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OCR for page 355
Investigation of Global and Local Flow Details
by a Fully Three-dimensional Seakeeping Method
V Bertra n (HSVA, Ger any)
H Yasu awa (Mitsubis i Hea y Industries, Japan)
Abstract
A fully three dimensional P nkine panel
method in the frequency domain is validated for local
pressures, motions, and added resistance Previous
formulae for added ---i Lance contained errors result
ing in large differences to e cperiments This has n w
been remedied The method is linearized with respect
to wave height The steady fl w contribution is cap
tured completely by solving the fully nonlinear w we
resistance problem first and linearizing the seakeep
ing problem around this solution The same grids
on the hull are taken for both steady and se keeping
computation On the free surface different grids are
used, either following quasi streamlined grids or re t
angular grids with cut out for the hull The results
from the steady solution are interpolated on the new
free surface grid The method is applied to various
test cases Motions are in good agreement with em
periment, but this is also the case for trip method
rmults Local pressurm, especially for sho ter wa:vm,
are much better predicted than by strip method The
added resi Lance is sensitive to higher derivatives of
the potential and a numerical differentiation of these
terms may be preferable to using highe~order panels
1. Introduction
The most commonly used tools to determine
seakeeping properties are based on strip theory The
strip method approach is cheap, fast, and for most
cases also quite accurate H wever, strip methods
do not perform so w 11 for high speed ships, full hull
form (tankers), ships with strong flare, and generally
for I w encounter frequencies which typically occur
in toll wing seas They are also questionable with
regard to local pressurm which are needed as input
for finite element analyzes
Approaches to improve =:
keeping properties should capture:
3 D effects are impo tant for I w encounter
frequencies and full hull form 3 D diflra tion
at the bow region of tankers contributes con
siderablJ to added resistance, [1]
Forward speed effect
Strip methods include forward speed by the
change in encounter frequency But forward
speed enters the ship motion problem in ad
ditional ways: the local steady flow field, the
steady w we pattern of the ship, and the change
of the hull form and wetted surface due to squat
(dynamic sink =- and trim)
We will present here a 3 d Pankine singuiaritJ
method (PSM) which captures all forward speed ef
fe ts The method is 'fully thre~dimensional', i e
both steady and unsteady flow contributions are cap
tured three dimensionally For a recent survey of
P nkine singularity methods for to ward speed sear
keeping, we refer to [1], [2]
2. Theory
2.1. Physical model
We consider a ship moving with mean speed
U in a harmonic wave of small amplitude h We
assume an ideal fl w Then the fundamental field
equation is Laplace's equation In addition, bound
11 art conditions are po tulat d:
I No water flows through the ship's surface
2 At the tmiling edge of the ship, the pressures
are equal on both sides (Kutta condition)
3 No water flow through the free surface (Kine
matic free su face condition)
OCR for page 356
4 There is atmospheric pressure at the free sur
face (Dynamic free so face condition) svatio
5 Far way from the ship, the di turbance caused
by the ship vanishes
6 Wwes created -- the ship move way from the
ship For certain combinations of frequency of
incident w we and speed of the ship, w wes cre
Ted -- the ship propagate only downstream
(Rzdiation condition)
7 Wwes created by the ship should les:ve zrtifi
cial boundaries of the computational domain
without reflection They may not reach the
ship gain (Open boundary condition)
8 Forces on the ship result in motions (Average
longitudinal forcm are subsumed to be counter
acted -- corresponding propulsive forces, i e
the werage speed U remains constant )
2.2. Mathematical model
All coordinate systems here are right handed
Czrtmian system The inertial 07 ye system moves
uniformly with velocity U x point in the dire tion
of the hod s mean velocity U. z points vertically
d w ward The Oxy sy tem is fixed at the body
and toll ws it motions When the body is at rest
position, i, y, z coincide with x, y, z The zngle
of encounter ~ between body and incident wwe is
defined such that ~ = 180° denotes head 22a and
= 90° beam sea
The body has 6 degrees of freedom for rigid
body motion We denote corresponding to the de
Frees of freedom:
7 ~ surge motion of O in x direction, relative to O
7 g way motion of O in y direction, relative to O
7 g he we motion of O in z direction, relative to O
7 ~ zngle of roll = zngle of rotation around x axis
7z zngle of pitch = zngle of rotation around y axis
7 g zngle of J w = zngle of rotation around z axis
The motion vector is f f and the rotational
motion vector ~ are given ---
f = 17 1.2 9,7 3} (1)
~ 17 ~ 7 z 7 2} = Len, 2. C~9} (2)
~11 IIIUbIUII~ ALL ~~UIII~U bU US ~111~11 AL ULU~L ~ Act)
Then for the 3 angles a\, the following zpprcxima~
tions are valid: 2in(c~,) = tan(~\) = a\, c02(c~,) = I
The theory has been described Other sxtensivelJ -'
[2] in the following, we will therefore only briefiJ
review the theory except in cases where cEznges to
[2] ju tiff z more detailed discussion
We decompose potential and free surface el
on into steady and time harmonic pa ts:
f202~(~ y z;t)=f(°)(~ y,z)+f(~)(~,y,z;t) (a)
= f(0) + ~2(f (~)(7 y 9)9 )
:202~ (a yip) = :(°)(~,y) + :(~)(~,y;t) (4)
= (o) + ~z(~(~)(~ y)2 ' )
The superposition principle can be used
within z linearized theory Therefore the radiation
problem for all 6 degrees of freedom of the rigid
body motions and the did action problem are solved
separately The total solution is z linear combination
of the solutions for each independent problem
The harmonic potential f(~) is divided into
the potential of the incident wave fit, the diffraction
potential Ad, and 6 radiation potentials it is onve
nient to divide fit = fW,6 + fw,~ and fd f + fS
into symmetrical and anti ymm2trical parts to tzLe
advantage of the (usual) geometrical symmetry:
f(~)=fw,6+fw,~+~f\7 +f7+f2
The conditions satisfied by the toady fi w potential
f(°) are:
The pa ticle acceleration in the toady fi w is:
.u if(°) = (Vf(°)V)Vf(°)
We define an acceleration ve tor if 9
if 9 = if(°) {O, o, get
For convenience we introduce an abbreviation:
B = 7 ~9 (Of (°)if9)
At the steady free surface:
Vf(°)if9 = 0
2(Vf( ))2 9~(0) = iU2
On the body surface:
7-OVf(°)(, = 0
Ths combined, linearized free surtacs condi
tion is at z = :(°):
( h32 + Bfh~)f(~)+
((2fh,~ + B)Vf(°) + if(°) + if 9)Vf(~) (6)
+Vf(°)(Vf(°)V)Vf(') = 0
Ths last term in (6) is sxplicitlJ written:
Vf(°)(Vf(°)V)Vf(') =
(f(0))2f(-) + (f(0))2f(~) + (f(0))2f(~)
+2 (f(o)ffo)f(~) + f(0)f(0)f(~) ~
(7)
OCR for page 357
With the abbreviation m = (~V)Vf(°) the boundary
condition at S(x-) = 0 is:
If (') + ri(m Sac L)
+c~(x x ( i ~~ L) + ~ x Vf (a)) = 0
The Kuttn condition requires that at the trailing edge
the pressures are equal on both sidm For monchulls,
this is automatically fulfilled on the centreplane for
the symmetric contributions Then only the anti Em
metric pressures have to vanish, compare (17):
P(f i + Vf (°)Vf i) = 0
This yield on point at the trailing edge:
fhlcf\+Vf(°)Vf\=0
(10)
For catamarans, the Kuttn condition requires for
both symmetric and anti ymmetric contributions,
that the pressures on both sides of the trailing edge
are the same This is enforced by selecting pairs of
collocation points at the trailing edge and matching
the pressures
The 2 unknown did action potentials and the
6 unknown radiation pot Steals are determined by
npprcximating the unkn wn potentials -- n super
position of n finite number of Pnnkine higher order
panels on the ship and above the free surface For
the anti ymmetric casm, in addition Thinrt elements
(semi infinite dipole strips on the plane y = 0), [2],
[3], are arranged and n Kuttn condition is imposed
on collocation points at the last column of collocation
points on the stern The I h s of the four system of
equations for the symmetrical cases and the I h s for
the four zJ tems of equations for the antisymmetrical
cases share the same coefficients each Thus four SJS
tems of equations can be solved simultaneously using
Gauss elimination
P diction and open boundary conditions are
fulfilled -- the 'shifiing' technique (adding one r w
of collocation points at the up tream end of the free
su face grid and one row of source elements at the
d wnstrenm end of the free surface grid), [4] This
technique w rks only well for r > 0 4, as also demon
strafed by [5] Elements use mirror im ges at y = 0
For the symmetrical cases, all mirror im ges have ~
same strength For the anti ymmetrical case, the ir
mirror images on the negative y se tor have negative
element strength of s me absolute magnitude
f =~m\p
m\ is the strength of the fth element, p the potential
of an element of unit strength including all mirror im
ages p is real for the Pnnkine element and complex
(8) Or the Thinrt elements
The same grid on the hull is used as for the
tends problem The grid on the free surface is :-e
at d new The quantities on the new grid are linearly
interpolated within the new grid from the valum on
the old grid Out ide the old grid in the far field,
all quantities are set to uniform flow on the new grid
The interpolation of results introduces only mall dif
Structured grids on the free su face ar
crated -- one of the foil wing techniques:
I The longitudinal grid lines follow quasi stream
lines am :- i the hull The transverse grid lines
are equidistantly spaced on lines y =const A
maximum entrance angle of 30° is kept which
results in zones not covered -- the grid near
the b w and stern of blunt ships
2 A rectangular grid is created consi ting of lines
x =const and y =const Panels within the war
terline are deleted
The first technique is well suited for slender ships, the
second technique better for blunt ships The second
technique, called 'cut out'technique was proposed for
the steady wave resistance problem by [6] and [7]
Both grid generation options are wnilable for both
tends and senkeeping fre~surtace grid generation
While it is in principle possible to switch from one
grid type to the other between steady and senkeeping
computations, it is recommended to use consistently
cut out grids for full hulls like tankers and the tan
dard grid option (quasi streamlines) for slender ships
Afier the potential f\ (I = 1 8) Ewe been
determined, only the motions I remain as unknowns
The forces F and moment M acting on the body
result from the body s weight and from integrating
the pressure over the instantaneous wetted surface S
The body s weight G is:
G = 10, 0, mg}T
or is the body s mass F and M are expressed in the
nertinl zJ tem ( f is the i ward unit ~ al vector):
F = J p(x pomps fS + G
(11) M = J Pi)
(12)
(13)
po) (X x if (X)) iS + xg x G (14)
OCR for page 358
xg is the center of gr witJ The prmsure is given bJ
Bernoulli's equation:
P(~) Po = p(°) + p(~) (15)
P (2(Vf( )(~))2 272 93) (16)
p (Vf(°)vf(~) + f(~)) (17)
(14) Jioid:
J p(o)~- dS + G = 0
s(o)
= Jp(o)(~-x~)ds+~09xG=
s(o)
p(o) =
p(~) =
Eqs (13) znd
F(°) =
M(°)
(18)
o (19)
The ship is in equilibrium for steadJ fl w Therefore
the teadJ forcm snd moment zre zll zero The first
order part give (r h s quantities sre n w zll functions
of ~):
s(o)
M(~)
:(p(~) + Vp(°) (o x x~ + 7~)1 ~- dS
x G (20)
= J [(P(~, + vp(o) (~ x ~- + 7~)] (i x ~) dS
s(°) The relatir
~g x (~ x G) (21) celeration is:
where (~ x x-) x ~ + x~ x (~ x ~) = ~ x (;~ x ~)
znd Eqs (18) znd (19) (steadJ equilibrium) hs:ve been
used Note: Vp(°) = pd9 The difierence between
instantaneous wetted su face znd werage wetted sur
face till hss not to be considered ss the steadJ pres
sure p(°) is small in the region of difierence
The in tationsrJ prmsure is divided intc
part due to the incident w e, radiation znd difirac
tion:
M(~) / (pw + pa)~ x ~) dS
p(~) = pw + pa
6
+ ~, p\7 \ (22)
\=t
Again the incident w we znd difimction contributions
can be decomposed into sJmmetrical znd zntisJm
metrical parts:
pW = pW,6 + pW,~ (23)
pd = pd,6 + pd,~ = p7 + ps (24)
Using the unit motion potentials znd the pressure
equation (17) the pressure parts p\ zre derived:
P\ = r(f2 + Vf (°)Vf\)
P = P(f2 +Vf(°)VfW) (25)
p = r(f2 + Vf(°)Vf;)
The individual terms in the integmls (20) znd (21)
zre exprmsed in terms of the motions 7 \, using the
vector identitJ (~ x 0d9 = ~(x~ x d9):
F(r) = J (pW + pd )~ dS
s(o)
+~ (J pM-dS: 7\
\=t :(o) J
+ J p(7-d9 + 0 (i x i ))i dS
s(o)
~xG
+ ~ | J P (~ X ~ d~ 7 \
\=t )
yiv~/
~g x (~ x G)
+ J p(7-d9+cr(~xi ))(~:
s(o)
~n bstvesn forcm
(26)
(27)
~i)dS
. moment znd motion zc
F(~) = m(zl22 + cr22 x xg)
M(~) = m(~g x 7~22) + 1cr22
I=
o~ O o~z
0 o9 0
o~z O oz
u Msss di tribution sJmmstrical in y is sssumed ~g
stc zrs ths moments of inertiz znd ths centrifugal
moments with rssps t to ths origin of ths bodJ fixed
Oxyz sJstsm:
0~ = J(y + z ) dm; 0~ = J y dm; stc
We introducs the zbbtsviations:
~h~, h2, hg}T = pi
~h~, hg, hg}T = px~ x d9
h7 = p '
hg = p,
Recall tEzt ths instationarJ prmsurs ~
p = p(lhlef\+Vf(°)Vf\)
(28)
(29)
~.
(30)
(31)
(32)
(33)
(34)
ontribution is:
(35)
OCR for page 359
Then w can rewrite (26), (27) and (28):
W(2~22 + C22 X X-g) O X G (36)
+~ ~ J p\+ h\)~- dS) ~\ = 0
=t :(o) J
m(x~g x 2222) /C22
(p\+h\)~x~d? j ~\
O O O O W O
O O O W O O
O O O O O O
O O O ZgW O O
O O O O ~gW O
O O O ~gW O O
2~t ~
2~2 1
2~9 ~ (38)
4 1
~9 1
~9 )
Th~ msss tOrmS m(2222 + C22 X 4 ) nd m(39 x J
2122) /022 contributs:
~2
m~t2Ml2~,2~2,2t
with M bsing
t9,24,2~9,2~9} (39)
1 0 0
0 1 0
O 0 1
O zg O
Z9 0 is
O is 0
O zg O
0 g9 0
k~ O k~
O g O
k2z o k2
Ths fores in x dirsction is given bJ:
/ p(~) po)~r dS
(41)
Ths integral over ths wstted surfacs can bs sxprsssed
ss a doubls integral over a bodJ fitted curvilinsar co
(37) ordinats sJstsm Ons coordinats foil ws rather longi
tudinal linss from stsrn to b w, ths other coordinats
foil ws ths hull contour from ths frss su facs d wn to
ths kesl Ons of ths longitudinal coordinats linss fol
low ths contour of ths stsadJ ws:vs profils and this is
ths 'zero' lins for ths other 'ssction' coordinats This
modifisd watsrlins contour C accounts also for tsadJ
trim and sink gs and difisrs usuallJ ths still watsr
lins contour Ths contour lins C splits at ths stsrn
and both sidss run from stsrn to b w Ths'ssction'
coordinats runs from ths actual frss surfacs Z to ths
kml K Then we can rs writs anJ integral over ths
w tted surfacs ss:
K
I 1
(42)
= J J ds de + J J ds de (43)
z
= J J ds de+ J
C o s(o)
dS (44)
Ws can thus split ths integration into ons integral
(40) crzer ths s:vemgs wstted surfacs S(°) and a corrsction
doubls integml if ws aPPIJ this to ths sscond order
tims wsrags longitudinal fores on ths ship, ws ob
tain:
wEsrs ths mdii of inertia h ws besn introduced
G~ = mk_ stc F( ) = J [p(t, + vp(o) (c x x~ + ~] (c x i ~ (4 i)
Combining Eqs (36) and (37) Jislds a lin
is quickiJ solved using Gaus.
2.3. Added resistance
~ slimination
Following a similar approach ss for ths fir t
order forcss, a formula for ths dded rssi tancs can C
bs derived that usss onlJ quantitiss computed so far
Ths added rssistancs is ths negativs tims wsraged
vslus of ths x component of ths sscond order fores
If tt and t2 arO tims harmonic quantitiss, ths tims
wsrags of tit2 is 2 lls(t~t2), wEsrs t2 is ths conjugats
complex of t2
lo)
+ [p(2) + Vp(~)(C X X~ + 22)] ~t dS
J J [p(o) + p(t, +vp(o)(~ x ~+~,] ~t ds de
c 0
'ombining (20) and (28), Jislds:
J [(P',+vp(,(~X~+~]~ dS (46)
= c x G + m(z222 + C22 x ~09)
OCR for page 360
Thus:
J [P(~)
s(o)
+ Vp(°)(c x ~- + ~)] (c x ~)~ d
2Re~mg~3 mh3~[~2(~3 cr2~3)
cr3 (~2 + cr3~3 crtzg)]}
The second order prersure p(2) is:
p(2) = 9(Vf(~))2 rf2
second orde
potential vanishes in the time a:ver ge:
We develop p(°)(z~) in z TZJIOr series zround z~ = 0
(a:vemge free surtace = steadJ free surtace):
P()(Z)=P()(O)+P2, (O)Z +PN dNZ (56)
= [P~ (O) + (~IP( ) + ~2P( ))~3]Z~
As z'znd z point both d wnwards, the derivation is
intercEzngable The'teadJ'pressure is zero zt the
'steadJ'free su face Thus:
(48) |P(°) dz' = [p(°)(O) + (~P(°) + ~2P(°))r/3] I z~ dz'
. ~ z2 p(O)(o) + (~p(°) + ~2P( ))~3] (57)
p = (Vf(~))2 (49) The p(~) term is simple A TZJIOr series gives
P( )(3 ) ~ P(~)(O) Then:
The term Vp(~) involvm zgain second derivatives of
the potential on the hull:
vp(~) = p[Vf(~) + v
(Vf (°) Vf (~) )] (50)
Z is the first order difierence between
(steadJ) znd instantaneous wa:ve profile on .
with:
n werage
n tR~ R~lil
Z = :( ) (~3 ~z~ +~dY
:(~)= f()+Vf(°)Vf(~) I (
ai P~
(51)
(52)
The curvilinear 'section' coordinate s can be zpprcxi
mated to first order in the vicinitJ of the steadJ w we
profile bJ z tangential straight line:
s ~ Z ~T~
z~ is z vertical coordinate with origin zt the height of
the steadJ su face pointing downwards
Let ~' be z modified normal:
~ ~/~ ~ ~ ~/V~ ~
~ = ~2/~ = ~2/~f ~ ~ Th
~3/~ ~3/V~ )
(bd)
Let N be the unit normal on the contour in the ~ y
plane Then
, ~ , ~
N ~t :~ + ~z y
I P(~) dz = p(~)z
J
SimilarlJ the other first order quantitJ Vp(°)(c x ~+
) is simpiJ multiplied bJ Z in the integration over
z'
F(2) =T~+Tz+T3
Tr= 2Tk~ J [ PVf(~)Vf(~)*
s(o)
+Vp(~)~(c x ~ + ~)1 ~ d~J
Thus eventuallJ we get for the time weraged
second order longitudinal force:
(53)
(58)
Tz = 2Tle ( J (2Z[P(~)(O) + (iP(°) + ~zP(°))
~e] +P( ) +VP( )(cr x~+~)) Z*~ de}
3= 2Tk{mghrh3 mh3~[~z(~3 cr2 3)
cr3(~2 + crSi 3 CrtZ3)]}
e zdded resistance is:
T(~w = F( ) (59)
The integrals in the T\ zre evaluated numeri
callJ over the starboard half onlJ znd multiplied
55 bJ 2 for sJmmetrical/sJmmetrical znd zntisJmmet
( ) rical/zntisJmmetrical prmsure normal combinations
onlJ (AntisJ mmetri cal /SJ mmetrical combi nati ons
Jioid zero contributions ) The decomposition into
OCR for page 361
sJmmetrical and antisJmmetrical pa ts complicates
evaluating the added resistance The individual
terms are decomposed into sJmmetrical and antisJm
metrical parts:
f(~),5 = fif ~ + f3< 3
f(~),~=f2<2+f~<~
(Vp(~))6
=
(Vp(~))~
p
with
+fsf 5+f7+fW,6
+fSf 5+f8+fw,~
(60)
(61)
= p(ih,~f(~)~6 + Vf(°)Vf(~),~)
= p(ihl~f(~)~+Vf(°)Vf() )
fhl~f( ) +
f(?f(~),6 + f(?f(~),6 + f(o)f(
f(o)f(~)~6 + f(o)f(~)~6 + f(o)f(
fhl~f ) +
f (?f (~) ~ + f (°)f (~) ~ + f (°)f (~),~
f (o)7 (~)~ + f (°)f (~)~ + f (°)f (~),~
f(o)f(~),6 + f(o)f(~),6 + f(o)f(~),6]
f(o)f(~),6 + f(o)f(~),6 + f(o)f(~),6
fh3~f( ) +
f(?f(~) ~ + g?f(~) ~ + f(°~)f(~)~
f(O)f(~),~ + f(O)f(~),~ + f(O)f(~),~
f(O)f(~)~6 + f(O)f(~)~6 +
f(O)f(~)~6 + f(O)f(~)~6 +
fh3~f( ) +
f(O)f(~),~ + f(O)f(~),~ + f(O)f(~),~+
f (O)f (~),~ + f (O)f (~),~ + f (o) f (~),~ J
Note that the second component of each of these vec
tors contains y derivativm of the 'other' potential tc
ensure consistentiJ sJmmetrical (i e f (y) = f ( y))
and antisJmmetrical (f(y) = f ( y)) beha:vior Ths
second derivativm of the harmonic potentials are ne
glected in the exprmsion for Vp(~) for simpticitJ ('des
peration rather than phJsical insight')
We introduce the abbreviation h = c x x-+ il
h=h +h
~ f sz + < ~ ~
= ~ f si f ~z+<2 ~ +
~ fsi +f3 )
Z Z6 Z~
Z6 fh~f( ) +Vf( )Vf( )' f +< i
b),6 + I
p(~),6 ~
Pc: i
Z~ ~(~)~
(62)
(63)
), ~
64)
(69)
p(~),~
P~i
P taining onlJ sJmmetrical term
= 2 ~e (J (2 (Z6Z6~* +
P~ (°) + (~P(°) + ~2P(°))~3]
+(p( ) + p(°)h~ + p(°)h i + P(°)h: )Z6~*
+ p(~) ~ + p(°)h~ + p(°)h2 + p(°)h3)Z~)
7 ~ fc}
2 ( J [ (Vf(
+VfO VfO )
+(Vp(~))6h + (Vp(~))~h ]~ dS}
2 ~eTmg 9~3 mh3~(~2(< 3 ~2~3)
~3(< 2 + ~3i 3 ~23))}
Due to sJmmetrJ, the above integrals are twice the
value of the integrals over the tarboard half onlJ
3. Applications
3.1. Local premures
So far, applications of the present PSM were
shown onlJ for relativelJ high Froude numbers, which
for most angles of encounters and w we lengths of in
terest result in sufficientiJ high r values For these
cases, good agreement with experiments for motions
w demonstmted, [2], [3], [8] [11] Numerical studies
showed that the influence of the ste dJ flow on the
result for motions is signiflcant, for moderate wa:ve
( ) lengths, but negligible for short and long w wes This
w explained bJ pureiJ numerical investigations of
7) local pressures A research cooperation allowed n w
to investigate local pressures for a VLCC, Table 1,
at F~ = 0 131 The exa t geometrJ of the test case
~8) is confldential The pressures are the amplitudes of
the pressure fluctuation, i e pressures without hJdro
tatic and steadJ hJdrodJnamic pressures
The tanker was discretised using 495 ele
(7o)
3 in T~ and T2 Jioids:
(71)
OCR for page 362
ments First the steady fully nonlinear wave resin
tance problem was solved The grid on the free
su face was generated using the 'cut out' technique
This technique generates a structured grid consist
ing of rectangular elements Element which are par
tiallJ or totally inside the hull are then eliminated
and then the ship ing technique is applied This tech
nique is known to give better result for full hulls than
streamlining a grid around the hull The fully nonlin
ear method used 3 iterations which reduced the error
at the free surface -- 4 orders of magnitude
The same grid for the hull was employed for
the seakeeping computations (at the dynamic trim
and sink he) Pressure integrations considered only
the area submerged in the steady case The free sur
face in the seakeeping computations was discretised
with typically 1400 elements Again the 'cut out'
technique was employed and the steady rmults inter
polated from the'steadJ' grid to the 'unsteady' grid
Test computations for two wave lengths with free
su face grids involving approximately 4200 elements
yielded rmults that were only To different This may
be interpreted as that the coarser discretisation is suf
ficient
The computational results are compared to
measurements of [12] and MHI strip method result
The strip method is based on standard STF method
with Lewis section representation, but includes an
empirical correction [13] The rmults include mo
tions and pressures on the hull at a location x =
0 078Lg~, (23 95m behind midships) The motions
agree rather well for both head sea and oblique sea
with ~ = 150°, Figs I and 2 H wever, strip method
also predicts hes:ve and pitch motions w 11 in fa t,
for long w wes strip methods gives better results than
the PSM This is not surprising Strip methods are
known to predict he we and pitch motions well for
usual ships and ship speeds The present PSM uses
the shifting technique which deteriorates in perfor
mance for r < 0 4 0 5 - .- and J w are also well
predicted, the maximum of the roll motion is under
predicted This may be due to the deterioration of
th e ship i ng technique, as for a fast contai nership with
Fin = 0 275 Be tram [2] obtained significant overpre
diction for roll resonance as expected for a method
that does not include empirical corrections for non
linear roll damping
Figs 3 and 4 compare pressures Starboard is
the weather side For head w wes the computed pres
sores are of course symmetrical to the midship plane
(90°) One point on the port side was then plotted
on its corresponding position on the starboard side
Pressures computed by thePSM greewell with mean
sured pressures for N/L < I 25 for ~ = 180° and
N/L < I 0 for ~ = 150° These limits correspond
for the investigated I w F oude number to r values
around 0 35 0 4 For sho t w wes, the computations
underpredict the pressures at the bottom of the ship
compared to measurements However, as the pres
sores should decay exponentially with depth like all
w e effect, for short waves the near zero values of
the computation appear to be more plausible and we
assume that they reflect in this case reality better
than the measured values For waves of moderate
length 0 5 < N/L < 0 75, measured and computed
pressures at the ship bottom agree well The trip
method results for pressures are worse for short w wes
N/L = 0 2, 0 3 where diffraction effects are stronger
than radiation effect
In summary, the PSM predicted pressures
and motions well, the trip method predicted pres
sores in short waver badly, but motions well The
PSM is currently limited in practice to approximately
r > 0 4 Unless techniques are developed to e tend it
to smaller r Values, the PSM will remain a research
tool of limited functionality We see hybrid methods
matching an inner PSM solution to an outer Green
function method or Fourier Kochin solution as most
promising approach to extend the method to I w r
valu m, but at present no such research is planned due
to lack of funds
Table 1: Test case VLCC
L99 307 00 m 29 4 333 m
B 54 00 m he 19 193 m
T 19 50 m k 73 987 m
CB 0 813 .: 76 750 m
KG 15 17 m ken 0 m
xg 10 045 m
3.2. Added resistance
We show hers applications to the ITTC tan
hard test case S 175 containership in head seas Com
put tions are compared to experiment of Mitsubishi
Heavy Industries Fig 5 shows results for Fin = 0 25
and Fig 6 for Fin = 0 3 For all three motions the
agreement is good in fact, the computational result
for hes:vs for long waves appear to he more plan
sibls, as they tend as expected monotonously to I
The discrepancies between experiments and compu
tations for the higher F ouds number Fin = 0 3 are
most likely due to nonlinear damping enacts in the
0 275, [2] showed that the
also correct captured, at Mast
OCR for page 363
for wa:ve lengths where the PAOs nre not npprcxi
matelJ zero
The ndded resistance is similarlJ well cap
tured The diflerences for the higher F oude number
nre explained bJ the diflerences in capturing the mo
tions The derived formuln seems to be debugged
n w nnd cnn be recommended for other'fullJ three
dimensional' methods
Ackn wledgment
We nre grnteful for the support of our
Jounger collengues Shuji Mizok ml nnd 'C wboJ
Tnnakn in preparntion of the results
Referencez
I Bertrnm, V
kine source methods
JnErbuch der Schiflbau
Springer, 1996, pp 411
znd Yasuk wn, H. "Pnn
for senkeeping problem,"
technischen Gesellschnfl,
425
2 Bertrnm, V, "Numerical invmtigntion of
stendJ flow efle ts in 3 d senkeeping computntions",
22 SJmp Naval HJdrodJn, Washington, 1998
3 Bertrnm, V and Thin t, G. "A Kuttn condi
tion for ship senkeeping computntionswith n Pnnkine
panel method", Ship TechnotogJ Pesenrch 45, 1998,
pp 54 63
4 Be trnm, V, "F lfilling open boundarJ nnd rndia~
tion condition in free surtace problem using Pnnkine
sources", Ship TechnotogJ Pesenrch 37, 1990, pp 47
52
5 Iwashita, H nnd Ito, A, "Senkeeping compu
tntions of n blunt ship capturing the influence of
thestendJ flow', Ship TechnotogJ Pmenrch 45, 1998,
pp 159 171
6 Jensen, G. "Berechnung der stntioni ren Poten
tinl tromung um ein Schifl unter Beruck ichtigung
der nichtlinenren Pnndbedingung nn der freien
Wasseroberfli che", IflS Peport 484, Univ Hnmburg,
1988
7 Nnk take, K nnd Ando, J. "P nkine source
method using rectangular panels on wnter su face",
11 Workshop Wnter Wwes nnd Flonting Bodim,
Hnmburg, 1996
8 Bertmm, V, "Vergleich ver
schiedener 3D Vertahren zur Berech
nung des SeeverEnltens von Schiflen",
JnErbuch Schiflbautechnische Gesellschnfl, Springer,
1997, pp 594 600
9 Be trnm, V nnd Thinrt
G. "A P nkine panel
method for ships in oblique wwes", Euromech 374,
Poitiers, 1998, pp 221229
10 Bertrnm, V nnd Thinrt, G. "F IIJ
three dimensional ship senkeeping computntions
with n surge corrected Pnnkine panel method",
J Mnrine Science nnd TechnologJ, 1998, pp 94 101
11 Bertrnm, V nnd Thint, G. "F IIJ 3 d
senkeeping computntions for renl ship geometries",
JnErbuch Schiflbautechnische Gmellschaft, Springer,
1998, pp 244 249
12 Tnniz wn, K, Tnguchi, H. Snrutn, T
nnd Wntanabe, I, "Experimental tudJ of wa:ve
pressure on VLCC running in short wa:vm",
J Soc N. w Arch Jnpan 174, 1993, pp 233 242
13 Mizoguchi, S. "Exciting forces on n high speed
contniner ship in regular oblique w es F equencJ
sele tions for calculating exciting forces bJ the strip
method ~', J Knnsni Soc Nw Arch Jnpan 187,
1982, pp 71 83
OCR for page 364
1 u3/h
10
05
10
05 _
I no
-
-
-t ~
t o - to
05 10
g 1: Motions fi
ut/h
02-
^^oo 0 ~
05
. u3/h
. us Ikh
05 10
Fig 2: Like Fig 1, but for ~ = 150°
o
. u 15 20 >/:
r VLCC, Fir = 0 131
10~
05:
01~
1 us /kh
o
o o
t
.
05 1 0 1 5 20 >/
1 u2 /h
1 0 1 5 2 0 >/: 0 5 1 0 1 5 2 0 >/
1 us /kh
,05 _
04l u / fir
03]
02~
01t
I ~
0 5 1 0 1 5 0 >/:
~ ° ~
05 1 0 1 5 20 >/
u6 /kh
lo
15 20 >/: 05 10 15 20 >/
OCR for page 365
P A/L=02 p A/L=03 Pt ' A/L=04
10- ~ 10- dl.. 10- ~oo
30° 120° 130° 30° 120° 130° ,~4^~
P A/L = 0 5 p ' A/L = 0 625 pt )/L = 0 75
1 0 , e'D O O. , , I O _ 0\ 1 0 - . —
50° 120° 180° 50° 120° 180° 50° 120° 180°
A/L = 1 0 P A/L = 1 25 pt A/L = 1 5
1 O- ~Oo ;°°°~°°°° ; °~ooooo
W~oo o ~ ~ -; .; · , , . ; ·; · . . .
50° 120° 180° 50° 120° 180° 50° 120° 180°
Fig 3: VLCC, F~ = 0 131, ~ = 180°; unste dJ prersure pt = p(~) /(pgh) nt x = 0 078Lp,, plotted
over circumference nngle; 90° = bottom, O starbonrd CWL; e exp, o RSM, str~p method
-
10-
A/L=02
50° 120° 130°
°o~o ~ o°
50° 120° 180°
P , A/L=I O
10- .
°.~0 0 0 0 0 oo~OOooo
50° 120° 180°
Fig 4: Like Fig 3, but for ~ = 150°
P , A/L = 0 3
10- O
. ^.^^^.^ ^ ^i
50° 120° 180°
P .
10- _
A/L = 0 525
~oO~Ooo::ooo
50° 120° 180°
P A/L = 1 25
1 0 - joO,OOO~o 0 0 0 0 0 0 o~ooooo
50° 120° 180°
p*. , A/L=04
1 0- Oo
O 000.
°~'o^~^ P~°
50° 120° 180°
p*, )/L = 0 75
1 O- °Oo
50° 120° 180°
P ~ A/L=1 5
50° 120° 180°
OCR for page 366
a
. h~W/pgLh2
3~ 0
.-
02- . .o
·tA-- o°°° °
05 10 15 20~/: 05
3 /h
-
05 _
o. O
. 3
o
3 t I :: ^ t
05 10 15 20 >/: 05 05
Fig 5: AOs for motions and added resistance for S175, Fin = 025,
3~ Row/pgLh2
nl-
05
03~
n a_
U ~
0 1-
~3/h
. t
05 10 15 20 >/
Fig 6: As Fig 5, but for Fin = 0 3
-
05 _
10
~ his /kh
15 20~/:
. 3
O U 1 O
I t .tSt. , , O . 0
05 10 15 20~/: 05 10 15
s /kh
, ~ .
3
. . . .
05 10 15 20 >/:
OCR for page 367
DISCUSSION
H Chun
Pus m Nctiorul University, Korea
1) I thi k that you solved iteratively the non-
lmear free surface boundary condition in Nat
case, how did you rre It the coefficients (such as
added mass, damping, ~estormg force) of the
motion equation?
2) You mentioned that She panel shift method
c m not meet the radiation condition for some
cases Do you have my clterrurtive idea to
satisfy the radiation condition for such case ?
AU? HORS'RRPLY
1) We heated She linearized (unsteady) ship
motion problem based on fully nonlinear steady
flown] Then, since the ship motion problem is
linearized, we c m defme all coefficients such as
added mass, damping, restoring force md
exciting forces Of course, the effect of steady
wave elevation md flow on the coefficients i
included in the computations
2) We see Bow tl led hybrid methods matching m
inner RSM solution to m outer Green function
method or Fourier-Kochin solution as most
promising approach to extend the method to low
values
DISCUSSION
M Kcshiwagi
Ky shu University, Jcp m
Fir fly I wish to commend you for completing
complicated calculations of the added resistance
with the pressure mteg~ation method I have c
couple of questions concerning the grids I
under t md that what you call 'cut - ou t' technique
is used for c tanker md possibly S-175
contamership as well Why do you thi k the
'cut-out'techmique goes w 11? If you use the
clterrurtive qucsi-steamline grid, how is the
result going to be? Does She computation break
down or are obtained results much different from
experiments? Ale there some criteria of which
grid should be used for given values of the b lock
coefficient md She Froude mmmber?
AU7 HORS'RRPLY
We used'cut-out'method for the computation of
LCC he reason is that qmtsi-ttretmlined
grids either give ve y distorted grids near the
bow md stern or do not locate collocation points
on She water surface near the ship ends Distorted
cells lead to problems with She radiation
condition he tut-out'medhod is robuster m
this respect for f 11 waterline forms We
recommend the cut-out method for ships with
block~oefficients clove 0 7
DISCUSSION
M Ohkusu
Ky shu University, Jcp m
Appmently, c pressure t Reducer is located on
th water line m your experiment This
t msducer is mtturally out of water some
duration during one period of the motion Then
the time histo y of the pressme measured will be
of not c sinusoidal but c truncated sinusoidal
So I wonder how you treated with the truncated
curve to derive your value of pressure if you
take She first harm onic components of f is curve,
you will obtain much smaller amplitude of the
pressure Nevertheless, your pressme It the
water line looks consistent with the pressure It
other locations
AU7 HORS'RRPLY
As you pointed out, w observedth time history
of c rmnctred simmsoidal curve m the pressure
measurement m She vicinity of he water Ime
From the time hi tory data, cmplit de was
defmed Is variation betw en zero-level md the
positive peck value[l2] So She experimental data
plotted m Figs 3 md 4 is not th first harm mic
components of She curve in the computations,
w do not take She truncated effect mto account
The reason why the calculated accuracy is
imufhcient in She vicinity of water Ime may be
due to treatment of the truncated effect
Representative terms from entire chapter:
added resistance
