, giving a minimum depth achievable by the
vortex of approximately S!2. Since this will
correspond to the highest value of induced surface
current by the vortices, SiL was chosen as the second
parameter.
The last parameter gives the relative scales of
the vortex strength to the wave velocity, and is
commonly called the Froude number. The Froude
number for this problem can be defined in several
ways depending on one's choice of length scale.
Previous works without ambient waves have utilized
the initial vortex separation following the definition
F=e of equation (21. Since it has been shown that He
induced surface current has a dominant influence on
the modification of waves, it would appear prudent to
define a Froude number based on the ratio of this
induced current and the group velocity of the ambient
waves. The maximum induced velocity (including
image vortex above free surface) is given by:
vi=2I~/7rS which occurs at D=S/2 (10)
494
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and occurs after the vortices have approached the free
surface and separated. The deep water group velocity
for the ambient waves is
Vg=0.5: 2gh (11)
The ratio V'/Vg will be used as the Froude number,
F. and can therefore be formed as shown below.
4r
V
i AS

4~;7 ~
Vg ~ ~ So (12)
27
with r = ~ r vie
So ~ ~
F=r~(gS3~1/2 P=A~ Q=S/\ R=S/D
where ~ is the vortex circulation, S is the initial
vortex pair separation, D is the initial vortex pair
depth, ~ is the ambient wavelength, and A is the
ambient wave amplitude. Substituting these relations
into the previously defined ALM boundary conditions
gives the following results:
Dvnamic:
Kinematic:
fit = ¢o~/2  A; 1l/`F2 Q3> (14)
ill = 4'y + 11§oyy ~Q~llx (15)
The corresponding deepwater wave potential using the
above nondimensional parameters takes the form:
law = P/(F Q3/2 `271;~1/2' e2nY
sin [2=  `2~1/2t/`F Q3/2~ (16)
The vortex potential is also nond~mensionalized and
given below in complex form:
p° = to + into  i 1 in [(ZZ1~/(ZZ211
i1 in [(ZZ 1~/(Zz 2~] (17)
where the "*" character represents the complex
conjugate.
FROUDE NUMBER VARIATION
The examination of Froude number influences
will consist of flow simulations with the vortices
started at a relatively deep position (S/D=1/31. This
value of S/D was chosen to m~nize surface
disturbances associated with the transients of the
vortices' impulsive start. The ambient wave
modifications would therefore be associated
predominantly with the velocity field generated by the
slowly moving vortices rather than the initial
condition wave motion generated by the vortices.
S/D=1/3 also corresponds well with the experimental
studies of Willmarth et.al.l°. S/\ is set to 1 to equate
Froude numbers based on different length scales. The
resulting surface profiles for F ranging between 0.1
and 0.6 are shown in figures 5 to 9. Since the time
scale of modification is much slower at F=0.1, the
profiles shown in figure 5 coincide with 2T intervals
rather than the IT intervals used at the other Froude
numbers. The vortex path is shown in figure 10. The
surface modification in these profiles involves a
stretching of waves in the center region above the
vortices at later times. The ambient waves also
appear to be shortened somewhat on the left or
"windward" side due to the opposite directions of vi
and vg. The wave shortening on the right or
"leeward" side is much less pronounced since vi and
vg are in the same direction.
The most obvious effect of varying F in these
cases is the change in time scale for the surface
disturbance to develop. The occurrence of ambient
wave modification at earlier times at higher F is due
primarily to the increase in relative speed of the
vortex pair moving towards the surface. To minimize
this attribute, surface wave profiles can be compared
between different F cases when the vortex pairs are at
common positions. Two common positions are
labeled "A" and "B" in figure 10. The corresponding
profiles from each F are collected and shown in
expanded vertical scale in figures 11 and 12. Case
"A" corresponds to a moderately deep vortex pair.
Only the higher F cases corresponding to relatively
stronger vortices have an influence at this depth,
verifying the notion that stronger vortices will
influence the surface from greater depths.
A more subtle and interesting difference is
observed as the vortex pair draws near the surface at
position "B". Here one notices the lengthening of
residual ambient waves in the central region above the
vortices. This increase in residual wave length is
presumably caused by the higher tangential velocity
gradients at the surface in cases where F is increased.
TO width of the disturbed region above the vortex
pair does not depend heavily on the value of F.
Figure 13 shows the vortexinduced current for
condition B without waves. For any threshold value
of v~/vg, one would expect from this figure that
increasing F would cause a greater increase in the
affected region width than the simulations actually
495
OCR for page 491
en
n
17.
.C
CL
·c
CR
C`i
lo
c
u
01
·C
x/x
Fig. 6 V~/Vg Variation: V~/Vg = 0.2
.
Fig. 7 V~/Vg Variation: V,/Vg = 0.4
x/x
V IV_ Variation: V /V. = 0.1 Fig. 9 VI/VF Variation: V~/V`~ = 0.6
12T
IT
OT
6T
AT
OT
Fig. 8 Vi/Vg Variation: Vi/Vg = 0.5
IT
IT
OT
x/x
Fig. 10 Vi/Vg Variation: Vortex Paths
show in figure 4b. The reduced dependency of
affected region width on F suggests that the time
scale of the free surface interaction is relatively short,
and that using slowly varying current distribution
theories such as the conservation of wave action is
inadequate for treatment of problems of this type.
This further emphasizes the inherent transient nature
of the wave modification by vortices of the scale used
to model ship wakes.
An additional note on the variation of F is the
transition occurring around values of 0.5 of the
surface disturbance directly above the vortex pair. At
values above 0.5, the creation of a center "humps
appears similar to center humps calculated by Telste9
at higher Froude numbers without ambient waves. It
should be noted that in this case the vortices have not
deviated from their rigid wall paths. This indicates a
transition Froude number value of 0.5 above which
the vortex pair begins dominating the surface profile.
INITIAL GEOMETRIC CONDITIONS
The initial position of the vortex pair is
equally important to its strength in defining the form
of ambient wave modification. As noted previously,
the ratio of induced surface velocity to ambient wave
group velocity is highly dependent on the depth of the
vortex. Time scales for the vortex pair to approach
the surface from large depths is also obviously
dependent on the initial depth and separation distance.
496
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En
02
O ~
0.2
7 
_
\/
. ~
b
bL
r
\J ~\J
\J ~
_ v L
5 4
Fig. 11 Vi/Vg
no
0.2
V
\J
\J
\J
V
~ V
Fig. 12 Vi/Vg
~ 74 i~
~ ~ ~ ,
Err ;T~ ~ ~
8 
r
_ J V
3 2 1 0
X/\
Variation: Position"A"
2
od
4
_ _
~ r
. v
3 2
VL
/ V
fY
\
1
~
A
' ~1
1 o
X/)
Lx Position A l
h.~
I I I F ~
IA h ^/1
W1 1 F
NVI F ~
3 4 5 6 7
\ /
0.4
0.2
0.1
'' F=Vi/Vg
ax P
\
A
)
~ ~ Vortex Ptadi~n B ~ ~
~ 1 ~ ^ 1 ^ A ~ ~
red in
OCR for page 491
1 2 3 ~
2
3
_s
 ~3 2
S/D
· 0.5
· 2.0
· 6.0
1 0
x/)
Fig. 17 S/D Variation: Vortex Paths
current. This wave alteration is also noticed to be
almost symmetric about the line of symmetry for the
vortex paths. This indicates little net exchange of
energy between the vortices and the waves as they
pass through. As the depth is decreased, the vortex
induced surface current grows to the value of the
group velocity of the ambient waves, and essentially
stops the transfer of ambient wave energy from left to
right. Two important processes occur which deserve
further attention. First, the ambient waves are
shortened as they enter the region of influence of the
vortex pair by the opposing induced current. The
resultant decrease in ambient wavelength is
accompanied by a decrease in the absolute group
velocity of the waves. The result of this process, is
that ambient waves of higher speed may be halted by
vortices of smaller magnitude than that predicted from
v~/vg=1 .
S/X Variation
As mentioned in the S/D section above, initial
vortex pair location is important in examining the
influences on ambient surface waves. In addition to
specifying the portion of the path that the pair will be
started on, a particular path must be chosen. The
parameter S./\ is used as a path selection parameter
for its role in scaling the vortex path geometry to the
ambient wavelength. Three values of S/\ were
simulated with a constant value of S/D=6. S/\ values
were set to 3.0, 4.5, and 6.0. The upper limit on S/\
t....
. . . . . . .
TV van
~ ~ Wl1
For
J VV\/
1 ~ret ~ ~
~ lIV\J
V
r
7 B 5 ~ 3  2 1 o 1 2 3 4 s ~ 7
x/x
Fig. 18 S/\ Variation: Final Profiles t=8T
is caused by the size limitation of tile computational
domain, and the influence of partial wave reflections
on the right hand boundary. F varied between 0.06
and 0.125 to preserve a v'/v at the surface of 0.5.
The surface profiles after 818 for each condition are
shown in figure 18. Note the growth in modified
wave region size as S/\ is increased.
An additional process observed in the
simulations is the dynamics of waves caught in the
central influence region during the initial Notion of
the vortex pair. These surviving center waves may be
leftover from the initial ambient wave profile in this
region in the shorter tinge scales. These left over
waves are found in greater numbers as S/\ is
increased, as shown in figure 18. In the cases shown,
a wave blocking action is rapidly generated due to a
relatively large value of v'/vg. Wave energy may also
propagate into the center region when the vortex pairs
are initialized at low values of S/D. This energy
propagation is facilitated by the canceling influence of
the induced tangential velocity from each of the
counter rotating submerged vortices at low values of
S/D. The evolution of these center waves is
particularly complex in cases of moderate S/D. Initial
waves in this region are typically stretched by the
surface current gradient. The resulting increase in
wavelength gives rise to an increase in phase and
group velocity. Because the gravity wave dispersion
relation specifies that longer waves move at higher
velocity, die central region may be evacuated over a
shorter tine than that predicted by ambient wave
group velocity. Simulation results were not carried
far enough in time to show complete smoothing of the
central region due to the influence of partial wave
reflection from the right side boundary. Larger
surface grids and greater computer times will
therefore be necessary to validate this hypothesis.
WAVE STEEPNESS AND PHASE VARIATION
The wave steepness and phase with respect to
the vortex pair are additional parameters which result
from the problem formulation and enter the
calculations through the initial boundary condition on
the free surface and the time varying stream function
prescribed on the side boundaries. Simulation results
show no significant influence of these parameters on
the form of the interaction. In the case of wave
steepness, the waves and their modification can be
linearly scaled with AN for values less than 1/10
corresponding to the linearized ambient waves being
described. The relative phase of the ambient wave
was also found to have no effect on the general forth
of the wave modification at the S/X values studied
here.
498
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SPECTRAL ANALYSIS
In order to make the previous findings useful
in the study of real ocean flows, a method of analysis
must be developed which illuminates the impact of
wave modification on radar images. It was shown in
the introduction how the redected radar intensity is
proportional to the amount of Bragg waves present in
the region being illuminated by the radar. This
intensity is usually expressed as the backscattering
cross section per unit surface area, c,O. This cross
section is derived by Valenzuela1 in the well known
Bragg scattering expressioll:
o0 = 47~4cos48 Feed M(2krsin8) (18)
where: kr = radar w~avenumber and ~ = vertical
incidence angle of radar. Here Feel is a scattering
coefficient dependent on the radar polarization and
incidence angle, and M() is the component of the
wave spectra corresponding to the Bragg condition.
Remember, from the introduction, that the Bragg
condition is satisfied when kw (surface wavenumber
component in the look direction of the radar) is equal
to the argument of M() in the equation above. If ~
and kw are known constants, then DO is proportional
to Mel. This property indicates that a relative
variation in radar backscatter over different sections
of the water surface can be determined by the
variations in the Bragg wave spectral component
among the respective surface sections. Estimates of
the relative radar intensity image can therefore be
compiled by performing finite Fourier transforms of
the surface elevation over regions corresponding in
size to the resolution cell size of the radar. This will
be performed here by sweeping a window across the
simulated surface data and calculating the finite
Fourier transform at various window positions. The
Fourier transform component corresponding to the
Bragg waves in each window position is then
assigned to the midpoint location of the window. The
resulting distribution of Bragg wave components over
the surface is then contour plotted for comparison
with SAR imagery.
Method
The finite Fourier transform used to determine
the Bragg wave spectral content in a data window is
given by:
L/2
Y(k,L)=  1l(x)e~i2~k~dx
J
L/2
(19)
where L is the window width. The application of this
equation to discrete data can be performed using an
algorithm known as the Fast Fourier Transform, or
FFT. This cannon method is described in detail by
Bendat and Piersoll2. The For output coefficients
are complex in form providing phase information for
each wavenu~nber component. Since this phase was
shown to be unimportant in the previous chapter, the
absolute magnitude of each wavenu~nber component
is formed from the modulus of the real and imaginary
parts:
M(ki=(Y 2rki ~ A. 2`k~ 1/2 (20)
 I, treat v~, ~ lmag. '''"
where: FFT(k) is an output coefficient.
Before applying the scanning FFT window to
simulation results, several comments must be made
here to avoid misinterpretation of the method. The
regions used in computation of the spectra will be
overlapped here because of the finite extent of the
surface domain and the desire to have moderate sized
surface patches. If the patches are allowed to be too
small, the resolution of low wavenumber components
decreases, resulting in undistinguishable movements
of wave energy among the wavenumbers of concern.
This overlapping is not representative of radar
imaging, and therefore the results must not be
considered direct simulation of radar operation. The
general trends in the resultant component levels in the
Bragg wavenu~nber range are believed to be
somewhat representative of results obtained with radar
processing.
Application to Ship Wake
The case examined using this teclmique is a
simulation approximating the wake velocities from a
twin screw destroyer calculated by Sweenl3. It
should be noted that in simulating the full ship case,
the grid resolution was lowered considerably by
constraints in available computer memory. The grid
density is 32 points per ambient wavelength. The
ambient wavelength is set to 25 cm and corresponds
to the Bragg scattering wavelength for an "L band"
SAR operating at an incidence angle of 30 degrees.
The initial vortex separation and depth are 5 m and 2
m respectively. Gravitational acceleration is equal to
9.8 mls2, and the ambient wave amplitude is set to
2.5 cm. The vortex circulation strength, r, is chosen
to give approximately the same maximum surface
current reported by Sweenl3 for the given depth of
submergence. The resulting r value was 1.0. The
simulation domain extended to + 6.25 m in the x
direction with a depth of 12.5 m. The simulation was
carried out for 8 periods of the ambient wave (3.2
seconds) and the resultant profiles at each half period
are shown in figure 19. The corresponding
downstream distance spanned by the simulation is 25
m for a ship speed of 15 knots.
_
The FFT window size was chosen to be 2 m
wide, which is typical of SAR resolution cell
dimensions. The resultant spectral resolution is
3.14/cm in wavenumber giving components at
wavelengths of 22 cm, 25 cm, and 28 cm. The
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consecutive FFT windows were overlapped in the
analysis by 3 \. This overlap gives denser data for
examining the variations in Bragg component, though
only nonoverlapping sections would be directly
comparable to typical radar imaging techniques. The
Bragg component of each spectrum is assigned to the
location of the FFT window center. The resulting
contour plot of the magnitude of the Bragg
component versus space and time is given in figure
20. This plot indicates a reduction in Bragg wave
component in the center region as time progresses. In
order to understand the energy flow path in this
region, figure 21 is constructed. This figure shows
the progression of the spectra as the window is
traversed across the contour plot at the 2 second time
frame. The thick line indicates the wavenumber
associated with the Bragg wavelength. Figure 21
illuminates the shift in wave energy to lower
wavenumbers in the center region. Since these
wavenumber components do not satisfy the Bragg
scattering criterion, the resultant received energy at
the radar is decreased. This phenomenon of
wavenumber shifting due to induced surface currents
appears to produce a weaker radar return signal in the
region between the vortices. In the ocean, the wave
field can be described by the superposition of many
wavelength waves. PiersonMoskowitz spectral
models of the sea show a decreasing level of energy
as wavenumber is increased in the range of the L
Band Bragg scattering waves. The implication of this
is that though higher wavenumber (shorter) waves
will be stretched into the Bragg wavelength, their
original energy is lower. Thus one would expect that
a larger amount of energy was shifted out of the
Bragg sensitivity window than was being shifted in.
The inclusion of these higher wavenumbers would
however, decrease the magnitude of variation in M()
shown in figure 20. The importance of this action is
that it produces the proper trend in the surface
response in the wake region of a ship and may be a
significant contributor to the overall signature
production process for the dark centerline wake.
_~ .
:~
~ .
it_
or.
0 it__
~
25  20  1 S 1 0  5
0 5
x/\
10 15 20 25
Fig. 19 Ship Wake Simulated Surface Profiles
IT
M(k,~)
a
74A
is o
_ ..
_ ,,.
° Time (8ec) 3.2
Fig. 20 Contours of Bragg Wave Content
'it
k
kBragg
Fig. 21 Crosswake Variation in Wave Spectra,
t=2 see
SUMMA RY
The modification of surface waves by
submerged vortices has been explored here in a
straightforward progression from analytical
examination through numerical simulation of a simple
vortex ambient wave model. The motivation for
study of this problem is the determination of ship
wake impacts on synthetic aperture radar images
showing dark centerline regions behind ships moving
at sea. This process has led to several discoveries in
the nature of the interaction process between vortices
and ambient waves.
A perturbation analysis of the free surface
boundary conditions about the zero Froude number
base state shows that additional terms (not found in
the classical free surface linearization for small
amplitude waves) are needed to describe the first
order interaction of vortices with the free surface.
These terms show the importance of the vortex
induced surface current on the modification of
ambient waves. Example simulations show that
neglect of these induced current terms give much
smaller modification of surface waves.
The simulation program has been used to
examine the specific flow configuration of a vortex
pair impulsively started below a free surface
containing a sinusoidal ambient wave train. This
examination determines which parameters are
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significant in determiriing the form of the ambient
wave modification. It is found that the Proude
number of the vortices and their position relative both
to each other and the free surface play the dominant
roles in modifying ambient waves. The ambient wave
steepness and initial relative phase with respect to the
vortices have only secondary influence in the surface
modification.
The simulation is then extended to represent
the simple twin vortex wake model of a generic ship
wake. The results of this simulation are analyzed
using a scanning Fast Fourier Transform window on
the surface to obtain spatial distributions of the
spectral content of the surface. These spatial
distributions are filtered to extract the Bragg
scattering wave component and contour plotted. The
resulting image shows that the current induced shift in
wave energy to longer wavelengths can produce an
effect consistent with the observed dark centerline
wake regions of moving ships in radar images.
ACKNOWLEDGEMENT
This work was sponsored by the ONT
Independent Research Program administered by the
David Taylor Research Center.
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2. Peltzer, R., and W. Garrett and P. Smith, "A
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