Ocean Engineering 32 (2005) 241–271
www.elsevier.com/locate/oceaneng
Hydrodynamic properties of two vertical truncated
cylinders in waves
Y.H. Zhenga,*, Y.M. Shenb, Y.G. Youa, B.J. Wua, Liu Ronga
a
Guangzhou Institute of Energy Conversion, Chinese Academy of Sciences,
Guangzhou 510640, People’s Republic of China
b
State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology,
Dalian 116023, People’s Republic of China
Received 30 April 2004; accepted 23 September 2004
Available online 2 December 2004
Abstract
The radiation and diffraction boundary value problem arising from the interaction of linear water
waves with two vertical truncated cylinders is investigated by use of the method of separation of
variables and the method of matched eigenfunction expansion. Analytical expressions for the
radiated and diffracted potentials are obtained as infinite series of orthogonal functions. The
unknown coefficients in the obtained expressions are determined by use of the matched
eigenfunction expansion method. To verify the obtained expressions, the Green’s second identity
and the symmetry of the matrices for the added masses and damping coefficients are used. The results
show that the analytical expressions presented in this paper are correct. By use of the present
analytical solution, the hydrodynamic coefficients and wave forces for some specific cases are
calculated and the hydrodynamic effects of the cylinders’ radii on the hydrodynamic properties of the
cylinders are investigated which will supply some useful information for the design of this kind of
system.
q 2004 Elsevier Ltd. All rights reserved.
Keywords: Hydrodynamic property; Radiation; Diffraction; Water waves; Analytical method; Two vertical
truncated cylinders
* Corresponding author. Tel.: C86 20 87057612; fax: C86 20 87057597.
E-mail address: zhengyh@ms.giec.ac.cn (Y.H. Zheng).
0029-8018/$ - see front matter q 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.oceaneng.2004.09.002
242
Y.H. Zheng et al. / Ocean Engineering 32 (2005) 241–271
1. Introduction
It is well known that floating structures, such as ocean platforms, breakwaters, and
wave energy devices, are often used in ocean engineering. The hydrodynamic properties,
of which most important are the hydrodynamic coefficients, wave excitation forces, and
transmission and reflection coefficients, are of major interest of designers and many
researches have been carried out and lots of results have been obtained.
To analyze the hydrodynamic properties of floating structures, various methods, such as
the Boundary Element Method (BEM), the Finite Element Method (FEM), and some
analytical methods, can be used. Of all existing methods the most efficient are analytical
methods which are only applicable to some particular problems, for example, the problem
considered in the present paper. Numerical methods like BEM and FEM are suitable for
general problems, but the computational procedure is complex, the efficiency and the
accuracy are relatively lower compared with those of analytical methods. So the best
methods for some particular problems are analytical ones.
Up to now, many scholars have applied various analytical methods, such as the
matched eigenfunction expansion, the multipole expansion, and the multiple scattering
technique, to the study of hydrodynamic properties of floating structures. The often-used
analytical methods are the matched eigenfunction expansion method. The following are
some examples of application of these analytical methods to hydrodynamics of vertical
floating circular structures. By using the eigenfunction method, Black et al. (1971), Yeung
(1981), Sabuncu and Calisal (1981), Calisal and Sabuncu (1984, 1993), Williams et al.
(2000) and Bhatta and Rahman (2003) studied the radiation and/or diffraction by a single
floating circular cylinder and obtained theoretical results of hydrodynamic coefficients
and/or wave forces. Berggren and Johansson (1992) and Eidsmoen (1995) investigated the
heave radiation problem of a two-body axisymmetric system and calculated the heave
added masses and damping coefficients. Wu et al. (2004) explored the diffraction and
radiation problem for a cylinder over a caisson in water of finite depth and presented the
calculated results of hydrodynamic coefficients, wave forces and hydrodynamic effect of
the caisson on the hydrodynamics of the cylinder. Yeung and Sphaier (1989) determined
the radiation and diffraction properties of a floating vertical cylinder of finite draft in a
channel. Mciver and Bennett (1993) and Linton and Evans (1993) studied hydrodynamic
characteristics of a body in a channel by use of the multipole expansion. Simon (1982),
Kagemoto and Yue (1986), Williams and Demirbilek (1988), Williams and Abul-Azm
(1989) and Williams and Rangappa (1994) investigated the scattering and/or radiation
problems of horizontally arranged cylinder arrays by application of the multiple scattering
technique and plane-wave approximation or modified plane-wave technique.
In this paper, we consider the radiation and diffraction problem of a system consisting
of two coaxial cylinders arranged vertically. The problem has practical engineering
background, and part of which, namely cylinders in heave, was studied by Berggren and
Johansson (1992) and Eidsmoen (1995). They did not consider the sway and roll motions
and the scattering of water waves though Eidsmoen (1995) obtained the vertical force on
the cylinders by use of the general Haskind’s theorem. The information about the sway and
roll motions of the cylinders and wave scattering may be important for designers, so we
study the radiation and diffraction problem here. The method used here is the matched
Y.H. Zheng et al. / Ocean Engineering 32 (2005) 241–271
243
eigenfunction expansion which has been used by many investigators listed above.
Analytical expressions for the radiated and diffracted potentials are obtained. The
hydrodynamic coefficients and wave forces are calculated and the hydrodynamic effects of
the cylinders’ radii on the hydrodynamic properties are investigated.
2. Problem formulation and mathematical model
Here the diffraction and radiation including heave, sway and roll of linear water waves
by a system consisting of a floating circular cylinder and a submerged circular cylinder
will be studied. The two cylinders are coaxial with the same radius. The geometry of the
system and the arrangement of the right-hand Cartesian coordinate system Oxyz and the
cylindrical coordinate system Orzq are shown in Fig. 1. The origin of the coordinate
system is at the undisturbed water surface with the positive z pointing upwardly and the
positive x directing to the right. The floating cylinder, which is called hereafter cylinder 1,
occupies the space defined by r%R, 0%q%2p, Kd1%z, and the submerged cylinder
called as cylinder 2 hereafter occupies the space with r%R, 0%q%2p and Ke2%z%Ke1.
As usual we assume that the fluid is incompressible, inviscid and periodic, and the
motion of the fluid is irrotational. There exists a velocity potential
fðr; q; z; tÞZ Re½Fðr; q; zÞeKiut , where Re[ ] denotes the real part of the complex
expression, u is the wave angular frequency, t is the time and F is a time-independent
spatial velocity potential which satisfies the following three-dimensional Laplace equation
v2 F 1 v
vF
1 v2 F
r
C
Z0
(1)
C 2
2
r vr
vr
vz
r vq2
Fig. 1. Schematic of geometry.
244
Y.H. Zheng et al. / Ocean Engineering 32 (2005) 241–271
For linear water waves considered here, it is customary not to solve F directly, but make
the following decomposition if we consider only heave, sway and roll modes of radiation
F Z FI C FD C
2 X
3
X
FðJ;LÞ
R
(2)
JZ1 LZ1
where FI is the incident wave potential; FD is the diffracted potential; FðJ;LÞ
is the radiated
R
potential due to the motion of the cylinders. Here LZ1 stands for the heave, 2 for sway and
3 for roll. JZ1,2 represents cylinder 1 and cylinder 2, respectively. In the following
sections, L always ranges from 1 to 3 and J always ranges from 1 to 2 if they are not
specified particularly.
The solution of F is equivalent to the solving of FD and FðJ;LÞ
as FI is generally
R
prescribed beforehand. For linear water waves of amplitude A and angular frequency u
propagating along the positive x direction in water of depth h1, the time-independent
incident wave potential may be expressed by
N
igA cosh½kðz C h1 Þ X
mm Jm ðkrÞcosðmqÞ
(3)
u
coshðkh1 Þ
mZ0
pffiffiffiffiffiffi
where iZ K1; g is the gravitational acceleration; k is the wave number, which is
determined by the dispersion relation k tanh(kh1)Zu2/g; Jm( ) is the Bessel function of
order m; mm is a coefficient given by
(
1
m Z0
mm Z
2im mO 0
FI Z K
2.1. Mathematical model for radiated potentials
If the complex amplitudes of heave, sway and roll of cylinder J are assumed as AðJ;1Þ
R ,
ðJ;LÞ
and AðJ;3Þ
can be expressed as
R , respectively, the radiated potential FR
(
ðJ;1Þ
L Z1
KiuAðJ;1Þ
R 4R ðr; zÞ
ðJ;LÞ
FR Z
(4)
ðJ;LÞ
KiuAðJ;LÞ
R 4R ðr; zÞcos q L Z 2; 3
AðJ;2Þ
R
is a spatial quantity independent of q. For convenience, Eq. (4) can be
where 4ðJ;LÞ
R
expressed in a general form (LZ1,2,3)
ðJ;LÞ
FðJ;LÞ
Z KiuAðJ;LÞ
R
R 4R ðr; zÞcos½ð1 K d1;L Þq
(5)
where
(
dj;L Z
1
LZj
0
L sj
:
Substitution of Eq. (5) into Eq. (1) gives the following general partial differential
equation (LZ1,2,3)
Y.H. Zheng et al. / Ocean Engineering 32 (2005) 241–271
"
#
ð1 K d1;L Þ2 4ðJ;LÞ
v2 4ðJ;LÞ
1 v
v4ðJ;LÞ
R
R
R
r
C
Z0
K
2
r vr
vr
vz
r2
245
(6)
To obtain the unique solution to Eq. (6), it is necessary to specify the following boundary
conditions:
1. The free surface boundary condition
v4ðJ;LÞ
u2
R
K 4ðJ;LÞ
Z0
vz
g R
ðz Z 0; rO RÞ
(7)
2. The sea bed boundary condition
v4ðJ;LÞ
R
Z0
vz
ðz Z Kh1 Þ
(8)
3. The body surface conditions
v4ðJ;LÞ
R
Z dJ;1 ðd1;L K rd3;L Þ ðz Z Kd1 ; r% RÞ
vz
(9)
v4ðJ;LÞ
R
Z dJ;2 ðd1;L K rd3;L Þ ðz Z Ke1 ; r% RÞ
vz
(10)
v4ðJ;LÞ
R
Z dJ;2 ðd1;L K rd3;L Þ ðz Z Ke2 ; r% RÞ
vz
(11)
v4ðJ;LÞ
R
Z dJ;1 ½d2;L C ðz K z0 Þd3;L ðKd1 % z% 0; r Z RÞ
vr
(12)
v4ðJ;LÞ
R
Z dJ;2 ½d2;L C ðz K z0 Þd3;L ðKe2 % z%Ke1 ; r Z RÞ
vr
(13)
4. The radiation condition
"
#
pffiffi v4ðJ;LÞ
ðJ;LÞ
R
lim r
K ik4R
Z 0 r/N
r/N
vr
(14)
246
Y.H. Zheng et al. / Ocean Engineering 32 (2005) 241–271
It should be noted that (0,0,z0) is assumed to be the center of rotation in order to specify the
body surface boundary condition of roll motion.
2.2. Mathematical model for the diffracted potential
The governing equation and the corresponding boundary conditions for the diffracted
potential are expressed as follows
v2 FD 1 v
vFD
1 v2 F D
r
C
Z0
C
r vr
vr
r 2 vq2
vz2
(15)
vFD u2
K FD Z 0 ðz Z 0; rO RÞ
vz
g
(16)
vFD
Z0
vz
(17)
ðz Z Kh1 Þ
vFD
vF
ZK I
vn
vn
on S1 and S2
pffiffi vFD
K ikFD Z 0
lim r
r/N
vr
(18)
r/N
(19)
where n is the outward normal from the fluid; S1 and S2 are the submerged surfaces of
cylinders 1 and 2, respectively.
3. Solution method
To obtain the expressions for the radiated and diffracted potentials, we divide the fluid
domain into three subdomains I, II and III as shown in Fig. 1. Here the radiated potentials
ðJ;LÞ
ðJ;LÞ
in the three subdomains are denoted by 4ðJ;LÞ
R1 , 4R2 and 4R3 , and the diffracted potentials
are expressed by FD1, FD2 and FD3, respectively. The method of separation of variables is
applied in each subdomain to obtain the expressions for the radiated and diffracted
potentials with unknown coefficients. The unknown coefficients are then solved by use of
the matched eigenfunction expansion method.
3.1. Expressions for the radiated potentials in three subdomains
Application of the method of separation of variables may give the analytical
expressions for the radiated potentials in regions I, II and III as follows.
Y.H. Zheng et al. / Ocean Engineering 32 (2005) 241–271
247
3.1.1. Region I
4ðJ;LÞ
R1 Z
N
X
AðJ;LÞ
cos½ln ðz C h1 Þ
n
nZ1
R1Kd1;L ðln rÞ
R1Kd1;L ðln RÞ
(20)
where the eigenvalues are given by
l1 Z Kik; k is the wave number; k tanhðkh1 Þ Z u2 =g
ln tanðln h1 Þ Z Ku2 =g
nZ1
n Z 2; 3; .
(21)
(22)
and the radial function R1Kd1;L is given by
( ð1Þ
H1Kd1;L ðkrÞ n Z 1
R1Kd1;L ðln rÞ Z
K1Kd1;L ðln rÞ nO 1
(23)
ð1Þ
and K1Kd1;L are the first kind Hankel function and the second kind modified
where H1Kd
1;L
Bessel function of order 1Kd1,L, respectively.
3.1.2. Region II
ðJ;LÞ
ðJ;LÞ 1Kd1;L
4ðJ;LÞ
r
C
R2 Z 4R2P C B1
N
X
nZ2
BðJ;LÞ
cos½bn ðz C e1 Þ
n
I1Kd1;L ðbn rÞ
I1Kd1;L ðbn RÞ
(24)
where I1Kd1;L is the first kind modified Bessel function of order 1Kd1,L; 4ðJ;LÞ
R2P and bn are the
particular solution and eigenvalue of the radiation mode L of cylinder J in region II, whose
expressions are given by
8
ðz C e1 Þ2 K r 2 =2
ðz C d1 Þ2 K r 2 =2
>
>
d
K
dJ;2
LZ1
>
J;1
>
>
2h2
2h2
<
0
LZ2
(25)
4ðJ;LÞ
R2P Z
>
>
2
3
2
3
>
ðz C e1 Þ r K r =4
ðz C d1 Þ r K r =4
>
>
:K
dJ;1 C
dJ;2 L Z 3
2h2
2h2
bn Z ðn K 1Þp=h2
(26)
3.1.3. Region III
ðJ;LÞ
ðJ;LÞ 1Kd1;L
r
C
4ðJ;LÞ
R3 Z 4R3P C C1
N
X
nZ2
CnðJ;LÞ cos½gn ðz C h1 Þ
I1Kd1;L ðgn rÞ
I1Kd1;L ðgn RÞ
(27)
where 4ðJ;LÞ
R3P and gn are the particular solution and eigenvalue of the radiation mode L of
cylinder J in region III and the expressions for them are
248
Y.H. Zheng et al. / Ocean Engineering 32 (2005) 241–271
8
ðz C h1 Þ2 K r 2 =2
>
>
dJ;2
>
>
>
2h3
<
0
4ðJ;LÞ
R3P Z
>
>
>
ðz C h1 Þ2 r K r 3 =4
>
>
:K
dJ;2
2h3
L Z1
L Z2
(28)
L Z3
gn Z ðn K 1Þp=h3
(29)
, BðJ;LÞ
and CnðJ;LÞ in Eqs. (20), (24) and (27) are unknown and will be
The coefficients AðJ;LÞ
n
n
determined in Section 3.3.
3.2. Expressions for the diffracted potentials in three subdomains
Similarly, applying the method of separation of variables to the diffraction problem
satisfying Eqs. (15)–(19), one can easily derive the expressions for the diffracted potentials
in regions I, II and III as follows, respectively.
FD1 Z
N X
N
X
Am;n cos½ln ðz C h1 Þ
mZ0 nZ1
FD2 Z KFI C
N
X
(
mZ0
FD3 Z KFI C
N
X
mZ0
Rm ðln rÞ
cosðmqÞ
Rm ðln RÞ
(30)
)
Im ðbn rÞ
Bm;1 r C
Bm;n cos½bn ðz C e1 Þ
cosðmqÞ
Im ðbn RÞ
nZ2
(31)
)
Im ðgn rÞ
cos½gn ðz C h1 Þ
cosðmqÞ
Im ðgn RÞ
(32)
m
N
X
m
N
X
(
Cm;1 r C
Cm;n
nZ2
where Am,n, Bm,n and Cm,n are the coefficients to be solved in Section 3.3; ln, bn and gn are
defined by Eqs. (21), (22), (26) and (29).
3.3. Method for the unknown coefficients
The expressions for the radiated and diffracted potentials given in Sections 3.1 and 3.2
satisfy all the boundary conditions except those at the boundary rZR. The remaining
problem is to solve the unknown coefficients AðJ;LÞ
, BðJ;LÞ
and CnðJ;LÞ in the expressions for
n
n
the radiated potentials and Am,n, Bm,n and Cm,n (mZ0,1,2,.; nZ1,2,.) in the expressions
for the diffracted potentials. These unknown coefficients are determined by use of the
conditions of continuity of pressure and normal velocity at rZR. For the radiation problem
considered here, the conditions of continuity are the following
ðJ;LÞ
4ðJ;LÞ
R2 Z 4R1
K e1 % z%Kd1
(33)
ðJ;LÞ
4ðJ;LÞ
R3 Z 4R1
K h1 % z%Ke2
(34)
Y.H. Zheng et al. / Ocean Engineering 32 (2005) 241–271
v4ðJ;LÞ
R1
vr
8
dJ;1 ½d2;L C ðz K z0 Þd3;L >
>
>
>
>
ðJ;LÞ
>
>
> v4R2
<
vr
Z
dJ;2 ½d2;L C ðz K z0 Þd3;L >
>
>
>
>
ðJ;LÞ
>
>
>
: v4R3
vr
249
Kd1 % z% 0
Ke1 % z%Kd1
Ke2 % z%Ke1
(35)
Kh1 % z%Ke2
For the diffraction problem, the conditions of continuity are expressed by
FD2 Z FD1
r Z R; K e1 % z%Kd1
(36)
FD3 Z FD1
r Z R; K h1 % z%Ke2
(37)
8
vF
>
>
K I
>
>
vr
>
>
>
> vFD2
>
vFD1 < vr
Z
vF
>
vr
>
>
K I
>
>
vr
>
>
>
>
: vFD3
vr
r Z R; K d1 % z% 0
r Z R; K e1 % z%Kd1
(38)
r Z R; K e2 % z%Ke1
r Z R; K h1 % z%Ke2
If the method of matched eigenfunction expansion is applied at the boundary rZR, one
can easily obtain the following expressions
ðKd1
ðKd1
4ðJ;LÞ
ðR;
zÞcos½b
ðz
C
e
Þ
dz
Z
4ðJ;LÞ
(39)
i
1
R1
R2 ðR; zÞcos½bi ðz C e1 Þ dz
Ke1
ðKe2
Ke1
4ðJ;LÞ
R1 ðR; zÞcos½gi ðz C h1 Þ
Kh1
ð0
Kh1
ðKe2
dz Z
4ðJ;LÞ
R3 ðR; zÞcos½gi ðz C h1 Þ dz
(40)
Kh1
v4ðJ;LÞ
R1 ðr; zÞ
jrZR cos½li ðz C h1 Þdz
vr
ðKd1
Z
ðKe2 ðJ;LÞ
v4ðJ;LÞ
v4R3 ðr; zÞ
R2 ðr; zÞ
jrZR cos½li ðz C h1 Þdz C
jrZR cos½li ðz
vr
vr
Ke1
Kh1
ð0
dJ;1 ½d2;L C ðz K z0 Þd3;L cos½li ðz C h1 Þdz
C h1 Þdz C
Kd1
ðKe1
dJ;2 ½d2;L C ðz K z0 Þd3;L cos½li ðz C h1 Þdz
(41)
ðFD1 K FD2 Þcos½bi ðz C e1 ÞcosðjqÞdq dz Z 0
(42)
C
Ke2
ð 2p ðKd1
0
Ke1
250
Y.H. Zheng et al. / Ocean Engineering 32 (2005) 241–271
ð 2p ðKd1
0
ðFD1 K FD3 Þcos½gi ðz C h1 ÞcosðjqÞdq dz Z 0
(43)
Ke1
ð 2p ð 0
vFD1
jrZR cos½li ðz C h1 ÞcosðjqÞdq dz
0
Kh1 vr
ðKd1
ðKe1
ðKe2
ð 2p ð 0
vFI
vFD2
vFI
vFD3
C
C
C
Z
cos½li ðz C h1 Þdz
0
Kd1 vr
Ke1 vr
Ke2 vr
Kh1 vr
ð44Þ
!cosðjqÞdq
Substitution of the expressions derived above for 4ðJ;LÞ
and FD into Eqs. (39)–(44) yields
R
( ðJ;LÞ
N
X
Bi h2 R1Kd1;L i Z 1
ðJ;LÞ
AðJ;LÞ
Eðl
;
b
Þ
Z
P
C
(45)
j i
j
1i
BðJ;LÞ
h
=2
iO
1
jZ1
2
i
N
X
(
AðJ;LÞ
Eðlj ; gi Þ
j
Z PðJ;LÞ
2i
C
jZ1
CiðJ;LÞ h3 R1Kd1;L
iZ1
CiðJ;LÞ h3 =2
iO 1
ðJ;LÞ
DðLÞ
AðJ;LÞ
i
li Nðli Þ Z P3i
"
BðJ;LÞ
ð1 K d1;L ÞRKd1;L
1
C
C
N
X
(46)
#
BðJ;LÞ
Db ð1 K d1;L ; jÞ
j
Eðli ; bj Þ
jZ2
"
C
C1ðJ;LÞ ð1 K d1;L ÞRKd1;L
C
N
X
#
CjðJ;LÞ Dg ð1 K d1;L ; jÞ
Eðli ; gj Þ
jZ2
(47)
N
X
(
Am;j Eðlj ; bi Þ Z P4 ðm; iÞ C
jZ1
N
X
(
Am;j Eðlj ; gi Þ Z P5 ðm; iÞ C
jZ1
Bm;i h2 Rm
i Z1
Bm;i h2 =2
iO 1
Cm;i h3 Rm
iZ1
Cm;i h3 =2
iO 1
Am;i Dl ðm; iÞNl ðiÞ Z P6 ðm; iÞ C mR
C
N
X
mK1
sinðli h2 Þ
sinðli h3 Þ
Bm;1
C Cm;1
li
li
(48)
(49)
½Bm;j Db ðm; jÞEðli ; bj Þ C Cm;j Dg ðm; jÞEðli ; gj Þ
jZ2
where iZ1,2,.N; mZ0,1,2,.N and
(50)
Y.H. Zheng et al. / Ocean Engineering 32 (2005) 241–271
Eðlj ; bi Þ Z
ðKd1
251
cos½lj ðz C h1 Þcos½bi ðz C e1 Þdz
(51a)
cos½lj ðz C h1 Þcos½gi ðz C h1 Þdz
(51b)
Ke1
Eðlj ; gi Þ Z
ðKe2
Kh1
Nl ðiÞ Z
ð0
cos2 ½li ðz C h1 Þdz
(52)
Kh1
8 ð1Þ0
kHm ðkRÞ
>
>
< ð1Þ
Hm ðkRÞ
Dl ðm; iÞ Z
0
>
l
>
: i Km ðli RÞ
Km ðli RÞ
iZ1
(53a)
iO 1
Db ðm; jÞ Z
bj Im0 ðbj RÞ
Im ðbj RÞ
(53b)
Dg ðm; jÞ Z
gj Im0 ðgj RÞ
Im ðgj RÞ
(53c)
PðJ;LÞ
Z
1i
ðKd1
4ðJ;LÞ
R2P jrZR cos½bi ðz C e1 Þdz
(54a)
4ðJ;LÞ
R3P jrZR cos½bi ðz C e1 Þdz
(54b)
Ke1
PðJ;LÞ
Z
2i
ðKd1
Ke1
PðJ;LÞ
3i
ð0
Z
dJ;1 ½d2;L
Kd1
ðKe1
C ðz K z0 Þd3;L cos½li ðz C h1 Þdz
dJ;2 ½d2;L C ðz K z0 Þd3;L cos½li ðz C h1 Þdz
ðKd1 ðJ;LÞ ðKe2 ðJ;LÞ
v4R2P v4R3P
C
j cos½li ðz C h1 Þdz
rZR cos½li ðz C h1 Þdz C
vr
vr rZR
Ke1
Kh1
(54c)
C
Ke2
P4 ðm; iÞ Z
igkAmm Jm ðkRÞ ðK1ÞiK1 sinh½kðh1 K d1 Þ K sinh½kðh1 K e1 Þ
u coshðkh1 Þ
k2 C b2i
(55a)
P5 ðm; iÞ Z
igkAmm Jm ðkRÞ ðK1ÞiK1 sinh½kðh1 K e2 Þ
u coshðkh1 Þ
k2 C g2i
(55b)
252
Y.H. Zheng et al. / Ocean Engineering 32 (2005) 241–271
igkAmm Jm0 ðkRÞ
P6 ðm; iÞ Z
u coshðkh1 Þ
ð0
cosh½kðz C h1 Þcos½li ðz C h1 Þdz
(55c)
Kh1
, BðJ;LÞ
, CnðJ;LÞ , Am,n, Bm,n and
To obtain the numerical solutions to the coefficients AðJ;LÞ
n
n
Cm,n, it is necessary to choose the finite terms of the infinite series to carry out the
computation. If the first N and M!N terms of the infinite series are chosen for the radiated
and diffracted problems, respectively, one can obtain three sets and M sets of system of
equations, each set of system of equations consist of 3N equations with complex
coefficients and equal number of unknowns. Making some arrangements for these
equations yields
ðJ;LÞ
SðJ;LÞ
R X R Z FR
ðmÞ
SðmÞ
D X D Z FD
ðL Z 1; 2; 3; J Z 1; 2Þ
ðm Z 0; 1; .; MÞ
(56a)
(56b)
where
ðJ;LÞ
ðJ;LÞ
; AðJ;LÞ
; .; AðJ;LÞ
; BðJ;LÞ
; .; BðJ;LÞ
; C2ðJ;LÞ ; .; CNðJ;LÞ T ;
XR Z ½AðJ;LÞ
N ; B1
N ; C1
1
2
2
XD Z ½Am;1 ; Am;2 ; .; Am;N ; Bm;1 ; Bm;2 ; .; Bm;N ; Cm;1 ; Cm;2 ; .; Cm;N T ;
ðJ;LÞ
SðJ;LÞ
and SðmÞ
and FðmÞ
R
D are the coefficient matrices; FR
D are the right hand vectors of the
ðJ;LÞ
ðJ;LÞ
systems of equations. The elements in SR and FR can be computed from Eqs. (45)–(47),
ðmÞ
while SðmÞ
D and FD can be calculated by Eqs. (48)–(50), and their expressions are given in
Appendices A and B for convenience to readers, respectively.
The application of the solution method for the linear system of equations to Eq. (56)
will give the results to the unknowns in the expressions of the radiated and diffracted
potentials. So the radiated and diffracted potentials at any position in the fluid can be
computed, and the wave forces, added masses and damping coefficients can be calculated
from the linearized Bernoulli’s equation, which will be illustrated in Section 4.
4. Wave forces and hydrodynamic coefficients
4.1. Expressions for wave excitation forces
The wave excitation forces are computed from the incident and diffracted wave
potentials. The relation of the dynamic fluid pressure with the velocity potentials is
obtained from the Bernoulli’s equation first, and then the integration of the dynamic
pressure along the submerged surfaces of the cylinders is carried out subsequently, which
will yield the expression for the wave excitation force of cylinder J in direction L as follow
ðJ;LÞ
ðJ;LÞ Kiut
FWt
Z FW
e
(57)
Y.H. Zheng et al. / Ocean Engineering 32 (2005) 241–271
253
ðJ;LÞ
where FW
is the time-independent wave force which is computed by
ð
ðJ;LÞ
FW Z iruðFI C FD ÞnL ds
(58)
SJ
where SJ(JZ1,2) is the submerged surface of cylinder J; nL(LZ1,2,3) is the generalized
outward normal from the fluid with n1Znz, n2Znx, n3Z(zKz0)nxKxnz; nx and nz are the
components of the unit inward normal to the cylinders. For convenience to readers, the
ðJ;LÞ
expressions for FW
obtained from Eq. (58) are given in Appendix C.
It should be noted that the wave excitation force can also be calculated from the
incident and radiated wave potentials. The application of the Green’s second identity and
considering the boundary conditions for the radiated and diffracted potentials yield the
following expression
2
ð
6
ðJ;LÞ
FW
Z r iu4 FI
v4ðJ;LÞ
R
vn
SJ
ð
ds K 4ðJ;LÞ
R
ð
3
vFI
vFI 7
ds K 4ðJ;LÞ
ds5
R
vn
vn
S1
(59)
S2
ð1;LÞ
and the expressions for FW
obtained from Eq. (59) are given in Appendix D.
4.2. Expressions for hydrodynamic coefficients
The radiation forces can be computed from the radiated potentials due to the motions of
the cylinders. For cylinder I in K direction, it is expressed by
ðX
3 X
2
FRðI;KÞ Z r iu
SI
Z eKiut
FðJ;LÞ
eKiut nK ds
R
LZ1 JZ1
3 X
2
X
ðJ;LÞ
ðJ;LÞ
ðJ;LÞ
½u2 AðJ;LÞ
R CaðI;KÞ C iuAR CdðI;KÞ (60)
LZ1 JZ1
ðJ;LÞ
where CaðJ;LÞ
ðI;KÞ and CdðI;KÞ represent the added mass and damping coefficient of cylinder I in
direction K due to the motion mode L of cylinder J, respectively. The expressions for them
are
ð
ðJ;LÞ
ðJ;LÞ
CaðI;KÞ Z r Re½4ðJ;LÞ
(61)
R cos½ð1 K d1;L ÞqnK ds Z Re½rfðI;KÞ SI
ð
ðJ;LÞ
CdðJ;LÞ
Z
ru
Im½4ðJ;LÞ
R cos½ð1 K d1;L ÞqnK ds Z Im½rufðI;KÞ ðI;KÞ
SI
(62)
254
Y.H. Zheng et al. / Ocean Engineering 32 (2005) 241–271
where
ðJ;LÞ
fðI;KÞ
ð
Z 4ðJ;LÞ
cos½ð1 K d1;L ÞqnK ds
R
(63)
SI
which can be expressed in a matrix form as
2 ð1;1Þ ð1;2Þ ð1;3Þ ð2;1Þ ð2;2Þ
fð1;1Þ fð1;1Þ fð1;1Þ fð1;1Þ fð1;1Þ
6 ð1;1Þ ð1;2Þ ð1;3Þ ð2;1Þ ð2;2Þ
6f
6 ð1;2Þ fð1;2Þ fð1;2Þ fð1;2Þ fð1;2Þ
6
6 f ð1;1Þ f ð1;2Þ f ð1;3Þ f ð2;1Þ f ð2;2Þ
6 ð1;3Þ ð1;3Þ ð1;3Þ ð1;3Þ ð1;3Þ
f Z6
6 ð1;1Þ ð1;2Þ ð1;3Þ ð2;1Þ ð2;2Þ
6 fð2;1Þ fð2;1Þ fð2;1Þ fð2;1Þ fð2;1Þ
6
6 ð1;1Þ ð1;2Þ ð1;3Þ ð2;1Þ ð2;2Þ
6 fð2;2Þ fð2;2Þ fð2;2Þ fð2;2Þ fð2;2Þ
4
2
ð1;1Þ
fð2;3Þ
ð1;2Þ
fð2;3Þ
ð1;3Þ
fð2;3Þ
ð2;1Þ
fð2;3Þ
ð2;2Þ
fð2;3Þ
ð1;1Þ
fð1;1Þ
0
0
ð2;1Þ
fð1;1Þ
0
ð1;2Þ
fð1;2Þ
ð1;3Þ
fð1;2Þ
0
ð2;2Þ
fð1;2Þ
ð1;2Þ
fð1;3Þ
ð1;3Þ
fð1;3Þ
0
ð2;2Þ
fð1;3Þ
0
0
ð2;1Þ
fð2;1Þ
0
ð1;2Þ
fð2;2Þ
ð1;3Þ
fð2;2Þ
0
ð2;2Þ
fð2;2Þ
ð1;2Þ
fð2;3Þ
ð1;3Þ
fð2;3Þ
0
ð2;2Þ
fð2;3Þ
6
6 0
6
6
6 0
6
Z6
6 ð1;1Þ
6 fð2;1Þ
6
6
6 0
4
0
ð2;3Þ
fð1;1Þ
3
7
ð2;3Þ 7
fð1;2Þ
7
7
ð2;3Þ 7
fð1;3Þ 7
7
ð2;3Þ 7
7
fð2;1Þ
7
ð2;3Þ 7
fð2;2Þ 7
5
ð2;3Þ
fð2;3Þ
0
3
7
ð2;3Þ 7
fð1;2Þ
7
7
ð2;3Þ 7
fð1;3Þ 7
7
7
0 7
7
ð2;3Þ 7
7
fð2;2Þ
5
(64)
ð2;3Þ
fð2;3Þ
As we know, f is a symmetrical matrix, so are the matrices for the added masses and
damping coefficients, which can be used to verify indirectly the correctness of the
expressions for the radiated potentials. The expressions for the nonzero elements in f are
given in Appendix E for convenience to readers.
5. Results and discussions
5.1. Verification of the expressions for radiated and diffracted potentials
As stated in Section 1, the method used here is the same as that in Berggren and
Johansson (1992) where the method and the obtained expressions for the radiated
potentials of heave motion in each subregion were verified. The expressions obtained
here for the heave motion are the same as those in Berggren and Johansson (1992)
which can be used to verify indirectly the correctness of the diffracted potentials
because the wave forces including the vertical force, horizontal force and roll torque
can be obtained not only by the incident and diffracted potentials using Eq. (58), but
also by the incident and radiated potentials through Eq. (59). If the two results
obtained by Eqs. (58) and (59) for the vertical force are the same, the correctness of
Y.H. Zheng et al. / Ocean Engineering 32 (2005) 241–271
255
Table 1
Geometrical parameters for the calculation
Case no.
d1/h1
R/h1
e1/h1
e2/h1
1
2
3
4
5
6
0.1
0.1
0.1
0.1
0.1
0.1
0.2
0.2
0.2
0.2
0.5
1.0
0.25
0.4
0.7
0.4
0.4
0.4
0.35
0.5
0.8
0.5
0.5
0.5
Fig. 2. Dimensionless vertical forces on cylinders 1 and 2 of cases 1, 2 and 3. (a) Vertical force on cylinder 1 of
case 1. (b) Vertical force on cylinder 2 of case 1. (c) Vertical force on cylinder 1 of case 2. (d) Vertical force on
cylinder 2 of case 2. (e) Vertical force on cylinder 1 of case 3. (f) Vertical force on cylinder 2 of case 3.
256
Y.H. Zheng et al. / Ocean Engineering 32 (2005) 241–271
Fig. 3. Dimensionless horizontal forces on cylinders 1 and 2 of cases 1, 2 and 3. (a) Horizontal force on
cylinder 1 of case 1. (b) Horizontal force on cylinder 2 of case 1. (c) Horizontal force on cylinder 1 of case 2.
(d) Horizontal force on cylinder 2 of case 2. (e) Horizontal force on cylinder 1 of case 3. (f) Horizontal force on
cylinder 2 of case 3.
the diffracted potential is confirmed. After this, we can use the diffracted potential to
verify the correctness of the expressions for the radiated potentials of sway and roll
motions. In addition, the symmetry of the matrices for the added masses and damping
coefficients can also be used to verify indirectly the correctness of the expressions for
the radiated potentials of sway and roll motions, which will be illustrated in the
following.
Y.H. Zheng et al. / Ocean Engineering 32 (2005) 241–271
257
Fig. 4. Dimensionless torques on cylinders 1 and 2 of cases 1, 2 and 3 with z0Z0. (a) Torque on cylinder 1 of
case 1. (b) Torque on cylinder 2 of case 1. (c) Torque on cylinder 1 of case 2. (d) Torque on cylinder 2 of case 2.
(e) Torque of cylinder 1 of case 3. (f) Torque on cylinder 2 of case 3.
The geometrical parameters, shown in Table 1 of cases 1–3, are taken from Berggren
and Johansson (1992) where they were used for the calculation of the hydrodynamic
coefficients of heave motion. Here we use them to calculate wave forces and the
hydrodynamic coefficients of sway and roll motions which were not considered in
Berggren and Johansson (1992) and to verify the expressions for the radiated and
diffracted potentials.
258
Y.H. Zheng et al. / Ocean Engineering 32 (2005) 241–271
Fig. 5. Asymmetrical quantities of dimensionless added masses and damping coefficients. (a) Dimensionless
heave added mass. (b) Dimensionless heave damping. (c) Dimensionless surge added mass. (d) Dimensionless
surge damping. (e) Dimensionless pitch added mass. (f) Dimensionless pitch damping.
In our computations the first 30 terms and 30!30 terms in the infinite series of
the radiated and diffracted potentials are taken, respectively. The wave forces and
the hydrodynamic coefficients shown in all figures of the following sections are
nondimensionlized as follows (JZ1,2; LZ1,2,3)
(
F
ðJ;LÞ
Z
ðJ;LÞ
j=ðrgpR2 AÞ L Z 1; 2
jFW
ðJ;LÞ
j=ðrgpR3 AÞ L Z 3
jFW
Y.H. Zheng et al. / Ocean Engineering 32 (2005) 241–271
259
Fig. 6. Asymmetrical quantities of dimensionless added masses and damping coefficients. (a) Dimensionless
heave added mass. (b) Dimensionless heave damping. (c) Dimensionless surge added mass. (d) Dimensionless
surge damping. (e) Dimensionless pitch added mass. (f) Dimensionless pitch damping.
Ca1ðJ;LÞ
ðI;KÞ Z
8 ðJ;LÞ
>
CaðI;KÞ =ðrpR2 d1 Þ L Z 1; 2; K Z 1; 2
>
>
>
>
>
3
< CaðJ;LÞ
ðI;KÞ =ðrpR d1 Þ L Z 1; 2; K Z 3
3
>
>
CaðJ;LÞ
>
ðI;KÞ =ðrpR d1 Þ L Z 3; K Z 1; 2
>
>
>
: ðJ;LÞ
CaðI;KÞ =ðrpR4 d1 Þ L Z 3; K Z 3
260
Y.H. Zheng et al. / Ocean Engineering 32 (2005) 241–271
Fig. 7. Symmetrical quantities of dimensionless added masses and damping coefficients of case 1.
Cd1ðJ;LÞ
ðI;KÞ Z
8 ðJ;LÞ
>
CdðI;KÞ =ðrpR2 d1 uÞ
>
>
>
>
>
3
< CdðJ;LÞ
ðI;KÞ =ðrpR d1 uÞ
>
>
CdðJ;LÞ =ðrpR3 d1 uÞ
>
> ðI;KÞ
>
>
:
4
CdðJ;LÞ
ðI;KÞ =ðrpR d1 uÞ
L Z 1; 2; K Z 1; 2
L Z 1; 2; K Z 3
L Z 3; K Z 1; 2
L Z 3; K Z 3
Fig. 2(a) and (b) shows the vertical forces on cylinder 1 and 2 of case 1,
respectively. Fig. 2(c) and (d) gives the vertical forces of case 2, and the vertical
Y.H. Zheng et al. / Ocean Engineering 32 (2005) 241–271
261
Fig. 8. Symmetrical quantities of dimensionless added masses and damping coefficients of case 1.
forces of case 3 are illustrated in Fig. 2(e) and (f) where the solid lines and circles
represent the results calculated by use of Eqs. (58) and (59), respectively. It can be
seen from these figures that for all cases the results given by Eq. (58) are the same
as those calculated by use of Eq. (59), which illustrates that the expressions for
the diffracted potentials are correct due to the correctness of the expressions for the
heave motion.
Figs. 3 and 4 present the horizontal forces and the roll torques computed by Eqs.
(58) and (59), respectively. We can see the two results obtained by Eqs. (58) and (59)
are the same, which indicates that the expressions for the radiated potentials are
262
Y.H. Zheng et al. / Ocean Engineering 32 (2005) 241–271
Fig. 9. Hydrodynamic effects of the cylinders’ radii on excitation forces. (a) Vertical force on cylinder 1.
(b) Vertical force on cylinder 2. (c) Horizontal force on cylinder 1. (d) Horizontal force on cylinder 2. (e) Pitch
torque on cylinder 1. (f) Pitch torque on cylinder 2.
correct. To further verify the correctness of the expressions for the radiated potentials,
we compute the added masses and damping coefficients shown in Figs. 5–8. The
results of the diagonal elements in the matrices for the added masses and damping
coefficients are presented in Figs. 5 and 6, and the values of the non-diagonal
symmetrical elements are given in Figs. 7 and 8. It can be seen from Figs. 7 and 8
that the values of the symmetrical elements are all the same which indicates further
that the expressions for the radiated potentials are correct.
Y.H. Zheng et al. / Ocean Engineering 32 (2005) 241–271
263
Fig. 10. Hydrodynamic effects of the cylinders’ radii on added masses and damping coefficients.
(a) Dimensionless heave added mass. (b) Dimensionless heave damping. (c) Dimensionless surge added mass.
(d) Dimensionless surge damping. (e) Dimensionless pitch added mass. (f) Dimensionless pitch damping.
5.2. Hydrodynamic effects of the cylinders’ radii
In Section 5.1, we not only verified the correctness of the expressions given in
this paper, but also considered the hydrodynamic effects of the submerged positions
of cylinder 2, while in this section the hydrodynamic effects of the radius on the wave
force, added masses, damping coefficients will be considered. The geometrical parameters
are listed in Table 1 of cases 4–6.
264
Y.H. Zheng et al. / Ocean Engineering 32 (2005) 241–271
Fig. 11. Hydrodynamic effects of the cylinders’ radii on added masses and damping coefficients.
(a) Dimensionless heave added mass. (b) Dimensionless heave damping. (c) Dimensionless surge added mass.
(d) Dimensionless surge damping. (e) Dimensionless pitch added mass. (f) Dimensionless pitch damping.
The hydrodynamic effects of the cylinders’ radii on the dimensionless excitation forces
are shown in Fig. 9. For fixed d1/h1, e1/h1 and e2/h1, the lager the radii of the cylinders, the
smaller the dimensionless vertical and horizontal wave forces and the larger the
dimensionless roll torque on cylinder 1 and the smaller the dimensionless roll torque on
cylinder 2.
Figs. 10 and 11 illustrate the hydrodynamic effects of the cylinders’ radii on the
dimensionless added mass and damping coefficients. For fixed d1/h1, e1/h1 and e2/h1,
ð1;1Þ
ð1;3Þ
ð1;3Þ
ð2;1Þ
ð2;1Þ
ð2;3Þ
ð2;3Þ
ð2;2Þ
Ca1ð1;1Þ
ð1;1Þ , Cd1ð1;1Þ , Ca1ð1;3Þ , Cd1ð1;3Þ , Ca1ð2;1Þ , Cd1ð2;1Þ , Ca1ð2;3Þ , Cd1ð2;3Þ , and Cd1ð2;2Þ
Y.H. Zheng et al. / Ocean Engineering 32 (2005) 241–271
265
ð1;2Þ
ð2;2Þ
increase with the augment of the cylinders’ radii, while Ca1ð1;2Þ
ð1;2Þ , Cd1ð1;2Þ and Ca1ð2;2Þ
decrease.
In a sum, the rule of the influence of the cylinders’ radii on the hydrodynamic properties
is relatively simple and deserves attention of designers.
6. Conclusions
In order to analyze the dynamics of a system in waves, it is necessary to study the
hydrodynamic coefficients and wave forces so as to obtain some important information for
designers. Here the method of separation of variables and the method of matched
eigenfunction expansion are used to obtain the analytical expressions for the radiated and
diffracted potentials. By using the analytical expressions, the hydrodynamic coefficients
and wave forces are calculated for some specific examples, and the hydrodynamic effects
of the radius on the hydrodynamic properties of the cylinders are investigated, which may
provide some useful information for designers.
Acknowledgements
This research is supported by the Chinese Academy of Sciences Foundation under
Grant No. KGCX2-SW-305, the High Tech Research and Development (863) Program of
China under Grant No. 2003AA516010, Chinese National Science Fund for Distinguished
Young Scholars under Grant No. 50125924 and the National Natural Science Foundation
of China under grant Nos. 50379001 and 10332050.
ðJ;LÞ
ðJ;LÞ
Appendix A. Expressions for the elements in SR
and FR
SðJ;LÞ
Ri;j Z Eðlj ; bi Þ
(
SðJ;LÞ
Ri;NCi
Z
(A1)
Kh2 R1Kd1;L
iZ1
Kh2 =2
iO 1
(A2)
ðJ;LÞ
FRi
Z PðJ;LÞ
1i
(A3)
SðJ;LÞ
RNCi;j Z Eðlj ; gi Þ
(A4)
(
SðJ;LÞ
RNCi;2NCi
Z
ðJ;LÞ
FRNCi
Z PðJ;LÞ
2i
Kh3 R1Kd1;L
iZ1
Kh3 =2
iO 1
(A5)
(A6)
266
Y.H. Zheng et al. / Ocean Engineering 32 (2005) 241–271
SðJ;LÞ
R2NCi;i Z Dl ð1 K d1;L ; jÞNðli Þ
(
SðJ;LÞ
R2NCi;NCj
Z
Kð1 K d1;L ÞRKd1;L Eðli ; bj Þ j Z 1
KDb ð1 K d1;L ; jÞEðli ; bj Þ
(
SðJ;LÞ
R2Nþi;2Nþj
(A7)
¼
(A8)
jO 1
Kð1 K d1;L ÞRKd1;L Eðli ; gj Þ j ¼ 1
KDg ð1 K d1;L ; jÞ Eðli ; gj Þ
(A9)
jO 1
ðJ;LÞ
Z PðJ;LÞ
FR2NCi
3i
(A10)
where iZ1,2,.,N; jZ1,2,.,N; LZ1,2,3; JZ1,2. All the other elements in SðJ;LÞ
are zero.
R
ðmÞ
Appendix B. Expressions for the elements in SD
and FðmÞ
D
SðmÞ
Di;j Z Eðlj ; bi Þ
(
SðmÞ
Di;NCi
Z
(B1)
Kh2 Rm
iZ1
Kh2 =2
iO 1
(B2)
ðmÞ
FDi
Z P4 ðm; iÞ
(B3)
SðmÞ
DNCi;j Z Eðlj ; gi Þ
(B4)
(
SðmÞ
DNCi;2NCi
Z
Kh3 Rm
iZ1
Kh3 =2
iO 1
(B5)
ðmÞ
FDNCi
Z P5 ðm; iÞ
SðmÞ
D2NCi;i Z
(B6)
li Rm0 ðli RÞNl ðiÞ
Rm ðli RÞ
SðmÞ
D2NCi;NCj Z
8
<
(B7)
sin½li ðh1 K d1 Þ K sin½li ðh1 K e1 Þ
li
: KD ðm; jÞEðl ; b Þ
b
i j
SðmÞ
D2NCi;2NCj Z
KmRmK1
8
<
sinðli h3 Þ
l
: KD ðm; jÞEðl i; g Þ
g
i
j
KmRmK1
j Z1
(B8)
jO 1
j Z1
(B9)
jO 1
ðmÞ
FD2NCi
Z P6 ðm; iÞ
where iZ1,2,.,N; jZ1,2,.,N; mZ0,1,2,.,M. All the other elements in
(B10)
SðmÞ
D
are zero.
Y.H. Zheng et al. / Ocean Engineering 32 (2005) 241–271
ðJ;LÞ
Appendix C. Expressions for FW
obtained from Eq. (58)
"
#
N
B0;1 R2 X
RI1 ðbn RÞ
ð1;1Þ
FW Z 2pr iu
C
B0;n cosðbn h2 Þ
bn I0 ðbn RÞ
2
nZ2
"
N
ðC0;1 K B0;1 ÞR2 X
ð2;1Þ
C
Z 2pr iu
W0n
FW
2
nZ2
ð1;2Þ
ZKr ipuR
FW
ð0
N
X
267
(C1)
#
A1;n cos½ln ðz Ch1 Þdz CCF2
Kd1 nZ1
(C2)
sinhðkh1 ÞKsinh½kðh1 Kd1 Þ
k
(C3)
ð2;2Þ
FW
ZKr ipuR
ðKe1 X
N
A1;n cos½ln ðzCh1 ÞdzCCF2
Ke2 nZ1
sinh½kðh1 Ke1 Þ Ksinhðkh3 Þ
k
(C4)
ð1;3Þ
FW
ZKpr iuR
N
X
ð0
A1;n
ðzKz0 Þcos½ln ðzCh1 Þdz
Kd1
nZ1
"
#
N
B1;1 R4 X
B1;n R2 cosðbn h2 Þ I2 ðbn RÞ
C
Kpr iu
CCF2 CTk1
I1 ðbn RÞ
bn
4
nZ2
ð2;3Þ
ZKpr iuR
FWD
N
X
ð Ke1
A1;n
nZ1
(C5)
ðz Kz0 Þcos½ln ðz Ch1 Þdz
Ke2
(
)
N
ðC1;1 KB1;1 ÞR4 X
C
W1n CCF2 CTk2
Kpr iu
4
nZ2
(C6)
where
CF2 ZK
2 iprgARJ1 ðkRÞ
coshðkh1 Þ
CTk1 Z
Kz0 sinhðkh1 ÞCðd1 Cz0 Þsinh½kðh1 Kd1 Þ coshðkh1 ÞKcosh½kðh1 Kd1 Þ
K
k
k2
CTk2 Z
ðe2 Cz0 Þsinh½kðh1 Ke2 Þ Kðe1 Cz0 Þsinh½kðh1 Ke1 Þ
k
C
cosh½kðh1 Ke2 Þ Kcosh½kðh1 Ke1 Þ
k2
268
Y.H. Zheng et al. / Ocean Engineering 32 (2005) 241–271
W0n ZC0;n cosðgn h3 Þ
RI1 ðgn RÞ
RI ðb RÞ
KB0;n 1 n
gn I0 ðgn RÞ
bn I0 ðbn RÞ
W1n ZC1;n cosðgn h3 Þ
R2 I2 ðgn RÞ
R2 I2 ðbn RÞ
KB1;n
gn I1 ðgn RÞ
bn I1 ðbn RÞ
Appendix D. Expressions for Fð1;LÞ
obtained from Eq. (59)
W
ð1;1Þ
FW
Z
CF1 RJ1 ðkRÞcosh½kðh1 Kd1 Þ
k2
)
ðR ( 2
N
h2 Kr 2 =2 X
I0 ðbn rÞ
ð1;1Þ
CCB1
C
Bn cosðbn h2 Þ
J ðkrÞr dr
2h2
I0 ðbn RÞ 0
0
nZ1
ðR (
KCB2
0
ð R (X
N
CCB3
CCL1
)
N
Kr 2 =2 X
ð1;1Þ I0 ðbn rÞ
J ðkrÞr dr
C
Bn
2h2
I0 ðbn RÞ 0
nZ1
I ðg rÞ
Cnð1;1Þ cosðgn h3 Þ 0 n
I0 ðgn RÞ
nZ1
0
N
X
Að1;1Þ
n
ð 0
)
J0 ðkrÞr dr
Fðk; ln ;z; h1 ÞdzC
ðKe1
Fðk; ln ; z;h1 Þdz
ðD1Þ
Ke2
Kd1
nZ1
sinhðkh1 ÞKsinh½kðh1 Kd1 Þ
k
)
ðR (
N
X
I1 ðbn rÞ
ð1;2Þ
ð1;2Þ
CiCB1
B1 r C
Bn cosðbn h2 Þ
J ðkrÞr dr
I1 ðbn RÞ 1
0
nZ2
ð1;2Þ
FW
Z CF2
ðR (
KiCB2
0
ðR (
CiCB3
0
N
X
CCL2
nZ1
Bð1;2Þ
1 rC
C1ð1;2Þ r C
Að1;2Þ
n
ð 0
Kd1
N
X
I1 ðbn rÞ
Bð1;2Þ
n
I
1 ðbn RÞ
nZ2
)
J1 ðkrÞr dr
N
X
I ðg rÞ
Cnð1;2Þ cosðgn h3 Þ 1 n
I
1 ðgn RÞ
nZ2
Fðk;ln ; z;h1 Þdz C
ðKe1
Ke2
)
J1 ðkrÞr dr
Fðk;ln ; z; h1 Þdz
ðD2Þ
Y.H. Zheng et al. / Ocean Engineering 32 (2005) 241–271
ð1;3Þ
FW
ZKiCF1 cosh½kðh1 Kd1 Þ
269
R2 J2 ðkRÞ
CCF2 CTk1
k
)
ðR (
N
X
h22 r Kr 3 =4
I1 ðbn rÞ
ð1;3Þ
ð1;3Þ
CiCB1
K
CB1 r C
Bn cosðbn h2 Þ
J ðkrÞr dr
2h2
I1 ðbn RÞ 1
0
nZ2
ðR (
KiCB2
0
ðR (
CiCB3
CCL2
0
N
X
)
N
X
r3
ð1;3Þ
ð1;3Þ I1 ðbn rÞ
J ðkrÞr dr
CB1 r C
Bn
I1 ðbn RÞ 1
8h2
nZ2
C1ð1;3Þ r C
Að1;3Þ
n
nZ1
ð 0
N
X
I ðg rÞ
Cnð1;3Þ cosðgn h3 Þ 1 n
I
1 ðgn RÞ
nZ2
Fðk;ln ; z; h1 Þdz C
ð Ke1
)
J1 ðkrÞr dr
Fðk; ln ;z; h1 Þdz
ðD2Þ
Ke2
Kd1
where CB1 ZKCF1 sinh½kðh1 Kd1 Þ, CB2 ZKCF1 sinh½kðh1 Ke1 Þ, CB3 ZKCF1 sinhðkh3 Þ,
CL1 ZKCF1 RJ1 ðkRÞ, CL2 ZiCF1 R½J0 ðkRÞKJ2 ðkRÞ=2, CF1 Z2prgkA=coshðkh1 Þ,
Fðk; ln ;z; h1 ÞZcosh½kðzCh1 Þcos½ln ðzCh1 Þ.
Appendix E. Expressions for the nonzero elements in f
"
#
N
2
X
h 2 R2
R4
Bð1;1Þ
R
RI
ðb
RÞ
1
n
ð1;1Þ
ð1;1Þ
K
C
fð1;1Þ Z 2p
Bn cosðbn h2 Þ
C 1
bn I0 ðbn RÞ
4
16h2
2
nZ2
"
ð2;1Þ
fð1;1Þ
N
R4
Bð2;1Þ R2 X
RI ðb RÞ
C
Z 2p
C 1
Bð2;1Þ
cosðbn h2 Þ 1 n
n
b
16h2
2
n I0 ðbn RÞ
nZ2
(E1)
#
(E2)
(
ð1;1Þ
fð2;1Þ
N
2
X
R4
½C ð1;1Þ K Bð1;1Þ
RI ðg RÞ
1 R
C
C 1
Cnð1;1Þ cosðgn h3 Þ 1 n
gn I0 ðgn RÞ
16h2
2
nZ2
)
N
X
RI1 ðbn RÞ
K
Bð1;1Þ
n
b
n I0 ðbn RÞ
nZ2
Z 2p
(
ð2;1Þ
fð2;1Þ
Z 2p
ðh2 C h3 ÞR2 R4
K
4
16
2
1
1
½C ð2;1Þ K Bð2;1Þ
1 R
C
C 1
h2 h3
2
N
X
RI ðg RÞ
RI1 ðbn RÞ
C
Cnð2;1Þ cosðgn h3 Þ 1 n K
Bð2;1Þ
n
gn I0 ðgn RÞ nZ2
bn I0 ðbn RÞ
nZ2
N
X
ðj;lÞ
fð1;2Þ
Z KRp
N
X
nZ1
Aðj;lÞ
n
(E3)
sinðln h1 Þ K sin½ln ðh1 K d1 Þ
ln
)
(E4)
(E5)
270
Y.H. Zheng et al. / Ocean Engineering 32 (2005) 241–271
N
X
ðj;lÞ
fð2;2Þ
Z KRp
Anðj;lÞ
nZ1
N
X
ðj;lÞ
Z KpR
fð1;3Þ
nZ1
Anðj;lÞ
sin½ln ðh1 K e1 Þ K sin½ln ðh1 K e2 Þ
ln
ð0
(E6)
cos½ln ðz C h1 Þðz K z0 Þdz
Kd1
"
#
N
B1ðj;lÞ R4 X
R2 I2 ðbn RÞ
ðj;lÞ
ðj;lÞ
C
Bn cosðbn h2 Þ
Kp
K pfPð1;3Þ
b
4
I
ðb
RÞ
n
1
n
nZ2
(E7)
(
)
N 4
2
2
X
½C1ðj;lÞ KBðj;lÞ
R
R
I
ðg
RÞ
R
I
ðb
RÞ
2
n
2
n
1
C
KBðj;lÞ
Cnðj;lÞ cosðgn h3 Þ
n
gn I1 ðgn RÞ
bn I1 ðbn RÞ
4
nZ2
ð
N
Ke1
X
ðj;lÞ
Aðj;lÞ
cos½ln ðzCh1 ÞðzKz0 ÞdzKpfPð2;3Þ
(E8)
KpR
n
ðj;lÞ
ZKp
fð2;3Þ
nZ1
Ke2
ð1;2Þ
ð2;2Þ
ð1;2Þ
ð2;2Þ
ZfPð1;3Þ
ZfPð2;3Þ
ZfPð2;3Þ
Z0
fPð1;3Þ
where jZ1,2; lZ2,3.
8 4 2
R
R
>
>
K h2
<
8 6h2
ðj;3Þ
fPð1;3Þ
Z
6
>
R
>
:K
48h2
j Z1
j Z2
8
R6
>
>
<K
48h2
ðj;3Þ
fPð2;3Þ
Z
6
>
R
1
1
R4 ðh2 C h3 Þ
>
:
C
K
8
48 h2 h3
j Z1
j Z2
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