Water wave interaction with a floating porous cylinder

Ocean Engineering 27 (2000) 1–28www.elsevier.com/locate/oceaneng

Water wave interaction with a floating porouscylinder

A.N. Williams*, W. Li, K.-H. WangDepartment of Civil and Environmental Engineering, University of Houston, Houston,

TX 77204-4791, USA

Received 26 August 1998; accepted 21 October 1998

Abstract

The interaction of water waves with a freely floating circular cylinder possessing a side-wall that is porous over a portion of its draft is investigated theoretically. The porous side-wall region is bounded top and bottom by impermeable end caps thereby resulting in anenclosed fluid region within the structure. The problem is formulated based on potential flowand linear wave theory and assuming small-amplitude structural oscillations. An eigenfunctionexpansion approach is then used to obtain semi-analytical expressions for the hydrodynamicexcitation and reaction loads on the structure. Numerical results are presented which illustratethe effects of the various wave and structural parameters on these quantities. It is found thatthe permeability, size and location of the porous region may have a significant influence onthe horizontal components of the hydrodynamic excitation and reaction loads, while its influ-ence on the vertical components in most cases is relatively minor. 1999 Elsevier ScienceLtd. All rights reserved.

Keywords:Wave diffraction; Wave radiation; Porous structure; Hydrodynamics

1. Introduction

Recently, there has been a great deal of effort directed towards quantifying waveinteractions with porous ocean structures. Wave diffraction by a semi-porous cylin-

* Corresponding author. Tel.:1 1-713-743-4269; fax:1 1-713-743-4260; e-mail:[email protected]

0029-8018/00/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved.PII: S0029 -8018(98 )00078-X

2 A.N. Williams et al. /Ocean Engineering 27 (2000) 1–28

drical breakwater protecting an impermeable circular cylinder was investigated theor-etically by Darwiche et al. (1994) by an eigenfunction expansion approach. Williamsand Li (1998) later extended this analysis to deal with the case where the interiorcylinder is mounted on a large cylindrical storage tank.

There have been several two-dimensional studies of wave interaction with porousstructures. Chwang (1983) and Chwang and Li (1983) have studied the use of porousplates as wave makers. The problem of the reflection and transmission of small-amplitude waves by a vertical porous plate has been treated by Chwang and Dong(1984). The use of rigid and flexible porous structures as breakwaters has been inves-tigated theoretically by Twu and Lin (1991) and Wang and Ren (1993a), respectively.Wang and Ren (1993b) have also studied the wave-trapping effect due to a flexibleporous breakwater located in front of a vertical impermeable wall. Yu and Chwang(1994) investigated the interaction of surface waves with a submerged horizontalporous plate. Recently, the interaction of waves and a porous pontoon structure withan impermeable top layer, representing an idealized tension leg platform, was studiedanalytically by Lee and Ker (1997).

In this paper, the hydrodynamics of a freely floating circular cylinder with a side-wall that is porous over a portion of its length is investigated theoretically. Theporous side-wall region is bounded top and bottom by impermeable end caps, whichresults in an enclosed fluid region within the structure. The fluid domain is dividedinto three regions: the interior fluid region, the region beneath the cylinder and theexterior region extending to infinity in the horizontal plane. Under the assumptionsof linearized potential flow and small-amplitude structural oscillations, analyticalexpressions are obtained for the wave motion in each flow region based on an eigen-function expansion approach. The solutions in the three fluid domains are thenmatched using appropriate boundary conditions at the interfaces between them. Semi-analytical expressions for the hydrodynamic excitation and reaction loads (addedmass and radiation damping) on the structure are obtained. Numerical results arepresented which illustrated the effects of the various wave and structural parameterson these hydrodynamic quantities. It is found that the permeability, size and locationof the porous region may have a significant influence on the horizontal componentsof the hydrodynamic excitation and reaction loads, while its influence on the verticalcomponents in most cases is relatively minor.

2. Theoretical formulation

The geometry of the problem is shown in Fig. 1. A circular cylinder of radiusaand draftb is situated in water of depthd. Cylindrical polar coordinates (r, u, z) areemployed with the origin located at the intersection of the cylinder axes with thestill-water level. Part of the side-wall of the cylinder,2 h2 , z , 2 h1 is porousand solid end caps located atz 5 2 h1 andz 5 2 h2 define the interior fluid region.

It is customary in linearized diffraction analysis to decompose the problem intoscattering and radiation sub-problems. These correspond, respectively, to waves inci-dent upon a fixed structure, and the structure performing prescribed oscillatory

3A.N. Williams et al. /Ocean Engineering 27 (2000) 1–28

Fig. 1. Definition sketch.

motions in an otherwise quiescent fluid. In the scattering problem, the structure issubjected to a train of small amplitude regular waves of heightH and frequencyvpropagating in the positivex direction, while in the radiation problem the structureperforms prescribed oscillations of unit amplitude and frequencyv. In each case,the small-amplitude, irrotational motion of the inviscid, incompressible fluid may bedescribed in terms of a velocity potentialf(r, u, z, t) 5 Re[F(r, u, z) e−ivt], whereRe[ ] denotes the real part of a complex expression. The fluid velocity vectorq5 =f. Subsequently, the common time-dependencee−ivt will be dropped from alldynamic variables.

The fluid domain is divided into three regions: an exterior region (1) defined byr $ a, 2 d # z # 0; an interior region (2) defined by 0# r # a, 2 h2 # z # 2h1; and a region beneath the cylinder (3) defined by 0# r # a, 2 d # z # 2 b.Denoting the velocity potentials in each region j byFj, j 5 1, 2, 3, these potentialsmust satisfy Laplace’s equation in each flow region, namely

=2Fj 5 0 for j 5 1,2,3 (1)

These potentials are also required to satisfy appropriate boundary conditions onthe free-surface, and sea-bed, namely


g∂F1

∂z2 v2F1 5 0 on z 5 0 (2)

∂Fj

∂z5 0 on z 5 2 d for j 5 1,3 (3)

whereg is the acceleration due to gravity.On the solid surface of the structure,Ss, and porous surface of the structure,Sp,

continuity of normal velocity is imposed, that is

∂Fj

∂n5 v·n on Ss (4)

∂Fj

∂n5 v·n 2 W(u,z) on Sp(r 5 a, 2 h2 # z # 2 h1) (5)

where v is the structural velocity vector at any point onSs or Sp, and j 5 1, 2, or3 as appropriate. In Eq. (5),W(u,z) is the spatial component of the normal velocityw(u,z,t) of the fluid passing through the porous cylinder from region 1 to region 2,i.e. w(u,z,t) 5 Re[W(u,z)e−ivt]. The fluid flow passing through the porous cylinderwall is assumed to obey Darcy’s law. Hence, the porous flow velocity is linearlyproportional to the pressure difference across the thickness of the porous wall (see,for example, Taylor, 1956). Therefore, it follows that

W(u,z) 5g

mriv[F1(a,u,z) 2 F2(a,u,z)] on r 5 a, 2 h2 # z # 2 h1 (6)

wherem is the coefficient of dynamic viscosity andg is a material constant havingthe dimensions of length. Subsequently, the porosity of the breakwater will becharacterized by the dimensionless parameterG0 5 rvg/(mk0).

As stated above, it is convenient to decompose the total velocity potential intoincident, scattered and radiated components according to

Fj 5 djFI 1 FDj 1 O3

m 5 1

DmCmj (7)

whered1 5 1 andd2 5 d3 5 0, Dm denotes the structural displacement in the mthmode andCm

j denotes the radiation potential in mth mode in region j. The radiationmodes are numbered such that m5 1, 2 and 3 correspond to surge, heave and pitch,respectively. In Eq. (7),FI is the spatial component of the incident potential, given by

FI 5 2igH2v

coshk0(z 1 d)coshk0d

O`m 5 0

emimJm(k0r)cosmu (8)

wheree0 5 1 andem 5 2 for m $ 1, Jm() denotes the Bessel function of the firstkind of order m andk0 is the incident wavenumber which is related to the angularfrequency though the dispersion relation,

v2 5 gk0tanhk0d (9)


Finally, the diffracted and radiated components of the velocity potential in theexterior region must satisfy the usual radiation boundary condition, that is

limr→`

√r F ∂∂r

(FDj ,Cm

1 ) 2 ik0(FDj ,Cm

1 )G 5 0 for m 5 1,2,3 (10)

Finally, the velocity potentials in regions 1 and 3 are linked through the followingmatching conditions:

F1 5 F3 on r 5 a, 2 d # z # 2 b (11)

∂F1

∂r5

∂F3

∂ron r 5 a, 2 d # z # 2 b (12)

3. Analytical solution

The scattering and radiation problems will now each be addressed in turn. Thesolution to the scattering problem yields the exciting loads on the structure whilethe radiation problem leads to the added-mass and radiation damping components.

3.1. Scattering problem

A suitable form for the scattered velocity potential in region 1, which satisfies theappropriate free-surface, sea-bed and structural boundary conditions, can be writ-ten as

FD1 5 2

igH2v

O`m 5 0

O`n 5 0

AmnTmn(r)cosan(z 1 d)cosmu (13)

in which

Tmn(r) 5 H H(1)m (k0r)

H(1)m 9(k0a)

n 5 0 (14)

Tmn(r) 5 HKm(knr)K9m(kna)

n $ 1

whereH(1)m denotes the Hankel function of the first kind of order m andKm are the

modified bessel functions of the second kind of order m. the primes denote differen-tiation with respect to argument. The wavenumberskn, n 5 1, 2, … are the positivereal roots ofv2 1 gkntanknd 5 0. Also the wavenumbera0 5 ik0 and an 5 kn forn $ 1.

The diffracted velocity potential in region 2, which satisfies structural boundaryconditions, is given by

FD2 5 2

igH2v

O`m 5 0

O`n 5 0

BmnSmn(r)cosbn(z 1 h2)cosmu (15)


in which

Smn(r) 5 H rm

am 2 1 n 5 0 (16)

Smn(r) 5 HIm(bnr)I9m(bna)

n $ 1

whereIm are the modified bessel functions of the first kind of order m. The wavenum-bersbn 5 np/(h2 2 h1), for n $ 0.

A suitable form for the diffracted velocity potential in region 3, which satisfiesthe bottom boundary condition, sea-bed and appropriate structural boundary con-ditions, can be written as

FD1 5 2

igH2v

O`m 5 0

O`n 5 0

CmnRmn(r)cosgn(z 1 d)cosmu (17)

in which

Rmn(r) 5 H rm

am 2 1 n 5 0 (18)

Rmn(r) 5 HIm(gnr)I9m(gna)

n $ 1

where the wavenumbersgn 5 np/(d 2 b), for n $ 0.The unknown potential coefficentsAmn, Bmn andCmn are determined by first trun-

cating the infinite series for the velocity potentials in each of the fluid regions aftera finite number of terms. Then the structural boundary condition and the matchingconditions are applied and the resulting sets of algebraic equations are simplifiedutilizing the orthogonality properties of the vertical eigenfunctions in each of thefluid regions. The sets of algebraic equations for the potential coefficients may besolved by standard matrix techniques.

Once the velocity potential in each region has been determined, various quantitiesof engineering interest may be calculated. The hydrodynamic forces in thex and zdirections and the overturning moment about ay axis through the point (0,0,zc), maybe obtained by integrating the pressure distributions on the cylinder. The correspond-ing complex force and moment amplitudes and are given by

Fx 5 2rivhEp

0

E0

2 b

FD1 (a,u,z)cos(p 2 u)adudz (19)

2 Ep

0

E2 h1

2 h2

FD2 (a,u,z)cos(p 2 u)adudzj


Fz 5 2rivhEp

0

Ea

0

FD3 (r,u, 2 b)rdudr 2 E

p

0

Ea

0

FD2 (r,u, 2 h2)rdudr (20)

1 Ep

0

Ea

0

FD2 (r,u, 2 h1)rdudrj

My 5 2rivhEp

0

E0

2 b

FD1 (a,u,z)(z 2 zc)cos(p 2 u)adudz

2 Ep

0

E2 h1

2 h2

FD2 (a,u,z)(z 2 zc)cos(p 2 u)adudz (21)

1 Ep

0

Ea

0

[FD2 (r,u, 2 h1) 2 FD

2 (r,u, 2 h2)]cos(p 2 u)r2dudr

1 Ep

0

Ea

0

FD3 (r,u, 2 b)cos(p 2 u)r2dudrj

3.2. Radiation problem

A suitable form for the radiated velocity potential in region 1, which satisfies theappropriate free-surface, sea-bed and structural boundary conditions, can be writ-ten as

Cm1 5 O`

n 5 0

Amn Tm

n (r)cosan(z 1 d)cosmu (22)

for m 5 1, 2, 3, in which

Tmn (r) 5 HH(1)

n (k0r)H(1)9

n (k0a)n 5 0 (23)

Tmn (r) 5 HKn(knr)

K 9n(kna)

n $ 1

wheren 5 1 for m 5 1, 3 andn 5 0 for m 5 2.The radiated velocity potentials in region 2, which satisfy the structural boundary

conditions, are given by

Cm2 5 O`

n 5 0

{ Bmn Sm

n (r)cosbn(z 1 h2) 2 Lm2 }cosmu (24)


in which

Smn (r) 5 H rn

an 2 1 n 5 0 (25)

Smn (r) 5 HIn(bnr)

I9n(bna)

n $ 1

for m 5 1, 2, 3. The termLm2 in Eq. (24) represents the mode-dependent particular

solution and enables the radiated potentials to satisfy the non-homogeneous boundaryconditions on the body surface. The particular solutions for region 2 are defined by

L12 5 0 (26)

L22 5 z (27)

L32 5 2 rz (28)

Suitable forms for the radiated potentials in region 3, which satisfy the bottomboundary condition, radiation boundary condition and appropriate structural bound-ary conditions, are

Cm3 5 O`

n 5 0

{ Cmn Rm

n (r)cosgn(z 1 d) 1 Lm3 }cosmu (29)

for m 5 1, 2, 3, in which

Rmn (r) 5 H rn

an 2 1 n 5 0 (30)

Rmn (r) 5 HIn(gnr)

I9n(gna)

n $ 1

The particular solutions for region 3 are defined by

L13 5 0 (31)

L23 5 F(z 1 d)2 2

r2

2G 12(d 2 b)

(32)

L33 5 2

r2(d 2 b) F(z 1 d)2 2

r2

4G (33)

The unknown radiation potential coefficentsAmn ,Bm

n and Cmn are determined in a

similar manner to those in the scattering problem. The solution to the radiation prob-lem yields the added mass and radiation damping components. These quantities arerelated to the real and imaginary parts of the hydrodynamic reaction loads on thebody caused by prescribed body motions. The hydrodynamic force/moment in thex andz directions/about they axis through the point (0,0,zc), due to motion in modem ( 5 1, 2, 3) may be obtained by integrating the corresponding pressure distri-


butions over the cylinder. The complex reaction force and moment amplitudesFm

xR,Fm

zRandMm

yRdue to prescribed motion in mode m are given by

FmxR

5 2rv2HEp

0

E0

2 b

Cm1 (a,u,z)cos(p 2 u)adudz (34)

2 Ep

0

E2 h1

2 h2

Cm2 (a,u,z)cos(p 2 u)adudzJ

FmzR

5 2rv2HEp

0

Ea

0

Cm3 (r,u, 2 b)rdudr 2 E

p

0

Ea

0

Cm2 (r,u, 2 h2)rdudr (35)

1 Ep

0

Ea

0

Cm2 (r,u, 2 h1)rdudrJ

MmyR

5 2rv2HEp

0

E0

2 b

Cm1 (a,u,z)(z 2 zc)cos(p 2 u)adudz

2 Ep

0

E2 h1

2 h2

Cm2 (au,z)(z 2 zc)cos(p 2 u)adudz (36)

1 Ep

0

Ea

0

[Cm2 (r,u, 2 h1) 2 Cm

2 (r,u, 2 h2)]cos(p 2 u)r2dudr

1 Ep

0

Ea

0

Cm3 (r,u, 2 b)cos(p 2 u)r2dudrJ

The various added mass and radiation damping components are now defined as

ajm 5Re[Pm

j ]v2 bjm 5

Im[Pmj ]

vfor j 5 1,2,3 (37)

where, Pm1 5 Fm

xR, Pm

2 5 FmzR

, Pm3 5 Mm

yR, and Re[] and Im[] denote the real and

imaginary parts of a complex quantity, respectively.


Fig. 2. Comparison of dimensionless force and moment amplitudes obtained using the present approach[lines] with results of Garrett (1971) [symbols] ford/a 5 3/4, b/a 5 1/2, andG0 5 0. (———I———)horizontal force; (– – –r– – –) vertical force; (– – –j– – –) overturning moment.

4. Numerical results and discussion

The effects of various wave and structural parameters on the hydrodynamic excit-ing forces and moments and added mass and radiation damping components willnow be investigated. In particular, the permeability, size and location of the porousregion will be considered.

First, the correctness of the present formulation for the limiting case ofG0 5 0(impermeable-walled cylinder) is considered. Figs. 2 and 3 show a comparison of

Fig. 3. Comparison of dimensionless force and moment amplitudes obtained using the present approach[lines] with results of Garrett (1971) [symbols] ford/a 5 3/2, b/a 5 1, andG0 5 0. (———I———)horizontal force; (– – –r– – –) vertical force; (– – –j– – –) overturning moment.


Fig. 4. Comparison of dimensionless added mass components obtained using present approach [lines]with results of Sabuncu and Calisal (1981) [symbols] ford/a 5 3, b/a 5 1, andG0 5 0. (——I——)surge–surge; (– – –G– – –) heave–heave; (– – –j– – –) surge–pitch; (– – –r– – –) pitch–pitch.

the dimensionless hydrodynamic exciting force/moment amplitudes computed by thepresent approach and the values of Garrett (1971) for two different cylinder geo-metries. The moment center is at the cylinder base (0,0,2 b). The excitation forcesand moment are non-dimensionalized byrgHpa2 and rgHpa3, respectively. Thedimensionless wavenumber isk0a. Figs. 4 and 5 show a comparison of the dimen-sionless added mass and radiation damping components computed by the presentapproach and the values of Sabuncu and Calisal (1981). The pitch center is at thewaterline, i.e. at (0, 0, 0). The surge–surge and heave–heave added mass (radiation

Fig. 5. Comparison of dimensionless radiation damping components obtained using present approach[lines] with results of Sabuncu and Calisal (1981) [symbols] ford/a 5 3, b/a 5 1, and G0 5 0.(———I———) surge–surge; (– – –G– – –) heave–heave; (– – –j– – –) surge–pitch; (– – –r– – –)pitch–pitch.


Fig. 6. Variation of the dimensionless horizontal force amplitude with dimensionless wavenumber fordifferent porosity ford/a 5 3, b/a 5 2, h1/a 5 2/3 andh2/a 5 4/3. (———) G0 5 0; (– – –)G0 5 1;($$) G0 5 2.

damping) components are non-dimensionalized byrpa3 (rpa3v), the surge–pitchcomponents byrpa4 (rpa4v), and the pitch–pitch components byrpa5 (rpa5v).The dimensionless wavenumber isv2a/g. In all cases, excellent agreement is shownthroughout the frequency range of interest.

Following the above verification of the approach for the impermeable cylinder, aparametric study was carried out to determine the influence of the different character-istics of the porous region on the hydrodynamics of the structure. In presenting theresults of this parametric study, the same non-dimensionalization is adopted for theexciting forces and moment, and the various added mass and radiation damping

Fig. 7. Variation of the dimensionless vertical force amplitude with dimensionless wavenumber for dif-ferent porosity ford/a 5 3, b/a 5 2, h1/a 5 2/3 andh2/a 5 4/3. (———) G0 5 0; (– – –) G0 5 1;($$) G0 5 2.


Fig. 8. Variation of the dimensionless moment amplitude with dimensionless wavenumber for differentporosity ford/a 5 3, b/a 5 2, h1/a 5 2/3 andh2/a 5 4/3. (———) G0 5 0; (– – –)G0 5 1; ($$) G0

5 2.

components, however, in the following figures the dimensionless wavenumber isnow taken to bek0a. Both the moment center and the pitch center are now taken atthe base of the cylinder, i.e. at (0, 0,2 b). The influence of the cylinder porosityon the excitation loads is presented in Figs. 6–8. It can be see from the figures thatthe horizontal force and overturning moment decrease as the porosity parameterG0

increases. However, the effect on the vertical excitation force is minimal. The effectof the parameterG0 on the added mass and radiation damping components is shownin Figs. 9–16. In these cases, in order to provide a smooth transition of added-massfrom the impermeable to the porous condition, forG0 5 0 the interior region is also

Fig. 9. Variation of the dimensionless surge–surge added mass with dimensionless wavenumber for dif-ferent porosity ford/a 5 3, b/a 5 2, h1/a 5 2/3 andh2/a 5 4/3. (———) G0 5 0; (– – –) G0 5 1;($$) G0 5 2.


Fig. 10. Variation of the dimensionless heave–heave added mass with dimensionless wavenumber fordifferent porosity ford/a 5 3, b/a 5 2, h1/a 5 2/3 andh2/a 5 4/3. (———) G0 5 0; (– – –)G0 5 1;($$) G0 5 2.

Fig. 11. Variation of the dimensionless pitch–pitch added mass with dimensionless wavenumber fordifferent porosity ford/a 5 3, b/a 5 2, h1/a 5 2/3 andh2/a 5 4/3. (———) G0 5 0; (– – –)G0 5 1;($$) G0 5 2.

considered to be filled with fluid. It can be seen that forG0 5 1, 2, the surge–surgeand pitch–pitch added masses may be reduced to 20–30% of their impermeable-wallvalues over a wide frequency range. Similarly, the magnitude of the surge–pitchadded mass may be reduced to 30% of its impermeable-wall values. The heave–heave added mass component is only slightly influenced by the permeability of theporous region. The situation for the radiation damping is less clear, but there is asignificant increase in the amount of damping associated with both the horizontaland vertical motions, especially at high frequencies.


Fig. 12. Variation of the dimensionless surge–pitch added mass with dimensionless wavenumber fordifferent porosity ford/a 5 3, b/a 5 2, h1/a 5 2/3 andh2/a 5 4/3. (———) G0 5 0; (– – –)G0 5 1;($$) G0 5 2.

Fig. 13. Variation of the dimensionless surge–surge radiation damping with dimensionless wavenumberfor different porosity ford/a 5 3, b/a 5 2, h1/a 5 2/3 andh2/a 5 4/3. (———) G0 5 0; (– – –)G0 51; ($$) G0 5 2.

Figs. 17–19 present the influence of the size of the porous region on the excitationloads experienced by a semi-porous cylinder. It can be seen that as the extent of theporous region increases the horizontal force and overturning moment decrease, butthe vertical force increases, throughout the frequency range of interest. This increasein vertical force is associated with the ‘raising’ of the upper boundary of the interiorfluid domain closer to the free-surface as the extent of the porous region is increased.The corresponding influence of the size of the porous region on the added massand radiation damping components is presented in Figs. 20–27. At low frequencies


Fig. 14. Variation of the dimensionless heave–heave radiation damping with dimensionless wavenumberfor different porosity ford/a 5 3, b/a 5 2, h1/a 5 2/3 andh2/a 5 4/3. (———) G0 5 0; (– – –)G0 51; ($$) G0 5 2.

Fig. 15. Variation of the dimensionless pitch–pitch radiation damping with dimensionless wavenumberfor different porosity ford/a 5 3, b/a 5 2, h1/a 5 2/3 andh2/a 5 4/3. (———) G0 5 0; (– – –)G0 51; ($$) G0 5 2.

(wavenumbers), the surge–surge and pitch–pitch added mass components increaseas the extent of the porous region increases, while at higher frequencies(wavenumbers) the opposite is true. Similar behavior is observed in the heave addedmass although the high frequency trends are not so well defined. The surge–pitchadded mass increases with increasing porous domain size over the entire frequencyrange considered. As far as radiation damping is concerned, the most dramaticchanges with porous domain size are observed for the heave radiation damping whereanything from a two-fold to a five-fold increase in this quantity is obtained as theextent of the porous region is increased.


Fig. 16. Variation of the dimensionless surge–pitch radiation damping with dimensionless wavenumberfor different porosity ford/a 5 3, b/a 5 2, h1/a 5 2/3 andh2/a 5 4/3. (———) G0 5 0; (– – –)G0 51; ($$) G0 5 2.

Fig. 17. Variation of the dimensionless horizontal force amplitude with dimensionless wavenumber fordifferent sizes of the porous region ford/a 5 3, b/a 5 2 andG0 5 1. (———) h1/a 5 2/3, h2/a 5 4/3;(– – –) h1/a 5 1/3, h2/a 5 5/3;($$) h1/a 5 0, h2/a 5 2.

The influence on the hydrodynamic exciting loads of the location of the porousregion on the cylinder is presented in Figs. 28–30. In each case the size of the porousregion is kept constant. It can be seen that both the horizontal force and overturningmoment increase as the porous region is located further beneath the free-surface.The vertical force is essentially unaffected by the location of the porous region. Figs.31–38 present the corresponding influence of the location of the porous region onthe added mass and radiation damping components. At low frequencies(wavenumbers), the surge–surge added mass component increases as the porous


Fig. 18. Variation of the dimensionless vertical force amplitude with dimensionless wavenumber fordifferent sizes of the porous region ford/a 5 3, b/a 5 2 andG0 5 1. (———) h1/a 5 2/3, h2/a 5 4/3;(– – –) h1/a 5 1/3, h2/a 5 5/3;($$) h1/a 5 0, h2/a 5 2.

Fig. 19. Variation of the dimensionless overturning moment amplitude with dimensionless wavenumberfor different sizes of the porous region ford/a 5 3, b/a 5 2 andG0 5 1. (———) h1/a 5 2/3, h2/a 54/3; (– – –)h1/a 5 1/3, h2/a 5 5/3;($$) h1/a 5 0, h2/a 5 2.

region is located further beneath the free-surface, while at higher frequencies(wavenumbers) the opposite is true. The heave added mass is unaffected by thelocation of the porous region. The pitch–pitch added mass follows the opposite trendto the surge–surge added mass, namely its value at low frequencies is largest forporous regions located nearest the free-surface while at higher frequencies the largestvalues of this quantity are found for porous regions furthest from the free-surface.The surge–pitch added mass increases as the porous region is located further beneaththe free-surface over the entire frequency range of interest. Finally, the surge–surge


Fig. 20. Variation of the dimensionless surge–surge added mass with dimensionless wavenumber fordifferent sizes of the porous region ford/a 5 3, b/a 5 2 andG0 5 1. (———) h1/a 5 2/3, h2/a 5 4/3;(– – –) h1/a 5 1/3, h2/a 5 5/3;($$) h1/a 5 0, h2/a 5 2.

Fig. 21. Variation of the dimensionless heave–heave added mass with dimensionless wavenumber fordifferent sizes of the porous region ford/a 5 3, b/a 5 2 andG0 5 1. (———) h1/a 5 2/3, h2/a 5 4/3;(– – –) h1/a 5 1/3, h2/a 5 5/3;($$) h1/a 5 0, h2/a 5 2.

radiation damping is seen to increase as the porous region is located further beneaththe free-surface, while the heave radiation damping is essentially unaffected by thelocation of the porous region. Overall the influence of the location of the porousregion on the radiation damping components is not as significant as that observedin the added masses.


Fig. 22. Variation of the dimensionless pitch–pitch added mass with dimensionless wavenumber fordifferent sizes of the porous region ford/a 5 3, b/a 5 2 andG0 5 1. (———) h1/a 5 2/3, h2/a 5 4/3;(– – –) h1/a 5 1/3, h2/a 5 5/3;($$) h1/a 5 0, h2/a 5 2.

Fig. 23. Variation of the dimensionless surge–pitch added mass with dimensionless wavenumber fordifferent sizes of the porous region ford/a 5 3, b/a 5 2 andG0 5 1. (———) h1/a 5 2/3, h2/a 5 4/3;(– – –) h1/a 5 1/3, h2/a 5 5/3;($$) h1/a 5 0, h2/a 5 2.

5. Summary and conclusions

This paper has presented a theoretical study of the hydrodynamics of a freely-floating cylinder with a sidewall that is porous over a portion of its draft. Linearwave theory and small-amplitude structural oscillations have been assumed, andsemi-analytical expressions have been developed for the hydrodynamic excitationand reaction loads on the structure. Numerical results are presented which illustratedthe effects of the various wave and structural parameter on the hydrodynamic loads.


Fig. 24. Variation of the dimensionless surge–surge radiation damping with dimensionless wavenumberfor different sizes of the porous region ford/a 5 3, b/a 5 2 andG0 5 1. (———) h1/a 5 2/3, h2/a 54/3; (– – –)h1/a 5 1/3, h2/a 5 5/3;($$) h1/a 5 0, h2/a 5 2.

Fig. 25. Variation of the dimensionless heave–heave radiation damping with dimensionless wavenumberfor different sizes of the porous region ford/a 5 3, b/a 5 2 andG0 5 1. (———) h1/a 5 2/3, h2/a 54/3; (– – –)h1/a 5 1/3, h2/a 5 5/3;($$) h1/a 5 0, h2/a 5 2.

It is found that the permeability, size and location of the porous region may have asignificant influence on the horizontal components of the hydrodynamic excitationand reaction loads, while its influence on the vertical components in most cases isrelatively minor.


Fig. 26. Variation of the dimensionless pitch–pitch radiation damping with dimensionless wavenumberfor different sizes of the porous region ford/a 5 3, b/a 5 2 andG0 5 1. (———) h1/a 5 2/3, h2/a 54/3; (– – –)h1/a 5 1/3, h2/a 5 5/3;($$) h1/a 5 0, h2/a 5 2.

Fig. 27. Variation of the dimensionless surge–pitch radiation damping with dimensionless wavenumberfor different sizes of the porous region ford/a 5 3, b/a 5 2 andG0 5 1. (———) h1/a 5 2/3, h2/a 54/3; (– – –)h1/a 5 1/3, h2/a 5 5/3;($$) h1/a 5 0, h2/a 5 2.

Acknowledgement

This work has been funded in part by a grant from the Atlantia Corporation ofHouston, Texas. This support is gratefully acknowledged.


Fig. 28. Variation of the dimensionless horizontal force amplitude with dimensionless wavenumber fordifferent locations of the porous region ford/a 5 3, b/a 5 2 andG0 5 1. (———) h1/a 5 1/3, h2/a 51; (– – –)h1/a 5 2/3, h2/a 5 4/3;($$) h1/a 5 1, h2/a 5 5/3.

Fig. 29. Variation of the dimensionless vertical force amplitude with dimensionless wavenumber fordifferent locations of the porous region ford/a 5 3, b/a 5 2 andG0 5 1. (———) h1/a 5 1/3, h2/a 51; (– – –)h1/a 5 2/3, h2/a 5 4/3;($$) h1/a 5 1, h2/a 5 5/3.


Fig. 30. Variation of the dimensionless overturning moment amplitude with dimensionless wavenumberfor different locations of the porous region ford/a 5 3, b/a 5 2 andG0 5 1. (———) h1/a 5 1/3, h2/a5 1; (– – –)h1/a 5 2/3, h2/a 5 4/3;($$) h1/a 5 1, h2/a 5 5/3.

Fig. 31. Variation of the dimensionless surge–surge added mass with dimensionless wavenumber fordifferent locations of the porous region ford/a 5 3, b/a 5 2 andG0 5 1. (———) h1/a 5 1/3, h2/a 51; (– – –)h1/a 5 2/3, h2/a 5 4/3;($$) h1/a 5 1, h2/a 5 5/3.


Fig. 32. Variation of the dimensionless heave–heave added mass with dimensionless wavenumber fordifferent locations of the porous region ford/a 5 3, b/a 5 2 andG0 5 1. (———) h1/a 5 1/3, h2/a 51; (– – –)h1/a 5 2/3, h2/a 5 4/3;($$) h1/a 5 1, h2/a 5 5/3.

Fig. 33. Variation of the dimensionless pitch–pitch added mass with dimensionless wavenumber fordifferent locations of the porous region ford/a 5 3, b/a 5 2 andG0 5 1. (———) h1/a 5 1/3, h2/a 51; (– – –)h1/a 5 2/3, h2/a 5 4/3;($$) h1/a 5 1, h2/a 5 5/3.


Fig. 34. Variation of the dimensionless surge–pitch added mass with dimensionless wavenumber fordifferent locations of the porous region ford/a 5 3, b/a 5 2 andG0 5 1. (———) h1/a 5 1/3, h2/a 51; (– – –)h1/a 5 2/3, h2/a 5 4/3;($$) h1/a 5 1, h2/a 5 5/3.

Fig. 35. Variation of the dimensionless surge–surge radiation damping with dimensionless wavenumberfor different locations of the porous region ford/a 5 3, b/a 5 2 andG0 5 1. (———) h1/a 5 1/3, h2/a5 1; (– – –)h1/a 5 2/3, h2/a 5 4/3;($$) h1/a 5 1, h2/a 5 5/3.


Fig. 36. Variation of the dimensionless heave–heave radiation damping with dimensionless wavenumberfor different locations of the porous region ford/a 5 3, b/a 5 2 andG0 5 1. (———) h1/a 5 1/3, h2/a5 1; (– – –)h1/a 5 2/3, h2/a 5 4/3;($$) h1/a 5 1, h2/a 5 5/3.

Fig. 37. Variation of the dimensionless pitch–pitch radiation damping with dimensionless wavenumberfor different locations of the porous region ford/a 5 3, b/a 5 2 andG0 5 1. (———) h1/a 5 1/3, h2/a5 1; (– – –)h1/a 5 2/3, h2/a 5 4/3;($$) h1/a 5 1, h2/a 5 5/3.


Fig. 38. Variation of the dimensionless surge–pitch radiation damping with dimensionless wavenumberfor different locations of the porous region ford/a 5 3, b/a 5 2 andG0 5 1. (———) h1/a 5 1/3, h2/a5 1; (– – –)h1/a 5 2/3, h2/a 5 4/3;($$) h1/a 5 1, h2/a 5 5/3.

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Water wave interaction with a floating porous cylinder