WAVE LOADINGS ON A VERTICAL CYLINDER DUE

VOL. 18 NO. (1995) 151-170

WAVE LOADINGS ON A VERTICAL CYLINDER DUE

TO

HEAVE 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

WORDS

AND 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 six

degrees

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 as

surge, 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,

Meiand

Bray [2]

have calculated the waveforceson a truncated cylinderwhich either extends to the free surfaceorrestson the seabed.

Isaacson [6]

^extended

Garrett’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 and

Demirbilek [10].

Numericalresults fortheaddedmassand dampingcoefficientsofsemi-submergedtwo-dimensional heavingcylindersin wateroffinitedepthwerepresented byBai

[1].

^Heshowed that theaddedmass isbounded forallfrequenciesin water of finitedepth. He

’tudied

the limits of the addedmassand

damping 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 computations

o

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 a

vertical 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 level

SWL ),

^rmeasured radially from thez- axisand 0

Y

//A

FIGUBJB

^{1. Deqnition}

skecth.

from the positivex -axis.

For

an incompressible and inviscidfluid, and for small amplitudewave theory with irrotational motion, ^wecan introducea velocitypotential

d(r, ^O,

^z,

t).

^{This can be}

written as

+(,.,O,z,t) R[(,.,O,z)-"’].

From Bernoulli’s equationwegetpressure,

P(r, O,

z,

t),

^as

P -p---.

The forcecomponents

F., Fu, F.

^alongx,y,zdirections aregivenby

F [" [ 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 moment}

^M,

arising due^tothe forcesonthesideson the cylinder about sea-bottomand

M

arising due to the forces at the bottom of the.cylinder about z-axisare givenby

M. + 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

,

^where

e

^is the velocity potengial due to the

problemofan incidentwaveactingon thefixedcylinder, nd

,

^{dueto theredigion problem}

the cylinder forced tooscillete in oherwise stillwater. Thus can be^writgen

where Re

e

^-i’

^].

^Now by dividing the whole fluid domain into twodomns,

(a)

^inerir

domain region below the cylinderi.e. r -h

-; (b)

exterior domain region for r aand -h

O,

wewrite^ghevelocitypotentialfortheinteriordomain

i

and the velocity potential for theexterior domain

".

^hen

F ,F,, M., M

^{can be written the real}

fe-i,f,e-i, m.e -i, me

^-ierpecively where

f,

f,,m.,

m

^ere

ven

^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

⁰

a-gT;: 0

^{0 on z=0}

^r>_a

O

_{0 on z -h}

Oz

0 -ir

^{o z -b}

Oz o

07" ^{0 on r= a}

()

0<,<

(8)

b

<

z

<

0.

(9)

The boundary condition

(8)

^{can be}separated ^into two conditionsby writing^as

0Ca

Oz

^{0 on z -b 0}

< r<

a

(10)

0

Oz

^{-ia on z -b 0}

<

r

<

a.

(11)

Nowweconsider thisproblemby separating^itintotwoproblems,diffractionproblemandradiation problem. Thediffractionproblemwillgiveusthe exciting force and from theradiationproblem^we 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 potential

Cs

^where

+s)

must satisfy the radiation condition"

linoo ^V:{ O0s- ^iA0s}

⁰

where

A0

^isthewavenumber and

1

^{is theincidentwavevelocitypotential. Wesolvethisboundary}

valueproblem byconstructing the representation of

Ca

^{in theinterior domain} under the cylinder and theexteriordomain r

>

^{a inthe}followingsection. Letusassumeaproduct solution

(,

0,

z) Z(z)R(,.)

^os.0

m 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=O

^o +

n=l

.,.(.)

^o

^{.(z + )l}

^o.0

valid 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 will}

be 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 expansion

C:’,,(.,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 canbewrittenas

gA__cosh Ao(z + h)eixox

a cosh

Aoh

z__

_a ^osh_cosh

^{o(z + h)} ,,,i’",,,(o,.1

^cosm0

Aoh

_o

in which eo 1,e,,, 2,

(m > 1)

^and

Ao

^is the wavenumber. Here r and

Ao

^are related by the dispersion relationa

gAo

tanh

Aoh

^and

J,,,(Aor)

^isthe Bessel function of first kind andorderrn.

Because

of thepresenceofan objectweneed to considerthe scattering ofwaves. Therefore the appropriate satisfying^theradiationconditioncan be constructed from andisgivenby

where q,i’sarearbitrary constants.

Here H)(A0r)

^isthe Hankel functionof first kindof orderrn and

K,,, (Air)

^{isthemodifiedBesselfunction}of second kindand orderm.Also

A

^satisfiesthe relation

a

-gA

tan

j 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 symbol}

^k,,(= -)

^{used to}

representtheeigenvalues for theinterior solution. Thusthevelocity potentialisgivenby

_, e.,i"[{J.,(Aor) + q,,,oH)(Aor)}

^cosh

^Ao(z + ^h)

,,,=o cosh

Aoh

g,,(;)

^o

.( + h)

/ q,,,j

]cos

toO.

(15)

K(A.a)

^COS

The set offunctions

{cosh Ao(z + h),

^cos

A(z + h)},

1, 2, formsanorthogonalset definedin the interval -h

<

z

<

0 due totherelationa gA0tanh

Aoh

-gAj^tanAjh. Thus the orthonor- malset canbeconstructedprovided

where

zo(z) --N;o

^1/2oh

^ao(z + ⁾ z(z) g,

-1/2^o

⁽ + h)

(t6)

(17)

156 D. D. BHATTA

sinh

2Aoh]

Nxo ^[1 +

2Aoh

sin2Ajh

g [ ⁺ --;- ^I.

Thuswith these definitions

(15)

^{can be writtenas}

(8) (19)

where

B,, 4e,,i’".

^{This isvalid}for -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 follows

Multiplying

(21)

by

,j

0,1,2, andintegratingwithrespecttozfrom -h to0,^weobtn using^theorthogonal property

-h

h

Z.(z)(a,z)dz J,,(Aoa)+

q,,,,o ^o

Zo(0)

dz

h h

Zx(z):,(a z)dz

^q’j

x(z)

_dz.

Z(O)

^h

In

view of the

orthonormafity

of the set

{Zxo(z),Z(z)}

^{in -h z}

O,f m(*}dz

^d

fh (*)dz

^1,weobtain

zo(o)

j_ Zo()(,,,)d-

q,,o h

q,,o a

Zx.z.

^-....a,

z.

^dz

where 1, 2,

3.2 Determinationof the unknown coefficients.

To

preserve the continuity of the twosolutions atthe imaginary interfacer a,^{it is}required to satisfy

,(, z) :,(, ), (a)

Or Or

for -h

_<

z

<

-b. Alsobody surface

condii,

^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) +

%,0

H,(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)

yields

P"’’ h- b ^t,

B,(a,z)cosk.(z + h)dz

2B,,,

J.,(Aoa)

Zo(z

^cos

k.(z + h)dz

h

[ _z:(o)

qmj

/_-b

+ o ’.= ^{Z,(O z(z)}

^cos

^{k,(z +}

n 0,1,2,. Ifwedefine

(29)

/_- Z,(z)cos k,,(z + h)

"=

h-b h

Z(0)

^dz

wherev takes the valueAj,then

(29)

^{can berewrittenas}

Now simplifying^weget

p

2B,,,[J,,,(Aoa),,Xo +

j=O

(30)

r-]_b ^N-

^1/2cosh

^{Ao(Z + h)}

^cos

(h-)Zo(O)

^o

(- )"(h b)o

^sinh

Ao(h b)

(h ) o + .

^osh

^oh

nr(z + h)dz

h-b

n 0, 1,2, and

f-’

^_1/2

^{nr(z+ h)}

,x, (h b)Z.b(O

^.,-h

Nx

^cosAj(z

^{+ h)cos}

h b dz

.sin((h

b)j

+ nr) sin((h b)i

2(h b)

^cos

Ah

^Aj

+

^Aj -b

(-1)"(h b)jsini(h b)

((h b) ^.}

^o

h

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. Thisyields

Addingthese two equationsweget

h b

BIC,,,o,5,

o,

B, E qmJT"’mJ6".ir +

^mp,,o _2h

h

(31)

where

Sx.

^isKronecker delta. Inserting the expression ofp,,,,equation

(31)

^{canbewrittenas}

ICo6,o,. J,,,(Aoa)

^{h b}

Z,.(O)(mooo,- +

²

IC,,,,,o,,.r }

h

[7-/,,,i6, + h

^b

Z,(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)

^h

z,(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 valueproblemis

V2r

⁰

--a2r

^{0 at}

0 at Oz

Oz

0

_{0 at}

07"

z=0

on r

<

a and z -b

r a and b

<

z

<

O and theradiation condition

liAn v(--

r

-iA0,)

0 where

Ao

^isthe wavenumber. Weassumethat takes the form

(r,O,z) ,,(r,z)

^cos

n-’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/where

a,,,’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 partwewrite

where

Ckh

^and

^i,,p

satisfy the equations

(33)

^and

(34).

The boundary condition

(35)

^{can be} decomposedas

0i

_{0 on z=-b}

Oz

and

a o7 25

.

Oz

For homogeneous part, by method

o seperon o

vfib]es we

ge

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,

^and

Co

^areconstants to be determined fromthe givenconditions. Applying boundary conditions, weget

Bo

0,

Ao

_2(h-b), ^{and frown} the governing equation weget

2Ao + 4Co

^0.

Thus

Ao =-2Co-

_2(h-0"^{a. Hencetheparticularsolutionis}

am r

,,(,., ^z) ₂₍ [(z ^{+ ) F].}

Henceweget

, _--’(

^a,,,o

^r)., +

n=l^a

_I,,,(k,,a) ^l,,,(k.r)

^cos

^k.(z + ^h)+ _2(h-

^a,=

_b)[(z + h)2- 1" ^r2

Atr a wehave

(.,) ^{o +} .=, ^a.

^cos

^{k,,(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 thefunctionscos

k( + h)),

^{wegetanexpression}

for a,,, inthe following form

’"

^h-

’(a,)co,,(z + h)d-

where

and

am

^a

Ion, - (h ^{am [(} b) /_:[(z

³

^{b) +}

^a2

^h) - ^{)1 -]dz}

Forexteriorregion, theboundaryvalueproblemis

v(,z)- ,,(,z)

m

^o

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,(,)

^and

K,n

satisfy the radiation condition. Applying boundary conditions^wearriveatanexpression

,K,,,(s,’) O,],(r,z) ,,,oU,)(Ao,’)H,)(Ao a) Zxo(z) + .= ^{K,,(Aia) Zx,(z)}

where

Zx,(z)

^and

Nx,

^takethe forms defined in

(16), (17), (18)

^and

(19).

^j 0, 1,2,

Now

at

r a wehve

,:,(, )= z,,,z().

j=o

Multiplyingboth sides ofthisequation by andthen integratingboth sidesfrom-h to 0

(and

using the

orthogonal

propertyof the functions

Za(z)),

^{wegetanexpressionfor}

,i

ⁱⁿthe following form

j

=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 b}

Zxdz

Zx

^cos

^{k,.,(z + h)dz.}

L,,x

_h-b

h

Also

Z, cosk,,,

(z + h)dz

L’

_{h b}

h

h b cosh

A0u

^cosk,.,udu

(- 1)"N-o1/2(h ^b)Ao

^sinh

^o(h b)

(h b)2A02

-I-^n27r and

(40) (41)

(42)

(43)

(44)

162

where

Z,

^cos

k,,(z + h)dz

L,,,

_{h b}

h _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),

^wehave

G.,o+ mmo

2

G cosk,,(z+h)+

n=l j=O

for

^h

S

^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.. and}integratingin the regions of validity andadding^{them we}get

(48)

rnamo

_r h

G,.o +

₂ ^o,

,

^a,,,,,

^{G,.. L,,a,} +

_h b

Now

substitutingthevalues ofa,,,,,wegetasystemof equations

,

^{Ejflrnj X,,,}

j=0

0,1,2,

5,

EVALUATION

^OF

^{THE FORCES AND} MOMENTS.

The

horizontal

^{and theverticalforcesonthecylinderare}

calculated

^{from the}pressure obtained fromSernoulli’sequation^asmentionedin

(1), (2), (3)

^and

(4).

^{Since for}the radiation due^toheave m 0,contribution^to

f

^andrn8 will be from

Cd

^{only. Thus}

---0

f

⁰

^{J, ., tZ im+’} {Jm(’ka)Z,xo(O ^{Z,o(Z +}

^qmj

^Z(z) Z,(O)" ^,

^cos

^mO]

^cosOdOdz

-pgaA

=-b =o,=o

o

. Zx(z)

₁

2pgaA=_ =o[{J,(oazo(0) ⁺

^o

sin

ih

^sin

^{$(h b)} 2rpgaA[{J(oa)+qm} sinhh-sinh$(h-b)+

_qJ

a

^cos

ih ^]"

oa

^cosh

^oh =

Thus

where

(49)

D

rpga2A.

Thevertical forcecomponent

fz

^{canbewrittenas}

where

fzd

^and

f,r

^aregiven by’

and

where_{7oj 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=o

_qjz,(o)az

2rpga3A[J,(Aoa) ⁺

^q,o

^{ohsinh Aoh +}

^cosh

^{So(h b)}

cosh

oh (oa)

osh

oh + o(h )sinh o( )}

(oa)

q1

{cos$jh-cosSj(h-b) +hsin ^{1h 1(h- b)}

^sin

^1(h b)}].

m___ 2[J,(Aoa) ⁺

^q,o

^{Aohsinh Aoh +}

^cosh$o(h

^b)

Da cosh$oh

($oa)

cosh

oh + 2o(h b)

sinh

o(h b) (o)

q ^cos

Ah

^cos

^{A(h b)}

+ .= ^{( ()}

jh^sinAjh

$j(h b)sin $j(h

+

^-j.

Moment m

atthe bottomofthebody aboutz-axiscanbe

(51)

mb mbdq-_mbr where_tubaand _{mbr are}given by

m, ipa

(r, O,-b)rdrdO

=0

ipa

=o[(- ⁺ .=,P"

^cos

^k,,(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=O

^{oLo, +} _(=)*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 and

Fa

^isthe hydrostaticforce. Thuswehve

computed from hydrostatic force. Now theequationofmotion incomplexformbecomes

(M + )(-ia) + (-ia)u + t fza.

Thisyields

__

^x-

a(M +

I

+ i)"

Sincetheradiatedforce

Fr

^{canbedecomposedinto}components inphasewithacceleration and the velocity of the cylinderin thefollowingway

0 O

we cmawrite

L (u + i).

^{Thuswe have}

+i= S

whe

andS

ra(h b)p.

Thusweget

A Ao

^tanh

Aoh{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.

cients_qmjform 0andm 1.Tocomputethe horizontalexcitingforce,f.d,weneedtosolve the equation

(32)

^{form and thevertical}excitingforce, f,d, isevaluated using the solution ofthis equation when m 0. This infiniteorder systemismade finitetosolveit numericallyby writing

where

2)i.,.

^and

A,,,,

’saregiven by 2)j,

7-/,,,j8,, +

_, ^{v.q,,,. (54)}

j=O

A .,,00 Jo(oa)

r 0,1,...,N_nandj 0, 1,...,

Nn.

The complexmatrix equation

(48)

^{istruncatedas}following

., .o. ^Xot

j=O

where

’ti

^and

^X0

^{’saregiven by}

where

. .-:-_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)

^and

Jt0,

,4and ,t0tarevectorsoflength

(Np + ¹⁾

These systems of equations aresolved by usinga complex matrix inversion subroutine available in

IMSL

at

TUNS

cybersysteln. Weselect

N,

^{8 and}

N,,

12 whichare seemed to begood enough for the convergence of the solutions. Also wetake

N

^{20. Once}q0i qli and

0; ko,

omputth ods uig th xpssio

(), (), (0),() na ()

by truncating theinfinite series for theindicesj, n, andk at

N,, N,,,

and

N

respectively. The heaveamplitudeand the incidentwaveamplituderatiosaredepictedinFig. ^2asfunctionof

A0a.

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 ofthe

cylinderforcedtooscillate 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. Amplitude^of

/A.

AND M. RAHMAN

.6

h/ 00 b/ 100

IO----O

^{h/a 2C0b/a}

₁₀₀₁

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""

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1.

BAI,

K.

J.,

The added mass of two-dimensional cylin(lers heaving in water of finitedepth, .J. FluidMech.,81

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85-105.

2.

BLAC’K,

.1.

L., MEI,

C. C. and

BRAY,

C’.

G.,

Radiation and scattering of water waves by rigid bodies, J. Fluid Mech.,⁴⁶

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3.

GARRETT,

C’.J.R., Waveforces on acirculardock, J. FluidMech.,46

(1971),

129-139.

4.

GARRISON, C..J.,

Hydrodynamics of

Large

Objectsinthe

Sen;

Part I: Hydrodynamic Anal- ysis, JournalofHydronautics, 8,

(1974),

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GARRISON,

C.

J.,

Hydrodynamics of

Large

Objects in the

Sea;

Part II: Motion of Free- Floating Bodies, JournalofHydronautics, 9, 2

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ISAACSON, M.,

Waveforcesoncompound cylinders,Proc. Civil Engineeringin theOceans

IV, ASCE,

San Francisco,

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MCIVER,

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LINTON,

C.

M.,

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J. W. and

GILBERT,

J.

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SABUNCU,

T.and

CALISAL, S.,

Hydrodynamiccoefficients foravertical circularcylinders at finitedepth, Ocean

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WILLIAMS,

A. N. and

DEMIRBILEK, Z.,

Hydrodynamic interactions in floating cylinder arrays-I. Wavescattering, Ocean

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WILLIAMS, A.

N. and

ABUL-AZM,

A.

G.,

Hydrodynamic interactions in floating cylinder

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Wave radiation, Ocean

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YEUNG,

R.

W.,

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(1981),

^119-133.