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OCR for page 376
Application of a 3-D Time Domain Panel Method
to Ship Seakeenin~ Problems
H. Yasukawa (Nagasaki Experimental Tank, Mitsubishi Heavy Industries, Japan)
ABSTRACT
This paper describes outline of a 3-D time domain
panel method for ship seakeeping. The method in-
cludes nonlinearity of hull and free-surface bound-
ary conditions. To reduce the computational time
and the memory size, we employ 2 types of the free-
surface meshes: ship fitted mesh for near field and
space fixed regular mesh for far field. This also accel-
erates the computation of free-surface influence func-
tion. Hydrodynamic forces, ship motions in waves
and wave pressure on the hull surface are computed
for several ships. The results are compared with ex-
periments and the calculated results bar strip method.
The present method is validated for the hydrody-
namic forces, ship motions and wave pressures, and
they are much better predicted than by strip method.
1. Introduction
The most commonly used tools to determine seakeep-
ing performances are based on strip theory. The
. ~ . . . . ~ . . ~
such as actual container ships.
In addition, Iwashita pointed out that the fre-
quency domain approaches do not work so well in
the region where At_ Uw/g) is smaller than 0.5[4][5],
where U denotes ship speed, ~ the encounter fre-
quency and 9 the acceleration gravity. This means
that RAG is not obtained for a wide variety of wave
direction. That may be serious problem at the stage
of short-term prediction of the response.
We will present here a time domain 3-D panel
method including nonlinearity of hull and free-surface
boundary conditions for ship seakeeping. Time do-
main approach has no limitation about revalue and
is applicable to many ship and offshore problems
with large amplitude motions. Hydrodynamic force
and ship motion analyses using the similar nonlinear
time domain method have been made, for instance,
by Maskew[6] [7] [8], Beck et al. [9] [10] and Scorpio et
al.[11]. LAMP code is also well known for large am-
plitude ship motion analysis method in wavest12] [13~.
Generally, the time domain code requires consid-
str~p metnoct approach Is cheap, last, and tor most erable computational time for obtaining significant
cases also quite accurate. However, strip methods do solution. We employed some numerical techniques to
not perform so well for high-speed ships, ships with reduce the computational time with keeping the ac-
strong flare, and generally for low encounter frequen- curacy and the robustness. Outline of the method
cies which typically occur in following seas. will be presented in this paper. The present method
As new calculation methods to overcome the strip can be regarded as an extension of the method for un-
methods, frequency domain 3-D Rankine panel moth- steady wash analysis of a high speed vessel[14]. After
oafs were proposed. For a recent survey of Rankine that, we put a scheme of ship motion computation to
singularity methods for forward-speed seakeeping, we the original method[15]. To validate the method, hy-
refer to [1] and [2]. The methods include fully 3-D drodynamic forces, ship motions in waves and wave
effects of the flow and forward speed effects where pressure on the hull surface are computed for several
they are not taken into account in the strip method, ships. The results are compared with experiments
and we confirmed that local pressures, especially for and the calculated results by strip methodt16]. We
shorter waves, are much better predicted than by found that hydrodynamic forces, ship motions end lo-
strip method[3]. The frequency domain methods deal cal pressures are much better predicted than by strip
with linear ship motion problem based on the steady method. The present method is promising as a new
flow. Therefore, the methods may not be applicable ship design tool for more accurate prediction of sea-
to large amplitude motion for ships with strong flare keeping performances.
OCR for page 377
2. Basic Equations
2.1 Boundary value problem
Let us consider a ship advancing in a towing tank
as shown in Fig.1. The ship moves with speed U(t)
which varies as the function of time t. The coordinate
system fixed in the space is employed. The x-axis is
defined as direction from ship stern to the bow, y-
axis to port and z-axis vertically upward. The x—
y plane coincides with the still water surface. The
motion displacements with respect to surge, sway and
heave in the fixed coordinate system are defined as
(1,(2,T,3 respectively, and the vector A. Euler angles
with respect to roll, pitch and yaw are represented as
A, §, ~ respectively, and the vector Q. Incident waves
are generated by movement of flap angle /9 of wave
maker attached to tank wall. Deep water is assumed.
.~ ·~` ~ W.
Fig.1: Coordinate system
Incident Wave where
The perturbation velocity potential due to ship
moving in the tank is defined as oryx, y, z, t). Then, fib
has to fulfill the following boundary conditions:
,9t 9; 45~2Z;—2 (Vo) on z = o (1)
0L LIZ + ~Z2 ~—~ C Zinc—, 3 J) on z = 0 (2)
,)¢ = (v +w x r) r' on SH (3)
655 = vo on Sw
(4)
where V = (~/0x, 0/0y, 0/0z). ~ denotes wave ele-
vation. SH and Sw mean hull and tank wall surfaces
respectively.
Eqs.(1) and (2) are dynamic and kinematic free-
surface conditions respectively. For simplicity, free-
surface conditions including the 2nd order terms with
respect to o and ~ obtained by Taylor expansion at
z = 0 are employedt144.
Eq.(3) is hull surface condition, and has to be sat-
isfied on actual wetted surface SH. In eq.(3), r de-
notes coordinate of hull surface position, n the out-
ward normal vector of the hull. Further, v = (a, v, w)
means the ship velocity vector defined in the coor-
dinate system fixed to ship, and co = (p,q,r) the
angular velocity vector. It should be noted that v in-
cludes the component of ship velocity U(t). Relations
between v and A, and between w and Q are written
as follows:
v= LE(Q)~(,
a) = tH(Q)]Q
Here, means d/dt. The matrix E and matrix H are
written as:
Em, A, ¢) = feijI (7)
eat = cos ~ cos
em = cos ~ sin
el3 =—sin
e2l = sin ~ sin ~ cos ~—cos ~ sin
e22 = sin ~ sin ~ sin ~ + cos ~ cos
e23 = sin ~ cos
em = cos ~ sin ~ cos ~ + sin ~ sin
e32 = cos ~ sin ~ sin ~—sin ~ cos
e33 = cos ~ cos
0 —sin ~
H(~, 8, A) = O cos ~ cos ~ sin ~ (8,
—sin ~ cos ~ cos ~
Eq.(4) is boundary condition of tank wall surface,
and v0 means normal velocity on Sw. When gener-
ating the waves in the tank, we have to give a proper
value to vo. On the tank wall without wave maker,
we set v0 = 0. The detailed explanation is described
below.
The velocity potential ~ is represented using source
strength ~ as follows:
0(P) = /7 ~(Q~G(P;QjdS (9)
SH+SF+SW
G(P; Q) = )2 + ( _ yi)2 + (z _ zI)2 ( ~
OCR for page 378
P = (x, y, z) is filed point, and Q = (xl, Y1, zl) the respectively. These vectors and matrix are expressed
singular point. SF denotes the free-surface position as follows:
(z = 0~. Substituting eq.(9) to eqs.(3) and (4), the
following equations are obtained:
I2X O IF
TIJ= O IYY O (17)
!!SH+SF+SW (Q) Art dS = fH(P) (1l) IF O IZZ
where
I (P) _ ,{ (v +w x r) n for P on SH
v0 for P on Sw
(12)
Eq.(11) represents the boundary conditions on the
hull and tank wall surfaces.
2.2 Hydrodynamic forces and motion equa-
tions
Pressure on the ship hull surface is given from
Bernoulli's equation as:
p/p _ _ 0° _ ~ (V¢,)2 (13)
where p is density of water.
Hydrodynamic forces acting on the ship is obtained
by integrating p over the hull surface. Here, we define
that subscript 1, 2 and 3 denotes directions of surge,
sway and heave respectively, and subscript 4, 5 and
6 the directions of roll, pitch and yaw respectively.
Then, the hydrodynamic force/moment with respect
to i-direction Fit is written as:
/ 9 sin ~
FG = m —gcos~sin~ (18)
-gcos~cos~
~ qw—rv
FI = m | ru—pw (19)
\ pv—qu
~ (Izz—Iyy)qr—I~zpq
MI = (It—Izz)rp + I=Z(p2—r2) (20)
(IYY—Ix=)pq + I=zqr
2.3 Wave-maker and numerical absorbing
beach
Incident waves are generated by numerical wave mak-
ers (vertical flaps) on tank wall. Here we deal with
only regular waves. The flap starts to move from
vertical position, and the angle is given as follows:
O(t) = Go sin at (21)
where (3(t) denotes the actual flap angle, Go the am-
plitude of flap angle and ~ the frequency of flap mo-
tion.
Then, v0 in eq.(4) is represented ast174:
/7su [9 ,'~ + 2 (Ho) ] rlidS (~4) v0 = (d1 + Z) do (22)
where (rl~,~2,n3) — al, and (n4,ns,n6) _ r x n.
Solving the boundary value problem mentioned above
under proper initial conditions, velocity potential can
be obtained through the source strength. Calculating
velocity components and pressure on the ship hull
surface from the potential, unsteady hydrodynamic
forces are obtained from eq.(14~.
Motion equations of the ship are expressed as:
me + F] = FG + F (15)
LI] ~ + M] = M (16)
Here m denotes the ship's mass and tI] the matrix
with respect to moment of inertia of the hull. F and
M are vectors of hydrodynamic force and moment
respectively, defined as F = (F1,F2,F3) and M =
(F4, F5, F6). FG denotes force vector due to the hull
gravity force. F] and M] mean vectors of force and
moment due to centrifugal force acting on the ship
where do is the height of flap part (see Fig.1). For
simplicity, here, nonlinear boundary condition is not
adopted on the flap surface. Only v0 is given at ver-
tical fixed flap position in the computations.
As numerical absorbing beach technique, the
method proposed by Cointe et al. (184 where a
proper damping is added to free-surface conditions, is
adopted at only face-side wall against the wave mak-
ers. On tank-side, we do not impose any numerical
beach and allow of wave reflections.
3. Numerical Procedure
3.1 Flow of numerical computation
Numerical flow for calculating ship motions and hy-
drodynamic forces is as follows.
1. Accelerations of ship motions (( ,Q ), and
time derivatives of wave elevation and velocity
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Representative terms from entire chapter:
container ship
potential on free-surface ((k+l, Ok+ respec- 9. Hydrodynamic forces acting on hull are calcu-
tively at (k + 1~-th time step are assumed using lated using eq.~14~. The ¢~ term appears in
the values at k-th step. Here suffix t represents eq.~13) is evaluated as
time derivative.
2. According to Newmark's ~ method, ship motion
velocities and the displacements at (k+1~-th step 10.
are estimated by:
Ok+ Ok ^t (
The an and AH can be obtained by solving eqs.~32)
and (33) alternately until the solution is reached to
the convergence. To quickly solve the matrices tA]
and tD], we consider the use of SOR method which
is fast and simple.
Diagonal term of iD] is theoretically larger than
other terms, so we can solve this matrix easily using
SOR method. For tA], on the contrary, the diagonal
term is not always larger. This means that we do
not always obtain the solution. To certainly get the
solution, we place restrictions on free-surface panel
geometry with rectangular panels of the same size.
In this case, slightly larger diagonal can be achieved
in [A], and can solve the matrix by SOR method.
For faster computations, suitable relaxation factor for
each eqs.~32) and (33) should be chosen. For solving
eqs.~32), under-relaxation factor is effective.
3.3 Treatment of Resurface panels
Actually, we can not keep the same geometry and the
same size of the free-surface panels in the proximity
of ship hull because the hull has certain breadth. To
settle the problem, we employ the following treat-
ment:
.
2 different free-surface panels are used as shown
in Fig.2: rectangular panels fixed to the tank
(tank fixed panels) and arbitrary shaped panels
moving with the ship (moving panels).
· In the tank fixed panels, ignoring the presence
of the ship, regular rectangular panels are ar-
ranged. The free-surface panels are often ar-
ranged inside the hull on z = 0 plane.
· The moving panels are restricted within a small
region outside the ship's water line. The panel
geometry automatically changes so as to match
with boundary of the fixed free-surface panels.
The moving panels are treated as additional un-
known panels with source strength aF. Then,
the panels within the moving panel region are
double arranged. To avoid the trouble comes
from the double arrangement, we impose the
condition where source strength of the fixed pan-
els becomes zerot204.
In this treatment, we have to solve a new matrix with
respect to aF. However, increment of unknown vari-
ables is about several hundred, and the small-sized
matrix can be quickly solved by Gaussian elimina-
tion. That is no big difficulty.
The free-surface panel arrangement using regular
rectangular panels is also useful for efficient construc-
tion of matrix tA]. In this case, evaluation of influ-
,Tank Fixed F/S Panels
Fig.2: Treatment of free-surface panels
ence function with respect to (A] depends on only dis-
tance between 2 panels because the panel geometry
is all the same. Then, tA] can be constructed using
the influence functions induced by a row of panels
on the whole free-surface panels. We do not need
so large memory size for matrix [A] when employing
this treatment. Remarkable saving of the memory
size was achieved.
3.4 Ship's acceleration from the rest
In the present calculations, not forward thrust force
is given to surge force term but forward velocity is
forced at the motion equation of surge. This is corre-
sponding to simulating not self-propulsion test, but
resistance test in waves. Then, ship speed from the
rest U(t) is assumed to be given as follows:
~ Uo j6(t/to)5 - 15(t/to)4 + l°(t/to)3)
U(t) = ~ for t < to
Uo for t ~ to
(34)
Here, Uo denotes the target steady speed and to the
acceleration time until steady state. As to, we em-
ploy value of to~/377=15.7 to avoid the long period
oscillation of hydrodynamic forces appears in case of
rapid accelerationt214.
In acceleration period until reaching to the steady
ship speed, large damping is artificially added to the
force term of the motion equation. This roughly con-
strains the ship motion in the computations like a
cramp in the experiment. The treatment is useful
for reducing the computational time until reaching
to steady state of the ship motion.
4. Computations of Hydrodynamic
Forces
4.1 A modified Wigley hull with longitudinal
motions
First, the present method is validated for hydrody-
namic forces with respect to heave and pitch. The hy-
drodynamic forces acting on the hull with forced os-
cillation (radiation forces) and the restrained hull ad-
vancing in regular head waves (wave exciting forces)
are evaluated for a modified Wigley hull. The hull is
a mathematical form expressed as follows:
A' = ¢1—X,2~1 _ z'2y
where x' = x/(L/2), y' = y/(B/2), Z = Z/d
with L/B = 10.0 and B/d =1.6, where L, B and d de-
note the ship length, breadth and draft respectively.
Fig.3 shows panel arrangement of the hull surface.
Number of the hull panels is about 800 for a half body.
We employ the free-surface panel region with 12.5L
for the length and 1L for the half width. The number
of the free-surface panels is 5,000~= 250 x 20~. Cal-
culated results are compared with the experiments
conducted in Delft University of Technologyt224.
Oscillation frequency and V the ship's volume. The
time until t>/~=15.7 is acceleration zone and the
heave motion is given so as to gradually increase.
Where the non-dimensional time is up to 20, the
calculated forces look stable and regular sinusoidal
forces are obtained. Time averaged values of CF~
and CF3 are negative. This means that steady resis-
tance and sinkage forces are acting on the hull. The
averaged value of CF5 is almost zero.
By analyzing the time histories of hydrodynamic
forces, added mass and damping are obtained. Fig.5
shows comparison of added mass and damping coeffi-
cients. The air and bit mean added mass and damp-
ing respectively, with respect to i-th mode induced
by j-th motion, where i/j=3 means heave and i/j=5
the pitch. Horizontal axis in the figures means non-
~ .o _
~ Fn=0.3 /
~ . ~
o.o~ 10
of . . . . .
modified Wialev
-
Forced Heave Mode
w{(Ug)=3.86:
. . . i -
30t T(g/L)
, . . . . , . .]
Fig.3: Panel arrangement of modified Wigley hull.
This figure includes hull panels above the still water
surface.
Fig.4 shows time histories of hydrodynamic forces
excluding restoring force component for forced heave
with w~/~75=3.86. Froude number based on the
ship length En is 0.3. Amplitudes of the oscillation
(As, A5) are assumed to be 0.01L for heave and pitch
respectively. The hydrodynamic forces induced by
i-th motion are normalized as:
o.o, ,
J
- cO o
-0.01
o
t T(g/L)
! ~
t T(g/L)
u.u~ . . I ~ . . . ~ . . . . ~
002 ~~.
-0.04
0 10 20 30
10 20 tT(g/L)
F F3 F5 Fig.4: Computed time histories of ship speed
CFl'CF3= pVAc,,2' CF5= pVA w2l (36) (U/Uo), surge force(CFl), heave force(CF3), pitch
moment(CF5) and forced heave (63/L)
where F1,F3 and F5 mean surge force, heave force
and pitch moment acting the hull respectively, ~ the
a33lp V
0.80
06 -O. C -I--`
0.20 O ECal. (Delict)
- Strip Method o{(U5)
0.00 2 3 i 5
a351p VL
0.04
0.02 .. . . . . . . . ... . . . ..
-0 02 ~ - ~ O _ -
-0.06 ~ '' ...... -
~0.10 2 3 4 5w[(U' ) v.vv 2 3 4 5~(Ug)
a53/pVL b /pV~L
0.1C n53n-
0.0E
0.06
0.04
0.0z
O.OC
.0~
~ no
b~3/p VEIN
1 .00
0.80
0.60
0.40 _ 6 .
0.20 . ~ ---
0.00 2 3 4 ~ w{(U 1)
b351 p Vm L
0.10
0.08 - Strip Method
0.06
0.04
0.02
rid on
, . .
O Cal.
....... · Exp.(Delit)
Strip Method
3
u-
~. · Exp.(Delit)
· ".,,
,2 . ~ .. _
hA I , , , ,
0.00 _
-0.02
-0.04
-0.06
-0.08
~ 1^
_ ~ ~ .. ~
1 o cad. 1
............... · Exp.(Delit)
. ; Strip Method _
~ 2 3 4 5~1~(U~ it u 2 3 4 5w[(U~ I)
asgp VL
0.04
'D
0.03
non
_ . _
0.01 _
w[(Ug)
b551 p VL2w
. 002 'a'
001~ 849
0.00 2 3 4 5 0OO 2 3 4 5m{(Ug
Fig.5: Comparison of added mass and damping coef-
ficients for a modified Wigley hull (F~=0.3)
dimensional frequency of forced oscillation. Agree-
ment with experiments is satisfactory as a whole. Es-
pecially, calculated results of a33, b33, a35, a53 and
b55 show good agreement with experiments. How-
ever, small discrepancy is observed in terms of b35,
b53 and a55. On the other hand, calculation accuracy
by strip method is clearly worse than by the present
method.
Fig.6 shows comparison of wave exciting forces
Ei~i=3:heave force, ~=5:pitch moment). As the flap
angle of numerical wave maker, O0=0.4deg were se-
lected. Then, amplitude of incident waves ha was
about 0.0024-0.0030L. Calculated amplitude and
phase angle of heave force and pitch moment agree
well with experiments as a whole. In the region where
)/L is smaller than 1.0, however, pitch moment is
larger predicted. The reason may be due to lack
of free-surface panel density to capture the shorter
waves. The calculated results by strip method agree
well with experiments.
15.C _
~10.0 .
Q
5.0
0.q
modified WiqleY
. . . . . . . . . . . . . . . .
modified WiqleY
.L . . . · ' ! ' 1
~ ,0'~.'~- ..~.
L°f-
...
,....
---.. -]
| O Cal. |
· Exp.(Delh)
- - Strip Method
8.5 i 1;5 A /L 2
Fig.6: Comparison of wave exciting forces for a mod-
ified Wigley hull (F~=0.3)
4.2 A container ship with lateral motions
Next, the present method is validated for hydrody-
namic forces on the lateral motions such as sway, roll
and yaw. Calculations are carried out for a container
ship with Lpp/B = 6.72 and B/d = 2.74 as shown in
Table 1, and are compared with experimental data.
To obtain the hydrodynamic force coefficients on
the lateral motions, 3 forced oscillation tests such
as pure swaying, pure rolling and pure yawing tests
were conducted at towing tank, Nagasaki R & D cen-
ter, MHIt234. The size of the towing tank is 120m
in length, 6.1m in width and 3.65m in water depth.
The ship model was 3.0m in length, and has no bilge
keels and no rudder. Froude number based on Lpp
was 0.20. The amplitudes of forced oscillation in
pure swaying, pure rolling and pure yawing test were
0.0448B, 7.5deg and 2deg respectively. The forced
motions were given to center of gravity of the ship
model. The height of center of gravity was adjusted
so as to coincide with still water level.
Fig.7 shows panel arrangement of the hull. 990
panels for ship hull, 8,000~= 200 x 40) panels for free-
surface and 2,880 panels for tank walls were used in
the computations. The size of the numerical towing
tank (free-surface panel region) is 8Lpp in length and
2.03Lpp in width. The width and the water depth
was coincided with those of the actual towing tank.
Table 1: Principal dimensions of a container ship
Ship length Lpp
Breadth B
draft d
Displacement V
_ Model
3.000m
0.446m
l 0.163m
_ 121.77kg _
Ship l
175.0m
26.0m
9.5m
24,776ton
Fig.7: Panel arrangement of a container ship. This
figure includes hull panels above the still water sur-
face.
1
As an example, wave patterns generated by the
ship with pure swaying motion are shown in Fig.8.
Non-dimensional frequency is A' = 0.683 where A' _
w~,/B/~2g), and ~ = 0.501. Solid line means posi-
tive and dotted line the negative in wave elevations.
We see that the wave component generated by forced
sway motion laterally moves and reflects at the tank-
side. The unsymmetrical waves periodically appear
and are superimposed over the steady wave compo-
nent.
Figs.9-11 show comparison of added mass and
damping coefficients on the lateral motions. The aij
Fig.8: Computed wave patterns around a container
ship with pure swaying motion (Fn=0.2, T=0.501'
~'=0.683~. The phase angles against the swaying mo-
tion are 0, 45, 90 and 135deg from the top.
and bij mean added mass and damping with respect
to i-th mode induced by j-th motion, where i/j=2
means sway, i/j=4 the roll and i/j=6 the yaw. They
are non-dimensionalized as follows:
a22 = a22/(pV),
a42 = a42/(pVB),
a62 = a62/(pVB),
a24 = a24/(pVB),
a44 = a44/(pVB2),
a64 = a64/(pVB2),
a26 = a26/(pVLpp),
a46 = a46/(pVL2p),
a66 = a66/(pVL2p),
b22 = b22 ~7~/(pV)
b42 = b42~/(pVB)
b62 = b62~7~/(pvB)
b24 = b24~7~/(PVB)
b44 = b44~7~/(pVB2)
b64 = b64~7~/(pVB2)
b26 = b26~7~/(pvLpp)
b46 = b46~7~/(pvL2p)
b66 = b66~7~/(PVL2p)
Diagonal terms for the pure sway mode such as
a22 and b22 are in good agreement with experiments,
although some discrepancy is observed in coupling
terms b42 and b62. Added mass coefficients with re-
spect to pure yawing motion are also in good agree-
ment with experiments. However the damping coef-
ficients in yawing motion are uniformly smaller than
those in experiments. This may be due to viscous flow
effect on the hydrodynamic forces acting on the hull.
The present method treats the ship hull as non-lifting
body. To improve the accuracy of yaw damping, we
should include the lifting body effect due to change
of attack angle of the hull against the flow. Bertram
et al. calculated the ship motions including the lift-
ing body effect in frequency domain Rankine panel
method t244. In the pure rolling mode, calculation
accuracy becomes worse than that in other modes
as over all tendencies. Diagonal terms such as a44
and b44 are not in good agreement with experiments.
This is also due to viscous flow effect on the hydrody-
namic forces. To accurately simulate the lateral ship
motions, we need some empirical correction for b44 as
usually employed in strip method.
The results by strip method do not agree with ex-
periments except a22. In some coupling terms, ten-
dency versus forced frequency is quite different from
the experiments. The calculated accuracy by strip
method is clearly worse than by the present method.
5. Ship Motions in Heacl Waves for S-
175 Container Ship
Next, the present method is validated for ship mo-
tions in regular head waves. Calculations are carried
out for S-175 container ship as shown in Table 2, and
are compared with experimental data. The experi-
ments were conducted at seakeeping & maneuvering
basin, Nagasaki R & D center, MHI. The size of the
tank is l90m in length, 30m in width and 3.5m in
a'22
2.0 . , i l '
1.0 ~ ~ · ·
· Exp.
. ~ Stnp Method
on _ I . . . I . i
~0 0.2 0.4 0.6 0.8 ~,~~ 1
biz
1.0 _ . , . , . , . ,
. O Cal.
· Exp. ~ ~; ,.
- - - Strip Method ~ .~
. . .
· . ..
' . , · .~..- :"""', . . .
_.~)( ) 0.2 0.4 0.6 0~8 w'
0.5
r. n
O.
0.c
~ 1
0.1
0.0
~ ~ ~ ·'
-01 . i . , . . . , . , _~' ~.
0 0.2 0.4 0.6 0.8 ~,.~ 1 0 0.2 0.4 0.6 0.8 ~,~
a,42 a'62
0 10 _ O 0 2 0
0.00 _ ~ ~ ~ ~ -0 2 _
. , ; --,--- , --- -0.4
.05 —`0 6
0 0.2 0.4 0.6 0.8 c`' 1 0 0.2 0.4 0.6 0.8 `~, 1
bl62
1.0 . , I ' '
. O O ,
O.5 . O
—u.u~ O.O , j . i
0 0.2 0.4 0.6 0.8 `,' 1 0 0.2 0.4 0.6 0.8 `o 1
U.U~
0.0d
o.or
~.o,
Fig.9: Added mass and damping coefficients in pure swaying mode (Fn=0.2' 62a/B=0.0448)
a'26 a'46 a'66
0.2 1 1 ,- , 0.000 · 1 1 1 1 0.20 ,, . 1 ~ ' ' ':
. · ~ ~ ~ -0.001 (~~ '; 0. 15 ..
- O : -0.002 _ .. 0.10 ~ :---O ~
) ~ O Cal. I . -0.003 . ................. ' 0.05 ~ ~ e ~.
· - StriP Method | - . :
_ , -0 004 . 1 . 1 . 1 . 1 . o on . 1 . 1 . 1 1
O 0.2 0.4 0.6 0.8 w~ 1 0 0.2 0.4 0.6 0.8 ~~~ 1 0 0.2 0.4 0.6 0.8 ~~~ 1
b,26 bl46
0.2 . 1 1 1 1 ~ 0.001
· ·3 ~
. - - - - ~ - 1 ° °°°t
0 ... ~ -0.
''''''-0-----°- -"'.
l t
bl66
1 010
1 0.08p | O Cal.
| 0.06~-| - - Stnp Method
1 0.o4t
1 o~o2t
I o.oot
1 -0.020 0.2 0.4 0.6 0.8 `~' 1
Fig.10: Added mass and damping coefficients in pure yawing mode (Fn=0.2, ~a=2de=1
at24
0.1 0
E °
0.05p - - -- - --- - ---- ----- - --- --- - -- --- -
~ ·-. O
0.00F · ~ O O
~ .; ; '1'~''''''''''','~""' _
-O. 0R
0 0.2 0.4 0.6 0.8 ~,~~ 1
bl24
0.03
0.02p
0.01 g --- -- --- -- -- --- - i--- ·--
o.oo~ -._~ !.~ .o
0 01 t
-002
0 0.2 0.4 0.6 0.8 `~'
at44
] 0.041
3 0.02~---
3 o.ooF......
3 o 02h
3 -o.o4k
1 ~ ~~~21
......... . o I 11
| · Exp. 14
| - - - Strip Method | l
- -u.ucO 0.2 0.4 0.6 0.8 <~' 1
bl44
0.006L
0.004L
] 0.002t
I ^ n~rl
al64
°-°t . I . I . I . I . ~
—0.1t - - C,
~.2t .-.
~ . 1 . 1 . 1 . 1 . ~
0 0.2 0.4 0.6 0.8 `~, 1
bt64
0.O5L I I ! I
~ ao ~
o.ooF -~ .;;;..;.~; e. ............1
~ _0.05: O cat .. ·-3]
t .. i
- u uou~ ~ ~~ - - - ~ -010i - --- ~ --- - - -'- i ~ ~
1 0 0.2 0.4 0.6 0.8 `~, 1 0 0.2 0.4 0.6 0.8 <~ 1
Fig.11: Added mass and damping coefficients in pure rolling mode (Fn=0.2, ~a=7.5deg)
Table 2: Principal dimensions of S-175 container ship
Ship length Lpp
Breadth B
draft d
Displacement V
Model
3.500m
0.508m
O.l90m
. 193.57kg
Ship
75.0m
25.4m
9.5m
24,801ton
Table 3: Region of numerical towing tank and the
number of free-surface panels in the computations
Froude number
Range of A/Lpp
Tank Length
Tank Width
Number of Panels
0.15
0.7-1.3 1.5-2.0
8Lpp 8Lpp
2Lpp 3lpp
5,000 5,000
Fig.12: Panel arrangement of S-175 container ship.
This figure includes hull panels above the still water
surface.
water depth. The ship model used in the test was
3.5m in length and the radius of gyration key was
0.25Lpp. The tests were carried out at 2 Froude
numbers Fn = 0.15 and 0.25. There is no restoring
term with respect to surge. For certainly measur-
ing the surge motion in the tank test, we artificially
added the restoring force to the ship model using coil
springs. In the calculations, we put the restoring
force to the motion equation so as to coincide with
the tank test condition.
Fig.12 shows panel arrangement of the hull used
in the computations. The number of the panels is
about 800 for a half body. Table 3 shows the region of
numerical towing tank and number of the panels used
in the computations. The length of the numerical
tank is 8Lpp for Fn = 0.15 and lOLpp for Fn = 0.25.
5,000~= 250 x 20) panels were used for free-surface.
This number may be insufficient for capturing shorter
waves less than A/L = 0.5. For relatively long waves,
we confirmed no problem in view of practical uset264.
Flap angle of the wave maker was selected as
OO=l.Odeg so as to coincide with the amplitude of
incident waves in the tank test. The calculated am-
plitude was about 2-3m in fullscale.
No numerical wave absorbing beach was used at
the tank-side and we allowed the wave reflection in
the computations. We checked the tank-side effect on
the ship motion in waves by referring to Kashiwagi
et al.~254. Based on their study, we made a diagram
showing the minimum wave length for disappearing
the tank-side effect versus tank width (see Appendix).
From the diagram, we selected the tank width disap-
peering the tank-side effect.
lOLpp
2lpp
5,000
S-175 Container Ship
~ n _ I I I I L I
~= ~1 ~ r- `0
20 30 40 50 60 t {(g/L'
a
~P
.005
Fn=0.15 A /L=1.0 3
-0.011 ~ ~ . , , 1
20 30 40 50 60 70 80
t {(g/L)
O,Oc
-
_ C
~D
_O.Oc
20 30 40 50 60 70 80
t {(g/L)
l
20 30 40 50 60 70 80
t {(g/L)
Fig.13: Computed time histories of ship speed
(U/UO), surge ((l/l), heave (63/L), pitch (~) and
wave elevation (~/L) in regular head waves
Fig.13 shows the calculated time histories of ship
speed, ship motions and wave elevation at Fn = 0.15
and A/L = 1.0. The ship starts at non-dimensional
time T = 25 and reaches to given speed at around
T = 40. The ship motions gradually grow and at
around T = 50 reach to steady state. We see 2 com-
ponents of oscillation in (~/L: one is long period os-
cillation due to artificial restoring force term and the
other the surge motion oscillating with encounter fre-
quency. Such the behavior can be often seen in the
actual tank test results. Regular periodic oscillations
are obtained in 63/L and 8. Some fluctuation is ob-
served in amplitude of incident waves (~/L), however
this is negligible in view of practical purpose.
Fig.14 shows the computed wave patterns around
the ship in regular head waves. The wave component
generated by ship motion appears at the ship fore
part, and moves rearward and reflects at the tank-
side. Incident waves and steady waves components
are superimposed to the radiated waves. At present,
there is no validation data for wave computations,
but those look plausible.
Fig.15 shows snap shots of the ship attitude change
and wave profile at Fn = 0.15 and A/L = 1.0. Deck
wetness appears in the computations, however,
Frl 0.15' A/L Lo ~~ ~
hydrodynamic effect due to the deck wetness is ex-
cluded. No trouble occurred in such a large ship
motion case and it was confirmed that the present
method is robust.
By analyzing the time histories of the ship motions,
we obtained the amplitude and phase against the in-
cident wave. Figs.16 and 17 show the comparison of
amplitudes and phases of surge, heave and pitch at
Fn = 0.15 and 0.25. The amplitudes of surge and
heave, and pitch are non-dimensionalized by the am-
plitude of incident wave ha and the wave slope hat
respectively. The calculated results by strip method
are also plotted. In case of Fn = 0.15, the calculated
accuracy is satisfactory for surge. However, ampli-
tudes of heave and pitch are overestimated and are
rather close to the results by strip method. In case of
Fn = 0.25, calculations of amplitude and phase agree
well with experiments. The calculated accuracy by
strip method is not satisfactory for surge and heave.
Particularly, the heave amplitude by strip method is
too large in the range from 1.0 to 1.5 of A/Lpp.
Fr'—0.15' A/L - 1.0
Fig.14: Computed wave patterns around S-175 con-
tainer ship advancing in head waves. The phase an-
gles against incident wave are -9Odeg, -45deg, 0 and
45deg from the top.
Fig.15: Snap shots of ship attitude change and wave
profile in regular waves (one cycle from-9Odeg to -
90deg in phase angle)
,-175 Container Ship
o8t.
0.E
-
—0.4
~ SURGE
Fn-0.15, X =1 80deg
0 Cal.
. · Exp.
~ Strip Method
0.2
0. .5 1 ~ .
18C O O
~ ,~ a--~--0--~-0~ ) ~ <5
.~4, 0 .~.................................................................................................
~ ·e
-18 .5 i 1 5 A/L
. i ....
1 ~ 2/1 2
S-175 Container Ship S-175 Container Shi
. . . . , . . . . , .P. . .
HEAVE Fn=0.15, x=180deg
1.5 . 0 Cal.
· Exp.
....... Strip Method
_ 1.0
0.5
~ -.,,,e,.'', . .
°8 5 1 1.5 A/L .
180 :
44~ o , ·,· ~ ~ ~ 5-~-~ ~ (
—188 5 1 1.5 A/L
/
1 .5
PITCH
1 .0 ............
s
-
_
05 -- - io Fn=O.15, x=180deg
~ .
.. 0 Cal.
G. · Exp.
~ ........ Strip Method
O [. ~ . . . , . , . . . .
. .5 1 1 .5 A /L 2
360
a'
=180
: 000_,,_,,0,,,,,,,_____ ___.
O ..- ~ .- -- ---- -- ............
O"e
li ,,,,3.~ ~
IJ
05 1
- 7/L
Fig.16: Comparison of amplitudes and phases of surge, heave and pitch in regular head waves for S-175
container ship (Fn = 0.15)
S-175 Container Ship
~ ....,....,.-...
SURGE -
0.8
- - too. ~
—0.t
O.`
Fn=0.25, x=180deg
.
0 Cal.
· Exp.
Strip Method
°8 ;
.5 1 1 .5 A /L 2
180
~ O O Q ~ ~ .........
,~,, O ....................................................................................................
.' :
~ : ' :
-1 8Q
u.5 1 1.5 A /L 2
S-175 Container Shin _ S-175 Container Shin
1 .£
_ 1.t
co
~P
. .. _
0.5 C' 0 Cal.
. · Exp.
~ · ~ Strip Method
on' ~---..~.., ....i.... _
v.5 1 1.5 A /L 2
180 . . . . . . . . .
O' .........................................................................................
=, ; . ~
c~) r~ !..~
44e U _ . .e,,,,,~,., - ' ~
—1 8n
v.5 1 1.5 A /L 2 v.5
. ~ · '----- .
.~ O -
.i; c
Fn=0 25 x=180dea
, .~
1 .C
-
y
-
~D
O.c
fe
........ ~ ..........
. .
,,.e O
- -- --- ---r'. -° -- F'1=0~25, x~ =1 80deg ~
Cal. ~i
· Exp. 1 ~
~ ~ I Strip Method|4
V.vx.5 1 1.5 A /L 2
360, - . . . . .
a'
m18C
t
n~
;~ e-~--~---~------ -------1
1 1.5 A/L 2
Fig.17: Comparison of amplitudes and phases of surge, heave and pitch in regular head waves for S-175
container ship (Fn = 0.25)
6. Wave Pressures of a Large Container
Ship
Next, the present method is validated for wave pres-
sure in regular head waves. Calculations were carried
out for a 4,900TEU large container ship as shown in
Table 4, and were compared with experimental data.
The experiments were also conducted at seakeeping
& maneuvering basin, Nagasaki R & D center, MHI.
The pressure fluctuation was measured at 2 hull sec-
tions of square station (S.S.) 7 1/4 and 5 1/2 using
load-cell typed pickup. The pickup was installed to
the model on 4, 7, 10 and 13m water lines in fullscale.
Here, pi., P2, pa and p4 denote the pressures at 13, 10,
7 and 4m water lines respectively. In the test, ampli-
tude of the incident waves was corresponding to 2m
in fullscale. Radius of gyration of the model key was
0.25Lpp. The detailed test procedure and the ship
model used in the test were described in Ref.~274.
Table 4: Principal dimensions of a 4,900TEU large
container ship
Ship length Lpp
Breadth B
Depth D
draft d
ship speed U
Model
4.200m
0.652m
0.396m
0.204m
1.38m/s
.
Ship
258.0m
40.0m
24.3m
12.5m
_ 21kn
In computations, about 700 panels for a half body
were used. The length of the numerical tank is l OLpp
and free-surface panels were 5,000 (= 250 x 20~. Flap
angle of the wave maker was selected so as to coincide
with the amplitude of incident waves in the tank test.
Fig.18 shows comparison of amplitudes of heave
and pitch in regular waves. The calculated ampli-
tudes are larger than the experiments as a whole, al-
though the present calculation is much closer to the
experiment than by strip method.
F. =0.215, X =1 80deg
n ......
10 HEAVE . ~ --- ~10
: ,,.4e,
. ~ ,.
~ it- '' A..'
~ . . . . , . . . . , . . no
0.8 ,
Fn.0.215' X e1 80deg
.. .. I · ' ' · !
PITCH ....... -- - -
.~" ·
' O
...... .... .. .. . .. . .. ... . . ... .... . ... ....
.' ~ 0 Cal.
if · Exp.
~ Stnp Method
I 5 1 1.5 A IL u.~ i ~ . j . 1.5 A /L
Fig.18: Comparison of amplitudes of heave and pitch
in regular head waves for a large container ship
S.S.7 1/4 Cal. -------- Exp.
~ 1 1 1 1 1, 1 ! 1
mm _ j~ ~ ~ ~ ~ ~ _
Q o ~ ~ ~ -i Id/ -—~ ~
. ; ; ~
-2
60 61 62 63 64 65 66 67 68 t ~g/Lj70
_=
60 61 62 63 64 65 66 67 68 t ~ /L)70
, , ~ , , ~ ~ .. ~ ,
60 61 62 63 64 65 66 67 68 69 70
t Jr(g/L)
2r , , , , , , , I 1 1
Ct
s
Q C
~1
—2
60 61 62 63 64 65 66 67 68 tin /L)70
Fig.19: Comparison of time histories of hull pressures
on a large container ship (Fn = 0.215, A/Lpp = 1.1)
Fig.19 shows time histories of hull pressures at
S.S.7 1/4. Here, the pressures are set to be zero when
ship model is just floating without forward speed in
tank test. The calculated pressures are adjusted to
coincide with the test condition. Time history of Pi
has flat bottoms which occur due to exposure of the
pickup in the air. The P2 behaves the similar to Pi.
The present method can capture the nonlinear be-
havior due to the pickup exposure. Almost the sinu-
soidal history is obtained for pa. The history of p4
has round for the top and sharp for the bottom, and
the present method captures well such the behavior.
The calculated results are about 20% larger than the
experiments in absolute value of pressure peaks. This
copes with the situation that calculated ship motion
is larger than the experiment.
Fig.20 shows the comparison of pressure amplitude
distribution. In the figure, lip defines the hull surface
location of the given section: lip = 0 means the ship
bottom and lip = +9Odeg the hull sides of still wa-
ter line. The amplitudes were obtained by taking
rat
—90 - 60
2.OI
1.5
Q 1.0
n. .
o.o
_ _ 1 1
,, .,.,, 0 ECap. 1
.... -- Stri ~ Method |
_ ,, ,,, ,,,.,,,,, .,, ._.
_ ma. - ...., ,=_ ~
_ ) 0 0 0 0 0 Ol)Ooc~oo,
-30 0 30 60 90
~ p(deg)
S.S.7 1/2 l/L=O.9
. , ~ .
0, , . o'
~ ~LCL.~,''O~ C?- - - ~ or
. , +., . ,,, ~ ,.
. ',
—90 -60
, ,, ,, . . . _ . . _, ,
-30 0 30 60 90
~ p(deg)
~ .R .s 1/d ~ It =n ~ S.S.5 1/4 l/L=1.1 S.S.5 1/4 AIL=1.3
I, 1 1
2.0
1.5
Q. 1.0
0.5
0.0
2.(
-a
Q 1.t
0.!
O.`
5 ~ _ I I 1.5—
. .; -- . . . | Strip Method | Q 1.0 ......
5 _ Oo°c oco°° ; 0 0 ]° O 0 0 ~~ ~~ n r.
-90 -60 -30 0 30 60 90
6 p(deg)
S.S.7 1/2 A/L=1.1
1 _ , .
i _ , ~ L b ~ ~ ~
l - --I-
C)
i ., ...,..,.,,.~.,, .,,, .,, +,.........
—~0 -
·.O..C., ° ,
... , , ., . .,. ... , ... .,,, ... .,, ,, ;, .
o
.. . .. .. ... . .... ...
C
O _ . ' .,
-90 -60 -30 0 30 60 90
~ p(deg)
_ _ _ ' ' ' "'amp- Cr CJ 0~15 ~ ~~ - oO D
-90 So -30 0 30 60 90
~ p(deg)
S.S.7 1/2 A/L=1.3
2.0 .
1.5 ._
Q 1.0
0.5
0.0 _< 10
1 !
4~ IQO~~
IC-''''--'''-'',j-..,_ _o ~ ,- -''t'
1 1
'1-'---'------'--~---- - --------1-,_,,_,,----t'
1 1
1 ' 1-
90
._,,. _. I' °= o .. _,. _.~2 ' ',.'.'
' ?. .oOoOOOc
~ L
o
-
~n ~n 0 30 60
(deg)
Fig.20: Comparions of hull pressure amplitude distribution for a large container ship (Fn = 0.215)
the 1st order solution of Fourier analysis result of the
time histories. In the case of appearance of nonlin-
ear behavior as shown in the top of Fig.l9, the same
procedure was adopted for analysis. We also plot-
ted the results by strip method with correction for
considering the nonlinear behavior of pressure~285.
In the calculated distribution at S.S.5 1/4, notice-
able hump appears near Bp = +75deg. At A/L = 1.3
the calculated results agree well with experiments be-
cause the hump is clearly observed in the experiment.
At A/L = 0.9 and 1.1, however, such the hump is not
so remarkable in the experiments. Comparing with
the results by strip method, the present method im-
proves the prediction accuracy.
At S.S.7 1/2, the calculated pressure amplitude
turns down at the sides of the hull due to the nonlin-
ear effect as shown in Fig.l9. The calculated ampli-
tudes are about 10-20~o larger than the experiments.
As mentioned above, this comes from the larger ship
motions in the calculations. Strip method is overpre-
dicted as same as the present method.
In summary, the present method is better predicted
in amplitude of the pressure fluctuation at midship
section than strip method, and can capture the non-
linear behavior of the pressure appearing near the
ship hull sides at fore part.
7. Concluding Remarks
by means of the time domain code. All the results
except the wave patterns were compared with exper-
iments and the calculated results by strip method. As
a result, we found that hydrodynamic forces, ship mo-
tions and wave pressures are much better predicted
than by strip method. The present method is promis-
ing as a new ship design tool for more accurate pre-
diction of seakeeping performances.
However, the calculated accuracy of hydrodynamic
forces on lateral motions is not satisfactory. This is
due to viscous flow effect and we need improvement
of the method. For this purpose, simple empirical
correction may be useful as commonly employed in
strip method.
As the next step of the work, we will extend the
present method to large amplitude ship motion prob-
lem such as slamming.
Acknowledgement
We are grateful for the support of younger colleagues
Mr. S. Mizokami and 'Cowboy' Tanaka in prepara-
tion of the paper.
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" Wave Load on a Container Ship in Rough
Seas", J. of the Society of Naval Architects of
Japan, Vol. 189, 2001, pp. 181-192 (in Japanese).
t28] Toki, N., Fukushima, Y., Tozawa, S. and Wada,
Y., "On the Characteristics and Long-term Pre-
diction Procedure of Wave-induced Pressure
Fluctuation on a VLCC Hull", J. of the Soci-
ety of Naval Architects of Japan, Vol.176, 1994,
pp.375-385 (in Japanese).
Ory, and indicated the range of wave number dis-
appearing the tank-side effectt254. Based on their
achievement, we made a diagram showing the min-
imum wave length for disappearing the tank-side ef-
fect versus tank width as shown in Fig.21. We de-
cided the minimum width of the numerical tank from
the diagram.
, Fn=0.2,5,/ ,, ----'.'''
- " ~ Fn=0.20
1
O. . i . I
0 1 2
., ., ., 1
3 4 5
Fig.21: Minimum wave length for disappearing the
tank-side effect versus tank width
DISCUSSION
V. Bertram
ENSIETA, France
I enjoyed the paper very much. I congratulate
you to the significant progress achieved over the
last years.
For clarification: Eqs. (1) and (2) are formulated
at z-0. In a time-domain method, we should be
able to formulate them at the actual free surface
position. The good agreement of the
computations seem to indicate that for the
investigated cases, this simplification is justified.
Nevertheless, is this simplification in principle
necessary? Do you see major problems in
extending the approach to include a non-
linearized free-surface condition?
The new test case of a large container ship
described in chapter 6 is a very useful addition to
our suite of ship seakeeping test cases. Can I
encourage you to follow the good tradition of
MHI and have an English report on the
experiments with sufficient data given for the
world-wide community to use it as a validation
case?
AUTHOR'S REPLY
Our main purpose of the study is to apply 3D
time domain panel method to actual ship design.
In view of this, we gave precedence to
minimizing the computational time and keeping
the robustness of the code rather than the full-
nonlinear computations. The employment of the
2n~ order free-surface conditions on z=0 is useful
for dramatically reducing the computational time
because of no re-paneling the free-surface and no
re-constructing the base matrix with respect to
large number of free-surface panels. The
calculated accuracy is satisfactory for practical
use as mentioned in the paper.
Now we are planning another tank test to capture
nonlinear behavior of wave loads and wave
pressures for other modern container ship. This
test includes the cases of oblique waves and
irregular waves. These data may be useful as
validation data for various seakeeping prediction
methods. We will open the measured results
together with hull form data in near future.
