Ocean Engng,Vol.25,No.6,pp.385–394,1998
©1998Elsevier Science Ltd.All rights reserved Pergamon
Printed in Great Britain
0029–8018/98$19.00+0.00
PII:S0029–8018(97)00003–6
ANALYTICAL SOLUTIONS OF THE DIFFRACTION PROBLEM OF A GROUP OF TRUNCATED VERTICAL CYLINDERS
Oguz Yılmaz*†and Atilla Incecik‡§
†Istanbul Technical University,Faculty of Naval Architecture and Ocean Engineering,Maslak,Istanbul,
Turkey
‡University of Newcastle,Department of Marine Technology,Newcastle upon Tyne,U.K.
(Received12June1996;accepted infinal form10July1996)
Abstract—An exact analytical method is described to solve the diffraction problem of a group of
truncated vertical cylinders.In order to account for the interaction between the cylinders,Kagemoto
and Yue’s exact algebraic method is utilised.The isolated cylinder diffraction potential due to
incident waves is obtained using Garret’s solution and evanescent mode solutions are derived in
a similar manner.
Numerical results are presented for arrays of two and four cylinders.Comparisons between the
results obtained from the method presented here and those obtained from numerical methods show
excellent agreement.©1998Elsevier Science Ltd.
1.INTRODUCTION
The phenomenon of hydrodynamic interaction amongst a group of cylinders,for example between the columns of TLPs or semi-submersibles,has received considerable interest in recent years.Although numerical calculations through the use of Green’s function tech-niques are well established,their use may be expensive and cumbersome.An alternative method is to obtain semi-analytical solutions which take the hydrodynamic interactions into account.Recently,Linton and Evans(1990)improved on the direct matrix method of Spring and Monkmeyer(1974),wherein the amplitudes of the wave components around each body are solved simultaneously,to obtain simple expressions for force and free sur-face amplitudes.Another approach to the problem is the multiple scattering technique (Ohkusu,1974)in which successive scatters by each of the cylinders are introduced at each order.Kagemoto and Yue(19
86)combined the direct matrix method and the multiple scattering technique to obtain an exact algebraic method.In their interaction theory,the scattered wavefield around each body is expressed as a summation of cylindrical waves with undetermined amplitudes.By using addition theorems for Bessel functions,the scat-tered potential at one body is evaluated in the coordinate systems of other bodies.A set of linear algebraic equations which relates the total incident potential to the scattered potential is then solved simultaneously for all the unknown amplitude coefficients.
In this paper,the interaction theory of Kagemoto and Yue(1986)is used to obtain
*Presently at the Naval Architecture and Ocean Engineering Department,University of Glasgow,Glasgow,U.K., where this work was performed.
§Author to whom correspondence should be addressed.
385
386Oguz Yılmaz and Atilla Incecik
analytical solutions for the diffraction problem of truncated cylinders.The diffraction potential of an isolated cylinder due to incident waves is obtained using the solution of Garret(1971),and evanescent
mode solutions are derived in a similar manner to this solution.Numerical results are presented for arrays of two and four cylinders.Comparisons between the results obtained from the method presented in this paper and those obtained from numerical methods are excellent.
2.THEORETICAL DEVELOPMENT
An array of N truncated cylinders of equal radius a is placed in water of uniform depth d,the clearance beneath each cylinder is denoted by h.We will use N+1coordinate systems(r,␪,z)with the origin at the sea bed and the z axis positive upward.Local coordinates(r i,␪i,z),i=1,...,N,centred at the origin of each cylinder(x i,y i)are also used. The coordinate systems and the parameters used are depicted in Fig.1.
Assuming that thefluid is ideal and waves are of small amplitude,thefluid motion may be described by a velocity potential␾(x,y,z,t)=Re{␸(x,y,z)eϪi␻t}.This potential must satisfy Laplace’s equation together with boundary conditions at the free surface,on the body,at the sea bottom and a suitable radiation condition for the diffraction potential at infinity.
An incident wave potential for a plane wave of amplitude H,frequency␻and wave heading angle␤has the following form:
␸0j=gH␻cosh k0z
cosh k0d I j e i k0r j cos(␪jϪ␤)=
gH
␻
cosh k0z
cosh k0d
I j͸ϱn=Ϫϱe i nͩ␲2Ϫ␪j+␤ͪJ n(k0r j)
(1)
where I j=e i k0(x j cos␤+y j sin␤)is a phase factor associated with cylinder j and i is√Ϫ1. Wave number k0and frequency␻satisfy the dispersion relationship␻2=k0g tanh k0d with g being gravitational acceleration.J n is thefirst kind of Bessel function of order n.
Fig.1.Plan view of two cylinders and coordinate systems.
387
Analytical solutions of the diffraction problem By replacing n with Ϫn ,we rewrite Equation (1)as follows:
␸0j =gH ␻cosh k 0z cosh k 0d I j ͸ϱ
n =Ϫϱ
a I j (n )␺I j (n )(2)where a I j (n )=I j e i n ͩ␲2Ϫ␤ͪand ␺I j (n )=J n (k 0r j )e i n ␪j The general form of scattered wave field outside the immediate neighbourhood of body i can be expressed as a summation of cylindrical waves as follows:␸S i =gH ␻ͫ
cosh k 0z cosh k 0d ͸ϱn =ϪϱA 0n i H (1)n (k 0r i )e i n ␪i (3)
+
͸ϱm =1cos k m z ͸ϱn =ϪϱA mn i K (1)n (k m r i )e i n ␪i ͬ
where H (1)n and K n are,respectively,the n th order Hankel function of the first kind and modified Bessel function of the second kind,and the wave number k m ,m =1,2,...,is the positive real root of the dispersion equation ␻2=Ϫk m g tan k m d ,m Ͼ0correspond to the evanescent modes.
In order to express the scattered potential in the other cylinders’coordinate systems,addition theorems for Bessel functions (Abramowitz and Stegun,1964)will be used:
H (1)n (k 0r i )e i n ␪i =
͸ϱl =Ϫϱ
H (1)n +l (k 0L ij )J l (k 0r j )e i ␣ij (l +n )e i l (␲Ϫ␪j )(4a)K n (k m r i )e i n ␪i =
͸ϱ
l =ϪϱK n +l (k m L ij )I l (k m r j )e i ␣ij (l +n )e il (␲Ϫ␪j )(4b)
where I l is the l th order modified Bessel function of the first order.
By substituting Equations (4a)and (4b)in Equation (3)and replacing l by Ϫl ,we obtain
␸S i =gH ␻ͫ
cosh k 0z cosh k 0d
͸ϱn =ϪϱA 0n i ͸ϱl =ϪϱH (1)n Ϫl (k 0L ij )e i ␣ij (n Ϫl )J l (k 0r j )e i l ␪j (5)+͸ϱm =1cos k m z ͸ϱn =ϪϱA mn i ͸ϱl =ϪϱK n Ϫl (k m L ij )e i ␣ij (n Ϫl )(Ϫ1)l I l (k m r j )e i l ␪j ͬ
Equation (5)can be written in matrix notation as follows:␸S i =gH ␻ͫcosh k 0z cosh k 0d
A T i (n )T ij (n ,l )␺I j (l )+cos k m zA T i (m ,n )T ij (m ,n ,l )␺I j (m ,l )ͬ(6)where T ij (n ,l )=H (1)n Ϫl (k 0L ij )e i ␣ij (n Ϫl )and ␺I j (l )=J l (k 0r j )e i l ␪j for m =0(7a)
T ij (m ,n ,l )=K n Ϫl (k m L ij )e i ␣ij (n Ϫl )(Ϫ1)l and ␺I j (m ,l )=I l (k m r j )e
i l ␪j for m Ͼ0(7b)
388Oguz Yılmaz and Atilla Incecik
The total incident velocity potential near body j is the summation of the ambient incident wavefield and the scattered wavefield due to other cylinders:
␸I j=␸0j+͸N i=1(i j)A T i T ij␺I j=(a T j+͸N i=1(i j)A T i T ij)␺I j(8) According to Kagemoto and Yue(1986),the total incident and scattered waves for any body j are related to each other by the isolated body diffraction characteristics of that body,which will be denoted by B j,j=1,2,...,N:
A j(n)=
B j(n,l)[a j(l)+͸N i=1(i j)T T ij(n,l)A i(n)]for m=0(9) A similar equation can be written for mϾ0.Single body diffraction matrices B j are obtained by solving the diffraction problem of a truncated cylinder including both pro-gressive and evanescent modes.These solutions are given in the Appendix A and B j is given as follows:
B j(n,l)=J lЈ(k0a)
H lЈ(k0a)+
D(l)0ͱ2cosh(k0d)
H lЈ(k0a)ͱ1+sinh(2k0d)/(2k0d)for m=0(10a)
B j(m,n,l)=I lЈ(k m a)
K lЈ(k m a)+
D(l)mͱ2
K lЈ(k m a)ͱ1+sin(2k m d)/(2k m d)for mϾ0(10b)
Once B j is determined,A j can be obtained from Equation(9)andfirst order forces can be evaluated by integrating the total potential on the cylinder.The surge force on cylinder j is calculated as follows:
F x
j =Ϫi␳gH͵2␲␪=0
͵d
z=h
ͭͩa T j+͸
i=1
(i j)
A T i T ijͪ
B T j␺S j(a,␪j)N(z)+(a T j(11)
+͸N i=1i j A T i T ij)␺I j(a,␪j)N(z)
ͮa cos␪j d␪j d z
where
␺S j=H(l)l(k0r j)e i l␪j,N(z)=cosh(k0z)
cosh(k0d)
for m=0
␺S j=K l(k m r j)e i l␪j,N(z)=cos(k m z)for mϾ0
Thefirst term in Equation(11)is due to the diffraction effects,the second is due to the ambient incidentfield and scattering of all the other bodies.The heave force is evaluated in a similar manner,but the B j matrix used in the heave force calculation is different from the one used in the interaction theory and is given as follows:
389 Analytical solutions of the diffraction problem
B hj(n,l)=C(l)0
2a l
for m=0and B hj(m,n,l)=
C(l)m
I lͩm␲a h
truncated 带whereͪ(Ϫ1)m for mϾ0(12)
The derivation of these matrices is given in Appendix A.
In addition to this,the incident wave potential must be subtracted from thefirst term and integrated since the interior region potential obtained from the single body diffraction solution contains both incident and scattered wavefields:
F z
j =i␳gH͵2␲␪=0
͵a
r=0
ͭͩa T j+͸N
i=1
(i j)
A T i T ijͪ
B T hj␺S j(r j,␪j)N(h)
+ͩa T j+͸N i=1(i j)A T i T ijͪ␺I j(r j,␪j)N(h)(13)Ϫͩa T j+͸N i=1(i j)A T i T ijͪN(h)i l J l(k0r j)e i l␪jͮr j d␪j d r j
where␺S j=r͉l͉j e i l␪j for m=0and␺S j=ϪI l(m␲r/h)e i l␪j for mϾ0.
The pitch moment can be calculated by taking the moments of the forces about the free he integrand in Equation(11)should be multiplied by the lever(dϪz)and the integrand in Equation(13)by r i cos(␪).
3.NUMERICAL RESULTS AND DISCUSSION
Configurations chosen to validate the present method are depicted in Fig.2.Thefirst geometry is an array of two cylinders,the second an array of four cylinders.The distance between the cylinders is2.6m for geometry(a)and4m for geometry(b).
90unknowns were used for configuration(a)and180unknowns for configuration(b).
For both cases four evanescent modes were considered.Surge and heave forces shown
in
Fig.2.Configurations used in the calculations:(a)two truncated cylinders;(b)four truncated cylinders.

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