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OCR for page 1
Modern Seakeeping Computations for Ships
R. Beck (University of Michigan)
A Reed (David Taylor Model Basin, Carderock Division, Nava Surface Wa fare Center)
A'dSTRACT
Cm~ent computational methods tm sulvmg seakeepmg
p oblems ot ships wifh tmwad sped a~e ~ewiewed A
b~iet hisk~ical p ~sp tive is given k show fhe ink~de
p ndency md development ot fhe ditte~ent ship mmtion
themies fhat a~e p eendy bemg used Th se a~e placed
in conk zt by a discussion ot fhe tozommmy ot seakep
ing cumputations ~elative to fhe tully mnlmea~ mcom
p esible t~ee su~tace viscous fiow p oblem The state
ot the a~t in computational sekeepmg ot ships is dis
cussed ~ gene~al, fhe accu~acy ot the solution mu t
be bal mced agamst the computational ettmt The d
vanced codes give mm~e detailed md bette~ solutions, but
they ~equi~e sup ~ cumpuk~s m the equivalent Fully
md pa~tially mmnlmea~ in iscid computations tm wave
dit'fiaction, md added mass md dampmg a~e demibed
md a tew examples a~e p ovided k illushate fhe impact
ot the va~ious levels ot compleity ot fhe calculations
on fhe accu~acy ot ~esults cumpa~ed to exp dmental ~e
suits Finally,aseziesot tak otfhea~tissuesae~aised
computationally etficient mmmezical medhods, lage am
plitude mmtions md capsizmg, hmizontal plane mmtions
(cuuplmg betwen seakeepimg md maneuvetmg), finik
depfh in fhe litk ~ als, md validation md vesificatimm ot
codes tm eheme motions
I INTRODUCTION
Modesn seakeping computations ate used in all asp t
ot engineedng tm fhe maine envi~omment They have
become a t mda~d d sign k ol; fhey a~e used m sim
ulatms; md they a~e used op ~ationally k p edict fhe
motiom ot a vessel in ~eal time Mode~n seakep
ing computations a~e p ~tmmed usmg a wide va~iety ot
techmique fimm simple ship fhemy k ehemely com
pie tully mmnlmea~ mmste dy RANS computations To
cove~ all asp t w uld ~equue a book, mmt a shmt p~
p ~ Cunsequendy, we a~e going k limit the discus
sion to ships at tmwa~d ped This la~gely elimmak s
my discussion ot fhe computational k hmiques dewel
op d by fhe ottshme oil mdushy in mdez k compuk
wave loads md motions ot ottshme st~uctu~e We do
mmt w mt k minimie the conhibutions ot the ottshme
1
ind st y which have been subst mtial (some might even
a~gue fhat mmdezn computational techmiques have been
dkiven by fhe neds ot the ottshme indust~y), but the to
cus ot fhis symp smm is naval hydkodynamics with it
emphasis on ships at tmwa~d sped
Modesn seakeping computations a~e t'a~ fiom a m~
tu~e engmeedng science Theze a~e seve~al asp ts to
ship seakeepmg that m kc it one ot th mo t challeng
img p oblems in fhe madne hydkodynamics field it has
all fhe compleities ot wave ~esistance m maneuvezmg
p oblems wifh fhe addition ot unsteadmess due to in i
dent waves The ultimate goal, ot cuut se, is a unified th~
my ot ~ei tance, maneuvezmg, md seakeping Hisk ~i
cally md tm a vaiety ot ~esons, each ot the fields have
dewelop d indep ndently At p esent fhey a~e till sep~
~ ated md it will p obably be twenty yea~s betme compu
tations a~e huly unified Unto tunak Iy, deign p oblems
will mmt wait md designezs a~e constmdy pushing tm
bette~ computations in this pape~, we w mt k summ~
~ie fhe p esent state~t fhe at in seakeeping comput~
tions md then p mt out mym ~eseawh issues fhat ned
to be addkessed
The mym ditficulties m seakeeping cumputations
a~e the mmnlmeadties Theze a~e mmnlmea~ities associ
at d with the fiuid in fhe tmm ot viscosity md fhe ve
locity squaed te~ms in the p essu~e equation The t~ee
su~tace c mses mmnline behavim due k th nat ~e ot
the fiee su~tace bounda~y cunditions md the mmnlinea
behavim ot fhe mcident waves Finally, fhe body geom
eby otkn cmse mmnlmea~ hydkostatic ~esk~mg tmces
mdmmnline behavim atthebody/fiee su~taceink~sec
tion Ime The only good news is that bee mse ot tmwad
sped ships kndto be long md slendez with smmothva~i
ations along theu lengfh This geome~ic teatu~e ot typ
ical ships is the basis ot m my app ozimations that have
allowed a signific mt amount ot p ogtess to dak
Recendy, seakeping computations tm ships op
ating m fhe littmaMegion have become ot ink ~e t Ot
shme computations ae ofien done m finik depfh, but it is
ummsual tm ships Most themie md computations have
been tm infinik Iy deep wak ~ M my themies could be
exknded k finik depfh in a ~elatively shaighffmwad
mame~ Fm eample, ~eplacing the deep wak~ Gteen
tunction wifh a finite depfh G~en tun tion c m exk nd
OCR for page 2
Imea~ ip theo y ship mmtion p og~ams Unfi tunately,
bee mse ot the mmnlinea~ities associat with shallow w~
tes waves, cunent ettects, md mmn~nitmm bottom k
p g~aphy, the lin a~ p edictions may mmt be accutak
Sp ialized app oaches will cwentually have k be dewel
op d
Th oughout this pap ~ we shall assume that wates
is in mp essible md th density is const mt The com
p essibility ot wat ~ may be m imp ~tmt tack ~ m un
dezwate~ explosions md impact p oblems, but tm gen
ezal seakeeping studies the in omp essible assumption is
sutficient On the othez h md, wat ~ m the oce~m does
mmt have cunst mt d nsity Undez limit d cimumstances,
the ink ~nal waves that a~e set up bee mse ot the den
sity g~ dient m the wak ~ c m have m infiuence on ship
p ~tmmance Howeve~, m typical sit ations th density
va~iation in the vicinity ot the ship is negligible md the
constmt density assumption is justified
This pap ~ sta~t with m histmical ~eview ot ap
p oa~hes to seakeeping p edictions md a tozommmical
discussion ot the va~ ious app ozimations that a~e madeto
obtain hactable se keepmg p oblems tm solution This
is tollowed by a discussion ot cont mp ~a~y calculation
methods, which begms with a discussion ot the seakeep
ing viscous fiow md p k ntial fiow bounda~y value p ob
lems, va~ious app ozimations to the solution ot the p
t ntial fiow p oblem, examples ot some ot these solu
tions, md a discussion ot the dezived qu mtities st~uc
tu~al loads, g~een wak ~ on deck, md added ~esi tance in
waves Finally the pap ~ con ludes with a discussion ot
mym ~eseawh issues etficient mmmezical methods, la~ge
amplitude motions md capsizmg, hmimntal plane mm
tions (cuuplmg between seakeeping md maneuve~mg),
finik depth in the litk ~als, md vezification md valid~
tmn
2 BACKGROUND
This ~eview begins with m hi tmical ~wiew ot the com
putational app oaches k the seakeepmg ot ships These
a~e placed m cont t by a discussion ot se keepmg fiuid
dynamics p oblems as a tozommmy, sta~tmg with the
most gen~al in omp essible fiuid dynamics p oblems
md p og~essmg th ough a sequence ot app ozimations
md assumptions ~esulting in mm~e md mme ha~table
p oblems, which may m may mmt successtully mmdel the
physicaHeality
Hk~torioal App~ach to Seal~eping
The cumputation ot ship mmtions has a long his
to y sta~ting with Fmude's (F~oude IS61) migmal w ~k
on ~ollmg D tailed hisk ~ies ot the dewelopment c m
be tound m mmy sou~ces mcludmg Newmm (197S),
Mamo ( I 9S9) md Ogilvie ( 1977) Modezn cumputations
2
begmwithtw dewelopment mthel950's Thefi~stwas
the use ot ~ mdom p ocess theo y to dete~mme the statis
tics ot th ship ~esponses in a seaway The second was
thedewelopmentot Imea~ ship motion 6hemies to p edict
the ~esp nses ot the ship k ~egula~ waves
The seminal pape~ ot St D nis md Piezson (1953)
p op sed a method k p edict 6he statistics ot ship ~ -
sp nses k a ~ealistic seaway Using spechal medhods
dewelop d m othes fields, they ~elat d 6he sp hal den
sity ot ship ~esp nses to 6he mput oce~m wave sp hum
Tw assumptions a~e mitical 1) 6he sea su~tace is m ez
godic, G mssi m ~ mdom p ocesses wi6h zem me m md
2) the ship c m be ~ep esent d by a Imea~ sysk m The
fi~ t assumption em bles 6he p obability density fi n tion
ot 6he ship ~esp nses to be complekly chaack~ized by
the va~iance, which is simply 6he aea undez the sp
hal density ot 6he ~esp nse Once the p obability density
tunction tm a given ~esp nse is known, all th desi~ed
statistics otthe ~esp nse cm easily be dek~mined The
Imea~ system assumption allows 6he spechal density ot
my given ~esp nse k be tound by multiplymg the in i
dent wave sp tmm by the squa~e ot the ~esp nse ampli
tud op ~ak ~ (m RAO) ot 6he desued ~esp nse in othes
fields, the RAO is ofien called 6he hans6 fi n tion m
the Imea~ sysk m fi n tion At my smgle fiequ~cy, 6he
RAO is the amplitude md phase ot the desued ~esp nse
to ~egula~ in ident waves actmg on the vessel at the given
t~equency
In mdez k use 6he St D nis md Piezson app oach,
the mput wave sp tmm md 6he RAO's tm 6he vessel
must be known Havmg good wave spechal mtmmation
is mitical m mdez to obtain good ship ~esponse e timak s
Naval a~chik ts usually ~ely on ocea~mg~aphezs k p o
vid thismtmmation mdmmch~eseawhhasbeendon in
the aea New sak llik ha~king k hmiques a~e bemg d~
velop d 6hat will allow ~eal time wave sp hal e timak s
tm my p int in the oce~m Bee mse ot limit d space, we
will mmt discuss wave sp ha m this ~ewiew; it will be as
sumed that the necessa~y wave specha md m wave time
hi tmies ae available
The RAO's c m be det ~mmed eithes exp ~ imentally
m malytically Almmst all ot the malytic wmk has n
glect d viscosity md used p t ntial fiow Ezoept tm
some empi~ical viscuus cm~ections, seakeepmg compu
tations have all been p kntial fiow until app ozimat Iy
th la t five yea~s The 1950's saw the stat ot the d~
velopment ot malytic p ediction t hmiques The fi~ t
th ~ies built on 6he 6hin ship app ozimation ot Michell
(IS98) The thm ship app ozimation assumes that 6he
beam ot th ship is smalHelative k the leng6h md dEafi
The 6hin ship app ozimation was exammed mitically by
Pek~s mdSkkez(1957) Theyusedasyskmatiop ~t ~
bation p ocedu~e with 6he ship's beam md unsk ady mm
tions weze assumed to be ot 6he same small mdez ot mag
OCR for page 3
nit de The fi~st mdez themy was ~adhez hivial in that
it balanced hydkodynamic tmces due to 6he undistu~bed
incident wave p essu~e field (6he F~oude K ylov excitmg
tmce) mdthehydkostatic ~esk ~ing fi ces wi6h 6he ship's
mass times accelezation k~m Thus, to fi~st mdez the~e
is m unbou ded ~esunance m heave, pit h md ~oll b~
c mse ot 6he la~k ot hydkodynamic damping Newm m
(1961) avoided 6he shmk ommgs ot Pek ~s md Sk kez by
inhoducing ~efinements usmg a systematic exp msion in
multiple small paamet ~s md a mme accu~ak tak ment
ot the body bounda~y condition Comput d ~esult t~om
his theo y did not compa~e well wi6h exp dments
The p oblem is that typical ship hulls with both 6he
beam md dEaft small ~elative to the leng6h ae closes
to slendez b dies 6hm thm ships A16hough slendez
body theo y has be~ used m amdynamics smce Mu k
(19 4) st died 6he fiow amund auships, it was mmt until
the 1950's 6hat slendez body 6hemy was applied k ships,
fi~ t k sk ady tmwa~d motion md then to unsteady mm
tions Rigmous sle dezbody 6hemies wezemiginally d~
velop d by seve~al ~eseachezs (Joosen 1964, Newm m
1964, Newmm md Tuck 1964, Mmm 1966) using a
lomg wave assumption that 6he incident waveleng6h is
on 6he mdes ot the ship length Untmt nat Iy, as with
thin ship 6hemy, most mmnhivial hydkodynamic ettect
a~e highez mdez cumpa~ed k 6he F~oude K~ylov excitmg
tmce md the hydkostatic ~estming tmce Mmeoves, k
leading mdez the p edict d mmtions a~e mmn~eson mt b~
c mse the meztial tmce due k the b dy mass is ot highes
mdes
At the same time that the long wave sle dezbody
th ~ies weze being inve tigat d, m alk~native st~ip
slendez body was bemg alsu beimg studied K ~vm
K oukovsky (1955) (m a sequel by Kmvin Kmukovsky
md Jacobs (1957)) did 6he mitial w ~k Usmg a cumbi
nation ot slendez body 6hemy md go d physical msight,
they dewelop d a theo y tm heave md pit h 6hat was suit
able tm mmmezical computations on the newly emezgmg
digital compuk ~ s Ship themy was 6he fi~st ship mmtion
th ~y 6hat gave ~esults wi6h e~mugh engmeedng accu
~a~y that 6he p edict d motions we~e usefi I tm design
A modified ship 6hemy ot Ge~itsma md Beuk Imm
(1967) was show k give good ag cement with exp ~i
ments tm head seas Inthe lat 1960's mmecomp ehen
sive sthp themies we~e develop d by sewezal ~eseawhezs;
most widely cit d is Salvesen, et al (1970) Usmg a
combination ot madhematics md judicious assumptions,
these ~esea~chezs mgeniously anived at a tmm ot sthp
themy that k day is still 6he mmst widely used medhod tm
se keepmg computations ot ships
A madhematically consistent app oach to sthp th~
my was dewelop d by Ogilvie md Tuck (1969) h. ~e
Ogilvie (1977)] They made a shmt wavelength app oz
imation md ca~ded out a syskmatic malysis tm 6he
slendez body p oblem k det ~mme the added mass md
dampinginheave mdpiLh Atz~ospeedthe~esults~ -
duce to pu~e ship theo y M my ot 6he tmwad speed
conection tesms a~e simila~ to Salvesen, et al (1970)
butthe~ e also sume mt g al k~ms ove~ 6he t~ee su~
tace that make cwaluation ot 6he Ogilvie Tuck coetfi
cients ve y ditficult to cumpuk The ~ational app oach
to ship 6hemy alsu mvolves ch mges in the tmmulation
tm 6he ditfiaction excitmg tmces Bee mse ot the high
t~equency (sho t wavelength) ot 6he incident waves, 6he
ditfi action p t ntial is mm lomgez slowly va~ying along 6he
ship leng6h A solution must be sought as a p od et ot a
highly oscillak ~y longit dinal tunction times a slowly
va~ymg solution ot the Helmholtz equation T~oesch
(1976) exammed 6he case m mmn head seas Fm head
seas the p oblem is smgula~ md sp ial malysis is ~ -
quued (et Faltinsen 1972, Mamo md Sasaki 1974, m
Ogilvie 1977)
Ship 6hemy is a shmt waveleng6h theo y md
slendez body 6hemy is a long wavelength 6hemy At
t mpt have been m de k bddge th gap md find a th~
my 6hat was valid ove~ a widez t~equency ~ mge The m
tesp lation themy otMamo (1970) md 6he unified themy
ot Newm m (197S) a~e typical examples F ~ sho t wav~
lengths 6he ~esults ~educe k ship themy md tm Iong
wavelemgths 6he ~esults ot slendez body 6hemy a~e ~ecov
e~ed Thevelocityp t ntialintheimez~egionmcludesa
pa~ticula~ solution 6hat is equivalent k the sthp th ~y ~ -
suit md a homogmeous comp nent 6hat atk ~ mat hmg
wi6h the out ~ solutimm account tm mt ~actions along
the hull leng6h in a manr~ simila~ to long wave slendez
body 6hemy C mpa~isons wi6h exp dmentaMesult by
Sclavoumms (1990) have indficat d imp oved p edictions
~elative to ship themy p edictions Recent w ~k to be
p esent d by Kashiwagi, et al (2000) at this Symp
smm shows 6hat tm a VLCCI 6he unified md ship theo
~ies give essentially equivalent p edictions tm heave md
pik h motions at a vadety ot heading mgles The ve~tical
bendmg mmment at statimm five ot a containe~ ship a~e
slighUy bett ~ p edict d by unified theo y
The r~t dewelopment had k await th anival ot
tastes md la~gez compuk~s Usimg added mass md
damping tables, ship mmtion p edictions fiom K ~vm
K oukovsky's migmal st~ ip 6hemy could be calculat d by
h md Smce 6hen, 6he suphistication ot seakeepimg theo
~ies has pa~alleled the g~ow h ot computational p wes
At times the available cumputational p wez was g~eates
th m ou~ ability to use it p oductively md at othes times
~esea~chezs have been waiting tm la~ges md task ~ com
puk~s Today, 6he most advanced k hmiques a~e beyond
the capacity ot ~eadily available compuk~s md wide
sp cad ve~ification will have to await tudhez ir~eases
3
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in computational p wez
By the lae 1970's the Neumam Kelvin app oach
was tatmg to te used in 6he Neumam Kelvin ap
p oa~h the body bounday condition is applied on 6he
me~m p sition ot the exact body a~tace md 6he Im
emizedt~ tacebou dayconditionisused Theh~
ditiona app oach to sulvmg the Neumam Kelvin p ob
lem is to use bou day integ~al meth ds in which 6he
solution is tmmmiat d m t ~ms ot int g~als ot tund~
mental singulaities (sou~ce md dip les) ove~ 6he su~
tace sunoundmg the fiuid domain Nmmally, the int
g~a equation w uld have t be applied ove~ all su~taces
sunoundmg the fiuid domain How~wez, by combinmg
the tundamental singulaities wi6h othez malytic tune
tions, it is p ssible to dewelop G ~n fi nctions that sat
isty all the bounday conditions ot 6he p oblem except
on 6he body a~tace ~ 6his case, the goven~ing int
g~a equation need only be solved on the body a~tace
Fm wave p oblems, fiee su~tace G ~n tunctiom have
been e tablished tm m my ditte~ent cases (tm example
et Weh msen md Lait ne 1960, Newm m 19S5a, Telst
mdNoblesse 19S6) Ingene~a,6heg~eat ~ thecomplex
ity ot the p oblem, 6he mm~e ditficult it is to cwaluat 6he
G een fi nction Fm example, finit dep6h G~een tune
tions ae hades t cumput 6hm mfinit d pth G~een
tunctions; evaluation ot tmwad speed G ~n fi nctions
~equi~es mme etto t 6h m zem tmwad speed
Hess md Smith (1964) pionee~ed bounday element
methods tm fiows wi6hout a t~ee su~tace (equivalent t a
double body fiow wi6h a ~igid t~ee su~tace) Usmg just a
soumedishibution,6hey abdivideddhebody a~tacei to
N fiat quad ilat ~als oves which 6he sou~ce st~ength was
as amed cunst mt Satis ying 6he body bounday condi
tion atthe centes ot each quad ilat ~al (also called a mmde,
conhol, m collocation p int) ~esult d m a system ot N
Imea equations fi the unhmw sou~ce shengths By
k owmg 6he soume st~ength, the veboities md p essu~e
at each conhol p int c m easily be d t ~mmed The fiat
quad ilat ~als we~e ott n called pa~els md mmw the t ~m
"p mel medhods" has come to me m my solution t h
niq e in which the b dy a~tace ( md p ssibly othez su~
taces ot the p oblem) has b~n subdivided Highez mdes
pa~el methods irrvolve the use ot pa~els 6hat ae mmt fia
amd m smgulaity dist~ibution sheng hs 6hat ae mmt cun
st mt oves a pa~el A Gae~kin p ocedu~e c m be used to
satisty 6he mt g~al equaion m am inkg~acd sense oves
each pa~el
In pamel methods, tw tasks ~equue almost all ot
the computational ettmt The fi~st is setting up 6he m
fiuence mahiz 6hat ~equi~es mmitiple evaluations ot 6he
Gkeen tunction tm the p oblem The second is solvmg
the ~esultmg linea~ sysk m ot equations Fm small p ob
lems, duect solve~s such as L U decomp sition w ~k
fine As the p oblem becomes lages, am it ~ative t h
niq e such as GMRFS (Sa d amd Schultz 19S6) is mme
app op iae Duect solvezs ~equue on 6he mdez ot N3
op ~ations while it ~ative solvezs ae on 6he mdez ot N2
Howeve~, the~e is a set up time amd thus di~ect solvezs
w ~k bett ~ tm smal p oblems; the exact hade o6 p mt
dep nds on 6he comput ~ sysk m amd 6he sp ific p o
g~am Fm vesy lage p oblems mdez N medhods such as
tast mmitip le accelezation (et Scmpio amd Beck 199S)
m p e conect d Fast Fomiez T~amstmm (et K mg, et a
1999) may be nwessay A optimized mmmesica ap
p oa~h will bala e 6he mmmbes ot pa~els, the time spent
settmg up 6he mfiuence mahiz, amd 6he cost ot solvmg
the yst m ot Imea equaions in mdez to obtam a desi~ed
lewel ot accu~a~y
It should be p mt d out 6hat bounday eleme t
methods, while 6he mmst p pula, ae mmt 6he only medh
ods available to solve the Neumam Kelvin p oblem Ez
amples ot finik element m finik ditte~ence app oaches
aegivenbyBai,eta (1992)mWu,eta (1996) These
methods have significamtly mm~e unknow s, but the m~
hiz that mu t be in~t d is ve y spase The k tal com
putational ettmt amd accu~acy ot the solution ~elaive to
pa~el methods d~ends on 6he detals ot 6he code
The Neumam Kelvin app oach was fi~st used by
the ottshme ind st y sir~e ship theo y could mmt p s
sibly wmk fi 6he vessel geomet y typically used in
the expimation amd p oduction ot ottshme oil a d gas
The miginal codes used lowes mdez pa~el medhods amd
the ze~o speed, t~ tace Gkeen tunction m the t~—
quency domain Seweza commezcial codes ae avail
able, the fi~st p obably being Gadson (197S) amd 6he
most widely used is WAM T (K ~smeye~ et al 19S8)
The codes have been ext nded to ir~lude second mdes
me~m dkitt a d slowly vay tmces The ditficulty in ex
tending 6he ottshme wmk k ships is 6he tmwad speed
The tmwad speed t~ tace Gkeen fi nction in 6he
t~equency domam is exhemely ditficult k compuk The
fi~ tatkmptwasby Chamg (1977), wi6h lates wmkby ~
glis amd P ice (19SI), Guevel amd Bougis(19S2), Wu amd
Eatock Taylm (19S7), amd Iwashita amd Ohku a (1992)
Chen, et al (2000) ae p esentmg mme ~ecent w ~k a
this Symp slum
A alk~native k w ~king wi6h 6he t~equency
domain G~een tunctions is k w ~k m 6he time domam
The migmal w ~k on the time domam Gkeen fi nction
is medit d k Fi kelst m (1957) Fm fi lly linea p ob
lems at const mt m z~o tmwad speed, 6he time doman
amd fiequ~cy domam solutions ae ~elat d by F u~ies
hamstmms md ae,6hezetme, complementay (fi exam
pies ot time domam cumputations see Beck amd Magee
1990, Bingham, et al 1994, m Kmsmeyez amd Bingham
199S) Wmking m one domam m 6he othes might have
advamtages tm a paticula p oblem The time domam
~equi~es 6he cwaluation ot convolution mt g~als ovez all
4
OCR for page 5
p cwious time st ps; 6his takes both cumpuk~ time md
memo y 7he time d mam Gk~n fi nction is simila to
the ze~o speed t~equency domain Gkeen fi nction md it
cwaluation ~equi~es app ozimat Iy 6he same ammunt ot
etto t At zem speed 6he conventimmal fiequency domain
computations ae tastes beomse ot 6he corrvolution int
g~as 7hese mt g~als ~equi~e m my time st ps fi ad~
quak ~esulution whe~e as 6he t~equency domain ~equi~es
only a tew t~equencies How~wez, at tmwad speed 6he
situation is ~ sed ~ th time d mam the G~een tune
tion does mmt ch mge md ~mm time is app ozimaely 6he
same as ze~o tmwad speed in 6he t~equency domam,
thetmwad speed G~een tunction is much mm~ecomplex
wi6h g~eady mmeased comput ~ time
A inommsistent but tudhez ~efinement to 6he
Neuma Kelvin p oblem is to satis y the hull bm nd
ay condition on 6he exa~t wett d su~ta~e ot 6he body
while ~etaming 6he linea ized fi ee su~ tace bounday cun
dition 7 his body~act p oblem is a time va i mt linea
system md 6he fiequency domain md the time domams
ae mm longez simply ~elat d Ezoept tm sume ve y sim
pie cases, 6he body~a~t p oblem must be solved m 6he
time domain 7he hydkodynamic tmces a~tmg on a ves
sel unde~going smusoidal mmtions ae mm longez simply
smusoidal; the ~esult typically have a me m shitt with
thep esence ot second mdez md highez ha mmnics Beck
md Magee (1990), Magee (1994), Lm md Yue (1990),
m Shm, et al (1997) give examples ot 6his aB oach
7he Neumam Kelvm md body exa~t app oaches
ae Imemizations about the fiee sheam veboity 7his
is mmt 6he only p ssibility in the so caled doublebody
m 'Dawsmm's App oach" (Dawson 1977, Sclavoumms
1996), 6he linea~ization is about the doubl~body fiow
7he bounday conditions on the body ~emain 6he same
as in the Neumam Kelvin app oa~h but the t~ tace
bounday conditions ae signific mtly alt ~ed Beca se
the t~ee su~tace bounday cunditions ae a tunction ot
the geomehically d~endent double body fiow, a single
t~ tace G~een tunction is mm longes applicable 7he
~esultmt body vaue p oblem is typicaly solved usmg
a dist~ibution ot simple Rmkme sou~ces ovez bodh 6he
body md calm wak~ taces Nakos md Sclavoumms
(1990a, 1990b) ae examples ot 6he medhod applied to
seakeepmg p oblems Be~t~am (199S) gives a va iamt ot
the meth d m which he uses the calm wates t~ tace
amd p tential as the basis fiow As the body~act ap
p oa~h is a ~efinement ot the Neumam Kelvin medhod,
th weak scates hyp thesis ot Pawlowski (1992) is a fi ~
thes ~efinement ot 6he Dawson app oach Assuming 6he
ship dist ~ba e is smalMelative k 6he incident waves,
the Imea ization ot the ship ge~at d wave di tu~ba e
cam be done am nd 6he ambient wave p ofile with a
body~act condition on th ship hull Sclavoumms, et
al (1997), md Huamg md Sclavoumms (199S) have used
this method m the SWAN 2 cude
Both 6he body exact amd weak scak ~ app oaches to
seakeepmg computations heat the body bounday cun
dition p op ~Iy, but 6he t~ee smta~e bounday co dition
has been "linea~ized" m some sense A thud altesnative
is to keep 6he tully mmnlinea t~ee smta~e bounday cun
ditions Fully mmnlmea computations cam te p ~tmmed
in a va iety ot ways Fm sk ady tmwad motion, am ik ~
ative p ocedu~e c m be used in which 6he bounday cun
ditions ae initially applied on 6he calm wak ~ pla~e amd
the solution ik~at d u til the tully mmnlmea conditions
ae satisfied on the exact t~ee su~tace C nve~gence ot
the it ~ation p ocedu~e cam te a p oblem but successfi I
solutions have be~ obtamed by ammng odhezs Jensen, et
al ( I 9S9), Raven (1993, 199S), Scullen amd Tuck (1995),
amd Scullen (199S)
Fm unsteady p oblems, time st pping solutions
must be used Sp hal medhods have b~n applied to
wates wave p oblems amd k wave ditt~action by tw
dimensional amdm simple geomehies (see fi exam
pie Chapmam 1979, D mmezmmth, et al 19SS, Lm,
et al 1992) L nguet Higgins amd C kelet (1976) m
hoduced the ml ed Eulez Lag~ mge method tm solvmg
tw dimensional fi lly mmnlinea wates wave p oblems
In 6his time sk ppmg p ocedu~e tw majm tasks mu t be
complet d at ea~h time sk p The fi~st is to sulve a mized
bounday value m am Euleziam fiame The p kntial is
k own on the fiee su~tace amd the mm~mal velocities ae
k ownon 6he body su~tace fimm thebody bounday cun
dition ~ the Lag~amgiam phase, the tully mmnlmea t~ee
su~tace bounday cunditions ae used k hack the t~ee
su~tace amplit de amd 6he value ot 6he p tentia on 6he
t~ee su~tace The ~ igid body equations ot mmtion a e used
to updat the body p sitimm in pace amd the m ma v~
locity on the body su~ta~e is given by the body bm nd
ay condition The meth d has been applied to a wide
va iety ot tw amd th ee dimensiona wak ~ wave p ob
lems, both with amd without a body p esent Ammng 6he
~eseachezs who applied 6he meth d to tw dimensiona
p oblems ae Faltmsen (1977), Vmje amd B~ewig (19SI),
Bakez, et al (19S2), amd mme ~ecendy Gkosenbmgh
amd Yeung (19S8), amd C mt, et al (1990) Th ee
dimensional p oblems have been mvestigat d by Lin, tt
al (19S4), D mmezmubh amd Yue (19S7), Zhou amd Gu
(1990), Cao, tt al (1991), Scmpio, et a (1996), Beck
(1999), amd Sub~amami (2000)
Th ee dimensional, tully mmnlmea calculaiom ae
computationaly vesy ink nsive A comp omise app oach
is k sulve 6he mmnlinea~ p oblem in the moss fiow pla~e
amd pseudo time sk p 6he solution m 6he dow sheam di
~ection F ntame amd Tulm (199S) give a histo y ot
the medhod 6hat they call 2D + t The idea has been
used in pi ming boat p oblems tm mamy yeas Us
ing am app oach appaendy fi~st p op sed by Cummms
OCR for page 6
(1956), Ogilvie (1972) tudied 6he waves p oduced by
a fin ship's bow using a linea~ t e su~tace bounday
condition Chapm m (1976) used the tull mmnlin a t~e
su~tace bounda~y condition k investigak a yawed fiat
plate Only 6he dive~gent waves a~e simulat d by 6he
method md thus it is mmst aB op iat tm high s~ed
ships Yemmg md Kim (19S4) develop d a sp ial m
n ~egion G en tun tiondhatmets a linea~ t e smtace
bounda~y condition wi6h th tmwa~d sped k~ms The
im~ ~egion solution is then mat hed k the outemegion
to in lude both dive~ging md h msve~se wave systems
M mo md Song (1994) used fi 11y mmnlinea~ t e smtace
bounda~y mnditions m 6he moss fiow plane to in e ti
gak bow wave b~e king Mme ~ece t w ~k ot Wu, et al
(2000) applied 6he method k study deck wet ess
Tw p oblems wi6h the Eulez Lag~ mge medhod have
limit d its application As discove~ed by L nguet
Higgms md C kelet m the fi~st application ot 6he
method, the stability ottime skpping ot6he t~e smtace
c m be a p oblem The mmmezical t hmiques, panel size,
md time st p size must all balance m the t e smtace
c m become un table md 6he calculations b~eak dow
Smoodhing, ~eg~idding, md a~tificial damping have all
ben applied to hy md allewiat the p oblem The othes
mym ditficulty is wave b~eakmg Wave b~eakimg is
a nat ~al phe~mmemm 6hat occu~ y otkn but un
to tunak Iy c mses the Eulez Lag~ mge medhod to b~eak
down The mmst houblesome waves a~e 6he bow md
sk m waves ot high sped ships Any ~egion with a g~eat
deal ot fia~e will t nd k c mse ove tuming md h n e
b~eaking ot 6he local wave mest This local b~eaking may
have mm ettect on 6he global hydkodym mic tmces actmg
on 6he ship but c m c mse the computations to top
Bee mse ot 6he p oblems asseiated wi6h tully mmn
Imea~ computations, sewezal ~eseawhezs have ben ex
aminmg what we shall call "blendmg medhods " These
methods a~e a blend ot Imea~ md mmnlin a~ 6hemies in
these 6hemies the equations ot motion ae int g~at d in
th ti me do main, w ith 6he hydko tatic md F~ m de K y lov
tmces mt g at d ovez 6he exact wett d smtace The
added mass md damping a~e tound usmg a lin a~ th~
my, typically a ship app oa~h A detailed discussion ot
the dit6 ent themies md compa~isons with exp ~iment
c m be tound m 6he ISSC ~ep ~t on Ezbeme Hull Gi~des
L admg (ISSC 2000) Th blending themies ae used
bee mse they a~e tast md allow long time ~eomds to be
gene~at d with engme ing accu~acy
Finally, the mmst ~ecent app oach to se kepmg is
to solve 6he Reymmlds Avezaged Naviez Stok s equations
in the time domain (so called unsk ady RANS) This is
a n w a~ea ot ~eseawh md ~esults a~e just statmg k be
p esent d (et Wilsun, et al 1998 G ntaz, et al 1999)
Nmmally, RANS codes a~e iktat d until a skady stak
solution is obtained n unst dy RANS, it ~ation is still
used at each time skp but the global solution is made
time accu~ate by using a time steppmg medhod Not
e~mugh ~esult a~e yet available to anive at my con lu
sions md mmch mm~ k ~em ins k be don
Taxonom'Of Seakeping Cnmputations
At 6he p esent time, active ~eseawh m 6he aea ot
p edicting ship mmtions is contimmmg on panel medhods,
bodh tully mmnlinea~ md double body methods, blend
ing methods md 6he application ot unsk ady RANS F ~
design pu~p ses a naval awhik t has a wide choice ot
methods wi6h which k do seakeping computations; 6he
choices a~e mm longez limit d k ship 6hemy md it
dezivatives Howeve~, it should be p mt d out 6hat even
wi6h 6he av ilability ot a wide selectimm ot computational
methods, p obably SO p ment ot all design ~elat d cal
culations tm ships at tmwa~d s~ cd ae still m de usmg
ship theo y Ship 6heo y has 6he adv mtage otbeing ta t,
~eliable, md able to accommodat a wide ~ mge ot hull
tmms it is a meth d that is ha~d to ieat tm con entional
ships at mmdezak speds How~weg tm highez s~ed
vessels, highly mmn wallsided hull tmms, wave loads m
exheme mmtions, 6he compa~isons with exp ~iment a~e
much p mes; 6his has ben the p ima~y motivation tm 6he
dewelopment ot mm~e ad anced themies n mdez to hy
md put some ~elative mdez ink all 6he ditte~ent modem
se kepmg computational medhods md main hydkody
namics in gene~al, we p esent Figu~e I
The goveming equations m 6he fiuid tm 6he gen
ezal, th e dfimensional, in mp essible, const mt density
fiuid fiow p oblem a~e 6he mntmuity equation md 6he
th e comp nent ot 6he Naviez Stok s equations These
equations ~esult m a sysk m ot tou~, mupled mmnlin a
pa~tial ditfi ential equations tm the tou~ unlmowns ot
p essu~e md 6he th e comp nents ot velocity To ob
t in a unique sulution ~equi~es bounda~y conditions on
all su~taces su~munding 6he fiuid th wett d su~ta~e ot
the body, the t~ tace, the botk m, md th su~taces
at mfinity On sulid smta~es such as 6he body smtace
these a~e tw bounda~y conditions The fi~st is the kin
matic condfition ot mm fiow th ough 6he smtace And
the semnd is a mm slip condition on 6he tmgential v~
locity These a~e applied on 6he contmuously ch mgmg
wett d su~tace ot the vessel On the fie smtace 6heze is a
kin matic condition md a dynamic condition ot mnst mt
p essu~e with mm shea~ shess The fie su~tace bm nd
a~y conditions ae applied on the unlmown t e smtace
amplitude, which must alsu be dek~mined as pa~t ot
the solution On the botk m bounda~y tm finik dep h
these is a kin matic condition, m in infinit Iy dep w~
tes the distu~bance velocities must go k zem At mfin
ity, mcident waves a~e p esmibed md the~e is a ~adi~
tion mndition ot m tgomg waves on 6he ship g~at d
waves This gene~al p oblem is highly mmnlin a~ m both
6
OCR for page 7
Fig. I Tenors m; of hyd odynamics p oblems fm seakeeping
7
OCR for page 8
the goven~ing equations md bounda~y conditions; at the
p esent time, it is teyond the computatimmal tak ot th~
a~t Cunseque tly, app ozimations mmst be m de in m
dez to have a hactable m thematical p oblem Fm dis
cussion puq ses, we have hied to put all the ditte~ent
available computational k hmiques ink the b~oad t~am~
w ~k shown m Figu~e I Figu~e I cakgmizes the dit
t'esent app oaches that c m be taken k solve th gene~al
th ee dimensiom 1, mcomp essible, con t mt density m~
~me hydkodynamics p oblem While not all ot the boxes
a~e applicable tm seakeepmg computations, they have all
been kept tm cumplet ness ot the figute md k mdicak
that the~ e additional p ssibilities that might have ap
plication k seakeeping p oblems
The t hmiques to solve the gene~al th ee
dimensional poblem cm be divided mto tw majm
cakgmies visoms md inviscid tow appozimations
Viscous tow app ozimations atk mpt k model viscous
ettects by keeping some tmm ot the viscous tesms in
the Naviez Sk kes equations The biggest ditficulty is
the tu~bulence m the high Reymmids mmmbez t ows assu
ciat d with typical madne p oblems Duect Numezical
Simmiation DNS) solves the Naviez Sk kes equations
di~ecHy ir~ludmg tu~bul~ce DNS is so computation
ally intensive that it has only b~n applied to ve y simple
p oblems such as t ow in a ~ectmgula channel At the
oth ~ exhemeis Stok's tow whichkeepsonlythep es
su~e md viscous tesms in the Naviez Sk kes equations
Stok s' tow is essentially a ve y low Reymmids mmmbes
app ozimation, so it is uset I m lub~ication p oblems
md k model the swimming ot micsomg misms it is not
pa~ticulady usetul in high Reymmids mmmte~ seakeepmg
p oblems
High Reymmids mmmbez tows a~e cha~a~t dzed by
the viscous ettects bemg cmmfired k a ~egion nea~ the
body md a viscous wake Bounda~y layez app oz
imations give ~easonable ~esult up k the sepa~ation
p mt but c mmt be ca~dez t ~thez At p esent the~e
a~e tw methods to comput "avezage" viscous tow
La~ge Bddy Simulation LES) md Reymmids Avezaged
Naviez Stok s equations (RANS) Each app oach has
it shengths md weaknesses Tu~buleme mmdelmg tm
RANS ~equi~es ave~agmg ovez all velocity tuct ation
stat s The state~t the at is that RANS mmdels tail in
~egions ot signific mt misohopy, such as p ~tions ot the
fiow infiu~ced by ~igid mm slip boundaies md t~ee su~
taces LES methods model only small scale fiuct ations
while di~ecHy computmg the lage scale ones To the
exk nt that small scale fiuct ations a~e locally isohopic,
LES computations a~e p tentially mm~e acomat thm
RANS computations, but a~e achieved at signific mdy
mme cost (et D mmezmuth, et al 199S tm m exam
ple ot a LES calculation ot the st dy fiow about a ship
bow) RANS has typically b~n used to irrvestigak mt
~im fiows in duct md exk ~im fiows a~m nd bodies LES
has been used to study the ink ~ a~tions ot ditt'e~ent scales
ot t ~bulence in op n fiows such as occu~ in ship wakes
Reymm lds Ave~ aged N av iez Stokes equ ations a~ e d~
~ived by assuming that all the velocity comp ne ts c m
be app ozimat d by a me m comp nent plus a high y
oscillato y, small amplitude, ze~o me m comp nent that
~ - ese ts the t ~bulence These a~e sub titut d mto the
Naviez Stok s equations that a~e then time avezaged oves
a suitable time scale The ~esultmg equations tm the
me~m fiow a~e identical k the migmal Naviez Stok s
equations except tm the addition ot second mdez meztial
tesms in the oscillak ~y velocities that do mmt time ave~
age k ze~o These so~alled Reymmids shess k ~ ep
~esent the mfiuence ot the tu~bulence on the me m fiow
field Whi le theze a e mm mezou s mm mezic al mmdel s tm the
Reymmids shess k ~ ms, mmne ot them a~e entuely satistac
to y None ot the p esent tu~bulence mmdels c m p op
edy account tm the misuhopy ot the tu~bulence nea~ the
t~ee su~tace that c m have imp t mt ettect in the w kc
~egion RANS codes a~e stat ~t the a~t they a~e used
tm steady ~esistmce calculations md w ~k is p oceed
ing on unsk ady RANS that mcludes mcident waves md
ship motions (et Wilsun, et al 1998 md Gentaz, et al
1999) Yeung md his colleagues Yeung, et al 1998
Roddie~ et al 2000, Yeung, et al 2000) have stat d us
ing unsk ady RANS m ~oll damping computations in
the long k ~m, unsk ady RANS will p obably be mak hed
with fi lly mmnimea~ p kntial fiow cumputations in the
fiu field to give a complek solution
The boz labeled empidcal app ozimations undez the
viscous fiow branch m Figu~e I is included bee mse d~
signe~s mu t have mswezs md viscous fiow calcul~
tions otk n a~e mmt applicable m a~e k o computationally
exp nsive M my empaical methods have b~n dewel
op d m which theo y is used to dewelop a t~amewmk
with unknown coetficients that must be dek~mined by
exp ~iment md tull scale measmements Classic ex
amples a~e 1) The maneuvesing simmiation equations
that use stability dezivatives to e timak hydkodynamic
tmces 2)TheuseotMmison'sequationk appozimak
the wave excitmg tmces on cimula cyli dezs m ~egula
waves using m meztial coetficient tm the added mass et
tects md a dEag cuetficient tm the viscous comp nent
ot the load The coetficients a~e shongly dep ndent on
the t~equency o t the waves md the di amek ~ o t the cy I m
dez as exp essed m the Koolig m Ca~p nk ~ mmmbez (et
Sa~pkaya md Isaacson, 19SI) 3) The empaical ~oll
damping models that a~e used k e timak the mmease
in ~oll dampmg due to viscous ettects in p kntial fiow
ship motion calculations
Finally, on the viscous fiow side ot Figu~e I is
a special boz labeled Smooth Pa~ticle Hydkodynam
ics Smmoth Pa~ticle Hydkodynamics is a ~elatively new
g
OCR for page 9
t hmique to comput fiuid fiows, md it application
to seakeeping p oblems has mmt yet be~ det ~mmed
Monagh m, et al (1994) have used it to simmlat tw
dimensional fiee su~tace fiows Fontame, et al (2000)
show sume mt ~estmg ~esult tm the sloshing p oblem
md the p st b~eaking behavim ot wat ~ waves
he mviscid fiow mmdels neglect viscusity com
pletely md a~e indicat d by the second majm b~anch
ot Figu~e I in this case, the Navie~ Stok s equations
~educe t the Eule~ equations md some ot the bm nd
a~y conditions have to te mmdified Namely, the mm slip
bounda~y condition c m mm longe~ be met on dgid su~
taces such as the wett d su~tace ot the ship ~ addfition,
the bounda~y cundition ot ze~o shea~ ess on the t~ee
su~tace is mmt applicable Even this ~educed p oblem is
ve y ha~d to sulve md fi ~the~ simplifications a~e neces
sa~y Fm mtational fiows, m which vmticity is p ese t,
the vo ticity equations md vmt v meth ds c m be used
hese t hmiques have tound limit d application in ~oll
damping computations md sepa~at d fiows a~ound cu
cula~ cylmde~s
As indicat d by the second la~ge b~ mch undes Invis
cid Flow in Figu~e 1, the mmst widely used t hmique is
p t ntial fiow he vmt~ themems show that tm m m
viscid, const mt density fiuid sta~t d fiom ~est mm vo tic
ity c m be p esent ~ this case, the fiuid velocities c m be
w itten m tes m s o t the g~ adient o t a se al a veloc ity p ten
tial he govesnimg equation m the fiuid fiow is tound by
sub tit tion ot the g~adient ot the velocity p t ntial i to
the contmuity equation he ~esultmg Laplace equation
is a Imea~ pa~tial ditte~ential equation that depends only
on pace va~iables md is mdep ndent ot time Unique
solutions ot the Laplace equation ~equi~e bounda~y cun
ditions on all su~taces sunoundmg the fiuid domam
t g~ ating the Eules equation ~esults m the Benmulli equ~
tion that ~elat the p essu~e to the time de~ivative md
g~ dient ot the velocity p t ntial hus, the p t ntial
fiow assumption has allowed the p oblem t be ~educed
fimm solving tou~ coupled, r~nlmea~ pa~tial ditfi ential
equations t sulvmg a smgle linea~ patial ditfi ential
equation tm th velocity p t ntial he only mmlinea~i
ties lett in the p oblem a~e m the bounda~y condfitions
The kinematic body bounda~y condition may be
stat d such that at ea~h p int on the hull wett d smtace
the mm~mal velocity ot the wak ~ mu t equal the m mal
velocity ot the hu11 This cundition is linea except that is
must be applied on the exact wett d su~tace This leads to
a time vad mt sysk m tm which haditional Imea~ yst m
themy is mmt valid The majm mmnlineadties in the gen
e~al p tential fiow p oblem a~e in the t~ee su~tace bm nd
a~y conditions that irrvolve the qua~e ot the fiuid veloci
ties md p oducts ot the fiuid velocities with the unknow
t~ee smtace amplitude Cunsequently, the gen~al p
tential fiow p oblem with a t~ee su~tace is ve y ditficult
to solve md still fi ~thes simplifications have in the pa t
been tound necessa~y
The mmst obvious simplification is to eliminak all
the mmnlineaities by eliminating the t~ee su~tace The
boves on the ta lefi side ot Figute I ate use k mdicak
these mfinik fiuid p oblems ~finik fiuid p oblems a~e
usefi I in m my a~eas ot ma~me hydkodynamics mcludmg
subma~ k, p op lle~ w ~k, md the tudy ot fiow
amund app ndages Howeve~, in se keepmg ~eseawh
they ae ot little use except as mude app ovimations m
limitmg values ~ gene~al, the ettects ot the fiee smtace
a~e too imp t mt to neglect
Only ~ece tly has the comput ~ p we~ been avail
able that makes it teasible to atk mpt to sulve the tully
mmnlinea~ p oblem usmg the exa~t b dy bounda~y condi
tions md the fi lly mmnlmea~ t~ee smtace bounda~y cun
ditions As p eviously discussed, ~esults have been ob
tained tm a limit d m mbet ot hull tmms m mmdetak
seas The p inc ipal ditficu lties hese a~e mm me~ic al stab i l
ity ot the time steppmg method md the local b~eakmg
waves
In Figu~e 1, the boves unde~ the exact p t ntial fiow
p oblem ~ep esent the ditt'e~ent app ovimations that a~e
available today The g~eat t deg~ee ot app ovimation is
in the bov k the lefi md the least is mm the dght h md
side ~ gene~al, computational times inmease as one
moves k the ~ight but the~ e mm ha~d md ta t ~ules
Fm example, fiat ship theo y is simila~ k the Neumam
Kelvin p oblem m computatiom I ditficulty Tw sets ot
app ovimations have to be m de The fi~ st deals with the
t~ee smtace bounday conditions md the second withthe
body bm nda~y cunditions The tou~ set ot ve tical Imes
- ese t ditte~ent lewels ot app ovimation to the t~ee
su~tace bounday conditions The individual boves ae
ditfi ent k hmiques to meet the body bounday condi
tion
By fiu, the mmst widely used k hmique is to Im
emize the t~ee su~ta~e bounday condfition abmt the
t~ee st~eam velocity, Uo, amd satisty it on the calm wates
pla~e This allows the use ot the t~ee su~tace G~een tune
tions amd as discussed m the p cwious hi to y, mamy dit
t'esent themies have ~esult d The ditfi ent themies cam
be b~ok n dow ink a least tou~ basic app oaches It the
beam m dEatt is much smalle~ tham the length, the body
bounday conditimm cam be met on a fiat pla~e F ~ small
beam the body bounday condition cam be satisfied on
the cent qpla~e amd a thm ship themy ~e alt Thm ship
themy t nds k p oduce added mass amd damping coet
ficients tha ae too small amd is ~aely used Fm small
dEat, fiat ship themy satisfies the body bounday cun
dition mm the cam wak ~ pla~e The ~esulting equations
ae simila k lifimg su~tace themy m ae~odynamics A
fiat ship themy (et Lai 1994, Lai amd T~oesch 1995) has
been used to solve plammg boat p oblems whe~e cun
9
OCR for page 10
ve tiom I ship theo y t'ails
he na.t lewel ot app ozimation assumes that the
beam md dEatt ae smoothly vaying tunctimms md small
~elative k the length his ~esults in h msvezse desiv~
tives bemg m mdez ot magnitude lagez th m dezivatives
in the longit dinal di~ection in this case, the th ee
dimensional p oblem c m be ~educed to a sezies ot tw
dimensional p oblems in the h msve~se m "cmss fiow'
pla~e Dep nding on the as amed mdezs ot magnit de
ot tmwad speed md wave t~equency, ditfi ent themies
~esult Ship themy is a high fiequency theo y md tm
s lendez body them y the wavelength i s on the m dez o t the
ship length Unified theo y ii ks the two themies ink a
smgle themy valid tm a widez ~ mge ot t~equencies
In the Neumam Kelvin theo y the body bounday
condition is satisfied on the me m p sition ot the body
(ie thewettdsu~taceupk thecalmwak~lme) B~
c mse the body bounday condition is satisfied on the
me~m p sition ot the hull ~athes th m the exact wett d
su~tace, cc tam "m>" k~ms aise in the body bmnd
ay cundition he m: t tms ae ditficult k compuk
bee mse they involve highes dezivatives ot th cunst mt
tmwad speed p tu~batimm p kntial Fm this ~eason, a
tudhez simplification is otk n used in which the m: k ~ ms
ae app ozimat d usmg ju t the mgle ot attack cm~ec
tions that ae mdep ndent ot the tmwad speed p ten
tial Neumam Kelvin themy is tmly th ee dimensiona
md is typically sulved using pa~el methods in eithez the
t~equency domam m th time domain Neumam Kelvin
themy is widely used m the ottshme indust~y tm ot
shme shuct ~es such as semi's md T P's that ae high y
th ee dimensiom I it c m be exk nded to second mdes
me~m md slow dkitt tmces
Inthebody exact meth d, thebody bounday condi
tions ae applied on the exact wett d a~tace ot the body
while ~etaming the linea ized fiee su~ tace bounday cun
dition his ~esults in a time vaying Imea sysk m ~athes
th m a time inva i mt sysk m Cunsequently, the usual ap
pi ication o t ~ mdo m p oces s theo y wi I I mmt w ~ k md the
body exact p oblem is usually sulved in the time doman
usmg the time dep dent ke~ tunction
As p eviously discussed, the basis fiow tm the Im
emization ot the fiee su~tace bounday condition does
mmt have to be the t~ee st~eam in Dawsmm's method m
the double body tmmmlation, the Imemization is abmt
the double body fiow The ~e alting t~ee su~ta~e bm nd
ay condfitions ae applied on a k own p sition, but th y
irrvolve cumplex tunctions ot the usually mmmezically d~
tesmmed double body fiow Even th ugh the unknow
t~ tace di placement has been eliminacd amd the
t~ taceboundayconditionsaeappliedonak ow
su~tace, the ~emaming bounday value p oblem is still
ditficult k solve Ra kine sou~ce meth ds ae used with
soumes dishibut d ove~ both the fiee su~tace amd the
body a~ta~e The ~adiation conditions at the edge ot
the cu mputational do main mm st be c aefi l ly cmms ide~ed
to avoid wave ~efiection Sclavoumms, et al (1997) use am
absmbing bounday m SWAN 2 The weak scatt ~ tm
mulation goes one sk p fi ~thez amd applies the bounday
conditions on the mcide t wave distu~bed t~ tace
amd the mstmta~eous body wett d su~tace (et Huamg
amd Sclavoumms 199S)
The final two boxes ~ep esent solution k hmiques
thame tillmmnlinea~buthavebeen~educedinscop in
mdes to make them mme ha~table in the 2D + t meth
ods, the fi lly mmnlinea~ p oblem is solved in the moss
fiow pla~e with a hyp ~bolic maching used m the longi
tudfinal di~ection sta ting at the bo w The b lending meth
ods have little ~ational basis They ae am engmee~mg
solution that combmes the mmnlmemities that ae eas
ily comput d (typicaly mmnlinea~ hydko tatics a d the
F~oud - K ylov excitmg tmce) with Imea hydkodynam
ics Fm h ad seas it app as tha the p imay r~nlmea
ities a e the hydko tatic s md the F~ m de K y lov excitmg
tmce it a p og~am has these two comp ne ts conect,
p edictions tm Iage amplit de mmtions ae imp oved
3 CONTEtMPORARY CALCULATION
MEtTHODS
At the p esent time, th majmity ot design seakeepmg
computations tm ships at tmwad speed p obably still
irrvolve the use ot sh ip themy Tha does mmt meam to im
ply that the mme adva ed themies that we discussed in
sectimm 2 ae mmt imp ~tmt As ship geomet y becomes
mme complex amd the design speed in~eases, the d
va ed methods will find mm~e amd mme applications
This hamsitimm will be accelezat d only by the avalabil
ity ot cheapes amd t'ast ~ comput ts Th t hmica lik ~
at ~e on ship theo y, long wave slendezbody theo y amd
unified theo y is immense amd we shall mmt discuss them
amy tu~thez m this pap ~
Unstead' V's~us Flow
As just discussed m Sectfon 2, the viscous fiow
about a ship is govemed by the Naviez Stok s amd cunti
mmity equatfons
~\ + S~\
S~\ = 0
~\
whezethe~\,f = 1,2,3aethe~ ,y,amdz comp nent
ot the velocity, P is the p essu~e 9\ is the is the com
p nent ot the g~avitatfonal accelezatfon g in the ~\
di~ectfon, amd whe~e in acomda e with th Emst in
summatfon convention double subscsipt withm a t ~m
imply summation ove~ that mdex
S~ + pg\+ ~V ~\ (1)
10
(2)
OCR for page 11
quations (1) md (2) mmst satis y a kinematic md
a mm slip condition on the body These equations apply
to my viscous fiow, lamma~ m tu~bulent As we a~e m
tesest d m the fiow about ship with tmwa~d speed in
a seaway, th fiow will be tu~bulent ovez substmtially
all ot the ship's su~tace Thus, the velocity ~\ mmst ac
count tm th k tal velocity including the time dep ndent
comp nents at wave encountes fiequ~cy md the t ~bu
lent velocity comp ne ts with mmch task ~ iation both
t mp ~ally md spatially This ~esults in a computation
ally mhactable p oblem tm fiuid volumes the size ot a
ship on theoce m's su~tace
The commmnly accept d way m which this p ob
lem is made hactable is k decomp se the velocities md
p essu~es mto slowly va~ying md ~apidly va~ying com
p nent D ing this, m~e obtains ~ \ + ~\ md
P = P + P', wheze the "ovesbar' ~ - ese ts a Reymmlds
avezage taken ove~ a time/ patial scale la~ge ~elative to
the scale ot the tu~bulence md the p imed qu mtities ac
count tm the velocities md p essu~e at t ~bulent scales
Substit tmg this decomp sition ot the velocity md p es
su~e ink (1) md (2), one obtains the Reymmlds avezaged
Naviez Stok s (RANS) md contimmity equations
S~\ ~\ SP 2 Sr\>
3t + ~ S~ = :~ + P9\ + ~V
~\ o
~\
wheze r\> = ~\~ is the Reymmlds shess k nsm
The RANS equations must be sulved subject to
bounda~y conditions on the ship's h 11, the fiee smtace,
the t'a~ (fiom the ship) fiuid bounda~y, mdon the bottom
ot the fiuid domain As discussed eadiez, theze ae both
a kinematic md a "mm slip" condition on the hull sm tace
On the fiee su~tace, the~e is a kmematic condition ot mm
fiuid fiow th ough the su~tace, a dynamic condition that
~equi~es that the p essu~e equal the atmosphesic p essu~e,
md, assummg mm wind, a mm shea~ cundition On the ta
fiuid smta~e bounda~y, theze is eithez a mm di tu~bance
condition m a mm wave ~efiection condition, dep ndmg
on how t'a~ the t'a~ bommda~y is fiom the ship md how
lomg a time the simulation is bemg ~un On the bottom,
these is eithez a kinematic condition m the di tu~bance
must go k zem as th depth goes to mfinity
The mm shea~ condition on the fi ee su~ tace does not
me~m that theze a~e mm viscuus ettects at the t~ee smtace
D e to the natu~e ot wave fiow the~e is m i hezent nat
~al shea~, which, cwen tm small amplit de linea~ waves,
~esult m a thm viscous layez nea~ the fiee smtace (et
Mei 19S3) F ~tunately, the g~adients due k the wave
motion a~e small compa~ed k the g~ adients nea~ the hull
wheze the mm slip condition is applied Thus, the g~adi
ent m the waves may be neglect d with mm signific mt
consequences it one is only ink ~ested m the body tmces
md the fiow local to the ship The viscous nat ~e ot the
wave fiow is only imp t mt ove~ length scales g~eates
thm seve~al cha~act dstic wavelengths md time scales
g~eat ~ thm sewezal cha~act dstic wave pesi ds This
common wi dom may mmt hold it the waves a~e step
md theze ae signific mt mmnlineadties
quations (3) md (4) con tituk tou~ equations tm
13 unknowns, the th ee velocities, the p essu~e, md the
nine comp nent ot the Reymmlds shess k nsm Thus, the
equations ae mmt closed To obtam closu~e, the Reymmlds
shess t nsm is usually ~elat d to the me~m velocities by
m eddy viscosity This gene~ally involves the mhoduc
tion ot a~mthez vaiable such as the t ~bulent kmetic en
e~gy md mequation~elatmgthet ~bulentkmeticene~gy
to the me~m velocities md the eddy viscosity Sp iale
(1992) p ovides a su~vey otReymmlds shess mmdels
The use ot unskady RANS k solve 6he viscous
tmmulation ot 6he seakeepmg p oblem is m its nascent
stage at p esent Wilson, et al (199S) p esent 6he ~ -
suits o t RAN S s i mm l atimms tm both a Wig ley hu ll tm m
md DTMB Model 5415 fized in head seas at a single
F~oude mmmbez, wave t~equency, md wave elewation tm
each model The ~esults a~e la~gely mconclusive
Gentaz et al (1999) p esentthe ~esult ot RANS cal
culations ot tmced oscillation mmtimms tm a hemisphe~e
at zem speed, md tmced heave md pik h ot a Sezies 60
m del at a smgle F~oude mmmbez ovez a ~ mge ot wave
( ) t~equencies The added mass md dampmg p edictions
tm the Sezies 60 mmdel a~e show in Figmes 2 md 3
These ~esults include cumpadsons ot 6he RANS p edic
tions (called "P esent medh" on 6he figu~es) with tw
g~iddensities,agamstp t ntialfiowcalculations mdex
p dmental ~esult F ~ 6hese veztical plane motions, 6he
RANS md p tential fiow p edictions compa~e quik ~e~
sonably with the exp ~ime taMesults A r~t su~p ismg
~esult as p t ntial fiow methods have tm mmy yea~s
been p ovidmg ad quat p edictions ot vestical plane
motiom
Fm mommhull ships, ~oll is the mmde ot mmtion
wheze viscous ettect have 6he g~eak st significanc~
th ough viscous ~oll damping Although the ettmt ~ -
fiect dinWilson,etal (199S)dommtexplicitly mention
it this is 6he di~ection m which 6he RANS etto ts at 6he
Univezsity ot lowa a~e head d ~ Multihull vessels, such
as SWATH ships c m have signific mt viscous dampmg in
ve tical plane mmtions
Yeung, et al (1998 2000) md Roddiez, et al
(2000) p esent 6he ~esp nses, md added mass md damp
ing fiom both two dimensional exp ~ime ts md tw
dimensional RANS md p tential fiow calculations tm
~ectmgula~ cylindess fitt d with bilge keels Yeung
md his colleagues apply tw methods k 6he solution
11
OCR for page 35
dom seas to p edict the tme statisticaMesp nses ot the
ship in ~ mdom seas it may well be that mmmmch omatic
wave t tmg will easily elicit pa~amehic ~esmmmt ~ -
sp nses that a~e mmt ~ealistic in ~ mdom seas his would
be the case it the g~oup mmst cuntam a pa~ticula~ ly la~ge
mmmbez ot waves tm the specific exheme ~esp nse k oc
cut
A final issue ~elating to validation is that ot scale
ettects between mod I md tull scale measu~ement
While validation mmst almo t by definition be again t
m del scale exp ~iment, the tme issue at h md mmst be
to p edict the p ~tmmance ot tull scale ships m a se~
way he~ e conhibuk ~s to the equations ot mmtion
such as viscous mil dampmg that may have signific mt
va~iation fimm mmdel scale k sts to tull scale hials he
me~ms ot accounting tm these tactms in validation is r~t
clea~ he best solution w uld be to have a physics
based as opp sed to m empidcallybased mmdel tm
thephe~mmena, su thatthe mmdel mk matically account
tm the scale At p esent this is naive, but it may mdicak
a duection that cumputational mmdels mu t go
5 CONCLUSIONS
heze is a wide vaiety ot compuk~ cudes available
to pestmm se keepmg computations in low to mmdez
ak sea staks Ship theoy is till the mmst widely
used How~wez, m m my sit ations it does mmt give d
equak ~esults md mm~e advanced t hmiques mu t be
used Ship themy's p incipal weaknesses a~e the lack
ot th ee dimensional ettects, the mability k accou t tm
the abov~wates hull tmm, the tmwa~d speed cm~ec
tions, md the la~k ot viscuus ettects in themy, unsk ady
RANS withfi llymmnlinea~ fiee su~tacebounda~ycondi
tions c m account tm all ot these, but it is ve y mt nsive
computationally md tew ~esult a~e p esendy available
Potential fiow methods need only igmm~e the viscuus et
tects Advanced p kntial fiow methods have show
maked imp ovementoves shiptheo y p edictions
Fm p edictions in the littmals, new wave climatolo
gies a~e ~equi~ed, md comput ~ codes adapt d tm finik
depth md mmn~ni tm m botk m s a e needed Finit dep h
computations c m be amomplished by simply ch mgmg
the G~een tunction in existing seakeepmg codes
No codes p esently available c m adequak Iy p edict
the behavim ot ships m exheme seas mcluding g~n w~
tes on deck, bmaching, slamming md capsizimg This is
beyond state~t the at codes Much mm~e ~eseawh i to
b~eaking waves mdthedevelopmentotnew malytic md
mmmezical k hmiques needs to done
Theze is a m itical need tm mm~e validation data Ez
p dmental data, both model md tull scale, with e~mugh
detail md accu~a~y to be usetul tm validatimm tudies
is time consuming md co tly to obtam While these is
sutficient data tm the linea ~ mge (small amplitudes) in
head seas, tewe~ ~esults a~e accessible tm mmn head seas
Vezy limit d data ot validation quality tm the va~ iation ot
se keepmg ~esp nses with wave height is available F ~
exheme sea tak s m which the ~esponses a~e i he~endy
~ mdom, the p op ~ tmm ot validation data is mmt even
k own
ACKNOWLEDGMENTS
We ott'e~ ap logies k ~eseachezs in the seakeeping field
whose w ~k has mmt been cit d, we could mmt begm to
~et'e~mce all ot the wmks in the field To the ~e dez, we
u~ge you k see the m my ~et'e~mces in ou~ cit d w ~ks to
get a sense o t the tme b~ e dth o t the w ~ k m the field
R q's w ~k was supp ~t by g~ mts fimm the Otfice
ot Naval Reseawh C mputmg supp t tm the compu
tations ~efiect d m m my ot the figu~es came t~om the
U. S. Depa~tment ot D tense High Pestmmance C m
puting Modeznization P og~ am md the N atiom I Pa~ t ez
ship tm Advanced C mputational Infiashuct ~e
We owe thanks to Johm F. O'Dea, Kathezine Me
C~eight, F~ancis Noblesse, md Ed Rood who p ovided
discussions md suggestions on the conk nt ot the pap ~;
md Phil Aim m, L w Thomas, Woei Mm Lm, D mmis
Woolave~ md Ma~tm Dipp ~ who p ovided up to dak
data md ~efi ences tm 6he pap ~ Fmally, we w uld like
to th mk Vickie Kline, Kay Adams, T~aci Meadows md
Luella Millez who p ovided supp t tm the p epa~ation
ot the papes, md Suzanne Red who again wielded 6he
editm's p ncil
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APPENDIX—CORRECTION TO THE ADDED
RESISTANCE OF LIN AND REED (1976)
Tht te is tmt tm~ indhe~tmges otint g~ tion mthedefini
tion ot (Z:.F~), the mmp nentot added ~esistt e th t
tt ult fimm 6he mt tt tion ot 6he ~ diat on ditt~t tion
wtves wi6h themselves, givtn m Lin tmd Reed (1974) it
is cmtect d ht te:
(Z:.F~) 8 ~J~72 Jr
4~/2
l
4/2
+/ / 1
J~ ~ J~+~ ~
x Vi i(4) H(t +a,it)2
+ p / d~;(a)cosa
8t JO Vi+4vCosa
x H(t +~,:2)2
r r +n 1/2
(Z::.F~)= P / / + /
8t ~ 1/2 ~ n d+n
/ / 1
J~/2 J~+~o ~
dah l(~) cos~ H: + h ) 2
Vl + 4r cos a
p ~ t~ A t~ ~o
+8 Lf J~ A
J + J 1
~+~o ~+Ag
dah l(~) cos~ H: + h ) 2
Vl + 4r cosa
tm r 4/4, tesp tvely The con ttmt
iPtmdt i~app aingindhelimit otintg~ tionom
tt p nd to the zetos ot dP/d~ tot ha tmd t ~ A cot~t
sp nds t 6he zetos ot dP/da tm h2, whtve P(a) is dt
dved fimm:
cot P = [tan a ~ ]
smacosa
The l~chin tun ton H(a, h), ao, md ht ttedefined in
Lin md Reed (1976)
43
DISCUSSION
U. P. Bulgarelli
Istit to Ncziorurle per Studi ed Esperiff~ze di
A chitetturc Nacelle, Italy
I presume that c unified theory of resistance,
mu m m, Ed seckeeping is based on
unsteady R.~\ SE To do Nat one point should
be shessed much mme th m m the pest, the
treatment of the free surface chat is not yet
enough accm ate
AUTHOR'S REPLY
The mfhors appreciate the interest Ed
stimulating questions the discussers have
provided on this complicated subject As with
the paper, our responses will leave es m my
questions es Hey m wer
U. P. Bulgarelli points out Nat tree surface
resolution is c problem in RANS computations
For seckeeping cclcubtions in extreme sees it is
critical Nat m accurate fi ee-surface elevation be
predicted be mse he free surface will impact the
clove water p o non of the hull it remains to be
seen to what extent this issue m be resolved m
unsteady RANS open, this is still m issue for
steady RANS
DISCUSSION
L J. Doctors
University of South Wales, Australia
I would like to fir t saythat I enjoyed She
presentation very much As I have come to
expect from These two mthors over the years
In reference to slender hulls, you rater to She 2D
+ t (two-dimensiomtl plus time) method Do you
believe this approach will work for c catemmar~m
in which She vessel es c whole is not slender
even though the individual d m hulls may be
slender?
AUTHOR'S REPLY
L J. Doctors asks whether or not 'D + t methods
c m be applied to multi-hull vehicles which
overall me not slender even though the
mdi idual demi-hulls may be slender We are
not sure, but w Hi k there would be
deficiencies for two reasons First, the 'D + t
theory does not contain the tr msverse waves Ed
these surely will effect the unsteady loads on the
second hull Secondly, the second hull is in the
far field of She fir t hull in 'D + t Theo y w
would e. p et inaccuracies in the far-field wave
predictions that m t rn would lead to deficiencies
in the .. a ve load predictions on She second hull
DISCUSSION
L En
In tit to Superior Tecnico, Spain
The mthors definition of verification: 'The
dem onsh ction that the code is retsorLthh hug
free Ed that the output is mmmerically correct"
suggest the following comments:
Although w are aware That there is still c lot of
debate on She proper deli n On of verification,
the present d it con on does not mention the need
to qu notify the en or Ed or the uncertainty of the
verification procedure, which w believe to be
essential in such c process
It could also be mentioned in She paper that the
verif cation of c complex flow is not c trcight
fo ward exemise Ed Nat, m geneeal, it is very
costly Ed time consuming
AUTHOR'S REPLY
L E c tises the issue of mmmericcl error Ed un-
certainty m the verification process We agree
that She quantification of mmmericcl enors Ed
unce tcinty are imperative for verification Ed
validation of c code Whedher error analysis
belongs in verification or validation is not that
import mt She key is that is must be done in
reality, it probably should be part of both
processes
DISCUSSION
M Tulm
University of Calffomic, USA
I just w mt to con r~mh~e Bob Beck on His ve y
valuable paper Ed th mk him for its preparation
I say That particularly bee mse of She clear
emphasis mdfocushegavetok geamplit de
motions Ed She complexities of that regime He
mentioned SPH particle tracking methods)
which me in their bedimming I Hi k These hold
enormous potently for the future bee mse of their
impute sbi it to de tl with large su face
defommations,breckmg, plashing, mdvorticcl
structures He also gave emphasis to validation
Ed She importance of date it is trrilrmg how
much date w have in the smallest amplitude
regimes Ed so little (vat ally no yst matic
date) for very Urge ship motions
AUTHOR'S REPLY
M Tulin lauds SPH Ed endorses our
observation that thele is c pmcity of Urge
cmplit de motion date for validation Ed
verification purposes We are pleased that he
supports our position Nat much more validation
of large cmplit de ship m otions must be done
SPH hr. demonstrated intere ting
possibilities in r..-o-dimensional computations
Questions remain es to the validation of She SPH
predictions for physical qu mtities such es
pressure Ed particle velocity in addition, the
extension of She medhod to external fh~ee-
dimensiom~l flows could prove to be
probl matic
