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OCR for page 21
Ship Motions by a ThreeDimensional
Rankine Pane! Method
D. Nakos, P. Sclavounos (Massachusetts Institute of Technology, USA)
ABSTRACT
A Rankine Panel Method is presented for the solution of
the complete threedimensional steady and timeharmonic
potential flows past ships advancing with a forward veloc
ity. A new freesurface condition is derived, based on lin
earization about the doublebody flow and valid uniformly
from low to high Froude numbers.
Computations of the steady ship wave patterns reveal sig
nificant detail in the Kelvin wake a significant distance
downstream of the ship, permitted by the cubic order and
zero numerical damping of the panel method. The wave
pattern appears to be sensitive to the selection of the free
surface condition only for full ship forms.
The heave and pitch hydrodynamic coefficients exciting
forces and motions of a Wigley and a Series60 hull have
been evaluated in head waves over a wide range of fre
quencies and speeds. A robust treatment is proposed of
the mterms which are found to be critical importance
for the accurate solution of the problem. In all cases the
agreement with experiments is very satisfactory indicating
a significant improvement over strip theory, particularly in
the crosscoupling and diagonal pitch damping coefficients.
1. INTRODUCTION
Theoretical methods for the prediction of the seakeeping of
ships have evolved in three phases over the Dast 40 Years.
The first phase involved the development of strip theory,
and was followed by a series of developments in slender
body theory which formulated rationally the ship motion
problem and produced several refinements of striD theory.
The advent of powerful computers in the early 80's al
lowed the transition into the third and current phase of
seakeeping research which aims at the numerical solution
of the threedimensional problem. This paper presents our
progress in that direction.
The pioneering work of KorvinKroukovsky (19S5) stimu
lated a number of studies on the strip method which led to
the theory of Salvesen, Tuck and Faltinsen (1970~. Its pop
ularity to date arises from its satisfactory performance in
the prediction of the motions of conventional ships and its
computational simplicity. Well documented are however
its limitations in the prediction of the derived responses,
21
structural wave loads and in general the seakeeping char
acteristics of ships advancing at high Froude numbers te.g.
O'Dea and Jones (19833~.
The 60's and 70's witnessed several analytical studies aim
ing to extend the slenderbody theory of aerodynamics to
the seakeeping of Slender ships. The rational justification
of strip theory, as a method valid at high frequencies and
moderate Froude numbers, was presented by Ogilvie and
Tuck (1969~. This theory was extended to the diffrac
tion problem by Faltinsen (1971) and was further refined
by Maruo and Sasaki (1974~. The highfrequency restric
tion in earlier slendership theories was removed by the
unified theory framework presented by Newman (1978~.
Its extension to the diffraction problem was derived by
Sclavounos (1984) and applied to the seakeeping of ships
by Newman and Sclavounos (1980) and Sclavounos (1984~.
Subsequent slendership studies by Kim and Yeung (1984)
and Nestegard (1986), accounted directly for convective
forwardspeed wave effects near the ship hull and repre
sented the transition to numerical studies aiming at the
solution of the threedimensional shipmotion problem.
By the mid80's, the performance of slenderbody theory
for the seakeeping problem could only be validated from
experimental measurements. Moreover, it had become ev
ident that endeffects at high Froude numbers cannot be
modelled accurately by slenderbody approximations and
the need for a numerical solution of the complete three
dimensional had emerged. Early efforts towards this coal

by Chang (1977), Inglis and Price (1981) and Guevel and
Bougis (1982) were not conclusive because the significant
computational effort necessary for the evaluation of the
timeharmonic forwardspeed Green function limited the
total number of panels used on the ship surface. More
recently, King, Beck and Magee (1988) circumvented this
difficulty by solving the same problem in the time domain,
therefore making use of the zerospeed transient Green
function which is easier to evaluate.
The last decade witnessed the growing popularity of Rank
ine Panel Methods for the solution of the steady poten
tial flow past ships. The success of the early work of
G add (1976) and Dawson (1977) motivated several anal
ogous studies which concentrated upon the prediction of
the Kelvin wake and evaluation of the wave resistance. The
principal advantages of the method are twofold  the Rank
ine singularity is simple to treat computationally and the
distribution of panels over the free surface allows the en
forcement of more general freesurface conditions with vari
OCR for page 21
able coefficients. A drawback of Rankin~panel methods is
that they require about twice as many panels as methods
based on the distribution of wave singularities over the ship
surface alone. The resulting computational overhead is as
sociated with the solution of the resulting matrix equation,
but may not be significant if an outofcore iterative solu
tion method is available.
This paper outlines the solution of the threedimensional
timeharmonic ship motion problem by a Rankine Panel
Method. For the steady problem, the theory for the anal
ysis of the properties for such numerical schemes was in
troduced by Piers (1983) and generalized by Sclavounos
and Nakos (1988~. The extension of this numerical anal
ysis to the timeharmonic problem is presented in Nakos
and Sclavounos (1990~. In this reference the convergence
properties of a new quadraticspline scheme are derived,
which has been found to be accurate and robust for the so
lution of both steady and timeharmonic freesurface flows
in three dimensions. This scheme is applied in this paper
to the solution of the timeharmonic radiation/diffraction
potential flows around realistic ship hulls and the evalu
ation of the hydrodynamic forces and motions in regular
head waves.
A new threedimensional freesurface condition is derived,
using the doublebody flow as the base disturbance due to
the forward translation of the ship. This is shown to be
valid uniformly from low to high Froude numbers and over
the entire frequency range. Known lowFroudenumber
conditions for the steady problem, as well as the Neumann
Kelvin condition, are obtained as special cases. The ship
hull condition includes the mterms which are evaluated
from the solution of the threedimensional doublebody
flow. An important property of the solution scheme is that
the evaluation of the double gradients of the doublebody
flow is circumvented by an application of Stokes theorem.
Computations are presented of the steady wave patterns
trailing a fine Wigley model and a fuller Series60 hull.
The cubic order and zero numerical damping of the free
surface discretization allows the prediction of significant
detail of the Kelvin wake at a large distance downstream
of the ship. A comparison of the wave patterns obtained
form the NeummanKelvin and the more general double
body freesurface conditions reveals good agreement for
the Wigley hull, while evident differences appear in the
respective Series60 wakes.
Predictions of the heave and pitch addedmass and damp
ing coefficients and exciting forces are found to be in very
good agreement with experimental measurements both for
the Wigley and the Series60 hull. The contribution of
the complete mterms is found to be important, partic
ularly in the crosscoupling coefficients. The validity of a
more general set of TimmanNewman relations is observed
and conjectured in connection with freesurface conditions
based on the doublebody flow.
The heave and pitch motion amplitudes and phases pre
dicted by the present method are found in very good agree
ment with experiments and present an improvement over
strip theory.
2. THE BOUNDARY VALUE PROBLEM
Define a Cartesian coordinate system x = (x,y,z) fixed
on the ship which translates with a constant speed U.
The positive xdirection points upstream and the posi
tive zaxis upwards. The boundaryvalue problem will be
expressed relative to this translating coordinate system,
therefore the flow at infinity is a uniform stream and the
ship hull velocity is due to its oscillatory displacement from
its mean position.
The fluid is assumed incompressible and inviscid and the
flow irrotational, governed by a potential function ~(x,t)
which satisfies the Laplace equation in the fluid domain
V2~(X,t) = 0 ~
(2.1)
Over the wetted portion of the ship hull (B), the compm
nent of the fluid velocity normal to (B) is equal to the
corresponding component of the ship velocity VB, or
[} (x,t) = (VB · n)(x,t),
where the unit vector n points out of the fluid domain.
(2.2)
The fluid domain is also bounded by the free surface, de
fined by its elevation x = `(x, y, t) and subject to the kine
matic boundary condition,
(fit +V~I ~V) Liz~(x,y,t)] = 0 on z = ~(x~y~t)
(2.3)
The vanishing of the pressure on the free surface combined
with Bernoulli's equation, leads to the dynamic free surface
condition
`(x, y, t) =   (~' + 2V~ V4i  u2~1
2 J z= s (2.4)
The elimination of ~ from (2.3) and (2.4) leads to
(Ptt + 2V~ ~ V~t + 2 V\li ~ V(V~  V~) +g~z = 0 on z = ~ .
(2.5)
If the fluid domain is otherwise unbounded, the additional
condition must be imposed that at finite times the flow
velocity at infinity tends to that of the undisturbed stream.
Linearization of the free surface condition
Physical intuition suggests that linearization of the pre
ceding boundary value problem is justified when the dis
t,~rbance of the uniform incoming stream due to the ship
is in some sense small. Small disturbances may be justi
fied by geometrical slenderness, slow forward translation,
or a combination of the above. Fullshaped ships typically
advance at low speed and cause a small steady wave distur
bance. Fineshaped ships, on the other hand, often advance
at high Froude numbers. Yet the steady disturbances they
generate, is small if their geometry is sufficiently thin or
slender. Linearization may therefore be justified both at
low and high Froude numbers F, as long as it is tied to the
hull slenderness c. Linearization of the unsteady flow is
also supported by the assumption of a small ambient wave
amplitude.
The linearized free surface condition derived next is uni
formly valid between these two limits, and it~ validity is
22
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heuristically justified if the parameter eF2 is sufficiently
small. The details of the derivation outlined below are
given in Nakos (1990~. The total flow field ~(x,t) is de
composed into a basis flow ~(x), assumed to be of O(1), the
steady wave flow Add, and the unsteady wave flow ¢(x,t)
~(x, t) = ~ + ¢~ + ¢(x, t) . (2.6)
The doublebody flow is chosen as the basis flow, a selection
primarily motivated by the body boundary condition as
well as the simplifications it allows in the ensuing analysis.
Thus, ~ is subject to the rigid wall condition:
As = 0 , on z = 0 . (2.7)
The wave disturbances ¢, and `6 are superposed upon the
doublebody flow and are taken to be small relative to the
A. Linearization of (2.45), correct to leading order in
and ¢, leads to the conditions:
Vie VIVA V¢~+2V(V4) V~·V++g~z
 ~X2(V~ · V¢) =  2V(V~ · Vie) · V~
2 (U2Vie Vow, on z = 0
~ (~:E Y) = go (2Vq) Vie 2u2 +Vqi v¢~) ~
' + 2Vq} Vet + V4) V (V4} ~ V¢)
+V(V~ V~) V¢+gjx
~ + Vie V¢) = 0, on z
.. ..
<(x,y,t)=~('h+v~ v¢)z=o
=0~ (2.9)
for the steady and unsteady flows, respectively.
For slender/thin ships with ~ small, and for Froude num
bers of O(1), the uniform incident stream Up may be used
as the basis flow. In this case, (2.89) reduce to the well
known NeumannKelvin conditions. In the opposite limit
of blue ships with ~ of O(1) advancing at low Froude num
bers, (2.89) reduce to the conditions of slowship theory.
The condition (2.8) contains all terms present in Dawson's
(1977) condition, and it is closest to the one proposed by
Eggers (1981~. This property may explain the fact that,
even though Dawson's and Egger's conditions have been
derived as low Foude number approximations, they have
been found to perform satisfactorily over a wider range of
forward speeds.
Linearization of the body boundary condition
The linearization of the ship hull boundary condition may
also be derived from the decomposition (2.6). By defi
nition, the velocity potential of the doublebody flow is
subject to
04P = 0 , on (B) · (2.10)
Consequently, the steady wave flow also satisfies the homo
geneous condition
= 0 , on (B) , (2.11)
leaving the righthandside of (2.8) as the only forcing of
the steady wave problem.
The unsteady forcing due to the oscillatory motion of the
vessel is accounted for by the unsteady wave flow ¢. If a
is the oscillatory displacement vector measured from the
mean position of the vessel (B), it follows by substituting
of (2.6) in (2.2) that
B~¢ = ~ . n  V(~ + f) ~ n , on (B) (2 12)
Assuming that the magnitude of the displacement vector
a is small and comparable to the ambient wave amplitude,
the boundary condition (2.12) may be linearized about
the mean position of the hull surface iTimman and New
man(l962)],
B~ Ba n  [(a V)V~+(V~ V)~ n ,on ( )
(2.13)
The last term in (2.13) accounts for the interaction be
tween the steady and unsteady disturbances in a manner
consistent with the assumptions underlying the derivation
of the freesurface conditions (2.8). An alternative form
of (2.13) may be derived in terms of the rigidbody global
displacements (hi, (2, (3) and rotations (54, (5, (6), along
the axes (x, y, z) respectively,
An ~ ~ ,,` nj + gmj) , on (B), (2~14)
where my, j = 1, ...6, denote the socalled mterms tOgilvie
and Tuck (1969)~.
If the basis flow is approximated by the uniform stream the
only nonzero mterms are ma = Un3 and me = Un2,
which merely account for the 'angle of attack eEect' due
to yaw and pitch. This approximation of the mterms has
been employed in most previous studies of the ship motion
problem, consistently with the linearization steps leading
to the NeumannKelvin free surface boundary condition.
The performance of this linearization in practice will be
the subject of numerical experiments presented in section
7.
Frequency domain formulation of the unsteady problem
The unsteady excitation is due to an incident monochro
matic wave train. The frequency of the incident wave, as
viewed from the stationary frame is we, while in the trans
lating frame of reference x, the incident wave arrives at
23
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the frequency of encounter w. If ,6 is the angle between
the phase velocity of the incident wave and the forward
velocity of the ship, ~ is given by
~ = ~0  U ° costs . (2.15)
In the frame x, the velocity potential of the incident wave
of unit amplitude, in deep water, is given by the real part
of the complex potential ~0:
~o~x,t)=i 9 eg (~i2C0~iY~in') ire (2.16)
The linearity of the Boundary Value Problem that gov
erns the physical system, along with the form of the body
boundary condition (2.14), suggest the decomposition of
the wave flow as follows,
¢(x,t) = ~ {e Pablo + ('P7) + ~ (j~j~ } , (2.17)
where A is the amplitude of the incoming wave train, ~7
is the complex diffraction potential, and As, j = 1,...6,
are the complex radiation potentials due to the harmonic
oscillation of the ship in each of the six rigidbody degrees
of freedom, at frequency ~ and with unit amplitude.
Upon substitution of the linear decomposition into (2.9),
the free surface conditions for As, j = 1,...7, are derived.
It is important to point out that the free surface condition
for the diffraction problem is inhomogeneous, the forcing
arising from the interaction of the incoming wave train with
the doublebody flow. In the limit of slender/thin ships,
where the uniform stream may be taken as the basis flow,
this inhomogeneity vanishes.
3. THE HYDRODYNAMIC FORCES
Given the solution of the potential flow problem formu
lated in the preceding section, the hydrodynamic pressure
follows from Bernoulli's equation. Of particular interest,
in practice, is the pressure distribution on the ship wetted
surface and resultant forces and moments necessary for the
determination of the ship motions.
The pressure on the hull is given by
p= p [at + EVA V4i  ~u2 +9x] . (3.1)
~6 (B )
The unsteady portion of (3.1), correct to leading order in
¢, may be expressed as follows:
p = _ p (~ + Vim · V¢~(B)
p [(a V)~2 ](B) (3.2)
Under the assumption of small monochromatic motions at
the frequency of encounter A, the components of the un
steady force F = (F~, F2, F3) and moment M = (F4, F5, F6)
acting on the ship, accept the familiar decomposition
Fitt) = ~ feint Taxi + God (~2aijiwbijcalm ~ ,
{3.4]
where,
Xi = pal T/J. [iw($oO + $o7) + V4, V(`po + C°7~] nid'
(is)
aij = _ O2 ~ ~ / (imp + Vie Vail ni d8 )
(B)
bij = P ~ T  (its + V`P · V~j) ni do )
(B) (3.5)
cij = p /~(a.V)(,,v4! V~+9Z) ni ds,
J J
(B)
for i, j = 1, ..., 6.
The exciting forces Xi and the added mass and damping co
efficients, aij and bit are therefore functions of the forward
speed and the frequency of oscillation w. The restoring
coefficients ci`, on the other hand, include the classical hy
drostatic contribution augmented by a dynamic term due
to the gradients of the doublebody flow. The latter con
tribution depends linearly upon the deflection of the ship
surface from its mean position and quadratically on the
ship speed. It is therefore expected to be substantial at
high Froude numbers.
The equations governing the timeharmonic responses of
the ship follow from Newton's law. Using the definitions
(3.5) of the forces acting on the hull, the familiar sixdegree
of freedom system of equations is obtained
6
~ [w2(mij +aij)+i~bis +cij] fj =Xi, i= 1, ,6'
j=1
(3.6)
where mij is the ship inertia matrix, fj the complex ampli
tudes of the oscillatory ship displacements, and the restor
ing coefficients cij are modified to include the moments in
pitch and roll due to the corresponding displacement of the
center of gravity.
4. THE: INTEGRAL FORMULATION
Green's second identity is applied for the unknown poten
tials, A, ~ or As j = 1,...,7, using the Rankine source
potential,
G(x; x') = 2 ~ ~ A . (4.1)
as the Green function. The fluid domain is bounded by
the hull surface (B), the free surface (FS) and a cylindri
cal 'control' surface (SOO). The resulting integral equation
takes the form
24
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+(X)   ant, ~G(X;x')dx'+ll ¢(X') ~ i, Did
(FS) (FS)U (B )
 +~( , ~ G(X; X')dx' ,
(B)
x ~ (FS) U (B) · (4.2)
where ~ stands for any of the potentials it, +, As, j =
1,...,7, introduced in the preceding sections. The surface
integrals over the control surface (SO) can be shown to
vanish in the limit as (SOO ) is removed to infinity with
kept finite.
The derivatives of A, ¢' and Hi normal to the ship surface
(B) are known. The corresponding vertical derivative on
the free surface (FS) is replaced by the appropriate com
bination of the value and tangential convective derivatives,
according to the corresponding free surface condition.
Of particular interest is the treatment of the integral over
the ship hull which accounts for the mterms in the bound
ary condition (2.14). This is of the form:
i~mj G(xjx') do'
(B)
, j=1, ,6 . (4.3)
The evaluation of the mterms in (4.3) requires the com
putation of second order derivatives of the doublebody po
tential ~ on the ship hull. When it comes to the evaluation
of gradients of the solution potential, loworder panel meth
ods are known to be sensitive to discretization error, unless
their implementation and panel distribution is carefully se
lected. The evaluation of double gradients of the solution
are known to introduce serious difficulties, as illustrated by
Nestegard (1984) and Zhao and Faltinsen (1989).
Here, an alternative expression for the evaluation of the in
tegral (4.3) is derived by an application of Stokes' theorem.
Given that the basis flow ~ satisfies a zero flux condition
on the ship hull and the x = 0 plane, it follows that, for
j = 1,,6,
i/ mj G(X; X ) do =  // IV~(X ) · V~iG(X; X )] ni do
(B) (B)
(4.4)
The righthand side of (4.4) involves only first derivatives
of ~ on the hull, consequently it is clearly superior from
the computational standpoint.
The integral equation (4.2) will not accept unique solutions
unless a radiation condition is imposed enforcing no waves
upstream. In practice the solution domain of (4.2) on the
z0 plane will be truncated at a rectangular boundary
located at some distance from the ship where appropriate
'end conditions' will be imposed enforcing the radiation
condition. Due to the convective nature of the flow, the
condition at the upstream boundary is the most critical
and takes the form
(in  Ups ) ~ = (in  U,~8l ) ~ = o, (4.5)
where ~ stands for either the steady or the unsteady wave
disturbance. The origin and physical interpretation of these
two upstream conditions are discussed in detail in Sclavouno
and Nakos (1988) for a tw~dimensional steady flow, and
are extended to timeharmonic flows in Nakos (1990). It is
shown that both are necessary in order to ensure physically
meaningful numerical solutions of the steady and unsteady
problems. For ~ = wU/g > 1/4 no wave disturbance is
present upstream of the ship and the conditions (4.5) can
be shown to enforce this property of the flow. For ~ < 1/4
and with increasing E,roude numbers, the amplitude of the
waves upstream of the ship decreases relative to that of the
trailing wave pattern and conditions (4.5) perform well if
the truncation boundary is sufficiently removed from the
ship. No conditions are necessary on the transverse and
downstream truncation boundaries.
5. THE NUMERICAL SOLUTION ALGORITHM
The solution of integral equation (4.2) for the steady and
unsteady flows is obtained using a Panel Method. The sys
tematic methodology for the study of the numerical proper
ties of Rankine Panel Methods for free surface flows devel
oped in Sclavounos and Nakos (1988) led to the design of a
biquadratic splinecollocation scheme of cubic order, zero
numerical dissipation and capable to enforce accurately the
radiation condition (4.5~.
The boundary domain  including the ship hull and the
free surface solution domain  is discretized by a collection
of plane quadrilateral panels See Figure 1~. The unknown
velocity potential is approximated by the linear superposi
tion of biquadratic spline basis functions Bid, as follows
¢~ ~ ~ aj Bj(~,
(5.1)
where Bj is the basis function centered at the j'th panel
and at is the corresponding spline coefficient. By collocat
ing the integral equation (4.2) at the panel centroids and
enforcing the upstream condition (4.5), the discrete for
mutation follows in the form of a system of simultaneous
linear equations for the coefficients as. The relation (5.1)
provides a C1continuous representation of the velocity pm
tential and may be differentiated to give the velocity field
on the domain boundaries. The free surface elevation and
hydrodynamic pressure are evaluated using the relations
(2.89) and (3.12), respectively.
The error and stability analysis of the biquadratic spline
scheme is presented in Nakos and Sclavounos (1990~. It is
based on the introduction of a discrete dispersion relation
governing the wave propagation over the discretized free
surface. Comparison of the continuous and discrete dis
persion relations allows the rational definition of the con
sistency, order and stability properties of the numerical
solution scheme. It is shown that the numerical dispersion
is of O(h3) where h is the typical panel size and that no
numerical dissipation is present. Both are valuable prop
erties for the computation of ship wave patterns which are
not substantially distorted, damped or amplified by the
numerical algorithm.
Essential for the performance of the method is a stability
condition restricting the choice of the grid Froude number
Fh = U/~ relative the panel aspect ratio, c' = h=/hy,
where hr~hy are the panel dimensions in the streamwise
25
OCR for page 21
and transverse directions respectively. This condition, 6. STEADY AND UNSTEADY Stile WAVE
derived and discussed in detail in Nakos and Sclavounos
(1990), establishes 'stable' domains on the (Fh,~x) plane
with boundaries dependent on the frequency of oscillation.
For a given a Froude number, a stable discretization for the
highest frequency of oscillation is stable for all lowest fre
quencies. Therefore, no regridding of the ship hull and free
surface is necessary for the solution of the time harmonic
problem over a range of frequencies. The resulting complex
linear system is solved by an accelerated block GaussSiedel
iterative scheme which makes extensive use of outofcore
storage therefore permitting the use of discretizations with
several thousand panels.
Experimental verification of the convergence of the solu
tion algorithm has been established by comparing com
putations of 'elementary' flows around singularities and
thinstruts with analytical solutions iNakos and Sclavounos
(1990) and Nakos (1990~. The convergence of the hydro
dynamic addedmass and damping coefficients is discussed
in Section 7.
PATTERNS
The forwardspeed ship wave problems formulated in Sec
tion 2 have been solved for two hull forms using the nu
merical algorithm outlined in the preceding section. This
section presents converged computations of the steady and
time harmonic wave patterns around a Wigley and a Series
60 hull.
The Wigley model has parabolic sections and waterlines,
a lengthtobeam ratio L/B = 10 and beamtodraft ratio
B/T = 1.6. The grid used for the solution of the steady
problem consists of 40xlO panels on half the hull, providing
adequate resolution of the geometry, while the panels on
the free surface are aligned with those on the hull and have
a typical aspect ratio is c' = h~/hy = i. The grid Froude
number is Fh~6.3 · F. allowing an adequate resolution of
the steady wave flow for Froude numbers as low as F = 0.20
isee Nakos (1990~. The free surface domain is truncated
at a distance cup = 0.2L upstream of the bow and one ship
length downstream of the stern. The truncation in the
transverse direction is selected at You' =0.75L, so that the
entire wave sector is included in the computational domain.
The total number of panels in the grid is 2020.
Figure 1: Discretization of the free surface and the hull for a modified Wigley model, using 1110
panels on half the configuration.
26
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Figure 2 shows contour plots of the wave patterns resulting
from the steady forward translation of the Wigley model
at F = 0.25,0.35,0.40 . Predictions based on both the
NeumannKelvin and the doublebody linearizations are
presented. Due to the slenderness of this Wigley model,
the two wave fields agree well even at high speeds. Small
differences are visible along the diverging portion of the
wave system which originates from the stern, where the
NeumannKelvin solution tends to generate steeper waves,
particularly along the caustic. The opposite appears to be
true in the 'bow wave system'. For all Froude numbers,
the calculated wavelengths are not affected significantly by
n 's
_~ 75
1
the selected linearization.
The second ship tested is the Series60Cb = 0.6 hull which
is significantly fuller than the Wigley model, with length
tobeam and beamtodraft ratios L/B = 7.5 and B/T =
2.5, respectively. The principal characteristics of the grid
used for the computations are the same to those employed
for the Wigley model.
Figure 3 illustrates the wave patterns around the Series
60 model for F = 0.20,0.25,0.35, respectively. At low
speeds (F < 0.30) the amplitude of the generated waves
are comparable  if not smaller  than the ones computed
I\~] 1~1 1 ~ I ~ I ~ I I I I I 1 ~ I I y I
Double~Bod, J 
tleumannKel~rin
~/~ {/''~J I 1 1 /l I /l I I I ~ I I I
.50  1.00  0.50 0.00
~ I ~ I I I I ~ 1 1 1 1 1 1 1 ) I 1 _
_ ~_
F = 0.35
F = ~ 25
l~t~ble~Bod'
0 75
J
1 TO 1 .00  0.50 0.00 0.50
O.SO
o.oo
0.25
0. 75
0 75
~ = 0.4C
(/~/~//~< ~ J NeumannKelvin ~
 1.50  1.00 0.50 0.00 O.SO
I 0.00
0.25
0 75
Figure 2: Contour plots of the steady wave patterns due to the parabolic Wigley model advance ig
at Froude numbers F = 0.25, 0.35, 0.40.
27
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around the Wigley model, despite the increase in the 'full
ness' of the hull shape. For the Wigley model the bow and
sternwave systems are well formed while the correspond
ing wave pattern around the Series60 hull appears to be
more 'confused'.
Differences between the steady wave pattern computations
from the NeumannKelvin and doublebody linearizations
are here clearly noticeable. Again, significant discrepan
cies occur along the diverging portion of the sternwave
system, where the NeumannKelvin solution shows larger
amplitudes and shorter wavelengths. Moreover, the caustic
0.00
~ Do
0.25
0.7s
lines originating from the bow and stern appear at a larger
angle in the solution based on the doublebody lineariza
tion. The differences between the two solutions become
more pronounced as the speed increases, resulting in quite
different wave patterns at F=0.35 (see Figure 3c).
Figure 4 is a snapshot of the timeharmonic wave pattern
around a modified Wigley model translating at F = 0.2
and oscillating in heave at frequencies we = 3 and
7=5. The grid used for this flow field has the same
density as that in Figure 1. Both frequencies are over
critical (r = wU/g > 0.25), thus two wave systems appear
j I I I I 1 1 1 1/ 1 _
DoubleBod)r / ~
Neumann Kelvin ~
1.50  1.00  0.50 0.00 0.50
~ <~1\\t 1 \1 I\ I 141 1 1 1 1 1 1 1 ~ I ~
Double~Bod, /

_ 050
0.00
0.25
0.75
. )/,/, ], 1, J. ,1 , , , , , , ,), ~ ,
1.50  1 .00  0.50 0.00 0.50
I I I I I I I ~ I I
_ DoubleBody ~7 _
F = 0.35
0 75
0.75 
 1.50  1 too 0.50
0.so
0.00
Figure 3: Contour plots of the steady wave patterns due to the Series60cb =0.6 Yesse1 advancing
at Froude numbers F = 0.20, 0.25, 0.35.
28
OCR for page 21
downstream. At F = 0.3, the timeharmonic wave fields
around the modified Wigley model are illustrated in Fig
ure 5 and are obtained from the same grid as for F = 0.2.
For this larger Eroude number, the wavelengths appearing
in Figure 5 are larger than their counterparts of Figure 4,
although the general structure of the wave field is similar.
Figure 6 illustrates the wave patterns around the Series
6~C~ = 0.7 hull advancing at F = 0.2 and heaving at fre
quencies /=3 and =4. Relative to the cor
responding patterns generated by the Wigley hull, the di
verging wave system originating from the stern is more pro
nounced and is attributed to the more threedimensional
shape of the Series60 geometry. In all cases the steady
wave pattern has been removed.
Certain common features of these thre~dimensional time
harmonic wave patterns are worth emphasizing. The short
est wavelength scales are associated with the transverse
wave system which appears downstream of the stern and
propagates in the streamwise direction. Along the ship
length, on the other hand, the wave field is dominated
by relatively long divergent waves which propagate in the
transverse direction and tend to be become more two di
mensional as the frequency increases. This character of
the time harmonic wave pattern therefore appears to sum
port the 8,,e~,,~,~C I, ~nder~h~rlY thPr?rv Near +lle
ship hull the wave disturbance is convected primarily in the
transverse direction and becomes more focused as the free
quency increases. Its variation in the lengthwise direction
is gradual since cancellation effects appear to significantly
reduce the amplitude of the short transverse waves which
are clearly visible downstream of the stern.
Figure d: Snapshots of the timeharmonic wave patterns due to a modified Wigley model ad
vaDcing at F=0.20 while oscillating in heave at frequencies wp7ij=3.0,5.0.
29
OCR for page 21
7. HYDRODYNAMIC FORCES AND MOTIONS
IN HEAD WAVES
The unsteady hydrodynamic pressure on the hull is eval
uated from expression (3.2~. The restoring component of
the pressure which depends on the ship displacement and
the gradients of the steady flow has been neglected since
it been found to be small for the ship hulls and Froude
numbers considered below. The gradients of the steady
and timeharmonic potentials are obtained from the formal
differentiation of the spline representation of the velocity
potential (5.1~. Integration of the pressure over the hull
according to expressions (3.5), allows the determination of
the addedmass, damping coefficients and exciting forces
from expressions (3.5), and Response Amplitude Opera
tors from the solution of the linear system (3.6~. Only the
coupled heave and pitch modes of motion in head waves
are considered in this paper.
In order to establish the convergence of the solution algm
rithm, a systematic study of the effect of grid density on
the computations of the hydrodynamic coefficients was car
ried out for a modified Wigley model with L/B = 10 and
B/T = 1.6. The tim~harmonic wave flow was solved at a
E`roude number F = 0.3 for several frequencies of oscillation
in the range of practical interest w~ ~ [2.5, 5.0] .
The free surface domain was truncated at a distance 0.25L
upstream of the bow, 0.5L downstream of the stern and
L in the transverse direction. Four different grids were
considered, resulting in a systematic increase of the dim
cretization density on both the free surface and the hull.
These grids use 20, 30, 40 and 50 panels along the length
of the hull, respectively, while for all of them the aspect
ratio of the free surface panels is equal to 1.
Computations of the heave and pitch addedmass and damp
ing coefficients obtained from these grids, are illustrated in
Figure 7. The convergence rate is very satisfactory and
Figure 5: Snapshots of the timeharmonic wave patterns due to a modified Wigley model ad
vsacing at F0.30 while oscillating in heave at frequencies 7= 3.0, 5.0.
30
OCR for page 21
appears not to depend strongly on the frequency.
Having established the convergence of the numerical algo
rithm, the hydrodynamic coefficients and ship motions are
next compared to experimental measurements and strip
theory. A systematic set of experiments for a modified
Wigley hull were recently conducted by Gerritsma(1986~.
The diagonal heave and pitch addedmass and damping
coefficients at F = 0.3 are illustrated in Figure 8. The
experimental measurements are compared to strip theory
and the present method. The solid line, hereafter denoting
results from SWAN (ShipWaveANalysis), is based on the
doublebody free~surface condition (2.9) and the complete
treatment of the mterms. The NeummanKelvin curve is
obtained from the solution of the linearized problem using
the present Rankine panel method and is obtained by ap
proximating the steady flow by the uniform stream Up
both in the freesurface and body boundary conditions.
The agreement between SWAN and experiments is quite
satisfactory and represents an improvement over strip the
ory. For the diagonal coefficients, SWAN and the Neumman
Kelvin problem are in good qualitative and quantitative
agreement.
Significant differences between the three theoretical pre
dictions occur in the heave and pitch cross coupling co
efficients illustrated in Figure 9. These coefficients are
known to be sensitive to endeffects, therefore their ac
curate prediction requires the complete treatment of the
mterms which attain large values near the ship ends.
This is confirmed by the very good agreement between
SWAN and the experimental measurements. In spite of its
threedimensional character, the departure of the Neumman
Kelvin solution from the experiments is mainly attributed
to the incomplete treatment of the mterms.
Figure 6: Snapshots of the timeharmonic wave patterns due to the Serie~60 cb = 0.7 crewel
sd~rancing at F=0.20 while oscillating in heave at frequencies w~/~7;3.0,4.0.
31
OCR for page 21
AID
A
TIDE O
o
o
to
ID
I~ D
D  Cretan (A)
x D  cretizabon (B)
a ~ D~cretizatioI1 (C)
\ ~ Discretization (D)
~;m
MID
D~cretmation (a)
r. Discretization (B)
Discretization (C)
· o D~cret~ation (D)
i'_
Moo s.  ..oo
i:
4D O
I~ ~ o
~ D O
5.00 ~ ~
o . .
to
. . ·
°2.00 3.00 ~ 00
w47~;
5.00 6.00
Figure 7: Numerical convergence study for the heave and pitch hydrodynamic coefficients of a
modified Wigley model advancing at F = 0.3.
Of interest is also the observed symmetry of the experimen
tal measurements and the SWAN predictions of the cross
coupling coefficients. The modified Wigley hull is sym
metric fore and aft and a generalization of the Timman
Newman symmetry relations appears to hold. The origi
nal TimmanNewman relations were shown to be exact for
submerged vessels and the NeummanKelvin freesurface
condition. It is here conjectured that they are also exactly
valid for surface piercing vessels when the freesurface con
dition is based on the doublebody flow. No proof has yet
been attempted using the condition (2.9~.
32
Figure 10 compares experimental measurements with the
striptheory and SWAN and predictions for the heave and
pitch excitingforce and motion modulus and phase. The
pitch radius of gyration of the modified Wigley hull is k', =
0.25L, and the center of gravity is taken at x = y = z = 0.
The agreement of SWAN with the experiments is in all
cases very satisfactory. The stri~theory predictions have
been obtained from the MIT 5D Ship Motion program
which is regarded a standard striptheory code. The dim
crepancy between the striptheory and experimental heave
and pitch resonant frequencies, is attributed to the poor
prediction of the b55 and the cros~coupling coefficients by
strip theory (Figures 8 and 9~.
OCR for page 21
a
0 ·W
~'
 N
"ID
I~ D
lo
°2.00 3.00 4.00 5.00 B.00
0 Experimenb
_._ Strip Theory
SWAN
Neumann Kelvin
_;
lo
0 Experimen"
Strip Theory
SWAT
"~ N.`lmannKelvin
it.

\,
o
~ D O
no
0 · .
lo
0 . .
02.00 ~ Do ~ Do
w\~7;

5.00 6.00
Figure 8: Diagonal hydrodynamic coefficients in heave and pitch for a modified Wigley model
advancing at Eroude number F=0.3.
Figures 11 and 12 compare experiments with the strip the
ory and SWAN predictions of the heave and pitch added
mass and damping coefficients of the Series = 0.7
model, advancing at E`roude number F = 0.2. The experi
mental data are due to Gerritema, Beukelman and Gland
dorp (1974~. The performance of SWAN is in all cases very
satisfactory, offereing a significant improvement over strip
theory.
Due to the foreaft asymmetry of the Series60 model, the
TimmanNewman relations for the cross coupling coeffi
cients do not hold. It is interesting, however, to notice
that the curares corresponding to a35 and bs5 are very close
33
to being mirror images of the those corresponding to a63
and a53, respectively about a nonzero value. In strip the
ory, for example, it may be shown easily that a38a53
and b35  b53 are symmetric about the corresponding cm
efficients at zero forward speed (F=O), but no such proof
is yet available in three dimensions.
The Series 60 heave and pitch motion amplitude and phase
are shown in Figure 13. The agreement between theory and
experiments is again satisfactory for both striptheory and
SWAN, with a slight detuning of the striptheory predic
tions again attributed to its discrepancies with experiments
in the cross coupling coefficients and b55.
OCR for page 21
!D
D
f,>.,
b
0 Experiment
_. Strip Theory
 SWAN
~NeumannKelYin
.
o
~ , ~
10~ 
I trio
o
0 Experiments
Strip Theory
SWAN
Neumann Kelvin
cat _
o
'2.00 3.00 `.00
a _
s.oo 6.00 2.0 3.0 To
w~77
5.0 6.0
Figure 9: Cross coupling hydrodynamic coefficients between heave and pitch for a modified
Wigley model advancing at Froude number F=0.3.
8. CONCLUSIONS AND FUTURE WORK
A new threedimensional Rankine Panel Method method,
referred to as SWAN, has been developed for the solution of
the complete threedimensional steady and timeharmonic
shipmotion problem. Its principal attributes are:
· The use of a new freesurface condition based on the
doubl~body flow and valid uniformly from low to high
Froude numbers.
· The complete and accurate treatment of the mterms.
· A highorder nondissipative numerical algorithm for the
enforcement of the freesurface and radiation conditions.
34
OCR for page 21
.1 a
l ~
o
ID
o
Experiments
~Strip Theory
SWAN
j/
,s t.o
~ _
1
I
A/L
1:
0 Experiments
_. Stup Theory
SWAN
1?
//
1.~
. '~
.
2.0 2.5 ab.s t.0 t.5 2.0 2.5
A/L
Figure 10: Heave and pitch exciting forces and motions of a modified Wigley model advancing at
Froude number F=0.3 through regular head waves.
35
OCR for page 21
LID
~ 
o
o 1
lo
ID
a
lo
:D
o
lo
a\
Experiments
Stup Theory
SWAN
.`
0 ~
o
°~.oo 3.004.00
w\~7;
.. 1
.
~.  .
MID
lo
o
o
1 ~1 ~
1D ~
5 00 6 00 ~ 00
\ ~ Experiments
_._ Stup Theory
~ . SWAN
it.
it.
:
of
o HI  1  ~
\.
~\
O  .\
\
~\a
3 00 4 00
fit
s. go 6. go
Figure 11: Diagonal hydrodynamic coefficients in heave and pitch for the Serie~6~c.=0.7 vessel
advancing at Froude number F=0.2.
Computations of steady and timeharmonic ship wave pat
terns illustrate the capability of the method to resolve con
siderable detail in the wave disturbance and at a significant
downstream of the ship.
Predictions of the heave and pitch addedmass, damping
coefficients, exciting forces and motions of a Wigley and
the Serie~60 hull are found to be in very good agreement
with experiments and present a significant improvement
over strip theory. A complete treatment of the mterms
has been developed and found to be essential for the accu
rate prediction of the cros~coupling coefficients and ship
motions.
In summary, all important features of the threedimensional
timeharmonic flow around the ship appear to be well pre
dicted by the present method. This will permit the accu
rate prediction of the hydrodynamic pressure distribution,
wave loads, derived responses and addedresistance by di
rect use of the velocity potential and its gradients on the
ship hull and the free surface.
36
OCR for page 21
D

·
o
~ D
0
o
o
.

I _.
.
at\
'2.00 3.00
~°
~ D ~
.0 ID ~
0 .
4.00 5.oo 6.00 'ho
w~7i
0 Experiments
Strip Theory
SWAN
e=;
s.o no
w~7;
To 6.0
Figure 12: Crosscoupling hydrodynamic coefficients between heave and pitch for the Series60
Cb=0.7 newel advancing at Froude number F=0.2.
Future research towards the further development of the
present rankine panel method in the steady problem, will
concentrate upon the determination of the ship wave spec
trum from the available numerical data over the discretized
portion of the free surface. This information is useful
for the characterization of ships from their Kelvin wake
and the accurate and robust evaluation of the wave resin
lance. The proper implementation of the present numeri
cal scheme to hull forms with significant flare will also be
studied in both the steady and timeharmonic problems.
37
The application is also planned of the same method to the
prediction of the seakeeping properties of unconventional
ship forms (e.g. SWATH ships and SES's) the hydrody
namic analysis of which is particularly amenable by the
present threedimensional panel method.
OCR for page 21
too 
to
to
0
1,
~ to
~ o
to
i]
o
I',~
.:
I
~ Experiments
._ Strip Theory
SWAN
11
/ 11
1.~.~=
.,
l
In
l ), . .
o
°~.5
1~
_.
.0 1.5
A/L
2.0 2.5
Figure 13: Heave and pitch motions of the Series60cb =0.7 vessel advancing at Froude number
F = 0~2 through regular head waves.
38
OCR for page 21
9. ACKNOWLEDGEMENTS
This research has been supported by the Applied Hydrome
chanics Research Program administered by the Office of
Natural Research and the David Taylor Research Center (Con
tract: N001678~K0010) and by A. S. Veritas Research of
Norway. The majority of the computations reported in this
paper were carried out on the National Science Founda
tion Pittsburgh YMP Cray under the Grant OCE880003P.
This award is greatly appreciated. We are also indebted
to the Computer Aided Design Laboratory of the Depart
ment of Ocean Engineering at MIT for their assistance in
the preparation of the timeharmonic ship wave patterns
on their IRIS Workstation.
REFERENCES

Chang, M.S., 1977, 'Computations of threedimensional
ship motions with forward speed', 2nd International Con
ference on Numerical Ship Hydrodynamics, USA.
Dawson, C. W., 1977, 'A practical computer method for
solving shipwave problems', 2nd International Conference
on Numerical Ship Hydrodynamics, USA.
Eggers, K., 1981, 'NonKelvin Dispersive Waves around
NonSlender Ships', Schif3stechnik, Bd. 28.
Faltinsen, O., 1971, 'Wave Forces on a Restrained Ship in
HeadSea Waves', Ph.D. Thesis, University of Michigan,
USA.
Gadd, G. E., 1976, ' A method of computing the flow and
surface wave pattern around full forms', Mans. Roy. Asst.
Nav. Archit., Vol. 113, pg. 207.
Gerritsma, J., 1986, 'Measurments of Hydrodynamic Forces
and Motions for a modified Wigley Model', (unpublished).
Gerritsma, J., Beukelman, W., and Glansdorp, C. C., 1974,
'The effects of beam on the hydrodynamic characteristics
of ship hulls', 10th Symposium on Naval Hydrodynamics,
USA.
Guevel, P., and Bougis, J., 1982, 'Ship Motions with For
ward Speed in Infinite Depth', International Shipbuilding
Progress, No. 29, pp. 103117.
Inglis, R. B., and Price, W. G., 1981, 'A ThreeDimensional
Ship Motion Theory  Comparison between Theoretical
Predictions and Experimental Data of Hydrodynamic Co
efficients with Forward Speed', Transactions of the Royal
Institution on Naval A Tchitects, Vol.124, pp. 141157.
King, B. K., Beck, R. F., and Magee, A. R., 1988, 'Seakeep
ing Calculations with Forward Speed Using TimeDomain
Analysis', 17th Symposium on Naval Hydrodynamics, The
Netherlands.
KorvinKroukovsky, B. V., 1955, 'Investigation of ship mo
tions in regular waves', Soc. Nav. Archit. Mar. Eng.,
Trans. 6S, pp. 386435.
Maruo, H., and Sasaki, N., 1974, 'On the Wave Pressure
Acting on the Surface of an Elongated Body Fixed in Head
Seas', Journal of the Society of Naval Architects of Japan,
Vol. 136, pp. 3~42.
Nakos, D. E., 1990, 'Ship Wave Patterns and Motions by a
ThreeDimensional Rankine Panel Method', Ph.D. Thesis,
Mass. Inst. of Technology, USA.
Nakos, D. E., and Sclavounos, P. D., 1990, 'Steady and Un
steady Ship Wave Patterns', Journal of Fluid Mechanics,
Vol 215, pp. 265288.
Nestegard, A., 1984, 'End effects in the forward speed ra
diation problem for ships', Ph.D. Thesis, Mass. Inst. of
Technology, USA.
Newman, J. N., 1978, 'The theory of ship motions', Ad
vances in Applied Mechanics, Vol. 18, pp. 221283.
Newman, J. N., and Sclavounos, P. D., 1980, 'The Uni
fied Theory of Ship Motions', 13th Symposium on Naval
Hydrodynamics, Japan.
O'Dea, J. F., and Jones, H. D., 1983, 'Absolute and relative
motion measurments on a model of a highspeed contain
ership', Proceedings of the 20th ATTC, USA.
Ogilvie, T. F., and Tuck, E. O., 1969, 'A rational Strip
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of Naval Architecture and Marine Engineering, Univ. of
Michigan, USA.
Piers, W. J., 1983, 'Discretization schemes for the mod
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for hydrodynamic applications', NLR report TR83093L,
The Netherlands.
Salvesen, N., Tuck, E. O., and Faltinsen, O., 1970, 'Ship
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Sclavounos, P. D., 1984a, 'The Diffraction of FreeSurface
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Sclavounos, P. D., 1984b, 'The unified slenderbody the
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Sclavounos, P. D., and Nakos, D. E., 1988, 'Stability anal
ysis of panel methods for free surface flows with forward
speed', 17th Symposium on Naval Hydrodynamics, The
Netherlands.
Timman, R., and Newman, J. N., 1962, 'The coupled damp
ing coefficients of symmetric ships', Journal of Sh* Re
search, Vol. 5, No. 4, pp. 3455.
Yeung, R. W., and Kim, S. H., 1984, 'A New Development
in the Theory of Oscillating and Translating Slender Ships',
15th Symposium on Naval Hydrodynamics, Germany.
Zhao, R., and Faltinsen, O., 1989, 'A discussion of the
mterms in the wavecurrentbody interaction problem',
3rd International Workshop on Water Waves and Float
ing Bodies, Norway.
39
1
OCR for page 21
DISCUSSION
William R. McCreight
David Taylor Research Center, USA
Your predictions of added mass and damping for the Series 60 hull
are better than those for the Wigley hull, yet the motion predictions
are not as good. Could you describe the accuracy on the Series 60
excitingforce computations, which are not shown. If this does not
account for the discrepancy, what do you believe is the cause of this?
AUTHORS' REPLY
In response to Dr. McCreight's question we want to state that the
calculation of the heave/pitch exciting forces typically compare very
well with corresponding experimental data. Discrepancies between
the numerical and experimental results for the motions of the
Series60 may be partly attributed to the speed dependent portion of
the restoring force, which was not included in the presented
calculations. Additional differences may also arise due to ambiguities
about the appropriate values for the pitch moment of inertia and the
vertical position of the center of gravity, as well as about the location
of the point about which the heave/pitch motions are referenced.
DISCUSSION
Hoyte Raven
Maritime Research Institute Netherlands, The Netherlands
This paper is very interesting for me, in particular, as it addresses
some points studied in my paper. I have a question on the steady
wave resistance. You found differences in the remote wave pattern
between the Kelvin and the showship condition. These may,
however, be due to subtle changes in interference between wave
components. Did you find any substantial difference in wave
resistance? Secondly, as you noticed your free surface condition is
intermediate in form between those of Dawson and Eggers, 1979. I
have implemented your FSC in our code to make the same
comparisons as in my paper, and found that the result was also
intermediate for the Series 60 CB=0.60 model: the predicted Rw is
68% lower than with Dawson's condition, while Eggers is 20%
lower. For a full hull form, again the resistance is lower than
Dawson, but better behaved than Egger's condition. Ref. Raven,
H.C., Adequacy of Free Surface Conditions for the Wave Resistance
Problem,. this volume.
AUTHORS' REPLY
We would like to thank Dr. Raven for implementing and testing the
free surface condition proposed in this paper. The differences of the
wave patterns, as predicted by different free surface linearization
models are indeed reflected on the corresponding wave resistance
calculations. We strongly believe, however, that ~numerical"
evaluation of the relative performance of different linearization
models is still clouded due to the delicate nature of the underlying
calculations. The robustness of each scheme ought to be established
individually before comparison arguments can be stated. We are
currently working towards this direction by employing the
conservation of momentum as the selfconsistency criterion ([1]).
[1] Nakos, D.E., 1991, "Transverse Wave Cut Analysis by a
Rankine Panel Method, 6th Int. Workshop on Water Waves and
Floating Bodies, Woods Hole, MA, USA.
40