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OCR for page 231
24~ Symposium on Naval Hydrodynamics
Fukuoka, JAPAN, 8-13 July 2002
GENESIS OF DESIGN WAVE GROUPS IN EXTREME
SEAS FOR THE EVALUATION OF WAVE/STRUCTURE
INTERACTION
·.
GUNTHER F. CLAUSS (Technical University of Berlin, Dept. of Naval
Architecture and Ocean Engineering, Gennany)
ABSTRACT
For the design of safe and economic offshore
structures and ships the knowledge of the extreme
wave environment and related wave/structure
interactions is required. A stochastic analysis of these
phenomena is insufficient as local characteristics in
the wave pattern are of great importance for deriving
appropriate design criteria.
This paper describes techniques to synthesize
deterministic task-related 'rogue' waves or critical
wave groups for engineering applications. These
extreme events, represented by local characteristics
like tailored design wave sequences, are integrated in
a random or deterministic seaway with a defined
energy density spectrum. If a strictly deterministic
. . . .. . .
process Is estan~snect, cause and effect are clearly
related: at any position the non-linear surface
elevation and the associated pressure field as well as
the velocity and acceleration fields can be
determined. Also the point of wave/structure
interaction can be selected arbitrarily, and any test can
be repeated deliberately. Wave-structure interaction is
decomposable into subsequent steps: surface
elevation - wave kinematics and dynamics - forces on
structure components and the entire structure -
structure motions.
Firstly, the generation of linear wave groups is
presented. The method is based on the wave focussing
technique. In our approach the synthesis and up-
stream transformation of arbitrary wave packets is
developed from its so-called concentration point
where all component waves are superimposed without
phase-shift. For a target Fourier wave spectrum a
tailored wave sequence can be assigned to a selected
position. This wave train is linearly transformed back
to the wave maker and - by introducing the electro-
hydraulic and hydrodynamic transfer functions of the
wave generator - the associated control signal is
calculated. Based on this technique the seakeeping
behaviour of ships or offshore structures is efficiently
determined with just one single model test.
The generation of steeper and higher wave groups
requires a more sophisticated approach as propagation
velocity increases with wave height. With a semi-
empirical procedure the control signal of extremely high
wave groups is determined, and the propagation of the
associated wave train is calculated by iterative integration
of coupled equations of particle tracks. With this
deterministic technique 'freak' waves up to heights of
3.2m have been generated in a wave tank.
For many applications the detailed knowledge of the
nonlinear characteristics of the flow field is required, i.e.
wave elevation, pressure field as well as velocity and
acceleration fields. In this case a finite element method is
used to determine the velocity potential, which satisfies
the Laplace equation for Neumann and Dirichlet
boundary conditions.
So far, nonlinear wave groups in an ideal fluid have
been investigated. If viscous effects are also considered
an approach of transient viscous free surface flow
computation with RANSE/VOF solver is used. As an
application, an artificial reef- modeled as a submerged
permeable wall - has been analysed.
In general, extremely high 'rogue' waves or critical
wave groups are rare events embedded in a random
seaway. The most efficient and economical procedure to
simulate and generate such a specified wave scenario for
a given design variance spectrum is based on the
appropriate superposition of component waves or
waveless. As the method is linear, the wave train can be
transformed down-stream and up-stream between wave
board and target position. The desired characteristics like
wave height and period as well as crest height and
steepness are defined by an appropriate objective
function. The subsequent optimization of the initially
random phase spectrum is solved by a Sequential
Quadratic Programming method (SQP). The linear
synthetization of critical wave events is expanded to a
fully nonlinear simulation by applying the subplex
OCR for page 232
method. Improving the linear SQP-solution by the
nonlinear subplex expansion results in realistic
'rogue'-waves embedded in random seas.
As an illustration of this technique a reported
rogue wave - the Draupner 'mew Year Wave" is
simulated and generated in a physical wave tank. Also
a "Three Sisters" wave sequence with succeeding
wave heights HS...2Hs...Hs, embedded in an extreme
sea, is synthesized.
For investigating the consequences of specific
extreme sea conditions this paper analyses extreme
roll motions and the capsizing of a Ro-Ro vessel in a
severe storm wave group. In addition, the seakeeping
behaviour of a semisubmersible in the Draupner New
Year Wave, embedded in extreme irregular seas is
numerically and experimentally evaluated.
INTRODUCTION
Considering rogue wave events as rare
phenomena - according to Murphy's law - beyond our
present modelling abilities, Haver (2000) suggests
that freak waves (unexpected large crest height / wave
height, unexpected severe combination of wave
height and wave steepness, or unexpected group
pattern) should be defined as wave events which do
not belong to the population defined by a Rayleigh
model. To yield a sufficiently small contribution to
the overall risk of structural collapse, the structure
should withstand extreme waves corresponding to an
annual probability of exceedance of say 10-s _ 10-4 as
the Rayleigh model underpredicts the highest crest
heights indicating that real processes may be strictly
affected by higher order coefficients. In addition to
the ULS (ultimate limit state) based on a 100-year
design wave an ALS (accidental limit state) with a
return period of 10000 years is suggested. Based on
observations Faulkner (2000) suggests the freak or
abnormal wave height for survival design
He 2 2.5Hs . It is also recommended to characterize
wave impact loads so they can be quantified for
potentially critical seaways and operating conditions.
Present design methods should be complemented by
survival design procedures, i.e. two levels of design
wave climates are proposed:
The Operability Envelope which
corresponds to the best present design
practice
The Survivability Envelope based on
extreme wave spectra parameters which may
lead to episodic waves or wave sequences
(e.g. the Three Sisters) with extremely high
and steep crests.
Wave steepness, characterized by front and rear
steepness as well as by horizontal and vertical wave
asymmetries seems to be a parameter at least as important
as wave height (Kjeldsen an Myrhaug, 1979~.
A probability analysis of rogue wave data recorded at
North Alwyn from 1994 to 1998 reveals that these waves
are generally 50 % steeper than the significant steepness,
with wave heights HmaX > 2.3Hs (Wolfram et al, 2000~.
The preceding and succeeding waves have steepness
values around half the significant values while their
heights are around the significant height.
Steep-fronted wave surface profiles with significant
asymmetry in the horizontal direction excite extreme
relative motions at the bow of a cruising ship with
significant consequences on green water loading on the
fore deck and hatch covers of a bulk carrier (Drake,
1997~. Heavy weather damages caused by giant waves are
presented by Kjeldsen (1996), including the capsizing of
the semisubmersible Ocean Ranger. Faulkner and
Buckley (1997) describe a number of episodes of massive
damage to ships due to rogue waves, e.g. with the liners
Queen Elisabeth and Queen Mary. Haver and Anderson,
(2000) report on substantial damage of the jacket
platform Draupner when a giant wave (HmaX = 25.63m
with the crest height '7c = 18.5m hit the structure in 70m
water depth on January 1, 1995 (Fig. 1-top). Related to
the significant wave height H = 11.92m . the maximum
_ a
wave rises to HmaX = 2.15Hs with a crest of
77c = 0.72Hmax
Not as spectacular but still exceptional are wave data
from the Norwegian Frigg field - water depth 99.4m
(Hs = 8 49m' HmaX = 19.98m id = 12.24m ~ (Kjeldsen,
1990) and the Danish Corm field - water depth 40m
(Hs =6.9m, HmaX =17.8m, id ~13m) (Sand et al.,
2000~. Also remarkable are wave records of the Japanese
National Maritime Institute measured off Yura harbour at
a water depth of 43m (Hs = 5.09m, Hmax = 13 6m,
71C ~ 8.2m ~ (Mori et al., 2000) (Fig. 1-bottom).
All these wave data - with HmaX I Hs >2.15 and
id IHmaX > 0.6 - prove, that rogue waves are serious
events which should be considered in the design process.
Although their probability is very low they are physically
possible. It is a challenging question which maximum
wave and crest heights can develop in a certain sea-state
characterized by Hs and Tp. Concerning wave/structure
interactions, with respect to response based design loads
and motions or reliability based design: Is the highest
wave with the steepest crest the most relevant design
condition or should we identify critical wave sequences
embedded in an irregular wave train? In addition to the
global parameters Hs and Tp the wave effects on a
OCR for page 233
North Sea - Draupner jacket platform (Haver and Anderson, 2000) - New Year wave 01-01-95
Hs- 1 1.92m, HmaX = 25.63m = 2.15 HS; TIC = 18.5m = 0.72 HmaX (water depth 70m)
Japanese Sea - off Yura harbour - Japanese National Maritime Institute (Mori et al., 2000)
Hs- 5.09m, Hma,, = 13.6m = 2.67 Hs; ~c= 8.2m = 0.6 HmaX (water depth 43m)
Fig. 1 Rogue wave registrations
structure depend on superposition and the interaction
of wave components, i.e. on local wave
characteristics. Phase relations and nonlinear
interactions are key parameters to specify the relevant
surface profile at the structure. If wave kinematics
and dynamics are known, cause-effect relationships
can be detected.
This paper presents a numerical as well as an
experimental technique for the generation of design
wave sequences in extreme seas. Based on selected
global sea state data (Hs, Tp) the wave field is fitted
to predetermined characteristics at a target location,
such as wave heights, crest heights and periods of a
single or a sequence of extreme individual waves.
Starting with a linear approximation of the desired
wave train by optimizing an initially random phase
spectrum for a given variance spectrum we obtain an
initial guess for the wave board motion. This control
signal is systematically improved to fit the wave train
to the predetermined wave characteristics at target
location. Numerical and experimental methods are
complementing each other. If the fitting process is
conducted in a wave tank all nonlinear free surface
effects and even wave breaking are automatically
considered.
Firstly, the linear procedure is presented, and
illustrated by the generation of deterministic wave
packets as well as the synthesis of the above target wave
train into an irregular sea. Next, the nonlinear approach
with its experimental validation is presented. Finally, the
nonlinear fitting process of the target wave sequence
embedded in irregular seas is developed.
LINEAR TRANSIENT WAVE DESCRIPTION
The method for generating linear wave groups is
based on the wave focussing technique of Davis and
Zarnick ( 1964), and its significant development of
Takezawa and Hirayama (1976~. Clauss and Bergmann
(1986) recommended a special type of transient waves,
i.e. Gaussian wave packets, which have the advantage
that their propagation behaviour can be predicted
analytically. With increasing efficiency and capacity of
computer the restriction to a Gaussian distribution of
wave amplitudes has been abandoned, and the entire
process is now performed numerically (Clause and
Kuhnlein, 1995~. The shape and width of the wave
spectrum can be selected individually for providing
sufficient energy in the relevant frequency range. As a
result the wave train is predictable at any instant and at
any stationary or moving location. In addition, the wave
OCR for page 234
orbital motions as well as the pressure distribution
and the vector fields of velocity and acceleration can
be calculated. According to its high accuracy the
technique is capable of generating special purpose
transient waves.
A continuous real-valued wave record ((t) may
be represented in frequency domain by its complex
Fourier transform F(co) which is calculated by
Eq.~1~. Applying the inverse Fourier transformation,
Eq.~2), gives the original record ((t):
+~
F(~) = | ~ (t)e-ia't dt
_00
((I) = 2 |F(~)ei~ do') (2)
~00
where t represents the time and a' = 27f the angular
frequency. In polar notation, the complex Fourier
transform can be expressed by its amplitude and
phase spectrum:
F(~) = I F(co) leiargF(a)) (3)
In practice, it is necessary to adopt a discrete and
finite form of the Fourier transform pair described by
Eqs. (1) and (2):
N-1 ~4i
Fir/ = lit ~ ((k~t)e-i2'`rk/N ~ J
k=0
r = 0,1,2,. .., N /2
g(k/`t) = — ~ F(r/`Go)e i2~ rk/N (5)
~{ r=0
k = 0,1,2,. .., (N -1),
, . . .
superposition with/ \
random phase / \-
~r
I in-phase
superposition
(rare- but possible)
. . .~
extremely long registration of a severe irregular sea state
Fig. 2 Design wave as rare event of component wave
superposition.
where the values ((kAt) represent the available data
points of the discrete finite wave record, with At
denoting the sampling rate and /`a) = 2'z l(NAt) the
frequency resolution. The summation in Eqs. (4) and (5)
can be efficiently completed by the fast Fourier transform
(FFT) and its inverse algorithm (IFFT).
Extreme wave conditions in a 100-year design storm
arise from the most unfavourable superposition of
component waves of the related severe sea spectrum.
Fig.2 presents the simulation of an irregular sea state by
random phase superposition. As a rare - but possible -
event, a very high freak wave is observed. Freak waves
have been registered in standard irregular seas when
component waves accidentally superimpose in phase.
Extensive random time domain simulation of the ocean
surface for obtaining statistics of the extremes, however,
is very time consuming. In generating irregular seas in a
wave tank the phase shift is supposed to be random,
however, it is fixed by the control program on the basis of
a pseudo-random process: consequently, it is also a
deterministic parameter. Why should we wait for these
rare events if we can achieve these conditions by
intentionally selecting a suitable phase shift, and generate
a deterministic sequence of waves, which converge at a
preset concentration point? Assuming linear wave theory,
the synthesis and up-stream transformation of wave
packets is developed from this concentration point. At
this position all waves are superimposed without phase
shift resulting in a single high wave peak. From its
concentration point, the Fourier transform of the wave
train is transformed to the upstream position at the wave
board (Kuhnlein, 1997~.
The Fourier transform is characterized by the
amplitude spectrum and the related phase distribution.
During propagation the amplitude spectrum remains
invariant, however, the phase distribution and the related
shape of the wave train varies with its position. Fig. 3
(left hand side) shows a wave train as a function of time
at different positions and the related amplitude spectrum
of the complex Fourier transform. Note that the maximum
wave ellevation is deduced from the registration at the
concentration point, which starts with a deep trough,
followed by an extremely steep and high crest, and ends
symmetrically with a trough.
The wave train can be transformed from a function of
time at a fixed location xO to a function of space at a
given time to. Fig. 3 (right hand side) shows wave
elevations at selected moments. i.e. successive "photos"
of the water surface. At the concentration point the wave
is represented by a short and steep crest tapering off into
extremely long and shallow troughs at both sides.
As the process is strictly linear and deterministic,
wave groups can be analysed back and forth in time and
OCR for page 235
)~/
~ o -
_ 5
O
-5
~ O
.,,5 -5
5
~-5
O
u: -
Fourier-
~\ spectrum
, 1'LJUl 1~:1—
go - ~/~\ spectrum
c, - / \
/ \
~0.5 / \
3 : /wave frequency
O O i/ o~ [rad/s] \>
0 4 8 12
Pos. 1: wave packet converging no = 0 m
- Pos. 4: concentration point, no = 107 m
~ Pos. 5: wave packet diverging, no = 120 m
c' 1
-
_'
r , 5
~ O
I_ - 5
5
A_
O
-5
5
1
lo °1
~ - 51
-
5
a) °1-
_5
5
O
cn ,
;\ Fourier—
I\ spectrum
'.,
wave number
k~d/s~
To ~ 10 1.5
_
instant: to=200 s wave packet converging
I I I I ~ I I I I I I ~ ~ ~ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ~
~ ~ ~
- t=210 s
j ~ ~
, -t=220s _
t=225 s concentration point ~
,IIA ~
t=230 s wave Packet diverging
v0 100 200 t [Sl
-0 50 100 ~ Em 1
Fig. 3 Wave packet registration at different positions (left) as well as instantaneous wave profiles at selected instants
(right).
space. They also can be integrated into a specified
irregular sea.
Based on the linear wave packet technique the
seakeeping behaviour of ships or offshore structures
is efficiently determined with just one single model
test (Clause and Kuhnlein, 1995~. The testing
procedure is completely automated: Wave generation
as well as carriage are operating computer controlled
to ensure the predetermined interaction of wave train
and vessel. First, a wave train of about 100 m length
is generated. As each subsequent wave is slightly
longer (and faster) than the preceding one, the length
of the wave train is shrinking rapidly while
propagating along the tank. Shortly before the
concentration point, the cruising vessel meets the
wave train exactly at the predetermined position. As
shown in Fig. 4 this wave train excites heave and
pitch motions. To prove the linear behaviour of the
vessel in waves, spectra with different maximum
wave height have been used resulting in identical
RAOs, by amplitude and phase, with a resolution of
about 400 frequency points. Note that the seakeeping
tests for evaluating the entire ship motion behaviour
takes just 10 seconds, i.e. precise and highly resolved
results are achieved in a short time. In addition, the
ship starts and stops under still water conditions.
NONLINEAR TRANSIENT WAVE DESCRIPTION
The generation of higher and steeper wave
sequences, requires a more sophisticated approach as
propagation velocity increases with height. Consequently,
it is not possible anymore to calculate the wave train
linearly upstream back to the wave generator to determine
the (nonlinear) control signal of the wave board. To solve
this problem, Kuhnlein (1997) developed a semi-
empirical procedure for the evolution of extremely high
wave groups which is based on linear wave theory: the
propagation of high and steep wave trains is calculated by
iterative integration of coupled equations of particle
positions. With this deterministic technique "freak" waves
up to 3.2 m high have been generated in a wave tank
(Clause and Kuhnlein, 1997~. Fig. 5 shows the genesis of
this wave packet and presents registrations which have
been measured at various locations including the
concentration point at 84 m.
The associated wave board motion which has been
determined by the above semi-empirical procedure is the
key input for the nonlinear analysis of wave propagation.
As has been generally observed - at wave groups as well
as at irregular seas with embedded rogue wave sequences
- we register substantial differences between the
measured time series and the specified design wave train
at target location if a linearly synthesized control signal is
used for the generation of higher and steeper waves.
OCR for page 236
., 0.10
0.05
-o.oo
·_I
~ -o.os
0) -o. 10
a, :
0.10
0.05
~q o.oos
eO.004
L 0.003
C~ 0.002
~ , 0.001
_
pitch n~otion
_ of towed catamoran
~ -,,,, 1,,, 1 ~ ~ ~ ~ ~ ~ ~ ~ 1 ~ 1 _
50 160 170 180 190
model wave packet
~_-
. 1111~111116111161111~1111~1111~1111~1111~1111~111111111
- /~\ wave packets
~ ~/ ~ (dUlorent w. ~ heights)
/~
w.uvv ' I ' " " ' , ' , ~ , , , ~ I ~ , , , ~ I I I I I,,,, 1,,,,
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 B.0 9.0 10.0 11.0 12.0
Lq 0'005 L' ' ' " ' ' ' " ~
*
e O.004
0.003
~ 0.002
·L~ 0.001
n ono
~ 1.00
q)
> 0.50
r.
~>
—0.6
c,
0- _ 1 t
_ ~ I I ~ I ~ ~ ~ ~ I T I I I I ~ I I I ~
surface elevation
(registrat~on of wave group
~, .
at encounter position)
~ ~ 1 1 1 1 1 ~ 1 1~
. i ' ~ ~ ~ 1 ' ' ' 1 1 ' ' I ~ 1 ~ ~ ~ ~
encounter wave packet
(measured with a wave
, , , 1 , , 1
240 25C
. I ' ' ' I 'A' I I I ' ~ ~ ~ I ~ ~ ~ ,
~ = ~ _- ~ ~
decelerating vessel overrun
by its own stern wave system~
1,,,,1,,111 I , I ,1, ...
probe on board of the
moving carriage)
. 1,,,,1,,,,~,
50 160 170 lBO 190 200 210 220 230
- ' ' ' ' 1 1 ~ ~ I 1 ~ ~ ~ ' 1 ' ~ ' ~ 1
, _ heave motion
of towed cato m a ra n
T ~ T
~—~~
encounter
vesse l/wave
1 ~ ~ ~ 1 , . . .
-0.00
-0.05
-0.10
150 160 170 iBO 190 200 210 220 230 240 25C
3.0
2.0
1.0
0.0
-1.0
-2.0
-3.0
T r T r T l r l r
~'~—
time window
for evaluation
,,,, 1, . . .
_l . I I ~ ' ' I I ~ I ~_' I I I I ~ I I I I ~ I I I I 1 1 1 1 1 ~ r I I r 1~ 1 1 iT
~ / ~~ Vu=4m/s
0.0 1.0 2.0 3.0 4.0 6.0 B.0 7.0 8.0 9.0 10.0 11.0 lZ.0
1 ~tl 1 1 1 ~ ~ I 1 1 1 ~ 1 1 1 1 ~ 1 1 1 1 ~ I 1 1 1 ~ 1 1 1 1 1 1 1 1 1 ~ 1 1 1 1 ~ I 1 1 1
~ v=20.5kn
RAO heave
: `,
o.oo ~
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
~ =o
x~ c>
—·.V ~ 1 1 1 1 ~ I I I I ~ 1 1 1 1 1,, 1 1 ~ 1 ~ -
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
C) rrad/s
Fig. 4 Registrations, Fourier spectra, and transfer functions of a typical seakeeping test with a high-speed
catamaran in transient wave trains (model scale 1:7, Vm=4~0 m/s; full scale: V=20.5 kn; Fn=0.56~.
0.5:
0.0 r
. I I i I i I T i ~ ~ I ~ ~ ~ T I i i ~ i
Note: stern of th e vessel is lifted -
. ~mechanically by security rope
. ~
I , , , , 1
205 time [s] 210
i ~ I ~ ~ ~ I I ~ ~ ~ I I ~ ~ I I I l,~ T~ ~ I ~ I I I I I ~ I ~ I I I I I I I I I I I I I I
/~~ ~
~ 0.150 7
e ;
~ 0.100 _
_
_
0.050 _
2L -~ pitch
(modal soals)
U.U~V ,5- ,- , ~ ,I,,,,l, ~ ~ ~
0,0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0 11.0 12.0
Q r~r~ .
. .
V.VV
6.00 _
4.00 _
2.00
0.00 _
0.0
1.0
0.6 _
0.0
-0.5 ~ ~
: phase pitch ,~,
. ~ ~
-1.0 ,,,, 1,,,, 1,,,, I, I, I 1,, ', I I,,, i,,,, 1 1,, I ~ I l~, ~ \, · I I
0.0 0.5 1.0 1.5 2.0 2.5 3.0 q-.5 4.0 4.5 5.0
C) rrad/s
,,,,1-l,,'.1. . ,1,,.~. ,,.~
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
" ' ' ' " ' ' ' I I ' ' ' ' 1 ' ' ' ' I ' ' ' ' I ' ' ' ' I ' ' ' ' 1 ' ' ' ' ~ ' ' ' ',
(tuU ·~1 - ) _ .
~ RAO pitch ~
~ *
.
_~-
OCR for page 237
~ .' ~
-1.5
- registration at 3.6 in
0.5
-0.5
~ .5
0.5 .
_.
-1.S
-2.S
2.5
I.B
n ~
at 50.05 m
t [s]
L ~ I_LlA_L~ _l_~.- I_I.J_J ~.J ~_I.~_Il.l l_~ _l LJ_l_L.~ )_~ ~.~ _l_Ld_I_~.LI_t ~ 1_1 [.~_1
tO 20 30 40 .'50 '30 7n so so 100 tin 12t)
. ',,,,,,, I t [S] ~
—a.:)
O 10 20 30 40 50 60 70 SO So :CO 110 120
2.5
.._
0.5
-t)S
-IS
-2.5
n
1.5 _
(conc,eIltration
point is at 84 m)
,,,, 1: . ., I, . . . 1, I 1 1 a:, . I,,:, I,
J 10 20 3D 40 50 60
~ at 90.3 m (after hrt?akillg)
Fig. 5 Genesis of a 3.2 m rogue wave by deterministic superposition of component waves (water depth Ant m).
As illustrated in Fig. 6, however, the main
deviation is localized within a small range (Clause et
al., 2001~. This promising observation proves that it is
sufficient for only a short part of the control signal in
the time domain to be fitted. As a prerequisite,
however, the computer controlled loop in the
experimental generation process should imply
nonlinear wave theory and develop the wave
evolution by using a numerical time-stepping method.
The two dimensional fully nonlinear free surface
flow problem is analysed in time domain using
potential flow theory. Fig. 7 summarises the basic
equations and boundary conditions.
nno
=.02
_n net
0.03 t
.
Sudace elevations at target Ideation
—— A I ~ ~ ~ ~ l
.. ~ ·; I · - ~ Massured tree surtax elation .;
1 L—Tarot suna" aleva ~
Fig. 6 Comparison between target wave and
measuredlime series at target location.
27
OCR for page 238
1
7) ~
at//
Surface elevation and associated velocity potential ~
/- '/ ~ ~ ~ I//,,,,,,,,,,
A.
V2~ = 0
E -1
-
-2
-1
Fig. 7 Numerical wave tank (Steinhagen, 2001). s-2
Fig. 8 Finite element mesh for nonlinear analysis.
A finite element method developed by Wu and
Eatock Taylor (1994, 1995) is used to determine the
velocity potential, which satisfies the Laplace
equation for Neumann and Dirichlet boundary
conditions. The Neumann boundary condition at the
wave generator is introduced in form of the first time-
derivative of the measured wave board motion. To
develop the solution in time domain the forth order
Runge-Kutta method is applied. Starting from a finite
element mesh with 8000 triangular elements (401
nodes in x-direction, 11 nodes in z-direction, i.e. 4411
nodes) (see Fig. 8) a new boundary-fitted mesh is
created at each time step. Lagrangian particles
concentrate in regions of high velocity gradients,
leading to a high resolution at the concentration point.
This mixed Eulerian-Langrangian approach has
proved its capability to handle the singularities at
intersection points of the free surface and the wave
board. Fig. 9 shows wave profiles with associated
velocity potential as well as registrations at different
positions. Note that the pressure distribution as well
as velocity and acceleration fields including particle
tracks at arbitrary locations are deduced from the
velocity potential.
Fig. 10 presents numerical results as well as
experimental data to validate this nonlinear approach.
Excellent agreement of numerical and experimental
v.u~v ~
.........
4 1
50 60
1-
O 7~—-
7: : :: :::::: :
3 . . . . . . . . . . . . . . . :.
: :
1
SO 60
1
l l
n : 8.923 ~ An 7c AZ ,
. .. ... . . . . . . . .. . .
70 80 90 100 110 120 130 140
t=96258
1.64
, ,
l l
90 1X 110 120
50 60 70 80 90 100 110 120 130 140
x In m
Fin. 9 Nonlinear numerical simulation of transient waves.
results is observed. Note that all kinematic and dynamic
characteristics during wave packet propagation are
deduced from the velocity potential, i.e. registrations at
any position (top, left) with associated Fourier spectra,
wave profiles at arbitrary instants (top, right) as well as
velocity, acceleration and pressure fields.
Fig. 11 shows the maximum (crest) and minimum
(trough) surface elevation in the wave tank {maX and
(min as well as the difference, i.e. the wave height
~ - ~ . . Note the sudden rise of water level (crest and
max mm
trough) at the concentration point. Fig. 12 illustrates
numerically calculated orbital tracks of particles with
starting locations at x = 87m and at x = 126m, which is
very close to the concentration point. Generally, the
orbital tracks are not closed. Particles with starting
locations z > elm are shifted in the x-direction, and due
to mass conservation particles with lower z-coordinates
are shifted in the opposite direction.
Fig. 13 finally proofs that the technique for
generating nonlinear wave packets is adaptable to
different wave machines. The diagrams present results for
a two-flap wave generator, i.e. the angular motions (and
speed) of the lower and upper flaps as well as the
resulting wave group registration. Excellent agreement
between numerical and experimental results is observed.
Note that the short leading waves are generated by the
OCR for page 239
Registrations wave profiles ("photos")
.
E _ . at ~ . .
O '-- '2' V~
Ob
'~5 ~ ~ ~ ~ t~ q~ q4~ to
2
Follrier Nectar
v. ~
-
..~
Fig. 10 Wave packet registrations at different positions as well as instantaneous wave
profiles at selected instants - numerical calculations validated by experimental results
(Clause and Steinhagen, 1999).
!
upper flap. As the lower flap starts working, the
motion of the upper flap is reduced, and finally
oscillates anti-phase with the lower flap (Pakozdi,
2002, Hennig, 2001).
So far, nonlinear wave groups in an ideal fluid
have been considered. If viscous effects are also
considered and approach of transient viscous free
surface flow computation with (RANSE/VOF) solver
is used. As an application, an artificial reef -
modelled as a submerged permeable wall - has been
investigated: Using an unstructured grid, the
dissipation loss is explained by overtopping
phenomena and subsequent recirculation of the flow
locked in chambers between filter elements (Fig. 14~.
Jet flow between filter components is also fostering
high energy loss. Due to non-linear wave/filter
interactions long low-frequency incident waves with
substantial erosive impact are transformed into
irregular wave trains with high-frequency wave
.' —
"."
~ ·0.5~
\ (min ~
l l ! i I l l
~ 20 40 GO 80 100
Fig. 11 Maximum (crest) and minimum
(trough) surface elevations ~ (maX ~ {min ) as
well as wave height ~ _ ~
max mm
OCR for page 240
/
1~
ins
-1.s
·2.5
, , I , 1
125,5 126 126.5 127 127.5
xhm
Fig. 12 Particle tracks with starting location
at x=126m.
energy components, which cause less erosion to the
sea floor (Clause and Habel, 2000~.
0.02 _
~ O
.0.02
0.060
· _,,u.`l`elill~l!llltit |~.; _ _ lows flap angular motion ~
. . .
~ I 1
200 250 300 ~
-
Iower flap rotational speed
... .
41 u.~ ; ; ;
0 020 50 100 150 200 250 300 350
, 1 ,1 1 1 1 1
O
9~.= _
n me
upper flap angular niotion
._~ _ ,,_ .... ......... . ~ ~ a ....... .
. .
.
I 1 1
200 250 300 350
950
_.w
0 50 100 150 200 2SO
t [all
900
HSVA Rolls H TS079 registration at various positions
O.OQ ~ | _ S.88 m | . . . ~ A ~ I I A I l, ~ ~ I ~ d ~ !l . . ~ `. . i. A, , . . ~
JO 05 ~ ... ... ........ .... ... ' ' ' . \~ .,,~ ; .... ... ' ... ... ...
158 158 100 182 1B. 1 -
~ [S]
Fig 13 Motions of a two-flap wave generator and
related wave group registration comparing numerical
and experimental results.
Numencal simulations of
Wave I filter interactions ~ _
1. Submerged wall (WBAU) ~> " ~ I
2. Filter, 11% porosity (GWK)
.,
Fig. 14 Transient viscous computation of
artificial reefs (RANSE/VOF) - velocities due to
wave/filter interaction for submerged wall and
1 1% filter.
rNTEGRATION OF DESIGN WAVE GROUPS IN
IRREGULAR SEAS - LINEAR APPROACH
350
In general, extremely high 'rogue' waves or critical wave
groups are rare events embedded in a random seaway.
As long as linear wave theory is applied, the sea state
can be regarded as superposition of independent harmonic
waves, each having a particular direction, amplitude,
frequency and phase. For a given design variance
spectrum of an unidirectional wave train, the phase
spectrum is responsible for all local characteristics, e.g.
the wave height and period distribution as well as the
location of the highest wave crest in time and space. For
this reason, an initially random phase spectrum argF(~)
is optimized to generate the desired design wave train
with specified local properties. The phase values
~ = (me ~ p2 ~ pn)T are bounded by - adz < ~ < fez and are
initially determined from pi=2'z(Rj-O.S) where Rj
are random numbers in the interval 0 to 1 (Clause and
Steinhagen, 2000).
The set up of the optimization problem is illustrated
for a high transient design wave within a tailored group of
three successive waves in random sea. The crest front
steepness of the design wave in time domain at as
defined by Kjeldsen (1990):
2}Z Crest (6)
£ =
t gTr~seTzd
OCR for page 241
is maximized during the optimization process. acres,
denotes the crest height, Trise the time between the
zero-upcrossing and crest elevation, and To, the zero-
downcrossing period which includes the design wave.
The target zero-upcrossing wave heights of the
leading, the design and the trailing wave are defined
by H., H., and H. . The target locations in space and
time of the design wave crest height if,` are x~arge, and
tfarge`. These data define equality constraints. The
maximum values of stroke x velocity u and
m" ~ max ~
acceleration amaX of the wave board motion Aft)
define inequality constraints to be taken into account.
Hence the optimization problem is stated as
minimize f (,B) = - £,
-
g5 =
g6 =
g7 =
g7+j =
0 =
7+n+ j
subject to Al = H. I -Hl = 0,
g2 = Hi - Hd = 0,
g3 = Hi+l - H. = 0,
g4 ~ (Xtarge' ~ ttarge' ~ (d 0,
max { | Aft) | } Xmax — 0,
max { | Aft) | }- umax < 0,
max {|xb~t)|}-amax < 0'
-'z-,Bj < 0, j = 1,...,n
- Liz + Al < 0, j = 1,. .., n
where fain is the objective function to be
minimized. The general aim in constrained
optimization is to transform the problem into an
easier subproblem that can be solved, and is used as
the basis of an iterative process. A Sequential
Quadratic Programming (SQP) method is used which
allows to closely imitate Newton's method for
constrained optimization just as is done for
unconstrained optimization.
For evaluating the objective function and
constraints, the complex Fourier transform is
generated from the amplitude and phase spectrum.
Application of the IFFT algorithm yields the
associated time-dependent wave train at target
location. Zero-upcrossing wave and crest heights as
well as the crest front steepness c, of the design wave
are calculated. The motion of the wave board xb(t) is
determined by transforming the wave train at
x = x,arge, in terms of the complex Fourier transform
Forge to the location of the wave generator at
x = 0 and applying the complex hydrodynamic
transfer function Fhya,rO(~) which relates wave board
motion to surface elevation close to the wave generator:
xb(~t) = IFFT LF`arge'(O) F`ransfo) Fhy~ro(~)~] (~8)
with F'rans(~);~=exP(ikix~arge,) The maximum stroke of
the wave board is set to xmax = 2m, maximum velocity to
up = 1 .3m I s and maximum acceleration to
ama'` = 1.7m / s2 . The optimization terminates if the
magnitude of the directional derivative in search direction
is less than 10-3 and the constraint violation is less than
lo-2 .
In our example the design variance spectrum is
chosen to be the finite depth variant of the Jonswap
spectrum known as TMA spectrum (Bouws et al, 1985~:
~ ,~q) = ocq~s 1 2kd/ .(h`~2kd' e~ q y (~9'
where q = ~ / Lop = f / fp represents the normalized
frequency with respect to the peak frequency fp = 1/ Tp .
The Jonswap peak enhancement factor y is set to 3.3 and
the spectral width parameter of* to 0.07 for q < 1 and
0.09 for q > 1 with r = (q - 1~' / a* . The frequency-
dependent wave number k is calculated from the
dispersion relationship 602 = gk tanh~kd) where g is the
(7) acceleration due to gravity and d the water depth.
For the selected spectrum- significant wave height
Hs = 0.7m, peak period Tp = 4.43s, water depth
d = 5.5m - a high transient design wave within a tailored
group of three successive waves in random sea is
optimized. The target zero-upcrossing wave height of the
design wave is Hd = 2Hs with a maximum crest height
(~(Xtargett~arget)=06H~ =1.2Hs. Target location is at a
distance of x,arge, = lOOm from the wave generator, and
target time is ttarge~ = 80s . The heights of the leading and
the trailing waves adjoining the design wave are set to be
H`,=H,=Hs. Note that this wave sequence is quite
representative for rogue wave groups as has been proved
by Wolfram et al. (2000) who classified 114 extremely
high waves with their immediate neighbours out of
345245 waves collected between 1994 and 1998 of North
Alwyn.
As illustrated in Fig. 15, the optimization process
finds local minima, i.e. a number of different wave trains,
which depend on the initial phase values. Hence the
random character of the optimized sea state is not
completely lost.
OCR for page 243
determined from the nonlinear simulation in the
numerical wave tank.
The target wave characteristics define equality
constraints. The maximum values of stroke
xmax = 2m, velocity umax = 1 .7m I s, and acceleration
amaX=2.2mls2 of the wave board motion xB(t)
define inequality constraints to be taken into account.
The subplex minimization problem is formulated as
minimize f (C) = (
Hi,—HI,~arpe!
Hl,target )
Ti l—Tl~ta~et ~ ~ Hi—H2,ta~et ~
+ H +
1,target ~ 2,target
Ti—T2,ta et + c i— c,ta et +
( )( )
T2,target (;c,target
( )2 ( )2
to c,target H3,ta~et
Ti+l—T3 target ~ ~ ~ (XB (t))—(J(XB (t) initial )
3 target ) ~ ~(xB (t) initial ) )
(10)
subject to gl=maX{lXB(t)I}-Xmax < 0' (11)
g2 = max {| JOB (t) | }—UmaX < 0'
g3 = max {| xg(t) | }- amaX < 0,
where a(xB(t)) is the standard deviation of the wave
board motion.
Fig. 18 shows the improved wave board motion.
The zero-downcrossing characteristics of the wave
train are presented in Fig. 19. The target values of the
transient wave are significantly improved. Note that
the rogue wave sequence is exactly fitted, with
Hmax = 2Hs and '7c = 0 6HmaX As a result we obtain
a control signal of the wave generator which yields a
specified rogue wave sequence embedded in an
extreme irregular seaway characterized by the
selected global parameters Hs and Tp (Clause and
Steinhagen, 2001~.
Fig. 20 illustrates the evolution of this design
wave sequence, with registrations at 5 m, 50 m and
100 m (target position) behind the wave board (left
side) as well as wave profiles ("photos" of surface
elevation) at t = 75 s, 81 s (target) and 87 s (water
depth h = 5 m, Tp-3.13s).
The associated energy flux at the locations x=Sm,
50m and loom is shown in Fig. 21. As has been
expected the energy flux focuses at the target
position.
0.06 _
0.04
.c 0.02
~ O ~
K ~.02
~.04
Inn _
v.w
0.04
_ 0.02
~4
~ O
Ka) ~.02
.04
.06 , . .
0 5 10 15 20 25 30
t/T
p
Fig. 18 Comparison of optimized wave board
motions.
Wave board motion resulting from SQP optimization
From the velocity potential which has been
determined as a function of time and space all kinematic
and dynamic characteristics of the wave sequence are
evaluated. Fig. 22 presents the associated velocity,
acceleration and pressure fields (Steinhagen, 2001~. Note
that the effects of the three extremely high waves are
reaching down to the bottom.
Fourier spectrum
a 0.5 ~~ V VV\~
20 _
-A o
A -20` )
Be, . 1
0.5 1 1.5 2 2.5 3 `~/~. 3 5
Unwrapped and detrended phase spectrum P
—
~ :
~ . _
0.5 1 1.5 2 2.5 3 ~ / ~ 3.5
Surface elevation at target location x / h = 2n P
20 25 30 35 40 45 t / T
Zero-downcrossina wave heights P
~ ~ 1
~ 1 - ' ''Q 1 - ' t ; - .
s o ~ ~ I I ~ T I I I A| ~ ~ Q ~ ~ Q ~ 191 ~ I ~Q i~ in. 97
20 25 30 t / T 35 40 45
Wave height and crest structure P Wave height and period structure
0.7 0 ° 1 1.21 o
0.6 .
., 0.5 .
0.4 . _ _
0.3 ·O °
0.2 .
0 0.5
8°o Oo °
OOO O.3
g ~
80 '1
O 0 0 Go 0 O
Coo Oo O
0 0
....... ..°.
0 oo
0
1 on . . :,
1 1.5 2 0 0.5 1 1.5 2
H'H H/H~
Fig 19 Nonlinear wave train simulation with
predetermined wave sequence. Wave board motion
optimized with Subplex method.
OCR for page 244
Su~ace eleva~ion at x t h = 1
~.05
~.OS ..............
0.11 .....
0.05
~ O ~
0.05
1
10 IS 20 2S 30 n
Surface elevation as x ~ h = 10
-0.1 ~
n l r
nn.
.os
J)1
. . . .
· · . . .
5 10 15 20 2S 30
Surface devsdan 8t x / h = 20
~ ~ ~ .
Su~face elevation at t/T =24
p
:
. . . .
... ' ~ ~ 2 :
........... ~
.. ... .......
~ ~:Vi~0'~
_1
10 12 14 16 18 20 22 24 26 28 30
0.1 ~ ..
: ' :
0.05 .
~ O ~
.ns .
-°.l1 n
! !
..... ~ ,. q _ , ~
~--:'-~
. . .. . . .. . ... . . .. ... . . . . . .. . . .. . . . . . _ .. . . ... _ . . . _ . .. . . ... . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
· , · · . · . . ·
12 14 16 18 20 22 24 26 28 30
S=face devation at t i T = 28
p
. . . . . .
. . . . . . . . .
..... ~ ........... , ........... ~
. . . . . . . . .
: : : : : : ^: : :
· · · · · · 11- A
................. . . . . .. . ~
-'-°'W'.~\1'',:~.. \ iW
..... \ ,
: w: :
n1t
n.os~
-0.05 ~ ; : ; ; .
: : : : : :
0 5 10 IS 20 25 30 ~.ll 0 12 14 16 18 20 22 24 26 28 ~ O
tl p x/h
Fig. 20 Evolution of rogue wave sequence - registrations at x = 5m, SOm and 1OOm (left) as well as wave
profiles at t= 75s, 81s and 87 s (right hand side) (water depth h=Sm, Tp-3.13s).
Energy flux at x / h = 1
0.02
',~ 0.015
g
~,, 0.01
O.WS
0~021
',; 0.015
c~
~~. 0.01
E~
~ 0.005
.........
.........
. L
~ .
_~
5 10 15 20 25 30
ttTp
Fig. 21 Energy flux of nonlinear wave at x = S m, 50 m
and 100 m (target).
. .
. .
... ... .. . .. ... .. ...
:
. .
: :
. .. . . .. .. . .
:
. .
. .
. .
_~_^A~
042r
~1
~1
~41
~1
..8
Id 16 :IB ~ ~ 24
~7~ ~i~C ~elerml" b$/g
i,"
~ paruck velocity `'~J''~Igh)
~ .
1~1 t6 18 20
Dymm~c pm~sum PdJ(P~)
10
i1 1
1~, 1
1
10.1
In ns
14 ~16 18 20 ~;:2 ~ 26
xJh
Fig. 22 Kinematic and dynamic characteristics of rogue
wave sequence HS ... 2HS ... HS at target time t = 81 s.
OCR for page 245
Wave
X~et Gauge
Improved Control Signal
Fig. 23 Computer controlled experimental simulation
of tailored design wave sequences.
The above optimization method has also been
applied to generate the Yura wave and the New Year
wave (see Fig. 1) in the wave tank (Fig. 23). Firstly,
for the specified design variance spectrum, the SQP-
method yields an optimised phase spectrum which
corresponds to the desired wave characteristics at
target position. The wave generator control signal is
determined by transforming this wave train in terms
of the complex Fourier transform to the location of
the wave generator. The measured wave train at target
position is then iteratively improved by systematic
variation of the wave board control signal. To
-
synthesise the control signal wavelet coefficients are
used. The number of free variables is significantly
reduced if this signal is compressed by low-pass discrete
wavelet decomposition, concentrating on the high energy
band. Based on deviations between the measured wave
sequence and the design wave group at target location the
control signal for generating the seaway is iteratively
optimised in a fully automatic computer-controlled model
test procedure (Fig.23).
Fig. 24 presents the evolution of the Yura wave at a
scale of 1 : 112. The registrations show how the extremely
high wave develops on its way to the target position at
x=7m. As compared to full scale data the experimental
simulation is quite satisfactory.
The evolution of the Draupner New Year Wave is
shown in Fig. 25. Again the full scale data correlate quite
well with model test results at target position x=7.9m.
The wave tank tests illustrate how these extremely high
waves are developing from rather inconspicuous wave
trains and disperse shortly later. If wave/structure
interactions are investigated the tank tests allow for
considering memory effects. In addition, the mechanism
of nonlinear structure dynamics is evaluated, and cause-
effect relationship can be analysed.
. w ~
EWE x= 6 ''3 ~
~~ Rae ) ~ by\ ~ ~ It ~~ ~ ~ ~ ~ ~ ~ in< t~ .- sit ~ ~ ~ ~ if"
~ \ ~ ~ V ~ I- ~ ~ ~ ~~ it ~ ~ I- V, Y 'tJ-V Hi' ~ ~ ~ ~—~ \1
~ .
itt ~ I
,..
.. _ .. _ .. . . .
Fig. 24 Evolution of the Yura wave (scale 1:1 12)
(full scale wave data collected by National Maritime Research Institute, Japan (Mori et al., 2000)).
.A A ~ . ~WU ^. WOW:. 4 K.: :~.^ :~ 4.~ ~ -:~4 .'~ W~ _ ._ I ~ . ^ ~ ^ . - . ~ I .... ~ -
Fig. 25 Evolution of the New Year Wave (scale 1 :175)
(full scale wave data collected by Statoil (Haver and Anderson, 2000)).
OCR for page 246
COMPUTER CONTROLLED CAPSIZING
TESTS USING TAILORED WAVE
SEQUENCES
The technique of generating deterministic wave
sequences embedded in irregular seas is used to
analyse the mechanism of large roll motions with
subsequent capsizing or cruising ships (Clause and
Hennig, 2002~.
The parameters of the model seas - transient
wave elevation at ship center (moving frame)
- ~ , r I ! I I i I I 1 —- ~ --- --~- T T I j
.- Fj I-' tIl' IA ~
~'' ~f ~ ,~,:~' >; ' ,: ,~ K
178.60—T—~—
T'66.04 i—
.~,- r ship position
X 1~.o. h (x-coordinate'
[m],,: - ~ ~
1060 ~- -
1~$.00 t;;—~—.~
64 00 ~ ' ' ' ~ ' ' ' ' . - ~ . i .- i -- ~---- -.J--- -~- t_ ~ ~ i _ . . I
no..o 13Q0' 1~. 1,C.tO 13t4.Ctt
' ' I ' ~ ' I I ' ~--- T- ----------~-------'--- I I ' i ' I ' T
~. _
'3t' _
x _
L)e _
[m]' ,''
.,,.,~
~3t; _
_
IQt~
31Lt~
2G.0O
~ 1~-
[O] ~t4
~2tt4
~0
^16
11~C0 1~ -
26.t ~—I-—T~ I~ T--—T-—l~ T~ l-'
12Ce.to - ~v - ~t 4~
~3 ~t4 ~- ta get course
~t~ Y
.1S. - _
~tC.t. ::
t ~ 1 _
11 Ct.CO
"tO _
~C t4
2C.t~
~ 1`i.W
[°] {i.{O
mtt.'.
-2C.tO ,.
~i.60 _
J' t~
11 C,CO 1 ~OJR 1 S~.W
^~[ -— , 1,,, i,,, !
436
110.X =~ -
:~~ ship position ~
br-coordmate) N~%
. I. . 1 . , I 1 . ., ., . . 1, ,. ,,, . 1,, ,'
tS0D. 1.,QtO 13hCO
r--^i~~~—r-~~~i~T T-
.__.~._. - r- - --l----—r---l------l------------T------~-
roll motion
150.W 1r~40 10ti4CO
T -'r-—T-—r-.r.-..~._..1—]-—· -T--~- - .~--...
—~%,,~——Nx
~ Be = 1~
._ _._ ~.__
-
. ~
_
A
~_~~,, ,~_ .
1broachip'' with ~bsequer~t ca~sizipg ~4
~ .... .. , ~ ._ .. . .~ _,_ _.1 _ .~. ... ,.~
130^ 150.00 1~tC0 13(:40
r I I I r- ~--T-~-~
r-.
'. r
f `/'d '\~/ iSii _
17~C0 13~CO
t [S]
Fig. 26 Roll motion of the RO-RO vessel in a severe
storm wave train (Tp=14.6s, Hs=15.3m) at GM=1.36m
v=151m, z-manoeuvre with ,u = ~ 10°.
wave sequences consisting of random seas or regular
wave trains with an embedded deterministic high
transient wave - are systematically varied to
investigate the ship model response with regard to
metacentric height, model velocity, and course angle
for each of both ship types. The wave elevation at the
position of the ship model at any position in time and
space is calculated (and controlled by registrations during
model tests) in order to relate wave excitation to the
resulting roll motion.
Fig. 27 RO-RO vessel in a severe storm.
Fig. 26 presents a model test with a RO-RO vessel
(GM=1.36 m, natural roll period TR = 19.2 s, v = 15 kn) in
extremely high seas from astern (ITTC spectrum with
Hs=15.3 m, TP = 14.6 s, z-manoeuvre: target course
,u=110°~. The vessel broaches and finally capsizes as the
vessel roll exceeds 40 degrees and the course becomes
uncontrollable (Fig. 27~. Note that the wave elevation
refers to the ship center (moving frame), and has been
calculated from the registration at a stationary wave probe
- 10 meters in front of the wave board. Thus, relevant
wave elevation is directly related to the associated ship
motions.
DYNAMICS OF SEMISUBMERSIBLES IN ROGUE
WAVES
The method of synthesizing extremely high waves in
severe irregular seas is also applied to analyse the impact
of reported rogue waves on semisubmersibles. As the
procedure is strictly deterministic we can compare the
numerical (time-domain) approach and model test results
(Clause et al., 2002~.
For the numerical simulations the program TiMIT
(Time-domain investigations, developed at the
Massachusetts Institute of Technology) is used, a panel-
method program for transient wave-body interactions
(Korsmeyer et al, 1999) to evaluate the motions of the
semisubmersible. TiMIT performs linear seakeeping
analysis for bodies with or without forward speed. In a
OCR for page 247
first module the transient radiation and diffraction
problem is solved. The second module provides
results like the steady force and moment, frequency-
domain coefficients, response amplitude operators,
time histories of body response in a prescribed sea of
arbitrary frequency content on the basis of impulse-
response functions.
The drilling semisubmersible GVA 4000 has
been selected as a typical harsh weather offshore
structure to investigate the seakeeping behaviour in
rogue waves in time-domain. The wetted surface of
the body is discretized into 760 panels (Fig. 28~. The
number of panels is sufficient to simulate accurate
results.
Operatlon displacement
A=25940t
................. . ~:Q.~.iC: ...... ....... 1 _
`:4720 ~
1
'r ~ `.
i.,. ~ :'
(O 5' At)!
Fig. 28 Semisubmersible GVA 4000- main dimensions
and discretization of the wetted surface using 760
panels.
For validating TiMIT results of wave/structure
interactions in extreme seas the Draupner New Year
Wave (see Fig. 1) has been synthesized in a wave
tank at a scale of 1:81. Using the proposed wave
generation technique, the wave board signal is
calculated from the target wave sequence at the
selected wave tank location.
Fig. 29 presents the modelled wave train at target
location. For comparison the exact New Year Wave is
also shown to illustrate that we have not reached an
accurate agreement so far. However, this is not
detrimental since the associated numerical analysis is
based on the modelled wave train, registered at target
position.
Fig. 30 presents the modelled wave train as well
as the heave and pitch motions of the
semisubmersible comparing numerical results and
experimental data (scale 1:81~. The airgap as function
of time is also shown. Note that this airgap is quite
sufficient, even if the rogue wave passes the structure.
However, wave run-up at the columns (observed in
i ~ ~ . ~ 1
100 150 200 250 300 time [a] 350
Fig. 29 Comparison of model wave (scale 1:81) as
compared to the registered New Year Wave (Haver and
Anderson, 2000) presented as full scale data.
model tests) is quite dramatic, with the consequence that
green water will splash up to the platform deck.
As a general observation, the rogue wave is not
dramatically boosting the motion response. The
semisubmersible is rather oscillating at a period of about
14s with moderate amplitudes.
Related to the (modelled) maximum wave height of
HmaX = 23m we observe a maximum measured double
heave amplitude of 7m. The corresponding peak value
from numerical simulation is 8.6m. As a consequence, the
measured airgap is slightly smaller than the one from
numerical simulation. The associated maximum double
lo. pitch amplitudes compare quite well. Note that the impact
results in a sudden inclination of about 3°. Considering
the complete registration it can be stated that the
_ 20-
~ :
~ 10-
~ -
-
O~
! ~
:^ ire ryes
550 600 650 700 time [s] 750
1 0.0
calculated pitch motion ~ ~ I
5.0 -—memuret pitch motto" _ ~ ~L\~—
Boor ~~V~ \1~ Q^~
· ~ ~ ~ tJ `7 in V ~ ,
_~.o: I .,,, i, ., . I ....
550 600 650 700 Me [a] 750
815~
~ cc! - gap ~ | ! I
550 600 650 700 time [a] 750
Fig. 30 Results of numerical simulation and
experimental tests for semisubmersible GVA 4000:
Heave, pitch and airgap (measured at a scale 1 :81,
presented as full scale data).
OCR for page 248
.
numerical approach gives reliable results. At rogue
events the associated response is overestimated due to
the disregard of viscous effects in TiMIT calculations.
CONCLUSIONS
For the evaluation of wave-structure interactions
the relation of cause and effects is investigated
deterministically to reveal the relevant physical
mechanism. Based on the wave focussing technique
for the generation of task-related wave packets a new
technique is proposed for the synthetization of
tailored design wave sequences in extreme seas.
The physical wave field is fitted to predetermined
global and local target characteristics designed in
terms of significant wave height, peak period as well
as wave height, crest height and period of individual
waves. The generation procedure is based on two
steps: Firstly, a linear approximation of the desired
wave train is computed by a sequential quadratic
programming method which optimizes an initially
random phase spectrum for a given variance
spectrum. The wave board motion derived from this
initial guess serves as starting point for directly fitting
the physical wave train to the target parameters. The
Subplex method is applied to improve systematically
a certain time frame of the wave board motion which
is responsible for the evolution of the design wave
sequence. The discrete wavelet transform is
introduced to reduce significantly the number of free
variables to be considered in the fitting problem.
Wavelet analysis allows one to localize efficiently the
relevant information of the electrical control signal of
the wave maker in time and frequency domain.
As the presented technique permits the
deterministic generation of design rogue wave
sequences in extreme seas it is well suited for
investigating the mechanism of arbitrary
wave/structure interactions, including capsizing,
slamming and green water as well as other
survivability design aspects. Even worst case wave
sequences like the Draupner New Year Wave can be
modelled in the wave tank to analyse the evolution of
these events and evaluate the response of offshore
structures under abnormal conditions.
ACKNOWLEDGEMENTS
The fundamentals of transient wave generation
and optimization have been achieved in a research
project funded by the German Science Foundation
(DFG). Applications of this technique, i.e. the
significant improvement of seakeeping tests and the
analysis of wave breakers and artificial reefs in
deterministic wave packets have been funded by the
Federal Ministry of Education, Research and
Development (BMBF). Results are published in
outstanding PhD theses (J. Bergmann, W. Kuhnlein, R.
Habel, U. Steinhagen). The technique is further
developed to synthesise abnormal rogue waves in extreme
seas within the MAXWAVE project funded by the
European Union (contract number EVK-CT-2000-00026)
and to evaluate the mechanism of large roll motions and
capsizing of cruising ships (BMBF funded research
project ROLL-S). The author wishes to thank the above
research agencies for their generous support. He is also
grateful for the invaluable contributions of Dr.
Steinhagen, Dipl.-Ing. C. Pakozdi, Dipl.-Math. techn.
Janou Hennig and Dipl.-Ing. C. Schmittner.
REFERENCES
Bouws, E., Gunther, H., Rosenthal, W. and Vincent, C.
(1985~: Similarity ofthe wind wave spectrum in finite
depth water- 1. Spectral form, Journal of
Geophysical Research, 90(C1)
Clauss, G. and Bergmann J. (1986~: Gaussian wave
packets - a new approach to seakeeping tests of ocean
structures, Applied Ocean Research, 8~4)
Clauss, G. and Habel, R. (2000~: Artificial reefs for
coastal protection- transient viscous computation and
experimental evaluation, 27th International Conference
on Coastal Engineering (ICCE), Sydney, Australia
Clauss, G. and Hennig, J. (2001) Tailored transient wave
packet sequences for computer controlled seakeeping
tests, 20th International Conference on Offshore
Mechanics and Arctic Engineering (OMAE), Rio de
Janeiro, Brazil
Clauss, G. and Hennig, J. (2002) Computer controlled
capsizing tests using tailored wave sequences, 21 St
International Conference on Offshore Mechanics and
Arctic Engineering (OMAE), Oslo, Norway
Clauss, G. and Kuhnlein, W. (1995~: Transient wave
packets - an efficient technique for seakeeping tests of
self-propelled models in oblique waves, Third
International Conference on Fast Sea Transportation,
Lubeck-Travemunde, Germany
Clauss, G. and Kuhnlein, W. (1997~: Simulation of
Design Storm Wave Conditions with Tailored Wave
Groups, 7eh International Offshore and Polar
Engineering Conference (ISOPE), pp. 228-237.
Honolulu, Hawaii, USA.
Clauss, G., Pakozdi, C. and Steinhagen, U. (2001~:
Experimental Simulation of Tailored Design Wave
Sequences in Extreme Seas, 1 lth International
Offshore and Polar Engineering Conference (ISOPE),
Stavanger, Norway.
OCR for page 249
Clauss, G. and Steinhagen, U. (1999~: Numerical
Simulation of Nonlinear Transient Waves and its
Validation by Laboratory Data, 9th International
Offshore and Polar Engineering Conference
(ISOPE), Brest, France.
Clauss, G. and Steinhagen, U. (2000~: Optimization
of Transient Design Waves in Random Sea, 1 oth
International Offshore and Polar Engineering
Conference (ISOPE), Seatle, USA.
Clauss, G. and Steinhagen, U. (2001~: Generation and
Numerical Simulation of Predetermined Nonlinear
Wave Sequences in Random Seaways, 20th
OMAE Symposium, Rio de Janeiro, Brazil
Clauss, G., Schmittner, C. and Stutz, K. (2002~:
Time-domain investigations of a semisubmersible
in rogue waves, 21St International Conference on
Offshore Mechanics and Arctic Engineering
(OMAE), Oslo, Norway
Davis, M. and Zarnick, E. (1964~: Testing Ship
Models in Transient Waves, 5th Symposium on
Naval Hydrodynamics.
Drake K. (1997~: Wave profiles associated with
extreme loading in random waves, RINA
International Conference: Design and Operation
for Abnormal Conditions, Glasgow, Scotland
Faulkner (2000~: Rogue Waves- Defining Their
Characteristics for Marine Design, Rogue Waves
2000, Brest, France.
Faulkner, D. and Buckley, W. (1997~: Critical
survival conditions for ship design, RINA
International Conference: Design and Operation
for Abnormal Conditions, Glasgow, Scotland
Haver, S. (2000~: Some evidence of the existence of
socalled freak waves, Rogue Waves 2000, Brest,
France
Haver, S. and Anderson, O.J. (2000~: Freak Waves:
Rare Realizations of a Typical Population or
Typical Realization of a Rare Population?, 10th
International Offshore and Polar Engineering
Conference (ISOPE), Seattle, USA.
Kjeldsen, S.P. ( 19964: Example of heavy weather
damage caused by giant waves, Techno Marine,
Bull Of the society of Naval Architects of Japan,
No. 820
Kjeldsen, S.P. (1990~: Breaking Waves. Water Wave
Kinematic, Kluwer Academic Publisher, NATO
ASI Series, ISBN 0-7923-0638~, pp. 453-473.
Kjeldsen, S.P. and Myrhaug, D. (1997~: Breaking
waves in deep water and resulting wave forces,
Offshore Technology Conference, OTC 3646
Korsmeyer,F., gingham, H. and Newman, J.
(l999~:TiMIT- A panel-method program for transient
wave-body interactions, Research Laboratory of
Electronics, Massachusetts Institute of Technology
Kuhnlein, W. (1997~: Seegangsversuchstechnik mit
transienter Systemanregung, PhD Thesis, Technische
Universitat Berlin, D83.
Mori, N., Yasuda, T. and Nakayama, S. (2000~: Statistical
Properties of Freak Waves Observed in the Sea of
Japan, 10th International Offshore and Polar
Engineering Conference (ISOPE), Seattle, USA.
Nelder, J. and Mead, R. (1965~: A Simplex Method for
Function Minimization. Computer Journal, 7, pp. 308-
313.
Pakozdi, C. (2002~: Numerische Simulation nichtlinearer
transienter Wellengruppen - Bericht zum DFG-
Vorhaben Cl 35/30-1, Selbstverlag
Rowan, T. (19901: Functional Stability Analysis of
Numerical Algorithms, PhD Thesis, University of
Texas at Austin.
Sand, S.E., Ottesen, H.N.E., Klinting, P., Gudmestad
O.T. and Sterndorff, M.J. (19901: Freak Wave
Kinematics. Water Wave Kinematics, Kluwer
Academic Publisher, NATO ASI Series, ISBN 0-
7923-0638~, pp. S35-549.
Steinhagen, U. (2001): Synthesizing Nonlinear Transient
Gravity Waves in Random Seas, PhD Thesis,
Technische Universitat Berlin, D83.
Takezawa, S. and Hirayama, T. (1976~: Advanced
Experiment Techniques for Testing Ship Models in
Transient Water Waves, 1 1th Symposium on Naval
Hydrodynamics
Wolfram, J., Linfoot, B. and Stansell, P. (2000~: Long-
and Short-Term Extreme Wave Statistics in the North
Sea: 1994-1998, Rogue Waves 2000, Brest, France.
Wu, G-X., Eatock Taylor, R. (1994~: Finite Element
Analysis of Two-Dimensional Non-Linear Transient
Water Waves, Applied Ocean Research, 16~6), pp.
363-372.
Wu, G-X., Eatock Taylor, R. (1995~: Time Stepping
Solutions of Two-Dimensional Non-Linear Wave
Radiation Problem, Ocean Engineering, 22~8), pp.
785-798.
OCR for page 250
DISCUSSION
P. Kjeldsen
The Norwegian Maritime Academy, Norway
The wave generation technique described and
used in this work is not new, but has been
developed by Gunther Clauss and other
researchers for many years. Attention can here
be given to the paper " 2- and 3-dimensional
Deterministic Freak Waves" by S.P. Kjeldsen in
Proc. 18th International Conference on Coastal
Engineering, Cape Town, South Africa 1982.
Here it is shown that a very important parameter
that must be taken into consideration is the
directional spreading of the wave spectrum. The
norwegian technique that was made to generate
deterministic wave groups therefore was
designed to generate 3-dimensional freak waves.
In some cases these waves appeared also with 3-
dimensional breaking wave crests in coherence
with famous photos of such waves, see " A
Sudden Disaster - in Extreme Waves." by
Kjeldsen in Proc. Conference on RogueWaves,
Brest, France 2000.
The research presented by Gunther Clauss deals
only with 2-dimensional waves. It is fair to
assume that some of the severe marine accidents
that we have encountered appeared in complex
and confused 3-dimensional seas.
Another very important point that should be
considered here is to investigate if a freak wave
observed at sea is breaking or not. It is well
known that the largest risk for a capsizing event
is associated with a breaking wave. Here
attention should be given to a Norwegian
investigation of wave forces on platform legs in
deterministic freak waves. It was found that the
largest forces appeared in violent deep water
breaking waves with moderate wave heights. In
the same tests it was found that the highest
waves gave lower forces, see Kjeldsen ,T0rum,
Dean :" Wave Forces on Vertical Piles caused by
2- and 3-dimensional Breaking Waves." In Proc.
20th International Conference on Coastal
Engineering,Taipei, Taiwan 1986.
The last important point to consider is therefore
that it is not the wave height but the particle crest
velocity and acceleration in the crest of a
breaking wave that shall be measured
experimentally and documented properly from
wave tank tests. In Norway measurements of
both crest velocity and crest accelerations were
made with a wave-following current-meter-
technique, before capsizing test were undertaken,
see Kjeldsen " The Wave Follower Experiment "
in Proc. Of the Symposium on the Air-Sea
Interface, Radio and Acoustic Sensing,
Turbulence and Wave Dynamics. Marseille,
France, 1993.
I wish to congratulate Gunther Clauss with very
important research, and I hope he will continue
this work.
DISCUSSION
T. Hirayama
Yokohama National University, Japan
The author proposed the concept of the "task-
related design waves or extreme waves" and
showed concrete methods of synthesizing such
waves both in numerical and physical
simulation. This concept seems very valuable
like the concept of using the time history of E1
Centro earthquake for the assessment of
infrastructures in the field of civil engineering
and architecture. About the extreme waves, the
phenomenon become nonlinear, and the author
introduced nonlinear treatment and succeeded.
So the discusser wants to express his
congratulations to the authors.
My discussion is about the importance of 3-
dimensional effects of design waves or extreme
waves interacting with floating structures. The
examples shown in the present paper are all
related to long crested waves. Of course long
crested waves will be enough if we consider the
extreme waves near shore, because the crest lines
become parallel and long for this case. On the
other hand, if we consider the extreme waves in
deep ocean, the higher waves exceeding the
theoretical existing limit of 2-dimensional wave
steepness occurs by the simple superposition of
two wave system with different direction. This
will be called as triangular waves or pyramidal
waves. The time history measured at one point in
the ocean cannot detect if this is the result by
long crested wave or short crested wave. So, it
will be important to consider in which wave the
target phenomena is more severe in two
dimensional or three dimensional extreme wave
with similar time history of wave group.
Furthermore, the discusser wants to know if the
method proposed here is applicable to 3-
dimensional rogue waves or not. For reference, I
will refer to the picture showing both long
OCR for page 251
crested and short crested concentrated transient
wave. The pyramidal waves can be generated
even in a narrow and long experimental tank.
Ref. 1,3 are typical examples. We call this at
one-point concentrated Transient Water Waves.
In these waves, the velocity profile is not
uniform from wave surface to the sea bottom and
the capsizing phenomena will different from
long crested rogue waves. Here the non-linear
treatment about wave generation as the author is
not introduced.
REFERENCES
1. Hirayama,T. Ma.N,Harada,T. and Lee
J-H.~1995~: Application of Side Wall Reflection
Type Directional Wave Generator and Laser
Beam Type Wave Surface Probe-Especially on
the Linear Characteristics of the One Point
Concentrated Transient Water Waves-,
Proceedings of the Wave Generation'95~ pp60-
82
2. Hirayama T. et al.~1995~: Capsizing
and Restoration of a Sailing Yacht in Breaking
Isolated Triangular Transient Waves and
Breaking Long Crested Transient Waves, Journal
of Kansai Society of Naval Architects
Japan,No.223,pp59-66 (in Japanese)
3. Hirayama,T.(1997~: Modeling of
Multidirectional Waves in Naval Architectural
Field Proceedings IAHR Seminar on
Multidirectional Waves and their Interaction
with Structures(San Francisco),pp23 1-239
One Point Concentrated TWW(T=Osec, just
concentrated)(left).
In Line Concentrated TWW(just oncentrated)(right).
(from ref.3)
AUTHOR'S REPLY
First I would like to thank the discussants for
their highly competent and encouraging
comments. It is a special pleasure to present our
recent developments in wave focussing at the
ONR-Symposium in Japan as significant
contributions to this technique have been
achieved by S. Takezawa at the Yokohama
National University. As pointed out by
Tsuguikiyo Hirayama who continued these
research activities 3-dimensional effects play a
major role in wave-structure interactions as
shown with his pyramidal waves. Directional
spreading and the generation of 3-D freak waves
is also discussed by Peter Kjeldsen. In addition,
he emphasizes the importance of wave breaking
for the evaluation of platform forces, and
comments that wave kinematics play a key role.
As has been stated, our contributions to the
evolution of the wave focussing technique are
limited to long-crested linear and nonlinear
waves. Of course, 3-D effects are indispensable,
and I am grateful that both discussants are
engaged in this important research. I also agree
with Peter Kjeldsen that wave height is not the
most important parameter. However, not only
wave particle (crest) velocity and acceleration
but also wave steepness, wave sequence and
memory effects play a vital role for evaluating
wave impacts and the associated response of
offshore structures. Actually, this is the most
important aspect of our deterministic technique
with which all wave characteristics can be
evaluated in space and time. In conclusion, the
rogue wave story and the development of wave
focussing procedures as well as the deterministic
generation of extremely high waves and wave
sequences embedded in realistic seas must
continue to reveal the secrets of wave-structure
interaction and to improve the design of safer
ships and offshore structures.
DISCUSSION
C. M. Lee
Pohang University of Science and Technology,
Korea
It is an interesting paper which gives an
intelligent method for testing seakeeping quality
of ships and floating platforms under the extreme
wave environment. I suppose when we say the
extreme wave condition, it should mean for a
specific ship or platform since each marine
structure has its own resonant frequencies or
vulnerable load conditions. I don't think the
extreme wave condition not necessarily mean
only an impulse-type wave, thus to use the
terminology "extreme wave environment." We
OCR for page 252
should specify whether it is the amplitude,
frequency, and/or wave slope for swell type
waves or the significant height, model period,
and/or significant wave slope for irregular
waves. Not only that the wave heading angle
should be included as well. I would like to know
how the author define would the "extreme wave
environment" for a given ship or marine
structure?
DISCUSSION
Dr. Stephane Cordier
Bassin d'Essais des Carenes, France
Our experiences with force measurements during
transient wave tests, in particular in the case of
segmented models, stationary or with speed of
advance, shows that bending moment or shear
forces transfer function obtained in transient
wave tests match data obtained either through
regular wave or irregular wave tests. Can the
author suggest a method for choosing a wave
group which should be used for design of a
marine structure?
AUTHOR'S REPLY
I wish to thank Prof. Lee and Dr. Cordier for
their valuable comments and delicate questions.
As both are brilliant experts in this field, they
know that there is no pat solution, however, I
will discuss some aspects:
In fixed structures the maximum wave elevation
might be the most decisive parameter as the
lower decks should clearly raise above the
maximum wave crest with a sufficient air gap
because the impulsive peak forces increase
sharply if the wave hits the superstructure.
In case of flexible or floating stationary
structures a quick look at the (linear) response
amplitude operators reveals the neuralgic natural
frequencies which is the key for the evaluation of
wave/structure interaction. If resonance effects
play a major role, these investigations must be
supplemented by
analysis. In this case even the "real world" can
be simulated as has been presented in Figs. 29
and 30 showing the interaction of the giant New
Year Wave sequence with a semisubmersible.
For details see Clauss et al. (2002a). Note that
the highest wave in the "real world" wave train
(see Figs. 1 and 29) can be stretched or
compressed arbitrarily to vary frequency and
nonlinear time-domain
steepness. Also the preceding waves can be
designed specifically to introduce tailored
memory effects.
If the floating structure is cruising with a defined
speed and course, the selection of a critical
(response based) design wave group becomes
even more delicate. In general, a non-linear time-
domain analysis is required. As has been
sketched in Figs. 26 and 27 this procedure is
used to analyse the mechanism of capsizing.
Firstly, the physical mechanism of extreme roll
motions (sometimes with subsequent capsizing)
is investigated by the evaluation of the non-
linear cause-reaction relations of wave/structure
interactions. Next, dedicated computer-
controlled capsizing tests with deterministic
wave trains are carried out at model scale
embedding rogue wave sequences in severe seas.
Based on these results, we finally developed a
non-linear numerical method for simulating ship
motions in extreme seas. As a result of this
project which has been accomplished by our
institute in cooperation with the Hamburg Ship
Model Tank Basin (HSVA) and the Flensburger
Shipyard FSG, we determined polar plots with
limiting wave heights for the capsizing of a
specific vessel depending on its speed and course
(Clause et al., 2002b).
As shown in Fig. Al - left hand side - the most
critical regions of resonance motions as well as
of parametric rolling are clearly identified. Only
a change of trim by lm to stern (see right hand
side) reduces the capsizing risk considerably.
Consequently, the assessment of the seakeeping
behaviour of a floating structure requires a
highly complex procedure combining non-linear
numerical simulation methods validated by
deterministic seakeeping tests. As a result, safer
ships can be designed and loading conditions
optimized, improving ship operation and
navigation significantly.
In conclusion, the application of deterministic
wave sequences for the evaluation of
wave/structure interactions is recommended as
an additional tool in NWT investigations and
physical model tests. Aiming for a response
based design we may assume critical extreme
waves or wave sequences, and the analysis will
reveal whether we really succeeded in finding
the "extreme wave environment", i.e. systematic
variations are inevitable.
a) In detail, the method can be used as a
OCR for page 253
tool to analyse the mechanism of the
structure behaviour in waves because
the non-linear cause-reaction effects are
deduced from deterministically given
wave field characteristics like pressure
field, particle accelerations and
velocities as well as non-linear wave
elevation in space and time.
b) Wave trains can be designed
individually to investigate a specific
structure at a certain tank position, i.e.
some dedicated regular waves can
precede an extremely high wave or
wave group for simulating memory
effects. By stretching or compressing
the peak wave its frequency and slope
can be tuned accordingly. Also phase
relations between incident wave and
structure motions can be selected and
varied deterministically. Any test can be
repeated identically if a specific effect
is analyzed.
c)
Observed wave registrations, like the
extremely high New Year Wave
sequence (Fig. 1) can be generated in a
wave tank at a selected model scale.
Thus, the genesis of extreme events in
such wave groups can be analyzed in
space and time. Also, the seakeeping
behaviour of any structure can be
evaluated in such extreme
environments.
d) Finally, non-linear numerical methods
can be validated by dedicated
seakeeping model tests in deterministic
wave sequences. By systematic
simulations even the most critical wave
group may be identified.
I wish to thank all discussants for their excellent
comments and demanding questions.
REFERENCES
1. Clauss, G., Schmittner, C. and Stutz, K.
(2002a): Time-domain Investigation of a
Semisubmersible in Rogue Waves, In
Proceedings of 21 st Int. Conf. on Offshore
Mechanics and Arctic Engineering
(OMAE'02), Oslo, Norway, June 23-28
2002.
2. Clauss, G., Hennig, J., Kuhnlein, W., Brink,
K.E., Buhr, W., and Cramer, H. (2002b):
Entwicklung von Schiffen mit hoherer
Kentersicherheit durch deterministische
Analyse extremer Rollbewegungen in
schwerer See (Development of Safer Ships
with Reduced Capsizing Risk by
Deterministic Analysis of Extreme Roll
Motions in Severe Seas), Summer Meeting of
the German Society of Naval Architects
(STG), Flensburg/Gluckburg, Germany, May
21-24 2002.
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Representative terms from entire chapter:
wave sequences
