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OCR for page 993
Psper for Presertstior st the Twerh Third Symposium or Naval Hydrodyasmics
Vsl de Reuil, Fmace, September 17 22, 2000
Waves and Forces Caused by Oscillation of
a Floating Body Determined through
a Unified Nonlinear Shallow-Water Theory
Rupert Henn, Tao Jiang & Som Deo Sharma
(Institute of Ship Technology, Mercator University
D-47048 Duisburg, Germany)
ABSTRACT
Bcsed on c tmffed nonlinear shallow-water theo y,
wa~s md forces c msed by vertical oscillations of floct-
ing bodies m shall w water w re m mericclly it sti-
gated Whet es fhe classiccl set of Bottssmesq's eqtu-
tions for fhe ottter-field flow ffticclly tmder fhe ft e sur-
fcce was solved m c st mdard g id based on c complete
C mk Nicolson scheme, the t w set of Bottssmesq-t pe
eqtutions for the i mer-field fl w ve ticclly tmder fhe
body suricce was solved in c staggered g id ttsmg partly
c Crmk-Nicolson scheme md partly c f lly implicit
scheme For mcll oscillation tmplitttdes of the body, c
good cg ement betw en the tmified fheory md the Im-
ear wave theo y was cchieved for tmplit des of fhe wa~
elevation es w ll es for the wave fome H wever, fhere
was c phcse shfft m fhe wave force As the oscilktion
tmplit de mcrecsed, fhe tmified fheory sh wed strong
nonlmear effects on the wa~ elevation md particttlarly
on fhe wa~ fomes
INTRODUCTION
In the k st two decades c considerable tm otmt of research
effo t hcs been devoted to the tpplication of B ottssit sq
type eqtutions to variotts shall w water wave~el tted
problems
In simttlation of wave propagation from dep water
to shall w WCtff, c major cchievement was fhe exten-
sion of the cpplicability of Bottssmesq's eq mionr to
sho t waves in modercte w tter-depfh het :xist two
prmcipcl ways to obtcin this kind of modified Bottssi-
t sq's wave-models One possibility is to ttseamodffed
depfh-averaged horizontcl velocity m fhe classical for-
mtthtion, see Witting (1984), or to cdd high-order terms
to the ckssiccl formttlation, see hladsen et al (1991)
he par tmeter occurring indhese tt ctments c mbe fotmd
by tpplying c Pcde exp msion for the cssoci tted dispff-
sion relation of modffed Bottssmesq's eqtutions md of
the Imear wave theo y he other, c more mtiot~l, way
is to ttse c hori ontal velocity et m arbihary verticcl
level in deriving Bottssmesq-type eqtutions, see Nwog
(1993) md Sch oter (1995) he par tmeter reqtti cd
for defiming fhe ve ticcl level m cgam be rpff ified by
comparing the di persion relations betwefft the re mlt-
ing Imearized eqtutions md fhe lit ar wave theo y By
sttitable selfftion of fhe cssocicted par tmetff valt es fhe
m odff cd Bottssmesq's eqtutions are valid for c rctio of
wave leng h to water-depfh d wn to c valt e of 2, practi-
cclly to fhe deep-water region
In comptttation of waves get mted by ships, c sig-
nific mt conh ibtttion was fhe mclltsion of fhe ship's i fitt-
et e on fhe tmbient fiow As repo tedby Ji mg (2000b),
th e different cpproxim ttions wet cpplied to decling
with fhis ship-get r tted t arfield fiow For c slender
ship, fhe techmiqt e of matched c mptotic exp msions
was cpplied in fhe works of Pedersen (1988) md Ji mg
(1998) The so-celled techmiqt e of matched c mptotic
exp msions was first inhodt edby Tttck (1966) m c 1in-
earized version md then extfftded to forced KdV eqtu-
tionbyMei (1986) md to cKP eqtutionby M i & Choi
(1987) This nonlit ar ffrion was refimed by Chen &
Shmmc (1994) by ttsing c modffed KP eqtution md c
improved slenderbody fheory For c wall-sided ship, c
medhod based on fhe mass conrffv ttion Icw was derived
byEtekmetcl (1997) Forcfictship,cpresstm distri-
btttion proportiot~l to fhe loccl ship-dk tfi was cpplied by
Ji mg md Shmmc (1998) Due to the possibility of the
dirfft implementation of c pt sstm distribtttion on the
free suricce m Bottssit sq's eqtutions, wa~ get mtion
by crwlytical presstm distribtttions was f eqt ently st d-
iedbym myattfhors, see e g Wtt & Wtt (1982),E tekin
et cl (1986), md Pedffren (1988)
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To remove She Move restrictions on She hull fomm Ed
to extend the cppliccbi it of shall Rewater equations
of Boussinesq type for wavebody Interactions, c mi-
fied shall w water theory was recently derived by Ji mg
(2000a), which comprises c ck ssiccl set of Boussmesq's
equations for the omff-ti id flow vertically mder the free
smfae Ed c new set of nonlinear partial-d fferential
equations of Boussinesq type for She im r-ti id -I w
vertically Ed r the w ned hod -i~xhce in comparison
to She simplified approximations mentioned ohm e, this
new set of Boussinesq-type equations for She im anti id
flow sati fies the t mgenticl-fi w condition on the w t-
ted hod = nap Ed enables the direct cclcoktion of
the pressure distribution on it Thus, 6 is mff ed Theory
yields c theoretically consistent approximation of m my
wa~-related th ee-dimensiom~l problems in the horizon-
tal two-dimff shoal p me
In the present paper w apply 6 is mdied nonlin-
ear Theory to investigate waves Ed forces c Used by c
smfae-piercingbody oscilktmg m shallow water After
repo tmg the mom derivation procedure of the theo y,
c suitable m mericcl medhod will be implemented for
solving the miticl/bo mdary value problems governed
by the two sets of nonlinear particl-differenticl equa-
tions To meet She m mericcl dffficulties arising from
the nonlinear terms Ed the Imear high-order terms, w
apply cgam m implicit Crmk-Nicolson scheme m the
temporal Ed spatial discretizations To overcome the
difficulty c Used by the decoupling of the pressure on
the w tted body-smfae f om She depth-avemged conti-
mmity equation, She w 11-proven concept of c staggered
g id will be implemented in our computer code To store
the resulting diagonal parse matrix, c compressed di-
cgorurl storage fommat will be used The corresponding
large nonsymmeric linear equation system will then be
solved by me Us of the GORES method given by Sand
& Schult (1986) To verify 6 is miffed Theory Ed to
examine our m mericcl method, we investigate the wan
generation of c vertically oscillating cl cohr cylinder m
shall w WCtff For mall cmplit des of body motion,
the calculated wave profiles Ed wave forces will be
compared with Hose from She Imear wave theory The
nonlinear effects occurring et urger motion cmplit de
will also be discussed
THEORETICAL FORMULATION
Gen Description
Considering wave generctionby c surfae-piercingbody
oscilktmg m shallow water of varymg water depth
/1ii.1~. Of I Ed using prime ~ es superscript to denote all
dimensional q entities, c Cartesi m coordinate system
O~'lf,' is used to describe She velocity Ed pressure
field, where She plane O. I'd' is on the quiet f ee smfae
with She axis ,' positive upward
Assume that the fiuid is incompressible Ed inviscid,
the velocity components us. r'. At Ed pressure field It
c m be described by She contimmity equation
" ~'',11 . =0.
Ed Enler's equations
(1)
<' ]! <' <' ! ~ <' ,. t' <' ,! ,1 _ (2)
~ ]! <' ~ ! ~ ~ ,! (’ ~ ,! ,,1; ' ( )
(< ,! <' ’< ! ~ (< ,! (t ’< ,! .li tat! 0 (4)
in She who le -: i d dom cm Here m, of denote s the ace l -
erction due to g avity, . file water density, Ed ~ is the
independent time variable
The solution of the ah -. e governing equations is de-
temm ined by She foil wing bo mdary conditions:
o kinematic condition on She f ee smfae -' =
(~i.~. ,/. t I
~ ~ ~ ~ ~ . ~ ~ ~ ! ~ i ~ I! _
(5)
where the subscript f studs for the free surface,
less med to be differentiable m time Ed spa,
o dynamic condition on the f ee smfae :' =
~'( .'. I/. t I
If, = if. (6)
with She atmospheric pressure if on the f ee sur-
fa,
o t mgenticl-fi w c onditi on on the water b off om sur-
fa ~ /, ~ . . {/I
I I ~ I ~ ~ i / ~ ~ ~
(7)
where She subscript b st mds for She bottom sur-
fa, less med to be dffferentictle in space,
o t mgenticl-fi w condition on the w fled body sur-
fa _'=TONI. IJf.t')
~ ~ 6 ~ [] ! ~ ~ 6 at, ! ~ 6 ~ ~ 1
(8)
when the subscript s st mds for She body (ship)
surfime, nss med to be differentiable m time Ed
space,
OCR for page 995
md cppropri~te miti~l conditions
Using 6he two st mdard nondimensiomd parameters
in the shall w-water cpproximation
1t = = A''ii' md ~ = kA
~ 1,
with wave cmplitude (A. waVe length )' or wave m mbff
Ai, md the me m water-depth ’, 6he physiccl vari~bles
are nondimensiomdly sccled es follows, now denoted
without 6he primes ' for di tmction:
i.f l/l = k~i.l' '~) hori ontalcoordinates
ve ticcl coordinate
11 = ~ WCtff dep6h
t= ['\/y'T'f'
~ vi;;~
(~.= -
' ~ ~
~ = / 1: 1 I'I
I~ = ~!, pressure
This multiple scclmg leds to c nondimensiorul f -
moktion of the goveming equations:
It id' ’1j,)(<, = 0
<{,fid{~. ('<~:, - ,,2~<'<~.) - 1'. = 0
~ F:<'~ ~ ~{'},~(e'f'.)1':, = 0
<<''ei<~<' c,<~.:, - 1~: <<'<<',1~1). = 0. (12)
md of the corresponding bo mdary conditions:
o kmematic condition on the fre suricce _ =
(; i.t '1. ~ 1
(<'f = 1! [~!fid~f;.: - ('f’~l)]- (13)
o dynamic condition on the free smface ~ =
(i.- ,,.fi
mdependent time-vari~ole
ve ticcl velocity component
loccl ship dLcft
(9)
(10)
1h = 1~-
(1 1)
(14)
o tmgenticl-fl wconditiononthewaterbottomsur-
face _ =I~i.t -'11
(~Y =}t id~b/~: ’~11:,l-
o t mgenticl-fl w condition on 6he body suricce, =
-~(i''1.f)
<~., =_ ~ [~'e<,.T -~<'.~] (16)
(15)
Appro\imation of the Outer Field Flow
To descr~be the fl w m the outer fleld verticclly m-
der 6he fre suricce, 6he contmnity eq mion (9) is mte-
g cted from 6he water bottom /~ i.~'ml 1 to the free suricce
s’~.~/ f) Forcsingle-valuedfunctmnof~i.~ '/.f) md
/i i.~. ~/1 et ecch hori ontal point i.~ m/1 it yields:
/t ~ (~.: (':,1~1,(~f(~\ = 0. (17)
By mserting 6he kinematic condition on 6he fre suricce
(13) md the tmgenti~l-flow condition (15) on 6he water
bottom mto equation (17), C ve ticclly mteg cted conti-
mmity e qua t ion c m be derive d es
(!~' [(~;/'i"[ = 0 (Ig)
with the dep6h-averagedhori ontcl velocity components
W = id{- (-'l = . _ / /' 1~/~
md the two-dimensiomd m~bh-operctor ~ = i,~ ,, )
It is wor6hwhile to mention thm this verticclly mteg cted
contmnity eq mion m the outer fleld satisfles 6he conti-
mmity eq mion (9), the kmematic condition (13) on 6he
free suricce, md 6he t mgenti~l-fl w condition (15) on
the water bottom
By virt e of Kelvin's Theorem for inviscid fluids,
wave genemtion m be considered to be m inotstiorud
flow Fo I lowing c st mdard dff ivati on pr ocedure md ex-
ploiting 6he cbsff e of horizonbl vorticity-components,
see e g Ji mg (2000c), 6he outer velocity-fleld c m be
represented by 6he depth-avemged horizonbl velocity
w
(19)
(~.e')=li.i'-ll.ll
-It~i 6 - 2 )~(V ~)
-It~i2 - -~57[~7 il~W] - (~ic/~./~41- (20)
"'=-It2[,i\7 ~1-V ili~l]-()i/~4). (21)
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Representative terms from entire chapter:
vertically oscillating
To cpproximate fhe outer pressure-field Pi.l' IJ ~ f 1.
the ve tical moment m equation (12) is integ ated over
c local submerged depth (~;,1 Afier c rearr mge-
ment fhe outer pressure-field ca be fhen represented by
the depfh-averaged hori ontcl velocity w md fhe wave
elevation ’:
I) = I)f - ~ - ~
1' [ 2 ~'w'_57 C1,11,1]
(~(~.1~4). (22)
The relation betweff fhe depth~emged horizon-
tal velocity ll(.~ '~./1 md fhe wave elevation <~.t 'l./1
c m be obtamed by integ cting the horizonbl m oment m
equations (I O) md (I 1) over the water depth 16;11 1:
W. _ c(W . ~ )W _ ~;~I)f
-/t 2~[V (/lW,)]
_~/] ~(~ WI)_(~/~.~14~=0. (23)
Appro\imation of the inner Field Flow
To describe the fl w in the i mer field verticclly mder the
w tted body suricce, the contmnity equation (9) is now
integ cted from the water bottom /~ ’.~ .0) to the body sur-
fcce [~.f . tY f 1 For c smgle-~l~d f mction for fhe water
bottom /~ i.~ m~l md for fhe local body d cft Ti.l'm~. f) et
ecchhorizonbl pomt i.f . tY) it yields:
~ I
}t I ~.: - '':~1-1, - (<'6 - (~\ = 0, (24)
~ ~,
Insertmg eq mionr (15) md (16) into equation (24)
yields fhe following vertically integ cted contimmity
equation:
['6~' [(/1Tll1] = 0
where the depth~raged horizonbl velocity T is de-
fimed by
(~. ~ ) /] _ ~ ~ 11~/~.
Equation (25) satisfies fhe contimmity equation (9), fhe
tagenti~l-fi w condition of equation (15) on fhe we-
tff bottom suricce, md the tagenti~l-fi w condition of
equation (18) on the wetted suricce
Similar to the outer velocity-field, fhe im r velocity-
field c m also be representedby the correspondmg depfh-
averaged horizonbl velocity w
<' (lI = Wi.f. o-tl
6 - 2 )57~V wl
I ~ 2 -)~[V (/ib)] - (~/t ). (27)
(26)
<< =-/~[,(V Wl~ (/i=I] - (~/t ) (28)
Theim rpressure-fieldpC~ '~ ~ f)
~, . T? ,2.
1~=1~- ~ ~ 2 2 )
~.w~l_~t,1~7.~/,W,)
(~/~./~41
(29)
c m be obbined by integ ctmg fhe vertical moment m
equation (12) over fhe submerged depth ~~-), where
1~ denotes fhe pressure on fhe wettedbody smfae
The requi ed relation betwen depth~ragedhori-
zontcl velocity ll(.t . ~J f) ad pressure P6~.t . ~J_ f ) on fhe
wetted suricce m be derivedby integ ating fhe hori on-
tcl moment m equations (10) md (11) over the water
depth (/~T1:
r, -ei r.vlr-vl:~- ~
-1~21i 2 tV[~7. (/4w')]
-1~2 6 1i 1i v
for the outer-field fl w with dep6h-averaged hori motel
ve loc ity w md wa~ e levati on ~ es urJcn wns, md c new
set of nonlmear parti~l-differenti~l equations of Boussi-
nesq type
['6~ [(/iTllt] = o,
(34)
W. _ FiW . 57 )11\~116
~ 2 ~[~ ilil i)]
''2 6 \7iN7 l~'1
-~t 2 l\7[~7'(/~1-~i] - 3 ViV W}
_~Ti-~V r'-~7 i/4W')] = \~[ (35)
for the im r-field fl w with dep6h-averaged hori motel
velocity i md pressure P6 ating on the wetted body-
smfae es urJcoow s
Equation (34) would satisfy the t mgenti~l-fl w con-
dition on the w tted surfae exatly ffthe depth~vffcged
velocitiy w w re exat Smce w is cpproximated to
the lecding ordff of ~ md /R in the Boussmesq the-
ory, 6he new set of eq mions lecds to c Boussinesq's
cpproximation m the im ff fleld How ver, it is consis-
tent wi6h 6he B ous s me s q ' s cpproximat ion in 6he outer
fleld 7he resultmg mffed shallow-water theo y thus
yields c consistent approximation of m origirully 6 ee-
dimensiom~l problem in 6he reduced two-dimensiom~l
hori ontcl phne Moreover, it emibles the dirfft calcu-
htion of the pressure dish~bution on the wetted smfae
for the fl st time us mg c shal low-water wa~ the o y
In shallow water et con t mt water-depth /~ = 1,6his
mifledtheoycmbesimplifledto:
;! - ~ [i~ - IiI~] = o,
w' _ riw . 57 )w _ V;
-~' 3~7iy7 wlI=-VI)f.
for the outer-fleld fl w, md to:
-T'(V [(1 - t)1 ] = 0.
W. _ FiW. 57)W\~116
ii - T)2
~ 3 T'V(~7 1 `)
-~2il-ElVEV.w'=~t
(36)
(37)
(3g)
for the im r-fleld fl w
Couplhg Condfldons
7 he two sets of B oussinesq t pe equations cre coupledby
the cssoci~ted interfai~l conditions et 6he mst mt meous
waterline rJw = f i.~ w- f ) es m mst mt meous intersection
of 6he body surfae wi6h the free surfae
At locations without c ve ticcl sidewall near the we-
terline, i e, ~ =T. the mte fai~l couplmg conditions
reed:
I~i.i'w.'Iw.~) = I oi.i'w.'Iw.~).
1~6i.i'w. l/W.tl =l~i.tw. l/W.tl.
(4o)
(41)
where the subscripts lad O dffotedhe fl w inthe im ff
md OUtff fleld, respectively 7 he condition expressed by
equation (40) me ms c contmoous ch mge of 6he depth-
a~rage d hori motel velocity et the waterline
At locations with c ve ticcl sidewall near the water-
lme, i e, ~ > T. 6he intfffaicl couplmg conditions
reed:
(/lTw)Tl'~=l/l’)To,,.
(42)
1~6i.i'w. l/W-~W-~1 =l~i.i'w. t]W.[w.ll. (43)
where T~,, md To,, denote the depth~vffcged hori-
zontcl normal velocity component et 6he waterlme in
the imer md outer fleld, respectively 7he condition
expressed by equation (42) ensures mass conservation
th ough 6he waterline
NUMERICAL IMPLEMENTATION
Inthepresent tudyw cpplythe mffednonlmear6heory
to mvestigate the waves ge ffcted by c smfae-piercmg
circular cylmder oscilhting ve tica lly in shallow water of
constat water-depth /~ = /` G md 6he cssocicted forces
For c problem having axicl mmetry ctout 6he verti-
ccl axis _, the dimensiorul equations, now omittmg the
primes ~ for simplicity, m c cylmd iccl coordim~te systffm
corresponding to equations (36-39) reduce to:
(! i; 1] Gl(( ~ , i (~' O-
(44)
(!((, - .~'
i~ ~ ~ ~ ,'2` ~ ~ ~ I = 0- (45)
for the outer-fleld flow ~ > 1? md to
il, G t) it ~ ,, I t' ~ [! -
(46)
_ ~ 1 _ ~ 196 )
_ i / ~ G T'| ~ _ 1 I 1 1 2 1 )
11 G 1'1''~1 I 1 / 1 1 / / )
(11 G t)~' (1 I, ) yt' . (47)
for the i mer-field flow I < r Herein, ~ denotes the
dep6h-averaged radi~l velocity, I 6he radi~l coordincte,
md 11 6he radius of the subjected cylmder
The two sets of equations (44-45) md (46-47) ore
coupled by
[1 1 /] G T! )] I ~ 11 [1 1 /] G ~ 1] 1' /? (48)
1~ 1, ,, = 1'o 1, ,, -
with
1~01, ~r = L'.')~; - T1
{'( 2 T/' G 1 11 1 ~ I )] I ~ Ir
For the i mer-field flow the bo mdory condition et
= 0 reeds:
(49)
(so)
,tO.fl=O. (51)
For 6he outer-field fl w the linearived nom eflection con-
dition et 6he trmmation bo mdory reeds:
;~-~;: ;, =0.
~ ~ - 4;: 1, = 0,
(52)
(s3)
Theseequationsconespondtothew Il-k ow Sommer-
feld's condition md ore one-sided discreti:osd 6 oughout
our compubtions
Crank-Nicolson Scheme
Due to the relatively simple geometry of 6he horizonbl
sohtion domcin, c flmite dffference medhod c m be gen-
erally used to solve Boussinesq-type equations To meet
the m merical dffflculties cssoci~ted wi6h 6he nonlineor
tffm s md the Imeor high-order temm s of B oussinesq-type
equations, 6he Cr mk-Nicolson scheme is implemented
in temporcl md pati~l discretizations of the governmg
equations in 6he Outff fleld es foil ws:
~ ~ ~ _;, , ~ ~ ~ _ ~ ~ ~ _ ~ ~ ~ _ ~':~
Af ' 4Ar
1, ~ 1,21 1,, - 1'~,
(;i 4.hl
1; ~ / ~ G 1 2 i 0 - ( )
., - ., _ , r-, , - ., , ., , - .,L,
Af ' 46r
(, ~ _~' ~ _g, ~ _~l
-.'! 4,~
11 G . 1 _ 1
3 ~ Af
.,, _, ~ 1,' _ ,, _, ,~
2AfA,
2,' ,' - 2 ~ 1'''2~
AlA,.2
= 0. (55)
,' ~2,' ,'~
where Af md Al dff ote the time md spi~ step-si~
As sh woby Jimg (2000b) the ~oove implemented
scheme is generally sbble but lecds often to m mericcl
oscillations which c m be mppressed by me ms of global
md local flltering
Staggerffd Grid
The cpplication of 6he ctove Cr mk-Nicolson scheme for
the governmg equations m the im r fleld is not shaight-
forword Due to 6he decouplmg of 6he pressure /96 on
the w tted body-smfcce from the dep6h-averaged con-
tmnity equation, c speci~l trectment is requi cd For-
tunctely, 6he dff ouplmg effect is mathematicclly ormlo-
gous to that k ow f om Enler's equations Thus, the
well~stablished concept of c sbggered g id c m be cp-
plied to overcome this diflflculty A discretization based
on 6his concept is impiffmented for fhe governmg equa-
tions m fhe im r fleld Becimse of fhe lineority of fhe
vertically integ oted contmnity eq mion (46), it is dis-
creti:osd using c fully implicit schffme:
1 , , I, , ~ ,1 _ ,,: ~
il] G i l ( 6: 2,
.' I, - . j: '
(56)
i~t; = i :' i- ';lt ~
Theverticallyinteg otedmoment meq mion(47)is
discretized es
1; - 1 j: , 71; ~ (; ~ (; ~1 j:,
~ ~ 46r
_196; ~P6 ,!
/~r
3, ~ i 6'
1,, - 1;, - 1 jl, - 1;,
2~A,
2: l ~ l I ~ l I2` l <; I
~1
( 11 G T; | ~ (; _
3~: ~lL 2
~ I _ ~ :: I _ (, I _ ~ l~ l
46,
2: l I~ l Il~ l I 2` l ~ ;~ I
~ I _ ~ I _ ~ I _ ~ r~I
(11 G T; ) ~ ;( 2AtA1
~ I _ ~ l
(s7)
s t ~ = I .2.3. , ~ 1rI it i s worth m ent i onmg 6~t the
tffms ii ssoci~ted wi6h ~ s e discretized usmg the C lmk
Nicolson scheme lmd 6he pressure tffm usmg s fully im
plicit scheme N mffical test r ms htive sh wn 6~t 6his
treli tment mppressed the pressure oscilktions clmsed by
s complete C lmk Nicolson schffme h the Ih tter cs se
lm mderrellixation procedure wi6h s flictor of lipproxi
mlitely 0 6 hti d to be s dditiomdly pe fommed
Scheme for Coupling Conditions
At the mtersection ~ = ~ 1l,6he ve tica lly integ s ted outer
contimmity equation (44) is one sided discretized s s:
~ I _ ~ , ~ l II _ ~ l I _ ~ l I _ ~ l
~ ' 2~,
, , ': I _ ~ ': I _ ~ ~: _ (:
~;l 5 I ~ G 1 ~ i 0 ('
Intelid of usmg 6he verticlilly mteglited outer mo
ment m equation (45) ;; t ~ 1~. the coupling equation (48)
is discretized ;; s follows
I ~ G (; I (; I I ~ G <; )
2
= ~ l l; ~ /] G T'; |
;8)
(59)
Since 6he depth~vffliged contmnity equation (45) clm
be di ectly mteg lited for 6he given bo mdli y value
rC0.fl = 0, 6he value rll; ~ clm be determmed ei6her
by 6he limli Iyticli I solution
2 /1 G ~
or by the m mericli I integ ;; tion of equation (56) in ;; d
vance
7he required value of the pressure lit ;1~ clm be
obttiined from the couplmg condition (49) Since the
ve ticli lly integ s ted outer moment m equation (45) is
(60)
not used lit 6he g id point `1~. the high ordff temm
/~[1 ~ T/IGII(n " )]|, I? m equation (50) is ne
glected in 6he present st dy Consequently, 6he pressure
lit ;1r is lipproximlitedby
I'8l, =/"~kl, Til:l
Solution Method
(61)
7 he libove discretization scheme lecds to ii lineli ii Ige
brti ic equation ystem 7he corre ponding systffm mli -
tri isti pli semlihi whereonlydiligomdblockshtive
non :osroelements(so calleddi~gomdwithf mges) For
economy of storti ge, ii compressed dili gorul tortige for
mlit is used Since the pressure P6 lEppeli s only liS ii
spli ti~l derivli tive term in the verticli lly mteg ii ted im ff
m oment m equation,6he system mli tri thus hti s :osro cl
ements on 6he mli in dili gorul This implies thti t Cl;; ssicli I
itertition medhods, lik Jl; obi or Glmss Seidel, C;; mot
be ;; pplied Although the resulting linear equation sys
tem clmbe di ectly solved by usmg eliminction medhods,
such methods genera lly need s full storti ge of 6he system
mlihix lmd cons me s Ih ge computing time Thffe
fore,6he more s dvanced GMRFS (Genera lized Minimli I
Residual) method given by Sli~d & Schultz (1986) is
implemented to solve the resulting nonsymmetric Im
eli equation system usmg s compressed di~gomd tor
lige formlit To s celertite the convergence process, s
resbrted version of GMRPS is s pplied
RESULTS AND DISCUSSION
In the present st dy w lipply this mified 6heory to
simullite the ws s genertited by two verticli lly oscil
Ih ting bodies One corresponds to 6he horizonbl semi
submerged 2 D circulli cylmder with s rs duis 1? = 0.5
m, s s inve tiglited by Yu & Ursell (1961) lmd by Keil
(1974) The dkli fi lit rest reli ds
Ts = ~7 fOr 1t < -~
(62)
Thesecondistiverticlilcirculli cylmderwithtiradi i
1? = 2.3 m lmd wi6h s con tlmt dkli fi s t re t
Ts = 0.4 m The se tw o b o die s, b odh hti vmg s ws 11-
sided freeboli d over the still wliterline, li~e forced to
oscilllitehti moniclillymshtill wwliterofdepthll = I
m The remltmg instmttim ous loclil dklift mdff the
stillwli ter level is 6hen
~ = Ts(I cos ~.
(63)
where m is 6he forcmg frequency lmd rl the forcing s m
plitude
Note At for the ve tick circular cylinder She m mer-
ical implementation fommokted in c cylindkiccl coordi-
m~te ystem m be di ectly mplied For the hori on-
tal circular cylinder c m mericcl implementation cone-
spondmg to She governing equations m c Cartesim co-
ordincte system on the ve tica I plane c m be performed
The letter case was completely do merited by Jimg
(2000b)
To examme She numerical method proposed, w
st dy first the vertical ci char cylinder oscilktmg in
shall wwaterctaforcingperiod2'=2.3s Fig Icom-
pares the calcokted depth-avemged radial velocity wi6h
the crurlyticcl solution given by equation (60) There is
no visible difference in These two predictions Fmther-
m ore, since the bottom of the subject vertical cylinder is
fiat, the dynamic pressure cmbe amtlyticclly described
by
i /4 G T 21/1 G ~ ) ] ~ .{J; . (64)
where ~ takes the same value bodh for the crurlytical
solution md for She m mffica/ approximation
Fig 2 shows the comparison of She m mericclly ccl-
cokted pressure with the crurlyticcl solution Again,
there is no visible difference in these two predictions
After verffymg She m mericcl method, we apply our
computer code to compute the vertical cylinder oscilkt-
ing wi6h c small cmplit de n = 0.001 m m c rmge of
Basalt mbers ~ /~ G < 2 ~ c m be seen m Fig 3 that
withm the r mge of validity of the classical Boussmesq's
equations She amplitude ratio of wave forces determined
th ough She miffed nonlinear theory cg es w 11 wi6h
Ott from '.\ ~\~ r (1995) based on the Imear wave the-
ory he WANT forces were first computed m the
frequency domain in temms of added mass md damping
md Then >! thesi:D:d m the time domain ) As expected,
the iongff the waves are, the better the cg element Look-
ing n w et the time histories m Fig 4, w c m observe
Ott there is c phase shift bees en She two d fferent cclcu-
htions There maybe two possible reasons: (i) The two
c tlcnl Ill ons used different initial conditions '.\ A\ ~ r
yields only he asymptotic solution md our computation
assumed he motion started from re t (ii) The coupling
conditions have not yet been implemented perfectly or
they may be physics fly inexact We pi m to compare fur-
thff w ith he arulyt iccl s o hit ion der ive d by Ye mg ( 19 81 )
md wi6h physical m odel tests to ckrffy this phase shift
For c smell oscillating amplitude n = 0.001 m, Fig
5 compares She cclcohted amplitude ratio of the wave
elevation to the body motion wi6h those Gideon by Yu &
Ursell (1961) (solid line) md Keil (1974) (dashed line l,
both based on the linear wave theory for c hori motel
ci cular cylinder Since the results cited cg ee w 11 with
those from experiments, the good cg cement between
our nonlinear theo y md he linear wave theo y, et let t
withm he expected rmge of validity of Boussinesq's
equations, namely, waken mbers A /l G < 1.5, C m be m-
tewreted es c practical va/idmion
T ming now to he specific cdvmtage of he miffed
nonlinear shall w water Theory, namely, its capability to
acco mt for nonlinear elf - to of shallow-water waves,
Fig 6 shows mstmtmeousradicl wave profiles normcl-
i:D:d by the cmplit de of m otion of he vertical cylinder
for systematically Increasing amplitudes .: = 0.001 m
(dotted line l, c = 0.01 m (dashed line), md n = 0.1 m
(s o ii d I ine) As expecte d, She wave e levat ion decays w ith
increasing di tame from he body owing to dispersion
in the unrestricted hori ontal domain Hence, nonlinear
effects are more visible near he body They are better
seen m he re ponse forces since the dynam ic pressure is
proportiomtl to the wave elevation di ectly on the body
es describedby the coupling equation (50) Fig 7 com-
pares the th ee time hi tories of he re ponse force also
nom Shed by he cmplit de of motion Furthermore,
es no difference in the nom Prized wave elevation or re-
sponse force c m be noticed for oscillation cmplit des
smeller 6 m 0 01 m, linear theo y maybe safely used up
to .: < 0.01 m m he present co figuration
CONCLUSIONS
A mified nonlinear shtll3w water then y 03mprismg
two t ts of Bcussinesq t pe equaticus was introduced
to detffm me wows Ed forces o me d by cscilktions of
~ fixating body m shall w-water A m merited medhod
base d on he Cr mk-Nio 3 is on sohem e 0 3mb me d wi6h
fully implicit t h me was successfully implemented to
solve he governing equaticus in ~ taggered g id Fm
small motion tmplit des ~ good ag cement between the
miffed nonlinear chewy Ed linear wee then y was cb-
served for wave elevation Ed re pause force within the
r mge of validity of he classical B cussinesq's equations
H wever, the pret nt miffed then y did display nonlm-
esr effects for larger motion tmplit des Mme f mda-
menhlly,w own w03 fidently~pplythenew miffed
nonlmesr htlbw Water chewy to simulate wee hod
interactions
REFERENCES
Chen,X-N at Swarms, SD 1994: N3nlinesr6heory
of ~ mehio motion of ~ slender ship in ~ shallow
charnel Prop of he 206h Symp 3nNsval HydLcdy-
tmios, CA, USA, pp 386-407
Etekm,RC,Qim,ZM &Websusen,JV 1997: Up-
stre mm scliton ge Ration by ~ slender, vertical stn t
mdship: B3ussmesqequstions Prop cfthe -that
Offshore and Polar Engrng Conf., Honolulu, USA,
Vol. III, pp.238-246.
Ertekin, R.C., Webster, W.C. & Wehausen, J.V. 1986:
Waves caused by a moving disturbance in a shallow
channel of finite width. J. Fluid Mech., Vol. 169, pp.
275-292.
Jiang, T. 2000a: A unified nonlinear theory for ap-
proximations of wave-related problems in shallow
water. Submitted for publication in the Journal of
Ship Research.
Jiang, T. 1998: Investigation of waves generated by
ships in shallow water. Proc. of the 22nd Symp. on
Naval Hydrodynamics, Washington, D.C., USA.
Jiang, T. 2000b: Ship waves in shallow water. Habil-
itation Thesis,Mercator University Duisburg.
Jiang, T. & Sharma, S.D. 1998: Wavemaking of
Hat ships at transcritical speeds. Proc. of the 19th
Duisburger Colloquium, Institute of Ship Technol-
ogy, Mercator-Univ., Duisburg, Germany, pp.l71-
189 (in German).
Keil, H. 1974: The hydrodynamic forces due to
the perodic motion of 2-D bodies on the free surface
of shallow water. Institut fuer Schiffbau, Report No.
305 (in German).
Madsen, P.A., Murray, R. & Sorensen, O.R. 1991:
A new form of the Boussinesq equations with im-
proved linear dispersion characteristics. Coastal En-
grng., Vol. 15, pp. 371-388.
Mei, C.C. 1986: Radiation of solitons by slender bod-
ies advancing in a shallow channel. J. Fluid Mech.,
Vol. 162, pp.53-67.
Mei, C.C. & Choi, H.S. 1987: Forces on a slender ship
advancing near thecritical speed in a wide canal. J.
Fluid Mech., Vol. 179, pp. 59-76.
0.004
c~
Nwogu, O. 1993: Alternative form of Boussinesq
equations for nearshore wave propagation. J. Water-
way, Port, Coast. and Ocean Engrng., Vol. 119, pp.
618-638.
Pedersen, G. 1988: Three-dimensional wave pat-
terns generated by moving disturbances at transcrit-
ical speeds. J. of Fluid Mech., Vol. 196, pp. 39-63.
Saad, Y., & Schultz, M.H. 1986: A generalized
minimal residual algorithm of solving nonsymmet-
ric linear system. SIAM J. Sci. Stat. Comput., Vol.7,
No. 3.
Schroter,A. 1995: Nonlinear time-discretized seaway
simulation in shallow and deeper water. Institut fur
Stromungsmechanik und Elektronisches Rechnen im
Bauwesen der Universitat Hannover, Report No. 42
(in German).
Tuck, E.O. 1966: Shallow water Hows past slender
bodies. J. Fluid Mechanics, Vol. 26, pp. 81-95.
WAMIT. 1995: WAMIT user manual. Department of
Ocean Engineering, MIT.
Witting, J.M. 1984: A unified model for the evolution
of nonlinear water waves. J. Comp. Phys., Vol. 56,
pp. 203-236.
Wu, D.-M. & Wu, T.Y. 1982: Three-dimensionalnon-
linear long waves due to moving surface pressure.
Proc. of the 14th Symp. on the Naval Hydrodynam-
ics, Ann Arbor, USA, pp. 103-129.
Yu, Y.S. & Ursell, F. 1961: Surface waves generated
by an oscillating circular cylinder on water of finite
depth: theory and experiment. J. Fluid Mech., Vol.
11, pp. 529-551.
Yeung, R.W. 1981: Added mass and damping of a ver-
tical cylinder in finite-depth waters. Applied Ocean
Research, Vol. 3, No. 3, pp. 119-133.
radial velocity
o numerical solution
analytical solution
_'
0.0 0.5 1.0 1.5
2.0 r [m]
Fig.1 Comparison of numerical and analytical solutions for the depth-averaged radial velocity generated by a vertically
oscillating vertical circular cylinder
25
20-
15
10-
it.
dynamic pressure
- numerical solution
o analytical solution
- ~~.~
--~00(
l
1 1
0.5 1.0 1.5
it-`
W0`
~0
i I ,
2.0 r [m]
Fig. 2 Comparison of numerical and analytical solutions for the dynamic pressure on the subject cylinder
9-
8-
7-
6-
4-
3-
2-
O-
WAMIT
O unified theory
_ ~
few
-
C)~
O 1 kh 2
Fig. 3 Comparison of response-force amplitudes determined from the unified nonlinear shallow-water theory with
those from the linear wave theory for the subject cylinder
- - WAMTT 1 knifed thenrv
-200-
-300-
I I 1 I
0 2 4 6 8
l
t [s] 14
Fig. 4 Comparison of time histories of the response force determined from the unified nonlinear shallow-water theory
with that from the linear wave theory for the subject cylinder
-
’
or
1.0
0.8-
.
() 4_
o
Yu & Ursell (1961)
Keil (1974)
· unified theory
am''
1
khc 2
Fig. 5 Comparison of normalized wave-amplitudes determined from the unified nonlinear shallow-water theory with
those from the linear wave theory for a vertically oscillating horizontal circular cylinder
0.50-
0.25-
0.00-
-0.2
-0.50-
-0.7
-1.00-
a = 0.1 m
----- a=O.Olm
a= 0.001 m
l , 10 ~ ~ 40 r m 50
\\/ ~ V Id
Fig. 6 Comparison of normalized wave-profiles generated by a vertically oscillating vertical circular cylinder with a
forcing period 2.3 s
40
300-
200-
100-
~ O-
-100-
-200-
- 300-
a= 0.1 m ---- a= 0.01 m a= 0.001 m
0 2 4 6 8 10 12 14 t [s] 16
Fig. 7 Comparison of normalized response forces acting on the subject cylinder when vertically oscillating with
a forcing period 2.3 s
DISCUSSION
H B Bmgham
University of Demmark Denmark
I w ou I d I ike t o c o n mm Icte the mthor s on the
m ost successful dem on wren on I have yet seen of
the duect use of c Boussinesq-type method to
solve the unsteady ship motions problem I have
c comment Ed c question
Comme t: Yourdiscussionofthedisagre me t
between your results Ed \\ .'~tlr for the phase
of She radicti m force Fig 4), suggests chat there
is s me ambiguity cutout She definition of f is
qu Pity it is quite well defined Ed, es long es
your computations have reached c stecdy-state,
you will converge to the same mew r if you
solve the problem correctly You do not discuss
convergence, perhaps these results have not yet
converged
Question: Th two examples presentedhere are
special geometries m the sense that your
dismetizationfitfhemexactly Doyouexp ctc
fmite-difference impleme tation on c mmifomm
Cartesi m grid to be Cole to cope with c enersl
ship-like geomeh y, or will you need to develop c
boundary-fitted solution in this case?
AUTHOR'S RtlPLY
We greatly appreciate Dr Bingham's discussion
Regarding his comment, Fig g shows
our simulated time hi tories of the re ponse force
along with the corresponding WAMIT result
Graph (c) demonstrates be convergence of our
mmmerical recall with re pect to pace
discreri lotion Gr Oh h) compares our c mptotic
time history with Nat of WAMIT Agam, the
cg e merit is rem.ukrhle for th cmplit de
U fortunately, the phase shift betw en She two
different calculations persists Ed Thus c m not
be explained by possible k k of convergence,
neither m pace nor in time discretizatioa
Mmeover, our cdditiorurl simulations show that
the coupling conditions betw en the im r Ed
the outer flow field have some i flume on the
phase shift
We ogre Nat c uniform Cartesi m g id
will not workw 11 for c ship-lke geomeby
Cune fly, w me implementing om computer
program m c curvilinear grid Ed w hope to
present the new results et the next Symposium on
NavalHyd odynamics
F 100
it]
300
200
1 00
z 0
LL
- 1 00
-200
-300
~ 005m ~~~~~~
l
.--
34
~=0.01m ~~~ ~~=0.005m
WAS IT
- - - - · - Ar = 0.0 1 m
l
I _1
32
(a)
40 42 44
t [s]
Fig. 8: Time histories of the response force
(b)
