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OCR for page 21
Ship Motions by a Three-Dimensional
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 three-dimensional steady and time-harmonic
potential flows past ships advancing with a forward veloc-
ity. A new free-surface condition is derived, based on lin-
earization about the double-body 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 Series-60 hull have
been evaluated in head waves over a wide range of fre-
quencies and speeds. A robust treatment is proposed of
the m-terms 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 cross-coupling 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 three-dimensional problem. This paper presents our
progress in that direction.
The pioneering work of Korvin-Kroukovsky (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 slender-body 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 high-frequency restric-
tion in earlier slender-ship 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 slender-ship studies by Kim and Yeung (1984)
and Nestegard (1986), accounted directly for convective
forward-speed wave effects near the ship hull and repre-
sented the transition to numerical studies aiming at the
solution of the three-dimensional ship-motion problem.
By the mid-80's, the performance of slender-body theory
for the seakeeping problem could only be validated from
experimental measurements. Moreover, it had become ev-
ident that end-effects at high Froude numbers cannot be
modelled accurately by slender-body 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
time-harmonic forward-speed 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 zero-speed 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 free-surface conditions with vari
OCR for page 22
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 out-of-core iterative solu-
tion method is available.
This paper outlines the solution of the three-dimensional
time-harmonic 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 time-harmonic problem is presented in Nakos
and Sclavounos (1990~. In this reference the convergence
properties of a new quadratic-spline scheme are derived,
which has been found to be accurate and robust for the so-
lution of both steady and time-harmonic free-surface flows
in three dimensions. This scheme is applied in this paper
to the solution of the time-harmonic radiation/diffraction
potential flows around realistic ship hulls and the evalu-
ation of the hydrodynamic forces and motions in regular
head waves.
A new three-dimensional free-surface condition is derived,
using the double-body 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 low-Froude-number
conditions for the steady problem, as well as the Neumann-
Kelvin condition, are obtained as special cases. The ship-
hull condition includes the m-terms which are evaluated
from the solution of the three-dimensional double-body
flow. An important property of the solution scheme is that
the evaluation of the double gradients of the double-body
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 Series-60 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 Neumman-Kelvin and the more general double
body free-surface conditions reveals good agreement for
the Wigley hull, while evident differences appear in the
respective Series-60 wakes.
Predictions of the heave and pitch added-mass and damp-
ing coefficients and exciting forces are found to be in very
good agreement with experimental measurements both for
the Wigley and the Series-60 hull. The contribution of
the complete m-terms is found to be important, partic-
ularly in the cross-coupling coefficients. The validity of a
more general set of Timman-Newman relations is observed
and conjectured in connection with free-surface conditions
based on the double-body 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 x-direction points upstream and the posi-
tive z-axis upwards. The boundary-value 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. Full-shaped ships typically
advance at low speed and cause a small steady wave distur-
bance. Fine-shaped 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 double-body 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
double-body flow and are taken to be small relative to the
A. Linearization of (2.4-5), 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 (U2-Vie 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, (2.8-9) reduce to the well-
known Neumann-Kelvin conditions. In the opposite limit
of blue ships with ~ of O(1) advancing at low Froude num-
bers, (2.8-9) reduce to the conditions of slow-ship 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 double-body 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 right-hand-side 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 free-surface conditions (2.8). An alternative form
of (2.13) may be derived in terms of the rigid-body 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 so-called m-terms tOgilvie
and Tuck (1969)~.
If the basis flow is approximated by the uniform stream the
only non-zero m-terms are ma = Un3 and me = -Un2,
which merely account for the 'angle of attack eEect' due
to yaw and pitch. This approximation of the m-terms has
been employed in most previous studies of the ship motion
problem, consistently with the linearization steps leading
to the Neumann-Kelvin 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
OCR for page 24
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 rigid-body 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 double-body 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 (~2aijiwbij-calm ~ ,
{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 double-body 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 time-harmonic responses of
the ship follow from Newton's law. Using the definitions
(3.5) of the forces acting on the hull, the familiar six-degree
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
OCR for page 25
+(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 m-terms 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 m-terms in (4.3) requires the com-
putation of second order derivatives of the double-body po-
tential ~ on the ship hull. When it comes to the evaluation
of gradients of the solution potential, low-order 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 right-hand 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
z-0 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 time-harmonic 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
bi-quadratic spline-collocation 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 bi-quadratic 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 C1-continuous 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
(2.8-9) and (3.1-2), respectively.
The error and stability analysis of the bi-quadratic 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 26
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 Gauss-Siedel
iterative scheme which makes extensive use of out-of-core
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
thin-struts with analytical solutions iNakos and Sclavounos
(1990) and Nakos (1990~. The convergence of the hydro
dynamic added-mass and damping coefficients is discussed
in Section 7.
PATTERNS
The forward-speed 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 length-to-beam ratio L/B = 10 and beam-to-draft 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
OCR for page 27
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
Neumann-Kelvin and the double-body 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
Neumann-Kelvin 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 Series-60-Cb = 0.6 hull which
is significantly fuller than the Wigley model, with length-
to-beam and beam-to-draft 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 -
tleumann-Kel~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 Neumann-Kelvin ~
- 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
OCR for page 28
around the Wigley model, despite the increase in the 'full-
ness' of the hull shape. For the Wigley model the bow- and
stern-wave systems are well formed while the correspond-
ing wave pattern around the Series-60 hull appears to be
more 'confused'.
Differences between the steady wave pattern computations
from the Neumann-Kelvin and double-body linearizations
are here clearly noticeable. Again, significant discrepan-
cies occur along the diverging portion of the stern-wave
system, where the Neumann-Kelvin 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 double-body 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 time-harmonic 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 _
Double-Bod)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, /
-
_ 0-50
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
_ Double-Body ~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 Series-60-cb =0.6 Yesse1 advancing
at Froude numbers F = 0.20, 0.25, 0.35.
28
OCR for page 29
downstream. At F = 0.3, the time-harmonic 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 three-dimensional
shape of the Series-60 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 time-harmonic 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 30
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 time-harmonic 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 added-mass, 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 added-mass 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 time-harmonic wave patterns due to a modified Wigley model ad-
vsacing at F-0.30 while oscillating in heave at frequencies 7= 3.0, 5.0.
30
OCR for page 31
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 added-mass 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
double-body free~surface condition (2.9) and the complete
treatment of the m-terms. The Neumman-Kelvin 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 free-surface 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 end-effects, therefore their ac-
curate prediction requires the complete treatment of the
m-terms 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
three-dimensional character, the departure of the Neumman
Kelvin solution from the experiments is mainly attributed
to the incomplete treatment of the m-terms.
Figure 6: Snapshots of the time-harmonic 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 32
AID
A
TIDE O
o
o
to
I|D
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:
4|D 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 Timman-Newman relations were shown to be exact for
submerged vessels and the Neumman-Kelvin free-surface
condition. It is here conjectured that they are also exactly
valid for surface piercing vessels when the free-surface con-
dition is based on the double-body flow. No proof has yet
been attempted using the condition (2.9~.
32
Figure 10 compares experimental measurements with the
strip-theory and SWAN and predictions for the heave and
pitch exciting-force 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 5-D Ship Motion program
which is regarded a standard strip-theory code. The dim
crepancy between the strip-theory 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 33
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.`lmann-Kelvin
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 fore-aft asymmetry of the Series-60 model, the
Timman-Newman 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 non-zero value. In strip the-
ory, for example, it may be shown easily that a38-a53
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 strip-theory and
SWAN, with a slight detuning of the strip-theory predic-
tions again attributed to its discrepancies with experiments
in the cross coupling coefficients and b55.
OCR for page 34
!|D
|D
f,>.,
b
0 Experiment
_. Strip Theory
- SWAN
~Neumann-KelYin
.
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 three-dimensional Rankine Panel Method method,
referred to as SWAN, has been developed for the solution of
the complete three-dimensional steady and time-harmonic
ship-motion problem. Its principal attributes are:
· The use of a new free-surface condition based on the
doubl~body flow and valid uniformly from low to high
Froude numbers.
· The complete and accurate treatment of the m-terms.
· A high-order non-dissipative numerical algorithm for the
enforcement of the free-surface and radiation conditions.
34
OCR for page 35
.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 36
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 ~
1|D ~
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 time-harmonic 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 added-mass, 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 m-terms
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 three-dimensional
time-harmonic 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 added-resistance by di-
rect use of the velocity potential and its gradients on the
ship hull and the free surface.
36
OCR for page 37
||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: Cross-coupling hydrodynamic coefficients between heave and pitch for the Series-60-
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 time-harmonic 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 three-dimensional panel method.
OCR for page 38
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 Series-60-cb =0.7 vessel advancing at Froude number
F = 0~2 through regular head waves.
38
OCR for page 39
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: N00167-8~K-0010) 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 time-harmonic ship wave patterns
on their IRIS Workstation.
REFERENCES
-
Chang, M.-S., 1977, 'Computations of three-dimensional
ship motions with forward speed', 2nd International Con-
ference on Numerical Ship Hydrodynamics, USA.
Dawson, C. W., 1977, 'A practical computer method for
solving ship-wave problems', 2nd International Conference
on Numerical Ship Hydrodynamics, USA.
Eggers, K., 1981, 'Non-Kelvin Dispersive Waves around
Non-Slender Ships', Schif3stechnik, Bd. 28.
Faltinsen, O., 1971, 'Wave Forces on a Restrained Ship in
Head-Sea 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. 103-117.
Inglis, R. B., and Price, W. G., 1981, 'A Three-Dimensional
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Institution on Naval A Tchitects, Vol.124, pp. 141-157.
King, B. K., Beck, R. F., and Magee, A. R., 1988, 'Seakeep-
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Analysis', 17th Symposium on Naval Hydrodynamics, The
Netherlands.
Korvin-Kroukovsky, B. V., 1955, 'Investigation of ship mo-
tions in regular waves', Soc. Nav. Archit. Mar. Eng.,
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Maruo, H., and Sasaki, N., 1974, 'On the Wave Pressure
Acting on the Surface of an Elongated Body Fixed in Head
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Nakos, D. E., 1990, 'Ship Wave Patterns and Motions by a
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Nakos, D. E., and Sclavounos, P. D., 1990, 'Steady and Un-
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Nestegard, A., 1984, 'End effects in the forward speed ra-
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Newman, J. N., 1978, 'The theory of ship motions', Ad-
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Newman, J. N., and Sclavounos, P. D., 1980, 'The Uni-
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O'Dea, J. F., and Jones, H. D., 1983, 'Absolute and relative
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Ogilvie, T. F., and Tuck, E. O., 1969, 'A rational Strip
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Piers, W. J., 1983, 'Discretization schemes for the mod-
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Salvesen, N., Tuck, E. O., and Faltinsen, O., 1970, 'Ship
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Sclavounos, P. D., 1984a, 'The Diffraction of Free-Surface
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Sclavounos, P. D., 1984b, 'The unified slender-body the-
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Sclavounos, P. D., and Nakos, D. E., 1988, 'Stability anal-
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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. 34-55.
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
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39
1
OCR for page 40
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
exciting-force 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
Series-60 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 show-ship 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
6-8% 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 self-consistency 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
Representative terms from entire chapter:
strip theory
