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OCR for page 85
Nonlinear and Linear Motions
of a Rectangular Barge In a Perfect Fluid
R. Cointei, P. Geyer2, B. Kings, B. Molin2, M. Tramoni2
Assassin d'Essais des Carenes, France),
(2Institut Fran~ais du Petrole, France)
~ Abstract
The motion of a rectangular barge in beaten seas
is studied within the framework of potential
flow theory.
A simulation technique based on tile
Mixed Eulerian-Lagrangian ~netl~o`1 is de-
scribed. It allows the simulation of tl~e how and
tile resulting barge motions to lie performed
wilds either linear or fully nonlinear boun`lary
conditions on tile Idyll an`! on tile free surface.
Efficient artificial bou ndary conditions are i~n-
ple~nented tight allow nonlinear and linear si~n-
ulations to be performed over a large number of
wave periods. Results from linear frequency <1 -
main theories are recovered and nonlinear pl~e-
nomena are presented.
2 Introduction
Nu~nerical remodels based on linear potential
theory have proved the~nseIves to be efficient
tools to predict tile seakeeping behavior of
floating structures. As a matter of fact, very
often they have been found to perform surpris-
ingly well given all tile litnitations of the tI~e-
oretical framework (sn~all ~notion., small wave
stems, Coo viscous eire`..ts).
A known exception is tile roll Notion Of
ships ant] Urges at or near resonance, where
(litrraction-ra(liatio~ co(les are known to over-
predict the response. Addition of a supplemen-
tary damping (accounting for tile vortex sI~ed-
ding at the bilge corners) to the motion equa
85
tion usually notably improves the prediction.
For barges however, it has been argued tight
the nonlinearity of tile exciting forces ((rue to
the large variations of the wetted part of tl~e
Imp) should also be taken into account (e.g. see
Denise [1~. The relative importance of these
two factors (viscous damping and nonlinear ex-
citation) leas lee} to seine controversy.
In past years some attempts slave been
Inane at modeling the vortex sI~ecI`ling from
the bilge corners within linear potential flow
~nodels. Excellent agreement leas beets reported
witty available experi~nenta1 data id], t3] .
This success seems to rule out the possibil-
ity of the nonlinearity of tile potential loacli~g
playing an i~-nportant role Ail. It is our feel-
ing however that the publicized numerical a.n(l
experimental results are too scarce to entitle
any general conclusions to be dra.w~. Obvi-
ous~y, many parameters such as tile slope of
tile bilge corners (sI~a.rp or rounded), tile beam
over draft ratio, tile amount of potential da.rnp-
ing present at resonance, etc., are to Inlay a role.
(We are currently perform i Fig extensi; e exI,eri-
~nents to forge our own religions, by varying all
these parameters). It seems therefore to be of
interest to quantify the amount of nonlinearity
present in tl~e excitation forces, ant! tills is tile
aim of the numerics techie described in tile
present paper.
'ago investigate taxis problem we make use
of a time domain numerical model, Sin(l-
ba(l, based on the so-called Mixed Eulerian-
Lagrangian n~etho(l, which simulates a two
OCR for page 86
dimensional wavetank. Sindbad I' as been val-
idated on such problems as sloshing in tanks,
wave generation by piston type wave makers,
and wave diffraction over submerged cylinders,
t5i, ted, [7~. Two difficulties had to be solved
in order to extend its capabilities to the wave
response of a freely floating body:
· a means must be devised so that the
waves reflected by the body do not reflect again
on tile wavemaker and pollute tile incident
waves. In wave basins taxis problem is avoided
by locating the body far enough from the wave
maker, so that a steady state can be reached
prior to contamination of tile incident waves.
In the numerical mode! it is necessary to re-
strict the length of tile tank in order to limit
the size of the problem;
· the hydrodynamic loading on the hull
needs to be calculates! carefully in such a way to
permit a simultaneous integration of the body
motion equations.
Section 3 is devoted to a short descriptions
of bile mode} and to the way these two problems
have been solved.
Even though Sindbad can be run to solve
both the nonlinear and linear problems, it was
felt that a proper assessment of tile validity
of its results required some confrontation with
more traditional linear frequency domain mod-
els. Comparisons have been made with two
such models, one based on matching of eigen-
function expansions Or tile potential in tile three
sub-domains limited by a rectangular barge
(left, right, and underneatI~), and the artier on
a Rankine source distribution on the hull and
free surface. Both models are briefly described
in section 4.
The last part of tile paper (section 5) is
devoted to tile presentation Or some nu~neri-
cal results. First tile Midyear Otis Or Simian
is checked by recovering, once a steady state
Iran been reacts, tile frequency domain results
to a satisfactory accuracy. Results from non-
linear simulations are then presented for the
diffraction problem (fixed barge) and for the
diffraction-radiation problem.
3 Nonlinear and Linear Tran-
sient Solutions
3.1 Introduction
Since the work of Longuet- Higgins anti
Cokelet [8] that pioneered the Mixed Eulerian-
Lagrangian method (MEL), this method has
been widely used for the simulation of two-
dimensiona] free-surface flows; numerous i~n-
plementations exist. Many authors slave been
interested in the description of steep waves and
the MEL has proven very successful in describ-
ing the flow up to overturning, see for instance
ted. Vinje and Brevig {10] first modelled waves
interacting with rigid obstacles, either fixed, in
forced motion, or in free motion. This prob-
lem has since been addressed by several authors
(e.g. [11i, [12i, [13i, [14~. The motions of a
rectangular barge have been studied using act
implementation of the MEL at St John's [15~.
However, many of these studies were somewhat
qualitative and it appears that little attention
leas been given to the validation of tile resulting
codes by proper comparisons with other tI~eo-
ries and experiments. For instance, if computa-
tions of the free motion of floating bodies have
been performed, systematic comparisons witty
results from linear theory are almost nonexis-
tent. In our opinion, it is now time to validate
the MEL and to use it to provide quantitative
results.
An implementation of tile MEL at the
Institut Frangais du Petrole, and in cooper-
ation with DGA, led to the development of
the code Sindbac] (for Simulation nu~nerique
d'un bassin de houle). Developments concern-
ing this code are presently under way at both
organisms. Its application to the simulation
of two-dimensional flows in the presence of a
submerge or surface-pierci~g body in force(l
Otis Ails been `~scril~e(1 elsewhere ill solve
letai! (e g., tbi, A)
I~ tile next sections attention will focus on
tile new developments tilat Ilave been necessary
to (leal with a freely floating bo(ly of arbitrary
shape. First some indications concerning tile
principle of the simulation will be given.
86
OCR for page 87
i''' = ~ fe ~ + In 71, (~1)
v
s
rd
~ ~4
Wave maker & Floating Absorbing zone
absorbing zone boa,
Figure 1: Si~dbad wave basils geometry
3.2 Outline of tile MeLl~od
lee attain idea Of tItc n't''~erica.t procedure is to
caboose markers i''itia.lly at tl~e free surface I'm!
t`, f~,ll`,w tl~e'', ill their '''`~li<~.
We use ~ coortli''~te system (~,y). 'rile
x-axis coiticitiefi witIt tile refere'tce l~osilion Or
tile free surface a'td tile y-axis is oriented ver-
tiC?~ly t~l~wa.r`~s -see figure 1 rear geometric
`~e0~,iti`~fi. 1 I'~ Il''i`! is ass'''~! t<' I'n i'~-
l~ressit,le a.~' tI'n Ilt~w irrt~tali~.' so 1~;`t tail
velocity field v is given by:
v= V+, (1)
witty:
A~ = 0. (2)
'I've coal io't is l,crf~,r''le(! ill
t~ot'''4etI do''tai'~. Along rigid boundaries (x ~
i',,(1~), tI'~ '~,r''~.! Wily ill 1.~e {~'i(x). 'Ellis yields':
I9t 2 (~ s + 2 In p + city) <3~)
-
IFor the sake ol simplicity, we use llHitS Sll(h tl~at
the acceleration of gravity, .9, the specific mass of water,
p, and the sleuth of the tank, h, are eq''al to 1.
wipers D i8 used to implicate a 'bacterial deriva-
tive, ~ and n are vectors tangent a''~1 normal
to tile free surface, respectively, amuck ~ is an ar-
bitrary constant. Taxis co'~sta''t specifies tile
tangential Notion of tile remarkers: ¢ ~ 1 i(len-
tifies marker as particles while ¢ - O yields
a zero tangential Notion of tile remarkers. 'Levis
last cI'oice allows a current to lie si'~'ulaterl in
tile tank. For tile applications discussed I'ere,
= 1.
The pressure i8 assigned constant along
tile free surface; it call tI'us be included ill
tile function of time city. Witty an appropriate
choice of tile velocity potential, taxis r~'ctio~
can be taken equal to zero.
Tile kinematic co'~stra.i~t AI-O. associ-
ate(l witty tile boun(la.ry co''`litio'~ oil 1',,, per-
'~,its tile free surface l~o'''~la.ry co'~`litio'~s (3~-
(~1) to be expresser] as all evolutions equation
for (¢,x). 'finis stereos frolic tile fact tight if,
at a givers instant I, ~ is knower along plot)
anti ¢~' is known along l.~n~t), tilell 4>,, Call
be co''' l,'' te
OCR for page 88
curvilinear a6fiCifiSa fiO tint tile calculation of
tile Iliatrix e~elllelit,8 is rattler side a''`! vec-
torizes well.
Special care I,as to be taken at corners
of tl~e fluid I'oun~lary and Fore particularly at
tile intersections between tile free surface and a
piercing Wily. Billie '''.''''~rical treated limit is
used tilers is hase`l on a local asy''~l~totic a''al-
ysis in tile weakly `'o`~linea.r regime tight corre-
Si)On(~8 to a small accelerations (relative to grav-
ity) of tile bo`]y. 'l'llis n`~erical treatment is
discussed in snore details in t6] acid tot.
3.3 Wave Generations aback Absorp-
[io'~
3.3.1 Generalities
We are Oiler i'' lid tale ''~'ti`~s
of ~ Il~.`t I'd I,`~ly i'' 1 lie and aces'', s~'it-
ted to a givers incident wave field, over several
wave l,eri`~. Si''`e tare l,`~.ry `,r tall c''-
tire n~'i'! ~Iot'~ai'' liar to be discretize`l, artificial
boundaries must be introduced. I7or tile nu-
~nerical procedure to be eiHtcie'~t, tile Quill dm
'''~i'' sI~' l,~'as s''';~.~' an 1,~,~sil,l~'. 11 is 1~`r`~-
fore 'necessary to generate tile incipient waves
and to avoid their reflect iota oil tile artificial
bou''`laries introduced. IJitticulties ill reacl~i'~g
a steady-state, event for waves of spa. steep-
ness, probably explain wily con~pariso~s witty
linear res''lts I~aVt? ''fit t,~ ''lore syste',~ali-
ca,lly perfor'''e(l.
Wisest tile stea`ly-state li''ear sol.~ttio'~ is
c.o'~1l,'110tl, all ;J\( ill wave is l,~escril,`~' allli
a ra(liation condition is written tila.t fral~slllit-
te`l e'' d reflected waves '''test satisfy. Wrili~g ~
proper radiations conditions for tile second-order
problem llas Slid been a splatter or controversy.
For tile tra.nsie'~t problem, tile proper loel~av-
ior at infirmity can be acco~.~'te`! for by using
~r`,l,er emery I, bile (~'rec'' r''''`-
tion. 'Ellis al'I'roacIt requires a convolutio'' ifs
tinkle a''<] is o''ly 1~fisil'le if tile 1,rol,le'~' is li'~-
ear (at least ill all outer do'~a.i'~ exte''`li~'g to
i'~fi~ity).
I~t tile 3,l)fi~ICO Or Rely '~'e'~atically sat-
isfyi'~g answer yiel(li'~g perfectly transparent
boundary conditions' we slave chosen a l~rag-
'~atic solution sitar to that use`] for all ex
88
peri went i n a tank. 'l'lli8 approach 4OeS IlOt i ll-
volve ally I'ypotI~esis co'~cer''i'~g tile steeliness
of tile outgoing waves. Waves are generate by
a piston-type wave~naker and tile tank is closed
at tile other end by a vertical wall. Two ~la.~l,-
ing zones, one at each end Of tile tank, are used
for tile al~sorl~tio'` Or tile l~arasilic waVe8 that
are generated in tile tank see figure 1. 'Ibis
is equivalent to having a'' absorI,i'~g beacon at
one e'`d and an absorbing wave''~aker at tile
other en'l.
3.3.2 Absorbing beadle
Tiff absorbing beacon is a dig zone, si~-
ilar to that used in Al. Ill taxis zone, tile free
surface boundary conditions are '~o~lif~e~l lay
ad`li~'g a calming terser. We write:
'~` 2 () ¢l' + 2 ¢'r~ we're) (¢-¢'e)
(G)
- - ~ /~ ~ + din 7t-V(Xe) (x-ale), (7)
where tile s~ll)~ctil,' ~ (orresl)ol'(~s to tile r``rer-
e'tce co'~fi'guratio'~ for tile llui~! inhere, tile fluid
at rest).
Lie pri'~ciI,le of flus (Ia.~,I,i'~g zone is to
absorb tile incident wave energy before it call
reacts the wall. It may be seers intuitively tI'at,
if tile absorption is too weak, part or tile i'~ci-
~ent wave energy Will reacts tile wall a. be
rellecte(l. Inversely, if tile a.bsorptio't is too
st,rol~g, It Or tllP elIergy will lie rellectet! I,y
tile Handling zone itself.
'i'lle choice of tile tla'~l~i'~g coetH~cie'~t tJ(X)
is crucial to its efficiency. rrI'is coeib~cie~tt is
equal to zero except ill tile Wing zone (*0 <
x < x'). In taxis zone it is choose to be con-
ti~'uous an(l continuously `lilTere~tiable, anti is
"tilde." to a characteristic wave fre~l''e~'cy
atoll a characteristic wave number k:
I,(X)-tt~ [2 (x - xo)1 (8)
2~l3
no < x < x1 = To ~ k
walers car ant] ,6 are ~lililelisioll~ess I,a.ralileters.
OCR for page 89
OCR for page 93
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Representative terms from entire chapter:
fluid domain
3.3.3 Absorbi'~g Wave'~aker
'I'l~e al~sorbi'~g beacI' al~ows lo~'g silllu~atiOlis
to be perfor~ned in tI`e nu''~erical tank wI~en
tI'e generation a~d tIte propagation Of surrace
waves are ti~e o''ly co'~cerns. A ~lilficulty al'-
pears ror tI~e si''~ulation or tI`e rree '~otions Or a
body i'~ tI~e ta~k. Waves are l~artly rellecte~! 1,y
tIte bo~ly a,'~d tIte'' rellectetI back by tIte wave-
'''aker. In a real ta,'tk, tI'is 1'rot,le'~' is overco''`e
by locating tI'e tested body sufl~ciently far fro~n
tI'e wave'''aker. i'ar tI'e nu'''erical ta'tk, tItis
would i~ply too large a nun~ber of discretiza-
tion points. It is tI'erefore cr~tcial to develop
a, ~netI`o`! to avoi
OCR for page 90
'i'llifi C(lIl~tit)~t ('RI' D.~) t)O Wlit,~,011 11Sil~g '~n,tri
cia,' ''<,t,ati<~s O.fi:
·- _
M To= 1'. (12)
Eater forces al,I,lie~! to tile body, I', are
written as tile 8Ulll of I'ydro`lyna''~ic forces and
ogler external forces. No ~listi'~ctio~ is ''table
l~etwee'' I'y
In e~l''at,io'' (18), the velocity `` is givers lay
e~l~atio'ts (15) a''t! (1(~. 'I've a,ccelera,t,i`~' ax is
given by:
Hi - Ma-(p-YG) H-((X-AGO 82 (19)
if - YG + ((X-EGO ~-(F-YG) ~ .(20)
48 a consequence of tile8e e`luatiol~8, fin
·
~lel~e''`ls li''early o'' ~,r,. As ~ rest'lt, I~y~lrm
`lyna'''ic rorces a,l~l,lied to tI'e botly at a given
insta,nt t can fi~'ally i~e writte't as:
[7h = M XG + ·:o. (21)
'I'l~e vector [0 corresl~o'~Is to 1,l~e forces al~l,lie`!
to tI'e I'o~ly wI~e'' it8 accelera.tio'' is e~lua.' to
zero. 'l'l~e ~na,trix M' is si'~ila,r to a'~ a`~de`]
IliaSfi ''~atrix. It,s coell~cie'~l,s ca.~ I,e evaluat,e~!
by co'~p~ti'~g tI~e forces corresl~o'~di'~g to a, u'~it
a.~< olera,t,i~ ~,r ~ '' `~egree `,[ free`~.
It, sI~'l<1 l~e ''ote~l tl'~t i'~ order to e.sti-
''~a,te t\~e rigl~t l~an
right Or the Charge (l,) I lert (~' --- see fi'g-
'''~ ,l. 11 l'.~n aIr<~.~ly l,~t a1,l,li'`~' tt, tI'
salute l~rol'le''', see for i''sta'~ce Mel a'~' llIack
[lid. (Our '~atcJ~it~g l~roce`Jure `lilrers so~e-
wl,at fry tllI?ir,fi, a,11~\ is lit try (Barrett
jlBsi).
'lithe total Velocity l~ote''tial is writte'' ill
tile classical for''':
~ I'=!
where ~'-XG, x2-YG am ~3 = 49.
'flee ~liRraction ~,ote'~1 ial rev am ra~liatio'
~ote''tials ~`j ad'''it tl~e followi'~g ~leco'',l~osi-
tio'~:
· Do~nait~ 1~:
{B = bo ko Z eiko(~- b)
cosll kolI
00
~J~ ~ b~l COS k,,z ~-kn(~-b)
tt=1
·L)o't~ainC:
'Pc = cO - u e-ikO (2+b)
cosIt k~, /!
oo
1- ~ c,l cos k,~z ek1'(x+b) (24)
n=1
wI~ere ko,kn are tI~e roots Or tI~e eq~ation:
w2-gkO ta'tIt koII =-gkn tat' kulf
· D0~! ali I' A:
~A (~~~y'!) = (~' ~~~ ~! ~t R02 b (2~)
c`,sI~ A',x
~t ~, ~nl - - COS ~nZ
n-~ ct~sIi Anb
°° - si I'It At! x
~t >~ 47r~2 -; ~ ---~ ~ ~ t,8 ~ Z
wI'~?re A,, - ,''r//' a''<' ~' i.s ~ ''arlic''lar s<~-
I''ti`~' wI'i~ I' is 7~, r`,r tI'~ `lilEra' ti`~' a'~'
sway r;l`li.~ti~' l,~t'''Iin.~s, ;1~' is''`1~! f`, (Z2-
.~2~/2~! for I~eave a.~d to (~/2/~) (772/3_ Z2) rOr
rof I.
I~t tI'e l~revious exl~ressio'~s JJ is tI'e to-
tal water(leptI', b tI~e I,alf-wi`ItI~ of tI'e barge,
z - y + ll, a''`' I' = I1 - `1 tI'e water(lel)tI'
u'tUertteatI' tI~e I,arge, c! I,ei~'g tI'e drart.
,l'l~e coe[li' ie'~ts a,~ ~ ~ a,,2 ~ b,~, c,, a.re ~' ~ er-
'''i t'e`! I,y tr'' 't~ ~ ti'~g 1 I'~ i'~ fi '' i 1~? s`'rit's .~.' st,'',~`
fi~ite orders Na'Nb = NC, aI\~] ctl''ati'ig tI\e
~ote'~tial exl)a.~ 8iOllfi all `l t I~ci r x ~'eri vat i ves a t
~c-~ and ~--b. Con~si`leri'~g, rOr i'~sta'~ce,
tI`e (lifFraction proble'~, e~luati'~g ~n i- ~' a''~]
~,~ i~ ~-b yiel~s, after taki'~g a~lva~`tage Or
tI~e ortI~ogonati ty of tI~e cos ~n Z f'''~ ctio'~s over
t0 /~] by i'~tegrating in z:
_ ~ ~ _
A' + A2 - A 11 + 1 ~(26)
(22) wI`ere Ai = 7 (aOi, , aNai ),
B- T(bo,...,bNb)'a~] ~A is ~ Na Nb Illatrix.
Si~nilarly equati~g ~c + ~ an~] ~ at x =
-b ylel(ls:
A1-A2-~ C ~ 1~1 e 2ikoi (27)
siInilar consideratiolls on tile x derivatives
('23) in ~ = b an'' x--b, co'''t~i'~d witl~ i'~tegra
tions in z over to II], yiel(1:
L7=~(f~+~2~2~+ll2 (28)
C = ~ (fiAl -[2A2)- Il2 e-2ikob (29)
wilere 1; is a IV~ · Na Inatrix, a~l~] [~ f2 are two
rliago'~al rl~atrices.
Some Inanipuiation ot tI~ese 4 vectoria.t
_ ~
equatiolls yiei~s t~le filla' systeln ill 13 + C aIl(]
~ _
13 - C :
~ _
[~-~1~4](Li+C)
~ ~ll~l(l + e 2ikob) + 172(1 _ e-2ikob)
[l - ~ [2 '4](~ - C) = (31)
[2171(1-e 2tkob) + 1~2(1 ~ ~-2'kob)
TIle re8Oliltioll of tilis systell1 yiel(Is tlUtl
( ate(l series exl)~.llsiolls ror tile l~ot,elltin.ls il
ea(.ll (lolll;lil1 a.II(1 a.1I('w.s tile 11y(lro(lylla111ic
eilicielltS all(l tile (li~[ractioll [orces to be (Olil
p11 te(l.
4.2 Lil1ear Il~tegral-equatioll
rl'lle ra(liatiOn all(] (lilRractiol1 potetitialls are
rOUll(l l~y tile solutioll Or all illtegral e(~uatio
92
Isfx
~1 ~ ~ :~ ~ ~ ~ AL 7
SO /
SR
FiglI[e 4: (;~01~1Pt[lC dC1}I';tIO1'S R)[ 1.I'0 II1~[
il~trgrn.1 e~llln.1.1cl'
~- ~ ~ INS ~ ~ ~
141CI`1.51.y. '1'1~P "1~1~)n(:I1 1IS2~1 ~II0WS ticket 0[
~ ~J ~n~ [~1 WE ~ ~
(I(,IB1;~11 LIZ IR 1In(!(I R,r 1,II(! p1'()I)I{!IB1 S()I1I'.1~11;
~O r~ ~"d~n~ ~ ASHY ~ M~K
-~ ~ Bug Bun ~ US ~ 1~
. . ~ a, . ~ , . .
B@1)1)[O"@(:I' IS I[Ce (JI I[1~41I~r I[~(111~11C#~S ".11(' IS
1ISUf1Ii Air (:;1.147[1II (:;1.I(:1II;1.1~11 (A IIII(~;1[ I1y(I1-~1y
n~K Pay. To
1I11k11~)W1' 1~1~7111~;1.I ~, ()r am 1.~.k(!S tiff! him:
_ ~ ~(',) -1
-
o 0~1
1 / ~-~o
-1- / (~ (~} - (at)
. S" 0" J"
= /~
~ An 0~1
(:12)
WD~ am Gay ~n~- IS ~ GINO
ma= -~t
nI'`' I\, in 1.1Ill~l~l Ill 1~'ll
cl`>se to 8 s il1 tl~e slln.[l' fllers t:nse@ 'l Ile l~nl~r
w11.1' rt~lIll~lr~I `:q,['lr[R llnn n. 1~61~e r~l~lllIs `!~ll~n.
~ 1~ ~ ~d i~ n-~ ~1- in ~l is
tn 7.5 8.
Ilf~tlI 1.11e 1.[n.'lslr'`1, `:~,llll~lI1.n.1.1~,ll n.ll`I 1.llr
Oi8fIl-[lI11~1.~11 ~XllA.llSif)11 n.[P 1~[R>rlllr`1 i1' A
nl~ depth. In -rr ~ ~d Gnl~ ~Irpth rP
~cts1 ~ `1~1.1' o[ 15(} 1~' wns 1lse`I. 'I`lle illlen.r
~ 150.0r
i''l,'~gr;~.~-~'l'';~.~.i','' ''tt`1.~' in 1~'r~,r''''`~' ill illli- Z
''it'' 't<~1'll'.
'1'0 ~('011~lt [tar 801110 Vi8t'0118 dig and
to avoi<' very l'''~g tra'~sie''t l'l~e''o'''e''a i '' life
'''~steatly co'''l,''ta.lio'', ~ li''car It cm
ellicie''t e~lua.! to 5 1()5 kg..- (a','] corre
'',''''i''~ 1~, It's ~ .1, '?.'r,% Or ~ ril i' a.' <~;~.~, 1'i'~g)
WE ;~?~! tat Me e'1~.1.i''''~ ',[ I. .Si'''
IlolI-%ero tIl~ri7,~.' drift rorce is eXlleCte~ [Or
tl~e nonlinear co'''l,''tati~', a linear restraint i''
HWR.y WR.N D.lR(, ~. 1 llP ('t~rit'81~(.~(til~g 8tilr
''ees i~ e~l''7~.' t~, 5 1(~6 N''`-2 a''<' tI~e 'l;~1'i'~g
coellicie''t is e~lun.' lo 5 1()3 kg.~-~-~. 'J'l~is
Ien.~s to a. ''~t''ra.! 1~eri~' i', swny l;~.rger tI'a.
li ve ti'''es l t'~ 'ta.l '' ra.' ',eri<~] i '' r`~.
D.2 Li''en' C,'u'''~[ntio''
'l'l'n li'ten.r verni<~' ',r.cli'~.~' w;~s ''s~' '~, <~-
1,''1~' 11~' 1 r.~ni<~1 .~'~I,~tf;t' ''r ;~. '<~' I ;~;~.r
t'arge sui,'llillef~ to ;~.~' i'''i
E t;~).0
at
o 30.0
,-
~0.o
_1
ILL
30.0
°-60.0
1.00
) ,.
0.0 60.0
a. r-~-~ t-- ~-~- ~ ~ 1
120~0 180~0
TIME (s)
I;'ig''re 5 Mo'''e''t V8. ti'''e (li''ear si'''ulatio'')
0.200
' '- 1' -- ' ' ~1
240 0 300.0
F;1~11~.1 (rounds/ on) ~Sindbad {rounded corners)
_.IttIeg~
Sickbays. Due to `li faculties related to tile
sliarp corners3, oddly tile recta'~g~lar barge witty
rounded corners will be considered there.
Figure 11 shows the moment about the
center of gravity as a function of time for the
diffraction problem, the incident wave ampli-
tude being equal to 2 m (the wave period is still
equal to 8 fi and the saline discretization as in
the linear case in used). As in the linear case, a
steady state is reacI~ed rapidly. However, non-
linear eRects are very strong and tile presence
of Weigher harmonics is obvious. A Fourier anal-
ysis of the signal reveals that even the ampli-
tude of the first harmonic is not well predicted
by linear theory and tint the second harmonic
is twice as large in magnitude.
Figure 12 and 13 slew the sway and roll
displacements (XG and 8) as a function of time
for a coating barge restrained in sway. The
wave period is equal to 7.5 s (corresponding to
the resonance) en c! Else wave amplitude is equal
to 1 Gil. Tile saline discreti%ation as in tile li~-
ear case is used. A graphical representation of
the free-surface profiles during tile simulation is
sinews ill figure 14. 'rI'ere are much less trigger
I~ar''~o~ics Plan in tile diffraction case, the bo`ly
presur~ably acting as a "filter". The response
amplitude in roll is significantly modified, indi-
cating tint tile nonlinear potential flow effects
are important.
6 Conclusion
A simulation technique based on tile Mixed
Eulerian-Lagrangian method leas been de-
scribed. It allows the simulation of tile now and
tile resulting barge motions to be performed
with either linear or fully nonlinear boundary
conditions on the hull and on tile free sur-
face. Efficient artificial boundary conditions
have been implementer! tight allow long simu-
lations of difEraction-radiation probes.
Linear frequency domain results have been
recovere(l using tile Mishear version of tile code.
3~s shown before, the convergence is very poor for
the barge with sharp corners. Note moreover that the
singularity for the complex potential is expected to be
in Z2/3 yielding an infinite pressure due to the V¢. Vie
term.
Comparisons have been perfor~ne`] ',ot only for
global quantities, such as transfer functions,
but also for local quantities, sucks as pressures.
This linear version appears to be well validated.
Nonlinear simulations have been per-
formed, tasting for a large number of wave pe-
riods. Even though these nonlinear results are
difficult to validate, the accuracy of the linear
version of the code gives some confidence con-
cerning tile validity of the model.
References
[1] Denise, UP., 1983, "On the Roll Motion of
Barges," TRINA, Vol. 125.
[2] Standing, R.G., Cozens, P.D., ant!
Downie, M.J., 1988, "Prediction of Roll
Damping and Response of Ships and
Barges, Based on the Discrete Vortex
method," Proc. Int. Conf. on Computer
Mo(lelling in Ocean Eng., Venice.
t3] Downie, M.J., Graham, ].M.R., and
Zincing, X., 1990, "Tile Influence of Viscous
Effects on the Motion of a Belly Floating
in Waves," Proc. Nit Int. Chef. OllsI~ore
Mech. and Arctic Eng., Houston.
t4] Robinson, R.W., an(l Stod^dart, A.W.,
1987, "An Engineering Assessment of tile
Role of Non-linearities in Transportation
Barge Roll Response," TRINA, pp. 65-79.
Cointe, R., Molin, B., an(l Nays, P., 1988,
"Nonlinear and Second-or(ler rl'ransient
Waves in a Rectangular Tank," BOSS'88,
'Ron cll~ei m.
i6] Cointe, R., 1989, "Quelques Aspects de
la Simulation Nu~nerique d'un Canal a
Iloule," These (le Doctorat de I'Ecole Na-
tionale des Ponts et Chaussees, Paris (in
French).
[7] Cointe, R., 1990, "Numerical Si~nulation
of a Wavetank," Engineering Analysis
with Boundary Elements, special issue on
"Nonlinear Wave Analysis", to appear.
8] Longuet-~liggins, M.S., and Cokelet, E:.D.,
1976, "The Deformation of Steep Surface
96
600.0
of
4()0.0
200.C
250.50
250.00
249.50
r~
249.00
248.50
A)
248.00
247.50
C) I - - r ~r-~ - - I -
0 060~0
I - ~ -r ~I--- -r --
120 0 180~0
TIME (s)
. , ., . ~., , , . ~. . . .. . . .
240.0 30 ).0
I;'ig~lre 11: Mel vet ti '''e (~'o',li''ear Si'',''l?~ti`~)
_.Slr~dbad linear
_Si~dbod Nor~lir~ear
247.00 1 ,
0.0 1 00.0
0.150
0.050
L_
I O.0()O
J
C)
_ ().()5()
-0.100
TIME (s)
I;'igll re l'2: Sway V8. t illle
... Sludi~ad l.lr,~(lr
_Sir~dbad Nor~lir~ec~r
---em ~r---
200~0 300 0
0~ 1 50 ~1 ~-~-~~~~~- - - 1 1
0 0 1 00~0 200~0 300 0
TIME (s)
I;'ig~l to l 3. It oh vs. ti Ille
97
Y ~ It
,,1
OCR for page 99
DISCUSSION
J. Nicholas Newman
Massachusetts Institute of Technology, USA
This type of complete fully-nonlinear analysis seems to show very
different results for the roll response compared to the simplified
(hydrostatic) analyses described in the papers by Sanchez and Nayfeh
or Francescutto and Nabergoj. Perhaps this is due to the large beam-
draft ratio of this barge and its large damping? If so, it would be
very interesting to apply the Sindbad program to a vessel with smaller
roll damping.
AUTHORS' REPLY
The question of Prof. Newman is very interesting but we fear that we
cannot give a satisfactory answer at this stage of our study. Our
motivation for performing fully nonlinear simulations of the roll
motion of barges certainly stems from our desire to get a better
understanding of nonlinear phenomena involved. In particular, we
would like to use such simulations to assess the validity of models
using different levels of approximation. In this paper, however, we
describe the method that has been devised to perform the simulation
and we insist on the procedure that is used to validate it. We
understand that is somehow frustrating to discuss more the tool than
its results, but we strongly feel that nonlinear simulations can only be
useful if carefully validated.
The result we presented is for a barge having a large beam to draft
ratio, a very small radiation damping and an important roll-sway
coupling. It does not seem obvious to us why, in this particular case,
it should be very different from that of a simplified analysis, as long
as the roll-sway coupling is properly accounted for. The fully
nonlinear computation is still quite time consuming (it took several
hours on an Alliant FX80 for the example shown) and it cannot be
used for a systematic investigation in the phase plane. It is our
feeling that intermediate models might be better suited to check if the
analyses performed on a simple single degree of freedom system
apply to real situations.
We are currently performing experiments and we will use the Sindbad
program, as well as more less sophisticated simplified models, to
make comparisons. We shall report the results in the future and we
hope that we will then be able to answer this question more
satisfactorily.
DISCUSSION
Ronald W. Yeung
University of California at Berkeley, USA
I find it puzzling how you could eliminate the reflected waves in the
upstream zone by applying the damping layer to (-ok), ii being the
incident wave potential. The difficulty, as I see it, is that the
reflected wave from the body does not occupy the same fluid domain
as the incident waves since the problem is nonlinear. You must have
made additional assumptions in applying this procedure. Perhaps you
can clarify this point.
AUTHORS' REPLY
We thank Prof. Yeung for his question that will allow us to clarify
some points concerning the absorption mechanism we use. The
method we propose is based on physical considerations rather than on
a rigorous mathematical analysis. As a consequence, it is not clear
to us what is meant by "additional assumptions.. We substitute the
original problem with another problem with different boundary
conditions. As for a real tank with an absorbing beach or an
absorbing wavemaker, the only way to assess the efficiency of the
method is to perform tests to quantify the reflection.
As pointed out by Prof. Yeung, the reflected wave does not occupy
the same fluid domain as the incident wave. The definition of the
reflected wave that we give is therefore difficult to interpret is one
refers to the Eulerian velocity potential in the fluid domain.
However, as we use a Lagrangian specification for the time-stepping
procedure, there is no ambiguity for the implementation of the
method; we refer to the potentials attached to a marker labeled by its
abscissa xc along the reference position of the free surface (that
coincides with the x-axis). Similarly, the damping coefficient v(xe)
is that attached to the same marker.
99
