J Mar Sci Technol (1997) 2:135-147 Journal of

Marine Science and Technology �9 SNAJ 1997

The hydrodynamic forces acting on a cylinder array oscillating in waves and current

TAKESHI KINOSHITA 1, WEIGUANG BAO 1, and SHUNJI SUNAHARA 2

i Institute of Industrial Science, University of Tokyo, 7-22-1 Roppongi, Minato-ku, Tokyo 106, Japan 2Department of Ocean Engineering, University of Tokai, 3-20-10rido, Shimizu, Shizuoka 424, Japan

Abstract: The problem of the interaction of multiple cylin- ders oscillating in waves and slow current is considered. The interaction is represented by waves emitted from adjacent cylinders towards the cylinder under consideration. Wave drift forces and moment in the horizontal plane are calculated by the far-field method based on the conservation of momen- tum or angular momentum. A semianalytical formula for the calculation of the wave drift damping is then deduced. The conservation of the integrals in these formulae is proved. Special treatments to improve the accuracy of results are discussed. Comparisons between calculated results and ex- perimental measurements are made, showing satisfactory agreement. Effects of various combinations of current direc- tion and incident wave angle on the wave drift damping and damping moment are also examined.

Key words: wave drift damping, slow drift motion, floating structure, cylinder array

Introduction

Offshore structures, often consisting of vertical cylin- ders and pontoons, undergo slow drift motions when moored in ocean waves. Because the restoring force supplied by the mooring devices is quite weak, reso- nance with a large horizontal excursion may be trig- gered by slowly varying nonlinear wave excitation. Precise prediction of such motions, in which the wave drift damping plays a key role, is important in the design of the mooring system. Since the velocity of the drift motion is small compared with the typical phase veloc- ity of the incident waves, it is reasonable that the slow drift mot ion can be approximated by steady forward

Address correspondence to: T. Kinoshita Received for publication on Nov. 13, 1996; accepted on March 28, 1997

mot ion or equivalently by imposing a uniform current onto the wave field. Fur thermore , current up to l m / s is often observed in a sea environment in addition to the incident waves. Previous work shows that the drift force may be significantly increased by the current effect. 1.: Therefore , it is worth investigating the nonlinear inter- action of waves and current with multiple cylinders.

Considerable effort has been made in the study of the interaction between waves and slow current during the past decade. 2 4 Recently, Kinoshita and Bao 5 t reated the p rob lem of a truncated circular cylinder free to re- sponding to the linear wave excitation in all six oscilla- tion modes (surge, sway, heave, roll, pitch, and yaw).

In the present work, the interaction of an assembly of circular cylinders with regular waves and slow current is considered. The cylinders are free to responding to lin- ear wave excitation in all six mot ion modes but re- strained f rom drift motions. In the sections that follow, the boundary-value prob lem is formulated based on the potential-flow assumption. The solutions, which are ex- pressed in eigenfunction expansion, are discussed. The interaction among cylinders is represented by addi- tional waves emitted f rom adjacent cylinders towards the cylinders under consideration. The approaches of Kagemoto and Yue 6 and Linton and Evans 7 are ex- tended to the problem of interaction between waves and slow current. Hydrodynamic forces, including excit- ing forces, wave drift forces and wave drift damping, are given. The horizontal drift forces, as well as yaw mo- ment, are evaluated by the far-field method based on the conservation of m o m e n t u m and angular m o m e n t u m which are confirmed to agree with the results calculated by the near-field method by Kinoshita and Bao. 5 The wave drift damping is then deduced f rom these results by a semianalytical formula. The contribution f rom the second-order steady potential which is quadratic in wave ampli tude is also considered. Calculated examples are presented in various combinat ion of wave direction and current direction. Those results are also compared

136 T. Kinoshita et al.: Hydrodynamic forces on a cylinder array

with available experimental measurements to verify the present theory.

Formulat ion of the problem

The problem to be considered here is that of an assem- bly of circular cylinders with radius ap and draught dp floating in a water of depth h, where p = 1-P and P is the total number of cylinders. The cylinders do not neces- sarily extend to the sea bottom, i.e., dp < h, and they are not overlapped, which means the distance Spq between two cylinders, say cylinder p and cylinder q, is greater than the sum of radii, i.e., Spq > ap + aq. In addition to a uniform current of speed U, the cylinder is exposed to a long-crested regular wave of small amplitude with fre- quency coo and incident angle/3 referring to the positive x-axis. The cylinders are free to responding to the incoming waves, but restrained from drift motions. A right-handed coordinate system o-xyz is defined as a global coordinate. The plane z = 0 coincides with the still water free surface and the z-axis is positive up- wards. The x-axis points in the direction of uniform flow so that the current is moving in the positive x direction. In addition, a local coordinate system Op-XpypZp is also adopted for each cylinder; this local system is parallel to the global system, but the origin is located at the center of the cylinder (see Fig. 1).

The viscous effect is neglected and the flow is irrota- tional. Thus there exists a velocity potential q~r which satisfies the Laplace equation in the fluid domain V and the impermeable condition on the sea bottom z = -h.

Two assumptions are made in the present work. First, the amplitude of the incident wave is small corn-

I"

X p p Xp

X q ~ O"

Fig. 1. Definition of global and local coordinate systems

pared with the wavelength. Second, the current velocity U is small relative to the phase velocity of waves so that terms of the order of U 2, described as O(U2), are negligible.

To linearize the problem, the total potential ~ r is then decomposed to a uniform flow, a steady distur- bance potential r a first-order unsteady potential r and a second-order steady potential ~2), according to their order proportional to the incident wave slope e. The first-order unsteady potential r is in turn sepa- rated into radiation potentials and diffraction potential, i.e.,

q~r(x, t )= Ux + Ur Re -ia) ~jr

+

where Re means the real part will be taken and the encountering frequency is defined as co = COo + Uko cos/3, where k0 is the wave number and g is the acceleration of gravity. r ( j = 1-6) represents the radiation potentials corresponding to the six modes of oscillation motion (surge, sway, heave, roll, pitch, and yaw, respec- tively) and ~j is the complex amplitude of oscillation in mode j. 0~ ) indicates the diffraction potential which in- cludes a scattering potential ~1) and an incident wave potential q~(01). The latter is given by

r cosh 0(z+ )exp[i,,0r os(0_, )] cosh k 0 h

In Eq. (2), r and 0 are the cylindrical coordinates of the field point.

Here, two steady potentials exist. The first one, de- noted by -r is the disturbance of uniform flow caused by the body and not relevant to the wave. The other one is the second-order steady potential ~(2), which is of qua- dratic order in wave slope. The latter is used in the evaluation of wave drift damping.

Each potential decomposed in Eq. (1) satisfies the same governing equation and boundary condition at the sea bottom as the total potential. The boundary conditions on the free surface and the body surface satisfied by these potentials are exactly the same as in the case of a single cylinder, as described by Kinoshita and Bao. 5

Each mode of first-order unsteady potentials and motions are further expanded into a power series of %= o)oU/g, i.e.,

,f1,1 (3a) 'I -- ol'~ +

~ ) = g j +~ogj +

T. Kinoshita et al.: Hydrodynamic forces on a cylinder array 137

where the first number in the superscript indicates the order referring to the wave slope while the second num- ber refers to the order of %. If only one order is men- tioned, it will refer to the wave slope. This notation will be maintained throughout this article.

Substituting this expansion into the boundary condi- tions and collecting terms with the same power of %, we can obtain the boundary conditions satisfied by poten- tials with different orders. They take the same form as in the case of a single cylinder if expressed in local coordinates. Readers are referred to our previous work 5 for the details.

Solution to the problems

In this section, we discuss solutions to the boundary value problems for each order of potential. The diffrac- tion and radiation properties of each cylinder can be obtained by solving a single-cylinder problem as shown by Kinoshita and Bao. ~ The main idea to solve the prob- lem of interaction among multiple cylinders is that the interaction is regarded as some additional incident waves emitted from the adjacent cylinders towards the cylinder under consideration, say cylinder p. These waves are in turn diffracted by cylinder p. The diffracted waves, together with radiated waves from cylinder p in the radiation problem, advance to the adjacent cylin- ders, and so on. We are going to start with the steady disturbance potential, followed by the first-zeroth order and first-first order potentials.

Solution of the steady disturbance potential

Since the steady disturbance potential ~ decays quickly as the distance from the cylinder increases and the Kelvin waves are neglected in the present work, the interaction among multiple cylinders is not taken into account for this steady potential. This may be consid- ered inconsistent compared with the interaction of eva- nescent waves to be taken into account later in the first-order problems. However, in our experience, the disturbance to the uniform flow caused by a cylinder is small enough to be neglected when the distance from the cylinder is greater than three times the cylinder radius. Therefore, the corresponding solution of single cylinder is used near each cylinder.

Expressed in the local coordinates, the steady poten- tial caused by the disturbance of cylinder p to the uni- form flow can be written in a form as follows in the region of rp > ap:

~:apArRl(rp)Z(z)cosOp (4a)

where the diagonal matrix R~ and vector Z are given by

ap 0 "'" 0 re o :

" . "'. 0

o o

(4b)

Z / ~ [2c~ z+h z - - ( )

~ ) [2cos#[(z+h) (4c)

with Xm = rn~/h (m = 1, 2 . . . . ). K~ in Eq. (4b) is the modified Bessel function of the second kind of order 1.

The unknown coefficient vectors A in the expansions can be determined by solving the steady-flow problem for a single cylinder.

Solution of the first-zeroth order potential

The method to solve the first-zeroth order problem is an extension of the approach suggested by Kagemoto and Yue 6 as well as Linton and Evans 7 for the case of trun- cated cylinders�9

The total diffraction potential of first-zeroth order (j = D) in the vicinity of cylinder p can be written in the local coordinates centered at cylinder p as

= ~ P T ~ P Z inOp 0~ ~ Ao, [R,(rp)+ BT,,R,(rp) ] (z)e (5a)

where/~, and R, are diagonal matrices and Z(z) is a vector, which are defined respectively as follows:

0 0

0 :

" " . "'. 0

0 0

(5b)

0 0

0 :

"" "', 0

0 0

(5c)

l cosh ko(z +h)/coshkoh-

Z(z)= c~176 cos 2(z+ )/cosk h

(5d)

138 T. Kinoshita et al.: Hydrodynamic forces on a cylinder array

Here, J, and I~ are the Bessel function and the modified Bessel function of the first kind of order n, respectively, while H,, and K~ denote the Hankel function of the first- kind and the modified Bessel function of the second kind of order n, respectively. The term km is the real solution (when m = 0) or pure imaginary solution (m > 0) of the dispersion relation v0 = o)Zo/g = k tanh kh. The matrix BP, relates the incident wave amplitude to the amplitude of the wave diffracted by cylinder p, which can be obtained by solving a single cylinder problem and regarded as known hereafter. In the expression the superscript T denotes the transposed matrices or vec- tors. The unknown coefficient vector AP, represents the total wave amplitude moving towards cylinder p, which is the sum of the incident wave and waves emitted from the adjacent cylinders. Thus A~, satisfies the following linear algebraic equations:

=Ao~+~.Xi 'TPqB q~aq ( p = l - P) (6) A~. ~ p z., t. 71 "'or q=l l = - ~ q*P

where .40P, = (EPe ~"t~/2-p}, O, 0 , . . . ) r is the incident wave amplitude with E p = exp[ikorPCos(O~ - fl)] as a phase lag referrd to coordinate p. Here, the global location of cylinder p is denoted by (r p, 0g). The matrix Tt pq in Eq. (6) is given by

As soon as the interaction wave amplitude AjP (j = 1-6, D) is solved, the first-zeroth order potential can be obtained by Eqs. (5a) and (8) in the local coordi- nates. It should be mentioned that these forms ensure the satisfaction of the body boundary condition by the properties of B p (j = 1-7).

So far, all the derivations are made by infinite systems. To carry out numerical calculation, they must be truncated at some proper number for the subscripts l, m, and n according to the required accuracy.

Solution of the first-first order potential

The free surface condition satisfied by the first-first order potentials is no longer homogeneous. The inho- mogeneous terms on the right-hand side of the free surface condition [see Kinoshita and Bao, 5 Eqs. (10a) and (10c)] are known quantities once the steady distur- bance potential ~ and the first-zeroth order potentials q~l 1~ (j = 1-6, D) are solved. They can be divided into two parts

fP=f f i+ f~2 ( j= l -6 , D a n d p = l - P ) (10a)

with

', ,,(i,,s,,,) r I;'; = i

(7)

0 01 (-0"'r 0) .. "'. 0 " e4~ ~ ... 0 (-1)"K, ,(k,,,Seq)

which comes from Graf's addition theorem s for the Bessel function valid when rp < Spq. Here, (Spq, apq) is the location of cylinder p relative to the local coordinate q (see Fig. 1).

In a similar way, the radiation potential of mode j in the neighborhood of the cylinder p can be expressed as

~ p T ~ ~I;~ = =~_~[A~n R~(rp)+

-pr p P)Rn(r p)]z(z)e inG (j= 1-6) (8) Aj, B7n + B j,,

where the vector B p, (j = 1-6) is the radiated wave amplitude generated by oscillation of cylinder p in mode j and is determined in the problem of a single cylinder. The unknown interaction wave amplitude AjP can be obtained by solving the following linear algebraic equations:

~ s ( q T - q q ) ( p = l - P , j = l - 6 ) AjP = TFn q B7 n Ajz + B i t q=l l = - ~ q.o (9)

_ . 3 (1o ) f j P - 2 ( k 0 cos fl+t O---71~j p (10b)

f J = 2iV~p "V~I~ ~ --l~)jp" (10)02~p0z 2 (10C)

where the superscript or subscript p emphasizes that all the potentials are expressed in the local coordinates of cylinder p.

To solve the first-first order problem for each mode j (j = 1-6, D), a particular solution is sought which satis- fies the inhomogeneous free surface condition as well as the Laplace equation and boundary conditions at sea bottom and at far field. Regarding the particular solu- tion as a kind oL incident waves, a general solution is composed by adding the homogeneous solution to the particular solution. The general solution accounts for the interaction among cylinders in a similar way to the first-zeroth order problem.

The particular solution also has two parts, ~(H) and "t" jpl ~(11) corresponding to the inhomogeneous terms f~ jp2 ' and f~, respectively, which will be solved separately. To solve for ~(11) the method suggested by Emmerhoff "r jpl , and Sclavounos 2 is followed, i.e., a derivative operator is applied to the first-zeroth order potential:

~(11) ( ~____~) ~9~};0) (11) jpl =2 ko cos fl + i 3Vo

T. Kinoshita et al.: Hydrodynamic forces on a cylinder array 139

To obtain a particular solution for A(I1) the inho- "r jp2 ' mogeneous term f~ is regarded as a pressure dis- tribution over the free surface. It is expressed in a series expansion as well:

fP2 = ~ "P / ~ i,,o.,, (12) ]j2,,[rp ) e

The definition of f~,,(rp) was previously described by Kinoshita and Bao, s Eq. (18d).

Then, by using the Green function given by Wehausen and Laitone, 9 the particular solution around cylinder p can be obtained by the follwing integral over the free surface:

- dO' fr2(r', o')c(r , , z; r', O')r'dr"

1 s I P ( P ) ( )r'dr' (13) =-- ei'% f j2,, r g, rp, Z; r" 2n=-~ ap

where the Green function is also expanded into a series a s

with g,, given by

o(r. z; r')=

(14a)

(14b)

In the above expression, the matrices/~,, R,,, and Z have the same definition as before while the argument r> (or r<) indicates the larger (or smaller) one between r and r'. The coefficient vector C is defined in Eq. (20c) of our previous work. 5

This solution satisfies the ordinary Sommerfeld radia- tion condition due to the fast decay of the steady distur- bance potential as r increases.

It should be pointed out that the integral over the free surface in Eq. (13) has to be made with some caution. The integrand contains gradients of the first-zeroth

0 (1~ [see Eq. (10c)] which includes order potential _jp terms transferred from adjacent cylinders. The trans- form is only valid when the distance from cylinder p does not exceed the distance to the adjacent cylinder, i.e., rp < Spq (q = l -P , q ~p). However, the integral in Eq. (13) is to be extended to infinity, which will certainly violate this requirement when the integral dummy vari- able r' runs over Spq. Therefore another form of the addition theorem of Bessel functions should be used as rp > Spq, i.e.,

J, 0 . 0 ] i;;" = i ' - . o . e,i,-,,lo,,,,

. . . o

(15b)

After obtaining these particular solutions the general solution for the first-first order potential 0 (m (j = 1-6, -jp D) can be written as follows:

4; 2. =O,p, + - pT P "}- T

( p = I - P , I = I - 6 , D) (16)

where the vector PP represents the diffraction of the particular solutions 0~ (11) and 2,m) which are "r jpl "r jp2 regarded as a kind of incident waves. Meanwhile, the vector/5~ is the radiated wave amplitude due to the interaction between the oscillation of cylinder p and the steady flow as shown in the boundary condition on the body surface in Eq. (10b) in Kinoshita and Bao. 5 These quantities can be obtained by solving a single cylinder problem and regarded as known in the present work. In the case of the diffraction problem, PP, is trivial. The unknown interaction coefficient vector /ip can be obtained in a similar way to the first-zeroth order problem by solving the following linear algebraic equations:

ejPn ~-- ~ s wpq(In ~U7ll~qr ~Jll-:~Pa--"-JIJPP _{_ i~j~) (p= l-P, j= 1-6, D) q=II=-~ q~p

(17)

Forces and mot ions

In this section, methods to estimate forces acting on the cylinders will be discussed. The first-order forces, i.e., the added masses and damping coefficients as well as wave-exciting forces, are calculated by means of direct pressure integrating. The wave drift forces (moment) are evaluated by the far-field method based on con- servation of momentum. The wave drift damping or damping moment is then obtained by semianalytical formulation.

~ pq

tt = -~

where the diagonal matrix ~ pq Tt, is given by

(15a) Hydrodynamic forces and wave-exciting forces

Once the potentials are solved, the hydrodynamic forces can be evaluated by integrating the hydrody-

140 T. Kinoshita et al.: Hydrodynamic forces on a cylinder array

namic pressure over the mean wetted body surface of each cylinder�9 The hydrodynamic pressure is kept up to the order of wave slope�9

F • I ) = iwl.to _ A.ij �9 (lO) (1o)

p=, +r0Gsp )-UW.V(ajp ]nids

(i, j: 1-6) (18)

where/,tq and 2q are the added mass and damping coef- ficient, respectively, Sop denotes the mean wetted body surface of cylinder p at rest, and/9 is the density of water.

As with other quantities, we expand the added masses and damping coefficients into the first-zeroth order and first-first order according to r0. They are de- fined as follows:

l('OO]'giJ�9 (10) (10) p~lS S (10)'~ "4~ltiU,~ 1-6) (19a) -- Xi] = pio) o o ()jp (i, j= =

_ (lO) �9 (H) (H) . ~ e ( (11)_iv01 W.V~ajp /}nids I('O0]'Zi] -- ~biS = p,o,0LJs0.[*s.

(i, ] = 1-6) (19b)

To evaluate the wave-exciting forces, it is only re- quired to set j = D in Eq. (18) and integrate over the mean wetted surface:

For convenience in the evaluation of the wave drift damping later on, the motion amplitude is also expanded into a power series in T0 as mentioned previously�9 Each order of the motion amplitude is ob- tained by solving the corresponding order of motion equation, which is formed by grouping the terms with the same order in the equation of motion with respect to T0. The first-zeroth order equations of motion are the same as those in the ordinary wave problem without current:

6 X I _ ( . o 2 1 M i j J -it.- . ']-~liJ (10) , . (10)) -'('O0/bij '~-cij ~j / ] ~.(10)

= Fg~ = 1-6) (23a)

For the case of first-first order

~[ - 2/" (10) "~ . ~(10) ~ ~-(11) /-<,,o tM, , +,,,, ) - l (/~o.,#l,i/. + c i j ] ~ j

j = l L ~ /

= F ! n) + P,,,2,,(n) + im ~(11) G,J::(l~ L~O-iS o'~q j-s

2kocosfl[o)2(Mi ' (10)'~. ,(10)],00 ) v0 L , )+,c00,t,j jcj

(i = 1-6) (23b)

P FI~) _ P__g~Oto)o ~ fSo~[iog)(~ _UW. V$(~ ]nid s = ( i : 1-6)

(20)

which is also expanded according to the order of v0:

p~_l IS0p ~Dp~(10) .hi (i 1-6) (21a) F. 1~ = Pg~o as =

Wave drift force (moment) and wave drift damping The wave drift forces (moment) can be calculated either by t h e so-called near-field method or by the far-field method�9 In the present work, the far-field method is adopted to evaluate the drift forces or moment.

According to Grue and Palm, l~ the horizontal wave drift forces can be calculated by

,~P t" ((11) kocos]~ .(lO) F!ll):Pg~op~=)So~Oop_ + Vo COp

. ( l O ) ] , -• )ni s (i=1-6 / (21b)

Oscillation motions Once these first-order forces are obtained, the complex amplitude of the motions, i.e., s (j = 1-6), can be readily solved from the equation of motion:

~[-oj2(Mij...~-l.lij)-ioJ~ij-}-cij]~ll)=F! 1) (i=1-6) (22) l=l

where Mq and Q are corresponding components in the mass matrix and restoring matrix, respectively�9

[ ar ar * F!2)- p g ~ 2 Re Ss C V~ (1)- V~(1)*ni a x i a n

4v 0

a x i aJvl Ic< (p(1)~JO )*n i dl

+ ai 2 2imU fcc ~(1) a~ (1)* ] g ----~dl} (i= 1,2) (24)

where Sc is a control surface and Cc is its water line. The subscript i = 1 indicates the x (surge) direction and i = 2 means the y (sway) direction The term x~ denotes the coordinate x or y according to i = 1 or 2. The superscript asterisk (*) represents the complex conjugate and 6,.2 is the Kronecker delta function, i.e., ~2 = 0 if i r 2 and •i2 = 1 if i = 2.

T. Kinoshita et al.: Hydrodynamic forces on a cylinder array 141

The drift yawing moment is given by

a0(1 ) 00(1) * ~ 2 ) - pg~g Re -Is .ds

2v 0 , O0 On

/ )* + iwU Ic Y 04(1) 04(1) nl O0)dl g ~, On c~0

+2v ov o Is~ y On ~?0 n~ d (25)

[ G~0(10) O')0(11) * ff!2q-pgr ~ fs~ V0(10)'v0(ll):gFIi oTXi On

O~0(11)* 0"10(10) - ds + Ic ( v~ 0x~ On

+k o cos/34(1~176 fll + S~2i~c~ 4(.0) 04(1~ dl}

(i= 1, 2) (29a)

In Eqs. (24) and (25), 0 o) represents the total first- order unsteady potential omitting the time factor e -~ and normalized by g~o/i~, i.e.,

0(1) = 0(D1)(X)q 0)0)O 6 (1).,

which is further expanded into the power series of

0 (') = 0(1~ + r0001) (27a)

The first-zeroth and first-first order potentials are given as follows, respectively:

6 4(10) th(10)/-,r ..1-, '~'~ r(l~176 (27b)

=~'D ~ ] - - - o d . , ' ~ j ~'i L ~1 j=l

6((10) (11)_,.~(1')0510)(X) 0(11) = 0(oll)(X)"k-V0 ]~__ l ~j Oj (~]'r~j

k0 cos/3 (1o) (1o) _~ + v0 ~j 0j (*)) (27c)

Then the wave drift forces and moment are expanded into zeroth-order and first-order drift force referring to the order of Vo as with other quantities.

The zeroth order (it should be called the second- zeroth order) drift forces and moment are expressed as

I 00(1~ 00(1~ 7 P!2~ Pg~~ R e 4 v ~ Is~ V0(10)-v0(l~ O~i --~ 'Jds

+120IcO(l~176 ~ ( i= 1, 2) (28a) J

04(10) ~0(10) * } ff~20)_ pg~ Re -Is ds (28b)

2v 0 c O0 On

which are quantities without current effects. The contribution from the steady flow appears in the second-first order forces and moment as shown in the following:

04(10) a0(11) * if!2,)_ pg~2 Re -Is ds

2v 0 ~ O0 On

/ /* c?0(1~ 6~0(1~ 0(10)dl + ilc,: y On 00 nl

+zV~ -& ao nl ds (29b)

There are two parts contributing to the drift yaw moment. The first one, represented by the first two integrals in Eq. (29b), is the contribution from the first- order potentials, while the second one, given by the last integral in Eq. (29b), expresses the contribution from the second-order steady potential ~(2). These two parts will be evaluated separately. Grue and Palm( lm sug- gested that the contribution from the second-order steady potential can be calculated without explicitly solving this potential. By applying the Green identity and the boundary conditions satisfied by the second- order steady potential on the free surface and on the body surface, the last integral in Eq. (29b) can be evalu- ated as

( a /2t x 0at2t) , ,a tzt i y - -; , = L+,o t y + ds

1 / 20 10,, / : 2D0 ~sf(y+~)Im 4 (10) ~Z 2 ds

+ Iso (y + o) l< s

where S I denotes the free surface and So is the mean body surface. In Eq. (30), g represents the steady disturbance potential caused by a uniform flow in the y-direction while the second-order steady velocity V(2) is defined as the quantity on the right hand side of Eq. (12b) in our previous work. s

In addition to the advantage of simplicity, the far- field method has a conservative form as proved by Emmerhoff and Sclavounos 2 for the drift force in the surge direction. It can also be applied to the drift force

142 T. Kinoshita et al.: Hydrodynamic forces on a cylinder array

in the sway direction, as well as the first part of the drift yawing moment, i.e., the contribution from the first- order potentials given by the first two integrals Eq. (29b). The proof is given in the Appendix. By this ap- proach the integral used in the far-field method can be carried out along any control surface in the fluid domain as long as only the propagating wave modes are taken into account.

In the present work, the control surface is selected to coincide with the side wall of each cylinder but ex- tended to the sea bottom. The potentials used to calcu- late the drift forces (moment) are expressed in the corresponding local coordinates and only the propagat- ing wave terms in the eigenfunction expansion are re- mained. The total drift forces or moment are the sum of the integrals over the control surfaces encircling each cylinder.

The wave drift damping is a derivative of the wave drift force F} 2) with respect to U where U is set to zero.

= I 0ff12)] w--9-~ 2L) (i= 1, 2, or6) (31)

where i = 6 indicates the yawing moment. Since only the current in the x-direction is considered in the present work, calculated wave drift damping is that induced by the surge drift motion.

Special treatment of free surface integrals

In the calculation of the drift yaw moment due to the waves with slow current, it involves an integral of second-order derivative of the potential over the free surface (see Eq. (30)). This kind of integral also appears in the second part of the particular solution for the first-first problem. In the latter case, the integrand contains the second-order derivative of the steady disturbance potential. Since the solutions are all expressed in the eigenfunction expansion in the present work, higher order derivatives may cause problem of convergence, which will severely affect the accuracy of the calculation. Therefore, it would be better to transform these integrals to avoid higher order derivatives.

In the calculation of the drift yaw moment, the inte- gral given by

82000) * 11 = Im Is I (y + ~)9 (1~ Oz a ds (32)

causes a problem due to the second-order derivative with respect to z. This can be avoided by using the integral formula derived by Newman n as an extension of the Stoke's theorem, which is stated as

(33)

This is valid if V2~ = 0. The annular free surface S I is bounded by the circle Cy outside and by the circle Co inside. The normal derivative in the contour integrals is directed out of S( in the same plane and both contour integrals over C I and Co are evaluated in the positive direction (counterclockwise when viewed from above).

If we put ~t = ~10), and 0 = (Y + g ) 0 ~176 meanwhile, and choose Co as the water line on the body surface and Cr as a circle on the free surface at far field, it can readily be shown that

I 1 = Im{Is 10(1~ (y + ~). V200~ ds

-Ico+c,(Y+~) 00~ 80('~ dl} (34)

where V2 denotes a two-dimensional derivative opera- tor in the horizontal plane.

Similarly, the troublesome integral in the particular solution of the first-first problem is given by

I2 -- f,r (3s) 8z 2

where G denotes the Green function and ~ is the steady disturbance potential. If substituting ~ = ~ and ~ = G0 (~~ into Eq. (33), the above integral can be transformed to the following form:

12= Is ~ ( OO~ 2G + GV 200~ ) �9 V 2O dS - ico+c s G O(l~ O~ dl

(36)

In this way, the accuracy of these integrals is improved.

Calculated examples and experimental results

An assembly of four cylinders is taken as an example of calculations. The radius of each cylinder is 0.15m and draught is 0.30m. These cylinders are located at the corners of a rectangle. The distance between centers of cylinders in the x-direction is 10 times the radius, i.e., 1.5 m, while the distance in the y-direction is set to 5 times the radius, i.e., 0.75 m (see Fig. 2). The center of gravity is assumed to be located at the center of the rectangle and 0.15 m beneath the still water surface. The radius of gyration about the x, y, and z axes is set to be 0.43 m, 0.79 m, and 0.85 m, respectively.

T. Kinoshita et al.: Hydrodynamic forces on a cylinder array 143

r=l).15m

E q'-, I"-

k . Y ,, k )k

( ) ( 1.5m

)

Fig. 2. Arrangement of cylinder array used in calculation

T h e wa te r d e p t h is 1.8m. The cu r ren t ve loc i ty is de- t e r m i n e d by the given F r o u d e n u m b e r which is de f ined

as Fr = U/.,~ga where a is the rad ius of the cyl inders .

In the p r e sen t ca lcu la t ion the F r o u d e n u m b e r is set to Fr = 0, Fr = 0.05, or Fr = -0.05. T h e n o n d i m e n s i o n a l wave n u m b e r koa of the inc iden t wave goes f rom 0.2 to 1.2. In o r d e r to va l ida te the theory , e x p e r i m e n t s a re ca r r i ed out. T h e se t -up is de sc r ibed in ou r p rev ious work. 12

F i r s t of all, c a l cu la t ed resul ts and e x p e r i m e n t a l da t a of wave-exc i t ing forces or m o m e n t and wave dr if t forces or m o m e n t , w i thou t cu r r en t effects, a re p l o t t e d versus wave n u m b e r in Fig. 3 (a, b, and c) and Fig. 4 (a, b, and c), respec t ive ly . I t can be seen tha t they a re in g o o d ag reemen t .

T h r e e cases a re e x a m i n e d to show the effects of wave and cu r r en t d i rec t ions on h y d r o d y n a m i c forces. In the first case, the cu r ren t d i r ec t ion a co inc ides wi th the inc iden t wave angle /3 , i.e., a = /3 = 0 ~ 30 ~ 60 ~ or 90 ~ The wave dr if t d a m p i n g for the x-d i rec t ion , the y -d i rec - t ion and the yaw m o m e n t a re shown in Figs. 5a, 5b, and 5c, respec t ive ly . Because of symmet ry , x -d i r ec t ion wave drif t d a m p i n g vanishes w h e n a --/3 = 90 ~ whi le it is ze ro for the y -d i r ec t i on when a =/3 -- 0 ~ In e i the r case, the wave dr if t d a m p i n g m o m e n t in yaw vanishes . Expe r i - men ta l da t a a re also p r e sen t ed . I t can be o b s e r v e d tha t the ca l cu la t ed resul ts a re in g o o d a g r e e m e n t wi th the e x p e r i m e n t a l resul ts .

In the s econd case, the cu r r en t d i rec t ion is set to a = 0 ~ and the inc iden t wave angle var ies as/3 = 0 ~ 30 ~ 60 ~ or 90 ~ The ca l cua t ed wave dr i f t d a m p i n g and d a m p i n g m o m e n t a re shown in Figs. 6a, 6b, and 6c, respect ively �9 The wave dr if t d a m p i n g in the y -d i r ec t ion is t r ivial when the inc iden t waves are co inc iden t wi th or o r t h o g o n a l to the cu r ren t d i rec t ion ; only the resul ts for/3 = 30 ~ and 60 ~ are p lo t t ed in Fig. 6b. Simi lar ly , the wave dr if t d a m p i n g

F10 ~ )

Pg~o a2

1 5 . 0

1 0 . 0

5.0

0.0 0.2

I I Cal. Exp. I - - - - - = 3 0 o [ 5 = 3 0 I

! . - ' ~ - I D . . . . . /

0.4 0 . 6 0 . 8 1 . 0 koa

F20 ~ Cal. - - - f~=3o

Pg~0 a2 . . . . . . . . . . 60

1 5 . 0 . . . . . . 90

1 0 . 0

Exp. o [3=30 D =60

i ", 0/. /

5 .0 . . . . . . . D--, "-, /

)5~- D \ 0.0 ~, I

0.2 0.4

\

/

/

0 0 / /

/ xX /

/ . . . . . . . o . . . . ' \

~.r.7.1._.n ..... ~ i x /

0 . 6 0 . 8 1 . 0 koa b

I {1~ PgGa 3

40.0

3 0 . 0

2 0 . 0

1 0 . 0

I I Cal.

- - - 6=30 O

. . . . . . . . . . 60 o / \

/ \ / o o,

/ \ / [] \

c V . , . . n . . . . . . ~ . . . . . . . . . . . . . , \

& r " / , , �9 D",, \o

/,,"

Exp.

o 13=30

[] =60

/ /

/ /

/

m X " ~ . ........ , ,\,,�9 "" L

/ i L I I

0 .0 0 .2 0 .4 0 . 6 0 . 8 1 . 0 koa

F i g . 3 . a . Wave-exciting force in x-direction without" current effects under various incident wave angles (]3). b : Wave- exciting force in y-direction without current effects under vatious incident wave angles (/3). e. Wave-exciting yaw mo- ment without current effects under various incident wave angles (/3)

144 T. K i n o s h i t a e t al.: H y d r o d y n a m i c fo rces o n a cy l inde r a r r ay

b

E(2~ pg~a

3.0

2.0

1.0

0.0 0.2

F 2 (20)

pg~o~a 2.0

Cal. Exp. ~=0

. . . . 30 o {5=30

. . . . . . . . . . 60 D =60

I" 1

0 . "

/ ~. . . . ' '.... _

0.4 0.6 0.8 1.0 koa

1.5

1.0

0.5

0.0 0.2

I

[]

. . . . [] . -- . .

• : '

/ i

D ,' /

/ / / Q/

. . . . . ~ . . . . . . ~ . . . 2 / ,

0.4 0.6

o /

/ /

o /

/

Cal.

- - - 13=3o . . . . . . . . . = 6 0

[-I "

. / .~

O

[]

.0 0.8

Exp.

[3=30

=60

koa

p~o o~ 2o a

40.0

20.0

0.0

-20.0

Cal. Exp. I ct=l~=O . . . . 30 o ct=~.=30 . . . . . . . . . . 60 [] -60

~ o \

-40.0 I I 0.2 0.4 0.6

I I

I I

0.8 1 .o ,,k0a

-10.0

B2 Cal. Exp. ] Pt~ 2a - - -ct=~=30 o ct=fl=30

I 30.0 ......... =60 [] =60 - - - - = 9 0

20.0 f - x f - - . .

10.0 / ,,,, \ / ~- / o r , [] -~, . - - - . . . . . . . . . .

, F3-" \

0.0 - ~ ~

/ \ ,

f \, i

/ \, f ..

.: \

',,. ,/ ,, D ',

N..j ".

Xf I I I

-20.0 0.2 0.4 0.6 0.8 1.0 koa

/~6(2 o)

1.0

0.0

-1.0

-2.0

I I

", o/ \,,, \ g , / , , ' \

o,,"..K ..... / \

Cal. Exp.

- - ~ = 3 0 o ~ = 3 0

. . . . . . . . . =60 [] =60 -3.0

c 0.2 0.4 0 . 6

ul I / ,

[] .�9149 ',

. . " ~',,, ,, . / \

.'" 0 , / '. \

o / ",, \

"x ,'

\ \ \ \

I I

0.8 1.0 koa

Fig. 4. a. W a v e dr i f t fo rce in x - d i r e c t i o n w i t h o u t c u r r e n t ef- fec t s u n d e r va r ious i nc iden t w a v e ang les (/3). b. W a v e dr i f t f o r c e in y - d i r e c t i o n w i t h o u t c u r r e n t e f fec t s u n d e r va r ious inci- d e n t w a v e angles (fl). c . _Wave dr i f t yaw m o m e n t w i t h o u t c u r r e n t e f fec t s u n d e r varib"us i nc iden t w a v e ang les (/3)

B~ # O ) o ~ ' f a 2

50.0

0.0

-50.0

-100.0 0.2

Cal. Exp. I '

I - - - - - ~ = 1 ~ = 3 0 o ~=15=30 . . . . . . . . . . 60 [] =60

[]

...... Z-,~, . . . . o / o.x~-- " . . . . 2 h "..~ / ,, . . /

~...,,..~. ...... ~ \ ~ ,- o_. .....o \~.:.." ii. ~

" '( /

" :'~ /

[] \ / \ /

I I I I

0.4 0.6 0.8 1.0 koa

Fig. 5. a. W a v e dr i f t d a m p i n g in x - d i r e c t i o n u n d e r va r ious c u r r e n t d i r e c t i o n s ( a ) w h i c h a re c o i n c i d e n t wi th i n c i d e n t w a v e ang les (fl). b. W a v e dr i f t d a m p i n g in y - d i r e c t i o n u n d e r va r ious c u r r e n t d i r e c t i o n ( a ) w h i c h a re c o i n c i d e n t wi th i n c i d e n t w a v e angles (/3). c. W a v e dr i f t d a m p i n g m o m e n t in y a w u n d e r vari- ous c u r r e n t d i r e c t i o n s (tx) w h i c h a re c o i n c i d e n t wi th i n c i d e n t w a v e angles (/3)

T. Kinoshita et al.: Hydrodynamic forces on a cylinder array 145

B, POJo~Zoa

4O.O

20.0

0.0

-20.0

-40 .0

-60.0

/ , ~ X- ) ~ ' / %1 " ' ~ / " ....... 7 . .... ) g - : ' _ . " . . . \ \

. . . . 30

. . . . . . . . . . 60

. . . . . . 90 I L

0.2 0 .4 0 .6 0.8 1.0 koa

8 2

pr-o o~ ~o a 15.0

10.0

5 .0

0.0

Cal.

- - - [ 3 = 3 0

...... =60

__,---6 --- /

Exp. I o [3=30

fo

o / / \ ./"~-,,,,. / ,,

",,._."

-5.0

-10 .0 1 I I I

0.2 0.4 0 .6 0,8 1.0

", t /

/

\ o : , t / \ " , !

: i

i

', /

koa

&

p(.o o~o~a 2

50.0

0.0

-50.0

Cal. Exp.

- - - [ 3 = 3 0 o 13=30

" - ~ 6 0 / N

. . . . . . 90 / /

o , , g x , . ................ .v" k-~'~'-~.-'-=. =-:.=." ... . . . . \ " . % / / " ""'" - - 0 ~ > . ~ ,

, /

\ �9 \ /

/ x ,

/ /

/

- 1 0 0 . 0 I t I

0.2 0 .4 0 .6 0.8 1.0 koa

Fig. 6. a. Wave drift damping in x-direction under various incident wave angles (13) when current direction is fixed to 0 ~ b. Wave drift damping in y-direction under various incident wave angles (13) when current direction is fixed to 0 ~ c. Wave drift damping moment in yaw under various incident wave angles (13) when current direction is fixed to 0 ~

momen t in yaw has significant value only when/3 = 30 ~ 60 ~ and 90 ~ which are plot ted in Fig. 6c. In these figures, experimental results of 30 ~ incident waves are also presented. The agreement of calculation and ex- per iment is fairly good.

In the last case, the incident wave angle is fixed to/3 = 0 ~ and the current direction is changed as a = 0% 30 ~ 60 ~ , or 90 ~ . The wave drift damping and damping momen t are plotted in Figs. 7a, 7b, and 7c, respectively. Trivial values of wave drift damping for the x-direction when a = 90 ~ and for the y-direction and the damping m o m e n t in yaw when a = 0 ~ are omitted. It is interesting to observe that the wave drift damping momen t in yaw has the biggest peaks when the current is 90 ~ f rom the incident waves. It can also be seen that with changes of current direction, the wave drift damping has a similar tendency of fluctuation while the peak values are changed.

Concluding remarks

In the present work, the interaction p rob lem of multiple cylinders in waves and slow current is considered. The velocity potential is expanded into a double perturba- tion series of small parameters , i.e., the wave slope e and current velocity pa ramete r To. The solution is expressed in eigenfunction expansions which are convenient to represent interaction among multiple cylinders. The first-order hydrodynamic forces and moment s referring to e are obtained by integrating the corresponding pres- sure over the mean wetted body surface. The second- order forces and moments referring to e are calculated by the far-field method based on the conservation of m o m e n t u m and angular momen tum. The wave drift damping and damping momen t can be obtained by simple formulae with integrals having a proper ty of con- servation, which means that the control surface of the integrals can be placed anywhere in the fluid domain. Fur thermore , it is essential for maintaining the accuracy of the results that the second-order derivatives in the integrals over the free surface in these formulae should be replaced by first-order derivatives using proper inte- gral identity.

Numerical examples are presented. Some results are compared with the available exper imental data, showing satisfactory agreement . Effects of the current direction and the incident wave angle are shown in these examples. It can be observed that the greatest value for the wave drift damping m o m e n t in yaw is genera ted when the current is or thogonal to the inci- dent waves. The examples also show that the tendency of the fluctuation in wave drift damping is not greatly affected by a change in the current direction, which implies that the fluctuation in these quantities is mainly

146 T. Kinoshita et al.: Hydrodynamic forces on a cylinder array

pt.o 0 - ~ 0 a m

4 0 . 0

2 0 . 0

0 .0

- 2 0 . 0

- 4 0 , 0

0 . 2 0 .4 0 .6 0 ,8 1 . 0 koa

pW o~o~a

-4 ,0

-6 .0

-8 .0

- 1 0 . 0 Cat.

- - - - - ct=30 - 1 2 . 0

. . . . . . . . . . 60

. . . . . . 90 - 1 4 . 0

0 .2 0 .4

B6

,:,. : _2_.: .z-z.z,,, "x \

k', ~,

\',

\

k

\

\ /

\ /

. " l

\ , ' I -

X , ' l

" , , " / , , ,, ,

" + " /

/

I I I

0 . 6 0 . 8 1 . 0 koa

pOg o~2 az

2 0 . 0

10 .0

0 .0

- 1 0 . 0

- 2 0 . 0

-30 .0

Cal.

- - - - - a = 3 0

. . . . . . . . . 6 0

. . . . . . 90

r 'I T /.

, \ /,,--, ,

, , ,.\

/ , ' ,,,

7/ / " "~ 'a .!. \), ?/ \~,

b _ ~.r %>,/'

1. ..I

C 0 .2 0 .4 0 .6

fL" ~.:' '.:,~

': '/-" \:t ;7 ,~

\ v / :./ \\

~.\

\',,

\

0 . 8 1.0 koa

Fig. 7. a. Wave drift damping in x-direction under various current directions (t~) when incident wave angle is set to 0 ~ b. Wave drift damping in y-direction under various current directions (a) when incident wave angle is set to 0 ~ c. Wave drift damping moment in yaw under various current directions (a) when incident wave angle is set to 0 ~

caused by the interaction of waves with multiple cylinders.

A p p e n d i x

Equat ion (29a) and Eq. (29b), omitting terms involving the second-order steady potential, are in conservation form, i.e., their values are independent of the position of the control surface Sc. This can be shown for any two potentials ~ and ~ which satisfy nonpermissible condi- tion at the sea bo t tom and the following free surface condition at z = 0

~0 &

Dq/

- - - %@ = 0 ( A l a )

- v0r = 2 i ~ - + 2k 0 cosfl@ ( A l b )

respectively. It should be noted that the free surface condition satisfied by the propagat ing part of the first-first-order potential has the same form as in Eq. (Alb) .

By means of Gauss ' integral theorem, the validation of the following identity is readily shown:

Re V @ . V ~ * n , - ~ On 0x i

The superscript asterisk (*) denotes the complex conju- gate and

0 &

0 0 Ox, 03,

3 N

0 0 - y - g

i = 1)

( i - -2)

( ;--6)

(A2b)

while n i is the x- or y -componen t for i = 1 or 2, respec- tively, of the unit normal vector n pointing out of the enclosed fluid domain. When i -- 6, the component of normal vector is, defined as n 6 = X n 2 - - yn~.

The integral is carried out over a closed surface S = Scl + Sc2 + Sy + Sb where Sol and Sc: are two vertical circular cylinders extending f rom z = 0 to z = - h in the fluid domain, and S I denotes the annular strip on z = 0 between the intersecting contours C~ and C2. The annu- lar strip on the sea bo t tom z = - h is designated by Sb, the integral over which makes no contribution due to the nonpermissible condition satisfied by the velocity po- tentials and the fact that nt.2 = 0 there.

The integral over Sj., where r/1. 2 = 0 tOO, can be simpli- fied by using the free surface condition described by Eqs. (A la ) and (Alb) . The integrand on this surface may be reduced to the form

T. Kinoshita et al.: Hydrodynamic forces on a cylinder array 147

O~ Ol/s* Ol/s* a~ t Re cOxi & &i ~zz )

= R e - ~0--~-~ (I)01//* -2i ~ oxit " &

e( ~0+~,. +k0 cos ~O+*) = Re �9 Oxi

c9 &~ * O 0r *

+ , O~*

(A3a)

for i = 1 and 2. In a similar way, for the case of i = 6 ,

R e - & i OZ oaXi OZZ

{ a(,.,0+~* +~0cosnoo*) = Re Ox i

-,[• (<, 0+,/]/ Laxt ax, ) ax, t. ax)jj

+,v (yov+,)

(A3b)

where use has been made of the fact that the real part of the quantity

,y[V2~).V2~)*-k~)V2~)*l:iy[V2~).V2~)*-k2~)~) * ] (a3c)

is zero. By means of 2-dimensional Gauss' theorem, the integral over S I can be replaced by two line-integrals over C1 and C2, i.e.

Re Is, O+ 0gt* 0u/* o~r ]ds f OX i & OX i 6~ )

/;:12) -,0%-7n, -g-n,

R ,r Jc~+o~ \ an o'x i J

(A4)

where the fact that 1/6 = X n 2 - - yn~ = 0 on C~ and C 2 has been used.

Substituting Eq. (A4) into Eq. (A2a), we have

ReL,,+'2 (v+ v~'*"'- ax,a+ a~.*a, a~.*ox, eoa.~ )~'

Re{Ic,+(,[-(u.OIlt* +kocos,6~qJ*)ni

. (O0* O+*nild I = 0 + ax, ax ) J J

-,o - - , , , - ] l (,=1,2)

{ Reilc,+c: YO--~n ox, ) (A5)

If i = 1, the last line integral in Eq. (A5) vanishes and it is the same as derived by Emmerhoff and Sclavounos 2 for the drift force in the surge direction. When i = 2, the integrand in the last line integral of Eq. (A5) becomes

(&0* _ cgq~* / Oq~* (A6) 0 --g- , 0x "2)--~ By setting 0 and ~ to be O ('~ and elm, respectively, it proves Eqs. (29a) and (29b) are in a conservative form.

References

1. Matsui T, Lee SY, Sano K (1991) Hydrodynamic forces on a vertical cylinder in current and Waves (in Japanese). J Soc Nav Archit Jpn 170:277-287

2. Emmerhoff O J, Sclavounos PD (1992) The slow-drift motion of arrays of vertical cylinders. J Fluid Mech 242:31-50

3. Clark PJ, Malenica S, Molin B (1993) Steady forces due to onset flow and waves on wall-sided bodies in finite depth water. Proc OMAE 1:83 90

4. Bao W, Kinoshita T (1993) Hydrodynamic interaction among multiple floating cylinders in both waves and slow current. J Soc Nav Archit Jpn174:193-203

5. Kinoshita T, Bao W (1996) Hydrodynamic forces acting on a circular cylinder oscillation in waves and a small current. J Mar Sci Technol 1:155-173

6. Kagemoto H, Yue DKP (1986) Hydrodynamic interaction among multiple three dimensional floating bodies; an exact algebraic method. J Fluid Mech 166:189-209

7. Linton CM, Evans DV (1990) The interaction of waves with arrays of vertical circular cylinders. J Fluid Mech 215:549-569

8. Abramowitz M, Stegun IA (eds) (1965) Handbook of mathemati- cal functions. Dover, New York, pp 355-434

9. Wehausen JV, Laitone EV (1960) Surface wave. In: Flugge S (ed) Handbuch der physik. Springer, Heidelberg, pp 592~02

10. Grue J, Palm E (1993) The mean drift force and yaw moment on marine structures in waves and current. J Fluid Mech 250:121-145

11. Newman, JN (1993) Wave-drift damping of floating bodies in waves. J Fluid Mech 249:241-259

12. Kinoshita T, Sunahara S, Bao W (1995) Wave drift damping acting on multiple circular cylinders (model tests). Proc OMAE, 1-A:443-454