VOL. 18 NO. (1995) 151-170
WAVE LOADINGS ON A VERTICAL CYLINDER DUE
TOHEAVE MOTION
D.D.BHA’I-rAandM.RAHMAN DepLrtlnCntofApplied Mathematics
Technical UniversityofNovaScotia Halifax, B3J 2X4, Canada.
(Received December 17, 1992 and in revised form .June 6, 1993)
ABSTRACT. Wave forces and moments due to scattering and radiation for a vertical circular cylinderheavinginwateroffinitedepth arederived analytically. Thesearederivedfi’omthe total velocity potentialwhichcanbedecomposedastwo velocity potentials; oneduetoscatteringinthe presenceofanincidentwave onfixedstructure
(diffraction problem),
andtheother due to radiation by the heave motion on cahn water(radiation
problem). For each part, the velocity potentialis derivedby consideringtworegions,namely,interiorregionand exteriorregion. The complexmatrix equations aresolved numerically to determine the unknown coefficientstocomputethewaveloads.Somenumericalresultsarepresentedfor differentdepthto radius anddraft to radius ratios.
KEY
WORDSAND PHRSES.
Scattering,radiation, heavemotion, velocity potential.1992 AMS
SUBJECT CLASSIFICATION
CODES. 76B5.1.
INTRODUCTION.
Theestimation ofhydrodynamic forces on an offshore structure has received considerable at- tention from the designers. Accurate predictionof the wave loads exerted by surface waves on rigidstructures isabsolutelynecessary todesign offshore structures.
A
rigidfloatingstructuremay undergo sixdegrees
of freedom three translational and three rotational. Assuming a suitable coordinate system,OXYZ,
the translational motions in the x,y and z directions are referred assurge, sway and heave respectively; and therotational motions about x,y and z axes are referred
as
roll,
pitch andyawrespectively. Herezaxis is considered tobe vertically upwards fromits still water level. Often the structure is restrained to havefewer degreesof freedom due to the type of mechanical connection used to fasten it to the seafloor. The problem ofscattering of surface wavesbyacirculardockwascarriedout byMilesandGilbert[8]
andthen byGarrett[3].
Garrett presentedthe results for the horizontal andverticalforce andmomentonthe dock.Black,
MeiandBray [2]
have calculated the waveforceson a truncated cylinderwhich either extends to the free surfaceorrestson the seabed.Isaacson [6]
extendedGarrett’s
method forasubmergedtruncated cylinder sitting on the sea-bed. The hydrodynamic interactions due to wave scattering between thenumbers ofan arrayof stationary, truncatedcylindershavebeen investigated by Williams andDemirbilek [10].
Numericalresults fortheaddedmassand dampingcoefficientsofsemi-submergedtwo-dimensional heavingcylindersin wateroffinitedepthwerepresented byBai
[1].
Heshowed that theaddedmass isbounded forallfrequenciesin water of finitedepth. He’tudied
the limits of the addedmassanddamping coefficients forhigh and low frequencies.
Yeung [12]
presented aset of heoretical added massesanddampingcoefficientsforafloatingcircularcylinderinfinite-depthwater. Sabuncu and Calisal[9]
obtMned hydrodynamiccoefficientsforverticalcylindersat finitewaterdepth. Williams nd Abul-Azm[11]
investigatedthehydrodynamicinteractionsbetweenthe members ofanarrayof floating circularcylinderswhich occur whenone memberundergoes prescribed forced oscillations.Numerical resultsfor the added massof bodies heaving atlow frequency in wateroffinitedepth
werealsopresentedby Mclver and Linton
[7].
Garrison[4]
presenteda numericalmethod for the computationso
determinewaveexcitation forcesaswellasadded mass and dampingcoefficients for largeobjectsin wateroffinitedepth. He[5]
presented anumerical analysis for themotionof large free-floting bodies.Weassumethat thefluid isincompressible,the fluid motion is irrotational and thewaves areof small anaplitude. Hereweconsider the coefficientsrelatedto themotion with onedegreeoffreedom, namely, translational motion in thez direction, i.e. heave.
In
this paper we hvepresented the analyticM solution for theboundary value problem to evaluate the forces and the moments for avertical circular cylinderheavinginwater offinitedepth. Numericalresultsarealsopresented.
2.
MATHEMATICAL FORMULATION.
Weconsidernsurfacewaveof amplitude
A
incidenton vertical circularcylinderof rdiusain wateroffinitedepthh. The bodyisassumed to beheavingwithheave amplitude inthepresence ofincidentwavewith angular frequencya. Thewaveis paralleltox-axisatthetimeofincidence on the cylinder and propagatingalong+ve
direction. The draft of the cylinderinwateris b. The geometryis depictedin Figure 1.We
consider the cylindrical coordinate system(r,
0,z)
with z vertically upwards from the still water levelSWL ),
rmeasured radially from thez- axisand 0Y
//A
FIGUBJB
1. Deqnitionskecth.
from the positivex -axis.
For
an incompressible and inviscidfluid, and for small amplitudewave theory with irrotational motion, wecan introducea velocitypotentiald(r, O,
z,t).
This can bewritten as
+(,.,O,z,t) R[(,.,O,z)-"’].
From Bernoulli’s equationwegetpressure,
P(r, O,
z,t),
asP -p---.
The forcecomponents
F., Fu, F.
alongx,y,zdirections aregivenbyF [" [ P(a,O,z,t)acosOdzdO
Ja--=O
SillOdzd
F, P(r,O,-b,t)rdrdO
=0
respectively. Since the incident wave is parallel to x-axis at the timeof incidence, the nonzero horizontal componentis
F.
The momentM,
arising duetothe forcesonthesideson the cylinder about sea-bottomandM
arising due to the forces at the bottom of the.cylinder about z-axisare givenbyM. + h)P(a, ,z, t)acosSdzd
=o
M P(r, O,-,
respectively. Becauseof the linearity of the situation, the velocity potential can bedecomposed into two velocity potentials
e
and,
wheree
is the velocity potengial due to theproblemofan incidentwaveactingon thefixedcylinder, nd
,
dueto theredigion problemthe cylinder forced tooscillete in oherwise stillwater. Thus can bewritgen
where Re
e
-i’].
Now by dividing the whole fluid domain into twodomns,(a)
inerirdomain region below the cylinderi.e. r -h
-; (b)
exterior domain region for r aand -hO,
wewriteghevelocitypotentialfortheinteriordomaini
and the velocity potential for theexterior domain".
henF ,F,, M., M
can be written the realfe-i,f,e-i, m.e -i, me
-ierpecively wheref,
f,,m.,m
ereven
by-ipaa
{(a,
8,z) + :(a,,z)}
cosSdzd(1)
=0
B
ipa{(r,O,-b) + i(r,O,-b))rdrdO (2)
=0
m, -ipaa
(z + h){,(a,O,z) + :(a,O,z)}
cosOdzd#(3)
=0
m,
ioa {(r,O,-b) + (r,O,-b)}r’drdO (4)
=o respectively.
Nowthe boundary value problemto be solvedhere is
XT
0a-gT;: 0
0 on z=0r>_a
O
0 on z -hOz
0 -ir
o z -bOz o
07" 0 on r= a
()
0<,<
(8)
b
<
z<
0.(9)
The boundary condition
(8)
can beseparated into two conditionsby writingas0Ca
Oz
0 on z -b 0< r<
a(10)
0
Oz
-ia on z -b 0<
r<
a.(11)
Nowweconsider thisproblemby separatingitintotwoproblems,diffractionproblemandradiation problem. Thediffractionproblemwillgiveusthe exciting force and from theradiationproblemwe willgettheradiatedforcein termsof added massand dampingcoefficients.
3. DIFFRACTION PROBLEM.
In
thiscasethe velocity potentialsatisfiesthe governingequation(5)
and the boundary condi- tions(6), (7), (9),
and(10).
Also scattered potentialCs
where+s)
must satisfy the radiation condition"linoo V:{ O0s- iA0s}
0where
A0
isthewavenumber and1
is theincidentwavevelocitypotential. Wesolvethisboundaryvalueproblem byconstructing the representation of
Ca
in theinterior domain under the cylinder and theexteriordomain r>
a inthefollowingsection. Letusassumeaproduct solution(,
0,z) Z(z)R(,.)
os.0m 0,1,2, Now wepresentsolutions for theinteriorregion andexteriorregion.
3.1 Interiorandexteriorsolutions.
Using the method of separation ofvariables,a physically acceptable general solution for the interiorregion canbeconstructedasfollows
[_() ,(.)
, =.
m=Oo +
n=l.,.(.)
o.(z + )l
o.0valid for -h
_<
z_<
-b and r_<
a; wherep(n
0,1,2, ,m 0,1,2,....)
are arbitrarycon-stants. Herek,,
,
n 1,2, aretheeigenvalues aad/,,,(k,,r)
is the modifiedBesselfunction of first kind and order m. It is to be noted herein obtainingthis expression for thatwehave discarded the termsinvolving()’,ln(),
because oftheirsingularnaturenear theorigin. It willbe convenient later ifwedefine
(:i,, (,’, z) p,,,0(r__),,,+ p
I,,,(k,,a)
cos
k,,(z + h)
suchthat atr a it becomesahalf-range Fourier cosine series expansionC:’,,(.,z)
2
+ Ev o,,(z + )
defined in -It
<
z<
-b.Here
p’s
aretheFourier coefficients and these coefficients areobtainedfrom(13)
,i,,(a,z)cosk,,(z + h)dz
P h- b h
(14)
n 0,1,2,. with
ko
0.To obtain theexteriorsolution, the boundary conditions(6), (7), (9)
and
(11)
aretobe satisfied. The incidentwavepotential canbewrittenasgA__cosh Ao(z + h)eixox
a cosh
Aoh
z__
a oshcosho(z + h) ,,,i’",,,(o,.1
cosm0Aoh
oin which eo 1,e,,, 2,
(m > 1)
andAo
is the wavenumber. Here r andAo
are related by the dispersion relationagAo
tanhAoh
andJ,,,(Aor)
isthe Bessel function of first kind andorderrn.Because
of thepresenceofan objectweneed to considerthe scattering ofwaves. Therefore the appropriate satisfyingtheradiationconditioncan be constructed from andisgivenbywhere q,i’sarearbitrary constants.
Here H)(A0r)
isthe Hankel functionof first kindof orderrn andK,,, (Air)
isthemodifiedBesselfunctionof second kindand orderm.AlsoA
satisfiesthe relationa
-gA
tanj 1,2,3,... Thisequation has infinite number ofroots correspondingtoj=1,2, It is tobe noted here that wehave used the symbol
i
not to confuse with the symbolk,,(= -)
used torepresenttheeigenvalues for theinterior solution. Thusthevelocity potentialisgivenby
_, e.,i"[{J.,(Aor) + q,,,oH)(Aor)}
coshAo(z + h)
,,,=o cosh
Aoh
g,,(;)
o.( + h)
/ q,,,j
]cos
toO.(15)
K(A.a)
COSThe set offunctions
{cosh Ao(z + h),
cosA(z + h)},
1, 2, formsanorthogonalset definedin the interval -h<
z<
0 due totherelationa gA0tanhAoh
-gAjtanAjh. Thus the orthonor- malset canbeconstructedprovidedwhere
zo(z) --N;o
1/2ohao(z + ) z(z) g,
-1/2o( + h)
(t6)
(17)
156 D. D. BHATTA
sinh
2Aoh]
Nxo [1 +
2Aoh
sin2Ajhg [ + --;- I.
Thuswith these definitions
(15)
can be writtenas(8) (19)
where
B,, 4e,,i’".
This isvalidfor -h_<
z_< O,
r>_
a.Forconveniencelet usdefine
Then wehave
(20)
Alsowegetatr a
m=0
:,(=,z) a,.(o=)z() z()
Zo(0) + Z:q
=o Z(0)
This equationcaneily berecognized the expansion of
( (a, z)
intheorthonormalseriesdefined in -h z 0. Therefore the unknowncoefficientsreobtained followsMultiplying
(21)
by,j
0,1,2, andintegratingwithrespecttozfrom -h to0,weobtn usingtheorthogonal property-h
hZ.(z)(a,z)dz J,,(Aoa)+
q,,,,o oZo(0)
dzh h
Zx(z):,(a z)dz
q’jx(z)
dz.Z(O)
hIn
view of theorthonormafity
of the set{Zxo(z),Z(z)}
in -h zO,f m(*}dz
dfh (*)dz
1,weobtainzo(o)
j_ Zo()(,,,)d-
q,,o h
q,,o a
Zx.z.
-....a,z.
dzwhere 1, 2,
3.2 Determinationof the unknown coefficients.
To
preserve the continuity of the twosolutions atthe imaginary interfacer a,it isrequired to satisfy,(, z) :,(, ), (a)
Or Or
for -h
_<
z<
-b. Alsobody surfacecondii,
namely,;t" It=,
O, i.e.istobe satisfied. Using thegradientcondition
(23)
validin -h z -b, wehave-,,,,o
+ o,,
2.
l,,,(k,,)
H2)’(Aoa)}
s,,,[o{J,,,(o.) +
%,0,’,,(.)z,()
in -h z -b. Equation
(24)
yields(24)
(25)
Z,o (.)
H)’(Aoa)
} Ao{J,,,(Aoa) +
%,0H,(1)(Aoa) Zxo(O)
g’,.(.) z(z) o +
.=q"J
g,n(ja)Z(O)
in -b
N
0. Equations(25) and(26)
canberewritten in compactform follows"(26)
in -h
<
z<
-b.(27)
in -b
_<
.z_<
O.where
S,,,],moZ,% (z) Bm E qmj"]"(mjZ(z) (28)
X:.,o oJ:,(o) zo(o)
1C,,,,,-
k,,al’(k,,a)
o H(o.)Z(0)
$iaK($ia) K(a)Z(O)
Now
from thefunctionalmatching(22),
equation(14)
yieldsP"’’ h- b t,
B,(a,z)cosk.(z + h)dz
2B,,,
J.,(Aoa)
Zo(z
cosk.(z + h)dz
h
[ z:(o)
qmj
/_-b
+ o ’.= Z,(O z(z)
cosk,(z +
n 0,1,2,. Ifwedefine
(29)
/_- Z,(z)cos k,,(z + h)
"=
h-b hZ(0)
dzwherev takes the valueAj,then
(29)
can berewrittenasNow simplifyingweget
p
2B,,,[J,,,(Aoa),,Xo +
j=O
(30)
r-]_b N-
1/2coshAo(Z + h)
cos(h-)Zo(O)
o(- )"(h b)o
sinhAo(h b)
(h ) o + .
oshoh
nr(z + h)dz
h-b
n 0, 1,2, and
f-’
_1/2nr(z+ h)
,x, (h b)Z.b(O
.,-hNx
cosAj(z+ h)cos
h b dz.sin((h
b)j+ nr) sin((h b)i
2(h b)
cosAh
Aj+
Aj -b(-1)"(h b)jsini(h b)
((h b) .}
oh
n 0, 1,2, andj 1,2, Equations
(27)
and(28)
aredefined in two domains. The unknowns q,,,=’s
can be determined providedwemultiply by and integrate with respect toz over the region of validity. ThisyieldsAddingthese two equationsweget
h b
BIC,,,o,5,
o,B, E qmJT"’mJ6".ir +
mp,,o 2hh
(31)
where
Sx.
isKronecker delta. Inserting the expression ofp,,,,equation(31)
canbewrittenasICo6,o,. J,,,(Aoa)
h bZ,.(O)(mooo,- +
2IC,,,,,o,,.r }
h
[7-/,,,i6, + h
bZ,(Ol{m:o,o +
2’ ]Cmn...n.r..n,.i}lqm
j=0 n=l
i.e.
Djqmj A
j=o
whereDj, and
A,,, ’s
aregivenby(32)
n=l
A ,,,00
J,,,(A0a)
hz,(0){,00 +
2,,,,,,,,,, }.
n=l
Equation
(32)
isacomplexmatrixequation. The unknownsarethe coefficientsq,1’s.
Theinfinite matrix Dshould be truncated at certain term tosolve(32)
numericMly. Commercially available natrix solution routinescan be used to obtain the solution ofthemodified equation. Once these coefficientsareknown the diffractionproblemisconpletely known.4. RADIATION PROBLEM.
In
thiscasetheboundary valueproblemisV2r
0--a2r
0 at0 at Oz
Oz
0
0 at07"
z=0
on r
<
a and z -br a and b
<
z<
O and theradiation conditionliAn v(--
r-iA0,)
0 whereAo
isthe wavenumber. Weassumethat takes the form(r,O,z) ,,(r,z)
cosn-’O
Now weobtaintheinterior solution andexteriorsolution.
4.1 Interiorandexterior solutions.
Toobtain theinterior solution for
,
wewrite,,=0 ,/(r, z)
cos m/9. Expanding -ia in Fourier cosineseries,we canwrite -ia,,=o
a,,, cos m/wherea,,,’s
arethe Fouriercoefficients.Thenwehave
V
,,, --,,,
m 0(33)
OZ
0 at z=-h(34)
Oz
a,,, on z=-b(35)
whereV is 2-D Laplacian in randz. Decomposing
,,
into homogeneousandnonhomogeneous partwewritewhere
Ckh
andi,,p
satisfy the equations(33)
and(34).
The boundary condition(35)
can be decomposedas0i
0 on z=-bOz
and
a o7 25
.
Oz
For homogeneous part, by method
o seperon o
vfib]es wege
a
I,,(k,,r)
,,,, (,,,o ),,, +
,,=,
i.,(,,.)
cos,,(. + h)
wherea ’sareconstantsand k,,
.
Toobtain particularsolution,wesume,i,,p Ao,3 + Bor(z + h) + Co(z + h)
where
Ao, Bo,
andCo
areconstants to be determined fromthe givenconditions. Applying boundary conditions, wegetBo
0,Ao
2(h-b), and frown the governing equation weget2Ao + 4Co
0.Thus
Ao =-2Co-
2(h-0"a. Hencetheparticularsolutionisam r
,,(,., z) 2( [(z + ) F].
Henceweget
, --’(
a,,,or)., +
n=laI,,,(k,,a) l,,,(k.r)
cosk.(z + h)+ 2(h-
a,=b)[(z + h)2- 1" r2
Atr a wehave
(.,) o + .=, a.
cosk,,(z + h) + 2(h )[(z + hI2 " l- (36/
Multiplyingboth sides ofthis equation by cos
k.(z + h)
and then integratingbothsides from -h to -b(and
usingtheorthogonal property of thefunctionscosk( + h)),
wegetanexpressionfor a,,, inthe following form
’"
h-’(a,)co,,(z + h)d-
where
and
am
aIon, - (h am [( b) /_:[(z
3b) +
a2h) - )1 -]dz
Forexteriorregion, theboundaryvalueproblemis
v(,z)- ,,(,z)
mo
g-gT- o
o o
o, Oz o
07"at z=0
at at
z=-h
r=a and -b<z<0
(37)
(38)
(39)
where V is 2-D Laplacian in )" and z. For large argument
H,(,)
andK,n
satisfy the radiation condition. Applying boundary conditionswearriveatanexpression,K,,,(s,’) O,],(r,z) ,,,oU,)(Ao,’)H,)(Ao a) Zxo(z) + .= K,,(Aia) Zx,(z)
where
Zx,(z)
andNx,
takethe forms defined in(16), (17), (18)
and(19).
j 0, 1,2,Now
atr a wehve
,:,(, )= z,,,z().
j=o
Multiplyingboth sides ofthisequation by andthen integratingboth sidesfrom-h to 0
(and
using the
orthogonal
propertyof the functionsZa(z)),
wegetanexpressionfor,i
inthe following formj
=o,,,
4.2 Determinefion of theunknown ceNcients.
Mechingconditionsare
’ (=, ) ’ (=, ),
for -h
<
z<
-b. Alsobodysurface condition,namely,1,=
0,i.e.isto be satisfied. Dom theequation
(37)
andcondition(40)
/_- %(=, )
o,( + h)a
h b
/_-’ (=, )
o.( + h)a
=h"b
h-b
=o
where
Lox
h bZxdz
Zx
cosk,.,(z + h)dz.
L,,x
h-bh
Also
Z, cosk,,,
(z + h)dz
L’
h bh
h b cosh
A0u
cosk,.,udu(- 1)"N-o1/2(h b)Ao
sinho(h b)
(h b)2A02
-I-n27r and(40) (41)
(42)
(43)
(44)
162
where
Z,
cosk,,(z + h)dz
L,,,
h bh _L
h b cosAjucosk,,udu
-1) Nx, (h b)$ sinj(h b)
(h- )a -.
n=0,1,2,. ,andj=1,2,....
Nowfrom the gradientcondition
(41)
madbodysurfacecondition(42),
wehaveG.,o+ mmo
2
G cosk,,(z+h)+
n=l j=O
for
hS
z -b,Z,,,g,.z,(z) o fo,. o
=0
where
(45)
(46) (47)
Now multiplyingthe equations
(46)
and(47)
by-
h 0,1 2.. andintegratingin the regions of validity andaddingthem weget(48)
rnamo
r hG,.o +
2 o,,
a,,,,,G,.. L,,a, +
h bNow
substitutingthevalues ofa,,,,,wegetasystemof equations,
Ejflrnj X,,,j=0
0,1,2,
5,
EVALUATION
OFTHE FORCES AND MOMENTS.
The
horizontal
and theverticalforcesonthecylinderarecalculated
from thepressure obtained fromSernoulli’sequationasmentionedin(1), (2), (3)
and(4).
Since forthe radiation duetoheave m 0,contributiontof
andrn8 will be fromCd
only. Thus---0
f
0J, ., tZ im+’ {Jm(’ka)Z,xo(O Z,o(Z +
qmjZ(z) Z,(O)" ,
cosmO]
cosOdOdz-pgaA
=-b =o,=o
o
. Zx(z)
12pgaA=_ =o[{J,(oazo(0) +
osin
ih
sin$(h b) 2rpgaA[{J(oa)+qm} sinhh-sinh$(h-b)+
qJa
cosih ]"
oa
coshoh =
Thus
where
(49)
D
rpga2A.
Thevertical forcecomponent
fz
canbewrittenaswhere
fzd
andf,r
aregiven by’and
where7oj o(,-b) and ao -ia. Thus wehave
Now
wecomputethe momenton thesideofthebodyabou sea-bottom(50)
164
Hencewehave
-ipaa
(z + h)(a, O, z)
cosOdzdO=0
o
Zo(Z) z,(z),
2rpgaA
/_ (z + hl[J,(Aoa) zo(O +
j=oqjz,(o)az
2rpga3A[J,(Aoa) +
q,oohsinh Aoh +
coshSo(h b)
cosh
oh (oa)
osh
oh + o(h )sinh o( )}
(oa)
q1
{cos$jh-cosSj(h-b) +hsin 1h 1(h- b)
sin1(h b)}].
m___ 2[J,(Aoa) +
q,oAohsinh Aoh +
cosh$o(hb)
Da cosh$oh
($oa)
cosh
oh + 2o(h b)
sinho(h b) (o)
q cos
Ah
cosA(h b)
+ .= ( ()
jhsinAjh
$j(h b)sin $j(h
+
-j.Moment m
atthe bottomofthebody aboutz-axiscanbe(51)
mb mbdq-mbr wheretubaand mbr aregiven by
m, ipa
(r, O,-b)rdrdO
=0
ipa
=o[(- + .=,P"
cosk,,(h o)jr
cosmOdrdO(_1)o
2ripaa3[ 7 + (k.a)alo(k.a) {(k.a)I(k=a) k=alo(k=a)+ I
k’"o()=}]
(-1)-o.
{.a(.a) Zo(.a) +
(2 + 1)2()}]
k=O
and
2"
- ie 0 (,’,O,-).rddO
=0
,,=
. + 2(h : g{(h b) F}lrdr
2paaa(h_b)[{lO_3()}a + 2iina-I"6ao(h- b)
4raapgAo(h b)tanh Aohf[ + () + g 7oiLox,
j=O
(-1) (Z=o 7oL.a, )
--1(k.a)2
+ .= (k.a)2io(k.a) {k.all(k.a) Io(k.a)+
k=o
(2k + 1)2(kl) }1
Thusweget where
mb
(-1)"
D’-’ 4*[-{So(Aoa)o.o + qoio,} + (k,,7k,,a) {J(Aa)" +
j:O n:l
{,,=,(,,=) 0(,,) +
( + )=() }] + o(h )t,m
k=O
= ) (-)"(E%ooL,,, )
+( +
j=OoLo, + (=)*o(,,=)
(,,=)*
*{,,=,(,,=)- o(,,=)+
( + )=(),}]. ()
k=0
6.
DETERMINATION OF THE HEAVE AMPLITUDE AND INCIDENT WAVE AMPLITUDE
RATION.Fromthe equation of heavemotionweget
F, + F: + F
where
M
isthemsof displacedfluid,F
isthe radiatedforce,F:
isthe exciting forcein z-direction andFa
isthe hydrostaticforce. Thuswehvecomputed from hydrostatic force. Now theequationofmotion incomplexformbecomes
(M + )(-ia) + (-ia)u + t fza.
Thisyields
__
x-a(M +
I+ i)"
Sincetheradiatedforce
Fr
canbedecomposedintocomponents inphasewithacceleration and the velocity of the cylinderin thefollowingway0 O
we cmawrite
L (u + i).
Thuswe have+i= S
wheandS
ra(h b)p.
ThuswegetA Ao
tanhAoh{b + (h b)cT}
where
(53)
fzd
7.
NUMERICAL RESULTS.
The complexmatrix equation
(32)
is to be solvedin orderto determinetheunknown coeffi-166 D. D. BHATTA AND M.
cientsqmjform 0andm 1.Tocomputethe horizontalexcitingforce,f.d,weneedtosolve the equation
(32)
form and theverticalexcitingforce, f,d, isevaluated using the solution ofthis equation when m 0. This infiniteorder systemismade finitetosolveit numericallyby writingwhere
2)i.,.
andA,,,,
’saregiven by 2)j,7-/,,,j8,, +
_, v.q,,,. (54)
j=O
A .,,00 Jo(oa)
r 0,1,...,Nnandj 0, 1,...,
Nn.
The complexmatrix equation
(48)
istruncatedasfollowing., .o. Xot
j=O
where
’ti
andX0
’saregiven bywhere
. .-:-_bGO.6,, +
2, Go,,L,,,L,,.
n----1
N.
Xot gooLo., + Y] Zm, Go,,
(55)
a
g00
-2(h b)
2(-1)"
np
D,271.0,1,...,
N,
and j 0,1,Np.
Thus2)and
E
aresquare matricesof order(Np + 1)
andJt0,
,4and ,t0tarevectorsoflength(Np + 1)
These systems of equations aresolved by usinga complex matrix inversion subroutine available inIMSL
atTUNS
cybersysteln. WeselectN,
8 andN,,
12 whichare seemed to begood enough for the convergence of the solutions. Also wetakeN
20. Onceq0i qli and0; ko,
omputth ods uig th xpssio(), (), (0),() na ()
by truncating theinfinite series for theindicesj, n, andk at
N,, N,,,
andN
respectively. The heaveamplitudeand the incidentwaveamplituderatiosaredepictedinFig. 2asfunctionofA0a.
Non-dimensional x-component of thehorizontal force is depicted in Fig. 3. Fig. 4 presents the non-dimensional vertical force. The non-dimensional moment actingon the sideof the cylinderis shown inFig. 5. Fig. 6depictsthe non-dimensional momentactingatthe bottom of thecylinder.
Different depth toradius ratios consideredhereare 2.00 and 3.00with a combinationofdraft to radius ratios0.75, 1.00, and 1.25.
8.
CONCLUSIONS.
The waveloads for a vertical circular cylinder heaving in finite depth water in thepresence ofan incidentwavehave beencomputed in this paper. Analyticalsolutions for thetotal velocity potential is obtained by dividing the whole boundary value problem into two problems, namely, diffractionproblemofanincidentwave actingon the
xed
cylinder and radiationproblem ofthecylinderforcedtooscillate in otherwise stillwater. Mathematical solutions for theboundary value problems are obtained in two physical regions, namely, interior region and exterior region. The exciting force componentsareobtainedby solvingthediffractionproblem and the added massand dampingcoefficientsareobtainedby solvingthe radiationproblem. Then heaveresponse inducedby waveexcitation is determinedfrom the equation of]notionof thefloatingcylinder. Using Bernoulli’s equation,pressure iscomputedwhich isused tocompute thewaveloads. Results for differentdepth toradiusand draft toradius ratiosarepresented in variousfigures.
x----xh/a 200.b/a 100 s- Eh/a 300. b/a tOO
2 4 6
e----e Na-200.b/a-0.75 --.xh/a-200.b/a-125
2 4 6
FIGURE2. Amplitudeof
/A.
AND M. RAHMAN
.6
h/ 00 b/ 100
IO----O
h/a 2C0b/a1001
00 2 6
FIGURE3.Non-dimensionalhorizontal force.
Na-3.00 b/a-100[
Na 2.00. b/a 100
’-.-..._.._._._
0
Na 200 b/a 075
FIG4.Non-dimensiMertical force.
25
20
1.5
_e
h/a 200b/a 100
,P.0.
o,
,x.,,,x
FIGURE5. Non-dimensionalmoment,D,"
.75
.25
0
tOO
.75
.25
Q
IVa 3.00. b/ee 100 lVa 2.00.I:ga 1.00
IVa 2.00.b/a-075
FIGURE6.Non-dimensional moment,
5""
REFERENCES
1.
BAI,
K.J.,
The added mass of two-dimensional cylin(lers heaving in water of finitedepth, .J. FluidMech.,81(1977),
85-105.2.
BLAC’K,
.1.L., MEI,
C. C. andBRAY,
C’.G.,
Radiation and scattering of water waves by rigid bodies, J. Fluid Mech.,46(1971),
151-164.3.
GARRETT,
C’.J.R., Waveforces on acirculardock, J. FluidMech.,46(1971),
129-139.4.
GARRISON, C..J.,
Hydrodynamics ofLarge
ObjectsintheSen;
Part I: Hydrodynamic Anal- ysis, JournalofHydronautics, 8,(1974),
5-12.5.
GARRISON,
C.J.,
Hydrodynamics ofLarge
Objects in theSea;
Part II: Motion of Free- Floating Bodies, JournalofHydronautics, 9, 2(1975),
58-63.6.
ISAACSON, M.,
Waveforcesoncompound cylinders,Proc. Civil Engineeringin theOceansIV, ASCE,
San Francisco,(1979),
518-530.7.
MCIVER,
P. andLINTON,
C.M.,
The added mass of bodies heaving at low frequency in wateroffinitedepth, Applied Ocean Research, 13,1(1991),
12-17.8.
MILES,
J. W. andGILBERT,
J.F.,
Scatteringofgravity wavesby acirculardock, J. Fluid Mech.,34(1968),
783-793.9.
SABUNCU,
T.andCALISAL, S.,
Hydrodynamiccoefficients foravertical circularcylinders at finitedepth, OceanEngng.,
8(1981),
25-63.10.
WILLIAMS,
A. N. andDEMIRBILEK, Z.,
Hydrodynamic interactions in floating cylinder arrays-I. Wavescattering, OceanEngng.,
15, 6(1988),
549-583.11.
WILLIAMS, A.
N. andABUL-AZM,
A.G.,
Hydrodynamic interactions in floating cylinderarrays-II.
Wave radiation, OceanEngng.,
16,3(1989),
217-263.12.