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Ocean Engineering 31 (2004) 269–304 www.elsevier.com/locate/oceaneng

Hydrodynamic coefficients of a floating, truncated vertical cylinder in shallow water Y. Drobyshevski



ICON Engineering, 13 B Langham Street, Nedlands 6009, WA, Australia Received 14 March 2003; received in revised form 1 July 2003; accepted 23 July 2003

Abstract

The paper seeks to apply the methodology of matched asymptotic expansions to obtain analytical solutions for hydrod analytical hydrodynamic ynamic propert properties ies of a circu circular lar cylindr cylindrical ical platform (trun (truncated cated circular cylinder) in extremely shallow water. By matching potentials of the fluid flows outside and under the cylinder bottom with the inner flow near its edge the radiation problems are solved for heave, surge and pitch motions. Closed asymptotic formulae are derived for all hydrodynamic coefficients of the floating cylinder, which include the first-order terms in the the gap heigh height. t. Th Thee fo formu rmulae lae are di disc scus usse sed d and and show shown n to comp compar aree we well ll wit with h nu nume meri rica call results published in literature. # 2003 Elsevier Ltd. All rights reserved. Keywords: Shallow Keywords:  Shallow water; Waves; Circular cylinder; Matched asymptotic expansions

1. Introd Introductio uction n

A truncated circular cylinder represents a convenient generic shape for a variety of app applic licati ations ons in ocean ocean engine engineerin ering, g, with with man many y cyl cylind indric rical al str struct ucture uress oper operati ating ng under the effects of wave-induced loads and motions. On the other hand, owing to the particular domain geometry, which allows the Laplace equation to be solved by separation of variables, hydrodynamic properties of cylindrical bodies have rendered themselves to efficient theoretical solutions, thereby providing valuable information for comparison with model tests and benchmarking of the more general numeri num erical cal too tools. ls. Due to its sig signific nificant ant practi practical cal and the theore oretic tical al imp import ortanc ancee the pr prob oble lem m of wa wave ve in inte terac racti tion on wi with th a trun trunca cate ted d ci circ rcul ular ar cylin cylinde derr attr attrac acte ted d mu much ch 

Tel./fax: +61-8-9386-3997. +61-8-9386-3997. E-mail address: [email protected] address:  [email protected] (Y. Drobyshevski).

0029-8018/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.oceaneng.2003.07.003

 

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Y. Drobyshevski / Ocean Engineering 31 (2004) 269–304

interest and was studied by a number of authors. A methodology involving eigenfu func ncti tion on ex expa pans nsio ions ns of th thee ve velo loci city ty pote potent ntia iall in cyli cylind ndri rica call co coor ordi dina nate tess wa wass developed and implemented in various forms by   Miles and Gilbert (1968), Garret (1971), Yeung (1981), Sabuncu and Calissal (1981). (1981) . Extensive numerical results are given in these studies for hydrodynamic coefficients of circular cylinders in finite depth waters. In the present paper, attention is focused on extreme shallow water conditions, when the under-bottom clearance is small compared with the structure dimensions and wat water er dep depth, th, a sit situat uation ion fre freque quently ntly enc encoun ounter tered ed in marine marine oper operati ations ons.. The hydrodynamic problem is considered by the use of the method of matched asymptotic expansions, generally following an approach first developed by   Tuck (1971, 1975) to 1975)  to study flows through small apertures. The heave, surge and pitch radiation problems for a floating truncated cylinder are solved in a closed form, and asymptotic formulae are derived for all hydrodynamic coefficients. These formulae, which include the first-order terms in the gap height, provide good accuracy over a range of water depths and allow for straightforward numerical implementation. Further review, references and a solution of the two-dimensional problem for a rectangular structure are given in the previous paper by this author, which will be hereinafter referred to as (D). The material is set out as follows. In Section 2, the radiation problem is formulated for a circular vertical sided platform in extremely shallow water. General expressions are given for the inner and outer velocity potentials, with particular emphas emp hasis is on the rad radiati iation on solutio solutions ns for the ver vertic tical al bot bottomtom-mou mounted nted cyl cylinde inder, r, which will be further used in this study. In Section 3, the heave radiation problem is solved by constructing and matching potentials of the ‘‘outer’’ flows (outside and under the bottom of the cylinder) with the ‘‘inner’’ flow near the edge of the structure. After the radiation potential has been determined, heave hydrodynamic coefficients are obtained and the zero and infinite frequency limits of the heave added mass are examined. Sections 4 and 5 are devoted to the surge and pitch radiation pr prob oble lems ms,, wh whic ich h ar aree so solv lved ed fo foll llow owin ing g th thee same same pr proc oced edur ure. e. Fo Forr al alll mo mode dess of  motions, the added mass and damping coefficients are obtained by integrating the fluid flu id pres pressu sure re ov over er the the stru structu cture re su surf rfac ace, e, wh where ereas as th thee wa wave ve exci exciti ting ng fo forc rces es are are found by using the Haskind–Newman formula, which enabled to avoid solving the diffraction problem. The obtained formulae for the hydrodynamic coefficients are discussed and shown to compare well with numerical results by  Yeung (1981). (1981).

2. Formul Formulation ation of the radia radiation tion problem problem

 2.1. The bounda boundary ry value problem Consid Cons ider er a ci circ rcula ularr ve vert rtic ical al side sided d stru struct ctur uree (a tr trun unca cate ted d circ circula ularr cyli cylinde nder) r) of  radius   a   and draft   T  floating   floating in the inviscid incompressible fluid of depth   H . The origin of the coordinate system is fixed at the free surface level; the   z-axis points vertic ver tically ally dow downwa nwards, rds, as sho shown wn in   Fig. 1 1.. Surgen t , heave   f t   and pit pitch ch   w t

ðÞ

ðÞ

ðÞ

 

Y. Drobyshevski / Ocean Engineering 31 (2004) 269–304

271

Fig. 1. Coordinate system.

1,, are heremotions of the structure, positive directions of which are shown in Fig. in  Fig. 1 inafter denoted by indices   i   1  1;; 3; 5, respectively. The perturbed fluid motion due to small amplitude oscillations of the structure with unit velocity amplitudes can be described by the velocity potential:

 ¼  ¼

ðx; y; z; tÞ ¼ Reðuðx; y; zÞ  ei Þ; rt

ð1Þ where  r  denotes the circular frequency, and the time-independent potential  uðx; y; zÞ U

 

is the solution of the following boundary value problem:

 

@ 2 @ x2 @  @ z

  @ 2 @  y2

 þ  þ  r 2

 þ g

u x; y; 0

Þ ¼ 0

Þ ¼ 0

  everywh everywhere ere in the fluid

  on the free surface r  >  a ; z

 j j

on the seabed   z

 ¼ H 

 ¼  0

@ u @ n

ð

ð

@ u @ z

8<  ¼ :

  @ 2 u x; y; z @ z2

ð Þ ð Þ ð Þ  xcosðn; zÞ

cos n; x cos n; z zcos n; x

9= ;

 ¼ 0

 

ð2Þ

 

ð3Þ

  ;

8<  ¼ ¼ 9= :  ¼  ¼ ¼¼ ;

for

i   1 i   3 i   5

ð4Þ

on the body surface

ð5Þ

Equation (2) is the Laplace equation, (3) is the linearized boundary condition on the free surface, conditions (4) and (5) are the kinematic boundary conditions on the hull surface and the seabed. Here   n   denotes the unit normal vector pointing into the fluid, and  r  is the radius of the polar coordinates r; h  defined such that:

 ð Þ

 ¼ rcosh;

x

  y

 ¼ rsinh

 

ð6Þ

 

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Y. Drobyshevski / Ocean Engineering 31 (2004) 269–304

For the vertical circular cylinder with a flat rigid bottom, the boundary condition (5) can be rewritten in the following form:

8< Þ¼ :8< Þ¼: 

cosh

@ u @ r

  ¼ f  ðz; h ii  

@ u @ z

 ¼  g ðr; h i 

0 zcosh 0 1

9= 8<  ¼ ¼ 9=  ¼  ¼  ¼  ¼ ;9= :8<  ¼ ¼;9= ; :  ¼  ¼ ¼¼ ; ;

for

;

for

rcosh

i   1 i   3 i   5 i  i  i 

 

on the side surface

 1  3  5

on the bottom

 

ð7Þ ð8Þ

The boundary value problem (2)–(5) must be supplemented by the radiation condition, which requires the potential to include outgoing waves only far away from the structu structure. re. An essential assumption of this study is that the gap between the structure bottom and the seabed is small compared with the water depth, whereas the structure radius and the depth are of the same order of magnitude, that is:

 ¼  ¼

h=H   e ;   e5 1;

 

 ¼ a=H  ¼  ¼ Oð1Þ;

a



 ¼ Oð1Þ  ¼

  T =H 

ð9Þ

If the small parameter   e  tends to zero an outside observer in the far field sees the structure as a cylinder extending over the entire water depth, which oscillates with its bottom sliding against the seabed. The flow through the narrow under-keel gap resembles oscillating sources (sinks) distributed over the line of contact with the seabed, the strength of which is unknown. At the same time, it can be shown that the fluid motion under the structure bottom sufficiently far from the edges is nearly a twotwo-di dime mensi nsion onal al ho horiz rizon onta tall flo flow, w, wh whic ich h is also also un unkn know own. n. Ac Acco cordi rding ng to th thee method met hod of mat matche ched d asympt asymptoti oticc expa expansi nsions ons the these se app approx roximat imation ions, s, whi which ch inv involv olvee several unknowns, can be considered as the outer fields. Coming to the ‘‘inner field’’, i.e. to the immediate vicinity of the structure edge, it must be noted that for any given angle   h   (Fig. 2) 2) the essential fluid motion of  interest occurs in the vertical plane normal to the structure surface. Physically, it can be inferred from the fact that the cylinder radius is substantially larger than the underunder-bottom bottom cleara clearance nce   h   O , so that theslowly geometry the fluid domain  Oe e   varies with a characteristic dimension very in theof   h-direction. Hence, within the inner field the fluid has mainly to flow in the vertical plane normal to the cylinder surface, whereas a slow geometry change in the other   h-direction can only result in a constant fluid velocity in that direction, the potential of which is locally a constant. Therefore, an additive constant must be retained in the inner potential to provide a correct description of the flow that must properly match the outer solutions. Perhaps the more rigorous explanation follows from the fact that the potential is a harmonic function, and is therefore uniquely defined everywhere in a do doma main in by give given n va valu lues es of th thee fu func ncti tion on an and d it itss no norm rmal al de deri riva vati tive ve on th thee boundary. Although the general form of the inner potential can be constructed in a straightforward manner, it will involve some unknowns. All the unknowns can be determined and the outer and inner potentials can be completely defined only after their appropriate expansions are matched to ensure

 ¼ ððÞÞ

 

Y. Drobyshevski / Ocean Engineering 31 (2004) 269–304

273

Fig. 2. Inner zone flow.

that both solutions describe a physically continuous flow field. This will enable us to compute the velocity potential everywhere in the fluid and to evaluate the pressure and hydrodynamic forces on the structure. After essential common derivations, the analysis is carried out separately for heave, surge and pitch motions. Only the first-order asymptotic solution in the gap height   h   will be sought, so that all the formulae for hydrodynamic coefficients will be derived with the first-order terms in th thee ga gap p he heig ight ht   h   incl includ uded ed,, if no nott stat stated ed ot othe herw rwis ise. e. Th Thee te term rm O e2   (or O h=H  2 ) will be omitted in most of the expressions for the sake of brevity.

þ ðð

 þ ð Þ

ÞÞ

 2.2. The flow outside the structure In the outer filed, the flow outside the structure can be described by the following potential:

Þ ¼ u ðx; y; zÞ þ u ðx; y; zÞ:

u x; y; z

ð

1

 

2

ð10Þ

ð Þ

Here the pot Here potent ential ial   u1  y; z   corre correspon sponds ds to the bou bounda ndary ry con condit dition ion (7) imp imposed osed on the side surface of the cylinder, which now appears to be extended over the entire ent ire flui fluid d dep depth, th, whe wherea reass the potent potential ial   u2  y; z   descr describes ibes the flow thr throug ough h the under-bottom gap. Each of the two potentials can be sought in the following gen-

ð Þ

eral eral form form,, pe pert rtin inen entt to the the thre threee-dim dimen ensi sion onal al flu fluid id do domai main n wi with th vert vertic ical al si side ded d

 

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Y. Drobyshevski / Ocean Engineering 31 (2004) 269–304

boundaries:

ð

Þ ¼ G  ðx; yÞ  Z 

u x; y; z

0

ðÞ

0

1 X ð Þþ

ð Þ ð Þ

 

G m x; y  Z m z :

z

¼

m 1

ð11Þ

ð Þ ðm ¼ 1; 2; 3; Þ  make the complete orthogonal set of functions

Here   Z 0 z ; Z m z

.. .

satisfying defined as:boundary conditions on the free surface and on the seabed, which are

ð Þ ¼ N   = cosha ðz  H Þ;   N   ¼ ð1 þ sinh2u =ð2u ÞÞ=2; Z  ðzÞ ¼  N  =  cosa ðz  H Þ;   N   ¼ ð1 þ sin2u =ð2u ÞÞ=2; u  ¼  a H ;   u  ¼  a H ;   m ¼  1 ; 2; 3;   1 2

Z 0 z

0

0

m

m

0

1 2

m

m

0

0

m

0

   

0

m

m

.. .

m

ð12Þ ð13Þ ð14Þ

where  a 0   and   am  are the roots of the dispersion equations: 2

a0 tanha0 H  am tanam

  ¼¼ m;   ðm ¼ r =gÞ H  ¼  ¼ m;   m ¼ 1; 2; 3

ð15Þ ð16Þ

 

.. .

Functions   G 0 x; y   and   G m x; y   for   m

 1  1;; 2; 3; . . .  can be considered as Fourier

coefficients of decomposition (11), with the following inversion formulae: H    1 H  u x; y; z  Z 0 z dz; G 0 x; y H  0 H    1 H  u x; y; z  Z m z dz; G m x; y H  0

ð Þ ð Þ¼ ð

ð Þ  ¼ Þ ð Þ

ð Þ¼

Þ ð Þ

ð  ð  ð

 

ð17Þ

Substituting (11) into Laplace equation (2), using (17) and taking into account that the functions   Z kk   z  are orthogonal, one can show that these functions satisfy the following partial differential equations:

ðÞ

 

@ 2 @ x2

 þ  þ a

2 0

@ 2 @ x2

 þ  @ @  y   a

2 m

  @ 2 @  y2 2

2

 

ð Þ ¼ 0;

 

G 0 x; y

ð Þ ¼ 0;

G m x; y

 ¼ 1; 2; 3

  m

H  H 



 1 H 

H H  



 1 H 

 ¼ ð    ¼ ð  

@ G G m   @ n

0

0

@ u Z 0 z dz @ n C 

ð Þ  ¼ w ðn; gÞ;

@ u Z m z dz @ n C 

0

ð Þ  ¼ w ðn; gÞ: m

 

.. .

and bou boundar ndary y con condit ditions ions on the bod body y con contou tourr   C   : plane: @ G G 0  @ n

ð18Þ ð19Þ

  fx ¼ n; y ¼ gg   in the water 

ð20Þ

 

ð21Þ

Thus for a vertical sided structure the three-dimensional problem is reduced to a pair pa ir of eq equat uatio ions ns (18) (18) and (1 (19) 9),, in tw two o di dime mensi nsion ons. s. Th Thee el eleme ement ntar ary y sol solut utio ions ns,, which satisfy the radiation condition in the far field, are known to have the follow-

 

Y. Drobyshevski / Ocean Engineering 31 (2004) 269–304

275

ing form:

ð Þ ¼ A  H ð  Þ ða RÞ; G  ðRÞ ¼  B   K  ða RÞ:

G 0 R

2 0

m

0

ð22Þ ð23Þ

   

0

m

ð2Þ

Here   H 0   is the Ha Hank nkel el functi function on of the the seco second nd kind, kind,   K 0   is the modified Bessel fu func ncti tion on of the the se seco cond nd ki kind nd,,   A   and   B    are arbi arbitr trar ary y co cons nsta tant nts, s, an and d R 2 2    y  g is the distanc distancee between between the sin singula gularit rity y poi point nt n; g   and the x  n field point x; y . More general solutions can be obtained by distributing singulariti rities es (22) (22),, (23) (23) over over the the bo body dy conto contour ur.. Ap Appl plyi ying ng th thee Gr Gree een n th theo eore rem m it can can be show shown n that that thes thesee solu soluti tion onss sati satisf sfy y the the fo foll llow owing ing in inte tegr gral al equ equat atio ions, ns, whic which h are are equivalent to (18), (20) and (19), (21), respectively:

p  ffiðffi ffi  ffiffi ffi ffi Þffi ffi ffiþffi ffiðffi ffi  ffiffi ffi ffi Þffi ffi

 ð Þ

 ð Þ

ð Þ ¼  4i

G 0 x; y

þ "

ð Þ ð  Þ ða RÞ  G  ðn; gÞ 2

w0 n; g H 0

0

ð2 Þ  @ H H  

0

0

@ n

  ða RÞ 0



 1 2p

ð Þ¼

G m x; y

þ 

ð Þ ð

#

Þ



ð24Þ

 

 @ K K  0 am R ds:  G m n; g @ n

Þ  ð Þ  ð

wm n; g K 0 am R

ds;

 ¼

 

ð25Þ

Finally, it follows from (24) and (25) that functions   G m   vanish quickly as the field point moves away from the structure, whereas the far filed behaviour of   G 0 can be expressed in the form:

ð Þj

G 0 x; y r!1

Qða ; hÞ ið ¼   ip    e 8pa r 0

p=4Þ þ Oð1=rÞ;

a0 r

 ffiffi ffi ffi ffi ffi 0

 

ð26Þ

where the analogue of the Kochin function is introduced as follows:

 þ  Þ¼

ð

ð Þ ð

Þ ð Þ

w0 n; g  w n; g; h

Q a0 ; h



w n; g; h

ð

 exp ia0 ncosh

Þ¼

f ð

 þ



 @ w  G 0 n; g ds; @ n

 

 gsinh :

ð27Þ 28

Þg

 

ð Þ

One can conclude therefore that expressions (26)–(28) define the far filed behaviour of the potential (11) in the form of the radiated outgoing waves. It should be noted that the above results, which were essentially established by Haskind (Has( Haskind, 1973), 1973), are valid for any vertical sided structure in finite depth waters. For a vertical plate, circular and elliptical cylinders equations (24), (25) can be solved analytically. Considering a circular cylinder of the radius   a, let us assume that the normal derivatives on the boundary (20), (21) are given in the form:

ð

1 X Þ ¼  

w0 n; g

wm n; g

ð

an cosnh;

¼ 1

n 0

ð29Þ

bnðmÞ  cosnh:

¼ Þ¼X n 0

 

 

30

 

ð Þ

 

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Y. Drobyshevski / Ocean Engineering 31 (2004) 269–304

Assumi Assu ming ng simi simila larr pr pres esen enta tati tion onss fo forr th thee fu func ncti tion onss   G 0 x; y   and   G m x; y (m  1  1;; 2; 3; . . .), substituting them into the integral equations (24), (25) and using the addition theorem for Bessel functions (Abramowitz ( Abramowitz and Stegun, 1964), 1964), one can find:

ð Þ

 ¼

1

ð2 Þ ða

Þ cosnh; ð Þ ¼ ¼   k H ð  Þ ðk Þ  1   K  ða rÞ G  ðr; hÞ ¼  a bð  Þ    cosnh; k K 0 ðk Þ

G 0 r; h

 a

n 0

m

2 n

0

   

0

m

m

¼

31

0

n

m n

n 0

 ¼

0r

  H n

an

X X

ð Þ

n

m

ð Þ ð32Þ

 ¼

where   k0  a 0 a,   km  a m a, and the prime denotes the derivative. The above expressions enable us to construct all the required solutions for the perturbed fluid flow outside the structure. For example, the potential   u1  y; z , corresponding to the bottom-mounted cylinder involved in small  surge  motions with unit velocity amplitudes, can be readily obtained by using boundary condition (7) and expressions (20), (21):

ð Þ

w0 a; h

ð Þ¼

 1 H 

 1 H 

ð Þ¼

wm a; h

H  H 

ð  ð 

  ð Þ  ¼   sinhu=  cosh;

 

ð33Þ

  ð Þ  ¼   sinu =  cosh:

 

ð34Þ

0 1 2 u0 N 0

cosh  Z 0 z dz

0

H  H 

m 1 2 um N m

cosh  Z m z dz

0

Upon comparing the above formulae with (29), (30) and using (31), (32) one gets:

G 0 r; h

ð Þ¼

ð2 Þ ða rÞ 0  cosh; 0 ð 2Þ H    ðk Þ

 

ð35Þ

ð Þ   cosh; ð Þ

 

ð36Þ

  asinhu0 H 1 1=2 u0 N 0

k0

ð Þ ¼   asinu=

m 1 2 um N m

G m r; h

0

1

K 1 am r km K 10 km

and the surge radiation potential (11) is therefore given by:

(

)

ð2Þ 1 sinu K  ða rÞcosa ðz  H Þ   sinhu0 H 1   ða0 rÞcosha0 ðz  H Þ 1 m m m þ u1 ¼  a cosh: 0 0 ð 2 Þ ð k Þ k K  u N  u0 N 0 m m m m k0 H 1   ðk0 Þ 1 m¼1

X

ð37Þ

Following a similar procedure, the  pitch  potential   u1  can be found, which corresponds spo nds to the bot bottom tom-mo -mount unted ed cyl cylind inder er pitchin pitching g abo about ut the   y-axis in the waterplane:

(  ð

ðÞ  1Þ H    ða rÞcosha ðz  H Þ    u  ¼H  ð Þ0 u N  H    ðk Þ 1 ð1  cosu Þ K  ða rÞcosa ðz  H Þ cosh: þm¼1 u N  K 0 ðk Þ 1

2

2 1

coshu0 3 0

X

m

0

2 1

0

m

3 m

0

1

m

0

m

1

m

)

38

 

ð Þ

 

Y. Drobyshevski / Ocean Engineering 31 (2004) 269–304

277

Noting that for  heave   motions, the potential   u1  y; z   is identically zero, let us construct the potential   u2  y; z  describing the flow through the under-bottom gap. As the outer field observer sees the small under-bottom gap shrinking to zero, the normal fluid velocity on the structure can be described by the Dirac delta-function, which is zero everywhere on the body side surface but has a finite integral flux   q h at the bottom  z  H . Hence, boundary conditions (20), (21) can be written as:

ð Þ

ð Þ

ðÞ

  ¼1 w ða; hÞ ¼ H  0

H H   0

H  H 

 1 H 

ð Þ¼

wm a; h

0

 

ð Þ  ¼   qðhÞ=

@ u Z 0 z dz @ n C 

ð  ð 

1 2 HN 0

ð Þ  ¼   qðhÞ=

@ u Z m z dz @ n C 

;

1 2 HN m

:

 

ð39Þ

 

ð40Þ

Using (31) and (32) and further representing the strength function by the Fourier series:

1 X ð Þ¼

 

qn cosnh;

qh

¼

n 0

ð41Þ

the required potential can be found in the following form: u2

 ¼  H a

X1 ( qn

¼

n 0

)

ð2Þ 1 K  ða rÞcosa ðz  H Þ H n   ða0 rÞcosha0 ðz  H Þ n m m þ cosnh: 0 ð 2Þ0 N  k K  ð k Þ m m m N 0 k0 H n   ðk0 Þ n m¼1

X

ð42Þ

The ab The abov ovee ex expr pres essio sion n is su suit itab able le for for all all mod modes es of mo moti tions ons an and d wi will ll be us used ed below.

 2.3. Flow under the structure bottom As the under-bottom clearance is small compared with the dimensions of the structure, vertical coordinate z  H    is a small quantity. Therefore, the velocity

 ð   Þ

potential can be sought in the form of the series: z  H  uð1Þ x; y z u x; y; z  u ð0Þ x; y  O z  H  3 :

ð

Þ¼

ð Þ þ ð   Þ ð Þ þ ð   H Þ uð Þ ðx; yÞ þ ðð   Þ Þ   2

2

ð43Þ

Due to the boundary condition on the seabed (4) and the Laplace equation (2) the above expression reduces to: 2

 ðz  H Þ uðx; y; zÞ ¼  u ð Þ ðx; yÞ  2 0



@ 2 @ x2

 þ

  @ 2 uð0Þ x; y 2 @  y



3

ð Þ þ Oððz  H Þ Þ:   ð44Þ

Having applied condition (8) on the structure bottom to (44), one obtains the Poisson differential equation for the first-order potential:



@ 2

  @ 2

@ x2

@  y2

 þ

uð0Þ x; y



ð Þ¼

ð  Þ ;

  g x; y h

45

 

ð Þ

 

278

Y. Drobyshevski / Ocean Engineering 31 (2004) 269–304

which in polar coordinates is:



@ 2 @ r2

 þ

 1 @  r @ r

 þ

  1 @ 2 uð0Þ r; h 2 2 r @ h



ð Þ ¼   gðrh; h Þ :

ð46Þ

 

For a particular mode of motion and given boundary condition on the cylinder bottom, the right-hand side of (46) is defined and the equation can be solved without difficulties.

 2.4. The inner flow

f  ¼  ¼ g

To examine the flow near the edge of the structure r  a ; z  T    (Fig. 2), 2), one can directly follow the approach used for the two-dimensional problem (D); this will not be repeated here. It should be noted only that the introduction of local ‘‘stretched’’ coordinates with the   x1-axis normal to the cylinder surface transforms the side surface locally into a tangential plane. The essential fluid motion in the inner flied is therefore a two-dimensional flow in the vertical plane, normal to the cylinder, whereas the flow in the   h-direction is taken into account by an unknown additive constant. By analogy with the two-dimensional problem, the two expansions of the velocity potential near the structure edge are:

ð

Þjð

u x1 ; z

ð

Þjð

u x1 ; z

 Þ!1 ¼ ½U ðhÞ þ f ðH ; hÞ  ðx1  aÞ

x1 a

 2U pðhÞh ð1  log2Þ þ C ðhÞ;  ¼  2U ðhÞh logjwj  2U ðhÞh log   2h

 Þ!þ1

x1 a

p

p



 

ð47Þ

p

þ f ðH ; hÞðx   aÞ þ C ðhÞ; 1

 

ð48Þ

where

q   ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi  ¼ ð   Þ þ ð   Þ

w

x1

 a

2

  z

 H  2 :

 

ð49Þ

Here   U  h   and   C  h  are the unknown normal fluid velocity near the edge, and a common constant, respectively, and   f  H ; h  is the normal velocity on the side surface of the structure according to boundary condition (7).

ðÞ

ðÞ

ð

Þ

3. Heave

Due to the symmetry of heave motions, the unknown fluid flux through the under-bottom gap must be constant over the periphery of the structure, so that only the first term in the series (41) should be retained:

ð Þ ¼ q   q:

qh

0

 

ð50Þ

 

Y. Drobyshevski / Ocean Engineering 31 (2004) 269–304

279

Expression (42) therefore gives the potential:

Þ ¼   aq H 

ð

u r; h; z

(

)

ð2Þ 1 K  ða rÞcosa ðz  H Þ H 0   ða0 rÞcosha0 ðz  H Þ 0 m m þ : 0 ð 2Þ0 ð k Þ N  k K  m m m N 0 k0 H 0   ðk0 Þ 0 m¼1

X

  ð51Þ

For the purposes of matching the expansion of the potential (51) is required near the structure edge. Note that the series in (51) is not converging when   r  a   and z  H , and therefore needs special treatment; a similar situation occurred in the two-dimensional problem. This should be anticipated as the potential must involve the the lo loga gari rith thmic mic sing singul ular arit ity y ne near ar the the sour source ce point point,, an and d th ther eref efor oree th thee fo follo llowi wing ng expansion can be suggested:

 !

 !

Þj !! ¼   2qp   logjwj þ A þ Oðr Þ:

ð

u r; h; z

2

 r a z H 

 

ð52Þ

Here the distance   w  is defined by (49) and the constant   A  should be determined as:  2q  ¼   lim uðx; y; zÞ  logjwj: r!a p

 

A

z H 

!

Substituting (51) into (53) and applying the limit  r

ð53Þ

 ! a  one gets:

ð2 Þ ð

Þ Þ  ð Þ ð   Þ ð Þ  1 logjz  H j ð  þ Þ ð Þ p

H 0 k0 2q 0 u0  sinhu0 coshu0 H ð2Þ k0 0 1 cosa z  H  K  k m 0 m 2q   lim z!H  um  sinum cosum K 00 km m¼1  

 ¼ ð  þ þ

A

( X

)

 

:

ð54Þ

The above formulae can be simplified if the logarithmic singularity is extracted explicitly from the series in brackets. This can be conveniently done using the following identity (Abramowitz (Abramowitz and Stegun, 1964): 1964):

  ð  

log   2sin

  p

Þ  ¼ X1  ¼

z  H  2H 

m

ð   Þ

1   mp z  H  cos m H  1



 

:

ð55Þ  ! H ,

Adding Addi ng an and d su subt btra ract ctin ing g (5 (55) 5) from from (5 (54) 4) and and ap appl plyi ying ng th thee li limi mitt   z obtains:

A

 ¼

 2 q

"

ð2 Þ ðk

Þ  1 log   p  F  ðu  þ sinhu coshu Þ H ð  Þ0 ðk Þ þ p H  þ  

0

H 0

1

0

0

2 0

0

0

0

 #

;

one

56

 

ð Þ

 

280

Y. Drobyshevski / Ocean Engineering 31 (2004) 269–304

where

1 ð X Þ¼ 0

 ¼ F  ðmH ; a

F 0



0

¼

m 1

1 Þ ð Þ ðu  þ sinu

K 0 km K 0 km

m cosum

m

Þ þ



 1 : mp

ð57Þ

 

The constant   F 0, an analogue of the constant introduced in the two-dimensional problem (see equation (27) of (D)), is presented in  Fig. 3  as a function of the frequency par quency paramet ameter er   ma. Un Unli like ke its its tw twoo-di dime mens nsio ional nal com compa patr trio iot, t, cons consta tant nt (5 (57) 7) depends not only on the water depth, but also on the radius of the structure. Its zero ze ro freq freque uenc ncy y (m  0;   um  m p) an and d the in infin finit itel ely y la larg rgee fr freq eque uenc ncy y (m ; um   p 2m  1 =2) limits are given by the formulae:

 !

 ¼ ð   Þ

1 ð Þ  X ð Þ¼ 0 ð Þ þ ¼ 1 ð Þ X ð1Þ ¼ 0

F 0 0

F 0

 ¼

 1

p

m 1

 1

p

¼

m 1

K 0 km K 0 km

K 0 km K 0 km

 1

 ! 1

 1 ;   km m

 ¼ mpa;

2 2m  1

ð Þ ð   Þ þ



ð58Þ

 



 1 ;   km m

 ¼  pað2m  1Þ=2; 

 

ð59Þ

0 expressions follows from above for and are also presented 11.  It in  Table wide structures, when   a  a =H 41. and therefore 1   forthat all fre  K the 0 km =K 0 km quencies, quenci es, the consta constant nt  F 0  becomes equal to the constant in two dimensions.

 ¼

ð Þ ð Þ

Fig. 3. Constant F  Constant  F 0.

 

Y. Drobyshevski / Ocean Engineering 31 (2004) 269–304

281

Table 1 Zero and infinite frequency limits of the constant  F 0 a=H 

 

Zero frequency

1.0 2.0 3.0 4.0 5.0 6.0 8.0 10.0

Infinite frequency

             

0.0579 0.0227 0.0157 0.0000 0.0000 0.0000 0.0000 0.0000

0.2692 0.3442 0.3787 0.3919 0.4002 0.4060 0.4373 0.4373

Substituting (56) into equation (52), one obtains the following expansion of the potential potent ial outside the struct structure: ure:

Þj !! ¼   2pq  logjwj

ð

u x; y; z

 r a z H 

þ 2q

 1 p

log

  p



þ F   þ 0

  u0

ð2 Þ ð

H 0 k0 1  sinhu0 coshu0 H ð2Þ0 k

Þ

:

  ð60Þ

ð  þ Þ  ð Þ For the under-bottom flow, boundary condition (8) gives   gðr; hÞ ¼  1 . Due to the 0

" 

0

#

flow symmetry, the potential must be independent of the angle   h, so that: u r; h

ð Þj ¼ uða; hÞ ¼ u ;

ð61Þ

 

0



where  u 0  is an unknown constant. Equation (46) then becomes:



@ 2 @ r2

 þ

 ð Þ¼

 1 @  u r r @ r

 1  : h

 

ð62Þ

Assuming that the radial fluid velocity under the centre of the platform is continuous and zero, the solution can be sought in the following form:  Ar 2

u r

ð Þ¼

 B ;

63

þ

 

ð Þ

where   A   and   B  are   are constants. Upon substituting (63) into (62) and using (61) one gets: u r r2  a2 = 4h  u0 : 64

ð Þ¼ð  Þ ð Þþ

 

ð Þ

Finally, the expansion of the potential near the edge of the structure can be written as a Taylor series:

ð Þj ! ¼ u  þ  2ha  ðr  aÞ þ Oðr  aÞ :

u r

r

a

0

2

 

ð65Þ

The ou The oute terr po pote tent ntial ialss (5 (51) 1) an and d (64) (64) in invo volve lve cert certain ain un unkn known owns, s, wh whic ich h can can be now no w de dete term rmin ined ed by matc matchi hing ng expan expansi sion onss of th thes esee po poten tenti tial alss with with ap appro propr priat iatee expansions of the inner solution. The expansion of the outer flow potential (60) must match the expansion of the inner potential (48), whereas the under-bottom pote po tent ntial ial (65) (65) mu must st ma matc tch h ex expa pansi nsion on (47) (47) of th thee in inne nerr po pote tent ntial ial.. Ta Taki king ng in into to

 

282

Y. Drobyshevski / Ocean Engineering 31 (2004) 269–304

account that both the fluid velocity and the additive constant must be independent of the polar angle in the inner expansions (47), (48), i.e.   U  h  U ; C  h  C , the matching matchi ng condit conditions ions yield the follow following ing relatio relations: ns:

ð Þ

 ¼ 2hU ;

a

ð66Þ

 

 a

  ¼¼  þ p ð1  log2Þ;  ¼  a=2; q ¼  hU  ¼

C   u 0

 ¼ a

u0

(

ð Þ

 

ð2Þ H 0   ðk0 Þ   1   1 F 0 þ ðu0 þ sinhu0 coshu0Þ H ð2 Þ0 ðk Þ  p 0

0

ð67Þ ð68Þ

 

    ) 1

 log

  4h H 

:

  ð69Þ

According to (66), (68) the fluid volume displaced by the vertically moving structure, which creates the flux through the gap   h, is equal to the footprint area. As all the quantities involved in expressions (51) and (64) have been now determined, the potential can be computed for both the outside and the under-bottom regions, and all hydrodynamic coefficients can be obtained. Usin Us ing g ex expr pres essi sion onss (64) (64) and (69) (69),, on onee ca can n fin find d th thee ad adde ded d ma mass ss an and d radi radiati ation on damping, defined by the known general formulae: @ u j  @ u j    ds :   ds ;   kij   qr Re ui  lij  qRe ui  @  n @  n S  S 

ð 

 ¼ 



ð 

 ¼



 

ð70Þ

Here subscripts   i   and   j   denote denote modes of motion, and the integral is taken over the immersed surface of the structure   S . For the circular cylinder and heave motions these formulae give: 2p

l33

a

ð  ð  ð Þ

 ¼ qRe

0

2p

0

0

0

 

;

 q pra2 Im u0 :

ð  ð  ð Þ  ¼  þ     þ

 ¼

  Reðu

Þ

ð71Þ

a

dh u r rdr

 qr Im

k33

 ¼

dh u r rdr

2 2 a  q pa 8h



0

ð Þ

72

 

ð Þ

Having substituted (69) into the above formulae, one can represent the added mass and damping in the following closed form: l33

k33

¼

  a qpa3 8h

 

¼ ðu þ 0

1

p

1

  4h log H 

qpa3 r sinhu0 coshu0

2

Þ pk ðJ  ðk Þ þ N  ðk ÞÞ : 0

2 1

0



ð Þ ð Þ þ ð Þ ð Þ  F  ; ð þ Þð ð Þ þ ð ÞÞ ð73Þ N 0 k0 N 1 k0   J 0 k0 J 1 k0 N 12 k0 u0 sinhu0 coshu0 J 12 k0

2 1

0

 

0

ð74Þ

Here   J kk    and   N kk   are Bessel and Neumann functions, respectively, and the constant F 0   is given by (57). The limiting values of the added mass corresponding to the zero zero an and d the the in infini finite tely ly larg largee freq freque uenc ncies ies ca can n be fo found und di dire rectl ctly y fr from om (7 (73) 3) us usin ing g

 

Y. Drobyshevski / Ocean Engineering 31 (2004) 269–304

283

known properties of cylindrical cylindrical functions: l33

j!¼ r

0

 þ      

  a qpa3 8h

1

1

p

   þ  ð Þ

 a k0   log 2 2H 

  4h log H 



F 0 0

  ð75Þ

;

  a 1   4h ¼ q a  F  ð1Þ ;   ð76Þ þ 1  log p !1 8h p H  j where   C ¼ logc ¼ 0:577215   is the Euler constant. It can be seen that the heave 3

l33

0

r

   



...

added mass involves the logarithmic singularity and tends to infinity when the frequen qu ency cy appro approac ache hess ze zero ro.. It shoul should d be no note ted d th that at such such be beha havi viou ourr of th thee he heav avee (1981).. added mass and a similar asymptotic formula were established by   Yeung (1981) Thee la Th latt tter er ho howe wever ver in invo volve lvess a di diffe ffere rent nt co cons nsta tant nt term term be beca caus usee of a si simp mpli lifie fied d approach employed for matching the potentials, as recognized by Yeung. The exciting force on the structure can be conveniently obtained by using the 1977): ): Haskind–Newma Haskind –Newman n formul formula a (Newman, 1977

ð Þ ¼ iei

rt

X  j  t

þ 

@ ui  uW @ n

qr

 



@ uW ds:  u j  @ n

R

 

ð77Þ

Equation (77) involves the potential of oncoming waves   uW   and the radiation potential   u j  for a given mode of motion  j . The integration is performed over a control surface (a cylinder of the infinitely large radius) located in the far filed, which greatly simplifies the task as only the far field behavior of the radiation potential is needed. This can be found from (51):

ð Þj !1 ¼   acosha ðz  H Þð Þ0 ðu  þ sinhu coshu ÞH    ðk Þ

u r; h; z

r

0

0

0

2 n

0

0

s  ffiffi ffi ffi ffi

2 eiða0 rp=4Þ pa0 r

  



 þ O   1r

:

 

ð78Þ

The potential of oncoming waves propagating in the positive   x-direction is given by: uW r; h; z

ð

Þ ¼ i grr

0

cosha0 z  H   eia0 rcosh ; coshu0

ð79Þ

 

ð   Þ 

denotin ing g the the wa wave ve ampli amplitu tude de.. Su Subs bstit titut utin ing g (78) (78) and and (7 (79) 9) in into to (7 (77) 7),, an and d with   r0   denot using the method of stationary phase to evaluate the integral, one obtains the following expression for the exciting force:

ðÞ¼

X 3 t

 

2iqgr0 a2

ð2Þ k0 coshu0 H 1   ðk0 Þ

  2qgr0 a2 ð 1Þ X    ¼

eirt

¼ ðX ð  Þ þ iX ð  Þ Þ  ei 3

ð Þ ; k coshu ð ð Þ þ ð ÞÞ   2qgr a J  ðk Þ ðÞ  : X    ¼  k coshu ðJ  ðk Þ þ N  ðk ÞÞ 3

0

0

1

3

N 1 k0 J 12 k0  N 12 k0

0

0

2

0

1

12

0

2

1

3

rt

 

;

 

ð81Þ

0

12

0

ð80Þ

82

 

ð Þ

 

284

Y. Drobyshevski / Ocean Engineering 31 (2004) 269–304

This completes the solution of the heave radiation problem. To check the applicability range and accuracy of the above formulae comparison was made with pub(1981) presents  presents extensive lished numerical results, among which a paper by  Yeung (1981) information on the added mass and damping coefficients of circular cylinders in finite depth waters.   Figs. 4 and 5  show the added mass, wave radiation damping and the exciting force amplitude for cylinders with the radius to depth ratio  a =H  1:0 and 5.0 in the following non-dimensional form:

 ¼  ¼

 33 l

 ¼ q  pla H  ; 33 2

 

 k

 ¼ q  pak H r ;

33

33 2

 

 X  3

 ¼ q  gjpX a jr 3 2

: 0

 

ð83Þ

 ¼  ¼

Fig. 4. Heave added mass, radiation damping and exciting force amplitude, a amplitude,  a= =H   1  1::0.

 

Y. Drobyshevski / Ocean Engineering 31 (2004) 269–304

285

Fig. 5. Heave added mass, radiation damping and exciting force amplitude,  a=  a =H   5  5::0.

  ¼¼

Calculations have been performed for water depth to draft ratios   H =T   1 :11, 1.33, 4.00, for which numerical results are available in Yeung’s paper. It can be seen that the heave added mass is strongly affected by water depth, increasing rapidly as the under-bottom clearance decreases with the tendency being similar to that for the rectangular profile in two dimensions. For example, if a 100-m diameter structure is towed in a 10-m deep area with the under-keel clearance of 1 m, the heave added mass exceeds the physical mass of the structure by a factor of 35. Correspondingly, the heave natural period is about six times greater than it may be expect exp ected ed fro from m the str structu uctural ral inerti inertia a only. only. Sim Similar ilar to the two two-di -dimen mensio sional nal cas case, e, beyond the low frequency range where the added mass increases sharply, it varies slowly with the frequency, so that the formula for the infinite frequency limit (76)

 ¼  ¼

 

286

Y. Drobyshevski / Ocean Engineering 31 (2004) 269–304

can can be usef useful ul for for esti estima matio tion n pu purp rpose oses. s. Th Thee radi radiat atio ion n dampi damping ng an and d th thee exci excitin ting g force are seen to be weakly affected by the seabed proximity. Both figures show points that have been taken directly from curves provided by  Yeung (1981)  up to the best of this author’s skills and accuracy. It can be seen that the proposed formulae are in very good agreement with exact results, in particular for the wide cylinder (a=H   5  5::0) where the asymptotic formulae provide excellent accuracy even fo forr the the wa wate terr de dept pth h of   H =T   4  4::0. Such Such a surp surpri risi sing ng resu result lt can can pr prob obab ably ly be expl explai aine ned d by the the do domi mina natin ting g effe effect ct of the the un under der-b -bot otto tom m flo flow, w, wh whic ich h on only ly co conntributes to the vertical hydrodynamic forces on the structure and which appears to be captured well by the present theory.

 ¼

 ¼  ¼

4. Surge

Using general equation (10), let us represent the potential of the outside flow as a sum of the two components: the potential (37) describing the flow past an oscillating bottom-mounted cylinder and the potential (42) of the flow through the gap between the structure bottom and the seabed. The latter can be simplified if the source strength at the bottom is assumed in the following form, which is similar to the boundary condition (7):

ð Þ ¼ qcosh:

qh

 

ð84Þ

Retaining only the corresponding term in the general expression (42), one gets:

Þ ¼   aq H 

u2 r; h; z

ð

(

)

ð2Þ 1 K  ða rÞcosa ðz  H Þ H 1   ða0 rÞcosha0 ðz  H Þ 1 m m þ cosh: 0 0 ð 2Þ N  k K  ð k Þ m m m N 0 k0 H 1   ðk0 Þ 1 m¼1

X

ð85Þ

The expansion of the total potential near the structure edge ( r  a ,   z  H ) can be constructed in the same manner as it has been done for the heave potential.

 !

 !

Omitting detailed derivations, one can write the result as: r  a cosh u r; h; z  r !a  u 1 a; h; H 

ð

Þj ! ¼ ð

Þ þ ð   Þ

"

 þ

z



þ 2q   p1 logjwj þ  1p log   H p

#

ð2Þ H 1   ðk0 Þ   1  F 1 þ ðu0 þ sinhu0 coshu0 Þ H ð2 Þ0 ðk Þ cosh:   ð86Þ 1

0

Here the first two terms make the expansion for the potential (37) in the form of  its Taylor series, and terms in brackets make the expansion of (85). The new constant  F 1  has been introduced:

F 1

 ð Þ¼ 0ð ¼ X

 ¼ F  ðmH ; a 1

1



m 1

K 1 km K 1 km

1 Þ Þ ðu  þ sinu m

m cosum

Þ þ



 1 ; mp

 

ð87Þ

which is of the same form as   F 0   but involves modified Bessel functions with the

 

Y. Drobyshevski / Ocean Engineering 31 (2004) 269–304

287

Fig. 6. Constant F  Constant F 1.

index 1 rather than 0. The constant   F 1   is presented in   Fig. 6   as a function of  the fre freque quency ncy par paramet ameter er   ma. Th Thee ze zero ro an and d in infin finit itee fr freq eque uenc ncy y li limi mits ts of (8 (87) 7) are are given by:

1 ð Þ  X ð Þ¼ 0 ð Þ þ ¼ 1 ð Þ X ð1Þ ¼ 0

F 1 0

F 1

 1

p

m 1

 1

p

¼

m 1

K 1 km K 1 km

K 1 km K 1 km

 1

 1 ;   km m

2 2m  1

ð Þ ð   Þ þ

 ¼ mpa;

 





 1 ;   km m

ð88Þ

 ¼  pað2m  1Þ=2; 

 

ð89Þ

and are also presented in   Table 2. 2. It can be seen that the two constants,   F 1   and F 0, exhibit similar behaviour becoming increasingly close to each other for wide  structures, i.e. when   a  a =H 41.

 ¼

r; h  0 , so that For the und under-b er-bott ottom om flow flow,, to bou boundar ndary y cond conditi ition on (8) giv gives es   g has Poisson equation (46) reduces the Laplace equation, which the following

ð Þ¼

Table 2 Zero and infinite frequency limits of the constant  F 1 a=H 

 

Zero frequency

Infinite frequency

1.0 2.0 3.0 4.0 5.0 6.0 8.0

0.0693 0.0257 0.0171 0.0000 0.0000 0.0000 0.0000

           

0.2197 0.3250 0.3690 0.3859 0.3962 0.4031

10.0

0.0000

 

0.4373

0.4373

 

288

Y. Drobyshevski / Ocean Engineering 31 (2004) 269–304

general solution bounded everywhere within the footprint area:

1 X Þ¼ ð

rn An cosnh

u r; h

ð

 þ B  sinnhÞ:

¼

n 0

 

n

ð90Þ

Here   An   and   B n  are unknown constants. Given boundary condition (7) for surge motions, let us assume that all the constants are zero except of   A1, so that the potential (90) and its expansion near the edge of the structure are given by:

ð Þ ¼ A  rcosh;

 

u r; h

u r; h

ð91Þ 2

ð Þj ! ¼ Aacosh þ Aðr  aÞcosh þ Oððr  aÞ Þ: r

 

a

ð92Þ

All the required expressions have been now prepared, and one can carry out the matc ma tchi hing ng of ex expa pans nsion ionss of the the oute outerr an and d in inne nerr solu soluti tions ons.. Expr Expres essio sion n (8 (86) 6) must must matc ma tch h the the ex expa pansi nsion on of the the in inne nerr pote potent ntia iall (48), (48), wh where ereas as th thee expa expans nsio ion n of th thee underund er-bott bottom om pot potenti ential al (92 (92)) mus mustt mat match ch exp expans ansion ion (47) (47) of the inn inner er pot potent ential ial..

ð Þ ¼ U cosh   and   C ðhÞ ¼ C cosh, these conditions yield the follow-

Assuming that   U  h ing equations:

 þ  ¼ A;  þ

 

U   1

ð93Þ

 2Uh   ð1  log2Þ þ C  ¼  ¼ Aa; p

 

Uh

 ¼ q;

"

C   2 q

 ¼  ¼

ð94Þ

 

 1 p

log

  2h H 

þ F   þ 1

a; h; H    : þ u ðcos h

1  sinhu0 coshu0

u0

ð  þ

 Þ

1

 

ð95Þ ð2 Þ H 1   ðk0 Þ ð2Þ0 H  k 0

1

#

 Þ   ð Þ

ð96Þ

After some algebra, the flux strength  q  can be found in the form:

 ð S ð Þ  1=2Þ þ iS ð Þ   ; q ¼  ¼ Dð Þ þ iDð Þ 1

 q a



2

1

 ð S ð1Þ  1=2ÞDð1Þ þ S ð2Þ Dð2Þ ð 1Þ    ;  ¼ ReðqÞ ¼ q Dð1Þ2 þ Dð2Þ2 

2

qð Þ



 ¼ ImðqÞ ¼

  S ð2Þ Dð1Þ

S ð1Þ

Dð1Þ2

ð97Þ

 

2

 

ð98Þ

 

ð99Þ

 1=2 Dð2Þ

  ð þ D ð Þ Þ   22

;

 

Y. Drobyshevski / Ocean Engineering 31 (2004) 269–304

289

where the following notations have been used:   sinhu0  A S ð1Þ ¼  u0

 þ sinhu coshu 0

0

1 X þ ¼

m 1

ðu  þ m

 ¼   þ  

Þ

 1

ð Þ

Dð1Þ ¼ Dð2Þ ¼

  k0  Að2Þ k0 ; u0  sinhu0 coshu0

 þ p

 þ

  4h  log H 

1

 

 

 F 1

  k0 Að1Þ k0 ; u0  sinhu0 coshu0

 þ  þ

ð Þ

ð100Þ ð101Þ

 

     

 a 2h

ð Þ ð Þ

K 1 km sinum ;  sinum cosum km K 10 km

  sinhu0  Að2Þ k0 u0  sinhu0 coshu0 ;

2

S ð Þ

ð1Þ ðk0 Þ

ð Þ

 

 

ð102Þ ð103Þ

and the following combinations of Bessel functions have been introduced:



ð2Þ   H 1   ðxÞ  1 ð 1Þ A ðxÞ ¼  Re ð2Þ0 x H    ðxÞ   J  x J  x 1 1 J 1 x

1

!

 xJ  x 0  xJ 0 x

 N  x N  x  xN  x 1 1 0 N 1 x  xN 0 x 2  

¼ ð Þðð ðð ÞÞ ðð ÞÞÞÞ þþ ð ð ðÞðÞ ð Þ ð ÞÞ ð ÞÞ ; ðÞ   H    ðxÞ  1 ð Þ A ðxÞ ¼  Im ð Þ0 x H    ðxÞ   1 ¼    p2 ðJ  ðxÞ  xJ  ðxÞÞ þ ðN  ðxÞ  xN  ðxÞÞ  :



2



2 1 2 1

1

2

 

ð104Þ

!

0

2

1

0

2

 

ð105Þ

The above expressions for the flux strength    q   provide complete solution for the radiation surge potential, which is given by formulae (11), (37), (85) and (91). The added mass and damping coefficients can be now computed using general formulae (70). Similar to the two-dimensional problem (D), it is convenient to replace the integration over the immersed structure surface   S  by   by integration over the equivalent surface   S  (Fig. 7). 7). The latter consists of the cylindrical side surface extended down to the seabed, the horizontal bottom of the structure and a short connecting cylin

Fig. 7. Integration surface for evaluating added mass and radiation damping.

 

290

Y. Drobyshevski / Ocean Engineering 31 (2004) 269–304

der across the under-bottom gap. Hence, the integral in (70) can be rewritten as: H  H 

@ u j    ds ui  @  n  S 

ð   ¼

I iij j 

2p

0

ð

Þð Þ

ð  ð   þ

0

ð

0

0

Þ ð Þ

ui  r; h; H  g j  r; h dh

rdr

ui  a; h; z  f  j  z; h dh

ui  a; h; H   f  j  H ; h dh;

 ah



dz

 a

2p

a

2p

ð  ð   ¼

0

ð

ð 

Þð

106

Þ

 

ð Þ

where the functions   f  j ,   g j  are defined by boundary conditions (7), (8) and the three terms ter ms cor corres respon pond d to the thr three ee sur surfac facee comp compone onents nts ide identi ntified fied abo above. ve. As the gap height   h   is small compared with dimensions of the structure the last integral has been simplified, consistent with the present first-order theory. It should be noted that the first integral involves the outer potential given by the sum of (37) and (85), whereas the other two integrals are taken of the potential (91). Upon integrating and taking the real and imaginary parts according to (70), one obtains the following formulae for the added mass and damping coefficients:   l11

 11 l

 

 2

p 2

sinhu0 Að1Þ k0

ð Þ

ð2Þ  A   k0 q

ð1Þ   k0 q

  sinhu0

 k

11

Þð  Þ

ð Þþ

0

 

  sinhu0 Að2Þ k0   sinhu0 2 u0 u0  sinhu0 coshu0

  k11 qpa2 H r

2

sinum km  q  sinum cosum u u m¼1 m m

ð  þ



ð Þ

1

 þ k qð Þ þ

m

m

m

0

1

 

0

) 

1

Þ  a ð Þ K  ðk Þ   þ Þ ðK  ðk Þ þ k K  ðk ÞÞ 2H  q 1

1

 þ   a    hAð Þ ðk Þ   þ ;   ð107Þ qð Þ þ ð ÞÞ 2H  2H 

q a H  u0 coshu0 u0 u0 sinh 1 sinum sinum  km  K 1 km q  sinum cosum K 1 km  km K 0 km u u m 1 m m

    ¼ ¼ ð  þ ð Þ Þ  ¼X 1 ð   þ þ Þð ¼ ð  þ   ¼  ð  þ  ¼ 1 ðÞ X þ

ð2Þ ðk0 Þ

m

2

ð1Þ k0 ð 2Þ  A   k0 q Að2Þ k0

)



ð Þ ð Þ   ð108Þ

:

The surge added mass and damping for the bottom-mounted cylinder that oscillates sliding against the seabed can be obtained from the above expressions by letting    q  0 ,   h  0. The limiting value of the surge added mass (107) corresponding to the zero frequency is:

 ¼

0  11 l

 ¼

 ¼

  l011 qpa2 H 

 ¼

  2u0 u0  a0 a qð1Þ u0 u0  u0

Þ þ   a qð Þ þ   h  ¼  1 þ   h  þ 2a qð Þ :   ð109Þ ð  þ   ð  þ Þ H  H  H  H   

1

 

1

Having noted that for small frequencies the flux strength (98), (99) reduces to:  qð1Þ

 ! 1=Dð Þ; 1

 

qð2Þ



 ! 0;

 

ð110Þ

the low frequency limit of the surge added mass can be represented as: 

0

l11

  l011

 4

 ¼ qpa H  2

 1

 e   1

b e

 ¼  þ      ð Þ  þ  ¼

2

 O e ;

ð Þ

 

ð111Þ

 

Y. Drobyshevski / Ocean Engineering 31 (2004) 269–304

291

where

 ð Þ ¼  þ    

b e

 1

 e   1

 



ð  Þ þ   2    2e logð4eÞ; a pa pa

 2F 1 0 



 



ð112Þ



 ¼ a=H  2.. The The constant   F  ð0Þ  is defined by equation (87) and is also given in   Table 2  ¼ h=H ;

e

a

1

above expression gives the added mass coefficient of a circular cylinder moving in close proximity to the seabed; it is also applicable to a vertical cylinder of the height 2(H  h) moving in the channel between the two walls spaced at the distance 2H . Using expression (106) and the pitch boundary conditions on the structure (7), (8)  f 5 z; h  z cosh,   g5 r; h rcosh  one can obtain the following formulae for the surge–pitch coupling inertia  l 15  and the damping term  k 15 :



ð Þ¼

 15 l

ð Þ¼

  l 15 qpa2 H 2

15

coshu0  1 A 1 k0  2 u2 u0 sinhu0 coshu0   sinhu0 0

 Að2Þ k0 ð2Þ  k0 q Að1Þ k0

0 1

 þ k qð Þ  ð Þ  ¼ 1 ð1  cosu Þðsinu  þ k qð Þ Þ K  ðk Þ þ ðK  ðk Þ þ k K  ðk ÞÞ u ðu  þ sinu cosu Þ m¼1 þ 12 ah qð Þ þ 1   4aH   þ  H h ;   ð113Þ  A ð Þ ðk Þ   ð coshu   1ÞAð Þ ðk Þ   k ð Þ ð Þ  ¼ qpa H  r ¼ 2 u ðu  þ sinhu coshu Þ   sinhu  þ k q  þ k q Að Þ ðk Þ 1 ð1  cosu Þk qð Þ a ; þ u ðu  þ sinu cosu Þ ðK  ðk ÞK þð kk ÞK  ðk ÞÞ þ  2aH  qð Þ   1 þ  4Hh

X 

 k

ðÞ   ¼   ð ð  þ   Þ ð ÞÞ  m

2 m

 

m

15 2 2

X ¼

m 1

m

m

m

1

2

0

2 0

0

m

m

1

1

m



0

2

m

m

0

0

m

1

1



2

m

2 m

m

2

 

1

m

0

0

m

m

m

m

0

1

 

0



0



2

m

2

1 2

2

0 0

)



ð114Þ Finally, to obtain the surge exciting force from the Haskind–Newman formula (77), one should use the far field behaviour of the radiation potential:

ð

  ¼

Þj !1

u r; h; z

r

  sinhu0 k0

 þ q

 

2acosh  cosha0 z

 

ð   H Þ ðu  þ sinhu coshu ÞH ð  Þ0 ðk Þ 0

0

0

2 n

0

s  ffiffi ffi ffi ffi 2

    eið pa r 0



Þ;

a0 r 3p=4

  ð115Þ

wheree the wher the tw two o te term rmss in br brac acke kets ts co corr rres espo pond nd to th thee po pote tent ntia ials ls (3 (37) 7) an and d (85) (85),, re resp spec ecti tive vely ly.. Su Subs bsti titu tuti ting ng (1 (115 15)) an and d (7 (79) 9) in into to (77) (77),, an and d us usin ing g th thee me meth thod od of 

 

292

Y. Drobyshevski / Ocean Engineering 31 (2004) 269–304

stationary phase to evaluate the integral, one obtains the surge exciting force on the cylinder: 0 1

X 1 t

ð Þ ¼ X 

  ¼ ð ð  Þ þ ð Þ  þ q sinhu0   k0

t  1

X 1

1

ð2 Þ Þ  eirt ;

 iX 1

ð116Þ

 

where

X 10

ðtÞ ¼

  4qgr0 a3 m

eirt : 0 ð 2 Þ k3 H    ðk Þ 0

 

0

1

ð117Þ

Expression Expres sion (11 (117) 7) giv gives es the hori horizon zontal tal for force ce on the bot bottom tom-mo -mounte unted d cyl cylinde inder, r, usually referred to as the MacCamy and Fuchs solution, whereas (116) can be considered as a correction for the small under-bottom clearance. The real and imaginary parts of the exciting force (116) are:

  þ ð Þ   ð Þ ð Þ ð Þ þ ð Þ   þ ð Þ ðÞ

ð1Þ 0ð1Þ X    ¼  Re jX  j ¼  X    1

1

1

ð2 Þ ¼ Im jX  j ¼  X 0 1

X 1

1

1

1

  k0  q1 sinhu0

k0  q2 sinhu0

2  k0 q ; sinhu0

 

ð118Þ

 1   k0 q ; sinhu0

 

ð119Þ

0 2  X 1

0 2

 X 1   1

where the components of the force (117):

ð  Þ ¼  Re jX 0 j

01

X 1

1

¼

  4qgr0 ma J 1 k0 a20

ð ð Þ

ð  Þ ¼  Im jX 0 j

ð Þ  J  ðk Þ ð ÞÞ þ ðN  ðk Þ  k N  ðk ÞÞ  ;

k0 J 0 k0  k0 J 0 k0 2

1

0

1

0

0

0

0

 

2

ð120Þ

02

X 1

1

ðk Þ  N  ðk Þ  : ¼  4qgra ma ðJ  ðk Þ  k J k ðN  k ÞÞ þ ðN  ðk Þ  k N  ðk ÞÞ 0 2 0

0

1

0

0 0

0

0

0 2

1

1

0

0

0

0

0

2

 

ð121Þ

 Figs. The the added surge hydrodynamic coefficients is now(107), completed. completed. Figs. and 9   solution show thefor surge mass and damping coefficients (108) and the8 non-dimensional amplitude of the exciting force (116): X  j  ¼ 2 qjgaHr 1

 X  1

 

:

0

ð122Þ

Cylinders of the relative radius   a=H   1  1::0, 5.0 have been considered, for water depth to draft ratios   H =T   1  1::001, 1.25, 2.00. It can be seen that all hydrodynamic coefficients are affected by water depth, with the added mass coefficient approaching the expected value of 1.00 as both the under-bottom clearance and the frequency tend to zero. The agreement between the asymptotic formulae and results  1 :0 at the water by   Yeung (1981)   is generally good, except of the radius   a=H   1: depth   H =T   2  2::00 whe when n exp expres ression sion (10 (107) 7) und underer-pred predict ictss the add added ed mas masss giv giving ing

 ¼  ¼

 ¼  ¼

 ¼  ¼

 ¼

 

Y. Drobyshevski / Ocean Engineering 31 (2004) 269–304

293

Fig. 8. Surge added mass, radiation damping and exciting force amplitude, a amplitude,  a= =H   1  1::0.

  ¼¼

even negative values. Such a result is not surprising because of the large depth, which whi ch is cer certain tainly ly beyond beyond the exp expecte ected d range range of sma small ll und under-b er-bott ottom om cle cleara arance. nce. Interestingly, for wider structures, when   a=H   5:  5 :0, the above discrepancy is not evident and the agreement between the asymptotic and exact solutions is good for all water depths considered. Figs. considered.  Figs. 10 and 11 show 11  show the surge–pitch coupling inertia and damping coefficients in the non-dimensional form (113), (114). It should be noted that due to different coordinate systems used, the above coefficients are of  the opposite sign compared to those defined by   Yeung (19 (1981) 81).. For the relative radius   a=H   1  1::0 only the water depth   H =T   1:  1 :50 has been considered for com10), ), where the agreement with Yenug’s results is extremely poor. It is parison (Fig. (Fig. 10

 ¼  ¼

 ¼

 ¼

 

294

Y. Drobyshevski / Ocean Engineering 31 (2004) 269–304

Fig. 9. Surge added mass, radiation damping and exciting force amplitude, a amplitude,  a= =H   5  5::0.

  ¼¼

believed that for lesser depths the asymptotic formulae provide increasingly better accuracy, but unfortunately this is the minimum depth considered by  Yeung (1981) for the pitch–surge coupling coefficients. Very small absolute values of the coefficients may also account for the discrepancies. One can see from Fig. from  Fig. 11 11 that  that for the radius   a=H   5  5::0 the agreement is excellent for all depths considered, up to the depth to draft ratio   H =T   4  4::00. The comparison shows that the asymptotic formulae provide better accuracy for relatively wide structures a=H 41 , when the under-bottom clearance is small compared with the structure radius, even though it may be of the same order of magnitude as the draft. For relatively narrow structures (a=H   O 1 ) there is no such advantage, and it is essential for the clearance to be small compared to both the structure radius and the draft.

 ¼  ¼

 ¼ ð Þ

 ¼  ¼

 ð

Þ

 

Y. Drobyshevski / Ocean Engineering 31 (2004) 269–304

295

 ¼  ¼

Fig. 10. Surge–pitch coupling added mass and damping coefficients,  a=  a =H   1  1::0.

 ¼  ¼

Fig. 11. Surge–pitch coupling added mass and damping coefficients,  a=  a =H   5  5::0.

 

296

Y. Drobyshevski / Ocean Engineering 31 (2004) 269–304

5. Pitch

The pitch radiation potential can be now found following the same procedure. The potential of the outside flow can be sought as a sum of the two components (38) and (42), with the latter based on the strength of the under-bottom flow:

ð Þ ¼ qcosh:

qh

 

ð123Þ

It follows therefore that the general expression (85) for the potential   u2  used in the surge radiation problem remains fully applicable. The expansion of the total potential near the structure edge ( r  a ; z  H ) is of the similar form:

 !  !

u r; h; z

Þj !! ¼ u ða; h; H Þ þ H ðr  aÞcosh

ð

 r a z H 

1

"

þ 2q   p1 logjwj þ  1p log   H p

 þ

#

ð2Þ H 1   ðk0 Þ   1  F 1 þ ðu0 þ sinhu0 coshu0 Þ H ð2 Þ0 ðk Þ cosh: 1

0

124

ð Þ

Here the first two terms make the expansion for the potential (38) in the form of  its Taylor series, terms in brackets make the expansion of the potential (85), and the constan constantt  F 1  is given by (87). For the under-bottom flow, boundary condition (8) is   g5 r; h rcosh, and the solution of Poisson’s equation (46) can be sought in the form:

ð Þ¼

3

ð Þ ¼ ðAr þ BrÞcosh:

u r; h

 

ð125Þ

Here   A   and   B  are   are constants. Upon substituting (125) in (46), the potential can be written writte n as:

ð

 Þ¼

r   u0 a

u r; h

 þ

 a 2 r   1 8h

  r2 a2

   

 

cosh;

ð126Þ

where  u 0  is a constant, which gives the potential over the periphery of the cylinder:

ð Þ ¼ u cosh:

u a; h

 

0

ð127Þ

The expansion of (126) near the edge of the structure is: u r; h

ð Þj ! ¼ u cosh þ r

a

0

  u0 a

 



  a2 4h

2

ðr  aÞcosh þ Oððr  aÞ Þ:



 

ð128Þ

 

Y. Drobyshevski / Ocean Engineering 31 (2004) 269–304

297

Matching expression (124) with the expansion of the inner potential (48), and the expansion of the under-bottom potential (128) with expansion (47) of the inner potential, one gets the following four equations:

  a2

  u0

U   H 

;

129

 þ  ¼ a    4h  2Uh ð1  log2Þ þ C  ¼  ¼ u ; p

   

0

 ¼ q;

 

Uh

"  þ

 ¼  ¼

 1

  2h H  p  u 1 a; h; H    : cosh

C   2 q

ð Þ ð130Þ

log

þ ð

Þ

ð131Þ

ð2Þ H 1   ðk0 Þ   1  F 1 þ ðu0 þ sinhu0 coshu0Þ H ð2 Þ0 ðk Þ 0

1

#

 

ð132Þ

After some algebra, the unknown flux strength  q  can be found in the form:   a2 8hH 

    þ ðÞ  ¼  ¼ ð Þ þ ð Þ ð Þ          ð Þ    ð Þ þ ð Þ ð Þ ð Þ ¼ ð Þ ¼ ðÞ þ ðÞ     ðÞ ð Þ ð Þ  ð Þ    ð Þ ¼ ð Þ ¼  q aH 

q 

q

1

q

2





 

1

D1

 Re  q

 Im  q

  S 

 iD 2  

D

1

1 2

 

Dð1Þ2

 D

2 2

1

 1 2

1

  S 

2

þ Dð Þ

2 2

 1 2

1

  S  D

1

 iS  2

  a2 D1 8hH 

  S 

1

ð133Þ

;

 1 2

 S  2 D 2

;

  ð134Þ

  a2 D2 8hH 

;

  ð135Þ

where the newly introduced notations are:

1 X  þ

 ðcoshu0  1Þ  A S ð1Þ ¼ 

ð1Þ ðk0 Þ

 ðcoshu0  1Þ  A S ð2Þ ¼ 

ð2Þ ðk0 Þ

ð  þ sinhu coshu Þ

u0 u0

0

0

ð  þ sinhu coshu Þ :

u0 u0

0

0

¼

m 1

ð1  cosu Þ K  ðk Þ ðu  þ sinu cosu Þ k K 0 ðk Þ ;   ð136Þ m

um

m

m

1

m

 

m

m

1

m

ð137Þ

Constants   Dð1Þ   and   Dð2Þ   have been already defined by formulae (102), (103) and the combinations of Bessel functions   Að1Þ   and   Að2Þ   are given by (104), (105). The pitch added mass and damping coefficients can be now computed using expressions for the potential (11), (38), (123), (126) and formulae (70) and (106). Upon integrating and taking the real and imaginary parts according to (70), one obtains the

 

298

Y. Drobyshevski / Ocean Engineering 31 (2004) 269–304

following formulae: l  55

coshu   1  A ð Þ ðk Þ ð Þ    Þ ð Þ ð Þ  ¼ Þ coshu þ k q   k q Að Þ ðk Þ ð  þ 1 ð1  cosu Þðð1  cosu Þ=u  þ k qð Þ Þ K  ðk Þ u ðu  þ sinu cosu Þ þm¼1 ðK  ðk Þ þ k K  ðk ÞÞ a þ12 qð Þ  H a  þ  2H   þ  H h þ  8aH  247 ah H a  þ qð Þ ah ;   ð138Þ   l55 qpa2 H 3

X

  

 k

55

coshu0  1 Að1Þ k0  2 2 u0 u0  sinhu0 coshu0

  ¼   ð

m 2 m 2

1

m

m

m

0

0

1

1

m

2

2

1

0

2

2

1



 

m

m

0

m



1

coshu0  1 Að2Þ k0 2 2 u0 u0  sinhu0 coshu0

  ¼    ð

  k55 qpa2 H 3 r

 X

0

1

0

0

1

2

m

m

m

m

0

2

0

2 m

0

m

  Þ ð Þ Þ ð  þ   coshu   1  A ð Þ ðk Þ ð Þ ð Þ  u þ k q  þ k q Að Þðk Þ 1 ð1  cosu Þk qð Þ  a þ u ðu  þ sinu cosu Þ ðK  ðk ÞK þð kk ÞK  ðk ÞÞ þ  2aH  qð Þ   1 þ 4hH  m¼1

 ¼

0

1

0

m

m



2



0

2

1

1

m



m

 

m

m

2

m

0

2



)

:

139

ð Þ

Using expression (106) and the surge boundary conditions on the structure one can obtain formulae for the pitch–surge coupling inertia   l51  and the damping term k51 :  51 l

 ¼

  l51 qpa2 H 2

51

 ¼

ð Þ

ð  þ Þ   coshu   1  A ð Þ ðk Þ ð Þ ð Þ þ k q   k q Að Þðk Þ u sinu ðð1  cosu Þ=u  þ k qð Þ Þ K  ðk Þ u ðu  þ sinu cosu Þ ðK  ðk Þ þ k K  ðk ÞÞ m 1 1 a qð Þ þ   a  þ   h ;   2 H  H  4H 

 1 X þ ¼ þ   k

  sinhu0 Að1Þ k0  2 u0 u0  sinhu0 coshu0

  ¼ 0

0

1

0

2

m

m

m

m

m

m

0

1

0

m

2

0

1



1

m

1

m

m

m

0

m

2

  1



2

  sinhu0 Að2Þ k0 2 u0 u0  sinhu0 coshu0

  ¼ 

  k51 qpa2 H 2 r

ð Þ

ð  þ Þ   coshu   1  A ð Þ ðk Þ ð Þ ð Þ þ k q  þ k q Að Þðk Þ u sinu k qð Þ  a ð Þ K  ðk Þ   þ q K  K  2  sin u cos u u ð k Þ þ  k ð k ÞÞ Þ ð H    þ u ð m¼1

 1 X þ

0

0

1

0

2

m m

m

m

1

0

2

0

m

ð140Þ

2

0

1

m

1

m



m

m

 

0

m

2

)

:

  ð141Þ

According to the reciprocity relations, the above coefficients must be identical to the surge–pitch coupling terms defined by equations (113), (114). Numerical checks confirmed that the two sets of formulae give exactly the same results.



 

Y. Drobyshevski / Ocean Engineering 31 (2004) 269–304

299

To obtain the pitch exciting force (moment) using the Haskind–Newman formula (77), one should use the far field behaviour of the radiation potential:

ð

  ¼

  coshu0 u20

Þj !1

u r; h; z

r

  1 þ aq 





2

cosh  cosha ðz   ð u2 H þ sinh ð  H  Þ0 Þ u coshu ÞH    ðk Þ 0

0

0

2 n

0

0

2

 e

i a0 r 3p=4

s  ffiffi ffi ffi ffi   ð pa0 r



Þ;

  ð142Þ

where the two terms in brackets correspond to the potentials (38) and (85), respectively. Substituting (142) and (79) into (77), and using the method of stationary phase to evaluate the integral, one obtains the following expression for the pitch exciting moment: 0 5

ð Þ ¼ X 

X 5 t

 ð Þ  þ t  1

qk0 u0 coshu0  1  

 

 ¼ ð ð  Þ þ X 5

ð2 Þ Þ  eirt ;

1

 

iX 5

ð143Þ

where   4qgr0 a2 H 

X 50 t

ð Þ ¼ k H  2 0

 ð  Þ ð Þ

2 1

0

k0

coshu0  1 u0 coshu0

   ei

rt :



ð144Þ

 

Expression (144) gives the pitch moment about the   y-axis in the water plane acting ing on the the bo bott ttomom-mo moun unted ted cy cylin linde der, r, i.e. i.e. the the mo mome ment nt du duee to th thee po pote tent ntia iall (3 (38) 8) only, which is further adjusted by (143) to give the moment on the truncated cylinder. The real and imaginary parts of the exciting force are:

ð1Þ 01 X    ¼  Re jX  j ¼  X    5

5

5

ð2Þ 0ð1Þ X    ¼  Im jX  j ¼  X  5

5

  k0 u0  qð1Þ coshu0  1

 ð Þ  þ

5

1

 

k0 u0  qð2Þ coshu0  1

 þ

 ðÞ

0 2  X 5  

 

where

ð  Þ ¼  Re jX 0 j

ð 2Þ  k0 u0 q  ; coshu0  1

 

ð145Þ

ð1Þ    k0 u0 q ; coshu0  1

 

ð146Þ

ðÞ

0 2  X 5

 þ

1

 

 





0 1

X 5

5

¼

  4qgr0 a2 H  coshu0 k0 u0 coshu0

ð  Þ ¼  Im jX 0 j

ð

  1Þ

ðJ  ðk Þ  1

0

ð Þ  J  ðk Þ ð ÞÞ þ ðN  ðk Þ  k N  ðk ÞÞ  ; ð147Þ

k0 J 0 k0  k0 J 0 k0 2

1

0

1

0

0

0

0

2

0 2

X 5

5

¼

 4qgr0 a2 H  coshu0 k0 u0 coshu0

ð

  1Þ

J 1 k0

ð ð Þ

ð Þ  N  ðk Þ

k0 N 0 k0  k0 J 0 k0 2

1

0

N 1 k0

ð ÞÞ þ ð ð Þ 

 k0 N 0 k0

2

 :

ð ðÞÞ148Þ

 

300

Y. Drobyshevski / Ocean Engineering 31 (2004) 269–304

All pitch hydrodynamic coefficients have been now obtained. Before considering numerical results, let us examine the exciting moment (143) in more detail. Similar to the two-dimensional case, one can now estimate the frequency of the cancellation effect, at which the wave exciting moment becomes zero due to the opposite contributions of the fluid pressure on the structure side surface and the bottom. First, let us assume that the under-bottom clearance   h  is so small that the terms of  the order   O e   can be also ignored and the components of the source strength (134), (135) can be rewritten as:

ðÞ

 qð1Þ

 ¼  a4 ð1  ð2e=p1aÞÞlogð4eÞ þ OðeÞ;  



 

qð2Þ

 ¼ 0 þ OðeÞ:

 

ð149Þ

Substituting (149) into (143) one gets:

ð1Þ 0ð1Þ X     X    ¼  X    5

5

5

   1

a

k0 u0 4 coshu0  1

 

1  log 4e 2e=pa

  Þ ð1  ð

ð

ÞÞ ð Þ



:

 

ð150Þ

Therefore, the frequency at which the exciting moment vanishes can be approximately determined from the following equation: coshu0 u20

 1

  ¼

 1 a 4 H 

2

1 2e=p a log 4e  :

  ð   ð 1

ð151Þ

 

ÞÞ ð Þ

Alternatively, for a given frequency parameter   u0  the above equation defines the structure radius    a  a =H   at which the exciting moment becomes zero. Equation (151) is very similar to the corresponding equation in the two-dimensional case. When  H =T   1  the limiting solution of (151) is given by:

 ¼

  ! ! a=H  ¼  ¼ 2 ðchu    1Þ=u ;

p  ffiffi ffi ffi ffi ffi ffi ffi ffi ffi ffi p   ffi ffi ffi ffi  ffi ffi p   ¼   ¼   ¼  ¼ 0

ð152Þ

 

0

and the low frequency limit of (152) is:

a=H   2

1=2

2

 1 :41421 . . .

ð153Þ

 

p   2:8284 attracts the minimum pitch moment when subjected to long waves in  extremely  ¼ ¼ shallow water. Note that in the two-dimensional case the corresponding beam to Hence a circular cylinder with the diameter to draft ratio 2 a=T 

 2 2

 ffiffi

draft ratio was found to be 2.4495. It can be also shown that for a given wave fre  >   a 0   (where    quency the phase of the exciting moment changes such that if   a a0 a=H  0   is the solution of equation (151)) the exciting moment acts in phase with   <   a 0   the moment and the wave slope the wave slope at the origin, whereas for   a 12). ). are in anti-phase (Fig. (Fig. 12 Figs. 13 and 14  show the added mass and damping coefficients (138), (139) and the non-dim non-dimensio ensional nal exciti exciting ng force amplitude:

ð

 ¼

Þ

j  ¼ q gpjX a Hr

 X  5

5 2

 

:

0

ð154Þ

For the relative radius   a=H   1  1::0 the water depth   H =T   1:  1 :50 has been considere sid ered d for com compar parison ison (Fig. 13), 13), at wh whic ich h on only ly th thee qu qual alita itati tive ve ag agre reeme ement nt wi with th

 ¼

 ¼

 

Y. Drobyshevski / Ocean Engineering 31 (2004) 269–304

301

Fig. 12. Phase of the pitch exciting moment.

exact results can be seen due to the large water depth. For the same reason the frequency at which the amplitude of the exciting moment reaches zero is higher than could coul d be ex expe pect cted ed from from eq equa uati tion on (151 (151). ). Fo Forr th thee radi radius us   a=H   5:  5 :0 ho howe weve verr th thee agreement is very good and the asymptotic formulae are seen to be fully applicable up to the depth to draft ratio  H =T   4  4::00.

 ¼  ¼

 ¼  ¼

6. Concl Concluding uding rremarks emarks

The asy asympt mptotic otic for formula mulaee for the hyd hydrod rodyna ynamic mic coe coefficie fficients nts presen presented ted in thi thiss paper are closed and straightforward; calculations can be performed using spreadsheets or common programming packages capable of computing cylindrical functions. Some formulae, in particular for the low and high frequency limits of the added mass, are suitable for hand calculations. Series involved in the formulae conve verg rgee rapi rapidl dly; y; all all resu result ltss pr pres esen ente ted d abov abovee have have be been en ob obta tain ined ed wi with th 40 term termss retained in each series. The applicability range and the accuracy provided by the asymptotic formulae depends on the radius to draft ratio   a=T   and the relative under-bottom clearance e  h =H . The wider the structure is the greater the range of water depths is where the formulae can be applied with a negligibly small error. It can be concluded from the the co compa mpari riso son n that that for for the the re rela lati tive ve ra radi dius us   a=H   5 :0   the pro propos posed ed for formul mulae ae for for surg surgee hy hydr drody odyna nami micc coeffi coeffici cient entss give give ac accur curat atee resu results lts fo forr th thee de dept pth h to dr draf aftt ratio   H =T   2 :0   (relative clearance   e  0  0::50), whereas for heave and pitch motions they can be applied even for deeper waters, which are beyond the expected range of ‘‘small’’ under-bottom clearance. For narrower structures ( a=H   1:  1 :0) the range of applicable depths reduces and it is important for the under-bottom clearance to be smaller than the cylinder draft. Finally, it should be noted that knowing the radiation potential in the analytical form not only enables us to calculate the hydrodynamic coefficients (the benefit

 ¼

 ¼  ¼

 ¼  ¼

 ¼

 ¼  ¼

 

302

Y. Drobyshevski / Ocean Engineering 31 (2004) 269–304

Fig. 13. Pitch added mass, radiation damping and exciting force amplitude, a amplitude,  a= =H   1  1::0.

 ¼  ¼

exploited in this study), but brings some other advantages. For example, the firstorder potential can be used in further analytical calculation of the second-order forces on the floating body, which may also provide useful benchmarking examples. Another essential aspect is that the methodology used in this study can be read readil ily y ex exte tend nded ed to ve vert rtic ical al side sided d bodie bodiess of an arbi arbitr trar ary y pl plan an-f -form orm.. Al Alth thou ough gh purely analytical solutions will not be possible in this case, the hydrodynamic problem can be conveniently reduced to one linear integral equation with a weakly singul sin gular ar kern kernel el for formul mulate ated d over the wat waterp erplane lane con contou tour, r, thu thuss offer offering ing com comput putaa-

 

Y. Drobyshevski / Ocean Engineering 31 (2004) 269–304

303

Fig. 14. Pitch added mass, radiation damping and exciting force amplitude,  a=  a =H   5  5::0.

  ¼¼

tional advantages over the complete panel formulation in three dimensions. These developments are beyond the scope of this paper.

Acknowledgements

A part of this work was performed within a project undertaken by the Center for Oil and Gas Engineering, the University of Western Australia under sponsorship of the Mineral and Energy Research Institute of Western Australia (MERIWA). IW A). Fi Fina nanc ncial ial supp suppor ortt an and d coope coopera rati tion on pr prov ovid ided ed by spons sponsor ors— s—Wo Wood odsi side de Energy, Concrete Offshore Structures Industry, Over Arup & Partner are gratefully acknowledged.

 

304

Y. Drobyshevski / Ocean Engineering 31 (2004) 269–304

References Abramowitz, M., Stegun, I.A., 1964. Handbook on Mathematical Functions with Formulas, Graphs and Mathematical Tables. Dover Publications Inc, New York. Garret, C.J.R., 1971. Wave forces on a circular dock. Journal of Fluid Mechanics 46 (Part 1), 129–139. Haskind, M.D., 1973. Hydrodynamic Theory of Ship Motions. Science, Moscow (in Russian). Miles, J., Gilbert, F., 1968. Scattering of gravity waves by a circular dock. Journal of Fluid Mechanics 34 (Part 4), 783–793. Newman, J.N., 1977. Marine Hydrodynamics. MIT Press, Cambridge, MA. Sabuncu, T., Calisal, S., 1981. Hydrodynamic coefficients for vertical circular cylinders at finite depth. Ocean Engineering 8, 25–63. Tuck, E.O., 1971. Transmission of water waves through small apertures. Journal of Fluid Mechanics 49 (Part 1), 65–74. Tuck, E.O., 1975. Matching problems involving flow through small holes. Advances in Applied Mechanics 15, 89–158. Yeung, Yeun g, R.W., 1981. 1981. Added Added mass and damping damping of a vertical vertical cylinder cylinder in finite-de finite-depth pth waters. Applied Applied Ocean Research 3 (3), 119–133.

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