Wave interaction with a semi-porous cylindrical breakwater mounted on a storage tank

Ocean Engng,Vol. 25, Nos. 2–3, pp. 195–219, 1998 1997 Elsevier Science Ltd. All rights reservedPergamon Printed in Great Britain

0029–8018/98 $19.00+ 0.00

PII: S0029–8018(97)00006–1

WAVE INTERACTION WITH A SEMI-POROUS CYLINDRICALBREAKWATER MOUNTED ON A STORAGE TANK

A. N. Williams and W. LiDepartment of Civil and Environmental Engineering, University of Houston, Houston, TX 77204-4791,

U.S.A.

Abstract—The interaction of linear water waves with a semi-porous cylindrical breakwatersurrounding a rigid vertical circular cylinder mounted on a storage tank is investigated theoretically.The cylindrical breakwater structure is porous in the vicinity of the free-surface, while at somedistance below the water surface it becomes impermeable. Under the assumptions of linearizedpotential flow, the coupled problem of flow in the interior and exterior fluid regions is solved byan eigenfunction expansion approach. Analytical expressions are obtained for the wave motion inboth the interior and exterior flow regions. Numerical results are presented which illustrate theeffects of the various wave and structural parameters on the hydrodynamic loads and interior andexterior wave fields. It is found that for certain parameter combinations the semi-porous, cylindricalbreakwater may result in a significant reduction in the wave field and hydrodynamic forcesexperienced by the interior structure. 1997 Elsevier Science Ltd.

1. INTRODUCTION

Recently, the interaction of linear surface waves with a semi-porous cylindrical breakwaterprotecting an impermeable circular cylinder was investigated theoretically by Darwicheetal. (1994). The breakwater consisted of a bottom-mounted, surface-piercing structurewhich was porous in the vicinity of the free-surface while at some distance below thissurface it became impermeable. Under the assumptions of linearized potential flow, ana-lytical expressions were obtained for the wave motion in both the interior and exteriorflow regions. Numerical results were presented to illustrate the effects of the various waveand structural parameters on the hydrodynamic loads and interior and exterior wave fields.

Several other investigators have studied wave interaction with thin porous structures.The use of porous plate structures as wavemakers has been studied by Chwang (1983)and Chwang and Li (1983). 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 investigated theor-etically 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 flexible porous breakwaterlocated in front of a vertical impermeable wall. Yu and Chwang (1994) investigated theinteraction of surface waves with a submerged horizontal porous plate. All of these studieswere two-dimensional.

In the present paper, the analysis of Darwicheet al. (1994) is extended to deal withthe case where the interior cylinder is mounted on a storage tank. Again, under the assump-tions of linearized potential flow, analytical expressions are obtained for the wave motion

195

196 A. N. Williams and W. Li

in both the interior and exterior flow regions based on an eigenfunction expansionapproach. The solutions in the two fluid domains are then matched using the appropriateboundary conditions at the interface between them. Numerical results are presented toillustrate the effects of the various wave and structural parameters on the hydrodynamicloads and interior and exterior wave fields. It is found that for certain parameter combi-nations the semi-porous, cylindrical breakwater may result in a significant reduction inthe wave field and hydrodynamic forces experienced by the interior cylinder.

2. THEORETICAL FORMULATION

The geometry of the problem is shown in Fig. 1. A bottom-mounted, surface-piercing,impermeable, circular cylinder of radiusa is mounted on a cylindrical storage tank ofradiusb. A thin, semi-porous cylindrical breakwater of radiusb rests on the storage tankas shown. The three structures, interior cylinder, storage tank and breakwater, are concen-tric and situated in water of uniform depthd. The cylindrical breakwater is porous to adepthh1 beneath the still-water level, below this depth, to a depthh2, the breakwater isimpermeable. The height of the storage tank is (d 2 h1 2 h2). Cylindrical polar coordinates

Fig. 1. Definition sketch of semi-porous cylindrical breakwater mounted on a storage tank.

197Wave interaction with a breakwater

(r, u, z) are employed with the origin located at the intersection of the cylinder axis withthe still-water level.

The system is subject to a train of small amplitude regular waves of heightH andfrequencyv propagating in the positivex-direction (Fig. 1). The small-amplitude, irrot-ational motion of the inviscid, incompressible fluid may be described in terms of a velocitypotentialf(r, u, z; t) = Re[F(r, u, z)e2ivt], where Re[ ] denotes the real part of a complexexpression. The fluid velocity vectorq = =f. Subsequently, the common time-dependencee2ivt will be dropped from all dynamic variables.

The fluid domain is divided into two regions, an interior region denoted by 1 (a # r# b, 2 h2 # z # 0) and an exterior region 2 (r $ b, 2 d # z # 0). Denoting the velocitypotentials in the interior and exterior regions by regionF1 and F2, respectively, thesepotentials satisfy Laplace’s equation in each flow region, namely

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

These potentials are also required to satisfy appropriate boundary conditions on the free-surface, sea-bed and cylinder surfaces, namely

g∂Fj

∂z2 v2Fj = 0 onz = 0 for j = 1,2 (2)

∂F1

∂z= 0 on z = 2 h2 (3)

∂F2

∂z= 0 = 0 on z = 2 d (4)

∂Fj

∂r= 2 W(u, z) on r = b, 2 h1 # z # 0 for j = 1,2 (5)

∂Fj

∂r= 0 on r = b, 2 h2 # z # 2 h1 for j = 1,2 (6)

∂F1

∂r= 0 on r = a, 2 h2 # z # 0 (7)

∂F2

∂r= 0 on r = b, 2 d # z # 2 h2 (8)

where g is the acceleration due to gravity andW(u, z) is the spatial component of thenormal velocityw(u, z, t) of the fluid passing through the porous cylinder from region 2to region 1, i.e.w(u, z, t) = Re[W(u, z)e2ivt].

Finally, the scattered component of the velocity potential in the exterior region mustsatisfy the usual radiation boundary condition, that is

limr→`

ÎrS ∂∂r

(F2 2 FI) 2 ik0(F2 2 FI)D = 0 (9)

whereFI is the spatial component of the incident wave potential, given by


FI = 2igH2v

coshk0(z + d)coshk0d

O`m = 0

emimJm(k0r) cosmu (10)

wheree0 = 1 andem = 2 for m $ 1, Jm( ) denotes the Bessel function of the first kind oforder m and k0 is the incident wavenumber which is related to the angular frequencythrough the dispersion relation

v2 = gk0 tanhk0d (11)

The hydrodynamic pressurep(r, u, z; t) = Re[P(r, u, z)e2ivt] at any point in the fluid domainmay then be determined from

P = rivF(r, u, z) (12)

wherer is the fluid density.The fluid flow passing through the porous cylindrical breakwater is assumed to obey

Darcy’s law. Hence, the porous flow velocityw is linearly proportional to the pressuredifference across the thickness of the porous breakwater (see, for example, Taylor, 1956).Therefore, it follows that

W(u, z) =g

mriv[F2(b, u, z) 2 F1(b, u, z)] on r = b, 2 h1 # z # 0 (13)

wherem is the coefficient of dynamic viscosity andg is a material constant having thedimensions of length. Subsequently, the porosity of the breakwater will be characterizedby the dimensionless parameterG0 = rvg/(mk0).

3. ANALYTICAL SOLUTIONS

The velocity potential amplitudes in the interior and exterior regions are now expressedin the following forms:

Fj = 2igH2v

O`m = 0

Cjm(r, z) cosmu for j = 1,2 (14)

Suitable forms for the functionsCjm(r, z) which satisfy the appropriate free-surface, sea-

bed and radiation boundary conditions in region 1 and the free-surface and sea-bed con-ditions in region 2 are

C1m(r, z) = HBm0

Jm(m0r)Jm9(m0b)

+ Cm0

Ym(m0r)Ym9(m0b)Jf0(z) + O`

n = 1

HBmn

Im(mnr)Im9(mnb)

(15)

+ Cmn

Km(mnr)Km9(mnb)Jfn(z)

C2m(r, z) =

coshk0(z + d)coshk0d

emimJm(k0r) + Am0

H(1)m (k0r)

H(1)m 9(k0b)

F0(z) (16)

+ O`n = 1

Amn

km(knr)Km9(knb)

Fn(z)


in which Amn, Bmn andCmn, m,n = 0,1,..., are unknown potential coefficients,H(1)m ( ), Km( )

are the Hankel function of the first kind and the modified Bessel function of the secondkind, both of orderm, respectively, and primes denote differentiation with respect to argu-ment. The orthonormal vertical eigenfunctionsfn(z) andFn(z), n = 0,1,2,..., are defined by

f n(z) = HP21/20 coshm0(z + d) n = 0

P21/2n cosmn(z + d) n $ 1

(17a)

Fn(z) = HQ21/20 coshk0(z + d) n = 0

Q21/2n coskn(z + d) n $ 1

(17b)

in which

Pn =

12 F1 +

sinh 2m0d2m0d

G n = 0

12 F1 +

sin 2mnd2mnd

G n $ 1

(18a)

Qn =

12 F1 +

sinh 2k0d2k0d

G n = 0

12 F1 +

sin 2knd2knd

G n $ 1

(18b)

where the wavenumberm0 satisfies the dispersion relation in the interior region, namelyv2 = gm0 tanhm0h2, and themn and kn, n = 1,2,..., are the positive real roots ofv2 +gmn tanmnh2 = 0 andv2 + gkn tanknd = 0, respectively.

Applying the boundary condition, Equation (7), on the interior cylinder,r = a leads tothe following relationships between the potential coefficientsBmn andCmn:

Cm0 = 2Jm9(m0a)Ym9(m0a)

Ym9(m0b)Jm9(m0b)

Bm0 (19a)

Cmn = 2Im9(mna)Km9(mna)

Km9(mnb)Im9(mnb)

Bmn for n $ 1 (19b)

Utilizing these relationships, the expression forC1m(r, z) in Equation (13) may be modi-

fied to

C1m(r, z) = Bm0Dm0(m0r)f0(z) + O`

n = 1

BmnDmn(mnr)fn(z) (20)

in which

Dm0(m0r) =Jm(m0r)Jm9(m0b)

2Jm9(m0a)Ym9(m0a)

Ym(m0r)Jm9(m0b)

(21a)

Dmn(mnr) =Im(mnr)Im9(mnb)

Im9(mna)Km9(mna)

Km(mnr)Im9(mnb)

(21b)


Equations (5) and (6) imply continuity of radial velocity across the breakwater over theinterval [ 2 h2, 0]. Therefore

∂C1m(b, z)∂r

=∂C2

m(b, z)∂r

for 2 h2 # z # 0, m = 0,1,... (22a)

and over the interval [2 d, 2 h2]

∂C2m(b, z)∂r

= 0 for 2 d # z # 2 h2, m = 0,1,... (22b)

Combining Equations (22a) and (22b), substituting the expressions forC1m and C2

m fromEquations (15) and (16) and utilizing the orthogonality properties of the vertical eigen-functions leads to the following relationships between the potential coefficients:

Am0 = −emimQ1/2

0 Jm9(k0b)coshk0d

+1

k0dhm0Bm0E00Dm09(m0b) + O`

q = 1

mqBmqEq0Dmq9(mqb)j

(23a)

Amn =1

kndhm0Bm0E0nDm09(m0b) + O`

q = 1

mqBmqEqnDmq9(mqb)j for n $ 1 (23b)

Now reapplying the breakwater boundary conditions, Equations (22a) and (22b), and mak-ing use of Equations (5) and (6) with the right-hand side of Equation (5) given by Equation(13) yields

Bm0HDm0(m0b)S0p 2 m0Dm09(m0b)F m

ivrgE0p +

1k0d

H(1)m (k0b)

H(1)m 9(k0b)

E00R0p

+ O`q = 1

1kqd

Km(kqb)Kq9(kqb)

E0qRqpGJ + O`n = 1

BmnHDmn(mnb)Snp

2 mnDmn9(mnb)F m

ivrgEnp +

1k0d

H(1)m (k0b)

H(1)m 9(k0b)

En0R0p (24)

+ O`q = 1

1kqd

Km(kqb)Kq9(kqb)

EnqRqpGJ = 2emimQ1/2

0

coshk0dR0pHJm(k0b)

2 Jm9(k0b)H(1)

m (k0b)H(1)

m 9(k0b)Jvalid for m = 0,1,2,.... The quantitiesEnq, Rqp, Snp, q, n, p = 0,1,... are defined by

Enq = E0

2 h2

fq(z)Fn(z) dz for q,n $ 0 (25a)

Rqp = E0

2 h1

Fq(z)Fp(z) dz for q,p $ 0 (25b)


Snp = E0

2 h1

fn(z)fp(z) dz for n,p $ 0 (25c)

Truncating the infinite series in Equation (24) after a finite number of terms, and applyingthis equation forp = 0,1,2,... in turn (withm fixed), leads to a matrix equation for thepotential coefficientsBmn, n = 0,1,..., which may be solved by standard techniques. Thepotential coefficientsAmn andCmn may then be obtained from Equations (23a), (23b), (19a)and (19b), respectively. The value ofm may now be incremented and the procedurerepeated. In this manner the velocity potentials in each fluid region may be determined.

Various quantities of engineering interest may now be calculated. The wave profiles inboth the interior and exterior regions,hj(r, u, t) = Re[zj(r, u)e2 ivt], j = 1,2, can be obtainedfrom the linearized dynamic free-surface boundary condition, namely

hj = 21g

∂fj

∂ton z = 0 for j = 1,2 (26)

then

z1 =H2 O`

m = 0

hBm0Dm0(m0r)f0(0) + O`n = 1

BmnDmn(mnr)fn(0)j cosmu (27)

z2 =H2 O`

m = 0

HemimJm(k0r) + Am0

H(1)m (k0r)

H(1)m 9(k0b)

F0(0) + O`n = 1

Amn

Km(knr)Km9(knb)

(28)

Fn(0)J cosmu

Evaluating these quantities onr = a [Equation (27)] orr = b [Equation (28)] as appropriateyields the wave run-up on the interior cylinder or the exterior surface of the break-water, respectively.

The total hydrodynamic forces in the direction of wave propagation on the inner cylin-der, F1

x, and the outer cylinder (consisting of the semi-porous breakwater and the storagetank), F2

x, may be obtained by integrating the pressure distributions on the structures.Defining the complex force amplitudesj

x by Fjx(t) = Re[ j

xe2 ivt], j = 1,2, then

^1x = 2rivE

p

0

E0

2 h2

F1 cos(p 2 u)a du dz on r = a

= 2rgapH

2hB10D10(m0a)U0 + O`

n = 1

B1nD1n(mna)Unj (29a)

^2x = 2rivE

p

0

E0

2 d

F2 cos(p 2 u)b du dz


2 2rivEp

0

E0

2 h2

F1 cos(p 2 0)a du dz on r = b

= 2rgbpH

2 H2iQ1/20 J1(k0b)

coshk0dV0 + A10

H(1)1 (k0b)

H(1)1 9(k0b)

V0

+ O`n = 1

A1n

K1(knb)K19(knb)

Vn 2 B10D10(m0b)U0 2 O`n = 1

B1nD1n(mnb)UnJ (29b)

in which the quantitiesUn andVn, n = 0,1,..., are defined by

Un = E0

2 h2

fn(z) dz for n $ 0 (30a)

Vn = E0

2 d

Fn(z) dz for n $ 0 (30b)

4. NUMERICAL RESULTS

The effects of various wave and structural parameters on the interior and exterior wavefield and on the hydrodynamic forces experienced by both the breakwater and the interiorcylinder will now be investigated. The numerical results for these quantities will bepresented in terms of the water depth to wavelength ratio. This parameter is related to theso-called Chwang’s parameter (Chwang, 1983),Cw = g/v2d = (k0d tanhk0d)21.

The dimensionless hydrodynamic force amplitudes on the interior and exterior cylin-ders, u^j

xu/rgadH, for j = 1,2, are plotted for variousa/b values forh1/d = 0.33, h2/d =0.8, andG0 = 1 in Figs 2 and 3. It can be seen from Fig. 2 that, for alla/b values, thewave force on the inner cylinder increases initially as the frequency parameterk0dincreases, then decreases with further increase in this parameter. Also, it can be seen fromthe same figure that the hydrodynamic force on the inner cylinder increases as the radiusratio, a/b, increases. The hydrodynamic forces on the outer cylinder, shown in Fig. 3, eachexhibit a minimum in thek0d range considered. The results also show that the hydrodyn-amic force on the exterior cylinder exhibits the opposite trend to those on the inner cylin-der, namely that fork0d , 2.2–2.6 (depending ona/b) the wave force on the outer cylinderdecreases asa/b increases.

Figs 4 and 5 present the dimensionless hydrodynamic force amplitudes on the interiorand exterior cylinders for different values of the porosity parameterG0 for h1/d = 0.33,h2/d = 0.8, anda/b = 0.2. Fig. 4 shows that the wave force on the interior cylinder increasesas the breakwater porosity is increased. This force increases monotonically with frequencyparameterk0d and reaches a maximum in the range 2.5, k0d , 3.2, depending on thevalue ofG0. It can be seen from Fig. 5 that the wave force on the cylindrical breakwater(exterior cylinder) decreases as the porosity parameter increases. Again, the wave force


Fig. 2. Variation of the dimensionless hydrodynamic force amplitude on the interior cylinder,u^1xu/rgadH, with

k0d for different a/b for h1/d = 0.33,h2/d = 0.8 andG0 = 1. Notation: — – —a/b = 0.2; · · ·a/b = 0.4; - - - a/b= 0.6; ——— a/b = 0.8.

Fig. 3. Variation of the dimensionless hydrodynamic force amplitude on the exterior cylinder,u^2xu/rgadH, with

k0d for different a/b for h1/d = 0.33,h2/d = 0.8 andG0 = 1. Notation: — – —a/b = 0.2; · · ·a/b = 0.4; - - - a/b= 0.6; ——— a/b = 0.8.



k0d for different G0 for h1/d = 0.33,h2/d = 0.8 anda/b = 0.2. Notation: – · –G0 = 1; — – — G0 = 2; · · ·G0 =3; - - - G0 = 4; ——— G0 = 5.


k0d for different G0 for h1/d = 0.33,h2/d = 0.8 anda/b = 0.2. Notation: – · –G0 = 1; — – — G0 = 2; · · ·G0 =3; - - - G0 = 4; ——— G0 = 5.


amplitudes on the porous breakwater each exhibit a zero in thek0d range of interest. Inthe present case, this zero occurs for ak0d value approximately equal to 2.6.

The dimensionless hydrodynamic force amplitudes on the interior and exterior cylindersfor different values ofh2/d for h1/d = 0.33, a/b = 0.2 andG0 = 1.0 are shown in Figs 6and 7. The caseh2/d = 1 corresponds to a breakwater extending from the sea-bed tothe free-surface (Darwicheet al., 1994). It can be seen that the maximum dimensionlesshydrodynamic force amplitude on the interior cylinder increases ash2/d decreases (Fig.6). Also, the results shown in Fig. 7 indicate that the dimensionless force amplitude onthe exterior cylinder (which includes the storage tank) is essentially unaffected by thevalue of h2/d. Figs 8 and 9 present the dimensionless hydrodynamic force amplitude onthe inner and outer cylinders for differenth1/h2 for h2/d = 0.8, a/b = 0.2 andG0 = 1. Itcan be seen that fork0d , 2.5 the hydrodynamic force on the interior cylinder increaseswith increasingh1/h2, i.e. as the porous region of the cylindrical breakwater increases (Fig.8). However, the opposite is true of the force on the exterior cylinder, this quantitydecreases as the porous portion of the breakwater increases. It is also noted that fork0d. 2.5 the hydrodynamic force on the interior cylinder decreases with increasingh1/h2,although at these high frequencies there is very little difference between the force estimateson the interior cylinder for differenth1/h2.

The maximum dimensionless wave run-up amplitudes on the interior and exterior cylin-ders, namely max |zj(a, u)|/(H/2), j = 1,2, have also been calculated. Figs 10 and 11 showthe maximum dimensionless wave run-up on the interior and exterior cylinders for differenta/b for h1/d = 0.33,h2/d = 0.8 andG0 = 1. In general, the maximum run-up on the interiorcylinder oscillates with dimensionless frequency. The largest run-up maxima occur for thelargest values ofa/b, i.e. when the size of the interior fluid region is smallest. It is also


k0d for different h2/d for h1/d = 0.33,G0 = 1 anda/b = 0.2. Notation: – · –h2/d = 0.67; — – —h2/d = 0.75; · · ·h2/d = 0.8; - - - h2/d = 0.86; ——— h2/d = 1.



k0d for different h2/d for h1/d = 0.33,G0 = 1 anda/b = 0.2. Notation: – · –h2/d = 0.67; — – —h2/d = 0.75; · · ·h2/d = 0.8; - - - h2/d = 0.86; ——— h2/d = 1.


k0d for different h1/h2 for h2/d = 0.8, G0 = 1 anda/b = 0.2. Notation: – · –h1/h2 = 0.17; — – —h1/h2 = 0.33;· · ·h1/h2 = 0.5; - - - h1/h2 = 0.83; ——— h1/h2 = 1.



k0d for different h1/h2 for h2/d = 0.8, G0 = 1 anda/b = 0.2. Notation: – · –h1/h2 = 0.17; — – —h1/h2 = 0.33;· · ·h1/h2 = 0.5; - - - h1/h2 = 0.83; ——— h1/h2 = 1.

Fig. 10. Variation of the maximum dimensionless wave run-up amplitude on the interior cylinder,max uz1(a, u)u/(H/2), with k0d for different a/b for h1/d = 0.33, h2/d = 0.8 andG0 = 1. Notation: — – —a/b =

0.2; · · ·a/b = 0.4; - - - a/b = 0.6; ——— a/b = 0.8.


Fig. 11. Variation of the maximum dimensionless wave run-up amplitude on the exterior cylinder,max uz2(b, u)u/(H/2), with k0d for different a/b for h1/d = 0.33, h2/d = 0.8 andG0 = 1. Notation: — – —a/b =

0.2; · · ·a/b = 0.4; - - - a/b = 0.6; ——— a/b = 0.8.

noted thata/b = 0.2 is the only case in which the cylindrical breakwater actually reducesthe maximum wave run-up on the interior cylinder. The maximum wave run-up on theoutside of the exterior cylinder is presented in Fig. 11. It can be seen that as far as theexterior cylinder is concerned, the maximum run-up is always greater than the incidentwave amplitude throughout the frequency range of interest.

The influence of the porosity parameter on the maximum wave run-up amplitudes onthe interior and exterior cylinders is shown in Figs 12 and 13 forh1/d = 0.33,h2/d = 0.8anda/b = 0.2. It is first noted that, in general, the maximum run-up on the interior cylinderdecreases with increasing wave frequency and increases with increasing porosity (Fig. 12).Also, it can be seen that as far as the exterior cylinder is concerned, porosity does nothave a significant influence on the wave run-up on the outside face of that body.

Figs 14 and 15 present the maximum wave run-up amplitudes on the interior and exteriorcylinders for different values ofh2/d for h1/d = 0.33,a/b = 0.2 andG0 = 1. It can be seenfrom the figures that the value ofh2/d has no significant influence on the maximum waverun-up on both the interior and exterior cylinders. The values of the maximum wave run-up on the two cylinders for different values ofh1/h2 are shown in Figs 16 and 17. Themaximum run-up on the interior cylinder is seen to increase as the value ofh1/h2 increases.The largest variation in maximum run-up withh1/h2 is seen to occur fork0d , 2.75, themaximum run-up is also noted to be lower than the incident wave amplitude over mostof the frequency range of interest. Fig. 17 indicates that the maximum wave run-up onthe exterior cylinder is greater than the incident wave amplitude over the entire frequencyrange of interest. The actual maximum run-up value oscillates with frequency and increaseswith increasing wave frequency. The influence of the value ofh1/h2 on the maximum run-up on the exterior cylinder is not that significant.


Fig. 12. Variation of the maximum dimensionless wave run-up amplitude on the interior cylinder,max uz1(a, u)u/(H/2), with k0d for different G0 for h1/d = 0.33, h2/d = 0.8 anda/b = 0.2. Notation: – · –G0 =

1; — – — G0 = 2; · · ·G0 = 3; - - - G0 = 4; ——— G0 = 5.

Fig. 13. Variation of the maximum dimensionless wave run-up amplitude on the exterior cylinder,max uz2(b, u)u/(H/2), with k0d for different G0 for h1/d = 0.33, h2/d = 0.8 anda/b = 0.2. Notation: – · –G0 =

1; — – — G0 = 2; · · ·G0 = 3; - - - G0 = 4; ——— G0 = 5.


Fig. 14. Variation of the maximum dimensionless wave run-up amplitude on the interior cylinder,max uz1(a, u)u/(H/2), with k0d for different h2/d for h1/d = 0.33, G0 = 1 anda/b = 0.2. Notation: – · –h2/d =

0.67; — – —h2/d = 0.75; · · ·h2/d = 0.8; - - - h2/d = 0.86; ——— h2/d = 1.

Fig. 15. Variation of the maximum dimensionless wave run-up amplitude on the exterior cylinder,max uz2(b, u)u/(H/2), with k0d for different h2/d for h1/d = 0.33, G0 = 1 anda/b = 0.2. Notation: – · –h2/d =

0.67; — – —h2/d = 0.75; · · ·h2/d = 0.8; - - - h2/d = 0.86; ——— h2/d = 1.


Fig. 16. Variation of the maximum dimensionless wave run-up amplitude on the interior cylinder,max uz1(a, u)u/(H/2), with k0d for different h1/h2 for h2/d = 0.8, G0 = 1 anda/b = 0.2. Notation: – · –h1/h2 =

0.17; — – —h1/h2 = 0.33; · · ·h1/h2 = 0.5; - - - h1/h2 = 0.83; ——— h1/h2 = 1.

Fig. 17. Variation of the maximum dimensionless wave run-up amplitude on the exterior cylinder,max uz2(b, u)u/(H/2), with k0d for different h1/h2 for h2/d = 0.8, G0 = 1 anda/b = 0.2. Notation: – · –h1/h2 =

0.17; — – —h1/h2 = 0.33; · · ·h1/h2 = 0.5; - - - h1/h2 = 0.83; ——— h1/h2 = 1.


Figs 18–23 summarize the variation of the dimensionless free-surface amplitudes in thevicinity of the breakwater system for various combinations of wave and structural para-meters. These variations are shown in the form of half-tone contour plots. Fig. 18 presentsthe free-surface amplitudes fora/b = 0.2, h1/d = 0.33,h2/d = 0.8 andG0 = 1.0 for incidentwaves ofk0d = 1.75 (Cw = 0.61) andk0d = 3.5 (Cw = 0.29), respectively. The diffractionof the incident wave field by the breakwater is clearly shown. The results also indicatethat the free-surface amplitudes along the upstream side of the inner cylinder are smallerthan those around the upstream side of the outer cylinder (breakwater). For the higher-frequency case, the sequence of back-scattered and forward-scattered waves is clearlyobserved. The wave amplitudes in front of the inner cylinder are again smaller than thosein front of the outer cylinder. Comparing the two figures, it can be seen that, for a higher-frequency incident wave, the diffracted wave field becomes more pronounced and morewaves are observed in the annular region. Fig. 19 shows the dimensionless wave ampli-tudes for the same data as Fig. 18, except that hereG0 = 5. The results for this case aresimilar to those shown in Fig. 18, however, it can be seen that the overall wave amplitudeinside the annular region increases as the porosity of the exterior breakwater increases.Also, due to the larger porosity of the breakwater, the scattered wave field emanating fromthe exterior cylinder is less pronounced.

The plots of dimensionless free-surface amplitudes for a larger inner cylinder,a/b =0.4, are shown in Figs 20 and 21 forG0 = 1 andG0 = 5, respectively, again forh1/d =0.33,h2/d = 0.8 andk0d = 1.75 and 3.5. Similar wave patterns to those shown in Figs 18and 19 are observed in Figs 20 and 21. However, the wave amplitude inside the annularregion for a/b = 0.4 is, in general, greater than that fora/b = 0.2. Figs 22 and 23 showthe dimensionless free-surface elevation fora/b = 0.4, h1/d = 0.8, h2/d = 0.8 andG0 = 1andG0 = 5, respectively, again fork0d = 1.75 and 3.5, i.e. in these cases the porous portionof the breakwater extends from the free-surface to the top of the storage tank. It can beseen that increasing the porous depth so thath1 = h2 results in a back-scattered wave fieldwhich is less pronounced than that forh1 = 0.4h2 (i.e. h1/d = 0.33, h2/d = 0.8). Also, itis noted that the wave amplitude inside the annular region increases. This occurs for boththe low- and high-frequency wave cases considered.

5. CONCLUSIONS

This paper has presented a theoretical study on the interaction of linear water waveswith a cylindrical breakwater surrounding a rigid circular cylinder mounted on a storagetank. The breakwater consisted of a surface-piercing structure, porous in the vicinity ofthe free-surface while at some distance below the water surface it becomes impermeable.Assuming linearized potential flow, analytical expressions have been obtained for the wavemotion in both the interior and exterior flow regions. Numerical results have beenpresented which illustrate the effects of the various wave and structural parameters on thehydrodynamic loads and interior and exterior wave fields. It has been found that for certainparameter combinations the semi-porous, cylindrical breakwater may result in a significantreduction in the wave field and hydrodynamic forces experienced by the interior cylinder.


Fig. 18. Dimensionless free-surface amplitude,uzj(r, u)u/(H/2), j = 1,2, for a/b = 0.2, h1/d = 0.33,h2/d = 0.8, G0

= 1 and (a)k0d = 1.75, (b)k0d = 3.5.



= 5 and (a)k0d = 1.75, (b)k0d = 3.5.



= 1 and (a)k0d = 1.75, (b)k0d = 3.5.



= 5 and (a)k0d = 1.75, (b)k0d = 3.5.


Fig. 22. Dimensionless free-surface amplitude,uzj(r, u)u/(H/2), j = 1,2, for a/b = 0.4, h1/d = 0.8, h2/d = 0.8, G0

= 1 and (a)k0d = 1.75, (b)k0d = 3.5.


Fig. 23. Dimensionless free-surface amplitude,uzj(r, u)u/(H/2), j = 1,2, for a/b = 0.4, h1/d = 0.8, h2/d = 0.8, G0

= 5 and (a)k0d = 1.75, (b)k0d = 3.5.


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Wave interaction with a semi-porous cylindrical breakwater mounted on a storage tank