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OCR for page 413
24eh Symposium on Naval Hydrodynamics
Fukuoka, JAPAN, 8-13 July 2002
Nonlinear Green Water Effects
On Ship Motions and Structural Loads
Daniel A. Liut, Kenneth W. Weems, and Woei-Min Lin
(Science Applications international Corporation, USA)
ABSTRACT
Green water on deck can have very important,
nonlinear effects on ship motions and structural
loads. Such occurrences can not only affect the
global motions and loads on ships but also cause
damage on deck structures. In this paper, a novel
finite-volume strategy is presented to simulate three-
dimensional green-water events, mainly on ship
platforms. The equations of conservation of mass
and momentum are solved in the time domain.
Shallow-water assumptions are made, and viscous
effects are ignored. The current green-water method
was developed in the framework of the nonlinear 3-D
time-domain ship-motion simulation program LAMP
(Large Amplitude Motion Program) to account for
water-on-deck effects. The method can handle a
variety of boundary and initial conditions, and it is
capable of supporting arbitrary motions and general
geometries of a ship deck. This approach has been
validated with available experimental data and has
been successfully integrated with the LAMP System.
INTRODUCTION
The objective of the work presented in this paper was
to develop a sophisticated green-water-on-deck
model that could be integrated directly into a time-
domain ship motion calculation, thus allowing green-
water effects to be included in the calculation of
nonlinear ship motions and loads. In order to attain
this, an approach was selected in which the ship
motions and green-water-on-deck calculations run
concurrently in the time domain.
The ship motions code in the present effort
is the Large Amplitude Motions Program (LAMP)
System. At each time step of the LAMP calculation,
the ship motion and wave definition are used to
compute the relative motion at the deck edge (often
called the deck edge exceedance), the deck tilt, and
the deck acceleration, which are then passed to the
green-water-on-deck calculation. The green-water
calculation is subsequently updated after which the
green-water forces are passed back to the ship
motions calculation to be used in solving the
equations of motions and computing sectional loads.
f Ship Moron
Coincident WaYe
~BodyBound~y~ f DeckMotion ~
`~ Condition J (Edge Exceedanc:)
u
~:~.V"~2 I'm ~.~.~2'~ ;5'"~.~'~'F'~ ~~ ~7 :'~'~"~'~ '.~ ~~ i' .~ .~. i. ~~ ~~ ~ ~~' ~~.~ I've ~~.~"~'~'~'~"~"~'~>'~'~"~i ~~:~'~""'~
i''" ~~ ~~ ~~ :' ~S~ ~ 'o'er —~ T~ ~~ + :: ' '?~'~S'S~T'~
~:~ I ~ ~wi~oU~o ~~ ~~ ill ~~ :~_r~—ore ~~.~uv~ator~
: .i .~ I', - ~ .~i. . . ~,~i~:~.~:~ .: is. A,. _,. we. -,:,.:: -I .~^ .-
t ~~ ~~ - i, ~ -I ~ -a: ~ ~~ ill ,~ ~~ At , : ~~ ~ I,,. Eli ~~ . ~~ hi. ~~,~ ~~ ~.~.; ~~ ~ ~~ ~~, ~ .~: . ail
TO: :ii*: : :~ ~ ~~;~ A it ~T`.~: ~ ~~ to ~ i,~'~2~
lu~rn:~n slim sac .~: >~i ~ ; ~ in: all i, p:f~: -:: ¢° :.. ~ ::'
A At i! ,~ , i.~,.:, ~~;~i~~ it_ _~ j~j~,j~:,~ ~~ ~ ., ~~ if_ ., I, '2 2~ '.2.~ ,. .
I'd'','': j~.~'~"~-.,~it' SS:'Ni~i,~'~" a.' ii"'': Ti~:i"i~ ~j'SS~'~"~:~ Ajar ~ ~'~''i'~'LL.'S~''~ ~.~'S~S'~''"'.~:~i''?.
: i: ~,'S,~:~: :~: A: i": :'~''~'S'S,2~' ~',~':4 ~ ~ j j ~ ~~ ~ ~ ~ j ~ j' j i, ~ ~ ~ 'S ~ '. ~ ~ j ,. :''-: :_ ' ~ :: ' ' ~ :: S~ an>: ::
Ian ~~ i~ ~~ .~ ~. ci:~i~i-i~-~i~ ~~ >~; ~ cu .~hon~ ~~-~
. ~ .. -all i ~ j j ~~ ..~jj~,~,,.~.~.~ .,,, . ~i ..~ ~.,.~jjjjj ~ j j.jj ,.,~., j.j.~.~ jiW., jj.~ji j.~ ~.~j~,> ~~.~ ~
r ~ r _
Deck Pressure
Boundary Forces
Wave Forces
Extends Forces
Figure 1: Structure of the LAMP System with
Green Water On Deck
For this approach to be successful, the
green-water-on-deck calculation is required to be
reasonably fast, robust, and capable of calculating the
flow of a deck that is moving with large-amplitude
six-degrees-of-freedom motions. A multi-level
implementation was selected incorporating both a
semi-empirical "baseline" model and a general
solution of the 3-D shallow-water flow equations.
The former model is intended to provide a fast
estimate of green-water effects while the later
provides a higher fidelity solution of the water-on-
deck problem. This latter higher-fidelity model
constitutes the principle focus of the present paper.
OCR for page 414
The numerical strategy adopted for solving
the 3-D shallow-water flow equations is based on a
novel finite-volume approach. The flow is assumed
to be incompressible and viscous effects are ignored.
Given the complexity inherent in the shallow-water
problem compound by the fact that the calculations
would in many cases involve large-amplitude
motions, one of the major goals of the new
development was to achieve the robustness necessary
to handle the general situation of a platform that is
constantly moving and exchanging water with the
environment. With this in mind, a numerical method
was developed capable of assuring a stable solution.
This paper begins with a discussion of the
LAMP System and the incorporation of the green-
water calculations. In subsequent sections, the
solution to the shallow-water equations of mass and
motion are summarized, including some comments
related to the stability of the solution. Next the
alternative semi-empirical approach is described.
This is followed by a discussion of computational
results, including validation data. The paper is closed
with some final remarks.
THE LAMP SYSTEM
The LAMP System is a time-domain simulation
model specifically developed for computing the
motions and loads of a ship operating in extreme sea
conditions. With its general nonlinear time-domain
approach and solution of the 3-D flow field, it is well
suited for incorporating a nonlinear green-water-on-
deck calculation model.
Wave-Body Hydrodynamics
One of the most important features of the LAMP
System is a 3-D body-nonlinear approach for solving
the wave-body interaction problem in the time-
domain. The computational model is based on a
potential-flow "body-nonlinear" approach (tin and
Yue, 1990, 1993; and Lin et al., 1994~. In contrast to
the linear approach in which the body boundary
condition is satisfied on the portion of the hull under
the mean water surface, the body-nonlinear approach
satisfies the body boundary condition exactly on the
portion of the instantaneous body surface below the
incident wave surface. It is assumed that both the
radiation and diffraction waves are small compared to
the incident wave so that the free-surface boundary
conditions can be linearized with respect to the
incident-wave surface. A complete body boundary
condition is applied incorporating forward speed, the
ship motion (radiation), and the incident wave
(diffraction) effects. The solution of this nonlinear
wave-body using a "panel" method gives the velocity
potential over the hull surface. Bernoulli's equation
can then be used to compute the hull pressure
distribution including the second-order velocity
terms. The hydrostatic restoring force is also
computed on the wetted hull up to the incident wave.
In this formulation, both the body motions and the
incident waves can be large relative to the draft of the
ship.
Several variations of Lin and Yue's original
body-nonlinear approach have been developed and
are currently available in the LAMP System. In
addition to the general body-nonlinear approach
described above, a 3-D body-linear approach has
been implemented, which solves for the velocity
potential only on the mean wetted hull surface and
computes the hydrostatic restoring forces, along with
Froude-Krylov wave forces, from hull water-plane
quantities. An approximate body-linear formulation
has also been implemented, which combines the
body-linear solution of the disturbance potential with
body-nonlinear hydrostatic-restoring and Froude-
Krylov wave forces. This latter approach captures
the significant nonlinear effects of most ship-wave
problems at a fraction of the computation effort of the
general body-nonlinear formulation.
Mixed Source Formulation
To solve the ship-wave interaction problem described
above, a hybrid numerical approach has been
developed that uses both transient Green functions
and Rankine sources (tin et al., 1999~. This
approach has been implemented in the LAMP System
as the "mixed-source formulation." In the mixed-
source formulation, the fluid domain is split into two
domains as shown in Figure 2. The outer domain is
solved with transient Green functions distributed over
an arbitrarily shaped matching surface, while the
inner domain is solved using Rankine sources. The
advantage of this formulation is that Rankine sources
behave much better than transient Green functions
near the body and free surface juncture, and that the
matching surface can be selected to guarantee good
numerical behavior of the transient Green functions.
The transient Green functions satisfy both the
linearized free-surface boundary condition and the
radiation condition, allowing the matching surface to
be placed fairly close to the body. This numerical
scheme has resulted in robust motion and load
predictions for hull forms with non-wall-sided
geometries.
2
OCR for page 415
~ ~: /Sf
Figure 2: Mixed Source Formulation
Non-Pressure Forces
In order to calculate the time-domain six-degree-of-
freedom coupled motions for any ship heading and
speed, LAMP also includes models for non-pressure
forces including viscous roll damping, propeller
thrust, bilge keels, rudder and anti-rolling fins,
mooring cables, and other systems. For oblique-sea
cases, a PID (Proportional, Integral, and Derivative)
course-keeping rudder control algorithm and a rudder
servo model are implemented. Because of the time-
domain approach, these non-pressure force models
can include arbitrary nonlinear dependency on the
motions, etc. Adjustable viscous roll-damping
models are available that allow the roll damping to be
"tuned" to match experimental values by simulating
roll decay tests.
Equations of Motion
Once the hydrodynamic and non-pressure forces have
been computed, the general 6-DOF equations of
motion are solved in the time domain by either a
fourth-order Runge-Kutta algorithm or a predictor-
corrector scheme. Since the forces on the right-hand
side of the equations of motion include the
instantaneous added mass, an estimated added-mass
term is added to both sides of the equations of motion
to achieve numerical stability.
Sectional Loads
In addition to motions, LAMP calculates the time-
domain wave-induced main girder loads, including
the vertical and lateral shear forces and bending
moments, torsional moments, and compression
forces, at any cross-section along the length of the
ship. Structural loads can be computed using rigid-
body or finite-element beam models and can include
the whipping responses to bottom or flare slams as
well as wave induced loads (Weems, et al. 1998~.
Interface to Green-water Calculation
At each time step, LAMP uses the ship rigid body
motion, the incident wave definition, and the hull
pressure distribution to compute the relative motion
of the edge of the deck to the wave surface. The hull
pressure is used to predict the disturbance wave or
'pile-up" of the free surface due to the presence of
the ship. This relative wave height (or deck
exceedance) and its relative flow velocity are passed
to the green-water-on-deck calculation module in
order to define suitable inflow and outflow boundary
conditions. The ship rigid body motion, velocity, and
acceleration vectors are also passed in order to define
the tilt of the deck and inertial terms in the green-
water-on-deck equations.
Based on these data, the green-water-on-
deck calculation is then advanced to the current
LAMP time calculation. The deck pressure and edge
forces due to green water are passed back to LAMP
where they are integrated and added to the right hand
side of the equations of motion as well as being used
in the sectional-load calculations.
GREEN-WATER FORMULATION
Conservation Of Mass
Given a control volume CV and the corresponding
control surface CS enveloping it, Reynold's
Transport Theorem can be used to express the
principle of conservation of mass as
do at ipdVol+ J.p(vren)ds=0 (~1~)
where S is the surface of CS, Sol represents the
volume of CV, p is the fluid density, t is time, m is
the mass of fluid inside CV, vr is the flow velocity on
CS, and n is the normal-to-S unit vector given point
wise on S [n = Ox, y, z)].
The first step to solving the shallow-water
problem with the present strategy is to divide the
computational domain into a set of vertical
hexahedrals elements (close-volume elements with
quadrilateral faces), which are contiguously
connected, as shown in Figure 3.a.
3
OCR for page 416
~ ——~ —it i f~%
x -A—h~ ~ ~ `~
(a)
d, / ce ~d3
,
,
d2
(b)
~ 1
Figure 3: Finite volume element e with adjacent
elements qs. The subscript s characterizes each of the
four sides of element e. The corresponding
numbering convention for s is given in part b, where
the characteristic area Ce for a generic element e is
shown, along with each side do.
As seen in this figure, each element e has a
fluid elevation he' measured from the geometric
center of the base of the corresponding hexahedral
element to its top. Also, for each element, a
characteristic area Ce is defined, which is the
projection of the base area of element e onto a
surface normal to the z axis, which contains the
geometric center of the hexahedral's base. The
vertical axis z is set to be parallel to the acceleration
of gravity. Four lateral surfaces As define each side s
of each element e (s = 1, 2, 3, 4), which remain
always vertical. As shown in Figure 3.a, the
subscript qs denotes the adjacent element q to side s
of a given element e. If an element had a triangular
base, one of its four side surfaces As would be
collapsed to a line, a situation that is perfectly
acceptable with the present method. The normal
vectors to the lateral areas, n, are always normal to
the gravity vector. The flow velocity measured on
each lateral surface As is represented by the vectors
vr, which are always parallel to the undisturbed water
level. If the principle stated in equation (1) is applied
to a generic element e, conservation of mass can be
expressed as:
Ha [pCh]e+[p~,vr5 ens As ~ O (2,
where vr is the average normal velocity to each
lateral surface As of element e. Each element has a
characteristic flow velocity Ye, with two horizontal
components vx and vy. As stated above, the
platform for the shallow-water occurrence is allowed
to move with six degrees of freedom following the
deck of a ship. Thus, for each element, the base
horizontal surface Ce will typically be a function of
time. Regarding the differentials as finite differences
and taking into account the incompressibility
condition, equation (2) can then be written as:
[h/\C+C/\h]e+
[~(Vrx;vry~s·(nx;ny)s hs ds ~ —O (3)
s=1 e
where the bars account for average values between
time step k and k -1, and where do are the four sides
s corresponding to the horizontal surface Ce (see
Figure 3.b). The subscripts x and y indicate vector
components parallel to Ce in the corresponding x and
y directions (see Figure 3.a). Expanding the finite
differences, equation (3) can be written as:
[ - k+1 k+1 —k+1 ( k+1 k)1
t[~( - — t+1 (— — Jk+1 h—k+1 dk+l] ~ o (4)
where the superscript k is the time step counter, and
/\Ck+l=Ck+l-Ck. Rearranging terms and after
some numerical treatment, equation (4) can be
written in the compact form
Meje hekj+1 +~Mejq hq =W) (5)
s=1
where
4
OCR for page 417
M J (~`C kj_~ +1 + C kj_~ +1 1 (6)
[ at [ v c h]+ ~ Vs (VrS · nS ) As ~ = (13)
Mj ~tQkj,+1 (7) _ |yz (n dS)- | b dVol
eqS 8 eqS
W' hi' Ckj l+1 6kj_l-1 d At
fit ~ (Qk,-l +1 hki-l +1 + ~ Qk~~l +1 hkj-l +1 )
4 ~ (Qeqs hqs + Qees he )
s=
where
Beet =(Vx;vy)e.(nx;ny)e des
Qeq5 = (VX; vy )q ~ (nx; ny )e des
~eqS 3 42 Ye ~eg,~ 61leqs
(9)
(10)
(1 1)
The variable ~e iS the vertical acceleration of element
e (which is described in the next section), whereas
/~qS is the difference in fluid level between element
e and element qS. The superscript j is an iterative
counter; an iterative process is needed to solve the
algebraic equations defined by (5) since the
coefficients defined by (7) and (8) are a function of
the elevations h, and the other two sets of unknowns
given by the vectors v.
Conservation Of Momentum
CSe C ve
where ~ is the vertical acceleration given by
~ = g + a~ (14)
Thus the vertical acceleration ~ includes the
acceleration of gravity g and the vertical acceleration
az induced by the vertical motion of the platform
considered. The body accelerations represented by b
are the horizontal accelerations (normal to the gravity
vector) induced by the motion of the platform
combined with the effect of the local slope of the
platform ~nz (nX;ny) . Solving the integrals in
equation (13), they yield
[Ch3V+vh3C+vCah
+~Vs (vr, ·ns)As~ = (15)
_ [ r ~ h2 d n + b C h]
Regarding the differentials as finite differences,
equation (15) can be written as:
r—-
LC h ~v+v h /`C+ v C Ah
+^t~Vs (Vr·nS) h5d55 _
Given a control volume CV, Reynold's Transport s=~ s (16)
Theorem applied to the conservation of momentum
principle yields the following formulation:
dt ~t |vpdVol+ |pv~vr~n~dS (12)
CS CV
t- 2 ~ h5 ds nS-bC h &|
s=]
where, as before, bars account for average values
= Jp(-nds)- JpbdKol between time step h end k-1. Expanding the finite
differences, this expression can take the following
compact form:
where b is a horizontal acceleration vector (described
below), and p is pressure. If the same control volume
defined for the conservation of mass formulation is
considered for each element, and taking into account
the incompressibility assumption, equation (12) can
be written as:
Reie Vej l +~Rejq vgj =sej (17)
s=
where
s
OCR for page 418
~ (—kj '+1 h—~j+1 h—kj+1 ~Ckj l+1 (18)
+C—kj_,+1 Ah kj+l )
Ri =_ Dsj~
eqs 2
SJ = Ce j he i (Ve —bei-l At)
2 ~ VqS Ds Pe (20)
s=l
(19)
Ds = 2 (he deS Vre ·neS +hq5 deS VrqS ·neS ) (21)
The term pk; is the force term due to the hydrostatic
pressure, which is given by
—kj 1_kj + —2 j+ —kj + _kj i+
Pe = A Ye At ~ (heq5)k ds us (22)
s=1
where Max is the maximum value of the vertical
acceleration (which includes the acceleration of
gravity), and where 427max he is an over-
conservative estimate of the maximum velocity that
could flow between two adjacent elements.
Approximating Ce as Ce = de do, such as,
do = Min[Max(d~, d3 ), Max(d2, d4 )] (25)
where do, d2, d3, d4 are the sides of Ce as defined in
Figure 3.b, then equation (24) can be written as:
At < (h k+} _|^he |) ~ h 3/ 2 (26)
max max
where /\he =hei+i -trek. The fluid elevation hma,` is
the maximum value of fluid elevation for the
shallow-water assumptions to hold. A maximum
value of lkhe can be estimated using the conservation
of mass equation (see equation (3)) as
Ce |Ahe | < At Max t(42 Ymax he ~ ds he be (27)
As in the case of the equations of mass, the
system of equations defined by (17) is solved
iteratively for v since the coefficients given by
equations (18)-(22) are a function of v and h. In the
overall scheme, at each iteration j, first the fluid
elevations he are solved, next the x components of
the vectors ve, followed by the y components of the
same vectors. This process is repeated, within the 12 h3/2
same time step, until convergence is achieved. |/`he| < At v Ymax max (28)
Minimum Time Step
To attain stability in solving the conservation of
momentum equations, the main diagonal of the
corresponding set of algebraic equations must be
predominant at all times. For the formulation
proposed in this paper, the following conservative
criterion was adopted:
|Ree | > Max~Reqs ~
(23)
If the term Face in equation (18) is neglected, the
previous criterion can be expressed as:
Ce he + Ce (he —he )
At Maxt(~/2 Ymax h5 ~d5 he he
where similar assumptions as those taken into
consideration for equation (24) were adopted. If Ce is
defined as Ce = de do, and using the concept of
hma<, inequality (27) can be written as
de
Replacing inequality (28) in (26), the latter yields
At < hk+~ dG (29)
Ymax max
>
(24)
Evidently, if he can take any value between zero and
ham the only stable solution would be the trivial
solution in which At = 0. Therefore, to render the
calculation possible while at the same time stable, a
minimum value of fluid level hmin must be defined
such that when the fluid elevation of an element e
drops below this value, that element should be
considered dry. This minimum value will determine
both the speed and the precision of the calculation
Equation (29) can be rewritten in terms of hmin as:
6
OCR for page 419
At ~ hmin 2 ,~h3l2 (30)
Ymax max
It remains to be determined what would
constitute a good estimate for hum. To this end, a
suitable estimate for de must first be found. If the
computational grid has elements such that their
individual Ce areas are comparable, the ratio
de_LIN will be chosen as an estimate for the
smallest value of de (which will determine the
maximum time step to be used). The ratio L I N is
computed as:
~ ~ (Ni Nj ) (31)
where Li denotes lengths taken along the grid i
directions, Lj designates lengths measured along the
corresponding j directions (see Figure 3), and where
Ni is the i-wise grid dimension, whereas Nj is the j-
. . . .
wise grlc . c .lmenslon.
If hm`= were arbitrarily small, At could take
arbitrarily large values. But it is desirable that hmaX
be allowed to take the largest possible values to
extend as much as possible the stable range for the
computations of he. To this end, it is taken as a
criterion that hma,` be an order of magnitude larger
than de. This criterion was chosen to be
de/hmaX-_LINi/2, which ensures that de be an
order of magnitude smaller than hm`nc, whereas hm`= is
an order of magnitude smaller than L. This can be
expressed as
where
6= ~ (34)
Replacing equations (32) and (33) into inequality
(30), the latter yields
2 a/ 2Lrymax
which sets the maximum value of At that can be used
to ensure a stable solution. The conditions required
for this criterion can be summarized as:
1) The individual area of each element of the
computational grid must be of the same order of
magnitude.
2) The maximum fluid elevation hmaX = Ll~
should never be exceeded.
It can also be proven that the stability
condition of inequality (35) deduced for the
conservation-of-momentum is sufficient to satisfy the
stability of the conservation-of-mass equations as
well.
Semi-Empirical Model
hma~: = ~ L = '~ L (32)
61 621, i L (33) Fe=p:CH a, +VH al <37'
al ]e
As mentioned in the introduction, the green-water
calculation has been implemented as a multi-level
approach. In addition to the f~nite-volume method
described so far, an alternative statistics-based
calculation was developed. With this approach, an
expedite though coarse prediction of green-water
effects can be produced. In relation to the finite
volume approach, this alternative method provides a
means to predict green-water forces for those
elements in which the fluid level may exceed hen,, for
which shallow-water assumptions cease to be valid.
Statistical data can be used to estimate the
water-on-deck as a function of freeboard exceedance.
With this information, and following a similar
approach proposed by Buchner (1995), the forces on
deck can roughly be estimated as
F - Jp b dVol = a |V P dVol (36)
cv cv
which for each element can be expressed as
where V is the flow velocity on deck, and H is the
green-water elevation. Two of the three components
of V are the horizontal flow components, whereas the
third is given by dH/dt. The two horizontal
components of V and the elevation H are computed
by interpolation from the flow velocity and water
elevation on the boundary of the computational grid.
The relationship between H and the water level on
the grid boundary is obtained from statistical data.
Figure 4 illustrates such a relationship as reported by
Zhouetal. (1999~.
7
OCR for page 420
15
,.;.
,.;,0 so
.= ~ 10
cad ~ s
3 ~
o
. ,~
~ 0
o ~
Jeff
.—~ 1 1 1 ~
s 10 15
Exceedance of freeboard "meters]
Figure 4: Relation of water height on deck boundary
and freeboard exceedance (Zhou et al., 1999~.
VALIDATION
Part of the present effort has been directed toward
establishing the accuracy of the green-water model in
relation to both theoretical and experimental data. To
this end, as a first step, a linear analytical solution to
the shallow-water problem was considered. One
example of this solution is given by Stoker (1957),
who presents the evolution of the level of water
behind a dam after the dam is suddenly removed. In
one of the examples given by Stoker, a dam whose
initial water height is 10 meters is removed, and the
corresponding water profile is computed one second
after the event. The same case was modeled with the
present numerical model, using a finite-volume grid
of 41 elements in a row. Both the analytical and
numerical solutions are plotted in Figure 5, where a
good agreement between the solutions can be
observed. The main difference between both
solutions appears in the smoother backpropagation of
the surface perturbation computed by the non-linear
approach.
12
10
—8
~ 6
to 4
con
2
o
!
Liut et al.: Non-Linear Sol.
~ x Stocker: Linear Sol.
-90 -60 -30 0
Distance from dam (m)
, __~
: ==
t = 1
30 60 90
Figure 5: Comparison of present numerical model
with the linear analytical solution of a breaking dam.
As mentioned above, comparisons have also been
made with experimental results. In this regard, some
experiments produced by Zhou et al. (1999) were
considered. One of those experiments is depicted in
Figure 6. Part a of that figure is a schematic of the
experimental set up. A flap is located at a certain
distance from the back of the tank, separating a
region with water from an initially empty region.
Suddenly the flap is removed, and the water is
allowed to flow freely until hitting an impact plate
where the water bounces back. Different instances of
the water distribution can be seen in Figure 7. The
water level is constantly recorded at a probing point
1.525 meters ahead of the removable flap. Some of
the experiments done with this tank setup were
modeled with the numerical approach presented in
this paper. In Figure 6, one of these experiments
(experiment Nr. 4487001) is described. A 21-
element finite-volume grid was used for this
calculation. Figure 6.b compares the experimental
outcome, the current numerical calculation, and a
numerical calculation done by Zhou et al. (1999)
using a shallow-water model based on Glimm's
method (Grimm, 1965). As it can be observed in the
corresponding plots, a good agreement was obtained
with the experiment.
r Impact Flow
| plate Area
,
'
~ 1.525 m ~
. ~
Probe
Flap Reservoir
Area
0.6
_
0.4 - _ _ _ l
0) 0.3- ___
I l
1
<~' 0.2-
0.1
O -
--1---
l.OOOm
~ 1.200m 1
to
L L
i
~ _ · ' 1 1
—Zhou et al -- Simulation l l
~ Zhou et al -- Experiment l l
0 5 1 --- ut et al. - Simulation ~ i
L — | L ;N ~
I /:v] I I
,. 2.020 m
(a)
r
L
1 1 1
1 1 1
I I ~ I
1 1 1 1 1 1 1
0 1 2 3 4 5
Time (sec.)
(b)
6 7
Figure 6: Comparison of present numerical model
with the linear analytical solution of a breaking dam.
8
OCR for page 421
(I
z
Ins
-
·1
_os~
to
(I
Figure 7: Different stages of water distribution after
removing the flap of the experiment described in
Figure 6, as computed by the present finite-volume
scheme.
LAMP BASED CALCULATIONS
As mentioned in the introduction, the present
approach was developed to provide green-water-on-
deck calculations within the LAMP ship-motion and
load-calculation environment. The approach is fully
integrated in the LAMP System, and several studies
have been carried out in which green-water
calculations for different ships in a varied range of
sea conditions have yielded satisfactory results.
At every time step, LAMP calculates the
relative motion at the deck edge and passes it, along
with the rigid-body ship motions, to the green-water
calculation supervisor. This supervisor updates the
green-water calculation and returns the pressure
distribution over the deck plus any boundary forces.
The integrated pressure and boundary forces are
added to the hydrodynamic and other forces on the
right hand side of the equations of motions, which are
then integrated to get the ship new position. The
deck pressure and boundary forces are also included
in LAMP's loads calculations.
When setting up an integrated LAMP-green-
water problem, a variety of boundary conditions can
be defined over the green-water computational
domain. For example in Figure 8, a superstructure
comprising a central deckhouse and a full width
bulkhead are modeled using "infinitely high" walls
(i.e. always taller than the green-water maximum
level). Also, bulwarks have been modeled on both
sides of the ship. These bulwarks were set to have a
rather small height above the deck. The ship was set
to move in head-storm conditions. In the second and
third slides (slides b and c) it can be appreciated how
9
OCR for page 422
the water pours over the rim of the bow bulwarks. In
the last three slides (slides c-e), it can also be seen
that the boundary conditions defining the
superstructure effectively isolate the interior of the
superstructure from the incoming water.
~2 0-04
(a) 0.04 y
MY
~0
(C) 0.04 y
2
MY
~2 0
~ n no
, .
0.04,`
~ 0.02
z
MY
non
~-0.02 0 —.02 0-04
(e) 0.04 y
Figure 8: Shallow-water on deck with different
boundary conditions. Head-seas conditions.
2 The effect of shallow water on the motion
of ships has been an important part of the research
done with this numerical tool. In Figure 9 parts a and
b, the effect of green-water on deck is shown for a
CG47 cruiser sailing on regular and head-storm seas,
respectively. The wave height of the regular-sea
conditions is 14.32 meters, whereas the significant
wave height of the head-storm conditions is 9.75
meters. In both cases the ship speed is 10 knots. In
this figure, the results from the current finite-volume
model are shown together with the results from the
green-water semi-empirical model, and the effect that
a mere hydrostatic force would produce if the water
completely covered the deck. It can be observed that
all three models have the tendency to reduce the
vertical bending moment. This is the expected
behavior since the net downward forces due to the
10
OCR for page 423
green water partially offset the large hydrostatic
restoring moment generated by the bow
submergence.
1 .0E-04
8.0E-05
a)
6.0E-05
4.0E-05
Q) 2.0E-05
._
I) O.OE+OO
m -ME 05
~ ~.OE-05
Cat
-6.0E-05
a)
> -8.0E-05
-1 .OE-O'
O _
No Green Water Effects
Semi-empirical
—Finite-volume
—Hydrostatic
(a)
8.0E-05 -
6.0E-05-
~ 4.0E-05-
o
2.0E-05-
. - O.OE+OO-
a' -2.0E-05 -
m
<15 ~.OE-05-
: -6.0E-05 -
.OE-05-
1 no no -
No Green Water Effects
Semi-empirical
Finite-volume
Hydrostatic
(b)
Time ~
Figure 9: Shallow-water effect on the bending
moment of a CG47 cruiser ship (a) in regular sea and
(b) in a head storm sea. The bending moment is non-
dimensionalized by p g L4 p .
For these cases, the semi-empirical model was run
ignoring the flow vertical accelerations aHlOt. This
conservative restriction was used for comparisons
with the hydrostatic model. Thus the similar
magnitude of the maxima these two models exhibit.
In Figure 10 the effect of the green-water on
deck for the pitch motion is visualized. Though the
impact of the green water seems to be very small on
the ship pitch motion, the plots seem to indicate that
the green water does induce a lag and an increase in
the pitch amplitude, though both very small. This is
consistent with the increase of the pitch moment of
inertia of the ship induced by the extra water on deck.
2.0E-O, -
1.5E-01
1.0E-01
_%
~ 5.0E-02
-
° O.OE+OO
Q
-5.0E-02
-1.0E-01
-1.5E-01
0.0
No Green Water Effects
Semi-empirical
—Finit~volume
—Hydrostatic.
5.0 1~.0 15.0
Time (see)
] f
~~ V
Figure 10: Shallow-water effect on the pitch motion
15 of a CG47 cruiser ship in a head storm sea.
Finally, Figures 11 and 12 show results from
green-water-on-deck calculations for a large
(LBP=30O meters) modern containership. Figure 11
part a depicts the ship motion, incident wave surface,
and computed green-water elevations for the ship in
large, regular head seas. The wave height is 11.0
meters, the wavelength is 300 meters, and the ship
speed is 20 knots. The green-water surface is above
the incident wave surface because of the significant
predicted "pile-up" created by the ship's extreme
bow flare. Part b shows results for the same wave
but at a bow-quartering condition. The shape of the
green-water surface looks consistent with the
corresponding sea condition for each case. Note the
piling up of water at the front of the containers,
which is treated as a wall in the green-water
calculation.
11
OCR for page 424
(a)
(b)
Part c of the same figure depicts a head-sea
situation in which the combination of a large seaway,
the extensive bow and stern flare, and the ship's
dynamic properties result in a very large roll response
through a phenomena known as parametric rolling.
This takes place in large waves when the principle
wave encounter period is approximately one half of
the roll natural period. For this case, the wave
encounter period is 10.4 seconds, which corresponds
to a wave period of 13.9 seconds, whereas the roll
natural period is 22 seconds. The nonlinear coupling
between the heave, pitch, and roll degrees of freedom
channels a kinetic energy transfer from the pitch to
the roll mode.
An advantage of the current approach is that
it includes a direct calculation of the dynamic
pressure distribution over the deck. Figure 12 shows
the instantaneous deck pressure at one time step of
the regular head sea case shown in part a above. The
calculation of the deck pressure allows the approach
to be used for evaluating deck and hatch cover loads
directly or by using the pressure to create load data
sets for detailed structural analysis.
.. _ ~
. ~ ~
9.0909
8.4848
7.8788
7.2727
6.6667
6.0606
t 5.4545
4.8485
4.2424
3.6364
3.0303
2.4242
1.8182
1.2121
0.6061
0.0000
(c)
Figure 11: Water on deck for modern container ship:
(a) Large regular head seas
(b) Large regular oblique seas
(c) Parametric rolling condition
Figure 12: Pressure distribution on deck due to green
water. The pressure units are kN/m2.
12
OCR for page 425
FINAL REMARKS
The numerical model presented in this paper has
proven an effective tool to compute green-water
occurrences with shallow-water assumptions, for
static or moving platforms. The method has been
validated with theoretical solutions and experimental
results. Several applications have been tested using
the LAMP System as a motion and hydrodynamic
platform. The outcome of these applications
indicates that the technique is very suitable for
computing shallow-water-on-deck situations for a
variety of sea conditions, ship types and motions, and
different deck layouts. The calculation is robust and
accurate, with a reasonable computational effort. The
approach demands a fairly simple grid structure, and
offers a broad flexibility in defining different types of
boundary conditions.
ACKNOWLEDGEMENTS
The development of the LAMP System has been
supported by the U.S. Navy, the Defense Advanced
Research Projects Agency (DARPA), the U.S. Coast
Guard, the American Bureau of Shipping (ABS), and
SAIC. The green-water development has been
supported by the Office of Naval Research (ONR)
under program manager Dr. Patrick Purtell, by the
US Coast Guard under the Program manager Mr.
Peter Minnick, and by ABS under program manager
Dr. Yung Shin.
REFERNCES
Buchner, B., "On the Impact of Green Water Loading
on Ship and Offshore Unit Design,'' in Proceedings
of the Sixth Symposium on Practical Design of Ships
and Mobile Units, September 17-22. 1995. on.1.430-
1.443.
, ~~
Glimm, J., "Solutions in the Large for Nonlinear
Hyperbolic Systems of Equations," Communications
on Pure and Applied Mathematics, Vol. 18, 1965,
pp.697-715.
Lin, W.M., and Yue, D.K.P., "Numerical Solutions
for Large-Amplitude Ship Motions in the Time-
Domain," in Proceedings of the Eighteenth
Symposium of Naval Hydrodynamics, The
University of Michigan, U.S.A, 1990.
Lin, W.M., and Yue, D.K.P., "Time-Domain
Analysis for Floating Bodies in Mild-Slope Waves of
Large Amplitude," in Proceedings of the Eighth
International Workshop on Water Waves and
Floating Bodies, Newfoundland, Canada, 1993.
Lin, W.M., Meinhold, M., Salvesen, N., and Yue,
D.K.P., "Large-Amplitude Ship Motions and Wave
Loads for Ship Design," in Proceedings of the
Twentieth Symposium of Naval Hydrodynamics The
University of California, Santa Barbara, U.S.A.,
1994.
Lin, W.M., Zhang, S., Weems, K., and Yue, D.K.P.,
"A Mixed Source Formulation for Nonlinear Ship-
Motion and Wave-Load Simulations," in Proceedings
of the Seventh International Conference on
Numerical Ship Hydrodynamics, Nantes, France,
1999.
Stoker, J.J., "Water Waves," Pure and Applied
Mathematics, Vol. 9, The Mathematical Theory and
Applications, Institute of Mathematical Sciences,
New York University, U. S. A., 1 95 7, pp. 29 1-3 14.
Weems, K., Zhang, S., Lin, W.M., Shin, Y.S, and
Bennett, J., "Structural Dynamic Loadings Due to
Impact and Whipping," Proceedings of the Seventh
International Symposium on Practical Design of
Ships and Mobile Units, The Hague, The
Netherlands, 1998.
Zhou, Z.Q., De Kat, J.O., and Buchner, B. "A
Nonlinear 3-D Approach to Simulate Green Water
Dynamics on Deck", in Proceedings of 7th Numerical
Simulation Hydrodynamics, 1999.
13
OCR for page 426
DISCUSSION
Dr. Ole A. Hermundstad
MARINTEK, Norway
It is interesting to see a numerical method for
green water calculations being applied to ships
with forward speed.
The shallow water formulation is similar to that
applied in (1) and (2), but the present method
allows for triangular elements, which give some
more flexibility in the geometric modelling.
The authors use a 3D body-nonlinear method for
the global ship motion problem, while the quasi-
3D shallow water formulation is used for the
local green water domain. However, the paper
does not precisely describe the interaction
between the local and global domains along their
common border, namely the deck edge. The
water flow on the deck is strongly dependent on
the conditions at the boundaries. It is not clear
from the paper if the method assumes a nonzero
horizontal flow across the bulwark or not. From
the transverse waves on the green water surface
in Figure 8 it seems that the horizontal velocity is
zero and that the boundary conditions are similar
to those of a dam-breaking problem with time-
dependent reservoir height. If this is the case the
method will generally underestimate the velocity
of the flow on the deck and be unconservative if
it were to form input for load-calculations on
superstructure and deck-mounted equipment. If
the authors really use a nonzero horizontal
velocity it would be interesting to know how it is
obtained from the LAMP results.
Another issue that requires some more attention
is the flow at obstacles, such as the walls of the
superstructure. The shallow water method will
give a pile-up of water when the flow meets a
wall, but one of the fundamental assumptions of
the formulation is obviously violated in that case.
I.e. the vertical velocity component is no longer
small compared to the horizontal components.
Another model is therefore needed to predict
slamming pressures on the wall.
The method, as presented in the paper, may be
used to predict pressures on decks and hatch
covers as well as hull girder loads caused by
green water. However, it seems to need further
development before it can be used to calculate
slamming loads on superstructure and containers.
(1) Hellan, 0., Hermundstad, O.A. and
Stansberg, C.T. "Designing for wave impact on
bow and deck structures" Proceedings of the
1 1th ISOPE Conference, Stavanger, Norway,
2001.
(2) Stansberg, C.T., Hellan, 0., Hoff, J.R. and
Moe, V. "Green sea and water impact on FPSO:
Numerical predictions validated against model
tests". Proceedings of the 21St OMAE
Conference, Oslo, Norway, 2002.
AUTHORS' REPLY
We would like to thank Dr. Hermundstad for his
comments. The papers quoted in the discussion
have a valuable contribution to the green water
study. In our current model, a fully nonlinear set
of differential equations describing the shallow-
water phenomena is solved, with a special care
for different types of boundary conditions and
obstacles. The methodology uses a finite-
volume approach that solves the equations in the
time domain. Special cares were taken to satisfy
numerical stability requirements. From the
content of the aforementioned papers, it seems
that the methodologies used in their work
involve elaborate semi-empirical strategies that
do not share much in common with our physics-
based approach.
The green water model does take information of
the environment as computed by LAMP. To that
effect, LAMP provides the velocity and wave
elevation with respect to the deck as inputs to the
green-water element at the deck edge. This
information is passed to the green-water solver.
The type of information used depends on the
types of cases, such as the water flowing in or
out of the deck and its being higher or lower than
the water deck elevation computed fin LAMP on
the deck boundary.
Regarding the accuracy of the method when
green water moves over obstacles, it is true that
if an obstacle were relatively high with respect to
the local water elevation (on a given point of the
green-water grid) the substantial vertical
component of the fluid velocity at this location
would be ignored. This limitation is based on
the model utilized, which solves for the shallow-
water assumptions. To capture the vertical flow
components even a jet, a fully 3D calculation
would be needed. A 3D capability is currently
under development at SAIC.
OCR for page 427
In principle, we do not see any particular
restriction on computing impact loads, provided
the shallow-water assumptions are still valid.
For cases involving large vertical flow velocity
or formation of jet, the current approach cannot
be used to calculate the impact load accurately.
DISCUSSION
Allen Engle, Naval Surface Warfare Center
Carderock, USA
A green water prediction capability is a welcome
addition to the LAMP suite of programs. The
need to include such affects within a motions and
loads prediction methodology are brought out in
the paper. However, I might add that water on
deck is an issue not only for the conditions
mentioned, but must also be considered as part of
a rigorous treatment of the capsize problem,
where the combination of a ship operating in
stern quartering seas with water shipping on deck
can exacerbate an already precarious situation.
With this in mind I have the following questions
for the authors,
1. Does the current formulation allow for the
user to easily simulate the shipping of water on
deck for all headings?
2. At the 22nd Symposium on Naval
Hydrodynamics, a paper presented by Huang et.
al. incorporated a correction factor to account for
viscous and other effects not explicitly included
as part of the water shipping on deck problem.
For the treatment at hand it is stated that viscous
effects are ignored. Could the authors comment
on the degree to which the physics of the water
shipping problem dictates the need to account for
viscous effects?
3. Have the authors performed any
convergence studies to identify an appropriate
level of grid size for simulating the green water
problem? and are there any potential problems in
using different grid sizes in defining the hull and
the deck regions?
AUTHORS' REPLY
We would like to thank Mr. Engle for his
valuable discussions also.
1. Yes, it does! The current green water
approach allows for an easy definition of all
types of boundary conditions in addition to
obstacles inside the grid. LAMP takes care of
automatically providing the needed data for
different ship headings and sea conditions.
Nevertheless, the green-water model
implementation can work in a standalone mode,
in which case, through an appropriate input file,
the user can specify any arbitrary motion of the
grid and boundary characteristics, both in terms
of obstacles and water elevations, and regarding
incoming or exiting water flow.
2. The model currently has a methodology to
estimate viscous effects using a Von Karman
type of analysis, which will be the subject of
fixture research and validation.
3. Convergence and stability is one of the
major accomplishments of this work. From the
beginning of this development, effort was
directed to study the stability conditions of the
numerical method. As a result, the program
automatically computes the (maximum) time
step below which numerical stability is ensured.
It can also automatically generate its own grid
satisfying convergence requirements. Research
was also conducted to ensure the stability of the
solution beyond shallow-water assumptions.
The reason for this is that whereas for a given
ship and sea conditions it would be expected that
the resulting ship motion and interaction with the
environment (computed by LAMP) would
generate shallow-water conditions during most
part of the calculation, there could be particular
instances (during the calculation) when the
motion of the ship or some unexpectedly high
incidence waves could induce water elevations
on deck that could go beyond shallow-water
assumptions. In such cases the solution will still
remain stable, though at the expected expense of
accuracy. Note that a single run may encounter
several green-water occurrences. Even if in
some particular case, shallow-water assumptions
may be compromised, the solution would always
remain stable and it will still be accurate for all
the other cases. This model has been involved in
the study of the influence of water on deck on
ship motions. This is already available in the
work done by Vadim et al (1) quoted below/
(1) Vadim, B.L., Liut, D.A., Weems, K.M.,
Young, S.Y., "Non-Linear Ship-Roll Simulation
with Water-On-Deck," Proceedings of the 2002
Stability Workshop, Web Institute, New York,
Glen Cover, October 13-16, 2002.
Representative terms from entire chapter:
ship motions
