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OCR for page 143
Second Order Waves Generated by Ship Motions
M.Ohkusu, M.Yasunaga (RI ~M, Kyushu University, Japan)
ABSTRACT
We present measured contou maps of u
steady wave elevation generated by oscillate y
motion of a ship navigating at con t nt speed
A ship model u -.1 for this experiment is S175
The model is forced to pit h at two different
mp itude Ou computerized wave measu ing
system gives accu ate wave contou map of the
f md mental f equency nd the second Harmon
ics component separately Exi hence of the see
ond order wave whose lU/iO itude is 15 to 20 per
cent of the -i st order's is confi med. Pattern of
the second order wave is discu -.1 in the ight of
asymptotic nalysis nd a theoretical method of
predicting the second order wave is te ted in its
accu acy on the measured wave contou maps
1. INTRODUCTION
Rational umderst riding of se keeping nd
accu ate prediction of ship behaviors in rough
seas are required more th n ever Non inear the
ory of th ee dimensional flow of ships, moving at
fo ward peed in high waves, is indi pen able for
these to be feasible That is why variou non
inear theories nd computational method have
been proposed nd implemented However ex
perimental evidence directly suppo ting or dis
proving those theoretically sophisticated meth
od is scarce This is what I argued at 22nd
ONR Symposium as well ( Oh u u (1998) )
Ex mination of the accu acy of theories in
predicting measured wave field generated by ship
motion or the diff action of the incident wave
will be more appropriate l, ./UI e the accuracy
of the measu ement is expected to be more re i
able th n that of hyd odyn mic pressure nd the
less integ ation effect of the wave field th n hy
d odyn mic forces will be suitable for proving or
disproving non inear theories more vivid y Fu
the more the accu ate prediction of wave eleva
1
tion close to the hu surface is ~ucial for prac
tical pu pose su h as accu ate e 4imate of wave
load on bow fiare nd de k wetness Wave field
away f om the ship is related to d mping of the
ship motion nd added resist nce in waves
Hyd odyn mics of waves generated by ship
motion or the diff action of the incident waves
when the ship's adv nce peed is not zero is a
hallenging nd intere4ing ubject for its own
s ke ( for ex mple Cao, Shu tz nd Beck (1994)
) Although several authors have presented their
results on the computed wave field but exper
imental data obt ined umder the exactly con
trolled condition is very rare ( Oh u u nd Wen
(1996) ) Actually they are not visible cleaHy in
the tank test becau e other effect ike the Kelvin
wave pattern nd the incident waves are mingled
Yet we wish to see experimentally how those u
steady waves are generated, not at a pot but a
whole area aroumd the ship, nd how the non in
eanty is exhibited in the wave field
In this paper n ex mple of the second order
effect m nifesting itse f in the 4 uctu e of mea
su ed wave field generated by a ship motion is
presented We investigate hyd odyn mic harac
teristics of this non inear effect with asymptotic
nalysis nd the computation of the second or
der wave elevation by the so called 2 5D theory
2. MEASUREMENT OF THE SECOND
ORDER WAVES
An ob j ective of ou study is t o visu ize non
inear effect in the fiow (u teady f ee surface eL
evation ) aroumd a ship moving at fo ward speed
in sea waves A ship model is umdergoing n o~
cillato y motion nd towed at a con t nt speed
in the water tank; the wave field, e g the con
tou map of in t nt neou u teady f ee su face
elevation, aroumd the ship model is what we wish
OCR for page 144
to obt in experimemqll-.y
Accu ate measurement of osci ato -. y wave
field aroumd a ship model, whi h is doing osciL
lato -. y motion nd rum ing at con t nt forward
speed, is not tr ightfo ward One of the present
authors has already repo ted at m ny occasion
ou tech ique to re l-e the measu ement Gen
eral idea of this technique is briefly described
below ( refer Oh u u nd Wen (1996) for the
det iL )
Wave is recorded at several lo cation contin
uou i. du ing a rum of a ship model; the location
are fixed to the water lank nd set on a ine par
Abel to the ship model track These record are
transferred to time series of wave elevation at ev
ery location in the reference f me moving with
theshipmodelifthelocation are onthe ine par
Abel t o the ship mo del track The transformation
is based on the fact that the wave probes rea h
to n identical location in the moving reference
f me at different time in t nts The transfo ma
tion requires some mathematical computation at
every location; a computer mu t be involved with
the measuring system The measurement is t o b e
repeated at different ines of the wave probe lo
cations so that it covers some region aroumd the
ship model
Ou wave measu ing tech ique enables u to
obt in So, :~ nd ~ so on at every location (x,y)
in the expression of the wave field
:(~,Y,[) = ~o(x,Y)
+~(2,y)e~+h(2 y)e22~2+ (1
where the coordinate y tem moves with the av
erage position of the ship model The ~ y pl ne
coincides with the c m water su face nd the
positive z is vertica~ly upward; the origin is at
FP of the ship model nd the ~ is directs rear
ward; ~ is the f equency of the ship model's o~
cillato y motion
The fi st term on the right of (1) is the
steady wave elevation the domin nt part of
whi h is the Kelvin wave pattern It will cont in
possibly the higher order effect resuting f om
the interaction of the os illatory part The sec
ond term is the oscillato y part at the f md men
tal f equency nd the third is of the second har
monics M in part of the f md mental f equency
term will be the inear effect nd the second har
monics part will be the second order effect
The u teady waves might be interacted
with the steady part It is however impossible to
confi m it by experiment becau e the u teady
waves without the te dy part is not re istic to
be compared with the one with it as long as the
ship has forward peed; the u teady waves gen
erated by the ship motion at no forward speed
is physica~ly other thing th n what we are con
ce ned
Con i tency nd repeatab ity of the mea
su ed waves are mo t pe fect It enables u to
d aw accu ate wave contou map with the resu ts
of the repeated measurement The teady wave
component is hard y ffected by the existence of
the um teady motion; the steady f ee su face eL
evation measu ed when the ship model is given
oscillatory motion is in good ag eement with the
one when it is towed on c m water at the s me
speed with the motion uppressed One excep
tion we observed so far is a sma~l te dy depres
sion of the f ee su face in ve y f ont of bluff bow
produced by the effect of the wave diff action
S175 model was forced to pit h at two differ
ent mag itudes expecting the larger non inearity
with the larger mag itude of motion Ou com
puterized wave measu ing y tem enables u to
obt in the in t nt neou wave contou map of
the fi st harmonics ~ component nd the second
harmonics 2w component separately The former
is con idered to corre pond to the fi t order ef
fect nd the latter the second order effect x
mples of the measu ed wave contou map are
shown in Figs 1 nd 2
Figu e 1 is two snap shots of the contou
map of the fi st order :~e2~2, the upper part of
the figu e represents the real part of :~ (cos com
ponent ) nd the lower part the negative of the
imaginary part ( sin component ) Elevation is
displayed g adationa~ly black t~white as well as
in the contou map The elevation is norm L
ized by the mp itude of ve tical motion at FP;
this figu e is the resu t for larger mplitude of
the pit h, 0 00185 radi n This gives the vertical
motion at the bow as large as 7097o of the d aft
OCR for page 145
o
l l
I ~
I ~
I
. ~~ ~~
I
I
~~ ~1
-
=
II II II
^~=
#~7
S
o
I I I
. . .
o
o
13~->
{.
o
o
1
OCR for page 146
The ship peed is F i = 0 275 i nd the f equency t
~ of pit h norm ized by the ship length L i nd
the g avity con 4: nt 9 is 5 26 a, the f equency
norm ized by the ship speed U i nd 9, is 1 446
With Fig 1 i nd other result at smaller i:
p itude of pit h though not shown here, we may
conclude that the me. in ed wave contou map
at the fund: mental f equency, when its altitude
norm ized by the pit h mag itude, is identical
rem d ess of difference in mag itude of the pit h
motion It sugge as that Fig 1 shows ce t in y a
inei: effect
A wave . I-m is noticeable in Fig 1 The
ine connecting the peaks of the wave system
m kes about 21° with the r i xis Wave pattern
of Fig 1 most perfectly ag ees with the pre
dieted on inei: iviiumptioni i: i described later
Figu e 2 is the napshots of the in it: nta
nexus f ee su face elevation of :2ei2~2 at the
so me condition of f equency of pitch i nd the ship
speed, i nd with the so me no malization i: i Fig 1
While the contou map c: n be d awn with this
ci: ie, we display the wave elevation on y g ada
tionally; it appe ently exhibits the eke acted
tics of the wave pattern more di 4inctly ~ mea
su ed at h f the i mplitude, not shown here, does
not reveal so clei Iy the featu e of the wave pat
tern It uggests :~ will be of the second order
with re pect to the i mp itude of the motion
Existence of the second order waves, whose
mag itude is 15 to 20 percent of the fi 4 order's,
wi: i confi med. Wave pattern demon4rated in
Fig 2 is so distinct that it is cert in that it hi: i a
clei: physical mei: ing
Two wave system; i: e seen in Fig 2 One
wave is most ike the wave y 4 em seen in Fig 1;
the ine connecting the pealcs of the wave system
m kes about 21° with the r is, though the
wave length ( the dist: nce between two adjacent
pealcs ) is different f om the one in Fig 1 Other
wave ystem hi: i the ni:~rower i ngle w ke; the
pe k ine m kes 13 5° with the r is
We uppose that the second order waves
have two components The so ca~led boumd com
ponent whi h hi: i the s: me phi: ie peed i i the
fi st order wave ( with i n i nalogy of u idirec
tional 2D Stokes wave ), i nd the f ee wave com
ponent whi h satisfies the s: me dispersive rela
Ea h effect of those two components in the
mei: iu ed waves is to be sepi: ated by uti izing
the fact that their wave length i i: e different f om
ea h other The f ee wave component will be
m inly due to the non inei: ity in the ship body
condition i nd the boumd component is a re ult
of the non inei: ity in the f ee su face condition
(ag in with i n i nalogy with Stokes wave ) The
fo mer will correspond to the 13 5° w ke in Fig 2
The latter will be mi: if ested i: i the 21° w ke
This conjectu e will be confi med more qu: nti
tatively in the next section
3. ASYMPTOTIC CHARACTERISTCS
We study i: iymptotic hi: acteri 4ics of the
oscillatory wave pattern ( Eggers (1957), Be ker
(1958) ) :~ (r, y) at a dist: nce relatively fi: to
the wave length is given in the form of
~(x Y) = J F~(9)e~(~ ~+~tysin9) 9
+ / F~(9)e~(~2~cos9+~2ysin9)~9 (2)
k~,~(9) = 2 °~ 9(1+ 2rcos9+ ~),
K 9 U~
° U~' r
The direction 9 that does not give real k~,~ is to
be excluded f om the integ ation of (2)
When k~,~/~ is Li ge, method of sta
tioni: y phi: ie will give a good estimate of the
wave patte n (2) except at a cauitic ( Iwi: ihita
(1990) ): Wave elevation at a location (r,y) is
contributed by a discrete number of wave compo
n nts whose direction 9 i: e given i: i the soluti on
pkg~(rcos9+ysin9) = 0 (3)
Equation (3) is rewritten into
y cos R sin 9 f k 4
~ sin~ 9 + ~/~ or
OCR for page 147
y cos ~ sin ~
_ = for kit (5)
sine ~ ~/~ 01
y/r = t n So mu t be less th n a value S°l,2
if (4) or (5) has real solution ~ S°l,~ is the ngle
of the outer boumd of the wave system outside of
.1 i h the wave does not exist in the asymptotic
sen e C e Us nd trough of the kit wave sys
tern computed by (5) at Fn = 0 275, r = 1 446
is shown in Fig 3 The outer boumd at 21° is
seen eleqrlv The kl wave is imited within mu h
narrower region ( about 10° to the r is )
p K2 Ware (Fu=0.775,Tzu=1.445)
,_, -~—;~. 1
0 05 10 15 X 20
Fig 3 A ymptotic Wave Patte n
The wave length on the ine S°l,~ ( the dis
t nce of two adjacent peaks of wave along the
ine ) will be given
A I ,I 21
L kl,~ COS(iI,2 ~cI,2 j
where tic is the ngle of elementary waves con
tributing to the wave elevation along So = S°l,~,
whi h is the solution of (4) or (5) at y/r =
t nook,
When we look at Fig 1 in the ight of the
knowledge of asymptotic waves, the kit wave is
visible but the kl wave is not Actually the pat
tern shown in Fig 3 is ju 4 what we see in Fig 2
SO of Fig 3 is 21 1° nd /L O. 642 They are
the s me as the ngle shown in Fig 1 nd the
dist nce between two mark it confines that
the wave v. hem we see in Fig 1 is the k2 wave
system
We Judy a wave system of the narrow am
gle of 13 5° we see in Fig 2 in the ight of the
inear theo y The wave y hem in Fig 2 is of the
second order, yet the inear theory c n captu e
one featu e of it As expl ined in the previou
section, the wave field of the second order will
be interpreted to be composed of the f ee wave
component that satisfies the body non inear but
the f ee surface inear condition nd the boumd
component that sati lies the f ee su face non in
ear but the body inear condition At radiation
problem we may consider the former as a in
ear wave with the second order body boundary
condition at the f equency of 2w Natu all-. the
body condition is decided by the fi 4 order flow
but imposed on the equi ibrium position of the
body as long as the body surface vertically inter
sects the f ee surface A ymptotic haracteri Tics
of this wave may be nalyzed in the inear way
described above
Par meters determining this wave are Fn =
0 275 nd r = 2 892 These par meters give
SO = 13 5° nd the wave length on the ngle
is 0 268, .1 i h perfectly ag ee with the observed
on a wave system of the maller ngle w }e nd
prominent behind the ship in Fig 2 This will not
be coincidence becau e there is no other possibiL
ity giving the ngle nd the wave length exactly
We may conclude this wave is cau -.1 by the body
non inearity but with the inear f ee surface con
dition
Other wave system at the ngle S°~ = 21°
seen in Fig 2 will probably be the boumd compo
(6) nent originated f om the f ee su face non inear
ity
4. COMPARISON WITH THEORETI-
CAL PREDICTION
V idation of theoretical method is at
tempted in term of accu acy in predicting the
contou map of the fi st order nd the second or
der waves obt ined in ou experiment Here we
employ the 2 5D theory for this pu pose This
theo y is umderstood to be practical in providing
relatively 4able resu ts nd requiring less com
putational burden as far as a ship is slender nd
at moderate to high peed; f Iy non inear time
dom in computation is done often with this ap
OCR for page 148
._~ ~ ~.~ (10) are the -i st order
these computation, experimental v idation at condition for ¢~ nd :I, nd (11) nd (12) are
the higher order will be necessary, though it has the second order condition for ¢~ nd
never been attempted
Theoretical approa is we u e here for the
2 5D theo y is not a f y non inear time dom in
computation but a pe tu bation nalysis up to
the second order in the f equency dom in A
reason is to make the second order effect more
distinct; direct companson of the f 1. non inear
computational results in time dom in nd the
measu ed wave elevation might not necessarily
provide a qu ntitative info motion on the higher
order effect We have to recog i-e that there
is a demerit in the pe tu bation nalysis that it
does not predict really large non inear effect but
moderate non ineanty
Let ¢(x, y, z, t) nd :(x,y, z,t) represent re
spectively the velocity potential of the u teady
flow nd the u teady wave elevation generated
by the ship motion Condition of them to be
satisfied at the f ee su face z = ~ are
pros is If we are to be ce t in of the accu acy of Equation (9) nd
O 94+( t+U t)¢
+ Ff ¢N +
2 Lead)
0 = 94~ + ( t + U,9z) ¢~ (9)
0 = ( +U ~ )~t ~t (10)
)=942+( t+Ud )~+~( t+Ud )~2
2 [( Z)x ) ( Z)y ) ( ZJz ) ] ( )
o=: t+Ud9zJ~ Hz + ~ 8x
¢t :t ¢t 12
+ ~ y :t Hz] ( )
All the condition are to be sati fled on z = 0
We introduce here a ~ucial assumption so
that we may proceed to the next stage of naly
sis: the mag itude of the slenderness par meters
~Z)~: + ~Z)~: 1 (7) s is independent of that of ~ i e we ret in 52
y) ~ZJz) ~ term that are of the lowest order with respect
to s despite that practicaLy s > ~
We may as ume for ny flow qu ntity f
0 = ~ {) + u {) ~ ~ + Z)¢ Z): + Z)¢ Z): {~¢ (8)
~ t t ) Z)x Z)x Z)y y ZJz
Suppose the expansion
:=~+~2, ¢=~+~2
~f ~f
<< ~
Yet we ret in U8f/~x in the fl st nd second
order f ee su face condition above because of
high peed U of the ship
nd as ume that the second te ms ¢~ nd :~ are Fiom now on ou nalysis is conce ned with
of higher order th n the fl st term ¢~ nd :~ the oscillato y part of the flow i e we umder
with respect to the mag itude ~ of ship motion st nd here fter that ¢~ nd ~ represent the o~
It is becau e we simply assume that the steady cillato y part of the second order Ou nalysis
flow aroumd the ship is u if o m flow U into the ~ is in the f equency dom in nd everything of the
direction Then one c n flmd the fl st nd second fl st order is oscillating at the f equency ~ Con
order f ee su face condition by ub tituting the sequently it is tr ightfo ward to show
expansions into the f ee su face condition (7)
nd (8) to separate the fl st nd the second order
¢~(~>Y>Z)=~(~>y~z)e 2t~+2~2 (13)
OCR for page 149
¢2 (2 ,y, a) = ;2 (x, Y. z)e i~+2~t
:~(~>Y>z)=~1~(~>y~z)e id +~
:2 (2 ~ y, z) = ~13 (x, y, z)e ~ D ~+2Ud2
(14)
Substitution of those e pression into the
-i st nd second order f ee surface condition 13
spectively will derive the condition for ~g2 nd
0t,2
U ~2 = 9~3
zoo? - 2 1 F - t 9ql
,9z ZJz 2 L MY Y
U - t = gut
U8ql - t
~~ 33
u ~ 824t 1
2 L0~8z82]
4 [( y ) + ( ZJz ) ] ( )
?- 3.2 ] (20)
In deriving equation (19) nd (20) we have re
t ined the lowest order te ms of e mong the
term of 52 The s me reasoning lead to
~~t,2 + ~~t ~ = 0 (21)
The body boundary condition are derived
,9n = 5[(2 20)~+F~]r~,e't~ (22)
- 2 = 0 552(x 1 ~ 1 2 aZ2 (23)
The body condition (22) nd (23) are both im
posed on the su face of the ship at the equi ib
rium position rig is the no mat directed to the
fluid on the ship's sectional contou nd r2~ is
) the z component of the normal so is the ~ coor
15 dinate of the longitudinal center of mass aroumd
( ) whi h pit h occu s All the formu ation above
16) is umder the condition that the ship fo m is wall
sided i e the hu su face intersects vertica~ly the
z=Opl ne
Equation (19),(20) nd (23) uggest that
;~ nd ~1~ will be decomposed into tw parts, one
part satisfying the homogeneous f ee surface con
dition corresponding to (21) nd (20), nd the
17 body condition (22), nd other part satisfying
( ) the f ee su face condition (19) nd (20) nd the
body condition (23) with the right h nd side re
(1S) placed by O. The fo mer will be the f ee wave
part due to the body non inearity nd the latter
the boumd part due to the f ee su face non inear
ity
Ou computation of both parts revealed
that the wave elevation by the fo mer part is so
sma~l compared with the latter's effect W ile
this is true on y with the wave elevation close to
the ship nd the f ee wave part due to the body
non inearity is signiflc nt behind the ship model
as shown in the previou section, we may ignore
it in ou computation of the wave elevation rela
tively close to the ship We concentrate hereafter
ou computation on y on the boumd te m due to
the f ee surface non ineanty When we umder
st nd ;~ nd ~1~ st nd for this term, then the
body condition will be
O
(24)
The f ee su face condition (17) is integ ated
to mar h fo ward the value of ;~ on z = 0 in ~
direction The condition (18) updates the f ee
su face elevation ~1~ at new ~ Dete mination of
;~ on z = 0 nd ;~ on the ship' section at new
~ is by solving the two dimen ional boumdary
value problem for ;~ with ;~ on the ship section
nd ;~ on z = 0 given Subscript denotes the
differentiation into its direction
Numerical implementation of this approa h
for the fl st order solution is 4r ightfo ward ( for
ex mple Wen (1997), Faltin en nd Zhao (1991)
) The 2nd order Rumge Kutta scheme was u ed
to forward the solution f om on section to next
OCR for page 150
section; me ing Rep ~x in the ~ direction we
hosen to be L/SO Two dimensions l,,,lul.L. -
v3 ue problem at each sections PIXIE- when ;
is given on z = 0 3 nd ;~ given on the sections
contour, we 3 solved with segment size 4 x 103L
on z = 0 on one side 3 nd eve y section. contou
divided into 30 segments Behavior of ;~ on z =
O away f om the body surface ( y > 0 5L) is
approximated by -. I dipole behavior ( Wen
(1997) ) We employed rather c13 3sic3 initi3
condition ;~ = ~1~ = 0 at ~ = 0
fter ;~ End ~1l 3 e obtained, we solve for
;~ End ~1h in 3 most simile manner Equation
(19) End (21) 3 e integ ated to forw3 d ;~ End
~12
We need to eve uate the forcing terms due
to ;~ 3 nd ~1~ on the right hi nd side of these equa
tion3 We mu t ret-. onnumeric~ differentiation
for eve uating ;~ 3 3 ;~ is given on z = 0 We
eve uated ;~ by solving a new L,,~u~.L~ -. y v3 u
problem for ;~ when ;~ 3 e prescribed on both
the ship section 3 nd the f ee surface
The condition imposed at 13 ge y is deter
mined 3 3 follows The forcing terms behavior
at 13 ge y is 3 eatery nown because Oh's we
3 3 umed of ve tics dipole at the -i 4 order com
putation For ;~ we 3 flume the slowe 4 attenu
ation conceivable at 13 ge y i e we 3 flume the
some behavior 3 3 ;~
The sing 3 ity of ;~ at the intersection of
the body su face End z = 0 will be Z logZ
where Z is the complex coordinate with the ori
gin at the intersection ( Cointe, Mo in 3 nd Nays
(1988)) it let ds to the sing 3 ity log Z of ;~
We avoided the difficu ty in solving the boumd
3 y v3 ue problem in ea h crosssection3 plane
by collocating not at the intersection but at lo
cations very close to it
Figu 33 4 to 13 3 e the comp3 I=. n of
the measured End the computed wave elevation
Clews in cr is sections planes at severs differ
ent x/L The me. 3u ed is to en f om the con
tou map shown in Fig 1 Figu 33 at the left side
of each page show the wave elevation at the in
stint of hit = 0 ( cosine component ) End the
right figu es the wave elevation at the in 43 nt of
hit = 1r/2 ( sine component ) in those figu 33
Theoretics (AD) depicts the computed by th ee
limension3 End desing 3 ized Rely He panel
method ( Sclavoumos (1996) ) usmg the doubly
model flow 3 3 the L. 3ic Heady flow
First of 3 we see that the wave elevation
norms ized by the 3 mp itude of the ship motion
at ~ = 0 is 3 most identic3 de pite the differ
ence in the 3 mplitude ( white 3 nd black circles )
It imp ies that the measured Hoot be accounted
for by ine3 theori 33 While both the theoreti
c3 2 5D 3 nd 3D captu e the genera featu es of
the wave elevation, either of the theoretics does
not ucceed in predicting accu ately the height
of wave elevation; the me. 3u ed is higher in the
cr At 3 nd deeper in the trough the n the theoret
ice predictions it is not upposed to be caused
by the 2 5D's incorrect initi3 condition at ~ = 0
because even the 3D not Offering f om this de
feet f3i 3 to predict them
Figu e 14 is the contou map predicted by
the 3D computation Once age in we see that the
whole featu e of the theoretics ag e 33 perfectly
with that of the measured shown in Fig 1
In Fig 15 to 23 we comp3 e the theoreti
c3 prediction of the second order wave he32~2
given by the 2 5D theory with the me. 3u -i cor
re pending to the resu t shown in Fig 2 The for
mat for the comp3 I=. n is the so me 3 3 for the fi st
order except that the sine component represent
the wave elevaation at 2wt = 1r/2 It appe3 s
that the genera featu e of the wave elevation is
predicted by the 2 5D theo y It me. 3 that the
second order wave elevation observed close to the
ship might be the boumd component caused by
the f ee surface non ine3 ity 3 3 sugg cited in the
previous section
The relatively 13 ge wave elevation of the
second order is not predicted by ou theoretics
method It may not be su pricing because the
prediction of the fi st order wave is not so correct
in the mag itude of the wave
The theoretics resu t on the second order
wave elevation f3 behind the ship is not av ilakle
( the 2 5D theory is not capable of computing the
wave field behind the ship's Bern ) We c3 n not
confi m ou statement on the narrow 3 ngle we e
we see in Fig 2
OCR for page 151
5. CONCLUDING REMARKS
We presented in t nt neou contou maps
of the -i st order nd the second order waves gen
crated by a ship motion They are not visible on
u ual experiment at water tank Those contou
maps were made av ilable by ou computerized
tech ique for wave measu ement
V idation of a theoretical method was at
tempted in te ms of accu acy of predicting those
contou maps of the -i st order nd the second
order waves
Theoretical prediction of the -i st order wave
is not entirely accu ate de pite the slender hu
fo m of the ship model W ile pattern of the
wave contou is accu ately predicted, discrep
ncy exhibits itse f in the predicted wave height
being much mailer th n the measu ed wave
height near the hu surface The discrep ncy
sugge Is that ou theoretical model is to be im
proved since we be ieve the measu ed -i st order
wave is of "the -i st order"
Two components are clearly disting ished
in the second order measu -.1 wave pattern, one
satisfying the inear dispersive relation nd the
second order body condition, nd other satisf
ing the non inear f ee su face condition nd the
homogeneou body condition The latter upper
ently is domin nt in the wave elevation at the
location close to the hu surface
A theoretical approa h to compute the sec
ond order wave contou was proposed The pre
diction of general featu e of the wave contou
does not seem so inaccu ate However the mag
nitude of the measu ed wave is larger th n the
predicted This will be inevitable when we con
sider inaccu acy in the prediction of the fl 4 or
der; the second order wave was computed uti iz
ing the fl st order prediction in ou theoretical
approa h
Practical imp ication of ou resu t to sea
keeping study is to be 4udied in future
ACKNOWLEDGEMENT
The authors a knowledge the help of Prof
H. Iwashita, Hiroshima University, in provid
ing his computational resu t by Rankine p nel
method
REFERENCES
Becker E (195S) Das Wellenbild einer umter der
Oberfla he eines Stromes Schwerer Flu igkeit
Pu ierendenQuelle, Z. Angew ndte,Mathematik
umd Mechanik,Bd38
Cao YS, S hutz W nd Be k R (1994)
Immer ngle Wavepa kets in n Un je dy W ke,
Proc l9thSymposium on Naval Hyd odyn mics,
Seou
Cointe R. Mo in B nd Nays P (1988) Non inear
nd second order transient waves in a rect ngu
lar tank, proc BOSS'88, T ondheim
Eggers K (1957) Uber das Wellenbild einer
Pu ierenden Sto mg in T anslation, S iff umd
H. fen,Heft2
Faltinsen O M nd Zhao R (1991) Numerical pre
diction of ship motion at high fo ward speed,
Phil T ans Royal Soc Lond A 334
Iwasita H (1990) Green function method for ship
motion at forward peed, PhD Thesis, Kyu hu
University
Ohku u M (1998) V idation of Theoretical
Method for Ship Motion by Means of Experi
ment, Proc 22nd Symposium on Naval Hyd o
dyn mics, Washington DC
Ohku u M nd Wen G (1996) Radiation nd
Diff action Waves of a Ship at Fo ward Speed,
Proc 214 Symposium on Naval Hyd odynan~
ics, T ondheim
Sclavoumos P (1996) Computation of wave ship
interaction ,Ad~c~ces m Mczme Hydrodyncm~cs
edited by M Ohku u, Computational Me hanics
Pub ication UK
Wen G C (1997) Theoretical prediction of sea
keeping of high speed ships, PhD Thesis,
Kyu hu University
OCR for page 152
o6n(
0~4 0~3 0 8y/L0 7
(Ist order, Cos)
)0 01 02 03
Fig 4 Wa:ve elevation
5 2m model
X/L=0 32 (
Ist cos component )
Amp 00370m
Amp 0013 im
Theoretica1(3D)
-- Theoretica1(2 iD)
.I.-L ~ ~
Y/L
~os)
00 Ot 02 03 04 0 i 03 / 07
Fig 5 Wa:ve elevation (Ist order, Sin)
S175 2m model
03 X/L=
04
02
00
02
04
03
oo 01 02 03 04 0 i 03 07
Y/L
Fig 7 Wa:ve elevation (Ist order, Sin)
S175 2m model
~, X/L=040 ( Ist sin component )
oo 01 02 03 04 0 i 03 07
Y/L
Fig 8 Wa:ve elevation (Ist order, Cos)
oo 01 02 03 04 0 i 03 07
Y/L
Fig 9 Wa:ve elevation (Ist order, Sin)
OCR for page 153
0.4
0.2
0.0
-0.2
S175 2m mode!
X/L=0.52 ~ 1st cos component
0.6 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ',
· Al~p. 0.0370
04 o Al~p. 0.01851~
o Theoretica1~3D)
· ~ Theoretica1(2.5D)
-;; t :~+
-0.6 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
).0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Y/L
Fig. 10 Wave elevation (1st order, Cos)
S175 2m model
0 6 X/L=0.72 ~ 1st cos componen
· Al~p. 0.0370
o Al~p. 0.0185
_ ~1~ 1/~ $
_ ·Q
_ ~.
_ ~ ~. (
_ a, ~ ~
-0.4
-0.6 1 1 1 1 1 1 1 1
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Y/L
Fig. 12 Wave elevation (1st order, Cos)
1 11~T'Cti~) .
~ Theoretica1~2.5D)
1 1 1 1 _
6~. ~
1111 1111 1111 1111
y/T
O.O -
-n.4
S 175 2m mode!
X/L=0.52 ~ 1st sin component
0 . 6
1 · Al~p. 0.0370
04 1 0 Al~p. 0.0185
. 0 ~ Theoretica1(3D)
0.2 t L~ lt
n n ./ \~. I _~.~QQ_~QQ~ I
-(~2
-nri ~ I I I ~
.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Y/L
Fig. 11 Wave elevation (1st order, Sin)
u.u
0.4
0.2
0.0
-0.2
-0.4
-0.6 -
Fn= 0.275, Tau= 1.446
S17r
A
1 1 1 1 1
2m model
X/L=0.72 ~ 1st sin component ~
~ ~ ~ ~ I I I I I I I I I I I I I I I I I I I I I I I I I I
· Al~p. 0.0370
° Al~p. 0.01851~ ~
Tlleoretica1(3D) | |
~ I --- TlleoretiCal(2 5D) |1
~te~t~--~
1 1 1 1 1 1 1 1
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Y/L
Fig. 13 Wave elevation (1st order, Sin)
61st cos component~
............... 8175'' '
| i\ 1st sin component~
0.5 1.0 1.5
fig. 24 3D-Computed wave pattern ( 1st oder )
X/L 2.0
OCR for page 154
S175 2m model
X/L=0 24 ( 2nd cos component
<0 20 T T T T
O ti =
010 =
OOi =
000
O i =
010 =
Oi
n n n ~
S175 2m model
<0 20 T T T
O ti =
010 ~ ~
OOi . .
0 00 T ~
O i ~
010 =
Oi
020
00 0t
~u;~ ~ ~ . ~ <0 20
P Amp 0 0370m ~ F ~ v Amp 0 0370m
Theoretica1(2 iD) ~ O t i ~ ~ Theoretica1(2 i6
~ ~ ~ CC :~- W
_ ~ °ti ~ ~ ~
t 0 2 03 04 0 i 03 020 0 0 t 0 2 03 04 0 i 03 / 0
Wnve elevation (2nd order, Cm) Fig 16 Wn:ve elevation (2nd order, Sin)
S175 2m model
S175 2m model
Y /I n tn ~ PnA O~ O~m~n~nt t
0 20 T T T T
F~o ti =
010 =
OOi = .
000 .l
Oi
010 =
O i =
020 o~o .~o
P A m p 0 0 3 7 0 m
Theoretica1(2 iD) |
'---~ T'-'T~
u2 03 04 0 i 03 07
Y/L
Fig 19 Wnve elevation (2nd order, Cm)
S175 2m model
0 20 X/L=0 40 ( 2nd sin comDon~nt
Oti
010
OOi
000
OOi
010
Oti
020
v Amp 0~0370m
- - j Theoretica1(2 iD)
~ GH ' ~
oo of 02 03 04 0 i 03 07
Y/L
Fig 20 Wn:ve elevation (2nd order, Sin)
OCR for page 155
S175 2m model S175 2m model
X/L=0 52 ( 2nd cos component ) X/L=0 52 ( 2nd sin component
0 20 ~T1T ~ ~ <0 20 ~r ~T _
v Amp 0 0370m F v Amp 0 0370m
O t5 ~ ~ ~ Theoretica1(2 5D) ~ O t5 ~ ~ Theoretica1(2 5D)
OtO ~1 ,':,~ ~ ~ OtO :~ ~ .
t5 ~ ~ = = = = ~ ~005 ~ = ~ = = = ~
Ot5 = = Ot5 = =
020 ~ ~ ~ ~ ~ ~ ~ 020 ~ ~ ~ ~ ~ ~ ~
00 0 t 02 03 0 t 05 03 7 00 0 t 02 03 0 t 05 03 / 07
Flg 21 Wvve elevation (2nd order, Cm) Flg 22 Wa:ve elevation (2nd order, Sin)
S175 2m model
0 20 ~TT~ ~TT~ X/L=0 72 (
OtO = ~ =
005 ~ ~;
0 5 =
OtO =
O t5 ~ ~
n n n ~ n 0
Amp 00370
-- Theoretica1(2 5
-~ ' 2nd cos component )
<020
~ Fo t5
= OtO
= 005
~ 000
= 005
= OtO
Ot5
020
03 op n
Y/L
Fle 23 Wvve elevat
OCR for page 156
DISCUSSION
X B Churn
Bureau Veritas, France
I cm interested in the stecdy part of ship waves
you measured Fr m She second order theory,
Here is c contribution of second order to She
Beads pen. which maybe the difference of wave
pattern between that in exam water md the total
ready part in the un decdy m csmem nt
Is it tme? If so, base you my mfcrm Don
(results) on She stecdy part of measly meet in
unsteady te d ?
AUTHOR'S REPLY
We are too mtere bed m the second order
mteraction of She ready md un decdy ship wa.es
First of all w must know that the effect of the
unsteady waves upon She stecdy ones will be
observable but not vice verse Reason is that She
unsteady wa.es not under She inf uence of She
stecdy waves is not realistic for the case of our
concern i e Hen c ship has forward peed
While w have not deported the result
here but ce tamly w mesmed the ready ship
wave without the ship motion or mcident waves,
md compared it with She decdy wave component
of the .s.e motion when c ship is forced to
oscillate or in the incident waves What w have
found is chat She stecdy wave is hardly effected by
the mmstelrdy wave et least for the mcgnit de of
She oscillation w adopted m our mernnema~t
In other words the stecdy component of the wave
motion Hen She ship is oscilEdmg cg es wish She
heady wave et no ship motion
One exception is that She stecdy wave m
fi ont of very blmmt bow is depressed slightly when
She ship is m incident waves
Representative terms from entire chapter:
ship model
