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OCR for page 41
Numerical Solutions for Larg - Amplitude
Ship Motions In the Time Domain
W.-M. Lin (Science Applications International Corporation, USA)
D. Yue (Massachusetts Institute of Technology, USA)
ABSTRACT
A three-dimensional time domain approach is used
to study the large-amplitude motions and loads
of a ship in a seaway. In this approach, the ex-
act body boundary condition is satisfied on the
instantaneous wetted surface of the moving body
while the free-surface boundary conditions are lin-
earized. The problem is solved using a transient
free-surface Green function source distribution on
the submerged hull.
Extensive results are presented which validate and
demonstrate the efficacy of the method. These re-
sults include linear and larg~amplitude motion co-
efficients and diffraction forces with and without
forward speed, calm-water resistance and added-
resistance with waves and motions, the large-
amplitude motion history of a ship advancing in an
irregular seaway, as well as load distributions on
the changing submerged hull. Most of the large-
amplitude results we obtained are new and illus-
trate the importance of nonlinear effects associated
with the changing wetted hull. Of special signifi-
cance are the dramatic changes of the added mass,
the steady resistance, and sinkage and trim forces
as the motion amplitudes increase.
The present method is a major step forward in
the development of design and prediction tools for
ship motions and loads, and represents a signifi-
cant milestone towards a full~nonlinear capability
in the forseeable future.
1 INTRODUCTION
The accurate prediction of wave-induced motions
and hydrodynamic loads is of crucial importance
in ship design. In addition to concerns such as
4
efficiency and comfort, severe motions can limit
operability and affect safety, while extreme loads
may lead to structural failure. Thus the general
problem of a moving body interacting with waves
has been pursued actively since at least the time
of Froude (1868) and Michell (1898~.
Traditionally, the problem is linearized and formu-
lated in the frequency domain, by assuming the
motions to be small and time harmonic, and the
resulting boundary-value problem is solved using a
singularity distribution on the mean body bound-
ary. For zero speed problems, this approach is
quite successful and has become a standard tool
for the design of large offshore structures (e.g., Ko-
rsmeyer, et al, 1988~. In the presence of forward
speed, the so-called Neumann-Kelvin problem is
significantly more difficult due primarily to the
complexity of the corresponding Green function.
Thus, despite several earlier attempts (e.g., Chang,
1977; Inglis & Price, 1981; Guevel & Bougis, 1982),
a truly satisfactory numerical solution is as yet
unavailable. A promising variation due to Gadd
(1976) and Dawson (1977) is the use of Rankine
sources on the body surface as well as a portion
of the free surface on which more general quasi-
linearized free-surface conditions can be specified.
Such approaches have been developed actively in
the past 5 or 6 years (e.g., Chang & Dean, 1986;
Xia, 1986; Larsson, 1987; Boppe, et al, 1987;
Jensen, et al, 1988; Letcher, et al, 1989; Bertram,
1990; Nakos & Sclavounos, 1990), with increasingly
encouraging results. In all of these methods, how-
ever, the free surface and body geometry remain
fixed in the undisturbed positions, and geometric
nonlinearities are not included.
An alternative to the frequency-domain approach
is to formulate the time-domain initial-value prob-
lem (cf., Finkelstein, 1957; Cummins, 1962~. The
OCR for page 42
requisite time-dependent Green function which
satisfies the linearized free-surface boundary condi-
tion is simpler than the corresponding ones in the
frequency domain, yet is capable of describing ar-
bitrary (large-amplitude) motions when the proper
free-surface memory effects are included. While
linearized and even fully nonlinear time-domain re-
sults have been available for problems in two di-
mensions (or with vertical axisymmetry) for some
time, developments for three-dimensional problems
have been relatively recent. Such work include Ko-
rsmeyer (1988) for the linearized radiation problem
without forward speed, and Liapis (1986), Beck &
Liapis (1987), King (1987), King et al. (1988) for
the general linearized problem with constant for-
ward speed. For submerged bodies, results for lin-
earized free surface but large body motions have
been obtained by Ferrant (1988) and recently by
Beck & Magee (1990~. We remark that for lin-
earized (small-amplitude) motions with zero or
constant forward speed, these time-domain solu-
tions are formally related to the frequency-domain
results via Fourier transforms.
In this paper, we extend the time-domain approach
to arbitrary large-amplitude motions of a surface-
piercing body in a seaway. The exact body bound-
ary condition is applied on the instantaneous sub-
merged hull surface while a linearized free-surface
condition is used. This approximation can be jus-
tified in principle upon the assumptions of small
incident wave slopes and slenderness of the body
geometry in the directions of the (large-amplitude)
motions. The practical utility of this approach
must, in the final analysis, be demonstrated by the
validity and accuracy of its predictions. This is the
focus of much of the present work.
In a boundary-element approach, the submerged
body surface at each time step is divided into a
number of panels over which linearized transient
free-surface sources are distributed. In contrast
to earlier work, the problem is formulated in a
coordinate system fixed in space. This is clearly
necessary for the case of arbitrary large-amplitude
motions and excursions which is the primary ob-
jective of the present code. Under this formula-
tion, a general and concise waterline integral term
can be derived to account for arbitrary transla-
tions and distortions of the body waterplane, and
the diffraction problem can be included straight-
forwardly by adding the incident wave contribu-
tion to the body boundary condition. For gen
eral nonlinear calculations, the position and ori-
entation of the body is updated (by solving the
equations of motion or as prescribed) and the un-
derwater body surface is repanelized at each time
step. Since the body boundary condition is satis-
fied on the exact instantaneous hull, the so called
"m-term" effects associated with forward speed
(Ogilvie & Tuck, 1969) are automatically and ez-
actly included. For the special case of constant
forward speed and small oscillatory motions (the
linearized seakeeping problem), the traditional lin-
earization (and decoupling) of the latter is, how-
ever, less explicit in the earth-fixed (time-domain)
formulation. For these linear forward-speed cal-
culations, the quadratic terms are included in the
force calculations to account for the forward-speed
couplings but the m-terms are otherwise neglected
in the body boundary conditions.
Linear and large-amplitude computational results
are presented for a floating sphere, two Wigley
hulls, and the Series 60 (CB = 0.7) hull under-
going free or captive motions and with or with-
out forward speed or incident seas. The program
is applicable for general six-degree-of-freedom mo-
tions (without lift) but we restrict ourselves to
vertical plane motions in head seas in this paper.
For the linear problems without forward speed,
(time) impulse response functions are computed
from which the requisite motion coefficients are
obtained via Fourier transforms. For the nonlin-
ear cases and for problems with forward speed,
the bodies are started from rest and computations
typically continued until steady states (limit cy-
cles) are achieved. Where available, the results are
compared with other time- and frequency-domain
calculations and experimental data. In all cases,
the importance of the nonlinear effects due to ge-
ometry variation is identified.
2 MATHEMATICAL FORMULATION
We consider a general
three-dimensional body
floating on a tree surface and undergoing arbitrary
six-degree-of-freedom motion in the presence of in-
cident waves. An earth-fixed Cartesian coordinate
system is chosen with the x-y plane coincident with
the quiescent free surface, and z is positive upward.
The fluid is assumed to be homogeneous, incom-
pressible, inviscid and its motion irrotational. Sur-
face tension is not included and the water depth is
infinite.
42
OCR for page 43
The fluid motions can be described by a velocity Note that G° = 0, so that
potential
4iT(X,t) = ~(x,t) + 4~(x,t), (1)
where ~! is the incident wave potential, 4} = AT-
4~r the total disturbance potential, t is time, and x
is the position vector. In the fluid domain V(t),
satisfies Laplace's equation
V2~ = 0 . (2)
On the mean free surface Butt, we impose the lin-
earized condition
~` +g4iz = 0 on 7(t), t ~ 0, (3)
where 9 is the acceleration due to gravity. On the
instantaneous body boundary B(t), we require no
normal flux:
~ = Vn - ~, ~ on B(t), t ~ 0, (4)
where the unit normal vector to the body n is pos-
itive out of the fluid and Vn is the instantaneous
body velocity in the normal direction. For finite
time, the conditions at infinity, SOO, are
A, 4~' ~ O onSOO, t > 0, (5)
and the initial conditions at t = 0 are
4} = 4~` = 0 on {(t), t = 0. (6)
We introduce the transient free-surface Green func-
tion for a step-function source below the free sur-
face (see, e.g., Stoker, 1957~:
G(P,t;Q,T)=G°+Gf=~_ :1+
2 / [1 - cos(~(t-Tall ek(2+~) Jo~kR) dk
o
for P 76 Q. t > T. (7)
where P = (a, y, z) and Q = (6, 71, <) are the source
and held points, r = IP- Ql, r' = IP- Q'l' Q' =
((, 9, -a), R2 = (x _ ()2 + (y - 71)2, G° = 1/r - 1/r'
is the Rankine part of the Green function, Gf =
G-G° is the free-surface memory part, and JO
Bessel function of order zero. Eq. (7) satisfies the
following initial-lo oundary-value problem:
V2G = 0
G`` + gGz = 0
G. G' ~ 0
G = G'
in V(t), t > T
on{(t), t > T
for t > ~
= 0 onT(t), t = ~ .
GO = Gi =
-2 / ~sin(~(t-T))ek(Z+~) Jo~kR) dk . (8)
o
To obtain a boundary integral formulation for
A, we apply Green's identity to IRAQ, T) and
Gr(P,t;Q,T) in a fluid domain V(T) bounded by
~F(T), B(\T), SOO, and a small surface SP excluding
point P:
///V(~)~.VQG, - G7VQ.) dV
= // ~ Agent-G~nQ`) dS, (9)
sco +~(~+~(~)+SP
where V(~T), ~F(T), I3(~T) are respectively V(~),
jF(T), I3(T) with the exclusion of possible point P.
The left-hand side of (9) and the integrals over SOO
and SP on the right-hand side are all zero. In-
tegrating the resulting equation with respect to T
from 0 to t, we obtain
/° //:~(T)+~3(r)(~?G7nQ-G~r~nQ~) dS = 0.
(10)
To eliminate the integral over F(\T), it is neces-
sary to exchange the time and surface integrals
involving ~F(T). It should be noted that (To iS
time-dependent in the earth-fixed coordinate sys-
tem. Applying the linearized free-surface condition
(3) and the transport theorem we obtain finally
// (.G,nQ GT4in~ ~ dS
9 { Br //3F(T) T T) dS
- / (.G??-G7.T\)VN dL} , (11)
where (To iS the instantaneous intersection of
the body with the free surface (excluding possi-
ble point P), N the unit normal to P(T) on the
free surface, positive out of the fluid domain, and
VN the normal velocity of [(T) in the N direction.
Combining (10) and (11) and applying the free sur-
face condition on G and the initial conditions on
4i, we obtain
//() ~ S + /0 dT //~(T)(.G'nQ-G,tnQ ~ dS
43
OCR for page 44
+ -| do| (.G,, - G?~^,)VN dL = 0. (12)
The integral over the free surface /(T) can now be
eliminated by applying Green's identity again to
4~(Q,t) and G° in F(t) and combining the result
with (12). Finally, we have for P on B(t):
2~r4~(P,t)+/t (EGG-4inQG ) dS
/o {//~`T'(qiG7nQ-Snags) dS
+ 9 /rt ,(.G77-4~'G])VN dL}. (13)
The overhead bars on the integration surfaces are
dropped since the integration over SO on these sur-
faces gives no contribution. The waterline memory
integral over [(T) in (13) is the general form for ar-
bitrary large motions in the earth-fixed coordinate
system. For a submerged body, this term vanishes.
For the special case of horizontal planar motions
only, this integral reduces to a similar term formu-
lated in a body-fixed coordinate system given by
Liapis (1986~.
For large-amplitude problems, the evaluation of
the tangential velocities on the body is of critical
importance, and, in the absence of lift, a source
formulation is preferred. To obtain the equation
for the source distribution, we formulate the inte-
rior problem governed by an equation similar to
(13), which upon combining with (13) yields
'T {/Jrs(~( /°
[// ~ S 9 /r(T) aGf VNVndL] } (14)
where a(Q,T) is the source strength at point Q
at time T. and VN and Vn are related by VN =
V /N
Finally, we apply the body boundary condition for
P on B(t) to obtain:
= Vn(P, t)-VAMP, t) · no
47r {//~(t) aGnp dS + / do
+ [// dS-9 /r~ ~aGnp`VNVn dL] ~
(15)
Eq.~15) can be solved for the unknown UP, t)
given B(t), Vntt), Arty, and a(P,~) and 8(T) for
O < T < t. Once the source strength is found, ~ is
evaluated by (14), and the velocity on the body,
Vat, is obtained using a vector form of (15~.
The total pressure is given by Bernoulli's equation,
P = -Pi {~! + 2~V.T~2 + go), (16)
and the force on the body is obtained by inte-
grating (16) over the instantaneous submerged hull
B(t).
The formulation for the general arbitrary motions
problem is thus complete. For linear (small mo-
tions) problems without forward speed, the body
boundary condition is linearized by applying (15)
on the mean body position BO (the waterline term
is absent) and the quadratic term is neglected in
integrating (16) on BO for the forces. For the lin-
earized problem with forward speed (the linear sea-
keeping problem) in the present earth-fixed coor-
dinate system, the quadratic contributions are in-
cluded in the pressure integration (Eqs. 16 and
19 ~ to account for forward-speed effects, but the
body boundary condition (15) on BO is otherwise
not further expanded. The formulation is thus nei-
ther strictly linearized (in terms of the motion am-
plitudes) nor consistent (in that the associated m-
terms are not included). This is a consequence of
the fact that a formal decomposition of the prob-
lem in terms of the steady forward-speed distur-
bance and the (small) oscillatory disturbance.is
no longer straightforward in the earth-fixed time-
domain system. (In this system, the forward speed
is in fact a large-amplitude motion.) A more con-
venient formulation for the linearized constant for-
ward speed problem is, of course, to adopt a coor-
dinate system translating with the ship (c/., Liapis
& Beck, 1985~.
3 NUMERICAL METHOD
3.1 Solution of the Integral Equation
A panel method is used for the solution of the in-
tegral equation (15~. The body surface B(t) is di-
vided into No) quadrilateral elements over which
the source strength is assumed constant. Non-
planar quadrilaterals are mapped to planar ele-
ments by fitting to the corner points in a least-
squares sense (Hess & Smith, 1964). Similarly, the
44
OCR for page 45
body waterline I`(t) is divided into NW(t) straight
line segments on which the source strength c,* is
assumed to be the same as that of the adjacent
body panel. The equation is satisfied at collocation
points corresponding to null points of the panels. A
forward Euler scheme with constant time step, At,
is used to integrate forward in time and the con-
volution integral is computed using a trapezoidal
rule.
The discretized form of (15) is given by:
NM
~ aM 1l Go p (Pi, Q j; O) d S
i=l s
=-Or [V(Pi, t)-V.I(Pi, t)] · np
M-1 _N,m
-At ~ Em ~ aJ || G/pt(Pi, pi; t-T) dS
m=0 i=1 S
Nw
-~ aim | G/pt(Pi, Qk; t-T)VNkVnkdL
k=1 r.
for i = 1,2,...,NM. (17)
In the above, M and m are the indices for t and
T respectively where t = MAt and T = mat; i,
j are respectively the panel indices for collocation
points P and field points Q (k is the index for the
waterline field points); and so = 1/2 and em = 1
for m > 0. Note that the convolution summation
ends at m = M - 1 due to a property of Gf.
Eq. (17) is in the form of a system of linear equa-
tions:
NM
~ AijaM = Bi i = 1,2,...,NM, (18)
j=1
which can be solved for the unknown panel source
strengths, aM, by standard means.
3.2 Evaluation of the Free Surface Tran-
sient Green Function
The integrals involving the Rankine part of the
Green function G° are evaluated using a method
similar to Hess & Smith (1964~. Special efforts
have been made to completely vectorize these eval-
uations, resulting typically in several factors of sav-
ings on vector processors.
The numerically more time consuming task is the
evaluation of the memory term Gf and its deriva-
tives which, because of the convolution integrals,
must be evaluated a large number of times. There
has been much effort in recent years to develop
efficient and accurate numerical methods for cal-
culating Gf and its derivatives, including Newman
(1985, 1990), Liapis & Beck (1985), and Magee &
Beck (1989~. The present approach is an improve-
ment upon the method given in Newman (1985~.
The domain for Gf is divided into a number of
regions wherein, depending on the arguments, as-
cending series, asymptotic series or a combination
of these and two-dimensional economized (Cheby-
shev) polynomial approximations are utilized. The
final results are maintained to a minimal accuracy
of 6D for Gf and 5D for its first and second deriva-
tives. The entire procedure is completely vector-
ized and the average computing time required for
the evaluation of Gf and its two first and two sec-
ond derivatives is 0~1-3) microseconds on a Cray-
Y/MP depending on the relative frequency of eval-
uations in the different regions. Details can be
found in Lin & Yue (1990~.
The numerical integration of Gf and its deriva-
tives over the quadrilateral panels is performed
using two-dimensional Gauss-Legendre quadrature
after mapping the general quadrilaterals into unit
squares. A similar one-dimensional quadrature is
used for the waterline elements.
3.3 Implementation & Force Evaluation
The present method is general for arbitrary large-
amplitude six-degree-of-freedom motions which in
general requires the modelling of a changing un-
derwater body geometry. Depending on the type
and amplitude of the motions, different numerical
treatments are required:
1. Linear motions with zero or constant forward
speed In this case, the underwater geome-
try does not change with time and the matrix
Aij in (18), which represents the Rankine in-
fluences and depends only on the relative dis-
tances between Pi and Qj, is constant. Thus,
A'j needs to be evaluated and inverted only
once at the beginning of the computations.
Furthermore, since Gf and its derivatives de-
pend only on Pi - Qj~ and t - T. for zero or
constant horizontal velocity, the previous val-
ues of Gf can be reused at each time step.
4s
OCR for page 46
Thus only one new set of evaluations for Gf
and its derivatives is necessary at a new time
step, and the rest can be obtained from pre-
vious calculations. The main differences be-
tween zero and constant forward speed prob-
lems are the absence of the waterline contri-
bution in the former, and the need to include
quadratic terms in the force evaluation to ac-
count for forward-speed cross-coupling contri-
butions in the latter.
" Linear motions with arbitrary horizontal ex-
cursions The body moves with variable
speed on a straight course or perform arbi-
trary excursions in the horizontal plane. In
this case, the underwater body geometry still
does not change in time and Aij needs to
be evaluated and inverted only once. How-
ever, the fixed relationships of Pi-Qj~ and
t-T between time steps no longer exist and
evaluations of Gf and its derivatives have to
be done for different T,S at every time step.
This increases the computational burden sig-
nificantly.
" Arbitrary large-amplitude motions-In this
case, the underwater body geometry changes
with time, and both Aij and all values of Gf
and its derivatives must be reevaluated at each
time step.
For large-amplitude motions, a robust geometry
processing capability is essential and an automatic
repanelizer was developed for this task. The orig-
inal input geometry must now include the above-
mean-waterline portion. At each time step, the un-
derwater geometry is represented by a number of
vertical strips with each strip having a fixed num-
ber of panels along its girth. As the body moves,
its new location and orientation is updated in the
global coordinate system and the new waterline is
found from the intersection with the mean free sur-
face. The underwater portion of each strip is then
repanelized using a spline curve fitting. For sim-
plicity, the number of panels in each strip is kept
the same but any particular vertical strip may be
eliminated completely if it is out of the water.
The force evaluation follows from integration of
(16) over the instantaneous or mean submerged
hull for the nonlinear and linearized problems re-
spectively. The quadratic terms are included in the
nonlinear and forward-speed calculations and are
evaluated directly given the normal and tangential
velocities on the body. The calculation of It
is separated into two parts: B.~/~1t is given from
the incident waves, and B~/0t is evaluated by
/ By! Hi m Am _ Am-~
~ fit J i At -Vim · V.m , ( 19)
where Vim, the 'grid' velocity at point P., is equal
to ~ Am _ pSm~! )/At for the nonlinear problem, and
is simply the forward speed, U. in the linearized
forward-speed problem. We remark that because
of repanelization for the nonlinear problems, the
control points Pi in general do not represent the
same global points or the same body points. When
the number of panels changes between time steps,
(19) can not be used in a straightforward manner
and special care must be taken. This is not con-
sidered in the present code.
4 RESULTS
Numerical computations were performed for lin-
earized radiation and diffraction problems; large-
amplitude forced motions and free motions of a
floating body with and without forward speed and
the presence of ambient waves; as well as wave re-
sistance and added resistance problems. For sim-
plicity we limit ourselves to heave/pitch motions
in head seas although the present code is capable
of general six degree-of-freedom motions (without
lift). A sphere, two Wigley hull forms, and the
Series 60 (CB=0.7) hull were used for this study
The first Wigley form was used by Gerritsma
( 1988) in his seakeeping experiments and has
a beam-to-length ratio 2b/L=O.1, and draft-to-
length ratio H/L=0.0625. This is designated as
the ``WSK', hull hereafter. The half beam y is given
by:
Yb = (1 - X)~1 - Z)~1 + 0.2X) + Z(1 - Z4~1 - X)4
where X-(2:~/L)2 and Z-(z/H)2. The other
Wigley hull is commonly used for wave resistance
studies which we designate the "WRT" hull. This
hull has 2b/L=O.1, and 11/L=0.0625 with half
beam given by y/b=~1- X)~1- Z).
For convenience, unless otherwise noted, all quan-
tities in the following are nondimensionalized by
fluid density p, gravitational acceleration 9, and
body length L (or radius a for the sphere). The
panel numbers, N. indicated are always for half
46
OCR for page 47
Hulme (1982)
SAMP N=18
SAMP N=150
. . .
0.0 5.0 10.0 15.0 20.0
told
Fig. 1: Impulse response function for the heave
force on a hemisphere, V _ 2~ra3/3.
Of the (submerged) body. The present compu-
tational results are referred to as SAMP (Small-
Amplitude Motion Program) when the linear op-
tion (fixed underwater geometry) is used, and as
LAMP (karge-Amplitude Motion Program) when
the large-amplitude capability with changing sub-
merged geometry is employed. Where available,
numerical calculations from the linearized time-
domain method developed at the University of
Michigan (Magee & Beck, 1988) are included and
denoted as "Michigan". Strip theory results are
based on Salvesen, Tuck & Faltinsen (1970~. Those
used in Secs. 4.1 and 4.2 are taken from Magee &
Beck (1988~.
4.1 Linear Radiation Problem
Fig. (1) shows the impulse-response function, L(t),
for the heave force on a hemisphere, radius a, as
a function of time, obtained using SAMP with
At=0.05. Here L(~) is defined as
~ ~ P.//l30 (20)
for an impulse (delta function) acceleration at
t=0. The curve for Hulme (1982) is obtained from
Fourier transform of his analytic frequency-domain
results. The time-domain results show the charac-
teristic oscillations at larger times which are caused
by the presence of irregular frequencies of the in-
terior problem (Adachi & Ohmatsu, 1982~. Note
that the amplitudes of these spurious oscillations
decrease as the number of panels is increased.
0 r I
D ~
~ o
jo
Do °
-
-
~.
o
A
1
Hulme (1982)
· · · SAMP N=18
SAMP N=150
\~_ ~ ___
(~ 1
it,, 1
Tier a/g |
1
A33/PV
0.0 2.0 4.0 6.0 8.0
w2a/9
Fig. 2: Heave added-mass and damping coefficients
for a hemisphere.
The added mass and damping coefficients are re-
lated to the cosine and sine transforms respectively
of the impulse-response function. These are shown
in Fig. (2) as compared to the frequency-domain
result of Hulme (1982~. For illustration the loca-
tion of the lowest irregular frequency (w,2r~a/g
2.56, Hulme, 1983) is also indicated. It is seen
that the oscillations in Fig. (1) are directed related
to the irregular behavior near Micra although the
singular nature is not fully captured because of the
finite-time truncation of L(t) (at t=20 in this case).
We note again the smaller and more confined os-
cillations for larger N.
For radiation problems with forward speed, a
forced oscillatory motion is superimposed on a
step-function jump of the forward velocity to the
prescribed value. The force coefficients are then
obtained from Fourier transforms of the SAMP
force time histories after steady state is reached.
In the presence of forward speed, quadratic con-
tributions must be retained in the present space-
fixed formulation and the impulse-response func-
tion is not generally useful. This computational
inefficiency for the special case of linearized sea-
keeping is the main (in fact only) disadvantage of
the spaced-fixed formulation.
Figs. (3,4) show the heave and pitch diagonal
and off-diagonal added-mass and damping coeffi-
cients of the WSK hull at Froude number En --
U/~/~=0.2. The SAMP calculations use N=120
and At=0.1. These results are considered to have
converged in that selected calculations (not shown)
47
OCR for page 48
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OCR for page 50
using N=200 and At=0.05 show changes of less
than 1% in the force coefficients. As pointed out
earlier, the SAMP calculations do not contain the
contributions of the m-terms while the general
LAMP computations do. To gauge the importance
of these effects, we include in the figures the small-
amplitude limits of the large-amplitude LAMP re-
sults (see Sec.4.5~. These values are obtained from
extrapolations of the small finite-amplitude LAMP
results to zero amplitude and are denoted hereafter
as LAMPo. From Figs.~18), it is seen that LAMP
approaches the LAMPo limit very smoothly so that
the estimates are reliable.
For the diagonal coefficients (Fig.3), the LAMPo
and SAMP results are both reasonably satisfactory
although the m-term effects appear to play an ap-
preciable role as the relative frequency, To--U~/9,
decreases and especially for the pitch coefficients.
On the other hand, the Michigan results (which
include m-terms) using the same N show unex-
plained deviations for intermediate values of To
Not surprisingly, strip theory performs better for
larger To but appears to be affected significantly
only for Bss at lower frequencies.
The results are much more interesting for the off-
diagonal coefficients. For the added-mass terms
A3s and As3, the SAMP calculations compare well
to measured data. When effects of the m-term are
included, the deviations of LAMPo from SAMP
are primarily at lower frequencies but the com-
parison to experiments is otherwise not improved.
For the damping terms B3s and Bs3, the SAMP
results are inadequate, showing a large somewhat
constant shift, and indicate that the m-term effects
may play an important role for these coefficients.
Indeed, when the m-term effects are included, the
LAMPo and Michigan results are both reasonably
close to experimental measurements.
In summary, we see that the SAMP results are
satisfactory for the off-diagonal added mass coef-
ficients, and generally acceptable for the diagonal
coefficients. Taken together, the LAMPo predic-
tions provide overall good agreement with exper-
iments and are clearly superior. By comparing
SAMP and LAMPo results, the effect of the m-
terms is quantified and shown to be important for
pitch-related coefficients at lower frequencies and
especially for the off-diagonal damping terms. In
is noteworthy that since LAMP uses a linearized
free-surface condition, the present results suggest
that nonlinear free-surface terms such as those that
may be included in Dawson-type codes (e.g., Nakos
& Sclavounos, 1990) may otherwise not be as im-
portant in the prediction of the motion coefficients.
A well-known theoretical difficulty of the linear
seakeeping problem is the singularity at the critical
relative frequency To =1/4 (e.g., Dagan & Miloh,
1980~. Physically, this may be explained by the
simple fact that at To = 1/4 the group velocity
of the waves generated by the oscillatory motion
matches the ship speed U and resonance occurs. In
the frequency domain, the linearized result is un-
bounded while in the time-domain the resonance
is manifest as a slow unbounded growth of the so-
lution with time.
While actual predictions at the critical frequency
may not be contemplated in the context of lin-
earized theory, it is useful to be able to avoid
the undesirable growing solution in geneml time-
dependent simulations. An obvious idea (also sug-
gested by Beck & Magee, 1990) is to simply trun-
cate the duration of the memory effect convolution
terms in (15) by requiring t - T < TCU`, say. This is
illustrated in Fig. (5) which shows the time history
of the vertical force on a WSK hull starting from
rest with a heaving frequency of w=1.25 and a for-
ward speed corresponding to To = 1/4 (Fn=0~2~.
When the memory effect is not limited, the force
grows slowly without reaching steady-state. When
we set TCUI=1O' however, steady-state is quickly
reached resembling the results at non-critical To,S.
We point out that because of the relatively slow
growth and the effective truncation of TCU' by the
total duration of the simulation, the problem of
the critical frequency may not pose a severe limi-
tation in practice for general (not critically forced)
time-domain applications. The motion coefficients
obtained from SAMP calculations using different
values of TCU! are presented as separate symbols
on the To = 1/4 vertical in Figs. (3,4~. We cau-
tion that these results should not be considered as
'predictions' at the critical frequency despite the
fact that they are now finite and appear to give a
roughly continuous trend.
4.2 Linear Diffraction Problem
The presence of incident waves, ~I(x,t), can be
included into the governing equation (15) in a
straightforward manner. In practice, this is intro-
duced instantaneously at t=0 and the diffraction
ejects evaluated after the transients vanish typi
so
OCR for page 51
o-
o
o
to -
o
0
to
to
o-
| SAMP IV= 120
unlimited memory effect
memory effect truncated for t-T ~ TO
0.0 20.0 40.0 60.0
t~l~
Fig. 5: Peaks of the leave force history for a lose
hull at Fn=0.2, TO=1/4.
cally in a very short time. In the case of no forward
speed, the linear diffraction problem is validated
for a floating hemisphere. The SAMP results are
indistinguishable from the earlier computations of
Cohen (1986) and King (1987) and the compar-
isons are not shown here.
Figs. (6) shows the amplitude and phase of the
wave exciting forces on a WSK hull moving with
forward speed, Fn=0.2, in head seas. For the
SAMP results, N=120 and 30 time steps per inci-
dent wave period is used. The comparisons to the
time-domain results of Michigan, strip theory and
experiments are overall satisfactory. The Michi-
gan calculations underpredict the secondary hump
at high frequency although both of the 3D time-
domain calculations show better correlations to the
experiments than strip theory.
For a more realistic and complicated geometry, we
consider the Series 60 (CB=0.7) hull form for which
extensive experimental data are available. Figs. (7)
show a complete set of results for Fn=0.2 in head
seas. The SAMP results with N=180 and 250 have
clearly converged (30 time steps per incident wave
period are found to be more than adequate) and
compare very well with experimental data. Not
surprisingly, strip theory results are generally ac-
ceptable compared to SAMP predictions at higher
frequencies. On the other hand, the Michigan re-
sults do not indicate convergence and appear to un-
derpredict the amplitudes especially at the higher
frequency second hump.
4.3 Large-Amplitude Motions at Zero
Speed
As a first example, we study the large-amplitude
heaving of a (complete) sphere. The sphere is ini-
tially semi-submerged with its center at the ori-
gin and a forced heaving motion with zest) =
-Ahsin~t is imposed at t=0. We set w=1 and
use lV=150 and wAt = 2,r/30. As pointed out
earlier, at each time step, the submerged portion
of the sphere is repanelized, the Rankine influence
matrix reevaluated and solved, and the memory
terms recalculated from T=0 to ~ = t.
Fig. (8) shows the different components of the ver-
tical force on the sphere as a function of time for
the case of Ah/a=0.5. Steady-state (limit cycle) is
rapidly reached (within one period) for all the com-
ponents. For this geometry, the hydrostatic force
is a large part of the total. The inertia (-~/0t)
term shows distinct higher-harmonic components
while the quadratic (-~V.~2/2) component is pri-
marily at the second harmonic, as expected. Sur-
prisingly, these higher harmonic contributions ap-
pear to cancel closely so that the total force oscil-
lates primarily at forcing frequency together with
a negative (suction) steady component.
To validate the large-amplitude capability of
LAMP, systematic convergence tests were per-
formed for this case. A typical example is
Fig. (9) which shows the hydrodynamic force his-
tory (FHD) for varying N and w/`t for Ah/a=0.5.
The results are indistinguishable indicating that
IV=150 and w/`t = 2,r/30 are more than adequate.
Fig. (10) shows the time history of the vertical hy-
drodynamic forces on the sphere for different Ah/a.
As expected, the nonlinear curves approach the
linear one as Ah/a decreases. For larger heaving
amplitudes, the peaks become sharper and higher,
while the troughs become shallower. The nonlin-
ear effect, however, is relatively small even when
the heaving amplitude is 50% of the radius. This
was found also in earlier full~nonlinear simulations
for heaving axisymmetric bodies (Lin, et al, 1984;
Dommermuth & Yue, 1986~.
A detailed analysis of the frequency components of
the force can be obtained from Fourier transform
of the steady-state (limit-cycle) time history. For
convenience, we define the frequency components
of the vertical force:
F (`t) = OR (Lfo + feint+ f2ei2W'+
51
OCR for page 56
o
en ~
to
*
~ o ly
-~ m
1
lo
11 ~ .
1 .
to .
to - .
1
o
o
1
SAMP N=168
SLAW (Letcher et al, 1989)
xs~ Experiment (ITTC, 1985)
0.0 t.0 2.0 3.0 to 5.0
t/To
Fig. 12: Horizontal force time history for a WRT
hull moving at Fn=0~3.
to
o
o-
1
cut
to
*
(a) 0 \
~ 1
- 1 ~1
[q up
_i
11 1
~4
~ 0
C`2-
SAMP N=168
SLAW (Letcher et al, 1989)
i 1
0.0 1.0 2.0 3.0 4.0 5.0
t/To
Fig. 13: Vertical force time history for a WRT hull
moving at Fn=0~3.
to
*
,~ 0
~ ~-
~ 1
:a
~ 0
-I'q C`2- I
lo
11 o_
1
0 l
4,
SAMP N=168
SLAW (Letcher at al 10~\
. . . . .
0.0 1.0 2.0 3.0 4.0 5.0
t/To
Fig. 14: Pitch moment time history for a WRT hull
moving at Fn=0~3.
the pitch moment is very good while the two re-
sults differ by about 15% for the steady vertical
force.
To evaluate the dependence of SAMP predictions
on panel size and time steps, systematic conver-
gence tests were performed for different numbers
of panels and time steps for Fn=0~3 and 0.4. These
results are summarized in Tables 2 and 3. Three
different grid distributions-uniform, cosine and
'geometric' spacing-are considered in the lon-
gitudinal direction while the girth-wise grid sizes
are kept constant. The so-called 'geometric' spac-
ing is based on the criterion that the projection of
each panel on the y-z plane be constant. For the
parabolic WRT hull, this corresponds to a square
root grid distribution in the longitudinal direction.
From the tables, some indication of convergence is
observed, although a definite trend is still difficult
to obtain. This suggests that possibly larger num-
bers of panels may be required for more accurate
predictions at these Froude numbers. Comparing
the three grid distributions, the geometric grids
seem to give the best and most consistent results.
For the calculations in this and later sections, a
geometric grid with N=168 (28x6) and ~t=2'r/40
was used.
To obtain a complete resistance curve, the calcu-
lation of Fig. (12) was repeated for Froude num-
bers ranging from 0.2 to 0.45. The final results
are shown in Fig. (15) together with the ITTC ex-
perimental data. The overall comparison is quite
satisfactory.
Finally, we evaluate the possible effect of the start-
up velocity profile on the steady-state force predic-
tions. For illustration, three velocity profiles were
considered with V(t)=0 for t < 0, V(t) = U for
t > To, and for O < t < To: (i) impulsive start,
V(t) = U; (ii) ramp start, V(t) = Ut/T~; and (iii)
cosine-function start, V(t) = Uf1-cos(,rt/2T')~/2.
Fig. (16) shows the horizontal force time histories
for these three cases with T~/To=0.6. The initial
transients and the phases of the later oscillations
differ appreciably between (i) and (ii),tiii) but the
same steady-state value is reached. As expected,
the smoother ramp and cosine-function profiles
produce smaller transient and To oscillations (asso-
ciated with milder start-up disturbances) and are
preferred for practical resistance predictions using
SAMP.
56
OCR for page 57
Table 2: Wave resistance coefflaents Cat x 103 of the WRT hull at Fn=0~3
an a function of grid size anu distribution.
| At/2,r 1 uniform grid I cosine grid I geometric grid ~ |
1 1 20x6 1 28x6 1 36x6 1l 20x6 1 28x6 1 36x6 11 20x6 1 28x6 ~ 36x6
. l .. ,
1/20 2.53 2.84 1.70 l 2.82 1.99 1.48 1.20 1.48 2.50
1/30 2.50 1.37 1.58 l 1.82 1.87 0.94 1.57 1.62 1.60
-1/40 1.68 2.79 1.26 1.22 1.84 1.78 . 1.29 1.64 1.75
Table 3: Wave resistance coefficients Cc x 103 of the WRT hull at Fn=0~4
as a function of grid size and distribution.
| l~tj2,r 1 uniform grid I cosine grid I geomeh
36x6
2.07
1.94
2.09
lo
._
*
20-x6 1 28x6 1 36x6 1l 20x6 128x6 1 36x6 1l 20x6 1 28x6
1/20 1.32 1.56 1.53 ~ 1.43 1.62 1.71 1.42 1.68
lt30 0.33 2.50 1.96 ~ 1.68 1.98 2.12 2.24 2.07
1/40 4.22 1.24 2.24 1.61 2.33 2.10 2.05 2.15
0 0
cr)- . ~O
· SAMP N=168
Experiment(ITTC, 1985)
lo
N
0.15 0.20 0.25 0.30 0.35 0.40 0.45 050
Fn
Fig. 15: Wave resistance coefficient for the WRT
hull as a function of Froude number.
4.5 Large-Amplitude Motions with For-
ward Speed
A main objective of this work is to evaluate and
quantify the importance of large-amplitude mo-
tions on seakeeping characteristics. While the
change of such relevant quantities as added mass
(and hence natural frequencies and response ampli
Impulsivestart
Ramp start
Cosine function start.
rem I I r - -l
00 10 20 3.0 40 5.0
t /Lo
Fig. 16: Horizontal force time history for a TORT
hull moving at Fn=0~3 using different start-up ve-
locity profiles (T1 /To=0.6 ).
tudes) due to large-amplitude motions can be ex-
pected on physical grounds, and has been reported
in experiments (e.g., O'Dea & Troesch, 1987), the
precise magnitudes or even dependencies are as yet
not well known.
To address this problem, the LAMP program
was applied to the large-amplitude forced heav
57
OCR for page 58
A
°1
.
A
A l
*
t-:, A
1
to
o-
1
00 0.1 02 0.3 04 ()5
Ah/0
Figs. 17: Steady surge, heave and pitch forces and
moment on a heaving (frequency w=1.0, amplitude
Ah) WRT hull moving with constant forward speed
(Fn=0.2~. .: LAMP (N= 160~; - - - : 'calm-water'
(Ah=O) value obtained with SAMP.
ing of a WRT hull moving with constant forward
speed. Specifically we chose a Froude number
Fn=0.2, a heaving frequency w=1.0, and consid-
ered a range of heaving amplitudes correspond-
ing to Ah/B ~7.5 - 45% where H is the draft.
Figs.~17) show the added resistance, added sinkage
and added trim forces as a function of the heave
amplitude. For contrast, the 'calm-water' values
(cf. Sec.4.4) are also indicated. For small Ah/H,
the results are close to the calm-water values as
expected and the approach to this limit is very
smooth. As the amplitude increases, however, the
A
c ~
.
co
.
o
.
D
~o
~ ,
D
-
· ~"added mass"
.
0 0 0
"damping"
added m ~ ~,
.
0.0 0.1 0.2 0.3 0 4 0
Ah/0
Fig. 18: Excitation frequency components of the
limit-cycle forces and moment on a heaving ~
w=1.0, amplitude Ah ~ WRT hull moving with con-
stant forward speed (Fn=0.2~. ~ in phase with
acceleration ~ "added-mass" ); O out-of-phase with
acceleration ~ "damping" ~ coefficients using LAMP
(N=160). Also indicated are I, \7 : small-
amplitude limits (LAMPo); and ·, As: SAMP
added mass and damping coefficients.
added resistance increases rapidly and is as much
as 0~5) or more times the calm-water value for the
larger amplitudes considered. The picture is sim-
ilar for the mean vertical force and moments but
somewhat less dramatic.
By analysing the limit-cycle force and moment his-
tories for the frequency components at the forc-
ing frequency that are in phase and out of phase
respectively with the acceleration, the so-called
large-amplitude "added-mass" and "damping" co-
efficients with forward speed can again be identi-
fied (cf. Sec.4.3~. These diagonal and off-diagonal
s8
OCR for page 59
seakeeping coefficients are shown in Figs.~18) as
a function of Ah/H. For small amplitudes, the
approaches to the linear (zero amplitude) limits
are very smooth allowing accurate extrapolation
for the linearized LAMPo estimates. As noted
in Figs.~3,4), this small-amplitude limit of LAMP
differs from the linearized SAMP prediction (also A,
shown in Figs.18) because of the absence of the
effect of the m-terms in the latter.
As in the case for the heaving sphere with no for-
ward speed (Figs.ll), the large-amplitude added
mass shows a clear dependence on the amplitude
and decreases appreciably as Ah increases. As an
illustration, when the heave amplitude is ~40% of
the draft, A33 has decreased to about 60~o of the
small-amplitude value. For the present WRT hull,
the waterplane area changes very little with heave
amplitude and consequently the relevant heave
natural frequency would increase by ~20% com-
pared to the linear value in this case. This may
explain some of the "unexpected" dependencies of
the response amplitude functions on forcing ampli-
tude observed in experiments. As with the sphere
with no forward speed, the LAMP damping coef-
ficients here are less sensitive to heave amplitude
probably due to the fact that nonlinear radiation
mechanisms are absent in the present approxima-
tion.
4.6 Motions of a Floating Body and Un-
steady Loads
With the ability to determine the relevant hydro-
dynamic forces at any given instant, the complete
six degree-of-freedom motion history of the body
can be obtained by a direct integration of the dy-
namical equations. Thus LAMP can be employed
to study general time-dependent motions such as
in the case of a ship advancing in an irregular sea-
way. Indeed, one of the important applications is
the study of episodic events involving large loads
and motions in the time domain.
To validate the motions solver, we consider first
the decaying motions of a freely-floating sphere re-
leased from rest with a given initial vertical dis-
placement. As a test, we integrate the problem di-
rectly in time, although for the linearized problem
the solution can be obtained by a Laplace trans-
form involving the force impulse-response function
of Fig. (1) (Liapis, 1986~. Consistent with the open
time quadrature formula of (17), we employ an ex
59
SAMP
LAMP Ah/a=0.50
0.0 8.0 16.0 24.0 32 0
told
Fig. 19: Vertical displacement history of a float-
ing hemisphere released from rest with an initial
height Ah.
plicit multi-step scheme for the motion time inte-
gration.
First, SAMP computations were performed and
compared to the linearized calculations and small-
amplitude experiments of Liapis (1986~. The re-
sults are indistinguishable and very close to the
measurements and are not shown. Fig. ( 19)
shows the SAMP and LAMP (for initial ampli-
tude Ah/a=0.5) results with N=150 and At=0.1.
The difference in the (normalized) amplitudes is
small but it is of interest to note that the LAMP
oscillation periods are longer than the linearized
values by about 5%. This is caused by the com-
peting effects of added-mass decrease due to large-
amplitude motions (cf. Fig. (11~) which tends to
decrease the period, and the reduction in the av-
erage waterplane area (and hence the hydrostatic
force) which tends to increase the period of oscil-
lation. In this case, the latter effect is greater and
consequently an increase in natural period is ob-
served. This again underscores the importance of
nonlinear geometry effects.
An interesting application of the motion program
is to reexamine the calm-water resistance problem
of Sec.4.4 but this time allowing the body to freely
sink and trim until steady-state values are reached.
Starting from rest, the ship is held to a constant
forward speed but is otherwise free to heave and
pitch. In the large-amplitude simulation, the un-
derwater geometry is allowed to change as steady
OCR for page 60
o
,(-
o
lo
o GO
*
o
C\2
lo
lo -
Pree Body:
· LAMP N = 196
:~ Experiments
(ITTC, 1985; Bai, 1979,
Fixed Body:
O SAMP N=168
Experiments
(ITTC, 1985)
, . . . . . .
0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
Fn
Fig. 20: Wave resistance coefficient for the WRT
hull as a function of Froude number.
-
_`
~ °
+ f
_'
CO
lo. .~
C~2
lo
lo
~ S;nkage · LAMP JV=196
;~ Experiment
(ITTC, 1985)
4~1
trim
0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
Fr'
Fig. 21: Sinkage (a) and trim (b) of the WRT hull
as a function of Froude number. [A--draft at AP,
[F-draft at FP. Positive trim means bow up.
state is attained by an equilibrium between hydro-
static forces of the final geometry and the steady
vertical force and pitch moment.
Fig. (20) shows the free sinkage and trim wave
resistance of the WRT hull as compared to cor-
responding experimental measurements as well as
the fixed body resistance results of Fig. (15~. The
present LAMP calculations use At=21r/40 and
N-196 (28x 7) where an extra horizontal row of
panels is added to the grid of Fig. (15) to model
the extra draft due to linkage. From the figure,
the increase in wave resistance due to sinkage and
trim is clearly predicted. Comparisons between the
LAMP results and experiments are excellent.
The comparison for sinkage and trim is shown in
Figs. (21~. The LAMP predictions are again sat-
isfactory both in the magnitude and the forward
speed dependence. The results are somewhat bet-
ter for the sinkage and show a slight underpredic-
tion of the trim at higher speeds. Since sinkage
is the main reason for the resistance increase, the
under-prediction of trim at high En does not affect
the resistance prediction significantly.
Finally, we applied LAMP to study free vertical-
plane motions of a ship advancing in irregular head
seas. A time record of the free surface was con-
structed using a two-parameter Pierson-Moskowitz
spectrum with a significant wave height H~/3=10
feet and a modal period Tm=12 seconds with low
and high frequency cutoffs at 0.1 and 4.25 rad/sec
respectively.
A 400 foot long WRT hull with constant forward
speed corresponding to Fn=0.2 was first chosen.
Figure (22) shows the time history of the inci-
dent wave elevation at the ship center of gravity,
along with the pitch and heave displacements from
LAMP calculations (N=160, At=0.3526 seconds).
For contrast, strip theory predictions obtained by
superposition of the linear frequency-domain re-
sponses at the component incident frequencies are
also included. To be consistent with the strip the-
ory program used here, the LAMP calculations in-
clude the hydrostatic (and Froude-Krylov) forces
only up to the mean waterline. To remove the ef-
fects of starting transients, the LAMP simulation
has been started from rest at t=-10 seconds.
As can be seen in Fig. (22), the differences between
LAMP and strip theory are small for the pitch dis-
placement, with the exception of apparent mem-
ory (transient) effects around 10 c t < 60 and
60
OCR for page 61
to
cO
T
o
co
LAMP
Strin theory
Do
0
_
to
1
o
CD
1
0
0
0_
1 \ ~ 1/
I O ~BY'
(J ~
INJI
J ~I\
.' ~l:~` I' ~
1 1~\ " :t ~7 :1: ~
,1
v V V \`,1
V1
o
1
_ 1
30.0 60.0 ~ 90.0
t (see)
Fig. 22: Incident wave elevation (~) at the ship's center of gravity, pitch angle All, and heave displacement
(z) of a WRT hull at Fn=0.2 in irregular head seas.
6
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lo ~
o
~-
1
o
r\'
-
~o
~ o
~ - .
1
o -
ll
600
90.0
Fig. 23: Incident wave elevation (~) at the ship's center of gravity, pitch angle All, and heave displacement
(z) of a Series 60 (CB=0.7O) hull at Fn=0.2 in irregular head seas.
62
OCR for page 63
| ~ t=0.9305 second
v 1 0.2 ~ _ _ 1 4 _ - - - 1 V
-1 Z~.
t=4.4564 second
vim
Fig. 24: Non-dimensional dynamic pressure distribution on a WRT hull at two time instances.
80 < ~ < 90. The same is true for the heave dis-
placement except for a nearly constant downward
shift of the nonlinear result. This is caused by the
steady sinkage force associated with forward speed
which is present in LAMP but absent in the strip
theory (from Fig. 21, this is estimated at about
0.53 feet for a 400 foot hull). Overall, the rela-
tive good performance of the strip theory is not
unexpected for the mathematical WRT hull (with
relatively small geometry changes near the water
line) and for head seas.
For a more realistic geometry, we show in Fig. (23)
the corresponding case for a 400 foot long Series 60
(CB=O.7) hull at Fn=0~2. In this case, strip the-
ory overpredicts the pitch motion significantly, in
some instances by over 50% (e.g., near t=75 see).
For the heave displacement, there are appreciable
underpredictions of the troughs in addition to the
absence of the steady downward linkage. These de-
ficiencies of strip theory for the more complicated
Series 60 hull are consistent with the observations
of Frank & Salvesen (1970~.
As a final illustration, we display the pressure
distribution on the hull as output from LAMP.
Fig. (24) shows the dynamic pressure distribution
on the WRT hull at t=0.9305 and 4.4564 seconds
respectively corresponding to the case in Fig. (22~.
Note that the pressures are given on the actual sub-
merged surface. The complete unsteady loads on
the ship hull can now be obtained from the pres-
sure distribution and motion histories.
5 CONCLUSIONS
A time-domain method, LAMP (karge-Amplitude
Motion Program), was developed for the general
large-amplitude motions of a three-dimensional
surface-piercing body in a seaway. The body
boundary condition is satisfied exactly on the in-
stantaneous underwater body surface while a lin-
earized free-surface condition is used.
To validate the approach and evaluate its accu-
racy, the method was applied extensively to obtain
linearized motion coefficients for a number of dif-
ferent geometries with or without forward speed.
The results include added-mass and damping coef-
ficients, wave exciting forces and steady wave resis-
tance, sinkage and trim forces and moments. These
are compared to experimental measurements and
existing linear time- and frequency-domain calcu-
lations. The comparisons are overall satisfactory
for all the results and show that LAMP is equal
or superior to any of the existing computational
methods in terms of accuracy.
The main feature and purpose of LAMP, however,
is for general nonlinear large-amplitude motions.
To illustrate its effectiveness, we apply LAMP
to study the large-amplitude forced heaving of a
(complete) floating sphere; the large vertical-plane
motions and the free sinkage and trim of a Wigley
hull moving with constant forward speed; and the
general time-dependent large-amplitude motions of
a Series-60 ship advancing in an irregular seaway.
63
OCR for page 64
Some of the main findings are the importance
of (added) steady (and higher-harmonic) compo-
nents, and the modifications of the first-harmonic
(excitation frequency) motion coefficients. In the
first case, the presence of large-amplitude heaving
motions is shown to result in significant increases
of the wave resistance and steady sinkage and trim
forces. In the latter, the nonlinear "added-mass" is
found to decrease markedly with increasing heave
amplitude. The consequent reduction of inertia
and increase in natural frequency may have impor-
tant implications to the motion dynamics of the
ship. This may also explain some of the exper-
imentally observed dependencies on amplitude of
normalized motion response functions. When ap-
plied to general time-dependent motions in irregu-
lar waves, LAMP demonstrates the importance of
transient (memory) and nonlinear geometry effects
especially for realistic ship geometries where strip
theory is found to be inadequate. The road is now
laid for nonlinear simulations of extreme episodic
events and complete load and motion predictions.
The current version of LAMP is fully vectorized
for high-speed vector processors. For a nonlinear
(large-amplitude) simulation using 0~150-200) un-
knowns on the body and a similar number of time
steps, the typical CPU time on a single Cray Y-MP
processor is 0~1-2) hours. Further code optimiza-
tion may reduce this requirement by a small factor.
For applications involving significantly larger num-
ber of unknowns and time steps, the time domain
formulation may be particularly suited for parallel
algorithms on multiple processors.
Acknowledgement
This research was sponsored by the Office of Naval
Research, the U.S. Coast Guard, and the De-
fense Advanced Research Projects Agency. We are
grateful to Cray Research, Inc., for the use of their
Cray Y-MP/832 supercomputer. Some computa-
tions were also performed on the NSF Pittsburgh
Supercomputer Center Cray Y-MP. We thank M.
Meinhold and K. Weems for valuable technical and
graphical help.
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DISCUSSION
Pierre Ferrant
Sirehna S.A., France
I would like to make a comment on your results about the heaving
sphere. I do not completely agree with your analysis of the
behaviour of the damping coefficient which you show to remain
constant when amplitude is varied. My own results show that at least
for a submerged body, the nonlinear phenomena associated to the
body boundary condition are very frequency-sensitive. In fact, when
dealing with oscillatory motions, even in the time domain, one cannot
ignore the importance of frequency and this parameter must be varied
before drawing conclusions. I would therefore be very interested if
you could give result for the surface-piercing heaving sphere at lower
frequencies, say about wet R/g=0.4.
AUTHORS' REPLY
We have preliminary results for the heaving (surface-piercing) sphere
for normalized frequencies ranging from w ~ 0.4 to ~ 3 and
amplitudes rangin, from Ah/a ~ O to ~ 0.5 or higher. In contrast
to your results for the submerged sphere, the damping coefficient
remains relatively independent of amplitude for lower frequencies (w
< ~ 1) and shows some sensitivity only for intermediate frequencies.
The precise mechanisms for these dependencies (an amplitude and
frequency) are as yet not completely understood.
The following table lists the results for the frequency ~ = 0.4 you
su~gest~l. For comparison, the data for Fig. (1) at w = 1.0, as well
as w = 1.5 and 3.0 are also included. Again, we note that these
results are only preliminary.
Normalized damping coefficients for a (surface-piercinz) heaving
~here.
Ah/a
0.125
0.250
0.375
0.500
w=0.4 w= 1.0 w= 1.5 w=3.0
0.036 0.195 0.151 0.314
0.036 0. 195 0.154 0.3 12
0.035 0.195 0.170 0.312
0.035 0.194 0.185 0.316
66
Representative terms from entire chapter:
wrt hull
