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On diffraction and radiation problem for a cylinder over a caisson in water of finite depth
International Journal of Engineering Science 42 (2004) 1193–1213 www.elsevier.com/locate/ijengsci
On diffraction and radiation problem for a cylinder over a caisson in water of finite depth Bi-jun Wu a
a,b,*
, Yong-hong Zheng a, Ya-ge You a, Xiao-yan Sun a, Yong Chen
a
Guangzhou Institute of Energy Conversion, Chinese Academy of Sciences, Guangzhou 510070, China b University of Science and Technology of China, Hefei 230026, China Received 10 September 2003; accepted 3 December 2003
Available online
Abstract A caisson and a vertical circular cylinder floating on the surface of fluid compose a wave power device. The hydrodynamic fluid/solid interaction becomes more complex due to the caisson. In this paper, in order to investigate the effects of the caisson on the cylinder’s hydrodynamic coefficients and exciting forces, the expressions for velocity potentials are derived in the presence of an incident linear wave by use of an eigenfunction expansion approach. A set of theoretical added mass, damping coefficient and exciting force expressions are obtained. Two approaches are proposed to calculate the exciting forces and they match with each other well. Finally, Analytical results of added masses, damping coefficients and exciting forces are given for the different sizes of the caisson. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Analytical approach; Velocity potential; Wave force
1. Introduction A buoy riding in waves, connected to a caisson fixed on the bottom of the sea by a rigid rope, has been proposed as a device for extracting energy from waves. The extracting energy by the buoy is transferred to a liquid pump in the caisson through the rope. The pump converts the extracting energy to the electrical energy. Obviously, the caisson changes the direction of the
*
Corresponding author. Address: Guangzhou Institute of Energy Conversion, Chinese Academy of Sciences, Guangzhou 510070, China. Tel.: +86-20-87057612; fax: +86-20-87057597. E-mail address: [email protected] (B.-j. Wu). 0020-7225/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2003.12.006
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wave’s ray at or near the surface of the caisson and makes the fluid/solid interaction more complex. The efficiency of the converted wave energy and the estimation of hydrodynamic forces (or moments) for a wave power device has been received considerable attention from the design point of view. An accurate prediction of the wave loads exerted by waves on rigid power devices is absolutely necessary to design the wave power devices. Generally, numerical methods are engaged to deal with the diffraction and radiation problem for the buoy, but the calculating accuracy is low and the CPU time is long and programming is complicated. In order to render the device in a simpler style and present the common properties acting on the buoy by the caisson, both the buoy and the caisson are idealised as vertical cylinders. The analytical approach is used to deal with the diffraction and radiation problem for the device. Assuming suitable coordinates, the motion of the cylinder is simply decomposed three motions: heave, sway and roll. Many theoretical studies have been performed to analyze the wave motion and wave force on a structure. Miles and Gilbert [1] formulated the problem of the scattering of surface wave by a circular dock. Garrett [2] obtained the results for the horizontal and vertical forces and torque on the dock. Yeung [3] formulated a set of theoretical added masses and damping coefficients for a floating circular cylinder in water of finite depth. Williams and Abul-Azm [4] presented the hydrodynamic interactions between the members of an array of floating circular cylinders which occur when one member undergoes prescribed forced oscillations. Black et al. [6] calculated the wave faces on a truncated cylinder, which either protruding from the sea bottom or partially immersing in the free surface. Berggren and Johansson [7] and Eidsmoen [8] presented the heaving radiation problem of a two vertical cylinder system, where one of the cylinders is submerged and the other floats. Mei and Black [10] considered the scattering properties for bottom and surface obstacles of various proportions on two dimensions in water of finite depth. Kanoria [9] investigated the scattering problem of surface water waves by a thick submerged rectangular wall with a gap in finite depth water and then Kanoria [11] presented the scattering problem of surface water waves by a thick vertical slotted barrier of rectangular cross-section with an arbitrary number of slots of unequal lengths along the vertical direction on two dimensions. Bhatta and Rahman [5] presented the analytical solution for the boundary value problem to evaluate the wave loads for a vertical circular cylinder with sway, heave and roll motions in water of finite depth in the presence of an incident wave. Through the above literature review, we found that some of them focus on the floating vertical cylinder diffraction (scattering) or radiation with periodic motion in water of finite depth and the others on the fixed barriers or floating rectangular structures scattering on two dimensions. The radii of cylinders or the lengths of the barriers along the horizontal direction are the same. To the authors’ knowledge, there is no study on the diffraction and radiation problem for a cylinder over a caisson fixed on the bottom on three dimensions in water of finite depth by use of an analytical approach. The radius of the caisson is longer than or equal to that of the cylinder. Obviously, the diffraction and radiation problem of the cylinder over a caisson whose radius is longer than or equal to that of the cylinder is more complicated than that of the cylinder without a caisson in water of finite depth. By use of an eigenfunction expansion approach, the analytical solution for the radiation potentials due to heave, sway, and roll and for the diffracted potential due to the diffraction of an incident wave acting on the fixed cylinder has been presented. A set of theoretical added mass, damping coefficient and exciting force expressions have been proposed.
B.-j. Wu et al. / International Journal of Engineering Science 42 (2004) 1193–1213
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Two approaches have been presented to calculate the exciting forces and they match with each other well. Finally, analytical results of the added masses, the damping coefficients and the exciting forces are given for the different sizes of the caisson.
2. Mathematical model A vertical circular cylinder with radius R is parked in the fluid. The incompressible fluid motion is irrotational and the waves are of the small amplitude. A Cartesian coordinate system Oxyz as well as the cylindrical coordinate system Orhz is defined with the origin in the undisturbed free surface and the Z-axis positive upwards. The cylinder occupies the space defined by r 6 R, 0 6 h 6 2p ,d1 6 z 6 0. Under the cylinder there is a caisson with radius Rb , which is coaxial with the cylinder. The caisson occupies the space defined by r 6 Rb , 0 6 h 6 2p, h1 6 z 6 e1 , as shown as Fig. 1. For an incompressible fluid, and for small amplitude wave theory with irrotational motion, we can introduce a velocity potential Uðr; h; z; tÞ ¼ Re½/ðr; h; zÞ eixt where Re½ denotes the real part of a complex expression and /ðr; h; zÞ is the spatial part of the velocity potential. /ðr; h; zÞ must satisfy the following equation: o2 / 1 o2 / 1 o o/ r þ þ ¼0 oz2 r2 oh2 r or or
ð1Þ
From the linear water wave theory, the velocity potential / can be decomposed into the incident wave potential /i , the diffracted potential /d due to the diffraction of an incident wave acting on the fixed cylinder and the radiated potential /r due to the motion of the cylinder. For the present problem, the incident potential of a linear wave with unit amplitude propagating to the positive x-direction is /i ¼
1 ig cosh kðz þ h1 Þ X l Jm ðkrÞ cos mh; x coshðkh1 Þ m¼0 m
r
ð2Þ
y
θ x
R z
x
d1 I h1
h2
III I
e1
Rb Fig. 1. The sketch of the device and the definition of fluid subdomains.
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where lm ¼
2im 1
m>0 m¼0
pffiffiffiffiffiffiffi i ¼ 1: x is the wave angular frequency, g is the gravitational acceleration, k is the wave number, which is determined by the dispersion relation k tanhðkh1 Þ ¼ x2 =g, h1 is the water depth. In this paper, /d , /r and their derivations are the key parameters to be solved.
3. Radiation velocity potential 3.1. Radiation problem The cylinder is assumed to have motions with three degrees of freedom in the presence of incident wave. The motions are referred as heave, sway and roll, namely three modes. The radiation modes are numbered such that l ¼ 1, 2 and 3 corresponding to heave, sway and roll respectively. The amplitude of the lth mode is denoted by Alr . So the radiated velocity potential /r can be expressed as
/r ¼
3 X
/lr ðr; h; zÞ ¼
l¼1
3 X
fix½Alr ulr ðr; zÞ cosð1 d1l Þhg
ð3Þ
l¼1
where /lr or ulr (l ¼ 1; 2; 3) is the spatial part of the radiated velocity potential, t is the time. Evidently, there is a relation as follows /lr ðr; h; zÞ ¼ ixAlr ulr ðr; zÞ cosð1 d1l Þh
ð4Þ
where, in order to compose the corresponding expressions in compact for the radiated problems, we introduce following sign dkl ¼
1 0
k¼l k 6¼ l
By substituting Eq. (4) into Laplace’s equation, we can obtain o2 ulr 1 o oulr ð1 d1l Þ2 ulr þ ¼0 r r or oz2 or r2
ð5Þ
For the lth mode motion, the radiated potential is also required to satisfy appropriate boundary conditions and the equations for the boundary conditions are obtained as follows in compact for three mode motions
B.-j. Wu et al. / International Journal of Engineering Science 42 (2004) 1193–1213
oulr x2 ulr ¼ 0 oz g
ðz ¼ 0; r P RÞ;
1197
ð6Þ
oulr ¼0 oz
ðz ¼ h1 ; r P Rb ; z ¼ e1 ; r 6 Rb Þ
ð7Þ
oulr ¼0 or
ðh1 < z < e1 ; r ¼ Rb Þ;
ð8Þ
oulr ¼ d1l rd3l oz
ðz ¼ d1 ; r < RÞ;
oulr ¼ d2l þ ðz zc Þd3l ðd1 < z < 0; r ¼ RÞ or pffiffi oulr lim r ikulr ¼ 0 r!/ or
ð9Þ ð10Þ ð11Þ
where ðxc ; yc ; zc Þ ¼ ð0; 0; zc Þ is the coordinate of a point in the rotational axis which is parallel to the y-axis. 3.2. Solution In the solution procedure the fluid domain is divided into three subdomains denoted by I, II and II as indicated in Fig. 1. The radiation velocity potentials of three subdomains are denoted by ulr1 , ulr2 and ulr3 respectively. The approach of separation of variables is applied in each subdomain in order to obtain expressions for unknown function, i.e. the velocity potential. Expressions valid in each subdomain are obtained as infinite series of orthogonal functions. These expressions are developed to satisfy all boundary conditions except at the boundary joining the subdomains, i.e. at r ¼ Rb and r ¼ R. It then remains to determine the unknown coefficients in the series. This is done by imposing the condition of continuity of pressure and normal velocity at r ¼ Rb and r ¼ R. Mathematically this is fulfilled by matching the potentials and the normal derivatives of the potentials, respectively. The application of the approach of separation of variables yields the spatial potentials in each subdomain expressed by orthogonal series. In regions I, II and III, the radiated potentials are expressed respectively for l ¼ 1, 2 or 3 as follows [7] 1. Region I:
ulr1 ¼
/ X n¼1
2. Region II:
Aln
Rð1d1l Þ ðkn rÞ cos½kn ðz þ h1 Þ Rð1d1l Þ ðkn Rb Þ
ð12Þ
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ulr2
/ X Sð1d1l Þ ðcn rÞ Tð1d1l Þ ðcn rÞ ¼ Bln þ Cln cos½cn ðz þ e1 Þ Sð1d1l Þ ðcn RÞ Tð1d1l Þ ðcn RÞ n¼1
ð13Þ
3. Region III: ulr3 ¼ wlp þ Dl1 rð1d1l Þ þ
/ X
Dln
n¼2
Ið1d1l Þ ðbn rÞ cos½bn ðz þ e1 Þ Ið1d1l Þ ðbn RÞ
ð14Þ
in which, 2
wlp ¼
2
ðz þ e1 Þ r2 =2 ðz þ e1 Þ r r3 =4 d1l d3l 2h2 2h2
k1 ¼ ik; c1 ¼ ike ;
k tanhðkh1 Þ ¼ x2 =g ke tanhðke e1 Þ ¼ x2 =g
kn tanðkn h1 Þ ¼ x2 =g cn tanðcn e1 Þ ¼ x2 =g
bn ¼ ðn 1Þp=h2
n¼1
n ¼ 2; 3; . . .
n ¼ 1; 2; 3; . . .
ð15Þ ð16Þ ð17Þ ð18Þ
and the radial function R, S and T are given by ð1Þ
Rð1d1l Þ ðk1 rÞ ¼ Hð1d1l Þ ðkrÞ Rð1d1l Þ ðkn rÞ ¼ Kð1d1l Þ ðkn rÞ ð1Þ
Sð1d1l Þ ðc1 rÞ ¼ Hð1d1l Þ ðke rÞ
n¼1 n ¼ 2; 3; . . .
ð20Þ
n¼1
ð21Þ
Sð1d1l Þ ðcn rÞ ¼ Kð1d1l Þ ðcn rÞ n ¼ 2; 3; . . . ð2Þ
Tð1d1l Þ ðc1 rÞ ¼ Hð1d1l Þ ðke rÞ Tð1d1l Þ ðcn rÞ ¼ Ið1d1l Þ ðcn rÞ ð1Þ
ð19Þ
n¼1 n ¼ 2; 3; . . .
ð22Þ ð23Þ ð24Þ
ð2Þ
where Hð1d1l Þ and Hð1d1l Þ are the first and second kind Hankel function of order ð1 d1l Þ, namely 0 or 1, respectively; Ið1d1l Þ and Kð1d1l Þ are the first and second kind modified Bessel function of order ð1 d1l Þ respectively. The continuity of pressure and normal velocity for the radiated potentials are given as follows at r ¼ Rb ulr1 ¼ ulr2 e1 6 z 6 0 ou lr2 oulr1 e1 < z < 0 or ¼ or 0 h1 < z < e1
ð25Þ ð26Þ
B.-j. Wu et al. / International Journal of Engineering Science 42 (2004) 1193–1213
1199
and at r ¼ R ulr2 ¼ ulr3 e1 6 z 6 d1 oulr2 d þ ðz zc Þd3l d1 < z < 0 ¼ ou2llr3 e1 < z < d1 or or
ð27Þ ð28Þ
The boundary conditions above are satisfied over the z interval in a least square by multiplying both sides of them by a proper set of eigenfunctions in their belonging regions and then by integrating them over the corresponding interval at the boundaries r ¼ Rb and r ¼ R. The procedure above gives the following equations. Eq. (25): Z
0
ulr1 ðRb ; zÞ cos½ci ðz þ e1 Þ dz ¼
e1
Z
0
ulr2 ðRb ; zÞ cos½ci ðz þ e1 Þ dz
ð29Þ
e1
Eq. (26): Z
0
h1
Z 0 oulr1 ðr; zÞ oulr2 ðr; zÞ cos½ki ðz þ h1 Þ dz ¼ cos½ki ðz þ h1 Þ dz or or e1 r¼Rb r¼Rb
ð30Þ
Eq. (27): Z
d1
ulr2 ðR; zÞ cos½bi ðz þ e1 Þ dz ¼
e1
Z
d1
ulr3 ðR; zÞ cos½bi ðz þ e1 Þ dz
ð31Þ
e1
Eq. (28): Z
0
e1
Z 0 oulr2 ðr; zÞ ½d2l þ ðz zc Þd3l cos ci ðz þ e1 Þ dz cos ci ðz þ e1 Þ dz ¼ or d1 r¼R Z d1 oulr3 ðr; zÞ þ cos ci ðz þ e1 Þ dz or e1 r¼R
Introduce the following functions Z z2 Eðan ; bk ; ha ; hb ; z1 ; z2 Þ ¼ cos an ðz þ ha Þ cos bk ðz þ hb Þ dz
ð32Þ
ð33Þ
z1
Nðai ; ha ; z1 ; z2 Þ ¼
Z
z2
cos2 ½ai ðz þ ha Þ dz
ð34Þ
z1
in which fan ; n ¼ 1; 2; . . .g and fbk ; k ¼ 1; 2; . . .g are two different sets of eigenvalues. Applying the functions (33) and (34) to Eqs. (29)–(32), we obtain:
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Eq. (29): / X
Alj Eðkj ; ci ; h1 ; e1 ; e1 ; 0Þ ¼ ðBli Sali þ Cli Tali ÞN ðci ; e1 ; e1 ; 0Þ þ P1li
ð35Þ
j¼1
where Sali ¼ Sð1d1l Þ ðci Rb Þ=Sð1d1l Þ ðci RÞ
ð36Þ
Tali ¼ Tð1d1l Þ ðci Rb Þ=Tð1d1l Þ ðci RÞ
ð37Þ
P1li ¼ 0;
ð38Þ
Eq. (30): Ali Oli N ðki ; h1 ; h1 ; 0Þ ¼
/ X
ðBlj Palj þ Clj Qalj ÞEðcj ; ki ; e1 ; h1 ; e1 ; 0Þ þ P2li
ð39Þ
j¼1
where ( Oli ¼
ð1Þ0
( Palj ¼
ð1Þ0
ð1Þ
ke Hð1d1l Þ ðke Rb Þ=Hð1d1l Þ ðke RÞ 0 ðcj Rb Þ=Kð1d1l Þ ðcj RÞ cj Kð1d 1l Þ
( Qalj ¼
ð1Þ
i¼1 kHð1d1l Þ ðkRb Þ=Hð1d1l Þ ðkRb Þ 0 ðk R Þ=K ðk R Þ i ¼ 2; 3; . . . ki Kð1d i b ð1d1l Þ i b 1l Þ
ð2Þ0
ð2Þ
ke Hð1d1l Þ ðke Rb Þ=Hð1d1l Þ ðke RÞ 0 ðcj Rb Þ=Ið1d1l Þ ðcj RÞ cj Ið1d 1l Þ
ð40Þ
j¼1 j ¼ 2; 3; . . .
ð41Þ
j¼1 j ¼ 2; 3; . . .
ð42Þ
P2li ¼ 0
ð43Þ
Eq. (31): / X
ðBlj þ Clj ÞEðcj ; bi ; e1 ; e1 ; e1 ; d1 Þ ¼ Dli Uli N ðbi ; e1 ; e1 ; d1 Þ þ P3li
ð44Þ
j¼1
where Uli ¼
Rð1d1l Þ 1
i¼1 i ¼ 2; 3; . . .
ð45Þ
B.-j. Wu et al. / International Journal of Engineering Science 42 (2004) 1193–1213
P3li ¼
Z
"
ðz þ e1 Þ2 R2 =2 d1l cos bi ðz þ e1 Þ dz 2h2 e1 # Z d1 " Rðz þ e1 Þ2 R3 =4 þ d3l cos bi ðz þ e1 Þ dz 2h2 e1 d1
1201
#
ð46Þ
Eq. (32): ðBli Pbli þ Cli Qbli ÞN ðci ; e1 ; e1 ; 0Þ ¼
/ X
Dlj Wlj Eðbj ; ci ; e1 ; e1 ; e1 ; d1 Þ þ P4li
ð47Þ
j¼1
where ( Pbli ¼
ð1Þ0
( Qbli ¼ Wlj ¼
ð2Þ0
ð2Þ
ke Hð1d1l Þ ðke RÞ=Hð1d1l Þ ðke RÞ 0 ðci RÞ=Ið1d1l Þ ðci RÞ ci Ið1d 1l Þ
ð1 d1l ÞRd1l 0 ðbj RÞ=Ið1d1l Þðbj RÞ bj Ið1d 1l Þ
i¼1 i ¼ 2; 3; . . .
ð48Þ
i¼1 i ¼ 2; 3; . . .
ð49Þ
j¼1 j ¼ 2; 3; . . .
Z 0 R d1l cos ci ðz þ e1 Þ dz þ ¼ ½d2l þ ðz zc Þd3l cos ci ðz þ e1 Þ dz 2h2 d1 e1 # Z d1 " 2 3R =4 ðz þ e1 Þ2 d3l cos ci ðz þ e1 Þ dz þ 2h2 e1 Z
P4li
ð1Þ
ke Hð1d1l Þ ðke RÞ=Hð1d1l Þ ðke RÞ 0 ci Kð1d ðci RÞ=Kð1d1l Þ ðci RÞ 1l Þ
d1
ð50Þ
ð51Þ
In order to find a solution to coefficients Aln , Bln , Cln , and Dln , we take the first N terms and make some arrangements yield 3 sets of linear system of 4N complex equations and get an equal number of unknown coefficients. Organizing the equations in matrices is as follows Sl Xl ¼ Fl
ð52Þ
where Xl ¼ ½Al1 ; Al2 ; . . . ; AlN ; Bl1 ; Bl2 ; . . . ; BlN ; Cl1 ; Cl2 ; . . . ; ClN ; Dl1 ; Dl2 ; . . . ; DlN T ; Sl is the coefficient matrices and Fl is the right hand vectors. Slði;jÞ of Sl and Fli of Fl are given as follows according to Eqs. (35)–(51) (i; j ¼ 1; 2; . . . ; N ): Eq. (35): Slði;jÞ ¼ Eðkj ; ci ; h1 ; e1 ; e1 ; 0Þ
ð53Þ
Slði;N þiÞ ¼ Sali N ðci ; e1 ; e1 ; 0Þ
ð54Þ
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Slði;2N þiÞ ¼ Tali Nðci ; e1 ; e1 ; 0Þ
ð55Þ
Eq. (38): Fli ¼ P1li
ð56Þ
Eq. (39): SlðN þi;iÞ ¼ Oli Nðki ; h1 ; h1 ; 0Þ
ð57Þ
SlðN þi;N þjÞ ¼ Palj Eðcj ; ki ; e1 ; h1 ; e1 ; 0Þ
ð58Þ
SlðN þi;2N þjÞ ¼ Qblj Eðcj ; ki ; e1 ; h1 ; e1 ; 0Þ
ð59Þ
Eq. (43): FlðN þiÞ ¼ P2li
ð60Þ
Eq. (44): Slð2N þi;N þiÞ ¼ Eðcj ; bi ; e1 ; e1 ; e1 ; d1 Þ
ð61Þ
Slð2N þi;2NþjÞ ¼ Eðcj ; bi ; e1 ; e1 ; e1 ; d1 Þ
ð62Þ
Slð2N þi;3NþiÞ ¼ Uli N ðbi ; e1 ; e1 ; d1 Þ
ð63Þ
Eq. (46): Flð2N þiÞ ¼ P3li
ð64Þ
Eq. (47): Slð3N þi;N þiÞ ¼ Pbli N ðci ; e1 ; e1 ; 0Þ
ð65Þ
Slð3N þi;2NþiÞ ¼ Qbli N ðci ; e1 ; e1 ; 0Þ
ð66Þ
Slð3N þi;3NþjÞ ¼ Wlj Eðbj ; ci ; e1 ; e1 ; e1 ; d1 Þ
ð67Þ
Eq. (51): Flð3N þiÞ ¼ P4li ;
ð68Þ
By substituting the coefficients Xl into Eqs. (12)–(14), respectively, the radiated potentials ulr for l ¼ 1, 2, or 3 at any position in each subdomain can be computed.
B.-j. Wu et al. / International Journal of Engineering Science 42 (2004) 1193–1213
1203
4. Diffracted velocity potential 4.1. The governing equation and the boundary conditions The diffraction problem is due to diffraction in the presence of an incident wave on a fixed structure. The diffracted velocity potential Ud can be expressed as Ud ¼ Re½/d ðr; h; zÞ expðixtÞ, where the spatial part of the diffracted velocity potential /d is governed by the following boundary-value problem: o2 /d 1 o o/d 1 o 2 /d þ ¼0 r þ r or r2 oh2 oz2 or
ð69Þ
o/d x2 /d ¼ 0 oz g
ð70Þ
o/d ¼0 oz
ðz ¼ 0; r P RÞ
ðz ¼ h1 ; r P Rb Þ
oð/d þ /i Þ ¼0 oz
ð71Þ
ðz ¼ d1 ; r < R; z ¼ e1 ; r < Rb Þ
oð/d þ /i Þ ¼ 0 ðd1 < z < 0; r ¼ R; h1 < z < e1 ; r ¼ Rb Þ or pffiffi o/d lim r ik/d ¼ 0 r!/ or
ð72Þ ð73Þ ð74Þ
where /i is the incident velocity potential, i.e. Eq. (2). 4.2. Solution to the problem Similarly, application of the approach of separation of variables gives the diffracted velocity potentials in regions I, II and III, respectively. The diffracted velocity potentials are denoted by /d1 , /d2 and /d3 , respectively. They are as follows: /d1 ¼
/ X / X
Am;n
m¼0 n¼1
/d2
Rm ðkn rÞ cos½kn ðz þ h1 Þ cos mh Rm ðkn Rb Þ
/ X / X Sm ðcn rÞ Tm ðcn rÞ ¼ /i þ Bm;n þ Cm;n cos cn ðz þ e1 Þ cos mh Sm ðcn RÞ Tm ðcn RÞ m¼0 n¼1
/d3 ¼ /i þ
/ X m¼0
( Dm;1 rm þ
/ X n¼2
) Im ðbn rÞ cos½bn ðz þ e1 Þ cos mh Dm;n Im ðbn RÞ
ð75Þ
ð76Þ
ð77Þ
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B.-j. Wu et al. / International Journal of Engineering Science 42 (2004) 1193–1213
where kn , cn and bn are determined by Eqs. (16)–(18), respectively. The radial functions Rm , Sm and Tm are given by Rm ðk1 rÞ ¼ Hmð1Þ ðik1 rÞ ¼ Hmð1Þ ðkrÞ n ¼ 1
ð78Þ
Rm ðkn rÞ ¼ Km ðkn rÞ
ð79Þ
n ¼ 2; 3; . . .
Sm ðc1 rÞ ¼ Hmð1Þ ðic1 rÞ ¼ Hmð1Þ ðke rÞ Sm ðcn rÞ ¼ Km ðcn rÞ
n ¼ 2; 3; . . .
Tm ðc1 rÞ ¼ Hmð2Þ ðic1 rÞ ¼ Hmð2Þ ðke rÞ Tm ðcn rÞ ¼ Im ðcn rÞ
n¼1
ð80Þ ð81Þ
n¼1
n ¼ 2; 3; . . .
ð82Þ ð83Þ
where Hmð1Þ and Hmð2Þ are the first and second kind Hankel function of order m, respectively; Im and Km are the first and second kind modified Bessel function of order m, respectively. At r ¼ Rb , matching conditions are /d1 ¼ /d2 o/d1 ¼ or
(
e1 6 z 6 0 o/d2 or i o/ or
e1 < z < 0 h1 < z < e1
ð84Þ ð85Þ
similarly, at r ¼ R, the continuity conditions become /d2 ¼ /d3 o/d2 ¼ or
(
e1 6 z 6 d1 i o/ or
o/d3 or
d1 < z < 0 e1 < z < d1
ð86Þ ð87Þ
Hence we can get Am;n , Bm;n , Cm;n and Dm;n from Eqs. (84)–(87) by use of the same approach for the radiated unknown coefficients presented in Section 3.2. By substituting the coefficients Am;n , Bm;n , Cm;n and Dm;n into Eqs. (75)–(77), we can get the diffracted potential /d ðr; h; zÞ.
5. Wave forces and hydrodynamic coefficients 5.1. Wave excitation forces Wave excitation forces are the forces by the velocity potentials except radiated velocity potentials. Those potentials are generally composed of the incident velocity potential /i and the diffracted velocity potential /d . Those forces are
B.-j. Wu et al. / International Journal of Engineering Science 42 (2004) 1193–1213
Fj ¼ Fij þ Fdj ¼ iqx
Z
/i nj ds þ iqx S
Z /d nj ds
1205
ð88Þ
S
where S is the wetted surface of the cylinder, nj is the generalized normal with n1 ¼ nz , n2 ¼ nx and j þ nz~ n ¼ nx~i þ ny~ k is the unit normal vector pointing to the cylinder n3 ¼ ðz zc Þnx ðx xc Þnz , ~ at the cylinder surface. The values, x, y, z are the projections of vector ~ r, which is from an origin (0,0,0) to a surface element ds. We have listed the force expressions according to Eq. (88) as follows. The vertical and horizontal excited force for the incident potential are as follows respectively Fi1 ¼ 2pqg Fi2 ¼
cosh½kðh1 d1 Þ R J1 ðkRÞ coshðkh1 Þ k
2pqgiRJ1 ðkRÞ sinhðkh1 Þ sinh½kðh1 d1 Þ coshðkh1 Þ k
ð89Þ ð90Þ
The roll moment for the incident potential is Fi3 ¼ FiXh þ FiXv
ð91Þ
FiXh ¼
FiXv ¼
2pqgiRJ1 ðkRÞ zc sinhðkh1 Þ þ ðd1 þ zc Þ sinh½kðh1 d1 Þ coshðkh1 Þ k coshðkh1 Þ cosh½kðh1 d1 Þ k2 2pqgi cosh½kðh1 d1 Þ R2 J2 ðkRÞ coshðkh1 Þ k
ð92Þ ð93Þ
Above the Eq. (91), the first term is the moment effected by the horizontal force Fi2 and the second term is the moment effected by vertical force Fi1 . According to Eq. (88), we can get the vertical diffraction force acting on the cylinder. That is "
Fd1
# / D0;1 R2 X R I1 ðbn RÞ þ Fi1 ¼ 2piqx D0;n cosðbn h2 Þ bn I0 ðbn RÞ 2 n¼2
ð94Þ
The horizontal diffraction force is Fd2 ¼ ipqxR
/ X n¼1
ðB1;n þ C1;n Þ
sinðcn e1 Þ sinðcn h2 Þ Fi2 cn
ð95Þ
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The roll diffraction moment is Z 0 X / Fd3 ¼ piqxR ðB1;n þ C1;n Þ cos½cn ðz þ e1 Þðz zc Þ dz "
d1 n¼1
# / D1;1 R4 X D1;n cosðbn h2 Þ R2 I2 ðbn RÞ þ Fi3 pqix I1 ðbn RÞ bn 4 n¼2
ð96Þ
The diffraction force or moment can also be calculated from the incident potential /i and the radiated potential /r by use of Haskind relation. It is given by Z Z o/ Fdj ¼ iqx /d nj ds ¼ iqx /lr i ds on S S0 ! Z 2p Z R Z 2p Z 0 o/i o/i /lr2 R dz dh /lr3 r dr dh ¼ iqx or r¼R oz z¼d1 0 d1 0 0 ! Z 2p Z R Z 2p Z Rb o/i o/i /lr2 r dr dh þ /lr3 r dr dh þ iqx oz z¼e1 oz z¼e1 R 0 0 0 Z 2p Z e1 o/i /lr1 Rb dz dh ð97Þ þ iqx or r¼Rb h1 0 where S0 is the wetted surface of the cylinder and the caisson. According to the Haskind relation, the expressions of the diffraction forces are very complicated. We only list the expression for the vertical diffraction force as follows. ! Z 2p Z 0 Z 2p Z R o/i o/i u1r2 R dz dh u1r3 r dr dh Fd1 ¼ iqx or r¼R oz z¼d1 0 d1 0 0 ! Z 2p Z R Z 2p Z Rb o/i o/i u1r2 r dr dh þ u1r3 r dr dh þ iqx oz z¼e1 oz z¼e1 R 0 0 0 Z 2p Z e1 o/i u1r1 Rb dz dh þ iqx or r¼Rb 0 h1 # Z R" 2 / X h2 r2 =2 I0 ðbn rÞ J0 ðkrÞr dr þ D11 þ D1n cos ðbn h2 Þ ¼ C1a 2h2 I0 ðbn RÞ 0 n¼2 Z Rb X / S0 ðcn rÞ T0 ðcn rÞ B1n þ C1n J0 ðkrÞr dr þ C1b S0 ðcn RÞ T0 ð cn RÞ R n¼1 # Z R" / X r2 =2 I0 ðbn rÞ J0 ðkrÞr dr þ C1b þ D11 þ D1n 2h2 I0 ð bn RÞ 0 n¼2 Z 0 / X þ C2a ðB1n þ C1n Þ cosh ½k ðz þ h1 Þ cos ½cn ð z þ e1 Þ dz n¼1
þ C2b
/ X n¼1
Z
d1 e1
cosh ½k ð z þ h1 Þ cos ½kn ðz þ h1 Þ dz
An h1
ð98Þ
B.-j. Wu et al. / International Journal of Engineering Science 42 (2004) 1193–1213
1207
where C1a ¼
2pqgk sinh ½k ðh1 d1 Þ cosh ðkh1 Þ
C1b ¼
2pqgk sinh ½kðh1 e1 Þ cosh ðkh1 Þ
C2a ¼
2pqgkRJ1 ðkRÞ cosh ðkh1 Þ
C2b ¼
2pqgkRb J1 ðkRb Þ cosh ðkh1 Þ
For further calculation, some indefinite integral expressions will be used as follows: Z Z Z
r J0 ðkrÞr dr ¼ J1 ðkrÞ þ C k r3 J0 ðkrÞ dr ¼
r 2 r3 J2 ðkrÞ þ C J1 ðkrÞ 2 k k
I0 ðarÞJ0 ðbrÞr dr ¼
brI0 ðarÞJ1 ðbrÞ þ arI1 ðarÞJ0 ðbrÞ þC a2 þ b2
where a2 þ b2 6¼ 0, I0 and I1 are the first kind modified Bessel function of order zero, one, respectively, J0 , J1 and J2 are the first kind Bessel function of order zero, one and two, respectively. 5.2. Added masses and radiation damping coefficients The radiation force is the force due to the motion of a body. It can be calculated from the radiated potentials, that is Z
2
Flj ¼ iqx expðixtÞ /lr nj ds ¼ qx Alr expðixtÞ S ik lj ¼ x2 Alr llj þ expðixtÞ x
Z
ulr cos ½ð1 d1l Þhnj ds S
ð99Þ
where Flj denotes the force in lth mode along the direction j. We can get llj þ
iklj ¼q x
Z
ulr cos ½ð1 d1l Þhnj ds
ð100Þ
S
where llj is the added mass; klj is the damping coefficient. From the expressions for the radiated potentials and Eq. (100), the added masses and damping coefficients are given as follows.
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The added mass and damping coefficient expressions for heave are " # / X ik11 ðh2 þ 2D11 ÞR2 R4 RI1 ðbn RÞ ¼ 2pq l11 þ þ D1n cos ðbn h2 Þ bn I0 ð bn RÞ x 4 16h2 n¼2
ð101Þ
l12 þ
ik12 ¼0 x
ð102Þ
l13 þ
ik13 ¼0 x
ð103Þ
The added mass and damping coefficient expressions for sway are l21 þ
ik21 ¼0 x
ð104Þ
l22 þ
/ X ik22 sin ðcn e1 Þ sin ðcn h2 Þ ¼ pqR ðB2n þ C2n Þ cn x n¼1
ð105Þ
l23 þ
Z 0 / X ik23 ðB2n þ C2n Þ cos ½cn ðz þ e1 Þðz zc Þ dz ¼ pqR x d1 n¼1 " # / D21 R4 X R2 I2 ðbn RÞ pq D2n cos ðbn h2 Þ þ bn I1 ðbn RÞ 4 n¼2
ð106Þ ð107Þ
The added mass and damping coefficient expressions for roll are l31 þ
ik31 ¼0 x
ð108Þ
l32 þ
/ X ik32 sin ðcn e1 Þ sin ðcn h2 Þ ðB3n þ C3n Þ ¼ pqR cn x n¼1
ð109Þ
Z 0 / X ik33 ¼ pqR ðB3n þ C3n Þ cos ½cn ðz þ e1 Þð z zc Þ dz l33 þ x d1 n¼1 # " / h2 R4 R6 D31 R4 X R2 I2 ðbn RÞ pq þ D3n cos ðbn h2 Þ þ þ bn I1 ðbn RÞ 8 48h2 4 n¼2
ð110Þ
6. Numerical results The numerical work is involved in the choice of a programming tool and the number of terms used in infinite summations. The former 30 terms (N ¼ 30) are taken in the infinite summations to
B.-j. Wu et al. / International Journal of Engineering Science 42 (2004) 1193–1213
1209
Fh /w0
compute numerical results because the infinite summations have excellent truncation characteristics [3,7]. Those programs are implemented by Compag Visual Fortran 6.5 version. In the presence of an incident wave, as a check on the expressions and the program, a comparison is made between two approaches to calculate the exciting force. The comparison is shown in Fig. 2. In the Fig. 2, Fh is the vertical exciting force, Fh ¼ Fi1 þ Fd1 , w0 ¼ qgpR2 , Fh =w0 is the dimensionless force; f is the wave frequency, f ¼ x=2p. Case 1 indicates d1 ¼ 0:2h1 , e1 ¼ 0:45h1 , R ¼ 0:5h1 and Rb ¼ 1:5h1 ; case 2 is d1 ¼ 0:3h1 , e1 ¼ 0:5h1 , R ¼ 0:5h1 and Rb ¼ 3:5h1 . In the diagram, the line stands for the result of Eqs. (89) and (94) and the solid block stands for the result of Eqs. (89) and (98). Obviously, the results of two approaches match with each other well. In order to present the effect of the caisson on the hydrodynamic coefficients and the exciting forces, some examples are listed, as shown as in Figs. 3–10. In the Figs. 3–10, l is the added mass; m0 ¼ pqR2 d1 ; I ¼ m0 R2 =2; l=m0 is the dimensionless added mass; k is the damping coefficient; k=m0 is the dimensionless damping coefficient; similarly, l=I and k=I are the dimensionless coefficients; kR is the product of k and R. In those figures, the line including the sold block (circle, prism and rectangle) stands for the dimensionless added mass or the dimensionless damping coefficient or the exciting force in different cases. Figs. 3–8 show the dimensionless the added masses, the damping coefficients and the exciting forces (notice that Fr =w0 is of dimension) with Rb as a parameter for d1 ¼ 0:2h1 , e1 ¼ 0:3h1 , R ¼ 0:5h1 . It is interesting to note how the common
1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0
Case 1 Case 2
0
0.5
1
1.5
2
2.5
f
(a)
7 6.5 6 5.5 5 4.5 4 3.5 3 2.5 2
6
R b=0.5 h 1 R b=0.8 h 1
λ 11/m0
µ11/m0
Fig. 2. Exciting force.
R b=1.2 h 1 R b=2.0 h 1
R b=0.5 h 1
5
R b= 0.8 h 1
4
R b=1.2 h 1 R b=2.0 h 1
3 2 1
0
0.5
1
1.5
f
2
0
2.5 (b)
0
0.5
1
1.5
f
Fig. 3. Added mass and damping for heave motion.
2
2.5
B.-j. Wu et al. / International Journal of Engineering Science 42 (2004) 1193–1213 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
R b =0.5 h 1 R b =0.8 h 1 R b =1.2 h 1
λ 22 /m0
µ 22/m0
1210
R b =2.0 h 1
0
0.5
1
(a)
f
1.5
2
2.5
2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0
R b =0.5 h 1 R b =0.8 h 1 R b =1.2 h 1 R b =2.0 h 1
0
0.5
1
(b)
f
1.5
2
2.5
(a)
0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45
R b=0.5 h 1 R b=0.8 h 1 R b=1.2 h 1 R b=2.0 h 1
0
0.5
1
1.5
2
λ 33/I
µ 33 /I
Fig. 4. Added mass and damping for sway motion.
2.5
f
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
R b=0.5 h 1 R b=0.8 h 1 R b=1.2 h 1 R b=2.0 h 1
0
0.5
1
(b)
1.5
2
2.5
f
Fig. 5. Added mass and damping for roll motion.
2 R b=0.5 h 1
Fh /w0
1.6
R b=0.8 h 1 R b=1.2 h 1
1.2
R b=2.0 h 1
0.8 0.4 0
0
0.5
1
f
1.5
2
2.5
Fig. 6. Horizontal exciting force.
properties transforms to remarkable counterpart against the radius of the caisson from Figs. 3–8. The figures show that the added mass, the damping coefficient and the exciting force almost increase as Rb increases when the radius of the caisson is shorter than a special value at any frequency. They also show that the curves of the figures have several extrema as the wave frequency is at the low frequency and the radius of the caisson is at some special values. The added mass, the damping coefficient and the exciting force are in oscillation and remarkable at the low frequency. The curves become common at the relative high frequency. To the authors’ knowledge, those
B.-j. Wu et al. / International Journal of Engineering Science 42 (2004) 1193–1213
1211
1 R b=0.5 h 1
0.8
R b=0.8 h 1
Fs /w0
R b=1.2 h 1
0.6
R b=2.0 h 1
0.4 0.2 0
0
0.5
1
f
1.5
2
2.5
Fig. 7. Vertical exciting force.
0.5 R b=0.5 h 1
0.4
R b=0.8 h 1 R b=1.2 h 1
Fr/w0
0.3
R b=2.0 h 1
0.2 0.1 0 0
0.5
1
1.5
2
2.5
f
(a)
5.5 5 4.5 4 3.5 3 2.5 2 1.5 1
3 e1 =0.3 h 1
e1 =0.3 h 1
2.5
e1 =0.4 h 1 e1 =0.6 h 1
λ 11/m 0
µ11 /m 0
Fig. 8. Roll exciting moment.
e1 =0.9 h 1
e1 =0.4 h 1 e1 =0.6 h 1
2
e1 =0.9 h 1
1.5 1 0.5
0
4
8
12
kR
16
20
0 (b)
0
4
8
12
16
20
kR
Fig. 9. Added mass and damping.
properties are the first reported. In Figs. 9 and 10, they show the added mass, the damping coefficient and the exciting forces against the gap h2 between the cylinder and the caisson when the cylinder is in heaving motion for d1 ¼ 0:2h1 , Rb ¼ 0:6h1 , R ¼ 0:5h1 . It appears from Fig. 9 and 10 that the values of the curves increase with the decrease of the gap. This is a common property. In all words, the influence of the caisson on the motion of the cylinder is obvious. Of particular the added masses, the damping coefficients and the exciting forces are remarkable as the wave frequency is at the low frequency and the radius of the caisson is at some special values.
B.-j. Wu et al. / International Journal of Engineering Science 42 (2004) 1193–1213
Fh / w 0
1212
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
eb =0.3 h 1 e1 =0.4 h 1 e1 =0.6 h 1 e1 =0.9 h 1
0
2
4
6
8 10 12 14 16 18 20 kR
Fig. 10. Exciting force.
7. Conclusions In this paper, by use of an eigenfunction expansion approach, we have presented the velocity potential expressions of the diffraction and radiation problem for a cylinder over a caisson fixed on the bottom in water of finite depth in the presence of a linear incident wave. The radius of the caisson is longer than or equal to that of the cylinder. We have proposed a set of theoretical added mass, damping coefficient and exciting force expressions. In order to check the involved expressions and programs, we have used two approaches to calculate the exciting forces and the results that match with each other well have indicated that the involved expressions are corrected and the programs are trustworthy. Finally, the analytical results of the added masses, the damping coefficients and the exciting forces are obtained for the different sizes of the caisson. The results presented have shown that the influence of the caisson on the motion of the cylinder is obvious. Of particular the interest in the results shown is that the added masses, the damping coefficients and the exciting forces are remarkable as the wave frequency is at the low frequency and the radius of the caisson is at some special values.
Acknowledgements This work is supported by the National High Technology Research and Development Program of China (no. 2001AA516010) and the CAS Pilot Project of the National Knowledge Innovation Program (KIP) (no. KGCX2-SW-305).
References [1] J.W. Miles, J.F. Gilbert, Scattering of gravity waves by a circular dock, Journal of Fluid Mechanics 34 (1968) 783– 793. [2] C.J.R. Garrett, Wave forces on a circular dock, Journal of Fluid Mechanics 46 (1971) 129–139. [3] R.W. Yeung, Added mass and damping of a vertical cylinder in finite-depth water, Applied Ocean Research 3 (3) (1981) 119–133. [4] A.N. Williams, A.G. Abul-Azm, Hydrodynamic interactions in floating cylinder arrays––II. Wave radiation, Ocean Engineering 16 (3) (1989) 217–263.
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[5] D.D. Bhatta, M. Rahman, On scattering and radiation problem for a cylinder in water of finite depth, International Journal of Engineering Science 41 (2003) 931–967. [6] J.L. Black, C.C. Mei, C.G. Bray, Radiation and scattering of water waves by rigid bodies [J], J. Fluid Mech. 46 (Part 1) (1971) 151–164. [7] L. Berggren, M. Johansson, Hydrodynamic coefficients of a wave energy device consisting of a buoy and a submerged plate, Applied Ocean Research 14 (1992) 51–58. [8] H. Eidsmoen, Hydrodynamic parameters for a two-body axisymmetric system, Applied Ocean Research 17 (1995) 103–115. [9] M. Kanoria, Water wave scattering by a submerged thick wall with a gap, Applied Ocean Research 21 (1998) 69– 80. [10] C.C. Mei, J.L. Black, Scattering of surface waves by rectangular obstacles in water of finite depth, Journal of Fluid Mechanics 38 (Part 3) (1969) 499–511. [11] M. Kanoria, Water wave scattering by thick rectangular slotted barriers, Applied Ocean Research 23 (2001) 285– 298.
On diffraction and radiation problem for a cylinder over a caisson in water of finite depth Bi-jun Wu a
a,b,*
, Yong-hong Zheng a, Ya-ge You a, Xiao-yan Sun a, Yong Chen
a
Guangzhou Institute of Energy Conversion, Chinese Academy of Sciences, Guangzhou 510070, China b University of Science and Technology of China, Hefei 230026, China Received 10 September 2003; accepted 3 December 2003
Available online
Abstract A caisson and a vertical circular cylinder floating on the surface of fluid compose a wave power device. The hydrodynamic fluid/solid interaction becomes more complex due to the caisson. In this paper, in order to investigate the effects of the caisson on the cylinder’s hydrodynamic coefficients and exciting forces, the expressions for velocity potentials are derived in the presence of an incident linear wave by use of an eigenfunction expansion approach. A set of theoretical added mass, damping coefficient and exciting force expressions are obtained. Two approaches are proposed to calculate the exciting forces and they match with each other well. Finally, Analytical results of added masses, damping coefficients and exciting forces are given for the different sizes of the caisson. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: Analytical approach; Velocity potential; Wave force
1. Introduction A buoy riding in waves, connected to a caisson fixed on the bottom of the sea by a rigid rope, has been proposed as a device for extracting energy from waves. The extracting energy by the buoy is transferred to a liquid pump in the caisson through the rope. The pump converts the extracting energy to the electrical energy. Obviously, the caisson changes the direction of the
*
Corresponding author. Address: Guangzhou Institute of Energy Conversion, Chinese Academy of Sciences, Guangzhou 510070, China. Tel.: +86-20-87057612; fax: +86-20-87057597. E-mail address: [email protected] (B.-j. Wu). 0020-7225/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijengsci.2003.12.006
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wave’s ray at or near the surface of the caisson and makes the fluid/solid interaction more complex. The efficiency of the converted wave energy and the estimation of hydrodynamic forces (or moments) for a wave power device has been received considerable attention from the design point of view. An accurate prediction of the wave loads exerted by waves on rigid power devices is absolutely necessary to design the wave power devices. Generally, numerical methods are engaged to deal with the diffraction and radiation problem for the buoy, but the calculating accuracy is low and the CPU time is long and programming is complicated. In order to render the device in a simpler style and present the common properties acting on the buoy by the caisson, both the buoy and the caisson are idealised as vertical cylinders. The analytical approach is used to deal with the diffraction and radiation problem for the device. Assuming suitable coordinates, the motion of the cylinder is simply decomposed three motions: heave, sway and roll. Many theoretical studies have been performed to analyze the wave motion and wave force on a structure. Miles and Gilbert [1] formulated the problem of the scattering of surface wave by a circular dock. Garrett [2] obtained the results for the horizontal and vertical forces and torque on the dock. Yeung [3] formulated a set of theoretical added masses and damping coefficients for a floating circular cylinder in water of finite depth. Williams and Abul-Azm [4] presented the hydrodynamic interactions between the members of an array of floating circular cylinders which occur when one member undergoes prescribed forced oscillations. Black et al. [6] calculated the wave faces on a truncated cylinder, which either protruding from the sea bottom or partially immersing in the free surface. Berggren and Johansson [7] and Eidsmoen [8] presented the heaving radiation problem of a two vertical cylinder system, where one of the cylinders is submerged and the other floats. Mei and Black [10] considered the scattering properties for bottom and surface obstacles of various proportions on two dimensions in water of finite depth. Kanoria [9] investigated the scattering problem of surface water waves by a thick submerged rectangular wall with a gap in finite depth water and then Kanoria [11] presented the scattering problem of surface water waves by a thick vertical slotted barrier of rectangular cross-section with an arbitrary number of slots of unequal lengths along the vertical direction on two dimensions. Bhatta and Rahman [5] presented the analytical solution for the boundary value problem to evaluate the wave loads for a vertical circular cylinder with sway, heave and roll motions in water of finite depth in the presence of an incident wave. Through the above literature review, we found that some of them focus on the floating vertical cylinder diffraction (scattering) or radiation with periodic motion in water of finite depth and the others on the fixed barriers or floating rectangular structures scattering on two dimensions. The radii of cylinders or the lengths of the barriers along the horizontal direction are the same. To the authors’ knowledge, there is no study on the diffraction and radiation problem for a cylinder over a caisson fixed on the bottom on three dimensions in water of finite depth by use of an analytical approach. The radius of the caisson is longer than or equal to that of the cylinder. Obviously, the diffraction and radiation problem of the cylinder over a caisson whose radius is longer than or equal to that of the cylinder is more complicated than that of the cylinder without a caisson in water of finite depth. By use of an eigenfunction expansion approach, the analytical solution for the radiation potentials due to heave, sway, and roll and for the diffracted potential due to the diffraction of an incident wave acting on the fixed cylinder has been presented. A set of theoretical added mass, damping coefficient and exciting force expressions have been proposed.
B.-j. Wu et al. / International Journal of Engineering Science 42 (2004) 1193–1213
1195
Two approaches have been presented to calculate the exciting forces and they match with each other well. Finally, analytical results of the added masses, the damping coefficients and the exciting forces are given for the different sizes of the caisson.
2. Mathematical model A vertical circular cylinder with radius R is parked in the fluid. The incompressible fluid motion is irrotational and the waves are of the small amplitude. A Cartesian coordinate system Oxyz as well as the cylindrical coordinate system Orhz is defined with the origin in the undisturbed free surface and the Z-axis positive upwards. The cylinder occupies the space defined by r 6 R, 0 6 h 6 2p ,d1 6 z 6 0. Under the cylinder there is a caisson with radius Rb , which is coaxial with the cylinder. The caisson occupies the space defined by r 6 Rb , 0 6 h 6 2p, h1 6 z 6 e1 , as shown as Fig. 1. For an incompressible fluid, and for small amplitude wave theory with irrotational motion, we can introduce a velocity potential Uðr; h; z; tÞ ¼ Re½/ðr; h; zÞ eixt where Re½ denotes the real part of a complex expression and /ðr; h; zÞ is the spatial part of the velocity potential. /ðr; h; zÞ must satisfy the following equation: o2 / 1 o2 / 1 o o/ r þ þ ¼0 oz2 r2 oh2 r or or
ð1Þ
From the linear water wave theory, the velocity potential / can be decomposed into the incident wave potential /i , the diffracted potential /d due to the diffraction of an incident wave acting on the fixed cylinder and the radiated potential /r due to the motion of the cylinder. For the present problem, the incident potential of a linear wave with unit amplitude propagating to the positive x-direction is /i ¼
1 ig cosh kðz þ h1 Þ X l Jm ðkrÞ cos mh; x coshðkh1 Þ m¼0 m
r
ð2Þ
y
θ x
R z
x
d1 I h1
h2
III I
e1
Rb Fig. 1. The sketch of the device and the definition of fluid subdomains.
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where lm ¼
2im 1
m>0 m¼0
pffiffiffiffiffiffiffi i ¼ 1: x is the wave angular frequency, g is the gravitational acceleration, k is the wave number, which is determined by the dispersion relation k tanhðkh1 Þ ¼ x2 =g, h1 is the water depth. In this paper, /d , /r and their derivations are the key parameters to be solved.
3. Radiation velocity potential 3.1. Radiation problem The cylinder is assumed to have motions with three degrees of freedom in the presence of incident wave. The motions are referred as heave, sway and roll, namely three modes. The radiation modes are numbered such that l ¼ 1, 2 and 3 corresponding to heave, sway and roll respectively. The amplitude of the lth mode is denoted by Alr . So the radiated velocity potential /r can be expressed as
/r ¼
3 X
/lr ðr; h; zÞ ¼
l¼1
3 X
fix½Alr ulr ðr; zÞ cosð1 d1l Þhg
ð3Þ
l¼1
where /lr or ulr (l ¼ 1; 2; 3) is the spatial part of the radiated velocity potential, t is the time. Evidently, there is a relation as follows /lr ðr; h; zÞ ¼ ixAlr ulr ðr; zÞ cosð1 d1l Þh
ð4Þ
where, in order to compose the corresponding expressions in compact for the radiated problems, we introduce following sign dkl ¼
1 0
k¼l k 6¼ l
By substituting Eq. (4) into Laplace’s equation, we can obtain o2 ulr 1 o oulr ð1 d1l Þ2 ulr þ ¼0 r r or oz2 or r2
ð5Þ
For the lth mode motion, the radiated potential is also required to satisfy appropriate boundary conditions and the equations for the boundary conditions are obtained as follows in compact for three mode motions
B.-j. Wu et al. / International Journal of Engineering Science 42 (2004) 1193–1213
oulr x2 ulr ¼ 0 oz g
ðz ¼ 0; r P RÞ;
1197
ð6Þ
oulr ¼0 oz
ðz ¼ h1 ; r P Rb ; z ¼ e1 ; r 6 Rb Þ
ð7Þ
oulr ¼0 or
ðh1 < z < e1 ; r ¼ Rb Þ;
ð8Þ
oulr ¼ d1l rd3l oz
ðz ¼ d1 ; r < RÞ;
oulr ¼ d2l þ ðz zc Þd3l ðd1 < z < 0; r ¼ RÞ or pffiffi oulr lim r ikulr ¼ 0 r!/ or
ð9Þ ð10Þ ð11Þ
where ðxc ; yc ; zc Þ ¼ ð0; 0; zc Þ is the coordinate of a point in the rotational axis which is parallel to the y-axis. 3.2. Solution In the solution procedure the fluid domain is divided into three subdomains denoted by I, II and II as indicated in Fig. 1. The radiation velocity potentials of three subdomains are denoted by ulr1 , ulr2 and ulr3 respectively. The approach of separation of variables is applied in each subdomain in order to obtain expressions for unknown function, i.e. the velocity potential. Expressions valid in each subdomain are obtained as infinite series of orthogonal functions. These expressions are developed to satisfy all boundary conditions except at the boundary joining the subdomains, i.e. at r ¼ Rb and r ¼ R. It then remains to determine the unknown coefficients in the series. This is done by imposing the condition of continuity of pressure and normal velocity at r ¼ Rb and r ¼ R. Mathematically this is fulfilled by matching the potentials and the normal derivatives of the potentials, respectively. The application of the approach of separation of variables yields the spatial potentials in each subdomain expressed by orthogonal series. In regions I, II and III, the radiated potentials are expressed respectively for l ¼ 1, 2 or 3 as follows [7] 1. Region I:
ulr1 ¼
/ X n¼1
2. Region II:
Aln
Rð1d1l Þ ðkn rÞ cos½kn ðz þ h1 Þ Rð1d1l Þ ðkn Rb Þ
ð12Þ
1198
B.-j. Wu et al. / International Journal of Engineering Science 42 (2004) 1193–1213
ulr2
/ X Sð1d1l Þ ðcn rÞ Tð1d1l Þ ðcn rÞ ¼ Bln þ Cln cos½cn ðz þ e1 Þ Sð1d1l Þ ðcn RÞ Tð1d1l Þ ðcn RÞ n¼1
ð13Þ
3. Region III: ulr3 ¼ wlp þ Dl1 rð1d1l Þ þ
/ X
Dln
n¼2
Ið1d1l Þ ðbn rÞ cos½bn ðz þ e1 Þ Ið1d1l Þ ðbn RÞ
ð14Þ
in which, 2
wlp ¼
2
ðz þ e1 Þ r2 =2 ðz þ e1 Þ r r3 =4 d1l d3l 2h2 2h2
k1 ¼ ik; c1 ¼ ike ;
k tanhðkh1 Þ ¼ x2 =g ke tanhðke e1 Þ ¼ x2 =g
kn tanðkn h1 Þ ¼ x2 =g cn tanðcn e1 Þ ¼ x2 =g
bn ¼ ðn 1Þp=h2
n¼1
n ¼ 2; 3; . . .
n ¼ 1; 2; 3; . . .
ð15Þ ð16Þ ð17Þ ð18Þ
and the radial function R, S and T are given by ð1Þ
Rð1d1l Þ ðk1 rÞ ¼ Hð1d1l Þ ðkrÞ Rð1d1l Þ ðkn rÞ ¼ Kð1d1l Þ ðkn rÞ ð1Þ
Sð1d1l Þ ðc1 rÞ ¼ Hð1d1l Þ ðke rÞ
n¼1 n ¼ 2; 3; . . .
ð20Þ
n¼1
ð21Þ
Sð1d1l Þ ðcn rÞ ¼ Kð1d1l Þ ðcn rÞ n ¼ 2; 3; . . . ð2Þ
Tð1d1l Þ ðc1 rÞ ¼ Hð1d1l Þ ðke rÞ Tð1d1l Þ ðcn rÞ ¼ Ið1d1l Þ ðcn rÞ ð1Þ
ð19Þ
n¼1 n ¼ 2; 3; . . .
ð22Þ ð23Þ ð24Þ
ð2Þ
where Hð1d1l Þ and Hð1d1l Þ are the first and second kind Hankel function of order ð1 d1l Þ, namely 0 or 1, respectively; Ið1d1l Þ and Kð1d1l Þ are the first and second kind modified Bessel function of order ð1 d1l Þ respectively. The continuity of pressure and normal velocity for the radiated potentials are given as follows at r ¼ Rb ulr1 ¼ ulr2 e1 6 z 6 0 ou lr2 oulr1 e1 < z < 0 or ¼ or 0 h1 < z < e1
ð25Þ ð26Þ
B.-j. Wu et al. / International Journal of Engineering Science 42 (2004) 1193–1213
1199
and at r ¼ R ulr2 ¼ ulr3 e1 6 z 6 d1 oulr2 d þ ðz zc Þd3l d1 < z < 0 ¼ ou2llr3 e1 < z < d1 or or
ð27Þ ð28Þ
The boundary conditions above are satisfied over the z interval in a least square by multiplying both sides of them by a proper set of eigenfunctions in their belonging regions and then by integrating them over the corresponding interval at the boundaries r ¼ Rb and r ¼ R. The procedure above gives the following equations. Eq. (25): Z
0
ulr1 ðRb ; zÞ cos½ci ðz þ e1 Þ dz ¼
e1
Z
0
ulr2 ðRb ; zÞ cos½ci ðz þ e1 Þ dz
ð29Þ
e1
Eq. (26): Z
0
h1
Z 0 oulr1 ðr; zÞ oulr2 ðr; zÞ cos½ki ðz þ h1 Þ dz ¼ cos½ki ðz þ h1 Þ dz or or e1 r¼Rb r¼Rb
ð30Þ
Eq. (27): Z
d1
ulr2 ðR; zÞ cos½bi ðz þ e1 Þ dz ¼
e1
Z
d1
ulr3 ðR; zÞ cos½bi ðz þ e1 Þ dz
ð31Þ
e1
Eq. (28): Z
0
e1
Z 0 oulr2 ðr; zÞ ½d2l þ ðz zc Þd3l cos ci ðz þ e1 Þ dz cos ci ðz þ e1 Þ dz ¼ or d1 r¼R Z d1 oulr3 ðr; zÞ þ cos ci ðz þ e1 Þ dz or e1 r¼R
Introduce the following functions Z z2 Eðan ; bk ; ha ; hb ; z1 ; z2 Þ ¼ cos an ðz þ ha Þ cos bk ðz þ hb Þ dz
ð32Þ
ð33Þ
z1
Nðai ; ha ; z1 ; z2 Þ ¼
Z
z2
cos2 ½ai ðz þ ha Þ dz
ð34Þ
z1
in which fan ; n ¼ 1; 2; . . .g and fbk ; k ¼ 1; 2; . . .g are two different sets of eigenvalues. Applying the functions (33) and (34) to Eqs. (29)–(32), we obtain:
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Eq. (29): / X
Alj Eðkj ; ci ; h1 ; e1 ; e1 ; 0Þ ¼ ðBli Sali þ Cli Tali ÞN ðci ; e1 ; e1 ; 0Þ þ P1li
ð35Þ
j¼1
where Sali ¼ Sð1d1l Þ ðci Rb Þ=Sð1d1l Þ ðci RÞ
ð36Þ
Tali ¼ Tð1d1l Þ ðci Rb Þ=Tð1d1l Þ ðci RÞ
ð37Þ
P1li ¼ 0;
ð38Þ
Eq. (30): Ali Oli N ðki ; h1 ; h1 ; 0Þ ¼
/ X
ðBlj Palj þ Clj Qalj ÞEðcj ; ki ; e1 ; h1 ; e1 ; 0Þ þ P2li
ð39Þ
j¼1
where ( Oli ¼
ð1Þ0
( Palj ¼
ð1Þ0
ð1Þ
ke Hð1d1l Þ ðke Rb Þ=Hð1d1l Þ ðke RÞ 0 ðcj Rb Þ=Kð1d1l Þ ðcj RÞ cj Kð1d 1l Þ
( Qalj ¼
ð1Þ
i¼1 kHð1d1l Þ ðkRb Þ=Hð1d1l Þ ðkRb Þ 0 ðk R Þ=K ðk R Þ i ¼ 2; 3; . . . ki Kð1d i b ð1d1l Þ i b 1l Þ
ð2Þ0
ð2Þ
ke Hð1d1l Þ ðke Rb Þ=Hð1d1l Þ ðke RÞ 0 ðcj Rb Þ=Ið1d1l Þ ðcj RÞ cj Ið1d 1l Þ
ð40Þ
j¼1 j ¼ 2; 3; . . .
ð41Þ
j¼1 j ¼ 2; 3; . . .
ð42Þ
P2li ¼ 0
ð43Þ
Eq. (31): / X
ðBlj þ Clj ÞEðcj ; bi ; e1 ; e1 ; e1 ; d1 Þ ¼ Dli Uli N ðbi ; e1 ; e1 ; d1 Þ þ P3li
ð44Þ
j¼1
where Uli ¼
Rð1d1l Þ 1
i¼1 i ¼ 2; 3; . . .
ð45Þ
B.-j. Wu et al. / International Journal of Engineering Science 42 (2004) 1193–1213
P3li ¼
Z
"
ðz þ e1 Þ2 R2 =2 d1l cos bi ðz þ e1 Þ dz 2h2 e1 # Z d1 " Rðz þ e1 Þ2 R3 =4 þ d3l cos bi ðz þ e1 Þ dz 2h2 e1 d1
1201
#
ð46Þ
Eq. (32): ðBli Pbli þ Cli Qbli ÞN ðci ; e1 ; e1 ; 0Þ ¼
/ X
Dlj Wlj Eðbj ; ci ; e1 ; e1 ; e1 ; d1 Þ þ P4li
ð47Þ
j¼1
where ( Pbli ¼
ð1Þ0
( Qbli ¼ Wlj ¼
ð2Þ0
ð2Þ
ke Hð1d1l Þ ðke RÞ=Hð1d1l Þ ðke RÞ 0 ðci RÞ=Ið1d1l Þ ðci RÞ ci Ið1d 1l Þ
ð1 d1l ÞRd1l 0 ðbj RÞ=Ið1d1l Þðbj RÞ bj Ið1d 1l Þ
i¼1 i ¼ 2; 3; . . .
ð48Þ
i¼1 i ¼ 2; 3; . . .
ð49Þ
j¼1 j ¼ 2; 3; . . .
Z 0 R d1l cos ci ðz þ e1 Þ dz þ ¼ ½d2l þ ðz zc Þd3l cos ci ðz þ e1 Þ dz 2h2 d1 e1 # Z d1 " 2 3R =4 ðz þ e1 Þ2 d3l cos ci ðz þ e1 Þ dz þ 2h2 e1 Z
P4li
ð1Þ
ke Hð1d1l Þ ðke RÞ=Hð1d1l Þ ðke RÞ 0 ci Kð1d ðci RÞ=Kð1d1l Þ ðci RÞ 1l Þ
d1
ð50Þ
ð51Þ
In order to find a solution to coefficients Aln , Bln , Cln , and Dln , we take the first N terms and make some arrangements yield 3 sets of linear system of 4N complex equations and get an equal number of unknown coefficients. Organizing the equations in matrices is as follows Sl Xl ¼ Fl
ð52Þ
where Xl ¼ ½Al1 ; Al2 ; . . . ; AlN ; Bl1 ; Bl2 ; . . . ; BlN ; Cl1 ; Cl2 ; . . . ; ClN ; Dl1 ; Dl2 ; . . . ; DlN T ; Sl is the coefficient matrices and Fl is the right hand vectors. Slði;jÞ of Sl and Fli of Fl are given as follows according to Eqs. (35)–(51) (i; j ¼ 1; 2; . . . ; N ): Eq. (35): Slði;jÞ ¼ Eðkj ; ci ; h1 ; e1 ; e1 ; 0Þ
ð53Þ
Slði;N þiÞ ¼ Sali N ðci ; e1 ; e1 ; 0Þ
ð54Þ
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Slði;2N þiÞ ¼ Tali Nðci ; e1 ; e1 ; 0Þ
ð55Þ
Eq. (38): Fli ¼ P1li
ð56Þ
Eq. (39): SlðN þi;iÞ ¼ Oli Nðki ; h1 ; h1 ; 0Þ
ð57Þ
SlðN þi;N þjÞ ¼ Palj Eðcj ; ki ; e1 ; h1 ; e1 ; 0Þ
ð58Þ
SlðN þi;2N þjÞ ¼ Qblj Eðcj ; ki ; e1 ; h1 ; e1 ; 0Þ
ð59Þ
Eq. (43): FlðN þiÞ ¼ P2li
ð60Þ
Eq. (44): Slð2N þi;N þiÞ ¼ Eðcj ; bi ; e1 ; e1 ; e1 ; d1 Þ
ð61Þ
Slð2N þi;2NþjÞ ¼ Eðcj ; bi ; e1 ; e1 ; e1 ; d1 Þ
ð62Þ
Slð2N þi;3NþiÞ ¼ Uli N ðbi ; e1 ; e1 ; d1 Þ
ð63Þ
Eq. (46): Flð2N þiÞ ¼ P3li
ð64Þ
Eq. (47): Slð3N þi;N þiÞ ¼ Pbli N ðci ; e1 ; e1 ; 0Þ
ð65Þ
Slð3N þi;2NþiÞ ¼ Qbli N ðci ; e1 ; e1 ; 0Þ
ð66Þ
Slð3N þi;3NþjÞ ¼ Wlj Eðbj ; ci ; e1 ; e1 ; e1 ; d1 Þ
ð67Þ
Eq. (51): Flð3N þiÞ ¼ P4li ;
ð68Þ
By substituting the coefficients Xl into Eqs. (12)–(14), respectively, the radiated potentials ulr for l ¼ 1, 2, or 3 at any position in each subdomain can be computed.
B.-j. Wu et al. / International Journal of Engineering Science 42 (2004) 1193–1213
1203
4. Diffracted velocity potential 4.1. The governing equation and the boundary conditions The diffraction problem is due to diffraction in the presence of an incident wave on a fixed structure. The diffracted velocity potential Ud can be expressed as Ud ¼ Re½/d ðr; h; zÞ expðixtÞ, where the spatial part of the diffracted velocity potential /d is governed by the following boundary-value problem: o2 /d 1 o o/d 1 o 2 /d þ ¼0 r þ r or r2 oh2 oz2 or
ð69Þ
o/d x2 /d ¼ 0 oz g
ð70Þ
o/d ¼0 oz
ðz ¼ 0; r P RÞ
ðz ¼ h1 ; r P Rb Þ
oð/d þ /i Þ ¼0 oz
ð71Þ
ðz ¼ d1 ; r < R; z ¼ e1 ; r < Rb Þ
oð/d þ /i Þ ¼ 0 ðd1 < z < 0; r ¼ R; h1 < z < e1 ; r ¼ Rb Þ or pffiffi o/d lim r ik/d ¼ 0 r!/ or
ð72Þ ð73Þ ð74Þ
where /i is the incident velocity potential, i.e. Eq. (2). 4.2. Solution to the problem Similarly, application of the approach of separation of variables gives the diffracted velocity potentials in regions I, II and III, respectively. The diffracted velocity potentials are denoted by /d1 , /d2 and /d3 , respectively. They are as follows: /d1 ¼
/ X / X
Am;n
m¼0 n¼1
/d2
Rm ðkn rÞ cos½kn ðz þ h1 Þ cos mh Rm ðkn Rb Þ
/ X / X Sm ðcn rÞ Tm ðcn rÞ ¼ /i þ Bm;n þ Cm;n cos cn ðz þ e1 Þ cos mh Sm ðcn RÞ Tm ðcn RÞ m¼0 n¼1
/d3 ¼ /i þ
/ X m¼0
( Dm;1 rm þ
/ X n¼2
) Im ðbn rÞ cos½bn ðz þ e1 Þ cos mh Dm;n Im ðbn RÞ
ð75Þ
ð76Þ
ð77Þ
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B.-j. Wu et al. / International Journal of Engineering Science 42 (2004) 1193–1213
where kn , cn and bn are determined by Eqs. (16)–(18), respectively. The radial functions Rm , Sm and Tm are given by Rm ðk1 rÞ ¼ Hmð1Þ ðik1 rÞ ¼ Hmð1Þ ðkrÞ n ¼ 1
ð78Þ
Rm ðkn rÞ ¼ Km ðkn rÞ
ð79Þ
n ¼ 2; 3; . . .
Sm ðc1 rÞ ¼ Hmð1Þ ðic1 rÞ ¼ Hmð1Þ ðke rÞ Sm ðcn rÞ ¼ Km ðcn rÞ
n ¼ 2; 3; . . .
Tm ðc1 rÞ ¼ Hmð2Þ ðic1 rÞ ¼ Hmð2Þ ðke rÞ Tm ðcn rÞ ¼ Im ðcn rÞ
n¼1
ð80Þ ð81Þ
n¼1
n ¼ 2; 3; . . .
ð82Þ ð83Þ
where Hmð1Þ and Hmð2Þ are the first and second kind Hankel function of order m, respectively; Im and Km are the first and second kind modified Bessel function of order m, respectively. At r ¼ Rb , matching conditions are /d1 ¼ /d2 o/d1 ¼ or
(
e1 6 z 6 0 o/d2 or i o/ or
e1 < z < 0 h1 < z < e1
ð84Þ ð85Þ
similarly, at r ¼ R, the continuity conditions become /d2 ¼ /d3 o/d2 ¼ or
(
e1 6 z 6 d1 i o/ or
o/d3 or
d1 < z < 0 e1 < z < d1
ð86Þ ð87Þ
Hence we can get Am;n , Bm;n , Cm;n and Dm;n from Eqs. (84)–(87) by use of the same approach for the radiated unknown coefficients presented in Section 3.2. By substituting the coefficients Am;n , Bm;n , Cm;n and Dm;n into Eqs. (75)–(77), we can get the diffracted potential /d ðr; h; zÞ.
5. Wave forces and hydrodynamic coefficients 5.1. Wave excitation forces Wave excitation forces are the forces by the velocity potentials except radiated velocity potentials. Those potentials are generally composed of the incident velocity potential /i and the diffracted velocity potential /d . Those forces are
B.-j. Wu et al. / International Journal of Engineering Science 42 (2004) 1193–1213
Fj ¼ Fij þ Fdj ¼ iqx
Z
/i nj ds þ iqx S
Z /d nj ds
1205
ð88Þ
S
where S is the wetted surface of the cylinder, nj is the generalized normal with n1 ¼ nz , n2 ¼ nx and j þ nz~ n ¼ nx~i þ ny~ k is the unit normal vector pointing to the cylinder n3 ¼ ðz zc Þnx ðx xc Þnz , ~ at the cylinder surface. The values, x, y, z are the projections of vector ~ r, which is from an origin (0,0,0) to a surface element ds. We have listed the force expressions according to Eq. (88) as follows. The vertical and horizontal excited force for the incident potential are as follows respectively Fi1 ¼ 2pqg Fi2 ¼
cosh½kðh1 d1 Þ R J1 ðkRÞ coshðkh1 Þ k
2pqgiRJ1 ðkRÞ sinhðkh1 Þ sinh½kðh1 d1 Þ coshðkh1 Þ k
ð89Þ ð90Þ
The roll moment for the incident potential is Fi3 ¼ FiXh þ FiXv
ð91Þ
FiXh ¼
FiXv ¼
2pqgiRJ1 ðkRÞ zc sinhðkh1 Þ þ ðd1 þ zc Þ sinh½kðh1 d1 Þ coshðkh1 Þ k coshðkh1 Þ cosh½kðh1 d1 Þ k2 2pqgi cosh½kðh1 d1 Þ R2 J2 ðkRÞ coshðkh1 Þ k
ð92Þ ð93Þ
Above the Eq. (91), the first term is the moment effected by the horizontal force Fi2 and the second term is the moment effected by vertical force Fi1 . According to Eq. (88), we can get the vertical diffraction force acting on the cylinder. That is "
Fd1
# / D0;1 R2 X R I1 ðbn RÞ þ Fi1 ¼ 2piqx D0;n cosðbn h2 Þ bn I0 ðbn RÞ 2 n¼2
ð94Þ
The horizontal diffraction force is Fd2 ¼ ipqxR
/ X n¼1
ðB1;n þ C1;n Þ
sinðcn e1 Þ sinðcn h2 Þ Fi2 cn
ð95Þ
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B.-j. Wu et al. / International Journal of Engineering Science 42 (2004) 1193–1213
The roll diffraction moment is Z 0 X / Fd3 ¼ piqxR ðB1;n þ C1;n Þ cos½cn ðz þ e1 Þðz zc Þ dz "
d1 n¼1
# / D1;1 R4 X D1;n cosðbn h2 Þ R2 I2 ðbn RÞ þ Fi3 pqix I1 ðbn RÞ bn 4 n¼2
ð96Þ
The diffraction force or moment can also be calculated from the incident potential /i and the radiated potential /r by use of Haskind relation. It is given by Z Z o/ Fdj ¼ iqx /d nj ds ¼ iqx /lr i ds on S S0 ! Z 2p Z R Z 2p Z 0 o/i o/i /lr2 R dz dh /lr3 r dr dh ¼ iqx or r¼R oz z¼d1 0 d1 0 0 ! Z 2p Z R Z 2p Z Rb o/i o/i /lr2 r dr dh þ /lr3 r dr dh þ iqx oz z¼e1 oz z¼e1 R 0 0 0 Z 2p Z e1 o/i /lr1 Rb dz dh ð97Þ þ iqx or r¼Rb h1 0 where S0 is the wetted surface of the cylinder and the caisson. According to the Haskind relation, the expressions of the diffraction forces are very complicated. We only list the expression for the vertical diffraction force as follows. ! Z 2p Z 0 Z 2p Z R o/i o/i u1r2 R dz dh u1r3 r dr dh Fd1 ¼ iqx or r¼R oz z¼d1 0 d1 0 0 ! Z 2p Z R Z 2p Z Rb o/i o/i u1r2 r dr dh þ u1r3 r dr dh þ iqx oz z¼e1 oz z¼e1 R 0 0 0 Z 2p Z e1 o/i u1r1 Rb dz dh þ iqx or r¼Rb 0 h1 # Z R" 2 / X h2 r2 =2 I0 ðbn rÞ J0 ðkrÞr dr þ D11 þ D1n cos ðbn h2 Þ ¼ C1a 2h2 I0 ðbn RÞ 0 n¼2 Z Rb X / S0 ðcn rÞ T0 ðcn rÞ B1n þ C1n J0 ðkrÞr dr þ C1b S0 ðcn RÞ T0 ð cn RÞ R n¼1 # Z R" / X r2 =2 I0 ðbn rÞ J0 ðkrÞr dr þ C1b þ D11 þ D1n 2h2 I0 ð bn RÞ 0 n¼2 Z 0 / X þ C2a ðB1n þ C1n Þ cosh ½k ðz þ h1 Þ cos ½cn ð z þ e1 Þ dz n¼1
þ C2b
/ X n¼1
Z
d1 e1
cosh ½k ð z þ h1 Þ cos ½kn ðz þ h1 Þ dz
An h1
ð98Þ
B.-j. Wu et al. / International Journal of Engineering Science 42 (2004) 1193–1213
1207
where C1a ¼
2pqgk sinh ½k ðh1 d1 Þ cosh ðkh1 Þ
C1b ¼
2pqgk sinh ½kðh1 e1 Þ cosh ðkh1 Þ
C2a ¼
2pqgkRJ1 ðkRÞ cosh ðkh1 Þ
C2b ¼
2pqgkRb J1 ðkRb Þ cosh ðkh1 Þ
For further calculation, some indefinite integral expressions will be used as follows: Z Z Z
r J0 ðkrÞr dr ¼ J1 ðkrÞ þ C k r3 J0 ðkrÞ dr ¼
r 2 r3 J2 ðkrÞ þ C J1 ðkrÞ 2 k k
I0 ðarÞJ0 ðbrÞr dr ¼
brI0 ðarÞJ1 ðbrÞ þ arI1 ðarÞJ0 ðbrÞ þC a2 þ b2
where a2 þ b2 6¼ 0, I0 and I1 are the first kind modified Bessel function of order zero, one, respectively, J0 , J1 and J2 are the first kind Bessel function of order zero, one and two, respectively. 5.2. Added masses and radiation damping coefficients The radiation force is the force due to the motion of a body. It can be calculated from the radiated potentials, that is Z
2
Flj ¼ iqx expðixtÞ /lr nj ds ¼ qx Alr expðixtÞ S ik lj ¼ x2 Alr llj þ expðixtÞ x
Z
ulr cos ½ð1 d1l Þhnj ds S
ð99Þ
where Flj denotes the force in lth mode along the direction j. We can get llj þ
iklj ¼q x
Z
ulr cos ½ð1 d1l Þhnj ds
ð100Þ
S
where llj is the added mass; klj is the damping coefficient. From the expressions for the radiated potentials and Eq. (100), the added masses and damping coefficients are given as follows.
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B.-j. Wu et al. / International Journal of Engineering Science 42 (2004) 1193–1213
The added mass and damping coefficient expressions for heave are " # / X ik11 ðh2 þ 2D11 ÞR2 R4 RI1 ðbn RÞ ¼ 2pq l11 þ þ D1n cos ðbn h2 Þ bn I0 ð bn RÞ x 4 16h2 n¼2
ð101Þ
l12 þ
ik12 ¼0 x
ð102Þ
l13 þ
ik13 ¼0 x
ð103Þ
The added mass and damping coefficient expressions for sway are l21 þ
ik21 ¼0 x
ð104Þ
l22 þ
/ X ik22 sin ðcn e1 Þ sin ðcn h2 Þ ¼ pqR ðB2n þ C2n Þ cn x n¼1
ð105Þ
l23 þ
Z 0 / X ik23 ðB2n þ C2n Þ cos ½cn ðz þ e1 Þðz zc Þ dz ¼ pqR x d1 n¼1 " # / D21 R4 X R2 I2 ðbn RÞ pq D2n cos ðbn h2 Þ þ bn I1 ðbn RÞ 4 n¼2
ð106Þ ð107Þ
The added mass and damping coefficient expressions for roll are l31 þ
ik31 ¼0 x
ð108Þ
l32 þ
/ X ik32 sin ðcn e1 Þ sin ðcn h2 Þ ðB3n þ C3n Þ ¼ pqR cn x n¼1
ð109Þ
Z 0 / X ik33 ¼ pqR ðB3n þ C3n Þ cos ½cn ðz þ e1 Þð z zc Þ dz l33 þ x d1 n¼1 # " / h2 R4 R6 D31 R4 X R2 I2 ðbn RÞ pq þ D3n cos ðbn h2 Þ þ þ bn I1 ðbn RÞ 8 48h2 4 n¼2
ð110Þ
6. Numerical results The numerical work is involved in the choice of a programming tool and the number of terms used in infinite summations. The former 30 terms (N ¼ 30) are taken in the infinite summations to
B.-j. Wu et al. / International Journal of Engineering Science 42 (2004) 1193–1213
1209
Fh /w0
compute numerical results because the infinite summations have excellent truncation characteristics [3,7]. Those programs are implemented by Compag Visual Fortran 6.5 version. In the presence of an incident wave, as a check on the expressions and the program, a comparison is made between two approaches to calculate the exciting force. The comparison is shown in Fig. 2. In the Fig. 2, Fh is the vertical exciting force, Fh ¼ Fi1 þ Fd1 , w0 ¼ qgpR2 , Fh =w0 is the dimensionless force; f is the wave frequency, f ¼ x=2p. Case 1 indicates d1 ¼ 0:2h1 , e1 ¼ 0:45h1 , R ¼ 0:5h1 and Rb ¼ 1:5h1 ; case 2 is d1 ¼ 0:3h1 , e1 ¼ 0:5h1 , R ¼ 0:5h1 and Rb ¼ 3:5h1 . In the diagram, the line stands for the result of Eqs. (89) and (94) and the solid block stands for the result of Eqs. (89) and (98). Obviously, the results of two approaches match with each other well. In order to present the effect of the caisson on the hydrodynamic coefficients and the exciting forces, some examples are listed, as shown as in Figs. 3–10. In the Figs. 3–10, l is the added mass; m0 ¼ pqR2 d1 ; I ¼ m0 R2 =2; l=m0 is the dimensionless added mass; k is the damping coefficient; k=m0 is the dimensionless damping coefficient; similarly, l=I and k=I are the dimensionless coefficients; kR is the product of k and R. In those figures, the line including the sold block (circle, prism and rectangle) stands for the dimensionless added mass or the dimensionless damping coefficient or the exciting force in different cases. Figs. 3–8 show the dimensionless the added masses, the damping coefficients and the exciting forces (notice that Fr =w0 is of dimension) with Rb as a parameter for d1 ¼ 0:2h1 , e1 ¼ 0:3h1 , R ¼ 0:5h1 . It is interesting to note how the common
1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0
Case 1 Case 2
0
0.5
1
1.5
2
2.5
f
(a)
7 6.5 6 5.5 5 4.5 4 3.5 3 2.5 2
6
R b=0.5 h 1 R b=0.8 h 1
λ 11/m0
µ11/m0
Fig. 2. Exciting force.
R b=1.2 h 1 R b=2.0 h 1
R b=0.5 h 1
5
R b= 0.8 h 1
4
R b=1.2 h 1 R b=2.0 h 1
3 2 1
0
0.5
1
1.5
f
2
0
2.5 (b)
0
0.5
1
1.5
f
Fig. 3. Added mass and damping for heave motion.
2
2.5
B.-j. Wu et al. / International Journal of Engineering Science 42 (2004) 1193–1213 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
R b =0.5 h 1 R b =0.8 h 1 R b =1.2 h 1
λ 22 /m0
µ 22/m0
1210
R b =2.0 h 1
0
0.5
1
(a)
f
1.5
2
2.5
2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0
R b =0.5 h 1 R b =0.8 h 1 R b =1.2 h 1 R b =2.0 h 1
0
0.5
1
(b)
f
1.5
2
2.5
(a)
0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45
R b=0.5 h 1 R b=0.8 h 1 R b=1.2 h 1 R b=2.0 h 1
0
0.5
1
1.5
2
λ 33/I
µ 33 /I
Fig. 4. Added mass and damping for sway motion.
2.5
f
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
R b=0.5 h 1 R b=0.8 h 1 R b=1.2 h 1 R b=2.0 h 1
0
0.5
1
(b)
1.5
2
2.5
f
Fig. 5. Added mass and damping for roll motion.
2 R b=0.5 h 1
Fh /w0
1.6
R b=0.8 h 1 R b=1.2 h 1
1.2
R b=2.0 h 1
0.8 0.4 0
0
0.5
1
f
1.5
2
2.5
Fig. 6. Horizontal exciting force.
properties transforms to remarkable counterpart against the radius of the caisson from Figs. 3–8. The figures show that the added mass, the damping coefficient and the exciting force almost increase as Rb increases when the radius of the caisson is shorter than a special value at any frequency. They also show that the curves of the figures have several extrema as the wave frequency is at the low frequency and the radius of the caisson is at some special values. The added mass, the damping coefficient and the exciting force are in oscillation and remarkable at the low frequency. The curves become common at the relative high frequency. To the authors’ knowledge, those
B.-j. Wu et al. / International Journal of Engineering Science 42 (2004) 1193–1213
1211
1 R b=0.5 h 1
0.8
R b=0.8 h 1
Fs /w0
R b=1.2 h 1
0.6
R b=2.0 h 1
0.4 0.2 0
0
0.5
1
f
1.5
2
2.5
Fig. 7. Vertical exciting force.
0.5 R b=0.5 h 1
0.4
R b=0.8 h 1 R b=1.2 h 1
Fr/w0
0.3
R b=2.0 h 1
0.2 0.1 0 0
0.5
1
1.5
2
2.5
f
(a)
5.5 5 4.5 4 3.5 3 2.5 2 1.5 1
3 e1 =0.3 h 1
e1 =0.3 h 1
2.5
e1 =0.4 h 1 e1 =0.6 h 1
λ 11/m 0
µ11 /m 0
Fig. 8. Roll exciting moment.
e1 =0.9 h 1
e1 =0.4 h 1 e1 =0.6 h 1
2
e1 =0.9 h 1
1.5 1 0.5
0
4
8
12
kR
16
20
0 (b)
0
4
8
12
16
20
kR
Fig. 9. Added mass and damping.
properties are the first reported. In Figs. 9 and 10, they show the added mass, the damping coefficient and the exciting forces against the gap h2 between the cylinder and the caisson when the cylinder is in heaving motion for d1 ¼ 0:2h1 , Rb ¼ 0:6h1 , R ¼ 0:5h1 . It appears from Fig. 9 and 10 that the values of the curves increase with the decrease of the gap. This is a common property. In all words, the influence of the caisson on the motion of the cylinder is obvious. Of particular the added masses, the damping coefficients and the exciting forces are remarkable as the wave frequency is at the low frequency and the radius of the caisson is at some special values.
B.-j. Wu et al. / International Journal of Engineering Science 42 (2004) 1193–1213
Fh / w 0
1212
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
eb =0.3 h 1 e1 =0.4 h 1 e1 =0.6 h 1 e1 =0.9 h 1
0
2
4
6
8 10 12 14 16 18 20 kR
Fig. 10. Exciting force.
7. Conclusions In this paper, by use of an eigenfunction expansion approach, we have presented the velocity potential expressions of the diffraction and radiation problem for a cylinder over a caisson fixed on the bottom in water of finite depth in the presence of a linear incident wave. The radius of the caisson is longer than or equal to that of the cylinder. We have proposed a set of theoretical added mass, damping coefficient and exciting force expressions. In order to check the involved expressions and programs, we have used two approaches to calculate the exciting forces and the results that match with each other well have indicated that the involved expressions are corrected and the programs are trustworthy. Finally, the analytical results of the added masses, the damping coefficients and the exciting forces are obtained for the different sizes of the caisson. The results presented have shown that the influence of the caisson on the motion of the cylinder is obvious. Of particular the interest in the results shown is that the added masses, the damping coefficients and the exciting forces are remarkable as the wave frequency is at the low frequency and the radius of the caisson is at some special values.
Acknowledgements This work is supported by the National High Technology Research and Development Program of China (no. 2001AA516010) and the CAS Pilot Project of the National Knowledge Innovation Program (KIP) (no. KGCX2-SW-305).
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