Hydrodynamic coefficients of a wave energy device consisting of a buoy and a submerged plate

Applied Ocean Research 14 (1992) 51-58

'

t--

Hydrodynamic coefficients of a wave energy device consisting of a buoy and a submerged plate Larry Berggren & Mickey Johansson Department of Hydraulics, Chalmers University of Technology, 412 96 Gothenburg, Sweden and Dynomar AB, Villa Sommarbo, 42700 Billdal, Sweden

(Received 26 November 1990; accepted 18 July 1991) Hydrodynamic coefficientsof a wave energy device consisting of a buoy connected to a submerged plate, are presented. Both the buoy and the plate are idealised as vertical cylinders. For this two-body system the case with the buoy oscillating vertically and the case with the submerged plate oscillating vertically are treated. The coefficientsare solved by the method of matched eigenfunction expansions. Numerical results showing the plate's influence on the hydrodynamic coefficients of the buoy and vice versa are presented.

I INTRODUCTION

pressure. In a two-body system this pressure will not only be experienced by the body in motion but also by the fixed body. This will give an added mass and damping coefficient for the body in motion as well as cross-terms of added mass and damping for the fixed body. In the present paper added mass and potential damping including cross-terms are calculated for the cases with vertical motions of each of the two bodies. The nomenclature of added mass and potential damping is also retained throughout the report for the cross-terms, which should be kept in mind to avoid confusion. The added mass of both the fixed and moving bodies is then defined as a quantity, that multiplied with the acceleration of the moving body, gives the part of the hydrodynamic force on the body which is in phase with the acceleration of the moving body. In a similar way the potential damping is defined as a quantity, that multiplied by the velocity of the moving body, gives that part of the hydrodynamic force on the body which is in phase with the velocity of the moving body. The wave energy device is idealised as two vertical cylinders of equal diameter. For such a geometry the problem is suitably solved by the method of matched eigenfunction expansions, a technique in which the fluid domain is divided into subdomains. In cash subdomain an eigenfunction expansion of the velocity potential is constructed. The technique has been used for both twodimensional problems (see e.g., Refs 2, 3 and 4) and axisymmetric three-dimensional problems (see e.g., Refs 5 and 6).

A buoy riding in waves, connected to a submerged plate by an elastomeric hose, has been proposed as a device for extracting energy from waves. 1 The elastomeric hose acts as a pump that is driven by the relative motion between the buoy and the submerged plate. The submerged plate is moored to the sea floor. The concept according to Hagerman ~ is described below. During the passage of a wave crest, the buoy heaves up, stretching the hose. The helical pattern of steel reinforcing wires in the hose causes it to constrict as it is stretched, thereby reducing its internal volume. This forces sea water out of the hose pump, through a check valve, and into a collecting line to a turbine. After the wave crest has passed the buoy drops down into the succeeding trough and the hose pump returns to its original length, restoring its diameter to its unstretched value. This increase in internal volume draws water into the hose through another check valve, which is open to the sea (see Fig. 1), In order to analyse the dynamics of the wave energy device, properties such as hydrodynamic coefficients have to be calculated. The problem is here formulated linearly which means that the coefficients associated with the harmonic motion of one of the bodies are calculated assuming the other body as fixed. The radiate waves associated with the body in motion cause dynamic Applied Ocean Research 0141-1187/92/$05.00 © 1992 Elsevier Science Publishers Ltd. 51

L. Berggren, M. Johansson

52

__

i_

the velocity potential Re{-ivJ(~b(r, z) exp(-i~ot) I (i = ~---1, e) = angular frequency, g - gravitational acceleration and t = time), where the spatial part of the velocity potential 4b is governed by the following boundary-value problem:

02d)

/ - Hose Pump

1 0 (rOqb

~3z2 + - r ~ \

0q$

~o2

c?z

g

04, __

/ ..- Valves a~F C°nnect2ngH°se

,

0z

0~ 0z

A

2

DampingPlate

0q~ #z

-~-

~ooring

04 0z

(2)

-

1

(z =

-dl,

r < R)

(4)

-

0

(z =

--el,

r < R)

(5)

-

0

(z =

-ez,

r <

(6)

-

0

(-di

< z < 0, r =

0-/ = 0

The geometrical properties of the idealised wave energy device is defined in Fig. 2. A cartesian coordinate system Oxyz as well as a cylindrical coordinate system OrOz is defined with the origin in the undisturbed free surface and the z-axis positive upwards. The buoy occupies the space defined by r ~< R, 0 ~< 0 ~< 2re, z 1> - d~, and the submerged plate occupies the space defined by r ~< R, 0 ~< 0 ~< 2rr, - e 2 ~< z ~< - e l . Small oscillating vertical motions are assumed. The flow is considered irrotational and the fluid incompressible. If the amplitude of the vertical motion of the buoy is denoted by ~ then the flow is suitably described by

(z = O,r >~ a)

hi)

2 FORMULATION

PROBLEM

(I)

(z

g0

OF THE

q$ = 0

0

0

Fig. 1. The hose pump concept. The elastomeric hose acts as a pump, driven by the relative heaving motion between a buoy and a submerged plate.

0r

=

~/

0~

(3)

R)

R)

( - e 2 < z < - e l , r = R)

lim ,,/7 ( &~ ,r b _ ikq$) = 0 w h e n ( r ~ ~ )

(7)

(8) (9)

where k is the wave number. In the case for which the plate oscillates the boundary value problem will be changed only slightly. The changes appear in boundary conditions (4), (5) and (6). They become 0--z =

0

(z = - d , , r

1

(z =

< R)

(10)

a¢ &

--

--el,

r < R)

(ll)

o¢ 0---z- =

1 (z = - e 2 , r < R)

(12)

3 SOLUTION

I

1 t

Fig. 2. Geometrical properties of the wave energy device and definition of fluid subdomains.

In the solution procedure the fluid domain is divided into three subdomains as indicated in Fig. 2. The method of separation of variables is applied in each subdomain in order to obtain expressions for the 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 = 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 = R. Mathematically this is fulfilled

Hydrodynamics o f a wave energy device by matching the potentials and the normal derivatives of the potentials, respectively. The formulation starts from the potentials developed independently in each subdomain. Applying the method of separation of variables gives the spatial potentials in ach region expressed in terms of orthogonal series. In region I the potential becomes ~, R.(2.r) 41 "~" n=, A n c o s 2n(g --I- h i ) gn(2nR----" ~

(13)

where the eigenvalues are given by 21 = - i k where k is the wavenumber k t a n h k h l = e~2[g n = 1 2. t a n h 2 . h l = -~o2/g n = 2 , 3 . . .

Rl(2,r )

H0m(i2,r) K(2.r)

=

R. (2. r) =

=

H(ol)(kr)

=

a,(i)

~

n=l n=2,3

I°(~"r)

1)rc/h2

(Z +

w2t,

el) 2 --

(2 +

--

dl) 2 --

(22)

(t)1 =

(~3,

--hi <~ z ~ --e2

(23)

042

-d~ < z < 0

-~r

--el

<

Z <

-d

0

--e 2 <

Z <

--e l

=

(24)

... (15)

0¢3

- h i < z < --e2

(16)

The boundary conditions above above are satisfied over the z interval in a least-square sense by multiplying each side of the boundary condition by a proper set of eigenfunctions and then by integrating them over the interval in question. Matching at r = R is achieved by the integrals following below, where the index k = 1, 2 . . . .

(17)

Boundary condition 1, eqn (22):

f--e~' 4I(R' Z){COS flk(Z "91- et)}dz

Boundary condition 2, eqn (23):

r2/2

2h2

(18) i = 2

4,(R, z){cos

(19)

= f£°42{cos2,(z W

7. =

(n-

(20)

+ f h, -~-'r

where the particular solution is given by frO (z + h,) 2 - r2/2 2h,

i = 1

(26)

fOh, W 041 {COS ~k(Z "~- h , ) } d z

~(i) W3p + ~ C. cos 7.(z + hi) I°(7"r) n=l I0(7. R)

1)n/h,

+ h,)} dz

Boundary condition 3, eqn (24):

43

3p =

(25)

= f--elI 42(R, z){cos ]~k(Z + e,)}dz

= I

i = 1

In region III the potential and corresponding eigenvalues become

4(i)

1

r2/2

2h 2 =

- e l ~< z ~ - d ,

&

where I0 is the modified Bessel function of first kind of zeroth order (p implies particular solution). The superscript (i) of the particular solution is used to identify which one of the two bodies oscillates. The superscript 1 refers to the case when the buoy oscillates and 2 refers to the case when the plate oscillates. The particular solution becomes .h(i)

42,

04,

t//2P "~- n=l B n cos fln(Z + el) io([3.R~---

ft. = ( n -

41 =

(14)

where Ho(1) is the Hankel function of first kind and zeroth order and K 0 is the modified Bessel function of second kind and zeroth order. In region II the potential and corresponding eigenvalues become:

42

satisfies the linear free surface boundary condition, the impermeable bottom condition and the radiation condition when r --* oo. The remaining problem is mainly to determine the three sets of unknown coefficients {(A,, B,, C,), n = 1, 2, . . . }. The three sets are found by imposing the boundary conditions at r = R. The requirements of continuity of pressure and normal velocity give the following conditions:

'0

and the radial function R. is given by

53

+ hl) dz

-e2 043 {COS ~k(Z J[-

hl)}dz

(27)

Introduce the following integral functions (21)

i = 2

The potentials given above describe the flow in the respective region and satisfy all boundary conditions except those at r = R. For example, in region I the potential

E(~,, ilk, h,, h a, z I , z2) =

f?I cos ~.(z +

ha)

x cos ~k(z + hp)dz

(28)

N(~., ha, zl, z2) = ff2 (cos ot.(z + h.))2 dz I (29)

L. Berggren, M. Johansson

54

w h e r e { a . , n = 1,2 . . . . } and (ilk, k = 1,2 . . . . } are two different sets of eigenvalues. Now. rewrite the boundary conditions, eqns (25)-(27) and introduce the integral functions (28) and (29). The following three sets of equations are then obtained:

Let the elements in the system matrix be denoted by S, and the elements in the right-hand side matrix by C, where i, j = 1, 2 . . . . . 3N. The elements in the two matrices are given by the boundary conditions as follows below, with local indices n, k = 1, 2 . . . . . N:

Boundary condition 1:

Boundary condition 1:

Sk. =

~ A . E ( 2 . , ilk, el, el, --el, --dl)

E(2., ilk, hi, et, - e~, - d,)

Skis+k) = = Plk + flkN(flk, el, - e l ,

-dl)

(30)

where the component associated with the particular solution is given by :

elk

f-a, W'2pI.Jt~,, z)cos B~(z + e l ) d z _ eI .~,),0

(31)

Fk

:

-- N (flk, el, --el, - d r )

Plk

(38a) (38b) (38C)

Boundary condition 2"

S(N+k). = E(2., 7k, h,, hi, --hi, --e2)

(38d)

S~u+~)¢2N+k) =

(38e)

-- N(yk, hi, --hi, --e2)

Boundary condition 2: F(N+k)

~ A~E(2~, 7k, hi, hi, --hi, --e2)

=

(38f)

P2k

Boundary condition 3:

n=l

= PEk + CkN(yk, hi, --hi, --e2)

(32)

l"oq3. R ) - ft. ~ E(fl., Xk, e~. h,. - el,

(2N+kXN +n) :

where

-

-

d,)

(38g)

P2k

:

f-e2 _h I .~(OlO "P'3p\*~'~ Z) COS yk(Z + h l ) d z

(33)

I~(y. R) S(2N+k)(2N+n)

:

-- ~n - -

Boundary condition 3: AkDkN(2k,

hi, hi, -hi,

O) =

x E(fl., 2k, et, ht, - e , ,

P3k + ~

.=,

-al)

[~tlBn

+ ~ 7.c. .=l

S(2N+k)k = DkN(2k, h,, - h , , O)

I~(~,.R___~)

F(2N+k)

I00.R)

(34)

f - d l ~rh(i) ~ ~'-"W2p COS2k(Z

+ hl)dz +

f-e2 ,q,h(i) ,~,e3p -h, Or

× cos 2k(z + hl)dZ

(35)

- - e2)

(38h)

Io(fl,,R)

where the component associated with the particular solution is given by =

E(7., 2k, hi, - h i ,

g (& R__~)

× E(7., 7k, ht, - hi, -- e2)

P3k

x00.R)

=

(38i) (38j)

P3k

Solving the complex system of equations gives the unknown coefficients in the orthogonal series and thereby also the potentials valid in each region. The forces caused by the motion of the structure are calculated by integration of the dynamic pressure given by the Bernoulli equation. In order to be consistent with the linear formulation the pressure is given by p =

0~ -p-~- =

i o ) p ~ ~;)

(39)

and

Dk =

I kHCo°'(kR)/HCo~)(kR)

k=

1

[,~kI'~O, kR)/KoO, k R)

k = 2, 3 . . . .

(36)

In order to find a solution we must truncate the infinite series of orthogonal functions. Assume that N is the number of orthogonal functions considered. We then get a system of 3N complex equations and an equal number of unknown coefficients. Organizing the equations in matrices gives SX =

F

(37)

where X

=

ig {Al, A2 . . . . . O9

BN, C i , C2 . . . . .

CN} T

As, Bt, BE . . . . .

where ~;) is the motion of the oscillating body. The force at body j, caused by an oscillation of body i, can be written as one part proportional to the acceleration of body i and one part proportional to the velocity as follows:

F ";) =

-- A~°)( ~° -- B~;:)~~°

(40)

The introduced quantities, A ~ij) and B ~ij), are the added mass and potential damping, respectively. The integration of the pressure over the respective body yields the following expressions for the hydrodynamic coefficients: At'') + iB°') O9

2rip f h2R2 Z R4/4 L 4h2

N ' RI, (ft. R) ] + .=IE B. ( -- 1)" ~ -3

(41)

Hydrodynamics of a wave energy device A (12) -1t- --iB(12)co =

--

RI, (ft. R)

R4 + ~_j

A (21) +

-o9

2,0

2 / ~ p [ n=l ~ C, ( - 1)"-'RI,(~.R)~,.I0(3,.R)

B. i0(,8.R)fl.

= 2rip

.=l

¢,o ¢~ 1 , 8 "

]

is (~)

=

2rip

B.

co

[~

Present method Method according to Yeung

E (42)

fl.Io(.13,R)

+

T'~22 (43)

A (22) + - -

55

C,

( - l)"-'~r,(,.R)

.=,

1,6r

"~ 1 , 4 ' N ¢1 cO •~ 1 , 2 ' c

E

~

3,.Io(3,.R)

O Z

\

1,0"

\

\

0,8 0,6

(44)

0,0

0,5

1,

1,5

2,0

2,5

3,0

kR

The hydrodynamic quantities given by the expressions above are the main results and will be presented in the next section.

(a) 0,6 o) c

4 NUMERICAL RESULTS

\

E0,5

As an initial cheek on the result, a comparison is made between the present solution and a solution for a single buoy. 7 The solution for the single buoy is based on a similar approach 5 to the one used here. In the comparison, the draught of the buoy to the water depth ratio d, ]hi = 0.25, and the radius of the bouy to the water depth ratio R[hl = 0.5. In the solution given here the plate was kept close to the sea bottom (d2/h, = 0 and h3/hl = 0.01) in order to not influence the added mass and damping o f the oscillating buoy. The comparison is shown in Fig. 3(a) and (b). In the following figures and tables the added mass is non-dimensionalized by the mass o f the water displaced by a semi-immersed sphere with radius R and the potential damping is non-dimensionalized by the same factor multiplied by the angular velocity co, i.e. 3A(iJ)

a °j) = b(ij) -

-

(45)

2nR3p 3A(U)

(46)

2rtR3pco

The maximum relative deviation, for the added mass, between the present method and the single buoy is less than 1.5%. The agreement is even better when comparing the potential damping. In Yeung s an approximative low-frequency solution is presented. The approximation, which is reported to give reasonably accurate predictions when R/h, > 1, is given by:

a°')(kR ~ O) = ~

4(h, - d,)

- hi ( - 3 , + In 2 - In

"

kR)

1

(47)

Present metho ¢lr Method according to Yeung k

e- 0,4

\

c O "~ 0,2 ®

E

o Z

,,....

0,0'

"~

0,0

0,5

1,0

1,5

2,0

2,5

3,0

kR

(b) Fig. 3. Coefficients of (a) added mass and (b) potential damping for the buoy associated with vertical motion (d,]h, = 0.25, R/hl = 0"50, hz/hl = 0.74, dz/hl = 0 and N = 50). Comparison with the solution of a single floating cylinder.

b°l)(kR --* O) = 3rcR 8h] I 1 + (kR) z x

(;

- 3, + I n 2 - l n k R

)1

(48)

where 3, is the Euler constant (3' ~ 0.5772). Hence it can be seen that the added mass at low frequency becomes logarithmic singular while the damping approaches a constant value. In the range when the approximation is valid, the present method gives results in reasonable agreement with eqns (47) and (48), (see Table 1). The truncation characteristics of the present solution are reasonable. Typically, in a wide frequency range, an increase from 10 terms in the series solutions to 50 terms gives a maximum relative deviation for a ('') of 11.2%. An increase from 20 to 50 terms gives a maximum relative

L. Berggren, M. Johansson

56

Table 1. Comparison between the present method and Yeung's approximation for low frequency. R/h~ = 2, d~/h~ = O, dz/h~ = O, h~/h~ = 0"01

kR

Yeung

2,0! I

~=

Present method

,

3-2e -3.2e -3-2e -3.1e-

4 3 2 1

12.62 9-169 5.715 2.255

b(H)

2"356 2.356 2.344 1.826

a01)

12.84 9.393 5.929 2.395

0150 0300 0600

1,0

~

N a(l~)

a11, 62161 a11, h2161 a11, 62/61

b(l])

~

~ 0,5 .o c ~ o,o

2.356 2.356 2-347 1.982

Z

~-~---. . . . .

t

• -

a12, h2/hl ~ 0.600 a12, 62161 = 0300 a12, 62161 = 0.160

-

, Z]===

2_ 'r

-I-

I

I

5

6

L"

o,5

,

.........

1,0 1

2

3

~

4

7

kR

deviation of 3.9%. Finally, an increase from 30 to 50 terms gives a m a x i m u m relative deviation o f 0.5%. The magnitude o f the h y d r o d y n a m i c coefficients o f both the b u o y and the plate, as a consequence o f the m o t i o n s o f the b u o y and the plate, respectively, are s h o w n in Fig. 4(a)-(d). The different curves s h o w the different distances between the two bodies. The geometric configuration was given by d~/h~ = 0.1, R/h~ = 0.2 and d2/h~ = 0.1. The distance between the two bodies varied with h~/h] = 0.15, 0.3 and 0.6. It appears from Fig. 4(a) and (c) that the added mass of the plate is greater than that o f the buoy i.e. a ° ~) < a ~2~. This is due to the fact that the plate is surrounded by the fluid while the buoy only has fluid on its underside. On the other hand, potential damping o f the b u o y is greater than that o f the plate, i.e. b (H) > b (22), which is due to the fact that the submerged plate radiates less waves than the buoy. This was investigated numerically by integrating the pressure in phase with the velocity on the top face and bottom face of the plate, respectively. These integrated forces were found to be of the same magnitude but with opposite signs. In the diagram one can also see that the added mass approaches a constant value when kR grows to infinity and the potential approaches zero. The asymptotic value of the added mass in infinitely deep water is simply given by a(l~)(kR--* ~ ) = 2/n (see Miles)) Due to the presence o f the sea b o t t o m and the submerged plate, the asymptotic added mass will exceed the infinitely deep water value. Furthermore, from the numerical result it appears that a " n = a (-") and b ( m = b (~), respectively. That these relations are valid can be shown with Green's theorem by applying it on the surface that surrounds the fluid domain (the wet surfaces of the buoy, the free surface of the water, the bottom and a surface that connects the free surface and the b o t t o m surface) and the surface o f the plate. 9 If the plate oscillates close to the b u o y or close to the b o t t o m , the added mass will grow to infinity as the distances h~ and/or h3 approach zero. Also, this is in agreement with the potential theory (the potential theory however is simply a proper model for real conditions as long as hi/6 > 1 with i = 2, 3 and 6 = the thickness o f the Stokes' boundary layer~°).

(a) 0,4

"~"

I I ! t

O, 3 •

8 "~ -.~

_

0,2

.

~ -----0----

b11, h2/61 = 01"50 011, 62/61 - 0.300

i

011, 62/hl = 0.600

k E

o,1

"~ ®

~

.

0,1

i t

70 -0,2

i 1

2

~

~ -

0

i

i

|

g o,o~j~

~

I

.y

I

-

3

I

~'"

012, 62/61= 0600 012, 62/61 = 0300 b12, h2161 = 0160

l .... 1.... I

i

4

7

5

6

kR

(b) 2,0

16 "-

i,

.2,01 _l__ '- o3oo o15o]" " ""j

_i' - .22 - "-'

. . . . . . . . . .

1,0

,2,,i -

0

a22, 62/61

=

a21, 62/hl = 0.600 . . . . . . . a21, 62/61 = 0300 a21, h2/hl = 0150

0 600 --'--'--

] _--

0,6'

-----o.-,-- -

o

j 0,0,

0,5

-1 ,0

-7I 0

1

t

J

i

4

6

6

J__

lI 2

tI -

3

7

kR (C)

0,10 -'

io22, h21hl '~ 0156 022, h2/61 = 0300

] ]

022, 62/61 = 0 6 0 0 - - - ' |

i o,oo~

L _J___i__j_ _1

~..--

.

.

.

.

.

.

_0,051 '1 / "~ -0,1

0

Z

.....

: °2°0°--

021, 621hl = 0160

~0-0,15 -0,20

I .... 0

I • 1

4

5

6

7

kR

(d) Fig. 4. Added mass ((a) and (c)) and potential damping ((b) and (d)), (a) and (b) as a consequence of the motion of the buoy, (c) and (d) as a consequence of the motion of the plate.

Hydrodynamics of a wave energy device Table 2. Added m a s s as a consequence o f

diminishing h2/h I where

57

R / h i = 0"2, d l / h I =

d2/hl = 0"2, N = 30,

k R = 3"19 x 10 -3

h2 / h i

ao o

aO2)

a (2J)

a(zz)

3 R/16h 2

0-010 O0 0"001 O0 0"000 10 0"000 Ol

5'365 6 39"099 376"60 3 751 "6

- 4"586 3 - 38"291 - 375'79 -- 3 750-8

-- 4'256 3 - 38"291 - 375'79 - 3 750"8

5"349 9 39-081 376"59 3 751-6

3.750 0 37"500 375.00 3 750'0

It should be pointed out that some care must be taken when calculating the hydrodynamic coefficients for small values o f h2 and/or h 3 . As h2 and/or h 3 approach zero, the eigenvalues ft, and ~, grow to infinity. If we calculate the coefficients in eqn (22), one obtains

f~edI q ~ l ( R , "x

z){cos flk(z + el)}dz

I2,{sinh (hi -- dl) -- sinh(hl - el)}

n=l n>l (49)

f-el' 4~°m(R' Z){COS/~k(Z+

e,)}dz

result as in Ref. 11. Here, the added mass was estimated by calculating directly the kinetic energy of the fluid in the narrow space between the bottom of a circular platform and the sea bottom under the assumption of uniform, radial flow. A comparison was made between the general formulation (eqns (41)-(44)) and the narrow spaced approximation (eqns (53)-(56)) for decreasing values of the ratio hz/hl (see Table 2). The difference between a °~) and a (12)when h2/h~ = 0.01 is mainly caused by the contribution from the potential q~3to the added mass a (12). If we separate the added mass, a °2), into two parts, one for the upper side and one for the lower side of the submerged plate we get: a(ulp2)perside

=

~'sin_ 2gn(_n- 1) [ 2 g ( n - 1) + 1

(50) n > 1

=

a02) upper side

with h2/hl

=

0"01

-3751-6 with h2/h~

=

0.00001

--5"3292

The calculations indicate that when h2/h~ ~ 0 then ,.~(12) lao 1)[ = -uppersdr I

f--e~' dP~a~t(R' z){COS flk(Z + e,)}dz

=

f~ h-~R(2-/14) "-l+ h~/6

n = 1

[.(-n---- 1 - ~

n > 1

5 CONCLUSIONS (51)

where H o m = homogeneous solution and Part = particular solution. If we solve the matrices in eqn (37) for small values of h2, the terms B~ will approach B.

= R2/4h2

n = 1

"~h2

n>

(52)

1

and terms of higher order than 1 can be neglected for small values of h2. If we substitute this relation into eqn (41) and calculate the added mass, one obtains A0t) =

2 n o [ h 4 R2

1P

R4 ]

16h-----2+ ~

-

~zpR4 8hz

(53)

ACKNOWLEDGEMENTS This research has been carried out at the Department of Hydraulics, Chalmers University of Technology, Sweden. Financial support was provided by the National Energy Administration, Sweden (STEV).

In a similar way we get A 02) =

~pR4 8h2

(54)

A (21) =

~zpR4 8h2

(55)

A(22) =

8

When either h2 o r

This study treats the radiation problem of a two-body system, where one of the bodies is submerged and the other floats. The problem is solved by using the method of eigenfunction expansions. Reasonable agreement with existing solutions of a single cylinder is stated. The present solution converges acceptably when including 30 terms in the eigenfunction expansions (in the tested cases the relative deviation is less than 0.5% between a solution truncated at 30 terms and one truncated at 50 terms). A narrow space analysis is performed. The derived formulas are in agreement with formulas derived directly from energy relations.

REFERENCES

h2 + ~ h3

(56)

approaches zero, one gets the same

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58

L. Berggren, M. Johansson

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7. Johansson, M., Transient motion of large floating structures. Lic. Eng. Thesis, Report A:I4, Department of Hydraulics, Chalmers University of Technology, G6teborg, Sweden, 1987. 8. Miles, J.W. On surface-wave forcing by a circular disk, J. Fluid Mech., 175 (1987) 97-108. 9. Srokosz, M.A. & Evans, D.V., A theory for wave-power absorption by two independently oscillating bodies. J. Fluid Mech., 90(2) (1979) 337-62. 10. Mei Chiang, C., Effects on a narrow gap between a bottomseated structure and the sea-floor. Appl. Ocean Res., (1) (1987) 51-2. 11. Bergdahl, L. & P~ilsson, I., Marine operations of detachable production platform. Paper presented at 10th International Conference on Port and Ocean Engineering under Arctic Conditions, 12-16 June 1989.