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W-lfc
A NEW METHOD FOR THE PREDICTION OF
NON-LINEAR RELATIVE WAVE MOTIONS
Bas Buchner
Maritime Research Institute Netherlands
ABSTRACT
The problem of green water on the bow decks of floating offshore
production units in large survival waves is an important design
problem. An accurate description of the relative wave motions around
the bow is important for the prediction of green water on the deck. In
the paper a new method is presented to predict these relative wave
motions, which are highly non-linear. Input for the method is the
calculation of the relative wave motions in front of the bow with
linear (3D) diffraction theory. Based on this input the proposed
method is able to account for the non-linearities which occur as a
result of the non-linearity in the waves, the influence of the above
water hull shape (bow flare) and the effect of green water on deck on
the ship motions. The method is based on a modification of the
Raleigh distribution and allows a fast evaluation of the effect of the
above water hull shape (bow flare angle and freeboard height) on the
relative wave motions. The prediction of the extreme relative wave
motions can be used for the prediction of the required bow height for
dry fore decks or to determine the maximum freeboard exceedence as
input to the study of green water on the deck.
NOMENCLATURE
a-f
fb
H.
L
Me
M(
P(r)
P(r>R)
r
R
s
TP
z
P
T
parameters in new expression for probability of
exceedence of relative wave motions, dependent on
wave period and bow flare angle
freeboard height from still waterline to top bulwark in
m
significant wave height in m
ship length between perpendiculars in m
wave exciting moment in kNm
green water moment in kNm
probability of occurrence in percent/100
probability of exceedence in percent/100
relative wave motion at centreline bow in m
extreme value in relative motions in m
standard deviation
peak period of wave spectrum in s
local vertical motion in m
value in expression for probability of exceedence
factor in expression for probability of exceedence
co
£
X
=
=
=
wave frequency in rad/s
wave elevation in m
deep water wave length in m
INTRODUCTION
With the trend to use Floating Production Storage and Offloading
(FPSO) units in increasingly harsh environments, the problem of solid
green water on the bow (or deck wetness) becomes an important
design aspect. Due to the weathervaning characteristic of turret
moored systems the bow of an FPSO is always exposed to the waves.
Recent experience with FPSOs at the North Sea confirmed that green
water loading can cause serious damage in the bow region. Because
green water loading is a major aspect in the safety of FPSOs, it should
be taken into account in the design. This requires an optimum design
of the hull shape and the (protecting) structures at the deck.
However, due to the complex nature of the green water phenomena,
there are no state-of-the-art methods that can predict green water
accurately and efficiently. In recent years the author carried out
extensive research on the subject of green water and FPSOs
(Buchner, 1995/1996). The research focused in the first place on the
description of the occurring phenomena. Based on this research it is
presently investigated if new numerical methods can be used to
predict the complex and non-linear phenomena green water problem
(Buchner and Cozijn, 1997, Ortloff and Krafft, 1997). However, these
numerical methods still have significant limitations and drawbacks.
Their capability to give reliable predictions is not clear yet, they are
sometimes very sensitive to small changes in the input, they require
heavy computational equipment and they still need extensive
validation with model tests after their development. This type of
methods is not expected to be reliable design tools for green water
prediction on FPSOs for numbers of years.
To take into account the aspect of green water loading in new
designs of FPSOs for harsh environments, the Joint Industry Project
P(P)SO Green Water Loading' has been carried out with wide
support of the industry. The JIP was based on extensive model tests
and computations. It covered a wide range of topics: from relative
wave motion prediction to the impact loading on structures with
different structural shapes at the deck.
This paper -based on the JIP results- focuses on the input to the
green water problem: the prediction of the strong non-linear relative
wave motions around the bow. A new method will be presented
which can be used to predict the required bow height for dry
foredecks, or to predict the amount of water on the deck. The new
method is developed and validated with a large number of model tests
with a full elliptical bow. The wateriine of this bow is given in the
Figure 1 and its main particulars are given in Table 1. This full
elliptical bow represents the base case for both new designs as well as
traditional tanker bows. Most of the existing tankers can be seen as a
variation (in flare) of this base shape. The effect of the bow shape
above the still wateriine was investigated too. Keeping the underwater
part of the ships constant, bow flare angles (with the vertical) of 0, 10,
30 and 50 degrees were investigated. The body plan with a bow flare
of 30 degrees is shown in Figure 1. The bow flare angle was defined
in a plane perpendicular to the wateriine.
Tests were carried out in regular as well as irregular waves. Regular
wave tests were carried out for different wave lengths (X/L=0.75, 1.0
and 1.25) and wave heights (50-115% of a reference wave height. For
50% and 70% there is no water on the deck). Irregular wave tests
were carried out in a significant wave height of 13.5 m with spectral
peak periods 12,14 and 16 s (JONSWAP).
The relative wave motions (r) in this paper were measured with a
ship fixed vertical wave probe at the centreline of the ship, 1.0 m in
front of the bulwark (see Figure 1). The ship was moored in a
horizontal soft spring mooring system (stiffness in x-direction 430
kN/m) and was free to move in six degrees of freedom. The water
depth was 150 m.
s is the standard deviation of the linear motion response in the wave
spectrum.
OBSERVATION OF NON-LINEARITIES
However, it was found in this and previous studies (Buchner,
1995/1996) that the actual relative wave motions in large survival
waves are highly non-linear. Figure 2 shows the probability of
exceedence of the relative wave motions in front of the bow for a bow
flare angle of 30 degrees for the different irregular waves (Peak periods
Tp of 12, 14 and 16 s). In the figures both the measured probability of
exceedence as well as the linear narrow banded Rayleigh distribution
according to expression (2) are shown.
As can be seen from this Figure, the measured distribution deviates
significantly from the Rayleigh distribution. The Rayleigh distribution
underpredicts the probability of exceedence of small crests, but
overpredicts the probability of exceedence of the larger crests. The
difference between the Rayleigh and measured distributions is dependent on the wave period. In short waves the differences are larger than in
long waves. A close study of the measured distributions shows that they
all have a clear discontinuity at the freeboard height. This aspect is
important in the new method that is developed in this study. The nonlinearity is confirmed by the comparison between measured and
calculated pitch and relative wave motion RAOs in Figure 3. The figure
shows the calculated response as well as the measured response in
regular waves of different amplitudes (50%-l 15% of a reference wave
height of 100%). In general the measurements approach the calculations
when the wave amplitude decreases, which confirms the validity of
diffraction analysis for small wave heights.
The physical background of the observed non-linearities can be related
to the effect of:
RELATIVE WAVE MOTIONS
The relative wave motions around the bow are the input to the green
water problem. They should therefore be predicted accurately to
obtain a reliable prediction of green water occurrence and loading.
- die water on deck on the ship motions
- the above water hull shape at bow and stem
- die non-linearities in the waves
The relative motions of the waves with respect to the ship are
defined as the difference between the local vertical vessel motion z
and the local wave motions C, according to:
The effect of the water on deck on the ship motions
In (Buchner, 1995b) a study was carried out to investigate the
sensitivity of the pitch motions for water on the deck. The load of the
water on the deck has a large moment arm with respect to the centre
of gravity, resulting in a significant green water moment Mg. In this
study it was found that the pitch motion changes significantly as a
result of the green water in short waves, whereas the effect in the
longer waves is small. This is due to the ratio between the wave
exciting moment M 6 and the moment due to the green water Mg. This
ratio is large for the longer waves and small for the shorter waves.
These results are supported by the results of the present study.
r= ;-z
(1)
FPSO motions and relative wave motions are nowadays generally
calculated with linear diffraction analysis. Diffraction analysis is
based on the main assumptions of small sinusoidal waves and vessel
motions. The interaction between structure and fluid is limited to the
part below the still wateriine, which implies that the bow shape above
the wateriine is considered to be vertical ('wall sided', no bow flare).
The Response Amplitude Operators (RAOs) from diffraction theory
can be used to determine the linear response in an irregular wave
spectrum. Applying the assumptions of a narrow banded linear
motion response to Gaussian distributed waves, Ochi (1964)
developed the following expression for the probability of exceedence
P of a certain value R of a peak of the relative wave motions:
p ( r > R ) = exp
2sJ
(2)
Based on the measurements of the water heights on the deck at
three positions and the vertical acceleration of the deck at the related
positions, the green water moment M ( was estimated for three regular
waves with different ratios between wave length X and ship length L,
In the Table 2 the maximum values of M ( and the linear wave moment
M e are summarised. It will be clear from these numbers that the green
water moment can have a large effect on the ship motions, especially in
short waves. Its final effect is dependent on the phase of this moment
compared to the ship motions and wave exciting moment (Buchner,
1995b).
The effect of the above water hull shape at bow and stern
Another important aspect in the non-linearity of the relative wave
motions is the effect of the hull shape above the still wateriine in large
waves. When the waves and ship motions are large, the bow and stem
of the vessel are pushed into the water and the effect of the above water
shape cannot be neglected any more. The additional buoyancy and
wave loading result in an effect on the ship motions, such as pitch.
The above water bow flare also has a significant disturbance on the
wave panem around the bow. Figure 4 shows die wave contours in
front of the full elliptical bow with a bow flare of 30 degrees. This
figure clearly shows the disturbance of the wave in a ripple dial is
progressing away from the bow. This ripple is clearly due to the above
water hull shape and has its effect on the relative motions. The flared
wedge that is pushed in die water with a certain velocity results in a
swell up around the bow.
In Figure 5 the time traces are shown of die relative motion in front of
the bow in regular wave tests on the full elliptical bow with flare angles
of 10,30 and 50 degrees. It is clear from these time traces that the ripple
on the relative wave signal is increasing with the flare angle. With
hardly any flare (10 degrees) the time traces are almost sinusoidal, but
going to 30 and 50 degrees the disturbance in the main signal is
growing. It should be noted that the disturbance of the bow flare stops
at die moment that the relative motion comes above me deck edge.
After such a freeboard exceedence the ripple progresses away from the
bow and does not effect the relative motions around die deck edge any
more, see Figure 4. If we now consider the effect of this wave
disturbance on me peaks in die relative motion time traces in Figure 6
for different wave heights (30 degrees bow flare), we see that the effect
on the peak values for large freeboard exceedences is small. However,
for smaller maxima in the relative motions, which do not come (high)
above the deck edge, me effect of the bow flare is relatively large. We
see that for the largest amplitude the bow flare disturbance is a ripple
on the time trace, without an effect on the maximum amplitude.
However, for the smaller relative wave motions in the other lime
traces we see that the disturbance starts to effect the maximum
amplitudes. In irregular wave spectra something similar happens,
where the small maxima below die freeboard are increased by die
above water flare angle, whereas the larger maxima are not affected.
This is confirmed by the probability of exceedence plots of die
relative motions in Figure 2.
water hull shape decrease die relative wave motions in large waves,
whereas die non-linearity in die waves increases me relative wave
motions. This makes die description of the combined non-linearity even
more complex.
DEVELOPMENT OF NEW METHOD
The number of methods for die description of non-linearities in
literature is significant, but these metiiods are generally unable to
describe phenomena that have a significant discontinuity as the present
problem at die freeboard height. Therefore a new metiiod had to be
developed. The following requirements were considered in the
development of this method:
a
b
c
Based on diese requirements die new metiiod was developed. A
schematic overview of the steps die new method is given in Figure 8
and die summary below.
1.
2.
The effect of die bow flare on the wave disturbance generally
increases die relative wave motions. However, the effect of the
buoyancy and wave loads on die bow flare tends to reduce the ship
motions.
The effect of the non-llnearitv In the waves
The incoming waves can, as the input to the green water problem, be
an important source of non-linearities in die relative wave motion
result In Figure 7 the probability of exceedence is shown for a survival
wave spectrum (Hs=13.5 m, Tp=14 s). Also the theoretical line based
on the linear Rayleigh distribution is shown.
This Figure clearly shows dial for diis type of survival waves the
wave crests do not follow the Rayleigh distribution (Kriebel and
Dawson, 1993). In fact, me Rayleigh distribution underestimates die
probability of exceedence of extreme wave crests significanüy, see
Figure 7. This result is surprising because me measured relative wave
motions wiui respect to the vessel are all overestimated by the Rayleigh
distribution for large extremes, see Figure 2. This suggests that the
water on deck and die above water hull shape have an opposite effect
dian the non-linearities in the waves. The water on deck and the above
The method should allow a fast, flexible and accurate evaluation
of die relative wave motions to determine the required freeboard
height for dry foredecks or to determine the maximum
exceedence of the freeboard if water is allowed on die deck.
The method should be able to include die effect of the main
(underwater) hull shape, above water hull shape and freeboard
height. The discontinuity at die freeboard level has a large
effect on die distribution of the extreme relative motions.
The probability of exceedence of die relative motions in die
expression should continuously decrease with increasing relative
motions.
3.
For die (underwater) hull shape of die ship tiiat has to be
investigated, the Response Amplitude Operator (RAO) of the
relative wave motion in front of die bow is calculated widi linear
(3D) diffraction theory. In this method the ship motions are
determined together with me incoming, reflected (diffracted) and
radiated waves. Linear diffraction analysis is a reliable, validated
and fast metiiod that is widely available in die offshore industry.
The results presented in this study show differences in large
survival waves between measurements and diffraction
calculations, but still it is clear tiiat diffraction dieory predicts the
main trends correctly. Especially die results in small waves
confirm the validity of die mediod.
Based on the RAO from the linear diffraction calculations the
linear relative motion response in die applicable wave spectrum is
determined, with its standard deviation (s) and typical period (T).
Based on die linear standard deviation, die extreme relative wave
motions are now calculated based on die new non-linear
expression for the probability of exceedence, taking into account
the above bow flare angle, wave period and freeboard height.
In this way the present mediod is based on a decomposition of the
problem in the determination of die main ship and wave motions (with
linear dieory) and die correction for non-linearities as a result of local
flow effects and large waves.
Expression for relative motions below the freeboard
Based on the observation mat linear dieory predicts die relative
motion very well in small waves, it was decided to develop for step 3 a
modified Rayleigh distribution for die relative motions below die
freeboard. We start with the assumption that die relation between die
non-linear result r and die linear result rt can be expressed as die
following third order polynomial function:
f] = a . r + b . r 2 + c . r 3
(3)
This expression is always going through zero, which implies that for
small motions the linear result and non-linear result are approaching
each other as expected based on linear theory. An example is shown in
Figure 9 below for a scries of regular wave tests with different wave
heights. This expression is slightly different from what is usual in
perturbation methods, where the non-linearity is a function of the linear
result. The present formulation is, however, essentially the same and
allows a more efficient formulation of the method, see equation 6.
For narrow band linear systems the probability of occurrence p(r,) can
be expressed as (Kriebel and Dawson, 1993, Ochi, 1964):
p(ri)=-rexP
P(r>R) = l - |
a.r + b . r 2 + c . r 3
(a.r + b . r 2 + c . r 3 ) 2
.exp
2s'
(10)
.(a+2.b.r + 3 . c . r )
Which is equal to:
2
P(r > R) = exp (--*—).(a + b . R + c . R )
2.s 2
(11)
For a=l, b=0 and c=0 this reduces as expected to the Rayleigh
distribution in expression (2). Next the discontinuity around the
freeboard has to be accounted for.
(4)
2s'
The probability of exceedence is equal to the well known Rayleigh
distribution in this case, see expression (2). The probability of
occurrence of the non-linear amplitude p(r) can now be obtained from
expressions (3) and (4) by transformation of random variables, see for
instance Kriebel et al. (1993) and Tayfun (?):
Expression for relative motions above the freeboard
After an extensive evaluation of the problem it was decided to
develop an additional expression for extremes above the freeboard.
Starting point is the probability of exceedence at the freeboard level
(fb), which is expressed in a factor P as follows:
P(r > fb) = exp ( - - ^ - ) . ( a + b.fb + c.fb 2 ) = exp[P]
2.s 2
dr,
p(D = p(r,)-idr
(12)
(5)
For the probability of exceedence of relative motions above the
freeboard the following additional expression was developed:
Therefore we write:
P(r > R) = exp p+TfcR-fb).d+(R-fb) 2 .e + ( R - f b ) 3 . f ]
—!-=a + 2.b.r + 3.c.r'
dr
(13)
(6)
or
This results in the following expression for the probability of
occurrence of the non-linear relative motion r.
P ( r > R ) = exp
P(r) =
a.r + b.r + c.r
2
2
3 2
J
2s
(7)
l
.(a + 2.b.T + 3.c.r )
The probability of exceedence of a certain value R can now be
expressed as:
P(r>R) = ] a.r + b.r
+ c.r
.exp
(14)
3
+ (R-fb) .e + (R-fb) .f]
(a.r + b . r + c . r )
.exp
( - ^ T ) . ( a + b.ib + c.fb2) + T[(R-fb).d
2.s
(a.r + b . r 2 + c . r 3 ) 2
2sJ
(8)
.(a + 2.b.T + 3.c.r*)
To come to a closed form solution, this is converted according to:
P(r > R) = 1 - P(r < R)
This expression describes the non-linearity above the freeboard
again as a third order polynomial non-linearity. The matching of the
distribution above and below the freeboard is guaranteed with this
formulation, whereas the discontinuity at the freeboard level can be
taken into account. The factor x corrects for the fact that the
distribution above the freeboard in expressions (13) and (14) is
influenced by the value of p at the freeboard height, which is a
disadvantage of the present method, T is dependent on the ratio
between the standard deviation of the linear relative motion (s) and
the freeboard height (fb), which in the present test series was equal to
fb/s=1.3 At present a linear relation is assumed between T and fb/s
according to:
T=1.0 + 0.83.(—-1.3)
s
(9)
(15)
Using the standard deviation from linear theory (s) and the applicable
freeboard height, (fb) the expressions (11) and (14) can now be fitted
through the measured probability of exceedences of the relative
motions.
In this way the parameters a-f were determined for each bow flare
angle (0, 10, 30 and 50 degrees) and peak period of the spectrum. In
Figure 10 below an example is shown of a measured and fitted probability of exceedence for a bow flare angle of 30 degrees. This figure
makes clear that the chosen expressions are able to describe successfully the observed non-linear distributions with their different behaviour
above and below die freeboard.
Validation of developed expression
Going back to the sequence in Figure 8, die developed method should
now be able to predict the non-linearity in other situations with other
hull dimensions or underwater shapes, but with similar above water
flare angles. For this validation two cases were used with a bow flare of
30 degrees (Tp=14.0s):
• Case 1 with a different stem but with a same bow and bow flare angle
- Case 2 with an enlarged freeboard height
In Case 1 there is a significant variation in standard deviation s in the
linear diffraction theory motions, from 8.03 m for the original (tanker
type) stem to 7.17 m for the full (box type) stem. In Case 2 there was a
significant variation in freeboard height (fb) from 10.5 to 15.5 m, with
the same linear standard deviation (the situation below the still
waterline is identical).
Figure 14 the predicted and measured probabilities of exceedence are
shown for the 2.0 knots current situation. The predicted probability of
exceedence shows a very good comparison widi the measurement.
This confirms that the presently proposed method can be used to take
into account the effect of current on die relative motions of the bow.
APPUCATION
In die Joint Industry Project the expressions were determined for
bow flare angles of 0, 10, 30 and 50 degrees. Spectral peak periods of
12, 14 and 16 s were considered. As an example Figure 15 shows the
probability of exceedence based on expressions (11) and (14) for all
bow flare angles with a freeboard height of 10.5 m and a spectral
peak period of 14.0 s. The related linear Rayleigh distribution is
shown too.
With the new expressions for the probability of exceedence design
values for the maximum relative wave motions can be determined, such
as the Most Probable Maximum (MPM) value. The MPM value is the
relative wave motion value for which the following relation applies:
PCr>RMPM> = 17
< 16)
The predicted and measured distributions for Case 1 are shown in
Figure 11, together with the Rayleigh distribution. It shows a good
comparison between measurements and predictions. In Figure 12 a
comparison is made between the original distribution (10.5 m
freeboard) and die distribution with enlarged freeboard height from
Case 2 (15.5 m freeboard). This Figure confirms a general observation
in tests of FPSOs: if after a test with much green water on me deck an
enlarged freeboard is applied to prevent green water, the relative
motions seem the increase too. The enlarge freeboard height is not
completely effective. This is due to the reduced amount of water on
the deck, the reduced lowering of the wave height around the bow
due to a reduced flow onto the deck and the increased swell up effect
of the flare on the wave motions. In Figure 13 it is shown that the new
method is able to predict this effect of the increased freeboard
accurately. The comparisons in Case 1 and 2 make clear that the chosen
expression is able to predict the non-linear distribution of extremes with
a good accuracy.
N is die total number of relative motion maxima in the total storm
duration. Taking die situation in Figure 15 and assuming 100 wave
oscillations (N=100, P=0.01) we find the overview of Most Probable
Maximum relative motions and freeboard exceedences for this
situation in Table 3, P(r>MPM=0.01). It can be concluded that the
Rayleigh distribution in this case is conservative with respect to die
measurements. The results also show the influence of the bow flare
angle on the relative wave motions.
Effect of current
Current can have a significant increasing effect on the green water
on FPSOs (Buchner, 1995a). This is not due to the effect of the
current velocity on the flow onto die deck, but a result of the
increased relative wave motions in current (when it is collinear with
the waves). This effect is mainly due to the increased pitch motions in
current. These increased pitch motions are due to the increase in wave
length if the same earth fixed wave frequency is used in current
(Buchner, 1995a). Due to this increase in wave length, the wave
exciting forces on the tanker increase, whereas the wave period comes
closer to the pitch natural period of the F(P)SO. This combined effect,
taking into account the influence of the current on added mass and
damping, results in the increase of vessel motions. These effects can
be calculated with linear theory, as was shown for instance by
Huijsmans and Dallinga (1983) and Beck and Loken (1989).
CONCLUSION
Based on the results presented in this paper it can be concluded that
the developed meüiod is very useful for the prediction of the extreme
relative wave motions. These can be used for the prediction of the
required bow height for dry fore decks or to determine die maximum
freeboard exceedence as input to the study of the green water
behaviour on die deck. It allows a fast evaluation of the effect of the
above water hull shape (bow flare angle and freeboard height) on the
relative wave motions.
Using the linear RAOs in current from mis method, the linear
standard deviation was determined in a current speed of 2.0 knots.
This standard deviation was included in expressions (11) and (14) for
a bow flare angle of 30 degrees (determined without current). In
Widi die developed expressions it is possible to study the effect of
design decisions with respect to die above water hull shape (bow flare
angle and freeboard height) on the relative wave motions. It should be
noted that die bow flare angle not only influences the relative wave
motions. It also influences the relation between the exceedence of the
freeboard and die water height on die deck, the water velocity over
the deck and die impact loading on structures are influenced by the
bow flare angle. This will be studied and presented in a future paper.
ACKNOWLEDGEMENTS
The author wishes to thank the participants of the Joint Industry
Project 'F(P)SO Green Water Loading' for die cooperation in die project
and the permission for publication of mis paper ABB Offshore
Technology, Bluewater Engineering, BP, Chevron, Conoco, Exxon,
FMC Sofec, Germanischer Lloyd, Health & Safety Executive (HSE),
Maersk Contractors, Mobil, Samsung Heavy Industries, SBM, Gusto
Engineering, Shell and Texaco. Gert van Ballegoyen of MARIN is
acknowledged for his continuous support in die analysis of die results.
REFERENCES
Beck, F.R. and Loken, A.E., 1989. 'Three-dimensional Effects in
Ship Relative-Motion Problems," Journal of Ship Research, Volume
33, No. 4, pp. 261-268.
Huijsmans, R.H.M. and Dallinga, R.P., 1983, "Non-linear Ship
Motions in Shallow Water," International Workshop on Ship and
Platform Motions, Berkeley.
Buchner, B., 1995a, 'The Impact of Green Water on FPSO Design,"
OTC Paper 7698, OTC 1995, Houston.
Kriebel, D.L. and Dawson, T. H., 1993, "Nonlinearity in Wave Crest
Statistics," 2nd Intl. Symposium on Ocean Wave Measurement and
Analysis, pp. 6175.
Buchner, B., 1995b, "On the Impact of Green Water Loading on Ship
and Offshore Unit Design," PRADS "95, Seoul.
Ochi, M.K., 1964, "Extreme Behaviour of a Ship in Rough Seas
Slamming and Shipping of Green Water," Annual Meeting SNAME.
Buchner, B., 1996, 'The Influence of the Bow Shape of FPSOs on
Drift Forces and Green Water," OTC Paper 8073, OTC 1996,
Houston.
Ortloff, C.R. and Krafft, M.J., 1997, "Numerical Test Tank:
Simulation of Ocean Engineering Problems by Computational Fluid
Dynamics," OTC 8269, OTC 1997. Houston.
Buchner, B. and Cozijn, J.L., 1997, "An Investigation into the
Numerical Simulation of Green Water," Boss"97, Delft.
Tayfun, M.A., "Narrow Banded Nonlinear Waves.' Journal of
Geophysical Research, Vol. 85. No. C3, pp. 1548-1552.
Table 1 Main particulars of ship
Length between perpendiculars
Beam
Draft (even keel)
Freeboard at bow (including bulwark)
Displacement
CoG above keel
CoG forward of midships
Longitudinal radius of gyration
260.34
47.09
16.50
10.5
168870
14.14
5.69
69.80
m
m
m
m
t
m
m
m
Table 2 Estimated green water moment and linear wave moment as function of ratio wave
length to ship length
Ratio wave length to ship length
1.25
1.0
0.75
Estimated green water moment Mf in
kNm
7.13.10"
6.44. 106
4.82. 106
Linear wave moment
Me in kNm
2.82.10'
1.37.10 7
4.82.10 6
Table 3 Predicted most probable maximum relative wave motions and freeboard
exceedences (fb=10.5m) with different bow flare angles for a spectral peak period of 14.0s
MPM
Relative wave motions
MPM
Freeboard exceedence
Rayleigh
(linear)
24.4 m
Odeg.
lOdeg.
30 deg.
20.2 m
19.7 m
19.1 m
18.5 m
13.9 m
9.7 m
9.2 m
8.6 m
8.0 m
50 deg.
Flare angle
FULL ELLIPTICAL
14
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VL
T = 16.5 n
8-10
Rgure 1 Wateriine and body plan of full elliptical bow (30 degrees bow flare). The relative wave motion (r) is measured
with a vertical wave probe at the centreline of the ship, directly in front of the bow flare.
Tp=l2.0s, Hs=13.5m
T
Tp=14.0s. Hs=13.5m
'
"ers^^-r—
'
0.1
NT"*.
£ tneas»
3
0
£, nyKa) o.oit-
rayKa)j 0.01
o.ooi r-
0.001
1
*
—
"-.,
\
-,
-
1-
1
1
10
15
1
20
25
Tp=16.0s. Hs=l3.5m
T
O.lH
It
_. rayKa)o.oi H
0.001 r~
0
Relative motion amplitude in m
Rgure 2 Probability of exceedence of relative wave motions for different spectral peak periods (Tp=12,14 and 16 s) and a
bow flare of 30 degrees. The measured distributions (solid lines) and Rayleigh distributions (dotted lines) are shown.
—
CD
O
e.
Measured
Measured
Calculated
Measured nSV. w a v e ampl.
Measured WOV. w a v e ampl.
M e a s u r e d 6S*A w a v e a m p i
70V. w a v e a m p l
50V. w a v e a m p l
X,
/S\.
0.S-
I
/
0.0
K
05
1.0
1.5
W a v e frequency in r a d / s
Calculated
Measured 115% wave ampl
Measured ttOK wave empL
Measured BSH wave «mpt.
OS
Measured. 70% wave «mpL
Measured 50% wave ampl
1.0
Wave frequency h rad/s
Rgure 3 Pitch and relative wave motion Response Amplitude Operators (RAOs) calculated with linear diffraction theory
(solid lines) and measured in different regular wave heights. For each wave frequency different wave heights were tested
to investigate non-linearities. The wave height varied from 50 to 115 % of a reference wave height With 50 and 70% there
was no water on the deck.
5.0 m
Watcrlinc
Figure 4 Wave contours of green water flow onto the deck, visualised with a plate at the centreline of the deck. Water
contours are shown with time steps of 0.31 s for a bow flare angle of 30 degrees.
FLARE 50 deg
SECONDS
Figure 5 Time traces of relative wave motions for bow flare angles of 10,30 and 50 degrees
Wave
Waveampl. 70%
-rso
SECONDS
Figure 6 Time traces of relative wave motions for 30 degrees bow flare angle and different wave heights (70, 85 and 100%
of reference height)
Tp=14.0s. Hs=13.5m
I "^jj
Hi»*.) 0 * 1
0.0011-
Wave crest amplitude in m
Figure 7 Probability of exceedence of wave crests for a spectral peak period Tp=14 s. The measured distribution (solid
line) and Rayleigh distribution (dotted line) are shown (Hs=13.5 m).
\
Underwater hull shape
Wave spectrum
Bow flare angle, freeboard
i '
1'
RAO
Linear
*'
s,T
Motion response
in wave spectrum
w
Prediction
non-linear extremes
'mix
Figure 8 Schematic overview of the proposed method with input and output
£
linear(0
f(r)
§ 1'!
10 - '
Measured R value
Figure 9 Linear predicted relative wave motion versus measured relative wave motions for different regular wave
heights. A linear relation is indicated (solid line) as well as a description of the non-linearity with a third order polynomial
(dotted line)
—-r— — * * * !
1
^ v ^
o.i --
1
~*~ „
***- w
\ N
o
rayl(r)
0.01
3
polytot(r)
Vx
0.001 I
1
1
10
15
1
20
25
Figure 10 The new expression for the probability of exceedence of relative wave motions (dashed line) fitted to the
measurements (solid line) for the bow with 30 degrees bow flare and a spectral peak period of 14.0 seconds. Also the
Rayleigh distribution is shown (dotted line).
1
"
0.1
"
•
^
^
^
-
N^.
~ -
.
\\
rayKr)
„
0.01 -
pred(r)
VX
\*\
X
\
0.001 -
1
12.5
r
\
\
25
Figure 11 Comparison of predicted relative wave motion with new method (dashed line) with the measurements (solid
line) for the situation with modified full stern. Also the Rayleigh distribution is shown (dotted line)
highj
low0.01 -
0.001
12.5
Relative motion in m
Rgure 12 Probability of exceedence of relative wave motions with original freeboard height (10.5m, low, dotted line) and
extended freeboard height (15.5 m, high, solid line), both with 30 degrees bow flare angle
8
_
Relative wave motion in m
Figure 13 Prediction of relative wave motions with new method for situation with extended freeboard height (dashed line),
compared to measurements (solid line). Also the Rayleigh distribution is shown (dotted line).
cwrl..
Figure 14 Prediction of relative wave motions with new method for current speed of 2.0 knots (dashed line) compared to
the measurements (solid line). Also the Raylelgh distribution is shown (dotted line)
prcdO(r)
0.1 -
• * .
(/"""-•v.
predKXO
pred30(r)
'o -NN
\ v \
N*
0.01 -
predS0(r)
rayKO
Rayleigh
\
0
\
N'V
^\\N V
X 30
0.001
\\50
10
IS
20
25
Figure 15 Effect of bow flare angle on the relative wave motions in a wave spectrum with peak period 14.0 s and
significant height of 13.5 m (freeboard height 10.5 m). Bow flare angles of 0,10,30 and 50 degrees are shown, together
with the Rayleigh distribution